Stochastic mechanics in the context of the properties of living systems

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1 Mathematical and Computer Modelling, 2006, accepted Page1of16 Stochastic mechanics in the context of the properties of living systems E. Mamontov 1,2, K. Psiuk-Maksymowicz 1,3, and A. Koptioug* 1 Department of Physics, Faculty of Science, Gothenburg University, SE Gothenburg, Sweden, 2 yem@physics.gu.se, correspondence author; 3 psiuk@physics.gu.se; * Department of Information Technology and Media, Mid Sweden University, SE Östersund, Sweden, Andrei.Koptioug@miun.se Abstract: Many features of living systems prevent application of fundamental statistical mechanics (FSM) to study these systems. The present work focuses on some of these features. By discussing all the basic approaches of FSM, the work formulates the extension of the kinetic-theory paradigm (based on the reduced one-particle distribution function) that exhibits all of the considered properties of living systems. This extension appears to be a model within the generalized-kinetic theory developed by N. Bellomo and his co-authors. In connection with this model, the work also stresses some other features necessary for making the model relevant to living systems. A mathematical formulation of homeorhesis is also derived. An example discussed in the work is a generalized kinetic equation coupled with the probabilitydensity equation representing the varying component content of a living system. The work also suggests a few directions for future research. Key words: fundamental statistical mechanics, living system, homeorhesis, generalized kinetic theory, stochastic process AMS MSC codes: 37N25, 82C05, 82C31, 82C40, 91D10, 92B05 1. Introduction The minimal unit of life is a living cell. A system which does not include living cells can be termed a nonliving system. Nonliving systems are studied in most of areas of physics, chemistry, and engineering. These fields are endowed with a great variety of experimental methods and theoretical treatments, including fundamental statistical mechanics (FSM) (e.g., [1] [4]). The latter proved to be a powerful tool in analysis of problems in nonliving matter. Subsequently, there is an irresistible temptation to apply the first-principle mechanical theory to virtually every new problem, in particular, to those living-system problems that appear similar to the ones successfully resolved by the nonliving-matter sciences in the previous decades. Thus, one can expect that physicists would suggest FSM as the unified theory suitable to both nonliving and living systems. Along with this, there are other opinions. One of them comes from biology [5, p. C49]: Biological systems are very different from the physical or chemical systems analysed by statistical mechanics or hydrodynamics. Statistical mechanics typically deals with systems containing many copies of a few interacting components, whereas cells contain from million to a few copies of each of thousands of different components, each with very specific interactions. The authors of these issues continue their thought in the following way:... The macroscopic signals that a cell receives from its environment can influence which

2 Stochastic Mechanics for Living Systems Page 2 of 16 genes it expresses and thus which proteins it contains at any given time or even the rate of mutation of its DNA, which could lead to changes in the molecular structures of the proteins. This is in contrast to physical systems where, typically, macroscopic perturbations or higher-level structures do not modify the structure of the molecular components. The present work analyzes in what specific respect FSM does not allow for the above features and what stochastic mechanics instead of FSM can enable them. The work singles out the version of stochastic mechanics most relevant, among available alternatives, for studying living systems. Biologists emphasize many features of living systems which do not allow to apply FSM to study these systems (see Sections 1 and 2.1). The results of the present work are the following. By discussing all the basic approaches of FSM (see Sections ), the work formulates (see Section 2.6) the extension of the kinetic-theory paradigm (based on the reduced one-particle distribution function) that possesses all of the living-systems properties considered in the work. This extension appears to be a model within the generalized-kinetic theory developed by N. Bellomo and his co-authors (e.g., [6] [8]; see also [9] [12]). In connection with this model, some other features necessary for making the model relevant to living systems are also stressed (see Section 2.7 and Appendix). An example discussed in the work (see Section 3) is a generalized kinetic equation coupled with the probability-density equation which represents the varying component content of a living system. The work also suggests a few directions for future research. One of them is accounting homeorhesis, the key phenomenon of the living-system dynamics. A mathematical formulation of homeorhesis is derived in terms of a generic dynamical treatment, namely ordinary differential equations (see Appendix). 2. Analysis 2.1. Implications from biology: Formulation in terms of FSM The issues in the two citations in Section 1 can be formulated in terms of FSM. We consider a population of particles which occupies a bounded or unbounded set where and is the time. For all, set is assumed to be a domain, i.e. a set which is connected and open in the topology in. To apply various treatments below, it is assumed that, if, then the boundary of is piecewise smooth and the conditions at this boundary are available but generally depend on the solutions of the employed models. Also note that domain with the above properties is a Borel set. Let the population comprise components, i.e. (different if ) groups of identical particles, and let such that (1),, (2) be the number of the particles in the th component. The total number of the particles in the population is expressed as Obviously,. (3)

3 E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 3 of 16. (4) Equality corresponds to the case when every component comprises exactly one particle or, equivalently, all the particles are pairwise different. This can take place in a living system. Remark 1. It follows from the first citation in Section 1 that and in (1) and (2) are not limited to any specific intervals, i.e. (A) number in (1) can take any value from very low, such as unit or a few units, to a few millions (or greater), (B) every number in (2) can take any value from very low, such as unit or a few units, to a few thousands (or greater). These are important properties of a living-particle population. The second citation in Section 1 implies that, generally speaking, the numbers in Remark 1 depend on time, i.e. and Substituting (6) into (3), one obtains (5),. (6) that also allows for (5). The four sections below analyze if any specific model within FSM allows the properties in Remark 1 and features (5), (6) Phase-space models In FSM, each particle is described with the three-dimensional position vector in domain (see the beginning of Section 2.1) and the three-dimensional momentum vector. Thus, the phase-space for one particle is. Subsequently, the phase-space models in FSM considers the -particle population in the phase space where. The corresponding key notion is the phase-space probability density (e.g., [3, (3.7), (3.8), and p. 25)]) where variables and represent the position and momentum vectors, respectively, of the th particle. Density (8), as a function of and,, at every fixed and, is also the joint probability density of the positions and momenta of the particles at time. The phase-space models form the core of FSM. They include the Liouville equation (e.g., [3, (3.15)]), the formal master equation (e.g., [2, (VIII.2), (VIII.21)]), the molecular-dynamics equation systems, and other formalisms. Along with this, it is unclear how to use the phasespace settings under condition (7). Generally speaking, this prevents application of the phasespace models to living systems. To resolve the problem, one chooses a phase space such that its dimension is independent of varying number (7). The simplest way to do that is discussed in Section 2.3. (7) (8)

4 Stochastic Mechanics for Living Systems Page 4 of One-particle reduced distribution function: One-component population The simplest way to make the phase-space dimension independent of (cf., (7)) is by using the six-dimensional space for one particle as the phase space for the entire population. In this respect, the -dimensional space is reduced to the 6-dimensional one. However, the reduction can be done only under a special condition. This is considered below. Joint probability density (8) enables one to introduce functions with equalities,. (9) If density function at every fixed is symmetric with respect to all pairs (e.g., [3, (4.7)]) or, because of (9),, (10) then the -particle population can be described with the reduced one-particle distribution function (e.g., [3, (4.9)]) where identity holds since it, in view of (3) and (7), is equivalent to (13) which, in turn, is equivalent to condition (10). In kinetic theory, distribution function in (11) is granted as a solution of a kinetic equation (e.g., [3, p. 55]). In so doing, the terms on the right-hand side of (11) are determined by means of this function as follows (11) (12),. (14) The condition that all the particles are identical disagrees with feature (A) of living systems in Remark 1 and is eliminated in Section 2.5. The next section discusses why the most important extension of the above paradigm is also unsuitable to the living-systems modelling Two- and multi-particle reduced distribution functions: One-component population This section considers the particle population which is one-component, i.e. the case when (12) or (13) holds. The one-component settings in the previous section can be regarded as a quite particular case of a more refined treatment, the Bogoliubov Born Green Kirkwood Yvon (BBGKY) equation chain (e.g., [3, p. 45]). It is based on the reduced -particle distribution functions,. The case in Section 2.3 corresponds to. The case when is discussed in the present section. The well-known normalization condition for the reduced -particle distribution function at (e.g., [3, (4.14) and (4.18)]) is equivalent to relation (e.g., [3, p. 41]),. (15)

5 E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 5 of 16 One can show that the relative error of this approximation is inversely proportional to,. (16) Since the BBGKY description is traditionally regarded as the first-principle model, approximate equality (15) should hold with a very low relative error, say, on the order of tenths or hundredths of one percent, i.e.. Application of these values to (16) requires. This limitation sharply disagrees with the living-systems feature (B) in Remark 1 (see also (12)). Subsequently, the reduced -particle distribution functions at and any model employing at least one of them (e.g., the BBGKY equation chain or the so-called correlation functions (e.g., [3, Section 4.4])) are in general not suitable for describing living systems. Subsequently, one has to come back to the reduced one-particle distribution function in Section One-particle reduced distribution function: Multicomponent population The common extension of the identical-particle treatment in Section 2.3 to the case when the particles do not need to be identical involves the notion of a multicomponent population. Indeed, no matter if or, one can without a loss of generality assume that the first particles in the -particle population (cf., (4)) are pairwise different and any other particle is identical to one of the mentioned particles. In so doing, number in (2) and (6), turns out to be the total number of the particles identical to the above th particle where. The corresponding generalization of (11) is where (17),. (18) and are the reduced one-particle distribution functions for the components. In kinetic theory, distribution functions in (18) are obtained as a solution of a system of kinetic equations (e.g., [4, Section 3.2.2]). Note, however, that the derivation of this system from representation (17) is not always included in the corresponding textbooks. Solutions of the system are set in. Applying (5) to this set, one encounters the difficulties analogous to those described in Section 2.2. Thus, the above multicomponent treatment underlying (17) is generally not suitable to the living-systems modelling. The problem is overcome in the next section by leaving the scope of FSM One-particle reduced distribution function: Time-dependent number of components. Generalized distribution function As it is stressed above, the notion of a multicomponent population is inherently associated with the features of the particles to be either identical or different. The latter is resolved by comparing the component-specific values of the related parameters of the particles, for instance, the particle mass, size, shape, electric charge, or other physical quantities. They can be regarded as the entries of a vector variable, say, (where ) such that the th component,, corresponds to value of

6 Stochastic Mechanics for Living Systems Page 6 of 16 and all the values are pairwise different, i.e.,,. (19) The treatment below follows the single-distribution-function approach to multicomponent populations which is proposed in [10], [9]. According to it, functions in (18) can be regarded as values of single function where vector varies in a bounded or unbounded, generally time-dependent domain in, say,. This transforms (17) into. (20) In so doing, functions and in (18) are also determined in terms of function, namely where (cf., (14)),,, (21), (22). (23) Thus, the reduced one-particle distribution function (see (20)) for the multicomponent population and other basic characteristics of the components, for instance, (21) (23) are completely described by means of single distribution function. The latter can be determined in the way discussed in the next section. Now we shall analyze how distribution function enables one to allow for the living-systems feature (5). The above set can be regarded as a sample for a random variable defined on domain and described with a probability density, say,. In the simplest case, i.e. when the random variable is defined on and is discrete, namely,,, (24) where is the -dimensional Dirac delta-function, expression (20) is equivalent to. According to the approach of [10], [9], one applies the latter equality even when (5) holds and both domain and probability density are not necessarily of the particular forms shown in (24), i.e.. (25) In this way, the component number specifies the -dependence (5). The time-dependent component number can be interpreted in various ways. The discussions in [10, Sections 5 and 7], [9, Section 4.3 and Appendix] suggest to read it as the number of the modes of probability density. These modes may be viewed as the spread out, nonzero-widths and finite-height peaks analogous to those in (24). Along with this, we present a more general and somewhat more precise definition of. Definition 1. Let the components of a population of particles be described with vector parameter. Let also the number of these components be denoted with. We

7 term any path-connected subset of set E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 7 of 16 the component-parameter subset of the population. The particles in a component of the population are the ones corresponding to the values of in a component-parameter subset. Subsequently, number is the number of the component-parameter sets in set (26). If set (26) includes more than one component-parameter subset, then, obviously, all these subsets are mutually non-intersecting. This feature generalizes condition (19) which is valid in case of (24) when the component-parameter subsets are single-point sets. Importantly, Definition 1 is more general than the mode-based definitions developed in [10], [9]. Indeed, any collection of the population particles which is a component in the sense of the present definition comprises at least one collection which is a component in the sense of the previous definitions. Remark 2. As it is well-known [13, p. 1306] life is the principle or force by which animals and plants are maintained in the performance of their functions and which distinguishes by its presence animate from inanimate matter; the state of a material complex or individual characterized by the capacity to perform certain functional activities including metabolism, growth, reproduction, and some form responsiveness or adaptability. Probability density in (25) and its component characteristics in Definition 1 are the very quantities which can be the core in the representing of the above metabolism, growth, reproduction, and responsiveness or adaptability. Expression (25) shows that the entire multicomponent population is modeled in terms of a single, generalized reduced one-particle distribution function that describes not only stochastic position and momentum of a particle (represented with and, respectively) but also its stochastic parameter vector (represented with ) which determines the particle content of the population. In the context of (25) and (27), is the conditional distribution function for conditioned with value, (see (25)) is the marginal distribution function for, (see (23)) is the conditional probability density for conditioned with values and (see (22)), and is the marginal probability density for. Generalized distribution function (GDF) (27) is of the type underlying the generalized-kinetics (GK) theory developed by N. Bellomo and his co-authors (e.g., [6] [8]; see also [9] [12]). All the aforementioned issues can be summarized in the following way. Remark 3. Definition 1, single generalized reduced one-particle distribution function (27), and related expressions (21) (23), (25) enable one to take into account not only the features in Remark 1 and (6) but also (5), i.e. all the discussed properties of living systems. Among the stochastic-mechanics models considered in the present work, representation (27) and the related expressions are the only treatment that can allow for Remark 1 as well as (5) and (6). Subsequently, if a population of living particles can be modeled by a stochastic-mechanics approach, then the latter is based on (27). In other words, application of GDF (27) is the necessary condition for modelling living systems by stochastic mechanics. (26) (27)

8 Stochastic Mechanics for Living Systems Page 8 of 16 The next section derives one more condition that the model in Remark 3 should meet to properly describe the mechanics of the living-particle population Some other properties of living systems formulated in terms of the present model Stochastic variables and represent the mechanics and component content, respectively, of a population of living particles. The mechanical evolution described with can be recognized as the one of a living system only if the properties of the particles (e.g., those in Definition 1) influence the above evolution. This means that must depend on,. (28) The corresponding correction should be taken into account in (22), (23), (25), and (27). Because of homeorhesis [14, p. 32] (see also [15, p. 526] and [16]), GDF (see (27)) can not be stationary, i.e. independent of, in a system which is living. Along with this, a living system may in principle be mechanically stationary, i.e. the -independence of may be the case, at least as an approximation. In view of this and (27), probability density is always nonstationary, i.e. depends on. Paper [17] discusses the corresponding examples of experimental results. A mathematical reading of homeorhesis is developed in Appendix. Other details on the phenomenon in connection with (27) can be found in [9, Section 4.1]. Numerous observations of mature (non-embryonic) living systems show that, in any sufficiently short time intervals, living systems behave similar to nonliving bodies, namely those with the -independent. For instance, if the time needed for a driver to stop a car avoiding collision with an immobile obstacle is less than one second (the typical time for the driver reaction), then the collision is inevitable. In a short time interval when the mechanical-dynamics function noticeably varies, the driver action associated with remains unchanged. In specific terms, this means that, if a mature living system is modeled with (27), then,, (29) where set is described with (26), is the characteristic time of changes in,, and is the characteristic times of changes in (see (28)). The time scales in living systems are discussed, for instance, in [9, Section 4.4]. The above issues can be summarized as follows. Remark 4. If a living system is adequately described with the model in Remark 3, then: conditional distribution function depends on probability density, i.e. (28) holds; probability density depends on ; inequality (29) is valid. Moreover, the key phenomenon in the dynamics of a living system is homeorhesis. Its mathematical formulation in terms of ordinary differential equations (in Euclidean space or a function Banach space) is derived in Appendix. If any of the aforementioned properties does not hold, then the system is nonliving. The above summary lists the features to be accounted in development of specific models for GDF (28).

9 E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 9 of Generalized kinetic equation and related models. Directions for future research Conditional distribution function in (27), (28) can be obtained as a solution of the following generalized kinetic equation (GKE) [9, (3.15)],, (30) where scalar is the mass of a particle in the population component corresponding to and vector is the force acting on a particle and originated from the phenomena other than the collisions of the particles with the surrounding or each other. The term is the collision integral due to the collisions of the particles of one component, i.e. the component corresponding to, with the surrounding. The term is the collision integral due to the collisions of the particles of two different components, i.e. the ones corresponding to and. Specific examples of the collision integrals with a detailed discussion can be found, for instance, in [2, Section 3 of Chapter IX] or [3, Sections 7.2 and 7.3]. The three- and multi-component collisions are not included in GKE (30). Further details on this equation can be found [9, Section 3.3]. We only note that, in case of (24), GKE (30) becomes a common kinetic-equation system for a multicomponent particle population (cf., [4, Section 3.2.2]). Remark 5. GKE (30) is an example of a model that can provide the first feature of the living-system mechanics discussed in Remark 4. Note that, in the GK theory, conditional distribution function may depend not only on the entries of the component-content vector but also on other scalar parameters. The latter can be determined by means of the methods of the GK theory (see the references in the text above Remark 3). New effects can be accounted in GKE (30) by incorporating into the right-hand side a detailed treatment of many-body collisions in multicomponent populations of living particles. This presumes a detailed reading of the terms on the right-hand side of GKE (30) and complementing them with the corresponding extra terms. A possible way in this direction is discussed in [18], [19]. The general form of the model for probability density is equation [9, (3.16)],. (31) Equations (30) and (31) form the equation system for and. Solutions of this system supposedly have the second and third properties of the living-systems mechanics in Remark 4. These solutions should also comply with the mathematical implications from homeorhesis derived in Appendix. It follows from feature (A) in Remark 1 that equation (31) can be fairly complex. The fact that (31) must be regarded in conjunction with GKE (30) to make the model closed fully,

10 agrees with the well-known issues [5, p. C49]: Stochastic Mechanics for Living Systems Page 10 of the components of physical systems are often simple entities, whereas in biology each of the components is often a microscopic device in itself, able to transduce energy and work far from equilibrium. As a result, the microscopic description of the biological system is inevitably more lengthy than that of a physical system, and must remain so, unless one moves to a higher level of analysis. The form of function in (31) is still unknown. In spite of the fact that the model of [20] [22] takes into account certain crucial properties of living systems (see the text from (A.12) down to the end of Appendix), this model does not allow for (5) and employs the simplest version (24) of. In fact, it in part applies the nonliving-system paradigm. More research is needed to specify function. One of the fields for future study is associated with the fact that is assigned to represent the component structure due to biochemical reactions in the particle population. Subsequently, the results on the complex nature of these reactions (e.g., [23]) may contribute to derivations of function. A discussion on equation (31) is also included in [9, Section 3.4]. We only add a few issues. One of the simplest cases of equation (31) is the Kolmogorov forward, or Fokker-Planck, (KFFP) equation (e.g., [24] and the references therein). Research on the KFFP reading of (31) can be facilitated by a series of the recent results. For instance, the KFFP-based nonstationary invariant probability densities may appropriately describe homeorhesis (cf., the text below (28); see also Appendix). A fairly general treatment for these densities is reported in [25]. The work [24] includes a group of analytical-numerical methods developed for KFFPs with nonlinear coefficients in the high-dimensional case, i.e. when dimension of vector in (31) is high. Appendix. Dynamics of a living system in terms of ordinary differential equations The main distinguishing features of the dynamical behavior of a living systems are well known in biology since long ago [14, Chapter 2], [15] (see also [16] for a recent discussion). However, they have not been formulated in the language of mathematics yet. The present appendix derives the corresponding mathematical formulations in terms of ordinary differential equations (ODEs). The aforementioned features are the following. (i) A living system is an open system. (ii) A living system develops along the trajectory (in the space of the system states) that is completely determined by the internal properties of the system and its environment, i.e. the related exogenous signals. (iii) The trajectory called creode (the necessary route or path) [14, p. 32] corresponds to the most favored exogenous signals [14, p. 19]. The latter are also known as the formative drives (according to Aristotle s theory of epigenesis). (iv) The actual trajectory, i.e. the one corresponding to the actual rather than most favored exogenous signals, need not be the creode, but in the course of time tends to the latter, no matter what the signals are (in a certain, system-relevant range). This stability with respect to the exogenous signals is known as homeorhesis [14, p. 32]. (Homeorhesis is

11 E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 11 of 16 the time-dependent generalization of homeostasis. The latter term was coined by W. B. Cannon in 1926 who discussed homeostasis in detail later [26]. Remark A.1. In the behavior of any living system, one recognizes a purposeful component, however, without indicating the corresponding mechanism or specifying the terms. Resolving the latter problem is substantially contributed by Waddington s theory of homeorhesis. Indeed, it follows from feature (iv) that the homeorhetic mechanism indicates the creode trajectory (which is independent on the actual exogenous signal) as the purpose of a living system. This purpose is not invariable: its evolution is described with the creode dynamics. Remark A.2. Exogenous signals are parameters of the environment of a living system but are not parts of the system. They vary in time smoothly and are independent of the moments and ( ) of the birth and death of the system. We denote an exogenous signal with and regard it as a vector in the -dimensional Euclidean space, i.e. where. In view of Remark A.2, where is the set of the functions which have values in, and are defined and sufficiently smooth in the entire time axis. Waddington (see [14, p. 22]) suggested to regard the state-space of a living system as Euclidean space, say, ( ) and to describe the system with an ODE of a fairly general form in. The coordinates in the state space can include, for instance, concentrations (or volumetric number densities) of cells or molecules [14, Chapter 2 and Appendix]. We denote with a state vector of a system which may be living and describe the system with ODE, (A.1) where vector is the variable which represents the values of exogenous signals and function has common features, in particular, it is defined and sufficiently smooth in. Moreover, we assume that every solution of (A.1) exists for every. The entire time axis comprises three intervals where the system behaves differently. In intervals and, i.e. before the birth and after the death, the system is not living. It is living in the life interval, so is the system life span. Function in (A.1) is related to the nature of the living system whereas the environment is presented with function. In view of these issues and feature (ii), any trajectory of the living system and, thus, its birth and death moments and must be determined solely in terms of functions and, i.e. as a special solution of ODE (A.1), without involving any initial condition. This leads to the mathematical property formulated as follows. Property A.1. Equation (A.1) at any has the unique solution in, say,, i.e. (A.2) and. This solution describes the only system trajectory corresponding to exogenous signal, no matter if the system is living or nonliving. The trajectory also defines the birth and death moments and because of feature (ii) and the fact that they depend solely on the exogenous signals and the nature of the system. Feature (i) and Remark A.2 imply another assumption.

12 Stochastic Mechanics for Living Systems Page 12 of 16 Property A.2. If Property A.1 holds, then, at any, solution depends on if the system is living, i.e. in the life interval. Remark A.3. Property A.2 remains true if functions are independent of, i.e. ODE (A.1) is autonomous. In this specific respect, solution of autonomous ODE (A.1) in interval can be regarded as self-oscillating in a broad sense. In particular, it may be periodic, quasi-periodic, or almost-periodic. The living-system behaviors of this kind were mentioned by Waddington (see [14, p. 33]). We denote the most favored exogenous signal mentioned in feature (iii) with. It need not coincide with actual exogenous signal and hence is, generally speaking, hypothetical. (As explained in the text below (A.11), it is designed by the living system itself.) In view of feature (iii) and Property A.1, the creode is trajectory. It is the solution of ODE (A.1) under the condition that, i.e.. (A.3) The above issues lead to the following formulation of feature (iv) in the terms based on ODE (A.1). Remark A.4. Let the living system be considered only in a subinterval of the life interval located sufficiently far from both and, i.e. where the life fully performs. In this subinterval, the effects of both the birth and death moments and on any solution (which has Property A.1) can be approximately neglected. As a result, the living state can be regarded as the steady state, i.e. the one defined in the entire axis rather than in a finite interval. This allows to study asymptotic behavior by means of precise relations thereby avoiding use of semi-qualitative inequalities (such as much less or much greater ). Within the above steady-state approximation, one can mathematically formulate homeorhesis (see feature (iv)) as follows. For every sufficiently small, when vector is independent of time and, moreover, is zero. Relations more precisely express the mentioned limit behavior., (A.4) (A.5) It is still unclear in what specific respect exogenous signal involved in (A.5) is the most favored one. To elucidate that, one has to develop the corresponding mathematical formulation. The latter can be done in conjunction with feature (iv) as described in the theorem below. Theorem A.1. Let the assumptions mentioned in the text above Property A.1 and Property A.1 be valid. Let also functions and be such that is sufficiently small. We also assume that the steady-state approximation introduced in Remark A.4 is the case and the homeorhesis conditions (A.4) and (A.5) hold. Then the following assertions are valid.

13 E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 13 of 16 (1) Function is such that is independent of, more specifically. (A.6) (2) Both ODE (A.1) and ODE where (A.7) (A.8) have the unique solution in function set. Proof. Subtracting (A.3) from (A.2), one obtains equation or, equivalently,. (A.9) In view of (A.4) and (A.5), both the left-hand side of (A.9) and the term in the first bracket pair on the right-hand side of (A.9) tend to zero when. Since the term in the second bracket pair on the right-hand side of (A.9) is independent of, it is identically equal to zero, i.e. (A.6) is valid. This proves assertion (1). Application of (A.6) to (A.9) leads to (A.10) where the matrix is described with expression (A.8) which stems from the well-known formula. Assertion (2) follows from (A.10), Property A.1, and the meaning of functional set (see the text below Remark A.2 for the latter). This completes the proof. The limit relations (A.4) and (A.5) in the actual case when the steady-state approximation is not involved and the system is living only in interval can be interpreted as follows. The relaxation times of function associated with both asymptotic representations (A.4) and (A.5) are much less than the life span. Remark A.5. According to Theorem A.1, if ODE (A.1) properly describes a living system, then the hypothesis and all the assertions of the theorem must hold. This result and Property A.2 present a mathematical formulation of the conditions necessary for the living-system ODE-based modelling. They in particular show what features function in (A.1) should have. More research is needed to further specify this function. Remark A.5 summarizes the main result of the present appendix. In connection with this, we note the following two issues. Firstly, relation (A.6) that holds for every sufficiently small can be read as equality, at any sufficiently small. (A.11) and, thus, as the insensitivity of the creode to the variation of values of the exogenous signal (see (A.1)) in a neighborhood of. This fact contributes to the interpretation

14 Stochastic Mechanics for Living Systems Page 14 of 16 below. Theorem A.1. reveals the specific meaning of the term most favored applied to the creode solution or the corresponding exogenous signal (cf., feature (iii)). Indeed, the theorem shows the following. Homeorhesis implies that a living system chooses the hypothetical value of the actual exogenous-signal function such that this value determines the creode as the mode of the system which provides its insensitivity (A.6) (or (A.11)) to the changes of locally, in a neighborhood of. This insensitivity protects the living system from the influences of the environmental variations. In other words, the above ODE vision suggests a specific formulation (and the way to obtain it) for what is known as the wisdom of the body (cf., [26]). Secondly, living systems develop not only in time but also in space. However, the above results can also be interpreted in terms of the space-time evolution if ODE (A.1) is regarded in an appropriate function Banach space rather than in Euclidean space. The key task at any of these settings is a derivation of function in (A.1) which has the proper mathematical properties (cf., Remark A.5) and is informally meaningful, i.e. fully relevant to the living system underlying the model. More research on the ways leading to this function is needed. Even if one can not specify function with all of the aforementioned properties, one can apply assertion (2) of Theorem A.1 as a practical recipe to model a living system, namely: the system is described with ODE (A.7) where matrix is not determined by means of (A.8) but results from one or another approximation which is not related to function used in (A.8); the creode trajectory is extracted from the corresponding experimental data; the actual trajectory is determined as the unique solution of ODE (A.7) in the function set (cf., assertion (2) of the Theorem A.1). The above approach, not fully consistent but reasonably pragmatic, is used in [20] [22] to describe oncogenic hyperplasia and the effect of the related therapy or drugs on the formation and disintegration of hyperplastic tumors. The specific form of ODE (A.7) employed in this approach is where (A.12), (A.13) is the concentration of the cells, is the creode value of,, is the value of corresponding to the hyperplastic tumor, is the parameter of the cell population called the critical concentration of the cells,, is the cell parameter which presents the extent of the cell genotoxicity and which is coupled with, and is the lifetime of a cell at. Parameter is such that is proportional to. The actual cell-population concentration is, according to assertion (2) of Theorem A.1, determined as the unique solution of ODE (A.12) in the function set. If the solution is known at one time point, say

15 E. Mamontov, K. Psiuk-Maksymowicz, and A. Koptioug Page 15 of 16, (A.14) then at any can be evaluated from initial-value problem (A.12), (A.14). Importantly, when, then tends to either the creode or a tumor depending on if or, respectively. Thus, the switching of between sufficiently high and sufficiently low values initiates the formation or disintegration of the tumor (if the latter is formed). Also note that example (A.13) of matrix (A.8) allows to interpret oncogeny as a genotoxically activated (i.e. caused by or ) homeorhetic dysfunction, in a complete agreement with the related results in biomedicine (e.g., [27]). More generally, the form of function (see (A.8)) in ODE (A.7) describes not only the creode but also other stable states (i.e. homeorhetic dysfunctions) which can be approached or left by the living-system trajectory depending on the switching characteristics (such as the above critical concentration ). This important option substantially extends the capabilities of evolution equation (A.1) in modelling various aspects of living systems. Acknowledgement The present work was partly supported by the Instituto Nazionale di Alta Matematica Francesco Severi (Rome, Italy). The authors thanks the European Commission Marie Curie Research Training Network MRTN-CT Modelling, Mathematical Methods and Computer Simulation of Tumour Growth and Therapy ( for the full support of the second author. The authors are very grateful to N. Bellomo, Politecnico di Torino (Turin, Italy), for a series of stimulating and fruitful discussions during the research stay of the first author at Politecnico di Torino in autumn References 1. N. N. Bogolubov and N. N. Bogolubov, Jr., An Introduction to Quantum Statistical Mechanics, Gordon and Breach, Lausanne, (1994). 2. P. Résibois and M. De Leener, Classical Kinetic Theory of Fluids, John Wiley & Sons, New York, (1977). 3. R.Balescu, Statistical Dynamics: Matter out of Equilibrium, Imperial College Press, London, (1997). 4. R. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, John Wiley & Sons, New York, (1998). 5. L. H. Hartwell, J. J. Hopfield, S. Leibler, and A. W. Murray, From molecular to modular cell biology, Nature 402 (6761), C47-C52, (1999). 6. N. Bellomo and P.K. Maini, Preface and the Special Issue Multiscale Cancer Modelling A New Frontier in Applied Mathematics, Math. Models Methods Appl. Sci. 15, iii-viii, (2005). 7. A. Bellouquid and M. Delitala, Mathematical methods and tools of kinetic theory towards modelling complex biological systems, Math. Models Methods Appl. Sci. 15, , (2005). 8. N. Bellomo, A. Bellouquid, and M. Delitala, Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition, Math. Models Methods Appl. Sci. 14 (11), , (2004). 9. M. Willander, E. Mamontov, and Z. Chiragwandi, Modelling living fluids with the subdivision into the components in terms of probability distributions, Math. Models Methods Appl. Sci. 14 (10), , (2004). 10. N. Bellomo, E. Mamontov, and M. Willander, The generalized kinetic modelling of a multicomponent reallife fluid by means of a single distribution function, Mathl. Comput. Modelling 38 (5-6), , (2003). 11. N. Bellomo, A. Bellouquid, and E. De Angelis, The modelling of the immune competition by generalized kinetic (Boltzmann) models: Review and research perspectives, Mathl. Comput. Modelling 37 (1-2), 65-85, (2003). 12. L. Arlotti, N. Bellomo, and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Math. Models Methods Appl. Sci. 12 (4), , (2002).

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