Analytical Derivation of Moment Equations in Stochastic Chemical Kinetics.

The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments only provide limited information, especially when complex, non-linear dynamics are involved.

Model reduction of multiscale chemical langevin equations: a numerical case study.

Two very important characteristics of biological reaction networks need to be considered carefully when modeling these systems. First, models must account for the inherent probabilistic nature of systems far from the thermodynamic limit. Often, biological systems cannot be modeled with traditional continuous-deterministic models. Second, models must take into consideration the disparate spectrum of time scales observed in biological phenomena, such as slow transcription events and fast dimerization reactions. In the last decade, significant efforts have been expended on the development of stochastic chemical kinetics models to capture the dynamics of biomolecular systems, and on the development of robust multiscale algorithms, able to handle stiffness. In this paper, the focus is on the dynamics of reaction sets governed by stiff chemical Langevin equations, i.e., stiff stochastic differential equations. These are particularly challenging systems to model, requiring prohibitively small integration step sizes. We describe and illustrate the application of a semianalytical reduction framework for chemical Langevin equations that results in significant gains in computational cost.

SynBioSS: the synthetic biology modeling suite.

UNLABELLED: SynBioSS (Synthetic Biology Software Suite) is a suite of software for the modeling and simulation of synthetic genetic constructs. SynBioSS utilizes the registry of standard biological parts, a database of kinetic parameters, and both graphical and command-line interfaces to multiscale simulation algorithms. AVAILABILITY: SynBioSS is available under the GNU General Public License at http://synbioss.sourceforge.net.

An adaptive time step scheme for a system of stochastic differential equations with multiple multiplicative noise: chemical Langevin equation, a proof of concept.

Models involving stochastic differential equations (SDEs) play a prominent role in a wide range of applications where systems are not at the thermodynamic limit, for example, biological population dynamics. Therefore there is a need for numerical schemes that are capable of accurately and efficiently integrating systems of SDEs. In this work we introduce a variable size step algorithm and apply it to systems of stiff SDEs with multiple multiplicative noise. The algorithm is validated using a subclass of SDEs called chemical Langevin equations that appear in the description of dilute chemical kinetics models, with important applications mainly in biology. Three representative examples are used to test and report on the behavior of the proposed scheme. We demonstrate the advantages and disadvantages over fixed time step integration schemes of the proposed method, showing that the adaptive time step method is considerably more stable than fixed step methods with no excessive additional computational overhead.

Synthetic tetracycline-inducible regulatory networks: computer-aided design of dynamic phenotypes.

BACKGROUND: Tightly regulated gene networks, precisely controlling the expression of protein molecules, have received considerable interest by the biomedical community due to their promising applications. Among the most well studied inducible transcription systems are the tetracycline regulatory expression systems based on the tetracycline resistance operon of Escherichia coli, Tet-Off (tTA) and Tet-On (rtTA). Despite their initial success and improved designs, limitations still persist, such as low inducer sensitivity. Instead of looking at these networks statically, and simply changing or mutating the promoter and operator regions with trial and error, a systematic investigation of the dynamic behavior of the network can result in rational design of regulatory gene expression systems. Sophisticated algorithms can accurately capture the dynamical behavior of gene networks. With computer aided design, we aim to improve the synthesis of regulatory networks and propose new designs that enable tighter control of expression. RESULTS: In this paper we engineer novel networks by recombining existing genes or part of genes. We synthesize four novel regulatory networks based on the Tet-Off and Tet-On systems. We model all the known individual biomolecular interactions involved in transcription, translation, regulation and induction. With multiple time-scale stochastic-discrete and stochastic-continuous models we accurately capture the transient and steady state dynamics of these networks. Important biomolecular interactions are identified and the strength of the interactions engineered to satisfy design criteria. A set of clear design rules is developed and appropriate mutants of regulatory proteins and operator sites are proposed. CONCLUSION: The complexity of biomolecular interactions is accurately captured through computer simulations. Computer simulations allow us to look into the molecular level, portray the dynamic behavior of gene regulatory networks and rationally engineer novel ones with useful applications. We are able to propose, test and accept or reject design principles for each network. Guided by simulations, we develop a set of design principles for novel tetracycline-inducible networks.

Multiscale Hy3S: hybrid stochastic simulation for supercomputers.

BACKGROUND: Stochastic simulation has become a useful tool to both study natural biological systems and design new synthetic ones. By capturing the intrinsic molecular fluctuations of "small" systems, these simulations produce a more accurate picture of single cell dynamics, including interesting phenomena missed by deterministic methods, such as noise-induced oscillations and transitions between stable states. However, the computational cost of the original stochastic simulation algorithm can be high, motivating the use of hybrid stochastic methods. Hybrid stochastic methods partition the system into multiple subsets and describe each subset as a different representation, such as a jump Markov, Poisson, continuous Markov, or deterministic process. By applying valid approximations and self-consistently merging disparate descriptions, a method can be considerably faster, while retaining accuracy. In this paper, we describe Hy3S, a collection of multiscale simulation programs. RESULTS: Building on our previous work on developing novel hybrid stochastic algorithms, we have created the Hy3S software package to enable scientists and engineers to both study and design extremely large well-mixed biological systems with many thousands of reactions and chemical species. We have added adaptive stochastic numerical integrators to permit the robust simulation of dynamically stiff biological systems. In addition, Hy3S has many useful features, including embarrassingly parallelized simulations with MPI; special discrete events, such as transcriptional and translation elongation and cell division; mid-simulation perturbations in both the number of molecules of species and reaction kinetic parameters; combinatorial variation of both initial conditions and kinetic parameters to enable sensitivity analysis; use of NetCDF optimized binary format to quickly read and write large datasets; and a simple graphical user interface, written in Matlab, to help users create biological systems and analyze data. We demonstrate the accuracy and efficiency of Hy3S with examples, including a large-scale system benchmark and a complex bistable biochemical network with positive feedback. The software itself is open-sourced under the GPL license and is modular, allowing users to modify it for their own purposes. CONCLUSION: Hy3S is a powerful suite of simulation programs for simulating the stochastic dynamics of networks of biochemical reactions. Its first public version enables computational biologists to more efficiently investigate the dynamics of realistic biological systems.