Bifurcation analysis of bistable and oscillatory dynamics in biological networks using the root‐locus method

Most of the biological systems including gene regulatory networks can be described well by ordinary differential equation models with rational non‐linearities. These models are derived either based on the reaction kinetics or by curve fitting to experimental data. This study demonstrates the applicability of the root‐locus‐based bifurcation analysis method for studying the complex dynamics of such models. The effectiveness of the bifurcation analysis in determining the exact parameter regions in each of which the system shows a certain dynamical behaviour, such as bistability, oscillation, and asymptotically equilibrium dynamics is shown by considering two mostly studied gene regulatory networks, namely Gardner''s genetic toggle switch and p53 gene network possessing two‐phase (mono‐stable/oscillation) dynamics.Inspec keywords: oscillations, curve fitting, differential equations, bifurcation, genetics, nonlinear dynamical systemsOther keywords: nonlinearities, reaction kinetics, root‐locus‐based bifurcation analysis method, complex dynamics, exact parameter regions, dynamical behaviour, equilibrium dynamics, studied gene regulatory networks, p53 gene network, bistable dynamics, oscillatory dynamics, biological networks, root‐locus method, biological systems, ordinary differential equation models     

Insights from unifying modern approximations to infections on networks

Networks are increasingly central to modern science owing to their ability to conceptualize multiple interacting components of a complex system. As a specific example of this, understanding the implications of contact network structure for the transmission of infectious diseases remains a key issue in epidemiology. Three broad approaches to this problem exist: explicit simulation; derivation of exact results for special networks; and dynamical approximations. This paper focuses on the last of these approaches, and makes two main contributions. Firstly, formal mathematical links are demonstrated between several prima facie unrelated dynamical approximations. And secondly, these links are used to derive two novel dynamical models for network epidemiology, which are compared against explicit stochastic simulation. The success of these new models provides improved understanding about the interaction of network structure and transmission dynamics.     

Proximity of intracellular regulatory networks to monotone systems

Networks that contain only sign-consistent loops, such as positive feedforward and feedback loops, function as monotone systems. Simulated using differential equations, monotone systems display well-ordered behaviour that excludes the possibility for chaotic dynamics. Perturbations of such systems have unambiguous global effects and a predictability characteristic that confers robustness and adaptability. The authors assess whether the topology of biological regulatory networks is similar to the topology of monotone systems. For this, three intracellular regulatory networks are analysed where links are specified for the directionality and the effects of interactions. These networks were assembled from functional studies in the experimental literature. It is found that the three biological networks contain far more positive 'sign-consistent' feedback and feedforward loops than negative loops. Negative loops can be 'eliminated' from the real networks by the removal of fewer links as compared with the corresponding shuffled networks. The abundance of positive feedforward and feedback loops in real networks emerges from the presence of hubs that are enriched with either negative or positive links. These observations suggest that intracellular regulatory networks are 'close-to-monotone', a characteristic that could contribute to the dynamical stability observed in cellular behaviour.     

Functional tunability of biological circuits from additional toggle switches

In many complex regulatory networks with interlinked feedback loops, the simple core circuits are sufficient to achieve the specific biological functions of the whole networks, naturally raising a question: what is the role of the additional feedback loops. By investigating the effect of an additional toggle switch on the auto‐activation circuit responsible for competent switch in Bacillus subtilits and on the activator–repressor circuit responsible for cell cycle in Xenopus embryonic, the authors show that the additional toggle switch can elaborate the dynamical behaviour of both circuits. Specifically, the additional toggle switch in B. subtilits does not significantly affect the saturation level of the competent state but can tune the activation threshold (i.e. the minimal stimulus required to switch the system from the non‐competent state to the competent state). For the activator–repressor circuit in X. embryonic cell cycle, the additional toggle switch can tune the oscillation frequency but does not change the oscillation amplitude. The proposed detailed results not only provide guidelines to the engineering of synthetic genetic circuits, but also imply a significant fact that additional toggle switches in a complex network are not really redundant but play a role of tuning network functions.Inspec keywords: biochemistry, cellular biophysics, microorganismsOther keywords: functional tunability, biological circuits, toggle switches, complex regulatory networks, interlinked feedback loops, Bacillus subtilits, autoactivation circuit, activator‐repressor circuit, Xenopus embryonic     

Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity, which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network, and study how the frequency distributions of oscillatory modes of such a system depend on the average connectivity.     

SMILE: a novel procedure for subcellular module identification with localisation expansion

Computational clustering methods help identify functional modules in protein–protein interaction (PPI) network, in which proteins participate in the same biological pathways or specific functions. Subcellular localisation is crucial for proteins to implement biological functions and each compartment accommodates specific portions of the protein interaction structure. However, the importance of protein subcellular localisation is often neglected in the studies of module identification. In this study, the authors propose a novel procedure, subcellular module identification with localisation expansion (SMILE), to identify super modules that consist of several subcellular modules performing specific biological functions among cell compartments. These super modules identified by SMILE are more functionally diverse and have been verified to be more associated with known protein complexes and biological pathways compared with the modules identified from the global PPI networks in both the compartmentalised PPI and InWeb_InBioMap datasets. The authors’ results reveal that subcellular localisation is a principal feature of functional modules and offers important guidance in detecting biologically meaningful results.Inspec keywords: cellular biophysics, proteins, molecular biophysicsOther keywords: subcellular module identification, localisation expansion, computational clustering methods, protein‐protein interaction network, biological functions, protein interaction structure, protein subcellular localisation, subcellular modules, InWeb‐InBioMap datasets, subcellular localisation     

Gene networks reconstruction and time-series prediction from microarray data using recurrent neural fuzzy networks

Reverse engineering problems concerning the reconstruction and identification of gene regulatory networks through gene expression data are central issues in computational molecular biology and have become the focus of much research in the last few years. An approach has been proposed for inferring the complex causal relationships among genes from microarray experimental data, which is based on a novel neural fuzzy recurrent network. The method derives information on the gene interactions in a highly interpretable form (fuzzy rules) and takes into account the dynamical aspects of gene regulation through its recurrent structure. To determine the efficiency of the proposed approach, microarray data from two experiments relating to Saccharomyces cerevisiae and Escherichia coli have been used and experiments concerning gene expression time course prediction have been conducted. The interactions that have been retrieved among a set of genes known to be highly regulated during the yeast cell-cycle are validated by previous biological studies. The method surpasses other computational techniques, which have attempted genetic network reconstruction, by being able to recover significantly more biologically valid relationships among genes     

Editorial - Selected papers from the 2nd international symposium on optimization and systems biology

One of the major challenges for post-genomic biology is to understand how genes, proteins and small molecules interact to form cellular systems. It has been recognised that a complicated living organism cannot be fully understood by merely analysing individual components, and that interactions of those components or networks are ultimately responsible for an organism?s form and functions. Instead of analysing individual components or aspects of the organism, systems biology is the study of an organism, viewed as a dynamical and interacting network of biomolecules which give rise to a complicated life. With increasingly accumulated data from high-throughput technologies, molecular networks and their dynamics have been studied extensively from various aspects of living organisms. Many mathematical methods have been adopted in computational systems biology; in particular, optimisation and statistics play a key role in analysing and understanding biological mechanisms from system-wide viewpoints.     

Digital clocks: simple Boolean models can quantitatively describe circadian systems

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock''s oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.     

Simulation of genetic networks modelled by piecewise deterministic Markov processes

The authors propose piecewise deterministic Markov processes as an alternative approach to model gene regulatory networks. A hybrid simulation algorithm is presented and discussed, and several standard regulatory modules are analysed by numerical means. It is shown that despite of the partial simplification of the mesoscopic nature of regulatory networks such processes are suitable to reveal the intrinsic noise effects because of the low copy numbers of genes.     

Computational modelling of mitotic exit in budding yeast: the role of separase and Cdc14 endocycles

The operating principles of complex regulatory networks are best understood with the help of mathematical modelling rather than by intuitive reasoning. Hereby, we study the dynamics of the mitotic exit (ME) control system in budding yeast by further developing the Queralt''s model. A comprehensive systems view of the network regulating ME is provided based on classical experiments in the literature. In this picture, Cdc20–APC is a critical node controlling both cyclin (Clb2 and Clb5) and phosphatase (Cdc14) branches of the regulatory network. On the basis of experimental situations ranging from single to quintuple mutants, the kinetic parameters of the network are estimated. Numerical analysis of the model quantifies the dependence of ME control on the proteolytic and non-proteolytic functions of separase. We show that the requirement of the non-proteolytic function of separase for ME depends on cyclin-dependent kinase activity. The model is also used for the systematic analysis of the recently discovered Cdc14 endocycles. The significance of Cdc14 endocycles in eukaryotic cell cycle control is discussed as well.     

Hierarchical parameter estimation of GRN based on topological analysis

Reverse engineering of gene regulatory network (GRN) is an important and challenging task in systems biology. Existing parameter estimation approaches that compute model parameters with the same importance are usually computationally expensive or infeasible, especially in dealing with complex biological networks.In order to improve the efficiency of computational modeling, the paper applies a hierarchical estimation methodology in computational modeling of GRN based on topological analysis. This paper divides nodes in a network into various priority levels using the graph‐based measure and genetic algorithm. The nodes in the first level, that correspond to root strongly connected components(SCC) in the digraph of GRN, are given top priority in parameter estimation. The estimated parameters of vertices in the previous priority level ARE used to infer the parameters for nodes in the next priority level. The proposed hierarchical estimation methodology obtains lower error indexes while consuming less computational resources compared with single estimation methodology. Experimental outcomes with insilico networks and a realistic network show that gene networks are decomposed into no more than four levels, which is consistent with the properties of inherent modularity for GRN. In addition, the proposed hierarchical parameter estimation achieves a balance between computational efficiency and accuracy.Inspec keywords: biology computing, network theory (graphs), reverse engineering, graph theory, genetics, genetic algorithms, directed graphs, parameter estimationOther keywords: hierarchical parameter estimation, GRN, topological analysis, gene regulatory network, important task, computational systems biology, compute model parameters, complex biological networks, efficient information, model quality, parameter reliability, computational modelling, study divides nodes, priority levels, graph‐based measure, previous priority level, hierarchical estimation methodology obtains, computational resources, single time estimation, insilico network, realistic network show, computational efficiency