The study of how cells interact to produce tissue development, homeostasis, or diseases was, until recently, almost purely experimental. Now, multi-cell computer simulation methods, ranging from relatively simple cellular automata to complex immersed-boundary and finite-element mechanistic models, allow in silico study of multi-cell phenomena at the tissue scale based on biologically observed cell behaviors and interactions such as movement, adhesion, growth, death, mitosis, secretion of chemicals, chemotaxis, etc. This tutorial introduces the lattice-based Glazier–Graner–Hogeweg (GGH) Monte Carlo multi-cell modeling and the open-source GGH-based CompuCell3D simulation environment that allows rapid and intuitive modeling and simulation of cellular and multi-cellular behaviors in the context of tissue formation and subsequent dynamics. We also present a walkthrough of four biological models and their associated simulations that demonstrate the capabilities of the GGH and CompuCell3D.

A self-propelled particle model is introduced to study cell sorting occurring in some living organisms. This allows us to evaluate the influence of intrinsic cell motility separately from differential adhesion with fluctuations, a mechanism previously shown to be sufficient to explain a variety of cell rearrangement processes. We find that the tendency of cells to actively follow their neighbors greatly reduces segregation time scales. A finite-size analysis of the sorting process reveals clear algebraic growth laws as in physical phase-ordering processes, albeit with unusual scaling exponents. DOI: 10.1103/PhysRevLett.100.248702 PACS numbers: 87.17.ÿd, 05.65.+b, 64.75.ÿg Regeneration after tissue dissociation and reaggregation of some species of sponges, sea urchins, and hydras has been investigated in a series of experiments [1]. These phenomena involve cell sorting: Initially mixed cells form clusters, similarly to domain growth processes in physics. To explain cell sorting, Steinberg [2] proposed the differential adhesion hypothesis (DAH), which postulates that local rearrangements depend on the adhesion and motility properties of the different types of cells involved.Experiments have verified some of the DAH postulates. Surface tension measurements in chicken embryonic tissues [3] and determination of adhesive forces between pairs of Hydra cells [4] confirmed the relative magnitudes necessary to favor tissue envelopment: Ectodermic cells are less cohesive than endodermic cells. It was also shown that random membrane ruffling induces effective cell motion [5]. However, the coherent component of cellular flow must also be considered, as in viscoelastic fluids [6]. Remarkable experiments on Hydra performed in two spatial dimensions by Rieu et al. [7,8] discriminate between random and coherent motion contributions to cell segregation, showing that, during aggregate rounding or sorting, endodermic cell dynamics is dominated by the coherent behavior of ectodermic cells. Also, short-range spatial correlations corresponding to parallel displacement of adjacent cells were observed. The diffusive properties of cells were also measured. In all cases, diffusion was found to be normal only beyond a trapping time during which cells keep the same neighbors. This time is smaller for endodermic cells immersed in the ectoderm.Cell-sorting phenomena, such as Hydra regeneration from random cellular aggregates, have been simulated in the seminal work by Graner and Glazier in 1992 [9], later extended by Hogeweg and co-workers [10] (hereafter the GGH model). In the GGH model, cells are represented on a site-labeled lattice. A connected group of sites with the same label stands for a cell, and an energy function accounts for surface tension between adjacent cells while keeping the cell sizes fluctuating around specified targets.Monte Carlo simulations showed that sorting may occur, with less adhesive cells engulfing the more adhesive ones. However, this type of approach cannot easily account for locally coherent active cell mo...

Binaries and source code for Microsoft Windows, Linux and Solaris are available for download from http://sourceforge.net/projects/compucell/

The influence of surface tension and size on rounding of cell aggregates are studied using chick embryonic cells and numerical simulations based on the cellular Potts model. Our results show exponential relaxation in both cases as verified in previous studies using 2D Hydra cell aggregates. The relaxation time decreases with higher surface tension as expected from hydrodynamics laws. However, it increases faster than linearly with aggregate size. The results provide an additional support to the validity of the cellular Potts model for non-equilibrium situations and indicate that aggregate shape relaxation is not governed by the hydrodynamics of viscous liquids

Tumor cells and structure both evolve due to heritable variation of cell behaviors and selection over periods of weeks to years (somatic evolution). Micro-environmental factors exert selection pressures on tumor-cell behaviors, which influence both the rate and direction of evolution of specific behaviors, especially the development of tumor-cell aggression and resistance to chemotherapies. In this paper, we present, step-by-step, the development of a multi-cell, virtual-tissue model of tumor somatic evolution, simulated using the open-source CompuCell3D modeling environment. Our model includes essential cell behaviors, microenvironmental components and their interactions. Our model provides a platform for exploring selection pressures leading to the evolution of tumor-cell aggression, showing that emergent stratification into regions with different cell survival rates drives the evolution of less cohesive cells with lower levels of cadherins and higher levels of integrins. Such reduced cohesivity is a key hallmark in the progression of many types of solid tumors.

The actions of cell adhesion molecules, in particular, cadherins during embryonic development and morphogenesis more generally, regulate many aspects of cellular interactions, regulation and signaling. Often, a gradient of cadherin expression levels drives collective and relative cell motions generating macroscopic cell sorting. Computer simulations of cell sorting have focused on the interactions of cells with only a few discrete adhesion levels between cells, ignoring biologically observed continuous variations in expression levels and possible nonlinearities in molecular binding. In this paper, we present three models relating the surface density of cadherins to the net intercellular adhesion and interfacial tension for both discrete and continuous levels of cadherin expression. We then use then the Glazier-Graner-Hogeweg (GGH) model to investigate how variations in the distribution of the number of cadherins per cell and in the choice of binding model affect cell sorting. We find that an aggregate with a continuous variation in the level of a single type of cadherin molecule sorts more slowly than one with two levels. The rate of sorting increases strongly with the interfacial tension, which depends both on the maximum difference in number of cadherins per cell and on the binding model. Our approach helps connect signaling at the molecular level to tissue-level morphogenesis.

We present a stochastic cellular automaton model for the behavior of limb bud precartilage mesenchymal cells undergoing chondrogenic patterning. This "agent-oriented" model represents cells by points on a lattice that obey rules motivated by experimental findings. The "cells" follow these rules as autonomous agents, interacting with other cells and with the microenvironments cell activities produce. The rules include random cell motion, production and lateral deposition of a substrate adhesion molecule (SAM, corresponding to fibronectin), production and release of a diffusible growth factor ("activator," corresponding to TGF-beta) that stimulates production of the SAM, and another diffusible factor ("inhibitor") that suppresses the activity of the activator. We implemented the cellular automaton on a two-dimensional (2D) square lattice to emulate the quasi-2D micromass culture extensively used to study patterning in avian limb bud precartilage cells. We identified parameters that produce nodular patterns that resemble, in size and distribution, cell condensations in leg-cell cultures, thus establishing a correspondence between in vitro and in silico results. We then studied the in vitro and in silico micromass cultures experimentally. We altered the standard in vitro micromass culture by diluting the initial cell density, transiently exposing it to exogenous activator, suppressing the inhibitor, and constitutively activating fibronectin production. We altered the standard in silico micromass culture in each case by changing the corresponding parameter. In vitro and in silico experiments agreed well. We also used the model to test hypotheses for differences in the in vitro patterns of cells derived from chick embryo forelimb and hindlimb. We discuss the applicability of this model to limb development in vivo and to other organ development.

We perform extensive Potts model simulations of three-dimensional dry foam coarsening. Starting with 2.25 million bubbles, we have enough statistics to fulfill the three constraints required for the study of statistical scale invariance: first, enough time for the transient to end and reach the scaling state; then, enough time in the scaling state itself to characterize its properties; and finally, enough bubbles at the end to avoid spurious finite size effects. In the scaling state, we find that the average surface area of the bubbles increases linearly with time. The geometry ͑bubble shape and size͒ and topology ͑number of faces and edges͒, as well as their correlations, become constant in time. Their distributions agree with the data of the literature. We present an analytical model ͑universal, up to parameters extracted from the simulations͒ for a disordered foam minimizing its free energy, which agrees with the simulations. We discuss the limitations of the simulations and of the model.