Alan Veliz-Cuba

Steady state analysis of Boolean molecular network models via model reduction and computational algebra

BackgroundA key problem in the analysis of mathematical models of molecular networks is the determination of their steady states. The present paper addresses this problem for Boolean network models, an increasingly popular modeling paradigm for networks lacking detailed kinetic information. For small models, the problem can be solved by exhaustive enumeration of all state transitions. But for larger models this is not feasible, since the size of the phase space grows exponentially with the dimension of the network. The dimension of published models is growing to over 100, so that efficient methods for steady state determination are essential. Several methods have been proposed for large networks, some of them heuristic. While these methods represent a substantial improvement in scalability over exhaustive enumeration, the problem for large networks is still unsolved in general.ResultsThis paper presents an algorithm that consists of two main parts. The first is a graph theoretic reduction of the wiring diagram of the network, while preserving all information about steady states. The second part formulates the determination of all steady states of a Boolean network as a problem of finding all solutions to a system of polynomial equations over the finite number system with two elements. This problem can be solved with existing computer algebra software. This algorithm compares favorably with several existing algorithms for steady state determination. One advantage is that it is not heuristic or reliant on sampling, but rather determines algorithmically and exactly all steady states of a Boolean network. The code for the algorithm, as well as the test suite of benchmark networks, is available upon request from the corresponding author.ConclusionsThe algorithm presented in this paper reliably determines all steady states of sparse Boolean networks with up to 1000 nodes. The algorithm is effective at analyzing virtually all published models even those of moderate connectivity. The problem for large Boolean networks with high average connectivity remains an open problem.

Stochastic Models of Evidence Accumulation in Changing Environments

Organisms and ecological groups accumulate evidence to make decisions. Classic experiments and theoretical studies have explored this process when the correct choice is fixed during each trial. However, we live in a constantly changing world. What effect does such impermanence have on classical results about decision making? To address this question we use sequential analysis to derive a tractable model of evidence accumulation when the correct option changes in time. Our analysis shows that ideal observers discount prior evidence at a rate determined by the volatility of the environment, and the dynamics of evidence accumulation is governed by the information gained over an average environmental epoch. A plausible neural implementation of an optimal observer in a changing environment shows that, in contrast to previous models, neural populations representing alternate choices are coupled through excitation. Our work builds a bridge between statistical decision making in volatile environments and stochast...

Dimension Reduction of Large Sparse AND-NOT Network Models

In this manuscript we propose and implement a dimension reduction algorithm of AND-NOT networks for the purpose of steady state computation. Our method of network reduction consists in using steady state approximations that do not change the number of steady states. The algorithm is designed to work at the wiring diagram level without the need to evaluate or simplify Boolean functions. Also, our implementation of the algorithm takes advantage of the sparsity typical of discrete models of biological systems.The main features of our reduction algorithm are that it works at the wiring diagram level and it preserves the number of steady states. Furthermore, the steady states of the original network can be recovered from the steady states of the reduced network; thus, all steady states are found. Also, heuristic analysis and simulations show that it runs in polynomial time. We used our results to study AND-NOT network models of gene networks and showed that our algorithm greatly simplifies steady state analysis. Furthermore, our algorithm can handle sparse AND-NOT networks with up to 1,000,000 nodes.

Algebraic Models and Their Use in Systems Biology

Progress in systems biology relies on the use of mathematical and statistical models for system-level studies of biological processes. Several different modeling frameworks have been used successfully, including traditional differential-equation-based models, a variety of stochastic models, agent-based models, and Boolean networks, to name some common ones. This chapter focuses on discrete models, and describes a mathematical approach to the construction and analysis of discrete models which relies on combinatorics and computational algebraic geometry. The underlying mathematical concept is that of a polynomial dynamical system over a finite field. Examples are given of the advantages of this approach, and several applications are discussed.

Piecewise linear and Boolean models of chemical reaction networks.

Models of biochemical networks are frequently complex and high-dimensional. Reduction methods that preserve important dynamical properties are therefore essential for their study. Interactions in biochemical networks are frequently modeled using Hill functions (

Effects of cell cycle noise on excitable gene circuits

x^n/(J^n+x^n)

Evidence accumulation and change rate inference in dynamic environments

xn/(Jn+xn)). Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent