Homeorhesis and evolutionary properties of living systems: From ordinary differential equations to the active-particle generalized kinetics theory

• Advanced generalized-kinetic-theory (GKT) models for biological systems are developed for populations of active (or living) particles [1]-[5]. These particles are described with both the stochastic variables common in kinetic theory (such as time, the particle random location and velocity) and the stochastic variables related to the internal states of an active particle. Evolution of these states represents biological, ecological, or social properties of the particle behavior. Paper [6] analyzes a number of the well-known statistical-mechanics approaches and shows that the active-particle GKT (APGKT) is the only treatment capable of modelling living systems. Work [2] summarizes the significance of the notion of an active particle in kinetic models. This notion draws attention to the features distinguishing living matter from nonliving matter. They are discussed by many authors (e.g., [7]-[15], [1]-[3], [6], [16]-[18]). Work [11] considers a lot of differences between living and nonliving matters, and the limitations of the modelling approaches developed for nonliving matter. Work [6] mainly focuses on the comparison of a few theoretical mechanics treatments in terms of the key living-matter properties formulated in [15]. One of the necessary properties of the evolution of living systems is homeorhesis. It is, loosely speaking, a peculiar qualitative and quantitative insensitivity of a living system to the exogenous signals acting on it. The earlier notion, homeostasis, was introduced by W. B. Cannon in 1926 who discussed the phenomenon in detail later [7]. Homeorhesis introduced by C. H. Waddington [8, p. 32] generalizes homeostasis and is well known in biology [8], [9], [12]. It is an inherent part of mathematical models for oncogeny (e.g., [16]-[18], [6, Appendix]). Homeorhesis is also discussed in [3, Section 4] in connection with APGKT. Homeorhesis is documented in ecology (e.g., [11], [13, the left column on p. 675]) where it is one of the key notions of the strong Gaia theory, a version of the Gaia theory (e.g., [14, Chapter 8]). The strong Gaia theory “states that the planet with its life, a single living system, is regulated in certain aspects by that life” [14, p. 124]. The very origin of the name “Gaia” is related to homeorhesis or homeostasis [14, p. 118]. These notions are also used in psychology and sociology. If evolution of a system is not homeorhetic, the system can not be living. Work [6, Appendix] derives a preliminary mathematical formulation of homeorhesis in terms of the simplest dynamical systems, i.e. ordinary differential equations (ODEs). The present work complements, extended, and further specify the approach of [6, Appendix]. The work comprises the two main parts. The first part develops the sufficient conditions for ODE systems to describe homeorhesis, and suggests a fairly general structure of the ODE model. It regards homeorhesis as piecewise homeostasis. The model can be specified in different ways depending on specific systems and specific purposes of the analysis. An example of the specification is also noted (the PhasTraM nonlinear reaction-diffusion model for hyperplastic oncogeny [16]-[18]). The second part of the work discusses implementation of the above homeorhesis ODE model in terms of a special version [3] of APGKT (see above). The key feature of this version is that the components of a living population need not be discrete: the subdivision into the components is described with a general, continuous-discrete probability distribution (see also [6]). This enables certain properties of living matter noted in [15]. Moreover, the corresponding APGKT model presents a system of, firstly, a generalized kinetic equation for the conditional distribution function conditioned by the internal states of the population and, secondly, Ito's stochastic differential equations for these states. This treatement employs the results on nonstationary invariant diffusion stochastic processes [19]. The second part of the work also stresses that APGKT is substantially more important for the living-matter analysis than in the case of nonliving matter. One of the reasons is certain limitations in experimental sampling of the living-system modes presented with stochastic processes. A few directions for future research are suggested as well. REFERENCES: [1] Bellomo, N., Bellouquid, A. and Delitala, M., 2004, Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition, Math. Models Methods Appl. Sci., 14, 1683-1733. [2] Bellomo, N., 2006, New hot Paper Comments, Essential Science Indicators, http://www.esi-topics.com/nhp/2006 /may- 06-NicolaBellomo.html. [3] Willander, M., Mamontov, E. and Chiragwandi, Z., 2004, Modelling living fluids with the subdivision into the components in terms of probability distributions, Math. Models Methods Appl. Sci. 14, 1495-1520. [4] Bellomo, N. and Maini, P.K., 2005, Preface and the Special Issue “Multiscale Cancer Modelling-A New Frontier in Applied Mathematics”, Math. Models Methods Appl. Sci., 15, iii-viii. [5] De Angelis, E. and Delitala, M., 2006, Modelling complex systems in applied sciences: Methods and tools of the mathematical kinetic theory for active particles. Mathl Comput. Modelling, 43, 1310-1328. [6] Mamontov, E., Psiuk-Maksymowicz, K. and Koptioug, A., 2006, Stochastic mechanics in the context of the properties of living systems, Mathl Comput. Modelling, Article in Press, 13 pp. [7] Cannon, W.B., 1932, The Wisdom of the Body (New York: Norton). [8] Waddington, C.H., 1957, The Strategy of the Genes. A Discussion of Some Aspects of Theoretical Biology (London, George Allen and Unwin). [9] Waddington, C.H., 1968, Towards a theoretical biology, Nature, 218, 525-527. [10] Cotnoir, P.-A., 1981, La compétence environnementale: Une affaire d’adaptation. Séminaire en écologie behaviorale, Univeristé du Québec, Montralé. Available online at: http://pac.cam.org/culture.doc . [11] O’Neill, R.V., DeAngelis, D.L., Waide, J.B. and Allen, T.F.H., 1986, A Hierarchical Concept of Ecosystems, Princeton: Princeton Univ. Press). [12] Sauvant, D., 1992, La modélisation systémique en nutrition, Reprod. Nutr. Dev., 32, 217-230. [13] Christensen, N.L., Bartuska, A.M., Brown, J.H., Carpenter, S., D'Antonio, C., Francis, R., Franklin, J.F., MacMahon, J.A., Noss, R.F., Parsons, D.J., Peterson, C.H., Turner, M.G. and Woodmansee, R.G., 1996, The Report of the Ecological Society of America Committee on the Scientific Basis for Ecosystem Management, Ecological Applications, 6, 665-691. Available online at: http://www.esa.org/pao/esaPositions/Papers/ReportOfSBEM.php. [14] Margulis, L., 1998, Symbiotic Planet. A New Look at Evolution (Amherst: Sciencewriters). [15] Hartwell, L.H., Hopfield, J.J., Leibler, S. and Murray, A.W., 1999, From molecular to modular cell biology, Nature, 402, C47-C52. [16] Mamontov, E., Koptioug, A.V. and Psiuk-Maksymowicz, K., 2006, The minimal, phase-transition model for the cell- number maintenance by the hyperplasia-extended homeorhesis, Acta Biotheoretica, 54, 44 pp., (no. 2, May-June, accepted). [17] Psiuk-Maksymowicz, K. and Mamontov, E., 2005, The time-slices method for rapid solving the Cauchy problem for nonlinear reaction-diffusion equations in the competition of homeorhesis with genotoxically activated hyperplasia, In: European Conference on Mathematical and Theoretical Biology - ECMTB05 (July 18-22, 2005) Book of Abstracts, Vol.1 (Dresden: Center for Information Services and High Performance Computing, Dresden Univ. Technol.), p. 429 (http://www.ecmtb05.org/). [18] Psiuk-Maksymowicz, K. and Mamontov, E., 2006, The homeorhesis-based modelling and fast numerical analysis for oncogenic hyperplasia under radiation therapy, submitted. [19] Mamontov, E., 2005, Nonstationary invariant distributions and the hydrodynamic-style generalization of the Kolmogorov-forward/Fokker-Planck equation, Appl. Math. Lett. 18 (9) 976-982.

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