from a discrete to a continuous model of biological cell movement the process by which one may take a discrete model of a biophysical process and construct a continuous model based upon it is of mathematical interest as well as being of practical use in this work we take the extended potts model applied to biological cell movement to its continuous limit beginning with a single cell moving in one dimension on a lattice and obeying potts model rules of movement we develop an expression for the diffusion coefficient of a collection of noninteracting cells which depends explicitly on the potts model parameters we show how this coefficient varies when the potts parameters for cell membrane elasticity and cell medium adhesion are varied and perform computer simulations which support our theoretical result we explain the relationship between the probability of occupancy of lattice points and the density profile in the continuous limit and extend our analysis by including interactions between the cells in so doing we are able to develop a set of coupled ordinary differential equations showing the evolution of a density profile in the presence of significant cell cell adhesion and show how increases in the strength of this adhesion modulates diffusion in so doing we develop some insights into how continuous models of physical systems can be based upon discrete models which describe the same system

multiscale dynamics of biological cells with chemotactic interactions from a discrete stochastic model to a continuous description the cellular potts model cpm has been used for simulating various biological phenomena such as differential adhesion fruiting body formation of the slime mold dictyostelium discoideum angiogenesis cancer invasion chondrogenesis in embryonic vertebrate limbs and many others we derive a continuous limit of a discrete one dimensional cpm with the chemotactic interactions between cells in the form of a fokker planck equation for the evolution of the cell probability density function this equation is then reduced to the classical macroscopic keller segel model in particular all coefficients of the keller segel model are obtained from parameters of the cpm theoretical results are verified numerically by comparing monte carlo simulations for the cpm with numerics for the keller segel model

myopic random walkers and exclusion processes single and multispecies abstract a motility mechanism based on a simple exclusion process where the probability of movement of an agent depends on the number of unoccupied nearest neighbor sites is considered such interacting agents are termed myopic this problem is an extension of the famous blind or myopic ant in a labyrinth problem for the interacting agent models considered here each agent plays the role of an ant in a labyrinth where the paths of allowed sites though the labyrinth consist of sites not occupied by other agents we derive a nonlinear diffusion equation for the average occupancy of the discrete agent based model for myopic agents in contrast interacting blind agents have a constant probability of movement to each of their nearest neighbor sites giving rise to a linear diffusion equation insight into the various terms in the nonlinear diffusion coefficient is obtained from a study of multiple subpopulations of interacting myopic agents where an advection diffusion equation for each subpopulation is derived and from tracking an individual agent within the crowd where a motility coefficient is extracted averaged discrete simulation data compares very well with the solution to the continuum models we also compare the behavior of myopic and blind agents the myopic motility mechanism is biologically motivated to emulate information an individual cell gathers from environment cues the multispecies model developed and investigated here assists with the interpretation of experimental data involving the tracking subpopulations of cells within a total cell population

building macroscale models from microscale probabilistic models a general probabilistic approach for nonlinear diffusion and multispecies phenomena a discrete agent based model on a periodic lattice of arbitrary dimension is considered agents move to nearest neighbor sites by a motility mechanism accounting for general interactions which may include volume exclusion the partial differential equation describing the average occupancy of the agent population is derived systematically a diffusion equation arises for all types of interactions and is nonlinear except for the simplest interactions in addition multiple species of interacting subpopulations give rise to an advection diffusion equation for each subpopulation this work extends and generalizes previous specific results providing a construction method for determining the transport coefficients in terms of a single conditional transition probability which depends on the occupancy of sites in an influence region these coefficients characterize the diffusion of agents in a crowded environment in biological and physical processes

tumor growth instability and the onset of invasion motivated by experimental observations we develop a mathematical model of chemotactically directed tumor growth we present an analytical study of the model as well as a numerical one the mathematical analysis shows that i tumor cell proliferation by itself cannot generate the invasive branching behavior observed experimentally ii heterotype chemotaxis provides an instability mechanism that leads to the onset of tumor invasion and iii homotype chemotaxis does not provide such an instability mechanism but enhances the mean speed of the tumor surface the numerical results not only support the assumptions needed to perform the mathematical analysis but they also provide evidence of i ii and iii finally both the analytical study and the numerical work agree with the experimental phenomena

persistent exclusion processes inertia drift mixing and correlation in many biological systems motile agents exhibit random motion with short term directional persistence together with crowding effects arising from spatial exclusion we formulate and study a class of lattice based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions and use a mean field approximation to investigate relevant population level partial differential equations in the continuum limit we show that this model of a persistent exclusion process is in general well described by a nonlinear diffusion equation with reference to results presented in the current literature our results reveal that the nonlinearity arises from the combination of motion persistence and volume exclusion with linearity in terms of the canonical diffusion or heat equation being recovered in either the case of persistence without spatial exclusion or spatial exclusion without persistence we generalize our results to include systems of multiple species of interacting motion persistent walkers as well as to incorporate a global drift in addition to persistence these models are shown to be governed approximately by systems of nonlinear advection diffusion equations by comparing the prediction of the mean field approximation to stochastic simulation results we assess the performance of our results finally we also address the problem of inferring the presence of persistence from simulation results with a view to application to experimental cell imaging data

the np completeness of a tomographical problem on bicolored domino tilings in this paper we study the problem of reconstructing a bicolored domino tiling of a rectangular surface from its horizontal and vertical projections we use a reduction from the np complete problem numerical matching with target sums in order to prove that unless p np this task cannot be performed in polynomial time the reconstruction of monochromatic domino tilings is still open

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

field theoretic approach to metastability in the contact process a quantum field theoretic formulation of the dynamics of the contact process on a regular graph of degree z is introduced a perturbative calculation in powers of 1 z of the effective potential for the density of particles phi t and an instantonic field psi t emerging from the quantum formalism is performed corrections to the mean field distribution of densities of particles in the out of equilibrium stationary state are derived in powers of 1 z results for typical e g average density and rare fluctuation e g lifetime of the metastable state properties are in very good agreement with numerical simulations carried out on d dimensional hypercubic z 2d and cayley lattices

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

scaling limits for interacting diffusions we consider a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance we derive a nonlinear bulk diffusion equation for the density of the particle distribution on the circle the diffusion coefficient is determined as a function of density in terms of standard thermodynamical objects

universal renormalization group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics we uncover the dynamics at the chaos threshold m of the logistic map and find that it consists of trajectories made of intertwined power laws that reproduce the entire period doubling cascade that occurs for m m we corroborate this structure analytically via the feigenbaum renormalization group rg transformation and find that the sensitivity to initial conditions has precisely the form of a q exponential of which we determine the q index and the q generalized lyapunov coefficientl q our results are an unequivocal validation of the applicability of the nonextensive generalization of boltzmann gibbs statistical mechanics to critical points of nonlinear maps critical points of nonlinear maps offer a suitable playground for testing the validity of the nonextensive generalization of the boltzmann gibbs bg statistical mechanics proposed by tsallis over a decade ago 1 2 here we describe universal properties related to the dynamics of iterates at the onset of chaos in unimodal maps 3 which provide a literal confirmation of the generalized nonextensive theory to this end we employ the celebrated one dimensional logistic map f m x 512muxu 2 21 x 1 and the properties of its renormalization group rg fixed point to present evidence of previously unexposed scaling properties at the onset of chaos m5m at this state the most prominent of the map critical points the trajectories of the iterates exhibit an intricate structure that we describe and show is governed by the feigenbaum rg transformation 3 the domain of validity of bg statistical mechanics has been implicitly challenged by the proposal of its nonextensive generalization subsequent studies have offered experimental and numerical evidence that point out both the inadequacy of the standard bg statistics and the plausible competence of the generalized theory in describing various types of phenomena and systems this theoretical development represents an exceptional event in the long and trustworthy history of bg statistical mechanics however it is still in the process of being converted into a rigorously corroborated and fully understood fact the suggested circumstances under which the generalized theory is believed to be applicable at least with regards to nonlinear dynamical systems are those associated to a phase space with power law sensitivity to initial conditions to the consequent vanishing of the largest lyapunov exponent and to a fractal or multifractal geometrical structure 2 here we show that our results for the dynamics at the onset of chaos in unimodal maps constitute an unequivocal proof of the universal validity of the nonextensive statistics at such critical points in fact at the chaos threshold as well as at other critical points of the map the lyapunov exponent l 1 vanishes and the sensitivity to initial conditions j t for large iteration time t ceases to obey exponential behavior exhibiting instead a power law behavior 4 in order to describe the dynamics at such critical points the q exponential expression

lattice free models of directed cell motility directed cell migration often occurs when individual cells move in response to an external chemical stimulus cells can respond by moving in either the direction of increasing chemoattraction or decreasing chemorepulsion concentration many previous models of directed cell migration use a lattice based framework where agents undergo a lattice based random walk and the direction of nearest neighbour motility events is biased in a preferred direction such lattice based models can lead to unrealistic configurations of agents since the agents always move on an artificial lattice structure which is never observed experimentally we present a lattice free model of directed cell migration that incorporates two key features first agents move on a continuous domain with the possibility that there is some preferred direction of motion second to be consistent with experimental observations we enforce a crowding mechanism so that motility events that would lead to agent overlap are not permitted we compare simulation data from the new lattice free model with a more traditional lattice based model to provide additional insight into the lattice free model we construct an approximate conservation statement which corresponds to a nonlinear advection diffusion equation in the continuum limit the solution of this mean field model compares well with averaged data from the individual based model

models of collective cell spreading with variable cell aspect ratio a motivation for degenerate diffusion models continuum diffusion models are often used to represent the collective motion of cell populations most previous studies have simply used linear diffusion to represent collective cell spreading while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles in the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations furthermore there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate here we provide a link between individual based and continuum models using a multi scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice based exclusion process for round agents that occupy a single lattice site we find that the relevant continuum description of the system is a linear diffusion equation whereas for elongated rod shaped agents that occupy l adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation pme the exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles results suggest that when using continuum models to represent cell population spreading we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models

modeling tumor cell migration from microscopic to macroscopic models it has been shown experimentally that contact interactions may influence the migration of cancer cells previous works have modelized this thanks to stochastic discrete models cellular automata at the cell level however for the study of the growth of real size tumors with several million cells it is best to use a macroscopic model having the form of a partial differential equation pde for the density of cells the difficulty is to predict the effect at the macroscopic scale of contact interactions that take place at the microscopic scale to address this we use a multiscale approach starting from a very simple yet experimentally validated microscopic model of migration with contact interactions we derive a macroscopic model we show that a diffusion equation arises as is often postulated in the field of glioma modeling but it is nonlinear because of the interactions we give the explicit dependence of diffusivity on the cell density and on a parameter governing cell cell interactions we discuss in detail the conditions of validity of the approximations used in the derivation and we compare analytic results from our pde to numerical simulations and to some in vitro experiments we notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses supercooled liquids and jamming systems

nonlinear diffusion and exclusion processes with contact interactions exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level the average properties of an exclusion process may be described by a continuum model given by a partial differential equation we combine a general class of contact interactions with an exclusion process we determine that many different types of contact interactions at the agent level always give rise to a nonlinear diffusion equation with a vast variety of diffusion functions d c we find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models while strong interactions often show discrepancies between the two particularly when d c takes on negative values we present a measure to predict the goodness of fit between the discrete and continuous model and thus the validity of the continuum description of a motile contact interacting population of agents this work has implications for modeling cell motility and interpreting cell motility assays giving the ability to incorporate biologically realistic cell cell interactions and develop global measures of discrete microscopic data

distinguishing between mean field moment dynamics and stochastic descriptions of birth death movement processes abstract mathematical descriptions of birth death movement processes are often calibrated to measurements from cell biology experiments to quantify tissue growth rates here we describe and analyze a discrete model of a birth death movement process applied to a typical two dimensional cell biology experiment we present three different descriptions of the system i a standard mean field description which neglects correlation effects and clustering ii a moment dynamics description which approximately incorporates correlation and clustering effects and iii averaged data from repeated discrete simulations which directly incorporates correlation and clustering effects comparing these three descriptions indicates that the mean field and moment dynamics approaches are valid only for certain parameter regimes and that both these descriptions fail to make accurate predictions of the system for sufficiently fast birth and death rates where the effects of spatial correlations and clustering are sufficiently strong without any method to distinguish between the parameter regimes where these three descriptions are valid it is possible that either the mean field or moment dynamics model could be calibrated to experimental data under inappropriate conditions leading to errors in parameter estimation in this work we demonstrate that a simple measurement of agent clustering and correlation based on coordination number data provides an indirect measure of agent correlation and clustering effects and can therefore be used to make a distinction between the validity of the different descriptions of the birth death movement process

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

models of collective cell motion for cell populations with different aspect ratio diffusion proliferation and travelling waves continuum partial differential equation models are often used to describe the collective motion of cell populations with various types of motility represented by the choice of diffusion coefficient and cell proliferation captured by the source terms previously the choice of diffusion coefficient has been largely arbitrary with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual level properties of the cell motility mechanism in this work we provide a new link between individual level models which account for important cell properties such as varying cell shape and volume exclusion and population level partial differential equation models we work in an exclusion process framework considering aligned elongated cells that may occupy more than one lattice site in order to represent populations of agents with different sizes three different idealisations of the individual level mechanism are proposed and these are connected to three different partial differential equations each with a different diffusion coefficient one linear one nonlinear and degenerate and one nonlinear and nondegenerate we test the ability of these three models to predict the population level response of a cell spreading problem for both proliferative and nonproliferative cases we also explore the potential of our models to predict long time travelling wave invasion rates and extend our results to two dimensional spreading and invasion our results show that each model can accurately predict density data for nonproliferative systems but that only one does so for proliferative systems hence great care must be taken to predict density data with varying cell shape