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Register a new userA robust Lyapunov criterion for non-oscillatory behaviors in biological interaction networks
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We introduce the notion of non-oscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowneys result on non-existence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust non-oscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles where available methods are not able to provide any information about their qualitative global behavior.
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We present a framework to transform the problem of finding a Lyapunov function of a Chemical Reaction Network (CRN) in concentration coordinates with arbitrary monotone kinetics into finding a common Lyapunov function for a linear parameter varying system in reaction coordinates. Alternative formulations of the proposed Lyapunov function is presented also. This is applied to reinterpret previous results by the authors on Piecewise Linear in Rates Lyapunov functions, and to establish a link with contraction analysis. Persistence and uniqueness of equilibria are discussed also.
Inferring genetic networks from gene expression data is one of the most challenging work in the post-genomic era, partly due to the vast space of possible networks and the relatively small amount of data available. In this field, Gaussian Graphical Model (GGM) provides a convenient framework for the discovery of biological networks. In this paper, we propose an original approach for inferring gene regulation networks using a robust biological prior on their structure in order to limit the set of candidate networks. Pathways, that represent biological knowledge on the regulatory networks, will be used as an informative prior knowledge to drive Network Inference. This approach is based on the selection of a relevant set of genes, called the molecular signature, associated with a condition of interest (for instance, the genes involved in disease development). In this context, differential expression analysis is a well established strategy. However outcome signatures are often not consistent and show little overlap between studies. Thus, we will dedicate the first part of our work to the improvement of the standard process of biomarker identification to guarantee the robustness and reproducibility of the molecular signature. Our approach enables to compare the networks inferred between two conditions of interest (for instance case and control networks) and help along the biological interpretation of results. Thus it allows to identify differential regulations that occur in these conditions. We illustrate the proposed approach by applying our method to a study of breast cancers response to treatment.
Complexes of physically interacting proteins are one of the fundamental functional units responsible for driving key biological mechanisms within the cell. Their identification is therefore necessary not only to understand complex formation but also the higher level organization of the cell. With the advent of high-throughput techniques in molecular biology, significant amount of physical interaction data has been cataloged from organisms such as yeast, which has in turn fueled computational approaches to systematically mine complexes from the network of physical interactions among proteins (PPI network). In this survey, we review, classify and evaluate some of the key computational methods developed till date for the identification of protein complexes from PPI networks. We present two insightful taxonomies that reflect how these methods have evolved over the years towards improving automated complex prediction. We also discuss some open challenges facing accurate reconstruction of complexes, the crucial ones being presence of high proportion of errors and noise in current high-throughput datasets and some key aspects overlooked by current complex detection methods. We hope this review will not only help to condense the history of computational complex detection for easy reference, but also provide valuable insights to drive further research in this area.
A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input the state converges to a unique equilibrium point for any initial condition. We consider the problem of maximizing a weighted average of the systems output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a periodic optimal control problem which can be analyzed by means of the Pontryagin maximum principle or solved numerically via powerful software packages. We then apply our framework to a class of occupancy models that appear frequently in biological synthesis systems and other applications. We show that, perhaps surprisingly, constant inputs are optimal for various architectures. This suggests that the presence of non-constant periodic signals, which frequently appear in biological occupancy systems, is a signature of an underlying time-varying objective functional being optimized.
Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically; and the frequency spectrum of this chemical fluctuation carries valuable information about the mechanism and the dynamics of the intracellular reactions creating these biomolecules. Although recent advances in single-cell experimental techniques enable the direct monitoring of the time-traces of the biological noise in each cell, the development of the theoretical tools needed to extract the information encoded in the stochastic dynamics of intracellular chemical fluctuation is still in its adolescence. Here, we present a simple and general equation that relates the power-spectrum of the product number fluctuation to the product lifetime and the reaction dynamics of the product creation process. By analyzing the time traces of the protein copy number using this theory, we can extract the power spectrum of the mRNA number, which cannot be directly measured by currently available experimental techniques. From the power spectrum of the mRNA number, we can further extract quantitative information about the transcriptional regulation dynamics. Our power spectrum analysis of gene expression noise is demonstrated for the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. Additionally, we investigate how the non-Poisson reaction dynamics and the cell-to-cell heterogeneity in transcription and translation affect the power-spectra of the mRNA and protein number.
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