Mathematical modeling of biological processes.

We have already said that the mathematical approach to the study of certain phenomena real world usually begins with the creation of appropriate general concepts, i.e. from construction mathematical models, which have properties that are essential for us of the systems and processes that we study. We also mentioned the difficulties associated with the construction of such models in biology, difficulties caused by the extreme complexity of biological systems. However, despite these difficulties, the “model” approach to biological problems is now successfully developing and has already brought certain results. We will look at some models related to various biological processes and systems.

Speaking about the role of models in biological research, it is important to note the following. Although we understand the term “model” in an abstract sense - as a certain system logical concepts, and not as a real physical device, yet a model is something significantly more than a simple description of a phenomenon or a purely qualitative hypothesis, in which there is still enough room for various kinds of ambiguities and subjective opinions. Let us recall the following example, dating back to the rather distant past. At one time, Helmholtz, while studying hearing, put forward the so-called resonance theory, which looked plausible from a purely qualitative point of view. However, quantitative calculations carried out later, taking into account the real values ​​of the masses, elasticity and viscosity of the components that make up the auditory system, showed the inconsistency of this hypothesis. In other words, the attempt to transform a purely qualitative hypothesis into an exact model that allows its investigation by mathematical methods immediately revealed the inconsistency of the original principles. Of course, if we have built a certain model and even obtained good agreement between this model and the results of the corresponding biological experiment, this does not yet prove the correctness of our model. Now, if, based on the study of our model, we can make some predictions about the biological system that we are modeling, and then confirm these predictions with a real experiment, then this will be much more valuable evidence in favor of the correctness of the model.

But let's move on to specific examples.

2.Blood circulation

One of the first, if not the very first, work on mathematical modeling of biological processes should be considered the work of Leonhard Euler, in which he developed the mathematical theory of blood circulation, considering in a first approximation the entire circulatory system as consisting of a reservoir with elastic walls, peripheral resistance and a pump. These ideas of Euler (as well as some of his other works) were at first completely forgotten, and then revived in later works of other authors.

3. Mendel's laws

A fairly old and well-known, but nevertheless very remarkable model in biology is the Mendelian theory of heredity. This model, based on probability theoretical concepts, is that the chromosomes of the parent cells contain certain sets of characteristics, which during fertilization are combined independently and randomly. Subsequently, this basic idea underwent very significant clarifications; for example, it was discovered that different signs are not always independent of each other; if they are associated with the same chromosome, then they can only be transmitted in a certain combination. Further, it was discovered that different chromosomes do not combine independently, but there is a property called chromosome affinity, which violates this independence, etc. Currently, theoretical-probabilistic and statistical methods have very widely penetrated into genetic research and even the term “mathematical genetics” "received full citizenship rights. Currently, intensive work is being carried out in this area; many results have been obtained that are interesting both from a biological and from a purely mathematical point of view. However, the very basis of these studies is the model that was created by Mendel more than 100 years ago.

4. Muscle models

One of the most interesting objects of physiological research is muscle. This object is very accessible, and the experimenter can carry out many studies simply on himself, having only relatively simple equipment. The functions that a muscle performs in a living organism are also quite clear and definite. Despite all this, numerous attempts to construct a satisfactory model of muscle function have not yielded definitive results. It is clear that although a muscle can stretch and contract like a spring, their properties are completely different, and even to the very first approximation, a spring cannot be considered as a semblance of a muscle. For a spring, there is a strict relationship between its elongation and the load applied to it. This is not the case for a muscle: a muscle can change its length while maintaining tension, and vice versa, change the traction force without changing its length. Simply put, at the same length, a muscle can be relaxed or tense.

Among the various modes of operation possible for a muscle, the most significant are the so-called isotonic contraction (i.e., a contraction in which the muscle tension remains constant) and isometric tension, in which the length of the muscle does not change (both ends are fixed). Studying a muscle in these modes is important for understanding the principles of its operation, although under natural conditions muscle activity is neither purely isotonic nor purely isometric.

Various mathematical formulas have been proposed to describe the relationship between the speed of isotonic muscle contraction and the magnitude of the load. The most famous of them is the so-called characteristic Hill equation. It looks like

(P+a)V=b(P 0 -P),

- speed of contraction, a, b And P 0- permanent.

Other well-known formulas for describing the same relationship are Ober's equation

P = P 0 e- V⁄P ±F

and the Polissar equation

V=const (A 1-P/P 0 - B 1-P/P 0).

Hill's equation has become widespread in physiology; it gives a fairly good agreement with experiment for the muscles of a wide variety of animals, although in fact it represents the result of a "fit" rather than an inference from some model. Two other equations, which give approximately the same dependence over a fairly wide range of loads as the Hill equation, were obtained by their authors from certain ideas about the physicochemical mechanism of muscle contraction. There are a number of attempts to construct a model of muscle work, considering the latter as some combination of elastic and viscous elements. However, there is still no sufficiently satisfactory model that reflects all the main features of muscle work in various modes.

5. Neuron models, neural networks

Nerve cells, or neurons, are the “working units” that make up the nervous system and to which the animal or human body owes all its abilities to perceive external signals and control various parts of the body. A characteristic feature of nerve cells is that such a cell can be in two states - rest and excitation. In this, nerve cells are similar to elements such as radio tubes or semiconductor triggers, from which the logical circuits of computers are assembled. Over the past 15-20 years, many attempts have been made to model activities nervous system, based on the same principles on which the work of universal computers is based. Back in the 40s, American researchers McCulloch and Pitts introduced the concept of a “formal neuron,” defining it as an element (the physical nature of which does not matter) equipped with a certain number of “excitatory” and a certain number of “inhibitory” inputs. This element itself can be in two states - “rest” or “excitement”. An excited state occurs if the neuron receives a sufficient number of excitatory signals and there are no inhibitory signals. McCulloch and Pitts showed that with the help of circuits composed of such elements, it is possible, in principle, to implement any of the types of information processing that occur in a living organism. This, however, does not mean at all that we have thereby learned the actual principles of the nervous system. First of all, although nerve cells are characterized by the “all or nothing” principle, i.e. the presence of two clearly defined states - rest and excitation, it does not at all follow that our nervous system, like a universal computer, uses a binary digital code consisting of zeros and ones. For example, in the nervous system, frequency modulation apparently plays a significant role, that is, the transmission of information using the length of time intervals between impulses. In general, in the nervous system, apparently, there is no such division of information encoding methods into “digital” discrete) and “analog” (continuous), which is available in modern computer technology.

In order for a system of neurons to work as a whole, it is necessary that there be certain connections between these neurons: impulses generated by one neuron must arrive at the inputs of other neurons. These connections can have a correct, regular structure, or they can be determined only by statistical patterns and be subject to certain random changes. In currently existing computing devices, no randomness in connections between elements is allowed, however, there are a number of theoretical studies on the possibility of constructing computing devices based on the principles random connections between elements. There are quite serious arguments in favor of the fact that the connections between real neurons in the nervous system are also largely statistical, and not strictly regular. However, opinions on this matter differ.

In general, the following can be said about the problem of modeling the nervous system. We already know quite a lot about the peculiarities of the work of neurons, that is, those elements that make up the nervous system. Moreover, with the help of systems of formal neurons (understood in the sense of McCulloch and Pitts or in some other sense), simulating the basic properties of real nerve cells, it is possible to simulate, as already mentioned, very diverse ways of processing information. Nevertheless, we are still quite far from a clear understanding of the basic principles of the functioning of the nervous system and its individual parts, and, consequently, from creating its satisfactory model *.

* (If we can create some kind of system that can solve the same problems as some other system, this does not mean that both systems work according to the same principles. For example, you can numerically solve a differential equation on a digital computer by giving it the appropriate program, or you can solve the same equation on an analog computer. We will get the same or almost the same results, but the principles of information processing in these two types of machines are completely different.)

6. Perception of visual images. Color vision

Vision is one of the main channels through which we receive information about the outside world. Famous expression- it is better to see once than to hear a hundred times - this is also true, by the way, from a purely informational point of view: the amount of information that we perceive through vision is incomparably greater than that perceived by other senses. This importance of the visual system for a living organism, along with other considerations (specificity of functions, the possibility of conducting various studies without any damage to the system, etc.) stimulated its study and, in particular, attempts at a model approach to this problem.

The eye is an organ that serves as both an optical system and an information processing device. From both points of view, this system has a number of amazing properties. The eye's ability to adapt to a very wide range of light intensities and to correctly perceive all colors is remarkable. For example, a piece of chalk in a dimly lit room reflects less light than a piece of coal placed in a bright room. sunlight, nevertheless, in each of these cases we perceive the colors of the corresponding objects correctly. The eye conveys relative differences in illumination intensities well and even “exaggerates” them somewhat. Thus, a gray line on a bright white background seems darker to us than a solid field of the same gray. This ability of the eye to emphasize contrasts in illumination is due to the fact that visual neurons have an inhibitory effect on each other: if the first of two neighboring neurons receives a stronger signal than the second, then it has an intense inhibitory effect on the second, and the difference in the output of these neurons is the intensity is greater than the difference in the intensity of the input signals. Models consisting of formal neurons connected by both excitatory and inhibitory connections have attracted the attention of both physiologists and mathematicians. There are also interesting results and unresolved issues.

Of great interest is the mechanism by which the eye perceives different colors. As you know, all shades of colors perceived by our eyes can be represented as combinations of three primary colors. Usually these primary colors are red, blue and yellow colors, corresponding to wavelengths 700, 540 and 450 Å, but this choice is not unambiguous.

The “three-color” nature of our vision is due to the fact that the human eye has three types of receptors, with maximum sensitivity in the yellow, blue and red zones, respectively. The question is how do we distinguish between these three receptors? a large number of color shades, is not very simple. For example, it is not yet clear enough what exactly this or that color is encoded in our eye: the frequency of nerve impulses, the localization of the neuron that predominantly reacts to a given shade of color, or something else. There are some model ideas about this process of perception of shades, but they are still quite preliminary. There is no doubt, however, that here, too, systems of neurons connected to each other by both excitatory and inhibitory connections should play a significant role.

Finally, the eye is also very interesting as a kinematic system. A series of ingenious experiments (many of them were carried out in the laboratory of vision physiology of the Institute for Problems of Information Transmission in Moscow) established the following at first glance unexpected fact: if some image is motionless relative to the eye, then the eye does not perceive it. Our eye, when examining an object, literally “feels” it (these eye movements can be accurately recorded using appropriate equipment). The study of the motor apparatus of the eye and the development of corresponding model representations are quite interesting both in themselves and in connection with other (optical, informational, etc.) properties of our visual system.

To summarize, we can say that we are still far from creating completely satisfactory models of the visual system that well describe all its basic properties. However, a number of important aspects and principles of its operation are already quite clear and can be modeled in the form of computer programs for a computer or even in the form of technical devices.

7. Active medium model. Spread of excitation

One of the very characteristic properties of many living tissues, primarily nervous tissue, is their ability to excite and transfer excitation from one area to another. About once a second, a wave of excitement runs through our heart muscle, causing it to contract and drive blood throughout the body. Excitation along nerve fibers, spreading from the periphery (sensory organs) to the spinal cord and brain, informs us about the outside world, and in the opposite direction there are excitation commands that prescribe certain actions to the muscles.

Excitation in a nerve cell can occur on its own (as they say, “spontaneously”), under the influence of an excited neighboring cell, or under the influence of some external signal, say, electrical stimulation coming from some current source. Having passed into an excited state, the cell remains in it for some time, and then the excitement disappears, after which a certain period of cell immunity to new stimuli begins - the so-called refractory period. During this period, the cell does not respond to signals received by it. Then the cell returns to its original state, from which a transition to a state of excitation is possible. Thus, the excitation of nerve cells has a number of clearly defined properties, starting from which it is possible to construct an axiomatic model of this phenomenon. Further, to study this model, pure mathematical methods.

Ideas about such a model were developed several years ago in the works of I.M. Gelfand and M.L. Tsetlin, which were then continued by a number of other authors. Let us formulate an axiomatic description of the model in question.

By “excitable medium” we mean a certain set X elements (“cells”) with the following properties:

1.Each element can be in one of three states: rest, excitement and refractoriness;

2. From each excited element, excitation spreads through many elements at rest at a certain speed v;

3.If the item X hasn't been excited for some specific time T(x), then after this time it spontaneously goes into an excited state. Time T(x) called the period of spontaneous activity of the element X. This does not exclude the case when T(x)= ∞, i.e. when spontaneous activity is actually absent;

4. The state of excitement lasts for some time τ (which may depend on X), then the element moves for a while R(x) into a refractory state, after which a state of rest sets in.

Similar mathematical models arise in completely other areas, for example, in the theory of combustion, or in problems of the propagation of light in an inhomogeneous medium. However, the presence of a “refractory period” is a characteristic feature of biological processes.

The described model can be studied either by analytical methods or by implementing it on a computer. In the latter case, we are, of course, forced to assume that the set X(excitable medium) consists of a certain finite number of elements (in accordance with the capabilities of existing computer technology - on the order of several thousand). For analytical research it is natural to assume X some continuous variety (for example, consider that X- this is a piece of plane). The simplest case of such a model is obtained if we take X some segment (a prototype of a nerve fiber) and assume that the time during which each element is in an excited state is very short. Then the process of sequential propagation of impulses along such a “nerve fiber” can be described by a chain of ordinary first-order differential equations. Already in this simplified model, a number of features of the propagation process that are also found in real biological experiments are reproduced.

The question of the conditions for the occurrence of so-called fibrillation in such a model active medium is very interesting from both a theoretical and applied medical point of view. This phenomenon, observed experimentally, for example in the heart muscle, consists in the fact that instead of rhythmic coordinated contractions, random local excitations appear in the heart, devoid of periodicity and disrupting its functioning. The first theoretical study of this problem was undertaken in the work of N. Wiener and A. Rosenbluth in the 50s. Currently, work in this direction is being intensively carried out in our country and has already yielded a number of interesting results.

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 last lectures are devoted modern methods 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 a single 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 influence of the 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. Oscillations and periodic spatial distributions of the pH value and electric potential along cell membrane 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 processes in the photosynthetic membrane
Nonlinear natural scientific thinking and environmental consciousness
Stages of evolution of complex systems.

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Functioning of a complex biological system, including of cardio-vascular system, is the result of the interaction of its constituent elements and the processes occurring in it. It should be borne in mind that according to the general principle of an ascending hierarchy of types of movement (mechanical - physical - chemical - biological - social), the biological form of movement cannot be completely reduced to the mechanical, physical or chemical form of movement, and biological systems cannot be fully described from the standpoint of any one of these forms of movement. These forms of movement can serve as models biological form movement, that is, its simplified images.

It is possible to find out the basic principles of regulating the processes of a complex biological system by first constructing a mechanical, physical or chemical model of the system, and then constructing their mathematical models, that is, finding the mathematical functions that describe these models, including equations (creating mathematical models). The lower the hierarchy level, the simpler model, the more factors of the real system are excluded from consideration.

Modeling is a method in which the study of some complex object (process, phenomenon) is replaced by the study of its simplified analogue - a model. In biophysics, biology and medicine, physical, chemical, biological and mathematical models are widely used. For example, the flow of blood through vessels is modeled by the movement of fluid through pipes (physical model). A biological model is simple biological objects, convenient for experimental research, on which the properties of real, more complex biological systems are studied. For example, the patterns of occurrence and propagation of action potentials along nerve fibers were studied using a biological model—the giant squid axon.

A mathematical model is a set of mathematical objects and relationships between them, reflecting the properties and characteristics of a real object that are of interest to the researcher. An adequate mathematical model can only be built using specific data and ideas about the mechanisms of complex processes. After construction, the mathematical model “lives” according to its internal laws, the knowledge of which allows us to identify character traits the system under study (see diagram in Fig. 1.1.). The simulation results form the basis for managing processes of any nature.

Biological systems, in fact, are extremely complex structural and functional units.

Most often, mathematical models of biological processes are specified in the form of differential or difference equations, but other types of model representations are also possible. After the model is built, the task is reduced to studying its properties using mathematical deduction methods or through machine modeling.

When studying a complex phenomenon, several alternative models are usually proposed. The qualitative correspondence of these models to the object is checked. For example, they establish the presence of stable stationary states in the model and the existence of oscillatory modes. Model, the best way corresponding to the system under study is chosen as the main one. The selected model is specified in relation to the specific system under study. Numerical values ​​of the parameters are set based on experimental data.

The process of searching for a mathematical model of a complex phenomenon can be divided into stages, the sequence and interconnection of which is reflected in the diagram in Fig. 1.2.

Stage 1 corresponds to the collection of data available at the beginning of the study about the object being studied.

At stage 2, a basic model (system of equations) is selected from possible alternative models based on qualitative characteristics.

At stage 3, the model parameters are identified from experimental data.

At stage 4, the behavior of the model is verified using independent experimental data. To do this, it is often necessary to perform additional experiments.

If the experimental data taken to verify the model “does not fit” into the model, it is necessary to analyze the situation and put forward other models, study the properties of these new models, and then conduct experiments that allow one to draw a conclusion about the preference of one of them (step 5).

The stage of constructing a mathematical model (stage 2, Fig. 1.2) is the most important stage in mathematical modeling. Ideas about the mechanisms and laws that operate in the system and which are embedded in the mathematical model determine the framework of the modeling results. Thus, when modeling the functioning of the cardiovascular system based on ideas about the work of the heart from the standpoint of mechanics, we can build a mechanical-mathematical model.

When we talk about mathematical modeling of the dynamics of a complex biological system based on physical laws, we are entering the field of mathematical biophysics of complex systems. It was at the junction of three sciences: mathematics, physics and biology that in the last five decades there has been a qualitative leap in the mathematical description of the behavior of any system (physical, biological, economic).

It is common practice to measure physiological quantities as a function of time. To characterize such time dependencies, there are four basic mathematical concepts: stationary states, oscillations, chaos and noise. Steady state in mathematics can be related to the concept of homeostasis in physiology, for example, average arterial pressure is maintained constant in humans. During physical activity, the pressure increases, and after the cessation of physical activity, the pressure returns to the stationary level within a few minutes. Examples of oscillatory processes in the human body include: rhythms of the heartbeat, respiration and cell reproduction, cycles of sleep and wakefulness, insulin secretion, peristaltic waves in the intestines and ureter, electrical activity of the cerebral cortex and the autonomic nervous system, etc. It is known that even careful measurement of a physical or physiological quantity never produces an absolutely stationary or strictly periodic time relationship. There will always be fluctuations (deviations) around some fixed level or period of oscillation. In addition, there are systems so irregular that it is difficult to find an underlying stationary or periodic process. Such processes are considered in mathematics as either noise (relating to fluctuations) or chaos (the "highest degree" of order, the irregularity observed in a deterministic system). Chaos can also be observed in the complete absence of noise in the environment.

The basis of the mathematical model is a system of mathematical equations (formula 1.1). A dynamic mathematical model characterizes the behavior of a system over time, which can be described using physical concepts such as speed and acceleration. Dynamic models are described by systems of differential equations, which are subject to restrictions arising from the physical or physiological meaning of the accepted quantities:

where f 1 ,…, f n - some functions , x 1 ,…, x n– independent variables, P - dimension of phase space, a,…, e, etc. - parameters of differential equations.

Stationary stable states correspond to constant solutions of the equations of system 1.1 (Fig. 1. 3, A). Stationary vibrations of biological or physical quantities correspond to periodic solutions of the system of equations (Fig. 1.3, B). Irregular (aperiodic) time solutions of equations correspond to noise or chaos (Figure 1.3, B).

For some parameter values, it is possible to obtain several solutions, that is, the system can be in several stationary states (for example, in two states). The transition of a system, as a result of which it may find itself in one of the possible states, is called bifurcation. Typically, some states are stable, others are unstable. If two stable states are possible, then the system can jump from one state to another with a slight external influence, including fluctuations. This phenomenon is called bistability.

As an example of constructing a model of a periodic biological process, let us consider Volterra’s “predator-prey” mathematical model.

Voltaire's model

Let hares and lynxes live in some closed area. Hares eat plant foods, which are always available in sufficient quantities. Lynxes (predators) feed only on hares (prey). Let's denote the number of hares in this area by N 1, and the number of lynxes by N 2. N 1 and N 2 are functions of time.

Since the amount of food for hares is not limited, we can assume that in the absence of predators, their number would increase over time t in direct proportion to the number of available individuals:

Where a i– proportionality coefficient.

If only lynxes lived in this area, they would die out due to lack of food.

Over the past decades, there has been significant progress in quantitative (mathematical) description functions of various biosystems at various levels of life organization: molecular, cellular, organ, organismal, population, biogeocenological (ecosystem). 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. Models based on essential postulates about the principles of system functioning, which describe and explain a wide range of phenomena and express knowledge in a compact, formalized form, can be spoken of as biosystem theories. Construction of mathematical models(theories) 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 diagrams of various cells were crystallized, within which various physicochemical and biochemical processes occur in an orderly manner in space and time, forming very complex interweavings.

The second very important circumstance, which contributes to the involvement of the 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.

And finally, third condition What determined the success of mathematical modeling in biology was the development of powerful computing tools in the form of personal computers, supercomputers and information technologies. This is due to the fact that usually the processes that control one or another function of cells or organs are numerous, covered by loops of direct and feedback and are therefore described complex systems nonlinear equations with a large number of unknowns. 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. Although experimental facts are used as model postulates, the need for some assumptions and assumptions is an important theoretical component of modeling. These assumptions and assumptions are hypotheses, which can be subjected to experimental verification. Thus, models become sources of hypotheses, moreover, experimentally verifiable. 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 an increasingly deeper and more accurate understanding of the phenomenon:

• the experiment refines the model,
• the new model puts forward new hypotheses,
• experiment clarifies new model etc.

Currently field of mathematical modeling of living systems unites a number of different and already established traditional and more modern disciplines, the names of which sound quite general, so that it is difficult to strictly delimit the areas of their specific use. At present, specialized areas of application of mathematical modeling of living systems are developing particularly rapidly - mathematical physiology, mathematical immunology, mathematical epidemiology, aimed at developing mathematical theories and computer models of relevant systems and processes.

Like any scientific discipline, mathematical (theoretical) biology has its own subject, methods, methods and procedures of research. As subject of research are mathematical (computer) models of biological processes, which at the same time represent both an object of research and a tool for studying biological objects themselves. In connection with this dual essence of biomathematical models, they imply use of existing and development of new methods for analyzing mathematical systems(theories and methods of 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 to reproduce and analyze experimental data obtained in biological experiments. At the same time, one of the most important purposes of mathematical models (and theoretical 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 conducting the corresponding biological experiments.

The main research method and the use of complex models of biological systems is computational computer experiment, which requires the use of adequate calculation methods for the corresponding mathematical systems, calculation algorithms, development and implementation technologies computer programs, storage and processing of computer modeling results.

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 theory and practice biological science, and primarily on the results of full-scale experiments.

A method for describing biological systems using an adequate mathematical apparatus. Definition of math. apparatus that adequately reflects the operation of biological systems is a difficult task associated with their classification. The classification of biological systems by complexity (logarithm of the number of states) can be carried out using, for example, a scale on which simple systems systems with up to a thousand states are classified as complex - from a thousand to a million and very complex - over a million states. Second the most important characteristic biological system is a pattern expressed by the law of probability distribution of states. According to this law, it is possible to determine the uncertainty of its work according to K. Shannon and an assessment of the relative organization. Thus, biol. systems can be classified by complexity (maximum diversity or maximum possible uncertainty) and relative organization, i.e., degree of organization (see Biological systems organization).

Classification diagram of biosystems:

Simple systems;

Complex systems;

Very complex systems;

Probabilistic systems;

Probabilistic-deterministic systems;

Deterministic systems.

In Fig. a classification diagram of biosystems is shown on the axes of the maximum possible uncertainty characterizing the number of states of the system and determined by the logarithm of the number of states, and the level of relative organization - characterizing the degree of organization of the system. The diagram gives the names of the corresponding bands so that, for example, the area under the number 8 means “very complex probabilistically determined biosystems.” Experience in studying biosystems shows that if , calculated from the histogram of the distribution of deviations of the studied indicator from its mathematical expectation, lies in the range from 1.0 to 0.3, then we can consider that it is a deterministic biosystem. Such systems include internal control systems. organs, mainly hormonal (humoral) control systems. Neuron, internal organs spheres, metabolic systems according to certain parameters can also be classified as deterministic biosystems. Math. models of such systems are built on the basis of physico-chemical. relationships between elements or organs of the system. In this case, the dynamics of changes in input, intermediate and output indicators are subject to modeling. These are, for example, biophysical models of the nerve cell, the cardiovascular system, the blood sugar control system, and others. Math. the apparatus that adequately describes the behavior of such deterministic biosystems is the theory of diff. and integral equations. Based on math. models of biosystems, it is possible, using methods of automatic control theory, to successfully solve differential problems. diagnostics and treatment optimization. The field of modeling deterministic biosystems is most fully developed.

If the organization of biosystems in relation to the studied indicator (or system of indicators) lies in the range of 0.3 - 0.1, then the systems can be considered probabilistically determined. These include internal control systems. organs with a clearly expressed component of nervous regulation (for example, the pulse rate control system), as well as hormonal regulation systems in the case of pathology. As an adequate math. The device can serve as a representation of the dynamics of changes in diff indicators. equations with coefficients that obey certain distribution laws. Modeling of such biosystems is relatively poorly developed, although it is of significant interest for the purposes of medical cybernetics.

Probabilistic biosystems are characterized by an organization value R ranging from 0.1 to 0. These include systems that determine the interaction of analyzers and behavioral reactions, including learning processes during simple conditioned reflex acts and complex relationships between signals environment and body reactions. Adequate math. apparatus

for modeling such biosystems is the theory of deterministic and random automata interacting with deterministic and random environments, the theory of random processes.

Math. modeling of biosystems includes preliminary statistical processing of experimental results (see Biological research, mathematical methods), study of the complexity and organization of biosystems, selection of adequate mathematics. models and definition numerical values parameters math. models based on experimental data (see Biological cybernetics). The last problem is generally very difficult. For deterministic biological systems, the models of which can be represented by linear diff. equations, the determination of the best parameters of the model (coefficient of differential equation) can be carried out by the descent method (see Gradient method) in the space of model parameters, estimated by the integral of the squared error. In this case, it is necessary to apply the parameter descent procedure to minimize the functionality

where T is the period, the characteristic time for the indicator, y is the experimental curve of changes in the indicator of the biosystem, y is the solution of the math. models. If it is necessary to obtain the best (in the sense of the integral of the squared error) approximation math. models for the operation of a biosystem according to several indicators for various internal states of the biosystem or for various characteristic external influences, then it is possible, using the descent method in the space of model parameters, to minimize the sum of partial functionals . When using this procedure for selecting parameters, math. model, you can increase the probability of obtaining a single set of coefficients. models that correspond to the adopted structure. With the help of B. s. m.m. it is desirable to receive not only quantitative characteristics the work of biosystems, its elements and the characteristics of the relationship of elements, but also to identify the criteria for the operation of biosystems, to establish certain general principles their functioning. Lit.: Glushkov V. M. Introduction to cybernetics. K., 1964 [bibliogr. With. 319-322]; Modeling in biology and medicine, in. 1-3. K., 1965-68; Bush R., Mosteller F. Stochastic models of learning ability. Per. from English M., 1962. Yu. G. Antomonov.