Fig 1.
Hierarchical representation of cell decision-making.
(A) The cell’s genotype dictates the regulation of the phenotypic switch. This switch determines the cell’s reaction to its microenvironment. Subsequently, the cell interacts and shapes its microenvironment according to its phenotype. (B) In the mathematical model and the context of the go-or-grow dichotomy, the genotype is represented by the parameter κ which controls the phenotypic switch between the migratory and proliferating phenotypes dependent on the local cell density. After assuming either phenotype the cell influences the local microenvironment by either reducing the cell density (by migration) or increasing it (by proliferation). We neglect possible epigenetic changes by a persistent microenvironment on long time scales.
Fig 2.
The lattice-gas cellular automaton (LGCA) model.
(A) Left: The LGCA model is implemented on a one-dimensional lattice, where each node comprises two velocity channels for movement to the nearest-neighbor nodes and one rest channel for no movement. Each channel can be occupied by any number of cells. Cells in velocity channels (marked red) have the migratory phenotype, and cells in rest channels have the proliferative phenotype (blue). White channels denote the absence of tumor cells. Right: Schematic illustration of model dynamics: Cells switch between phenotypes depending on their genotype κ and the local tumor cell density. Proliferative cells divide with a constant rate α. Migratory cells randomly choose a new direction of movement. All cells die with a constant rate δ. (B) Cell migration. Each migratory cell (red) moves in the direction of its respective velocity channel. (C and D) Phenotypic plasticity. The phenotypic switch probability rκ depends on the tumor cell density in the microenvironment and the cell’s genotype κ. The sign of κ determines the switch regulation. (C) κ > 0 results in attractive behavior (proliferating phenotype triggered by high cell density), and κ < 0 (D) leads to repulsive behavior (cells switch to migratory phenotype if the local cell density becomes too high). The parameter θ indicates the cell density threshold, where the probability for either phenotype is 1/2.
Fig 3.
Phenotypic and genetic heterogeneity emergence.
Example realizations (A-F) and ensemble averages (G-L) for three parameter sets. Evolution of cell density profiles (A-C) and local switch parameter (D-F). (G-I) Distribution of proliferating (blue) and migrating (orange) cells at the endpoint. (J-L) Switch parameter κ distribution at the endpoint. Shaded regions represent one standard deviation. Negligible cell death (first column) results in attractive type cells (κ > 0) around the initial lattice node at x = L/2, with migrating phenotype cells outgrowing due to quick spread across the lattice (regime 1, Fig 4). High cell death and low switch threshold (second column) lead to predominantly proliferative phenotypes with attractive-type cells (κ > 0) dominating the tumor bulk and repulsive-type cells (κ < 0) invading the edge (regime 2, Fig 4). High cell death and high switch threshold (third column) result in a majority of proliferative cells but with a significant fraction of migrating cells throughout the tumor. Repulsive-type cells dominate the tumor, and weaker phenotype switch cells (κ ≈ 0) are more likely at the tumor edge (regime 3, Fig 4).
Fig 4.
Emerging heterogeneity and treatment responses depend on death rate and switch threshold as predicted by mean-field theory.
(A) Average switch parameter κ in the cell population. We can distinguish three different regimes. (1) If there is no cell death, the population is dominated by cells that can invade the surrounding tissue the fastest, which are cells with a switch parameter at zero or slightly below. (2) For a low switch threshold and non-zero death rate, the tumor is dominated by cells with an attractive go-or-grow strategy (κ > 0). However, there is considerable genetic heterogeneity, with repulsive cells at the tumor front, see Fig 3, second column. (3) With increasing switch threshold θ and death rate δ, there is a transition to the repulsive regime, where all cells in the tumor use the repulsive strategy (κ < 0). This transition can be predicted by mean-field theory (black dashed line, given by Eqs 6 and 7. (B) The phenotypic heterogeneity is largest near the transition line between evolutionary regimes. (C) The genetic heterogeneity shows a bimodal behavior, with a minimum at the transition line surrounded by local maxima. (D) The recurrence time is longest in regime (2) and minimal near the transition line as well as for low values of the switch threshold θ.
Fig 5.
Genetic and phenotypic heterogeneity predict treatment success.
Shown is the time to recurrence of synthetic tumors versus the genetic entropy (A) and phenotypic entropy (B). Higher phenotypic entropy is associated with lower recurrence times, while higher genetic entropy is associated with higher recurrence time, corresponding to better prognosis. Black dashed lines are linear regressions.