Cellular behaviours, such as motion, interactions, mechanics, etc., are described as energy terms in the overall system Hamiltonian. In our simulations we use three major terms to describe cell-cell adhesiveness, cell volume elastic constraint, and cell surface elastic constraint. The simulation itself consists of a series of random attempts of cells to extend their boundaries by copying the value of a pixel σ(i) to neighbouring site, see Fig. 6.5. Since each cell in the GGH model is represented as a collection of pixels i, at each step we randomly select a pixel i as a target pixel and randomly select one of its fourth-order neighbouring pixel i0as a source pixel. Then we attempt to change its index from σ(i) to the index σ0= σ (i0). If they belong to the same generalised cell, σ(i) = σ (i0), we do not need to copy the index. A successful index copy increases the
0 5 10 15 20
Figure 6.4: Dynamics of β -catenin, E-cadherin-β -catenin complex, and β -catenin-proteasome complex at attachment and detachment conditions based on the underlying ODEs using parameters in Table 6.1.
volume of the source cell and decreases the volume of the target cell by one pixel. At each pixel copy attempt we calculate the change in the system energy ∆E and accept pixel reassignment with probability P(∆E)
P(∆E) =
where Tmis a parameter representing the effective cell motility.
Energy dependent pixel copy probability and Hamiltonian terms constitute the essence of the GGH models. Although ∆E denotes change of the overall system en-ergy, in practice contributions to ∆E are almost always “local” i.e., in our calculations we only have to examine a small neighbourhood of pixels involved in the pixel-copy.
The simulation is subdivided in the so-called Monte Carlo Steps (MCS) which corre-spond to a unit of physical time. By convention, each MCS consists of one index-copy attempt for each pixel in the cell lattice. The conversion between MCS and physical time depends on model parameters. In a simple case for example, in Bionetsolver we
set timestepBionetwork to 0.03 and if Bionetsolver gets called every MCS then 1 MCS corresponds to 0.03 hours. In this chapter we do not specifically set a relationship between MCS and the physical time because in the computational simulations we also incorporate cell mitosis or cell division which in the process itself also requires another time convention. In the mitosis process we do not apply any intracellular pathway, but instead we use a built-in mitosis function provided by CompuCell3D. The physical distance is recovered by converting pixels into unit of length. This conversion is more straightforward than time conversion and in our simulations we set 1 pixel corresponds to 2 µm.
Unlike models where cellular behaviours are put by hand by specifying various parameters of the Hamiltonian, in our model the most critical parameters (i.e., concen-trations of E-cadherins, β -catenins, and E-cadherin-β -catenin complexes) are linked to the molecular pathways running inside each individual cell. This approach allows us to represent more faithfully the multiscale nature of tumour invasion. Biological cells (or tumour cells) change their phenotypic properties due to the dynamic events explained in the previous subsection which occur inside the cell, in the nucleus or cytoplasm, and on the cell membrane. We try to mimic the biology as closely as possible by linking phenomena occurring at the intracellular scales (E-cadherin/β -catenin dynamics) to processes which operate at the cell/multicell level (cellular adhesion).
To build the CC3D-Bionetsolver implementation, we provide four files and do the following. The pathway of E-cadherin and β -catenin interactions is written in an SBML file using JarnacLite (http://www.sys-bio.org/sbwWiki/sbw/jarnaclite).
The setup of Potts dimension, related CC3D plugins, and initial configuration are listed in an XML file. In addition to plugins, there are also modules called steppables which run either repeatedly after a defined intervals of Monte Carlo Steps or once at the be-ginning or the end of the simulation. Steppables typically define initial conditions,
alter cell states, update fields or output intermediate results. The Bionetsolver func-tions are called from within CC3D steppables. The CC3D steppables and Bionetsolver functions are written in a Python file. The main file that is called from the CC3D player to run the simulation is written in Python and it lists the core CC3D simulation objects and required steppables.
Figure 6.5: Schematic diagram showing the GGH representation of an index-copy attempt for two cells on a 2-dimensional square lattice. The “white” pixel (source) attempts to replace the
“grey” pixel (target). The probability of accepting the index copy is given by equation (6.19).
For the boundary energy (Hboundary) of the Hamiltonian (6.6) we use the
ContactLocalProductplugin that calculates the boundary energy based on the lev-els of E-cadherin expression per cell. We set two types of cells, namely “LowBeta-Cat” for cells having concentration of β -catenin below the threshold cT and “High-BetaCat” for cells with β -catenin higher than cT. The values of contact/boundary energy between the two types of cells, between the cells and the medium are spec-ified in the ContactLocalProduct plugin. The medium is the non-cell area. The
VolumeLocalFlex and SurfaceLocalFlex plugins are used for the volume con-straint (Hvolume) and the surface constraintHsurface), respectively. We use these plug-ins in the XML file to allow the target volume and the target surface to vary individu-ally for each cell during the simulation without having to explicitly specify the values of λvolume, Vt, λsurface, and St. For initialisation, in the steppable file we assign their dimensionless parameter values as listed in Table 6.2.
Table 6.2: CC3D parameter values
Parameter Definition Dimensionless Value
λvolume Cell’s inverse compressibility 0.5
Vt Cell’s target volume 1.2×cell volume
λsurface Cell’s inverse membrane compressibility 1.5
St Cell’s target surface area 400
The integration of CC3D and Bionetsolver in the model is illustrated in the flow chart, see Fig. 6.6. The simulations are started by adjusting cell shape and for simula-tions that require cell division we invoke the cell mitosis function, which is a built-in function of CC3D. After a certain time (denoted by T 1), we start the Bionetsolver for the numerical integration of the differential equations. To change the concentration of β -catenin at time T 2, we switch the values of k2and k+, where if the concentration of β -catenin inside each cell ([β ]) is higher than the concentration of β -catenin threshold (cT) the cell then detaches from the main tumour mass.
6.4 Computational Simulation Results
We modelled three different scenarios as follows: (1) detachment waves of β -catenin on a thin layer of epithelial cells, described in subsection 6.4.1; (2) tumour growth and
Figure 6.6: Flow chart of the integration of CC3D and Bionetsolver in the model.
detachment of cells from a layer of epithelial cells; and (3) tumour growth and detach-ment of cells in a multicellular tumour spheroid, both described in subsection 6.4.2.