Model on cell movement, growth, differentiation and de-differentiation: Reaction-diffusion equation and wave propagation

Abstract

We construct a model for cell proliferation with differentiation into different cell types, allowing backward de-differentiation and cell movement. With different cell types labeled by state variables, the model can be formulated in terms of the associated transition probabilities between various states. The cell population densities can be described by coupled reaction-diffusion partial differential equations, allowing steady wavefront propagation solutions. The wavefront profile is calculated analytically for the simple pure growth case (2-states), and analytic expressions for the steady wavefront propagating speeds and population growth rates are obtained for the simpler cases of 2-, 3- and 4-states systems. These analytic results are verified by direct numerical solutions of the reaction-diffusion PDEs. Furthermore, in the absence of de-differentiation, it is found that, as the mobility and/or self-proliferation rate of the down-lineage descendant cells become sufficiently large, the propagation dynamics can switch from a steady propagating wavefront to the interesting situation of propagation of a faster wavefront with a slower waveback. For the case of a non-vanishing de-differentiation probability, the cell growth rate and wavefront propagation speed are both enhanced, and the wavefront speeds can be obtained analytically and confirmed by numerical solution of the reaction-diffusion equations.

Regulatory Feedback Effects on Tissue Growth Dynamics in a Two-Stage Cell Lineage Model

“…The present work focused on a simple two-stage lineage cell model, it can also be extended to include lineage of multiple stages [3] or branched lineages [23] with cross-regulations across different lineages. The interplay between self-proliferation, differentiation and de-differentiation [16,24], cell-cell interactions [25,26] can be incorporated to in-vestigate the effects on the growth dynamics. With the present theoretical basis, more sophisticated clinical situations can be modeled with appropriate extension of the present model.…”

“…For simplicity and the purpose of illustrating the ideas, we consider a simple model of two-cell lineage to investigate the strategy of controlling a final-state system. It has been shown in [16,17] that in the continuum limit, different cell types in different stages of a lineage together with the diffusion of regulatory molecules can be modeled by coupled PDEs of the cell densities and regulatory molecule concentrations. And for the simple case of a two-stage lineage model with cell and molecule advection, the cell density and feedback molecule concentration can be modeled as [9]…”

Regulatory feedback effects on tissue growth dynamics in a two-stage cell lineage model

“…It has been shown in Refs. [15,16] that in the continuum limit, different cell types in different stages of a lineage together with the diffusion of regulatory molecules can be modeled by coupled PDEs of the cell densities and regulatory molecule concentrations. And for the simple case of a two-stage lineage model with cell and molecule advection, the cell density and feedback molecule concentration can be modeled as [9]…”

Measure-transmission metric and stability of structured population models

“…They include modeling differentiation switches via Markov chains or systems of ordinary differential equations (see [4,5,6]), modeling the inherent stochasticity via branching processes (see e.g. [7,8,9]), modeling delays via delay differential equations (see [10,11,12] and references therein), modeling spatial dynamics via discrete lattice models or reaction-diffusion equations (see [13,14]) and others.…”