Elena S. Dimitrova

Discretization of Time Series Data

An increasing number of algorithms for biochemical network inference from experimental data require discrete data as input. For example, dynamic Bayesian network methods and methods that use the framework of finite dynamical systems, such as Boolean networks, all take discrete input. Experimental data, however, are typically continuous and represented by computer floating point numbers. The translation from continuous to discrete data is crucial in preserving the variable dependencies and thus has a significant impact on the performance of the network inference algorithms. We compare the performance of two such algorithms that use discrete data using several different discretization algorithms. One of the inference methods uses a dynamic Bayesian network framework, the other-a time-and state-discrete dynamical system framework. The discretization algorithms are quantile, interval discretization, and a new algorithm introduced in this article, SSD. SSD is especially designed for short time series data and is capable of determining the optimal number of discretization states. The experiments show that both inference methods perform better with SSD than with the other methods. In addition, SSD is demonstrated to preserve the dynamic features of the time series, as well as to be robust to noise in the experimental data. A C++ implementation of SSD is available from the authors at http://polymath.vbi.vt.edu/discretization .

Parameter estimation for Boolean models of biological networks

Boolean networks have long been used as models of molecular networks, and they play an increasingly important role in systems biology. This paper describes a software package, Polynome, offered as a web service, that helps users construct Boolean network models based on experimental data and biological input. The key feature is a discrete analog of parameter estimation for continuous models. With only experimental data as input, the software can be used as a tool for reverse-engineering of Boolean network models from experimental time course data.

A Gröbner fan method for biochemical network modeling

Polynomial dynamical systems (PDSs) have been used successfully as a framework for the reconstruction, or reverse engineering of biochemical networks from experimental data. Within this modeling space, a particular PDS is chosen by way of a Gröbner basis, and using different monomial orders may result in different polynomial models. In this paper, we present a systematic method for selecting most likely polynomial models for a given data set, using the Gröbner fan of the ideal of the input data. We apply the method to reverse engineer two biochemical networks, a Boolean model of lactose metabolism in E. coli and a protein signal transduction network in S. cerevisiae and compare our results to those from two published network-reconstruction methods.

Nested canalyzing depth and network stability.

We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.

Molecular network control through boolean canalization.

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges to control in the wiring diagram of a network for avoiding undesirable state transitions. The method is based on identifying appropriate input-output combinations on undesirable transitions that can be modified using the edges in the wiring diagram of the network. Moreover, a method for estimating the number of changed transitions in the state space of the system as a result of an edge deletion in the wiring diagram is presented. The control methods of this paper were applied to a mutated cell-cycle model and to a p53-mdm2 model to identify potential control targets.

Probabilistic polynomial dynamical systems for reverse engineering of gene regulatory networks

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 un-certainty, 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.

Estimating the Volumes of the Cones in a Gröbner Fan

We present a stochastic method for estimating the relative volumes of the Gröbner cones of a Gröbner fan without computing the actual fan. The method is particularly useful when the dimension of the Gröbner fan is large and/or the volumes of several or all cones need to be estimated. A Macaulay 2 implementation for uniform sampling from the Gröbner fan is provided by the author.

STATISTICS OF SOME LOW-DIMENSIONAL CHAOTIC FLOWS

As a result of the recent finding that the Lorenz system exhibits blurred self-affinity for values of its controlling parameter slightly above the onset of chaos, we study other low-dimensional chaotic flows with the purpose of providing an approximate description of their second-order, two-point statistical functions. The main pool of chaotic systems on which we focus our attention is that reported by Sprott [1994], generalized however to depend on their intrinsic number of parameters. We show that their statistical properties are adequately described as processes with spectra having three segments all of power-law type. On this basis we identify quasi-periodic behavior pertaining to the relatively slow process in the attractors and approximate self-affine statistical symmetry characterizing the fast processes.

System Identification for Discrete Polynomial Models of Gene Regulatory Networks

Abstract 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.

Polynomial algebra reveals diverging roles of the unfolded protein response in endothelial cells during ischemia-reperfusion injury

The unfolded protein response (UPR) – the endoplasmic reticulum stress response – is found in various pathologies including ischemia–reperfusion injury (IRI). However, its role during IRI is still unclear. Here, by combining two different bioinformatical methods – a method based on ordinary differential equations (Time Series Network Inference) and an algebraic method (probabilistic polynomial dynamical systems) – we identified the IRE1α–XBP1 and the ATF6 pathways as the main UPR effectors involved in cells adaptation to IRI. We validated these findings experimentally by assessing the impact of their knock‐out and knock‐down on cell survival during IRI.