Hysteretic and graded responses in bacterial two-component signal transduction

Spatiotemporal Patterning Enabled by Gene Regulatory Networks

Spatiotemporal pattern formation plays a key role in various biological phenomena including embryogenesis and neural network formation. Though the reaction-diffusion systems enabling pattern formation have been studied phenomenologically, the biomolecular mechanisms behind these processes have not been modeled in detail. Here, we study the emergence of spatiotemporal patterns due to simple, synthetic and commonly observed two- and three-node gene regulatory network motifs coupled with their molecular diffusion in one- and two-dimensional space. We investigate the patterns formed due to the coupling of inherent multistable and oscillatory behavior of the toggle switch, toggle switch with double self-activation, toggle triad, and repressilator with the effect of spatial diffusion of these molecules. We probe multiple parameter regimes corresponding to different regions of stability (monostable, multistable, oscillatory) and assess the impact of varying diffusion coefficients. This analysis offers valuable insights into the design principles of pattern formation facilitated by these network motifs, and it suggests the mechanistic underpinnings of biological pattern formation.

Exploring the mono-/bistability range of positively autoregulated signaling systems in the presence of competing transcription factor binding sites

Binding of transcription factor (TF) proteins to regulatory DNA sites is key to accurate control of gene expression in response to environmental stimuli. Theoretical modeling of transcription regulation is often focused on a limited set of genes of interest, while binding of the TF to other genomic sites is seldom considered. The total number of TF binding sites (TFBSs) affects the availability of TF protein molecules and sequestration of a TF by TFBSs can promote bistability. For many signaling systems where a graded response is desirable for continuous control over the input range, biochemical parameters of the regulatory proteins need be tuned to avoid bistability. Here we analyze the mono-/bistable parameter range for positively autoregulated two-component systems (TCSs) in the presence of different numbers of competing TFBSs. TCS signaling, one of the major bacterial signaling strategies, couples signal perception with output responses via protein phosphorylation. For bistability, competition for TF proteins by TFBSs lowers the requirement for high fold change of the autoregulated transcription but demands high phosphorylation activities of TCS proteins. We show that bistability can be avoided with a low phosphorylation capacity of TCSs, a high TF affinity for the autoregulated promoter or a low fold change in signaling protein levels upon induction. These may represent general design rules for TCSs to ensure uniform graded responses. Examining the mono-/bistability parameter range allows qualitative prediction of steady-state responses, which are experimentally validated in the E. coli CusRS system.

A mathematical model for strigolactone biosynthesis in plants

Strigolactones mediate plant development, trigger symbiosis with arbuscular mycorrhizal fungi, are abundant in 80% of the plant kingdom and help plants gain resistance to environmental stressors. They also induce germination of parasitic plant seeds that are endemic to various continents, such as Orobanche in Europe or Asia and Striga in Africa. The genes involved in the early stages of strigolactones biosynthesis are known in several plants. The regulatory structure and the latter parts of the pathway, where flux branching occurs to produce alternative strigolactones, are less well-understood. Here we present a computational study that collects the available experimental evidence and proposes alternative biosynthetic pathways that are consistent with that evidence. Then, we test the alternative pathways through in silico simulation experiments and compare those experiments to experimental information. Our results predict the differences in dynamic behavior between alternative pathway designs. Independent of design, the analysis suggests that feedback regulation is unlikely to exist in strigolactone biosynthesis. In addition, our experiments suggest that engineering the pathway to modulate the production of strigolactones could be most easily achieved by increasing the flux of β-carotenes going into the biosynthetic pathway. Finally, we find that changing the ratio of alternative strigolactones produced by the pathway can be done by changing the activity of the enzymes after the flux branching points.

Real-time detection of response regulator phosphorylation dynamics in live bacteria

Bacteria utilize two-component system (TCS) signal transduction pathways to sense and adapt to changing environments. In a typical TCS, a stimulus induces a sensor histidine kinase (SHK) to phosphorylate a response regulator (RR), which then dimerizes and activates a transcriptional response. Here, we demonstrate that oligomerization-dependent depolarization of excitation light by fused mNeonGreen fluorescent protein probes enables real-time monitoring of RR dimerization dynamics in live bacteria. Using inducible promoters to independently express SHKs and RRs, we detect RR dimerization within seconds of stimulus addition in several model pathways. We go on to combine experiments with mathematical modeling to reveal that TCS phosphosignaling accelerates with SHK expression but decelerates with RR expression and SHK phosphatase activity. We further observe pulsatile activation of the SHK NarX in response to addition and depletion of the extracellular electron acceptor nitrate when the corresponding TCS is expressed from both inducible systems and the native chromosomal operon. Finally, we combine our method with polarized light microscopy to enable single-cell measurements of RR dimerization under changing stimulus conditions. Direct in vivo characterization of RR oligomerization dynamics should enable insights into the regulation of bacterial physiology.

The inverse correlation between robustness and sensitivity to autoregulation in two-component systems

Two-component systems (TCS) are signal transduction systems in bacteria and many other organisms that relay the sensory signal to genetic components. TCS consist of two proteins: a histidine kinase and a response regulator that the histidine kinase activates. This seemingly simple machinery can generate complex regulatory dynamics that enables the level of gene expression that matches the input signal: many TCS response regulators act on their own genes as transcription factors, resulting in a positive autoregulation mechanism. This regulation, in return, modulates the transcription factor activity as a function of the input signal. Positive autoregulation does not necessarily result in positive feedback. Sensitivity to autoregulation is quantified as the output level amplification resulting from the positive autoregulation mechanism. Another structural property of these systems is formally characterized as "robustness": in a robust TCS, the output of the system is solely a function of the input signal. Thus, a robust TCS remains insensitive to fluctuations in the concentrations of its protein components and, this way, maintains the precision in the output transcription factor activity in response to input stimulus. In this paper, we show with a formal model that TCS operate on a spectrum of inverse correlation between robustness and sensitivity to autoregulation. Our model predicts that the modulation by positive autoregulation is a function of loss in TCS robustness, for example, by spontaneous dephosphorylation of the histidine kinase. Consequently, the loss in robustness provides a proportional modulation by positive autoregulation to widen the response range with a scaled amplification of the output. At the other end of the spectrum, in the presence of a strictly robust TCS machinery, amplification of the transcription factor activity by autoregulation is diminished. We show that our results are in agreement with published experimental results. Our results suggest that these TCS evolve to converge at a trade-off between robustness and positive autoregulation.

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.

Maximization of information transmission influences selection of native phosphorelay architectures

Phosphorelays are signal transduction circuits that sense environmental changes and adjust cellular metabolism. Five different circuit architectures account for 99% of all phosphorelay operons annotated in over 9,000 fully sequenced genomes. Here we asked what biological design principles, if any, could explain selection among those architectures in nature. We began by studying kinetically well characterized phosphorelays (Spo0 of Bacillus subtilis and Sln1 of Saccharomyces cerevisiae). We find that natural circuit architecture maximizes information transmission in both cases. We use mathematical models to compare information transmission among the architectures for a realistic range of concentration and parameter values. Mapping experimentally determined phosphorelay protein concentrations onto that range reveals that the native architecture maximizes information transmission in sixteen out of seventeen analyzed phosphorelays. These results suggest that maximization of information transmission is important in the selection of native phosphorelay architectures, parameter values and protein concentrations.

The interplay of spatial organization and biochemistry in building blocks of cellular signalling pathways

Biochemical pathways and networks are central to cellular information processing. While a broad range of studies have dissected multiple aspects of information processing in biochemical pathways, the effect of spatial organization remains much less understood. It is clear that space is central to intracellular organization, plays important roles in cellular information processing and has been exploited in evolution; additionally, it is being increasingly exploited in synthetic biology through the development of artificial compartments, in a variety of ways. In this paper, we dissect different aspects of the interplay between spatial organization and biochemical pathways, by focusing on basic building blocks of these pathways: covalent modification cycles and two-component systems, with enzymes which may be monofunctional or bifunctional. Our analysis of spatial organization is performed by examining a range of 'spatial designs': patterns of localization or non-localization of enzymes/substrates, theoretically and computationally. Using these well-characterized in silico systems, we analyse the following. (i) The effect of different types of spatial organization on the overall kinetics of modification, and the role of distinct modification mechanisms therein. (ii) How different information processing characteristics seen experimentally and studied from the viewpoint of kinetics are perturbed, or generated. (iii) How the activity of enzymes (bifunctional enzymes in particular) may be spatially manipulated, and the relationship between localization and activity. (iv) How transitions in spatial organization (encountered either through evolution or through the lifetime of cells, as seen in multiple model organisms) impacts the kinetic module (and pathway) behaviour, and how transitions in chemistry may be impacted by prior spatial organization. The basic insights which emerge are central to understanding the role of spatial organization in biochemical pathways in both bacteria and eukaryotes, and are of direct relevance to engineering spatial organization of pathways in bottom-up synthetic biology in cellular and cell-free systems.

Trade-offs between effectiveness and cost in bifunctional enzyme circuit with concentration robustness

A fundamental trade-off in biological systems is whether they consume resources to perform biological functions or save resources. Bacteria need to reliably and rapidly respond to input signals by using limited cellular resources. However, excessive resource consumption will become a burden for bacteria growth. To investigate the relationship between functional effectiveness and resource cost, we study the ubiquitous bifunctional enzyme circuit, which is robust to fluctuations in protein concentration and responds quickly to signal changes. We show that trade-off relationships exist between functional effectiveness and protein cost. Expressing more proteins of the circuit increases concentration robustness and response speed but affects bacterial growth. In particular, our study reveals a general relationship between free-energy dissipation rate, response speed, and concentration robustness. The dissipation of free energy plays an important role in the concentration robustness and response speed. High robustness can only be achieved with a large amount of free-energy consumption and protein cost. In addition, the noise of the output increases with increasing protein cost, while the noise of the response time decreases with increasing protein cost. We also calculate the trade-off relationships in the EnvZ-OmpR system and the nitrogen assimilation system, which both have the bifunctional enzyme. Similar results indicate that these relationships are mainly derived from the specific feature of the bifunctional enzyme circuits and are not relevant to the details of the models. According to the trade-off relationships, bacteria take a compromise solution that reliably performs biological functions at a reasonable cost.

Two-Component Sensing and Regulation: How Do Histidine Kinases Talk with Response Regulators at the Molecular Level?

Perceiving environmental and internal information and reacting in adaptive ways are essential attributes of living organisms. Two-component systems are relevant protein machineries from prokaryotes and lower eukaryotes that enable cells to sense and process signals. Implicating sensory histidine kinases and response regulator proteins, both components take advantage of protein phosphorylation and flexibility to switch conformations in a signal-dependent way. Dozens of two-component systems act simultaneously in any given cell, challenging our understanding about the means that ensure proper connectivity. This review dives into the molecular level, attempting to summarize an emerging picture of how histidine kinases and cognate response regulators achieve required efficiency, specificity, and directionality of signaling pathways, properties that rely on protein:protein interactions. α helices that carry information through long distances, the fine combination of loose and specific kinase/regulator interactions, and malleable reaction centers built when the two components meet emerge as relevant universal principles.

Phosphatase activity tunes two-component system sensor detection threshold

Two-component systems (TCSs) are the largest family of multi-step signal transduction pathways in biology, and a major source of sensors for biotechnology. However, the input concentrations to which biosensors respond are often mismatched with application requirements. Here, we utilize a mathematical model to show that TCS detection thresholds increase with the phosphatase activity of the sensor histidine kinase. We experimentally validate this result in engineered Bacillus subtilis nitrate and E. coli aspartate TCS sensors by tuning their detection threshold up to two orders of magnitude. We go on to apply our TCS tuning method to recently described tetrathionate and thiosulfate sensors by mutating a widely conserved residue previously shown to impact phosphatase activity. Finally, we apply TCS tuning to engineer B. subtilis to sense and report a wide range of fertilizer concentrations in soil. This work will enable the engineering of tailor-made biosensors for diverse synthetic biology applications.

Two-component systems are a major family of signal transduction pathways and a rich source of sensors for biotechnology. Here, the authors develop a general method for rationally tuning two-component system input detection thresholds via specific point mutations in sensor histidine kinase proteins.

Hill kinetics as a noise filter: the role of transcription factor autoregulation in gene cascades

An intuition based on deterministic models of chemical kinetics is that population heterogeneity of transcription factor levels in cells is transmitted unchanged downstream to the target genes. We use a stochastic model of a two-gene cascade with a self-regulating upstream gene to show that, counter to the intuition, there is no simple mapping (bimodal to bimodal, unimodal to unimodal) between the shapes of the distributions of transcription factor numbers and target protein numbers in cells. Due to the presence of the two regulations, the system contains two nonlinear transfer functions, defined by the Hill kinetics of transcription factor binding. The transfer function of the regulator can "interfere" with the transfer function of the target, converting the bimodal input into a unimodal output or vice versa. We show that this effect can be predicted by a geometric construction. As an example application of the method, we present a case study of a system of several downstream genes of different sensitivities, controlled by a common transcription factor which also regulates its own transcription. We show that a single regulator can induce qualitatively different patterns (binary or graded) of responses to a signal in different downstream genes, depending on whether the sensitivity regions of the transfer functions of the upstream and downstream genes overlap or not. Alternatively, the same model can be interpreted as describing a single downstream gene that has different sensitivities in different cell lines due to mutations. Our model shows, therefore, a possible kinetic mechanism by which different genes can interpret the same biological signal in a different manner.

A Tug-of-War Mechanism for Pattern Formation in a Genetic Network

Synthesizing spatial patterns with genetic networks is an ongoing challenge in synthetic biology. A successful demonstration of pattern formation would imply a better understanding of systems in the natural world and advance applications in synthetic biology. In developmental systems, transient patterning may suffice in order to imprint instructions for long-term development. In this paper we show that transient but persistent patterns can emerge from a realizable synthetic gene network based on a toggle switch. We show that a bistable system incorporating diffusible molecules can generate patterns that resemble Turing patterns but are distinctly different in the underlying mechanism: diffusion of mutually inhibiting molecules creates a prolonged "tug-of-war" between patches of cells at opposing bistable states. The patterns are transient but longer wavelength patterns persist for extended periods of time. Analysis of a representative small scale model implies the eigenvalues of the persistent modes are just above the threshold of stability. The results are verified through simulation of biologically relevant models.

Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration

Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks.

Molecular and cellular factors control signal transduction via switchable allosteric modulator proteins (SAMPs)

Background

Rap proteins from Bacilli directly target response regulators of bacterial two-component systems and modulate their activity. Their effects are controlled by binding of signaling peptides to an allosteric site. Hence Raps exemplify a class of monomeric signaling receptors, which we call switchable allosteric modulator proteins (SAMPs). These proteins have potential applications in diverse biomedical and biotechnical settings, but a quantitative understanding of the impact of molecular and cellular factors on signal transduction is lacking. Here we introduce mathematical models that elucidate how signals are propagated though the network upon receptor stimulation and control the level of active response regulator.

Results

Based on a systematic parameter analysis of the models, we show that key features of the dose-response behavior at steady state are controlled either by the molecular properties of the modulator or the signaling context. In particular, we find that the biochemical activity (i.e. non-enzymatic vs. enzymatic) and allosteric properties of the modulator control the response amplitude. The Hill coefficient and the EC₅₀ are controlled in addition by the relative ligand affinities. By tuning receptor properties, either graded or more switch-like (memory-less) response functions can be fashioned. Furthermore, we show that other contextual factors (e.g. relative concentrations of network components and kinase activity) have a substantial impact on the response, and we predict that there exists a modulator concentration which is optimal for response amplitude.

Conclusion

We discuss data on Rap-Phr systems in B. subtilis to show how our models can contribute to an integrated view of SAMP signaling by combining biochemical, structural and physiological insights. Our results also suggest that SAMPs could be evolved or engineered to implement diverse response behaviors. However—without additional regulatory controls—they can generate rather variable cellular outputs.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0274-3) contains supplementary material, which is available to authorized users.

Coordination of cell decisions and promotion of phenotypic diversity in B. subtilis via pulsed behavior of the phosphorelay

The phosphorelay of Bacillus subtilis, a kinase cascade that activates master regulator Spo0A ~ P in response to starvation signals, is the core of a large network controlling the cell's decision to differentiate into sporulation and other phenotypes. This article reviews recent advances in understanding the origins and purposes of the complex dynamical behavior of the phosphorelay, which pulses with peaks of activity coordinated with the cell cycle. The transient imbalance in the expression of two critical genes caused by their strategic placement at opposing ends of the chromosome proved to be the key for this pulsed behavior. Feedback control loops in the phosphorelay use these pulses to implement a timer mechanism, which creates several windows of opportunity for phenotypic transitions over multiple generations. This strategy allows the cell to coordinate multiple differentiation programs in a decision process that fosters phenotypic diversity and adapts to current conditions.

Deciphering Parameter Sensitivity in the BvgAS Signal Transduction

To understand the switching of different phenotypic phases of Bordetella pertussis, we propose an optimized mathematical framework for signal transduction through BvgAS two-component system. The response of the network output to the sensory input has been demonstrated in steady state. An analysis in terms of local sensitivity amplification characterizes the nature of the molecular switch. The sensitivity analysis of the model parameters within the framework of various correlation coefficients helps to decipher the contribution of the modular structure in signal propagation. Once classified, the model parameters are tuned to generate the behavior of some novel strains using simulated annealing, a stochastic optimization technique.

Stochastic simulation of prokaryotic two-component signalling indicates stochasticity-induced active-state locking and growth-rate dependent bistability

Signal transduction by prokaryotes almost exclusively relies on two-component systems for sensing and responding to (extracellular) signals. Here, we use stochastic models of two-component systems to better understand the impact of stochasticity on the fidelity and robustness of signal transmission, the outcome of autoregulatory gene expression and the influence of cell growth and division. We report that two-component systems are remarkably robust against copy number fluctuations of the signalling proteins they are composed of, which enhances signal transmission fidelity. Furthermore, we find that due to stochasticity these systems can get locked in an active state for extended time periods when (initially high) signal levels drop to zero. This behaviour can contribute to a bet-hedging adaptation strategy, aiding survival in fluctuating environments. Additionally, autoregulatory gene expression can cause two-component systems to become bistable at realistic parameter values. As a result, two sub-populations of cells can co-exist-active and inactive cells, which contributes to fitness in unpredictable environments. Bistability proved robust with respect to cell growth and division, and is tunable by the growth rate. In conclusion, our results indicate how single cells can cope with the inevitable stochasticity occurring in the activity of their two-component systems. They are robust to disadvantageous fluctuations that scramble signal transduction and they exploit beneficial stochasticity that generates fitness-enhancing heterogeneity across an isogenic population of cells.

A survey of HK, HPt, and RR domains and their organization in two-component systems and phosphorelay proteins of organisms with fully sequenced genomes

Two Component Systems and Phosphorelays (TCS/PR) are environmental signal transduction cascades in prokaryotes and, less frequently, in eukaryotes. The internal domain organization of proteins and the topology of TCS/PR cascades play an important role in shaping the responses of the circuits. It is thus important to maintain updated censuses of TCS/PR proteins in order to identify the various topologies used by nature and enable a systematic study of the dynamics associated with those topologies. To create such a census, we analyzed the proteomes of 7,609 organisms from all domains of life with fully sequenced and annotated genomes. To begin, we survey each proteome searching for proteins containing domains that are associated with internal signal transmission within TCS/PR: Histidine Kinase (HK), Response Regulator (RR) and Histidine Phosphotranfer (HPt) domains, and analyze how these domains are arranged in the individual proteins. Then, we find all types of operon organization and calculate how much more likely are proteins that contain TCS/PR domains to be coded by neighboring genes than one would expect from the genome background of each organism. Finally, we analyze if the fusion of domains into single TCS/PR proteins is more frequently observed than one might expect from the background of each proteome. We find 50 alternative ways in which the HK, HPt, and RR domains are observed to organize into single proteins. In prokaryotes, TCS/PR coding genes tend to be clustered in operons. 90% of all proteins identified in this study contain just one of the three domains, while 8% of the remaining proteins combine one copy of an HK, a RR, and/or an HPt domain. In eukaryotes, 25% of all TCS/PR proteins have more than one domain. These results might have implications for how signals are internally transmitted within TCS/PR cascades. These implications could explain the selection of the various designs in alternative circumstances.

Chromosomal Arrangement of Phosphorelay Genes Couples Sporulation and DNA Replication

Genes encoding proteins in a common regulatory network are frequently located close to one another on the chromosome to facilitate co-regulation or couple gene expression to growth rate. Contrasting with these observations, here, we demonstrate a functional role for the arrangement of Bacillus subtilis sporulation network genes on opposite sides of the chromosome. We show that the arrangement of two sporulation network genes, one located close to the origin and the other close to the terminus, leads to a transient gene dosage imbalance during chromosome replication. This imbalance is detected by the sporulation network to produce cell-cycle coordinated pulses of the sporulation master regulator Spo0A∼P. This pulsed response allows cells to decide between sporulation and continued vegetative growth during each cell cycle spent in starvation. The simplicity of this coordination mechanism suggests that it may be widely applicable in a variety of gene regulatory and stress-response settings. VIDEO ABSTRACT.

Probing entrainment of Ostreococcus tauri circadian clock by green and blue light through a mathematical modeling approach

Most organisms anticipate daily environmental variations and orchestrate cellular functions thanks to a circadian clock which entrains robustly to the day/night cycle, despite fluctuations in light intensity due to weather or seasonal variations. Marine organisms are also subjected to fluctuations in light spectral composition as their depth varies, due to differential absorption of different wavelengths by sea water. Studying how light input pathways contribute to circadian clock robustness is therefore important. Ostreococcus tauri, a unicellular picoplanktonic marine green alga with low genomic complexity and simple cellular organization, has become a promising model organism for systems biology. Functional and modeling approaches have shown that a core circadian oscillator based on orthologs of Arabidopsis TOC1 and CCA1 clock genes accounts for most experimental data acquired under a wide range of conditions. Some evidence points at putative light input pathway(s) consisting of a two-component signaling system (TCS) controlled by the only two histidine kinases (HK) of O. tauri. LOV-HK is a blue light photoreceptor under circadian control, that is required for circadian clock function. An involvement of Rhodopsin-HK (Rhod-HK) is also conceivable since rhodopsin photoreceptors mediate blue to green light input in animal circadian clocks. Here, we probe the role of LOV-HK and Rhod-HK in mediating light input to the TOC1-CCA1 oscillator using a mathematical model incorporating the TCS hypothesis. This model agrees with clock gene expression time series representative of multiple environmental conditions in blue or green light, characterizing entrainment by light/dark cycles, free-running in constant light, and resetting. Experimental and theoretical results indicate that both blue and green light can reset O. tauri circadian clock. Moreover, our mathematical analysis suggests that Rhod-HK is a blue-green light receptor and drives the clock together with LOV-HK.

Mutually repressing repressor functions and multi-layered cellular heterogeneity regulate the bistable Salmonella fliC census

Bistable flagellar and virulence gene expression generates specialized Salmonella subpopulations with distinct functions. Repressing flagellar genes allows Salmonella to evade caspase-1 mediated host defenses and enhances systemic colonization. By definition, bistability arises when intermediate states of gene expression are rendered unstable by the underlying genetic circuitry. We demonstrate sustained bistable fliC expression in virulent Salmonella 14028 and document dynamic control of the distribution, or single-cell census, of flagellar gene expression by the mutually repressing repressors YdiV and FliZ. YdiV partitions cells into the fliC-OFF subpopulation, while FliZ partitions cells into the fliC-HIGH subpopulation at late time points during growth. Bistability of ΔfliZ populations and ydiV-independent FliZ control of flagellar gene expression provide evidence that the YdiV-FliZ mutually repressing repressor circuit is not required for bistability. Repression and activation by YdiV and FliZ (respectively) can shape the census of fliC expression independently, and bistability collapses into a predominantly intermediate population in the absence of both regulators. Metered expression of YdiV and FliZ reveals variable sensitivity to these regulators and defines conditions where expression of FliZ enhances fliC expression and where FliZ does not alter the fliC census. Thus, this evolved genetic circuitry coordinates multiple layers of regulatory heterogeneity into a binary response.

Phosphate sink containing two-component signaling systems as tunable threshold devices

Synthetic biology aims to design de novo biological systems and reengineer existing ones. These efforts have mostly focused on transcriptional circuits, with reengineering of signaling circuits hampered by limited understanding of their systems dynamics and experimental challenges. Bacterial two-component signaling systems offer a rich diversity of sensory systems that are built around a core phosphotransfer reaction between histidine kinases and their output response regulator proteins, and thus are a good target for reengineering through synthetic biology. Here, we explore the signal-response relationship arising from a specific motif found in two-component signaling. In this motif, a single histidine kinase (HK) phosphotransfers reversibly to two separate output response regulator (RR) proteins. We show that, under the experimentally observed parameters from bacteria and yeast, this motif not only allows rapid signal termination, whereby one of the RRs acts as a phosphate sink towards the other RR (i.e. the output RR), but also implements a sigmoidal signal-response relationship. We identify two mathematical conditions on system parameters that are necessary for sigmoidal signal-response relationships and define key parameters that control threshold levels and sensitivity of the signal-response curve. We confirm these findings experimentally, by in vitro reconstitution of the one HK-two RR motif found in the Sinorhizobium meliloti chemotaxis pathway and measuring the resulting signal-response curve. We find that the level of sigmoidality in this system can be experimentally controlled by the presence of the sink RR, and also through an auxiliary protein that is shown to bind to the HK (yielding Hill coefficients of above 7). These findings show that the one HK-two RR motif allows bacteria and yeast to implement tunable switch-like signal processing and provides an ideal basis for developing threshold devices for synthetic biology applications.

Two-component signaling systems are found in bacteria, fungi and plants. Their modular structures make them ideal targets for de novo engineering through synthetic biology. Here, we explore the signal-response relationship arising from a common two-component system, where a single HK phosphotransfers reversibly to two separate output RRs. We show that under the experimentally observed parameters, this motif implements a sigmoidal signal-response relationship, whereby one of the RRs acts as a phosphate sink towards the other. We identify two mathematical conditions on the system parameters that are necessary for sigmoidality and define key parameters that control threshold levels and sensitivity. We confirm these findings experimentally by in vitro reconstitution of the “one HK-two RR” motif found in S. meliloti. Particularly, we show that the level of sigmoidality in this system can be experimentally controlled by the amount of sink RR and also through an auxiliary protein, CheS. These findings show that the one HK-two RR motif can open the way to the design of novel threshold systems in synthetic biology.

Modulating the frequency and bias of stochastic switching to control phenotypic variation

Mechanisms that control cell-to-cell variation in gene expression ('phenotypic variation') can determine a population's growth rate, robustness, adaptability and capacity for complex behaviours. Here we describe a general strategy (termed FABMOS) for tuning the phenotypic variation and mean expression of cell populations by modulating the frequency and bias of stochastic transitions between 'OFF' and 'ON' expression states of a genetic switch. We validated the strategy experimentally using a synthetic fim switch in Escherichia coli. Modulating the frequency of switching can generate a bimodal (low frequency) or a unimodal (high frequency) population distribution with the same mean expression. Modulating the bias as well as the frequency of switching can generate a spectrum of bimodal and unimodal distributions with the same mean expression. This remarkable control over phenotypic variation, which cannot be easily achieved with standard gene regulatory mechanisms, has many potential applications for synthetic biology, engineered microbial ecosystems and experimental evolution.

Reciprocal regulation as a source of ultrasensitivity in two-component systems with a bifunctional sensor kinase

Two-component signal transduction systems, where the phosphorylation state of a regulator protein is modulated by a sensor kinase, are common in bacteria and other microbes. In many of these systems, the sensor kinase is bifunctional catalyzing both, the phosphorylation and the dephosphorylation of the regulator protein in response to input signals. Previous studies have shown that systems with a bifunctional enzyme can adjust the phosphorylation level of the regulator protein independently of the total protein concentrations – a property known as concentration robustness. Here, I argue that two-component systems with a bifunctional enzyme may also exhibit ultrasensitivity if the input signal reciprocally affects multiple activities of the sensor kinase. To this end, I consider the case where an allosteric effector inhibits autophosphorylation and, concomitantly, activates the enzyme's phosphatase activity, as observed experimentally in the PhoQ/PhoP and NRII/NRI systems. A theoretical analysis reveals two operating regimes under steady state conditions depending on the effector affinity: If the affinity is low the system produces a graded response with respect to input signals and exhibits stimulus-dependent concentration robustness – consistent with previous experiments. In contrast, a high-affinity effector may generate ultrasensitivity by a similar mechanism as phosphorylation-dephosphorylation cycles with distinct converter enzymes. The occurrence of ultrasensitivity requires saturation of the sensor kinase's phosphatase activity, but is restricted to low effector concentrations, which suggests that this mode of operation might be employed for the detection and amplification of low abundant input signals. Interestingly, the same mechanism also applies to covalent modification cycles with a bifunctional converter enzyme, which suggests that reciprocal regulation, as a mechanism to generate ultrasensitivity, is not restricted to two-component systems, but may apply more generally to bifunctional enzyme systems.

Bacteria often use two-component systems to sense and respond to environmental changes, which involves autophosphorylation of a sensor kinase and phosphotransfer to a cognate response regulator. However, despite conservation of this ‘classical’ scheme there exist substantial variations in the mechanism of phosphotransfer among systems. Also, many sensor kinases exhibit phosphatase activity raising the question whether such a bifunctional architecture enables special regulatory properties in the response behavior to input signals. According to previous studies, classical two-component systems are unlikely to produce sigmoidal response curves (ultrasensitivity) if the sensor protein is bifunctional. Here, I argue that this is not necessarily true if the input stimulus (allosteric effector) reciprocally affects multiple activities of the sensor kinase, as it seems to be common for bifunctional enzymes. To this end, I propose and analyze an extension of the experimentally well-supported Batchelor-Goulian model which shows that ultrasensitivity requires a high-affinity effector and saturation of the phosphatase activity. The underlying mechanism involves sequestration of the effector by the sensor kinase which restricts the occurrence of ultrasensitivity to sufficiently low effector concentrations. Hence, this operating regime might be useful to sense effector limitations or to amplify weak input signals.

Role of functionality in two-component signal transduction: a stochastic study

We present a stochastic formalism for signal transduction processes in a bacterial two-component system. Using elementary mass action kinetics, the proposed model takes care of signal transduction in terms of a phosphotransfer mechanism between the cognate partners of a two-component system, viz., the sensor kinase and the response regulator. Based on the difference in functionality of the sensor kinase, the noisy phosphotransfer mechanism has been studied for monofunctional and bifunctional two-component systems using the formalism of the linear noise approximation. Steady-state analysis of both models quantifies different physically realizable quantities, e.g., the variance, the Fano factor (variance/mean), and mutual information. The resultant data reveal that both systems reliably transfer information of extracellular environment under low external stimulus and in a high-kinase-and-phosphatase regime. We extend our analysis further by studying the role of the two-component system in downstream gene regulation.

Stochastic analysis of biochemical reaction networks with absolute concentration robustness

It has recently been shown that structural conditions on the reaction network, rather than a 'fine-tuning' of system parameters, often suffice to impart 'absolute concentration robustness' (ACR) on a wide class of biologically relevant, deterministically modelled mass-action systems. We show here that fundamentally different conclusions about the long-term behaviour of such systems are reached if the systems are instead modelled with stochastic dynamics and a discrete state space. Specifically, we characterize a large class of models that exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space, i.e. the system undergoes an extinction event. If the time to extinction is large relative to the relevant timescales of the system, the process will appear to settle down to a stationary distribution long before the inevitable extinction will occur. This quasi-stationary distribution is considered for two systems taken from the literature, and results consistent with ACR are recovered by showing that the quasi-stationary distribution of the robust species approaches a Poisson distribution.

Evolutionary tuning of protein expression levels of a positively autoregulated two-component system

Cellular adaptation relies on the development of proper regulatory schemes for accurate control of gene expression levels in response to environmental cues. Over- or under-expression can lead to diminished cell fitness due to increased costs or insufficient benefits. Positive autoregulation is a common regulatory scheme that controls protein expression levels and gives rise to essential features in diverse signaling systems, yet its roles in cell fitness are less understood. It remains largely unknown how much protein expression is 'appropriate' for optimal cell fitness under specific extracellular conditions and how the dynamic environment shapes the regulatory scheme to reach appropriate expression levels. Here, we investigate the correlation of cell fitness and output response with protein expression levels of the E. coli PhoB/PhoR two-component system (TCS). In response to phosphate (Pi)-depletion, the PhoB/PhoR system activates genes involved in phosphorus assimilation as well as genes encoding themselves, similarly to many other positively autoregulated TCSs. We developed a bacteria competition assay in continuous cultures and discovered that different Pi conditions have conflicting requirements of protein expression levels for optimal cell fitness. Pi-replete conditions favored cells with low levels of PhoB/PhoR while Pi-deplete conditions selected for cells with high levels of PhoB/PhoR. These two levels matched PhoB/PhoR concentrations achieved via positive autoregulation in wild-type cells under Pi-replete and -deplete conditions, respectively. The fitness optimum correlates with the wild-type expression level, above which the phosphorylation output saturates, thus further increase in expression presumably provides no additional benefits. Laboratory evolution experiments further indicate that cells with non-ideal protein levels can evolve toward the optimal levels with diverse mutational strategies. Our results suggest that the natural protein expression levels and feedback regulatory schemes of TCSs are evolved to match the phosphorylation output of the system, which is determined by intrinsic activities of TCS proteins.

Simplifying biochemical models with intermediate species

Mathematical models are increasingly being used to understand complex biochemical systems, to analyse experimental data and make predictions about unobserved quantities. However, we rarely know how robust our conclusions are with respect to the choice and uncertainties of the model. Using algebraic techniques, we study systematically the effects of intermediate, or transient, species in biochemical systems and provide a simple, yet rigorous mathematical classification of all models obtained from a core model by including intermediates. Main examples include enzymatic and post-translational modification systems, where intermediates often are considered insignificant and neglected in a model, or they are not included because we are unaware of their existence. All possible models obtained from the core model are classified into a finite number of classes. Each class is defined by a mathematically simple canonical model that characterizes crucial dynamical properties, such as mono- and multistationarity and stability of steady states, of all models in the class. We show that if the core model does not have conservation laws, then the introduction of intermediates does not change the steady-state concentrations of the species in the core model, after suitable matching of parameters. Importantly, our results provide guidelines to the modeller in choosing between models and in distinguishing their properties. Further, our work provides a formal way of comparing models that share a common skeleton.

The architecture of a prototypical bacterial signaling circuit enables a single point mutation to confer novel network properties

Even a single mutation can cause a marked change in a protein's properties. When the mutant protein functions within a network, complex phenotypes may emerge that are not intrinsic properties of the protein itself. Network architectures that enable such dramatic changes in function from a few mutations remain relatively uncharacterized. We describe a remarkable example of this versatility in the well-studied PhoQ/PhoP bacterial signaling network, which has an architecture found in many two-component systems. We found that a single point mutation that abolishes the phosphatase activity of the sensor kinase PhoQ results in a striking change in phenotype. The mutant responds to stimulus in a bistable manner, as opposed to the wild-type, which has a graded response. Mutant cells in on and off states have different morphologies, and their state is inherited over many generations. Interestingly, external conditions that repress signaling in the wild-type drive the mutant to the on state. Mathematical modeling and experiments suggest that the bistability depends on positive autoregulation of the two key proteins in the circuit, PhoP and PhoQ. The qualitatively different characteristics of the mutant come at a substantial fitness cost. Relative to the off state, the on state has a lower fitness in stationary phase cultures in rich medium (LB). However, due to the high inheritance of the on state, a population of on cells can be epigenetically trapped in a low-fitness state. Our results demonstrate the remarkable versatility of the prototypical two-component signaling architecture and highlight the tradeoffs in the particular case of the PhoQ/PhoP system.

A mutation can cause significant changes to a protein's function. Since proteins often act together in genetic circuits to control various cellular processes, mutant proteins can lead to unexpected consequences for system-level behavior. In this study, we describe a remarkable example of this phenomenon in a mutant of a well-studied bacterial circuit. PhoQ and PhoP are the primary regulatory proteins in a circuit that responds to low magnesium. The wild-type (unmutated) network responds to environmental signals in an analog or graded manner. In contrast, the mutant responds to signals in an OFF-or-ON or digital fashion. Moreover, the distribution of OFF and ON cells is strongly influenced by how cells were cultured in the past. These remarkable changes can be traced to features of the wiring diagram of the PhoQ/PhoP circuit. Since these features are shared among a broad class of bacterial signaling circuits, we suggest that other circuits may show similar remarkable properties when mutated.

A bistable hysteretic switch in an activator-repressor regulated restriction-modification system

Restriction–modification (RM) systems are extremely widespread among bacteria and archaea, and are often specified by mobile genetic elements. In type II RM systems, where the restriction endonuclease (REase) and protective DNA methyltransferase (MTase) are separate proteins, a major regulatory challenge is delaying expression of the REase relative to the MTase after RM genes enter a new host cell. Basic understanding of this regulation is available for few RM systems, and detailed understanding for none. The PvuII RM system is one of a large subset in which the central regulatory role is played by an activator–repressor protein (called C, for controller). REase expression depends upon activation by C, whereas expression of the MTase does not. Thus delay of REase expression depends on the rate of C-protein accumulation. This is a nonlinear process, as C also activates transcription of its own gene. Mathematical modeling of the PvuII system led to the unexpected predictions of responsiveness to a factor not previously studied in RM system control—gene copy number—and of a hysteretic response. In this study, those predictions have been confirmed experimentally. The results may apply to many other C-regulated RM systems, and help explain their ability to spread so widely.

Split histidine kinases enable ultrasensitivity and bistability in two-component signaling networks

Bacteria sense and respond to their environment through signaling cascades generally referred to as two-component signaling networks. These networks comprise histidine kinases and their cognate response regulators. Histidine kinases have a number of biochemical activities: ATP binding, autophosphorylation, the ability to act as a phosphodonor for their response regulators, and in many cases the ability to catalyze the hydrolytic dephosphorylation of their response regulator. Here, we explore the functional role of "split kinases" where the ATP binding and phosphotransfer activities of a conventional histidine kinase are split onto two distinct proteins that form a complex. We find that this unusual configuration can enable ultrasensitivity and bistability in the signal-response relationship of the resulting system. These dynamics are displayed under a wide parameter range but only when specific biochemical requirements are met. We experimentally show that one of these requirements, namely segregation of the phosphatase activity predominantly onto the free form of one of the proteins making up the split kinase, is met in Rhodobacter sphaeroides. These findings indicate split kinases as a bacterial alternative for enabling ultrasensitivity and bistability in signaling networks. Genomic analyses reveal that up 1.7% of all identified histidine kinases have the potential to be split and bifunctional.

Biochemical Systems Theory: A Review

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

Coupling between feedback loops in autoregulatory networks affects bistability range, open-loop gain and switching times

Biochemical regulatory networks governing diverse cellular processes such as stress-response, differentiation and cell cycle often contain coupled feedback loops. We aim at understanding how features of feedback architecture, such as the number of loops, the sign of the loops and the type of their coupling, affect network dynamical performance. Specifically, we investigate how bistability range, maximum open-loop gain and switching times of a network with transcriptional positive feedback are affected by additive or multiplicative coupling with another positive- or negative-feedback loop. We show that a network's bistability range is positively correlated with its maximum open-loop gain and that both quantities depend on the sign of the feedback loops and the type of feedback coupling. Moreover, we find that the addition of positive feedback could decrease the bistability range if we control the basal level in the signal-response curves of the two systems. Furthermore, the addition of negative feedback has the capacity to increase the bistability range if its dissociation constant is much lower than that of the positive feedback. We also find that the addition of a positive feedback to a bistable network increases the robustness of its bistability range, whereas the addition of a negative feedback decreases it. Finally, we show that the switching time for a transition from a high to a low steady state increases with the effective fold change in gene regulation. In summary, we show that the effect of coupled feedback loops on the bistability range and switching times depends on the underlying mechanistic details.

Cross-talk and information transfer in mammalian and bacterial signaling

In mammalian and bacterial cells simple phosphorylation circuits play an important role in signaling. Bacteria have hundreds of two-component signaling systems that involve phosphotransfer between a receptor and a response regulator. In mammalian cells a similar pathway is the TGF-beta pathway, where extracellular TGF-beta ligands activate cell surface receptors that phosphorylate Smad proteins, which in turn activate many genes. In TGF-beta signaling the multiplicity of ligands begs the question as to whether cells can distinguish signals coming from different ligands, but transduced through a small set of Smads. Here we use information theory with stochastic simulations of networks to address this question. We find that when signals are transduced through only one Smad, the cell cannot distinguish between different levels of the external ligands. Increasing the number of Smads from one to two significantly improves information transmission as well as the ability to discriminate between ligands. Surprisingly, both total information transmitted and the capacity to discriminate between ligands are quite insensitive to high levels of cross-talk between the two Smads. Robustness against cross-talk requires that the average amplitude of the signals are large. We find that smaller systems, as exemplified by some two-component systems in bacteria, are significantly much less robust against cross-talk. For such system sizes phosphotransfer is also less robust against cross-talk than phosphorylation. This suggests that mammalian signal transduction can tolerate a high amount of cross-talk without degrading information content. This may have played a role in the evolution of new functionalities from small mutations in signaling pathways, allowed for the development of cross-regulation and led to increased overall robustness due to redundancy in signaling pathways. On the other hand the lack of cross-regulation observed in many bacterial two-component systems may partly be due to the loss of information content due to cross-talk.

Two component systems: physiological effect of a third component

Signal transduction systems mediate the response and adaptation of organisms to environmental changes. In prokaryotes, this signal transduction is often done through Two Component Systems (TCS). These TCS are phosphotransfer protein cascades, and in their prototypical form they are composed by a kinase that senses the environmental signals (SK) and by a response regulator (RR) that regulates the cellular response. This basic motif can be modified by the addition of a third protein that interacts either with the SK or the RR in a way that could change the dynamic response of the TCS module. In this work we aim at understanding the effect of such an additional protein (which we call "third component") on the functional properties of a prototypical TCS. To do so we build mathematical models of TCS with alternative designs for their interaction with that third component. These mathematical models are analyzed in order to identify the differences in dynamic behavior inherent to each design, with respect to functionally relevant properties such as sensitivity to changes in either the parameter values or the molecular concentrations, temporal responsiveness, possibility of multiple steady states, or stochastic fluctuations in the system. The differences are then correlated to the physiological requirements that impinge on the functioning of the TCS. This analysis sheds light on both, the dynamic behavior of synthetically designed TCS, and the conditions under which natural selection might favor each of the designs. We find that a third component that modulates SK activity increases the parameter space where a bistable response of the TCS module to signals is possible, if SK is monofunctional, but decreases it when the SK is bifunctional. The presence of a third component that modulates RR activity decreases the parameter space where a bistable response of the TCS module to signals is possible.

Speed, sensitivity, and bistability in auto-activating signaling circuits

Cells employ a myriad of signaling circuits to detect environmental signals and drive specific gene expression responses. A common motif in these circuits is inducible auto-activation: a transcription factor that activates its own transcription upon activation by a ligand or by post-transcriptional modification. Examples range from the two-component signaling systems in bacteria and plants to the genetic circuits of animal viruses such as HIV. We here present a theoretical study of such circuits, based on analytical calculations, numerical computations, and simulation. Our results reveal several surprising characteristics. They show that auto-activation can drastically enhance the sensitivity of the circuit's response to input signals: even without molecular cooperativity, an ultra-sensitive threshold response can be obtained. However, the increased sensitivity comes at a cost: auto-activation tends to severely slow down the speed of induction, a stochastic effect that was strongly underestimated by earlier deterministic models. This slow-induction effect again requires no molecular cooperativity and is intimately related to the bimodality recently observed in non-cooperative auto-activation circuits. These phenomena pose strong constraints on the use of auto-activation in signaling networks. To achieve both a high sensitivity and a rapid induction, an inducible auto-activation circuit is predicted to acquire low cooperativity and low fold-induction. Examples from Escherichia coli's two-component signaling systems support these predictions.

Different times call for different measures. Therefore, cells adjust their protein levels depending on their environment. Upon the detection of certain environmental signals, transcription factors are activated, which activate or inhibit the production of specific sets of proteins. As it turns out, these transcription factors often also stimulate their own production. Indeed, such self-regulation is a common motif in signal–response systems of many organisms, including bacteria, animals, plants and viruses–but its function is not well understood. We have used mathematical models to study its benefits and drawbacks. On the one hand, calculations show that self-regulation can be a very useful tool if the cell needs to respond in a sensitive way to changes in its environment, or if it is supposed to respond only if the signal exceeds a threshold level. On the other hand, these benefits come at a cost: self-regulation severely slows down the cell's response to changes in the environment. We have analyzed how the cell can benefit from the advantages of self-regulation, while mitigating the drawbacks. This leads to strict design constraints that examples from the bacterium E. coli indeed seem to obey.

Non-transcriptional regulatory processes shape transcriptional network dynamics

Information about the extra- or intracellular environment is often captured as biochemical signals that propagate through regulatory networks. These signals eventually drive phenotypic changes, typically by altering gene expression programmes in the cell. Reconstruction of transcriptional regulatory networks has given a compelling picture of bacterial physiology, but transcriptional network maps alone often fail to describe phenotypes. Cellular response dynamics are ultimately determined by interactions between transcriptional and non-transcriptional networks, with dramatic implications for physiology and evolution. Here, we provide an overview of non-transcriptional interactions that can affect the performance of natural and synthetic bacterial regulatory networks.

Modeling the light- and redox-dependent interaction of PpsR/AppA in Rhodobacter sphaeroides

Facultative photosynthetic bacteria switch their energy generation mechanism from respiration to photosynthesis depending on oxygen tension and light. Part of this transition is mediated by the aerobic transcriptional repressor PpsR. In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized by the redox- and blue-light-sensitive flavoprotein AppA which results in a unique phenotype: the repression of photosynthesis genes at intermediate oxygen levels and high light intensity, which is believed to reduce the risk of photooxidative stress. To analyze the underlying mechanism we developed a simple mathematical model based on the AppA-dependent reduction of a disulfide bond in PpsR and the light-sensitive complex formation between the reduced forms of AppA and PpsR. A steady-state analysis shows that high light repression can indeed occur at intermediate oxygen levels if PpsR is reduced on a faster timescale than AppA and if the electron transfer from AppA to PpsR is effectively irreversible. The model further predicts that if AppA copy numbers exceed those of PpsR by at least a factor of two, the transition from aerobic to anaerobic growth mode can occur via a bistable regime. We provide necessary conditions for the emergence of bistability and discuss possible experimental verifications.

Bistable responses in bacterial genetic networks: designs and dynamical consequences

A key property of living cells is their ability to react to stimuli with specific biochemical responses. These responses can be understood through the dynamics of underlying biochemical and genetic networks. Evolutionary design principles have been well studied in networks that display graded responses, with a continuous relationship between input signal and system output. Alternatively, biochemical networks can exhibit bistable responses so that over a range of signals the network possesses two stable steady states. In this review, we discuss several conceptual examples illustrating network designs that can result in a bistable response of the biochemical network. Next, we examine manifestations of these designs in bacterial master-regulatory genetic circuits. In particular, we discuss mechanisms and dynamic consequences of bistability in three circuits: two-component systems, sigma-factor networks, and a multistep phosphorelay. Analyzing these examples allows us to expand our knowledge of evolutionary design principles networks with bistable responses.

Analysis of operating principles with S-system models

Operating principles address general questions regarding the response dynamics of biological systems as we observe or hypothesize them, in comparison to a priori equally valid alternatives. In analogy to design principles, the question arises: Why are some operating strategies encountered more frequently than others and in what sense might they be superior? It is at this point impossible to study operation principles in complete generality, but the work here discusses the important situation where a biological system must shift operation from its normal steady state to a new steady state. This situation is quite common and includes many stress responses. We present two distinct methods for determining different solutions to this task of achieving a new target steady state. Both methods utilize the property of S-system models within Biochemical Systems Theory (BST) that steady states can be explicitly represented as systems of linear algebraic equations. The first method uses matrix inversion, a pseudo-inverse, or regression to characterize the entire admissible solution space. Operations on the basis of the solution space permit modest alterations of the transients toward the target steady state. The second method uses standard or mixed integer linear programming to determine admissible solutions that satisfy criteria of functional effectiveness, which are specified beforehand. As an illustration, we use both methods to characterize alternative response patterns of yeast subjected to heat stress, and compare them with observations from the literature.

The interplay of multiple feedback loops with post-translational kinetics results in bistability of mycobacterial stress response

Bacterial persistence is the phenomenon in which a genetically identical fraction of a bacterial population can survive exposure to stress by reduction or cessation of growth. Persistence in mycobacteria has been recently linked to a stress-response network, consisting of the MprA/MprB two-component system and alternative sigma factor sigma(E). This network contains multiple positive transcriptional feedback loops which may give rise to bistability, making it a good candidate for controlling the mycobacterial persistence switch. To analyze the possibility of bistability, we develop a method that involves decoupling of the network into transcriptional and post-translational interaction modules. As a result we reduce the dimensionality of the dynamical system and independently analyze input-output relations in the two modules to formulate a necessary condition for bistability in terms of their logarithmic gains. We show that neither the positive autoregulation in the MprA/MprB network nor the sigma(E)-mediated transcriptional feedback is sufficient to induce bistability in a biochemically realistic parameter range. Nonetheless, inclusion of the post-translational regulation of sigma(E) by RseA increases the effective cooperativity of the system, resulting in bistability that is robust to parameter variation. We predict that overexpression or deletion of RseA, the key element controlling the ultrasensitive response, can eliminate bistability.

Response dynamics of phosphorelays suggest their potential utility in cell signalling

Phosphorelays are extended two-component signalling systems found in diverse bacteria, lower eukaryotes and plants. Only few of these systems are characterized, and we still lack a full understanding of their signalling abilities. Here, we aim to achieve a global understanding of phosphorelay signalling and its dynamical properties. We develop a generic model, allowing us to systematically analyse response dynamics under different assumptions. Using this model, we find that the steady-state concentration of phosphorylated protein at the final layer of a phosphorelay is a linearly increasing, but eventually saturating function of the input. In contrast, the intermediate layers can display ultrasensitivity. We find that such ultrasensitivity is a direct result of the phosphorelay biochemistry; shuttling of a single phosphate group from the first to the last layer. The response dynamics of the phosphorelay results in tolerance of cross-talk, especially when it occurs as cross-deactivation. Further, it leads to a high signal-to-noise ratio for the final layer. We find that a relay length of four, which is most commonly observed, acts as a saturating point for these dynamic properties. These findings suggest that phosphorelays could act as a mechanism to reduce noise and effects of cross-talk on the final layer of the relay and enforce its input–response relation to be linear. In addition, our analysis suggests that middle layers of phosphorelays could embed thresholds. We discuss the consequence of these findings in relation to why cells might use phosphorelays along with enzymatic kinase cascades.

How important is the phosphatase activity of sensor kinases?

In two-component signaling systems, phosphorylated response regulators (RRs) are often dephosphorylated by their partner kinases in order to control the in vivo concentration of phospho-RR (RR approximately P). This activity is easily demonstrated in vitro, but these experiments have typically used very high concentrations of the histidine kinase (HK) compared to the RR approximately P. Many two-component systems exhibit exquisite control over the ratio of HK to RR in vivo. The question thus arises as to whether the phosphatase activity of HKs is significant in vivo. This topic will be explored in the present review.

Adaptable functionality of transcriptional feedback in bacterial two-component systems

A widespread mechanism of bacterial signaling occurs through two-component systems, comprised of a sensor histidine kinase (SHK) and a transcriptional response regulator (RR). The SHK activates RR by phosphorylation. The most common two-component system structure involves expression from a single operon, the transcription of which is activated by its own phosphorylated RR. The role of this feedback is poorly understood, but it has been associated with an overshooting kinetic response and with fast recovery of previous interrupted signaling events in different systems. Mathematical models show that overshoot is only attainable with negative feedback that also improves response time. Our models also predict that fast recovery of previous interrupted signaling depends on high accumulation of SHK and RR, which is more likely in a positive feedback regime. We use Monte Carlo sampling of the parameter space to explore the range of attainable model behaviors. The model predicts that the effective feedback sign can change from negative to positive depending on the signal level. Variations in two-component system architectures and parameters may therefore have evolved to optimize responses in different bacterial lifestyles. We propose a conceptual model where low signal conditions result in a responsive system with effectively negative feedback while high signal conditions with positive feedback favor persistence of system output.

Bacteria have evolved various mechanisms for surviving unpredictable changes and stresses in the environment, such as nutrient limitation. One common survival mechanism is the two-component system, where a sensor protein responds to a particular type of stress by activating a regulator in the cell. These regulators can in turn activate genes that produce proteins for stress-appropriate responses. The activated regulator often positively regulates transcription of its own operon containing the sensor and regulator genes leading to a feedback loop. This is interesting, because positive feedback is usually associated with a slower response time than negative feedback and therefore negative feedback would often be selected for by evolution. Here we analyze a mathematical model to study the interplay of this feedback and postranslational mechanisms regulating two-component system signaling. We found that modulation of regulator activity by its operon partner can lead to overall negative feedback to result from autoactivation. This happens if (1) the sensor can both activate and deactivate the regulator, and (2) there is some reaction resulting in regulator activation independently of its cognate sensor. As a result our model predicts that two-component systems may be capable of flexibly switching between positive and negative feedback depending on different circumstances, allowing for appropriate responses in a variety of conditions.

Two-component signaling circuit structure and properties

Various modeling and experimental studies have analyzed the reactions, interconnections, and motifs in two-component systems, with an eye toward understanding their physiological implications and the differences between alternative designs. Examples where recent progress has been made include aspects of autoregulation, signal integration in branched pathways, cross-talk suppression, and cross-regulation via connector proteins.

Identifying quantitative operation principles in metabolic pathways: a systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses

BACKGROUND

Optimization methods allow designing changes in a system so that specific goals are attained. These techniques are fundamental for metabolic engineering. However, they are not directly applicable for investigating the evolution of metabolic adaptation to environmental changes. Although biological systems have evolved by natural selection and result in well-adapted systems, we can hardly expect that actual metabolic processes are at the theoretical optimum that could result from an optimization analysis. More likely, natural systems are to be found in a feasible region compatible with global physiological requirements.

RESULTS

We first present a new method for globally optimizing nonlinear models of metabolic pathways that are based on the Generalized Mass Action (GMA) representation. The optimization task is posed as a nonconvex nonlinear programming (NLP) problem that is solved by an outer-approximation algorithm. This method relies on solving iteratively reduced NLP slave subproblems and mixed-integer linear programming (MILP) master problems that provide valid upper and lower bounds, respectively, on the global solution to the original NLP. The capabilities of this method are illustrated through its application to the anaerobic fermentation pathway in Saccharomyces cerevisiae. We next introduce a method to identify the feasibility parametric regions that allow a system to meet a set of physiological constraints that can be represented in mathematical terms through algebraic equations. This technique is based on applying the outer-approximation based algorithm iteratively over a reduced search space in order to identify regions that contain feasible solutions to the problem and discard others in which no feasible solution exists. As an example, we characterize the feasible enzyme activity changes that are compatible with an appropriate adaptive response of yeast Saccharomyces cerevisiae to heat shock

CONCLUSION

Our results show the utility of the suggested approach for investigating the evolution of adaptive responses to environmental changes. The proposed method can be used in other important applications such as the evaluation of parameter changes that are compatible with health and disease states.

ResDE-dependent regulation of enterotoxin gene expression in Bacillus cereus: evidence for multiple modes of binding for ResD and interaction with Fnr

In the food-borne pathogen Bacillus cereus F4430/73, the production of major virulence factors hemolysin BL (Hbl) and nonhemolytic enterotoxin (Nhe) is regulated through complex mechanisms. The two-component regulatory system ResDE is involved in the activation of hbl and nhe transcription. Here, the response regulator ResD and the sensor kinase ResE were overexpressed and purified, and autophosphorylation of ResE and transphosphorylation of ResD by ResE were demonstrated in vitro. ResD is mainly monomeric in solution, regardless of its phosphorylation state. ResD was shown to interact directly with promoter regions (p) of the enterotoxin regulator genes resDE, fnr, and plcR and the enterotoxin structural genes nhe and hbl, but with different affinities. Binding of ResD to pplcR, pnhe, and phbl was not dependent on the ResD phosphorylation status. In contrast, ResD phosphorylation significantly increased interactions between ResD and presDE and pfnr. Taken together, these results showed that phosphorylation of ResD results in a different target expression pattern. Furthermore, ResD and the redox activator Fnr were found to physically interact and simultaneously bind their target DNAs. We propose that unphosphorylated ResD acts as an antiactivator of Fnr, while phosphorylated ResD acts as a coactivator of Fnr. Finally, our findings represent the first molecular evidence of the role of ResDE as a sentinel system capable of sensing redox changes and coordinating a response that modulates B. cereus virulence.