Reverse-engineering of polynomial dynamical systems

2011

Elucidating the structure and/or dynamics of gene regulatory networks from experimental data is a major goal of systems biology. Stochastic models have the potential to absorb noise, account for uncertainty , and help avoid data overfitting. Within the frame work of probabilistic polynomial dynamical systems, we present an algorithm for the reverse engineering of any gene regulatory network as a discrete, probabilistic polynomial dynamical system. The resulting stochastic model is assembled from all minimal models in the model space and the probability assignment is based on partitioning the model space according to the likeliness with which a minimal model explains the observed data. We used this method to identify stochastic models for two published synthetic network models. In both cases, the generated model retains the key features of the original model and compares favorably to the resulting models from other algorithms.

2009

This paper gives a review of tools for the system identification of dynamic models for gene regulatory networks, using the modeling framework of polynomial dynamical systems over finite fields.

arXiv: Molecular Networks, 2018

Computation biology helps to understand all processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a reasonable time. For the last few years there has been a growing interest in biological theory connected to finite fields: the algebraic modeling tools used up to now are based on Grobner bases or Boolean group. Let n variables representing gene products, changing over the time on p values. A Polynomial dynamical system (PDS) is a function which has several components; each one is a polynom with n variables and coefficient in the finite field Z/pZ that model the evolution of gene products. We propose herein a method using algebraic separators, which are special polynomials abundantly studied in effective Galois theory. This approach avoids heavy calculations and provides a first Polynomial model in linear time.

Bioinformatics/computer Applications in The Biosciences, 2010

Motivation: An increasing number of discrete mathematical models are being published in systems biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks, and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. Results: This paper provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for model analysis.

Annals of the New York Academy of Sciences, 2007

We consider the problem of reverse-engineering dynamic models of biochemical networks from experimental data using polynomial dynamical systems. In earlier work, we developed an algorithm to identify minimal wiring diagrams, that is, directed graphs that represent the causal relationships between network variables. Here we extend this algorithm to identify a most likely dynamic model from the set of all possible dynamic models that fit the data over a fixed wiring diagram. To illustrate its performance, the method is applied to simulated time-course data from a published gene regulatory network in the fruitfly Drosophila melanogaster.

BMC Systems Biology, 2014

Background: The inference of gene regulatory networks (GRNs) from experimental observations is at the heart of systems biology. This includes the inference of both the network topology and its dynamics. While there are many algorithms available to infer the network topology from experimental data, less emphasis has been placed on methods that infer network dynamics. Furthermore, since the network inference problem is typically underdetermined, it is essential to have the option of incorporating into the inference process, prior knowledge about the network, along with an effective description of the search space of dynamic models. Finally, it is also important to have an understanding of how a given inference method is affected by experimental and other noise in the data used. Results: This paper contains a novel inference algorithm using the algebraic framework of Boolean polynomial dynamical systems (BPDS), meeting all these requirements. The algorithm takes as input time series data, including those from network perturbations, such as knockout mutant strains and RNAi experiments. It allows for the incorporation of prior biological knowledge while being robust to significant levels of noise in the data used for inference. It uses an evolutionary algorithm for local optimization with an encoding of the mathematical models as BPDS. The BPDS framework allows an effective representation of the search space for algebraic dynamic models that improves computational performance. The algorithm is validated with both simulated and experimental microarray expression profile data. Robustness to noise is tested using a published mathematical model of the segment polarity gene network in Drosophila melanogaster. Benchmarking of the algorithm is done by comparison with a spectrum of state-of-the-art network inference methods on data from the synthetic IRMA network to demonstrate that our method has good precision and recall for the network reconstruction task, while also predicting several of the dynamic patterns present in the network. Conclusions: Boolean polynomial dynamical systems provide a powerful modeling framework for the reverse engineering of gene regulatory networks, that enables a rich mathematical structure on the model search space. A

2016

Discrete models of gene regulatory networks have gained popularity in computational systems biology over the last dozen years. However, not all discrete network models reflect the behaviors of real biological systems. In this work, we focus on two model selection methods and algebraic geometry arising from these model selection methods. The first model selection method involves biologically relevant functions. We begin by introducing k-canalizing functions, a generalization of nested canalizing functions. We extend results on nested canalizing functions and derived a unique extended monomial form of arbitrary Boolean functions. This gives us a stratification of the set of n-variable Boolean functions by canalizing depth. We obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions. We characterize the set of k-canalizing functions as an algebraic variety in...

Journal of Theoretical Biology, 2004

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. The complexity of the algorithm is quadratic in the number of variables and cubic in the number of time points. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster.

Bulletin of Mathematical Biology, 2014

Identification of models of gene regulatory networks is sensitive to the amount of data used as input. Considering the substantial costs in conducting experiments, it is of value to have an estimate of the amount of data required to infer the network structure. To minimize wasted resources, it is also beneficial to know which data are necessary to identify the network. Knowledge of the data and knowledge of the terms in polynomial models are often required a priori in model identification. In applications, it is unlikely that the structure of a polynomial model will be known, which may force data sets to be unnecessarily large in order to identify a model. Furthermore, none of the known results provides any strategy for constructing data sets to uniquely identify a model. We provide a specialization of an existing criterion for deciding when a set of data points identifies a minimal polynomial model when its monomial terms have been specified. Then, we relax the requirement of the knowledge of the monomials and present results for model identification given only the data. Finally, we present a method for constructing data sets that identify minimal polynomial models.

2005

We study the Finite State Linear Model (FSLM) for modelling gene regulatory networks proposed by A.Brazma and T.Schlitt in [4]. The model incorporates biologically intuitive gene regulatory mechanism similar to that in Boolean networks, and can describe also the continuous changes in protein levels. We consider several theoretical properties of this model; in particular we show that the problem whether a particular gene will reach an active state is algorithmically unsolvable. This imposes some practical difficulties in simulation and reverse engineering of FSLM networks. Nevertheless, our simulation experiments show that sufficiently many of FSLM networks exhibit a regular behaviour and that the model is still quite adequate to describe biological reality.