Li-Ping Tian

M-matrix-based stability conditions for genetic regulatory networks with time-varying delays and noise perturbations

Stability is essential for designing and controlling any dynamic systems. Recently, the stability of genetic regulatory networks has been widely studied by employing linear matrix inequality (LMI) approach, which results in checking the existence of feasible solutions to high-dimensional LMIs. In the previous study, the authors present several stability conditions for genetic regulatory networks with time-varying delays, based on M-matrix theory and using the non-smooth Lyapunov function, which results in determining whether a low-dimensional matrix is a non-singular M-matrix. However, the previous approach cannot be applied to analyse the stability of genetic regulatory networks with noise perturbations. Here, the authors design a smooth Lyapunov function quadratic in state variables and employ M-matrix theory to derive new stability conditions for genetic regulatory networks with time-varying delays. Theoretically, these conditions are less conservative than existing ones in some genetic regulatory networks. Then the results are extended to genetic regulatory networks with time-varying delays and noise perturbations. For genetic regulatory networks with n genes and n proteins, the derived conditions are to check if an n × n matrix is a non-singular M-matrix. To further present the new theories proposed in this study, three example regulatory networks are analysed.

Robust and Global Delay-Dependent Stability for Genetic Regulatory Networks With Parameter Uncertainties

The study of stability is essential for designing or controlling genetic regulatory networks, which can be described by nonlinear differential equations with time delays. Much attention has been paid to the study of delay-independent stability of genetic regulatory networks and as a result, many sufficient conditions have been derived for delay-independent stability. Although it might be more interesting in practice, delay-dependent stability of genetic regulatory networks has been studied insufficiently. Based on the linear matrix inequality (LMI) approach, in this study we will present some delay-dependent stability conditions for genetic regulatory networks. Then we extend these results to genetic regulatory networks with parameter uncertainties. To illustrate the effectiveness of our theoretical results, gene repressilatory networks are analyzed.

Iterative Linear Least Squares Method of Parameter Estimation for Linear-Fractional Models of Molecular Biological Systems

Based on statistical thermodynamics principle or Michaelis-Menten kinetics equation, the models for biological systems contain linear fractional functions as reaction rates which are nonlinear in both parameters and states. Generally it is challenging to estimate parameters nonlinear in a model although there have been many traditional nonlinear parameter estimation methods such as Gauss-Newton iteration method and its variants. However, in a linear fractional model both the denominator and numerator are linear in the parameters. Based on this observation, we develop an iterative linear least squares method for estimating parameters in biological system modeled by linear fractional function. The basic idea is to transfer optimizing a nonlinear least squares objective function into iteratively solving a sequence of linear least squares problems. The developed method is applied to a linear fractional function and an auto-regulatory gene network. The simulation results show the superior performance of the proposed method over some existing algorithms.

State Observer Design for Delayed Genetic Regulatory Networks

Genetic regulatory networks are dynamic systems which describe the interactions among gene products (mRNAs and proteins). The internal states of a genetic regulatory network consist of the concentrations of mRNA and proteins involved in it, which are very helpful in understanding its dynamic behaviors. However, because of some limitations such as experiment techniques, not all internal states of genetic regulatory network can be effectively measured. Therefore it becomes an important issue to estimate the unmeasured states via the available measurements. In this study, we design a state observer to estimate the states of genetic regulatory networks with time delays from available measurements. Furthermore, based on linear matrix inequality (LMI) approach, a criterion is established to guarantee that the dynamic of estimation error is globally asymptotically stable. A gene repressillatory network is employed to illustrate the effectiveness of our design approach.

Nonlinear Model-Based Method for Clustering Periodically Expressed Genes

Clustering periodically expressed genes from their time-course expression data could help understand the molecular mechanism of those biological processes. In this paper, we propose a nonlinear model-based clustering method for periodically expressed gene profiles. As periodically expressed genes are associated with periodic biological processes, the proposed method naturally assumes that a periodically expressed gene dataset is generated by a number of periodical processes. Each periodical process is modelled by a linear combination of trigonometric sine and cosine functions in time plus a Gaussian noise term. A two stage method is proposed to estimate the model parameter, and a relocation-iteration algorithm is employed to assign each gene to an appropriate cluster. A bootstrapping method and an average adjusted Rand index (AARI) are employed to measure the quality of clustering. One synthetic dataset and two biological datasets were employed to evaluate the performance of the proposed method. The results show that our method allows the better quality clustering than other clustering methods (e.g., k-means) for periodically expressed gene data, and thus it is an effective cluster analysis method for periodically expressed gene data.

Alternating Constraint Least Squares Parameter Estimation for S-System Models of Biological Networks

S-system models for biological systems are derived from the generalized mass action law and are typically a group of nonlinear differential equations. Estimation of parameters in these models from experimental measurements is thus a nonlinear problem. In principle, all algorithms for nonlinear optimization can be used to estimate parameters in molecular biological systems, for example, Gauss-Newton iteration method and its variants. However, these methods do not take the special structures of biological system models into account and thus are not efficient. In this paper, we propose an alternating constraint least squares method for estimating parameters in S-system model by taking use of their special structure and the biological meaning of parameters. To investigate its performance, the alternating constraint least squares method is applied to a biological system and is compared with other parameter estimation methods. Simulation results show the good performance of the proposed estimation method.

Disease Gene Prediction by Integrating PPI Networks, Clinical RNA-Seq Data and OMIM Data

Disease gene prediction is a challenging task that has a variety of applications such as early diagnosis and drug development. The existing machine learning methods suffer from the imbalanced sample issue because the number of known disease genes (positive samples) is much less than that of unknown genes which are typically considered to be negative samples. In addition, most methods have not utilized clinical data from patients with a specific disease to predict disease genes. In this study, we propose a disease gene prediction algorithm (called dgSeq) by combining protein-protein interaction (PPI) network, clinical RNA-Seq data, and Online Mendelian Inheritance in Man (OMIN) data. Our dgSeq constructs differential networks based on rewiring information calculated from clinical RNA-Seq data. To select balanced sets of non-disease genes (negative samples), a disease-gene network is also constructed from OMIM data. After features are extracted from the PPI networks and differential networks, the logistic regression classifiers are trained. Our dgSeq obtains AUC values of 0.88, 0.83, and 0.80 for identifying breast cancer genes, thyroid cancer genes, and Alzheimers disease genes, respectively, which indicates its superiority to other three competing methods. Both gene set enrichment analysis and predicted results demonstrate that dgSeq can effectively predict new disease genes.

Identifying disease genes from PPI networks weighted by gene expression under different conditions

The identification of disease genes is an essential issue to decipher the mechanisms of complex diseases. Many existing methods combine machine learning algorithms and network information to predict disease genes and are based on the ‘guilt by association’ assumption, where disease genes are considered to be close to each other in a biomolecular network. Although these methods have gained many novel findings, most of them ignored the edge dynamic changes of biomolecular networks under different conditions when only utilizing the ‘guilt by association’ principle, which will limit their performance. To address this problem, we propose an algorithm that combines the ‘guilt by association’ and the ‘guilt by rewiring’ of biomolecular networks at the same time. The difference of gene co-expression between case and control samples are first processed to obtain the edge dynamic changes (rewiring) of biomolecular networks through weighting the edges of protein-protein interaction (PPI) networks. Then, features are extracted from the weighted PPI network. Finally, a logistic regression is adopted to identify the disease genes. The algorithm achieves AUC values of 0.95, 0.90 and 0.92 on the identification of breast-cancer-related, lung-cancer-related and schizophrenia-related genes, respectively. Two new schizophrenia-related genes are also found from the ranked unknown genes list.

New global stability conditions for genetic regulatory networks with time-varying delays

The study of the global stability is essential for designing and controlling genetic regulatory networks. Most existing results on this issue are based on linear matrix inequality (LMI) approach, which results in checking the existence of feasible solutions to high dimensional LMIs. In our previous study, we present several stability conditions for genetic regulatory networks with time-varying delays, based on M-matrix theory and the non-smooth Lyapunov function. In this paper, we design a smooth Lyapunov function and employ M-matrix theory to derive new stability conditions for genetic regulatory networks with time-varying delays. Theoretically, these conditions are less conservative than existing ones in some cases. For genetic regulatory networks with n genes and n proteins, these conditions become to check if an n×n matrix is an M-matrix, which is much easier than existing results. To illustrate the effectiveness of our theoretical results, two genetic regulatory networks are analyzed.

Globally delay-independent stability of ring-structured genetic regulatory networks

Modeling genetic regulatory networks in terms of differential equations with time delays provides a powerful tool for understanding gene regulatory processes in living organisms. In this paper we studied the globally delay-independent stability ring-structured genetic regulatory networks. We first present a sufficient condition for globally delay-independent stability of such genetic regulatory networks, based on the M-matrix theory. Then this sufficient condition is reduced to determine if the roots of a polynomial lie in the right complex plane. Finally, autoregulatory network and repressilatory network are employed to illustrate the theorems developed in this study.