A computational algebra approach to the reverse engineering of gene regulatory networks

Abstract

This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. An analysis of the complexity of the algorithm is performed. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster. r

Boolean monomial control systems

“…Since PathSim is a stochastic agentiii based computer simulation, Dr. Laubenbacher's idea was to use the average output of PathSim as data to construct (or to "reverse engineer") a deterministic, time discrete dynamical system over a suitable finite field. To this end, he and some of his graduate students had developed some specific methods 1 [65], [35]. This approach needed to be analyzed and tested, so, I performed the mathematical analysis of the reverse engineering method described in [65].…”

On the Algebraic Geometry of Polynomial Dynamical Systems

“…Modifications of the algorithm in [20] have been constructed. The algorithm in [13] starts with only data as input and computes all possible minimal wiring diagrams of polynomial models that fit the given data and outputs a most likely one, based on one of several possible model scoring methods.…”

“…Another approach to the problem of dependency of the model selection process in [20] on the chosen term order is taken in [6]. The algorithm there uses the Gröbner fan of the ideal of points as a computational tool to find a most likely wiring diagram.…”

Algebraic Methods in Mathematical Biology

“…At the genomic level, biology is essentially digital, and so it is not surprising that combinatorics has been used very successfully there, most spectacularly in connection with the Human Genome Project, but also in other areas, such as the study of secondary RNA structures (Bakhtin and Heitsch 2009). Searching the PubMed database of literature related to biomedical research, one now finds papers utilizing Lie algebras (Sanchez et al 2006), the Riemann Mapping Theorem (Hurdal and Stephenson 2009), and methods from algebraic topology (Singh et al 2008) and algebraic geometry (Wang et al 2005;Laubenbacher and Stigler 2004), to name a few topics not traditionally found in the applied mathematics repertoire. As the biological challenges evolve, there is every reason to expect that more and more fields of mathematics will be in a position to contribute new points of view and new approaches.…”

“…Here the collection of siphons is encoded as an ideal, a particular kind of algebraic object in a polynomial ring, which can then be studied using theoretical tools from algebraic geometry. Another biological problem that can be translated into a problem in computational algebraic geometry is the inference of gene regulatory networks from experimental data (Laubenbacher and Stigler 2004). Here, the polynomials arise as equations that govern the evolution of a time-discrete dynamical system over a finite state space.…”