Modeling Subtilin Production in Bacillus subtilis Using Stochastic Hybrid Systems

The genetic network regulating the biosynthesis of subtilin in Bacillus subtilis is modeled as a stochastic hybrid system. The continuous state of the hybrid system is the concentrations of subtilin and various regulating proteins, whose productions are controlled by switches in the genetic network that are in turn modeled as Markov chains. Some preliminary results are given by both analysis and simulations. 1 Background of Subtilin Production In order to survive, bacteria develop a number of strategies to cope with harsh environmental conditions. One of the survival strategies employed by bacteria is the release of antibiotics to eliminate competing microbial species in the same ecosystem [15]. It is observed that the production of antibiotics in the cells is affected by not only the environmental stimuli (e.g. nutrient levels, aeration, etc.) but also the local population density of their own species [12]. Therefore, the physiological states of the cell and the external signals both contribute to the regulation of antibiotic synthesis. Our study focuses on the subtilin, an antibiotic produced by Bacillus subtilis ATCC 6633, because the genetics of subtilin is known and its biosynthetic pathways are well characterized [2, 7, 11]. We briefly describe the production process of subtilin in B. subtilis. It is shown in [19] that the production is controlled by two independent mechanisms. When the foods are abundant, the population proliferates and the cells produce very little amount (non-lethal dose) of subtilin. However, when the foods become scarce, the production of subtilin picks up as follows. First, sigma-H (SigH), a sigma factor that regulates gene expression, enables the production of SpaRK (SpaR and SpaK) proteins by binding to the promoter regions of their genes (spaR and spaK). The membrane-bound SpaK protein senses the extracellular subtilin accumulating in the environment as the cell colony becomes large, and activates the SpaR protein. The activated SpaR (SpaR∼p) in turn directs the productions of the subtilin structural peptide SpaS, the biosynthesis complex SpaBTC which modifies SpaS to yield the final product subtilin, and the immunity machinery SpaIFEG which protects the cell against the killing effect of ? This research is partially supported by the National Science Foundation under Grant No. EIA-0122599. Fig. 1. Schematic representation of subtilin biosynthesis, immunity, and regulation in Bacillus subtilis. Subtilin prepeptide SpaS is modified, cleaved, and translocated across the cell membrane by the subtilin synthetase complex SpaBTC. The genes of subtilin are organized in an operon-like structure (spaBTC, spaS, spaIFEG, and spaRK) so that each functional unit is transcribed together. The extra-cellular subtilin functions as pheromone that regulates its own synthesis via an autoinduction feedback pathway. subtilin. See Fig. 1 for a schematic representation of the biosynthesis process of subtilin in B. subtilis. In this work, a simplified version of the production process is adopted for ease of study. Namely, we ignore the dynamics in the post-translational processing of SpaS by SpaBTC to form mature subtilin and the signal transduction between SpaK and SpaR. Hence, the amount of SpaS is assumed to be equivalent to the amount of subtilin released by the cell and the SpaK and SpaR proteins are considered as one protein species. From the above description, a dynamical model of subtilin production consists of two parts: discrete events and continuous dynamics. The discrete events include the initiations and terminations of transcriptions of various genes due to the binding and unbinding of their transcription regulators to their promoter regions, while the continuous dynamics include the accumulations and degradations of the protein species after the expressions of their genes are being switched on and off, respectively. Thus, hybrid systems can be a suitable choice for such a model. Furthermore, in cellular networks involving coupled genetic and biochemical reactions, the expressions of genes are intrinsically non-deterministic, as is evidenced by the random fluctuations (noise) in the concentrations of protein species in the cell population, even in an isogenic culture [6]. One reason for the stochasticity in gene expressions is the small copy number of interacting molecular species (e.g. regulatory proteins and genes) in the relatively large cell volume [13, 14, 16], since in a chemical system with extremely low concentrations of reacting species, a reaction (collision of the reacting molecules) occurs in a short time interval and is best viewed as a probabilistic event [9]. This is also the case in our example, as an individual B. subtilis cell has only one copy of each spa gene for the corresponding spa protein. Techniques such as stochastic differential equations and Monte Carlo algorithms are often used to model and simulate such biological systems with noise (see [17] for a review). Although stochastic algorithms are computationally involved, they produce a more realistic and complete description of the time-dependent behavior of biochemical systems than deterministic algorithms. In this paper, we adopt such a stochastic point of view and propose a stochastic hybrid systems approach to analyze the dynamics of the spa genes in the subtilin system. We also present simulation results of the subtilin regulatory network to demonstrate the distinctive behaviors of the systems using the deterministic and stochastic modeling formalisms. 2 Stochastic Hybrid Systems A framework first proposed by the engineering community to model systems that exhibit both continuous dynamics and discrete state changes, hybrid systems have in recent years found increasingly wide applications in many practical fields. In particular, applications of hybrid systems in modeling of biological systems can be found in [8, 1], to name a few. However, most of the models proposed in the literature so far are deterministic, and are not suitable to model system with inherent randomness. This is the case, for example, in cellular processes modeling, especially when the number of participating cells is not large enough and random fluctuations within single cells cannot be ignored. Hence one needs to extend the framework to a more general class of hybrid systems with built-in randomness, namely, stochastic hybrid systems. There have already been some work on stochastic hybrid systems (see, for example, [10, 3]). In this section we shall present a simple model suitable for our main example, the production of subtilin, to be given in the next section. Basically, the state of the stochastic hybrid system consists of two parts, a continuous one and a discrete one. The discrete state has a finite number of possible values, and evolves randomly according to a Markov chain. The continuous state, on the other hand, takes value in a certain Euclidean space, and evolves deterministically according to some ordinary differential equations. The dynamics of the discrete and the continuous states are coupled in the following sense: on one hand, the transition probabilities of the Markov chain for the discrete state depend on the value of the continuous state at the moment of jump; on the other hand, the differential equations governing the evolution of the continuous state are different when the discrete state takes on different values. More formally, we consider the following model of a stochastic hybrid system. Its state (q, x) consists of a discrete part (mode) q taking values in a finite set Q and a continuous part x taking values in a Euclidean space R, n ≥ 1. Both q and x are functions of time t and their dynamics are specified in the following. – (Discrete Dynamics) The discrete state q follows a Markov chain that jumps at epochs t = 0, ∆, 2∆, . . . of constant interval ∆ > 0 with transition proba-