A three-dimensional model of cell movement in multicellular systems

Abstract

A mathematical model for cell movement in multicellular systems has been developed that allows us to simulate and visualize, in three dimensions, individual cell movements in a number of multicellular systems. These include cell movement during aggregation and slug stage of Dictyostelium discoideum, embryogenesis, limb formation and wound healing.

The model is quite adaptable to a number of systems, due to the way it is designed. The building blocks of the model are individual cells, where each cell has certain given properties that are not necessarily the same for all cells. The basic properties are that a cell can deform under force (either stretch or compress), while conserving its volume, it adheres to other cells and it can generate an active motive force. The response of a cell depends on its internal parameter state, and on the information it receives from its external environment, which includes neighbor cells, the extracellular matrix and chemical signals. The net force on a cell is calculated by summing up all the forces that a cell experiences at its surroundings. Each cell is then moved and deformed according to the equations of motion and deformation. Finally, the net movement of all the cells gives the collective movement of the entire tissue.

Here we introduce this model and show examples of its applications and compare the results with experimental data. In the first simulations, we show how different cell types can be sorted out based solely on differences in adhesion. We compare our results to cell sorting experiments done by Steinberg and co-workers [R.A. Foty, C.M. Pfleger, G. Forgacs, M.S. Steinberg, Development 122 (1996) 1611–1620; M.S. Steinberg, Reconstruction of tissues by dissociated cells, Science 141 (1963) 3579] using values for adhesion within the range of the experimental values, and show that the model reproduces the experiments very well. We also present results from simulations of Dictyostelium movements. We first modeled the aggregation stage, where cells are aggregating chemotactically, towards a signaling center, in response to cAMP waves. In these simulations one can observe stream formation and how the mound arises due to the inward motion of the cells towards the signaling center. Later we studied simulations of 2D slugs, and compared them to observations of 2D slugs done by Bonner [J.T. Bonner, Proc. Natl. Acad. Sci. USA 95 (1998) 9355–9359].

Introduction

Cell and tissue movement plays a vital role throughout the lifespan of many organisms. Bacteria and other single-cell organisms find food and avoid repellents by chemotaxis. Coordinated movements of cells and tissues occur throughout early embryonic development, and special terminology such as gastrulation and invagination is used to describe them.

In multicellular systems the relative movement of cells can lead to sorting out of different cell types and formation of specific structures and patterns, such as during gastrulation [1], [10], cancer cell invasion into tissues [39], the development of Dictyostelium discoideum [3], limb regeneration [43], in wound healing [1], and white blood cell movements through blood vessels [23]. In these multicellular systems the collective motion can be quite different than the motion of isolated individuals. Since the combined effect of cell–cell interactions, and production and propagation of chemotactic signals are often very complex, it is not always obvious what underlying mechanism is responsible for the observed patterns. Thus, there is a need to develop models of cell movements in multicellular systems that will give insight into the mechanism governing this complex behavior. There have been several different approaches used to model the collective motion of cells in multicellular systems:

• A cellular automata approach [34], in which formal rules for cell–cell interactions and for the reactions of the cells to their chemical environment are postulated. Simulations of such models show patterns that resemble experimental observations, but limited insight is gained from such models because they do not incorporate the physics of cell–cell interactions and cell–cell signaling.

• Models designed to simulate cell sorting [7], [20], [24], [25], [38], [41]. One class are fluid type models where the cell aggregate is modeled as a mixture of two fluids. Here the cells have zero size, so the sorting time only depends on the “temperature” (randomness) of the fluid mixture. The problem with these models is that the size and stiffness of the cells are ignored, so they really verify only the theoretical predictions regarding mixing of two fluids. Another class of models use some kind of Potts model [17], [18], [34] to represent the energy between the cells, and cells exchange position with higher probabilities if that exchange is energetically favorable. The problem with Potts type models is that they fail to explain how the cells exchange positions and in some sense they assume that the cell knows beforehand which positions are energetically favorable.

• Continuum approaches such as those used by Odell and Bonner [27] and by Vasiev et al. [45] to model the slug movement in Dictyostelium discoideum. In the Odell–Bonner model, cells respond to cAMP chemotactically, as they do during the aggregation phase, and the active component of the propulsive force enters as a contribution to the stress tensor. Active stresses are derived from an active relative velocity field set up at the cell surface. Each amoeba is thought of as a “cortical tractor” pulling the surrounding viscous fluid past itself by rotating its membrane at different rates on different parts of the surface. While this is an important attempt at modeling the motion of the slug, there are limitations to the model. Firstly, it is known that cytoplasm is not a simple Newtonian fluid, as they assume [26], [33]. Secondly, the properties of the extracellular fluid, which are critical for the model, are not known. Thirdly, traction forces play no role in the model. Finally, no molecular or cell-based mechanism is proposed that gives rise to the constitutive relations used in the model, and thus one cannot relate them to properties of the cells. Vasiev et al. incorporate the cAMP dynamics, via a FitzHugh–Nagumo model, into a continuum model of mound formation in Dictyostelium. These authors begin with the Navier–Stokes equations for the velocity field and simply add forces intended to model those arising from chemotaxis. While this approach can lead to solutions that apparently reflect aggregation, it is completely phenomenological and no procedure is given to connect the forces postulated with experimentally measurable quantities such as the force exerted by a single cell.

One major problem with most of these models is that they fail to properly balance the active locomotive forces generated by the cells, or the models do not even include them. Some models have an active force that propels the cell, but there is no discussion on where that force is applied. Also most of these models are two-dimensional and therefore often do not adequately describe movement in multicellular systems. The three-dimensional models are cellular automata models that do not contain much information about individual cell characteristics and forces between cells.

The model we present here is a three-dimensional mathematical model of cell movement and cell–cell interactions which allows us to analyze and gain insight into the various processes that govern cell movement inside multicellular systems. To make it feasible to model the motion of tissues comprising thousands of cells, we stipulate that all cells are ellipsoids, and thus restrict the admissible deformations to those which only change the relative lengths of the semi-axes but conserve the cell volume. These computational cells have characteristics and responses that correspond to those observed in real cells. These characteristics include the stiffness of the cell, cell adhesion, locomotive force generation and response to environmental cues. Most cell types have common characteristics like, how they move and interact, deform, exert forces onto other cells and move in response to chemical or mechanical cues, but their exact responses differ. For instances the stiffness of a cell can vary by up to two orders of magnitude; a fibroblast has stiffness 0.6 mdyn/μm whereas a red blood cell has a stiffness of 10⁻³ mdyn/μm [5]. The adhesion can also differ for different type of cells [30], the surface tension for a limb bud aggregate is 20 dyn/cm but it is 1.6 dyn/cm for Neural retina aggregate [15]. The locomotive force a cell applies also varies more than an order of magnitude [19], [21], [22], [28], [42]. In our model, these values can be changed for each cell individually depending on what cell types we are trying to model, thus making the model very flexible, and applicable to a large number of different multicellular systems. Since our model is based on individual cells and the movement and deformation of the cells is calculated directly from all the forces acting on the cells from the equations of motion and deformation, it does not have many of the limitations of the other models mentioned above. This feature allows us to study the effect that finite cell size, the stiffness of the cells and random movement, has on the rate and completeness of sorting.

Before designing the model, it is important to know how an individual cell moves. When an ameoboid cell moves, either randomly or in response to a chemotactic signal, it sends out pseudopods, one of which eventually dominates and attaches it to the surroundings [46], [48]. This determines the direction of motion, as the cell realigns its axis towards the pseudopod, and the rest of the cell body is “pulled” towards the attached pseudopod. The realignment is not achieved by rotation but by disassembly and reassembly of several proteins [43], [44]. The establishment of a new pseudopod and realignment of the axis does take some time [46], [48], but once established the cell becomes polarized and moves in the new direction for up to several minutes; for Dictyostelium new pseudopods form about once every 3 min [44]. When the cell senses a chemical gradient, most of the new pseudopods are formed towards the gradient, but in the absence of a gradient the probability of lateral pseudopods is increased [13], [44]. Once the cell moves in a gradient, it becomes polarized and is even less likely to form lateral pseudopods.

The extensions of pseudopods and the retraction of the rest of the cell body require active force generation, and this force is applied at the site of pseudopod attachment (Fig. 1a). When the cell is moving on a surface, the pseudopod attachment will be onto the surface and the applied force is transmitted directly to the ground. However, when the cell is inside a multicellular aggregate, it must attach the pseudopod to another cell (Fig. 1b). It is important to realize that when a cell attaches its pseudopod to another cell, and pulls itself forward, the other cell experiences an equal force in the opposite direction. So if that cell is not firmly attached to the ground it is pulled backwards; for a good discussion see [27]. Only forces that are transmitted down to the ground give rise to any net movement of the multicellular system and therefore a complete model of cell movements must balance all the forces. Neglecting to balance the forces has been the main fault of many previous models. In 2D it is not as important since one can assume that each cell is in direct contact with the surface where it applies the active force. However this does not hold true in 3D where the active force most often is applied on a neighbor cell, and this force also affects the motion of that cell. These interactions can have a significant effect on the motion of the system as a whole.