Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays

Design of Oscillatory Networks through Post-Translational Control of Network Components

Many essential functions in biological systems, including cell cycle progression and circadian rhythm regulation, are governed by the periodic behaviors of specific molecules. These periodic behaviors arise from the precise arrangement of components in biomolecular networks that generate oscillatory output signals. The dynamic properties of individual components of these networks, such as maturation delays and degradation rates, often play a key role in determining the network's oscillatory behavior. In this study, we explored the post-translational modulation of network components as a means to generate genetic circuits with oscillatory behaviors and perturb the oscillation features. Specifically, we used the NanoDeg platform-A bifunctional molecule consisting of a target-specific nanobody and a degron tag-to control the degradation rates of the circuit's components and predicted the effect of NanoDeg-mediated post-translational depletion of a key circuit component on the behavior of a series of proto-oscillating network topologies. We modeled the behavior of two main classes of oscillators, namely relaxation oscillator topologies (the activator-repressor and the Goodwin oscillator) and ring oscillator topologies (repressilators). We identified two main mechanisms by which non-oscillating networks could be induced to oscillate through post-translational modulation of network components: an increase in the separation of timescales of network components and mitigation of the leaky expression of network components. These results are in agreement with previous findings describing the effect of timescale separation and mitigation of leaky expression on oscillatory behaviors. This work thus validates the use of tools to control protein degradation rates as a strategy to modulate existing oscillatory signals and construct oscillatory networks. In addition, this study provides the design rules to implement such an approach based on the control of protein degradation rates using the NanoDeg platform, which does not require genetic manipulation of the network components and can be adapted to virtually any cellular protein. This work also establishes a framework to explore the use of tools for post-translational perturbations of biomolecular networks and generates desired behaviors of the network output.

Universality in kinetic models of circadian rhythms in [Formula: see text]

Biological evolution has endowed the plant Arabidopsis thaliana with genetically regulated circadian rhythms. A number of authors have published kinetic models for these oscillating chemical reactions based on a network of interacting genes. To investigate the hypothesis that the Arabidopsis circadian dynamical system is poised near a Hopf bifurcation like some other biological oscillators, we varied the kinetic parameters in the models and searched for bifurcations. Finding that each model does exhibit a supercritical Hopf bifurcation, we performed a weakly nonlinear analysis near the bifurcation points to derive the Stuart-Landau amplitude equation. To illustrate a common dynamical structure, we scaled the numerical solutions to the models with the asymptotic solutions to the Stuart-Landau equation to collapse the circadian oscillations onto two universal curves-one for amplitude, and one for frequency. However, some models are close to bifurcation while others are far, some models are post-bifurcation while others are pre-bifurcation, and kinetic parameters that lead to a bifurcation in some models do not lead to a bifurcation in others. Future kinetic modeling can make use of our analysis to ensure models are consistent with each other and with the dynamics of the Arabidopsis circadian rhythm.

Mathematical Modelling of p53 Signalling during DNA Damage Response: A Survey

No gene has garnered more interest than p53 since its discovery over 40 years ago. In the last two decades, thanks to seminal work from Uri Alon and Ghalit Lahav, p53 has defined a truly synergistic topic in the field of mathematical biology, with a rich body of research connecting mathematic endeavour with experimental design and data. In this review we survey and distill the extensive literature of mathematical models of p53. Specifically, we focus on models which seek to reproduce the oscillatory dynamics of p53 in response to DNA damage. We review the standard modelling approaches used in the field categorising them into three types: time delay models, spatial models and coupled negative-positive feedback models, providing sample model equations and simulation results which show clear oscillatory dynamics. We discuss the interplay between mathematics and biology and show how one informs the other; the deep connections between the two disciplines has helped to develop our understanding of this complex gene and paint a picture of its dynamical response. Although yet more is to be elucidated, we offer the current state-of-the-art understanding of p53 response to DNA damage.

Stability and bifurcation analysis of a fractional-order single-gene regulatory model with delays under a novel PDα control law

In this paper, we propose a novel fractional-order proportional-derivative (PD) strategy to achieve the control of bifurcation of a fractional-order gene regulatory model with delays. The stability theory of fractional differential equations proved that with delays, some explicit conditions for the local asymptotical stability and Hopf bifurcation are given for the controlled fractional-order genetic model. It is demonstrated that the fractional-order gene regulatory model becomes controllable by adjusting the control gain parameters. In addition, the effect of fractional-order parameter on the dynamical behaviors is shown. Finally, numerical simulations are carried out to testify the validity of the main results and the availability of the fractional-order PD controller.

Bifurcation manifolds in predator-prey models computed by Gröbner basis method

Many natural processes studied in population biology, systems biology, biochemistry, chemistry or physics are modeled by dynamical systems with polynomial or rational right-hand sides in state and parameter variables. The problem of finding bifurcation manifolds of such discrete or continuous dynamical systems leads to a problem of finding solutions to a system of non-linear algebraic equations. This approach often fails since it is not possible to express equilibria explicitly. Here we describe an algebraic procedure based on the Gröbner basis computation that finds bifurcation manifolds without computing equilibria. Our method provides formulas for bifurcation manifolds in commonly studied cases in applied research - for the fold, transcritical, cusp, Hopf and Bogdanov-Takens bifurcations. The method returns bifurcation manifolds as implicitly defined functions or parametric functions in full parameter space. The approach can be implemented in any computer algebra system; therefore it can be used in applied research as a supporting autonomous computation even by non-experts in bifurcation theory. This paper demonstrates our new approach on the recently published Rosenzweig-MacArthur predator-prey model generalizations in order to highlight the simplicity of our method compared to the published analysis.

Bifurcation and Oscillatory Dynamics of Delayed Cyclic Gene Networks Including Small RNAs

It has been demonstrated in a large number of experimental results that small RNAs (sRNAs) play a vital role in gene regulation processes. Thus, the gene regulation process is dominated by sRNAs in addition to messenger RNAs and proteins. However, the regulation mechanism of sRNAs is not well understood and there are few models considering the effect of sRNAs. So it is of realistic biological background to include sRNAs when modeling gene networks. In this paper, sRNAs are incorporated into the process of gene expression and a new differential equation model is put forward to describe cyclic genetic regulatory networks with sRNAs and multiple delays. We mainly investigate the stability and bifurcation criteria for two cases: 1) positive cyclic genetic regulatory networks and 2) negative cyclic genetic regulatory networks. For a positive cyclic genetic regulatory network, it is revealed that there may exist more than one equilibrium and the multistability can appear. Sufficient conditions are established for the delay-independent stability and fold bifurcations. It is found that the dynamics of positive cyclic gene networks has no bearing on time delays, but depends on the biochemical parameters, the Hill coefficient and the equilibrium itself. For a negative cyclic genetic regulatory network, it is proved that there exists a unique equilibrium. Delay-dependent conditions for the stability are derived, and the existence of Hopf bifurcations is examined. Different from the delay-independent stability of positive gain networks, the stability of equilibrium is determined not only by the biochemical parameters, the Hill coefficient and the equilibrium itself, but also by the total delay. At last, three illustrative examples are provided to validate the major results.

PCA based population generation for genetic network optimization

A gene regulatory network (GRN) represents a set of genes and its regulatory interactions. The inference of the regulatory interactions between genes is usually carried out using an appropriate mathematical model and the available gene expression profile. Among the various models proposed for GRN inference, our recently proposed Michaelis-Menten based ODE model provides a good trade-off between the computational complexity and biological relevance. This model, like other known GRN models, also uses an evolutionary algorithm for parameter estimation. Considering various issues associated with such population based stochastic optimization approaches (e.g. diversity, premature convergence due to local optima, accuracy, etc.), it becomes important to seed the initial population with good individuals which are closer to the optimal solution. In this paper, we exploit the inherent strength of principal component analysis (PCA) in a novel manner to initialize the population for GRN optimization. The benefit of the proposed method is validated by reconstructing in silico and in vivo networks of various sizes. For the same level of accuracy, the approach with PCA based initialization shows improved convergence speed.

Approximation of state variables for discrete-time stochastic genetic regulatory networks with leakage, distributed, and probabilistic measurement delays: a robust stability problem

This work predominantly labels the problem of approximation of state variables for discrete-time stochastic genetic regulatory networks with leakage, distributed, and probabilistic measurement delays. Here we design a linear estimator in such a way that the absorption of mRNA and protein can be approximated via known measurement outputs. By utilizing a Lyapunov-Krasovskii functional and some stochastic analysis execution, we obtain the stability formula of the estimation error systems in the structure of linear matrix inequalities under which the estimation error dynamics is robustly exponentially stable. Further, the obtained conditions (in the form of LMIs) can be effortlessly solved by some available software packages. Moreover, the specific expression of the desired estimator is also shown in the main section. Finally, two mathematical illustrative examples are accorded to show the advantage of the proposed conceptual results.

Finite-Time Stability Analysis of Reaction-Diffusion Genetic Regulatory Networks with Time-Varying Delays

This paper is concerned with the finite-time stability problem of the delayed genetic regulatory networks (GRNs) with reaction-diffusion terms under Dirichlet boundary conditions. By constructing a Lyapunov-Krasovskii functional including quad-slope integrations, we establish delay-dependent finite-time stability criteria by employing the Wirtinger-type integral inequality, Gronwall inequality, convex technique, and reciprocally convex technique. In addition, the obtained criteria are also reaction-diffusion-dependent. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results.

Stability and Neimark-Sacker Bifurcation Analysis in a Genetic Network with Delay

This paper investigates a genetic model with delay. The stability, direction, and bifurcation periodic solution is derived by using the center manifold theorem and normal form theory. Numerical simulations illustrate the theoretical results.

Steady-State-Preserving Simulation of Genetic Regulatory Systems

A novel family of exponential Runge-Kutta (expRK) methods are designed incorporating the stable steady-state structure of genetic regulatory systems. A natural and convenient approach to constructing new expRK methods on the base of traditional RK methods is provided. In the numerical integration of the one-gene, two-gene, and p53-mdm2 regulatory systems, the new expRK methods are shown to be more accurate than their prototype RK methods. Moreover, for nonstiff genetic regulatory systems, the expRK methods are more efficient than some traditional exponential RK integrators in the scientific literature.

Feedback, Mass Conservation and Reaction Kinetics Impact the Robustness of Cellular Oscillations

Oscillations occur in a wide variety of cellular processes, for example in calcium and p53 signaling responses, in metabolic pathways or within gene-regulatory networks, e.g. the circadian system. Since it is of central importance to understand the influence of perturbations on the dynamics of these systems a number of experimental and theoretical studies have examined their robustness. The period of circadian oscillations has been found to be very robust and to provide reliable timing. For intracellular calcium oscillations the period has been shown to be very sensitive and to allow for frequency-encoded signaling. We here apply a comprehensive computational approach to study the robustness of period and amplitude of oscillatory systems. We employ different prototype oscillator models and a large number of parameter sets obtained by random sampling. This framework is used to examine the effect of three design principles on the sensitivities towards perturbations of the kinetic parameters. We find that a prototype oscillator with negative feedback has lower period sensitivities than a prototype oscillator relying on positive feedback, but on average higher amplitude sensitivities. For both oscillator types, the use of Michaelis-Menten instead of mass action kinetics in all degradation and conversion reactions leads to an increase in period as well as amplitude sensitivities. We observe moderate changes in sensitivities if replacing mass conversion reactions by purely regulatory reactions. These insights are validated for a set of established models of various cellular rhythms. Overall, our work highlights the importance of reaction kinetics and feedback type for the variability of period and amplitude and therefore for the establishment of predictive models.

Rhythmic behavior is omnipresent in biology and has many crucial functions. In cells the activation levels and abundances of signaling molecules such as NF-κB, p53, EGFR or calcium repeatedly increase and decrease in response to stimuli. Such a dynamic behavior can also be observed monitoring the concentrations of mRNAs and proteins in the circadian clock and the cell cycle. Period and amplitude which are the time span between peaks and the peak height, respectively, as well as their variabilities are important features of oscillations. The circadian period is very stable allowing for proper time keeping, whereas in calcium signaling the period is very variable encoding different stimulation strengths. Our goal is to examine the origin of differences in sensitivities of periods and amplitudes using a computational approach. We use prototype oscillators and demonstrate that they can be used to derive general principles that explain the degree of robustness in period and amplitude for a set of commonly used models of cellular oscillators. Our findings imply that the robustness of oscillating systems can be influenced by feedback type and kinetic properties to which special attention should be paid when designing mathematical models of cellular rhythms.

Time-Delayed Models of Gene Regulatory Networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

Splitting strategy for simulating genetic regulatory networks

The splitting approach is developed for the numerical simulation of genetic regulatory networks with a stable steady-state structure. The numerical results of the simulation of a one-gene network, a two-gene network, and a p53-mdm2 network show that the new splitting methods constructed in this paper are remarkably more effective and more suitable for long-term computation with large steps than the traditional general-purpose Runge-Kutta methods. The new methods have no restriction on the choice of stepsize due to their infinitely large stability regions.

State estimation for delayed genetic regulatory networks based on passivity theory

This paper is concerned with the state estimation problem for delayed genetic regulatory networks (GRNs) based on passivity analysis approach. The main purpose of the problem is to design the estimator to approximate the true concentrations of the mRNA and protein through available measurement outputs. Time-varying delays are explicitly assumed to be non-differentiable and constraint on the derivative of the time-varying delay is less than one can be removed. Based on the Lyapunov-Krasovskii functionals involving triple integral terms, using some integral inequalities and convex combination technique, a delay-dependent passivity criterion is established for GRNs in terms of linear matrix inequalities (LMIs) that can efficiently be solved by any available LMI solvers. Finally, numerical examples and simulation are presented to demonstrate the efficiency of the proposed estimation schemes.

Stability switches and multistability coexistence in a delay-coupled neural oscillators system

In this paper, we present a neural network system composed of two delay-coupled neural oscillators, where each of these can be regarded as the dynamical system describing the average activity of neural population. Analyzing the corresponding characteristic equation, the local stability of rest state is studied. The system exhibits the switch phenomenon between the rest state and periodic activity. Furthermore, the Hopf bifurcation is analyzed and the bifurcation curve is given in the parameters plane. The stability of the bifurcating periodic solutions and direction of the Hopf bifurcation are exhibited. Regarding time delay and coupled weight as the bifurcation parameters, the Fold-Hopf bifurcation is investigated in detail in terms of the central manifold reduction and normal form method. The neural system demonstrates the coexistence of the rest states and periodic activities in the different parameter regions. Employing the normal form of the original system, the coexistence regions are illustrated approximately near the Fold-Hopf singularity point. Finally, numerical simulations are performed to display more complex dynamics. The results illustrate that system may exhibit the rich coexistence of the different neuro-computational properties, such as the rest states, periodic activities, and quasi-periodic behavior. In particular, some periodic activities can evolve into the bursting-type behaviors with the varying time delay. It implies that the coexistence of the quasi-periodic activity and bursting-type behavior can be obtained if the suitable value of system parameter is chosen.

An oscillatory criterion for a time delayed neural ring network model

The effects of delays on dynamical networks and the stability analysis of time delayed systems have received a notable attention over the past decades. In this paper, the effects of delays on the oscillatory properties of a neural ring networks model are considered. The existence of oscillations for a specific type of recurrent neural network with time delays between neural interconnections is investigated. By using Chafee's closed orbit theory, some sufficient conditions for permanent oscillations are obtained. Simple and practical criteria for selecting the range of parameters in this network model are also derived. Among other things, the solutions that we provide can be applied to various activation functions. A few computer simulations are presented to support our analysis. The present study can be applied to analyze under which conditions a ring network could be exploited as an oscillatory pattern generator.

Improvements in the reconstruction of time-varying gene regulatory networks: dynamic programming and regularization by information sharing among genes

METHOD

Dynamic Bayesian networks (DBNs) have been applied widely to reconstruct the structure of regulatory processes from time series data, and they have established themselves as a standard modelling tool in computational systems biology. The conventional approach is based on the assumption of a homogeneous Markov chain, and many recent research efforts have focused on relaxing this restriction. An approach that enjoys particular popularity is based on a combination of a DBN with a multiple changepoint process, and the application of a Bayesian inference scheme via reversible jump Markov chain Monte Carlo (RJMCMC). In the present article, we expand this approach in two ways. First, we show that a dynamic programming scheme allows the changepoints to be sampled from the correct conditional distribution, which results in improved convergence over RJMCMC. Second, we introduce a novel Bayesian clustering and information sharing scheme among nodes, which provides a mechanism for automatic model complexity tuning.

RESULTS

We evaluate the dynamic programming scheme on expression time series for Arabidopsis thaliana genes involved in circadian regulation. In a simulation study we demonstrate that the regularization scheme improves the network reconstruction accuracy over that obtained with recently proposed inhomogeneous DBNs. For gene expression profiles from a synthetically designed Saccharomyces cerevisiae strain under switching carbon metabolism we show that the combination of both: dynamic programming and regularization yields an inference procedure that outperforms two alternative established network reconstruction methods from the biology literature.

AVAILABILITY AND IMPLEMENTATION

A MATLAB implementation of the algorithm and a supplementary paper with algorithmic details and further results for the Arabidopsis data can be downloaded from: http://www.statistik.tu-dortmund.de/bio2010.html.

Fixed point characterization of biological networks with complex graph topology

MOTIVATION

Feedback circuits are important motifs in biological networks and part of virtually all regulation processes that are needed for a reliable functioning of the cell. Mathematically, feedback is connected to complex behavior of the systems, which is often related to bifurcations of fixed points. Therefore, several approaches for the investigation of fixed points in biological networks have been developed in recent years. Many of them assume the fixed point coordinates to be known, and an efficient way to calculate the entire set of fixed points for interrelated feedback structures is highly desirable.

RESULTS

In this article, we consider regulatory network models, which are differential equations with an underlying directed graph that illustrates independencies among variables. We introduce the circuit-breaking algorithm (CBA), a method that constructs one-dimensional characteristics for these network models, which inherit important information about the system. In particular, fixed points are related to the zeros of these characteristics. The CBA operates on the graph topology, and results from graph theory are used in order to make calculations efficient. Our framework provides a general scheme for analyzing network models in terms of interrelated feedback circuits. The efficiency of the approach is demonstrated on a model for calcium oscillations based on experiments in hepatocytes, which consists of several interrelated feedback circuits.

Boolean versus continuous dynamics on simple two-gene modules

We investigate the dynamical behavior of simple modules composed of two genes with two or three regulating connections. Continuous dynamics for mRNA and protein concentrations is compared to a Boolean model for gene activity. Using a generalized method, we study within a single framework different continuous models and different types of regulatory functions, and establish conditions under which the system can display stable oscillations. These conditions depend only on general features such as the ratio of the relevant time scales, the degree of cooperativity of the regulating interactions, and the logical structure of the interactions. Our results combine and generalize the findings of several disconnected previous studies.

On multistability of delayed genetic regulatory networks with multivariable regulation functions

Many genetic regulatory networks (GRNs) have the capacity to reach different stable states. This capacity is defined as multistability which is an important regulation mechanism. Multiple time delays and multivariable regulation functions are usually inevitable in such GRNs. In this paper, multistability of GRNs is analyzed by applying the control theory and mathematical tools. This study is to provide a theoretical tool to facilitate the design of synthetic gene circuit with multistability in the perspective of control theory. By transforming such GRNs into a new and uniform mathematical formulation, we put forward a general sector-like regulation function that is capable of quantifying the regulation effects in a more precise way. By resorting to up-to-date techniques, a novel Lyapunov-Krasovskii functional (LKF) is introduced for achieving delay dependence to ensure less conservatism. New conditions are then proposed to ensure the multistability of a GRN in the form of linear matrix inequalities (LMIs) that are dependent on the delays. Our multistability conditions are applicable to several frequently used regulation functions especially the multivariable ones. Two examples are employed to illustrate the applicability and usefulness of the developed theoretical results.