MATHEMATICAL MODELS IN BIOLOGY
T.I. Volynkina
D. Skripnikova student
Federal State Educational Institution of Higher Professional Education "Oryol State Agrarian University"
Mathematical biology is the theory of mathematical models of biological processes and phenomena. Mathematical biology refers to applied mathematics and actively uses its methods. The criterion of truth in it is mathematical proof; the most important role is played by mathematical modeling using computers. Unlike purely mathematical sciences, in mathematical biology purely biological problems and problems are studied using the methods of modern mathematics, and the results have a biological interpretation. The tasks of mathematical biology are the description of the laws of nature at the level of biology and the main task is the interpretation of the results obtained during research. An example is the Hardy-Weinberg law, which proves that a population system can be predicted based on this law. Based on this law, a population is a group of self-sustaining alleles in which natural selection provides the basis. Natural selection itself is, from a mathematical point of view, an independent variable, and a population is a dependent variable, and a population is considered a number of variables that influence each other. This is the number of individuals, the number of alleles, the density of alleles, the ratio of the density of dominant alleles to the density of recessive alleles, etc. Over the past decades, there has been significant progress in the quantitative (mathematical) description of the functions of various biosystems at various levels of life organization: molecular, cellular, organ, organismal, population, biogeocenological. Life is determined by many different characteristics of these biosystems and processes occurring at the appropriate levels of system organization and integrated into a single whole during the functioning of the system.
The construction of mathematical models of biological systems became possible thanks to the exceptionally intensive analytical work of experimenters: morphologists, biochemists, physiologists, specialists in molecular biology, etc. As a result of this work, the morphofunctional schemes of various cells were crystallized, within which various physicochemical processes occur in an orderly manner in space and time and biochemical processes that form a very complex interweaving.
The second circumstance contributing to the involvement of mathematical apparatus in biology is the careful experimental determination of the rate constants of numerous intracellular reactions that determine the functions of the cell and the corresponding biosystem. Without knowledge of such constants, a formal mathematical description of intracellular processes is impossible.
The third condition that determined the success of mathematical modeling in biology was the development of powerful computing tools in the form of personal computers and supercomputers. This is due to the fact that usually the processes that control a particular function of cells or organs are numerous, covered by feedforward and feedback loops and, therefore, described by systems of nonlinear equations. Such equations cannot be solved analytically, but can be solved numerically using a computer.
Numerical experiments on models capable of reproducing a wide class of phenomena in cells, organs and the body allow us to evaluate the correctness of the assumptions made when constructing the models. Experimental facts are used as postulates of the models; the need for certain assumptions and assumptions is an important theoretical component of the modeling. These assumptions and assumptions are hypotheses that can be tested experimentally. Thus, models become sources of hypotheses, and, moreover, experimentally verifiable ones. An experiment aimed at testing a given hypothesis can refute or confirm it and thereby help refine the model. This interaction between modeling and experiment occurs continuously, leading to a deeper and more accurate understanding of the phenomenon: experiment refines the model, a new model puts forward new hypotheses, experiment refines the new model, and so on.
Currently, mathematical biology, which includes mathematical theories of various biological systems and processes, is, on the one hand, already a sufficiently established scientific discipline, and on the other hand, one of the most rapidly developing scientific disciplines, combining the efforts of specialists from various fields knowledge - mathematicians, biologists, physicists, chemists and computer science specialists. A number of disciplines of mathematical biology have been formed: mathematical genetics, immunology, epidemiology, ecology, a number of sections of mathematical physiology, in particular, mathematical physiology of the cardiovascular system.
Like any scientific discipline, mathematical biology has its own subject, methods, methods and procedures of research. As a subject of research, mathematical (computer) models of biological processes arise, which at the same time represent both an object of research and a tool for studying biological systems themselves. In connection with this dual essence of biomathematical models, they imply the use of existing and development of new methods for analyzing mathematical objects (theories and methods of the relevant branches of mathematics) in order to study the properties of the model itself as a mathematical object, as well as the use of the model for reproducing and analyzing experimental data obtained in biological experiments. At the same time, one of the most important purposes of mathematical models (and mathematical biology in general) is the ability to predict biological phenomena and scenarios for the behavior of a biosystem under certain conditions and their theoretical justification before (or even instead of) conducting corresponding biological experiments.
The main method for studying and using complex models of biological systems is a computational computer experiment, which requires the use of adequate calculation methods for the corresponding mathematical systems, calculation algorithms, technologies for developing and implementing computer programs, storing and processing the results of computer modeling. These requirements imply the development of theories, methods, algorithms and computer modeling technologies within various areas of biomathematics.
Finally, in connection with the main goal of using biomathematical models to understand the laws of functioning of biological systems, all stages of the development and use of mathematical models require mandatory reliance on the theory and practice of biological science.
Hardly any biologist denies the need to use mathematical methods in biological research, in particular for other population analyses. However, in understanding the place of mathematical analysis in biology, there are different, sometimes opposing points of view. Some believe that the most important task is “knowing the behavior of a population as a statistical aggregate” (Beverton and Holt, 1957; Graham, 1956). According to this point of view, the task of a biologist is reduced to statistical analysis and is limited to the establishment of various correlative connections. The theoretical basis for this point of view is the statement of Bertrand Russell that “biological laws... like the laws of quantum theory are discrete and statistical laws” (Russell, 1957, p. 69).
Others proceed from the fact that mathematical analysis in biology, including population studies, is necessary, but only as an intermediate, and not the final stage of research. This second point of view is based on the idea of the specificity of the forms of motion of matter. In population analysis, this direction sees the final task of research in identifying the adaptive essence and understanding the causes of a biological phenomenon. From these positions we approach the use of mathematical models when studying patterns of population dynamics.
Mathematical modeling is a method by which it is possible to identify the mechanism of a process and understand its structural features - to establish the parameters of the analyzed population. Mathematical modeling in the presence of large digital material allows the use of computing and modeling devices for faster and more reliable processing of the material and for a more comprehensive and objective analysis of the collected data.
A very important task, which allows the widespread use of mathematical models, is the development of a methodology and the preparation of forecasts for fluctuations in the number and possible catches of commercial fish, as well as the calculation of optimal operating regimes for commercial fish, such regimes that would ensure regular receipt of the largest amount of fish from year to year. products of the highest quality. Currently, a huge amount of time and effort is spent on performing these tasks, especially on making forecasts of possible catches of individual commercial fish, and the results are not always sufficiently accurate. Therefore, it is extremely important to simplify and mechanize as much as possible the processes of making forecasts and calculating the operating regime of commercial fish stocks, while ensuring high accuracy of these calculations.
The use of high-speed electronic computers for research purposes allows us to significantly expand the scope of research and approach the development of issues of population ecology, the solution of which was impossible before the advent of computers.
Mathematical modeling method
The widespread use of computers in all areas of research, including ichthyological research, makes it possible to greatly speed up research and achieve high accuracy of the results obtained.
However, in order to be able to use computers in population analysis, it is necessary to create programs that correctly reflect the course of the process of interest to us. This is, first of all, a set of rules and instructions for converting quantities of interest to us (process algorithm), which can include dependencies both in the form of equations and directly in the form of tables and graphs. However, to obtain a “working” mathematical model of the process, it is necessary that it be based on those causal connections, on those internal contradictions that reflect the actual essence of the development of a biological phenomenon, and not on external random connections that obey only statistical laws and do not reflect the essence of the phenomenon. And it is natural that both here and abroad (Regier, 1970) in population analysis, models are increasingly being used, which are based on the idea of a population as a self-regulating open system, built on the principle of feedback - plus or minus interaction.
The presence of connections of different signs in a closed loop under certain conditions ensures the relative stability of the system (Menshutkin, 1971).
By a mathematical model I mean a mathematical expression of the quantitative side of the course of a particular process or phenomenon, including the dynamics of the number and biomass of animal populations. In almost every biological study, we directly or indirectly use mathematical models. For example, the numerical expression of the average and amplitude of the number of rays in the fin of a fish already represents the simplest mathematical model of the fin. In relation to mathematical models of population dynamics, it seems to me that we need to understand equations or systems of equations that reflect the quantitative side of the process of population dynamics and allow us to predict the further course of the phenomenon. Naturally, the question arises of what place mathematical modeling should occupy in the study of population dynamics and how to contribute to the success of biological research through the use of mathematical models.
The processes occurring in the organic world - those internal contradictions that drive development, are mainly deterministic in nature and belong both to the group of processes of continuous action with varying intensity (i.e., magnitude and speed), and to the group of discrete processes. These are processes that determine the course of a phenomenon. But any natural phenomenon is a complex interweaving of internal and external contradictions; the latter seem to create the environment in which the phenomenon occurs. If processes reflecting the internal contradiction of living things belong to the category of deterministic processes of discrete or continuous action, then external influences are, as a rule, discrete in nature and are not connected with the population by a clear feedback. When starting to build a mathematical model of a population, it is necessary to take all this into account.
As is known (Nikolsky, 1959), using the mathematical method, it is possible to identify the mechanism of a phenomenon, but not to reveal its adaptive essence. However, knowledge of the mechanism of a biological phenomenon is absolutely necessary to understand its essence, and if the method of mathematical modeling can help clarify the mechanism of the phenomenon - in our case, the mechanism of population dynamics - then it should be used to the maximum.
Varley (1962), speaking in a discussion on the applicability of mathematical models in population research, depicted the place of a mathematical model in population research as follows:
However, a theoretical model can be used for practical purposes only after it has been tested to determine its parameters in nature and has been transformed from a theoretical model into a working one. The actual theoretical model in Varley’s understanding is not a mathematical model reflecting the course of a phenomenon, but a working hypothesis based on preliminary biological observations, which makes it possible to organize a study to determine the initial parameters. The latter make it possible to create a working model suitable for predicting the quantitative side of the course of a phenomenon, i.e. Varley’s “theoretical model” is those biological principles that should be the basis of the working model.
Closer to the process of using computers and mathematical models in developing the problem of population dynamics is the scheme proposed by D.I. Blokhintsev (1964) for the work of a modern physicist: 1) measurement (set of facts); 2) processing of the received information (on a computer); 3) conclusions (building working hypotheses); 4) checking them on calculating machines; 5) building theories (prediction for the future).
I think that measurement (selection of facts) should also be preceded by a hypothesis based on a general methodology.
In this regard, it is more correct, as suggested by D. N. Horafas (1967), to begin research using models and computers by stating the problem. This author proposes the following sequence of operations: 1) defining the problem; 2) finding the main variables; 3) determining the relationships between these variables and system parameters; 4) formulation of a hypothesis regarding the nature of the conditions being studied; 5) construction of a mathematical or any other model; 6) conducting or planning experiments; 7) hypothesis testing; 8) evaluation of the hypothesis depending on the outcome of the experiments; 9) acceptance or rejection of the hypothesis and formulation of conclusions; 10) forecasting the further development of systems taking into account their interaction; 11) development of a course of action; 12) transition to the model refinement stage, making the necessary adjustments.
D. N. Khorafas’s scheme, as it seems to me, is close to the scheme proposed by D. I. Blokhintsev, but it introduces a number of clarifications that may be useful in population analysis.
Thus, in studies in the field of population dynamics, mathematical modeling should provide a clearer understanding of the process, mainly about its quantitative side. Mathematical modeling should simplify the process of long-term forecasting of population dynamics and, finally, guarantee a reliable calculation of the operating regime of populations - the regime that ensures the greatest productivity of the population. The practical task posed to biologists and mathematicians in the field of constructing mathematical models is the creation of a model that would make it possible to automate the long-term forecast service and use computer technology in calculating the optimal modes of exploitation of game animals.
It seems to me that the following is the course of biological research into population dynamics and the place of mathematical modeling in it. Based on the understanding of the available factual material, a working hypothesis of the phenomenon is created; On the basis of this working hypothesis, a research program is built that provides materials that reveal both the causes and the mechanism of the phenomenon. These materials should also provide the possibility of constructing a mathematical model of the course of the phenomenon. Thus, there are two stages in creating a mathematical model. The first (theoretical model in the Varley scheme) - a working hypothesis based on the collected facts is formalized in the form of an equation of varying complexity; The vast majority of mathematical models belong to this type of model. The second stage - based on testing the working hypothesis, a working model is created, suitable for practical calculations for prognostic and operational purposes. Both theoretical and working models are always based on one or another set of theoretical concepts, and the closer these theoretical concepts are to the laws operating in nature, the more correct and effective the created mathematical model will be.
Despite the diversity of living systems, they all have the following specific features that must be taken into account when constructing models.
- 1. Complex systems. All biological systems are complex, multicomponent, spatially structured, and their elements have individuality. When modeling such systems, two approaches are possible. The first is aggregated, phenomenological. In accordance with this approach, the defining characteristics of the system are identified (for example, the total number of species) and the qualitative properties of the behavior of these quantities over time are considered (stability of a stationary state, the presence of oscillations, the existence of spatial heterogeneity). This approach is historically the most ancient and is characteristic of the dynamic theory of populations. Another approach is a detailed consideration of the elements of the system and their interactions, the construction of a simulation model, the parameters of which have a clear physical and biological meaning. Such a model does not allow analytical research, but with good experimental study of the fragments of the system, it can give a quantitative prediction of its behavior under various external influences.
- 2. Reproducing systems (capable of autoreproduction). This most important property of living systems determines their ability to process inorganic and organic matter for the biosynthesis of biological macromolecules, cells, and organisms. In phenomenological models, this property is expressed in the presence in the equations of autocatalytic terms that determine the possibility of growth (in non-limited conditions - exponential), the possibility of instability of the stationary state in local systems (a necessary condition for the emergence of oscillatory and quasistochastic modes) and the instability of a homogeneous stationary state in spatially distributed systems ( condition of spatially inhomogeneous distributions and autowave regimes). An important role in the development of complex spatiotemporal regimes is played by the processes of interaction of components (biochemical reactions) and transfer processes, both chaotic (diffusion) and associated with the direction of external forces (gravity, electromagnetic fields) or with the adaptive functions of living organisms (for example, movement cytoplasm in cells under the influence of microfilamepts).
- 3. Open systems that constantly allow flows of matter and energy to pass through them. Biological systems are far from thermodynamic equilibrium and are therefore described nonlinear equations. Linear Onsager relations connecting forces and flows are valid only near thermodynamic equilibrium.
- 4. Biological objects have a complex multi-level regulation system. In biochemical kinetics, this is expressed in the presence of feedback loops in circuits, both positive and negative. In the equations of local interactions, feedbacks are described by nonlinear functions, the nature of which determines the possibility of the occurrence and properties of complex kinetic regimes, including oscillatory and quasistochastic ones. This type of nonlinearity, when taking into account the spatial distribution and transport processes, is determined by patterns of stationary structures (spots of various shapes, periodic dissipative structures) and types of autowave behavior (moving fronts, traveling waves, leading centers, spiral waves, etc.).
- 5. Living systems have complex spatial structure. A living cell and the organelles it contains have membranes; any living organism contains a huge number of membranes, the total area of which amounts to tens of hectares. Naturally, the environment within living systems cannot be considered homogeneous. The very emergence of such a spatial structure and the laws of its formation represent one of the problems of theoretical biology. One of the approaches to solving such a problem is the mathematical theory of morphogenesis.
Membranes not only separate different reaction volumes of living cells, but also separate living from non-living (environment). They play a key role in metabolism, selectively passing the flow of inorganic ions and organic molecules. The primary processes of photosynthesis take place in the membranes of chloroplasts - storing light energy in the form of energy of high-energy chemical compounds, which are subsequently used for the synthesis of organic matter and other intracellular processes. The key stages of the respiration process are concentrated in the membranes of mitochondria; the membranes of nerve cells determine their ability to conduct nerves. Mathematical models of processes in biological membranes constitute an essential part of mathematical biophysics.
Existing models are mainly systems of differential equations. However, it is obvious that continuous models are not able to describe in detail the processes occurring in such individual and structured complex systems as living systems. In connection with the development of the computational, graphical and intellectual capabilities of computers, simulation models built on the basis of discrete mathematics, including models of cellular automata, play an increasingly important role in mathematical biophysics.
6. Simulation models of specific complex living systems, as a rule, take into account the available information about the object as much as possible. Simulation models are used to describe objects at various levels of organization of living matter - from biomacromolecules to models of biogeocenoses. In the latter case, models should include blocks describing both living and “inert” components. A classic example of simulation models are models molecular dynamics, in which the coordinates and momenta of all atoms that make up the biomacromolecule and the laws of their interaction are specified. A computer-calculated picture of the “life” of a system allows us to trace how physical laws are manifested in the functioning of the simplest biological objects - biomacromolecules and their environment. Similar models, in which the elements (building blocks) are no longer atoms, but groups of atoms, are used in modern computer technology for the design of biotechnological catalysts and drugs that act on certain active groups of membranes of microorganisms, viruses, or perform other targeted actions.
Simulation models are created to describe physiological processes, occurring in vital organs: nerve fiber, heart, brain, gastrointestinal tract, bloodstream. They play “scenarios” of processes occurring normally and in various pathologies, and study the influence of various external influences, including medications, on the processes. Simulation models are widely used to describe plant production process and are used to develop an optimal regime for growing plants in order to obtain maximum yield or obtain the most evenly distributed ripening of fruits over time. Such developments are especially important for expensive and energy-intensive greenhouse farming.
For a long period of time, biology was a descriptive science, poorly suited to predicting observed phenomena. With the development of computer technology, the situation has changed. At first, the most used methods in biology were the methods of mathematical statistics, which made it possible to correctly process experimental data and evaluate certain significance for making certain decisions and drawing conclusions. Over time, when the methods of chemistry and physics entered biology, they began to use complex mathematical models that made it possible to process data from real experiments and predict the course of biological processes during virtual experiments.
Models in biology
Modeling of biological systems is the process of creating models of biological systems with their characteristic properties. The object of modeling can be any of the biological systems.
Biology uses modeling of biological structures, functions and processes at the molecular, subcellular, cellular, organ-systemic, organismal and population-biocenotic levels of organization of living organisms. Modeling is also applied to various biological phenomena, living conditions of individuals, populations, and ecosystems.
Definition 1
Biological systems are very complex structural and functional units.
Computer and visual modeling of biological components is used. There are a huge number of examples of such biological models. Here are some examples of biological models:
There is a rapidly increasing importance of computer simulation models in almost all areas of biology. Computer modeling is used for the analysis of calculated data, which includes image processing, for the analysis of nucleotide sequences encoding a gene and individual proteins, for computer training in modern biology, etc. By conducting “virtual” experiments on personal computers, it is possible to control all variables and influencing factors, which allows for the analysis of biological systems and the development of physical models for the components of these systems, which cannot be carried out in real experiments.
Main types of models in biology
Biological laboratory animal models reproduce certain conditions or diseases that occur in animals or humans. Their use makes it possible to study, during experiments, the mechanisms of occurrence of a given condition or disease, its course and outcome, and influence its course. Examples of biological models are artificially induced genetic disorders, infectious process, intoxication, reproduction of hypertensive and hypoxic conditions, malignant neoplasms, hyperfunction or hypofunction of certain organs, neuroses and emotional states.
To create biological models, the genetic apparatus is influenced, microbial infection is used, toxins are introduced, individual organs are removed, etc. Physicochemical models reproduce biological structures, functions or processes by chemical or physical means and are usually a close approximation of the biological phenomenon that is being modeled.
Significant progress has been made in creating models of the physical and chemical conditions of existence of living organisms, their organs and cells. For example, solutions of inorganic and organic substances (solutions of Ringer, Locke, Tyrode, etc.) have been selected that imitate the internal environment of the body and support the existence of isolated organs or cultured cells inside the body.
Note 1
Modeling of biological membranes allows one to study the physicochemical basis of ion transport processes and the influence of various factors on it. Using chemical reactions that occur in solutions in a self-oscillatory mode, oscillatory processes characteristic of many biological phenomena are modeled.
Mathematical models (descriptions of the structure, connections and patterns of functioning of living systems) are built on the basis of experimental data or represent a formalized description of a hypothesis, theory or open pattern of any biological phenomenon and require further experimental verification. Different versions of such experiments determine the boundaries of the use of mathematical models and provide material for its further adjustment. Testing a mathematical model of a biological phenomenon on a personal computer makes it possible to predict the nature of changes in the biological process under study under conditions that are difficult to reproduce experimentally.
Mathematical models make it possible to predict in individual cases certain phenomena that were previously unknown to the researcher. For example, the model of cardiac activity proposed by the Dutch scientists van der Pol and van der Mark, based on the theory of relaxation oscillations, showed the possibility of a special disturbance of the heart rhythm, which was subsequently discovered in humans. A mathematical model of physiological phenomena is also the model of nerve fiber excitation, which was developed by English scientists A. Hodgkin and A. Huxley. There are logical and mathematical models of the interaction of neurons, built on the basis of the theory of nerve networks, which were developed by American scientists W. McCulloch and W. Pits.
The book consists of lectures on mathematical modeling of biological processes and is written based on the material of courses taught at the Faculty of Biology of Moscow State University. M. V. Lomonosov.
24 lectures outline the classification and features of modeling living systems, the basics of the mathematical apparatus used to build dynamic models in biology, basic models of population growth and interaction of species, models of multistationary, oscillatory and quasistochastic processes in biology. Methods for studying the spatiotemporal behavior of biological systems, models of autowave biochemical reactions, propagation of a nerve impulse, models of coloring animal skins, and others are considered. Particular attention is paid to the concept of the hierarchy of times, which is important for modeling in biology, and modern concepts of fractals and dynamic chaos. The latest lectures are devoted to modern methods of mathematical and computer modeling of photosynthesis processes. The lectures are intended for undergraduates, graduate students and specialists who want to become familiar with the modern foundations of mathematical modeling in biology.
Molecular dynamics.
Throughout the history of Western science, the question has been whether, knowing the coordinates of all atoms and the laws of their interaction, it is possible to describe all the processes occurring in the Universe. The question has not found its unambiguous answer. Quantum mechanics established the concept of uncertainty at the micro level. In lectures 10-12 we will see that the existence of quasi-stochastic types of behavior in deterministic systems makes it almost impossible to predict the behavior of some deterministic systems at the macro level.
A corollary to the first question is the second: the question of “reducibility.” Is it possible, knowing the laws of physics, i.e., the laws of motion of all atoms that make up biological systems, and the laws of their interaction, to describe the behavior of living systems. In principle, this question can be answered using a simulation model, which contains the coordinates and velocities of movement of all atoms of any living system and the laws of their interaction. For any living system, such a model must contain a huge number of variables and parameters. Attempts to model using this approach the functioning of elements of living systems - biomacromolecules - have been made since the 70s.
Content
Preface to the second edition
Preface to the first edition
Lecture 1. Introduction. Mathematical models in biology
Lecture 2. Models of biological systems described by one first-order differential equation
Lecture 3. Population growth models
Lecture 4. Models described by systems of two autonomous differential equations
Lecture 5. Study of the stability of stationary states of second-order nonlinear systems
Lecture 6. The problem of fast and slow variables. Tikhonov's theorem. Types of bifurcations. Disasters
Lecture 7. Multistationary systems
Lecture 8. Oscillations in biological systems
Lecture 9. Models of interaction of two types
Lecture 10. Dynamic chaos. Models of biological communities
Examples of fractal sets
Lecture 11. Modeling microbial populations
Lecture 12. Model of the effect of a weak electric field on a nonlinear system of transmembrane ion transport
Lecture 13. Distributed biological systems. Reaction-diffusion equation
Lecture 14. Solving the diffusion equation. Stability of homogeneous stationary states
Lecture 15. Propagation of a concentration wave in systems with diffusion
Lecture 16. Stability of homogeneous stationary solutions of a system of two equations of the reaction-diffusion type. Dissipative structures
Lecture 17. Belousov-Zhabotinsky reaction
Lecture 18. Models of propagation of nerve impulses. Autowave processes and cardiac arrhythmias
Lecture 19. Distributed triggers and morphogenesis. Animal skin coloring patterns
Lecture 20. Spatiotemporal models of species interaction
Lecture 21. Fluctuations and periodic spatial distributions of pH and electrical potential along the cell membrane of the giant algae Chara corallina
Lecture 22. Models of photosynthetic electron transport. Electron transfer in a multienzyme complex
Lecture 23. Kinetic models of photosynthetic electron transport processes
Lecture 24. Direct computer models of processes in the photosynthetic membrane
Nonlinear natural scientific thinking and environmental consciousness
Stages of evolution of complex systems.
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