Applications of Integration in Biomedical Science. by William T. Self UCF EXCEL Applications of Calculus

Transcription

1 Applications of Integration in Biomedical Science by William T. Self UCF EXCEL Applications of Calculus

2 Calculus Topic: Defining area under the curve Topic #1: Approximating rectangles One possible method for estimating area under a given curve (or function) is the use of approximating rectangles This is a simple method, but has limitations in its ability to accurately define the area

3 Calculus concept # 1 Approximating Rectangles Section 5.1 #1: By reading values from the given graph of f (shown on the next slide) use three rectangles to find a lower estimate for the area under the given graph of f from x=0 to x=6. In each case sketch the rectangles that you use.

4 Approximating rectangles Reminder: 3 rectangles Lower limit From x=0 to x= y = f(x) Answers: A) 17 B) 19 C) 21 D) 20 E) 28 y x

5 Approximating rectangles The use of this technique is inadequate to determine the area under a curve since it can overestimate and underestimate this area This section of the applications course will introduce you to concepts and methods in biomedical science that rely on calculus to determine the quantity of compounds and macromolecules

6 Applications of Integration in Biomedical Science Some of the future courses (that you may take) that this will be relevant: MCB 3020 General Microbiology BSC 3403C Quantitative Biological Methods MCB 4414 Microbial Metabolism BCH 4053 Biochemistry I BCH 4054 Biochemistry II

7 Life Its existence on Earth Time Line for Planet Earth Prokaryotes Eukaryotes Prokaryotes involved in formation of the biosphere required for plant & animal survival

8 Life Cellular level eukaryotic cell prokaryotic cell

9 What are cells made of (E.coli )? CHNOPS: Carbon Hydrogen Nitrogen Oxygen Phosphorus Sulfur Adenosine triphosphate - ATP

10 Biological Macromolecules

11 Trace Elements Human composition (complements of Dept. of Energy) Dry weight % Carbon 61.7 Nitrogen 11.0 Oxygen 9.3 Hydrogen 5.7 Calcium 5.0 Phosphorus 3.3 Potassium 1.3 Sulfur 1.0 Chlorine 0.7 Sodium 0.7 Magnesium 0.3 Trace amounts of B, F, Si, V, Cr, Mn, Fe, Co, Cu, Zn, Se, Mo, Sn, I. There are some arguments as to the importance of other trace elements

12 Biological Cells Complex mixtures Basics: DNA, RNA: Polymers of nucleic acids encode proteins Proteins: Polymers of amino acids can be structural or act as enzymes Lipids: Polymers of carbon structural components of cell membranes

13 Biological Cells Complex mixtures A given cell will have thousands of different proteins, RNA molecules and metabolites present under a particular growth condition How do we define the role of each individual protein (for example)? First we must purify this protein away from all other components, then study it in a test tube (in vitro)

14 Protein Purification Proteins are polymers of amino acids Protein sequence defines the chemical composition Each protein has unique size, charge and shape

15 Chromatography separation of mixtures Chromatography in general is the separation of compounds from mixtures using a Solid phase and a mobile phase Typically the solid phase is stationary, and held in place in a column The mobile phase (usually aqueous) moves through the solid phase and carries the sample

16 Chromatography separation of mixtures Samples separate from each other on the column due to differences in their unique properties: 1.) net charge 2.) hydrophobicity 3.) size 4.) specific affinity

17 Chromatography separation of mixtures Types of chromatography used in protein purification: 1.) Ion Exchange 2.) Gel filtration 3.) Hydrophobic 4.) Affinity

18 Types of chromatography Protein separations 1.) Ion exchange: The solid phase has a strong or weak charged group (e.g. strong positive charge) If a protein has a net negative charge (anionic), it will bind to a column that has a cationic group (positive charge). Each protein will have a slightly different net charge and thus mixtures of proteins can be separated based on net charge

19 Types of chromatography Protein separations 2.) Gel filtration Proteins will separate based on size, due to pores present in beads in the solid phase. The pores define the separation capabilities of the media (e.g. 30,000 MW to 3,000,000 MW)

20 Types of chromatography Protein separations 3.) Hydrophobic Interaction Chromatography Based on binding of hydrophobic amino acids (such as leucine, isoleucine) that are usually buried but occasionally present on the surface Common groups on the stationary phase are phenyl groups or carbon chains

21 Types of chromatography Protein separations 4.) Affinity Chromatography Generally, proteins can be engineered to contain tags at their ends that will bind to a certain group (e.g. metal). This tag is usually unique in the mixture and thus a tagged protein can be purified quite readily from a cell extract using this procedure. The use of protein tags has revolutionized the study of proteins in enzymes in the wake of the era of molecular biology and cloning.

22 How does this relate to Calculus??? To find and determine the quantity of a given protein, or other molecule of interest, we follow the elution of these molecules using a detector. This pattern is essentially a continuous function from one time period to the next as follows: Samples eluting become a series of peaks that can be followed and quantified by area under the curve

23 Calculus concept # 2 Limitations of approximating rectangles Section 5.1 #2 Use 4 rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x =6. Three questions left and right endpoints and finally midpoints

24 Calculus Topic: Defining area under the curve 10 Reminder: Left endpoints Answers: A) B) C) D) E) y y = f(x) x

25 Calculus Topic: Defining area under the curve 10 Reminder: Right endpoints Answers: A) B) C) D) E) y y = f(x) x

26 Calculus Topic: Defining area under the curve 10 Reminder: Midpoints Answers: A) B) C) D) E) y y = f(x) x

27 Calculus Topic: Defining area under the curve Which of the three techniques is best? Why? Could there be a better way based on your current knowledge of calculus?

28 Applications of Integration in Biomedical Science In addition to protein purification, chromatography (and area under the curve) has many other uses in biomedical science Some issues to discuss: 1.) Arsenic (and other contaminants) in drinking water 2.) Drug testing (e.g. steroid use) 3.) Bioterrorism detection of explosives 4.) Pesticides in agriculture and consumer use

29 Applications of chromatography How do we determine the presence of a pesticide present in a lake, river or stream? How do we quantify such a compound? Why does this quantization matter?

30 Drug testing front lines Websites for discussion: tions/standard_urine_testing_es.html ckground/2007/bg_sports_drug_testing.pdf

31 Drug testing front lines Recent article in the journal Nature outlines issues in drug testing for anabolic steroids

32 Example of LC-MS analysis Overview of typical HPLC setup: Detector is typically a mass spectrometer that can predict the mass of eluting compounds

33 Example of LC-MS analysis Agilent Technologies example of LC profile of steroids

34 Gas chromatography (GC) Gas chromatography: Similar to HPLC, with the exception that the mobile phase is a gas Sample is either a gas or is derivatized to a volatile form to allow for separation in a gas mobile phase Column has a liquid stationary phase which is bound to an inert support phase that is solid This form of chromatography is most common in analytical analysis of pesticides and lipid analysis.

35 GC typical set up Typical GC setup courtesy of Waters, Inc.

36 Explosives GC-MS example Small amounts of explosives can be buried in compounds that mask their presence in samples GC-MS can uncover readily

37 Calculus concept #3 Fundamental theorem of calculus The fundamental theorem of calculus states: (Part 1) g(x) = f (t )dt where f is a continuous function on [ a,b] and x varies between a and b.

38 Fundamental Theorem of Calculus y y = f(t) area = g(x) a x b x

39 Fundamental Theorem of Calculus Part 2 states: If f is continuous on [a,b] then: f (x ) dx = F (b ) F (a ) Essentially, for purposes of defining area under the curve, the difference in the antiderivative of f between two points [a,b ] on the curve (assuming a continuous function) is equal to the area of that curve to the x-axis This is the most critical application (in biological sciences) of the fundamental theorem

40 Fundamental Theorem of Calculus Insert clicker question Integration (Alvaro figure)

41 Fundamental Theorem of Calculus

42 Biomedical Science - review What are the three most abundant elements in the human body (dry weight analysis)? A.) Hydrogen, Nitrogen and Calcium B.) Carbon, Nitrogen and Hydrogen C.) Magnesium, Carbon and Nitrogen D.) Carbon, Oxygen and Hydrogen E.) Carbon, Selenium and Magnesium

43 Methods of detection in chromatography After separation (HPLC, GC, etc.) we must identify and quantify a molecule of interest Some of the commons ways to find and quantify molecules: 1.) UV-visible spectroscopy 2.) Mass spectrometry 3.) Flame ionization (FID) 4.) Thermal conductivity (TCD)

44 Methods of detection in chromatography These abbreviations lead to the multitude of common analytical techniques: LC-MS (Liquid chromatography detection by mass spectrometry GC-MS, etc. All are still based on the fundamental concepts of chromatography, and all can use integration of peak area to determine the quantity of an eluted sample

45 Methods of detection in chromatography 1.) UV-visible spectroscopy Functional groups in a molecule can absorb light at a given wavelength Aromatics and metalcomplex ligands are common groups in biological samples that absorb light in UV or visible range

46 Methods of detection in chromatography DNA absorbs light at approximately 260 nanometers Courtesy Biocompare

47 Methods of detection in chromatography Proteins absorb light at approximately 280 nanometers Due to tryptophan and tyrosine residues

48 Methods of detection in chromatography HPLC analysis of purines A purine metabolizing enzyme was tested for its substrate specificity (which compounds it acts on) using HPLC analysis Each substrate and product elutes at a different time from reverse phase HPLC (hydrophobic stationary phase) Purines followed by UV-vis*

49 Methods of detection in chromatography 2.) Mass spectrometry Mass spectrometry determines the overall predicted molecular weight of a molecule based weighing its charge to mass ratio Molecules are charged in an ion source, then accelerated to a high speed. They are then passed through a magnetic field and their trajectory is altered by this field, dependent on their charge to mass ratio

50 Methods of detection in chromatography The particles are then detected and their composition can be predicted based on this charge to mass ratio Image courtesy of USGS Other information on the sample is generally needed to be able to identify and confirm the molecule of interest

51 Methods of detection in chromatography 3.) Flame Ionization Detection (FID) FID is commonly used in GC applications, and is based on burning of the sample FID is very good at detecting hydrocarbons and other carbon containing molecules 4.) Thermal Conductivity Detection (TCD) TCD is commonly used to detect gases (hydrogen) when carried in an inert gas (argon) TCD is based on changes in thermal conductivity useful since it can detect nearly any compound

52 Use of calculus in Biomedical Science - Review What characteristic of proteins is useful in gel filtration chromatography? A.) Affinity for ligands B.) Net charge C.) Hydrophobicity D.) Size E.) Sequence

53 Proteomics cutting edge use of chromatography Cancer diagnosis: Current techniques Example: Breast cancer Mammogram Ultrasound Biopsy Genetic screening Expensive, labor intensive and usually only detect cancer at later stages (not when first forming)

54 Proteomics cutting edge use of chromatography Proteomics: The proteome is defined as the set of proteins present in the cell under a given growth condition The complement of proteins changes in different cell types (tissues) and under different conditions (stress, infection, disease) Genetic variability also is displayed in the proteome

55 Proteomics cutting edge use of chromatography Proteomics in Cancer diagnosis: Using reverse phase chromatography to follow the proteome of a clinical sample (e.g. serum), one can obtain a profile of the peptides that are present in a patient Analysis of hundreds of patients, both ill and healthy, allow for patterns to emerge in this analysis

56 Proteomics cutting edge use of chromatography Above is a sample chromatogram of the peptides in serum of an ovarian cancer patient Biomarkers of Ovarian Cancer, Gynecologic Oncology 88, S25 S28 (2003) doi: /gyno

57 Proteomics cutting edge use of chromatography Proteomic analysis to diagnose cancer: In a study published in 2002 using a blinded set of samples, the proteomic pattern correctly predicted 36 (95%, 95% confidence interval [CI] = 82% to 99%) of 38 patients with prostate cancer, while 177 (78%, 95% CI = 72% to 83%) of 228 patients were correctly classified as having benign conditions. Serum proteomic patterns for detection of prostate cancer. J Natl Cancer Inst Oct 16;94(20):