Modeling Uniaxial Nonuniform Cell Proliferation

Abstract

Growth in biological systems occurs as a consequence of cell proliferation fueled by a nutrient supply. In general, the nutrient gradient of the system will be nonconstant, resulting in biased cell proliferation. We develop a uniaxial discrete cellular automaton with biased cell proliferation using a probability distribution which reflects the nutrient gradient of the system. An explicit probability mass function for the displacement of any tracked cell under the cellular automaton model is derived and verified against averaged simulation results; this displacement distribution has applications in predicting cell trajectories and evolution of expected site occupancies.

Appendix

Reformulating the cellular automaton as a time inhomogeneous Markov chain provides an alternative method for calculating the cell displacement distribution numerically. This time inhomogeneous Markov chain is formed by the sequence of random variables $\{K_n\}_{n\ge 0}$. The transition matrix at some iteration $n\ge 0$ is given by the $(n+1)\times (n+2)$ matrix

$$\begin{aligned} T_n = \begin{bmatrix} F(0,n)&\quad S(0,n)&\quad 0&\quad \dots&\quad 0&\quad 0 \\ 0&\quad F(1,n)&\quad S(1,n)&\quad \dots&\quad 0&\quad 0 \\ \vdots&\quad \vdots&\quad \vdots&\quad \ddots&\quad \vdots&\quad \vdots \\ 0&\quad 0&\quad 0&\quad \dots&\quad F(n,n)&\quad S(n,n) \end{bmatrix}. \end{aligned}$$

(13)

The probability that a fixed agent $X \in \{1,\dots ,L\}$ is displaced $k \in \{0,\dots ,N\}$ times in N iterations is then

$$\begin{aligned} {\mathbb {P}}(K_N = k;X,L) = [T_0T_1\dots T_{N-1}]_{1,k+1}. \end{aligned}$$

(14)