Reconstruction of multimodal distributions for hybrid moment-based chemical kinetics

Abstract

The stochastic dynamics of biochemical reaction networks can be accurately described by discrete-state Markov processes where each chemical reaction corresponds to a state transition of the process. Due to the largeness problem of the state space, analysis techniques based on an exploration of the state space are often not feasible and the integration of the moments of the underlying probability distribution has become a very popular alternative. In this paper the focus is on a comparison of reconstructed distributions from their moments obtained by two different moment-based analysis methods, the method of moments (MM) and the method of conditional moments (MCM). We use the maximum entropy principle to derive a distribution that fits best to a given sequence of (conditional) moments. For the two gene regulatory networks that we consider we find that the MCM approach is more suitable to describe multimodal distributions and that the reconstruction of marginal distributions is more accurate if conditional distributions are considered.

Minimal moment equations for stochastic models of biochemical reaction networks with partially finite state space

“…The results revealed that the implications of a partially finite state space had not been completely understood in earlier studies and that the number of classical moment equations that are really needed is smaller than what was previously claimed. 11,12,35 The results of this paper can also be seen as the basis for a hybrid approach for the analysis of stochastic models of biochemical reaction networks. The binary variable representation of the network, and the derivation of moment equations from it, automatically keeps the full probability distribution over the finite part of the state space while resorting to low-order moments over the infinite part.…”

“…Together with the fact that the classical moments can be recovered from the conditional moments, one reaches the mathematically quite unintuitive conclusion that more properties of the underlying probability distribution can be captured with a smaller number of equations. 11,12 In this paper, I will show that this result is only obtained because the N M, L classical moment equations contain many equations that are redundant. Specifically, while Eq.…”

“…Recently, it has been argued that in such cases, the stochastic dynamics of the network can be described more accurately with fewer equations through the use of conditional moments. 11,12 Here, I show that the result that less a) Electronic mail: jruess@ist.ac.at equations are needed is only true if the specific structure of the reaction network is not taken into account in the derivation of the classical unconditional moment equations. I provide a formula to determine how many classical moment equations are minimally needed to compute all moments up to any desired order L, and show that, for all L, this number is smaller than the number of conditional moment equationsin line with the mathematical intuition that more accurate descriptions of the whole underlying probability distribution cannot be obtained with less equations.…”

Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions

“…Historically related to the discrete models of Stuart Kauffman [11] and René Thomas [12], process hitting attempts to address problems of scalability in classical modeling methods while maintaining the highest degree of expressiveness possible. Formally a subclass of asynchronous automata, it relies on large degrees of abstraction to describe the system as a whole.…”

“…In the most general sense, modeling approaches can be thought of as being either quantitative or qualitative. Quantitative methods, such as ordinary differential equations or the chemical master equation, are widespread in the literature [1][2][3][4][5][6][7][8][9][10][11]; when the model is well developed, the detail therein can be incredibly informative. However, these methods are not well suited for all applications.…”