Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion

Abstract

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.

Figure 1. Schematic illustration of a lattice-based representation of cells in the GGH model (left figure) and a lattice-free representation in the centre-based model (right figure).

Figure 2. Plots showing a sequence of the disruption of a layer of epithelial cells due to an increase in the -catenin concentration inside the cells.

After all cells have detached from the layer of cells or from each other (EMT), -catenin concentrations eventually drop, causing cells that are close to each other to undergo re-attachment (MET) while other cells that are not close remain as mesenchymal cells. Colours of the cells correspond to concentration of -catenin.

Figure 3. Plots of -catenin, E-cadherin--catenin complex, and proteasome--catenin concentrations for a simulation in which cells undergo epithelial-mesenchymal transition (EMT) and subsequently recover by mesenchymal-epithelial transition (MET).

The cells reattach to adjacent cells and thereby reform an epithelial layer. The cycle of detachment and reattachment occurs about times until MCS.

Figure 4. Plots of -catenin, E-cadherin--catenin complex, and proteasome--catenin concentrations for a typical cell undergoing epithelial-mesenchymal transition.

Figure 5. Plots showing the results of a simulation of tumour growth and local invasion (detachment) from a layer of cells.

The tumour grows rapidly from a single layer and eventually EMT events are observed to occur. Cell colour represents -catenin concentration.

Figure 6. Plot of a cross sectional view showing the spatial distribution of -catenin concentration inside cells from the simulation of tumour growth from a layer of cells.

Cells in the centre of the tumour mass have a large number of binding neighbours, hence the concentration of -catenin is lower than the cells at the outer layer of tumour mass that have fewer binding neighbours and a high concentration of free -catenin.

Figure 7. Plots showing the effect of varying the parameter on the number of cells that detach from a primary tumour mass in a layer configuration.

The value of was varied between high, intermediate and low values and the number of cells that detach and migrate a certain distance from the tumour mass was monitored.

Figure 8. Plots showing the results of multicellular tumour spheroid simulations.

The tumour grows from a single cell placed in the middle of a cubic lattice.

Figure 9. Position of cell ID with respect to the centre of a cubic lattice of size pixels during simulations of MTS using the following parameter values for : , , and .

Tumour radius is apparent from the horizontal portion of the cell position time-courses. In each case (for all three parameter values) this occurs at a pixel value of .

Figure 10. Plots showing the number of cells removed from MTS simulations using different values of (corresponding to different levels of invasiveness).

Figure 11. Comparison between our computational results with experimental data.

Images showing experimental data of MTS growth patterns in low collagen concentration (top left figure), a less invasive pattern, and in high collagen concentration (top right figure), a more invasive pattern. Our computational simulation results (bottom right figure with and bottom left figure with ) are comparable to the experimental data. The simulation results were taken at MCS. Reprinted from Biophysical Journal, 89/1, L. Kaufman, C. Brangwynne, K. Kasza, E. Filippidi, V. Gordon, T. Deisboeck, and D. Weitz, Glioma expansion in collagen I matrices: analyzing collagen concentration-dependent growth and motility patterns, 635–650, Copyright (2005), with permission from Elsevier [OR APPLICABLE SOCIETY COPYRIGHT OWNER].

Figure 12. Comparison of -catenin detachment wave simulations based on the centre model of (left figure) and our CC3D-Bionetsolver simulation results (right figure).

Figure 13. Schematic diagram showing the GGH representation of an index-copy attempt for two cells on a 2-dimensional square lattice.

The “white” pixel (source) of cell with attempts to replace the “grey” pixel (target) of cell with . The probability of accepting the index copy is given by equation (1). Bold lines denote boundaries of the cells. Pixel colour denotes cell type. Notice that in GGH simulations we typically have multiple cells with different id but belonging to the same type .

Figure 14. A schematic diagram of the E-cadherin and interactions with -catenin.