Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation

Resonant noise amplification in a predator-prey model with quasi-discrete generations

Predator-prey models have been shown to exhibit resonance-like behaviour, in which random fluctuations in the number of organisms (demographic noise) are amplified when their frequency is close to the natural oscillatory frequency of the system. This behaviour has been traditionally studied in models with exponentially distributed replication and death times. Here we consider a biologically more realistic model, in which organisms replicate quasi-synchronously such that the distribution of replication times has a narrow maximum at some T > 0 corresponding to the mean doubling time. We show that when the frequency of replication f = 1 / T is tuned to the natural oscillatory frequency of the predator-prey model, the system exhibits oscillations that are much stronger than in the model with Poissonian (non-synchronous) replication and death. These oscillations lead to population instability and the extinction of one of the species much sooner than in the case of Poissonian replication. The effect can be explained by resonant amplification of coloured noise generated by quasi-synchronous replication events.

Stochastic modeling of the mRNA life process: A generalized master equation

The mRNA life cycle is a complex biochemical process, involving transcription initiation, elongation, termination, splicing, and degradation. Each of these molecular events is multistep and can create a memory. The effect of this molecular memory on gene expression is not clear, although there are many related yet scattered experimental reports. To address this important issue, we develop a general theoretical framework formulated as a master equation in the sense of queue theory, which can reduce to multiple previously studied gene models in limiting cases. This framework allows us to interpret experimental observations, extract kinetic parameters from experimental data, and identify how the mRNA kinetics vary under regulatory influences. Notably, it allows us to evaluate the influences of elongation processes on mature RNA distribution; e.g., we find that the non-exponential elongation time can induce the bimodal mRNA expression and there is an optimal elongation noise intensity such that the mature RNA noise achieves the lowest level. In a word, our framework can not only provide insight into complex mRNA life processes but also bridge a dialogue between theoretical studies and experimental data.

Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

Analytical and Numerical Treatment of Continuous Ageing in the Voter Model

The conventional voter model is modified so that an agent's switching rate depends on the 'age' of the agent-that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

Generalized fluctuation-dissipation theorem for non-Markovian reaction networks

Intracellular biochemical networks often display large fluctuations in the molecule numbers or the concentrations of reactive species, making molecular approaches necessary for system descriptions. For Markovian reaction networks, the fluctuation-dissipation theorem (FDT) has been well established and extensively used in fast evaluation of fluctuations in reactive species. For non-Markovian reaction networks, however, the similar FDT has not been established so far. Here, we present a generalized FDT (gFDT) for a large class of non-Markovian reaction networks where general intrinsic-event waiting-time distributions account for the effect of intrinsic noise and general stochastic reaction delays represent the impact of extrinsic noise from environmental perturbations. The starting point is a generalized chemical master equation (gCME), which describes the probabilistic behavior of an equivalent Markovian reaction network and identifies the structure of the original non-Markovian reaction network in terms of stoichiometries and effective transition rates (extensions of common reaction propensity functions). From this formulation follows directly the solution of the linear noise approximation of the stationary gCME for all the components in the non-Markovian reaction network. While the gFDT can quickly trace noisy sources in non-Markovian reaction networks, example analysis verifies its effectiveness.

Effects of bidirectional phenotype switching on signal noise in a bacterial community

Cells can sense and process various signals. Noise is inevitable in the cell signaling system. In a bacterial community, the mutual conversion between normal cells and persistent cells forms a bidirectional phenotype switching cascade, in which either one can be used as an upstream signal and the other as a downstream signal. In order to quantitatively describe the relationship between noise and signal amplification of each phenotype, the gain-fluctuation relationship is obtained by using the linear noise approximation of the master equation. Through the simulation of these theoretical formulas, it is found that the bidirectional phenotype switching can directly generate interconversion noise which is usually very small and almost negligible. In particular, the bidirectional phenotype switching can provide a global fluctuating environment, which will not only affect the values of noise and covariance, but also generate additional intrinsic noise. The additional intrinsic noise in each phenotype is the main part of the total noise and can be transmitted to the other phenotype. The transmitted noise is also a powerful supplement to the total noise. Therefore, the indirect impact of bidirectional phenotype switching is far greater than its direct impact, which may be one of the reasons why chronic infections caused by persistent cells are refractory to treat.

A spatio-temporal model to reveal oscillator phenotypes in molecular clocks: Parameter estimation elucidates circadian gene transcription dynamics in single-cells

We propose a stochastic distributed delay model together with a Markov random field prior and a measurement model for bioluminescence-reporting to analyse spatio-temporal gene expression in intact networks of cells. The model describes the oscillating time evolution of molecular mRNA counts through a negative transcriptional-translational feedback loop encoded in a chemical Langevin equation with a probabilistic delay distribution. The model is extended spatially by means of a multiplicative random effects model with a first order Markov random field prior distribution. Our methodology effectively separates intrinsic molecular noise, measurement noise, and extrinsic noise and phenotypic variation driving cell heterogeneity, while being amenable to parameter identification and inference. Based on the single-cell model we propose a novel computational stability analysis that allows us to infer two key characteristics, namely the robustness of the oscillations, i.e. whether the reaction network exhibits sustained or damped oscillations, and the profile of the regulation, i.e. whether the inhibition occurs over time in a more distributed versus a more direct manner, which affects the cells' ability to phase-shift to new schedules. We show how insight into the spatio-temporal characteristics of the circadian feedback loop in the suprachiasmatic nucleus (SCN) can be gained by applying the methodology to bioluminescence-reported expression of the circadian core clock gene Cry1 across mouse SCN tissue. We find that while (almost) all SCN neurons exhibit robust cell-autonomous oscillations, the parameters that are associated with the regulatory transcription profile give rise to a spatial division of the tissue between the central region whose oscillations are resilient to perturbation in the sense that they maintain a high degree of synchronicity, and the dorsal region which appears to phase shift in a more diversified way as a response to large perturbations and thus could be more amenable to entrainment.

Consensus, polarization, and coexistence in a continuous opinion dynamics model with quenched disorder

A model of opinion dynamics is introduced in which each individual's opinion is measured on a bounded continuous spectrum. Each opinion is influenced heterogeneously by every other opinion in the population. It is demonstrated that consensus, polarization and a spread of moderate opinions are all possible within this model. Using dynamic mean-field theory, we are able to identify the statistical features of the interactions between individuals that give rise to each of the aforementioned emergent phenomena. The nature of the transitions between each of the observed macroscopic states is also studied. It is demonstrated that heterogeneity of interactions between individuals can lead to polarization, that mostly antagonistic or contrarian interactions can promote consensus at a moderate opinion, and that mostly reinforcing interactions encourage the majority to take an extreme opinion.

Time-delayed and stochastic effects in a predator-prey model with ratio dependence and Holling type III functional response

In this article, we derive and analyze a novel predator-prey model with account for maturation delay in predators, ratio dependence, and Holling type III functional response. The analysis of the system's steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state. Demographic stochasticity in the model is explored by means of deriving a delayed chemical master equation. Using system size expansion, we study the structure of stochastic oscillations around the deterministically stable coexistence state by analyzing the dependence of variance and coherence of stochastic oscillations on system parameters. Numerical simulations of the stochastic model are performed to illustrate stochastic amplification, where individual stochastic realizations can exhibit sustained oscillations in the case, where deterministically the system approaches a stable steady state. These results provide a framework for studying realistic predator-prey systems with Holling type III functional response in the presence of stochasticity, where an important role is played by non-negligible predator maturation delay.

Effects of noise and time delay on E2F's expression level in a bistable Rb-E2F gene's regulatory network

The bistable Rb-E2F gene regulatory network plays a central role in regulating cellular proliferation-quiescence transition. Based on Gillespie's chemical Langevin method, the stochastic bistable Rb-E2F gene's regulatory network with time delays is proposed. It is found that under the moderate intensity of internal noise, delay in the Cyclin E synthesis rate can greatly increase the average concentration value of E2F. When the delay is considered in both E2F-related positive feedback loops, within a specific range of delay (3-13) hr , the average expression of E2F is significantly increased. Also, this range is in the scope with that experimentally given by Dong et al. [65]. By analysing the quasi-potential curves at different delay times, simulation results show that delay regulates the dynamic behaviour of the system in the following way: small delay stabilises the bistable system; the medium delay is conducive to a high steady-state, making the system fluctuate near the high steady-state; large delay induces approximately periodic transitions between high and low steady-state. Therefore, by regulating noise and time delay, the cell itself can control the expression level of E2F to respond to different situations. These findings may provide an explanation of some experimental result intricacies related to the cell cycle.

Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression

Gene expression is a complex biochemical process involving multiple reaction steps, creating molecular memory because the probability of waiting time between consecutive reaction steps no longer follows exponential distributions. What effect the molecular memory has on metastable states in gene expression remains not fully understood. Here, we study transition paths of switching between bistable states for a non-Markovian model of gene expression equipped with a self-regulation. Employing the large deviation theory for this model, we analyze the optimal transition paths of switching between bistable states in gene expression, interestingly finding that dynamic behaviors in gene expression along the optimal transition paths significantly depend on the molecular memory. Moreover, we discover that the molecular memory can prolong the time of switching between bistable states in gene expression along the optimal transition paths. Our results imply that the molecular memory may be an unneglectable factor to affect switching between metastable states in gene expression.

Modeling Protein Aggregation Kinetics: The Method of Second Stochasticization

The nucleation of protein aggregates and their growth are important in determining the structure of the cell's membraneless organelles as well as the pathogenesis of many diseases. The large number of molecular types of such aggregates along with the intrinsically stochastic nature of aggregation challenges our theoretical and computational abilities. Kinetic Monte Carlo simulation using the Gillespie algorithm is a powerful tool for modeling stochastic kinetics, but it is computationally demanding when a large number of diverse species is involved. To explore the mechanisms and statistics of aggregation more efficiently, we introduce a new approach to model stochastic aggregation kinetics which introduces noise into already statistically averaged equations obtained using mathematical moment closure schemes. Stochastic moment equations summarize succinctly the dynamics of the large diversity of species with different molecularity involved in aggregation but still take into account the stochastic fluctuations that accompany not only primary and secondary nucleation but also aggregate elongation, dissociation, and fragmentation. This method of "second stochasticization" works well where the fluctuations are modest in magnitude as is often encountered in vivo where the number of protein copies in some computations can be in the hundreds to thousands. Simulations using second stochasticization reveal a scaling law that correlates the size of the fluctuations in aggregate size and number with the total number of monomers. This scaling law is confirmed using experimental data. We believe second stochasticization schemes will prove valuable for bridging the gap between in vivo cell biology and detailed modeling. (The code is released on https://github.com/MYTLab/stoch-agg.).

Noisy signal propagation and amplification in phenotypic transition cascade of colonic cells

Like genes and proteins, cells can use biochemical networks to sense and process information. The differentiation of the cell state in colonic crypts forms a typical unidirectional phenotypic transitional cascade, in which stem cells differentiate into the transit-amplifying cells (TACs), and TACs continue to differentiate into fully differentiated cells. In order to quantitatively describe the relationship between the noise of each compartment and the amplification of signals, the gain factor is introduced, and the gain-fluctuation relation is obtained by using the linear noise approximation of the master equation. Through the simulation of these theoretical formulas, the characters of noise propagation and amplification are studied. It is found that the transmitted noise is an important part of the total noise in each downstream cell. Therefore, a small number of downstream cells can only cause its small inherent noise, but the total noise may be very large due to the transmitted noise. The influence of the transmitted noise may be the indirect cause of colon cancer. In addition, the total noise of the downstream cells always has a minimum value. As long as a reasonable value of the gain factor is selected, the number of cells in colonic crypts will be controlled within the normal range. This may be a good method to intervene the uncontrollable growth of tumor cells and effectively control the deterioration of colon cancer.

Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional turbulent chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional laminar chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [Hart, Roy, Müller-Bender, Otto, and Radons, Phys. Rev. Lett. 123, 154101 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.154101], where it is demonstrated that laminar chaos is a robust phenomenon, which can be observed in experimental systems. We provide a time series analysis toolbox for the detection of robust features of laminar chaos. We benchmark our toolbox by experimental time series and time series of a model system which is described by a nonlinear Langevin equation with time-varying delay. The benchmark is done for different noise strengths for both the experimental system and the model system, where laminar chaos can be detected, even if it is hard to distinguish from turbulent chaos by a visual analysis of the trajectory.

Filtering and inference for stochastic oscillators with distributed delays

Motivation

The time evolution of molecular species involved in biochemical reaction networks often arises from complex stochastic processes involving many species and reaction events. Inference for such systems is profoundly challenged by the relative sparseness of experimental data, as measurements are often limited to a small subset of the participating species measured at discrete time points. The need for model reduction can be realistically achieved for oscillatory dynamics resulting from negative translational and transcriptional feedback loops by the introduction of probabilistic time-delays. Although this approach yields a simplified model, inference is challenging and subject to ongoing research. The linear noise approximation (LNA) has recently been proposed to address such systems in stochastic form and will be exploited here.

Results

We develop a novel filtering approach for the LNA in stochastic systems with distributed delays, which allows the parameter values and unobserved states of a stochastic negative feedback model to be inferred from univariate time-series data. The performance of the methods is tested for simulated data. Results are obtained for real data when the model is fitted to imaging data on Cry1, a key gene involved in the mammalian central circadian clock, observed via a luciferase reporter construct in a mouse suprachiasmatic nucleus.

Availability and implementation

Programmes are written in MATLAB and Statistics Toolbox Release 2016 b, The MathWorks, Inc., Natick, Massachusetts, USA. Sample code and Cry1 data are available on GitHub https://github.com/scalderazzo/FLNADD.

Supplementary information

Supplementary data are available at Bioinformatics online.

Chemical Langevin Equation for Complex Reactions

The stochastic dynamical behaviors of an elementary reaction system can be investigated by the chemical Langevin equation (CLE). However, most of the reactions in engineering belong to complex reactions. It is not appropriate to describe the random evolution process of a complex chemical reaction system by directly using CLE because the foundation of the deviation of CLE is the equilibrium equation of the number of molecules in elementary reaction systems. In the study, the chemical Langevin equation for complex reactions (CLE-CR) is proposed based on the random process theory by introducing the extent of reactions to express the reaction rates of complex reactions. The reaction rates of complex reactions are regarded as some random variables following Poisson distribution. To illustrate the essential consistency of CLE-CR and CLE, the physical meaning of the propensity function in CLE is comprehensively discussed. A numerical example from chemical engineering is employed to demonstrate the effectiveness of CLE-CR and the solving procedure. The results show that CLE-CR can be conveniently applied into engineering to investigate the stochastic dynamical behaviors of complex reaction systems, giving the probabilistic information of the concentration evolution of chemical constituents.

Intrinsic noise, Delta-Notch signalling and delayed reactions promote sustained, coherent, synchronized oscillations in the presomitic mesoderm

Using a stochastic individual-based modelling approach, we examine the role that Delta-Notch signalling plays in the regulation of a robust and reliable somite segmentation clock. We find that not only can Delta-Notch signalling synchronize noisy cycles of gene expression in adjacent cells in the presomitic mesoderm (as is known), but it can also amplify and increase the coherence of these cycles. We examine some of the shortcomings of deterministic approaches to modelling these cycles and demonstrate how intrinsic noise can play an active role in promoting sustained oscillations, giving rise to noise-induced quasi-cycles. Finally, we explore how translational/transcriptional delays can result in the cycles in neighbouring cells oscillating in anti-phase and we study how this effect relates to the propagation of noise-induced stochastic waves.

Markovian approaches to modeling intracellular reaction processes with molecular memory

Many cellular processes are governed by stochastic reaction events. These events do not necessarily occur in single steps of individual molecules, and, conversely, each birth or death of a macromolecule (e.g., protein) could involve several small reaction steps, creating a memory between individual events and thus leading to nonmarkovian reaction kinetics. Characterizing this kinetics is challenging. Here, we develop a systematic approach for a general reaction network with arbitrary intrinsic waiting-time distributions, which includes the stationary generalized chemical-master equation (sgCME), the stationary generalized Fokker-Planck equation, and the generalized linear-noise approximation. The first formulation converts a nonmarkovian issue into a markovian one by introducing effective transition rates (that explicitly decode the effect of molecular memory) for the reactions in an equivalent reaction network with the same substrates but without molecular memory. Nonmarkovian features of the reaction kinetics can be revealed by solving the sgCME. The latter 2 formulations can be used in the fast evaluation of fluctuations. These formulations can have broad applications, and, in particular, they may help us discover new biological knowledge underlying memory effects. When they are applied to generalized stochastic models of gene-expression regulation, we find that molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch.

Frequency spectrum of chemical fluctuation: A probe of reaction mechanism and dynamics

Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically, and the frequency spectrum of this chemical fluctuation carries valuable information about the dynamics of the reactions creating these biomolecules. Recent advances in single-cell techniques enable direct monitoring of the time-traces of the protein number in each cell; however, it is not yet clear how the stochastic dynamics of these time-traces is related to the reaction mechanism and dynamics. Here, we derive a rigorous relation between the frequency-spectrum of the product number fluctuation and the reaction mechanism and dynamics, starting from a generalized master equation. This relation enables us to analyze the time-traces of the protein number and extract information about dynamics of mRNA number and transcriptional regulation, which cannot be directly observed by current experimental techniques. We demonstrate our frequency spectrum analysis of protein number fluctuation, using the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. We also discuss how the dynamic heterogeneity of transcription and translation rates affects the frequency-spectra of the mRNA and protein number.

Bayesian inference of distributed time delay in transcriptional and translational regulation

MOTIVATION

Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks.

RESULTS

We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth-death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy.

AVAILABILITY AND IMPLEMENTATION

Accompanying code in R is available at https://github.com/cbskust/DDE_BD.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Stochastic fluctuations and quasipattern formation in reaction-diffusion systems with anomalous transport

Many approaches to modeling reaction-diffusion systems with anomalous transport rely on deterministic equations which ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasipatterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the formation of both deterministic and stochastic patterns. Our analysis shows that the combination of subdiffusion and intrinsic stochasticity can reduce the threshold ratio of the effective diffusion coefficients required for pattern formation to a greater degree than either effect on its own.

Topological time-series analysis with delay-variant embedding

Identifying the qualitative changes in time-series data provides insights into the dynamics associated with such data. Such qualitative changes can be detected through topological approaches, which first embed the data into a high-dimensional space using a time-delay parameter and subsequently extract topological features describing the shape of the data from the embedded points. However, the essential topological features that are extracted using a single time delay are considered to be insufficient for evaluating the aforementioned qualitative changes, even when a well-selected time delay is used. We therefore propose a delay-variant embedding method that constructs the extended topological features by considering the time delay as a variable parameter instead of considering it as a single fixed value. This delay-variant embedding method reveals multiple-timescale patterns in a time series by allowing the observation of the variations in topological features, with the time delay serving as an additional dimension in the topological feature space. We theoretically prove that the constructed topological features are robust when the time series is perturbed by noise. Furthermore, we combine these features with the kernel technique in machine learning algorithms to classify the general time-series data. We demonstrate the effectiveness of our method for classifying the synthetic noisy biological and real time-series data. Our method outperforms a method that is based on a single time delay and, surprisingly, achieves the highest classification accuracy on an average among the standard time-series analysis techniques.

Chemical Continuous Time Random Walks

Kinetic Monte Carlo methods such as the Gillespie algorithm model chemical reactions as random walks in particle number space. The interreaction times are exponentially distributed under the assumption that the system is well mixed. We introduce an arbitrary interreaction time distribution, which may account for the impact of incomplete mixing on chemical reactions, and in general stochastic reaction delay, which may represent the impact of extrinsic noise. This process defines an inhomogeneous continuous time random walk in particle number space, from which we derive a generalized chemical master equation. This leads naturally to a generalization of the Gillespie algorithm. Based on this formalism, we determine the modified chemical rate laws for different interreaction time distributions. This framework traces Michaelis-Menten-type kinetics back to finite-mean delay times, and predicts time-nonlocal macroscopic reaction kinetics as a consequence of broadly distributed delays. Non-Markovian kinetics exhibit weak ergodicity breaking and show key features of reactions under local nonequilibrium.

Identifying stochastic oscillations in single-cell live imaging time series using Gaussian processes

Multiple biological processes are driven by oscillatory gene expression at different time scales. Pulsatile dynamics are thought to be widespread, and single-cell live imaging of gene expression has lead to a surge of dynamic, possibly oscillatory, data for different gene networks. However, the regulation of gene expression at the level of an individual cell involves reactions between finite numbers of molecules, and this can result in inherent randomness in expression dynamics, which blurs the boundaries between aperiodic fluctuations and noisy oscillators. This underlies a new challenge to the experimentalist because neither intuition nor pre-existing methods work well for identifying oscillatory activity in noisy biological time series. Thus, there is an acute need for an objective statistical method for classifying whether an experimentally derived noisy time series is periodic. Here, we present a new data analysis method that combines mechanistic stochastic modelling with the powerful methods of non-parametric regression with Gaussian processes. Our method can distinguish oscillatory gene expression from random fluctuations of non-oscillatory expression in single-cell time series, despite peak-to-peak variability in period and amplitude of single-cell oscillations. We show that our method outperforms the Lomb-Scargle periodogram in successfully classifying cells as oscillatory or non-oscillatory in data simulated from a simple genetic oscillator model and in experimental data. Analysis of bioluminescent live-cell imaging shows a significantly greater number of oscillatory cells when luciferase is driven by a Hes1 promoter (10/19), which has previously been reported to oscillate, than the constitutive MoMuLV 5’ LTR (MMLV) promoter (0/25). The method can be applied to data from any gene network to both quantify the proportion of oscillating cells within a population and to measure the period and quality of oscillations. It is publicly available as a MATLAB package.

Technological advances now allow us to observe gene expression in real-time at a single-cell level. In a wide variety of biological contexts this new data has revealed that gene expression is highly dynamic and possibly oscillatory. It is thought that periodic gene expression may be useful for keeping track of time and space, as well as transmitting information about signalling cues. Classifying a time series as periodic from single cell data is difficult because it is necessary to distinguish whether peaks and troughs are generated from an underlying oscillator or whether they are aperiodic fluctuations. To this end, we present a novel tool to classify live-cell data as oscillatory or non-oscillatory that accounts for inherent biological noise. We first demonstrate that the method outperforms a competing scheme in classifying computationally simulated single-cell data, and we subsequently analyse live-cell imaging time series. Our method is able to successfully detect oscillations in a known genetic oscillator, but it classifies data from a constitutively expressed gene as aperiodic. The method forms a basis for discovering new gene expression oscillators and quantifying how oscillatory activity alters in response to changes in cell fate and environmental or genetic perturbations.

Reaction–diffusion with stochastic decay rates

Microscopic physical and chemical fluctuations in a reaction–diffusion system lead to anomalous chemical kinetics and transport on the mesoscopic scale. Emergent non-Markovian effects lead to power-law reaction times and localization of reacting species.

Effects of cell cycle noise on excitable gene circuits

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Noise Decomposition Principle in a Coherent Feed-Forward Transcriptional Regulatory Loop

Coherent feed-forward loops exist extensively in realistic biological regulatory systems, and are common signaling motifs. Here, we study the characteristics and the propagation mechanism of the output noise in a coherent feed-forward transcriptional regulatory loop that can be divided into a main road and branch. Using the linear noise approximation, we derive analytical formulae for the total noise of the full loop, the noise of the branch, and the noise of the main road, which are verified by the Gillespie algorithm. Importantly, we find that (i) compared with the branch motif or the main road motif, the full motif can effectively attenuate the output noise level; (ii) there is a transition point of system state such that the noise of the main road is dominated when the underlying system is below this point, whereas the noise of the branch is dominated when the system is beyond the point. The entire analysis reveals the mechanism of how the noise is generated and propagated in a simple yet representative signaling module.

Stochastic modeling of biochemical systems with multistep reactions using state-dependent time delay

To deal with the growing scale of molecular systems, sophisticated modelling techniques have been designed in recent years to reduce the complexity of mathematical models. Among them, a widely used approach is delayed reaction for simplifying multistep reactions. However, recent research results suggest that a delayed reaction with constant time delay is unable to describe multistep reactions accurately. To address this issue, we propose a novel approach using state-dependent time delay to approximate multistep reactions. We first use stochastic simulations to calculate time delay arising from multistep reactions exactly. Then we design algorithms to calculate time delay based on system dynamics precisely. To demonstrate the power of proposed method, two processes of mRNA degradation are used to investigate the function of time delay in determining system dynamics. In addition, a multistep pathway of metabolic synthesis is used to explore the potential of the proposed method to simplify multistep reactions with nonlinear reaction rates. Simulation results suggest that the state-dependent time delay is a promising and accurate approach to reduce model complexity and decrease the number of unknown parameters in the models.

Stochasticity in the miR-9/Hes1 oscillatory network can account for clonal heterogeneity in the timing of differentiation

Recent studies suggest that cells make stochastic choices with respect to differentiation or division. However, the molecular mechanism underlying such stochasticity is unknown. We previously proposed that the timing of vertebrate neuronal differentiation is regulated by molecular oscillations of a transcriptional repressor, HES1, tuned by a post-transcriptional repressor, miR-9. Here, we computationally model the effects of intrinsic noise on the Hes1/miR-9 oscillator as a consequence of low molecular numbers of interacting species, determined experimentally. We report that increased stochasticity spreads the timing of differentiation in a population, such that initially equivalent cells differentiate over a period of time. Surprisingly, inherent stochasticity also increases the robustness of the progenitor state and lessens the impact of unequal, random distribution of molecules at cell division on the temporal spread of differentiation at the population level. This advantageous use of biological noise contrasts with the view that noise needs to be counteracted.

Generating functionals and Gaussian approximations for interruptible delay reactions

We develop a generating functional description of the dynamics of non-Markovian individual-based systems in which delay reactions can be terminated before completion. This generalizes previous work in which a path-integral approach was applied to dynamics in which delay reactions complete with certainty. We construct a more widely applicable theory, and from it we derive Gaussian approximations of the dynamics, valid in the limit of large, but finite, population sizes. As an application of our theory we study predator-prey models with delay dynamics due to gestation or lag periods to reach the reproductive age. In particular, we focus on the effects of delay on noise-induced cycles.

Fluctuation and noise propagation in phenotypic transition cascades of clonal populations

Quantitative modeling of fluctuations of each phenotype is a crucial step towards a fundamental understanding of noise propagation through various phenotypic transition cascades. The theoretical formulas for noise propagation in various phenotypic transition cascades are derived by using the linear noise approximation of master equation and the logarithmic gain. By virtue of the theoretical formulas, we study the noise propagation in bidirectional and unidirectional phenotypic transition cascades, respectively. It is found that noise propagation in these two phenotypic transition cascades evidently differs: In the bidirectional cascade, a systemic random environment is provided by a correlated global component. The total noise of each phenotype is mainly determined by the intrinsic noise and the transmitted noise from other phenotypes. The intrinsic noise enlarged by interconversion through an added part shows a novel noise propagation mechanism. However, in the unidirectional cascade, the random environment of each downstream phenotype is provided by upstream phenotypes. The total noise of each downstream phenotype is mainly determined by the transmitted noises from upstream phenotypes. The intrinsic noise and the conversion noise can propagate in both bidirectional and unidirectional phenotypic transition cascades.

Modeling delay in genetic networks: from delay birth-death processes to delay stochastic differential equations

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.

Simulating non-Markovian stochastic processes

We present a simple and general framework to simulate statistically correct realizations of a system of non-Markovian discrete stochastic processes. We give the exact analytical solution and a practical and efficient algorithm like the Gillespie algorithm for Markovian processes, with the difference being that now the occurrence rates of the events depend on the time elapsed since the event last took place. We use our non-Markovian generalized Gillespie stochastic simulation methodology to investigate the effects of nonexponential interevent time distributions in the susceptible-infected-susceptible model of epidemic spreading. Strikingly, our results unveil the drastic effects that very subtle differences in the modeling of non-Markovian processes have on the global behavior of complex systems, with important implications for their understanding and prediction. We also assess our generalized Gillespie algorithm on a system of biochemical reactions with time delays. As compared to other existing methods, we find that the generalized Gillespie algorithm is the most general because it can be implemented very easily in cases (such as for delays coupled to the evolution of the system) in which other algorithms do not work or need adapted versions that are less efficient in computational terms.

Spatiotemporally complete condensation in a non-Poissonian exclusion process

We investigate a non-Poissonian version of the asymmetric simple exclusion process, motivated by the observation that coarse graining the interactions between particles in complex systems generically leads to a stochastic process with a non-Markovian (history-dependent) character. We characterize a large family of one-dimensional hopping processes using a waiting-time distribution for individual particle hops. We find that when its variance is infinite, a real-space condensate forms that is complete in space (involves all particles) and time (exists at almost any given instant) in the thermodynamic limit. The mechanism for the onset and stability of the condensate is rather subtle and depends on the microscopic dynamics subsequent to a failed particle hop attempt.

Transcriptional delay stabilizes bistable gene networks

Transcriptional delay can significantly impact the dynamics of gene networks. Here we examine how such delay affects bistable systems. We investigate several stochastic models of bistable gene networks and find that increasing delay dramatically increases the mean residence times near stable states. To explain this, we introduce a non-Markovian, analytically tractable reduced model. The model shows that stabilization is the consequence of an increased number of failed transitions between stable states. Each of the bistable systems that we simulate behaves in this manner.