Efficient moment matrix generation for arbitrary chemical networks

Proc. Natl. Acad. Sci. U.S.A.

2013

103

106

Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higherorder moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.stochastic models | information theory | entropy maximization | statistical mechanics T he fabric of all things living is discrete and noisy, individual molecules in perpetual random motion. However, humans, in our effort to understand and manipulate the biological cosmos, have historically perceived and modeled nature as large collections of molecules with behaviors not far from an expected average. Mathematical models, founded on such determinism, may be excellent approximations of reality when the number of molecules is very large, approaching the limit of an infinitely sized molecular population (1-5). Of course, the size of biomolecular systems is far from infinite. And we know that the behavior of a few molecules fluctuating from the average in unexpected ways may forever seal the fate of a living organism. It has thus been commonly recognized that models of small, evolving molecular populations better account for the noisy, probabilistic nature of outcomes (6-8).The most complete model of stochastically evolving molecular populations is one based on the master probability equation (9). The "master" in the name reflects the all-encompassing nature of an equation that purports to govern all possible outcomes for all time. Because of its ambitious character, the master equation has remained unsolved for all but the simplest of molecular interaction networks, even though it is now over seven decades since the first master equations were set up for chemical systems (10,11). Herein we ...

“…Generating the matrices in Eq. 2 can be performed either analytically (12,13) or numerically (14). For linear systems with only zeroth or first-order reactions, A′ is empty.…”

mentioning

confidence: 99%

A closure scheme for chemical master equations

Smadbeck

Proc. Natl. Acad. Sci. U.S.A.

2013

103

106

“…On the right, μ ’ is the vector of higher-order moments, and the corresponding matrix A’ . Generating the matrices in Equation 2 can be performed either analytically [36, 37] or numerically [38]. For linear systems with only 0 th or 1 st -order reactions, A’ is empty.…”

Section: Methods For Multiscale Models Of Biomolecular Systemsmentioning

confidence: 99%

Chemical Master Equation Closure for Computer-Aided Synthetic Biology

Smadbeck

Methods in Molecular Biology

2014

SUMMARY With inexpensive DNA synthesis technologies, we can now construct biological systems by quickly piecing together DNA sequences. Synthetic biology is the promising discipline that focuses on the construction of these new biological systems. Synthetic biology is an engineering discipline, and as such, it can benefit from mathematical modeling. This chapter focuses on mathematical models of biological systems. These models take the form of chemical reaction networks. The importance of stochasticity is discussed and methods to simulate stochastic reaction networks are reviewed. A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks for over seventy years. With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized. We anticipate that models based on the closure scheme described herein may assist in rationally designing synthetic biological systems.

“…Therefore, it is a common method in probability theory and statistics to employ moment generating functions instead of working directly with probability distributions. The mathematical details behind this transformation will be omitted for simplicity, but they can be found in other references (Smadbeck and Kaznessis, 2012; Sotiropoulos and Kaznessis, 2010). …”

Section: Theoretical Backgroundmentioning

confidence: 99%

Maximum entropy prediction of non-equilibrium stationary distributions for stochastic reaction networks with oscillatory dynamics

Constantino

Chemical Engineering Science

2017

Many chemical reaction networks in biological systems present complex oscillatory dynamics. In systems such as regulatory gene networks, cell cycle, and enzymatic processes, the number of molecules involved is often far from the thermodynamic limit. Although stochastic models based on the probabilistic approach of the Chemical Master Equation (CME) have been proposed, studies in the literature have been limited by the challenges of solving the CME and the lack of computational power to perform large-scale stochastic simulations. In this paper, we show that the infinite set of stationary moment equations describing the stochastic Brusselator and Schnakenberg oscillatory reactions networks can be truncated and solved using maximization of the entropy of the distributions. The results from our numerical experiments compare with the distributions obtained from well-established kinetic Monte Carlo methods and suggest that the accuracy of the prediction increases exponentially with the closure order chosen for the system. We conclude that maximum entropy models can be used as an efficient closure scheme alternative for moment equations to predict the non-equilibrium stationary distributions of stochastic chemical reactions with oscillatory dynamics. This prediction is accomplished without any prior knowledge of the system dynamics and without imposing any biased assumptions on the mathematical relations among species involved.