Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli?

Transcription-replication interactions reveal bacterial genome regulation

Organisms determine the transcription rates of thousands of genes through a few modes of regulation that recur across the genome1. In bacteria, the relationship between the regulatory architecture of a gene and its expression is well understood for individual model gene circuits2,3. However, a broader perspective of these dynamics at the genome scale is lacking, in part because bacterial transcriptomics has hitherto captured only a static snapshot of expression averaged across millions of cells4. As a result, the full diversity of gene expression dynamics and their relation to regulatory architecture remains unknown. Here we present a novel genome-wide classification of regulatory modes based on the transcriptional response of each gene to its own replication, which we term the transcription-replication interaction profile (TRIP). Analysing single-bacterium RNA-sequencing data, we found that the response to the universal perturbation of chromosomal replication integrates biological regulatory factors with biophysical molecular events on the chromosome to reveal the local regulatory context of a gene. Whereas the TRIPs of many genes conform to a gene dosage-dependent pattern, others diverge in distinct ways, and this is shaped by factors such as intra-operon position and repression state. By revealing the underlying mechanistic drivers of gene expression heterogeneity, this work provides a quantitative, biophysical framework for modelling replication-dependent expression dynamics.

The roles of nucleoid-associated proteins and topoisomerases in chromosome structure, strand segregation, and the generation of phenotypic heterogeneity in bacteria

How to adapt to a changing environment is a fundamental, recurrent problem confronting cells. One solution is for cells to organize their constituents into a limited number of spatially extended, functionally relevant, macromolecular assemblies or hyperstructures, and then to segregate these hyperstructures asymmetrically into daughter cells. This asymmetric segregation becomes a particularly powerful way of generating a coherent phenotypic diversity when the segregation of certain hyperstructures is with only one of the parental DNA strands and when this pattern of segregation continues over successive generations. Candidate hyperstructures for such asymmetric segregation in prokaryotes include those containing the nucleoid-associated proteins (NAPs) and the topoisomerases. Another solution to the problem of creating a coherent phenotypic diversity is by creating a growth-environment-dependent gradient of supercoiling generated along the replication origin-to-terminus axis of the bacterial chromosome. This gradient is modulated by transcription, NAPs, and topoisomerases. Here, we focus primarily on two topoisomerases, TopoIV and DNA gyrase in Escherichia coli, on three of its NAPs (H-NS, HU, and IHF), and on the single-stranded binding protein, SSB. We propose that the combination of supercoiling-gradient-dependent and strand-segregation-dependent topoisomerase activities result in significant differences in the supercoiling of daughter chromosomes, and hence in the phenotypes of daughter cells.

Random Walk Approximation for Stochastic Processes on Graphs

We introduce the Random Walk Approximation (RWA), a new method to approximate the stationary solution of master equations describing stochastic processes taking place on graphs. Our approximation can be used for all processes governed by non-linear master equations without long-range interactions and with a conserved number of entities, which are typical in biological systems, such as gene regulatory or chemical reaction networks, where no exact solution exists. For linear systems, the RWA becomes the exact result obtained from the maximum entropy principle. The RWA allows having a simple analytical, even though approximated, form of the solution, which is global and easier to deal with than the standard System Size Expansion (SSE). Here, we give some theoretically sufficient conditions for the validity of the RWA and estimate the order of error calculated by the approximation with respect to the number of particles. We compare RWA with SSE for two examples, a toy model and the more realistic dual phosphorylation cycle, governed by the same underlying process. Both approximations are compared with the exact integration of the master equation, showing for the RWA good performances of the same order or better than the SSE, even in regions where sufficient conditions are not met.

Spatiotemporal Coupling of DNA Supercoiling and Genomic Sequence Organization-A Timing Chain for the Bacterial Growth Cycle?

In this article we describe the bacterial growth cycle as a closed, self-reproducing, or autopoietic circuit, reestablishing the physiological state of stationary cells initially inoculated in the growth medium. In batch culture, this process of self-reproduction is associated with the gradual decline in available metabolic energy and corresponding change in the physiological state of the population as a function of "travelled distance" along the autopoietic path. We argue that this directional alteration of cell physiology is both reflected in and supported by sequential gene expression along the chromosomal OriC-Ter axis. We propose that during the E. coli growth cycle, the spatiotemporal order of gene expression is established by coupling the temporal gradient of supercoiling energy to the spatial gradient of DNA thermodynamic stability along the chromosomal OriC-Ter axis.

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.

Phenotypic heterogeneity in persisters: a novel 'hunker' theory of persistence

Persistence has been linked to treatment failure since its discovery over 70 years ago and understanding formation, nature and survival of this key antibiotic refractory subpopulation is crucial to enhancing treatment success and combatting the threat of antimicrobial resistance (AMR). The term 'persistence' is often used interchangeably with other terms such as tolerance or dormancy. In this review we focus on 'antibiotic persistence' which we broadly define as a feature of a subpopulation of bacterial cells that possesses the non-heritable character of surviving exposure to one or more antibiotics; and persisters as cells that possess this characteristic. We discuss novel molecular mechanisms involved in persister cell formation, as well as environmental factors which can contribute to increased antibiotic persistence in vivo, highlighting recent developments advanced by single-cell studies. We also aim to provide a comprehensive model of persistence, the 'hunker' theory which is grounded in intrinsic heterogeneity of bacterial populations and a myriad of 'hunkering down' mechanisms which can contribute to antibiotic survival of the persister subpopulation. Finally, we discuss antibiotic persistence as a 'stepping-stone' to AMR and stress the urgent need to develop effective anti-persister treatment regimes to treat this highly clinically relevant bacterial sub-population.

Improved estimations of stochastic chemical kinetics by finite-state expansion

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

Exact Maximal Reduction Of Stochastic Reaction Networks By Species Lumping

MOTIVATION

Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortunately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user.

RESULTS

We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks.

AVAILABILITY

The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu.

Allocation of gene products to daughter cells is determined by the age of the mother in single Escherichia coli cells

Gene expression and growth rate are highly stochastic in Escherichia coli. Some of the growth rate variations result from the deterministic and asymmetric partitioning of damage by the mother to its daughters. One daughter, denoted the old daughter, receives more damage, grows more slowly and ages. To determine if expressed gene products are also allocated asymmetrically, we compared the levels of expressed green fluorescence protein in growing daughters descending from the same mother. Our results show that old daughters were less fluorescent than new daughters. Moreover, old mothers, which were born as old daughters, produced daughters that were more asymmetric when compared to new mothers. Thus, variation in gene products in a clonal E. coli population also has a deterministic component. Because fluorescence levels and growth rates were positively correlated, the aging of old daughters appears to result from both the presence of both more damage and fewer expressed gene products.

Measuring transcription at a single gene copy reveals hidden drivers of bacterial individuality

Single-cell measurements of mRNA copy-number inform our understanding of stochastic gene expression [1-3], but these measurements coarse-grain over the individual copies of the gene, where transcription and its regulation stochastically take plasce [4, 5]. Here we combine single-molecule quantification of mRNA and gene loci to measure the transcriptional activity of an endogenous gene in individual Escherichia coli bacteria. Interpreted using a theoretical model for mRNA dynamics, the single-cell data allows us to obtain the probabilistic rates of promoter switching, transcription initiation and elongation, mRNA release and degradation. Unexpectedly, we find that gene activity can be strongly coupled to the transcriptional state of another copy of the same gene present in the cell, and to the event of gene replication during the bacterial cell cycle. These gene-copy and cell-cycle correlations demonstrate the limits of mapping whole-cell mRNA numbers to the underlying stochastic gene activity, and instead highlight the contribution of previously hidden variables to the observed population heterogeneity.

Chemical Transformation Motifs-Modelling Pathways as Integer Hyperflows

We present an elaborate framework for formally modelling pathways in chemical reaction networks on a mechanistic level. Networks are modelled mathematically as directed multi-hypergraphs, with vertices corresponding to molecules and hyperedges to reactions. Pathways are modelled as integer hyperflows and we expand the network model by detailed routing constraints. In contrast to the more traditional approaches like Flux Balance Analysis or Elementary Mode analysis we insist on integer-valued flows. While this choice makes it necessary to solve possibly hard integer linear programs, it has the advantage that more detailed mechanistic questions can be formulated. It is thus possible to query networks for general transformation motifs, and to automatically enumerate optimal and near-optimal pathways. Similarities and differences between our work and traditional approaches in metabolic network analysis are discussed in detail. To demonstrate the applicability of the mathematical framework to real-life problems we first explore the design space of possible non-oxidative glycolysis pathways and show that recent manually designed pathways can be further optimized. We then use a model of sugar chemistry to investigate pathways in the autocatalytic formose process. A graph transformation-based approach is used to automatically generate the reaction networks of interest.

Modulation of first-passage time for bursty gene expression via random signals

The stochastic nature of cell-specific signal molecules (such as transcription factor, ribosome, etc.) and the intrinsic stochastic nature of gene expression process result in cell-to-cell variations at protein levels. Increasing experimental evidences suggest that cell phenotypic variations often depend on the accumulation of some special proteins. Hence, a natural and fundamental question is: How does input signal affect the timing of protein count up to a given threshold? To this end, we study effects of input signal on the first-passage time (FPT), the time at which the number of proteins crosses a given threshold. Input signal is distinguished into two types: constant input signal and random input signal, regulating only burst frequency (or burst size) of gene expression. Firstly, we derive analytical formulae for FPT moments in each case of constant signal regulation and random signal regulation. Then, we find that random input signal tends to increases the mean and noise of FPT compared with constant input signal. Finally, we observe that different regulation ways of random signal have different effects on FPT, that is, burst size modulation tends to decrease the mean of FPT and increase the noise of FPT compared with burst frequency modulation. Our findings imply a fundamental mechanism that random fluctuating environment may prolong FPT. This can provide theoretical guidance for studies of some cellular key events such as latency of HIV and lysis time of bacteriophage λ. In conclusion, our results reveal impacts of external signal on FPT and aid understanding the regulation mechanism of gene expression.

Robustness of a model microbial community emerges from population structure among single cells of a clonal population

Microbial populations can withstand, overcome and persist in the face of environmental fluctuation. Previously, we demonstrated how conditional gene regulation in a fluctuating environment drives dilution of condition-specific transcripts, causing a population of Desulfovibrio vulgaris Hildenborough (DvH) to collapse after repeatedly transitioning from sulfate respiration to syntrophic conditions with the methanogen Methanococcus maripaludis. Failure of the DvH to successfully transition contributed to the collapse of this model community. We investigated the mechanistic basis for loss of robustness by examining whether conditional gene regulation altered heterogeneity in gene expression across individual DvH cells. We discovered that robustness of a microbial population across environmental transitions was attributable to the retention of cells in two states that exhibited different condition-specific gene expression patterns. In our experiments, a population with disrupted conditional regulation successfully alternated between cell states. Meanwhile, a population with intact conditional regulation successfully switched between cell states initially, but collapsed after repeated transitions, possibly due to the high energy requirements of regulation. These results demonstrate that the survival of this entire model microbial community is dependent on the regulatory system's influence on the distribution of distinct cell states among individual cells within a clonal population.

Intrinsic noise and deviations from criticality in Boolean gene-regulatory networks

Gene regulatory networks can be successfully modeled as Boolean networks. A much discussed hypothesis says that such model networks reproduce empirical findings the best if they are tuned to operate at criticality, i.e. at the borderline between their ordered and disordered phases. Critical networks have been argued to lead to a number of functional advantages such as maximal dynamical range, maximal sensitivity to environmental changes, as well as to an excellent tradeoff between stability and flexibility. Here, we study the effect of noise within the context of Boolean networks trained to learn complex tasks under supervision. We verify that quasi-critical networks are the ones learning in the fastest possible way -even for asynchronous updating rules- and that the larger the task complexity the smaller the distance to criticality. On the other hand, when additional sources of intrinsic noise in the network states and/or in its wiring pattern are introduced, the optimally performing networks become clearly subcritical. These results suggest that in order to compensate for inherent stochasticity, regulatory and other type of biological networks might become subcritical rather than being critical, all the most if the task to be performed has limited complexity.

Origins of transcriptional transition: balance between upstream and downstream regulatory gene sequences

By measuring individual mRNA production at the single-cell level, we investigated the lac promoter’s transcriptional transition during cell growth phases. In exponential phase, variation in transition rates generates two mixed phenotypes, low and high numbers of mRNAs, by modulating their burst frequency and sizes. Independent activation of the regulatory-gene sequence does not produce bimodal populations at the mRNA level, but bimodal populations are produced when the regulatory gene is activated coordinately with the upstream and downstream region promoter sequence (URS and DRS, respectively). Time-lapse microscopy of mRNAs for lac and a variant lac promoter confirm this observation. Activation of the URS/DRS elements of the promoter reveals a counterplay behavior during cell phases. The promoter transition rate coupled with cell phases determines the mRNA and transcriptional noise. We further show that bias in partitioning of RNA does not lead to phenotypic switching. Our results demonstrate that the balance between the URS and the DRS in transcriptional regulation determines population diversity.

By measuring individual mRNA production at the single-cell level, we investigated the lac promoter transcriptional transition during cell growth phases. In exponential phase, variation in transition rate generates two mixed phenotypes producing low and high numbers of mRNAs by modulating the burst frequency and size. Independent activation of the regulatory gene sequence does not produce bimodal populations at the mRNA level, while it does when activated together through the coordination of upstream/downstream promoter sequences (URS/DRS). Time-lapse microscopy of mRNAs for lac and a lac variant promoter confirm this observation. Activation of the URS/DRS elements of the promoter reveals a counterplay behavior during cell phases. The promoter transition rate coupled with cell phases determines the mRNA and transcriptional noise. We further show that bias in partitioning of RNA does not lead to phenotypic switching. Our results demonstrate that the balance between URS and DRS in transcription regulation is determining the population diversity.

Stochastic gene expression with delay

The expression of genes usually follows a two-step procedure. First, a gene (encoded in the genome) is transcribed resulting in a strand of (messenger) RNA. Afterwards, the RNA is translated into protein. We extend the classical stochastic jump model by adding delays (with arbitrary distributions) to transcription and translation. Already in the classical model, production of RNA and protein comes in bursts by activation and deactivation of the gene, resulting in a large variance of the number of RNA and proteins in equilibrium. We derive precise formulas for this second-order structure with the model including delay in equilibrium.

Mosaic physiology from developmental noise: within-organism physiological diversity as an alternative to phenotypic plasticity and phenotypic flexibility

A key problem in organismal biology is to explain the origins of functional diversity. In the context of organismal biology, functional diversity describes the set of phenotypes, across scales of biological organization and through time, that a single genotype, or genome, or organism, can produce. Functional diversity encompasses many phenomena: differences in cell types within organisms; physiological and morphological differences among tissues and organs; differences in performance; morphological shifts in external phenotype; and changes in behavior. How can single genomes produce so many different phenotypes? Modern biology proposes two general mechanisms. The first is developmental programs, by which single cells and their single genomes diversify, via relatively deterministic processes, into the sets of cell types, tissues and organs that we see in most multicellular organisms. The second general mechanism is phenotypic modification stemming from interactions between organisms and their environments – modifications known either as phenotypic plasticity or as phenotypic flexibility, depending on the time scale of the response and the degree of reversibility. These two diversity-generating mechanisms are related because phenotypic modifications may sometimes arise as a consequence of environments influencing developmental programs. Here, I propose that functional diversity also arises via a third fundamental mechanism: stochastic developmental events giving rise to mosaics of physiological diversity within individual organisms. In biological systems, stochasticity stems from the inherently random actions of small numbers of molecules interacting with one another. Although stochastic effects occur in many biological contexts, available evidence suggests that they can be especially important in gene networks, specifically as a consequence of low transcript numbers in individual cells. I briefly review known mechanisms by which organisms control such stochasticity, and how they may use it to create adaptive functional diversity. I then fold this idea into modern thinking on phenotypic plasticity and flexibility, proposing that multicellular organisms exhibit ‘mosaic physiology’. Mosaic physiology refers to sets of diversified phenotypes, within individual organisms, that carry out related functions at the same time, but that are distributed in space. Mosaic physiology arises from stochasticity-driven differentiation of cells, early during cell diversification, which is then amplified by cell division and growth into macroscopic phenotypic modules (cells, tissues, organs) making up the physiological systems of later life stages. Mosaic physiology provides a set of standing, diversified phenotypes, within single organisms, that raise the likelihood of the organism coping well with novel environmental challenges. These diversified phenotypes can be distinct, akin to polyphenisms at the organismal level; or they can be continuously distributed, creating a kind of standing, simultaneously expressed reaction norm of physiological capacities.

Single-molecule methods for studying gene regulation in vivo

The recent emergence of new experimental tools employing sensitive fluorescence detection in vivo has made it possible to visualize various aspects of gene regulation at the single-molecule level in the native, intracellular context. In this review, we will first describe general considerations for in vivo, single-molecule fluorescence detection of DNA, mRNA, and protein molecules involved in gene regulation. We will then give an overview of the rapidly evolving suite of molecular tools available for observing gene regulation in vivo and discuss new insights they have brought into gene regulation.

Aptamer based strategy for cytosensing and evaluation of HER-3 on the surface of MCF-7 cells by using the signal amplification of nucleic acid-functionalized nanocrystals

The electrochemical detection of cell lines of MCF-7 (human breast cancer) has been reported, using magnetic beads for the separation tool and high-affinity DNA aptamers for signal recognition. The high specificity was obtained by using the magnetic beads and aptamers, and the good sensitivity was realized with the signal amplification of DNA capped CdS or PbS nanocrystals. The ASV (anodic stripping voltammetry) technology was employed for the detection of cadmic cation and lead ions, for electrochemical assay of the amount of the target cells and biomarkers on the membrane of target cells, respectively. This electrochemical method could respond to as low as 100 cells mL(-1) of cancer cells with a linear calibration range from 1.0×10(2) to 1.0×10(6) cells mL(-1), showing very high sensitivity. Moreover, the amounts of HER-3 which were overexpressed on MCF-7 cells were calculated correspond to be 3.56×10(4) anti-HER-3 antibody molecules. In addition, the assay was able to differentiate between different types of target and control cells based on the aptamers and magnetic beads used in the assay, indicating the wide applicability of the assay for early and accurate diagnose of cancers.

Simulation of stochastic kinetic models

A growing realization of the importance of stochasticity in cell and molecular processes has stimulated the need for statistical models that incorporate intrinsic (and extrinsic) variability. In this chapter we consider stochastic kinetic models of reaction networks leading to a Markov jump process representation of a system of interest. Traditionally, the stochastic model is characterized by a chemical master equation. While the intractability of such models can preclude a direct analysis, simulation can be straightforward and may present the only practical approach to gaining insight into a system's dynamics. We review exact simulation procedures before considering some efficient approximate alternatives.

Classes of fast and specific search mechanisms for proteins on DNA

Problems of search and recognition appear over different scales in biological systems. In this review we focus on the challenges posed by interactions between proteins, in particular transcription factors, and DNA and possible mechanisms which allow for fast and selective target location. Initially we argue that DNA-binding proteins can be classified, broadly, into three distinct classes which we illustrate using experimental data. Each class calls for a different search process and we discuss the possible application of different search mechanisms proposed over the years to each class. The main thrust of this review is a new mechanism which is based on barrier discrimination. We introduce the model and analyze in detail its consequences. It is shown that this mechanism applies to all classes of transcription factors and can lead to a fast and specific search. Moreover, it is shown that the mechanism has interesting transient features which allow for stability at the target despite rapid binding and unbinding of the transcription factor from the target.

Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it.

Mathematical descriptions of biochemical networks: stability, stochasticity, evolution

In this paper, we review some fundamental aspects, as well as some new developments, in the emerging field of network biology. The focus of attention is placed on mathematical approaches to conceptual modeling of biomolecular networks with special emphasis on dynamic stability, stochasticity and evolution.

Power spectra methods for a stochastic description of diffusion on deterministically growing domains

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state.

Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity, which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network, and study how the frequency distributions of oscillatory modes of such a system depend on the average connectivity.

Using a single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression

Fluorescent and luminescent proteins are often used as reporters of transcriptional activity. Given the prevalence of noise in biochemical systems, the time-series data arising from these is of significant interest in efforts to calibrate stochastic models of gene expression and obtain information about sources of nongenetic variability. We present a statistical inference framework that can be used to estimate kinetic parameters of gene expression, as well as the strength and half-life of extrinsic noise from single fluorescent-reporter-gene time-series data. The method takes into account stochastic variability in a fluorescent signal resulting from intrinsic noise of gene expression, kinetics of fluorescent protein maturation, and extrinsic noise, which is assumed to arise at transcriptional level. We use the linear noise approximation and derive an explicit formula for the likelihood of observed fluorescent data. The method is embedded in a Bayesian paradigm, so that certain parameters can be informed from other experiments allowing portability of results across different studies. Inference is performed using Markov chain Monte Carlo. Fluorescent reporters are primary tools to observe dynamics of gene expression and the correct interpretation of fluorescent data is crucial to investigating these fundamental processes of cellular life. As both magnitude and frequency of the noise may have a dramatic effect on the cell fitness, the quantification of stochastic fluctuation is essential to the understanding of how genes are regulated. Our method provides a framework that addresses this important question.

Discreteness-induced slow relaxation in reversible catalytic reaction networks

Slowing down of the relaxation of the fluctuations around equilibrium is investigated both by stochastic simulations and by analysis of master equation of reversible reaction networks consisting of reactions between a pair of resource and the corresponding high-energy product that works as a catalyst for another resource-product reaction. As the number of molecules N is decreased, the relaxation time to equilibrium is prolonged due to the deficiency of catalysts, as demonstrated by the amplification compared to that by the continuum limit. This amplification ratio of the relaxation time is represented by a scaling function as h=N exp(-βV), and it becomes prominent as N becomes less than a critical value h ∼ 1, where β is the inverse temperature and V is the energy required to the transformation from resources to the corresponding products.

How subdiffusion changes the kinetics of binding to a surface

Under molecular crowding conditions, biopolymers have been reported to subdiffuse, (r(2)(t)) approximately = t(alpha), with 0 <alpha < 1. Here we study the exchange dynamics of such a subdiffusing particle with a reactive boundary using a continuous time random walk approach. We derive the generalized boundary condition and consider the unbinding from the boundary. An ensuing weak ergodicity breaking has profound consequences for material exchange between the boundary and bulk. We discuss the effects in biological contexts such as gene regulation or membrane-bulk exchange processes. We also suggest various methods to experimentally probe the subdiffusive behavior.

Bayesian inference of biochemical kinetic parameters using the linear noise approximation

Background

Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.

Results

We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo.

Conclusion

The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods.

Self-organized criticality of a catalytic reaction network under flow

Self-organized critical behavior in a catalytic reaction network system induced by smallness in the molecule number is reported. The system under a flow of chemicals is shown to undergo a transition from a stationary to an intermittent reaction phase when the flow rate is decreased. In the intermittent reaction phase, two temporal regimes with active and halted reactions alternate. The number frequency of reaction events at each active regime and its duration time are shown to obey a universal power law with the exponents 4/3 and 3/2, respectively, independently of the parameters and network structure. These power laws are explained by a one-dimensional random-walk representation of the number of catalytically active chemicals. Possible relevance of the result to reaction dynamics in artificial and biological cells is briefly discussed.

Robust stochastic chemical reaction networks and bounded tau-leaping

The behavior of some stochastic chemical reaction networks is largely unaffected by slight inaccuracies in reaction rates. We formalize the robustness of state probabilities to reaction rate deviations, and describe a formal connection between robustness and efficiency of simulation. Without robustness guarantees, stochastic simulation seems to require computational time proportional to the total number of reaction events. Even if the concentration (molecular count per volume) stays bounded, the number of reaction events can be linear in the duration of simulated time and total molecular count. We show that the behavior of robust systems can be predicted such that the computational work scales linearly with the duration of simulated time and concentration, and only polylogarithmically in the total molecular count. Thus our asymptotic analysis captures the dramatic speedup when molecular counts are large, and shows that for bounded concentrations the computation time is essentially invariant with molecular count. Finally, by noticing that even robust stochastic chemical reaction networks are capable of embedding complex computational problems, we argue that the linear dependence on simulated time and concentration is likely optimal.

Conditions of appreciable influence of microRNA on a large number of target mRNAs

In eukaryotic cells, messenger RNAs (mRNAs) can be regulated by microRNAs (miRNAs) via association and subsequent degradation. Each miRNA is now believed to potentially have hundreds of target mRNAs. Employing a generic kinetic model with physically reasonable parameters, we have quantified the mutual influence of miRNA and mRNAs in the case when the number of target mRNAs is large (e.g., 100). The decrease in the population of mRNAs due to interaction with miRNA is found to be appreciable (about 1.5-2-fold) only if the rate of the miRNA synthesis is very high. In the absence of the miRNA-mRNA interaction, it should be sufficient to maintain the miRNA population in the order of 10(4) per cell. In addition, the average mRNA population should not be too high (lower than or comparable to 100 for each kind of mRNA). For lower miRNA synthesis rates, the significant influence of miRNA on mRNAs is only possible provided that the average mRNA population is very low (of the order of 10). These general findings are complemented by a brief discussion of some relevant recent experimental results.

Potential role of phenotypic mutations in the evolution of protein expression and stability

Phenotypic mutations (errors occurring during protein synthesis) are orders of magnitude more frequent than genetic mutations. Consequently, the sequences of individual protein molecules transcribed and translated from the same gene can differ. To test the effects of such mutations, we established a bacterial system in which an antibiotic resistance gene (TEM-1 beta-lactamase) was transcribed by either a high-fidelity RNA polymerase or its error-prone mutant. This setup enabled the analysis of individual mRNA transcripts that were synthesized under normal or error-prone conditions. We found that an increase of approximately 20-fold in the frequency of transcription errors promoted the evolution of higher TEM-1 expression levels and of more stable enzyme variants. The stabilized variants exhibited a distinct advantage under error-prone transcription, although under normal transcription they conferred resistance similar to wild-type TEM-1. They did so, primarily, by increasing TEM-1's tolerance to destabilizing deleterious mutations that arise from transcriptional errors. The stabilized TEM-1 variants also showed increased tolerance to genetic mutations. Thus, although phenotypic mutations are not individually subjected to inheritance and natural selection, as are genetic mutations, they collectively exert a direct and immediate effect on protein fitness. They may therefore play a role in shaping protein traits such as expression levels, stability, and tolerance to genetic mutations.

Patterns of stochastic behavior in dynamically unstable high-dimensional biochemical networks

The question of dynamical stability and stochastic behavior of large biochemical networks is discussed. It is argued that stringent conditions of asymptotic stability have very little chance to materialize in a multidimensional system described by the differential equations of chemical kinetics. The reason is that the criteria of asymptotic stability (Routh-Hurwitz, Lyapunov criteria, Feinberg’s Deficiency Zero theorem) would impose the limitations of very high algebraic order on the kinetic rates and stoichiometric coefficients, and there are no natural laws that would guarantee their unconditional validity. Highly nonlinear, dynamically unstable systems, however, are not necessarily doomed to collapse, as a simple Jacobian analysis would suggest. It is possible that their dynamics may assume the form of pseudo-random fluctuations quite similar to a shot noise, and, therefore, their behavior may be described in terms of Langevin and Fokker-Plank equations. We have shown by simulation that the resulting pseudo-stochastic processes obey the heavy-tailed Generalized Pareto Distribution with temporal sequence of pulses forming the set of constituent-specific Poisson processes. Being applied to intracellular dynamics, these properties are naturally associated with burstiness, a well documented phenomenon in the biology of gene expression.

Genetic noise control via protein oligomerization

BACKGROUND

Gene expression in a cell entails random reaction events occurring over disparate time scales. Thus, molecular noise that often results in phenotypic and population-dynamic consequences sets a fundamental limit to biochemical signaling. While there have been numerous studies correlating the architecture of cellular reaction networks with noise tolerance, only a limited effort has been made to understand the dynamic role of protein-protein interactions.

RESULTS

We have developed a fully stochastic model for the positive feedback control of a single gene, as well as a pair of genes (toggle switch), integrating quantitative results from previous in vivo and in vitro studies. In particular, we explicitly account for the fast binding-unbinding kinetics among proteins, RNA polymerases, and the promoter/operator sequences of DNA. We find that the overall noise-level is reduced and the frequency content of the noise is dramatically shifted to the physiologically irrelevant high-frequency regime in the presence of protein dimerization. This is independent of the choice of monomer or dimer as transcription factor and persists throughout the multiple model topologies considered. For the toggle switch, we additionally find that the presence of a protein dimer, either homodimer or heterodimer, may significantly reduce its random switching rate. Hence, the dimer promotes the robust function of bistable switches by preventing the uninduced (induced) state from randomly being induced (uninduced).

CONCLUSION

The specific binding between regulatory proteins provides a buffer that may prevent the propagation of fluctuations in genetic activity. The capacity of the buffer is a non-monotonic function of association-dissociation rates. Since the protein oligomerization per se does not require extra protein components to be expressed, it provides a basis for the rapid control of intrinsic or extrinsic noise. The stabilization of regulatory circuits and epigenetic memory in general is of direct implications to organism fitness. Our results also suggest possible avenues for the design of synthetic gene circuits with tunable robustness for a wide range of engineering purposes.

Modelling and analysis of gene regulatory networks

Gene regulatory networks have an important role in every process of life, including cell differentiation, metabolism, the cell cycle and signal transduction. By understanding the dynamics of these networks we can shed light on the mechanisms of diseases that occur when these cellular processes are dysregulated. Accurate prediction of the behaviour of regulatory networks will also speed up biotechnological projects, as such predictions are quicker and cheaper than lab experiments. Computational methods, both for supporting the development of network models and for the analysis of their functionality, have already proved to be a valuable research tool.