A Hill type equation can predict target gene expression driven by p53 pulsing

Many factors determine target gene expression dynamics under p53 pulsing. In this study, I sought to determine the mechanism by which duration, frequency, binding affinity and maximal transcription rate affect the expression dynamics of target genes. Using an analytical method to solve a simple model, I found that the fold change of target gene expression increases relative to the number of p53 pulses, and the optimal frequency, 0.18 h-1 , from two real p53 pulses drives the maximal fold change with a decay rate of 0.18 h-1 . Moreover, p53 pulses may also lead to a higher fold change than sustained p53. Finally, I discovered that a Hill-type equation, including these effect factors, can characterise target gene expression. The average error between the theoretical predictions and experiments was 23%. Collectively, this equation advances the understanding of transcription factor dynamics, where duration and frequency play a significant role in the fine regulation of target gene expression with higher binding affinity.

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.

Synchronizing Genetic Oscillators by Signaling Molecules

The authors examine collective rhythms in a general multicell system with both linearly diffusive and nondiffusive couplings. The effect of coupling on synchronization through intercellular signaling in a population of Escherichia coli cells is studied. In particular, a synchronization solution is given through the auxiliary individual system for 2 types of couplings. The sufficient conditions for the global synchronization of such a coupled system are derived based on the Lyapunov function method. The authors show that an appropriate design of the coupling and the inner-linking matrix can ensure global synchronization of the coupled synthetic biological system. Moreover, they demonstrate that the dynamics of an individual cell with coupling and without coupling may be qualitatively different; one is oscillatory, and the other is steady state. The change from a nonoscillatory state to an oscillatory one is induced by appropriate coupling, which also entrains all cells to synchronization. These results establish not only a theoretical foundation but also a quantitative basis for understanding the essential cooperative dynamics, such as collective rhythms or synchronization, in a population of cells.

Delays induce novel stochastic effects in negative feedback gene circuits

Stochastic models of reaction networks are widely used to depict gene expression dynamics. However, stochastic does not necessarily imply accurate, as subtle assumptions can yield erroneous results, masking key discrete effects. For instance, transcription and translation are not instantaneous processes-explicit delays separate their initiation from the appearance of their functional products. However, delays are often ignored in stochastic, single-gene expression models. By consequence, effects such as delay-induced stochastic oscillations at the single-cell level have remained relatively unexplored. Here, we present a systematic study of periodicity and multimodality in a simple gene circuit with negative feedback, analyzing the influence of negative feedback strength and transcriptional/translational delays on expression dynamics. We demonstrate that an oscillatory regime emerges through a Hopf bifurcation in both deterministic and stochastic frameworks. Of importance, a shift in the stochastic Hopf bifurcation evidences inaccuracies of the deterministic bifurcation analysis. Furthermore, noise fluctuations within stochastic oscillations decrease alongside increasing values of transcriptional delays and within a specific range of negative feedback strengths, whereas a strong feedback is associated with oscillations triggered by bursts. Finally, we demonstrate that explicitly accounting for delays increases the number of accessible states in the multimodal regime, and also introduces features typical of excitable systems.

The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models

To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m-repressilator is deducible from the mass action law and, in the limit n→∞, it is a Boolean type model. With this embedding and with delays, if m is odd and n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n>1, there is always a choice of parameters for which the m-repressilator model has bistability.

Systematic computation of nonlinear cellular and molecular dynamics with low-power CytoMimetic circuits: a simulation study

This paper presents a novel method for the systematic implementation of low-power microelectronic circuits aimed at computing nonlinear cellular and molecular dynamics. The method proposed is based on the Nonlinear Bernoulli Cell Formalism (NBCF), an advanced mathematical framework stemming from the Bernoulli Cell Formalism (BCF) originally exploited for the modular synthesis and analysis of linear, time-invariant, high dynamic range, logarithmic filters. Our approach identifies and exploits the striking similarities existing between the NBCF and coupled nonlinear ordinary differential equations (ODEs) typically appearing in models of naturally encountered biochemical systems. The resulting continuous-time, continuous-value, low-power CytoMimetic electronic circuits succeed in simulating fast and with good accuracy cellular and molecular dynamics. The application of the method is illustrated by synthesising for the first time microelectronic CytoMimetic topologies which simulate successfully: 1) a nonlinear intracellular calcium oscillations model for several Hill coefficient values and 2) a gene-protein regulatory system model. The dynamic behaviours generated by the proposed CytoMimetic circuits are compared and found to be in very good agreement with their biological counterparts. The circuits exploit the exponential law codifying the low-power subthreshold operation regime and have been simulated with realistic parameters from a commercially available CMOS process. They occupy an area of a fraction of a square-millimetre, while consuming between 1 and 12 microwatts of power. Simulations of fabrication-related variability results are also presented.

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.

Mean square exponential and robust stability of stochastic discrete-time genetic regulatory networks with uncertainties

This paper aims to analyze global robust exponential stability in the mean square sense of stochastic discrete-time genetic regulatory networks with stochastic delays and parameter uncertainties. Comparing to the previous research works, time-varying delays are assumed to be stochastic whose variation ranges and probability distributions of the time-varying delays are explored. Based on the stochastic analysis approach and some analysis techniques, several sufficient criteria for the global robust exponential stability in the mean square sense of the networks are derived. Moreover, two numerical examples are presented to show the effectiveness of the obtained results.

Self-sustained collective oscillation generated in an array of nonoscillatory cells

Oscillations are ubiquitous phenomena in biological systems. Conventional models of biological periodic oscillations usually invoke interconnecting transcriptional feedback loops. Some specific proteins function as transcription factors, which in turn negatively regulate the expression of the genes that encode these "clock proteins." These loops may lead to rhythmic changes in gene expression in a cell. In the case of multicellular tissue, collective oscillation is often due to the synchronization of these cells, which manifest themselves as autonomous oscillators. In contrast, we propose here a different scenario for the occurrence of collective oscillation in a group of nonoscillatory cells. Neither periodic external stimulation nor pacemaker cells with intrinsically oscillator are included in the present system. By adopting a spatially inhomogeneous active factor, we observe and analyze a coupling-induced oscillation, inherent to the phenomenon of wave propagation due to intracellular communication.

The impact of time delays on the robustness of biological oscillators and the effect of bifurcations on the inverse problem

Differential equation models for biological oscillators are often not robust with respect to parameter variations. They are based on chemical reaction kinetics, and solutions typically converge to a fixed point. This behavior is in contrast to real biological oscillators, which work reliably under varying conditions. Moreover, it complicates network inference from time series data. This paper investigates differential equation models for biological oscillators from two perspectives. First, we investigate the effect of time delays on the robustness of these oscillator models. In particular, we provide sufficient conditions for a time delay to cause oscillations by destabilizing a fixed point in two-dimensional systems. Moreover, we show that the inclusion of a time delay also stabilizes oscillating behavior in this way in larger networks. The second part focuses on the inverse problem of estimating model parameters from time series data. Bifurcations are related to nonsmoothness and multiple local minima of the objective function.

Synchronization of genetic oscillators

Synchronization of genetic or cellular oscillators is a central topic in understanding the rhythmicity of living organisms at both molecular and cellular levels. Here, we show how a collective rhythm across a population of genetic oscillators through synchronization-induced intercellular communication is achieved, and how an ensemble of independent genetic oscillators is synchronized by a common noisy signaling molecule. Our main purpose is to elucidate various synchronization mechanisms from the viewpoint of dynamics, by investigating the effects of various biologically plausible couplings, several kinds of noise, and external stimuli. To have a comprehensive understanding on the synchronization of genetic oscillators, we consider three classes of genetic oscillators: smooth oscillators (exhibiting sine-like oscillations), relaxation oscillators (displaying jump dynamics), and stochastic oscillators (noise-induced oscillation). For every class, we further study two cases: with intercellular communication (including phase-attractive and repulsive coupling) and without communication between cells. We find that an ensemble of smooth oscillators has different synchronization phenomena from those in the case of relaxation oscillators, where noise plays a different but key role in synchronization. To show differences in synchronization between them, we make comparisons in many aspects. We also show that a population of genetic stochastic oscillators have their own synchronization mechanisms. In addition, we present interesting phenomena, e.g., for relaxation-type stochastic oscillators coupled to a quorum-sensing mechanism, different noise intensities can induce different periodic motions (i.e., inhomogeneous limit cycles).

Dynamic properties of a delayed protein cross talk model

In this paper we investigate how the inclusion of time delay alters the dynamical properties of the Jacob-Monod model, describing the control of the beta-galactosidase synthesis by the lac repressor protein in E. coli. The consequences of a time delay on the dynamics of this system are analysed using Hopf's theorem and Lyapunov-Andronov's theory applied to the original mathematical model and to an approximated version. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations. A critical value of time delay, which depends on the values of the model parameters, is analytically established. Around this critical value, the properties of the system change drastically, allowing under certain conditions the emergence of stable limit cycles, that is self-sustained oscillations. In addition, the features of the end product repression play an essential role in the characterisation of these limit cycles: if cooperativity is considered in the end product repression, time delay higher than the mentioned critical value induce differentiated responses during the oscillations, provoking cycles of all-or-nothing response in the concentration of the species.

Stochastic suppression of gene expression oscillators under intercell coupling

This paper examines the dynamics of an ensemble of hysteresis-based genetic relaxation oscillators, focusing on the influence of noise and cell-to-cell coupling on the appearance of new dynamical regimes. In particular, we show that control of the coupling strength and noise can effectively change the dynamics of the system leading to behaviors such as clustering, synchronous and asynchronous oscillations, and suppression. Moreover, under certain conditions an optimal amount of noise can lead to increased order in the system. The results obtained are correlated with relevant biological processes that occur in living organisms.

Synthetic gene oscillators by negative feedback networks

We aim at developing a new methodology to model and design periodic oscillators of gene regulatory networks with multiple genes, proteins and time delays, by using multiple time-scale networks (MTN). Multiple time scale properties are exploited to simplify the model according to singular perturbation theory. We show that a MTN has no stable equilibria but stable periodic orbits. Finally, we design a biologically plausible gene oscillator by cI and Lac genes, to demonstrate our theoretical results.

The minimal, phase-transition model for the cell-number maintenance by the hyperplasia-extended homeorhesis

Oncogenic hyperplasia is the first and inevitable stage of formation of a (solid) tumor. This stage is also the core of many other proliferative diseases. The present work proposes the first minimal model that combines homeorhesis with oncogenic hyperplasia where the latter is regarded as a genotoxically activated homeorhetic dysfunction. This dysfunction is specified as the transitions of the fluid of cells from a fluid, homeorhetic state to a solid, hyperplastic-tumor state, and back. The key part of the model is a nonlinear reaction-diffusion equation (RDE) where the biochemical-reaction rate is generalized to the one in the well-known Schlögl physical theory of the non-equilibrium phase transitions. A rigorous analysis of the stability and qualitative aspects of the model, where possible, are presented in detail. This is related to the spatially homogeneous case, i.e. when the above RDE is reduced to a nonlinear ordinary differential equation. The mentioned genotoxic activation is treated as a prevention of the quiescent G0-stage of the cell cycle implemented with the threshold mechanism that employs the critical concentration of the cellular fluid and the nonquiescent-cell-duplication time. The continuous tumor morphogeny is described by a time-space-dependent cellular-fluid concentration. There are no sharp boundaries (i.e. no concentration jumps exist) between the domains of the homeorhesis- and tumor-cell populations. No presumption on the shape of a tumor is used. To estimate a tumor in specific quantities, the model provides the time-dependent tumor locus, volume, and boundary that also points out the tumor shape and size. The above features are indispensable in the quantitative development of antiproliferative drugs or therapies and strategies to prevent oncogenic hyperplasia in cancer and other proliferative diseases. The work proposes an analytical-numerical method for solving the aforementioned RDE. A few topics for future research are suggested.

Detection of cellular rhythms and global stability within interlocked feedback systems

In this paper, we show how to detect cellular rhythm and its global stability by extending the techniques from the recently developed theory of monotone systems. We establish theoretical results for globally asymptotic stability with consideration of delay by a discrete map. The relationship between positive, negative elements and delay in a general class of interlocked feedback networks can be understood in a system level. Moreover, the correspondence of attractors between a network and its reduced map is obtained and can be used to detect cellular rhythm, and further control the dynamics of the network. We show that global cellular rhythms can always be obtained, thereby enhancing robustness against perturbations of initial conditions and avoiding chaotic oscillations or complete abolishment of oscillations. In this paper, we focus on analyzing the circadian oscillator in Drosophila as an example to detect the occurrence of cellular rhythm and its global stability.

A model for the circadian rhythm of cyanobacteria that maintains oscillation without gene expression

An intriguing property of the cyanobacterial circadian clock is that endogenous rhythm persists when protein abundances are kept constant either in the presence of translation and transcription inhibitors or in the constant dark condition. Here we propose a regulatory mechanism of KaiC phosphorylation for the generation of circadian oscillations in cyanobacteria. In the model, clock proteins KaiA and KaiB are assumed to have multiple states, regulating the KaiC phosphorylation process. The model can explain 1), the sustained oscillation of gene expression and protein abundance when the expression of the kaiBC gene is regulated by KaiC protein, and 2), the sustained oscillation of phosphorylated KaiC when transcription and translation processes are inhibited and total protein abundance is fixed. Results of this work suggest that KaiA and KaiB strengthen the nonlinearity of KaiC phosphorylation, thereby promoting the circadian rhythm in cyanobacteria.

Construction of genetic oscillators with interlocked feedback networks

To understand how a gene regulatory network functioning as an oscillator is built, a precise mathematical description of the network and its dynamical properties are developed in this paper. Our approach is based on analyzing the effect of interactions between smaller subnetworks with simple dynamics. We relate an oscillatory behavior of the network to the destabilization phenomenon of a steady state caused by the interactions in a simple discrete map. When source of the instability of the steady state rather than the oscillatory behavior itself is considered, linear stability analysis and feedback control theory can be employed. Moreover, the amplitudes robust against change in delay can also be obtained from the discrete map. The main ideas are illustrated by constructing a genetic oscillator termed a "repressilator" and analyzing the occurrence of cellular rhythms, although the theoretical results hold for a general class of biological systems. The method can be directly applied to design, construct genetic oscillators, and further control their dynamics, even for large-scale networks.

Noise-induced cooperative behavior in a multicell system

MOTIVATION

All cell components exhibit intracellular noise on account of random births and deaths of individual molecules, and extracellular noise because of environment perturbations. Gene regulation in particular, is an inherently noisy process with transcriptional control, alternative splicing, translation, diffusion and chemical modification reactions, all of which involve stochastic fluctuations. Such stochastic noises may not only affect the dynamics of the entire system but may also be exploited by living organisms to actively facilitate certain functions, such as cooperative behavior and communication.

RESULTS

We have provided a general model and an analytic tool to examine the cooperative behavior of a multicell system with both intracellular and extracellular stochastic fluctuations. A multicell system with a synthetic gene network is adopted to demonstrate the effects of noises and coupling on collective dynamics. These results establish not only a theoretical foundation but also a quantitative basis for understanding essential roles of noises on cooperative dynamics, such as synchronization and communication among cells.

Dynamics of gene regulatory networks with cell division cycle

This paper focuses on modeling and analyzing the nonlinear dynamics of gene regulatory networks with the consideration of a cell division cycle with duplication process of DNA, in particular for switches and oscillators of synthetic networks. We derive two models that may correspond to the eukaryotic and prokaryotic cells, respectively. A biologically plausible three-gene model ( lac, tetR, and cI ) and a repressilator as switch and oscillator examples are used to illustrate our theoretical results. We show that the cell cycle may play a significant role in gene regulation due to the nonlinear dynamics of a gene regulatory network although gene expressions are usually tightly controlled by transcriptional factors.

Modelling periodic oscillation of biological systems with multiple timescale networks

In this paper, we aim to develop a new methodology to model and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using multiple timescale networks (MTN). Fast reactions constitute a positive feedback-loop network (PFN), while slow reactions consist of a cyclic feedback-loop network (CFN), in MTN. Multiple timescales are exploited to simplify models according to singular perturbation theory. We show that a MTN has no stable equilibrium but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of MTNs with only one PFN, and then generalise the result to MTNs with multiple PFNs. Finally, we design a biologically plausible gene regulatory network by the cI and Lac genes, to demonstrate the theoretical results. Since there is less restriction on the network structure of a MTN, it can be expected to apply to a wide variety of areas on the modelling, analysing and designing of biological systems.

Feedback loops in the signal paths controlling gene expression

Theoretical and empirical studies support the concepts suggesting emission of electromagnetic fields by living cells. Experiments performed in our laboratory indicate, in addition, tight links between attributes of these fields and the phase of cell division cycle, i.e., processes governed by gene transcription. This work supplements the now quite extensive literature concerning gene networks in that it identifies the well-known feedback mechanisms in exact topological and quantitative terms. Special diagrammatic apparatus is employed for this purpose. The simple system comprising one gene and one transcription factor (binding as a dimer to the gene regulatory region), controlled via phosphorylation process by external signal, is analyzed to demonstrate the approach. The diagram representative of the system's dynamics contains a feedback loop; the so-called transmission function of this feature of diagram topology is shown to control the dependences of gene product concentration on the stimulus intensity.

Modeling genetic switches with positive feedback loops

In this paper, we develop a new methodology to design synthetic genetic switch networks with multiple genes and time delays, by using monotone dynamical systems. We show that the networks with only positive feedback loops have no stable oscillation but stable equilibria whose stability is independent of the time delays. In other words, such systems have ideal properties for switch networks and can be designed without consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces. Therefore, we can ensure that the designed switches function correctly even with uncertain delays. 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 design tractable even for large-scaled systems. Finally, to demonstrate our theoretical results, we show biologically plausible examples by designing a synthetic genetic switch with experimentally well investigated lacI, tetR, and cI genes for numerical simulation.