Reverse Engineering Discrete Dynamical Systems from Data Sets with Random Input Vectors

Model construction of Boolean network via observed data

In this paper, a set of data is assumed to be obtained from an experiment that satisfies a Boolean dynamic process. For instance, the dataset can be obtained from the diagnosis of describing the diffusion process of cancer cells. With the observed datasets, several methods to construct the dynamic models for such Boolean networks are proposed. Instead of building the logical dynamics of a Boolean network directly, its algebraic form is constructed first and then is converted back to the logical form. Firstly, a general construction technique is proposed. To reduce the size of required data, the model with the known network graph is considered. Motivated by this, the least in-degree model is constructed that can reduce the size of required data set tremendously. Next, the uniform network is investigated. The number of required data points for identification of such networks is independent of the size of the network. Finally, some principles are proposed for dealing with data with errors.

Comparison of Co-Temporal Modeling Algorithms on Sparse Experimental Time Series Data Sets

Multiple approaches for reverse-engineering biological networks from time-series data have been proposed in the computational biology literature. These approaches can be classified by their underlying mathematical algorithms, such as Bayesian or algebraic techniques, as well as by their time paradigm, which includes next-state and co-temporal modeling. The types of biological relationships, such as parent-child or siblings, discovered by these algorithms are quite varied. It is important to understand the strengths and weaknesses of the various algorithms and time paradigms on actual experimental data. We assess how well the co-temporal implementations of three algorithms, continuous Bayesian, discrete Bayesian, and computational algebraic, can 1) identify two types of entity relationships, parent and sibling, between biological entities, 2) deal with experimental sparse time course data, and 3) handle experimental noise seen in replicate data sets. These algorithms are evaluated, using the shuffle index metric, for how well the resulting models match literature models in terms of siblings and parent relationships. Results indicate that all three co-temporal algorithms perform well, at a statistically significant level, at finding sibling relationships, but perform relatively poorly in finding parent relationships.

A trade-off between sample complexity and computational complexity in learning Boolean networks from time-series data

A key problem in molecular biology is to infer regulatory relationships between genes from expression data. This paper studies a simplified model of such inference problems in which one or more Boolean variables, modeling, for example, the expression levels of genes, each depend deterministically on a small but unknown subset of a large number of Boolean input variables. Our model assumes that the expression data comprises a time series, in which successive samples may be correlated. We provide bounds on the expected amount of data needed to infer the correct relationships between output and input variables. These bounds improve and generalize previous results for Boolean network inference and continuous-time switching network inference. Although the computational problem is intractable in general, we describe a fixed-parameter tractable algorithm that is guaranteed to provide at least a partial solution to the problem. Most interestingly, both the sample complexity and computational complexity of the problem depend on the strength of correlations between successive samples in the time series but in opposing ways. Uncorrelated samples minimize the total number of samples needed while maximizing computational complexity; a strong correlation between successive samples has the opposite effect. This observation has implications for the design of experiments for measuring gene expression.

List-decoding methods for inferring polynomials in finite dynamical gene network models

MOTIVATION

The problem of reverse engineering the dynamics of gene expression profiles is of focal importance in systems biology. Due to noise and the inherent lack of sufficiently large datasets generated via high-throughput measurements, known reconstruction frameworks based on dynamical systems models fail to provide adequate settings for network analysis. This motivates the study of new approaches that produce stochastic lists of explanations for the observed network dynamics that can be efficiently inferred from small sample sets and in the presence of errors.

RESULTS

We introduce a novel algebraic modeling framework, termed stochastic polynomial dynamical systems (SPDSs) that can capture the dynamics of regulatory networks based on microarray expression data. Here, we refer to dynamics of the network as the trajectories of gene expression profiles over time. The model assumes that the expression data is quantized in a manner that allows for imposing a finite field structure on the observations, and the existence of polynomial update functions for each gene in the network. The underlying reverse engineering algorithm is based on ideas borrowed from coding theory, and in particular, list-decoding methods for so called Reed-Muller codes. The list-decoding method was tested on synthetic data and on microarray expression measurements from the M(3D) database, corresponding to a subnetwork of the Escherichia coli SOS repair system, as well as on the complete transcription factor network, available at RegulonDB. The results show that SPDSs constructed via list-decoders significantly outperform other algebraic reverse engineering methods, and that they also provide good guidelines for estimating the influence of genes on the dynamics of the network.

AVAILABILITY

Software codes for list-decoding algorithms suitable for direct application to quantized expression data will be publicly available at the authors' web-pages.

Reverse engineering time discrete finite dynamical systems: a feasible undertaking?

With the advent of high-throughput profiling methods, interest in reverse engineering the structure and dynamics of biochemical networks is high. Recently an algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is a top-down approach using time discrete dynamical systems. One of its key steps includes the choice of a term order, a technicality imposed by the use of Gröbner-bases calculations. The aim of this paper is to identify minimal requirements on data sets to be used with this algorithm and to characterize optimal data sets. We found minimal requirements on a data set based on how many terms the functions to be reverse engineered display. Furthermore, we identified optimal data sets, which we characterized using a geometric property called "general position". Moreover, we developed a constructive method to generate optimal data sets, provided a codimensional condition is fulfilled. In addition, we present a generalization of their algorithm that does not depend on the choice of a term order. For this method we derived a formula for the probability of finding the correct model, provided the data set used is optimal. We analyzed the asymptotic behavior of the probability formula for a growing number of variables n (i.e. interacting chemicals). Unfortunately, this formula converges to zero as fast as , where and . Therefore, even if an optimal data set is used and the restrictions in using term orders are overcome, the reverse engineering problem remains unfeasible, unless prodigious amounts of data are available. Such large data sets are experimentally impossible to generate with today's technologies.