Dynamics of Biological Systems
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1 Dynamics of Biological Systems Part I - Biological background and mathematical modelling Paolo Milazzo (Università di Pisa) Dynamics of biological systems 1 / 53
2 Introduction The recent developments in biology have produced a huge amount of data about the structure of living matter; consider as an example the success of the Human Genome Project Less is known about the versatile functions that cells and their components show. In the last few years the scientific interest has started to move from structures to functionalities The complexity of the cellular processes has stimulated the growth of a new paradigm, that moves from the classical reductionist approach to a system level understending of life Such a paradigm is called systems biology Paolo Milazzo (Università di Pisa) Dynamics of biological systems 2 / 53
3 Introduction A better understanding of the funcitoning of cellular processes may allow: a better undertanding of diseases the development of more effective drugs the development of preventive and early diagnosis techniques Mathematical and computational modelling may contribute to the study of cellular processes with simulation tools that, based on data from laboratory experiments, could be used to: validate hypotheses suggest further experiments predict the effect of some treatments In the future, treatment of diseases will be based on patient-specific therapies simulation tools capable to predict the effect of some therapy on a specific patient will be essential Paolo Milazzo (Università di Pisa) Dynamics of biological systems 3 / 53
4 Outline of the talk 1 Introduction 2 Biological Background Elements of cell biology Examples of cellular processes 3 Mathematical modelling The mass action kinetics of chemical reactions The Michealis-Menten kinetics The logistic function Advantages and disadvantages of ODE modelling of biological systems Paolo Milazzo (Università di Pisa) Dynamics of biological systems 4 / 53
5 Cells: complex systems of interactive components Two classifications of cell: prokaryotic eukaryotic Main actors: membranes proteins DNA/RNA ions, macromolecules,... Interaction networks: metabolic pathways signaling pathways gene regulatory networks Paolo Milazzo (Università di Pisa) Dynamics of biological systems 5 / 53
6 The DNA The DNA is: a molecule structured as a string over an alphabet of four elements (nucleic acids, bases) denoted A,T,C,G DNA forms double-stranded helices: Base pairing: A T,C G The complement of a string is obtained by replacing A with T and C with G, and viceversa Two complementary strings form a helic Paolo Milazzo (Università di Pisa) Dynamics of biological systems 6 / 53
7 Proteins A gene is a substring of the DNA some genes are the source code of proteins A protein is: a molecule structured as a string over an alphabet of twenty elements (amino acids) Proteins have complex 3D structures related with their functions: Catalysis of chemical reactions (enzymes) Transport Structure... Paolo Milazzo (Università di Pisa) Dynamics of biological systems 7 / 53
8 The central dogma of Molecular Biology Schematically, in cells we have this flux of information: DNA transcription RNA translation Protein Where the RNA is a molecule structured as a string over the alphabet A,U,C,G (similar to that of DNA) It is essentially a copy of the DNA (this motivates the terminology of transcription) Both transcription and translation can be regulated in order to synthesize proteins only when necessary Paolo Milazzo (Università di Pisa) Dynamics of biological systems 8 / 53
9 On the importance of membranes Eukariotic cells have a large number of compartments separated by membranes this allows a more efficient cell management (based on divide et impera ) Membranes have (at least) two fundamental functions: as separators of compartments as channels of communication between compartments Paolo Milazzo (Università di Pisa) Dynamics of biological systems 9 / 53
10 The main classes of cellular processes Gene regulation networks: are collection of genes in a cell which interact with each other (indirectly through their RNA and protein expression products) and with other substances in the cell, thereby governing the rates at which genes in the network are transcribed into RNA. Metabolic pathways: are series of chemical reactions occurring within a cell aimed at transforming some initial molecule into another product. Cell signalling pathways: (or signal transduction pathways) are series of chemical reactions occurring within a cell aimed at converting some chemical stimulus into a specific cellular response. Signal transduction starts with a signal to a receptor, and ends with a change in cell behavior. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 10 / 53
11 Outline of the talk 1 Introduction 2 Biological Background Elements of cell biology Examples of cellular processes 3 Mathematical modelling The mass action kinetics of chemical reactions The Michealis-Menten kinetics The logistic function Advantages and disadvantages of ODE modelling of biological systems Paolo Milazzo (Università di Pisa) Dynamics of biological systems 11 / 53
12 Example of gene regulation network: the lac operon E. coli is a bacterium often present in the intestine of many animals. It is one of the most completely studied of all living things. As most bacteria, E.coli is often exposed to a constantly changing physical and chemical environment, and reacts to changes in its environment through changes in the kinds of enzymes it produces. In order to save energy, bacteria do not synthesize degradative enzymes unless the substrates (e.g. lactose) for these enzymes are present in the environment. This result is obtained by controlling the transcription of some genes into the corresponding enzymes. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 12 / 53
13 Example of gene regulation network: the lac operon Two enzymes are mainly involved in lactose degradation: the lactose permease, which is incorporated in the membrane of the bacterium and actively transports the sugar into the cell, and the beta galactosidase, which splits lactose into glucose and galactose. The bacterium produces also the transacetylase enzyme, whose role in the lactose degradation is marginal. The sequence of genes in the DNA of E. coli which produces the described enzymes, is known as the lactose operon. The lactose operon consists of six genes: The first three genes of the operon (i, p and o) regulate the production of the enzymes, the last three (z, y and a), called structural genes, are transcribed (when allowed) into the mrna for beta galactosidase, lactose permease and transacetylase, respectively. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 13 / 53
14 Example of gene regulation network: the lac operon The regulation process is as follows: Gene i encodes the lac Repressor, which, in the absence of lactose, binds to gene o (the operator). Transcription of structural genes into mrna is performed by the RNA polymerase enzyme, which usually binds to gene p (the promoter) and scans the operon from left to right by transcribing the three structural genes z, y and a into a single mrna fragment. When the lac Repressor is bound to gene o, it becomes an obstacle for the RNA polymerase, and the transcription of the structural genes is not performed. On the other hand, when lactose is present inside the bacterium, it binds to the Repressor and this cannot stop anymore the activity of the RNA polymerase. In this case the transcription is performed and the three enzymes for lactose degradation are synthesized. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 14 / 53
15 Example of gene regulation network: the lac operon DNA i p o z y a mrna proteins R lac Repressor beta-gal. permease transacet. i p o z y a a) RNA Polimerase R NO TRANSCRIPTION i p o z y a b) RNA Polimerase R LACTOSE TRANSCRIPTION Paolo Milazzo (Università di Pisa) Dynamics of biological systems 15 / 53
16 Example of metabolic pathway: the glycolysis pathway Glycolysis is the metabolic pathway that converts glucose, C6H12O6, into pyruvate, CH3COCOO- and H+. The free energy released in this process is used to form the high energy compounds, ATP (adenosine triphosphate) and NADH (reduced nicotinamide adenine dinucleotide). Glycolysis is a definite sequence of ten reactions involving intermediate compounds. The intermediates provide entry points to glycolysis. For example, most monosaccharides, such as fructose, glucose, and galactose, can be converted to one of these intermediates. The intermediates may also be directly useful. For example, the intermediate dihydroxyacetone phosphate is a source of the glycerol that combines with fatty acids to form fat. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 16 / 53
17 Example of metabolic pathway: the glycolysis pathway Paolo Milazzo (Università di Pisa) Dynamics of biological systems 17 / 53
18 Example of signalling pathway: the EGF pathway A classical example of biological system is the EGF signal transduction pathway. If EGF proteins are present in the environment of a cell, they must be interpreted as a signal from the environment meaning that new cells are needed. A cell recognizes the EGF signal from the environment because it has on its membrane some EGF receptor proteins (EGFR), which are transmembrane proteins (they have some intra cellular and some extra cellular domains). Paolo Milazzo (Università di Pisa) Dynamics of biological systems 18 / 53
19 Example of signalling pathway: the EGF pathway The signalling pathway is as follows: One of the extra cellular domains binds to one EGF protein in the environment, forming a signal receptor complex on the membrane. This causes a conformational change on the receptor protein that enables it to bind to another one signal receptor complex. The formation of the binding of the two signal receptor complexes (called dimerization) causes the phosphorylation (addition of some phosphate groups PO 4 ) of some intra cellular domains of the dimer. This causes the internal domains of the dimer to be recognized by proteins that are in the cytoplasm, which bind to the dimer, enabling a chain of protein protein interactions inside the cell. This chain of interactions finally activate some proteins which bind to the DNA and stimulate synthesis of proteins for cell proliferation. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 19 / 53
20 Example of signalling pathway: the EGF pathway Paolo Milazzo (Università di Pisa) Dynamics of biological systems 20 / 53
21 The cell cycle The cell cycle is a series of sequential events leading to cell replication via cell division. It consists of four phases: G 1, S, G 2 and M. G 1 and G 2 are gap phases in which the cell prepares itself to enter phases S and M, respectively S is a synthesis phase, in which DNA is replicated M is a mitosis phase, in which the cell segregates the duplicated sets of chromosomes between daughter cells and then divides. The duration of the cell cycle depends on the type of cell (e.g a human normal cell takes approximately 24 hours to perform a cycle). Paolo Milazzo (Università di Pisa) Dynamics of biological systems 21 / 53
22 The cell cycle Paolo Milazzo (Università di Pisa) Dynamics of biological systems 22 / 53
23 Outline of the talk 1 Introduction 2 Biological Background Elements of cell biology Examples of cellular processes 3 Mathematical modelling The mass action kinetics of chemical reactions The Michealis-Menten kinetics The logistic function Advantages and disadvantages of ODE modelling of biological systems Paolo Milazzo (Università di Pisa) Dynamics of biological systems 23 / 53
24 Background The modelling of the dynamics of biological systems is essentially based on the modelling of the dynamics (kinetics) of some (bio)chemical reactions The modelling of chemical reactions using deterministic rate laws has proven extremely successful in both chemistry and biochemistry for many years This deterministic approach has its core the law of mass action, an emprirical law giving a simple relation between reaction rates and molecular component concentrations Given knowledge of inital concentrations, the law of mass action provides a complete picture of the component concentrations at all future time points Paolo Milazzo (Università di Pisa) Dynamics of biological systems 24 / 53
25 Keywords Some definitions (mostly from Wikipedia...) chemical solution: a homogeneous mixture composed of two or more substances (molecules) biochemical solution: chemical solution in which substances have an organic nature (proteins, DNA, etc...) chemical reactions: a process that leads to the transformation of one set of chemical substances to another biochemical reactions: chemical reactions in which substances have an organic nature (proteins, DNA, etc...) kinetics: the study of (bio)chemical reaction rates (which govern the dynamics of (bio)chemical solutions) rate: frequency of an event (e.g. a reaction) over time concentration: average quantity of molecules of a solution in a volume unit deterministic model: model which exhibit a single (usually average) behaviour (dynamics) stochastic model: model which takes probabilistic/stochastic behaviour (dynamics) into account Paolo Milazzo (Università di Pisa) Dynamics of biological systems 25 / 53
26 The mass action kinetics of chemical reactions Usual notation for chemical reactions: l 1 S l ρ S ρ k k 1 l 1P l γp γ where: S i, P i are molecules (reactants) l i, l i are stoichiometric coefficients k, k 1 are the kinetic constants Examples of chemical reactions: 2H 2 + O 2 k k 1 2H 2 O E + S k 1 ES k 2 E + P k 1 k 2 Paolo Milazzo (Università di Pisa) Dynamics of biological systems 26 / 53
27 The mass action kinetics of chemical reactions Usual notation for chemical reactions: l 1 S l ρ S ρ k k 1 l 1P l γp γ The kinetics is described by the law of mass action: The rate of a reaction is proportional to the products of the concentrations of the participating molecules. The rate of a reaction (the left-to-right direction of the reversible reaction above) is defined as follows: k[s 1 ] l1 [S ρ ] lρ similarly, the rate of the opposite reaction (the right-to-left direction) is k 1 [P 1 ] l 1 [Pγ ] l γ Paolo Milazzo (Università di Pisa) Dynamics of biological systems 27 / 53
28 The mass action kinetics of chemical reactions For example, the rates of 2H 2 + O H 2 O are 5[H 2 ] 2 [O 2 ] for the left-to-right direction 0.5[H 2 O] 2 for the right-to-left direction Similarly, the rates of are E + S ES E + P 0.1[E][S] for the formation of the complex ES from E and S 1000[ES] for the unbinding of ES into E and S 0.3[ES] for the unbinding of ES into E and P 0.01[E][P] for the formation of the complex ES from E and P Paolo Milazzo (Università di Pisa) Dynamics of biological systems 28 / 53
29 The mass action kinetics of chemical reactions Usual notation for chemical reactions: l 1 S l ρ S ρ k k 1 l 1P l γp γ A reversible reaction is said to be in dynamic equilibrium when the rates of its two directions are the same, namely: k[s 1 ] l1 [S ρ ] lρ = k 1 [P 1 ] l 1 [Pγ ] l γ When the dynamic equilibrium is reached, often the chemical solution seems stable (it seems that nothing is happening inside it). Instead, reactions happen continuously, compensating each other. It is easy to see that at the equilibrium we have: k = [P 1] l 1 [Pγ ] l γ k 1 [S 1 ] l 1 [Sρ ] lρ Paolo Milazzo (Università di Pisa) Dynamics of biological systems 29 / 53
30 Ordinary differential equations (ODEs) Reaction rates can be used to construct a mathematical model (based on ordinary differential equations - ODEs) of the dynamics of a set of chemical reactions Ordinary differential equations (ODEs) are mathematical equations for un unkonwn function of one or of several variables that relates the values of the function itself and its derivatives. In other words, an ODE describes (by means of derivatives) how a variable of an unknown function changes over time The unknown function could be obtained by integrating the corresponding ODEs (often not feasible) Paolo Milazzo (Università di Pisa) Dynamics of biological systems 30 / 53
31 Ordinary differential equations (ODEs) Some examples: dx = 2 Paolo Milazzo (Università di Pisa) Dynamics of biological systems 31 / 53
32 Ordinary differential equations (ODEs) Some examples: dx = 2 Paolo Milazzo (Università di Pisa) Dynamics of biological systems 32 / 53
33 Ordinary differential equations (ODEs) Some examples: dx = 2x + 1 Paolo Milazzo (Università di Pisa) Dynamics of biological systems 33 / 53
34 Ordinary differential equations (ODEs) Some examples: dx 1 = 0.04x 1 dx 2 = 0.04x x 2 dx 3 = 0.03x 2 Paolo Milazzo (Università di Pisa) Dynamics of biological systems 34 / 53
35 Numerical solution of ODEs Since ODEs often cannot be solved analytically, numerical integration (or simulation) techniques are usually applied to find approximate solutions. The simplest numerical integration algorithm is Euler s method. Euler s method approximates a continuous function x = f (t) on 0 t L (where L is a constant representing the simulation time) which is a solution of the differential equation dx = F (x, t) with initial condition x 0 = α (where α is a given constant). The interval 0 t L is then divided into n equal parts, each of length h, with h very small (the smaller it is, the more precise the approximation will be). We obtain the points t i = ih with i = 0, 1,..., n. Euler s method consist in computing values x i with i = 0, 1,..., n obtained by the following recurrence relation: x 0 = α x i + 1 = y i + hf (x i, t i ) with i = 0, 1,..., n 1 Paolo Milazzo (Università di Pisa) Dynamics of biological systems 35 / 53
36 The mass action kinetics of chemical reactions The ODE model of a set of chemical reactions contains one differential equation for each molecular species From these two reactions: l 1 S l ρ S ρ k k 1 l 1 P l γp γ we obtain the following mathematical model: = l 1 k 1 [P 1 ] l 1 [Pγ ] l γ l 1 k[s 1 ] l1 [S ρ ] lρ d[s 1 ]. d[s ρ] =... = l 1 k[s 1] l1 [S ρ ] lρ l 1 k 1[P 1 ] l 1 [Pγ ] l γ d[p 1 ]. d[p γ] =... Paolo Milazzo (Università di Pisa) Dynamics of biological systems 36 / 53
37 The mass action kinetics of chemical reactions 2H 2 + O H 2 O ODEs obtained from mass action kinetics: d[h 2 ] = 2 0.5[H 2 O] 2 2 5[H 2 ] 2 [O 2 ] d[o 2 ] = 0.5[H 2 O] 2 5[H 2 ] 2 [O 2 ] d[h 2 O] = 2 5[H 2 ] 2 [O 2 ] 2 0.5[H 2 O] 2 E + S ES E + P ODEs obtained from mass action kinetics: d[e] = 1000[ES] + 0.3[ES] 0.1[E][S] d[es] = 0.1[E][S] [E][P] 1000[ES] 0.3[ES] d[s] d[p] = 1000[ES] 0.1[E][S] = 0.3[ES] 0.01[E][P] Paolo Milazzo (Università di Pisa) Dynamics of biological systems 37 / 53
38 The mass action kinetics of chemical reactions We can now simulate the dynamics of the reactions (from given initial concentrations) by using numerical integration d[h 2] 5 = 2 0.5[H 2 O] 2 2 5[H 2 ] 2 [O 2 ] d[o 2H 2 + O H 2 O 2] = 0.5[H 2 O] 2 5[H 2 ] 2 [O 2 ] d[h 2O] = 2 5[H 2 ] 2 [O 2 ] 2 0.5[H 2 O] 2 with initial concentrations [H 2 ] 0 = [O 2 ] 0 = 100 and [H 2 O] 0 = 0. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 38 / 53
39 Other kinetic models of chemical reactions In some cases the mass action kinetics could be considered too low-level, and introduce unneccesary details in the models of some phenomena. Two examples of more abstract kinetic models are: the Michaelis-Menten kinetics of enzyme-catalysed reactions the logistic function for reactions subject to saturation These kinetic models are approximations of the mass action kinetics that could be considered reasonable under certain conditions and that make models simpler (hence, easier to be simulated numerically) Paolo Milazzo (Università di Pisa) Dynamics of biological systems 39 / 53
40 Outline of the talk 1 Introduction 2 Biological Background Elements of cell biology Examples of cellular processes 3 Mathematical modelling The mass action kinetics of chemical reactions The Michealis-Menten kinetics The logistic function Advantages and disadvantages of ODE modelling of biological systems Paolo Milazzo (Università di Pisa) Dynamics of biological systems 40 / 53
41 The Michaelis-Menten kinetics The Michaelis-Menten kinetics describes the dynamics of irreversible enzyme-catalysed reactions such as: E + S k 1 ES k 2 E + P k 1 The assumptions under which the kinetics is an acceptable approximation of the mass action kinetics are the following: the whole enzyme-catalysed reaction is irreversible, namely the reaction E + P k 2 ES has a very small kinetic constant k 2 and can be ignored rates of E + S k 1 k 1 ES as given by the mass action kinetics are much higher than the rate of ES k 2 E + P. Hence, the first two reactions can be considered in dynamic equilibrium. Paolo Milazzo (Università di Pisa) Dynamics of biological systems 41 / 53
42 The Michaelis-Menten kinetics E + S k1 ES k2 E + P k 1 by the dynamic equilibrium assumption we have k 1 [E][S] = (k 1 + k 2 )[ES] since the enzyme is not consumed we have [E] 0 = [E] + [ES] constant hence, k 1 ([E] 0 + [ES])[S] = (k 1 + k 2 )[ES] hence, (k 1 + k 2 )[ES] k 1 [ES][S] = k 1 [E] 0 [S] hence, [ES] = k1[e]0[s] (k 1+k 2 k 1[S] is the concentration of the complex ES at the dynamic equilibrium let K M = k 1+k2 k 1, we obtain [ES] = [E]0[S] K M +[S] we know that d[p] = k 2 [ES], hence, d[p] let V max = k 2 [E] 0, we finally obtain: d[p] that is the Michaelis-Menten equation. = k2[e]0[s] K M +[S] = V max[s] K M + [S] Paolo Milazzo (Università di Pisa) Dynamics of biological systems 42 / 53
43 The Michaelis-Menten kinetics In the Michaelis-Menten equation d[p] = V max[s] K M + [S] we have only two parameters: K M (called Michaelis-Menten constant) that corresponds to k 1+k 2 k 1 V max that corresponds to k 2 [E] 0 and is the max speed of production of P The concentrations of the enzyme [E] and [ES] are no longer necessary variables in the ODE model of the considered reactions Paolo Milazzo (Università di Pisa) Dynamics of biological systems 43 / 53
44 The Michaelis-Menten kinetics E + S k 1 ES k 2 E + P k 1 ODEs obtained from mass action kinetics: d[e] = k 1 [ES] + k 2 [ES] k 1 [E][S] d[es] = k 1 [E][S] k 1 [ES] k 2 [ES] d[s] d[p] = k 1 [ES] k 1 [E][S] = k 2 [ES] ODEs obtained from Michaelis-Menten kinetics: { d[s] d[p] Vmax [S] = = K M +[S] Vmax [S] K M +[S] Paolo Milazzo (Università di Pisa) Dynamics of biological systems 44 / 53
45 The Michaelis-Menten kinetics E + S k 1 ES k 2 E + P k 1 Numerical simulation with k 1 = 0.1, k 1 = 1000, k 2 = 0.3 and with initial concentrations [E] 0 = 100, [S] 0 = 10000, [ES] 0 = [P] 0 = 0: Mass Action Michaelis-Menten Paolo Milazzo (Università di Pisa) Dynamics of biological systems 45 / 53
46 Outline of the talk 1 Introduction 2 Biological Background Elements of cell biology Examples of cellular processes 3 Mathematical modelling The mass action kinetics of chemical reactions The Michealis-Menten kinetics The logistic function Advantages and disadvantages of ODE modelling of biological systems Paolo Milazzo (Università di Pisa) Dynamics of biological systems 46 / 53
47 The logistic function The logistic funciton finds common application in ecology to describe population growth in an enviroment with limited resources A differential equation in which the logistic function is used is as follows: dp = αp(1 P K ) where α is a constant representing the growth rate and K is the carrying capacity, namely the maximal sustainable value of P For example, with α = 2 and K = 1000, starting from P = 5 we obtain Paolo Milazzo (Università di Pisa) Dynamics of biological systems 47 / 53
48 The logistic function In general, the logistic function can be used to descibe the dynamics of something subject to an exponential growth combined with a saturation effect An example of such kind of dynamics in (bio)chemistry are autocatalytic reactions, namely reactions such as A + B k 2B Another biological example that can often be described with a logistic function is tumour growth Paolo Milazzo (Università di Pisa) Dynamics of biological systems 48 / 53
49 The logistic function A + B k 2B ODEs obtained from mass action kinetics: { d[a] d[b] = k[a][b] = k[a][b] ODEs obtained by using the logistic function: { d[b] = k[b][a] 0 (1 [B] [A] 0 +[B] 0 ) where [A] 0 and [B] 0 are constants representing the initial concentrations of A and B, respectively Paolo Milazzo (Università di Pisa) Dynamics of biological systems 49 / 53
50 The logistic function A + B k 2B Numerical simulation with k = 0.01, [A] 0 = 100 and [B] 0 = 10: Mass Action (A and B) Logistic function (only B) Paolo Milazzo (Università di Pisa) Dynamics of biological systems 50 / 53
51 Outline of the talk 1 Introduction 2 Biological Background Elements of cell biology Examples of cellular processes 3 Mathematical modelling The mass action kinetics of chemical reactions The Michealis-Menten kinetics The logistic function Advantages and disadvantages of ODE modelling of biological systems Paolo Milazzo (Università di Pisa) Dynamics of biological systems 51 / 53
52 Advantages of ODEs We have seen that ODEs can be used to describe the dynamics of biological systems. Advantages: ODEs offer tools for reasoning about the functioning of biological systems (validation of hypotheses, suggestions for new laboratory experiments, etc...) models based on ODEs can be used to make predictions ODEs can be also studied analytically the theory of ODEs is well-established (ODEs are the same in all books) a lot of rather efficient numeric solvers are available Paolo Milazzo (Università di Pisa) Dynamics of biological systems 52 / 53
53 Disadvantages of ODEs We have seen that ODEs can be used to describe the dynamics of biological systems. Disadvantages: When the complexity of the modelled system increases, ODE models become difficult to manage ODEs are continuous and deterministic. This makes them unsuitable to describe systems in which components occur in small numbers and subject to stochastic events (e.g. systems involving interaction with the DNA) Next step: stochastic modelling and simulation... Paolo Milazzo (Università di Pisa) Dynamics of biological systems 53 / 53
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