Translated from Izvestiya Rossiiskoi Akademii Nauk, Seriya Biologicheskaya,

No. 2, pp. 320-323, March-April, 1993.

THEORETICAL BIOLOGY: SEARCH FOR EQUATION OF GENERALIZED MOVEMENT

A. P. Levich

UDC 574/578

Since 1974 regional and interregional workshops on theoretical biology have been held with participation of scientists from the universities of Tartu, Saint Petersburg, and Moscow. The problem of theoretical biology has always been discussed at these meetings and is not yet exhausted leaving the room for more than one generations of scientists.

The participants of the discussion can, in first approximation, be divided into those supporting the purely methodical status of theoretical biology, those who favor identification of theoretical biology with various types of sign, formal simulation, and apologists of the existence of specific subject of theoretical biology.

The "methodists" define theoretical biology as a set of methods for obtaining biological results using, figuratively speaking, the pencil and the paper. Thus theoretical biology includes all methods of processing, analysis, generalization, and prediction of the results of biological experiments and observations. The "model" status of theoretical biology appears to he a special case of its methodic status.

The problem vision of theoretical biology implies the search for the nature of life. What is the difference between the living and the inorganic? Can the phenomenon of life be reduced to complex interactions of complex molecules? Are all essences of the world generating the phenomenon of life reliably recorded and reproduced in scientific laboratories? Naturally, the problem of the nature of life can be solved only in an experimental study. But the pathway to crucial experiments runs through the theoretical comprehension of the natural laws. Two main complementary types of such laws can be distinguished: laws of form (scientific classifications) and laws of change (dynamic approach).

For me, just as for Bauer (1935), theoretical biology is interesting by its search for pathways towards the theory of generalized movement and for sources of non-equilibrium of the living matter.

METHODOLOGY OF CONSTRUCTION OF THE DYNAMIC THEORY

The dynamic theory of any fragment of the reality includes some obligatory components whose development appears, openly or implicitly, as stages of the theory construction (Levich, 1989).

Component *O *consists in description of the idealized structure of an elementary object of the theory.

Component *S *consists in enumeration of admissible states of the theory objects. In other words, component S is a space of states of the studied system.

Component *C *fixes the pathways of changes of the objects by introducing sequences of admissible states of processes in the theory.

Component *T* is the method of measurement of changes. The method consists in the choice of a natural object, whose changes are taken as the standard ones and this standard acquire the status of a part in the theory. Let us note that "there is no method of measurement of time, which would be more correct than the other; the method adopted is just more convenient" (Poincare, 1898).

Component *L *is the formulation of the law of changes which isolates the real generalized movement of objects among all possible changes in the space of states.

Component *I* is a set of interpreting procedures — agreements about comparison of formal constructs of the theory with the objective reality and rules of comparison of these notions with experimentally measured values. The interpreting procedures transform a formal theoretical scheme to the science about reality.

I will provide examples of components of some dynamic theories.

Classical mechanics considers material points with their positions and velocities, such as the planets of the solar system, as elementary objects. The space of states is a six-dimensional phase space of coordinates and velocities. The movement along the trajectory is parametrized by the standard movement of the Earth in the gravitation field of the Sun or by the movement of a pendulum in the gravitation field of the Earth. The laws of changes for weak fields and low velocities are determined by Newton's or, in the general case, Einstein's movement equations.

Quantum mechanics considers amplitudes of probabilities for states of microobjects, e.g., energetic states of the atom, as elementary objects. Changes in the space of states are set by the vector trajectories in the infinite Gilbert space. Electromagnetic oscillations of the reference atoms serve as the clock. The law of changes is set by the movement equations, such as Shredinger, Maxwell, or Shirak ones. The quantum mechanics apparatus has as a formal object complex-significant wave functions and operators acting on them. Transition to notions of macrophysical reality is realized according to the postulated rules: the wave function square is the possibility to detect microparticles in a certain point of the space and the operator value is the quantitative value of physical characteristic corresponding to the operator. Observation of probability distributions requires interference experiments with the passage of particles through obstacles, while the energetic characteristics of atoms are determined through distances between the lines of experimental atomic spectra.

In embryology the living cell plays the role of elementary object and cell divisions appears as canonical changes. The space of states is described by morphological features of archetypes of zoological classifications. The interval between identical phases of cleavage divisions can be conveniently taken as a "unit of embryological time," which is species-specific and depends on environmental conditions (Detlaf and Detlaf, 1982). Development of fundamental equations of the "ontogenetic movement" is the task of theoretical embryology.

In ecology of communities the population is an object. Changes are summed from processes of individual birth and death. The space of states is a set of all possible vectors (n_{1}, n_{2}, .., n_{w}), where n_{i} is the number of a population of the i-th species, a component of the community. The set is limited by environmental resources accessible to organisms. Parametrization of changes in the community may be based on measurement of the amount of substrate-energy resources assimilated by the community generating the metabolic time of the ecosystem (Levich, 1989). The population time in ecology can be measured also by the number of generations (Abakumov, 1969). The law of generalized movement of a particular community in the space of its states as function of metabolic time is set by the so-called formula of species structure (Levich, 1980).

THE CONSTRUCTION OF TIME IS REQUIRED

The aim of any dynamic theory is to* *create component *L*, i.e., the law of generalized movement. This aim cannot he achieved without correct development of components *O, C, S, T, *and *L*. Components *T *and *C *are the least reflected by theoreticians. Their creation constitutes in fact the task of development of explicit time construction in each of the studied areas. The properties of the world necessitating component *T* of the theory consist in possible difference of the clocks we establish in the reference systems for describing changes in natural objects. But the modern theoretical naturalism is permeated by parametrization of changes with the help of exclusively physical clocks (the above mentioned specially selected embryological and ecological examples are an exception, rather than the rule).

Component *L *of the theory, the law of movement, is a description of changes of the studied object through changes of standard clocks (Sharov, 1989). Therefore the chances to discover and express explicitly the law of changes depend on the degree of adequacy of components *C *and *T *to the studied processes. It is possible that difficulties in derivation of movement equations in many areas of biology are related to non-coordination between physical methods of time measurement and non-physical nature of the studied laws.

OUTLINE OF DYNAMIC THEORY IN ECOLOGY OF COMMUNITIES

Details of development of the theory have already been published (Levich, 1980, 1982). Here we will follow the realization of the above proposed postulates on the example of ecology of phytoplankton communities.

All natural systems are hierarchic. For example, molecules, cells, organisms, populations, communities, associations of communities, and the biosphere may serve elements of the biological hierarchy levels. Fragments of biological hierarchy, communities consisting of populations of several phytoplankton species, are proposed as an elementary object for the theory in ecology of communities.

The general obligatory phenomenon of replacement of elements at every level of the system hierarchy exists in all natural systems. The general process occurs also at all levels of biological hierarchy. Was it this process that Georges Cuvier described: "Life represents a more or less rapid, a more or less complex vortex, whose direction is constant and which always captures molecules with certain properties; individual molecules continuously penetrate into it and come out of it, so that the shape of the living body is more important for it than its substance. Until this movement occurs in the body, it is alive, it lives" (Cuvier, 1817; cit. from Vernadskii, 1967, p. 65). Changes in the community are formed out of processes of cell division, their elimination from the community (lysis, eating, sedimentation, transfer), introduction of individuals in the community from outside, as well as out of processes of consumption by the cells of molecular substrates and energy from the environment.

The space of system states is represented by all possible vectors of the community state {n_{1}, n_{2}, ..., n_{w}), where n_{i} are numbers of each of w species of the community. The set of states is limited by the amount of substrate-energy resources of the environment consumed by species of the community and accessible to them:

where L^{k} is the amount of resource k available in the medium and _{;} is the requirement of species i in resource k.

Construction of the metabolic time (Levich, 1989) suggests parametrization of changes in natural systems by a multicomponent value including the amounts of elements replaced during the general process at each of the levels of the system hierarchy. For our hierarchy these components are: number ΔW of replaced species, amounts An, of replaced cells in each population, and amounts ΔL^{k }of resources consumed by the community. "Conjugation of technology of time measurement with fundamental processes, such as cell division or metabolism, suggests that the biological clock is not an addition to the biosystem building, but rather the building itself" (Voitenko, 1985, p. 74). The multicomponent pattern of the system's metabolic time poses the problem of choice of the standard level and standard object. Measurement of time of the hierarchic system by replacing elements of the deepest level makes the corresponding temporal scale the most universal and the least discrete. However the set of biogenic substances at the molecular level of the biological hierarchy assimilated by phytoplankton is big enough and even the preference of the substrates limiting cell division only preserves ambiguity when choosing a factor for standard metabolic clock. This seemingly pure technical problem have deep theoretical roots. The metabolic time is determined by the number of replacements of the elements. The notion "number of elements" is strictly explained in the set theory by cardinal numbers, e.g., natural numbers, for structureless objects, while any application of the formal methods for description of natural systems is based on structurized mathematical objects. Therefore correct count of elements of the structurized sets requires generalization of the quantity notions. The required generalization for arbitrary mathematics structures is given by the functor method of structure comparison (Sharov, 1977; Levich, 1982) in the mathematical theory of categories. The theory of categories is a convenient language for description of natural systems. The structure (axiomatics) of objects of a category determines the elementary object of the theory. Objects of the category characterize the system states, while morphisms determined the admissible methods of changing states. Each system state corresponds to its numerical invariant, cardinal number of the image according to the functor representation of structures. This invariant generalizes the notion "number of elements" to structurized objects and, while parametrizing the evolution of states, plays the role of categorical time of systems. Calculation of the invariants of mathematical structures suggests that in special cases these invariants coincide with the Boltzmann's form of representation of the system entropy, there fore the functor comparison of the structures leads to generalization of construction of the entropy of natural systems described by arbitrary mathematical structures. Thus, the use of invariants of the mathematical structure for construction of component *T* of the theory leads to the entropic parametrization of time.

The functor comparison of the structures produces an easy solution of the problem of development of component *L* of the theory: the law of change of the states is formulated as an extreme principle, which postulates evolution of the set state of the system to the state with an extreme value of the invariant or generalized entropy. Subsequently, calculus variations allows a ready use of the principle for derivation of the equations of generalized movement. It is essential that the functionals (invariants of structures) initial for the variational problem are strictly calculated, rather than postulated.

Application of the theoretical-categorical methodology to simulation of algocoenoses leads to the formula of species structure:

(2)

where

(3)

and λ^{k} are Lagrange factors depending on the resources L^{k} limiting the community growth and unambiguously searched when solving the variational problem with limitations in the form inequalities (1)

Thus, the formula of species structure describes dynamics of the species' numbers in the metabolic time of the system which is determined by the amount of limiting resources L^{k} consumed by the community. The formula of species structure describes adequately the species' abundance in coenoses, predicts effective methods of controlling the structure of coenoses, and makes it possible to study the fundamental principles of ecology of communities.

Biological Faculty, Moscow, State University, Moscow, Russia. Translated from Izvestiya Rossiiskoi Akademii Nauk, Seriya Biologicheskaya, No. 2, pp. 320-323, March-April, 1993. Original article submitted March 3, 1992.

1062-3590/93/2002-0266$12.50 ©1993 Plenum Publishing Corporation

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