Modelling Chemotactic Motion of Cells in Biological Tissues

Developmental processes in biology are underlined by proliferation, differentiation and migration of cells. The latter two are interlinked since cellular differentiation is governed by the dynamics of morphogens which, in turn, is affected by the movement of cells. Mutual effects of morphogenetic and cell movement patterns are enhanced when the movement is due to chemotactic response of cells to the morphogens. In this study we introduce a mathematical model to analyse how this interplay can result in a steady movement of cells in a tissue and associated formation of travelling waves in a concentration field of morphogen. Using the model we have identified four chemotactic scenarios for migration of single cell or homogeneous group of cells in a tissue. Such a migration can take place if moving cells are (1) repelled by a chemical produced by themselves or (2) attracted by a chemical produced by the surrounding cells in a tissue. Furthermore, the group of cells can also move if cells in surrounding tissue are (3) repelled by a chemical produced by moving cells or (4) attracted by a chemical produced by surrounding cells themselves. The proposed mechanisms can underlie migration of cells during embryonic development as well as spread of metastatic cells

Temperature expressions and ergodicity of the Nosé-Hoover deterministic schemes

Thermostats are dynamic equations used to model thermodynamic variables in molecular dynamics. The applicability of thermostats is based on the ergodic hypothesis. The most commonly used thermostats are designed according to the Nos\'e-Hoover scheme, although it is known that it often violates ergodicity. Here, following a method from our recent study \citep{SamoletovVasiev2017}, we have extended the classic Nos\'e-Hoover scheme with an additional temperature control tool. However, as with the NH scheme, a single thermostat variable is used. In the present study we analyze the statistical properties of the modified equations of motion with an emphasis on ergodicity. Simultaneous thermostatting of all phase variables with minimal extra computational costs is an advantage of the specific theoretical scheme presented here

A statistical interpretation of the logistic equation

The logistic equation is one of the established paradigms in modelling population growth. Here we propose a statistical interpretation of the logistic equation. This interpretation is based on modelling the population-environment relationship, the mathematical theory of which we discuss in detail. By applying this theory, we obtain stochastic evolutionary equations, for which the logistic equation is a limiting case. The prospect of modifying logistic population growth is discussed.Comment: 12 pages, 4 figure

Stochastic thermostats and temperature expressions

Abstract Molecular dynamics (MD) is in the core of fundamental research for a range of disciplines in natural sciences and is known for its applications in the design of new functional materials and the drug discovery. MD simulations are performed under certain thermodynamic conditions, typically at fixed temperature and pressure. The thermodynamic variables in the MD are modeled using equations that are called thermostats. Many different thermostats have been proposed. Recently (Samoletov A and Vasiev B 2017 J. Chem. Phys. 147 204106), we have shown that a range of thermostats can be derived in the framework of a unified approach based on the fundamental principles of statistical physics, so that the relevant dynamic schemes are based on the concept of temperature expression (in short, ϑ-expression). However, only a few specific ϑ-expressions have been used so far and reported in the literature. In this paper, we are using a wider set of ϑ-expressions and their mathematical properties that allow us to modify the known and offer new thermostats with improved computational efficiency and ergodicity. We focus on the Nosé-Hoover-Langevin stochastic scheme and extend it with additional temperature control tools. Simultaneous thermostatting of all phase space variables with minimal additional computational costs is an advantage of the modified dynamics.</jats:p

Coordination of Cell Differentiation and Migration in Mathematical Models of Caudal Embryonic Axis Extension

Vertebrate embryos display a predominant head-to-tail body axis whose formation is associated with the progressive development of post-cranial structures from a pool of caudal undifferentiated cells. This involves the maintenance of active FGF signaling in this caudal region as a consequence of the restricted production of the secreted factor FGF8. FGF8 is transcribed specifically in the caudal precursor region and is down-regulated as cells differentiate and the embryo extends caudally. We are interested in understanding the progressive down-regulation of FGF8 and its coordination with the caudal movement of cells which is also known to be FGF-signaling dependent. Our study is performed using mathematical modeling and computer simulations. We use an individual-based hybrid model as well as a caricature continuous model for the simulation of experimental observations (ours and those known from the literature) in order to examine possible mechanisms that drive differentiation and cell movement during the axis elongation. Using these models we have identified a possible gene regulatory network involving self-repression of a caudal morphogen coupled to directional domain movement that may account for progressive down-regulation of FGF8 and conservation of the FGF8 domain of expression. Furthermore, we have shown that chemotaxis driven by molecules, such as FGF8 secreted in the stem zone, could underlie the migration of the caudal precursor zone and, therefore, embryonic axis extension. These mechanisms may also be at play in other developmental processes displaying a similar mode of axis extension coupled to cell differentiation

On the heterogeneity of human populations as reflected by mortality dynamics

The heterogeneity of human populations is a common consideration in describing and validating their various age-related features. Heterogeneity, in particular, amongst other factors, is used to explain the variability of mortality rates across the lifespan and deviations from an exponential growth at young and very old ages. A mathematical model that combines the population heterogeneity with the assumption that the mortality of each constituent subpopulation increases exponentially with age, has recently been shown to successfully reproduce the entire mortality pattern across the lifespan as well as its evolution over time. Furthermore, the analysis of time-evolution of the mortality pattern, performed by fitting the model to actual data of consecutive periods, confirms the applicability of the compensation law of mortality to each subpopulation and concludes on the evolution of the population towards homogenisation. In this work we aim to show that the heterogeneity of human populations is not only a convenient consideration for fitting mortality data but is indeed the actual structure of the population as reflected by the dynamics of its mortality over age and time. In particular, we demonstrate that the model of heterogeneous populations fits mortality data better than most of the other models if the data are taken for the entire lifespan and better than all other models if we consider only old ages. Also, we show that the model can reproduce seemingly contradicting observations in late-life mortality dynamics namely deceleration, levelling-off and mortality decline. Assuming that heterogeneity is reflected in genetic variations within the population, using Swedish mortality data for 20th century we show that the homogenisation of the population, observed in the model fits, can be associated with the evolution of allele frequencies

Self-organized Vortex State in Two-dimensional Dictyostelium Dynamics

We present results of experiments on the dynamics of Dictyostelium discoideum in a novel set-up which constraints cell motion to a plane. After aggregation, the amoebae collect into round ''pancake" structures in which the cells rotate around the center of the pancake. This vortex state persists for many hours and we have explicitly verified that the motion is not due to rotating waves of cAMP. To provide an alternative mechanism for the self-organization of the Dictyostelium cells, we have developed a new model of the dynamics of self-propelled deformable objects. In this model, we show that cohesive energy between the cells, together with a coupling between the self-generated propulsive force and the cell's configuration produces a self-organized vortex state. The angular velocity profiles of the experiment and of the model are qualitatively similar. The mechanism for self-organization reported here can possibly explain similar vortex states in other biological systems.Comment: submitted to PRL; revised version dated 3/8/9