Finite State Projection Method Research Articles

A finite state projection method for steady-state sensitivity analysis of stochastic reaction networks.

Consider the standard stochastic reaction network model where the dynamics is given by a continuous-time Markov chain over a discrete lattice. For such models, estimation of parameter sensitivities is an important problem, but the existing computational approaches to solve this problem usually require time-consuming Monte Carlo simulations of the reaction dynamics. Therefore, these simulation-based approaches can only be expected to work over finite time-intervals, while it is often of interest in applications to examine the sensitivity values at the steady-state after the Markov chain has relaxed to its stationary distribution. The aim of this paper is to present a computational method for the estimation of steady-state parameter sensitivities, which instead of using simulations relies on the recently developed stationary finite state projection algorithm [Gupta et al., J. Chem. Phys. 147, 154101 (2017)] that provides an accurate estimate of the stationary distribution at a fixed set of parameters. We show that sensitivity values at these parameters can be estimated from the solution of a Poisson equation associated with the infinitesimal generator of the Markov chain. We develop an approach to numerically solve the Poisson equation, and this yields an efficient estimator for steady-state parameter sensitivities. We illustrate this method using several examples.

Understanding the finite state projection and related methods for solving the chemical master equation

The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP.

Computational study of p53 regulation via the chemical master equation

A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1–16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.

Modern Numerical Methods for Continuous Time Markov Chains of High-Dimensions

The aim of the talk is to present modern methods for computing large Continuous Time Markov Chains (CTMC) which arise in Biology. These modern methods have theoretically and experimentally overcome key obstacles of dimension and computationinherent in high dimensional CTMC [1,2,3]. Continuous Time Markov Chains (CTMC) are a key tool in describing discretely interacting biological systems. Biological processes such as RNA transcription, signalling cascades and catalysis are key examples where the system is being driven by independent inhomogeneous Poisson processes. Researchers have used realisation based simulation methods, such as the Stochastic Simulation Algorithm (SSA), forin-silico observations of these systems. Even though simulations are cheap and quick to implement, it has been shown that they do not always guarantee the capture of critical features such as bi-modality, without multiple realisations. In this talk, we will be presenting the Optimal Finite State Projection method, proposed by Sunkara and Hegland [1] and the Hybrid method proposed by Jahnke andSchutte independently [2,3]. For the two methods, we will discuss their respective error bounds and computation times for particular examples.

DNA looping increases the range of bistability in a stochastic model of the lac genetic switch

Conditions and parameters affecting the range of bistability of the lac genetic switch in Escherichia coli are examined for a model which includes DNA looping interactions with the lac repressor and a lactose analogue. This stochastic gene–mRNA–protein model of the lac switch describes DNA looping using a third transcriptional state. We exploit the fast bursting dynamics of mRNA by combining a novel geometric burst extension with the finite state projection method. This limits the number of protein/mRNA states, allowing for an accelerated search of the model's parameter space. We evaluate how the addition of the third state changes the bistability properties of the model and find a critical region of parameter space where the phenotypic switching occurs in a range seen in single molecule fluorescence studies. Stochastic simulations show induction in the looping model is preceded by a rare complete dissociation of the loop followed by an immediate burst of mRNA rather than a slower build up of mRNA as in the two-state model. The overall effect of the looped state is to allow for faster switching times while at the same time further differentiating the uninduced and induced phenotypes. Furthermore, the kinetic parameters are consistent with free energies derived from thermodynamic studies suggesting that this minimal model of DNA looping could have a broader range of application.

Parallelising the finite state projection method

Many realistic mathematical models of biological and chemical systems, such as enzyme cascades and gene regulatory networks, need to include stochasticity. These systems are described as Markov processes and are modelled using the Chemical Master Equation. The Chemical Master Equation is a differential-difference equation (continuous in time and discrete in the state space) for the probability of a certain state at a given time. The state space is the population count of species in the system. A successful method for computing the Chemical Master Equation is the Finite State Projection Method. We give a new algorithm to distribute the Finite State Projection Method method onto multi-core systems. This method is called the Parallel Finite State Projection method. This article also analyses the theory needed for parallelisation of the Chemical Master Equation. References cmepy, Jan 2011. https://github.com/fcostin/cmepy K. Burrage, M. Hegland, S. MacNamara, and R.B. Sidje. A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modeling of biological systems. In: Langville, A. N., Stewart, W. J. (Eds.), Proceedings of the 150th Markov Anniversary Meeting, Boson Books, pp. 21--38, 2006. S. Engblom. Numerical Solution Methods in Stochastic Chemical Kenetics. PhD thesis, Uppsala University, 2008. M. Hegland, A. Hellander, and P. Lotstedt. Sparse grid and hybrid methods for the chemical master equation. BIT Numerical Mathematics, 48(2):265--283, 2008. doi:10.1007/s10543-008-0174-z M. Khammash and B. Munsky. The finite state projection algorithm for the solution of the chemical master equation. Journal of Chemical Physics, 124(044104):1--12, 2006. doi:10.1063/1.2145882 S. MacNamara, A.M. Bersani, K. Burrage, and R.B. Sidje. Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation. Journal of Chemical Physics, 129(095105):1--13, 2008. doi:10.1063/1.2971036 V. Sunkara and M. Hegland. An optimal finite state projection method. Procedia Computer Science, 1(1):1579--1586, 2010. ICCS 2010. http://www.sciencedirect.com/science/article/pii/S187705091000178X, doi:10.1016/j.procs.2010.04.177

An optimal Finite State Projection Method

Abstract It is well known that many realistic mathematical models of biological and chemical systems, such as enzyme cascades and gene regulatory networks, need to include stochasticity. These systems can be described as Markov processes and are modelled using the Chemical Master Equation (CME). The CME is a differential-difference equation (continuous in time and discrete in the state space) for the probability of certain state at a given time. The state space is the population count of species in the system. A successful method for computing the CME is the Finite State Projection Method (FSP). In this paper we will give the mathematical background of the FSP and propose a new addition to the FSP which guaranties our approximation to have optimal order.

The diffusive finite state projection algorithm for efficient simulation of the stochastic reaction-diffusion master equation.

We have developed a computational framework for accurate and efficient simulation of stochastic spatially inhomogeneous biochemical systems. The new computational method employs a fractional step hybrid strategy. A novel formulation of the finite state projection (FSP) method, called the diffusive FSP method, is introduced for the efficient and accurate simulation of diffusive transport. Reactions are handled by the stochastic simulation algorithm.

A multiple time interval finite state projection algorithm for the solution to the chemical master equation

At the mesoscopic scale, chemical processes have probability distributions that evolve according to an infinite set of linear ordinary differential equations known as the chemical master equation (CME). Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed finite state projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds. For many non-trivial systems, the original FSP technique has been shown to yield accurate approximations to the CME solution. Other systems may require a projection that is still too large to be solved efficiently; for these, the linearity of the FSP allows for many model reductions and computational techniques, which can increase the efficiency of the FSP method with little or no loss in accuracy. In this paper, we present a new approach for choosing and expanding the projection for the original FSP algorithm. Based upon this approach, we develop a new algorithm that exploits the linearity property of super-position. The new algorithm retains the full accuracy guarantees of the original FSP approach, but with significantly increased efficiency for some problems and a greater range of applicability. We illustrate the benefits of this algorithm on a simplified model of the heat shock mechanism in Escherichia coli.

The finite state projection algorithm for the solution of the chemical master equation

This article introduces the finite state projection (FSP) method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation (CME). Unlike Monte Carlo methods such as the stochastic simulation algorithm (SSA) or tau leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and tau leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than tau leaping methods.