Dynamics analysis of gene regulatory networks

Pioneering theoretical work on gene regulatory networks (GRNs) has anticipated the emergence of postgenomic research and provided a mathematical framework for the current description and analysis of complex regulatory mechanisms. In biochemical networks, the rates of reaction of substrates, enzymes, factors or products have attracted considerable attention in correspondence with changes in concentrations. The dynamical behaviours of genes, proteins and metabolites can be modelled by a series of differential equations, in which the detailed understanding of different behaviours exhibited by a gene regulatory network could be explored. The GRN diagrams that resemble complex electrical circuits are generated by the connectivity of genes and proteins. Recently, applying control theory to study biology is fast becoming an interesting and exciting idea, although there exist large differences in culture, approach and the tools used in these two fields.

The complexity of GRNs poses many challenges for scientists and engineers. In particular, the biological systems have apparently become dependent on the complex infrastructure of GRNs to such an extent that it is difficult to analyse and control these networks thoroughly with our current capabilities. Therefore, there is an urgent need for research into modelling, analysis of behaviours, systems theory, synchronisation and control in GRNs. Numerous fundamental questions have been addressed about the connections between GRN structure and dynamic properties including stability, bifurcations, controllability and other observable aspects. However, some major problems have not been fully investigated, such as the behaviour of stability, synchronisation and chaos control for GRNs, as well as their applications in, for example, systems biology and bioinformatics.

GRNs have already become an ideal research area for control engineers, mathematicians, computer scientists and biologists to manage, analyse and interpret functional information from real-world networks. Sophisticated computer system theories and computing algorithms have emerged or been exploited in the general area of dynamic analysis of GRNs, such as analysis of algorithms, artificial intelligence, automata, computational complexity, computer security, concurrency and parallelism, data structures, knowledge discovery, DNA and quantum computing, randomisation, semantics, symbol manipulation, numerical analysis and mathematical software. In the past decade, a large amount of research results have been available in the literature on the topics that include, but are not limited to the following aspects of GRNs: (1) systems and control analysis of GRNs; (2) parameter identification of GRNs; (3) dynamic analysis of regulatory motifs such as repressors and circadian oscillators; (4) robustness and fragility analysis of GRNs; and (5) methods and algorithms for GRN analysis.

This Special Issue aims to bring together the latest approaches to understanding gene regulatory networks from a dynamic system perspective. We have solicited submissions to this Special Issue from control engineers, mathematicians, biologists and computer scientists. After a rigorous peer review process, nine articles have been selected that provide overviews, solutions, or early promise, to manage, analyse and interpret functional information from gene regulatory networks. These articles have covered both the practical and theoretical aspects of gene regulatory networks in the broad areas of dynamical systems, artificial intelligence, mathematics, statistics, operational research and engineering.

The modelling of gene regulatory networks is necessary to describe the manner in which cells execute and control normal function and how abnormal function results from a breakdown in regulation. Hence, gene regulatory networks are critical to translational genomics, whose aim is to develop therapies based on the disruption or mitigation of aberrant gene function contributing to the pathology of a disease. Two basic intervention approaches have been considered for gene regulatory networks in the context of probabilistic Boolean networks (PBNs), external control and structural intervention. In the article ‘Stationary and structural control in gene regulatory networks: basic concepts’ by Dougherty, Pal, Qian, Bittner and Datta, the fundamental aspects of stationary and structural intervention in Markovian gene regulatory networks, in particular, PBNs, are reviewed. Various issues regarding regulatory intervention are addressed. This review article mainly consists of five parts, i.e. the introduction of PBNs, the stationary control problem for PBNs, the stationary control problem in a Melanoma network, the structural intervention problem and the structural intervention in a Melanoma network. Some discussions on future research directions are included in the conclusion section.

GRNs can be broadly defined as groups of genes that are activated by particular signals and stimuli, and once activated, orchestrate their operation to regulate certain biological functions, such as metabolism, development and the cell cycle. These gene networks execute stored programs to adjust the organism to its natural milieu, in addition to emergency programs that deal with extraordinary and novel growth conditions. The logic and physical implementation of these genetic programs include nonlinear interactions, time delays, positive and negative feedback, and crosstalk. With mathematical models, one can study the effect of perturbations (such as environmental stressors) and assess the causes of disease (merely the failure of these regulatory networks) or the effects of therapeutic drugs. In the article ‘Modelling and analysis of gene regulatory network using feedback control theory’ by El-Samad and Khammash, the mathematical methods commonly used in modelling gene networks are reviewed. It is emphasised through example that all of these approaches are complementary in their ability to uncover patterns of biological behaviours. Throughout the exposition, it is demonstrated that biological complex behaviours can best be probed and understood through the use of ideas and principles from feedback control theory. As an illustration, a list of behaviours and characteristics are presented that stem directly from the presence of feedback in gene regulatory networks and biological examples are discussed where this behaviour is present.

Biological systems and the models developed to describe them are often complex due to the large number of components involved and the nonlinearity of the interactions between them. The size and complexity of these models pose new difficulties. For instance, the inverse problem of finding parameter values gives rise to nonlinear and high-dimensional optimisation problems, and is hampered by incomplete and noisy data. Furthermore, all conclusions from numerical simulation are ‘local’ in the sense that they are only valid for finite choices of parameter values and initial conditions. In systems of significant size and nonlinearity it is difficult (and often impossible) to move from such local conclusions to more global claims. A key question is: ‘given limited, qualitative information, what can we say about the behaviour of biological systems?’ In the article ‘Graphical methods for analysing feedback in biological networks – a survey’ by Radde, Bar and Banaji, the focus is on analytical approaches to make claims about robust behaviour based on model structure. It is particularly appealing if such claims can be made from diagrams commonly used by experimental biologists. Such diagrams contain information on which components of a system interact with each other, whether the interactions are activatory or inhibitory, etc. While they take a variety of forms, they can often be seen mathematically as generalised graphs: graphs or multigraphs with additional structures (e.g. edges may have a sign and/or direction and/or weight). Being visual, they are more intuitive to understand, and moreover many efficient algorithms exist for their analysis. At the heart of this review is the natural association between cycles in these diagrams and feedback control, of central importance in biology.

In recent years, dynamical systems described by differential equations (also called connectionist model, linear transcription model, additive regulation model, etc.) have frequently been exploited to model gene networks, where the main idea is to use an updated rule based on a weighted sum of inputs. In particular, the stability analysis problem for genetic regulatory networks has recently stirred increasing research interest. When modelling gene networks, parameter uncertainties and stochastic disturbances are arguably two of the challenging issues that impact on the model quality. On the one hand, a mathematical model can by no means exactly represent the real gene network. On the other hand, it may be impossible to model the behaviour of the system exactly with a purely deterministic model. Given the unavoidable parameter uncertainties and stochastic disturbances, an interesting problem of biological significance is therefore to estimate the variable values (usually concentrations of mRNA and proteins). In the article ‘Robust state estimation for stochastic genetic regulatory networks’ by Liang and Lam, the robust state estimation problem is dealt with for GRNs with stochastic disturbances and norm-bounded uncertainties. The aim is to design a linear state estimator such that, for the admissible norm-bounded uncertainties and stochastic noise disturbance, the dynamics of the estimation error system is stochastically stable. By using the linear matrix inequality (LMI) technique, sufficient conditions are first derived to ensure the desired estimation performance for the GRNs. Then, the estimator gain is characterised in terms of the solution to a set of LMIs, which can be easily solved by using available software packages. A three-node gene network is presented to show the effectiveness of the proposed design procedures.

GRNs explain the interactions between genes and proteins to form complex systems that perform complicated biological functions. A first classification divides GRNs into two main groups, specifically the Boolean model (or discrete model) and the differential equation model (or continuous model). An important issue in GRNs consists of establishing stability of equilibrium points for a differential equation model. In fact, stability is related to the ability of an organism to robustly regulate its function in spite of the presence of changes that move the state of the organism away from equilibrium. Unfortunately, this is a difficult issue since GRNs are nonlinear systems. In particular, they are characterised by sums or products of saturation functions, and to determine whether an equilibrium point of such a system is globally asymptotically stable is known to be a NP-hard problem. In the article ‘Polynomial relaxation-based conditions for global asymptotic stability of equilibrium points of genetic regulatory networks’ by Chesi, a possible solution for this problem is proposed. In particular, GRNs with SUM or PROD regulatory functions are considered. It is shown that sufficient conditions for global asymptotical stability of equilibrium points of these networks can be obtained in terms of LMI feasibility tests, which amount to solving convex optimisation. These conditions are derived by searching for a Lyapunov function for the equilibrium point, and are constructed through the use of suitable polynomial relaxations. The advantage of these conditions is that their conservatism can be decreased by increasing the degree of the polynomial relaxations, since no approximation of the nonlinearities present in the GRNs is introduced.

Models of gene regulatory networks are often derived from statistical thermodynamics principles or the Michaelis–Menten kinetics equation. As a result, the models contain rational reaction rates which are nonlinear in both parameters and states. 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. In the article ‘Estimation of parameters in rational reaction rates of molecular biological systems via weighted least squares’ by Wu, Mu and Shi, a weighted least-squares method is presented for estimating the parameters in GRNs with linear fractional reaction rates. The presented methods make use of the special structure of the linear fractional models: both the denominator and the numerator are linear in parameters. By carefully designing the weight matrix, the parameter estimation in the linear fractional reaction rates which are essentially nonlinear in parameters is transformed into solving two linear least squares problems. Surprisingly, the estimate of parameters can be analytically expressed in observation data. Compared with the traditional Gauss–Newton method and its variants, the presented method does not need any initial estimate of parameters and the iterations. Two illustrated examples show that the presented method outperforms over the traditional Gauss–Newton method in terms of the relative estimation errors.

The systems biology approach for investigating biochemical systems includes quantitative data-based mathematical modelling and their subsequent analysis. The underlying idea is that this methodology will be a suitable strategy to gain insight into the behaviour of biochemical systems, reducing the experimental effort necessary and making accessible the investigation of pathway modifications not reachable with current experimental techniques. A well established modelling framework for signalling pathways is that of ordinary differential equations (ODEs), describing spatio-temporal changes of protein concentrations. When the mathematical models include numerous equations with nonlinear equations kinetics and control structures such as feedback loops, the complex dynamics of the system can only be elucidated with the support of more sophisticated tools like sensitivity analysis, model reduction and bifurcation analysis. In recent years, several researchers suggested that mathematical modelling could be used to investigate differences between healthy and pathologic configurations of biological pathways. In the article ‘Integration of sensitivity and bifurcation analysis to detect critical processes in a model combining signalling and cell population dynamics’ by Nikolov, Lai, Liebal, Wolkenhauer and Vera, the aim is to use mathematical modelling in combination with some of the aforementioned system theoretic tools to detect the processes that control the dynamics of critical properties in the investigated system. In this case, the investigated system is a multilevel model accounting for the effect of Epo-mediated activation of the JAK2-STAT5 signalling pathway in erythropoiesis. The strategy is not only to use several of these methods, but also to make them work in a coordinated way such as information generated in one step feeding into the definition of further analysis, thus helping to refine the structure of the mathematical model.

Similar to other dynamic systems, GRNs have stability as their key property. The term ‘stability’ mentioned here aims at the unique equilibrium point while the term ‘multistability’ is concerned with the coexistence of multiple steady states in response to a single set of external inputs. Multistability, the capacity to achieve multiple internal states in response to a single set of external inputs, plays an important role in gene circuit design in synthetic biological systems because it satisfies the minimal requirement for the networks to possess memory where the state of the networks stores information about its past. When forced by a transient stimulus into one state or the other, such a system remains in that state after the transient stimulus has been removed. Multistability has certain unique properties which are not shared by other mechanisms of integrative control and it almost certainly plays an essential role in the dynamics of living cells and organisms. In the article ‘Multistability of genetic regulatory networks’ by Pan, Zhang and Liu, two research issues are addressed. One is the modelling of time delayed GRNs with multistability and parameter uncertainties, and the other is the robust multistability analysis of a GRN with time-varying delays and parameter uncertainties. The delayed uncertain GRNs are modelled as switched systems with interval time-varying delays and parameter uncertainties, which share the similarity with Reset-Set latch and relay in electrical engineering. Methods in switched systems are applied. An important feature of the model proposed here is that this model can describe a GRN with multiple steady states (multistability) rather than only two steady states (bistability). Another feature lies in the fact that this model serves as a more practical description of the physical system by introducing time delays and parameter uncertainties.

It is clear today that mathematical models and tools are required to analyse the complex GRNs. The analysis of periodic solutions in piecewise affine gene network models is well characterised in the special case where all decay rates are supposed equal. With nonuniform decay rates, however, very few results are known, and the techniques developed for homogeneous decay rates cannot be generalised. This lack of results is unfortunate, since real gene networks are known to involve very distinct degradation rates, both among mRNAs and proteins. Moreover, the question is also relevant from a more mathematical viewpoint, since systems with uniform decay rates display appreciably simpler dynamics than systems with distinct decays. Actually, in the uniform decay rate setting, trajectories are locally straight lines, whereas distinct decays lead to pieces of exponential curves. In the article ‘Limit cycles in piecewise-affine gene network models with multiple interaction loops’ by Farcot and Gouze, the attention is focused on periodically visited sequences of boxes, but generalise the authors’ previous results to a setting where the main assumption is only this alignment of successive focal points, regardless of the interaction structure. A theorem about periodic solutions of piecewise linear models of gene regulatory networks is presented and proved in this article. This theorem relates discrete transitions between regular domains of these systems and their actual solutions. It does so under hypotheses of a local nature (alignment of pairs of successive focal points), hence allowing applications to a large variety of examples. The alignment condition is related to the interaction structure of the system only locally, allowing complicated global interaction graphs to be handled within the present framework.

This Special Issue is a timely reflection of the research progress in the area of dynamics analysis of GRNs. Finally, we wish to acknowledge all authors for their efforts in submitting high-quality articles. We are also very grateful to the reviewers for their thorough and on-time reviews of the papers. Last, but not least, our deepest gratitude goes to Professor Peter Fleming, Editor-in-Chief of International Journal of Systems Science for his consideration, encouragement, and advice to publish this special issue.