Quantitative Microscopic Techniques for Monitoring Dynamic Processes in Microarrays


 Christopher Flowers
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1 Quantitative Microscopic Techniques for Monitoring Dynamic Processes in Microarrays Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties, in het openbaar te verdedigen op 12 maart 22 om 13:3 uur door Lennert Richard VAN DEN DOEL natuurkundig ingenieur geboren te Vlissingen.
2 Dit proefschrift is goedgekeurd door de promotoren Prof. dr. ir. L.J. van Vliet, Prof. dr. I.T. Young Samenstelling promotiecommissie: Rector Magnificus, Prof. dr. ir. L.J. van Vliet Prof. dr. I.T. Young Prof. dr. ir. J.J.M. Braat Prof. dr. G.W.K. van Dedem Prof. dr. T.W.J. Gadella Prof. dr. J. Greve Prof. dr. J.G. Korvink voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, promotor Technische Universiteit Delft Technische Universiteit Delft Universiteit van Amsterdam Universiteit Twente Albert Ludwig University Freiburg, Duitsland Advanced School for Computing and Imaging This work was carried out in the ASCI graduate school. ASCI dissertation series number 75. ISBN c 22, L.R. van den Doel, all rights reserved.
3 You play, you win. You play, you lose. YOU PLAY. Jeanette Winterson, The Passion
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5 Voor Thomas & Jolanda Voor Rosanne, fonkelende ster, mijn kleine engel R
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7 Contents 1 Labsonachip Combinatorial Chemistry Human Genome Project Intelligent Molecular Diagnostic Systems Development of a Microarray reader Outline of this thesis 5 Part I Quantitative Reading of Microarrays 2 Yeast as a Cell Factory: the Glycolytic Pathway Yeast: Saccharomyces cerevisiae The Glycolytic Pathway Summary 13 3 Conventional Microscopebased Microarray Reader Microarray Fabrication Nanoliter Liquid Handling Conventional Microscopy Concepts Infinitycorrected Optical System 2 vii
8 viii CONTENTS Köhler Illumination Characterization of the Princeton Versarray CCD camera Readout noise Dark current Linearity and Sensitivity Summary 33 4 Quantitative Reading of Microarrays Limiting Factors of the Instrument Detection Limit of the Microarray Reader Improving the Detection Limit (for 1 and N measurements!) Readout Noise Limited Detection Ideal Detection System Single Element Detection System with Readout Noise Many Pixels Detection System with Readout Noise Stray Light Limited Detection Minimum Detectable Signal for Different Configurations Modifications to the Microscope System Conclusions and Discussion 48 5 Monitoring Enzymecatalyzed Reactions in Microarrays Imaging with the modified optical system NADH Calibration Monitoring Enzymecatalyzed reactions Conclusions 64 Part II Monitoring Dynamic Liquid Behavior in Micromachined Vials 6 Dynamic Liquid Behavior in Micromachined Vials Introduction Liquid pinning: ring stains from coffee droplets Modeling dynamic liquid behavior Impedancebased liquid volume sensor Survey of Microscopic Techniques Conventional Fluorescence Microscopy Confocal Fluorescence Microscopy InterferenceContrast Microscopy 82
9 CONTENTS ix 6.3 Conclusions 86 7 Electromagnetic Theory of InterferenceConstrast Microscopy Introduction Phase Difference Intensity of Electric Field Comparison between Simulation and Experimental Results Alternative Description for Intensity of Electric Field Summary 99 8 Temporal Phase Unwrapping Algorithm Phase Unwrapping: Spatial vs. Temporal Temporal Phase Unwrapping Algorithm Computation of Reference Frame Precision and Accuracy of Phase Estimation Simultaneously Sampling in Space and in Time Summary Experimental Results Introduction Experimental Setup Monitoring Evaporation in Square Micromachined Vials Lateral Scaling of Height Profiles Evaporation Rate in Square Vials Monitoring Evaporation in Circular Micromachined Vials Radial Scaling of Height Profiles Evaporation Rate in Circular Vials Calibration of the Electronic Volume Sensor NanometerScale Height Measurements Conclusions and discussion 14 Appendix Deegan s Theory on DiffusionLimited Evaporation 14 References 145 Summary 149 Samenvatting 151 Dankwoord 155
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11 1 Labsonachip One of the most widely exploited fields of research during the last decade is that of combinatorial chemistry. The stated goal of this approach is to drastically reduce the time and cost that is required for obtaining new lead compounds. Currently, many samples are tested against a large number of targets in microarrays for highthroughput screening. In the future, this technology will lead to a miniaturized labonachip. The Delft University of Technology is financing an interdisciplinary research program, Intelligent Molecular Diagnostic Systems, in which analytical labsonchips are being developed in order to gain insight into the metabolic regulation of the glycolytic pathway in yeast cells. 1.1 COMBINATORIAL CHEMISTRY As early as the 195s and 196s a technique to synthesize different chains of peptides 1 on a solid phase was developed. Before that time peptide chains could be synthesized only in solution. The main advantage of solid phase peptide synthesis over traditional solution phase chemistry is that after each cycle of adding a peptide to the chain, the solid carrier beads remain insoluble and excess reagents and unreacted materials can be rinsed away. This allows for stepbystep purification through simple washing steps. This technique was developed by Merrifield and published in 1963 [1]. In 1984 he won the Nobel Prize in Chemistry for this research. 1 a peptide yields two or more amino acids on hydrolysis. Peptides are formed by the loss of water from the NH 2 and COOH groups of adjacent amino acids. Peptides form the constituent parts of proteins. There are 2 naturally occurring amino acids from which proteins are synthesized. 1
12 2 LABSONACHIP Combinatorial chemistry is a technique that adapts this technique of peptide synthesis for the construction of large libraries of structurally related molecules. These libraries can be constructed as a mixture of all these related molecules in the same reaction vessel or all these related molecules can be constructed in separate vessels individually by parallel synthesis. After construction, the library is screened for receptorinhibitor activity. The active compounds are possible candidates for drug development. Highthroughput screening (HTS) is the screening and testing of hundreds to thousands of compounds in a relatively short period of time: construction of the library and screening should take approximately the same amount of time. Combinatorial chemistry requires adaptation of classical synthesizing techniques, automization of liquid handling, and miniaturization of reaction vessels for the construction of large compound libraries. Highthroughput screening requires sensitive detectors to monitor the chemical activities in all vessels. This screening generates enormous amounts of data. These screening results must be integrated with biological data and structural information of the library. 1.2 HUMAN GENOME PROJECT Similar developments can be seen in the field of genetics. In October 199 the United States Department of Energy and the National Institutes of Health joined forces in the Human Genome Project. Two of the goals of this project are to discover all the 3 to 35 thousand genes of the human genome and to determine the 3 billion nucleotides of the complete human genome sequence. In February 21 the working draft versions of the human genome were published in Nature [2] and Science [3] Much effort in the Human Genome Project is put into miniaturization, e.g. the fabrication of highdensity combinatorial arrays of oligomers (short chains of nucleotides). The oligomers are fabricated on an array by photolithographic techniques: in N processing steps 4 N different oligomers (4 different nucleotides) with a length of N nucleotides can be fabricated. These arrays allow for largescale hybridization assays. Now that the draft version of the human genome is available, the new challenges are to find out what the function is of all genes, how all the genes are regulated, to gain understanding of dynamic living systems, and to model complex biological systems. 1.3 INTELLIGENT MOLECULAR DIAGNOSTIC SYSTEMS In the context of these developments, the Delft University of Technology is financing an interdisciplinary research program for development of miniaturized analytical labsonchips for the quantification of a variety of chemical compounds in a short period of time using a minimum amount of reagents. Furthermore, the system should allow for interpretation of the acquired quantitative data in combination with expert knowledge and historical data. One can think of various applications for such a system:
13 INTELLIGENT MOLECULAR DIAGNOSTIC SYSTEMS 3 Quality control in the food industry: the presence of residues from pesticides, antibiotics, or hormones is unwanted, but the detection poses enormous problems due to the required sensitivity and also to the enormous variety of chemical compounds present. Forensic research: a maximum amount of information must be obtained out of a minimum amount of sample in a short period of time. This requires not only efficient screening for a large number of compounds, but also interpretation of the gathered data. Medical diagnostics: more and more tests are carried out to obtain as complete as possible a picture of every patient who is hospitalized or treated polyclinically. A system which is capable of carrying out a multitude of assays and of interpreting the results is likely to lead to significant costreductions as well as to improvement in the quality of medical care. Epidemiology: this requires fast and massive screening using small sample volumes from many individuals. Bioprocess industry: controlling or improving fermentation processes requires rapid tests on the activities of several biochemical compounds, such as enzymes. Based on the interpretation of the results from the tests the fermentation processes can be tuned or further optimized. The last application is being used in a demonstrator system within the framework of this interdisciplinary research program. This demonstrator system will be built around silicon based microarrays containing reaction vials with volumes on the order of a few nanoliter. Each vial will be equipped with sensors and actuators to monitor and control a number of important parameters. In this demonstrator all substrates involved in the twelve reactions of the glycolytic pathway, which is the conversion of glucose to ethanol and carbondioxide, will be put into small reaction vessels. All these reactions are enzymecatalyzed reactions. The twelve enzymes are acquired from cellfree extracts of yeast cells taken from the wellconditioned environment of a chemostat. Yeast is a unicellular eukaryotic organism, which is frequently used as a model system for other higher eukaryotes, including man. One of the fermentation processes taking place in eukaryotic organisms is the fermentation of glucose. Differences in the environment of the yeast and or genetic modifications of the yeast will result in differences in enzyme activity levels along this pathway. Enzyme activity levels are monitored by measuring fluorescence levels of NADH. This fluorescent molecule is involved in all the reactions of the glycolytic pathway. The aim is to gain insight in the metabolic regulation of this pathway by correlating the experimental data with the environmental conditions and or genetic modifications. There are a number of challenges involved in this research program: Chemistry must be adapted to nanoliter scale. Substrates and cofactors have to be deposited in the vials. After drying of the solvent, the solute must be kept stable in the vials during storage. When using these prepared microarrays for measurements, the solutes must be dissolved again. The sample solution with the enzymes must be put into the vials without any degeneration of the enzymes. Fast evaporation of the solution must be avoided to guarantee that the desired reactions are actually taking place.
14 4 LABSONACHIP Liquid handling needs to be modified to the nanoliter range. Ultra small volumes must be fast and automatically dispensed in (sub)nanoliter vials in an accurate, precise, and reproducible manner. A system must be developed that allows for pipetting solutions containing various substrates at different concentrations. This system must be automated since it will be used to prepare hundreds of identical microarrays. Furthermore, techniques must be developed to deposit the sample solution in all vials in a massively parallel manner. A labonachip is more than just a miniaturized microtiter plate: microarrays will contain vials with builtin intelligence, i.e. integrated actuators and sensors: every vial will be equipped with a temperature sensor and a heater to control the temperature, a sensor to measure the phvalue of the solution, a detector to measure the volume of the injected solution, etc. The biochemical reactions will produce many fluorescent signals simultaneously. Therefore, sensitive detectors and detection methods are necessary in order to perform quantitative measurements during monitoring of all the biochemical reactions. Highthroughput screening generates enormous amounts of data. In order to extract useful information, it is essential to develop dedicated data analysis systems. Furthermore, data interpretation systems must be developed to derive new insight into the underlying biochemical principles. 1.4 DEVELOPMENT OF A MICROARRAY READER The research that will be presented in this thesis focuses on the development of a microarray reader: a detection system to monitor all the biochemical reactions in all vials of the microarray. This microarray reader will be built around a widefield microscope system equipped with a scientific grade CCD camera and a motorized stage for scanning the microarray. With low power optics, i.e. low magnification, low Numerical Aperture, and therefor a low lightgathering power, a microarray with approximately a hundred vials can be imaged entirely onto the CCD camera. This allows for parallel reading of the microarray. Wide fieldofview imaging, however, is at the expense of absolute sensitivity. With high power optics only a single vial per image can be acquired. This implies a better sensitivity, but requires sequential scanning of the microarray. The research questions we will try answer in this research are What are the limiting factors of our microarray reader? What determines the lowest concentration of fluorescent molecules that can be detected with our system? It will turn out that stray light, originating in the (excitation) optical path of the microscope is the limiting factor. Given this limiting factor, what are the possibilities to improve the detection limit of our system? If stray light is the limiting factor, a better detector does not help, since it will increase the signal level as well as the noise level, leaving the SignaltoNoise Ratio unchanged.
15 OUTLINE OF THIS THESIS 5 What is the dynamic range of concentrations of fluorescent NADH molecules that can be detected with our system under limitation of stray light? Given this stray light limitation, what modifications can we apply to our system to improve the scan rate of the microarray? The enzyme activity levels will be derived from the production or consumption rate of NADH. Given the scan rate of our microarray reader, what is the dynamic range of enzyme activity levels that can be monitored with our system? 1.5 OUTLINE OF THIS THESIS This thesis consists of two parts. The first part of this thesis presents the research into the development of a microscopebased microarray reader. This research fits completely within the framework of the interdisciplinary research program to develop analytical labsonchips. The second part of this thesis presents research that has been performed to monitor dynamic liquid behavior in the vials of our microarrays. Evaporation of liquid samples is a key problem in the development of microarray technology. The research in the second part of this thesis presents a type of microscopy, interferencecontrast microscopy, that allows for quantitative monitoring of the airliquid interface during drying of a liquid sample in a vial. One of the most striking results of this research is that the evaporation rate is linearly proportional to the perimeter of the vial and not to the area of the airliquid interface, as one would intuitively expect. This research also presents calibration results of an electronic liquid volume sensor, that will be integrated in our labonachip. Quantitative Reading of Microarrays Chapter 2 presents in more detail the biochemical challenge for which microarray technology can provide the right analytical tools: exploring the metabolic regulation of the glycolytic pathway in yeast cells. Chapter 3 describes the fabrication process of the microarrays and the different liquid handling strategies. Furthermore, some important concepts of microscopy will be introduced. Finally, this chapter presents the characteristics of the CCD camera used in the experiments. This chapter tries to provide sufficient background knowledge in order to fully understand the modifications we will apply to our microscope system to improve its performance. Chapter 4 describes in detail what the limiting factors are of measuring low levels of fluorescence in our microarrays. The limiting factors are either readout noise in the detector or stray light originating in the (excitation) optical path. Given these limitations, we have modeled various detection systems to investigate what the strength is of the minimum detectable signals. These models amount to modifications to our microscope system that we have applied to improve its performance. Finally, Chapter 5 presents the experimental results of reading microarrays with our modified conventional microscope based microarray reader. The functionality of our microarray reader to monitor reactions of the glycolytic pathway is demonstrated by monitoring NADH producing and consuming enzymatic reactions in our microarrays at various conversion rates.
16 6 LABSONACHIP Monitoring Dynamic Liquid Behavior in Micromachined Vials Chapter 6 looks from different points of view at the process of evaporation to motivate this research and explains the spatial and temporal sampling requirements for quantitative monitoring of the evaporation process in the vials of our microarrays. Interferencecontrast microscopy is a type of microscopy that satisfies these sampling requirements. Chapter 7 presents a classical electromagnetic theory to describe the generation of the dynamic fringe patterns, which are observed with interferencecontrast microscopy. Chapter 8 introduces a temporal phase unwrapping algorithm that computes the dynamic height profiles of the meniscus from these fringe patterns. This algorithm analyzes the interferograms pointbypoint in time without using any spatial information. Chapter 9 presents experimental results of monitoring the evaporation process in micromachined vials with the technique of interferencecontrast microscopy. We have monitored the evaporation process in square vials as well as in circular shaped vials of different sizes. Furthermore, this chapter will show experimental results of the calibration of a dedicated electronic volume sensor.
17 Part I Quantitative Reading of Microarrays
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19 2 Yeast as a Cell Factory: the Glycolytic Pathway This chapter presents a biochemical challenge for which microarray technology can provide the right analytical tools: exploring the metabolic regulation of the glycolytic pathway in yeast cells. Yeast, a eukaryotic cell, is a model system for other higher eukaryotes, including man. The glycolytic pathway converts glucose to ethanol and carbon dioxide. This metabolic pathway, consisting of 12 enzymecatalyzed reactions, is part of a larger compartmented metabolic network, which is interlinked at various levels. Studying this basic process in yeast is required not only to optimize this fermentation process. The insights gained can also be used for other applications, such as the production of pharmaceutical (human) proteins by yeast. One century of research aimed at increasing the glycolytic capacity (rate of alcoholic fermentation) has been large unsuccessful. Microarray technology allows for doing quantitative measurements, cheap and fast, in a massive parallel manner: instead of monitoring a single enzymatic reaction, as performed in the past century, it is possible to monitor all the different reactions simultaneously in microarrays and extract large amounts of information from all these measurements. The central theme in this research program is the development of analytical tools for an integral analysis of the metabolic regulation of this pathway, rather than applying metabolic engineering to the yeast cell itself. 2.1 YEAST: Saccharomyces cerevisiae The yeast Saccharomyces cerevisiae is probably better known as brewers or bakers yeast. In the past 8 years, this fungus has played an important role in the production and conservation of food due to its ability to ferment glucose to ethanol and carbondioxide. Nowadays, this yeast is used as a multipurpose cell factory ; it is not only used for the production of ethanol 9
20 1 YEAST AS A CELL FACTORY: THE GLYCOLYTIC PATHWAY Fig. 2.1 An image of a budding Saccharomyces cerevisiae cell acquired with electron microscopy. and carbondioxide, beer and wine, but also for glycerol, various flavors, different enzymes and vitamins, pharmaceutical (human) proteins, etc. Yeast, which is a unicellular eukaryote 1, has become a unique and powerful model system for biological research, because it has some attractive features: yeast is cheap and easy to cultivate, yeast has short generation times, detailed genetic and biochemical knowledge of yeast has accumulated over the past 1 years, and, molecular techniques for genetic manipulation can be easily applied to the yeast genome. This organism, therefore, provides a highly suitable system to study basic biological processes that are relevant for many other higher eukaryotes, including man. The complete genome sequence of Saccharomyces cerevisiae [4] is known: the complete genome consists of 16 chromosomes with approximately 12 million nucleotides and about 64 genes. The function of 4% of the genes of S. cerevisiae is still unknown. Figure 2.1 shows an image of Saccharomyces cerevisiae acquired with electron microscopy. In this image the organism is in the state of budding (cell reproduction). In the past century many researchers have tried to improve the yeast strain, producing relatively more desired product and less (unwanted) byproducts, to make it stronger, to let it remain productive under many different conditions, and to make it faster, thereby increasing the rate of production of the product of interest. The most important technique to accomplish this is 1 A eukaryotic organism has its genetic information contained in a separate cellular compartment: the nucleus of the cell. In addition to their nuclear genomes, all eukaryotic cells contain small additional extranuclear genomes, which are contained in the mitochondria (and in plastids in photosynthesis performing eukaryotes). The mitochondria are selfautonomous, self reproducing organelles within the cytoplasm of eukaryotic cells that are bounded by two membranes. These organelles are responsible for the energy conversion of most of the cellular energy metabolites into adenosine triphosphate (ATP).
21 THE GLYCOLYTIC PATHWAY 11 metabolic engineering: improving cellular activities by manipulation of enzymatic, transport and regulatory functions of the cell with the use of recombinantdna technology. Simply speaking, one modifies some nucleotides or even a complete gene in the genome of the yeast cell. These modifications at the DNA level are transcribed onto RNA. The modified RNA is translated into a modified protein. The modified protein has a different structure, a different function and a different regulation with respect to the unmodified protein. Due to these modifications it will alter the entire metabolic process of interest: metabolic pathways are not simple linear pathways, but interlinked metabolic networks. There is regulation of enzyme activity at various levels: at the transcription level from DNA to RNA, at the translation level from RNA to the protein, covalent modifications of proteins might occur, and finally, there is activity regulation by substrates, products, and effectors. Furthermore, the control of flux through pathways is not confined to a single bottleneck enzyme, but is shared by multiple enzymes. Finally, metabolism is compartmented in eukaryotic cells; metabolic processes take place in different areas within the yeast cell. In this research program we want to provide analytical tools based on microarrays to monitor the effects of modifications to the yeast genome and the changes in the environment of the yeast in a systematic way. We will focus, in particular, on the metabolic regulation of the glycolytic pathway. One of the approaches to study this metabolic regulation is to vary environmental parameters and / or introduce mutations in the yeast genome. After that, one tries to relate these measured parameters (RNA transcript level, protein activity, metabolite concentration) to the quantity to be optimized, e.g. the glycolytic capacity. The central research themes of this program are focusing on the biochemical challenge to perform the enzymatic reactions of the glycolytic pathway in microarrays, the technological challenge to construct labsonachip in which the reactions can take place, to monitor the enzymatic reactions in these microarrays, and the bioinformatic challenge to derive new insights on the regulation of this metabolic process from the acquired data. The research described in this part of this thesis focuses on developing a quantitative microarray reader for monitoring the different reactions of this pathway in microarrays. 2.2 THE GLYCOLYTIC PATHWAY The yeast Saccharomyces cerevisiae is wellknown for its alcoholic fermentation. The biochemical process of the fermentation from glucose to ethanol and carbondioxide is known as the glycolytic pathway, which is shown in Figure 2.2. This pathway consists of 12 enzymecatalyzed reactions. Table 2.1 lists all twelve enzymes involved in the glycolytic pathway. This table shows in what type of assay these enzymes are involved and if the enzyme activity is measured via a direct reaction or via a coupled reaction. An example of a direct reaction is the following: glucose + ATP HXK G6P + ADP + photon, (2.1) where G6P is glucose6phosphate, and HXK is the enzyme hexokinase. In this reaction the bioluminescent substrate ATP (adenosine triphosphate, bioluminescence wavelength: 58nm) is converted to ADP (adenosine diphosphate). The rate of bioluminescence of the ATP is a measure for the enzyme activity. It is also possible to measure the activity of hexokinase via
22 12 YEAST AS A CELL FACTORY: THE GLYCOLYTIC PATHWAY glucose ATP hexokinase ADP glucose6phosphate phosphoglucose isomerase fructose6phosphate ATP phosphofructokinase ADP fructose1,6bisphosphate triosep fructosebisp isomerase aldolase dihydroxy glyceraldehyde3phosphate acetonphosphate NAD glyceraldehyde3p NADH dehydrogenase 1,3diphosphoglycerate ADP phospho ATP glyceratekinase 3phosphoglycerate phospho glyceratemutase 2phosphoglycerate lactate dehydrogenase NADH NAD lactate enolase phosphoenolpyruvate ADP pyruvatekinase ATP pyruvate pyruvate CO2 decarboxylase aceetaldehyde NADH alcohol NAD dehydrogenase ethanol Fig. 2.2 This figure shows the Glycolytic pathway: the fermentation of glucose to ethanol. This pathway consists of 12 enzymecatalyzed reactions. two coupled reactions (the procduct of the first reaction is a substrate in the second reaction): glucose + ATP G6P + NADP HXK G6P + ADP G6P DH gluconolactone + NADPH, (2.2) where NADP is Nicotinamide Adenine Dinucleotide Phosphate. In these two reactions, the generation of the fluorescent product NADPH (excitation wavelength 36nm, emission wavelength 45nm) is monitored. Note that the first reaction must be the conversion rate limiting reaction. Otherwise, the activity of the enzyme glucose6phosphatedehydrogenase (G6PDH) is measured. The last column in Table 2.1 gives the specific activity of the enzymes in a well conditioned environment of a chemostat, where the reactions are aerobic, i.e. in the presence of oxygen, and where the reactions are glucose limited, i.e. the conversion of glucose to ethanol is fully determined by the amount of glucose and not by other factors. The enzyme activity is expressed in units per milliliter (U/ml): an enzyme activity of 1U/ml means that 1µMol of substrate in 1ml solution is converted in 1 minute of time.
23 SUMMARY 13 Table 2.1 List of enzymes involved in the Glycolytic pathway. The third column gives the type of assay. The fourth column shows if the enzyme activity is measured via a direct reaction or via a coupled reaction. The last column gives the specific enzyme activities measured in a chemostat, where the culture in under aerobic glucose limited conditions [5]. The unit U is defined in the text. Enzyme Abbrev. Assay Reaction Specific activity U/ml hexokinase HXK ATP direct 1.7 phosphoglucoseisomerase PGI NADPH coupled 2.8 phosphofructokinase PFK ATP direct.3 fructosebispadolase FBA NADH coupled 1. triosepisomerase TPI NADH coupled 52.5 glyceraldehyde3pdehydrogenase GAPDH NADH coupled phosphoglyceratekinase PGK ATP direct 7.4 phosphoglyceratemutase PGM ATP coupled 6.8 enolase ENO ATP coupled.7 pyruvatekinase PYK ATP direct 2.9 pyruvatedecarboxylase PDC NADH coupled.6 alcoholdehydrogenase ADH NADH direct 9.1 The experimental procedure to monitor the effects of an adjustment to the stable and wellcontrolled environment of a chemostat to the glycolytic pathway consists of the following steps. Two different samples are taken from the fermentor. From both samples the cellfree extract is extracted. These cellfree extracts are added to microarrays containing all required substrates for the enzymatic reactions. The substrates are available in two different concentrations to check whether or not the reactions are enzymelimited. The enzyme activity levels of all reactions taking place in the microarray are observed by monitoring the generation or decrease in signal level from either ATP or NAD(P)H. Bioluminescence has the great advantage, that is does not require any excitation source. It produces, however, a very weak signal, even weaker than fluorescence. Therefore, all the assays of the glycolytic pathway will be designed in such a way that the enzyme activity can be measured via the NAD(P)H production or consumption. The measured enzyme activity levels indicate the effects of the modified environmental conditions or modifications to the yeast genome. 2.3 SUMMARY Microarray technology provides useful tools for a study of the regulation of the glycolytic pathway in yeast cells. The glycolytic pathway consists of 12 enzymecatalyzed reactions. These reactions take place on microarrays where all required substrates are already available. The enzymes are acquired from two different cellfree extracts from yeast cells from the
24 14 YEAST AS A CELL FACTORY: THE GLYCOLYTIC PATHWAY stable and wellconditioned environment of a chemostat. The enzyme activities are related to the conversion rate of either the bioluminescent molecule ATP or the fluorescent molecule NADPH. The generation of these products or the consumption of these substrates will be monitored for each reaction in time. The next chapter will present the fabrication process of the microarrays and the different liquid handling strategies. The microarray reader for acquisition of these signals will be build around a widefield microscope system. The next chapter will also present some basic concepts of quantitative microscopy. A good understanding of these concepts is necessary to understand the modifications we will apply to our microscope system to improve its performance. Finally, we will present results of a number of experiments that will bring us some steps closer to the desired goal.
25 3 Conventional Microscopebased Microarray Reader In the previous chapter the biochemical challenge of this research was introduced: development of a microarray reader to monitor enzymecatalyzed reactions in microarrays. This microarray reader will be built around a conventional microscope system equipped with a scientific grade CCD camera. This chapter will present the fabrication process of the microarrays and different liquid handling strategies. Furthermore, we will introduce two concepts of modern microscopy: the infinitycorrected optical system of a microscope and the concept of Köhler illumination. Finally, we will present the characteristics of the CCD camera used in the experiments. This chapter tries to provide sufficient background knowledge in order to fully understand the modifications we will apply to our microscope system to improve its performance. 3.1 MICROARRAY FABRICATION The microarrays that will be used in this research are manufactured in silicon at DIMES (Delft Institute for Microelectronics and Submicron Technology, Delft University of Technology, NL2628 CD Delft, the Netherlands). With conventional silicon fabrication techniques a variety of different geometries can be manufactured. In our experiments we used either square of circularly shaped vials with different sizes and depths. Table 3.1 gives an overview of the volumes for typical dimensions of the different vials. Fabrication process. The silicon wafers are first covered with a thin layer of silicon nitride. On top of this layer a thin film of photo resist is deposited. This film is exposed through the masking pattern of the microarrays. After removal of the exposed photo resist, the wafer is 15
26 16 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER This table gives an overview of the volumes in nl of the different vials used in the experi Table 3.1 ments. depth square vial circular vial (diameter) µm 3 3µm 2 4 4µm 2 3µm 4µm etched using plasma etching. During this process, the silicon is removed at the locations of the vials. The remaining photo resist is removed by rinsing the surface with acetone. Both wet and dry etching techniques can be employed on the silicon wafers for the realization of the vials. Wet etching is performed in potassium hydroxide (KOH). Because of the crystal structure of silicon, this technique results in an anisotropic etching of the silicon. The silicon wafers have a < 1 > surface plane orientation. The etching rate for the < 111 > crystal planes is much slower in KOH solution than for the other crystal planes (typically by a factor of 5). Therefore, wet etching of the silicon results in vials with the shape of a truncated pyramid, which has < 111 > planes as side walls for circular as well as for square etching masks. These two types of vials are shown in Figure 3.1(a) and 3.1(b) respectively. In the case of a circular window, underetching of the silicon nitride will take place resulting in free hanging silicon nitride in the corners of the vial, as can be seen in Figure 3.1(a). The angle between the < 111 > planes and the < 1 > plane is about 54.7 degree. Dry etching is conducted by Reactive Ion Etching (RIE), which results in an anisotropic etching. The RIE etched vials have a cylindrical or cubic shape after etching with a circular or square masking pattern respectively. These vials are shown in Figure 3.1(c) and 3.1(d). These two types of vials have been used in most of our experiments. Microarrays can also be manufactured in glass. Glass wafers are then covered with a thin layer of polysilicon, which is then processed to form the etching mask. Wet etching is conducted in a hydrofluoric acid (HF) solution. Glass is an amorphous material, and thus isotropic for etching. This process results in a cilindrical or cubic shape with rounded edges and corners. These shapes are shown in Figure 3.1(e) and 3.1(f). In the initial phase of this research glass microarrays have been used in experiments, but, from a "labonachip" pointofview, silicon microarrays are to be preferred, since they allow for easy integration of electronic sensors. The microarrays used in the experiments typically contain 5 5 vials. The centertocenter distance ranges from 8µm to 12µm. The dimensions of the microarray itself are identical for all microarrays 2 1cm 2. In Figure 3.2 one of the microarrays is shown. From these dimensions follows that the vial density is in the range from 7 to 15 vials per square centimeter. The relatively large centertocenter distance is needed for a dedicated filling method (not described in this thesis). With this manual filling method even smaller vials could be filled. In the next section, however, an automated liquid handling procedure will be described for which the minimal vial width is about 3µm.
27 NANOLITER LIQUID HANDLING 17 (a) Silicon, circular, KOH (b) Silicon, square, KOH etched (c) Silicon, circular, RIE etched (d) Silicon, square, RIE etched (e) Glass, circular, HF etched (f) Glass, square, HF etched Fig. 3.1 This figure shows six different types of vials that can be fabricated with standard silicon fabrication techniques. 3.2 NANOLITER LIQUID HANDLING One of the difficulties involved in the miniaturization of highthroughput screening technologies is liquid handling. Fluid volumes on the order of a nanoliter need to be injected into the vials on the microarray. Within the framework of this research program, different nanoliter liquid handling strategies have been developed. In this section we will briefly discuss three techniques. Manual liquid handling with Eppendorf Transjector An Eppendorf Transjector 5246 (Eppendorf, NethelerHinz GmbH, Hamburg, GE) is used to manually inject liquid samples into the vials. This machine is primarily used for In Vitro Fertilization (IVF), in which sperm cells are injected through the cell membrane into an oval cell. In order to inject liquid volumes into a vial, a special capillary the Femtotip II is used. The Femtotip II is about 5cm long and the outlet of the capillary has an inner diameter of less than.5µm. The capillary tapers off to the outlet. In order to inject a small liquid sample into a vial, the Transjector builds up a sudden injection pressure. This pressure is held during the injection time. After injection the pressure is exponentially released to the compensation pressure. It
28 18 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER 4 1 cm DIOCIMDS 4 2 cm Fig. 3.2 A 2 1cm 2 microarray etched in silicon with two arrays of 5 5 vials. The centertocenter distance is 8µm. Fig. 3.3 The experimental setup for filling microarrays manually with the Eppendorf Transjector. The Femtotip II is placed at an angle under a low magnification objective (working distance 1cm). On the right a "Dutch Dime" is placed (diameter is 15mm). is necessary that the tip of the capillary touches the surface of the vial in order to release the liquid drop from the tip and to get the liquid in the vial. This kind of liquid handling obviously does not meet our demands. With this device it is only possible to fill one vial at a time with just one liquid sample. Furthermore, to avoid fast evaporation of the liquid sample, a mixture of glycerol and water (1 : 1, v/v) must be used. Figure 3.3 shows the experimental setup for manually filling the vials with a solution. Automated liquid handling: Electro Spray Dispensing Besides this manual liquid handling technique an automated liquid handling technique has been developed in this research program: Electro Spray Dispensing [6]. The piston of a syringe is driven by a precision displacement pump. This guarantees a wellcontrolled liquid flow at the outlet of a capillary (inner diameter 6µm) connected to the syringe. A microarray is placed on a motorized xyzstage at a distance of about 25µm under the outlet of the capillary. The metal capillary
29 CONVENTIONAL MICROSCOPY CONCEPTS 19 Fig. 3.4 A liquid droplet is formed by an aerosol of small liquid droplets from the capillary, induced by a strong electric field. is connected to a high voltage, where the microarray is grounded. This potential difference induces a large electric field. This electric field produces an aerosol of small droplets of the conductive liquid, which are accelerated towards the microarray. This forms uniform spots with a diameter on the order of 2µm. Figure 3.4 shows the outlet of the capillary and the formation of a droplet in the microarray. The spraying can be stopped and resumed instanteneously by increasing and decreasing the distance between the array and the outlet of the capillary. The increased distance lowers the electric field strength. This liquid handling technique allows for noncontact, accurate and reproducible dispensing of ultrasmall liquid volumes [6]. Coarse massive parallel filling In the case of manual filling as well as in the case of electrospraying fast evaporation of the liquid sample occurs. To avoid fast evaporation, a third technique has been used. A droplet of about 2µl is pipetted on the microarray and spread over it, such that all wells are filled. Then a microscope slide is placed on top of the microarray. The microarray and the microscope slide are pressed firmly together to remove the excess liquid. The Van der Waals forces exerted on the microarray and microscope slide prevent evaporation of the liquid in the vials. This technique allows for massive parallel filling of practically all vials of a microarray. 3.3 CONVENTIONAL MICROSCOPY CONCEPTS The previous sections described how the microarrays can be fabricated and how they can be filled. The next sections will describe how the fluorescent signals generated in the vials of the microarrays can be detected. Our microarray reader is based upon a Zeiss Axioskop microscope system equipped with a scientific grade CCD camera. In this section we will present two concepts of modern quantita
30 2 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER Object plane Objective Intermediate image plane d=45 mm f tube =15 mm Fig. 3.5 This figure shows the image forming beam path in a finitecorrected optical system. tive microscopy: the infinitycorrected optics and Köhler illumination. These concepts form the basis of the modifications we will introduce in the next chapter to improve the performance of our microarray reader Infinitycorrected Optical System Nowadays, all major microscope manufacturers provide infinitycorrected optical systems for their microscopes. In this section we will explain the concept of infinitycorrected optics and its advantages with respect to a finitecorrected optical system. Finitecorrected optical system. In a finitecorrected optical system the objective forms an image at the primary image plane at a finite distance from the objective. Figure 3.5 shows a beam in the image forming path of a finitecorrected optical system. The wellknown standards for microscopes with a finitecorrected optical system are the following: The parfocal distance d equals 45mm, as indicated in Figure 3.5. The parfocal distance is a pure mechanical dimension. It is the distance between the object plane and the objective shoulder. When the objective is changed, the image focus is more or less retained (parfocality). The tube length f tube equals 15mm, as indicated in Figure 3.5. The tube length is the distance between the objective shoulder and the primary or intermediate image plane. The mechanical tube length is 16mm. This is the distance between the objective shoulder and the shoulder of the body tube that supports the ocular. The objective thread size equals.8 = 2.32mm, (W.8 1/36 ). With these standard dimensions the objectives from different microscope manufacturers could be exchanged between different microscopes. One of the major shortcomings of this approach is that the design for intermediate accessories, such as fluorescence filters is rather difficult, since they have to be placed in the image forming bundle between the objective and the
31 CONVENTIONAL MICROSCOPY CONCEPTS 21 Object plane Objective Tube lens Intermediate image plane f obj parallel optical path f tube Fig. 3.6 This figure shows the image forming beam path in a infinitycorrected optical system. image plane. These intermediate accessories may introduce magnification errors. An infinitycorrected optical system overcomes this difficulty. Infinitycorrected optical system. In an infinitycorrected optical system the objective forms an image at infinite distance from the objective. This means that the exit bundle of the objective is a parallel bundle of rays. A second (fixed) lens must be placed after the objective to form an image at the primary image plane. This lens is called the tube lens. Figure 3.6 shows a beam in the image forming path of an infinitycorrected optical system. It is a common misconception that the tube lens can be placed at an infinite distance from the objective. This can be easily seen in Figure 3.6: after passing through the objective, light from an object on the optical axis propagates parallel to this axis towards the tube lens. Light coming from the periphery of the object forms also a parallel bundle of rays, but propagates at an angle with respect to the optical axis towards the tube lens. Because of this angle there are instances where these rays of light can no longer be captured by the tube lens. This occurs if the tube lens is placed too far from the objective. As a conclusion, the term infinity implies that light after passing through the objective forms a bundle of parallel rays. In an infinitycorrected optical system the intermediate accessories can be placed in the parallel optical path without affecting the overall magnification. The overall magnification M for an infinitycorrected optical system, as shown in Figure 3.6. is given by M = f tube f obj, (3.1) where f tube and f obj are the focal lengths of the tube lens and the objective. Table 3.2 shows the dimensions of four major microscope manufacturers that they have applied in their infinitycorrected optical system. Zeiss called its infinitycorrected optical system Infinity Colorcorrected System, (ICS), Olympus Universal Infinity System, (UIS), Nikon Chrome Free Infinity (CFI6), where 6 refers to the adapted parfocal distance of 6mm, and Leica Harmonic Component System (HCS), which is the successor of the DELTA infinitycorrected optical system.
32 22 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER Dimensions of the infinitycorrected optical systems of four major microscope manufac Table 3.2 turers parfocal distance tube focal length objective thread mm mm Leica HCS 45 2 M Nikon CFI6 6 2 M Olympus UIS W.8 1/36 Zeiss ICS W.8 1/36 ) This objective thread has a diameter of 25mm Köhler Illumination The illumination scheme of Köhler [7] for a conventional microscope was published in 1893, and is still used in modern microscopy. Figure 3.7(a) shows the excitation and emission paths in an unfolded infinitycorrected epiillumination fluorescence microscope system. This means that the objective and condenser as indicated in Figure 3.7(a) are the same optical component, and that the condenser aperture has the same size as the back focal plane of the objective, i.e. the exit pupil of the objective. The optical path of the excitation light is drawn from the light source (Hglamp) to the image plane. Right before the second tube lens, at the position of the emission filter, the excitation light is blocked. This is the reason why the rays of the excitation light are drawn as dashed lines. The emission light originates at the object plane, because in that plane fluorescence is generated. Only the cone of light that is captured by the objective and imaged onto the image plane is drawn. By reversal of rays, the propagation of the emission light from the object plane towards the light source can also be drawn. These rays are drawn as dashed lines. In Figure 3.7(a) both tube lenses are identical, but there are two separate tube lenses in the microscope. The size of the condenser aperture determines the extent of the light source that is being used for excitation. The size of the field stop determines the size of the fieldofview. The fieldofview is homogeneously illuminated. Effects of closing the field stop. Figure 3.7(b) shows a microscope adjusted to Köhler illumination, identical to Figure 3.7(a), but here the aperture of the field stop is smaller. The following can be seen in Figure 3.7(b): 1. Closing the field stop results in a smaller fieldofview. Compare the image planes in Figure 3.7(a) and Figure 3.7(b). 2. The contribution of the light originating from a single point of the light source to the intensity of the excitation light in the object plane (in terms of excitation power per unit area) does not change. Only the total illuminated area gets smaller. The total extent of the light source is used for illumination of the fieldofview. 3. Since the total excitation power per unit area remains constant, the probability to generate fluorescence in the fieldofview remains constant as well.
33 CONVENTIONAL MICROSCOPY CONCEPTS 23 Collector Tubelens Condenser Excitation filter Objective Tubelens Emission filter Hg lamp Fieldstop Condenser Aperture Object plane Backfocal plane Image plane (a) This figure shows the optical paths of the excitation light and the emission light in a conventional epiillumination fluorescence microscope with an infinitycorrected optical system. This microscope system is adjusted to Köhler illumination. Collector Tubelens Condenser Excitation filter Objective Tubelens Emission filter Hg lamp Fieldstop Condenser Aperture Object plane Backfocal plane Image plane (b) This figure shows a microscope adjusted to Köhler illumination. The difference with Figure 3.7(a) is that the aperture of the field stop is smaller. Collector Tubelens Condenser Excitation filter Objective Tubelens Emission filter Hg lamp Fieldstop Condenser Aperture Object plane Backfocal plane Image plane (c) This figure shows a microscope adjusted to Köhler illumination. The difference with Figure 3.7(a) is that the aperture of the condenser is smaller. Fig. 3.7
34 24 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER 4. The collection efficiency of the objective, in terms of the maximum half opening angle of the cone of light that can be collected, is unaltered. The intensity of the collected emission light (in terms of power per unit area) remains constant. Effect of closing the condenser aperture. Figure 3.7(c) shows a microscope adjusted to Köhler illumination, identical to Figure 3.7(a), but here the aperture of the condenser is smaller. In practice, this is not possible, since an objective has a fixed Numerical Aperture, and thus the size of the back focal plane is fixed. As explained before, in fluorescence microscopy the condenser is the objective. This implies that condenser aperture equals the back focal plane of the objective and cannot be changed. Figure 3.7(c), however, clearly illustrates what happens: 1. The illuminated fieldofview does not change. 2. The contribution of the excitation light, that passes the condenser aperture, to the intensity of the excitation light in the object plane does not change. A smaller fraction of the extent of the light source, however, passes the condenser aperture. As a result the total excitation power in the object plane is smaller. 3. Since the total excitation power per unit area is smaller, the probability to generate fluorescence in the fieldofview is smaller. 4. Since the condenser aperture is related to the size of the back focal plane of the objective, the maximum half angle of the cone of light that can be collected gets also smaller. This implies that the collection efficiency of the objective gets smaller and fewer emission photons are captured. As a conclusion, in order to have the highest collection efficiency, high power objectives are needed. This implies a fieldofview on the order of the size of a single vial. As a consequence, the microarray should be readout sequentially, i.e. well per well. As described in the previous chapter, the fluorescence signals are in the shorter wavelength part of the spectrum. This will generate a large amount of stray light. Stray light originates in the excitation path. This noise contribution can be minimized by closing the field stop as much as possible. Köhler illumination gives rise to the maximum lateral resolution. We are, however, not interested in spatial resolution. With critical (nonuniform) illumination an image of the light source is focussed on to the object plane. This gives a relatively small area of the fieldofview that is intensely illuminated. Critical illumination increases the excitation power in a small region. 3.4 CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA The second component of our conventional microscope based microarray reader is a CCD camera. Our CCD camera is the Princeton Versarray 512B Backilluminated CCD camera with a Tektronic DB CCD chip. The CCD array consists of pixels. Each pixel is 24 24µm 2. The CCD array has a 1% fill factor. The physical dimensions of the CCD chip are mm 2. This camera has three different gain settings: low ( 1 2 ), medium (1 ), and high (2 ). This CCD camera offers the possibility for readout at 1kHz
35 CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA 25 or at 1MHz and has a 16 bit analogtodigital converter. The operating temperature can be set from room temperature down to 5 C. The CCD camera is controlled via an ST133 Controller from Roper Scientific. Images are collected and analyzed using Matlab. Matlab communicates with the CCD camera via the VinView software of Roper Scientific. In this section we will present the most important characteristics of this CCD camera. For these experiments we have used a number of wellknown procedures [8] Readout noise Readout noise is the lower noise bound of the CCD camera. It has the following properties: readout noise originates in the preamplifier of the CCD camera, is independent of the signal and the integration time, is additive Gaussian noise, defined in terms of the RMS error, depends on the readout rate, and is assumed to be stationary, and ergodic, i.e. the noise of all pixels is independent, identical and constant in time, such that the variances of the readout noise per pixel can be added. The Princeton Versarray offers the possibility to collect the data at two readout rates: 1kHz and 1MHz. Reading out the camera means that the shutter was kept closed during data collection, and that an integration time of seconds was used. The electronic gain was set to low. The CCD camera was cooled to 4 C during this experiment. Readout rate = 1kHz. In this experiment, each pixel of the CCD array is read out 1 times. This implies, that for each pixel a histogram of the readout levels can be constructed. Figure 3.8 shows four of these histograms for four different pixels of the CCD array. The histogram in Figure 3.8(a) corresponds to pixel (8,8) on the CCD array, the histogram in Figure 3.8(b) to pixel (54,8), the histogram in Figure 3.8(c) to pixel (8,54), and the histogram in Figure 3.8(c) to pixel (54,54). Gaussian distributions defined by the sample mean and the sample variance are overlaid to each of the histograms. These histograms indicate that the readout noise of the pixels is normally distributed. Each and every pixel, however, has a different average readout level (µ) and a slightly different readout noise level (σ): the readout levels vary from 12.9#ADU (AnalogtoDigital Unit) to 19.1#ADU (the first column of the CCD has an average readout level of #ADU ). The readout noise levels have an average value of σ = 1.14#ADU (the lowest readout noise level is.98#adu ). Figure 3.9(b) shows the readout noise levels of all pixels. The lowest value in this image corresponds to the value of the 5% percentile (σ = 1.6) and the highest value in this image corresponds to the value of the 95% percentile (σ = 1.28). This image shows that the readout noise has a positive gradient towards the lower half of the CCD array. This gradient is introduced by dark current. The total readout time of the full chip is approximately 2.5
36 26 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER relative frequency.35 µ= σ= relative frequency µ= σ= readout level of Versarray [#ADU] readout level of Versarray [#ADU] (a) (8,8) (b) (54,8) relative frequency µ= σ=1.142 relative frequency µ= σ= readout level of Versarray [#ADU] readout level of Versarray [#ADU] (c) (8,54) (d) (54,54) Fig. 3.8 Histograms of readout levels of four different pixels of the CCD array. The curves are Gaussian distributions defined by the mean µ and standard deviation σ. seconds. During this readout time dark current is collected as well. This dark current accounts fully for this gradient 1. Readout rate = 1MHz. The same experiment is repeated at the high readout rate. Figure 3.1 shows the histograms corresponding to the same pixels as in Figure 3.8. Again, each and every pixel has a different average readout level and a different readout noise level: the readout levels vary from 19.#ADU to 3.2#ADU (the first column of the CCD has an average readout level of #ADU ). The readout noise levels have an average value of σ = 2.19#ADU (the lowest readout noise level is 1.96#ADU ). Figure 3.11(a) shows the readout level of all pixels. The lowest value in this image corresponds to the value of the 5% percentile (26.#ADU ), and the highest value to the 95% percentile (27.4#ADU ). 1 The next section will show that the dark current at 4 C is on average 1.5#ADU /s/pixel (low gain). During the readout time of 2.5s 3.8#ADU of dark current is collected at the lowest rows of the CCD array. The extra variance due to this dark current equals the electronic gain G times the variance of the dark current. The former equals.1#adu /e as will be shown in Section 3.4.3, the latter equals the average value of the dark current, since this a Poisson process. Thus, this extra variance is =.38. The overall standard deviation σ due to readout noise and dark current is = 1.14#ADU.
37 CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA 27 (a) (b) Fig. 3.9 These images show the readout levels (left) and the readout noise level (right) of all pixels. The range in the left image is ( ADU ). The range in the right image is ( ADU ). The readout noise has a positive gradient towards the lower, righthand corner of the image. Figure 3.11(b) shows the readout noise levels of all pixels. The lowest value in this image corresponds to the value of the 5% percentile (σ = 2.11) and the highest value in this image corresponds to the value of the 95% percentile (σ = 2.27). This image shows the spatial distribution of the readout noise level per pixel. This spatial distribution is a Gaussian distribution with a standard deviation of.9#adu. The spatial distribution of the readout noise in a single read out image has a Gaussian distribution with a standard deviation of 2.17#ADU. The temporal distribution equals the spatial distribution. This implies that for a high readout level the readout noise is ergodic; reading out the CCD array at 1MHz introduces no spatially varying noise due to dark current Dark current Dark current consists of electrons induced by (thermal) vibrations, which cannot be distinguished from photoelectrons. The production of dark current yields a Poisson distribution and is dependent on the operating temperature of the CCD and the integration time. The dark current is proportional to exp ( ) E kt, with k the Boltzmann constant ( J/K), and T the absolute temperature in K: as a rule of thumb, the dark current doubles with a temperature increase of 6K. This implies that dark current can be drastically reduced by cooling. To measure the dark current flux in e /s/µm 2 the Princeton Versarray was set to the low readout rate (1kHz). With forced air and builtin Peltier cooling the Versarray can be cooled down to about 5 C. In this experiment the shutter was kept closed and series of 1 images
38 28 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER relative frequency.15.1 µ= σ=2.11 relative frequency.15.1 µ= σ= readout level of Versarray [#ADU] readout level of Versarray [#ADU] (a) (8,8) (b) (54,8) relative frequency.15.1 µ= σ=2.181 relative frequency.2 µ= σ= readout level of Versarray [#ADU] readout level of Versarray [#ADU] (c) (8,54) (d) (54,54) Fig. 3.1 Histograms of readout levels of four different pixels of the CCD array. The curves are Gaussian distributions defined by the mean µ and standard deviation σ. were taken at different temperatures in the range from 2 C to 4 C with steps of 2 C. For each operating temperature two integration times were used:.s, and 16.s. The series of images acquired with an exposure time of.s corresponds to reading out the CCD array. Images of the sample mean per pixel of each series were computed for each series. For each temperature, images of the dark current flux in #ADU /s were computed from the sample mean acquired with an exposure time of.s and 16.s. Figure 3.12 shows the image of the dark current rate at 4. C. The lowest value in this image corresponds to the 5% percentile (1.4#ADU /s) and the highest value to the 95% percentile (1.83#ADU /s). This implies that the dark current is larger than the readout noise for integration times longer than a second. The gradient of the dark current rate in Figure 3.12 indicates that the temperature of the CCD array is not uniform: the temperature on the left side is lower than the temperature on the right side of the CCD array. This is due to nonuniform performance of the Peltier cooling element(s). For all different temperatures in this experiment, images of the dark current rate as shown in Figure 3.12 are computed. The average of the dark current rate and the standard deviation of the dark current rate for these temperatures are shown in Figure 3.13.
39 CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA 29 (a) (b) Fig These images show the readout levels (left) and the readout noise level (right) of all pixels. The range in the left image is ( ADU ). The range in the right image is ( ADU ). Fig This image shows the spatial distribution of the dark current rate in #ADU /s. The lowest value corresponds to the value of the 5% percentile (1.4#ADU /s), the highest value to the value of the 95% percentile (1.83#ADU /s). The dark current rate on the right side of the CCD array is higher than on the left side Linearity and Sensitivity Linearity. To measure the linearity of the photometric response of the Princeton Versarray, the CCD camera was mounted on a brightfield microscope system. The microscope was adjusted to Köhler illumination. The fieldofview was empty. After the light level in the
40 3 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER Dark current flux (#ADUs/pixel/s) Temperature (degrees Celsius) Fig This graph shows the average dark current rate in #ADU /s in the temperature range from 4 C to 2 C. Table 3.3 Mean and standard deviation of the regression coefficient R 2 over the CCD array. Gain µ σ low medium high object plane was adjusted to comfortable viewing by the human eye (light intensity not too bright, not too dim), a neutral density filter with a transmission of 1% was placed right after the halogen lamp. Series of images were acquired at the three different gain settings of the camera with different exposure times ranging from.1s to 1.s. Each series consisted of 25 images. The average photometric response at a certain integration time and gain was computed. Figure 3.14 shows the photometric response for a single pixel for the three different gain settings. This figure shows that in the low gain mode just half of the dynamic range of the camera can be used. This is the reason why the low gain is given as.5. For each pixel on the CCD array the linearity of the photometric reponse is computed as the regression coefficient R 2 to the data, i.e. the photometric response as a function of the exposure time. Table 3.3 shows the average regression coefficient of the CCD array and the standard deviation of the regression coefficient for the three diffent gain settings. Images showing the regression coefficient for all pixels suggest artefacts in the respons of the CCD camera due to the nonuniform illumination. These artefacts are not present: the values of the regression coefficient differ only in the fifth decimal.
41 CHARACTERIZATION OF THE PRINCETON VERSARRAY CCD CAMERA 31 mean grey value [1 4 ADUs] HIGH MEDIUM LOW exposure time [s] Fig This graph shows the average photometric response as a function of the integration time for the three different gain settings of the CCD camera for a single pixel. Sensitivity. The relative sensitivity or electronic gain G in grey level per photoelectron (#ADU /e ) can be measured using statistical methods under the assumption that the CCD camera is photon limited. That is, Poisson noise, due to the quantum nature of light, is the dominating noise source. The grey level I of a pixel is proportional to a number of collected photoelectrons N e according to I = G N e, (3.2) where G is the electronic gain. The number of photoelectrons N e is drawn from a Poisson distribution: the variance of the number of photoelectrons Var[N e ] acquired in an integration time T equals the expected value of the number of photoelectrons E[N e ] acquired in an integration time T. From this follows: { E[I] = G E[N e ] Var[I] = G 2 Var[N e ] = G 2 (3.3) E[N e ] = G E[I] The sensitivity G follows now as: G = Var[I], (3.4) E[I] where E[I] can be computed as the average value of a pair of images and Var[I] can be computed as Var[I] = 1 2 Var[I 1 I 2 ]. (3.5) A series of pairs of images is acquired at different light intensities. The light intensity is varied by placing different neutral density filters in the optical path. In each image the same uniform
42 32 CONVENTIONAL MICROSCOPEBASED MICROARRAY READER variance [14 ADUs2] HIGH:.36 ADU/e MEDIUM:.18 ADU/e LOW:.9 ADU/e mean grey value [14 ADUs] Fig This graph shows the measured pixel variance Var[I] as a function of the average value E[I]. The slope of these curves is the electronic gain or relative sensitivity. region was selected for computation of the pixel variance Var[I] and average value E[I]. This experiment is repeated for the three different gain settings. The results of these measurements are shown in Figure Table 3.4 gives the value of the electronic gain for three settings of the CCD camera. Table 3.4 The electronic gain G in #ADU /e for the different gains. gain setting electronic gain #ADU /e LOW.9 MEDIUM.18 HIGH.36 The same analysis can be used to compute the maximum signaltonoise ratio SNR: ( E 2 ) [I] SNR = 1 log = 1 log (G E[N e ]). (3.6) Var[I] Figure 3.16 shows the SNR as a function of the mean grey value, which is based on the same images as in the previous experiment. The curves in this graph show the theoretical SNR for the three different electronic gains G. The image based SNR is computed using the image
43 SUMMARY 33 SNR [db] pixel based SNRs (Photon limited) LOW MEDIUM HIGH 45 4 image based SNR mean grey value [1 4 ADUs] Fig This graph shows the SNRs as a function of the average intensity computed in a uniform region of an image for the three different gains. The curves are the Photon limited SNRs for these gains. The image based SNR is independent of the electronic gain. variance, instead of the pixel variance. This computation is independent of the electronic gain G. Here the variance is limited by the variation in pixel gain. 3.5 SUMMARY This chapter described the how the microarrays are fabricated and what methods have been developed in order to fill these microarray with solutions containing fluorescent molecules. The microarrays can be read out with a conventional micoscope system equipped with a scientific grade CCD camera. High power objectives amount to the maximum excitation and collection efficiency. This implies sequential scanning of the microarray. Closing the field stop reduces the contribution of stray light. A backilluminated CCD with an average Quantum Efficiency of approximately 7% in the visible part of the spectrum guarantees an efficient collection of the low wavelength emission photons of NADH. In the next chapter, we will look in detail to fluorescence measurements with this experimental setup.
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45 4 Quantitative Reading of Microarrays Chapter 2 introduced the biological challenge of this research program: development of an analytical tool based on microarrays for integral analysis of metabolic or regulatory networks. The demonstrator application for our developed microarray technology is the glycolytic pathway, i.e. the production of ethanol and carbon dioxide from glucose by yeast. This analytical tool is based on microarrays fabricated in silicon. The fabrication process as well as the different nanoliter liquid handling procedures are described in the previous chapter. A conventional microscope system equipped with a scientific grade CCD camera is used to measure the fluorescent signals generated in the biochemical reactions in all vials of the microarray. The most important concepts of quantitative microscopy are also described in the previous chapter. This chapter will describe in detail what the limiting factors are for measuring low levels of fluorescence in these microarrays, and what modifications to our microscope system we have implemented in order to overcome this limitation. 4.1 LIMITING FACTORS OF THE INSTRUMENT Reading the microarray, i.e. measuring fluorescence levels from all the vials, is hampered by many unwanted factors from: (a) the sample (biological / biochemical variations), (b) the optical system, and (c) the detector. The latter two characterize the instrument and are independent of the sample. These variations should be smaller than the biological / biochemical variations. There are two possible limiting factors (noise sources) that determine the detection limit of the system, i.e. the lowest fluorescence concentration that can be distinguished from a blank (control) solution. The first limiting factor of the detection limit is stray light. Stray light originates in the optical (excitation) system by reflections (flare) at airlens interfaces and 35
46 36 QUANTITATIVE READING OF MICROARRAYS scattering at diffuse surfaces within the optical system (glare). Other contributions to stray light are (Raman) scattering in the solvent and autofluorescence in the objective. Furthermore, there are contributions to the fluorescent signal from nonspecific (auto)fluorescence of the solvent, or ingredients. Not only the deterministic part of stray light is important, but in particular the stochastic part of the stray light. This contribution of stray light is not only due to Poisson statistics, but more likely to variations of the interfaces of the microscope slide, the bottom of the microarray, and airbubbles in the solvent. The second limiting factor is at the detection side of the microscope system: readout noise. Readout noise originates in the preamplifier of the CCD camera. The readout noise is a lower bound of the camera noise. Minimizing the largest noise contributions will result in a lower detection limit of the microscope system. 4.2 DETECTION LIMIT OF THE MICROARRAY READER We will take a closer look at fluorescence measurements to determine the detection limit. We assume that the concentration of fluorescent molecules is low to avoid serious attenuation of the excitation light deeper in the sample and to avoid reabsorption of the emitted fluorescence: in other words a linear relation between the excitation power and the emission power. Figure 4.1(a) shows the response of a detection system as a function of the concentration of the fluorescent particles. The dashed lines indicate the distribution of the blank solution. The solid lines indicate the distributions of the different fluorescent solutions. These distributions imply that the blank solution as well as the fluorescent solutions are measured with a certain precision and accuracy. Both precision and accuracy are defined by the system, i.e. we do not consider here the biological variation. The accuracy of the fluorescent solutions equals the accuracy of the blank solution. This means that a possible bias (background signal) can be removed by subtracting the average blank signal from the fluorescent signals. This does not correct for the stochastic variation. We assume in this figure that the probability density functions of the blank solution and the fluorescent solutions can be estimated extrinsically during a calibration procedure, i.e. each concentration can be independently measured many times on the same microarray (builtin redundancy). The results of this calibration procedure are a probability density function of the instrument s response to a blank solution and probability density functions of the instrument s responses to various fluorescent solutions. Figure 4.1(b) shows the definition of the detection limit. We define the minimum detectable signal such that 97.5% of all blank solutions will be classified as blank and only 2.5% will be erroneously treated as originating from the fluorescence process to be measured. This means that the strength of the minimum detectable signal equals the mean signal of the blank solution plus two times the standard deviation of the signal of the blank solution. If we choose the minimum detectable signal as the mean of the blank plus three times the standard deviation of the blank, then in 99.5% of the measurements of a blank solution, these measurements will be classified as a blank solution. As a consequence, the probability that a fluorescent solution with a concentration corresponding to the detection limit will produce a signal larger than the minimum detectable signal is 5%. When we measure the fluorescent signal of a solution in a vial, we need to estimate the underlying concentration of fluorescent particles. In other words, which concentration (and
47 DETECTION LIMIT OF THE MICROARRAY READER 37 Detector response [#ADUs] mean of fluorescent solution distribution of fluorescent solution kσ blank mean of blank (C=) solution distribution of blanco solution ConcentrationC of dye molecule [M] (a) Due to instrumental variations the response to a concentration of fluorescent particles has a certain distribution. Detector response [#ADUs] expected response to fluorescent solution kσ blank minimum detectable signal mean of blank (C=) solution distribution of blanco solution ConcentrationC of dye molecule [M] detection limit (b) The minimum detectable signal equals average response of the blank plus k times the standard deviation of measuring the blank. Fig. 4.1 Fluorescence response model.
48 38 QUANTITATIVE READING OF MICROARRAYS corresponding expected flux λ) has the highest probability of producing this measurement. For a single measurement this amounts to assigning the concentration according to the estimated calibration curve. In statistical terms, for one measurement as well as for many measurements, we select the molecular concentration that maximizes the a posteriori probability that a fluorescent solution λ generates this response: argmax λ P (λ X) = N p λ (x i ), (4.1) assuming equal a priori probabilities. Note that averaging measurements results in a probability density function with a variance (of the sample mean) that is inversely proportional to the number of measurements. For an infinite number of measurements, the sample mean equals the expectation value of the probability density function. As a conclusion, repeating the number of measurements will result in a lower detection limit. In theory, it is possible to distinguish between a blank solution and a solution with a single fluorescent particle under the assumption that a very large number ( ) of measurements can be performed. Note that an exact measurement does not contribute to an exact interpretation of the underlying biology / biochemistry, since they will still be limited by the biological variation. i= Improving the Detection Limit (for 1 and N measurements!) From Figure 4.1(a) the following possibilities to improve the detection limit can be derived. 1. Reduce the variation of measuring the blank solution. The variation of the blank has contributions of stray light (i.e. scatter, autofluorescence, reflections, and nonspecific fluorescence) and detector noise (i.e. dark current, and readout noise). Contributions of straylight can be minimized by closing the field stop as far as possible and only illuminate (a fraction of) a single vial. The noise contributions of the detector can be minimized by cooling the CCD and imaging the emission photons onto as few pixels as possible. e.g. using onchip binning. 2. Improve the sensitivity of the detection system. This is achieved by using high power objectives. The area of the field of view is proportional to 1/M 2, where M is the magnification of the objective. As a result, the excitation light intensity per unit area in the fieldofview is proportional to M 2. The detection (emission) efficiency is mostly influenced by the Numerical Aperture of the objective. The light gathering power of an objective in Köhler illumination is proportional to NA 2 /M 2 [9]. Under the assumption that there is a linear relation between excitation and emission energy, we can say that the intensity of the emission light is proportional to M 2. Combining these results, it follows that the light gathering power of an objective (with epiillumination adjusted to Köhler illumination) is proportional to NA 2. In the case of readout noise limited detection, the Quantum Efficiency of the detector is of equal importance to the readout noise. In the case of stray light limited detection, a better detector does not help, since it will increase the noise level as well as the signal level, leaving the SignaltoNoise Ratio unchanged. This means that the graph in Figure 4.1(a) will only be scaled in the vertical direction.
49 READOUT NOISE LIMITED DETECTION 39 Reducing the variation of measuring the fluorescent solutions implies a better accuracy. The measured variation of the fluorescent solutions contains contributions of stray light and detector noise. A third contribution to the variation of the fluorescent solutions is photon noise. Photon noise obeys a Poisson distribution due to the quantum nature of light. Repeating identical biochemical processes in different vials will generate the same Poisson process. A fourth contribution is caused by differences in injected sample volume. Differences in volume, and therefore different amounts of ingredients, will result in a different observed emission rate λ. If the differences of liquid volumes could be taken into account, e.g. with a dedicated electronic liquid volume sensor as presented in the second part of this thesis, or could be neglected, then at high fluorescent concentrations the limiting factor of the variation is photon shot noise. For low concentrations, the same argument holds for both the fluorescence solutions and the blank solution. In the following sections we will present possible modifications to overcome the limiting factors of our conventional microscope system and to improve the performance of the detection system in terms of the detection limit. 4.3 READOUT NOISE LIMITED DETECTION In Figure 4.1 it is assumed that the response of the detection system is linearly proportional to the emission power, and that fluorescence is a linear process, i.e. the emission power depends linearly on the excitation power. This assumption is true for low excitation power and low concentrations of fluorescing particles [1]. Furthermore, this figure suggests that the variation of the fluorescence measurements grows linearly with increasing concentrations. This linear behavior is not true, and this figure is only drawn this way for clarity. In the following sections we will estimate the strength of the minimum detectable signal in the case that readout noise is the limiting factor. In the previous section a definition for the minimum detectable signal was derived: the mean of the blank µ plus k times the standard deviation σ of the blank distribution (I min = µ blank + kσ blank ). Another statistic of interest is the coefficient of variation CV = σ/µ. We will show that the CV can be improved either by repeating the measurement n times in the same vial with individual measurement time T (n T ), or by increasing the measurement time T to nt (1 nt ). The result of this improvement in CV, however, has no biological / biochemical meaning, since it only implies a better estimate of the biochemical process in that single vial, characterized by the emission rate λ vial in that vial. It does not say anything about the biological variation of the process involved, characterized by the probability density function of λ Ideal Detection System First, we will consider the case that we have an ideal imaging and detection system: there are no external noise sources, the system consists of perfect components, and each emitted photon that enters the detection system is detected. This means true photon counting. The quantum nature of light shows that the number of photons emitted in a fixed time interval T varies. This number obeys a Poisson distribution where σ 2 = µ. Therefore, the exact
50 4 QUANTITATIVE READING OF MICROARRAYS concentration of fluorescent molecules cannot be measured: the coefficient of variation for a single measurement with this ideal detection system yields: CV single ideal = σ µ = λt λt = 1 λt, (4.2) where λ is the emission rate of the fluorescent solution in photons per second. If we want to discriminate a fluorescent solution from a blank solution (λ = ) with this ideal system within a measurement time of T = 1s, then only a single photon must be detected. If we want to detect this single photon with a probability of 99%, then the probability of not detecting a single photon is 1%: P(n = ; λt ) = e λt (λt ) n = e λt =.1. (4.3) n! In terms of Poisson statistics this translates to an emission rate λ = 4.61photons/s. This single measurement has a CV = 47%, which is much larger than the biological variation. Repeating this measurement n times (n T ) or increasing the measurement time T to 1 nt, changes the coefficient of variation into: CV n T 1 nt ideal = CV ideal = 1 CV single n ideal = 1, (4.4) nλt the CV improves with the square root of the number of measurements. With n = 87 the CV of these measurements becomes 5%. Note that either repeating the measurement n times or increasing the measurement time n times amounts to the same improvement of CV due to the absence of detector noise Single Element Detection System with Readout Noise Next, we will replace the ideal detector with a more realistic detector: a detector with readout noise. Readout noise originates in the preamplifier of the sensor (photodiode or CCD camera). The readout noise is independent of the signal and the integration time, but it depends on the readout rate. Readout noise is additive Gaussian noise with zero mean. The readout noise of the sensor, in units of electrons, is in terms of the RMS error at a certain readout frequency, e.g. the Photometrics Series 2 CCD camera has a readout noise level σ r = 11.7e [11] and the Princeton Versarray 512B backilluminated CCD camera has a readout noise level of σ r = 12.4e as shown in the previous chapter. This translates to the same number of photon equivalents, assuming a Quantum Efficiency of 1%. To avoid that reading out the camera results in a negative number of photons, a constant bias n bias is added to all signals: reading a blank solution will not result in zero photons, but in a number of photons related to the normal distribution of the readout noise with a constant bias added. The probability density function related to the blank solution is a gaussian distribution with mean n bias and standard deviation σ r : N(µ = n bias, σ = σ r ). In the case of reading a fluorescent solution, there is first the probability that the fluorescent solution generates n fl photons given an emission rate λ, and secondly the probability that n d "photons" are detected, given that n fl photons were emitted. To be more precise: each value
51 READOUT NOISE LIMITED DETECTION 41 Blank: + n bias Readout noise N(,σ r ) N(n bias,σ r ) Output: n=n(n bias,σ r ) SINGLE MEASUREMENT!!! n bias Bias: n bias n bias Fluorescence: P(n fl ;λt) n fl + n bias +n fl Readout noise N(,σ r ) N(n bias +n fl,σ r ) Output: n=n(n bias +n fl,σ r ) Fig. 4.2 Reading a blank solution with a detector with readout noise results in drawing from the distribution N(n bias, σ r). Reading a fluorescent solution is first drawing from a Poisson distribution defined by the emission rate λ and the measurement time T, and then drawing from the readout noise distribution N(n bias + n fl ). of n fl has a probability of occurring equal to P (n fl ; λt ). The distribution N(n bias + n fl ; σ r ) of the readout noise is centered around n = n bias + n fl and weighted with this probability. Figure 4.2 tries to make this clear for a single measurement. Since the readout noise does not depend on the signal itself, this implies that the overall probability density function of reading a fluorescent solution is the convolution of a Poisson distribution given by the emission rate λ and the measurement time T and a normal distribution defined by the RMS error of the readout noise σ r : p(n; λt, σ r ) = p(n fl ; λt ) N(n bias ; σ = σ r ), (4.5) where is the convolution operator. The probability P (n d n fl ) is a drawing from this distribution. For λt large enough, this convolution results in an approximately normal distribution with mean n bias + λt and standard deviation σ f = σ 2 r + λt : N(µ = n bias + λt, σ = σ 2 r + λt ). In this case, the absolute minimum detectable signal equals n bias + kσ r. We are, however, only interested in the relative minimum detectable signal kσ r, i.e. we subtract the average blank level. This translates to an emission rate λ = 36photons/s, given k = 3, T = 1s, and σ r = 12e. The CV of a single measurement with this nonideal singleelement detector is σ CV single single = 2 r + λt, (4.6) λt which is 37.3%. Repeating this experiment n times (n T ) yields CV n T single = 1 CV single n single = 1 σ 2 r + λt. (4.7) n λt With n = 56 measurements it follows that CV = 5%. Increasing the measurement time T to 1 nt yields σ CV 1 nt single = 2 r + nλt. (4.8) nλt
52 42 QUANTITATIVE READING OF MICROARRAYS With n = 14.2 follows that CV = 5%. A longer measurement time is superior to repeated measurements in terms of CV Many Pixels Detection System with Readout Noise Finally, the point sensor is replaced by a CCD camera, which is built as a 2D array of photosensitive elements. Each element can be characterized as a point sensor. In this case, the photon detector is a CCD camera consisting of a large number of pixels, M. Each and every pixel suffers from readout noise. We assume here that the readout noise is stationary: statistical properties do not change over time, and therefore do not change from measurement to measurement, and ergodic: statistical properties derived from an ensembe of measurements (= #pixels) are identical to the properties derived from a single element in a series of measurements. i.e. the noise of all pixels is independent, identical and constant in time, such that the variances of the readout noise per pixel can be added: σ 2 r,m = M σr 2 = Mσr. 2 (4.9) m=1 Reading a blank solution will result in a normal distribution with mean M n bias and standard deviation M σ r : N(µ = M n bias, σ = M σ r ). The fluorescent signal is spread over all pixels. This results in a spatiotemporal Poisson distribution. The variance of the fluorescent signal summed over all pixels σf,m 2 is given by M M ( σf,m 2 = σf 2 = σr 2 + λt ) = Mσr 2 + M i=1 i=1 M i=1 λt M = Mσ2 r + λt, (4.1) where λt/m means that the total emission photon flux λt is spread over M pixels. Reading a fluorescent solution many times results in a normal distribution N(µ = M n bias + λt, σ = Mσr 2 + λt ). For this detector the relative minimum detectable signal equals k Mσ = λt = photons with k = 3, and M = 1 4 and σ r = 12e. The CV of this single measurement with this CCD camera yields: CV single M = which is 33.4%. Repeating this experiment n times yields Mσ 2 r + λt, (4.11) λt CV n T M = 1 CV single n M = 1 Mσ 2 r + λt. (4.12) n λt With n = 45 it follows that CV = 5%. Increasing T to 1 nt yields Mσ CV 1 nt 2 M = r + nλt. (4.13) nλt
53 STRAY LIGHT LIMITED DETECTION 43 CV [%] T = 1 s σr = 12e M = 14 pixels Ideal system (Poisson limited) λ=4.6 photons / s increased T Single element detector λ = 36 photons / s Many (1 4 ) element detector λ = 36x 13 photons / s #repeats #repeats, c.q. increase T Fig. 4.3 The different CV for different configurations as a function of the number of repeated measurements or increased measurement time. With n = 6.7 it follows that CV = 5%. As a conclusion, measuring low levels of fluorescence requires multiple measurements in order to achieve a sufficient CV. Furthermore, the minimum detectable signal can be lowered by imaging the fluorescent signal from the vial onto as few pixels as possible. Thus, measuring very low levels of fluorescence is at the expense of spatial resolution. Further on in this chapter we will show through modifications of the emission path of a fluorescence microscope, how this can be achieved. Furthermore, we will estimate the minimum detectable signal in photons for different configurations based on the model presented in this section. Figure 4.3 summarizes these conclusions: with an ideal detector very low levels of fluorescence can be detected at the expense of precision. A CCD camera, on the other hand, can detect fluorescence much more precise, but with a much lower sensitivity. 4.4 STRAY LIGHT LIMITED DETECTION In the case of stray light limited detection a better detector (higher QE, lower readout noise) does not result in a better detection limit: the detector yields not only a better response to the signal, but also to the noise, i.e. the SignaltoNoise Ratio will not improve. This means that Fig. 4.1(a) will only be scaled in the vertical direction and the detection limit will not shift to lower concentrations. If stray light (with a photon generation rate λ stray photons/s) is the limiting factor, then measuring a blank solution with a measurement time T will result in an average response of λ stray T photons, neglecting the bias n bias or the number of sensitive elements M. This is again a Poisson process: the variance of the blank measurements equals λ stray T. The
54 44 QUANTITATIVE READING OF MICROARRAYS strength of the absolute minimum detectable signal becomes I min = λ stray T + k λ stray T. Since the detection method is only based on counting photons, it is not possible to distinguish between a photon from a stray light source or a photon coming from the fluorescence process of interest. Measuring a fluorescent solution will result in an absolute average response of (λ + λ stray )T photons with a standard deviation equal to the square root of that number. The coefficient of variation for a single measurement under limitation of stray light is (λ + λ CV single stray )T stray =. (4.14) λt In this expression the average signal is corrected for the average blank response. Repeating the measurement or increasing the measurement time results in the same improvement of the coefficientofvariation as in the case of an ideal detector: CV n T stray = CV 1 nt stray = 1 CV single n stray = 1 (λ + λ stray )T, (4.15) n λt This expression for the CV is independent of any detector parameters. Note that we only include the deterministic component of the stray light. The amount of stray light is already minimized by using high power objectives, dedicated filter blocks and closing the field stop as far as possible. Another option to gain signal with respect to stray light is to use dyes with a much higher quantum yield. This can also be achieved by using other solvents. To avoid fast evaporation from open nanoliter reactors we have used a mixture of glycerol / water in the past. Practically all dyes, however, have a higher quantum yield in an aqueous solution. The massive parallel coarse filling procedure covers the vials and allows for using the appropriate buffer (aqueous solution) as solvent. This will result in a better quantum yield. Stray light could be avoided completely by detecting luminescence instead of fluorescence, where luminescence light is emitted in the absence of excitation light. Furthermore, the limiting factor of stray light can be overcome by exploiting other techniques, such as Fluorescence (Cross)Correlation Spectroscopy [12]. With this technique the fluorescence signal from a confocal volume is detected as a function of time and the correlation spectrum is computed. This technique is limited by Raman scattering of the solvent. The dynamic range of these techniques is from 1 9 M down to 1 12 M [12]. In principle, lower concentrations can be detected with this technique, but the measurement time is inversely proportional to the concentration of the fluorescent dye. 4.5 MINIMUM DETECTABLE SIGNAL FOR DIFFERENT CONFIGURATIONS We will now compute the strength of the minimum detectable signal for different configurations of uor system in the case of readout noise limited detection. These configurations differ in the optical emission path and in the type of detector. The excitation part of all configurations is the same. This implies that the generated emission photon flux is the same for all configurations. The area of each vial on the microarray is 2 2µm 2. From these examples we can predict the improvement of the detection limit for our new configuration, if readout noise limits the detection.
55 MODIFICATIONS TO THE MICROSCOPE SYSTEM The configuration used for the first experiments consisted of a Zeiss 2 /.75 FLUAR objective, a 1 camera mount, and the Photometrics Series 2 CCD camera (no binning). The vial of a microarray is imaged onto M = pixels (pixel size = µm 2 ). The readout noise per pixel is σ r = 11.7e. This translates to a minimum detectable signal of I min = k Mσ r = e with k = 3. NAD(P)H emits around 45nm. The Quantum Efficiency of this CCD camera is approximately 15% at this wavelength. The minimum detectable signal converts then to photons. The CV of this measurement is 5%. 2. The new configuration consists of a 2 /.75 objective. The tube lens is replaced by an inverted 5 objective. The overall magnification is 4. The Photometrics (no binning) is replaced by the Princeton Versarray back illuminated CCD camera (no binning). The vial will be imaged onto M = pixels. The readout noise per pixel is σ r = 12.4e. This translates to a minimum detectable signal k Mσ r = e. That is equivalent to photons (QE without binning. In theory this will translate to a detection limit which is a factor of 8 lower. The CV of this measurement is 24% 3. Another interesting configuration is the following: the optical configuration remains identical, but instead of the Princeton Versarray CCD camera, a photo multiplier tube is mounted on the microscope. The limiting factor for a PMT in photon counting mode is the dark current. The dark current is expressed in Equivalent Noise Input (ENI). A typical range for the ENI is W. This translates to.3 to 3 thousand photons per second [13]. For short integration times the minimum detectable signal for a PMT is on the same order of magnitude as for the new configuration. Experiments to be presented in the next chapter will show that our new configuration is, unfortunately, still operating under the limitation of stray light. The new configuration does not result in an improved detection limit, but due to the better collection efficiency, the fluorescence signals can be acquired much faster. This translates to a higher readout rate of the microarray. 4.6 MODIFICATIONS TO THE MICROSCOPE SYSTEM Figure 4.4 shows the modifications to our microscope system in terms of the excitation and emission light beams. Figure 4.5 shows photographs of the new configuration. The modifications consist of the following: 1. The amount of stray light is already minimized by closing the field stop as far as possible, such that only (a fraction of) a single vial is illuminated. Note, that this has no effect on the excitation power: only that excitation light is blocked, that would not lead to excitation anyway. We have chosen a (relatively) high magnification objective (2 /.75), such that the excitation power per unit area is large. The field of view is just a single vial. This implies a sequential scanning of the microarray. 2. The high NA objective implies a high collection efficiency. Furthermore, the trinocular tube with the eyepieces and the relay optics (M = 1 ) has been replaced by an inverted
56 46 QUANTITATIVE READING OF MICROARRAYS Excitation part Emission part Detection part Collector lens Tube lens 2x/.75 Condenser Fluorescing sample 2µm 2x/.75 Inverted 5x/.25 onchip binning Readout Hglamp Field stop Excitation filter Object plane Emission filter WD CCDchip Shutter (WD=9.3mm) Fig. 4.4 The excitation and emission beams of our modified conventional microscope system. The closed field stop minimizes the influence of stray light. The 2 /.75 objectives ensures large excitation power per unit area, and high collection efficiency. The inverted 5 /.25 objective enlarges the photon flux by a factor of 25. The CCD camera with large pixels (and onchip binning) implies a high sensitivity at the expense of spatial resolution. 5 /.25 Zeiss FLUAR objective. This enlarges the emission photon flux in photons per unit area by a factor of 25: we are not interested in spatial resolution, only in sensitivity. Both objectives are infinitycorrected objectives. This means that the exit bundle of the objectives is a parallel bundle. Placing the infinitycorrected objectives backtoback creates a proper imaging system. The size of the back focal plane of the 5 objective is a little smaller than the extent of the tube lens. The consequence is that the effective NA of the system is a little smaller. Another consequence of the smaller extent of the 5 objective is vignetting [14]. This can be avoided by closing the field stop far enough. In principal, instead of a 5 objective, an identical 2 objective could be used. Because of the distance between the mechanical shutter and the CCD array, 1cm, a very large working distance is necessary to make a sharp image. The 5 objective has a working distance WD = 9.3mm. 3. The scientific grade CCD camera with small pixels is replaced by a CCD camera with large pixels. This implies that only one tenth, the ratio of the pixel sizes, of the photon flux is necessary to get the same increase in signal level per pixel, given that both cameras have the same overall conversion factor from photons to gray level. This gain can be further increased by using onchip binning. This gain in signal at reduced readout noise level implies a better SNR.
57 MODIFICATIONS TO THE MICROSCOPE SYSTEM 47 Princeton Versarray 512B Back Illuminated CCD Zeiss Axioskop INVERTED Zeiss FLUAR 5x /.25 WD = 9.3 mm Zeiss FLUAR 2x /.75 (a) Nikon F to C mount adapter Tube for INVERTED Zeiss FLUAR 5x /.25 with Nikon Fmount (Replaces trinocular tube) Zeiss FLUAR 2x /.75 (b) Fig. 4.5 The upper photo shows the microscope: the trinocular tube is removed and replaced by an inverted 5 /.25 Zeiss FLUAR objective. A dedicated mount connects the Princeton Versarray CCD camera to the microscope. The lower photo shows the camera mount and inverted objective in detail.
58 48 QUANTITATIVE READING OF MICROARRAYS 4.7 CONCLUSIONS AND DISCUSSION This chapter discusses the measurement of absolute fluorescence signals in a conventional microscope system equipped with a scientific grade CCD camera. Two limiting factors for the detection limit can be distinguished: stray light originating in the excitation part of the microscope system or readout noise in the photon detection part of the system. This chapter showed possible modifications to a conventional microscope system to improve the sensitivity in terms of the detection limit. This chapter showed that measuring very low levels of fluorescence is at the expense of spatial resolution. Measuring low levels of fluorescence requires multiple measurements in order to achieve a sufficient CV. The modifications to our microscope system consisted of a lower overall magnification by replacing the tube lens in the microscope with an inverted 5 objective and replacing the detector with a new CCD camera with large pixels and a higher Quantum Efficiency. In the case of readout noise limited detection, these modifications should result in a detection limit which is about two orders of magnitude better than our unmodified configuration. In the case of stray light limited detection, these modifications result in a faster scanning repetition rate of the microarray. The next chapter will show the performance of our modified microarray reader, and will demonstrate the functionality of this system by monitoring NADH consuming and producing enzymecatalyzed reactions in our microarrays.
59 5 Monitoring Enzymecatalyzed Reactions in Microarrays This chapter presents results of a number of experiments that have been performed with the microarray reader described in the previous chapter. First, we show that the modified optical system produces sharp images. Secondly, we determine the dynamic range of our detection system for the fluorescent molecule NADH. Furthermore, we have monitored one NADH producing and one NADH consuming enzymecatalyzed reaction in our microarrays. The experimental results demonstrate that our microarray reader can be used to monitor all the reactions in the glycolytic pathway. In the end, this system can be used as an analytical tool to gain insight in the metabolic regulation of the glycolytic pathway. 5.1 IMAGING WITH THE MODIFIED OPTICAL SYSTEM. First, we will show that our modified microscope system produces sharp images. Note that we are not interested in spatial resolution: replacing the tube lens with an inverted 5 /.25 FLUAR objective introduces aberrations. The inverted 5 objective demagnifies the image. Therefore, ordinary test patterns for microscopy are not suitable. As a simple test pattern a series of lines each with a thickness of.1mm and 5 divisions per millimeter (linetoline spacing is.2mm) was printed on plain paper (8gr/m 2 ) by a HP laser jet 5m at 6dpi. This test pattern was placed under the microscope and observed with transillumination. Figure 5.1(a) shows an image of this test pattern acquired with the modified microscope system. As can be seen in Figure 5.1(a) the fieldofview of the camera is much larger than the fieldofview of the optical system. Figure 5.1(b) shows the intensities along a line in the image of Figure 5.1(a). From this figure we conclude the following: 49
60 5 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS Practically no photons reach the detector in the area outside the fieldofview of the optical system. The fieldofview is not homogeneously illuminated. This nonhomogeneity is also present in an unmodified microscope system adjusted to Köhler illumination [15]. A rough estimation of the sampling density follows from the distance between the two outer minima in Figure 5.1(b). The physical distance is 1.2mm. In the image this corresponds to a distance of 19 pixels. The sampling density is approximately 91mm 1. Given the pitch of 24µm, it follows that the overall magnification of the modified optical system is approximately 2.2. One would expect an overall magnification of 4. The working distance of the 5 objective is "only" 9.3mm. The distance between the mechanical shutter of the CCD camera and the CCD array itself is larger than this working distance. According to Equation 3.1, the overall magnification in an infinitycorrected optical system was defined as the ratio of the focal lengths of the two lenses in the system. Our system effectively uses longer focal lengths. This amounts to a smaller ratio, and thus a lower overall magnification. The diameter of the fieldofview of the optical system corresponds to 181 pixels or 2.1mm. 5.2 NADH CALIBRATION As mentioned in Chapter 2 all enzymatic reactions of the glycolytic pathway can be monitored via the production or consumption of the fluorescent molecule NADH. In some reactions the enzyme activity is measured directly via NADH, in other reactions it is measured via two coupled reactions. An example of a direct reaction is the reaction in Expression 2.1, the reactions in Expression 2.2 are an example of coupled reactions. First, we have performed experiments to determine the dynamic range of NADH concentrations that we can detect with our microarray reader. The detection system we have used in the experiments is the microarray reader described in Section 4.6 with a 2 /.75 FLUAR objective and a 5 /.25 FLUAR objective placed backtoback, both from Zeiss. Filterset 1 from Zeiss (excitation: BP 365 / 12, beam splitter: FT 395, and emission: LP 397). The Princeton Versarray 512B CCD camera is mounted on the microscope with an inhouse built camera mount. Solutions of NADH were prepared by serial dilution in Millipore demineralized water. A microarray with vials with dimensions 2 2 2µm 3 is filled with one of the NADH solutions using the massive parallel filling technique described in Section 3.2. With this technique a microscope slide is shifted over the microarray to avoid fast evaporation; this slide serves as a cover slip. For the experiments we used a microscope slide cut from a Pyrex wafer. The thickness of this microscope slide is.5mm. This material has a transmittance of almost 95% for light with a wavelength of 36nm. As a result, reflections of the excitation light are much less on this material than on an ordinary microscope slide. All 25 vials of the microarray are read sequentially. In order to minimize the contribution of stray light the field stop is closed as far as possible. As a result only a fraction of the vial is illuminated. Figure 5.2 shows an
61 NADH CALIBRATION 51 (a) Intensity in % line position in pixels (b) Fig. 5.1 The upper image shows an image of a test pattern with lines (linetoline distance is.2mm). The fieldofview of the camera is much larger than the fieldofview of the optical system. The bottom graph shows the intensities along a line in the upper image.
62 52 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS 4µm Fig. 5.2 This image shows a vial of a microarray filled with an NADH solution. The spot of the emission light is smaller than the extend of the vial. Transillumination is used to make the edges of the vial visible. image of a single vial etched in silicon with dimensions of 4 4µm 2. For acquisition of this image epiillumination as well as transillumination is used. The silicon wafer is not completely opaque, but a little transparent. The light from the halogen lamp makes the edges of the vial visible and also causes the bright spot in the left region. The bright spot in the vial is the emission light generated by the NADH molecules in the vial. It is clear that the size of the illumination spot is smaller than the size of the vial. Figure 5.3(a) shows the results of these measurements. The data points are the average intensities of the 25 vials, the error bars correspond to a 3σinterval. From this graph, we conclude that there is a linear response for NADH concentrations up to a concentration of 1mMolar NADH. The gain setting of the CCD camera in this experiments was set to low ( 1 2 ). The dynamic range of the CCD camera with this gain setting is approximately #ADU. From the graph in Figure 5.3(a) it is not possible to determine the detection limit for NADH. Figure 5.3(b) shows the same data on a logarithmic scale. The average response of the blank solution is subtracted from the other data points. The minimum detectable signal is drawn in this graph as three times the standard deviation of the blank solution. The minimum detectable signal crosses the calibration curve at a concentration of 5µMolar. This NADH concentration is the detection limit of our microarray reader. The dynamic range of NADH ranges from 5µMolar to 1mMolar. As expected, the minimum detectable signal is far above the readout noise of the detector. We conclude that the microarray reader is operating under limitation of stray light.
63 NADH CALIBRATION 53 Detector response [1 #ADUs] Volume vial = 2 x 2 x 2 µm 3 Exposuretime =.1 s (no Binning, 1x gain) Cover slide = Pyrex ±3σ interval Concentration NADH [µmolar] (a) Detector response [1 #ADUs] Volume vial = 2 x 2 x 2 µm 3 Exposuretime =.1 s (no Binning, 1x gain) Cover slide = Pyrex Datapoints: averaged over 25 different vials ±3σ interval minimum detectable signal 3σblank Detection Concentration NADH [µmolar] limit = 5 µmolar (b) Fig. 5.3 The upper graph shows the calibration for NADH measured with our microarray reader. The lower graph shows the same calibration curve on a logarithmic scale. The detection limit for NADH is 5µMolar.
64 54 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS 5.3 MONITORING ENZYMECATALYZED REACTIONS The following enzymecatalyzed re Glucose6phosphatedehydrogenase (G6PDH). action has been monitored in our microarrays: G6P + NAD G6P DH Gluconolactone + NADH, (5.1) where G6P is glucose6phosphate, and G6PDH is the enzyme glucose6phosphatedehydrogenase. In this reaction the fluorescent molecule NADH is produced. A cocktail of G6P, NAD, and activator MgCl 2, containing equal concentrations of G6P and NAD is prepared. The initial concentration is 2mM. The concentration MgCl 2 in the cocktail is 5mM. Solutions with different enzyme concentrations of G6PDH are prepared:.4u/ml,.2u/ml,.1u/ml, and.5u/ml. An enzyme concentration of 1U/ml equals an amount of enzyme molecules that convert 1µMol of substrate in 1ml solution per minute of time. The reaction will be initialized by mixing two equal volumes of the cocktail and the enzyme solution. After mixing, the substrate concentrations as well as the enzyme concentrations are reduced by a factor of 2. After bulk mixing, 5µl will be spread over the microarray. The microarray is shifted under a Pyrex microscope slide. After attachment, the microarray and the microscope slide are pressed firmly together for about half a minute. Reading out the microarray will start approximately two minutes after initialization of the reaction. A single scan of the 25 vial array takes on the order of half a minute. The exposure time / integration time per vial is.1s. The microarray will be continuously readout for a chosen number of scans. The raw results consists of an average intensity in #ADU measured in an image I n out of an image sequence n = 1, 2,, N. The image sequence, as well as the time are stored. Figure 5.4 shows the acquired image sequence of a single vial during the reaction. This image sequence consists of 1 images. Figure 5.5 shows an acquired image sequence of a single vial in which evaporation occurs during the reaction. The latter measurement sequence is not used in the analysis. We assume that the enzymecatalyzed reaction is performed under pseudofirstorder conditions. This means that the following equation describes the NADH concentration: [NADH(t)] = [NAD()] { 1 exp ( t τ )}. (5.2) At t = no NADH has been produced and as t all NAD is converted to NADH and the concentration NADH at the endpoint of the reaction equals the initial concentration of
65 MONITORING ENZYMECATALYZED REACTIONS 55 Fig. 5.4 reaction. This figure shows an acquired image sequence of a single vial during an enzymecatalyzed Fig. 5.5 This figure shows an acquired image sequence of a single vial in which evaporation occurs during an enzymecatalyzed reaction.
66 56 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS intensity in 1 #ADUs time in s Fig. 5.6 This figure shows for 5 5 vials the acquired signals of the above mentioned reaction with an enzyme concentration of.25u/ml and the fitted pseudofirstorder reaction model. The italic numbers in the graphs are the values of the characteristic time τ in seconds NAD: [NADH( )] = [NAD()]. Figure 5.6 shows for 5 5 vials the acquired signals of the above mentioned reaction with an enzyme concentration of.25u/ml and the fitted firstorder reaction model. A nonlinear fit procedure (Matlab) using the LevenbergMarquardt optimization method has been used to fit this model to the data. Two extra parameters have been added to the model in Equation 5.2: a vertical offset and a time shift t. The italic numbers in the graphs are the estimated characteristic times τ for this reaction. This characteristic time τ indicates the amount of time in which the concentration NAD is reduced by a factor e: in a time τ after initialization of the reaction the NAD concentration is only 37% of the initial NAD concentration, and after 3τ, the NAD concentration is 5% of the initial concentration, and the NADH concentration has reached a level of 95% of the endpoint of the reaction. The following four tables (Table ) give the estimated characteristic times τ in 5 5 vials of a microarray. For all four experiments the same microarray has been used. After each experiment the microarray is cleaned to avoid crosscontamination. The final enzyme concentrations in the reactions are.25u/ml,.5u/ml,.1u/ml, and.2u/ml. The numbers between brackets are the values of the χ 2 merit function of the nonlinear fit procedure. Table 5.5 summarizes the experimental results from the previous four tables. The graph in 1 Figure 5.7 shows the values of the reciprocal of the measured characteristic times, τ as a
67 MONITORING ENZYMECATALYZED REACTIONS 57 function of the enzyme concentration [G6PDH]. This graph shows that our measurements correspond to the expected behavior of this reaction. Table 5.1 This table shows the measured characteristic times τ in s for the reaction in Equation 5.1. The enzyme concentration is.25u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit (39.9) 1614 (56.2) 1633 (7.1) 159 (34.3) 1594 (45.2) 1474 (42.2) 1529 (71.2) 1548 (46.1) 1646 (91.6) 1642 (79.3) 1542 (45.1) (66.1) 1628 (41.8) 1643 (63.) 1718 (65.1) 151 (33.8) 1554 (59.9) 1553 (32.5) 158 (53.2) 1297 (39.3) 1511 (49.7) 1521 (33.6) 166 (65.5)  Table 5.2 This table shows the measured characteristic times τ in s for the reaction in Equation 5.1. The enzyme concentration is.5u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit (6.8) 1149 (9.9) 19 (1.5) 1133 (9.2) 1229 (13.8) 1181 (5.4) 1265 (8.1) 1234 (7.) 1263 (9.3) 123 (8.2) 1169 (7.2) 114 (11.4) 124 (1.9) 1226 (12.9) 123 (12.) 1223 (9.7) 1231 (8.1) 133 (8.7) 1241 (13.9) 1171 (12.7) 1146 (8.6) 1242 (5.7) 1196 (14.3) 1147 (13.1) 1136 (11.2) Table 5.3 This table shows the measured characteristic times τ in s for the reaction in Equation 5.1. The enzyme concentration is.1u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit. Measurements are shown in Figure (15.8) 693 (7.) 145 (6.8) (7.9) 481 (13.4) 723 (9.3) 947 (13.23) 1166 (5.8) 1257 (6.2) 525 (8.5) 78 (8.4) 83 (7.5) (11.5) (7.1) 744 (7.7) 814 (7.3) 789 (1.5)
68 58 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS Table 5.4 This table shows the measured characteristic times τ in s for the reaction in Equation 5.1. The enzyme concentration is.2u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit. 351 (5.8) 346 (7.) 362 (6.) 378 (9.6) 374 (6.3) 363 (11.2) 355 (1.6) 311 (9.7) 358 (9.8) 351 (5.8) (8.3) 342 (5.) 359 (13.9) 396 (12.5) 458 (4.5) 333 (34.8) 337 (12.1) 334 (13.4) 267 (19.7) (6.5) 485 (49.4) 183 (23.7) 29 (23.8) Table 5.5 This table summarizes the results from the tables above: the first column gives the enzyme concentration, the second column gives the number of monitored reactions in a 5 5 microarray, the third column gives the mean characteristic time µ τ, the fourth column gives the standard deviation of the measured characteristic times σ τ, and the last column gives the coefficient of variation of the experiment CV τ Enzyme conc. reactions µ τ σ τ CV τ U/ml (out of 25) s s % /τ [s 1 ] G6PDH enzyme concentration [U/ml] Fig. 5.7 This graph shows the reciprocal of the measured characteristic times, 1 τ enzyme concentration [G6PDH]. as a function of the
69 MONITORING ENZYMECATALYZED REACTIONS 59 intensity in 1 #ADUs time in s Fig. 5.8 This figure shows for 5 5 vials the acquired signals of the above mentioned reaction with an enzyme concentration of.1u/ml and the fitted pseudofirstorder reaction model. The italic numbers in the graphs are the values of the characteristic time τ in seconds. Evaporation corrupted the experiment in 8 out of 25 vials.
70 6 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS Lactatedehydrogenase (LDH). monitored in our microarrays: The following enzymecatalyzed reaction has also been pyruvate + NADH LDH lactate + NAD, (5.3) where LDH is the enzyme lactatedehydrogenase. In this reaction the fluorescent molecule NADH is consumed. A cocktail of pyruvate, and NADH, containing equal concentrations of pyruvate and NADH is prepared. The initial concentration is 2mM. A stock solution with an enzyme concentration of.6u /ml is used to prepare solutions with different enzyme concentrations of LDH. The table below shows the amounts of cocktail solution, enzyme solution and H 2 O that are added together to initialize the above reaction. Table 5.6 This table shows the amounts of cocktail solution, enzyme solution, and H 2O that are added together to initialize the enzymecatalyzed reaction. The enzyme concentration in the stock solution is.6u /ml. cocktail H 2 O enzyme final enzyme conc. µl µl µl U/ml After bulk mixing, 2µl will be spread over a 2cm long region along the long edge of a Pyrex microscope slide. The microarray is shifted under the wetted part of the Pyrex microscope slide. After attachment, the microarray and the microscope slide are pressed firmly together for about half a minute. Reading out the microarray will start approximately three minutes after initialization of the reaction. A single scan of the 25 vials of the array takes on the order of half a minute. The exposure time / integration time per vial is.1s. The raw results consists of an average intensity in #ADU measured in an image I n out of an image sequence n = 1, 2,, N. The image sequence, as well as the time are stored. We again assume that the enzymecatalyzed reaction is performed under pseudofirstorder conditions. This means that the following equation yields for the NADH concentration: ( [NADH(t)] = [NADH()] exp t ). (5.4) τ
71 MONITORING ENZYMECATALYZED REACTIONS 61 intensity in 1 #ADUs time in s Fig. 5.9 This figure shows for 5 5 vials the acquired signals of the above mentioned reaction with an enzyme concentration of.1u /ml and the fitted pseudofirstorder reaction model. The italic numbers in the graphs are the values of the characteristic time τ in seconds. At t = no NADH has been consumed and as t all NADH is consumed: [NADH( )] =. Figure 5.9 shows for 5 5 vials the acquired signals of the above mentioned reaction with an enzyme concentration of.1u/ml and the fitted firstorder reaction model. One extra parameter to incorporate a vertical offset has been added to the model in Equation 5.4. The italic numbers in the graphs are the estimated characteristic times τ for this reaction. The following four tables (Table ) give the estimated characteristic times τ in 5 5 vials of a microarray. For all four experiments the same microarray has been used. After each experiment the microarray has been cleaned. The final enzyme concentrations in the reactions are.25u/ml,.5u/ml,.1u/ml, and.2u/ml. The numbers between brackets are the values of the χ 2 merit function of the nonlinear fit procedure. Table 5.11 summarizes the experimental results from the previous four tables. The graph in Figure 5.1 shows the values of the reciprocal of the measured characteristic times, 1 τ as a function of the enzyme concentration [LDH].
72 62 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS Table 5.7 This table shows the measured characteristic times τ in s for the reaction in Equation 5.3. The enzyme concentration is.25u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit (85.1) 1169 (135) 1132 (66.4) 924 (81.5) 1466 (78.7) 1732 (14) 124 (12) 1381 (89.4) 1133 (81.8) 1484 (8.7) (95.1) 1381 (15) 143 (69.6) 1154 (91.9) 1363 (77.) 1235 (61.2) 1248 (69.9) 1491 (66.) 1192 (8.8) 181 (124) 169 (84.6) 1215 (63.4) 1488 (53.3) Table 5.8 This table shows the measured characteristic times τ in s for the reaction in Equation 5.3. The enzyme concentration is.5u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit. 993 (32.5) 915 (33.6) 99 (26.9) 951 (35.8) 17 (27.4) 926 (27.6) 941 (23.9) 956 (31.4) 958 (23.6) 862 (36.6) 953 (26.8) 889 (27.8) 822 (3.1) 95 (33.1) 894 (36.6) 936 (33.1) 941 (3.3) 942 (28.1) 925 (29.7) 124 (25.6) 173 (26.6) 169 (23.8) 14 (22.8) 1114 (3.1) 1362 (57.6) Table 5.9 This table shows the measured characteristic times τ in s for the reaction in Equation 5.3. The enzyme concentration is.1u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit. 718 (8.1) 764 (1.4) 758 (11.9) 749 (1.2) 739 (12.4) 638 (9.8) 681 (11.1) 638 (23.5) 682 (8.7) 77 (8.6) 558 (19.) 611 (11.5) 589 (15.) 627 (9.3) 641 (14.7) 495 (19.9) 53 (23.5) 498 (22.1) 5 (23.1) 536 (22.2) 517 (22.8) 546 (17.) 572 (2.) 469 (37.5) 588 (13.7) Table 5.1 This table shows the measured characteristic times τ in s for the reaction in Equation 5.3. The enzyme concentration is.2u/ml. The numbers betweens between brackets are the values of the χ 2 merit function of the nonlinear fit (15.7) 21 (33.7) 193 (25.9) (11.3) (3.9) (13.) (18.8) (17.3)  25 (19.) 212 (19.) (21.4) 
73 MONITORING ENZYMECATALYZED REACTIONS 63 Table 5.11 This table summarizes the results from the above tables: the first column gives the enzyme concentration, the second column gives the number of monitored reactions in a 5 5 microarray, the third column gives the mean characteristic time µ τ, the fourth column gives the standard deviation of the measured characteristic times σ τ, and the last column gives the coefficient of variation of the experiment CV τ Enzyme conc. reactions µ τ σ τ CV τ U/ml (out of 25) s s % /τ [s 1 ] LDH enzyme concentration [U/ml] Fig. 5.1 This graph shows the reciprocal of the measured characteristic times, 1 as a function of the τ enzyme concentration [LDH].
74 64 MONITORING ENZYMECATALYZED REACTIONS IN MICROARRAYS 5.4 CONCLUSIONS We have shown that our modified microscope system produces sharp images. The overall magnification is approximately two times. By closing the field stop as far as possible, the amount of stray light is minimized and only a fraction of the vial is illuminated. The dynamic range of our system for NADH ranges from 5µM to 1mM. Our system has a linear response in this range. To demonstrate the functionality of our system, we monitored two enzymecatalyzed reactions at different conversion rates. One reaction produces NADH, and one reaction consumes NADH. The enzyme concentrations are given as enzyme activity levels: identical enzyme concentrations for two different enzymes amount to identical conversion rates, regardless if the reaction consumes or produces NADH. For this reason it was expected that the curves in Figure 5.7 and Figure 5.1 would have the same slope. Inspection of these curves shows that this is not the case. Possible causes for this deviation are that (a) the stock solution of one of the enzymes contained a different concentration, or (b) the difference in temperature in which the reactions were monitored. The reaction with G6PDH was monitored at 3 C, and the reaction with LDH at 26 C. We have used a pseudofirstorder kinetic model to describe the reaction. Although this model describes our experimental data very well, it is, chemical speaking, perhaps not the right model. Our model assumes that the reactions reach an endpoint, where all NADH is either produced or consumed. Other chemical models do not assume a "oneway" reaction, but a reversible ("twoway") enzymecatalyzed reaction: in that case the reaction reaches an equilibrium. A study to other chemical models falls outside the scope of this thesis.
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79 Part II Monitoring Dynamic Liquid Behavior in Micromachined Vials
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81 6 Dynamic Liquid Behavior in Micromachined Vials Evaporation is a key problem in the development of microarray technology, especially in the case of microarrays with open reactors. In order to establish completed (bio)chemical reactions and to be assured of sufficient readout time for acquiring all fluorescent signals from all vials on a microarray, the evaporation process should be extended as long as possible. In previous chapters successful solutions to overcome the problem of evaporation have been demonstrated. In this part of this thesis we will look in close detail at the phenomenon of evaporation in micromachined subnanoliter vials. We will not only describe the process of evaporation qualitatively in terms of some underlying physical principles, we will also introduce a microscopic technique, interferencecontrast microscopy, that allows quantitative monitoring of the evaporation process. First, we will look from different points of view to the process of evaporation to motivate this research. 6.1 INTRODUCTION Liquid pinning: ring stains from coffee droplets One would expect that the evaporation process of liquid droplets was described many years ago. On the contrary, just recently some underlying principles of the dynamic process of evaporation have been published, e.g. by Deegan et al in Nature [16] and in Physical Review [17]. Figure 6.1 shows the drying of a coffee droplet. Initially, the coffee particles were homogeneously distributed in the droplet, but during evaporation they become more and more concentrated at the edge, leaving a ring like stain. These rings are also found after drying of spotted DNA materials on microarray structures for Comparative Genomic Hybridization 71
82 72 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS "The thrill of the spill" (New Scientist) [18] (a) t = min (b) t = 1min (c) t = 2min (d) t = 3min (e) t = 4min (f) t = 5min Fig. 6.1 This figure shows six images of a drying coffee droplet. It can be seen that the droplet is pinned to its initial position, and that it leaves a ringlike stain after drying. (CGH). According to Deegan, ring formation as shown in Figure 6.1 occurs if and only if the following two conditions are satisfied: 1. the contact line of the drop is pinned to its initial position, and 2. evaporation takes place at the edge of the drop. On smooth teflon, no pinning of the contact line of the droplet occurs, and the drying droplet will contract: no ring will be formed after drying. Covering the drop with a lit with a small hole over the center, reduces the evaporation at the edge. Although there is contact line pinning, the deposit will be uniformly distributed, due to the reduced evaporation at the edge. Contact line pinning. In a static situation, without evaporation, the contact angle between the solvent and the surface on which the droplet lies, is determined by the balance between the surface tension of the solvent, which tries to pull the droplet into a sphere, and the adhesive forces, which try to attract the liquid to the surface. If the droplet lies on a perfectly flat and smooth surface, the edges of an evaporating droplet have to move inwards to maintain the required contact angle. This is shown in Figure 6.2(a). But real surfaces are not that smooth.
83 INTRODUCTION 73 (a) A contracting droplet on smooth surface. (b) A pinned droplet. Fig. 6.2 The top figure shows an evaporating droplet on a flat surface, which contracts to maintain the required contact angle. The bottom figure shows a drying droplet pinned to its initial position. The loss of liquid at the edge is replenished by liquid from the bulk of the droplet. If the surface is just a little bit rough, when the edge loses some liquid, it only has to move a little bit inwards to find a position, where the roughness of the surface gives it the right contact angle again. This way, it moves so slowly inwards that it is effectively pinned in place. This pinning occurs for many different solvents and many different surfaces. Pinning on the underlying surface prevents the droplet from shrinking. This implies that the footprint of the droplet remains constant. The evaporated liquid at the edge is replenished by liquid from the bulk of the droplet. This means that there is a flow of liquid moving outwards to the edge of the droplet. This is shown in Figure 6.2(b). The driving force of the outward liquid flow is the elasticity of the surface, which tries to minimize the surface area. As the solvent is evaporating at the edge, the outward flow of the solvent with particles, like in a droplet of coffee, will deposit the particles at the edge, which make the surface even rougher. As a result the edge is even more firmly pinned. Finally, when all the liquid has evaporated, the (coffee) particles will form a ring of stain. Deegan s model of diffusion limited evaporation. Deegan states that the functional form of the local evaporation rate along the airliquid interface depends on the rate limiting step of the evaporation process. If the ratelimiting step is the transfer rate across the liquidvapor boundary, then the local evaporation rate j(r) is constant. This implies that the total evaporation rate is proportional to the area of the airliquid interface. If the ratelimiting step is the diffusive relaxation of the saturated vapor layer immediately above the drop, then j(r) is strongly enhanced towards the edge of the drop. In this case the total evaporation rate is proportional to a characteristic length of the droplet. Deegan concludes that the evaporation process in his experiments is diffusion limited.
84 74 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS The diffusion goes from the saturated vapor layer (u s ) immediately above the droplet to the ambient vapor pressure (u ). The evaporation process will rapidly attain this steady state. The diffusion equation reduces then to Laplace s equation: 2 u = D u t =, j = D u, (6.1) where u is the mass of vapor per unit volume, and D is the diffusion constant for vapor in air. This boundary value problem has an electric analogon, if u is replaced by the electric potential V and j(r) with the electric field E. Futhermore, Deegan assumes that the pinned droplets have the shape of a spherical cap during evaporation (with a constant footprint of radius R). The solution to this boundary problem is well approximated by [17] j f(λ) j(r) = ( 1 ( ) ) r 2 λ ˆn, λ = π 2Θ c 1 2π 2Θ c 2, (6.2) R where Θ c is the contact angle of the droplet with the underlying surface, and ˆn the unit normal vector of the airliquid interface. From this expression follows that the rate of mass loss j(r) diverges towards the edge of the droplet (r R). Section will show observations of an evaporating Rhodamine solution in a micromachined vial. These observations show that the liquid is pinned in place to the edge of the vial and that the fluorescent particles first move towards the edges of the vial and finally towards the four corners of the square vial during the evaporation. This implies that there is evaporation at the edge of the vial. These observations in micromachined vials are in agreement with the observations of Deegan Modeling dynamic liquid behavior In a collaborative research project with the Institute for Microsystem Technology of the Albert Ludwigs University Freiburg and the Electronic Instrumentation Laboratory of the Delft University of Technology, Korvink et al have developed a physical model that describes the evaporation process in micromachined vials in terms of the shape of the airliquid interface during evaporation, the local evaporation rate, and the flow field in the liquid. In this section we will only summarize the approach to compute this phenomenon. Qualitatively speaking, the following processes take place after injection of a liquid sample in a vial. It is assumed that after injection the liquid "instantaneously" assumes its equilibrium shape s(t). In the next instant a fraction of the liquid evaporates. According to Korvink, the driving force of evaporation is the nonequilibrium that arises between the liquid phase and the gaseous environment: if the partial pressure of the gas molecules is not at its equilibrium value, it will lead to either evaporation or condensation. Korvink assumes that the current density of molecules through the gasliquid interface j is fully determined by the difference of the equilibrium partial pressure p s in the saturated gas phase and the actual value of the vapor pressure p d [19, 2]: j = 1 M α c (p s p d ) ˆn, (6.3) 2π RT
85 INTRODUCTION 75 where α c is the condensation coefficient of the material, M is the molar mass of the material, R is the gas constant, and T is the absolute temperature. 1 The above equation is only valid for a flat boundary separating two infinite halfspaces of the liquid phase and the gas phase. For a convex, respectively a concave gasliquid interface the vapour pressure increases respectively decreases. The vapour pressure p s of the saturated gas above a curved surface is a function of the mean curvature κ [19, 2]: ( ) 2mγlg p s (κ) = p s () exp k B T ρ κ, (6.4) where p s () is the vapour pressure for a flat surface with curvature κ =, m is the molecular mass, γ lg is the liquidgas surface tension, k B is Boltzmann s constant, and ρ is the mass density. This implies that the shape s(t) of the gasliquid interface defines the mass flux j from the liquid across the meniscus to the gaseous environment. The mass flux j varies with position, because of the spacevarying shape s(t). In the same instant of evaporation, the remaining liquid volume must compensate for its content loss by adjusting its surface shape s(t) 2. This means that there is a liquid flow inside the liquid phase to replenish the nonuniform loss of mass over the droplet surface. In order to avoid solving for the fullycoupled evaporation, surface evolution, and fluid flow, it is assumed that the surface evolution is sufficiently slow compared to the evaporation and the fluidflow. Choosing a timestep small enough effectively decouples the surface evolution from the fluid flow. This leads to the following (simplified) algorithm: 1. Given the mass M(t) of the liquid in the vial, compute the fluid shape s(t) at instant t. 2. Given the fluid shape s(t), compute the mean curvature κ(t) of the surface elements. The mean curvature is related the vapour pressure p s (κ) of the saturated gas above the surface according to Equation 6.4. The vapour pressure p s (κ) in turn is related to the mass flux density j according to Equation Compute the space varying mass flux density j(r) on the liquid surface. Integrate j(r) over the total surface S to compute the total mass change in the interval t. 4. Compute the remaining mass of the liquid M(t + t) = M(t) + t jd s. 5. Return to step 1, as long as boundary conditions are met. The algorithm has two computationally intensive operations: the computation of the instantaneous surface shape s(t) from the NavierStokes equations, and the computation of the space 1 In other words, Korvink assumes that the ratelimiting step of the evaporation is the transfer of molecules through the interface. As explained in the previous section, Deegan assumes that the ratelimiting step of the evaporation is the diffusion from the saturated vapor layer immediately above the airliquid interface. Our experiments will prove that Deegan s assumption is right. 2 The airliquid interface of a liquid sample in a circular shaped vial is a spherical cap as will be shown in Section9.4. This means that the curvature κ is constant over the airliquid interface. This implies a constant local evaporation rate j(r) and the total evaporation rate will be proportional to R 2 and varies as a function of time, when the meniscus changes from convex to concave. Our experiments will show that this is not the case. 3 Since this model describes the interface transfer limited evaporation, the actual value of the vapor pressure p d =
86 76 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS Fig. 6.3 This figure shows the initial surface s() of the liquid volume. This computation did not include the boundary condition that the liquid was pinned to the edge of the vial. varying mass flux j(s). Figure 6.3 shows the initial surface s() after injection of the liquid. The computation of the surface s() did not use the boundary condition that the liquid was pinned in place at the boundary of the vial. Verification of the computational results based on the model above, especially the shape s(t) of the airliquid interface, requires quantitative measurements of the physical quantities of interest. These quantities can be derived from microscopic image sequences acquired during evaporation of the liquid in the vials Impedancebased liquid volume sensor In the first part of this thesis, it has become clear that an electronic volume sensor is an important sensor for a labonachip. Such a sensor does not only indicate the presence of liquid in a vial, moreover it measures the amount of liquid in the vial. Silicon integrated microarrays make it possible to use an electrical method for detecting the presence and amount of liquid in a vial. Conventional silicon fabrication techniques enable the use of planar electrodes. This opens up the possibility for complete integration of the sensor electrodes and the electronics for a dedicated electronic volume sensor. Figure 6.4 shows a first generation of a microarray with integrated electrodes on the bottom of each of the twelve different vials with sizes ranging from µm 3 up to µm 3 and volumes from 6pl to 1.5nl. When liquid is dispensed in the vials, the liquid alters the impedance between the electrodes. The impedance can be correlated to the volume of the liquid. A schematic crosssection of such a reactor is shown in Figure 6.5. A silicon wafer is covered with a 2µm thick layer of SiO 2. Aluminium electrodes with a height of 3nm are patterned onto this layer. A 5nm thick cover of Si x N y is fabricated for electrical insulation. Finally, the vials are formed in a 6.µm thick layer of SiO 2. An aluminium layer was sputtered on the backside of the wafer to form an Ohmic contact. Grounding of the Si wafer reduces the effect of external electrical interferences, and minimizes parasitic current between the electrodes through the silicon substrate. The electrical equivalent circuit of Figure 6.5 is shown in Figure 6.6. In this figure C f and R f are the capacitance and the resistance of the liquid. The current through the Si substrate is negligible, because of the small capacitance and the grounding of this substrate. From Figure 6.6 follows that the capacitances C SixN y,
87 INTRODUCTION 77 Fig. 6.4 This photograph shows a first generation of a microarray with integrated electrodes at the bottom of each of the twelve vials. The area of the array is 5 5mm 2, the area of the different wells varies from 1 1µm 2 to 5 5µm 2. The bright areas within the dark rectangles and circle are the aluminium electrodes. the capacitance C f, and the resistance R f of the liquid contribute to the impedance Z x : R 2 ( ) 2 f 2 Z x = 1 + ω 2 Cf 2 +, (6.5) R2 f ωc SixN y where ω = 2πf, and f is the frequency of the AC current. With a floating impedance method the impedance Z x can be measured. Such a method minimizes the contribution of parasitic components. The coaxial cables used for connection are the main causes for parasitic capacitances. An AC voltage source operating at 2kHz in combination with a currenttovoltage converter is used to measure the current. Optical measurements of the remaining amount of liquid in the vial during evaporation can be used for calibration of the impedance measurements of this electronic volume sensor. In Section 9.5 we will show experimental results of this electronic volume sensor. The previous sections illustrate the need for quantitative measurements especially of the shape of the airliquid interface during evaporation. These measurements should allow verification
88 78 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS Cf SiO2 polymer SixNy SiO2 liquid CSixNy left electrode CSiO2 Rf R Si CSixNy CSiO 2 right electrode Si RSi RSi Al Alelectrodes Fig. 6.5 This figure shows a crosssection through a vial with two aluminium electrodes patterned on the bottom. Fig. 6.6 This figure shows the electric equivalent of a vial equipped with two electrodes filled with liquid. of the underlying physical model. A straightforward measurement that follows from this is the remaining liquid volume in the vial. The measurements of the liquid volume enable calibration of an electronic liquid sensor. First, we will give a survey of microscope techniques that could be used to perform the required measurements. In the following chapters we will focus on a specific microscope technique in combination with a signal processing algorithm that produces the required quantities of interest. 6.2 SURVEY OF MICROSCOPIC TECHNIQUES Optical monitoring of the evaporation process in subnanoliter vials requires a type of microscopy that allows reconstruction of the shape of the liquid volume, especially from the airliquid interface, during evaporation. This implies that the threedimensional volume needs to be properly sampled in space (both lateral and axial) as well as in time Conventional Fluorescence Microscopy Consider the case that a vial is filled with a homogeneous fluorescent solution. The fluorescent signal of the particles indicate the presence of the liquid. The total fluorescent intensity is a measure for the initial liquid volume in the vial. The total fluorescent intensity does not change, however, during evaporation, and is therefore not a good measure. A second drawback is that with a conventional widefield microscope a threedimensional volume is imaged onto a twodimensional image sensor and depth information is lost. It is not straightforward to retrieve the threedimensional shape of the liquid from these intensity differences. As a conclusion, video microscopy with a conventional microscope fulfills only the temporal sampling requirement. Figure 6.7 shows five instants of the evaporation proces of a Rhodamine solution in a square vial. The left parts in Figure 6.7 show the recorded images, whereas the right parts show the intensity distribution of the recorded images. At t =, just after injection of the liquid, the signal from the vial is uniform, besides some geometrical effects along the sidewalls and in
89 SURVEY OF MICROSCOPIC TECHNIQUES 79 (a) t = min (b) t = 5min (c) t = 1min (d) t = 15min (e) t = 2min Fig. 6.7 A series of images showing the evaporation process of ethyleneglycolwater (9%/1%, v/v) in a vial at different times. Note that the images show the fluorescent intensity profile, not the shape of the meniscus. the corners of the vial. This implies that the meniscus of the liquid is virtually flat and the vial is completely filled. After filling of the vial, the liquid is pinned to the edge of the vial. During evaporation, the pinning of the liquid ensures that the evaporation from the edge is replenished by liquid from the center part of the liquid in the vial [17]. This can be seen qualitatively in Figure 6.7: the fluorescent signal gets weaker in the center, until the bottom of the vial is reached. The signal disappears from that area at some instant between 1 and 15min. While the evaporation continues, the liquid is stuck in the four corners of the vial. At these spots the concentration of fluorescing particles is furhter increasing. Finally, when the solvent is completely evaporated, the particles loose their fluorescent behavior and the signal disappears. In Figure 6.8 the average fluorescence signal from the entire vial and the signals from the center, from one of the sidewalls and from one of the corners of the vial are shown
90 8 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS intensity in #ADUs Intensity in corner Average vial intensity 2 Intensity in centre Intensity at sidewall time in minutes Fig. 6.8 The average fluorescence signal from the vial, the signals from the center of the vial, the sidewalls of the vial and the corner of the vial as a function of time. It can be seen that, besides some geometrical effects, the average signal of the vial does not decrease during the evaporation process. as a function of time. It can be seen in this graph that the average fluorescence signal from the vial remains almost constant during evaporation. The slight increase of the signal in the beginning is caused by geometrical effects, due to the shape of the vial. These effects are less than 1% of the signal. Furthermore, it can be seen that the signal in the corners of the vial increases steadily, because of the accumulation of the fluorescent particles at these spots. Again, conventional fluorescence microscopy allows a qualitative description of the evaporation process, but it is not a suitable technique to do quantitative measurements of the shape of the meniscus Confocal Fluorescence Microscopy A second wellestablished method to analyze threedimensional microvolumes is confocal fluorescence microscopy. This kind of microscopy, however, fulfills only the spatial sampling requirement. In a confocal microscope system a pinhole in the excitation path causes point illumination 4, whereas a second pinhole in front of the detector prevents outoffocus light from reaching the detector. This way only light from a small volume in the sample is collected. As a result an image of a threedimensional volume can be acquired by scanning a stack of slices at different depths. Due to the timeconsuming scanning the temporal resolution of confocal microscopy is poor. The left part of Figure 6.9 shows different crosssections of threedimensional confocal images of a vial filled with an evaporating fluorescent solution. These figures clearly show the shape of the vial. The right part of Figure 6.9 shows the 4 In practice, a confocal microscope is equipped with a focused laser beam, which is focused sufficiently small for point illumination
91 SURVEY OF MICROSCOPIC TECHNIQUES 81 height in µm 54 o column pixel position row pixel position (a) t = min height in µm column pixel position row pixel position (b) t = 3min height in µm column pixel position row pixel position (c) t = 6min Fig. 6.9 The left figures show crosssections through confocal images of an evaporating liquid sample in a square vial. The shape of the meniscus is clearly visible. The vial has the shape of a truncated pyramid. The right figures show the shape of the meniscus computed from the confocal images. threedimensional shapes of the meniscus at the same instants as the left part. These shapes of the meniscus are computed by measuring at each lateral position the axial position where the fluorescent signal vanishes, since this indicates the position of the airliquid interface.
92 82 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS InterferenceContrast Microscopy A third type of microscopy that complies with the temporal sampling requirement as well as with the spatial sampling requirement is interferencecontrast microscopy. Unlike the first two types of microscopy, in which the observed intensity is proportional to the local concentration of the fluorescent molecules in the sample, the measured contrast in interferencecontrast microscopy is an extrinsic property of the sample under observation. The evaporation process of the liquid is observed with epiilluminated imaging with a narrow band light source. Part of the incident light reflects at the air liquid interface. Another part reflects at the bottom of the vial and interferes with the first reflected part at the air liquid interface. This results in constructive and destructive interference as a function of the distance to the bottom of the vial. Evaporation of the liquid changes the profile of the meniscus. As a result of this the interference patterns are dynamic. Imaging of the meniscus profile yields a fringe pattern in which the isophotes correspond to isoheight curves of the meniscus. Figure 6.1 shows three images of an acquired image sequence of the evaporation process in one of the vials. The instant t = s corresponds to an image with a perfectly flat meniscus. In Section 8.3 we will discuss how this instant is measured. Furthermore, Figure 6.1 shows the height profiles of the meniscus along one half of a diagonal of the vial, which resulted in the observed interferograms. These profiles are computed with the algorithm to be described in Section 8.2. Description of dynamic fringe patterns. The following observations are made from the interferogram sequence acquired during evaporation of the liquid. These observations can also be seen in Figure This figure shows the time evolution of the fringe pattern along a line through the center of the vial. 1. First, the meniscus is convex and the fringes propagate towards the center of the vial and disappear. This indicates that the meniscus becomes flat. After the final fringe has disappeared, i.e. when the meniscus is perfectly flat, the fringes reappear in the center of the vial and propagate towards the sidewalls of the vial. During evaporation more and more fringes are generated. This implies that the meniscus becomes more concave. This is shown in the right part of Figure 6.1. This can also be seen in Figure Starting from the instant when the meniscus is flat and no fringes are present, until the moment that the liquid reaches the bottom and the liquid film breaks, 29 bright fringes appear in the center of the vial. We will prove that the number of observed fringes depends on the wavelength of the incident light and the refractive index of the liquid (and weakly on the angle of incidence). The left part of Figure 6.12 shows the time evolution of the fringes during evaporation in the center of the vial, recorded with a wavelength of 5nm and 29 bright fringes can be counted, whereas the right part of of Figure 6.12 shows the time evolution of the fringes in the same vial, now recorded with a wavelength of 6nm and 34 bright fringes can be counted. 3. Just before the instant at which the liquid film breaks at the bottom of the vial, a dark fringe originates in the center of the vial. As will be explained in Chapter 7 this is related to the difference of the refractive index of the liquid and the refractive index of the bottom.
93 SURVEY OF MICROSCOPIC TECHNIQUES t =53 (a) t = 53s t =97 (b) t = 97s t =164 (c) t = 164s Fig. 6.1 The left figures show the dynamic interference patterns as recorded in a 6.µm deep vial. The right graphs show the onedimensional height profiles (computed as explained in Section 8.2) along a diagonal.
94 84 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS sidewall of vial center of vial sidewall of vial time t=s bottom of vial Fig This figure shows the time evolution of the fringes along a line through the center of the vial. In the center of the vial most modulations occur, whereas along the sidewalls of the vial no modulations occur at all.
95 SURVEY OF MICROSCOPIC TECHNIQUES 85 t=: meniscus is flat number of bright fringes counted from t= bottom of the vial 5 nm 6 nm Fig The left figure shows the time evolution of the fringe pattern in the center of the vial recorded with a wavelength of 5nm, whereas the right figure shows the time evolution of the fringe pattern in the center of the same vial, now recorded with 6nm.
96 86 DYNAMIC LIQUID BEHAVIOR IN MICROMACHINED VIALS 4. The contrast between the dark and bright fringes increases in time, i.e. the contrast increases with decreasing height. This can be seen in Figure 6.1. On the other hand, this implies that for larger heights of the liquid, the fringe patterns will not be observable, because of too little contrast. This will be further explained in Section The spatial frequency of the fringes increases during evaporation. At a certain moment this results in spatial undersampling. Spatial phase unwrapping algorithms that analyze the fringe patterns will not be successful in these regions due to aliasing. As will be discussed later, our temporal phase unwrapping algorithm will unwrap most of these regions successfully. 6.3 CONCLUSIONS In previous chapters we have shown successful solutions to overcome the problem of evaporation. There are, however, a number of motivations to study the process of evaporation in detail. Monitoring the evaporation process requires sufficiently high spatial sampling as well as temporal sampling. Conventional fluorescence video microscopy complies only with the temporal sampling requirement, and confocal fluorescence microscopy only with the spatial sampling requirement. The technique of interferencecontrast microscopy complies with both sampling requirements. In the next chapter we will present an electromagnetic theory to describe the generation of the fringe pattern at the meniscus of the thin liquid sample. In Chapter 8 we will describe a signal processing algorithm to analyze these fringe patterns and to derive the height profiles of the meniscus. In Chapter 9 we will present results of a number of experiments we have performed with this technique.
97 7 Electromagnetic Theory of InterferenceConstrast Microscopy This chapter presents a classical electromagnetic theory to describe the generation of the dynamic fringe patterns, which are observed during evaporation of a liquid sample in a 6µm deep micromachined vial. The fringe patterns will be described as modulations of the intensity of the electric field as a function of the height of the airliquid interface. This model explains most of our observations as described in Chapter 6. This model is based on the classical theory of the generation of fringes of equal thickness in thin films [21] and it shows analogies to the model of fluorescence interferencecontrast microscopy as presented by Lambacher and Fromherz [22]. 7.1 INTRODUCTION The optical model of interferencecontrast microscopy is based on the classical theory of the generation of fringes of equal thickness in thin films [21]. According to this theory interference patterns are observed due to an optical path difference between two parts of an incident plane wave. One part of the incident plane wave is reflected from the upper surface of the thin film, whereas the other part is reflected from the lower surface of the thin film. Lambacher and Fromherz [22] expanded this model to describe microscopic observations of interference patterns generated by a fluorescent cyanine dye embedded in a Langmuir Blodgett film. The dye is spaced from a reflecting silicon surface by a layer of silicon dioxide. Their optical model takes the distance dependent excitation of the fluorescent dye as well as the distance dependent emission of the dye into account. Lambacher and Fromherz call this type of microscopy fluorescence interferencecontrast microscopy. This chapter presents an electromagnetic model to describe the observations of interference patterns in subnanoliter 87
98 88 ELECTROMAGNETIC THEORY OF INTERFERENCECONSTRAST MICROSCOPY vials etched in silicon dioxide with a typical depth of 6µm as shown in Chapter 6. This model is an extension of the classical theory and shows analogies to the model of Lambacher and Fromherz. The observed interference patterns show modulations of the intensity of the electric field as a function of the height d of the airliquid interface (the meniscus) of the liquid in the vial. In this chapter an expression for the intensity of the electric field will be derived. This derivation takes into account that the interference patterns are observed via epiillumination of the liquid sample in the vial with an incoherent unpolarized narrowband light source. Furthermore, the incident light has a limited angle of incidence. The derivation of an expression for the intensity of the electric field consists of the following steps. The first step derives an expression for the phase difference between the part of the incident plane wave that is reflected at the airliquid interface and the part of the incident plane wave that is reflected at the bottom of the vial. The second step is to derive an expression for the sum of the electric field vectors of these parts. In the third step the electric field vector will be integrated analytically over all angles of polarization. Then, two weight functions are introduced to take the different wavenumbers and angles of incidence into account. This results in an expression for the intensity of the electric field, which can only be evaluated numerically. 7.2 PHASE DIFFERENCE The optical model of interferencecontrast microscopy is shown in Figure 7.1. The microscope is adjusted to Köhler illumination: this implies incoherent illumination, and the incident light consists of plane waves. This plane wave is incident at an angle Θ with respect to the normal of the airliquid interface. After refraction at the airliquid interface, this plane wave propagates towards the reflecting bottom of the vial at an angle Θ in. The direct part of the incident plane wave interferes with the part of the incident plane wave that is reflected at the bottom of the vial. Whether this interference is constructive or destructive depends on the height d with respect to the bottom of the vial and the angle of incidence Θ. Thus, interference occurs at every height d, but, as will be explained further on, an image sensor will only observe the interference at the airliquid interface. First, the airliquid interface is considered to be parallel to the silicon dioxide bottom of the vial, as shown in Figure 7.1. The optical path difference OPD between the direct part and the reflected part of the incident plane wave at a height d above the reflecting bottom of the vial is given by OPD = n liq (ABCD FD) n air (EF ), (7.1) where n liq and n air are the refractive indices of the liquid (ethyleneglycol, n liq = 1.432), respectively the air. In Figure 7.1 it can be seen that the optical path length AB equals FD. The OPD reduces then to OPD = n liq (BCD) n air (EF ). (7.2) The optical path length BCD equals twice the optical path length BC. With cos(θ in ) = the OPD can be rewritten as OPD = 2 n liq d cos(θ in ) n air (EF ), (7.3) d BC
99 PHASE DIFFERENCE 89 E out,1 E out,2 E out,3 E out,4 plane of incidence d F E D E E γ E Θ in B incident plane wave Θ o A air Ethyleneglycol Z reflecting interface  mirror C X Y bottom of well (silicon dioxide) Fig. 7.1 A plane wave, incident at an angle Θ, is refracted at the airliquid interface of the liquid sample in the vial. After refraction, the direct part of this plane wave interferes with the part of this plane wave that is reflected from the bottom of the vial. Whether this interference is constructive or destructive depends on the height d with respect to the bottom of the vial. The arrows indicate the different parts of the electric field: E is the part of the electric field in the plane of incidence (xzplane), and E is the part of the electric field perpendicular to the plane of incidence (yaxis). where Θ in is the angle of incidence in the liquid sample. Furthermore, from Figure 7.1 follows sin(θ ) = EF DB, where DB equals 2 d tan(θ in). The angle Θ is the angle of incidence in the air. Inserting this in Equation 7.3 yields OPD = Applying Snellius law n air sin(θ ) = n liq sin(θ in ) yields OPD = 2n liqd cos(θ in ) 2n air d sin(θ ) tan(θ in ). (7.4) 2 n liq d cos(θ in ) 2 n liqd sin 2 (Θ in ) = 2 n liq d cos(θ in ). (7.5) cos(θ in ) The assumption in this derivation is that the airliquid interface is parallel to the bottom of the vial. We will now prove that Formula 7.5 is also valid, when the airliquid interface is inclined at an angle α with respect to the bottom of the vial. This situation is shown in Figure 7.2. Again, the optical path difference is given by Equation 7.1. Furthermore, AB equals FD and with Snellius law follows n liq (BB ) = n air (EF ). The optical path difference reduces then to OPD = n liq (B CD). (7.6)
100 9 ELECTROMAGNETIC THEORY OF INTERFERENCECONSTRAST MICROSCOPY incident plane wave E θ air Ethylene glycol α A B B' θ in F D d C reflecting bottom silicon dioxide Fig. 7.2 An incident plane wave is refracted at the airliquid interface, which is inclined at an angle α with respect to the bottom of the vial. The optical path difference between both parts of this plane wave is given by Equation 7.5. From Figure 7.2 follows cos(2θ in ) = cos 2 (Θ in ) sin 2 (Θ in ) = B C CD. Inserting this in Equation 7.6 yields where CD follows from cos(θ in ) = OPD = n liq (1 + cos(2θ in ))(CD), (7.7) OPD = d CD. This leads to n liqd cos(θ in ) (cos(2θ in) + 1), (7.8) which reduces to the optical path difference given by Equation 7.5. This proves that the optical path difference between the direct part and the reflected part of the incident plane wave given by Equation 7.5 does not depend on the angle α between the airliquid interface and the bottom of the vial. Again, the optical path difference causes interference between the direct part and the reflected part of the incident plane wave as a function of the height d above the bottom of the vial. The modulations at a height d in the liquid, however, cannot be observed. The reason is that in a microscope system the outgoing rays, indicated in Figure 7.1 as E out,i (i = 1, 2, 3, 4), are observed in different uncorrelated points on an image sensor. An image sensor mounted on a microscope can only observe interference, if two rays combine at a single point on the image sensor, which seem to originate from the same point, but have different optical paths. According to Figure 7.1, the rays E out,2 and E out,4 originate from the same point A, but these rays are observed at different points on an image sensor. The rays E out,2 and E out,3 have different optical paths, but they do not originate from the same point, nor are they observed at the same position on an image sensor. When d is extended to the airliquid interface, however, these two rays shift towards each other and end up in the same position. The image sensor mounted on the microscope will observe this as interference. This implies that the observed
101 INTENSITY OF ELECTRIC FIELD 91 interference patterns originate at the airliquid interface and that the modulations inside the liquid cannot be observed at all. The phase difference Φ in corresponding to the optical path difference given by Equation 7.5 is Φ in = 4π n liq d cos(θ in ), (7.9) λ where λ is the wavelength of the incident light. In terms of the wavenumber k = 2π λ, this phase difference becomes Φ in = 2 n liq k d cos(θ in ). (7.1) 7.3 INTENSITY OF ELECTRIC FIELD Electric Field Vectors. As indicated in Figure 7.1, the electric field vectors of the incident plane wave are split into vectors normal to the plane of incidence( E ), and vectors parallel to the plane of incidence ( E ). Using the notation from Figure 7.1, the electric field vector of the incident plane wave E i and the electric field vector E r after reflection at the airliquid interface are given in Cartesian coordinates by E i = E E r = E cos(γ ) cos(θ ) sin(γ ) cos(γ ) sin(θ ) cos(γ ) cos(θ )r 1 sin(γ )r 1 cos(γ ) sin(θ )r 1, (7.11), (7.12) where E is the amplitude of the electric field of the incident plane wave. The angle γ is the angle of polarization with respect to the plane of incidence. The angle Θ is the angle of incidence of the incident plane wave. The coefficients r ij and r ij are the Fresnel coefficients for reflection at the interface between media i and j for respectively the parallel component and the normal component with respect to the plane of incidence. The index is for the air, 1 for the liquid in the vial and 2 for the silicon dioxide under the bottom of the vial. The electric field vector E trt after refraction at the airliquid interface, reflection at the bottom of the vial and again refraction at the liquidair interface is given in Cartesian coordinates by E trt = E e iφin cos(γ ) cos(θ )t 1 r 12 t 1 sin(γ )t 1 r 12 t 1 cos(γ ) sin(θ )t 1 r 12 t 1, (7.13) where Φ in is the phase difference as defined in Equation 7.1. The coefficients t ij and t ij are the Fresnel coefficients for transmission through the interface between media i and j for the parallel component and the normal component with respect to the plane of incidence.
102 92 ELECTROMAGNETIC THEORY OF INTERFERENCECONSTRAST MICROSCOPY For completeness, we give here the definitions of the Fresnel coefficients [23] based on the notations from Figure 7.1: r ij = n i cos(θ i ) n j cos(θ j ) n i cos(θ i ) + n j cos(θ j ), (7.14) r ij = n j cos(θ i ) + n i cos(θ j ) n j cos(θ i ) + n i cos(θ j ), (7.15) t ij = t ij = 2n i cos(θ i ) n i cos(θ i ) + n i cos(θ j ), (7.16) 2n i cos(θ i ) n j cos(θ i ) + n i cos(θ j ), (7.17) where n i and n j are the refractive indices of the two media, and Θ i and Θ j are the angle of incidence, and the angle of refraction. These two angles are related by Snellius law. The sum of the two parts of the reflected plane waves follows as E = E cos(γ ) cos(θ )(r 1 + t 1 r 12 t 1 eiφin ) sin(γ )(r 1 + t 1 r 12 t 1 e iφin ) cos(γ ) sin(θ )(r 1 + t 1 r 12 t 1 eiφin ). (7.18) The intensity of the electric field E 2 describes the observed interference as a function of the height d for a single incident plane wave with a certain angle of polarization, angle of incidence, and wavenumber. The sum of all these singlequantum modulations gives rise to the observed interference patterns [22]. Angle of Polarization. The first step to compute the intensity of the electric field is to integrate E 2 analytically over all angles of polarization. The result of this computation can be split into an offset, which is not important, and a modulating part: E 2 γ = OFFSET { }} { π(r r 1 + t 2 1 r 2 12 t t 1 r 12 t 1 ) + (7.19) 2π(r1 t 1 r12 t 1 + r 1 t 1 r 12 t 1 ) cos(φ in). } {{ } MODULATING PART The amplitude term preceding the cosine function shows very little dependency on the angle of incidence Θ in in the interval of interest [,.2]rad. Transmission Wavenumber Spectrum. The second step to evaluate the intensity of the electric field is performed by averaging over all wavenumbers. For the experiments, a narrow bandpass filter (6FS15, Andover Corporation, Salem, NH, USA) with a central wavelength λ c = 62.3nm and a full width at half maximum (FWHM) of 9.7nm is placed in the illumination path. The transmission spectrum of this filter S(k, σ k ) (normalized in amplitude) in terms of the wavenumber k can be approximated by a flattened Gaussian function: S(k, σ k ) = (1 + (k k c) 2 ) exp ( (k k c) 2 ), (7.2) 2σ 2 k 2σ 2 k
103 INTENSITY OF ELECTRIC FIELD 93 relative transmission FWHM wavenumber k[µm1 ] Fig. 7.3 The filled curve shows the relative transmission of the measured spectrum of the narrow bandpass filter. The black curve shows the relative transmission of the approximated spectrum as defined by Equation 7.2. with σ k =.443µm 1 corresponding to the FWHM of the filter. Figure 7.3 shows the relative transmission of the measured spectrum and of the approximated spectrum. Angle of Incidence. Finally, we will introduce a weight function for the angle of incidence. Assume that the microscope system used for the experiments is adjusted to Köhler illumination [9]. This implies that the backfocal plane of the objective is a congruent plane of the light source. Furthermore, we assume that the backfocal plane is uniformly filled. See Figure 7.4. Each point in the back focal plane corresponds to a certain angle of incidence Θ, according to the relation r tan(θ ), with r the distance to the optical axis in the back focal plane, and Θ the angle of incidence. All points with the same distance to the optical axis correspond to the same angle of incidence. The probability that an incident plane wave has an angle of incidence smaller than the angle Θ is proportional to πr 2, which is a circular fraction of the back focal plane. The probability that the angle of incidence of a plane wave is smaller than the maximum angle of incidence Θ max is, of course one with r = R the radius of the back focal plane. The cumulative probability density function equals then ( r R )2. The probability density function that a certain incident plane wave has an angle of incidence Θ is then equal to 2r R 2, the derivative of the cumulative probability density function, which is proportional to tan(θ ). This defines the weight function for the different angles of incidence. The halogen lamp used for the experiments, however, does not fill the back focal plane of the objective (Zeiss FLUAR 2 /.75) completely 1. Given the definition of the Numerical 1 Note that in interferencecontrast microscopy, this does not affect the resolving power of the optical system: the period length of the modulations, which is reciprocal to the resolution is only determined by the angle between the
104 94 ELECTROMAGNETIC THEORY OF INTERFERENCECONSTRAST MICROSCOPY R R r' r' optical axis Θ' back focal plane (front view) back focal plane (side view) objective object plane Fig. 7.4 Each point in the back focal plane corresponds to a certain angle of incidence Θ, according to the relation r tan(θ ), with r the distance to the optical axis in the back focal plane, and Θ the angle of incidence. Aperture (NA = n air sin(θ,max )) and a light source large enough to fill the back focal plane, the following relation yields the maximum angle of incidence: tan(θ,max ) = NA n 2air NA2. (7.21) The halogen lamp fills only onefifth of the diameter of the back focal plane. In that case, tan(θ,max ) equals onefifth of the right hand side of Formula As said before, the angles of incidence Θ and Θ in are related by Snellius law. The maximum angle of incidence in the liquid Θ in,max follows then as Θ in,max = arcsin n air sin arctan 1 NA. (7.22) n liq 5 n 2 air NA2 With these two weight functions for the wavenumber and the angle of incidence of the incident light, the intensity of the electric field at a height d above the bottom of the vial can be computed as: E(d) 2 = Θin,max Θ in = k= tan(θ in )S(k, σ k ) E 2 γ dθ in dk, (7.23) airliquid interface and the liquidsio 2 interface. Even with just normal incidence the same high frequency pattern can be generated. The highest frequency, that an objective can image onto a sensor is determined by the illumination wavelength and the Numerical Aperture of the objective and is defined by f c = 2NA. In transmission microscopy, λ an incompletely filled back focal plane would, of course, seriously affect the resolution.
105 COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL RESULTS 95 where S(k, σ k ) is defined in Formula 7.2 and E 2 γ in Formula The maximum angle of incidence Θ in,max follows from Formula COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL RESULTS Expression 7.23 is evaluated numerically as a function of the height d above the bottom of the vial with the following set of parameters: 1. central wavenumber of the transmission filter: k c = 1.4µm 1, 2. width of the transmission spectrum of the filter: σ k =.443µm 1, 3. refractive index of the liquid in the vial: n liq = , 4. refractive index of the silicon dioxide bottom: n sil = 1.46, and 5. maximum angle of incidence: Θ in,max =.155rad. The result of this computation is shown in Figure This computation is compared to measurements in a digital recording of an evaporating ethyleneglycol sample. In each frame of this recording the intensity in the center of the vial is measured. The modulating intensity, measured as a function of time, is unwrapped with the algorithm that will be presented in Chapter 8. This temporal phase unwrapping algorithm gives the height of the liquid as a function of time. A parametric plot of the measured modulating intensity against the computed height of the liquid is also shown in Figure 7.5. Recall the observations mentioned in Chapter6. The second observation was that 29 bright fringes appeared in the center of the vial from the moment that the meniscus was perfectly flat (t = s). This can be checked in Figure 7.5. The third observation was that the final fringe just above the bottom of the vial is a dark fringe. Our model agrees with this. Furthermore, the fourth observation was that the contrast increases with decreasing height. This can also be seen in Figure 7.5. Finally, as can be seen in Figure 7.5 the period of the measured modulations equals the period of the simulated modulations from Formula The amplitude of the measured modulations, however, drops much faster than that of the simulated modulations. We think that the poor agreement between our model and the experiments is due to outoffocus light. The focal plane for the experiments is the bottom of the vial. This implies that in the beginning of the recording, when the meniscus is convex, the meniscus is not focused and contains large outoffocus contributions. During the evaporation, the meniscus gets more and more in focus and the contributions of the outoffocus light get smaller and finally disappear. The contributions of the outoffocus light result in a reduced contrast for large values of d. 2 The bottom of the vial is covered by a layer of uniform thickness of Si xn y. The incident light is reflected at the Si xn y SiO 2 interface. The refractive index of Si xn y is approximately 2.1, which is larger than the refractive index of the SiO 2 bottom of the vial (n sil = 1.46). The Fresnel coefficients in Equations 7.17 dictate that no phase shift of πrad occurs at reflection on this interface. Of course, this layer introduces an additional constant optical path difference. This extra OPD almost equals an integer number of periods for the wavelengths used in the experiments. Therefore it is not taken into account and we subtract a phase shift of πrad from the simulation results to remove the extra phase shift introduced by reflection on the liquid SiO 2 interface used in the model.
106 96 ELECTROMAGNETIC THEORY OF INTERFERENCECONSTRAST MICROSCOPY Simulated modulations Time t in s Measured modulations Intensity t=6 s. t= s. t=3 s. t=6 s. t=3 s. t=12 s. t=9 s. t=21 s. t=18 s.t=15 s. Height d in µm Fig. 7.5 This graph shows the intensity of the electric field as a function of the height d above the reflecting bottom of the vial. One curve is based on evaluation of Expression 7.23, whereas the other curve has been experimentally acquired. The moment t = s corresponds to the moment in the recording of the interference pattern where the meniscus is perfectly flat.
107 ALTERNATIVE DESCRIPTION FOR INTENSITY OF ELECTRIC FIELD ALTERNATIVE DESCRIPTION FOR INTENSITY OF ELECTRIC FIELD The simulation results in Figure 7.5 indicate that the modulations can be written as a single cosine function with varying amplitude A(d) cos(φ(d)), where the argument Φ(d) is a function that defines the period of the modulations, and the amplitude A(d) is a function that defines the envelope of the modulations. Length of Modulations. The argument of the cosine function Φ(d) is, of course, related to the phase difference as defined in Equation 7.1. The different wavenumbers can be replaced by a single effective wavenumber k eff and the different angles of incidence can be replaced by a single effective angle of incidence Θ eff in. This yields Φ(d) = 2 n liq k eff d cos(θ eff in ), (7.24) From this equation follows, that the length of the modulations d, i.e. the change in the height d corresponding to a phase difference Φ(d) = 2πrad equals d = 2n liq k eff 2π cos(θ eff (7.25) in ). Because of the symmetry of the transmission spectrum S(k, σ k ) around k = k c, we conclude that the period of the modulations is proportional to k c. This implies that the effective wavenumber k eff = k c. The effective angle of incidence Θ eff in corresponds to the effective radius of the backfocal plane 1 2 2R, thus Θ eff in = 2 1 2Θin,max. For small values of d, the length of the modulations depends weakly on the angle of incidence [21]. For large values of d, however, both the angular distribution and the spectral bandwidth will have a significant effect on the fringe visibility. Finally, we compared the positions of the zerocrossings in our measured modulations with the predicted positions of the zerocrossings 2k+1 4 d, k {, 1, 2, 3,...}. The error between the measured positions of the zerocrossings and the predicted positions has a standard deviation of 2.5nm, which equals approximately one percent of the length of the modulations. Remember that there is a bias as explained in footnote 2. Here we are only interested in the precision of the measurements. Envelope of Modulations. In order to find an expression for the envelope or the amplitude factor A(d) of the modulations, we will look in closer detail at Equation First, we will only consider the integral dependent on the wavenumber k and omit the dependency on the angle of incidence Θ in. Insert Equation 7.1 and the modulating part of Equation 7.19 into Equation For clarity, we replace the cosine function by a complex exponential function. This yields { ) ) } X(d) = R (1 + k2 k= 2σk 2 exp ( k2 2σk 2 exp (j2n liq k d cos(θ in )) dk, (7.26) where X( ) just indicates a function of the height d, and R{ } is the real part. From a mathematical point of view the modulations are caused by the fact that k c. In order to
108 98 ELECTROMAGNETIC THEORY OF INTERFERENCECONSTRAST MICROSCOPY compute the envelope, we must set k c =. The following example illustrates this point of view: Consider the following function f mod (d): f mod (d) = exp ( (x x ) 2 ) exp (jxd) dx, where the subscript mod indicates that this is an modulating function. Apply the variable transformation x = x x. This yields: f mod (d) = = exp (jx d) 2σ 2 x exp ( x 2 2σx 2 ) exp (j(x + x )d) dx exp = exp (jx d) f env (d), ( x 2 2σ 2 x ) exp (jx d) dx where the subscript env indicates that this function is the envelope of the function f mod (d). This means that the modulating function can be described by a complex exponential function with period 2π x and amplitude function f env (d). Figure 7.6 shows the real part of the modulating function f mod (d) and its corresponding envelope f env (d) in the interval d = [, 5] with x = 2π and σ x = π f mod (d) f env (d) d 1 Fig. 7.6 This figure shows the real part of the modulating function f mod (d) and its corresponding envelope f env (d) in the interval d = [, 5] with x = 2π and σ x = π. 1 Setting k c = yields the following: { ) ) } X(d) = R (1 + k2 k= 2σk 2 exp ( k2 2σk 2 exp (j2 n liq cos(θ in ) d k) dk. (7.27) Note that we have replaced the integral range from (, ) to (, ), and omitted the correcting factor of 1 2. We recognize the function X(d) as the Fourier transform of the transmission spectrum. Note that k and d are physical quantities with reciprocal units. Recalling the omitted dependency on the angle of incidence Θ in, results in the envelope A(d) corresponding to the modulations E(d) 2 from Equation This shows that the envelope of the modulations is a Fourier transform of the wavenumber spectrum (weighted with the angle of
109 SUMMARY 99 incidence Θ in ): in a similar manner as in a Fourier Transform Spectrometer [24] the produced interferogram encodes the spectrum of the light. Neglecting constant factors, the envelope A(d) is given by, A(d) = Θin,max Θ in = F (Θ in ) tan(θ in ) σ k (7.28) (4 n 2 liq d 2 σ 2 k cos 2 (Θ in ) 3) exp( 2 3 n 2 liq d 2 σ 2 k cos 2 (Θ in ))dθ in, where F ( ) stands for the amplitude term of all Fresnel coefficients in Equation An extra factor of 3 has been added to the exponential term for proper scaling. This analysis shows that the amplitude of the modulations decreases as a function of the height d. As a consequence, the modulations will not be observable above some height 3. Furthermore, from Fourier analysis we know that the Fourier transform of S(k, σ k ) can be broadend by narrowing this function. Ultimately, with monochromatic light, the fringe patterns would be visible up to large heights d. 7.6 SUMMARY This chapter presents a classical electromagnetic theory to describe the generation of fringe patterns in the vials during the evaporation of liquid in the vial. The main concept of this theory is that one part of an incident plane wave, that is reflected on the airliquid interface, interferes with another part of the same incident plane wave, that is reflected on the bottom of the vial. The phase difference between both parts of this plane wave is given by Φ in = 2 n liq k d cos(θ in ). (7.29) The fringe patterns are observed as modulations of the intensity of the electric field. The computation of the intensity of the electric field takes into account the angle of polarization, the transmission spectrum, the angle of incidence, and the maximum angle of incidence. The intensity of the electric field can be computed by numerical evaluation of the following equation: E(d) 2 = Θin,max Θ in = k= tan(θ in )S(k, σ k ) E 2 γ dθ in dk, (7.3) where S(k, σ k ) is defined in Formula 7.2 and E 2 γ in Formula The maximum angle of incidence Θ in,max follows from Formula The electromagnetic model agrees with experimental results. The period of the modulations is given by: d = 2π 2n liq k c cos( Θin,max ). (7.31) With k c = 1.43µm 1, n liq = and Θ in,max =.155rad follows d = 211.6nm. 3 Under the assumption that the optical system can still spatially resolve the modulations.
110
111 8 Temporal Phase Unwrapping Algorithm The dynamic interferograms as shown in Chapter 6 can be regarded as dynamic contour maps of the airliquid interface: each isophote curve in an interferogram is an isoheight curve. In this chapter we will present a temporal phase unwrapping algorithm that computes the dynamic height profiles of the meniscus from these contour maps. This algorithm analyzes the interferogram pointbypoint in time without using any spatial information. Analysis of this algorithm will show that this approach has a less severe spatial sampling requirement than a spatial phase unwrapping algorithm. 8.1 PHASE UNWRAPPING: SPATIAL VS. TEMPORAL The acquired interferograms I[m, n; t] as shown in Figure 6.1 and Figure 6.11 can be described by: I[m, n; t] = A[m, n; t] cos(φ[m, n; t]) + B[m, n; t] + N[m, n; t], (8.1) where A[ ] and B[ ] are the space and time varying amplitude and background. The signal N[m, n; t] is an additive noise signal. The argument Φ[m, n; t] is the phase map of the interferogram. The absolute or unwrapped phase map is related to the height d of the liquid via Φ[m, n; t] = 4πn liq cos( Θin,max ) d[m, n; t]. (8.2) λ c 11
112 12 TEMPORAL PHASE UNWRAPPING ALGORITHM This equation is identical to Equation 7.24 in Chapter 7. The length of a single period d follows from this equation as λ c d = 2n liq cos( Θin,max ). (8.3) Due to the cosine operator in the interferograms, the unwrapped phase map must be retrieved from the wrapped or relative phase map ˆΦ[m, n; t]: ˆΦ[m, n; t] = Φ[m, n; t] mod 2π. (8.4) The goal of an unwrapping algorithm is first to derive the wrapped phase map ˆΦ[m, n; t] from the observed interferogram I[m, n; t] by measuring the relative phase at every time. The second step is to retrieve the absolute phase map Φ[m, n; t] with respect to a reference point by adding 2π to the relative phase at every located 2π discontinuity. Section 8.3 will describe how this reference point is determined. Several algorithms exists that unwrap 2D phase maps derived from a static interference pattern, e.g. [25, 26]. These algorithms compare phase values at neighboring pixels. Spatial unwrapping algorithms might suffer from the following three problems. 1. Phase errors originating in regions with a high noise level propagate outside these regions and corrupt the remaining part of the image. 2. The fringe pattern must be sampled at a spatial frequency that is at least twice the highest frequency in the interferogram to meet the requirements of the Nyquist sampling theorem. One of our observations in Chapter 6 is that the images are undersampled in regions with a steep meniscus. In Section 8.5 we will treat the aspects of sampling in detail. 3. A spatial unwrapping algorithm will fail when true discontinuities are present in the phase map. These discontinuities are observed as segmented fringes and are caused by height differences. Experimental results to be shown in Chapter 9 will prove that our temporal phase unwrapping algorithm can easily deal with these discontinuities. To avoid the problems mentioned above we propose to analyze the acquired interferograms in time, pointbypoint as will be described in Section 8.2. Temporal phase unwrapping was described earlier, e.g. by Huntley and Saldner[27]. Temporal phase unwrapping with a fourstep interferometer. Huntley and Saldner used a fourstep interferometer to generate four intensity images at phase steps of π 2 : I k [m, n; t] = A[m, n; t] cos(φ[m, n; t] + Φ k ) + B[m, n; t] + N[m, n; t], where Φ k = (k 1)π, k = 1, 2, 3, 4. 2
113 TEMPORAL PHASE UNWRAPPING ALGORITHM 13 A phase map is directly computed from these four images according to the following formula: ( ) I42 [m, n; t] Φ[m, n; t] = arctan, I 13 [m, n; t] where Thus, I ij [m, n; t] = I ij [t] = I i [m, n; t] I j [m, n; t]. I 42 [t] = A[t](cos(Φ[t] + 3π 2 ) cos(φ[t] + π 2 )) = A[t](sin(Φ[t]) + sin(φ[t])) = 2A[t] sin(φ[t]) With this method the phase difference between two successive phase maps is straightforward to compute: Thus, Φ[m, n; t] = Φ[m, n; t] Φ[m, n; t 1] ( ) I42 [t] I 13 [t 1] I 13 [t] I 42 [t 1] = arctan. I 13 [t] I 13 [t 1] + I 42 [t] I 42 [t 1] Φ[m, n; t] ( ) cos(φ[t]) sin(φ[t 1]) sin(φ[t]) cos(φ[t 1]) = arctan sin(φ[t]) sin(φ[t 1]) + cos(φ[t]) cos(φ[t 1]) ( ) sin(φ[t] Φ[t 1]) = arctan cos(φ[t] Φ[t 1]) With the method of Huntley and Saldner the phase map is computed directly from a series of interferograms with known phase steps. However, in our case, the interferograms are not recorded with known phase steps between successive images and it is not possible to derive the phase map directly from successive images. The phase difference between two successive images measured at different position is a continuum due to the space varying evaporation of the liquid during the experiments. As can be seen in Figure 6.1 and Figure 9.3, the total height change in the center of the vial is maximal. This leads to a maximal phase difference at that position in two successive images. On the other hand, at the border of the vial there is practically no height change: the phase difference is minimal. 8.2 TEMPORAL PHASE UNWRAPPING ALGORITHM In the remainder of this section the spatial dependency of the physical quantities in Equation 8.1 will be dropped: I[t] = I[m, n; t] and so forth. The square brackets indicate that the signals are discrete signals.
114 14 TEMPORAL PHASE UNWRAPPING ALGORITHM In order to compute the phase map, we have to estimate the phase at every point in our signal. The phase of a discrete signal can be estimated by applying a discrete Fourier transform. To avoid discontinuities in the periodic representation of the signal, a window function must be used before applying the Fourier transform. In our algorithm, this window function has a fixed width. The bandwidth of our signal varies spatially. Subsampling the signal with a spatially varying subsampling factor results in an approximately uniform bandwidth. The signal will be subsampled in such a way that approximately two periods of the subsampled signal fall within the width of the window function. This avoids that the low frequency component of interest gets mixed with the DC component of the signal. Figures 8.1(a) 8.2(b) show the successive steps of the temporal phase unwrapping algorithm. The following recipe gives a short overview of the different steps. 1. Subtract the background B[t] from the interferogram I[t]. The result of this computation is denoted as I [t], which equals A[t] cos(φ[t]) + N[t]. See figure 8.1(a). 2. Compute a subsampling factor f sub in order to subsample I [t] in such a way that approximately two periods of the subsampled interferogram, denoted as I sub [t] fall into a K w number of data points wide interval. 3. Multiply a K w number of data points wide region of I sub [t] with a K w number of data points wide window function (see Figure 8.1(b)) and get the wrapped phase from the frequency spectrum. The wrapped phase is denoted as ˆΦ sub [t]. See Figure 8.2(a). 4. Retrieve the absolute phase, Φ sub [t], by unwrapping the wrapped phases. Interpolate the absolute phase to its original length. This is denoted as Φ[t]. Finally, convert the absolute phase to the height of the meniscus as a function of time. The height of the meniscus is denoted as d[t]. See figure 8.2(b). Background removal. The first step in the computation of the phase map Φ[t] from the interferogram I[t] is to subtract the background B[t] from the interferogram. The time dependent background is estimated by lowpass filtering I[t] with a Gaussian kernel with a very large standard deviation. The result of this operation is denoted as I [t] and is shown in Figure 8.1(a). The signal I [t] consists of the cosine function with varying amplitude and the additive noise term. Subsampling. The next step is to subsample the low frequency signal I [t] with a factor f sub. The temporal sampling density is constant, but the temporal bandwidth varies spatially as can be seen in Fig To satisfy the Nyquist criterion in the time dimension over the entire image, the signal in time is oversampled: the oversampling rate increases towards the sidewalls of the vial. We propose to use a window function with a fixed width of 128 points and to subsample the low frequency signal I [t]. This subsampling is performed in such a way that approximately two periods of I sub [t] fall within the width of a window W [t]. The window function is defined by ( ) 2 ( ) 4 ( ) W [t] = t K w t K w Kw 2 (t 2 exp )2 2 σ t 8 σ t 2σt 2, (8.5)
115 TEMPORAL PHASE UNWRAPPING ALGORITHM 15 Intensity in #ADU's Frame number (a) This graph shows the background corrected data I [t] in the center point of the vial in the frame series Intensity in #ADU's Window function (b) This graph shows the window function W [t] and the product I sub [t]w [t]. Fig. 8.1 The first two steps of the temporal phase unwrapping algorithm. where σ t = 16., and K w is the fixed width of the window. The subsampling factor f sub is defined as the ratio between the average length of the modulations in I [t] and the required average length of the modulations within the width K w of the window. The former equals the number of data points N in I [t] divided by the number of periods in I [t], where each period has two zerocrossings ZC, and the latter simply equals K w /2. Thus, the subsampling factor f sub yields: [ ] 2 2 N I [t] f sub =, (8.6) K w #ZC where [ ] is the round operator. In order to estimate the number of zerocrossings #ZC in I [t] we apply a lowpass filter in the time dimension with a variable width σ that suppresses the noise sufficiently, but does not corrupt the signal. This implies that the width of the filter must be tuned in such a way that it removes all zerocrossings introduced by the noise, but not the zerocrossings of the signal. First, the number of zerocrossings #ZC is counted in I [t]. Then I [t] is filtered
116 16 TEMPORAL PHASE UNWRAPPING ALGORITHM Phase π π 2 flat meniscus π Frame number π calibrated phase wraps (a) This graph shows the estimated phases in the frame series Height in µm Frame number (b) This graph shows the height profile, which follows by unwrapping of the estimated wrapped phases. Fig. 8.2 The last two steps of the temporal phase unwrapping algorithm. with a Gaussian kernel G[t; σ = 2] and the number of zerocrossings is counted again. This operation removes zerocrossings due to the high frequency components of the noise. This filtering is repeated with the size of G[t; σ] increased by a factor of 2 until the number of zerocrossings does not change after two successive filterings. In this situation the noise is suppressed sufficiently and only true zerocrossings are present in the signal. The number of zerocrossings can vary from 1 at the sidewall of the vial to 84 in the center of the vial. Under assumption that the evaporation rate remains constant 1, the average number of periods within K w can be computed with Formula 8.6. As a result of the resampling the average number of modulations within K w of the subsampled interferogram varies from 1.6 to 2.4 modulations. The subsampled signal is denoted as I sub [t]. This way of subsampling is necessary to avoid that F{I sub [t]w [t]} is mixed with the DC component of the frequency spectrum. The operator 1 In Chapter 9 we will show that this assumption is true. Furthermore, this can al ready be seen in Figure 8.2(b), which shows the height of the liquid as a function of time.
117 COMPUTATION OF REFERENCE FRAME 17 F{ } denotes the Fourier transform. The window function W [t] and the product I sub [t]w [t] are shown in Figure 8.1(b). Note that the discrete Fourier transform (DFT) is a least squares algorithm for a number of uniformly spaced frequencies k 2π N, k = {, 1,..., N 1} [28]. The aforementioned subsampling strategy guarantees a proper match between the actual frequency and two times the fundamental frequency 2π/N of the discrete frequencies of the DFT. Phase estimation. The third step in the algorithm is to compute the 1D wrapped phase map ˆΦ sub [t]. One way to do this is to apply an FFT algorithm to get a Fourier transform of a windowed part of I sub [t]. The window function avoids discontinuities in the periodic representation of I sub [t] computed with the discrete Fourier transform. The phase to be estimated equals the phase of the bin with the maximum amplitude in the discrete frequency spectrum. For each point I sub [t ] the wrapped phase ˆΦ sub [t ] is computed by multiplying a 128 wide window W [t] with I sub [t + t ] around the instant t and computing the Fast Fourier Transform of this product. The wrapped phase at the instant t follows from the discrete frequency spectrum: ˆΦ sub [t ] = Phase{W [t]i sub [t ]}. The wrapped phase map, denoted as ˆΦ sub [t] is shown in Figure 8.2(a). Unwrapping and scaling. After these three steps the 1D wrapped phase map is computed. Next, the wrapped phase map is calibrated with respect to a reference point as described in Section 8.3. In this calibration step the number of observed phase wraps is set to zero at the instant where the meniscus is flat, i.e. we start counting phase wraps from this instant on. The final step of the algorithm is to unwrap the wrapped phase map. The unwrapping is done with an unwrapping operator found in literature [26]. Finally, the subsampled unwrapped phase map Φ sub [t] is interpolated by a factor f sub to the original length of I[t]. The phase map Φ[t] is scaled with a factor d 2π, with d given in Equation 8.3, to an absolute height in micrometers with respect to the flat meniscus. The result of this unwrapping algorithm is the height of the liquid d[m, n; t] in a single data point [m, n] as a function of time. The height d[t] with respect to the bottom of the vial is shown in Figure 8.2(b). 8.3 COMPUTATION OF REFERENCE FRAME It is not possible to calibrate all wrapped phase maps at the final image of the image sequence, because the liquid does not reach the bottom of the vial at every point at the same time. The wrapped phase maps can be calibrated, however, when the meniscus is flat. At that instant an interference pattern is absent, because the meniscus is parallel to the bottom of the vial. In this time interval the OPD is approximately independent of the position. Although no fringe pattern is being observed, the measured intensity over the entire vial is influenced by the actual phase difference. This means that the intensity over the entire vial shows the same modulation in time due to the timevarying OPD when the meniscus changes from convex to concave. This has been monitored and is shown in Figure 8.3. The bottom graph shows the modulation of the average intensity as a function of time, whereas the top graph shows the variation of the average intensity (measured after background correction). The frame with the smallest variation in average intensity corresponds to the one with the flat meniscus. In this
118 18 TEMPORAL PHASE UNWRAPPING ALGORITHM StDev(#ADU's) Average (#ADU's) Frame number Fig. 8.3 The frame, where the meniscus is perfectly flat, is determined by a minimum variation in intensity. This frame is used to trigger all relative height profiles by setting the number of counted 2πdiscontinuities to zero in this frame. figure, the standard deviation σ frame of the intensity distribution of frame i is computed as σ frame = 1 M NM 1 m=1 n=1 N (I i [m, n] I[m, n] t= ) 2, (8.7) where I[m, n] t is the timeaverage over 6 frames in the time window where the meniscus is almost flat. From Figure 8.3 follows that the meniscus becomes perfectly flat in frame 14. The time t = in Figure 7.5 corresponds to this frame number. All wrapped phase maps are calibrated by setting the number of observed phase wraps at that instant to zero. The labels of the calibrated phase wraps are indicated in Figure 8.2(a). The instant of the flat meniscus defines the absolute height of the vial. The absolute height of the vial follows from Figure 7.5: the height of the vial is 6.13µm. 8.4 PRECISION AND ACCURACY OF PHASE ESTIMATION Accuracy. The phase estimation is biased. Figure 8.4 shows the bias of the phase estimation of the temporal phase unwrapping algorithm. The bias of the phase estimate is computed by unwrapping cos(φ[t]) with unit amplitude and no noise. The argument of the cosine function Φ[t] is given by α 2π K w (t Kw 2 ), where α defines the number of periods of Φ[t] within K w. The number of periods varies from 1.5 periods at the bottom of this graph to 2.5 periods at the top. Recall that the number of periods within K w varies approximately from 1.6 to 2.4 periods due to the subsampling. The bias is less than.1rad. It can be seen in Figure 8.4 that the bias of the phase estimation gets very large for windows smaller than 1.58 periods. The bias in these regions is at least one order of magnitude larger than in the remaining part of this figure. From simulations follows that
119 PRECISION AND ACCURACY OF PHASE ESTIMATION 19 Bias in rad.1.5 number of periods α in K w phase = 2π phase = π unbiased phase phase = phase = π phase = 2π Fig. 8.4 This figure shows the bias of the phase estimation for a cosine function with a varying number of periods α in K w. The cosine function has unit amplitude and no noise. The bias is less than.1rad. In the region below α = 1.58 the bias is at least one order of magnitude larger. 1. varying the amplitude A[t] as a function of the time t (A[t] cos(φ[t])) results in a larger bias. 2. The bias increases when the number of periods within K w deviates more from two. 3. The bias of the phase estimation with exactly 2 periods within K w is minimal. 4. The phase estimation Φ = is only unbiased when the amplitude A[t] is constant. The shape of the regions below 1.58 periods is maintained. Precision. So far, we have not taken the additive noise term of Equation 8.1 into account. Figure 8.1(a) shows a part of the interferogram in the center of the vial. The amplitude A[t] is on the order of 7ADUs (AnologtoDigital Units). The standard deviation of the noise σ N is estimated by σ N = 1 T (I[t] I[t] G[t; σ = 2.]) T 1 2, (8.8) t=1 where G[ ; σ] is a Gaussian kernel with standard deviation σ and is the convolution operator. A typical noise level for the measurements presented in this part of this thesis is σ N =.77. The SignaltoNoise Ratio (SNR) is on the order of 1. The variation of the phase estimation is measured by applying the algorithm to a series of 1 artificially generated interferograms with A = 1, B =, and N[t] drawn from a Gaussian distribution with a standard deviation
120 11 TEMPORAL PHASE UNWRAPPING ALGORITHM RMS error of phase estimation (rad) Standard Deviation of additive Gaussian noise SNR = 1 SNR < 1 SNR > Amplitude A of interferogram (#ADU) Fig. 8.5 This graph shows the RMS error of the phase estimation as a function of the amplitude of the interferogram with a constant noise level. Even when the amplitude equals the standard deviation of the noise, the phase can still be estimated with an RMS error of.16. If the SNR is below 1, the RMS error increases rapidly. of.1 (SNR=1). The RMS error of the phase estimation is.15rad. We conclude that for this SNR our phase estimation is very precise. Our observation that the contrast increases during evaporation (See also Figure 8.1(a)) suggests, under the assumption that the noise level remains constant in time, that the SNR increases during evaporation. For most regions this statement holds. For some regions, however, the gain in amplitude is reduced by spatial undersampling. Note that the optical spatial resolution ) is not the limiting factor, but the sampling density related to the video camera and the acquisition software. It is important to determine the lowest required SNR for a proper estimation of the phase map. To investigate this empirically, a series of 1 interferograms were artificially generated with A =.1 1., B =, and N[t] drawn from a Gaussian distribution with a standard deviation of.5. For each series the RMS error of the phase estimation is measured. The results of these simulations are shown in Figure 8.5. The RMS error of the phase estimation is below.16rad, when the SNR is higher than 1. Below this SNR the RMS error increases rapidly. ( 2NA λ 8.5 SIMULTANEOUSLY SAMPLING IN SPACE AND IN TIME In the first part of this chapter three possible problems from which spatial unwrapping algorithms might suffer were mentioned. One of these problems is sampling. In this section we will discuss in detail the aspects of sampling. From a signal processing point of view there is a fundamental advantage to do the unwrapping in time.
121 SIMULTANEOUSLY SAMPLING IN SPACE AND IN TIME 111 The Nyquist sampling criterion dictates that the fringe pattern needs to be sampled at least at twice the highest frequency in the interferogram to enable reconstruction of the analog fringes. Here, however, we are not interested in reconstruction of the fringes, but in reconstruction of the meniscus that produced this fringe pattern. Image acquisition with a CCD camera with a 1% fill factor for the photosensitive sites, however, is not just sampling the signal with a 2D impulse train. Image acquisition with a CCD camera can be described in two steps: first, the continuous spatial signal is convolved with the transfer function of the pixel. In the ideal case, each pixel on the CCD element has a uniform sensitivity. This implies that the transfer function of a pixel is a block function. Convolution of the signal with the pixel transfer function is equal to a multiplication of the Fourier transform of the signal and the Fourier transform of the pixel transfer function in the frequency domain. The Fourier transform of the pixel transfer function is a sincfunction. This sinc function has its first zerocrossing at the sampling frequency. The second step in image acquisition is to sample the convolved signal with a 2D impulse train. As long as the fringe pattern is spatially sampled at a frequency above the Nyquist frequency, then spatial reconstruction of the fringe pattern is possible with this kind of acquisition. In our experiments, the fringe patterns are sampled in space as well as in time. For this moment, we assume that our temporal sampling satisfies the Nyquist criterion. The left image of Fig. 8.6 shows a onedimensional fringe pattern generated at the meniscus interface as a function of the contact angle of the meniscus as indicated in Fig. 9.3: from the bottom of this figure upwards the contact angle increases. This results in a steeper meniscus, which is assumed in this example to be straight, and therefore a spatially denser fringe pattern. We assume in this example that the fringe pattern is generated with monochromatic light, such that the amplitude of the fringe pattern is constant for different heights. For three different contact angles of the meniscus the amplitude of the fringe pattern is shown. The right image shows the effects of imaging the left image onto a CCD camera. First, the left figure is convolved with a uniform filter, the pixel transfer function. As a result of this convolution, the amplitude decreases with increasing frequency, and at a certain frequency (due to a larger contact angle), the amplitude of the fringe pattern becomes negative. This phase jump of π radians is visualized in region 3 in the right image. Secondly, the filtered fringe pattern is sampled with a 1D impulse train, indicated by the dots.
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