### abstract ###
Competence is a transiently differentiated state that certain bacterial cells reach when faced with a stressful environment.
Entrance into competence can be attributed to the excitability of the dynamics governing the genetic circuit that regulates this cellular behavior.
Like many biological behaviors, entrance into competence is a stochastic event.
In this case cellular noise is responsible for driving the cell from a vegetative state into competence and back.
In this work we present a novel numerical method for the analysis of stochastic biochemical events and use it to study the excitable dynamics responsible for competence in Bacillus subtilis.
Starting with a Finite State Projection solution of the chemical master equation, we develop efficient numerical tools for accurately computing competence probability.
Additionally, we propose a new approach for the sensitivity analysis of stochastic events and utilize it to elucidate the robustness properties of the competence regulatory genetic circuit.
We also propose and implement a numerical method to calculate the expected time it takes a cell to return from competence.
Although this study is focused on an example of cell-differentiation in Bacillus subtilis, our approach can be applied to a wide range of stochastic phenomena in biological systems.
### introduction ###
Competence is the ability of a cell, usually a bacterium, to bind and internalize transforming exogenous DNA.
Under stressful environments, such as nutrient limitations, some cells enter competence while other cells commit irreversibly to sporulation.
Entry in competence is a transient probabilistic event that facilitates copying of the exogenous DNA CITATION, CITATION.
It has been shown that among a group of cells only a randomly chosen fraction enters in competence CITATION, CITATION.
Proper modeling and correctly accounting for noise in the model of this phenomenon is crucial to understanding the underlying biological explanation.
The few cells that enter competence express a high concentration of the key regulator ComK, which activates hundreds of genes, including the genes encoding the DNA-uptake and recombination systems CITATION CITATION.
Competence is understood as a bistability pattern CITATION, CITATION and the nonlinear system describing the competence regulatory circuit is an excitable dynamical system.
Auto-activation of the regulator ComK is responsible for the bistable response in competence development.
Auto-activation of ComK, is essential and can be sufficient to generate a bistable expression pattern CITATION CITATION.
Specifically, the concentration of an inducer must cross a certain threshold to start the positive feedback.
Different experimental studies concluded that an auto activation of ComK is the only needed factor for bistability to occur in the expression of this protein CITATION, CITATION, CITATION.
In CITATION, Smits et.
al discuss the factors that determine the required threshold for the activation of ComK and deduce that other transcription factors can raise or lower the threshold.
Although many proteins are involved in the regulation of competence, there are two main proteins that play a major role.
S el et al. CITATION propose a deterministic model driven by an additive noise to describe the dynamics of competence regulation.
We use the reduced order Stochastic Differential Equation model presented in CITATION to develop a discrete stochastic model for competence.
Calculating the probability and the expected time for entering and returning from competence, requires solving for the splitting probabilities and the first moment passage time.
The problem of calculating the first passage time has been studied heavily in the literature for the stochastic difference equations, Fokker Planck equations and some special cases of the CME.
For a detailed treatment of this topic see CITATION CITATION and references therein.
Researchers usually use Monte-Carlo simulations to calculate the distribution of the first passage time when working with he CME.
We propose in this work, an alternative approach that makes it possible to calculate the states in which the system will be as time evolves.
The main idea here is to aggregate regions of the state space over which specie evolve into absorbing states.
This technique is useful in analytically computing the distribution of the first passage time, by providing a way to deal with the infinite dimension of the state space over which the system evolves.
The contributions of this paper are threefold.
First, it provides a new method to calculate exact probabilities of biological phenomena where transient behaviors such as competence, which is the topic we chose to study here, occur.
Second, it shows how to calculate sensitivities of the probabilities of passing to the transient state with respect to the system's parameters.
Third, it gives a methodology to calculate the expected time that it takes a cell to return from its transient state.
All these methods can be used to analyze any biological system that has the characteristic of switching between two states, while staying for a while in the unstable state.
In this paper we start by describing the chemical reactions and the deterministic model.
We then generate the Chemical Master Equation of our proposed discrete stochastic model.
The CME characterizes the evolution of the probability density of the different discrete states.
We simulate it using the Stochastic Simulation Algorithm and show how the solution can be approximated using the Finite State Projection method.
We then conduct a sensitivity analysis studying the effect that the various system parameters have on the probability with which a cell enters in competence.
This analysis shows the usefulness of our proposed numerical method in analyzing the roles of the different affinity, transcription and degradation rates, etc., in driving the cellular switching.
Finally, we analyze the roles of these parameters in determining the expected time a cell stays in competence.
