MATHEMATICAL MODELING OF THE ORGANISM

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Скачать бесплатно! Научная работа на тему MATHEMATICAL MODELING OF THE ORGANISM. Аудитория: ученые, педагоги, деятели науки, работники образования, студенты (18-50). Minsk, Belarus. Research paper. Agreement.

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By Vassily NOVOSELTSEV, Dr. Sc. (Tech.), V. A. Trapeznikov Institute of Management Problems, RAS

This country was first introduced to cybernetics back in the early 1960s. At the time the USSR held top positions in the military and aerospace areas for which different types of product management technologies were developed. It was common for Soviet scientists to view a living organism as a simple analog of complex technological "monitoring systems". They believed that constant body temperature, arterial pressure, and blood sugar level indicate the presence in the organism of a mechanism sustaining (controlling) a particular parameter. For each vital variable they assumed an optimum value, while physiology was reduced to an "executive function" identifying and eliminating deviations from a concrete level.

Initially scientists were quite happy with such a simplistic concept of the organism and mathematical models based thereupon. They were used for engineering artificial life support technologies which, as early as the 1970s, included a number of complex systems, such as facilities for monitoring human activities in extreme conditions (in outer space, at large sea depths and in hazardous industrial environment), different types of medical systems for supporting vital functions (artificial kidney, artificial heart) as well as means of controlling the blood sugar level, hemodialysis equipment, and so forth. These concepts and models had other medical applications too, e.g., the selection and dosage of drugs (pharmacokinetics), tomography, electroencephalography, and so forth.

However, in time the gap between the simplistic mathematical models of the organism and the complex control processes within it became increasingly apparent. Thus, all attempts of our researchers to apply pharmacokinetics to the study of the effect produced by potent toxic preparations encountered problems.

The classic model of the organism as a series of "chambers" through which the administered drug is distributed evenly, irrespective of the dose, just would not work. Now it proved good in describing the process when a negligible quantity of an agent entered the human body, which was but of little significance for toxicology. The fact is that a toxic agent affects all organs, and the organism attempts to arrest its action even before homeostasis is disrupted completely*. A similar


Articles in this rubric reflect the opinion of the author. -Ed.

* Homeostasis-relative dynamic equilibrium of the chemical composition and characteristics of the intercellular medium and stability of essential physiological functions of the organism. -Ed.

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The organism struggling for life.

scenario is observed in any other critical situations, e.g. in serious injuries.

So, to get to the bottom of things and resolve the problem of protecting and/or restoring different functional systems of the organism, scientists needed mathematical models simulating not just individual processes but reproducing the organism in its entirety. That would make it possible to study the organism's behavior in critical conditions, including intervital ones, i.e. those on the brink of death. Simultaneously, we could charter ways of further technological development relative to artificial life support systems.

WHAT KEEPS THE WHEELS OF LIFE GOING

The organism can struggle for survival as long as its cells are alive. Drugs sustain the body's suppressed functions, while life support facilities compensate for serious dysfunctions.

Cells "have no idea" of what is going on in the organism at the macrolevel. They can only "sense" changes that occur in their own selves and in the direct environment. Unless cell structures are disturbed, and if the chemical composition of the intercellular medium remains steady (homeostasis), the cells keep viable. They just have to produce enough energy with the use of the available "fuel" and oxidizer and get "raw materials" for their "self-repair" and "reproductive activity".

Thus viewed, the integral organism emerges as an aggregate model of the cell and physiological systems supporting vital functions. The "fuel", primarily, glucose and fats, gets into the organism with food via the gastrointestinal tract, and the oxidizer is inhaled with air through the lungs. The blood circulation system is responsible for their delivery to the cell. To prevent "the production waste" from contaminating the cell, the blood stream washes it away into "purification systems". Poisons that can be utilized by the organism are processed by the liver, the rest is eliminated through kidneys. The thermoregulation system sustains the heat balance.

There is nothing new to such methods of simulating physiological systems. Moreover, they can incorporate virtually all presently available approaches. A cell model, essentially, serves as an integrating element, and for it the energy consumption and supply balance as well as the current energy resource should be determined.

An integral model of the organism representing both the cell and the physiological systems responsible

A cell living in the integral organism.

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Energy model of the cell.

for corresponding processes inside it has proved to be viable, and has been successfully applied, in particular, to the analysis of acute food and chemical poisonings of the human organism, as evidenced by the model of acute poisoning with the death-cup (Amanita phalloides) , which we have developed in collaboration with specialists of the Information and Consulting Toxico-logical Center of the Ministry of Health of the Russian Federation in the mid-1990s, in the period of mass mushroom poisoning in Russia. Later we applied the same approach in creating models of gas-poisoning (ammonia and chlorine poisoning).

MATHEMATICAL MODELING OF THE ORGANISM, OF ITS LIFE AND AGEING

The mainstream of mathematical modeling of the integral organism is represented by ageing analysis and life expectancy projections. To describe these processes, the so- called human or animal life history is used-representing changes in the death rate as a function of age, and also showing the start of the reproductive function, the annual distribution of reproductive events and so on.

The recent tendency has been to explain ageing by the accumulation of oxidative wear in the organism. The gist of this theory is that the oxygen consumption required for life support is accompanied by the generation in the human or animal organism of harmful highly toxic substances-free radicals of ionized oxygen, hydrogen peroxide and hydroxyl. From 1 to 3 percent of the entire O 2 amount assimilated by a living organism is transformed into such substances in spite of the enzyme protection system keeping free radicals off.

Currently, it is assumed that the rate of ageing and life expectancy correlates with the rate of metabolism (consumption of matter and energy in the process of active functioning of the organism) and the efficiency of anti-oxidant protection. Since both energy requirement and oxidative wear accumulation can be measured in the same units, the life history may be represented by a quite straightforward and easy-to- interpret scheme. The consumption of oxygen by body cells is determined by its demand, its ingress being described on the basis of generalized Pick's law. The law says: the inflow of O 2 into living matter is proportionate to the difference between the atmospheric and intra-cellular pressure and depends on the properties of barriers which the oxygen flow encounters in passing from the atmosphere to mitochondria*. In the process of its consumption cells sustain oxidative wear, and the accumulation of this damage in time determines the ageing of the organism. Accordingly, the homeostatic capability declines, and the cell energy resource is exhausted.

The above scheme is too approximate to be used for describing the life cycles of humans and animals. It

Simulation of lethal ammonia poisoning.


* Mitochondria-animal and plant cell organelles where redox reactions take place. The number of mitochondria in a cell varies from a few ones to thousands. - Ed.

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A biont's life history.

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Life cycle of the DROSOPHILA MELANOGASTER female.

is mainly applied to the building of biological models of lower organisms, primarily, the fruit fly Drosophila and the nematode Caenorhabditis elegans.

DROSOPHILA'S LIFE STORY

The energy expenditure by a living organism is aimed at two interconnected objectives: reproduction and functional activity. As for the oxygen consumption by the fruit fly (Drosophila), it may be described as a sum of two components: the basal level reflecting energy consumption for sustaining the soma (this term applies to all animal or plant cells, with the exception of sex cells) and energy expenditure for reproduction (breeding and oviposition). The accumulation of oxidative damage (wear) decreases the homeostatic capability virtually exponentially, and the exhaustion of the cell resource correlates with a certain "critical age". Once reaching it, the Drosophila female starts laying fewer eggs, the reason why its organism consumes less oxygen. However, having hit the bottom, the energy resource of its organism is restored, and the flies who have survived the crisis will enjoy "happy old days".

MODELING AND EVOLUTIONARY OPTIMALITY

Contemporary evolutionary theories assume that all currently existing organisms have survived and ousted their competitors because their genotypes have proved best in reproduction. In other words, only those genotypes have survived which allowed their carriers to beget the maximum number of descendants. That is a species capable of producing numerous offspring in a unit of time but short-living is as far from the optimum as is one that lives long but invests too little energy in reproduction. Although tested by time and illustrated with numerous examples, these postulates are yet to be confirmed experimentally. The gap is largely filled by a mathematical modeling of the organism's life cycle through numerical experiments involving the distribution of reproductive and somatic resources.

A graph point formed by the intersection of the respective curves and corresponding to a genotype at the succession of generations under the effect of natural selection will slide through the "evolutionary landscape" and tend to a possible maximum with its value limited only by the structural and functional characteristics of the organism.

So, the analysis of evolutionary optimality by the methods of mathematical modeling falls into three parts. First, we should build a model in which successful breeding will depend on the available resource and the mode of its distribution among reproductive and somatic functions. Next, we should study a variety of all possible model options and build an "evolutionary land-

"Evolutionary landscape". Reproductive success of the Drosophila female as a function of the reproductive and somatic resource.

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Human life expectancy. To assess the maximum potential of the organism we have used P. Astrand's data which he has obtained in a case study of 380 athletes. Assuming the minimum oxygen demand of the young within 2-3 "basal exchange levels" of 70-year-olds, the estimated human life expectancy maximum should range from 138 to 169 years.

scape" for a particular organism. And last, we should find the nature of "development constraints" to identify the theoretically attainable optimum point. If this point corresponds to the parameter values of an average statistical organism in a population, the concept of its evolutionary efficiency (optimality) will get substantial experimental support.

IS THE DROSOPHILA'S ORGANISM OPTIMAL?

Taking a model of the Drosophila's life cycle, we just could not resist the temptation of verifying the optimality of its organism. According to experimental data obtained for the well-studied population of the Drosophila's strain (Wayne state), the power of the organism of an average statistical female in the period of maximum oviposition reaches 149.7 meal O 2 per day, while reproduction consumes a little more than half of this energy (42.5 percent). As a result, in 44 days of life it lays 1,150 eggs. Is it good or bad? Is there room for improvement through lower or, conversely, higher general energy expenditure? Theoretical answers to such questions are well known. Thus, Thomas Kirkwood's theory of "one-time soma" states (Health Institute for the Aged, Newcastle, Britain) that investments into somatic functions should not be too big, since the Drosophila's organism is used only once for the transfer of genes from one generation to another.

Another common approach to the optimality theory has been offered by L. Partridge and N. Burton (Halton Laboratory, University College, London), who demonstrated that the optimum of the "evolutionary landscape" is located in the coalescence point of the "development constraints" curve and the line of the maximum reproductive success level.

Mathematical modeling allows to obtain an "evolutionary landscape" for the real Drosophila population on the basis of both theories. It will be enough therefore to represent all conceivable combinations of the life cycle parameters of the Drosophila female, something that presents no difficulties (see p. 56).

To verify our theory we just have to identify the nature of "development constraints", plot the corresponding curve, find the theoretical optimum point and compare it with the experimental one. Here it is convenient to express evolutionary constraints in terms of reproductive effectiveness. It is obvious that the lower "the overhead charges" and the higher the portion of the general resource expended for reproduction, the better the design of the organism is. That means that the sought indicator for any genotype can be graphically depicted as a beam departing from the origin of coordinates. This is the "line of evolutionary constraints".

Now in accordance with Th. Kirkwood's idea we can follow the change of reproductive success moving along the beam from left to right. It is obvious that the costs of self-preservation will be growing along with the proportional increase of reproduction costs. As for the level of reproduction, it will initially grow too, but on reaching the experimental point it will be dipping. That means that the experimental point coincides with that of the theoretical maximum!

Similar results are obtained in digital experiments in keeping with the Partridge- Burton theory. In this case the difference between the reproductive effectiveness of genotypes is shown by turning the beam around the origin of coordinates. The higher the effectiveness, the steeper the beam. If for simplicity

Pages. 57


we assume the amount of general resource as constant, the optimum reproductive success proves to be 1,150 eggs. This is the value which we have determined for the experimental population under study.

Thus the mathematical modeling of the Drosophila's organism argues for the evolutionary optimality of its actual genotype which has been studied in depth.

LIFE EXPECTANCY LIMITS OF ANIMALS AND MAN

The life cycle of a female fly proves: its average life expectancy is optimal for the conditions simulated during the experiment. In other words, both the reduction and the increase of its life span results in the falling number of the average statistical fly's offspring. No doubt, a change in the environmental conditions (temperature, food calorie or reproduction possibilities) tells upon the duration of the Drosophila's life cycle. However, mathematical modeling takes account of these factors and allows prognostication.

Unfortunately, we are yet unable to simulate human life history in as much detail as in the case of animals. Not only because the human organism is much more complex. The point is that in their everyday life humans have long since stopped to abide by the criteria which determined the formation of their organisms in the course of evolution. For example, it is most unlikely to expect a contemporary woman to regard the number of born children as the only token of her success in life.

However, the analysis of animal "life story" may be conclusive in some aspects for humans. For example, the gerontological literature often postulates that human physiological functions decline gradually with age. Still, simulation of ageing in the fauna (with the mechanism similar to that of man) demonstrates that it is rather described by an exponential function, gradually slowing down with age. The analysis of digital test results also argues in favor of the effectiveness of the energy-based approach for human life expectancy prognosis: theoretically, the limit of longevity will be reached when even the maximum exertion of the organism's resources will not be enough to meet the minimum requirements of the ageing organism. The required digital (statistical) material on the subject is available in the literature.

Say, the maximum oxygen demand (MOD) indicator is often used for assessing man's work efficiency. As for survivability of the organism, it is often associated with the minimal oxygen demand (mOD). If, in accordance with the above ideas, we are to approximate the MOD downward trend (curve) with age, and compare the minimal demand with the amount of mOD, the intersection of these curves will give the sought value for the limit of human life expectancy, that is between 138 and 169 years (see p.57). Naturally, this result should be treated just as an illustration of the idea, since both the data used and the very methods need substantial refinement. Still, this example is quite good for demonstrating the potential of the mathematical modeling of the organism.

Illustrations supplied by the author.



Опубликовано 15 сентября 2018 года

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