Mathematical modelling of regulatory mechanisms of cell groups functioning at norm and at unregulated cell division

. This paper presents a method for modeling organs and tissues based on the equations of the functional unit of organs and tissues (FUOT) and mathematical models for the main cellular functions: division, growth, differentiation, performance of specific functions, aging, apoptosis and natural death. The elements of FUOT are cells. The analysis shows that there are short circuits of regulation, which in specific realizations of cell development correspond to stable or partially stable cellular structures. This can occur as a normal phenomenon in the form of an adaptive response to stressful influences, and as an abnormal pathological condition. Such partial FUOT can be called minifuot. The study of the proliferative minifuot equations show the presence of a nonzero stable equilibrium position, and a stable limit cycle in the first quadrant. The results of these studies can be useful in studying the mechanisms of cell groups functioning at norm and at unregulated cell division.


Introduction
Theoretical developments regarding the structural and functional organization of living systems are very diverse and are carried out from different points of view [1][2][3]. In our opinion, the "principles of mathematical biology" put forward by N. Rashevsky are the most constructive [1]: the theory of organismic sets, the principle of "biological epimorphism" and relational biology, which constitute the principle of the topological representation of both biological and social "organisms" [1]. These principles can be used to model the regularities of the spatial organization of biosystems, evolutionary changes in organisms and the structural and functional organization of cellular systems in plant and animal organisms.
In the last years of the last century, much attention was devoted to the analysis of the principle of hierarchy of block organization of living systems at various levels: molecular, cellular, and supra-cellular [2][3]. It was clear that the spontaneous organization of molecular systems cannot lead during the Earth's existence to the observed structural and functional organizations of living systems. Only the formation of individual elementary acts, functions and structures with their further consolidation in the course of evolution and their use to build the next stages of organization of living things could provide the observed level of complex, hierarchical structural and functional organization of living systems [2].
On the other hand (taking into account the evolutionary series of increasingly complex cellular structures), the epimorphism of the totality of biological functions of organisms and the identity of their components, with the existence of a wide variety of spatial organization and functional activity of organs and tissues of animals and plant organisms, leads to the idea of the existence of universal functional unit of organs and tissues of organisms that perform the basic set of elementary functions of living systems (renewal, specialization, metabolism with the environment, fulfillment of specific functions, aging, natural death, apoptosis) -a functional unit of organs and tissues, the formations from which constitute the organs and tissues of a multicellular organism.
Worldwide development of the theory and practice of the regulatory mechanisms functioning of living systems at the main hierarchical levels of the organization is connected with its successful application to biology, medicine and agriculture, because it allows to choose the most effective ways for diseases prophylaxis and treatment, for agricultural techniques of cultivation and plant selection, to creation various biotechnology products.The works, devoted to different types of mathematical modeling of living system regulatory mechanisms, by B. Goodwin, J.Smith, M. Eigen, V.A. Ratner, E.E. Selkova, D.S. Chernavsky (subcellular level); Antomonov, Sendov, R. Tsanev (cellular level); L.I. Lischetovich, Y. Kibardin, K.K. Dzhanseitov (organ-tissue level); N. Rashevsky, A.M. Molchanov, G.I. Marchuk (organismal level) and other are discovered basics mechanisms of biosystem regulation at considered levels, permitting using mathematical modeling to solve medical and biological problems. However, to date, there is no united approach to create mathematical models and effective methods for the quantitative analyzing regulatory mechanisms of living systems, taking into account spatial and temporal organization.

Methods
We offer the following definition: a connected set (on the space and (or) time) of cells can be taken as a functional unit of the considered organ or tissue if it contains dividing M, buffer B1, differentiating D, performing specific functions S1, S2, ..., Sn and aging B2 cell groups, functioning in an interconnected manner as a whole (Fig. 1). The presence of a certain resting phase after the phase of cell division, during which cell growth occurs, the formation of intracellular structures and the choice of the further path of life (division or differentiation) has led to the isolation of a separate pool of B1 with buffer cells. The same zone of buffer cells B2 is highlighted, after the zone of specific functions.
These cells also perform adaptive functions and, if necessary, can, through dedifferentiation, proceed to the repeated performance of specific functions.
Mathematical models of the basic cellular functions of FUOT are built on the basis of equations for the regulatorika of living systems [4][5][6][7]. Regulatorika is the science that involves the study of interconnected activity of regulatory mechanisms based on the ORASTA concept which consists of the operator-regulator OR (capable to accept, recycle and transfer signals) and ASTA (active system with time average, carrying out a feedback loop in system for finite time). Using ORASTA the functional-differential equations taking into account stimulating and inhibiting interactions, temporal relations, combined feedback and cooperativity in considered processes are developed [6]: and with initial conditions here ( ) are the sizes characterizing quantity of a signals, developed by -th OR at the time moment ; ℎ are a time intervals necessary for -th OR activity changing under theth OR activity influence; 0 , are parameters of formation and disintegration speeds ofth signal, accordingly; ( ) are continuous, positive initial functions. OR together with ASTA constitute a regulatory system -ORASTA (Fig. 2).
The equations for the model of cell growth are constructed on the basis of the equations for the regulatorika of molecular-genetic systems and the equations for the biosynthetic activity of cells. The basic equations of the model for cell differentiation are based on the model of the regulation of living systems, taking into account the polynucleotide competition of alternative metabolic pathways [4][5]. Models for fulfilment of the specific functions of the considered FUOT can be built on the basis of the activity of specific organs and tissues, a quantitative description of their function. The equations for the cells aging model (cells from the B2 phase) can be constructed on the basis of regulatorika laws for the molecular-genetic system with taking into account metabolic activity. Natural cell death in the framework of the FUOT equations is modeled by taking into account the rate of cell elimination from the cellular community.
It seems that in biosystems, in addition to the main chain, there are a short regulatory chains,which correspond to stable or partially stable cellular structures in specific realizations of cell development. This can occur both as a normal phenomenon in the form of an adaptive response to stressful influences, and as an abnormal pathological condition. Such partial FUOT can be called minifuot. Let us consider minifuots found in multicellular organisms.
Rapid proliferating cells under normal conditions are found mainly in the early phase of the development of the organism, in the epithelial tissues of animals and in the cambial tissues of plant organisms. In this case, the B1M transition has turned from weak (indicated on the FUOT diagram ( Figure 1) by a dotted line) into a solid one, indicated by a solid arrow (Fig. 3). This case also includes an unequal division of the egg cells at initial stages of embryonic development. In the case of extreme influences (mainly during resection, trauma, etc.), the minifuot arises in rapidly renewing tissues as an adaptive reaction, which leads to rapid tissue regeneration.
The connection between the buffer cells and the specialized cells is lost. Let's analyze this case. In some cases, this minifuot can occurs in normal tissues and organs. This leads to the formation of outgrowths, and in the case of stabilization of minifuots, may give rise to the formation of autonomous, proliferating groups of cells, very similar in characteristics to the cellular communities in cancer development.
Minifuot with an inferior specialization. These minifuots are rarely found in normal conditions. For example, the community of lymphocytes, without completing specialization, passes into the circulating blood and finally specializes in contact with alien cells or viruses. Basically, this minifuot occurs in the case of a cellular disease of the body. An analysis based on the schemes of cell relationships (see Fig. 4) shows that in this case, immature cells are formed in these tissues and this area atrophies, unable to perform its normal functions. This is especially often observed, apparently, under skin diseases.  (1) consists of terms expressing the rates of "multiplication" and "death".
Let us consider the case when cells from B1 to M do not arrive directly. Since the multiplication of cells requires the presence of at least one cell in M, the rate of reproduction is a homogeneous function of 1 ( ). Assuming that the specific rate of cell reproduction depends only on 2 ( ), we can write Obviously, a trivial equilibrium exists. For non-trivial equilibria, we have Then there exist 1 , 2 in the first quadrant, which are the phase coordinates of the equilibrium position for (3), if the initial value of the specific reproduction rate is greater than the value , i.e. 0 > . Indeed, in this case we can determine 2 from the first equation and, substituting this value into the second equation, we find 1 . If 0 < , then we have only a trivial equilibrium position. Therefore, = 0 / is a characteristic parameter of the equations. At = 1, we have a bifurcation of equilibrium positions. If > 1, then system (3) has, in addition to zero, and a positive equilibrium position. Let us consider the nature of the behavior of solutions near the critical points (0,0) and ( 1 , 2 ). Near a trivial equilibrium position, we can write It is clearly, that if < 1, then the point (0,0) is stable, at > 1unstable, and at = 1neutral. Let   Here, for = 0 > 1we have two equilibrium positions: trivial and nontrivial 0 , determined from The qualitative analysis shows the stability of the trivial equilibrium at < 1 and loss of stability at > 1.
Let us consider the solutions characteristics of (5). Linearization of the equation near a nontrivial equilibrium position leads to the equation According to the Hayes criterion [8], the roots of (7) are negative if and only if 1.
> −1; where is the root of the equation = . The first condition is fulfilled, since > 0, and the second condition is also fulfilled due to the accepted condition < 0 ( is a positively decreasing function). Analysis of the third inequality shows that, for certain values of the coefficients, it may not be fulfilled, and then the equilibrium position 0 loses stability. Let's consider this case in more detail. Let − = 1 for simplicity. Then the third condition takes the form − 0 − 1 < 2,2 and the stability condition is If this condition is violated, then there is a limit cycle, since the trivial equilibrium position is unstable and solutions of (5) are bounded. Thus, with the monotonically decreasing functions of the specific reproduction rate considered, the proliferative minifuot has a stable trivial equilibrium position, which, with an increase in the parameter P, allows bifurcation to unstable trivial and nontrivial critical points. The nontrivial equilibrium point, under certain conditions, can lose stability and lead to the appearance of a limit cycle. The results of the qualitative analysis show that the role of the delay is significant in this case.

Discussion
The method for modelling organs and tissues based on the allocation of a functional unit of organs and tissues (FUOT) makes it possible to simulate the main modes of the considered process and to identify regulatory mechanisms and laws of cellular communities functioning, has a prognostic ability. Since functional-differential equations with delay have an innate tendency to the presence of an oscillatory mode of solutions, their use as equations of FUOT models is reasonable and justified.
Qualitative studies of the proliferative minifuot show the presence of a nonzero stable equilibrium position, and sometimes a stable limit cycle in the first quadrant. Biologically, this means that if a cellular system of enhanced proliferation has been formed, then it can function for an arbitrarily long time. The results of these studies can be useful in studying the mechanisms of functioning of cell groups in normal conditions and during tumors development.