Stability analysis of a multiscale model of cell cycle dynamics coupled with quiescent and proliferating cell populations

Abstract

In this paper, we perform a mathematical analysis of our proposed nonlinear, multiscale mathematical model of physiologically structured quiescent and proliferating cell populations at the macroscale and cell-cycle proteins at the microscale. Cell cycle dynamics (microscale) are driven by growth factors derived from the total cell population of quiescent and proliferating cells. Cell-cycle protein concentrations, on the other hand, determine the rates of transition between the two subpopulations. Our model demonstrates the underlying impact of cell cycle dynamics on the evolution of cell population in a tissue. We study the model's well-posedness, derive steady-state solutions, and find sufficient conditions for the stability of steady-state solutions using semigroup and spectral theory. Finally, we performed numerical simulations to see how the parameters affect the model's nonlinear dynamics.

Fig 1. Model schematics.

In the macro-scale, two subpopulations are proliferating and quiescent cells with various transition effects given by χ, τ, γ, and μ functions. At the bottom, the microscale is represented with all four protein states and their interactions which are explained using legends in the bottom left. The feedback from the macroscale, in the form of growth factors g_f, manipulates the cell-cycle (microscale). The feedback loop is closed by the rate χ (corresponds to the rate of cells transitioning from proliferating to quiescent phase), determined by the protein dynamics at the microscale.

Fig 2. Evolution of microscale proteins from the cell-cycle.

Cyclin $\mathsf{D}-\mathsf{\text{CDK}}\mathsf{4}/\mathsf{6}$ shows a complete activation and degradation within a full cycle. The concentration of transcription factor $\mathsf{E}\mathsf{2}\mathsf{F}$ is elevated since Retinoblastoma protein $\mathsf{\text{Rb}}$ is inactivated with the rise in Cyclin $\mathsf{D}-\mathsf{\text{CDK}}\mathsf{4}/\mathsf{6}$ complex. Similarly, protein $\mathsf{p}\mathsf{21}$ elevates near the end of the cell-cycle to help in the degradation of the Cyclin’ complex.

Fig 3. Cell number density distribution.

(a) quiescent and (b) proliferating cell populations.

Fig 4. Behavior of total cell population, growth factors and gamma function.

(a) N(t) achieves steady-state. (b) Growth-factors g_f decreasing with increase in cell population. (c) Gamma function γ declines as the total cell population achieves steady-state.

Fig 5. Trivial steady-state solution.

(a) Total population of cells N(t) decays to zero. (b) Growth-factors g_f remain maximum due to decline in cell count. (c) Gamma function γ increasing to its maximum value due to less number of cell.

Fig 6. Unstable behavior.

(a) Total cell population N(t) grows exponentially with time thus depicting an unstable behavior. (b) Growth-factors are increasing with rising total population of cells. However, as the change in N(t) larger and larger, the change in growth factors is negligible. (c) Gamma function is also declining.