Noise-based switches and amplifiers for gene expression

Simulated maximum likelihood method for estimating kinetic rates in gene expression

MOTIVATION

Kinetic rate in gene expression is a key measurement of the stability of gene products and gives important information for the reconstruction of genetic regulatory networks. Recent developments in experimental technologies have made it possible to measure the numbers of transcripts and protein molecules in single cells. Although estimation methods based on deterministic models have been proposed aimed at evaluating kinetic rates from experimental observations, these methods cannot tackle noise in gene expression that may arise from discrete processes of gene expression, small numbers of mRNA transcript, fluctuations in the activity of transcriptional factors and variability in the experimental environment.

RESULTS

In this paper, we develop effective methods for estimating kinetic rates in genetic regulatory networks. The simulated maximum likelihood method is used to evaluate parameters in stochastic models described by either stochastic differential equations or discrete biochemical reactions. Different types of non-parametric density functions are used to measure the transitional probability of experimental observations. For stochastic models described by biochemical reactions, we propose to use the simulated frequency distribution to evaluate the transitional density based on the discrete nature of stochastic simulations. The genetic optimization algorithm is used as an efficient tool to search for optimal reaction rates. Numerical results indicate that the proposed methods can give robust estimations of kinetic rates with good accuracy.

Dual feedback loops in the GAL regulon suppress cellular heterogeneity in yeast

Transcriptional noise is known to be an important cause of cellular heterogeneity and phenotypic variation. The extent to which molecular interaction networks may have evolved to either filter or exploit transcriptional noise is a much debated question. The yeast genetic network regulating galactose metabolism involves two proteins, Gal3p and Gal80p, that feed back positively and negatively, respectively, on GAL gene expression. Using kinetic modeling and experimental validation, we demonstrate that these feedback interactions together are important for (i) controlling the cell-to-cell variability of GAL gene expression and (ii) ensuring that cells rapidly switch to an induced state for galactose uptake.

External noise and feedback regulation: steady-state statistics of auto-regulatory genetic network

The steady-state statistics of a single gene auto-regulatory genetic network with the additive external Gaussian white noises is investigated. The main result shows that the negative feedback will result in that the mRNA noise has a positive contribution to the protein noise, but the positive feedback will result in that the mRNA noise has a negative contribution to the protein noise. If there is no feed back, then the contribution of mRNA noise to protein noise is always positive. On the other hand, the analysis and numerical simulations of linear and nonlinear feedback show that it is possible that the negative feedback increases, but the positive feedback decreases, the protein noise.

Noise in genetic and neural networks

Both neural and genetic networks are significantly noisy, and stochastic effects in both cases ultimately arise from molecular events. Nevertheless, a gulf exists between the two fields, with researchers in one often being unaware of similar work in the other. In this Special Issue, we focus on bridging this gap and present a collection of papers from both fields together. For each field, the networks studied range from just a single gene or neuron to endogenous networks. In this introductory article, we describe the sources of noise in both genetic and neural systems. We discuss the modeling techniques in each area and point out similarities. We hope that, by reading both sets of papers, ideas developed in one field will give insight to scientists from the other and that a common language and methodology will develop.

Frequency domain analysis of noise in simple gene circuits

Recent advances in single cell methods have spurred progress in quantifying and analyzing stochastic fluctuations, or noise, in genetic networks. Many of these studies have focused on identifying the sources of noise and quantifying its magnitude, and at the same time, paying less attention to the frequency content of the noise. We have developed a frequency domain approach to extract the information contained in the frequency content of the noise. In this article we review our work in this area and extend it to explicitly consider sources of extrinsic and intrinsic noise. First we review applications of the frequency domain approach to several simple circuits, including a constitutively expressed gene, a gene regulated by transitions in its operator state, and a negatively autoregulated gene. We then review our recent experimental study, in which time-lapse microscopy was used to measure noise in the expression of green fluorescent protein in individual cells. The results demonstrate how changes in rate constants within the gene circuit are reflected in the spectral content of the noise in a manner consistent with the predictions derived through frequency domain analysis. The experimental results confirm our earlier theoretical prediction that negative autoregulation not only reduces the magnitude of the noise but shifts its content out to higher frequency. Finally, we develop a frequency domain model of gene expression that explicitly accounts for extrinsic noise at the transcriptional and translational levels. We apply the model to interpret a shift in the autocorrelation function of green fluorescent protein induced by perturbations of the translational process as a shift in the frequency spectrum of extrinsic noise and a decrease in its weighting relative to intrinsic noise.

Stochastic Transcriptional Regulatory Systems with Time Delays: A Mean-Field Approximation

Modeling transcriptional regulation with time delays is an important problem of computational cell biology. In this paper, we propose a computational tool for studying transcriptional regulation in single cells based on a mean-field approximation method. The main idea is to replace the occurrence probabilities of the underlying transcriptional events by their mean values and use appropriately chosen additive noise terms to model statistical variations not accounted by this approximation. The proposed methodology allows us to characterize the transient and steady-state behavior of transcriptional regulation. Moreover, it provides a rather simple and computationally attractive tool for rapid statistical characterization of the dynamic behavior of a nonlinear transcriptional regulatory system with time delays.

Stochastic models for regulatory networks of the genetic toggle switch

Bistability arises within a wide range of biological systems from the lambda phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

Transcriptional noise and cellular heterogeneity in mammalian macrophages

Transcriptional noise is known to play a crucial role in heterogeneity in bacteria and yeast. Mammalian macrophages are known to exhibit cell-to-cell variation in their responses to pathogens, but the source of this heterogeneity is not known. We have developed a detailed stochastic model of gene expression that takes into account scaling effects due to cell size and genome complexity. We report the results of applying this model to simulating gene expression variability in mammalian macrophages, demonstrating a possible molecular basis for heterogeneity in macrophage signalling responses. We note that the nature of predicted transcriptional noise in macrophages is different from that in yeast and bacteria. Some molecular interactions in yeast and bacteria are thought to have evolved to minimize the effects of the high-frequency noise observed in these species. Transcriptional noise in macrophages results in slow changes to gene expression levels and would not require the type of spike-filtering circuits observed in yeast and bacteria.

The limits of reductionism in medicine: could systems biology offer an alternative?

In the first of a two part series, Ahn and colleagues discuss the reductionist approach pervading medicine and explain how a systems approach (as advocated by systems biology) may complement reductionism.

Predicting stochastic gene expression dynamics in single cells

Fluctuations in protein numbers (noise) due to inherent stochastic effects in single cells can have large effects on the dynamic behavior of gene regulatory networks. Although deterministic models can predict the average network behavior, they fail to incorporate the stochasticity characteristic of gene expression, thereby limiting their relevance when single cell behaviors deviate from the population average. Recently, stochastic models have been used to predict distributions of steady-state protein levels within a population but not to predict the dynamic, presteady-state distributions. In the present work, we experimentally examine a system whose dynamics are heavily influenced by stochastic effects. We measure population distributions of protein numbers as a function of time in the Escherichia coli lactose uptake network (lac operon). We then introduce a dynamic stochastic model and show that prediction of dynamic distributions requires only a few noise parameters in addition to the rates that characterize a deterministic model. Whereas the deterministic model cannot fully capture the observed behavior, our stochastic model correctly predicts the experimental dynamics without any fit parameters. Our results provide a proof of principle for the possibility of faithfully predicting dynamic population distributions from deterministic models supplemented by a stochastic component that captures the major noise sources.

Stimulus-response curves of a neuronal model for noisy subthreshold oscillations and related spike generation

We investigate the stimulus-dependent tuning properties of a noisy ionic conductance model for intrinsic subthreshold oscillations in membrane potential and associated spike generation. Upon depolarization by an applied current, the model exhibits subthreshold oscillatory activity with an occasional spike generation when oscillations reach the spike threshold. We consider how the amount of applied current, the noise intensity, variation of maximum conductance values, and scaling to different temperature ranges alter the responses of the model with respect to voltage traces, interspike intervals and their statistics, and the mean spike frequency curves. We demonstrate that subthreshold oscillatory neurons in the presence of noise can sensitively and also selectively be tuned by the stimulus-dependent variation of model parameters.

Phenotypic variation in bacteria: the role of feedback regulation

To survive in rapidly changing environmental conditions, bacteria have evolved a diverse set of regulatory pathways that govern various adaptive responses. Recent research has reinforced the notion that bacteria use feedback-based circuitry to generate population heterogeneity in natural situations. Using artificial gene networks, it has been shown that a relatively simple 'wiring' of a bacterial genetic system can generate two or more stable subpopulations within an overall genetically homogeneous population. This review discusses the ubiquity of these processes throughout nature, as well as the presumed molecular mechanisms responsible for the heterogeneity observed in a selection of bacterial species.

Enhancement of internal-noise coherence resonance by modulation of external noise in a circadian oscillator

A circadian oscillator driven by external noise and internal noise has been studied by use of the chemical Langevin equation method. When the system is near a Hopf bifurcation and driven by internal noise only, it is found that the coherence resonance phenomenon can be induced by the internal noise. When the system is simultaneously driven by internal and external noise, it is found that external-noise coherence resonance can be suppressed by internal noise, while internal-noise coherence resonance can be enhanced by modulation of the external noise intensity in a certain range of noise intensity. Another interesting result is that the external noise can regulate the optimal system size when the internal-noise coherence resonance occurs.

On the distribution of state values of reproducing cells

Characterizing a cell state by measuring the degree of gene expression as well as its noise has gathered much attention. The distribution of such state values (e.g., abundances of some proteins) over cells has been measured, and is not only a result of intracellular process, but is also influenced by the growth in cell number that depends on the state. By incorporating the growth-death process into the standard Fokker-Planck equation, a nonlinear temporal evolution equation of distribution is derived and then solved by means of eigenfunction expansions. This general formalism is applied to the linear relaxation case. First, when the growth rate of a cell increases linearly with the state value x, the shift of the average x due to the growth effect is shown to be proportional to the variance of x and the relaxation time, similar to the biological fluctuation-response relationship. Second, when there is a threshold value of x for growth, the existence of a critical growth rate, represented again by the variance and the relaxation time, is demonstrated. The relevance of the results to the analysis of biological data on the distribution of cell states, as obtained for example by flow cytometry, is discussed.

Multistable and multistep dynamics in neutrophil differentiation

Background

Cell differentiation has long been theorized to represent a switch in a bistable system, and recent experimental work in micro-organisms has revealed bistable dynamics in small gene regulatory circuits. However, the dynamics of mammalian cell differentiation has not been analyzed with respect to bistability.

Results

Here we studied how HL60 promyelocytic precursor cells transition to the neutrophil cell lineage after stimulation with the differentiation inducer, dimethyl sulfoxide (DMSO). Single cell analysis of the expression kinetics of the differentiation marker CD11b (Mac-1) revealed all-or-none switch-like behavior, in contrast to the seemingly graduated change of expression when measured as a population average. Progression from the precursor to the differentiated state was detected as a discrete transition between low (CD11b^Low) and high (CD11b^High) expressor subpopulations distinguishable in a bimodal distribution. Hysteresis in the dependence of CD11b expression on DMSO dose suggests that this bimodality may reflect a bistable dynamic. But when an "unswitched" (CD11b^Low) subpopulation of cells in the bistable/bimodal regime was isolated and cultured, these cells were found to differ from undifferentiated precursor cells in that they were "primed" to differentiate.

Conclusion

These findings indicate that differentiation of human HL60 cells into neutrophils does not result from a simple state transition of a bistable switch as traditionally modeled. Instead, mammalian differentiation appears to be a multi-step process in a high-dimensional system, a result which is consistent with the high connectivity of the cells' complex underlying gene regulatory network.

On the attenuation and amplification of molecular noise in genetic regulatory networks

Background

Noise has many important roles in cellular genetic regulatory functions at the nanomolar scale. At present, no good theory exists for identifying all possible mechanisms of genetic regulatory networks to attenuate the molecular noise to achieve regulatory ability or to amplify the molecular noise to randomize outcomes to the advantage of diversity. Therefore, the noise filtering of genetic regulatory network is an important topic for gene networks under intrinsic fluctuation and extrinsic noise.

Results

Based on stochastic dynamic regulation equation, the intrinsic fluctuation in reaction rates is modeled as a state-dependent stochastic process, which will influence the stability of gene regulatory network, especially, with low concentrations of reacting species. Then the mechanisms of genetic regulatory network to attenuate or amplify extrinsic fluctuation are revealed from the nonlinear stochastic filtering point of view. Furthermore, a simple measure of attenuation level or amplification level of extrinsic noise for genetic regulatory networks is also introduced by nonlinear robust filtering method. Based on the global linearization scheme, a convenient method is introduced to measure noise attenuation or amplification for each gene of the nonlinear stochastic regulatory network by solving a set of filtering problems, which correspond to a set of linearized stochastic regulatory networks. Finally, by the proposed methods, several simulation examples of genetic regulatory networks are given to measure their robust stability under intrinsic fluctuations, and to estimate the genes' attenuation and amplification levels under extrinsic noises.

Conclusion

In this study, a stochastic nonlinear dynamic model is developed for genetic regulatory networks under intrinsic fluctuation and extrinsic noise. By the method we proposed, we could determine the robust stability under intrinsic fluctuations and identify the genes that are significantly affected by extrinsic noises, which we call the weak structure of the network. This method will be potential for robust gene circuit design in future, on which a drug design could be based.

Stochastic transcription initiation: Time dependent transcription rates

The noise in the central process such as transcription, replication and translation of the genomic DNA is very important since it can directly affect the phenotypic and behavioral aspects of an organism as well as the entire cellular function. Here we develop a model on the transcription process based on the assumption that the initiation of the transcription is a stochastic event and the transcription rates may be time dependent random quantities. We derive the central measure properties i.e. mean and the variance, of the distribution of the transcription rates. Our results show that the Fano factor which is a measure of deviation from the Poisson distribution associated with the fluctuations in the number of mRNA molecules deviates from unity due to the randomness in the transcription rates. However when the RNA polymerase molecule searches for the promoter sequences on the DNA lattice by random jumps, the Fano factor approaches the Poisson limit as the jump size associated with the RNA polymerase increases. Since the jump size associated with dynamics of RNAP molecule is positively correlated with the degree of super coiling of DNA, we argue that the super coiled or close-packed structure of DNA might have evolved to keep the noises at the transcriptional level in a minimum.

A hidden Markov model for transcriptional regulation in single cells

We discuss several issues pertaining to the use of stochastic biochemical systems for modeling transcriptional regulation in single cells. By appropriately choosing the system state, we can model transcriptional regulation by a hidden Markov model (HMM). This opens the possibility of using well-known techniques for the statistical analysis and stochastic control of HMMs to mathematically and computationally study transcriptional regulation in single cells. Unfortunately, in all but a few simple cases, analytical characterization of the statistical behavior of the proposed HMM is not possible. Moreover, analysis by Monte Carlo simulation is computationally cumbersome. We discuss several techniques for approximating the HMM by one that is more tractable. We employ simulations, based on a biologically relevant transcriptional regulatory system, to show the relative merits and limitations of various approximation techniques and provide general guidelines for their use.

The role of log-normal dynamics in the evolution of biochemical pathways

The study of the scale-free topology in non-biological and biological networks and the dynamics that can explain this fascinating property of complex systems have captured the attention of the scientific community in the last years. Here, we analyze the biochemical pathways of three organisms (Methanococcus jannaschii, Escherichia coli, Saccharomyces cerevisiae) which are representatives of the main kingdoms Archaea, Bacteria and Eukaryotes during the course of the biological evolution. We can consider two complementary representations of the biochemical pathways: the enzymes network and the chemical compounds network. In this article, we propose a stochastic model that explains that the scale-free topology with exponent in the vicinity of gamma approximately 3/2 found across these three organisms is governed by the log-normal dynamics in the evolution of the enzymes network. Precisely, the fluctuations of the connectivity degree of enzymes in the biochemical pathways between evolutionary distant organisms follow the same conserved dynamical principle, which in the end is the origin of the stationary scale-free distribution observed among species, from Archaea to Eukaryotes. In particular, the log-normal dynamics guarantees the conservation of the scale-free distribution in evolving networks. Furthermore, the log-normal dynamics also gives a possible explanation for the restricted range of observed exponents gamma in the scale-free networks (i.e., gamma > or = 3/2). Finally, our model is also applied to the chemical compounds network of biochemical pathways and the Internet network.

Graded and binary responses in stochastic gene expression

Recently, several theoretical and experimental studies have been undertaken to probe the effect of stochasticity on gene expression (GE). In experiments, the GE response to an inducing signal in a cell, measured by the amount of mRNAs/proteins synthesized, is found to be either graded or binary. The latter type of response gives rise to a bimodal distribution in protein levels in an ensemble of cells. One possible origin of binary response is cellular bistability achieved through positive feedback or autoregulation. In this paper, we study a simple, stochastic model of GE and show that the origin of binary response lies exclusively in stochasticity. The transitions between the active and inactive states of the gene are random in nature. Graded and binary responses occur in the model depending on the relative stability of the activated and deactivated gene states with respect to that of mRNAs/proteins. The theoretical results on binary response provide a good description of the 'all-or-none' phenomenon observed in an eukaryotic system.

Noise in gene expression: origins, consequences, and control

Genetically identical cells and organisms exhibit remarkable diversity even when they have identical histories of environmental exposure. Noise, or variation, in the process of gene expression may contribute to this phenotypic variability. Recent studies suggest that this noise has multiple sources, including the stochastic or inherently random nature of the biochemical reactions of gene expression. In this review, we summarize noise terminology and comment on recent investigations into the sources, consequences, and control of noise in gene expression.

Delay-induced stochastic oscillations in gene regulation

The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale.

An evolutionary relationship between genetic variation and phenotypic fluctuation

The relevance of phenotype fluctuations among clones (i.e., organisms with identical genes) to evolution has recently been recognized both theoretically and experimentally. By considering the stability of the distributions of genetic variations and phenotype fluctuations, we derive a general inequality between the phenotype variance due to genetic differences and the intrinsic phenotype variance of clones. For a given mutation rate, an approximately linear relationship between the two is obtained which elucidates the consistency between the fundamental theorem of natural selection by Fisher and the evolutionary fluctuation-response relationship (fluctuation dissipation theorem) proposed recently. A general condition for the error catastrophe is also derived as the violation of the inequality, which sets up the limit to the speed of stable evolution. All of these theoretical results are confirmed by a numerical evolution experiment of a cell that consists of a catalytic reaction network. Based on the relationships proposed here, relevance of the phenotypic plasticity to evolution as well as the genetic assimilation is discussed.

Phosphatases modulate the bistable sporulation gene expression pattern in Bacillus subtilis

Summary Spore formation in the Gram-positive bacterium Bacillus subtilis is a last resort adaptive response to starvation. To initiate sporulation, the key regulator in this process, Spo0A, needs to be activated by the so-called phosphorelay. Within a sporulating culture of B. subtilis, some cells initiate this developmental program, while other cells do not. Therefore, initiation of sporulation appears to be a regulatory process with a bistable outcome. Using a single cell analytical approach, we show that the autostimulatory loop of spo0A is responsible for generating a bistable response resulting in phenotypic variation within the sporulating culture. It is demonstrated that the main function of RapA, a phosphorelay phosphatase, is to maintain the bistable sporulation gene expression. As rapA expression is quorum regulated, it follows that quorum sensing influences sporulation bistability. Deletion of spo0E, a phosphatase directly acting on Spo0A approximately P, resulted in abolishment of the bistable expression pattern. Artificial induction of a heterologous Rap phosphatase restored heterogeneity in a rapA or spo0E mutant. These results demonstrate that with external phosphatases, B. subtilis can use the phosphorelay as a tuner to modulate the bistable outcome of the sporulating culture. This shows that B. subtilis employs multiple pathways to maintain the bistable nature of a sporulating culture, stressing the physiological importance of this phenomenon.

Stripping Bacillus: ComK auto-stimulation is responsible for the bistable response in competence development

In Bacillus subtilis competence for genetic transformation develops only in a subpopulation of cells in an isogenic culture. The molecular mechanisms underlying this phenotypic heterogeneity are unknown. In this study, we stepwise simplify the signal transduction cascade leading to competence, yielding a strain devoid of all regulatory inputs for this process that have been identified so far. We demonstrate that auto-stimulation of ComK, the master regulator for competence development, is essential and in itself can be sufficient to generate a bistable expression pattern. We argue that transcriptional regulation determines the threshold of ComK to initiate the auto-stimulatory response, and that the basal level of ComK (in a wild-type strain governed by MecA-mediated proteolytic control) determines the fraction of cells that reach this threshold, and thus develop competence.

Stochasticity in gene expression: from theories to phenotypes

Genetically identical cells exposed to the same environmental conditions can show significant variation in molecular content and marked differences in phenotypic characteristics. This variability is linked to stochasticity in gene expression, which is generally viewed as having detrimental effects on cellular function with potential implications for disease. However, stochasticity in gene expression can also be advantageous. It can provide the flexibility needed by cells to adapt to fluctuating environments or respond to sudden stresses, and a mechanism by which population heterogeneity can be established during cellular differentiation and development.

Nonlinear software sensor for monitoring genetic regulation processes with noise and modeling errors

Nonlinear control techniques by means of a software sensor that are commonly used in chemical engineering could be also applied to genetic regulation processes. We provide here a realistic formulation of this procedure by introducing an additive white Gaussian noise, which is usually found in experimental data. Besides, we include model errors, meaning that we assume we do not know the nonlinear regulation function of the process. In order to illustrate this procedure, we employ the Goodwin dynamics of the concentrations [B. C. Goodwin, (Academic, New York, 1963)] in the simple form recently applied to single gene systems and some operon cases [H. De Jong, J. Comput. Biol. 9, 67 (2002)], which involves the dynamics of the mRNA, given protein and metabolite concentrations. Further, we present results for a three gene case in coregulated sets of transcription units as they occur in prokaryotes. However, instead of considering their full dynamics, we use only the data of the metabolites and a designed software sensor. We also show, more generally, that it is possible to rebuild the complete set of nonmeasured concentrations despite the uncertainties in the regulation function or, even more, in the case of not knowing the mRNA dynamics. In addition, the rebuilding of concentrations is not affected by the perturbation due to the additive white Gaussian noise and also we managed to filter the noisy output of the biological system.

Enhancement of cellular memory by reducing stochastic transitions

On induction of cell differentiation, distinct cell phenotypes are encoded by complex genetic networks. These networks can prevent the reversion of established phenotypes even in the presence of significant fluctuations. Here we explore the key parameters that determine the stability of cellular memory by using the yeast galactose-signalling network as a model system. This network contains multiple nested feedback loops. Of the two positive feedback loops, only the loop mediated by the cytoplasmic signal transducer Gal3p is able to generate two stable expression states with a persistent memory of previous galactose consumption states. The parallel loop mediated by the galactose transporter Gal2p only increases the expression difference between the two states. A negative feedback through the inhibitor Gal80p reduces the strength of the core positive feedback. Despite this, a constitutive increase in the Gal80p concentration tunes the system from having destabilized memory to having persistent memory. A model reveals that fluctuations are trapped more efficiently at higher Gal80p concentrations. Indeed, the rate at which single cells randomly switch back and forth between expression states was reduced. These observations provide a quantitative understanding of the stability and reversibility of cellular differentiation states.

Ubiquity of log-normal distributions in intra-cellular reaction dynamics

The discovery of two fundamental laws concerning cellular dynamics with recursive growth is reported. Firstly, the chemical abundances measured over many cells were found to obey a log-normal distribution and secondly, the relationship between the average and standard deviation of the abundances was found to be linear. The ubiquity of these laws was explored both theoretically and experimentally. By means of a model with a catalytic reaction network, the laws were shown to exist near a critical state with efficient self-reproduction. Additionally, by measuring distributions of fluorescent proteins in bacteria cells, the ubiquity of log-normal distribution of protein abundances was confirmed. Relevance of these findings to cellular function and biological plasticity is briefly discussed.

Stochastic simulations of the origins and implications of long-tailed distributions in gene expression

Gene expression noise results in protein number distributions ranging from long-tailed to Gaussian. We show how long-tailed distributions arise from a stochastic model of the constituent chemical reactions and suggest that, in conjunction with cooperative switches, they lead to more sensitive selection of a subpopulation of cells with high protein number than is possible with Gaussian distributions. Single-cell-tracking experiments are presented to validate some of the assumptions of the stochastic simulations. We also examine the effect of DNA looping on the shape of protein distributions. We further show that when switches are incorporated in the regulation of a gene via a feedback loop, the distributions can become bimodal. This might explain the bimodal distribution of certain morphogens during early embryogenesis.

Noise propagation in gene networks

Accurately predicting noise propagation in gene networks is crucial for understanding signal fidelity in natural networks and designing noise-tolerant gene circuits. To quantify how noise propagates through gene networks, we measured expression correlations between genes in single cells. We found that noise in a gene was determined by its intrinsic fluctuations, transmitted noise from upstream genes, and global noise affecting all genes. A model was developed that explains the complex behavior exhibited by the correlations and reveals the dominant noise sources. The model successfully predicts the correlations as the network is systematically perturbed. This approach provides a step toward understanding and manipulating noise propagation in more complex gene networks.

Stochastic fluctuations in gene expression far from equilibrium: Ω expansion and linear noise approximation

The Omega expansion of the master equation is used to investigate the intrinsic noise in an autoregulatory gene expression system. This Omega expansion provides a mesoscale description of the system and is used to analyze the effect of feedback regulation on intrinsic noise when the system state is far from equilibrium. Using the linear noise approximation, analytic results are obtained for a single gene system with linear feedback that is far from equilibrium. Additionally, analytic expressions are obtained for nonlinear systems near equilibrium. Simulations of such autoregulatory reaction schemes with nonlinear feedback show that during the approach to equilibrium the noise is not always reduced by the strength of the feedback. This is contrary to results seen in the equilibrium limit which show decreased noise with feedback strength. These results demonstrate that the behavior of linearized systems near equilibrium cannot be readily applied to systems far from equilibrium and highlight the need to explore nonequilibrium domains in mesoscopic systems.

Optimizing genetic circuits by global sensitivity analysis

Artificial genetic circuits are becoming important tools for controlling cellular behavior and studying molecular biosystems. To genetically optimize the properties of complex circuits in a practically feasible fashion, it is necessary to identify the best genes and/or their regulatory components as mutation targets to avoid the mutation experiments being wasted on ineffective regions, but this goal is generally not achievable by current methods. The Random Sampling-High Dimensional Model Representation (RS-HDMR) algorithm is employed in this work as a global sensitivity analysis technique to estimate the sensitivities of the circuit properties with respect to the circuit model parameters, such as rate constants, without knowing the precise parameter values. The sensitivity information can then guide the selection of the optimal mutation targets and thereby reduce the laboratory effort. As a proof of principle, the in vivo effects of 16 pairwise mutations on the properties of a genetic inverter were compared against the RS-HDMR predictions, and the algorithm not only showed good consistency with laboratory results but also revealed useful information, such as different optimal mutation targets for optimizing different circuit properties, not available from previous experiments and modeling.

Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations

Stochastic effects in biomolecular systems have now been recognized as a major physiologically and evolutionarily important factor in the development and function of many living organisms. Nevertheless, they are often thought of as providing only moderate refinements to the behaviors otherwise predicted by the classical deterministic system description. In this work we show by using both analytical and numerical investigation that at least in one ubiquitous class of (bio)chemical-reaction mechanisms, enzymatic futile cycles, the external noise may induce a bistable oscillatory (dynamic switching) behavior that is both quantitatively and qualitatively different from what is predicted or possible deterministically. We further demonstrate that the noise required to produce these distinct properties can itself be caused by a set of auxiliary chemical reactions, making it feasible for biological systems of sufficient complexity to generate such behavior internally. This new stochastic dynamics then serves to confer additional functional modalities on the enzymatic futile cycle mechanism that include stochastic amplification and signaling, the characteristics of which could be controlled by both the type and parameters of the driving noise. Hence, such noise-induced phenomena may, among other roles, potentially offer a novel type of control mechanism in pathways that contain these cycles and the like units. In particular, observations of endogenous or externally driven noise-induced dynamics in regulatory networks may thus provide additional insight into their topology, structure, and kinetics.

The evolution of signaling complexity suggests a mechanism for reducing the genomic search space in human association studies

The size complexity of the human genome has been traditionally viewed as an obstacle that frustrates efforts aimed at identifying the genetic correlates of complex human phenotypes. As such complex phenotypes are attributed to the combined action of numerous genomic loci, attempts to identify the underlying multi-locus interactions may produce a combinatorial sum of false positives that drown out the real signal. Faced with such grim prospects for successfully identifying the genetic basis of complex phenotypes, many geneticists simply disregard epistatic interactions altogether. However, the emerging picture from systems biology is that the cellular programs encoded by the genome utilize nested signaling hierarchies to integrate a number of loosely coupled, semiautonomous, and functionally distinct genetic networks. The current view of these modules is that connections encoding inter-module signaling are relatively sparse, while the gene-to-gene (protein-to-protein) interactions within a particular module are typically denser. We believe that each of these modules is encoded by a finite set of discontinuous, sequence-specific, genomic intervals that are functionally linked to association rules, which correlate directly to features in the environment. Furthermore, because these environmental association rules have evolved incrementally over time, we explore theoretical models of cellular evolution to better understand the role of evolution in genomic complexity. Specifically, we present a conceptual framework for (1) reducing genomic complexity by partitioning the genome into subsets composed of functionally distinct genetic modules and (2) improving the selection of coding region SNPs, which results in an increased probability of identifying functionally relevant SNPs. Additionally, we introduce the notion of 'genomic closure,' which provides a quantitative measure of how functionally insulated a specific genetic module might be from the influence of the rest of the genome. We suggest that the development and use of theoretical models can provide insight into the nature of biological systems and may lead to significant improvements in computational algorithms designed to reduce the complexity of the human genome.

Noisy signal amplification in ultrasensitive signal transduction

Because intracellular processes are inherently noisy, stochastic reactions process noisy signals in cellular signal transduction. One essential feature of biological signal transduction systems is the amplification of small changes in input signals. However, small random changes in the input signals could also be amplified, and the transduction reaction can also generate noise. Here, we show theoretically how the abrupt response of ultrasensitive signal-transduction reactions results in the generation of large inherent noise and the high amplification of input noise. The inherently generated noise propagates with amplification through intracellular molecular network. We discuss how the contribution of such transmitted noise can be shown experimentally. Our results imply that the switch-like behavior of signal transduction could be limited by noise; however, high amplification reaction could be advantageous to generate large noise, which would be essential to maintain behavioral variability.

Intrinsic noise, gene regulation and steady-state statistics in a two-gene network

The intrinsic noise in a two-gene network model is analysed. The technique of the Fokker-Planck approximation is used to investigate the statistics of noise when the system state is near a stable equilibrium. This is called also the steady-state statistics. The relative size of noise is measured by the Fano factor that is defined as the ratio of the variance to the mean. Our main result shows that in general, the noise control in a two-gene network might be a very complicated process, but for the repressor-repressor system that is a very important case in investigating the genetic switch, the relative size of noise, i.e. the Fano factor, must be bigger than one for both the repressor proteins.

Modeling genetic networks from clonal analysis

In this report a systematic approach is used to determine the approximate genetic network and robust dependencies underlying differentiation. The data considered is in the form of a binary matrix and represent the expression of the nine genes across the 99 colonies. The report is divided into two parts: the first part identifies significant pair-wise dependencies from the given binary matrix using linear correlation and mutual information. A new method is proposed to determine statistically significant dependencies estimated using the mutual information measure. In the second, a Bayesian approach is used to obtain an approximate description (equivalence class) of network structures. The robustness of linear correlation, mutual information and the equivalence class of networks is investigated with perturbation and decreasing colony number. Perturbation of the data was achieved by generating bootstrap realizations. The results are refined with biological knowledge. It was found that certain dependencies in the network are immune to perturbation and decreasing colony number and may represent robust features, inherent in the differentiation program of osteoblast progenitor cells. The methods to be discussed are generic in nature and not restricted to the experimental paradigm addressed in this study.

The sonic hedgehog signaling system as a bistable genetic switch

Sonic hedgehog (Shh) controls critical cellular decisions between distinct fates in many systems, particularly in stem cells. The Shh network functions as a genetic switch, and we have theoretically and computationally analyzed how its structure can endow it with the ability to switch fate choices at a threshold Shh concentration. The network is composed of a positive transcriptional feedback loop embedded within a negative signaling feedback loop. Specifically, positive feedback by the transcription factor Gli, which upregulates its own expression, leads to a switch that can adopt two distinct states as a function of Shh. However, Gli also upregulates the signaling repressor Patched, negative feedback that reins in the strong Gli autoregulatory loop. Mutations that have been associated with cancer are predicted to yield an irreversible switch to a high Gli state. Finally, stochastic simulation reveals the negative Patched feedback loop serves a critical function of dampening Gli fluctuations to reduce spontaneous state switching and preserve the network's robust, switch-like behavior. Tightly linked positive and negative feedback loops are present in many signaling systems, and the Shh system is therefore likely representative of a large set of gene regulation networks that control stem cell fate throughout development and into adulthood.

Self-consistent proteomic field theory of stochastic gene switches

We present a self-consistent field approximation approach to the problem of the genetic switch composed of two mutually repressing/activating genes. The protein and DNA state dynamics are treated stochastically and on an equal footing. In this approach the mean influence of the proteomic cloud created by one gene on the action of another is self-consistently computed. Within this approximation a broad range of stochastic genetic switches may be solved exactly in terms of finding the probability distribution and its moments. A much larger class of problems, such as genetic networks and cascades, also remain exactly solvable with this approximation. We discuss, in depth, certain specific types of basic switches used by biological systems and compare their behavior to the expectation for a deterministic switch.

Fluctuations-induced switch in the gene transcriptional regulatory system

Based on the kinetic model of genetic regulation system proposed by Am. J. Physiol. 274, c531 (1998)], the effects of fluctuations in the degradation reaction rate and the synthesis reaction rate of the transcription factor have been investigated through numerical computation and analysis theory. In the case of uncorrelated noises, it is shown that only the fluctuation of degradation reaction rate can induce a switch process, and the mean first passage time (MFPT) from the high concentration state to the low concentration one is decreased when the noise intensity of degradation reaction rate is increased. In the case of correlations between noises, a switch process can also be induced by the cross-correlation intensity between noises and by the fluctuation of the synthesis reaction rate in the genetic regulatory system. It is found that, under large cross-correlation intensity, a successive switch process (i.e., "on" --> "off" --> "on," which we call the reentrance transition or twice switch ) occurs with an increase of noise intensities, and a critical noise intensity exists at which the MFPT of the switch process is the largest. While the system is initially in the high concentration state with an increase of the cross correlation, the stationary probability distribution (SPD) of the transcription factor activator monomer concentration at the low concentration state is increased, yet the MFPT is increased due to the decreasing of the SPD of the transient states between the two steady stable states.

Bifurcation analysis of a model of the budding yeast cell cycle

We study the bifurcations of a set of nine nonlinear ordinary differential equations that describe regulation of the cyclin-dependent kinase that triggers DNA synthesis and mitosis in the budding yeast, Saccharomyces cerevisiae. We show that Clb2-dependent kinase exhibits bistability (stable steady states of high or low kinase activity). The transition from low to high Clb2-dependent kinase activity is driven by transient activation of Cln2-dependent kinase, and the reverse transition is driven by transient activation of the Clb2 degradation machinery. We show that a four-variable model retains the main features of the nine-variable model. In a three-variable model exhibiting birhythmicity (two stable oscillatory states), we explore possible effects of extrinsic fluctuations on cell cycle progression.

Constructive effects of fluctuations in genetic and biochemical regulatory systems

Biochemical and genetic regulatory systems that involve low concentrations of molecules are inherently noisy. This intrinsic stochasticity has received considerable interest recently, leading to new insights about the sources and consequences of noise in complex systems of genetic regulation. However, most prior work was devoted to the reduction of fluctuation and the robustness of cellular function with respect to intrinsic noise. Here, we focus on several scenarios in which the inherent molecular fluctuations are not merely a nuisance, but act constructively and bring about qualitative changes in the dynamics of the system. It will be demonstrated that in many typical situations biochemical and genetic regulatory systems may utilize intrinsic noise to their advantage.

Noise, delays, robustness, canalization and all that

A biological system such as a developing embryo can withstand many perturbations. What is the basis of this robustness both against noise and mutation? Recent advances in modeling may throw new light on this old problem. First, recent theoretical and experimental work clearly demonstrates the importance of noise and time delays for the proper functioning of genetic networks: noise and delays are simply part of the normal operating constraints. By contrast, sweeping statements have been made recently about a so-called 'robustness' of biological processes, based on work that neglects noise and delays completely. I submit that studying the stability of complex biological systems with such omissions is an unnecessary, inadequate and potentially disastrous simplification. I review the existing alternatives and propose using them to construct a modeling framework that overcomes all serious limitations.

Multivariate analysis of noise in genetic regulatory networks

Stochasticity is an intrinsic property of genetic regulatory networks due to the low copy numbers of the major molecular species, such as, DNA, mRNA, and regulatory proteins. Therefore, investigation of the mechanisms that reduce the stochastic noise is essential in understanding the reproducible behaviors of real organisms and is also a key to design synthetic genetic regulatory networks that can reliably work. We use an analytical and systematic method, the linear noise approximation of the chemical master equation along with the decoupling of a stoichiometric matrix. In the analysis of fluctuations of multiple molecular species, the covariance is an important measure of noise. However, usually the representation of a covariance matrix in the natural coordinate system, i.e. the copy numbers of the molecular species, is intractably complicated because reactions change copy numbers of more than one molecular species simultaneously. Decoupling of a stoichiometric matrix, which is a transformation of variables, significantly simplifies the representation of a covariance matrix and elucidates the mechanisms behind the observed fluctuations in the copy numbers. We apply our method to three types of fundamental genetic regulatory networks, that is, a single-gene autoregulatory network, a two-gene autoregulatory network, and a mutually repressive network. We have found that there are multiple noise components differently originating. Each noise component produces fluctuation in the characteristic direction. The resulting fluctuations in the copy numbers of the molecular species are the sum of these fluctuations. In the examples, the limitation of the negative feedback in noise reduction and the trade-off of fluctuations in multiple molecular species are clearly explained. The analytical representations show the full parameter dependence. Additionally, the validity of our method is tested by stochastic simulations.