Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Stochastic Oscillations Induced by Intrinsic Fluctuations in a Self-Repressing Gene

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.     

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. III. Integrodifferential equations]

The Hodgkin - Huxley system of equations is reduced to single integral-differential equation in neglection of slow variables dynamics. Two limiting cases of fast and slow sodium activation processes are considered. The first case leads to a nonlinear differential equation for the potential, the second one - to an ordinary differential equation with a known source as a function of coordinate. Such a simplification is due to approximation of steady-state sodium activation variable with the help of Heviside function. The validity of this approximation is discussed; the corresponding error is estimated by calculation of the second approximation for the source function.     

Modeling the precision and robustness of Hunchback border during Drosophila embryonic development

During anterior-posterior axis specification in the Drosophila embryo, the Hunchback (Hb) protein forms a sharp boundary at the mid-point of the embryo with great positional precision. While Bicoid (Bcd) is a known upstream regulator for hb expression, there is evidence to suggest that Hb effectively filters out “noisy” data received from varied Bcd gradients. We use mathematical models to explore simple regulatory networks which filter out such noise to produce a precise Hb boundary. We find that in addition to Bcd and Hb, at least one freely evolving protein is necessary. An automated search yields a number of examples of three-protein networks exhibiting the desired precision. In all such networks, Hb diffuses much slower than the third protein. In addition, the action of Hb on the third protein is the opposite of the action of the third protein on hb (i.e. if Hb activates the third protein, then the third protein inhibits hb expression, and vice versa). Most of the discovered systems satisfy the known biological properties, that Bcd activates hb, and that Hb activates its own expression. We find that all network topologies satisfying these constraints arise among the networks exhibiting the desired precision. Investigating the dynamics of these networks, we find that under a general class of non-uniform initial conditions, Bcd can be eliminated from the system and the spatiotemporal evolution of these two proteins alone is sufficient to recapture the dynamics. We hypothesize that Bcd is needed only to spatially disturb the gradient of the third protein, and then becomes unnecessary in the further evolution of the Hb border. This provides a possible explanation as to why the Hb dynamics are robust under perturbations of the Bcd gradient. Under this hypothesis, other proteins would be able to assume the role of Bcd in our simulations (possibly in the case of evolutionary divergences or a redundancy in the process), with the only constraint that they act to positively regulate hb.     

On learning dynamics underlying the evolution of learning rules

In order to understand the development of non-genetically encoded actions during an animal’s lifespan, it is necessary to analyze the dynamics and evolution of learning rules producing behavior. Owing to the intrinsic stochastic and frequency-dependent nature of learning dynamics, these rules are often studied in evolutionary biology via agent-based computer simulations. In this paper, we show that stochastic approximation theory can help to qualitatively understand learning dynamics and formulate analytical models for the evolution of learning rules. We consider a population of individuals repeatedly interacting during their lifespan, and where the stage game faced by the individuals fluctuates according to an environmental stochastic process. Individuals adjust their behavioral actions according to learning rules belonging to the class of experience-weighted attraction learning mechanisms, which includes standard reinforcement and Bayesian learning as special cases. We use stochastic approximation theory in order to derive differential equations governing action play probabilities, which turn out to have qualitative features of mutator-selection equations. We then perform agent-based simulations to find the conditions where the deterministic approximation is closest to the original stochastic learning process for standard 2-action 2-player fluctuating games, where interaction between learning rules and preference reversal may occur. Finally, we analyze a simplified model for the evolution of learning in a producer–scrounger game, which shows that the exploration rate can interact in a non-intuitive way with other features of co-evolving learning rules. Overall, our analyses illustrate the usefulness of applying stochastic approximation theory in the study of animal learning.     

The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright–Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection–diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.     

Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal instability

In this paper, we derive analytic solutions of stochastic mutation-selection networks that describe early events of cancer formation. A main assumption is that cancer is initiated in tissue compartments, where only a relatively small number of cells are at risk of mutating into cells that escape from homeostatic regulation. In this case, the evolutionary dynamics can be approximated by a low-dimensional stochastic process with a linear Kolmogorov forward equation that can be solved analytically. Most of the time, the cell population is homogeneous with respect to relevant mutations. Occasionally, such homogeneous states are connected by 'stochastic tunnels'. We give a precise analysis of the existence of tunnels and calculate the rate of tunneling. Finally, we calculate the conditions for chromosomal instability (CIN) to precede inactivation of the first tumor suppressor gene. In this case, CIN is an early event and a driving force of cancer progression. The techniques developed in this paper can be used to study arbitrarily complex mutation-selection networks of the somatic evolution of cancer.     

An investigation into the statistical properties of TB episodes in a South African community with high HIV prevalence

Continuous differential equations are often applied to small populations with little time spent on understanding uncertainty brought about by small-population effects. Despite large numbers of individuals being latently infected with Mycobacterium tuberculosis (TB), progression from latent infection to observable disease is a relatively rare event. For small communities, this means case counts are subject to stochasticity, and deterministic models may not be appropriate tools for interpreting transmission trends. Furthermore, the nonlinear nature of the underlying dynamics means that fluctuations are autocorrelated, which can invalidate standard statistical analyses which assume independent fluctuations.Here we extend recent work using a system of differential equations to study the HIV-TB epidemic in Masiphumelele, a community near Cape Town in South Africa [Bacaër, et al., J. Mol. Biol. 57(4), 557-593] by studying the statistical properties of active TB events. We apply van Kampen's system-size (or population-size) expansion technique to obtain an approximation to a master equation describing the dynamics. We use the resulting Fokker-Planck equation and point-process theory to derive two-time correlation functions for active TB events. This method can be used to gain insight into the temporal aspect of cluster identification, which currently relies on DNA classification only.     

Deterministic epidemiological models at the individual level

In many fields of science including population dynamics, the vast state spaces inhabited by all but the very simplest of systems can preclude a deterministic analysis. Here, a class of approximate deterministic models is introduced into the field of epidemiology that reduces this state space to one that is numerically feasible. However, these reduced state space master equations do not in general form a closed set. To resolve this, the equations are approximated using closure approximations. This process results in a method for constructing deterministic differential equation models with a potentially large scope of application including dynamic directed contact networks and heterogeneous systems using time dependent parameters. The method is exemplified in the case of an SIR (susceptible-infectious-removed) epidemiological model and is numerically evaluated on a range of networks from spatially local to random. In the context of epidemics propagated on contact networks, this work assists in clarifying the link between stochastic simulation and traditional population level deterministic models.     

Interdependency and hierarchy of exact and approximate epidemic models on networks

Over the years numerous models of \(SIS\) (susceptible \(\rightarrow \) infected \(\rightarrow \) susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established models that enables this derivation. We also derive a general result for the exact differential equations for the expected number of an arbitrary motif directly from the Kolmogorov/master equations and conclude with a comparison of the performance of the different closed systems of equations on networks of varying structure.     

On the long-run sensitivity of probabilistic Boolean networks

Boolean networks and, more generally, probabilistic Boolean networks, as one class of gene regulatory networks, model biological processes with the network dynamics determined by the logic-rule regulatory functions in conjunction with probabilistic parameters involved in network transitions. While there has been significant research on applying different control policies to alter network dynamics as future gene therapeutic intervention, we have seen less work on understanding the sensitivity of network dynamics with respect to perturbations to networks, including regulatory rules and the involved parameters, which is particularly critical for the design of intervention strategies. This paper studies this less investigated issue of network sensitivity in the long run. As the underlying model of probabilistic Boolean networks is a finite Markov chain, we define the network sensitivity based on the steady-state distributions of probabilistic Boolean networks and call it long-run sensitivity. The steady-state distribution reflects the long-run behavior of the network and it can give insight into the dynamics or momentum existing in a system. The change of steady-state distribution caused by possible perturbations is the key measure for intervention. This newly defined long-run sensitivity can provide insight on both network inference and intervention. We show the results for probabilistic Boolean networks generated from random Boolean networks and the results from two real biological networks illustrate preliminary applications of sensitivity in intervention for practical problems.     

Charge transport in DNA model with vibrational and rotational coupling motions

The dynamics of the Peyrard-Bishop model for vibrational motion of DNA dynamics, which has been extended by taking into account the rotational motion for the nucleotides (Silva et al., J. Biol. Phys. 34, 511–519, 2018) is studied. We report on the presence of the modulational instability (MI) of a plane wave for charge migration in DNA and the generation of soliton-like excitations in DNA nucleotides. We show that the original differential-difference equation for the DNA dynamics can be reduced in the continuum approximation to a set of three coupled nonlinear equations. The linear stability analysis of continuous wave solutions of the coupled systems is performed and the growth rate of instability is found numerically. Numerical simulations show the validity of the analytical approach with the generation of wave packets provided that the wave numbers fall in the instability domain.     

Reaction diffusion model of the enzymatic erosion of insoluble fibrillar matrices

Predicting the time course of in vivo biodegradation is a key issue in the design of an increasing number of biomedical applications such as sutures, tissue analogs and drug-delivery devices. The design of such biodegradable devices is hampered by the absence of quantitative models for the enzymatic erosion of solid protein matrices. In this work, we derive and simulate a reaction diffusion model for the enzymatic erosion of fibrillar gels that successfully reproduces the main qualitative features of this process. A key aspect of the proposed model is the incorporation of steric hindrance into the standard Michaelis-Menten scheme for enzyme kinetics. In the limit of instantaneous diffusion, the model equations are analogous to the standard equations for enzymatic degradation in solution. Invoking this analogy, the total quasi-steady-state approximation is used to derive approximate analytical solutions that are valid for a wide range of in vitro conditions. Using these analytical approximations, an experimental-theoretical method is derived to unambiguously estimate all the kinetic model parameters. Moreover, the analytical approximations correctly describe the characteristic hyperbolic dependence of the erosion rate on enzyme concentration and the zero-order erosion of thin fibers. For definiteness, the analysis of published experimental results of enzymatic degradation of fibrillar collagen is demonstrated, and the role of diffusion in these experiments is elucidated.     

Autocatalytic networks with translation

We consider the kinetics of an autocatalytic reaction network in which replication and catalytic actions are separated by a translation step. We find that the behaviour of such a system is closely related to second-order replicator equations, which describe the kinetics of autocatalytic reaction networks in which the replicators act also as catalysts. In fact, the qualitative dynamics seems to be described almost entirely be the second-order reaction rates of the replication step. For two species we recover the qualitative dynamics of the replicator equations. Larger networks show some deviations, however. A hypercyclic system consisting of three interacting species can converge toward a stable limit cycle in contrast to the replicator equation case. A singular perturbation analysis shows that the replication-translation system reduces to a second-order replicator equation if translation is fast. The influence of mutations on replication-translation networks is also very similar to the behavior of selection-mutation equations.     

First return maps for the dynamics of synaptically coupled conditional bursters

The pre-Bötzinger complex (preBötc) in the mammalian brainstem has an important role in generating respiratory rhythms. An influential differential equation model for the activity of individual neurons in the preBötc yields transitions from quiescence to bursting to tonic spiking as a parameter is varied. Further, past work has established that bursting dynamics can arise from a pair of tonic model cells coupled with synaptic excitation. In this paper, we analytically derive one- and two-dimensional maps from the differential equations for a self-coupled neuron and a two-neuron network, respectively. Using a combination of analysis and simulations of these maps, we explore the possible forms of dynamics that the model networks can produce as well as which transitions between dynamic regimes are mathematically possible.     

Robust filtering for stochastic genetic regulatory networks with time-varying delay

This paper addresses the robust filtering problem for a class of linear genetic regulatory networks (GRNs) with stochastic disturbances, parameter uncertainties and time delays. The parameter uncertainties are assumed to reside in a polytopic region, the stochastic disturbance is state-dependent described by a scalar Brownian motion, and the time-varying delays enter into both the translation process and the feedback regulation process. We aim to estimate the true concentrations of mRNA and protein by designing a linear filter such that, for all admissible time delays, stochastic disturbances as well as polytopic uncertainties, the augmented state estimation dynamics is exponentially mean square stable with an expected decay rate. A delay-dependent linear matrix inequality (LMI) approach is first developed to derive sufficient conditions that guarantee the exponential stability of the augmented dynamics, and then the filter gains are parameterized in terms of the solution to a set of LMIs. Note that LMIs can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures.