Probability distributions for multimeric systems

Dimensionality reduction via path integration for computing mRNA distributions

Inherent stochasticity in gene expression leads to distributions of mRNA copy numbers in a population of identical cells. These distributions are determined primarily by the multitude of states of a gene promoter, each driving transcription at a different rate. In an era where single-cell mRNA copy number data are more and more available, there is an increasing need for fast computations of mRNA distributions. In this paper, we present a method for computing separate distributions for each species of mRNA molecules, i.e. mRNAs that have been either partially or fully processed post-transcription. The method involves the integration over all possible realizations of promoter states, which we cast into a set of linear ordinary differential equations of dimension [Formula: see text], where M is the number of available promoter states and [Formula: see text] is the mRNA copy number of species j up to which one wishes to compute the probability distribution. This approach is superior to solving the Master equation (ME) directly in two ways: (a) the number of coupled differential equations in the ME approach is [Formula: see text], where [Formula: see text] is the cutoff for the probability of the jth species of mRNA; and (b) the ME must be solved up to the cutoffs [Formula: see text], which must be selected a priori. In our approach, the equation for the probability to observe n mRNAs of any species depends only on the the probability of observing [Formula: see text] mRNAs of that species, thus yielding a correct probability distribution up to an arbitrary n. To demonstrate the validity of our derivations, we compare our results with Gillespie simulations for ten randomly selected system parameters.

A Hybrid of the Chemical Master Equation and the Gillespie Algorithm for Efficient Stochastic Simulations of Sub-Networks

Modeling stochastic behavior of chemical reaction networks is an important endeavor in many aspects of chemistry and systems biology. The chemical master equation (CME) and the Gillespie algorithm (GA) are the two most fundamental approaches to such modeling; however, each of them has its own limitations: the GA may require long computing times, while the CME may demand unrealistic memory storage capacity. We propose a method that combines the CME and the GA that allows one to simulate stochastically a part of a reaction network. First, a reaction network is divided into two parts. The first part is simulated via the GA, while the solution of the CME for the second part is fed into the GA in order to update its propensities. The advantage of this method is that it avoids the need to solve the CME or stochastically simulate the entire network, which makes it highly efficient. One of its drawbacks, however, is that most of the information about the second part of the network is lost in the process. Therefore, this method is most useful when only partial information about a reaction network is needed. We tested this method against the GA on two systems of interest in biology - the gene switch and the Griffith model of a genetic oscillator—and have shown it to be highly accurate. Comparing this method to four different stochastic algorithms revealed it to be at least an order of magnitude faster than the fastest among them.