Stochastic binary modeling of cells in continuous time as an alternative to biochemical reaction equations

We have developed a coarse-grained formulation for modeling the dynamic behavior of cells quantitatively, based on stochasticity and heterogeneity, rather than on biochemical reactions. We treat each reaction as a continuous-time stochastic process, while reducing each biochemical quantity to a binary value at the level of individual cells. The system can be analytically represented by a finite set of ordinary linear differential equations, which provides a continuous time course prediction of each molecular state. Here we introduce our formalism and demonstrate it with several examples.

Figure 1

A schematic representation of two possible approximations for interpreting bulk assays. (a) A typical output of a bulk time course experiment. (b) Cells behave stochastically and digitally, as assumed in our formulation. (c) Every cell behaves coherently and continuously, as assumed in the deterministic biochemical reaction modeling.

Figure 2

Examples of systems with two nodes. (a) A system with an interaction. The initial condition is $P_{TF}=1$. The black dashed line in the plot shows the exact solution of the master equation ($\tau =1$), while the others are the result of Monte Carlo simulations performed 1, 10, and 100 times, respectively. (b) A bistable system and the corresponding plots (${\tau }_1={\tau }_2=1$), as in (a).

Figure 3

A model of the TNF–NF-$\kappa$B system. (a) The network diagram. The number on each edge is the time-scale parameter ($\min$) for the interaction. The small circle at the end of the arrow from I$\kappa$B to NF-$\kappa$B indicates that the interaction occurs when the I$\kappa$B is inactivated. (b) The predicted time courses of our model for WT. (c) The change of the NF-$\kappa$B activation level for WT, I$\kappa$B KO, and A20 KO.