A General Modeling Strategy for Gene Regulatory Networks with Stochastic Dynamics

Modeling stochasticity in gene regulatory networks is an important and complex problem in molecular systems biology. To elucidate intrinsic noise, several modeling strategies such as the Gillespie algorithm have been used successfully. This article contributes an approach as an alternative to these classical settings. Within the discrete paradigm, where genes, proteins, and other molecular components of gene regulatory networks are modeled as discrete variables and are assigned as logical rules describing their regulation through interactions with other components. Stochasticity is modeled at the biological function level under the assumption that even if the expression levels of the input nodes of an update rule guarantee activation or degradation there is a probability that the process will not occur due to stochastic effects. This approach allows a finer analysis of discrete models and provides a natural setup for cell population simulations to study cell-to-cell variability. We applied our methods to two of the most studied regulatory networks, the outcome of lambda phage infection of bacteria and the p53-mdm2 complex.

Bistability arises within a wide range of biological systems from the lambda phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

We develop a new methodology to design synthetic genetic switch networks with multiple genes and time delays, by using monotone dynamical theory. We show that the networks with only positive feedback loops have no stable oscillation except equilibria whose stability is also independent of the time delays. Such systems have ideal properties for switch networks and can be designed with out consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces due to the independence to time delays. Specifically, we first prove the basic properties of the genetic networks composed of only positive feedback loops, and then propose a procedure to design the switches, which drastically simplifies analysis of the switches and makes theoretical analysis and designing tractable even for large scale systems. Finally, we demonstrate our theoretical results by designing a biologically plausible synthesized genetic switch with experimentally well investigated lacI, tetR, and cI genes.

From biochemical reaction principles, a genetic regulatory network can be described by a group of nonlinear differential equations with time delays. Previous studies have investigated delay-independent stability of genetic regulatory networks with time delays. However, if it is delay-independently stable, a genetic regulatory network loses other interesting properties such as oscillation. In this paper, we provide a computational method for computing the maximal delay interval over which the genetic regulatory network maintains stability, and beyond which the network will not be stable. Furthermore we prove that when its delay is exactly the maximal delay the network is oscillated. In addition, the formula for calculating the oscillation period is presented. The autoregulatory genetic network in zebrafish is used as an example to illustrate the presented method. The oscillation period calculated from our method is very close to that observed from the real-life of a zebrafish, which indicates the effectiveness of our method.

In this paper, we investigate the oscillatory expression in Escherichia coli mediated by microRNA with time delays and reaction-diffusion terms. First of all, the integrated effects of delays and diffusions are first introduced into the genetic regulatory networks involving microRNAs, and a general model of genetic regulatory networks is then formulated. Second, two functional issues on gene regulatory networks, i.e. stability and oscillation of such model, are addressed in detail, and an explicit algorithm determining the properties of periodic oscillation is also presented. We demonstrate that the oscillatory expression of Escherichia coli is not only crucially dependent on the transcriptional and translational delays, but also heavily influenced by the diffusion coefficients. The conclusion is practically verified by a lot of biological experiments and observations. We also find that if the diffusion coefficients of miRNA, mRNA, and protein are suitably small, it can predict that inhomogeneous periodic oscillations can occur unless there only exhibits spatially homogeneous periodic oscillations. The obtained results indicate that the effects of transcriptional and translational delays are essential factors for designing or controlling genetic regulatory networks, in the meantime the functions of reaction-diffusion must beconsidered. Finally, numerical examples are presented to illustrate and visualize theoretical results.

Bifurcation control of a novel fractional-order gene regulatory network with incommensurate order and time delay

Inferring gene regulatory network (GRN) from the microarray expression data is an important problem in Bioinformatics, because knowing the GRN is an essential first step in understanding the inner workings of the cell and the related diseases. Time delays exist in the regulatory effects from one gene to another due to the time needed for transcription, translation, and to accumulate a sufficient number of needed proteins. Also, it is known that the delays are important for oscillatory phenomenon. Therefore, it is crucial to develop a causal gene network model, preferably as a function of time. In this paper, we propose an algorithm CLINDE to infer causal directed links in GRN with time delays and regulatory effects in the links from time-series microarray gene expression data. It is one of the most comprehensive in terms of features compared to the state-of-the-art discrete gene network models. We have tested CLINDE on synthetic data, the in vivo IRMA (On and Off) datasets and the [1] yeast expression data validated using KEGG pathways. Results show that CLINDE can effectively recover the links, the time delays and the regulatory effects in the synthetic data, and outperforms other algorithms in the IRMA in vivo datasets.

Gene regulatory networks are essential for understanding the mechanisms underlying genetic disorders. However, traditional models based on deterministic Hill functions (without fluctuations) often fail to capture critical stochastic effects, particularly in small systems in which molecular fluctuations play a significant role. In this study, we introduced a modified Hill function derived from the stochastic dynamics of ligand–receptor interactions. By performing a second-order expansion of the chemical master equation, we derived a set of ordinary differential equations governing the mean concentrations and covariances. This framework allows the identification of effective reaction rates using Hill functions that include stochastic correction terms that account for fluctuations in ligand–receptor binding. We applied this approach to canonical models, such as the Toggle Switch, Repressilator, and Activator–Repressor Clock, and showed that the Hill function with stochastic corrections provides a more accurate description of system dynamics with intrinsic fluctuations. Moreover, the proposed method offers substantial computational advantages over direct stochastic simulations using the Gillespie algorithm. Our results demonstrate that this approximation captures the essential features of stochastic biochemical systems more accurately than models that use deterministic Hill functions, offering a powerful tool for studying the role of fluctuations in complex reaction networks and uncovering potential emergent behaviors driven by intrinsic fluctuations in the system.

Constructing gene regulatory networks from microarray data using GA/PSO with DTW

In existing works, the filters designed for delayed genetic regulatory networks contain time delay. If the time delay is unknown, the filters do not work in practical applications. In order to overcome the shortcoming in such existing works, this paper investigates the filter design problem of genetic regulatory networks with unknown constant time delay, and a novel adaptive filter is introduced, which can estimate online not only unknown network parameters but also the unknown time delay. By the Lyapunove approach, it is shown that the estimating errors asymptotically converge to the origin. Finally, simulation results are presented to illustrate the effectiveness of the proposed new design method.

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 interested in practice, delay-dependent stability of genetic regulatory networks has been studied insufficiently. Based on linear matrix inequality (LMI) approach, in this study we will present some delay-dependent stability conditions for genetic regulatory networks. To illustrate the effectiveness of our theoretical results, a gene repressillatory network is analyzed.

Impulsive effects on fractional order time delayed gene regulatory networks: Asymptotic stability analysis

Stochastic Simulation Algorithm (SSA) is crucial for modeling biochemical reactions and gene regulatory networks. Traditional SSA is characterized by Markovian property and cannot naturally model systems with time delays. Several algorithms have already been designed to handle delayed reactions, yet few easy-to-use implementations exist. To address these challenges, we have developed DelaySSA, an R package that implements currently available algorithms for SSA with or without delays. Meanwhile, we also provided Matlab and Python versions to support wider applications. We demonstrated its accuracy and validity by simulating two classical models: the Bursty model and Refractory model. We then tested its capability to simulate the RNA Velocity model, where it successfully reproduced both the up- and down-regulation stages in the phase portrait. Finally, we extended its application to simulate a gene regulatory network of lung cancer adeno-to-squamous transition (AST) and qualitatively analyzed its bistability behavior by approximating the Waddington's landscape. Modeling the therapeutic intervention of a SOX2 degrader as a delayed degradation reaction, AST is effectively blocked and reprogrammed back to the adenocarcinoma state, providing a useful clue for targeting drug-resistant AST in the future. Taken together, DelaySSA is a powerful and easy-to-use software suite, facilitating accurate modeling of various kinds of biological systems and broadening the scope of stochastic simulations in systems biology.

Investigation of oscillation accumulation triggered genetic switch in gene regulatory networks

Interaction among the gene products contributes to the regulation of gene expression.