Frequency selectivity, multistability, and oscillations emerge from models of genetic regulatory systems

Positive receptor feedback during lineage commitment can generate ultrasensitivity to ligand and confer robustness to a bistable switch

Cytokines and lineage-specific transcription factors are critical molecular effectors for terminal differentiation during hematopoiesis. Intrinsic transcription factor activity is often believed to drive commitment and differentiation, whereas cytokine receptor signals have been implicated in the regulation of cell proliferation, survival, and differentiation. In erythropoiesis, recent experimental findings provide direct evidence that erythropoietin (Epo) can generate commitment cues via the erythropoietin receptor (EpoR); specifically, EpoR signaling leads to activation of the transcription factor GATA-1, which then triggers transcription of erythrocyte-specific genes. In particular, activated GATA-1 induces two positive feedback loops in the system through the enhanced expression of both inactive GATA-1 and EpoR, the latter of which is externally regulatable by Epo. Based upon this network architecture, we present a mathematical model of GATA-1 activation by EpoR, which bidirectionally links a lineage-specific receptor and transcription factor. Our deterministic model offers insight into stimulus-response relationships between Epo and several downstream effectors. In addition to the survival signals that EpoR provides, steady-state analysis of our model suggests that receptor upregulation during lineage commitment can also generate ultrasensitivity to Epo and bistability in GATA-1 activity. These system-level properties can induce a switch-like characteristic during differentiation and provide robustness to the mature state. The topology also suggests a novel mechanism for achieving robust bistability in a purely deterministic manner without molecular cooperativity. The analytical solution of a generalized, minimal model is provided and the significance of each of the two positive feedback loops is elucidated through bifurcation analysis. This network topology, or variations thereof, may link other receptor-transcription factor pairs and may therefore be of general relevance in cellular decision-making.

Monotone bifurcation graphs

In recent work by Angeli and the authors, it was shown that the stability and global behaviour of strongly monotone dynamical systems may be profitably studied using a technique that involves feedback decompositions into 'well-behaved' subsystems. The present paper generalizes the approach, so that it applies to a far larger class of systems. As an illustration, the techniques are used in the analysis of a nine-variable autoregulatory transcription network. Also, extensions to delay and reaction diffusion systems are considered.

Dynamic properties of a delayed protein cross talk model

In this paper we investigate how the inclusion of time delay alters the dynamical properties of the Jacob-Monod model, describing the control of the beta-galactosidase synthesis by the lac repressor protein in E. coli. The consequences of a time delay on the dynamics of this system are analysed using Hopf's theorem and Lyapunov-Andronov's theory applied to the original mathematical model and to an approximated version. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations. A critical value of time delay, which depends on the values of the model parameters, is analytically established. Around this critical value, the properties of the system change drastically, allowing under certain conditions the emergence of stable limit cycles, that is self-sustained oscillations. In addition, the features of the end product repression play an essential role in the characterisation of these limit cycles: if cooperativity is considered in the end product repression, time delay higher than the mentioned critical value induce differentiated responses during the oscillations, provoking cycles of all-or-nothing response in the concentration of the species.

Noise-induced switches in network systems of the genetic toggle switch

Background

Bistability, the capacity to achieve two distinct stable steady states in response to a set of external stimuli, arises within biological systems ranging from the λ phage switch in bacteria to cellular signal transduction pathways in mammalian cells. On the other hand, more and more experimental evidence in the form of bimodal population distribution has indicated that noise plays a very important role in the switching of bistable systems. However, the physiological mechanism underling noise-induced switching behaviors remains to be fully understood.

Results

In this paper, we investigate the effect of noises on switching in single and coupled genetic toggle switch systems in Escherichia coli. In the case of the single toggle switch, we show that the multiplicative noises resulting from stochastic fluctuations in degradation rates can induce switching. In the case of the toggle switches interfaced by a quorum-sensing signaling pathway, we find that stochastic fluctuations in degradation rates inside cells, i.e., intracellular noises, can induce synchronized switching, whereas the extracellular noise additive to the common medium can not only entrain all the individual systems to switch in a synchronous manner but also enhance this ordering behavior efficiently, leading a robust collective rhythm in this interacting system.

Conclusion

These insights on the effect of noises would be beneficial to understanding the basic mechanism of how living systems optimally facilitate to function under various fluctuated environments.

A theoretical study on activation of transcription factor modulated by intracellular Ca2+ oscillations

This work presents both deterministic and stochastic models of genetic expression modulated by intracellular calcium (Ca2+) oscillations, based on macroscopic differential equations and chemical Langevin equations, respectively. In deterministic case, the oscillations of intracellular Ca2+ decrease the effective Ca2+ threshold for the activation of transcriptional activator (TF-A). The average activation of TF-A increases with the increase of the average amplitude of intracellular Ca2+ oscillations, but decreases with the increase of the period of intracellular Ca2+ oscillations, which are qualitatively consistent with the experimental results on the gene expression in lymphocytes. In stochastic case, it is found that a large internal fluctuation of the biochemical reaction can enhance gene expression efficiency specifically at a low level of external stimulations or at a small rate of TF-A dimer phosphorylation activated by Ca2+, which reduces the threshold of the average intracellular Ca2+ concentration for gene expression.

Dynamics of a minimal model of interlocked positive and negative feedback loops of transcriptional regulation by cAMP-response element binding proteins

cAMP-response element binding (CREB) proteins are involved in transcriptional regulation in a number of cellular processes (e.g., neural plasticity and circadian rhythms). The CREB family contains activators and repressors that may interact through positive and negative feedback loops. These loops can be generated by auto- and cross-regulation of expression of CREB proteins, via CRE elements in or near their genes. Experiments suggest that such feedback loops may operate in several systems (e.g., Aplysia and rat). To understand the functional implications of such feedback loops, which are interlocked via cross-regulation of transcription, a minimal model with a positive and negative loop was developed and investigated using bifurcation analysis. Bifurcation analysis revealed diverse nonlinear dynamics (e.g., bistability and oscillations). The stability of steady states or oscillations could be changed by time delays in the synthesis of the activator (CREB1) or the repressor (CREB2). Investigation of stochastic fluctuations due to small numbers of molecules of CREB1 and CREB2 revealed a bimodal distribution of CREB molecules in the bistability region. The robustness of the stable HIGH and LOW states of CREB expression to stochastic noise differs, and a critical number of molecules was required to sustain the HIGH state for days or longer. Increasing positive feedback or decreasing negative feedback also increased the lifetime of the HIGH state, and persistence of this state may correlate with long-term memory formation. A critical number of molecules was also required to sustain robust oscillations of CREB expression. If a steady state was near a deterministic Hopf bifurcation point, stochastic resonance could induce oscillations. This comparative analysis of deterministic and stochastic dynamics not only provides insights into the possible dynamics of CREB regulatory motifs, but also demonstrates a framework for understanding other regulatory processes with similar network architecture.

Intracellular delay limits cyclic changes in gene expression

Based on previously published experimental observations and mathematical models for Hes1, p53 and NF-kappaB gene expression, we improve these models through a distributed delay formulation of the time lag between transcription factor binding and mRNA production. This description of natural variability for delays introduces a transition from a stable steady state to limit cycle oscillations and then a second transition back to a stable steady state which has not been observed in previously published models. We demonstrate our approach for two models. The first model describes Hes1 autorepression with equations for Hes1 mRNA production and Hes1 protein translation. The second model describes Hes1 repression by the protein complex Gro/TLE1/Hes1, where Gro/TLE1 is activated by Hes1 phosphorylation. Finally, we discuss our analytical and numerical results in relation to experimental data.

Synchronization in coupled cells with activator-inhibitor pathways

The functional dynamics exhibited by cell collectives are fascinating examples of robust, synchronized, collective behavior in spatially extended biological systems. To investigate the roles of local cellular dynamics and interaction strength in the spatiotemporal dynamics of cell collectives of different sizes, we study a model system consisting of a ring of coupled cells incorporating a three-step biochemical pathway of regulated activator-inhibitor reactions. The isolated individual cells display very complex dynamics as a result of the nonlinear interactions common in cellular processes. On coupling the cells to nearest neighbors, through diffusion of the pathway end product, the ring of cells yields a host of interesting and unusual dynamical features such as, suppression of chaos, phase synchronization, traveling waves, and intermittency, for varying interaction strengths and system sizes. But robust complete synchronization can be induced in these coupled cells with a small degree of random coupling among them even where regular coupling yielded only intermittent synchronization. Our studies indicate that robustness in synchronized functional dynamics in tissues and cell populations in nature can be ensured by a few transient random connections among the cells. Such connections are being discovered only recently in real cellular systems.

Optimal transition paths of stochastic chemical kinetic systems

We present a new framework for finding the optimal transition paths of metastable stochastic chemical kinetic systems with large system sizes. The optimal transition paths are identified, in terms of reaction advancement coordinates, to be the most probable paths according to large deviation theory for the limiting dynamics governed by stochastic differential equations. Dynamical equations for the optimal transition paths are obtained using the variational principle. A multiscale minimum action method is proposed as a numerical scheme to solve the optimal transition paths. Applications to the toggle switch model are presented.

Stochastic kinetics description of a simple transcription model

We study a stochastic model of transcription kinetics in order to characterize the distributions of transcriptional delay and of elongation rates. Transcriptional delay is the time which elapses between the binding of RNA polymerase to a promoter sequence and its dissociation from the DNA template strand with consequent release of the transcript. Transcription elongation is the process by which the RNA polymerase slides along the template strand. The model considers a DNA template strand with one promoter site and n nucleotide sites, and five types of reaction processes, which we think are key ones in transcription. The chemical master equation is a set of ordinary differential equations in 3(n) variables, where n is the number of bases in the template. This model is too huge to be handled if n is large. We manage to get a reduced Markov model which has only 2n independent variables and can well approximate the original dynamics. We obtain a number of analytical and numerical results for this model, including delay and transcript elongation rate distributions. Recent studies of single-RNA polymerase transcription by using optical-trapping techniques raise an issue of whether the elongation rates measured in a population are heterogeneous or not. Our model implies that in the cases studied, different RNA polymerase molecules move at different characteristic rates along the template strand. We also discuss the implications of this work for the mathematical modeling of genetic regulatory circuits.

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.

Effect of the MAPK cascade structure, nuclear translocation and regulation of transcription factors on gene expression

The mitogen activated protein kinase (MAP kinase) cascade system represents a highly conserved prototype of signal transduction by enzyme cascades. One of the best-studied properties of the MAPK system is its ability to convert graded input stimulus to switch-like all-or-none responses. Previous theoretical studies have centered on quantifying dual phosphorylated MAPK as a final output response and have not incorporated its influence on the regulation of gene expression. The main objective of the current work is to understand the regulatory effect of positive feedback loop embedded in the MAPK cascade, nuclear translocation of active MAPK, phosphorylation and activation of nuclear target proteins on the regulation of specific gene expression. To achieve this objective, we have simulated the MAPK cascade system, which resembles Hog1p activation pathway in yeast, at steady state. Thus, the input signal to the MAPK system is correlated with gene expression as a final system-level output response. The steady state simulation results suggest that other than regulating the signal propagation through cascades, the nuclear translocation of activated MAPK and subsequent regulation of gene expression represent one of the key modes to control the threshold level of response. This work proposes that, it is essential to consider the compartmental distributions of signaling species and the corresponding regulatory mechanisms of gene expression to study the system-level performance of signaling modules such as the MAPK cascade. Such an analysis will relate the extracellular cues to the final phenotypic response by capturing the mechanistic details of the signaling pathway.

New directions in incidence-dose modeling

Many cellular responses are quantal; that is, they either take place or they do not. Examples of "either-or" responses include cell replication, differentiation and apoptosis. Surprisingly, induction of suites of genes and coordinated phenotypic changes in cells are also often quantal, where embedded molecular circuitry creates on-off switches. Mechanistic incidence-dose (ID) models need to account for the quantal characteristics of cellular switches that contribute, in turn, to dose thresholds and to the incidence of biological responses in individuals. Interdisciplinary systems biology approaches create mechanistic ID models based on: (i) detailed knowledge of the cellular circuitry controlling signal transduction; (ii) evolving biological modeling tools describing cellular circuits and their perturbations by chemicals and (iii) high throughput, high coverage "omic" screens for examining cell signaling pathways and biological responses. These interdisciplinary approaches should produce novel, quantitative ID models for biological responses and greatly improve the biological basis of safety and risk assessments.

The use of oscillatory signals in the study of genetic networks

The structure of a genetic network is uncovered by studying its response to external stimuli (input signals). We present a theory of propagation of an input signal through a linear stochastic genetic network. We found that there are important advantages in using oscillatory signals over step or impulse signals and that the system may enter into a pure fluctuation resonance for a specific input frequency.

Inference of signaling and gene regulatory networks by steady-state perturbation experiments: structure and accuracy

One of the fundamental problems of cell biology is the understanding of complex regulatory networks. Such networks are ubiquitous in cells and knowledge of their properties is essential for the understanding of cellular behavior. In earlier work (Kholodenko et al. (PNAS 99: 12841), it was shown how the structure of biological networks can be quantified from experimental measurements of steady-state concentrations of key intermediates as a result of perturbations using a simple algorithm called "unravelling". Here, we study the effect of experimental uncertainty on the accuracy of the inferred structure (i.e. whether interactions are excitatory or inhibitory) of the networks determined using the unravelling algorithm. We show that the accuracy of the network structure depends not only on the noise level but on the strength of the interactions within the network. In particular, both very small and very large values of the connection strengths lead to large uncertainty in the inferred network. We describe a powerful geometric tool for the intuitive understanding of the effect of experimental error on the qualitative accuracy of the inferred network. In addition, we show that the use of additional data beyond that needed to minimally constrain the network not only improves the accuracy of the inferred network, but also may allow the detection of situations in which the initial assumptions of unravelling with respect to the network and the perturbations have been violated. Our ideas are illustrated using the mitogen-activated protein kinase (MAPK) signaling network as an example.

Control of internal and external noise in genetic regulatory networks

Positive and negative feedback loops, for example, where a protein regulates its own transcription, play an important role in many genetic regulatory networks. Such systems will be subject to internal noise, which occurs due to the small number of molecules taking part in some reactions. This paper examines the effect of feedback loops on noise levels. Error growth techniques from nonlinear dynamics are used to estimate the variance of a system around a steady-state attractor. It is shown that variablity due to intrinsic stochasticity is directly linked to the stability of the steady state, and therefore to the system's resistance to external perturbations. The methods are demonstrated for a number of simple systems, including a genetic switch with homo-dimerizing regulatory protein, and an oscillator.

Continuous-time identification of gene expression models

One objective of systems biology is to create predictive, quantitative models of the transcriptional regulation networks that govern numerous cellular processes. Gene expression measurements, as provided by microarrays, are commonly used in studies that attempt to infer the regulation underlying these processes. At present, most gene expression models that have been derived from microarray data are based in discrete-time, which have limited applicability to common biological data sets, and may impede the integration of gene expression models with other models of biological processes that are formulated as ordinary differential equations (ODEs). To overcome these difficulties, a continuous-time approach for process identification to identify gene expression models based in ODEs was developed. The approach utilizes the modulating functions method of parameter identification. The method was applied to three simulated systems: (1) a linear gene expression model, (2) an autoregulatory gene expression model, and (3) simulated microarray data from a nonlinear transcriptional network. In general, the approach was well suited for identifying models of gene expression dynamics, capable of accurately identifying parameters for small numbers of data samples in the presence of modest experimental noise. Additionally, numerous insights about gene expression modeling were revealed by the case studies.

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.

Feedback loops in the signal paths controlling gene expression

Theoretical and empirical studies support the concepts suggesting emission of electromagnetic fields by living cells. Experiments performed in our laboratory indicate, in addition, tight links between attributes of these fields and the phase of cell division cycle, i.e., processes governed by gene transcription. This work supplements the now quite extensive literature concerning gene networks in that it identifies the well-known feedback mechanisms in exact topological and quantitative terms. Special diagrammatic apparatus is employed for this purpose. The simple system comprising one gene and one transcription factor (binding as a dimer to the gene regulatory region), controlled via phosphorylation process by external signal, is analyzed to demonstrate the approach. The diagram representative of the system's dynamics contains a feedback loop; the so-called transmission function of this feature of diagram topology is shown to control the dependences of gene product concentration on the stimulus intensity.

Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems

It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex. Here, we show that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this open-loop, feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable mitogen-activated protein kinase cascade.

Multistability in the lactose utilization network of Escherichia coli

Multistability, the capacity to achieve multiple internal states in response to a single set of external inputs, is the defining characteristic of a switch. Biological switches are essential for the determination of cell fate in multicellular organisms, the regulation of cell-cycle oscillations during mitosis and the maintenance of epigenetic traits in microbes. The multistability of several natural and synthetic systems has been attributed to positive feedback loops in their regulatory networks. However, feedback alone does not guarantee multistability. The phase diagram of a multistable system, a concise description of internal states as key parameters are varied, reveals the conditions required to produce a functional switch. Here we present the phase diagram of the bistable lactose utilization network of Escherichia coli. We use this phase diagram, coupled with a mathematical model of the network, to quantitatively investigate processes such as sugar uptake and transcriptional regulation in vivo. We then show how the hysteretic response of the wild-type system can be converted to an ultrasensitive graded response. The phase diagram thus serves as a sensitive probe of molecular interactions and as a powerful tool for rational network design.

Importance of input perturbations and stochastic gene expression in the reverse engineering of genetic regulatory networks: insights from an identifiability analysis of an in silico network

Gene expression profiles are an increasingly common data source that can yield insights into the functions of cells at a system-wide level. The present work considers the limitations in information content of gene expression data for reverse engineering regulatory networks. An in silico genetic regulatory network was constructed for this purpose. Using the in silico network, a formal identifiability analysis was performed that considered the accuracy with which the parameters in the network could be estimated using gene expression data and prior structural knowledge (which transcription factors regulate which genes) as a function of the input perturbation and stochastic gene expression. The analysis yielded experimentally relevant results. It was observed that, in addition to prior structural knowledge, prior knowledge of kinetic parameters, particularly mRNA degradation rate constants, was necessary for the network to be identifiable. Also, with the exception of cases where the noise due to stochastic gene expression was high, complex perturbations were more favorable for identifying the network than simple ones. Although the results may be specific to the network considered, the present study provides a framework for posing similar questions in other systems.

Prediction and measurement of an autoregulatory genetic module

The deduction of phenotypic cellular responses from the structure and behavior of complex gene regulatory networks is one of the defining challenges of systems biology. This goal will require a quantitative understanding of the modular components that constitute such networks. We pursued an integrated approach, combining theory and experiment, to analyze and describe the dynamics of an isolated genetic module, an in vivo autoregulatory gene network. As predicted by the model, temperature-induced protein destabilization led to the existence of two expression states, thus elucidating the trademark bistability of the positive feedback-network architecture. After sweeping the temperature, observed population distributions and coefficients of variation were in quantitative agreement with those predicted by a stochastic version of the model. Because model fluctuations originated from small molecule-number effects, the experimental validation underscores the importance of internal noise in gene expression. This work demonstrates that isolated gene networks, coupled with proper quantitative descriptions, can elucidate key properties of functional genetic modules. Such an approach could lead to the modular dissection of naturally occurring gene regulatory networks, the deduction of cellular processes such as differentiation, and the development of engineered cellular control.

Motifs, modules and games in bacteria

Global explorations of regulatory network dynamics, organization and evolution have become tractable thanks to high-throughput sequencing and molecular measurement of bacterial physiology. From these, a nascent conceptual framework is developing, that views the principles of regulation in term of motifs, modules and games. Motifs are small, repeated, and conserved biological units ranging from molecular domains to small reaction networks. They are arranged into functional modules, genetically dissectible cellular functions such as the cell cycle, or different stress responses. The dynamical functioning of modules defines the organism's strategy to survive in a game, pitting cell against cell, and cell against environment. Placing pathway structure and dynamics into an evolutionary context begins to allow discrimination between those physical and molecular features that particularize a species to its surroundings, and those that provide core physiological function. This approach promises to generate a higher level understanding of cellular design, pathway evolution and cellular bioengineering.

Genetic "code": representations and dynamical models of genetic components and networks

Dynamical modeling of biological systems is becoming increasingly widespread as people attempt to grasp biological phenomena in their full complexity and make sense of an accelerating stream of experimental data. We review a number of recent modeling studies that focus on systems specifically involving gene expression and regulation. These systems include bacterial metabolic operons and phase-variable piliation, bacteriophages T7 and lambda, and interacting networks of eukaryotic developmental genes. A wide range of conceptual and mathematical representations of genetic components and phenomena appears in these works. We discuss these representations in depth and give an overview of the tools currently available for creating and exploring dynamical models. We argue that for modeling to realize its full potential as a mainstream biological research technique the tools must become more general and flexible, and formal, standardized representations of biological knowledge and data must be developed.

Untangling the wires: a strategy to trace functional interactions in signaling and gene networks

Emerging technologies have enabled the acquisition of large genomics and proteomics data sets. However, current methodologies for analysis do not permit interpretation of the data in ways that unravel cellular networking. We propose a quantitative method for determining functional interactions in cellular signaling and gene networks. It can be used to explore cell systems at a mechanistic level or applied within a "modular" framework, which dramatically decreases the number of variables to be assayed. This method is based on a mathematical derivation that demonstrates how the topology and strength of network connections can be retrieved from experimentally measured network responses to successive perturbations of all modules. Importantly, our analysis can reveal functional interactions even when the components of the system are not all known. Under these circumstances, some connections retrieved by the analysis will not be direct but correspond to the interaction routes through unidentified elements. The method is tested and illustrated by using computer-generated responses of a modeled mitogen-activated protein kinase cascade and gene network.

Localized neuronal outgrowth induced by long-term sensitization training in aplysia

Biophysical, biochemical, and morphological studies have implicated sensory neurons as key sites of plasticity in the formation and retention of the memory of long-term sensitization in Aplysia californica. This study examined the effects of different sensitization training protocols on the structure of sensory neurons mediating the tail-siphon withdrawal reflex. A 4 d training period produced a robust localized outgrowth in these sensory neurons observed 24 hr after the end of training. These changes are consistent with previous results in siphon sensory neurons (Bailey and Chen, 1988a). In contrast, 1 d of sensitization training, which has been shown to effectively induce long-term behavioral sensitization and synaptic facilitation (Frost et al., 1985; Cleary et al., 1998), is not associated with morphological changes in tail sensory neurons at either 24 hr or 4 d after training. Similarly, a single treatment with the growth factor TGF-beta, which also induced facilitation, did not alter sensory neuron morphology. The different effectiveness of the two protocols was not simply a reflection of the number of stimuli presented, because a 1 d massed training protocol did not produce sensitization 24 hr after training, nor did it induce neuronal outgrowth. These results suggest that extensive sensitization training is required to induce neuronal outgrowth in tail sensory neurons, indicating that the memory of long-term sensitization induced by 1 d of training is mechanistically different from that induced by 4 d of training. Moreover, the induction of a form of long-term sensitization associated with neuronal outgrowth does not appear to be a function of the amount of stimulation but does appear to be dependent on the temporal spacing of the stimulation over multiple days.

Control motifs for intracellular regulatory networks

A number of technological innovations are yielding unprecedented data on the networks of biochemical, genetic, and biophysical reactions that underlie cellular behavior and failure. These networks are composed of hundreds to thousands of chemical species and structures, interacting via nonlinear and possibly stochastic physical processes. A central goal of modern biology is to optimally use the data on these networks to understand how their design leads to the observed cellular behaviors and failures. Ultimately, this knowledge should enable cellular engineers to redesign cellular processes to meet industrial needs (such as optimal natural product synthesis), aid in choosing the most effective targets for pharmaceuticals, and tailor treatment for individual genotypes. The size and complexity of these networks and the inevitable lack of complete data, however, makes reaching these goals extremely difficult. If it proves possible to modularize these networks into functional subnetworks, then these smaller networks may be amenable to direct analysis and might serve as regulatory motifs. These motifs, recurring elements of control, may help to deduce the structure and function of partially known networks and form the basis for fulfilling the goals described above. A number of approaches to identifying and analyzing control motifs in intracellular networks are reviewed.

Synchronizing genetic relaxation oscillators by intercell signaling

The ability to design and construct synthetic gene regulatory networks offers the prospect of studying issues related to cellular function in a simplified context; such networks also have many potential applications in biotechnology. A synthetic network exhibiting oscillatory behavior has recently been constructed [Elowitz, M. B. & Leibler, S. (2000) Nature (London) 403, 335-338]. It has also been shown that a natural bacterial quorum-sensing mechanism can be used in a synthetic system to communicate a signal between two populations of cells, such that receipt of the signal causes expression of a target gene [Weiss, R. & Knight, T. F. (2000) in DNA6: Sixth International Meeting on DNA-Based Computers, June 13-17, 2000, Leiden, The Netherlands]. We propose a synthetic gene network in Escherichia coli which combines these two features: the system acts as a relaxation oscillator and uses an intercell signaling mechanism to couple the oscillators and induce synchronous oscillations. We model the system and show that the proposed coupling scheme does lead to synchronous behavior across a population of cells. We provide an analytical treatment of the synchronization process, the dominant mechanism of which is "fast threshold modulation."

Interaction between amount and pattern of training in the induction of intermediate- and long-term memory for sensitization in aplysia

In Aplysia, three distinct phases of memory for sensitization can be dissociated based on their temporal and molecular features. A single training trial induces short-term memory (STM, lasting <30 min), whereas five trials delivered at 15-min intervals induces both intermediate-term memory (ITM, lasting >90 min) and long-term memory (LTM, lasting >24 h). Here, we explore the interaction of amount and pattern of training in establishing ITM and LTM by examining memory for sensitization after different numbers of trials (each trial = one tail shock) and different patterns of training (massed vs. spaced). Under spaced training patterns, two trials produced STM exclusively, whereas four or five trials each produced both ITM and LTM. Three spaced trials failed to induce LTM but did produce an early decaying form of ITM (E-ITM) that was significantly shorter and weaker in magnitude than the late-decaying ITM (L-ITM) observed after four to five trials. In addition, E-ITM was induced after three trials with both massed and spaced patterns of training. However, L-ITM and LTM after four to five trials require spaced training: Four or five massed trials failed to induce LTM and produced only E-ITM. Collectively, our results indicate that in addition to three identified phases of memory for sensitization--STM, ITM, and LTM--a unique temporal profile of memory, E-ITM, is revealed by varying either the amount or pattern of training.

Bistability in the JNK cascade

BACKGROUND

Important signaling properties, like adaptation, oscillations, and bistability, can emerge at the level of relatively simple systems of signaling proteins. Here, we have examined the quantitative properties of one well-studied signaling system, the JNK cascade. We experimentally assessed the response of JNK to a physiological stimulus (progesterone) and a pathological stress (hyperosmolar sorbitol) in Xenopus laevis oocytes, a cell type that is well-suited to the quantitative analysis of cell signaling. Our aim was to determine whether JNK responses are graded (Michaelian) in character; ultrasensitive in character, resembling the responses of cooperative enzymes; or bistable and all-or-none in character.

RESULTS

The responses of JNK to both progesterone and sorbitol were found to be essentially all-or-none. Individual oocytes had either very high or very low JNK activities, with few oocytes possessing intermediate levels of JNK activity. Moreover, JNK activation was autocatalytic, indicating that the JNK cascade is either embedded in or downstream of a positive feedback loop. JNK also exhibited hysteresis, a form of biochemical memory, in its response to sorbitol. These findings indicate that the JNK cascade is part of a bistable signaling system in oocytes.

CONCLUSIONS

In Xenopus oocytes, JNK responds to physiological and pathological stimuli in an all-or-none manner. The JNK response shows all the hallmarks of a bistable response, including strong positive feedback and hysteresis. Bistability is a recurring theme in the biochemistry of oocyte maturation and early embryogenesis; the Mos/MEK/p42 MAPK cascade also exhibits bistable responses, and the Cdc2/cyclin B system is hypothesized to be bistable as well. However, the mechanisms underpinning the positive feedback and bistability in the three cases are different, suggesting that evolution has repeatedly converged upon bistability as a way of producing digital responses.

Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion

Feedback is a ubiquitous control mechanism of gene networks. Here, we have used positive feedback to construct a synthetic eukaryotic gene switch in Saccharomyces cerevisiae. Within this system, a continuous gradient of constitutively expressed transcriptional activator is translated into a cell phenotype switch when the activator is expressed autocatalytically. This finding is consistent with a mathematical model whose analysis shows that continuous input parameters are converted into a bimodal probability distribution by positive feedback, and that this resembles analog-digital conversion. The autocatalytic switch is a robust property in eukaryotic gene expression. Although the behavior of individual cells within a population is random, the proportion of the cell population displaying either low or high expression states can be regulated. These results have implications for understanding the graded and probabilistic mechanisms of enhancer action and cell differentiation.

Computational studies of gene regulatory networks: in numero molecular biology

Remarkable progress in genomic research is leading to a complete map of the building blocks of biology. Knowledge of this map is, in turn, setting the stage for a fundamental description of cellular function at the DNA level. Such a description will entail an understanding of gene regulation, in which proteins often regulate their own production or that of other proteins in a complex web of interactions. The implications of the underlying logic of genetic networks are difficult to deduce through experimental techniques alone, and successful approaches will probably involve the union of new experiments and computational modelling techniques.

Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible

Xenopus oocyte maturation is an example of an all-or-none, irreversible cell fate induction process. In response to a submaximal concentration of the steroid hormone progesterone, a given oocyte may either mature or not mature, but it can exist in intermediate states only transiently. Moreover, once an oocyte has matured, it will remain arrested in the mature state even after the progesterone is removed. It has been hypothesized that the all-or-none character of oocyte maturation, and some aspects of the irreversibility of maturation, arise out of the bistability of the signal transduction system that triggers maturation. The bistability, in turn, is hypothesized to arise from the way the signal transducers are organized into a signaling circuit that includes positive feedback (which makes it so that the system cannot rest in intermediate states) and ultrasensitivity (which filters small stimuli out of the feedback loop, allowing the system to have a stable off-state). Here we review two simple graphical methods that are commonly used to analyze bistable systems, discuss the experimental evidence for bistability in oocyte maturation, and suggest that bistability may be a common means of producing all-or-none responses and a type of biochemical memory. (c) 2001 American Institute of Physics.

Designer gene networks: Towards fundamental cellular control

The engineered control of cellular function through the design of synthetic genetic networks is becoming plausible. Here we show how a naturally occurring network can be used as a parts list for artificial network design, and how model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics. We first review the relevant work on synthetic gene networks, highlighting the important experimental findings with regard to genetic switches and oscillators. We then present the derivation of a deterministic model describing the temporal evolution of the concentration of protein in a single-gene network. Bistability in the steady-state protein concentration arises naturally as a consequence of autoregulatory feedback, and we focus on the hysteretic properties of the protein concentration as a function of the degradation rate. We then formulate the effect of an external noise source which interacts with the protein degradation rate. We demonstrate the utility of such a formulation by constructing a protein switch, whereby external noise pulses are used to switch the protein concentration between two values. Following the lead of earlier work, we show how the addition of a second network component can be used to construct a relaxation oscillator, whereby the system is driven around the hysteresis loop. We highlight the frequency dependence on the tunable parameter values, and discuss design plausibility. We emphasize how the model equations can be used to develop design criteria for robust oscillations, and illustrate this point with parameter plots illuminating the oscillatory regions for given parameter values. We then turn to the utilization of an intrinsic cellular process as a means of controlling the oscillations. We consider a network design which exhibits self-sustained oscillations, and discuss the driving of the oscillator in the context of synchronization. Then, as a second design, we consider a synthetic network with parameter values near, but outside, the oscillatory boundary. In this case, we show how resonance can lead to the induction of oscillations and amplification of a cellular signal. Finally, we construct a toggle switch from positive regulatory elements, and compare the switching properties for this network with those of a network constructed using negative regulation. Our results demonstrate the utility of model analysis in the construction of synthetic gene regulatory networks. (c) 2001 American Institute of Physics.

Noise-based switches and amplifiers for gene expression

The regulation of cellular function is often controlled at the level of gene transcription. Such genetic regulation usually consists of interacting networks, whereby gene products from a single network can act to control their own expression or the production of protein in another network. Engineered control of cellular function through the design and manipulation of such networks lies within the constraints of current technology. Here we develop a model describing the regulation of gene expression and elucidate the effects of noise on the formulation. We consider a single network derived from bacteriophage lambda and construct a two-parameter deterministic model describing the temporal evolution of the concentration of lambda repressor protein. Bistability in the steady-state protein concentration arises naturally, and we show how the bistable regime is enhanced with the addition of the first operator site in the promotor region. We then show how additive and multiplicative external noise can be used to regulate expression. In the additive case, we demonstrate the utility of such control through the construction of a protein switch, whereby protein production is turned "on" and "off" by using short noise pulses. In the multiplicative case, we show that small deviations in the transcription rate can lead to large fluctuations in the production of protein, and we describe how these fluctuations can be used to amplify protein production significantly. These results suggest that an external noise source could be used as a switch and/or amplifier for gene expression. Such a development could have important implications for gene therapy.

Effects of macromolecular transport and stochastic fluctuations on dynamics of genetic regulatory systems

To predict the dynamics of genetic regulation, it may be necessary to consider macromolecular transport and stochastic fluctuations in macromolecule numbers. Transport can be diffusive or active, and in some cases a time delay might suffice to model active transport. We characterize major differences in the dynamics of model genetic systems when diffusive transport of mRNA and protein was compared with transport modeled as a time delay. Delays allow for history-dependent, non-Markovian responses to stimuli (i.e., "molecular memory"). Diffusion suppresses oscillations, whereas delays tend to create oscillations. When simulating essential elements of circadian oscillators, we found the delay between transcription and translation necessary for oscillations. Stochastic fluctuations tend to destabilize and thereby mask steady states with few molecules. This computational approach, combined with experiments, should provide a fruitful conceptual framework for investigating the function and dynamic properties of genetic regulatory systems.