Synchronized dynamics and non-equilibrium steady states in a stochastic yeast cell-cycle network

Applying the mathematical circulation theory of Markov chains, we investigate the synchronized stochastic dynamics of a discrete network model of yeast cell-cycle regulation where stochasticity has been kept rather than being averaged out. By comparing the network dynamics of the stochastic model with its corresponding deterministic network counterpart, we show that the synchronized dynamics can be soundly characterized by a dominant circulation in the stochastic model, which is the natural generalization of the deterministic limit cycle in the deterministic system. Moreover, the period of the main peak in the power spectrum, which is in common use to characterize the synchronized dynamics, perfectly corresponds to the number of states in the main cycle with dominant circulation. Such a large separation in the magnitude of the circulations, between a dominant, main cycle and the rest, gives rise to the stochastic synchronization phenomenon.     

The impact of bottlenecks on microbial survival,adaptation, and phenotypic switching in host–pathogen interactions

Microbial pathogens and viruses can often maintain sufficient population diversity to evade a wide range of host immune responses. However, when populations experience bottlenecks, as occurs frequently during initiation of new infections, pathogens require specialized mechanisms to regenerate diversity. We address the evolution of such mechanisms, known as stochastic phenotype switches, which are prevalent in pathogenic bacteria. We analyze a model of pathogen diversification in a changing host environment that accounts for selective bottlenecks, wherein different phenotypes have distinct transmission probabilities between hosts. We show that under stringent bottlenecks, such that only one phenotype can initiate new infections, there exists a threshold stochastic switching rate below which all pathogen lineages go extinct, and above which survival is a near certainty. We determine how quickly stochastic switching rates can evolve by computing a fitness landscape for the evolutionary dynamics of switching rates, and analyzing its dependence on both the stringency of bottlenecks and the duration of within‐host growth periods. We show that increasing the stringency of bottlenecks or decreasing the period of growth results in faster adaptation of switching rates. Our model provides strong theoretical evidence that bottlenecks play a critical role in accelerating the evolutionary dynamics of pathogens.     

Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments

The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as, for instance, the fluctuations in ribosome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment''s effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach.     

Coherent activation of a synthetic mammalian gene network

A quantitative analysis of naturally-occurring regulatory networks, especially those present in mammalian cells, is difficult due to their high complexity. Much simpler gene networks can be engineered in model organisms and analyzed as isolated regulatory modules. Recently, several synthetic networks have been constructed in mammalian systems. However, most of these engineered mammalian networks have been characterized using steady-state population level measurements. Here, we use an integrated experimental-computational approach to analyze the dynamical response of a synthetic positive feedback network in individual mammalian cells. We observe a switch-like activation of the network with variable delay times in individual cells. In agreement with a stochastic model of the network, we find that increasing the strength of the positive feedback results in a decrease in the mean delay time and a more coherent activation of individual cells. Our results are important for gaining insight into biological processes which rely on positive feedback regulation.     

The effect of coupled stochastic processes in a two-state biochemical switch

Cell signaling pathways consist of multiple connections of different types of gene, mRNA and protein networks. It is not a trivial task to follow the signals flowing through these networks. The difficulty comes from considering the entire biological structure as a single network without breaking it into connected modules. The study of these networks simplifies if the complex system is reduced to a hierarchy of interconnected modules. Out of many potential modules, a specific one, namely the Goldbeter–Koshland switch, was encountered by the authors during their study of the Mammalian Heat Shock Response Network (MHSRN) where the switch acts as a stress sensor. Usually, only the steady state behavior of the switch is studied, in which the phosphorylated protein is given as a function of the enzyme concentration. Experimental results show that the heat shock response is still present 20 h after the temperature stress had ended. Thus, it is useful to analyze the transient behavior of the switch that couples the environment to the MHSRN. A stochastic model for the switch is proposed using the Master Equation which is subsequently transformed into an equation for the factorial cumulant generating function. This generating function can be easily read from a graphical representation of the stochastic switch. The second order approximation of the equation for the factorial cumulant generating function is solved and the time dependence of the transient regime of the mean and standard deviation is readily obtained. Using the mean and standard deviation of the switch’s output as a function of the stochastic input signals that represent the environment, we classify the switches according to different criteria. The switches differ by the numerical values of the parameters that characterize the switch’s chemical reactions. The classifying criteria will distinguish the switches by the levels of the response for a given transition time and by the sensitivity of the response to the enzyme levels. It is also found that the environment can drastically change the response of the switch, which has important biological consequences.     

Synchronization and stochastic resonance of the small-world neural network based on the CPG

According to biological knowledge, the central nervous system controls the central pattern generator (CPG) to drive the locomotion. The brain is a complex system consisting of different functions and different interconnections. The topological properties of the brain display features of small-world network. The synchronization and stochastic resonance have important roles in neural information transmission and processing. In order to study the synchronization and stochastic resonance of the brain based on the CPG, we establish the model which shows the relationship between the small-world neural network (SWNN) and the CPG. We analyze the synchronization of the SWNN when the amplitude and frequency of the CPG are changed and the effects on the CPG when the SWNN’s parameters are changed. And we also study the stochastic resonance on the SWNN. The main findings include: (1) When the CPG is added into the SWNN, there exists parameters space of the CPG and the SWNN, which can make the synchronization of the SWNN optimum. (2) There exists an optimal noise level at which the resonance factor Q gets its peak value. And the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the noise intensity. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

Representational switching by dynamical reorganization of attractor structure in a network model of the prefrontal cortex

The prefrontal cortex (PFC) plays a crucial role in flexible cognitive behavior by representing task relevant information with its working memory. The working memory with sustained neural activity is described as a neural dynamical system composed of multiple attractors, each attractor of which corresponds to an active state of a cell assembly, representing a fragment of information. Recent studies have revealed that the PFC not only represents multiple sets of information but also switches multiple representations and transforms a set of information to another set depending on a given task context. This representational switching between different sets of information is possibly generated endogenously by flexible network dynamics but details of underlying mechanisms are unclear. Here we propose a dynamically reorganizable attractor network model based on certain internal changes in synaptic connectivity, or short-term plasticity. We construct a network model based on a spiking neuron model with dynamical synapses, which can qualitatively reproduce experimentally demonstrated representational switching in the PFC when a monkey was performing a goal-oriented action-planning task. The model holds multiple sets of information that are required for action planning before and after representational switching by reconfiguration of functional cell assemblies. Furthermore, we analyzed population dynamics of this model with a mean field model and show that the changes in cell assemblies' configuration correspond to those in attractor structure that can be viewed as a bifurcation process of the dynamical system. This dynamical reorganization of a neural network could be a key to uncovering the mechanism of flexible information processing in the PFC.     

An interaction switch predicts the nested architecture of mutualistic networks

Nested architecture is distinctive in plant-animal mutualistic networks. However, to date an integrative and quantitative explanation has been lacking. It is evident that species often switch their interactive partners in real-world mutualistic networks such as pollination and seed-dispersal networks. By incorporating an interaction switch into a novel multi-population model, we show that the nested architecture rapidly emerges from an initially random network. The model allowing interaction switches between partner species produced predictions which fit remarkably well with observations from 81 empirical networks. Thus, the nested architecture in mutualistic networks could be an intrinsic physical structure of dynamic networks and the interaction switch is likely a key ecological process that results in nestedness of real-world networks. Identifying the biological processes responsible for network structures is thus crucial for understanding the architecture of ecological networks.     

Dynamics of Competition between Subnetworks of Spiking Neuronal Networks in the Balanced State

We explore and analyze the nonlinear switching dynamics of neuronal networks with non-homogeneous connectivity. The general significance of such transient dynamics for brain function is unclear; however, for instance decision-making processes in perception and cognition have been implicated with it. The network under study here is comprised of three subnetworks of either excitatory or inhibitory leaky integrate-and-fire neurons, of which two are of the same type. The synaptic weights are arranged to establish and maintain a balance between excitation and inhibition in case of a constant external drive. Each subnetwork is randomly connected, where all neurons belonging to a particular population have the same in-degree and the same out-degree. Neurons in different subnetworks are also randomly connected with the same probability; however, depending on the type of the pre-synaptic neuron, the synaptic weight is scaled by a factor. We observed that for a certain range of the “within” versus “between” connection weights (bifurcation parameter), the network activation spontaneously switches between the two sub-networks of the same type. This kind of dynamics has been termed “winnerless competition”, which also has a random component here. In our model, this phenomenon is well described by a set of coupled stochastic differential equations of Lotka-Volterra type that imply a competition between the subnetworks. The associated mean-field model shows the same dynamical behavior as observed in simulations of large networks comprising thousands of spiking neurons. The deterministic phase portrait is characterized by two attractors and a saddle node, its stochastic component is essentially given by the multiplicative inherent noise of the system. We find that the dwell time distribution of the active states is exponential, indicating that the noise drives the system randomly from one attractor to the other. A similar model for a larger number of populations might suggest a general approach to study the dynamics of interacting populations of spiking networks.     

Coupled Reversible and Irreversible Bistable Switches Underlying TGFβ-induced Epithelial to Mesenchymal Transition

Epithelial to mesenchymal transition (EMT) plays an important role in embryonic development, tissue regeneration, and cancer metastasis. Whereas several feedback loops have been shown to regulate EMT, it remains elusive how they coordinately modulate EMT response to TGF-β treatment. We construct a mathematical model for the core regulatory network controlling TGF-β-induced EMT. Through deterministic analyses and stochastic simulations, we show that EMT is a sequential two-step program in which an epithelial cell first is converted to partial EMT then to the mesenchymal state, depending on the strength and duration of TGF-β stimulation. Mechanistically the system is governed by coupled reversible and irreversible bistable switches. The SNAIL1/miR-34 double-negative feedback loop is responsible for the reversible switch and regulates the initiation of EMT, whereas the ZEB/miR-200 feedback loop is accountable for the irreversible switch and controls the establishment of the mesenchymal state. Furthermore, an autocrine TGF-β/miR-200 feedback loop makes the second switch irreversible, modulating the maintenance of EMT. Such coupled bistable switches are robust to parameter variation and molecular noise. We provide a mechanistic explanation on multiple experimental observations. The model makes several explicit predictions on hysteretic dynamic behaviors, system response to pulsed stimulation, and various perturbations, which can be straightforwardly tested.     

Transient information flow in a network of excitatory and inhibitory model neurons: Role of noise and signal autocorrelation

We investigate the performance of sparsely-connected networks of integrate-and-fire neurons for ultra-short term information processing. We exploit the fact that the population activity of networks with balanced excitation and inhibition can switch from an oscillatory firing regime to a state of asynchronous irregular firing or quiescence depending on the rate of external background spikes. We find that in terms of information buffering the network performs best for a moderate, non-zero, amount of noise. Analogous to the phenomenon of stochastic resonance the performance decreases for higher and lower noise levels. The optimal amount of noise corresponds to the transition zone between a quiescent state and a regime of stochastic dynamics. This provides a potential explanation of the role of non-oscillatory population activity in a simplified model of cortical micro-circuits.     

Steady-state expression of self-regulated genes

MOTIVATION: Regulatory gene networks contain generic modules such as feedback loops that are essential for the regulation of many biological functions. The study of the stochastic mechanisms of gene regulation is instrumental for the understanding of how cells maintain their expression at levels commensurate with their biological role, as well as to engineer gene expression switches of appropriate behavior. The lack of precise knowledge on the steady-state distribution of gene expression requires the use of Gillespie algorithms and Monte-Carlo approximations. Methodology: In this study, we provide new exact formulas and efficient numerical algorithms for computing/modeling the steady-state of a class of self-regulated genes, and we use it to model/compute the stochastic expression of a gene of interest in an engineered network introduced in mammalian cells. The behavior of the genetic network is then analyzed experimentally in living cells. RESULTS: Stochastic models often reveal counter-intuitive experimental behaviors, and we find that this genetic architecture displays a unimodal behavior in mammalian cells, which was unexpected given its known bimodal response in unicellular organisms. We provide a molecular rationale for this behavior, and we implement it in the mathematical picture to explain the experimental results obtained from this network.     

Predicting the order in which contacts are broken during single molecule protein stretching experiments

We combine two methods to enable the prediction of the order in which contacts are broken under external stretching forces in single molecule experiments. These two methods are Gō-like models and elastic network models. The Gō-like models have shown remarkable success in representing many aspects of protein behavior, including the reproduction of experimental data obtained from atomic force microscopy. The simple elastic network models are often used successfully to predict the fluctuations of residues around their mean positions, comparing favorably with the experimentally measured crystallographic B-factors. The behavior of biomolecules under external forces has been demonstrated to depend principally on their elastic properties and the overall shape of their structure. We have studied in detail the muscle protein titin and green fluorescent protein and tested for ten other proteins. First, we stretch the proteins computationally by performing stochastic dynamics simulations with the Gō-like model. We obtain the force-displacement curves and unfolding scenarios of possible mechanical unfolding. We then use the elastic network model to calculate temperature factors (B-factors) and compare the slowest modes of motion for the stretched proteins and compare them with the predicted order of breaking contacts between residues in the Gō-like model. Our results show that a simple Gaussian network model is able to predict contacts that break in the next time stage of stretching. Additionally, we have found that the contact disruption is strictly correlated with the highest force exerted by the backbone on these residues. Our prediction of bond-breaking agrees well with the unfolding scenario obtained with the Gō-like model. We anticipate that this method will be a useful new tool for interpreting stretching experiments.     

Differential control of active and silent phases in relaxation models of neuronal rhythms

Rhythmic bursting activity, found in many biological systems, serves a variety of important functions. Such activity is composed of episodes, or bursts (the active phase, AP) that are separated by quiescent periods (the silent phase, SP). Here, we use mean field, firing rate models of excitatory neural network activity to study how AP and SP durations depend on two critical network parameters that control network connectivity and cellular excitability. In these models, the AP and SP correspond to the network's underlying bistability on a fast time scale due to rapid recurrent excitatory connectivity. Activity switches between the AP and SP because of two types of slow negative feedback: synaptic depression—which has a divisive effect on the network input/output function, or cellular adaptation—a subtractive effect on the input/output function. We show that if a model incorporates the divisive process (regardless of the presence of the subtractive process), then increasing cellular excitability will speed up the activity, mostly by decreasing the silent phase. Reciprocally, if the subtractive process is present, increasing the excitatory connectivity will slow down the activity, mostly by lengthening the active phase. We also show that the model incorporating both slow processes is less sensitive to parameter variations than the models with only one process. Finally, we note that these network models are formally analogous to a type of cellular pacemaker and thus similar results apply to these cellular pacemakers. Action Editor: Misha Tsodyks     

Coupled Reversible and Irreversible Bistable Switches Underlying TGFβ-induced Epithelial to Mesenchymal Transition

Epithelial to mesenchymal transition (EMT) plays an important role in embryonic development, tissue regeneration, and cancer metastasis. Whereas several feedback loops have been shown to regulate EMT, it remains elusive how they coordinately modulate EMT response to TGF-β treatment. We construct a mathematical model for the core regulatory network controlling TGF-β-induced EMT. Through deterministic analyses and stochastic simulations, we show that EMT is a sequential two-step program in which an epithelial cell first is converted to partial EMT then to the mesenchymal state, depending on the strength and duration of TGF-β stimulation. Mechanistically the system is governed by coupled reversible and irreversible bistable switches. The SNAIL1/miR-34 double-negative feedback loop is responsible for the reversible switch and regulates the initiation of EMT, whereas the ZEB/miR-200 feedback loop is accountable for the irreversible switch and controls the establishment of the mesenchymal state. Furthermore, an autocrine TGF-β/miR-200 feedback loop makes the second switch irreversible, modulating the maintenance of EMT. Such coupled bistable switches are robust to parameter variation and molecular noise. We provide a mechanistic explanation on multiple experimental observations. The model makes several explicit predictions on hysteretic dynamic behaviors, system response to pulsed stimulation, and various perturbations, which can be straightforwardly tested.