COMPUCELL, a multi-model framework for simulation of morphogenesis

MOTIVATION

CompuCell is a multi-model software framework for simulation of the development of multicellular organisms known as morphogenesis. It models the interaction of the gene regulatory network with generic cellular mechanisms, such as cell adhesion, division, haptotaxis and chemotaxis. A combination of a state automaton with stochastic local rules and a set of differential equations, including subcellular ordinary differential equations and extracellular reaction-diffusion partial differential equations, model gene regulation. This automaton in turn controls the differentiation of the cells, and cell-cell and cell-extracellular matrix interactions that give rise to cell rearrangements and pattern formation, e.g. mesenchymal condensation. The cellular Potts model, a stochastic model that accurately reproduces cell movement and rearrangement, models cell dynamics. All these models couple in a controllable way, resulting in a powerful and flexible computational environment for morphogenesis, which allows for simultaneous incorporation of growth and spatial patterning.

RESULTS

We use CompuCell to simulate the formation of the skeletal architecture in the avian limb bud.

AVAILABILITY

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

Modeling the interplay of generic and genetic mechanisms in cleavage, blastulation, and gastrulation

Early development of multicellular organisms is marked by a rapid initial increase in their cell numbers, accompanied by spectacular morphogenetic processes leading to the gradual formation of organs of characteristic shapes. During morphogenesis, through differentiation under strict genetic control, cells become more and more specialized. Morphogenesis also requires coordinated cell movement and elaborate interactions between cells, governed by fundamental physical or generic principles. As a consequence, early development must rely on an intricate interplay of generic and genetic mechanisms. We present the results of computer simulations of the first nontrivial morphogenetic transformations in the life of multicellular organisms: initial cleavages, blastula formation, and gastrulation. The same model, which is based on the physical properties of individual cells and their interactions, describes all these processes. The genetic code determines the values of the model parameters. The model accurately reproduces the major steps of early development. It predicts that physical constraints strongly influence the timing of gastrulation. Gastrulation must occur prior to the appearance of dynamical instability, which would destabilize and eventually derail normal development. Within our model, to avoid the instability, we suddenly change the values of some of the model parameters. We interpret this change as a consequence of specific gene activity. After changing the physical characteristics of some cells, normal development resumes, and gastrulation proceeds.

Cell sorting is analogous to phase ordering in fluids

Morphogenetic processes, like sorting or spreading of tissues, characterize early embryonic development. An analogy between viscoelastic fluids and certain properties of embryonic tissues helps interpret these phenomena. The values of tissue-specific surface tensions are consistent with the equilibrium configurations that the Differential Adhesion Hypothesis predicts such tissues reach after sorting and spreading. Here we extend the fluid analogy to cellular kinetics. The same formalism applies to recent experiments on the kinetics of phase ordering in two-phase fluids. Our results provide biologically relevant information on the strength of binding between cell adhesion molecules under near-physiological conditions.

Trafficking and signaling through the cytoskeleton: a specific mechanism

A specific mechanism for the intracellular translocation of nonvesicle-associated proteins is proposed. This movement machinery is based on the assumption that the cytoskeleton represents an interconnected network of filamentous macromolecules, which extends over the entire cytoplasm. Diffusion along the filaments provides an efficient way for movement and with this, for signal transduction, between various intracellular compartments. We calculate the First Passage Time (FPT), the average time it takes a signaling molecule, diffusing along the cytoskeleton, to arrive from the cell surface to the nucleus for the first time. We compare our results with the FPT of free diffusion and of diffusion in the permeating cytoplasm. The latter is hindered by intracellular organelles and the cytoskeleton itself. We find that for filament concentrations even below physiological values, the FPT along cytoskeletal filaments converges to that for free diffusion. When filaments are considered as obstacles, the FPT grows steadily with filament concentration. At realistic filament concentrations the FPT is insensitive to local modifications in the cytoskeletal network, including bundle formation. We conclude that diffusion along cytoskeletal tracks is a reliable alternative to other established ways of intracellular trafficking and signaling, and therefore provides an additional level of cell function regulation.

Viscoelastic properties of living embryonic tissues: a quantitative study

A number of properties of certain living embryonic tissues can be explained by considering them as liquids. Tissue fragments left in a shaker bath round up to form spherical aggregates, as do liquid drops. When cells comprising two distinct embryonic tissues are mixed, typically a nucleation-like process takes place, and one tissue sorts out from the other. The equilibrium configurations at the end of such sorting out phenomena have been interpreted in terms of tissue surface tensions arising from the adhesive interactions between individual cells. In the present study we go beyond these equilibrium properties and study the viscoelastic behavior of a number of living embryonic tissues. Using a specifically designed apparatus, spherical cell aggregates are mechanically compressed and their viscoelastic response is followed. A generalized Kelvin model of viscoelasticity accurately describes the measured relaxation curves for each of the four tissues studied. Quantitative results are obtained for the characteristic relaxation times and elastic and viscous parameters. Our analysis demonstrates that the cell aggregates studied here, when subjected to mechanical deformations, relax as elastic materials on short time scales and as viscous liquids on long time scales.

Viscosity and elasticity during collagen assembly in vitro: relevance to matrix-driven translocation

In order to better understand the gelation process associated with collagen assembly, and the mechanism of the in vitro morphogenetic phenomenon of "matrix-driven translocation" [S.A. Newman et al. (1985) Science, 228, 885-889], the viscosity and elastic modulus of assembling collagen matrices in the presence and absence of polystyrene latex beads was investigated. Viscosity measurements at very low shear rates (0.016-0.0549 s(-1)) were performed over a range of temperatures (6.9-11.5 degrees C) in a Couette viscometer. A magnetic levitation sphere rheometer was used to measure the shear elastic modulus of the assembling matrices during the late phase of the gelation process. Gelation was detected by the rapid increase in viscosity that occurred after a lag time tL that varied between O and approximately 500 s. After a rise in viscosity that occurred over an additional approximately 500 s, the collagen matrix was characterized by an elastic modulus of the order of several Pa. The lag time of the assembly process was relatively insensitive to differences in shear rate within the variability of the sample preparation, but was inversely proportional to the time the sample spent on ice before being raised to the test temperature, for test temperatures > 9 degrees C. This suggests that structures important for fibrillogenesis are capable of forming at 0 degrees C. The time dependence of the gelation process is well-described by an exponential law with a rate constant K approximately 0.1 s(-1). Significantly, K was consistently larger in collagen preparations that contained cell-sized polystyrene beads. From these results, along with prior information on effective surface tension differences of bead-containing and bead-lacking collagen matrices, we conclude that changes in matrix organization contributing to matrix-driven translocation are initiated during the lag phase of fibrillogenesis when the viscosity is < or = 0.1 Poise. The phenomenon may make use of small differentials in viscosity and/or elasticity, resulting from the interaction of the beads with the assembling matrix. These properties are well described by standard models of concentrated solutions.

Surface tensions of embryonic tissues predict their mutual envelopment behavior

During embryonic development, certain tissues stream to their destinations by liquidlike spreading movements. According to the ‘differential adhesion hypothesis’, these movements are guided by cell-adhesion-generated tissue surface tensions (sigmas), operating in the same manner as surface tensions do in the mutual spreading behavior of immiscible liquids, among which the liquid of lower surface tension is always the one that spreads over its partner. In order to conduct a direct physical test of the ‘differential adhesion hypothesis’, we have measured the sigmas of aggregates of five chick embryonic tissues, using a parallel plate compression apparatus specifically designed for this purpose, and compared the measured values with these tissues' mutual spreading behaviors. We show that aggregates of each of these tissues behave for a time as elasticoviscous liquids with characteristic surface tension values. Chick embryonic limb bud mesoderm (sigma = 20.1 dyne/cm) is enveloped by pigmented epithelium (sigma = 12.6 dyne/cm) which, in turn, is enveloped by heart (sigma = 8.5 dyne/cm) which, in turn, is enveloped by liver (sigma = 4.6 dyne/cm) which, in turn, is enveloped by neural retina (sigma = 1.6 dyne/cm). Thus, as predicted, the tissues' surface tension values fall in the precise sequence required to account for their mutual envelopment behavior.

Biological specificity and measurable physical properties of cell surface receptors and their possible role in signal transduction through the cytoskeleton

It is proposed that the binding specificities of cell adhesion molecules are manifested in their measurable physical properties. A method specifically designed to measure the interfacial tension of cell aggregates is described. With the introduction of a statistical mechanical model, the measured values of tensions for aggregates consisting of genetically engineered cells with controlled adhesive properties are used to obtain information on the strength of individual receptor-ligand bonds. The strength of binding must depend on the receptor and its ligand and reflects the amino acid sequence of the binding proteins. Many of the cell surface receptors, being transmembrane proteins, are attached to the various macromolecular networks of the cytoskeleton; therefore, it is suggested that their ligation and ensuing conformational change may substantially affect the mechanical state of the cytoskeletal assemblies. Since these assemblies are believed to actively participate in intracellular signaling by transmitting signals from the cell membrane into the nucleus, the cell adhesion molecules may influence signaling in a predictable way through their measurable physical characteristics. In particular, varying bond strength at the cell surface may lead to differential gene regulation.

Wetting, percolation and morphogenesis in a model tissue system

Artificial tissues constructed of cells or polystyrene beads suspended in a solution of type I collagen will, under appropriate conditions, protrude into regions of similar matrices lacking particles, but containing the extracellular glycoprotein fibronectin. This phenomenon has been termed "matrix-driven translocation". Conditions required for the effect include the presence of heparin-like molecules on the cell or bead surfaces, appropriate concentrations of particles and collagen, and physiological ionic strength and pH. Here we consider the idea that the driving force for the concerted movement of matrix and suspended particles is the thermodynamically spontaneous spreading or wetting behavior of two immiscible fluids bounded by common substrata. Wetting theory is shown to be capable of accounting for the behavior of this model system, but this analysis requires that the two matrix regions constitute separate phases at thermodynamic coexistence. We show that one plausible mechanism for the generation of separate phases is the formation of a percolation network of collagen fibers on a lattice of cells or beads. It is argued that the concepts of wetting and percolation apply to properties in common between the model system and living tissues, and may therefore be used to provide a physical account of aspects of tissue morphogenesis.