Roles of noise in single and coupled multiple genetic oscillators

Quantifying Landscape-Flux via Single-Cell Transcriptomics Uncovers the Underlying Mechanism of Cell Cycle

Recent developments in single-cell sequencing technology enable the acquisition of entire transcriptome data. Understanding the underlying mechanism and identifying the driving force of transcriptional regulation governing cell function directly from these data remains challenging. This study reconstructs a continuous vector field of the cell cycle based on discrete single-cell RNA velocity to quantify the single-cell global nonequilibrium dynamic landscape-flux. It reveals that large fluctuations disrupt the global landscape and genetic perturbations alter landscape-flux, thus identifying key genes in maintaining cell cycle dynamics and predicting associated functional effects. Additionally, it quantifies the fundamental energy cost of the cell cycle initiation and unveils that sustaining the cell cycle requires curl flux and dissipation to maintain the oscillatory phase coherence. This study enables the inference of the cell cycle gene regulatory networks directly from the single-cell transcriptomic data, including the feedback mechanisms and interaction intensity. This provides a golden opportunity to experimentally verify the landscape-flux theory and also obtain its associated quantifications. It also offers a unique framework for combining the landscape-flux theory and single-cell high-through sequencing experiments for understanding the underlying mechanisms of the cell cycle and can be extended to other nonequilibrium biological processes, such as differentiation development and disease pathogenesis.

Stochastic epigenetic dynamics of gene switching

Epigenetic modifications of histones crucially affect eukaryotic gene activity, while the epigenetic histone state is largely determined by the binding of specific factors such as the transcription factors (TFs) to DNA. Here, the way in which the TFs and the histone state are dynamically correlated is not obvious when the TF synthesis is regulated by the histone state. This type of feedback regulatory relation is ubiquitous in gene networks to determine cell fate in differentiation and other cell transformations. To gain insights into such dynamical feedback regulations, we theoretically analyze a model of epigenetic gene switching by extending the Doi-Peliti operator formalism of reaction kinetics to the problem of coupled molecular processes. Spin-1 and spin-1/2 coherent-state representations are introduced to describe stochastic reactions of histones and binding or unbinding of TFs in a unified way, which provides a concise view of the effects of timescale difference among these molecular processes; even in the case that binding or unbinding of TFs to or from DNA is adiabatically fast, the slow nonadiabatic histone dynamics gives rise to a distinct circular flow of the probability flux around basins in the landscape of the gene state distribution, which leads to hysteresis in gene switching. In contrast to the general belief that the change in the amount of TF precedes the histone state change, flux drives histones to be modified prior to the change in the amount of TF in self-regulating circuits. Flux-landscape analyses shed light on the nonlinear nonadiabatic mechanism of epigenetic cell fate decision making.

Landscape and flux govern cellular mode-hopping between oscillations

Recently, a "mode-hopping" phenomenon has been observed in a NF-κB gene regulatory network with oscillatory tumor necrosis factor (TNF) inputs. It was suggested that noise facilitates the switch between different oscillation modes. However, the underlying mechanism of this noise-induced "cellular mode-hopping" behavior remains elusive. We employed a landscape and flux approach to study the stochastic dynamics and global stability of the NF-κB regulatory system. We used a truncated moment equation approach to calculate the probability distribution and potential landscape for gene regulatory systems. The potential landscape of the NF-κB system exhibits a "double ring valley" shape. Barrier heights from landscape topography provide quantitative measures of the global stability and transition feasibility of the double oscillation system. We found that the landscape and flux jointly govern the dynamical "mode-hopping" behavior of the NF-κB regulatory system. The landscape attracts the system into a "double ring valley," and the flux drives the system to move cyclically. As the external noise increases, relevant barrier heights decrease, and the flux increases. As the amplitude of the TNF input increases, the flux contribution, from the total driving force, increases and the system behavior changes from one to two cycles and ultimately to chaotic dynamics. Therefore, the probabilistic flux may provide an origin of chaotic behavior. We found that the height of the peak of the power spectrum of autocorrelation functions and phase coherence is correlated with barrier heights of the landscape and provides quantitative measures of global stability of the system under intrinsic fluctuations.

Exploring the Underlying Mechanisms of the Xenopus laevis Embryonic Cell Cycle

The cell cycle is an indispensable process in proliferation and development. Despite significant efforts, global quantification and physical understanding are still challenging. In this study, we explored the mechanisms of the Xenopus laevis embryonic cell cycle by quantifying the underlying landscape and flux. We uncovered the Mexican hat landscape of the Xenopus laevis embryonic cell cycle with several local basins and barriers on the oscillation path. The local basins characterize the different phases of the Xenopus laevis embryonic cell cycle, and the local barriers represent the checkpoints. The checkpoint mechanism of the cell cycle is revealed by the landscape basins and barriers. While landscape shape determines the stabilities of the states on the oscillation path, the curl flux force determines the stability of the cell cycle flow. Replication is fundamental for biology of living cells. We quantify the input energy (through the entropy production) as the thermodynamic requirement for initiation and sustainability of single cell life (cell cycle). Furthermore, we also quantify curl flux originated from the input energy as the dynamical requirement for the emergence of a new stable phase (cell cycle). This can provide a new quantitative insight for the origin of single cell life. In fact, the curl flux originated from the energy input or nutrition supply determines the speed and guarantees the progression of the cell cycle. The speed of the cell cycle is a hallmark of cancer. We characterized the quality of the cell cycle by the coherence time and found it is supported by the flux and energy cost. We are also able to quantify the degree of time irreversibility by the cross correlation function forward and backward in time from the stochastic traces in the simulation or experiments, providing a way for the quantification of the time irreversibility and the flux. Through global sensitivity analysis upon landscape and flux, we can identify the key elements for controlling the cell cycle speed. This can help to design an effective strategy for drug discovery against cancer.

Effects of Collective Histone State Dynamics on Epigenetic Landscape and Kinetics of Cell Reprogramming

Cell reprogramming is a process of transitions from differentiated to pluripotent cell states via transient intermediate states. Within the epigenetic landscape framework, such a process is regarded as a sequence of transitions among basins on the landscape; therefore, theoretical construction of a model landscape which exhibits experimentally consistent dynamics can provide clues to understanding epigenetic mechanism of reprogramming. We propose a minimal gene-network model of the landscape, in which each gene is regulated by an integrated mechanism of transcription-factor binding/unbinding and the collective chemical modification of histones. We show that the slow collective variation of many histones around each gene locus alters topology of the landscape and significantly affects transition dynamics between basins. Differentiation and reprogramming follow different transition pathways on the calculated landscape, which should be verified experimentally via single-cell pursuit of the reprogramming process. Effects of modulation in collective histone state kinetics on transition dynamics and pathway are examined in search for an efficient protocol of reprogramming.

Effects of promoter leakage on dynamics of gene expression

Background

Quantitative analysis of simple molecular networks is an important step forward understanding fundamental intracellular processes. As network motifs occurring recurrently in complex biological networks, gene auto-regulatory circuits have been extensively studied but gene expression dynamics remain to be fully understood, e.g., how promoter leakage affects expression noise is unclear.

Results

In this work, we analyze a gene model with auto regulation, where the promoter is assumed to have one active state with highly efficient transcription and one inactive state with very lowly efficient transcription (termed as promoter leakage). We first derive the analytical distribution of gene product, and then analyze effects of promoter leakage on expression dynamics including bursting kinetics. Interestingly, we find that promoter leakage always reduces expression noise and that increasing the leakage rate tends to simplify phenotypes. In addition, higher leakage results in fewer bursts.

Conclusions

Our results reveal the essential role of promoter leakage in controlling expression dynamics and further phenotype. Specifically, promoter leakage is a universal mechanism of reducing expression noise, controlling phenotypes in different environments and making the gene produce generate fewer bursts.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0157-z) contains supplementary material, which is available to authorized users.

Multi-stable dynamics of the non-adiabatic repressilator

The assumption of the fast binding of transcription factors (TFs) to promoters is a typical point in studies of synthetic genetic circuits functioning in bacteria. Although the assumption is effective for simplifying the models, it becomes questionable in the light of in vivo measurements of the times TF spends searching for its cognate DNA sites. We investigated the dynamics of the full idealized model of the paradigmatic genetic oscillator, the repressilator, using deterministic mathematical modelling and stochastic simulations. We found (using experimentally approved parameter values) that decreases in the TF binding rate changes the type of transition between steady state and oscillation. As a result, this gives rise to the hysteresis region in the parameter space, where both the steady state and the oscillation coexist. We further show that the hysteresis is persistent over a considerable range of the parameter values, but the presence of the oscillations is limited by the low rate of TF dimer degradation. Finally, the stochastic simulation of the model confirms the hysteresis with switching between the two attractors, resulting in highly skewed period distributions. Moreover, intrinsic noise stipulates trains of large-amplitude modulations around the stable steady state outside the hysteresis region, which makes the period distributions bimodal.

The potential and flux landscape theory of ecology

The species in ecosystems are mutually interacting and self sustainable stable for a certain period. Stability and dynamics are crucial for understanding the structure and the function of ecosystems. We developed a potential and flux landscape theory of ecosystems to address these issues. We show that the driving force of the ecological dynamics can be decomposed to the gradient of the potential landscape and the curl probability flux measuring the degree of the breaking down of the detailed balance (due to in or out flow of the energy to the ecosystems). We found that the underlying intrinsic potential landscape is a global Lyapunov function monotonically going down in time and the topology of the landscape provides a quantitative measure for the global stability of the ecosystems. We also quantified the intrinsic energy, the entropy, the free energy and constructed the non-equilibrium thermodynamics for the ecosystems. We studied several typical and important ecological systems: the predation, competition, mutualism and a realistic lynx-snowshoe hare model. Single attractor, multiple attractors and limit cycle attractors emerge from these studies. We studied the stability and robustness of the ecosystems against the perturbations in parameters and the environmental fluctuations. We also found that the kinetic paths between the multiple attractors do not follow the gradient paths of the underlying landscape and are irreversible because of the non-zero flux. This theory provides a novel way for exploring the global stability, function and the robustness of ecosystems.

On the dephasing of genetic oscillators

The digital nature of genes combined with the associated low copy numbers of proteins regulating them is a significant source of stochasticity, which affects the phase of biochemical oscillations. We show that unlike ordinary chemical oscillators, the dichotomic molecular noise of gene state switching in gene oscillators affects the stochastic dephasing in a way that may not always be captured by phenomenological limit cycle-based models. Through simulations of a realistic model of the NFκB/IκB network, we also illustrate the dephasing phenomena that are important for reconciling single-cell and population-based experiments on gene oscillators.

Circadian clocks optimally adapt to sunlight for reliable synchronization

Circadian oscillation provides selection advantages through synchronization to the daylight cycle. However, a reliable clock must be designed through two conflicting properties: entrainability to synchronize internal time with periodic stimuli such as sunlight, and regularity to oscillate with a precise period. These two aspects do not easily coexist, because better entrainability favours higher sensitivity which may sacrifice regularity. To investigate conditions for satisfying the two properties, we analytically calculated the optimal phase-response curve with a variational method. Our results indicate an existence of a dead zone, i.e. a time period during which input stimuli neither advance nor delay the clock. A dead zone appears only when input stimuli obey the time course of actual solar radiation, but a simple sine curve cannot yield a dead zone. Our calculation demonstrates that every circadian clock with a dead zone is optimally adapted to the daylight cycle.

Time scales in epigenetic dynamics and phenotypic heterogeneity of embryonic stem cells

A remarkable feature of the self-renewing population of embryonic stem cells (ESCs) is their phenotypic heterogeneity: Nanog and other marker proteins of ESCs show large cell-to-cell variation in their expression level, which should significantly influence the differentiation process of individual cells. The molecular mechanism and biological implication of this heterogeneity, however, still remain elusive. We address this problem by constructing a model of the core gene-network of mouse ESCs. The model takes account of processes of binding/unbinding of transcription factors, formation/dissolution of transcription apparatus, and modification of histone code at each locus of genes in the network. These processes are hierarchically interrelated to each other forming the dynamical feedback loops. By simulating stochastic dynamics of this model, we show that the phenotypic heterogeneity of ESCs can be explained when the chromatin at the Nanog locus undergoes the large scale reorganization in formation/dissolution of transcription apparatus, which should have the timescale similar to the cell cycle period. With this slow transcriptional switching of Nanog, the simulated ESCs fluctuate among multiple transient states, which can trigger the differentiation into the lineage-specific cell states. From the simulated transitions among cell states, the epigenetic landscape underlying transitions is calculated. The slow Nanog switching gives rise to the wide basin of ESC states in the landscape. The bimodal Nanog distribution arising from the kinetic flow running through this ESC basin prevents transdifferentiation and promotes the definite decision of the cell fate. These results show that the distribution of timescales of the regulatory processes is decisively important to characterize the fluctuation of cells and their differentiation process. The analyses through the epigenetic landscape and the kinetic flow on the landscape should provide a guideline to engineer cell differentiation.

Embryonic stem cells (ESCs) can proliferate indefinitely by keeping pluripotency, i.e., the ability to differentiate into any cell-lineage. ESCs, therefore, have been the focus of intense biological and medical interests. A remarkable feature of ESCs is their phenotypic heterogeneity: ESCs show large cell-to-cell fluctuation in the expression level of Nanog, which is a key factor to maintain pluripotency. Since Nanog regulates many genes in ESCs, this fluctuation should seriously affect individual cells when they start differentiation. In this paper we analyze this phenotypic fluctuation by simulating the stochastic dynamics of gene network in ESCs. The model takes account of the mutually interrelated processes of gene regulation such as binding/unbinding of transcription factors, formation/dissolution of transcription apparatus, and histone-code modification. We show the distribution of timescales of these processes is decisively important to characterize the dynamical behavior of the gene network, and that the slow formation/dissolution of transcription apparatus at the Nanog locus explains the observed large fluctuation of ESCs. The epigenetic landscapes are calculated based on the stochastic simulation, and the role of the phenotypic fluctuation in the differentiation process is analyzed through the landscape picture.

Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Physical and biological systems are often involved with coupled processes of different time scales. In the system with electronic and atomic motions, for example, the interplay between the atomic motion along the same energy landscape and the electronic hopping between different landscapes is critical: the system behavior largely depends on whether the intralandscape motion is slower (adiabatic) or faster (nonadiabatic) than the interlandscape hopping. For general nonequilibrium dynamics where Hamiltonian or energy function is unknown a priori, the challenge is how to extend the concepts of the intra- and interlandscape dynamics. In this paper we establish a theoretical framework for describing global nonequilibrium and nonadiabatic complex system dynamics by transforming the coupled landscapes into a single landscape but with additional dimensions. On this single landscape, dynamics is driven by gradient of the potential landscape, which is closely related to the steady-state probability distribution of the enlarged dimensions, and the probability flux, which has a curl nature. Through an example of a self-regulating gene circuit, we show that the curl flux has dramatic effects on gene regulatory dynamics. The curl flux and landscape framework developed here are easy to visualize and can be used to guide further investigation of physical and biological nonequilibrium systems.

Towards a whole-cell modeling approach for synthetic biology

Despite rapid advances over the last decade, synthetic biology lacks the predictive tools needed to enable rational design. Unlike established engineering disciplines, the engineering of synthetic gene circuits still relies heavily on experimental trial-and-error, a time-consuming and inefficient process that slows down the biological design cycle. This reliance on experimental tuning is because current modeling approaches are unable to make reliable predictions about the in vivo behavior of synthetic circuits. A major reason for this lack of predictability is that current models view circuits in isolation, ignoring the vast number of complex cellular processes that impinge on the dynamics of the synthetic circuit and vice versa. To address this problem, we present a modeling approach for the design of synthetic circuits in the context of cellular networks. Using the recently published whole-cell model of Mycoplasma genitalium, we examined the effect of adding genes into the host genome. We also investigated how codon usage correlates with gene expression and find agreement with existing experimental results. Finally, we successfully implemented a synthetic Goodwin oscillator in the whole-cell model. We provide an updated software framework for the whole-cell model that lays the foundation for the integration of whole-cell models with synthetic gene circuit models. This software framework is made freely available to the community to enable future extensions. We envision that this approach will be critical to transforming the field of synthetic biology into a rational and predictive engineering discipline.

Growth-rate-dependent dynamics of a bacterial genetic oscillator

Gene networks exhibiting oscillatory dynamics are widespread in biology. The minimal regulatory designs giving rise to oscillations have been implemented synthetically and studied by mathematical modeling. However, most of the available analyses generally neglect the coupling of regulatory circuits with the cellular "chassis" in which the circuits are embedded. For example, the intracellular macromolecular composition of fast-growing bacteria changes with growth rate. As a consequence, important parameters of gene expression, such as ribosome concentration or cell volume, are growth-rate dependent, ultimately coupling the dynamics of genetic circuits with cell physiology. This work addresses the effects of growth rate on the dynamics of a paradigmatic example of genetic oscillator, the repressilator. Making use of empirical growth-rate dependencies of parameters in bacteria, we show that the repressilator dynamics can switch between oscillations and convergence to a fixed point depending on the cellular state of growth, and thus on the nutrients it is fed. The physical support of the circuit (type of plasmid or gene positions on the chromosome) also plays an important role in determining the oscillation stability and the growth-rate dependence of period and amplitude. This analysis has potential application in the field of synthetic biology, and suggests that the coupling between endogenous genetic oscillators and cell physiology can have substantial consequences for their functionality.

The potential and flux landscape theory of evolution

We established the potential and flux landscape theory for evolution. We found explicitly the conventional Wright's gradient adaptive landscape based on the mean fitness is inadequate to describe the general evolutionary dynamics. We show the intrinsic potential as being Lyapunov function(monotonically decreasing in time) does exist and can define the adaptive landscape for general evolution dynamics for studying global stability. The driving force determining the dynamics can be decomposed into gradient of potential landscape and curl probability flux. Non-zero flux causes detailed balance breaking and measures how far the evolution from equilibrium state. The gradient of intrinsic potential and curl flux are perpendicular to each other in zero fluctuation limit resembling electric and magnetic forces on electrons. We quantified intrinsic energy, entropy and free energy of evolution and constructed non-equilibrium thermodynamics. The intrinsic non-equilibrium free energy is a Lyapunov function. Both intrinsic potential and free energy can be used to quantify the global stability and robustness of evolution. We investigated an example of three allele evolutionary dynamics with frequency dependent selection (detailed balance broken). We uncovered the underlying single, triple, and limit cycle attractor landscapes. We found quantitative criterions for stability through landscape topography. We also quantified evolution pathways and found paths do not follow potential gradient and are irreversible due to non-zero flux. We generalized the original Fisher's fundamental theorem to the general (i.e., frequency dependent selection) regime of evolution by linking the adaptive rate with not only genetic variance related to the potential but also the flux. We show there is an optimum potential where curl flux resulting from biotic interactions of individuals within a species or between species can sustain an endless evolution even if the physical environment is unchanged. We offer a theoretical basis for explaining the corresponding Red Queen hypothesis proposed by Van Valen. Our work provides a theoretical foundation for evolutionary dynamics.

Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations

We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.

The energy pump and the origin of the non-equilibrium flux of the dynamical systems and the networks

The global stability of dynamical systems and networks is still challenging to study. We developed a landscape and flux framework to explore the global stability. The potential landscape is directly linked to the steady state probability distribution of the non-equilibrium dynamical systems which can be used to study the global stability. The steady state probability flux together with the landscape gradient determines the dynamics of the system. The non-zero probability flux implies the breaking down of the detailed balance which is a quantitative signature of the systems being in non-equilibrium states. We investigated the dynamics of several systems from monostability to limit cycle and explored the microscopic origin of the probability flux. We discovered that the origin of the probability flux is due to the non-equilibrium conditions on the concentrations resulting energy input acting like non-equilibrium pump or battery to the system. Another interesting behavior we uncovered is that the probabilistic flux is closely related to the steady state deterministic chemical flux. For the monostable model of the kinetic cycle, the analytical expression of the probabilistic flux is directly related to the deterministic flux, and the later is directly generated by the chemical potential difference from the adenosine triphosphate (ATP) hydrolysis. For the limit cycle of the reversible Schnakenberg model, we also show that the probabilistic flux is correlated to the chemical driving force, as well as the deterministic effective flux. Furthermore, we study the phase coherence of the stochastic oscillation against the energy pump, and argue that larger non-equilibrium pump results faster flux and higher coherence. This leads to higher robustness of the biological oscillations. We also uncovered how fluctuations influence the coherence of the oscillations in two steps: (1) The mild fluctuations influence the coherence of the system mainly through the probability flux while maintaining the regular landscape topography. (2) The larger fluctuations lead to flat landscape and the complete loss of the stability of the whole system.

Landscape and global stability of nonadiabatic and adiabatic oscillations in a gene network

We quantify the potential landscape to determine the global stability and coherence of biological oscillations. We explore a gene network motif in our experimental synthetic biology studies of two genes that mutually repress and activate each other with self-activation and self-repression. We find that in addition to intrinsic molecular number fluctuations, there is another type of fluctuation crucial for biological function: the fluctuation due to the slow binding/unbinding of protein regulators to gene promoters. We find that coherent limit cycle oscillations emerge in two regimes: an adiabatic regime with fast binding/unbinding and a nonadiabatic regime with slow binding/unbinding relative to protein synthesis/degradation. This leads to two mechanisms of producing the stable oscillations: the effective interactions from averaging the gene states in the adiabatic regime; and the time delays due to slow binding/unbinding to promoters in the nonadiabatic regime, which can be tested by forthcoming experiments. In both regimes, the landscape has a topological shape of the Mexican hat in protein concentrations that quantitatively determines the global stability of limit cycle dynamics. The oscillation coherence is shown to be correlated with the shape of the Mexican hat characterized by the height from the oscillation ring to the central top. The oscillation period can be tuned in a wide range by changing the binding/unbinding rate without changing the amplitude much, which is important for the functionality of a biological clock. A negative feedback loop with time delays due to slow binding/unbinding can also generate oscillations. Although positive feedback is not necessary for generating oscillations, it can make the oscillations more robust.

Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation

Circadian rhythms with a period of ~24 h, are natural timing machines. They are broadly distributed in living organisms, such as Neurospora, Drosophila, and mammals. The underlying natures of the rhythmic behavior have been explored by experimental and theoretical approaches. However, the global and physical natures of the oscillation under fluctuations are still not very clear. We developed a landscape and flux framework to explore the global stability and robustness of a circadian oscillation system. The potential landscape of the network is uncovered and has a global Mexican-hat shape. The height of the Mexican-hat provides a quantitative measure to evaluate the robustness and coherence of the oscillation. We found that in nonequilibrium dynamic systems, not only the potential landscape but also the probability flux are important to the dynamics of the system under intrinsic noise. Landscape attracts the systems down to the oscillation ring while flux drives the coherent oscillation on the ring. We also investigated the phase coherence and the entropy production rate of the system at different fluctuations and found that dissipations are less and the coherence is higher for larger number of molecules. We also found that the power spectrum of autocorrelation functions show resonance peak at the frequency of coherent oscillations. The peak is less prominent for smaller number of molecules and less barrier height and therefore can be used as another measure of stability of oscillations. As a consequence of nonzero probability flux, we show that the three-point correlations from the time traces show irreversibility, providing a possible way to explore the flux from the observations. Furthermore, we explored the escape time from the oscillation ring to outside at different molecular number. We found that when barrier height is higher, escape time is longer and phase coherence of oscillation is higher. Finally, we performed the global sensitivity analysis of the underlying parameters to find the key network wirings responsible for the stability of the oscillation system.

A perturbation analysis of rate theory of self-regulating genes and signaling networks

A thorough kinetic analysis of the rate theory for stochastic self-regulating gene networks is presented. The chemical master equation kinetic model in terms of a coupled birth-death process is deconstructed into several simpler kinetic modules. We formulate and improve upon the rate theory of self-regulating genes in terms of perturbation theory. We propose a simple five-state scheme as a faithful caricature that elucidates the full kinetics including the "resonance phenomenon" discovered by Walczak et al. [Proc. Natl. Acad. Sci. U.S.A. 102, 18926 (2005)]. The same analysis can be readily applied to other biochemical networks such as phosphorylation signaling with fluctuating kinase activity. Generalization of the present approach can be included in multiple time-scale numerical computations for large biochemical networks.

Approximation scheme based on effective interactions for stochastic gene regulation

Since gene regulatory systems sometimes contain only a small number of molecules, these systems are not described well by macroscopic rate equations; a master equation approach is needed for such cases. We develop an approximation scheme for dealing with the stochasticity of the gene regulatory systems. Using an effective interaction concept, original master equations can be reduced to simpler master equations, which can be solved analytically. We apply the approximation scheme to self-regulating systems with monomer or dimer interactions, and a two-gene system with an exclusive switch. The approximation scheme can recover the bistability of the exclusive switch adequately.

Potential landscape and probabilistic flux of a predator prey network

Predator-prey system, as an essential element of ecological dynamics, has been recently studied experimentally with synthetic biology. We developed a global probabilistic landscape and flux framework to explore a synthetic predator-prey network constructed with two Escherichia coli populations. We developed a self consistent mean field method to solve multidimensional problem and uncovered the potential landscape with Mexican hat ring valley shape for predator-prey oscillations. The landscape attracts the system down to the closed oscillation ring. The probability flux drives the coherent oscillations on the ring. Both the landscape and flux are essential for the stable and coherent oscillations. The landscape topography characterized by the barrier height from the top of Mexican hat to the closed ring valley provides a quantitative measure of global stability of system. The entropy production rate for the energy dissipation is less for smaller environmental fluctuations or perturbations. The global sensitivity analysis based on the landscape topography gives specific predictions for the effects of parameters on the stability and function of the system. This may provide some clues for the global stability, robustness, function and synthetic network design.

Stochastic approaches in systems biology

The discrete and random occurrence of chemical reactions far from thermodynamic equilibrium, and low copy numbers of chemical species, in systems biology necessitate stochastic approaches. This review is an effort to give the reader a flavor of the most important stochastic approaches relevant to systems biology. Notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy-to-follow presentation of a stochastic framework for modeling subcellular biochemical systems. In particular, we make an effort to show how the notion of propensity, the chemical master equation (CME), and the stochastic simulation algorithm arise as consequences of the Markov property. Most stochastic modeling reviews focus on stochastic simulation approaches--the exact stochastic simulation algorithm and its various improvements and approximations. We complement this with an outline of an analytical approximation. The most common formulation of stochastic models for biochemical networks is the CME. Although stochastic simulations are a practical way to realize the CME, analytical approximations offer more insight into the influence of randomness on system's behavior. Toward that end, we cover the chemical Langevin equation and the related Fokker-Planck equation and the two-moment approximation (2MA). Throughout the text, two pedagogical examples are used to key illustrate ideas. With extensive references to the literature, our goal is to clarify key concepts and thereby prepare the reader for more advanced texts.

A comparative analysis of synthetic genetic oscillators

Synthetic biology is a rapidly expanding discipline at the interface between engineering and biology. Much research in this area has focused on gene regulatory networks that function as biological switches and oscillators. Here we review the state of the art in the design and construction of oscillators, comparing the features of each of the main networks published to date, the models used for in silico design and validation and, where available, relevant experimental data. Trends are apparent in the ways that network topology constrains oscillator characteristics and dynamics. Also, noise and time delay within the network can both have constructive and destructive roles in generating oscillations, and stochastic coherence is commonplace. This review can be used to inform future work to design and implement new types of synthetic oscillators or to incorporate existing oscillators into new designs.

Oscillation, cooperativity, and intermediates in the self-repressing gene

Biological oscillators are vital to living organisms, which use them as clocks for time-sensitive processes. However, much is unknown about mechanisms which can give rise to coherent oscillatory behavior, with few exceptions (e.g., explicitly delayed self-repressors and simple models of specific organisms' circadian clocks). We present what may be the simplest possible reliable gene network oscillator, a self-repressing gene. We show that binding cooperativity, which has not been considered in detail in this context, can combine with small numbers of intermediate steps to create coherent oscillation. We also note that noise blurs the line between oscillatory and non-oscillatory behavior.

Potential and flux landscapes quantify the stability and robustness of budding yeast cell cycle network

Studying the cell cycle process is crucial for understanding cell growth, proliferation, development, and death. We uncovered some key factors in determining the global robustness and function of the budding yeast cell cycle by exploring the underlying landscape and flux of this nonequilibrium network. The dynamics of the system is determined by both the landscape which attracts the system down to the oscillation orbit and the curl flux which drives the periodic motion on the ring. This global structure of landscape is crucial for the coherent cell cycle dynamics and function. The topography of the underlying landscape, specifically the barrier height separating basins of attractions, characterizes the capability of changing from one part of the system to another. This quantifies the stability and robustness of the system. We studied how barrier height is influenced by environmental fluctuations and perturbations on specific wirings of the cell cycle network. When the fluctuations increase, the barrier height decreases and the period and amplitude of cell cycle oscillation is more dispersed and less coherent. The corresponding dissipation of the system quantitatively measured by the entropy production rate increases. This implies that the system is less stable under fluctuations. We identified some key structural elements for wirings of the cell cycle network responsible for the change of the barrier height and therefore the global stability of the system through the sensitivity analysis. The results are in agreement with recent experiments and also provide new predictions.

Importance of cell variability for calcium signaling in rat airway myocytes

Calcium signaling controls several essential physiological functions in different cell types. Hence, it is not surprising that different aspects of Ca(2+) dynamics are in the focus of in-depth and extensive investigations. Efforts concentrate on the development of proper theoretical models that would provide a unified description of Ca(2+) signaling. Remarkably, experimentally recorded Ca(2+) signals exhibit a rather large diversity, which can be observed irrespective of the cell type, measuring techniques, or the nature of the signal. Our goal in the present study therefore is to present a theoretical explanation for the variability observed in experiments, whereby we focus on caffeine-induced Ca(2+) responses in isolated airway myocytes. By employing a stochastic model, we first test whether the observed variability can be attributed to intrinsic fluctuations that are a common feature of biochemical reactions that govern Ca(2+) signalization. We find that stochastic effects, within ranges that correspond to actual conditions in the cell, are far too modest to explain the large diversity observed in experimental data. Foremost, we reveal that only cell variability in theoretical modeling can appropriately describe the observed diversity in single-cell responses.

Robustness and coherence of a three-protein circadian oscillator: landscape and flux perspectives

Three-protein circadian oscillations in cyanobacteria sustain for weeks. To understand how cellular oscillations function robustly in stochastic fluctuating environments, we used a stochastic model to uncover two natures of circadian oscillation: the potential landscape related to steady-state probability distribution of protein concentrations; and the corresponding flux related to speed of concentration changes which drive the oscillations. The barrier height of escaping from the oscillation attractor on the landscape provides a quantitative measure of the robustness and coherence for oscillations against intrinsic and external fluctuations. The difference between the locations of the zero total driving force and the extremal of the potential provides a possible experimental probe and quantification of the force from curl flux. These results, correlated with experiments, can help in the design of robust oscillatory networks.

Stable, precise, and reproducible patterning of bicoid and hunchback molecules in the early Drosophila embryo

Precise patterning of morphogen molecules and their accurate reading out are of key importance in embryonic development. Recent experiments have visualized distributions of proteins in developing embryos and shown that the gradient of concentration of Bicoid morphogen in Drosophila embryos is established rapidly after fertilization and remains stable through syncytial mitoses. This stable Bicoid gradient is read out in a precise way to distribute Hunchback with small fluctuations in each embryo and in a reproducible way, with small embryo-to-embryo fluctuation. The mechanisms of such stable, precise, and reproducible patterning through noisy cellular processes, however, still remain mysterious. To address these issues, here we develop the one- and three-dimensional stochastic models of the early Drosophila embryo. The simulated results show that the fluctuation in expression of the hunchback gene is dominated by the random arrival of Bicoid at the hunchback enhancer. Slow diffusion of Hunchback protein, however, averages out this intense fluctuation, leading to the precise patterning of distribution of Hunchback without loss of sharpness of the boundary of its distribution. The coordinated rates of diffusion and transport of input Bicoid and output Hunchback play decisive roles in suppressing fluctuations arising from the dynamical structure change in embryos and those arising from the random diffusion of molecules, and give rise to the stable, precise, and reproducible patterning of Bicoid and Hunchback distributions.

Investigating the two-moment characterisation of subcellular biochemical networks

While ordinary differential equations (ODEs) form the conceptual framework for modelling many cellular processes, specific situations demand stochastic models to capture the influence of noise. The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). While stochastic simulations are a practical way to realise the CME, analytical approximations offer more insight into the influence of noise. Towards that end, the two-moment approximation (2MA) is a promising addition to the established analytical approaches including the chemical Langevin equation (CLE) and the related linear noise approximation (LNA). The 2MA approach directly tracks the mean and (co)variance which are coupled in general. This coupling is not obvious in CME and CLE and ignored by LNA and conventional ODE models. We extend previous derivations of 2MA by allowing (a) non-elementary reactions and (b) relative concentrations. Often, several elementary reactions are approximated by a single step. Furthermore, practical situations often require the use of relative concentrations. We investigate the applicability of the 2MA approach to the well-established fission yeast cell cycle model. Our analytical model reproduces the clustering of cycle times observed in experiments. This is explained through multiple resettings of M-phase promoting factor (MPF), caused by the coupling between mean and (co)variance, near the G2/M transition.

Robustness, dissipations and coherence of the oscillation of circadian clock: potential landscape and flux perspectives

Finding the global probabilistic nature of a non-equilibrium circadian clock is essential for addressing important issues of robustness and function. We have uncovered the underlying potential energy landscape of a simple cyanobacteria biochemical network, and the corresponding flux which is the driving force for the oscillation. We found that the underlying potential landscape for the oscillation in the presence of small statistical fluctuations is like an explicit ring valley or doughnut shape in the three dimensional protein concentration space. We found that the barrier height separating the oscillation ring and other area is a quantitative measure of the oscillation robustness and decreases when the fluctuations increase. We also found that the entropy production rate characterizing the dissipation or heat loss decreases as the fluctuations decrease. In addition, we found that, as the fluctuations increase, the period and the amplitude of the oscillations is more dispersed, and the phase coherence decreases. We also found that the properties from exploring the effects of the inherent chemical rate parameters on the robustness. Our approach is quite general and can be applied to other oscillatory cellular network.

PACS Codes: 87.18.-h, 87.18.Vf, 87.18.Yt

Potential landscape and flux framework of nonequilibrium networks: robustness, dissipation, and coherence of biochemical oscillations

We established a theoretical framework for studying nonequilibrium networks with two distinct natures essential for characterizing the global probabilistic dynamics: the underlying potential landscape and the corresponding curl flux. We applied the idea to a biochemical oscillation network and found that the underlying potential landscape for the oscillation limit cycle has a distinct closed ring valley (Mexican hat-like) shape when the fluctuations are small. This global landscape structure leads to attractions of the system to the ring valley. On the ring, we found that the nonequilibrium flux is the driving force for oscillations. Therefore, both structured landscape and flux are needed to guarantee a robust oscillating network. The barrier height separating the oscillation ring and other areas derived from the landscape topography is shown to be correlated with the escaping time from the limit cycle attractor and provides a quantitative measure of the robustness for the network. The landscape becomes shallower and the closed ring valley shape structure becomes weaker (lower barrier height) with larger fluctuations. We observe that the period and the amplitude of the oscillations are more dispersed and oscillations become less coherent when the fluctuations increase. We also found that the entropy production of the whole network, characterizing the dissipation costs from the combined effects of both landscapes and fluxes, decreases when the fluctuations decrease. Therefore, less dissipation leads to more robust networks. Our approach is quite general and applicable to other networks, dynamical systems, and biological evolution. It can help in designing robust networks.

Effects of the DNA state fluctuation on single-cell dynamics of self-regulating gene

A dynamical mean-field theory is developed to analyze stochastic single-cell dynamics of gene expression. By explicitly taking account of nonequilibrium and nonadiabatic features of the DNA state fluctuation, two-time correlation functions and response functions of single-cell dynamics are derived. The method is applied to a self-regulating gene to predict a rich variety of dynamical phenomena such as an anomalous increase of relaxation time and oscillatory decay of correlations. The effective "temperature" defined as the ratio of the correlation to the response in the protein number is small when the DNA state change is frequent, while it grows large when the DNA state change is infrequent, indicating the strong enhancement of noise in the latter case.