Dynamic regulation of the tryptophan operon: a modeling study and comparison with experimental data

Boolean Models of the Transport, Synthesis, and Metabolism of Tryptophan in Escherichia coli

The tryptophan (trp) operon in Escherichia coli codes for the proteins responsible for the synthesis of the amino acid tryptophan from chorismic acid, and has been one of the most well-studied gene networks since its discovery in the 1960s. The tryptophanase (tna) operon codes for proteins needed to transport and metabolize it. Both of these have been modeled individually with delay differential equations under the assumption of mass-action kinetics. Recent work has provided strong evidence for bistable behavior of the tna operon. The authors of Orozco-Gómez et al. (Sci Rep 9(1):5451, 2019) identified a medium range of tryptophan in which the system has two stable steady-states, and they reproduced these experimentally. In this paper, we will show how a Boolean model can capture this bistability. We will also develop and analyze a Boolean model of the trp operon. Finally, we will combine these two to create a single Boolean model of the transport, synthesis, and metabolism of tryptophan. In this amalgamated model, the bistability disappears, presumably reflecting the ability of the trp operon to produce tryptophan and drive the system toward homeostasis. All of these models have longer attractors that we call "artifacts of synchrony", which disappear in the asynchronous automata. This curiously matches the behavior of a recent Boolean model of the arabinose operon in E. coli, and we discuss some open-ended questions that arise along these lines.

Differential roles of transcriptional and translational negative autoregulations in protein dynamics

Cells continuously respond to stimuli to function properly by employing a wide variety of regulatory mechanisms that often involve protein up or down regulations. This study focuses on dynamics of a protein with negative autoregulations in E. coli, and assumes that the input signal up-regulates the protein, and then the protein down-regulates its own production via 2 distinct types of mechanisms. The mathematical models describe the dynamics of mRNA and protein for 3 scenarios: (i) a simplistic model with no regulation, (ii) a model with transcriptional negative autoregulation, and (iii) a model with translational negative autoregulation. Our analysis shows that the negative autoregulation models produce faster responses and quicker return times to the input signals compared to the model with no regulation, while the transcriptional autoregulation model is the only model capable of producing oscillatory dynamics. The stochastic simulations predict that the transcriptional autoregulation model is the noisiest followed by the simplistic model, and the translational autoregulation model has the least noise. The noise level depends on the strength of inhibition. Furthermore, the transcriptional autoregulation model filters out the noise in the input signal for longer periods of time, and this time increases as the strength of the feedback gets stronger.

A Molecular Dynamic Model of Tryptophan Overproduction in Escherichia coli

Several deterministic models simulate the main molecular biology interactions among the numerous mechanisms controlling the dynamics of the tryptophan operon in native strains. However, no models exist to investigate bacterial tryptophan production from a biotechnological point of view. Here, we modified tryptophan models for native production to propose a biotechnological working model that incorporates the activity of tryptophan secretion systems and genetic modifications made in two reported E. coli strains. The resultant deterministic model could emulate the production of tryptophan in the same order of magnitude as those quantified experimentally by the genetically engineered E. coli strains GPT1001 and GPT1002 in shake flasks. We hope this work may contribute to the rational development of biological models that define and include the main parameters and molecular components for designing and engineering efficient biotechnological chassis to produce valuable chemicals.

Robustness: linking strain design to viable bioprocesses

Microbial cell factories are becoming increasingly popular for the sustainable production of various chemicals. Metabolic engineering has led to the design of advanced cell factories; however, their long-term yield, titer, and productivity falter when scaled up and subjected to industrial conditions. This limitation arises from a lack of robustness - the ability to maintain a constant phenotype despite the perturbations of such processes. This review describes predictable and stochastic industrial perturbations as well as state-of-the-art technologies to counter process variability. Moreover, we distinguish robustness from tolerance and discuss the potential of single-cell studies for improving system robustness. Finally, we highlight ways of achieving consistent and comparable quantification of robustness that can guide the selection of strains for industrial bioprocesses.

Excitable dynamics through toxin-induced mRNA cleavage in bacteria

Toxin-antitoxin (TA) systems in bacteria and archaea are small genetic elements consisting of the genes coding for an intracellular toxin and an antitoxin that can neutralize this toxin. In various cases, the toxins cleave the mRNA. In this theoretical work we use deterministic and stochastic modeling to explain how toxin-induced cleavage of mRNA in TA systems can lead to excitability, allowing large transient spikes in toxin levels to be triggered. By using a simplified network where secondary complex formation and transcriptional regulation are not included, we show that a two-dimensional, deterministic model captures the origin of such toxin excitations. Moreover, it allows to increase our understanding by examining the dynamics in the phase plane. By systematically comparing the deterministic results with Gillespie simulations we demonstrate that even though the real TA system is intrinsically stochastic, toxin excitations can be accurately described deterministically. A bifurcation analysis of the system shows that the excitable behavior is due to a nearby Hopf bifurcation in the parameter space, where the system becomes oscillatory. The influence of stress is modeled by varying the degradation rate of the antitoxin and the translation rate of the toxin. We find that stress increases the frequency of toxin excitations. The inclusion of secondary complex formation and transcriptional regulation does not fundamentally change the mechanism of toxin excitations. Finally, we show that including growth rate suppression and translational inhibition can lead to longer excitations, and even cause excitations in cases when the system would otherwise be non-excitable. To conclude, the deterministic model used in this work provides a simple and intuitive explanation of toxin excitations in TA systems.

In Vitro Analysis of Predicted DNA-Binding Sites for the Stl Repressor of the Staphylococcus aureus SaPIBov1 Pathogenicity Island

The regulation model of the Staphylococcus aureus pathogenicity island SaPIbov1 transfer was recently reported. The repressor protein Stl obstructs the expression of SaPI proteins Str and Xis, latter which is responsible for mobilization initiation. Upon Φ11 phage infection of S. aureus. phage dUTPase activates the SaPI transfer via Stl-dUTPase complex formation. Our aim was to predict the binding sites for the Stl repressor within the S. aureus pathogenicity island DNA sequence. We found that Stl was capable to bind to three 23-mer oligonucleotides, two of those constituting sequence segments in the stl-str, while the other corresponding to sequence segment within the str-xis intergenic region. Within these oligonucleotides, mutational analysis revealed that the predicted binding site for the Stl protein exists as a palindromic segment in both intergenic locations. The palindromes are built as 6-mer repeat sequences involved in Stl binding. The 6-mer repeats are separated by a 5 oligonucleotides long, nonspecific sequence. Future examination of the interaction between Stl and its binding sites in vivo will provide a molecular explanation for the mechanisms of gene repression and gene activation exerted simultaneously by the Stl protein in regulating transfer of the SaPIbov1 pathogenicity island in S. aureus.

Investigating Evolutionary Dynamics of RHA1 Operons

Grouping genes as operons is an important genomic feature of prokaryotic organisms. The comprehensive understanding of the operon organizations would be helpful to decipher transcriptional mechanisms, cellular pathways, and the evolutionary landscape of prokaryotic genomes. Although thousands of prokaryotes have been sequenced, genome-wide investigation of the evolutionary dynamics (division and recombination) of operons among these genomes remains unexplored. Here, we systematically analyzed the operon dynamics of Rhodococcus jostii RHA1 (RHA1), an oleaginous bacterium with high potential applications in biofuel, by comparing 340 prokaryotic genomes that were carefully selected from different genera. Interestingly, 99% of RHA1 operons were observed to exhibit evolutionary events of division and recombination among the 340 compared genomes. An operon that encodes all enzymes related to histidine biosynthesis in RHA1 (His-operon) was found to be segmented into smaller gene groups (sub-operons) in diverse genomes. These sub-operons were further reorganized with different functional genes as novel operons that are related to different biochemical processes. Comparatively, the operons involved in the functional categories of lipid transport and metabolism are relatively conserved among the 340 compared genomes. At the pathway level, RHA1 operons found to be significantly conserved were involved in ribosome synthesis, oxidative phosphorylation, and fatty acid synthesis. These analyses provide evolutionary insights of operon organization and the dynamic associations of various biochemical pathways in different prokaryotes.

Cis-Antisense Transcription Gives Rise to Tunable Genetic Switch Behavior: A Mathematical Modeling Approach

Antisense transcription has been extensively recognized as a regulatory mechanism for gene expression across all kingdoms of life. Despite the broad importance and extensive experimental determination of cis-antisense transcription, relatively little is known about its role in controlling cellular switching responses. Growing evidence suggests the presence of non-coding cis-antisense RNAs that regulate gene expression via antisense interaction. Recent studies also indicate the role of transcriptional interference in regulating expression of neighboring genes due to traffic of RNA polymerases from adjacent promoter regions. Previous models investigate these mechanisms independently, however, little is understood about how cells utilize coupling of these mechanisms in advantageous ways that could also be used to design novel synthetic genetic devices. Here, we present a mathematical modeling framework for antisense transcription that combines the effects of both transcriptional interference and cis-antisense regulation. We demonstrate the tunability of transcriptional interference through various parameters, and that coupling of transcriptional interference with cis-antisense RNA interaction gives rise to hypersensitive switches in expression of both antisense genes. When implementing additional positive and negative feed-back loops from proteins encoded by these genes, the system response acquires a bistable behavior. Our model shows that combining these multiple-levels of regulation allows fine-tuning of system parameters to give rise to a highly tunable output, ranging from a simple-first order response to biologically complex higher-order response such as tunable bistable switch. We identify important parameters affecting the cellular switch response in order to provide the design principles for tunable gene expression using antisense transcription. This presents an important insight into functional role of antisense transcription and its importance towards design of synthetic biological switches.

The utility of simple mathematical models in understanding gene regulatory dynamics

In this review, we survey work that has been carried out in the attempts of biomathematicians to understand the dynamic behaviour of simple bacterial operons starting with the initial work of the 1960's. We concentrate on the simplest of situations, discussing both repressible and inducible systems and then turning to concrete examples related to the biology of the lactose and tryptophan operons. We conclude with a brief discussion of the role of both extrinsic noise and so-called intrinsic noise in the form of translational and/or transcriptional bursting.

Analysis of L-glutamic acid fermentation by using a dynamic metabolic simulation model of Escherichia coli

Background

Understanding the process of amino acid fermentation as a comprehensive system is a challenging task. Previously, we developed a literature-based dynamic simulation model, which included transcriptional regulation, transcription, translation, and enzymatic reactions related to glycolysis, the pentose phosphate pathway, the tricarboxylic acid (TCA) cycle, and the anaplerotic pathway of Escherichia coli. During simulation, cell growth was defined such as to reproduce the experimental cell growth profile of fed-batch cultivation in jar fermenters. However, to confirm the biological appropriateness of our model, sensitivity analysis and experimental validation were required.

Results

We constructed an l-glutamic acid fermentation simulation model by removing sucAB, a gene encoding α-ketoglutarate dehydrogenase. We then performed systematic sensitivity analysis for l-glutamic acid production; the results of this process corresponded with previous experimental data regarding l-glutamic acid fermentation. Furthermore, it allowed us to predicted the possibility that accumulation of 3-phosphoglycerate in the cell would regulate the carbon flux into the TCA cycle and lead to an increase in the yield of l-glutamic acid via fermentation. We validated this hypothesis through a fermentation experiment involving a model l-glutamic acid-production strain, E. coli MG1655 ΔsucA in which the phosphoglycerate kinase gene had been amplified to cause accumulation of 3-phosphoglycerate. The observed increase in l-glutamic acid production verified the biologically meaningful predictive power of our dynamic metabolic simulation model.

Conclusions

In this study, dynamic simulation using a literature-based model was shown to be useful for elucidating the precise mechanisms involved in fermentation processes inside the cell. Further exhaustive sensitivity analysis will facilitate identification of novel factors involved in the metabolic regulation of amino acid fermentation.

A general model for toxin-antitoxin module dynamics can explain persister cell formation in E. coli

Toxin-Antitoxin modules are small operons involved in stress response and persister cell formation that encode a “toxin” and its corresponding neutralizing “antitoxin”. Regulation of these modules involves a complex mechanism known as conditional cooperativity, which is supposed to prevent unwanted toxin activation. Here we develop mathematical models for their regulation, based on published molecular and structural data, and parameterized using experimental data for F-plasmid ccdAB, bacteriophage P1 phd/doc and E. coli relBE. We show that the level of free toxin in the cell is mainly controlled through toxin sequestration in toxin-antitoxin complexes of various stoichiometry rather than by gene regulation. If the toxin translation rate exceeds twice the antitoxin translation rate, toxins accumulate in all cells. Conditional cooperativity and increasing the number of binding sites on the operator serves to reduce the metabolic burden of the cell by reducing the total amounts of proteins produced. Combining conditional cooperativity and bridging of antitoxins by toxins when bound to their operator sites allows creation of persister cells through rare, extreme stochastic spikes in the free toxin level. The amplitude of these spikes determines the duration of the persister state. Finally, increases in the antitoxin degradation rate and decreases in the bacterial growth rate cause a rise in the amount of persisters during nutritional stress.

Bacterial persistence plays an important role in many chronic infections. Persisters are subpopulations of bacteria which are tolerant to biological stresses such as antibiotics because they are in a dormant, non-dividing state. Toxin-antitoxin (TA) modules play a pivotal role in persister generation and bacterial stress response. These small genetic loci, ubiquitous in bacterial genomes and plasmids, code for a toxin that slows down or halts bacterial metabolism and a corresponding antitoxin that regulates this activity. In order to further unravel the intricate autoregulation of TA modules and their role in persister cell formation, we built stochastic models describing the transcriptional regulation including conditional cooperativity. This is a complex mechanism in which the molar ratio between both proteins determines whether the toxin will behave as a co-repressor or as a de-repressor for the antitoxin. We found that the necessary protein production and therefore the energetic cost decreases with increased binding site number. Finally, these models allow us to simulate the formation of persister cells through rare, stochastic increases in the free toxin level. We believe that our analysis provides a fresh view and contributes to our understanding of TA regulation and how it may be related to the emergence of persisters.

Synthetic gene circuits for metabolic control: design trade-offs and constraints

A grand challenge in synthetic biology is to push the design of biomolecular circuits from purely genetic constructs towards systems that interface different levels of the cellular machinery, including signalling networks and metabolic pathways. In this paper, we focus on a genetic circuit for feedback regulation of unbranched metabolic pathways. The objective of this feedback system is to dampen the effect of flux perturbations caused by changes in cellular demands or by engineered pathways consuming metabolic intermediates. We consider a mathematical model for a control circuit with an operon architecture, whereby the expression of all pathway enzymes is transcriptionally repressed by the metabolic product. We address the existence and stability of the steady state, the dynamic response of the network under perturbations, and their dependence on common tuneable knobs such as the promoter characteristic and ribosome binding site (RBS) strengths. Our analysis reveals trade-offs between the steady state of the enzymes and the intermediates, together with a separation principle between promoter and RBS design. We show that enzymatic saturation imposes limits on the parameter design space, which must be satisfied to prevent metabolite accumulation and guarantee the stability of the network. The use of promoters with a broad dynamic range and a small leaky expression enlarges the design space. Simulation results with realistic parameter values also suggest that the control circuit can effectively upregulate enzyme production to compensate flux perturbations.

Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops

Feedback controls are central to cellular regulation. Negative-feedback mechanisms are well known to underline oscillatory dynamics. However, the presence of multiple negative-feedback mechanisms is common in oscillatory cellular systems, raising intriguing questions of how they cooperate to regulate oscillations. In this work, we studied the dynamical properties of a set of general biochemical motifs with dual, nested negative-feedback structures. We showed analytically and then confirmed numerically that, in these motifs, each negative-feedback loop exhibits distinctly different oscillation-controlling functions. The longer, outer feedback loop was found to promote oscillations, whereas the short, inner loop suppresses and can even eliminate oscillations. We found that the position of the inner loop within the coupled motifs affects its repression strength towards oscillatory dynamics. Bifurcation analysis indicated that emergence of oscillations may be a strict parametric requirement and thus evolutionarily tricky. Investigation of the quantitative features of oscillations (i.e. frequency, amplitude and mean value) revealed that coupling negative feedback provides robust tuning of the oscillation dynamics. Finally, we demonstrated that the mitogen-activated protein kinase (MAPK) cascades also display properties seen in the general nested feedback motifs. The findings and implications in this study provide novel understanding of biochemical negative-feedback regulation in a mixed wiring context.

Multistability and oscillations in genetic control of metabolism

Genetic control of enzyme activity drives metabolic adaptations to environmental changes, and therefore the feedback interaction between gene expression and metabolism is essential to cell fitness. In this paper we develop a new formalism to detect the equilibrium regimes of an unbranched metabolic network under transcriptional feedback from one metabolite. Our results indicate that one-to-all transcriptional feedback can induce a wide range of metabolic phenotypes, including mono-, multistability and oscillatory behavior. The analysis is based on the use of switch-like models for transcriptional control and the exploitation of the time scale separation between metabolic and genetic dynamics. For any combination of activation and repression feedback loops, we derive conditions for the emergence of a specific phenotype in terms of genetic parameters such as enzyme expression rates and regulatory thresholds. We find that metabolic oscillations can emerge under uniform thresholds and, in the case of operon-controlled networks, the analysis reveals how nutrient-induced bistability and oscillations can emerge as a consequence of the transcriptional feedback.

Convergent transcription in the butyrolactone regulon in Streptomyces coelicolor confers a bistable genetic switch for antibiotic biosynthesis

cis-encoded antisense RNAs (cis asRNA) have been reported to participate in gene expression regulation in both eukaryotic and prokaryotic organisms. Its presence in Streptomyces coelicolor has also been reported recently; however, its role has yet to be fully investigated. Using mathematical modeling we explore the role of cis asRNA produced as a result of convergent transcription in scbA-scbR genetic switch. scbA and scbR gene pair, encoding repressor–amplifier proteins respectively, mediates the synthesis of a signaling molecule, the γ-butyrolactone SCB1 and controls the onset of antibiotic production. Our model considers that transcriptional interference caused by convergent transcription of two opposing RNA polymerases results in fatal collision and transcriptional termination, which suppresses transcription efficiency. Additionally, convergent transcription causes sense and antisense interactions between complementary sequences from opposing strands, rendering the full length transcript inaccessible for translation. We evaluated the role of transcriptional interference and the antisense effect conferred by convergent transcription on the behavior of scbA-scbR system. Stability analysis showed that while transcriptional interference affects the system, it is asRNA that confers scbA-scbR system the characteristics of a bistable switch in response to the signaling molecule SCB1. With its critical role of regulating the onset of antibiotic synthesis the bistable behavior offers this two gene system the needed robustness to be a genetic switch. The convergent two gene system with potential of transcriptional interference is a frequent feature in various genomes. The possibility of asRNA regulation in other such gene-pairs is yet to be examined.

Dynamic analysis of the KlGAL regulatory system in Kluyveromyces lactis: a comparative study with Saccharomyces cerevisiae

The GAL regulatory system is highly conserved in yeast species of Saccharomyces cerevisiae and Kluyveromyces lactis. While the GAL system is a well studied system in S. cerevisiae, the dynamic behavior of the KlGAL system in K. lactis has not been characterized. Here, we have characterized the GAL system in yeast K. lactis by developing a dynamic model and comparing its performance to its not-so-distant cousin S. cerevisiae. The present analysis demonstrates the significance of the autoregulatory feedbacks due to KlGal4p, KlGal80p, KlGal1p and Lac12p on the dynamic performance of the KlGAL switch. The model predicts the experimentally observed absence of bistability in the wild type strain of K. lactis, unlike the short term memory of preculturing conditions observed in S. cerevisiae. The performance of the GAL switch is distinct for the two yeast species although they share similarities in the molecular components. The analysis suggests that the whole genome duplication of S. cerevisiae, which resulted in a dedicated inducer protein, Gal3p, may be responsible for the high sensitivity of the system to galactose concentrations. On the other hand, K. lactis uses a bifunctional protein as an inducer in addition to its galactokinase activity, which restricts its regulatory role and hence higher galactose levels in the medium are needed to trigger the GAL system.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s11693-011-9082-7) contains supplementary material, which is available to authorized users.

Distinct noise-controlling roles of multiple negative feedback mechanisms in a prokaryotic operon system

Molecular fluctuations are known to affect dynamics of cellular systems in important ways. Studies aimed at understanding how molecular systems of certain regulatory architectures control noise therefore become essential. The interplay between feedback regulation and noise has been previously explored for cellular networks governed by a single negative feedback loop. However, similar issues within networks consisting of more complex regulatory structures remain elusive. The authors investigate how negative feedback loops manage noise within a biochemical cascade concurrently governed by multiple negative feedback loops, using the prokaryotic tryptophan (trp) operon system in Escherechia coli as the model system. To the authors knowledge, this is the first study of noise in the trp operon system. They show that the loops in the trp operon system possess distinct, even opposing, noise-controlling effects despite their seemingly analogous feedback structures. The enzyme inhibition loop, although controlling the last reaction of the cascade, was found to suppress noise not only for the tryptophan output but also for other upstream components. In contrast, the Repression (Rep) loop enhances noise for all systems components. Attenuation (Att) poses intermediate effects by attenuating noise for the upstream components but promoting noise for components downstream of its target. Regarding noise at the output tryptophan, Rep and Att can be categorised as noise-enhancing loops whereas Enzyme Inhibition as a noise-reducing loop. These findings suggest novel implications in how cellular systems with multiple feedback mechanisms control noise. [Includes supplementary material].

Design of regulation and dynamics in simple biochemical pathways

Complex regulation of biochemical pathways in a cell is brought about by the interaction of simpler regulatory structures. Among the basic regulatory designs, feedback inhibition of gene expression is the most common motif in gene regulation and a ubiquitous control structure found in nature. In this work, we have studied a common structural feature (delayed feedback) in gene organisation and shown, both theoretically and experimentally, its subtle but important functional role in gene expression kinetics in a negatively auto-regulated system. Using simple deterministic and stochastic models with varying levels of realism, we present detailed theoretical representations of negatively auto-regulated transcriptional circuits with increasing delays in the establishment of feedback of repression. The models of the circuits with and without delay are studied analytically as well as numerically for variation of parameters and delay lengths. The positive invariance, boundedness of the solutions, local and global asymptotic stability of both the systems around the unique positive steady state are studied analytically. Existence of transient temporal dynamics is shown mathematically. Comparison of the two types of model circuits shows that even though the long-term dynamics is stable and not affected by delays in repression, there is interesting variation in the transient dynamical features with increasing delays. Theoretical predictions are validated through experimentally constructed gene circuits of similar designs. This combined theoretical and experimental study helps delineate the opposing effects of delay-induced instability, and the stability-enhancing property of negative feedback in the pathway behaviour, and gives rationale for the abundance of similar designs in real biochemical pathways.

Functional characterization of alternate optimal solutions of Escherichia coli's transcriptional and translational machinery

The constraint-based reconstruction and analysis approach has recently been extended to describe Escherichia coli's transcriptional and translational machinery. Here, we introduce the concept of reaction coupling to represent the dependency between protein synthesis and utilization. These coupling constraints lead to a significant contraction of the feasible set of steady-state fluxes. The subset of alternate optimal solutions (AOS) consistent with maximal ribosome production was calculated. The majority of transcriptional and translational reactions were active for all of these AOS, showing that the network has a low degree of redundancy. Furthermore, all calculated AOS contained the qualitative expression of at least 92% of the known essential genes. Principal component analysis of AOS demonstrated that energy currencies (ATP, GTP, and phosphate) dominate the network's capability to produce ribosomes. Additionally, we identified regulatory control points of the network, which include the transcription reactions of sigma70 (RpoD) as well as that of a degradosome component (Rne) and of tRNA charging (ValS). These reactions contribute significant variance among AOS. These results show that constraint-based modeling can be applied to gain insight into the systemic properties of E. coli's transcriptional and translational machinery.

Integration of transcriptional inputs at promoters of the arabinose catabolic pathway

BACKGROUND

Most modelling efforts of transcriptional networks involve estimations of in vivo concentrations of components, binding affinities and reaction rates, derived from in vitro biochemical assays. These assays are difficult and in vitro measurements may not approximate actual in vivo conditions. Alternatively, changes in transcription factor activity can be estimated by using partially specified models which estimate the "hidden functions" of transcription factor concentration changes; however, non-unique solutions are a potential problem. We have applied a synthetic biology approach to develop reporters that are capable of measuring transcription factor activity in vivo in real time. These synthetic reporters are comprised of a constitutive promoter with an operator site for the specific transcription factor immediately downstream. Thus, increasing transcription factor activity is measured as repression of expression of the transcription factor reporter. Measuring repression instead of activation avoids the complications of non-linear interactions between the transcription factor and RNA polymerase which differs at each promoter.

RESULTS

Using these reporters, we show that a simple model is capable of determining the rules of integration for multiple transcriptional inputs at the four promoters of the arabinose catabolic pathway. Furthermore, we show that despite the complex and non-linear changes in cAMP-CRP activity in vivo during diauxic shift, the synthetic transcription factor reporters are capable of measuring real-time changes in transcription factor activity, and the simple model is capable of predicting the dynamic behaviour of the catabolic promoters.

CONCLUSIONS

Using a synthetic biology approach we show that the in vivo activity of transcription factors can be quantified without the need for measuring intracellular concentrations, binding affinities and reaction rates. Using measured transcription factor activity we show how different promoters can integrate common transcriptional inputs, resulting in distinct expression patterns. The data collected show that cAMP levels in vivo are dynamic and agree with observations showing that cAMP levels show a transient pulse during diauxic shift.

Cycling expression and cooperative operator interaction in the trp operon of Escherichia coli

Oscillatory behaviour in the tryptophan operon of an Escherichia coli mutant strain lacking the enzyme-inhibition regulatory mechanism has been observed by Bliss et al. but not confirmed by others. This behaviour could be important from the standpoint of synthetic biology, whose goals include the engineering of intracellular genetic oscillators. This work is devoted to investigating, from a mathematical modelling point of view, the possibility that the trp operon of the E. coli inhibition-free strain expresses cyclically. For that we extend a previously introduced model for the regulatory pathway of the tryptophan operon in Escherichia coli to account for the observed multiplicity and cooperativity of repressor binding sites. Thereafter we investigate the model dynamics using deterministic numeric solutions, stochastic simulations, and analytic studies. Our results suggest that a quasi-periodic behaviour could be observed in the trp operon expression level of single bacteria.

Multiple feedback loop design in the tryptophan regulatory network of Escherichia coli suggests a paradigm for robust regulation of processes in series

Biological networks have evolved through adaptation in uncertain environments. Of the different possible design paradigms, some may offer functional advantages over others. These designs can be quantified by the structure of the network resulting from molecular interactions and the parameter values. One may, therefore, like to identify the design motif present in the evolved network that makes it preferable over other alternatives. In this work, we focus on the regulatory networks characterized by serially arranged processes, which are regulated by multiple feedback loops. Specifically, we consider the tryptophan system present in Escherichia coli, which may be conceptualized as three processes in series, namely transcription, translation and tryptophan synthesis. The multiple feedback loop motif results from three distinct negative feedback loops, namely genetic repression, mRNA attenuation and enzyme inhibition. A framework is introduced to identify the key design components of this network responsible for its physiological performance. We demonstrate that the multiple feedback loop motif, as seen in the tryptophan system, enables robust performance to variations in system parameters while maintaining a rapid response to achieve homeostasis. Superior performance, if arising from a design principle, is intrinsic and, therefore, inherent to any similarly designed system, either natural or engineered. An experimental engineering implementation of the multiple feedback loop design on a two-tank system supports the generality of the robust attributes offered by the design.

On the functional diversity of dynamical behaviour in genetic and metabolic feedback systems

Background

Feedback regulation plays crucial roles in the robust control and maintenance of many cellular systems. Negative feedbacks are found to underline both stable and unstable, often oscillatory, behaviours. We explore the dynamical characteristics of systems with single as well as coupled negative feedback loops using a combined approach of analytical and numerical techniques. Particularly, we emphasise how the loop's characterising factors (strength and cooperativity levels) affect system dynamics and how individual loops interact in the coupled-loop systems.

Results

We develop an analytical bifurcation analysis based on the stability and the Routh- Hurwitz theorem for a common negative feedback system and a variety of its variants. We demonstrate that different combinations of the feedback strengths of individual loops give rise to different dynamical behaviours. Moreover, incorporating more negative feedback loops always tend to enhance system stability. We show that two mechanisms, in addition to the lengthening of pathway, can lower the Hill coefficient to a biologically plausible level required for sustained oscillations. These include loops coupling and end-product utilisation. We find that the degradation rates solely affect the threshold Hill coefficient for sustained oscillation, while the synthesis rates have more significant roles in determining the threshold feedback strength. Unbalancing the degradation rates between the system species is found as a way to improve stability.

Conclusion

The analytical methods and insights presented in this study demonstrate that reallocation of the feedback loop may or may not make the system more stable; the specific effect is determined by the degradation rates of the newly inhibited molecular species. As the loop moves closer to the end of the pathway, the minimum Hill coefficient for oscillation is reduced. Furthermore, under general (unequal) values of the degradation rates, system extension becomes more stable only when the added species degrades slower than it is being produced; otherwise the system is more prone to oscillation. The coupling of loops significantly increases the richness of dynamical bifurcation characteristics. The likelihood of having oscillatory behaviour is directly determined by the loops' strength: stronger loops always result in smaller oscillatory regions.

A new model for investigating the evolution of transcription control networks

Biological systems show unbounded capacity for complex behaviors and responses to their environments. This principally arises from their genetic networks. The processes governing transcription, translation, and gene regulation are well understood, as are the mechanisms of network evolution, such as gene duplication and horizontal gene transfer. However, the evolved networks arising from these simple processes are much more difficult to understand, and it is difficult to perform experiments on the evolution of these networks in living organisms because of the timescales involved. We propose a new framework for modeling and investigating the evolution of transcription networks in realistic, varied environments. The model we introduce contains novel, important, and lifelike features that allow the evolution of arbitrarily complex transcription networks. Molecular interactions are not specified; instead they are determined dynamically based on shape, allowing protein function to freely evolve. Transcriptional logic provides a flexible mechanism for defining genetic regulatory activity. Simulations demonstrate a realistic life cycle as an emergent property, and that even in simple environments lifelike and complex regulation mechanisms are evolved, including stable proteins, unstable mRNA, and repressor activity. This study also highlights the importance of using in silico genetics techniques to investigate evolved model robustness.

Stochastic analysis of the GAL genetic switch in Saccharomyces cerevisiae: modeling and experiments reveal hierarchy in glucose repression

Background

Transcriptional regulation involves protein-DNA and protein-protein interactions. Protein-DNA interactions involve reactants that are present in low concentrations, leading to stochastic behavior. In addition, multiple regulatory mechanisms are typically involved in transcriptional regulation. In the GAL regulatory system of Saccharomyces cerevisiae, the inhibition of glucose is accomplished through two regulatory mechanisms: one through the transcriptional repressor Mig1p, and the other through regulating the amount of transcriptional activator Gal4p. However, the impact of stochasticity in gene expression and hierarchy in regulatory mechanisms on the phenotypic level is not clearly understood.

Results

We address the question of quantifying the effect of stochasticity inherent in these regulatory mechanisms on the performance of various genes under the regulation of Mig1p and Gal4p using a dynamic stochastic model. The stochastic analysis reveals the importance of both the mechanisms of regulation for tight expression of genes in the GAL network. The mechanism involving Gal4p is the dominant mechanism, yielding low variability in the expression of GAL genes. The mechanism involving Mig1p is necessary to maintain the switch-like response of certain GAL genes. The number of binding sites for Mig1p and Gal4p further influences the expression of the genes, with extra binding sites lowering the variability of expression. Our experiments involving growth on various substrates show that the trends predicted in mean expression and its variability are transmitted to the phenotypic level.

Conclusion

The mechanisms involved in the transcriptional regulation and their variability set up a hierarchy in the phenotypic response to growth on various substrates. Structural motifs, such as the number of binding sites and the mechanism of regulation, determine the level of stochasticity and eventually, the phenotypic response.

Biosynthesis of the Aromatic Amino Acids

This chapter describes in detail the genes and proteins of Escherichia coli involved in the biosynthesis and transport of the three aromatic amino acids tyrosine, phenylalanine, and tryptophan. It provides a historical perspective on the elaboration of the various reactions of the common pathway converting erythrose-4-phosphate and phosphoenolpyruvate to chorismate and those of the three terminal pathways converting chorismate to phenylalanine, tyrosine, and tryptophan. The regulation of key reactions by feedback inhibition, attenuation, repression, and activation are also discussed. Two regulatory proteins, TrpR (108 amino acids) and TyrR (513 amino acids), play a major role in transcriptional regulation. The TrpR protein functions only as a dimer which, in the presence of tryptophan, represses the expression of trp operon plus four other genes (the TrpR regulon). The TyrR protein, which can function both as a dimer and as a hexamer, regulates the expression of nine genes constituting the TyrR regulon. TyrR can bind each of the three aromatic amino acids and ATP and under their influence can act as a repressor or activator of gene expression. The various domains of this protein involved in binding the aromatic amino acids and ATP, recognizing DNA binding sites, interacting with the alpha subunit of RNA polymerase, and changing from a monomer to a dimer or a hexamer are all described. There is also an analysis of the various strategies which allow TyrR in conjunction with particular amino acids to differentially affect the expression of individual genes of the TyrR regulon.

A bistable gene switch for antibiotic biosynthesis: the butyrolactone regulon in Streptomyces coelicolor

Many microorganisms, including bacteria of the class Streptomycetes, produce various secondary metabolites including antibiotics to gain a competitive advantage in their natural habitat. The production of these compounds is highly coordinated in a population to expedite accumulation to an effective concentration. Furthermore, as antibiotics are often toxic even to their producers, a coordinated production allows microbes to first arm themselves with a defense mechanism to resist their own antibiotics before production commences. One possible mechanism of coordination among individuals is through the production of signaling molecules. The gamma-butyrolactone system in Streptomyces coelicolor is a model of such a signaling system for secondary metabolite production. The accumulation of these signaling molecules triggers antibiotic production in the population. A pair of repressor-amplifier proteins encoded by scbA and scbR mediates the production and action of one particular gamma-butyrolactone, SCB1. Based on the proposed interactions of scbA and scbR, a mathematical model was constructed and used to explore the ability of this system to act as a robust genetic switch. Stability analysis shows that the butyrolactone system exhibits bistability and, in response to a threshold SCB1 concentration, can switch from an OFF state to an ON state corresponding to the activation of genes in the cryptic type I polyketide synthase gene cluster, which are responsible for production of the hypothetical polyketide. The switching time is inversely related to the inducer concentration above the threshold, such that short pulses of low inducer concentration cannot switch on the system, suggesting its possible role in noise filtering. In contrast, secondary metabolite production can be triggered rapidly in a population of cells producing the butyrolactone signal due to the presence of an amplification loop in the system. S. coelicolor was perturbed experimentally by varying concentrations of SCB1, and the model simulations match the experimental data well. Deciphering the complexity of this butyrolactone switch will provide valuable insights into how robust and efficient systems can be designed using "simple" two-protein networks.

Binding cooperativity in phage lambda is not sufficient to produce an effective switch

In the wild-type phage lambda, binding of CI to O(R)2 helps polymerase bound to P(RM) transition from a closed to open complex. Activators on other promoters increase the polymerase-DNA binding energy, or affect both the binding energy and the closed-open transition probability. Using a validated mathematical model, we show that these two modes of upregulation have very different effects on the promoter function. We predict that if CI(2) bound to O(R)2 produced equal increase in RNAP-DNA binding constant (compared to wild-type increase in the closed-open transition probability), the lysogen would be significantly less stable.

Component-Based Modelling of RNA Structure Folding

RNA structure is fundamentally important for many biological processes. In the past decades, diverse structure prediction algorithms and tools were developed but due to missing descriptions in clearly defined modelling formalisms it’s difficult or even impossible to integrate them into larger system models. We present an RNA secondary structure folding model described in ml-Devs, a variant of the Devs formalism, which enables the hierarchical combination with other model components like RNA binding proteins. An example of transcriptional attenuation will be given where model components of RNA polymerase, the folding RNA molecule, and the translating ribosome play together in a composed dynamic model.

Control of Streptococcus pyogenes virulence: modeling of the CovR/S signal transduction system

The CovR/S system in Streptococcus pyogenes (Group A Streptococcus, or GAS), a two-component signal transduction/transcription regulation system, controls the expression of major virulence factors. The presence of a negative feedback loop distinguishes the CovR/S system from the majority of bacterial two-component systems. We developed a deterministic model of the CovR/S system consisting of eight delay differential equations. Computational experiments showed that the system possessed a unique stable steady state. The dynamical behavior of the system showed a tendency for oscillations becoming more pronounced for longer but still biochemically realistic delays resulting from reductions in the rates of translation elongation. We have devised an efficient procedure for computing the system's steady state. Further, we have shown that the signal-response curves are hyperbolic for the default parameter values. However, in experiments with randomized parameters we demonstrated that sigmoidality of signal-response curves, implying a response threshold, is not only possible, but seems to be rather typical for CovR/S-like systems even when binding of the CovR response regulator protein to a promoter is non-cooperative. We used sensitivity analysis to simplify the model in order to make it analytically tractable. The existence and uniqueness of the steady state and hyperbolicity of signal-response curves for the majority of the variables was proved for the simplified model. Also, we found that provided CovS was active, the system was insensitive to changes in the concentration of any other phosphoryl donor such as acetyl phosphate.

Analytical study of the multiplicity of regulatory mechanisms in the tryptophan operon

In this paper we study the stability of a previously introduced model for the tryptophan operon regulatory pathway. For this, we make use of the second Lyapunov's method. The results obtained for the wild-type and for a couple ofin-silico mutant bacterial strains allow a deeper understanding of the multiplicity of regulatory mechanisms in this operon. In particular, we confirm that enzyme inhibition and transcription attenuation strengthen the system stability, the effect of transcription attenuation being much shorter than that of enzyme inhibition. Furthermore, the analysis here presented provides some insights about how enzyme inhibition affects the system stability.

Towards an understanding of lineage specification in hematopoietic stem cells: a mathematical model for the interaction of transcription factors GATA-1 and PU.1

In addition to their self-renewal capabilities, hematopoietic stem cells guarantee the continuous supply of fully differentiated, functional cells of various types in the peripheral blood. The process which controls differentiation into the different lineages of the hematopoietic system (erythroid, myeloid, lymphoid) is referred to as lineage specification. It requires a potentially multi-step decision sequence which determines the fate of the cells and their successors. It is generally accepted that lineage specification is regulated by a complex system of interacting transcription factors. However, the underlying principles controlling this regulation are currently unknown. Here, we propose a simple quantitative model describing the interaction of two transcription factors. This model is motivated by experimental observations on the transcription factors GATA-1 and PU.1, both known to act as key regulators and potential antagonists in the erythroid vs. myeloid differentiation processes of hematopoietic progenitor cells. We demonstrate the ability of the model to account for the observed switching behavior of a transition from a state of low expression of both factors (undifferentiated state) to the dominance of one factor (differentiated state). Depending on the parameter choice, the model predicts two different possibilities to explain the experimentally suggested, stem cell characterizing priming state of low level co-expression. Whereas increasing transcription rates are sufficient to induce differentiation in one scenario, an additional system perturbation (by stochastic fluctuations or directed impulses) of transcription factor levels is required in the other case.

A microarray data-based semi-kinetic method for predicting quantitative dynamics of genetic networks

Background

Elucidating the dynamic behaviour of genetic regulatory networks is one of the most significant challenges in systems biology. However, conventional quantitative predictions have been limited to small networks because publicly available transcriptome data has not been extensively applied to dynamic simulation.

Results

We present a microarray data-based semi-kinetic (MASK) method which facilitates the prediction of regulatory dynamics of genetic networks composed of recurrently appearing network motifs with reasonable accuracy. The MASK method allows the determination of model parameters representing the contribution of regulators to transcription rate from time-series microarray data. Using a virtual regulatory network and a Saccharomyces cerevisiae ribosomal protein gene module, we confirmed that a MASK model can predict expression profiles for various conditions as accurately as a conventional kinetic model.

Conclusion

We have demonstrated the MASK method for the construction of dynamic simulation models of genetic networks from time-series microarray data, initial mRNA copy number and first-order degradation constants of mRNA. The quantitative accuracy of the MASK models has been confirmed, and the results indicated that this method enables the prediction of quantitative dynamics in genetic networks composed of commonly used network motifs, which cover considerable fraction of the whole network.

A simple framework to describe the regulation of gene expression in prokaryotes

Based on the bimolecular mass action law and the derived mass conservation laws, we propose a mathematical framework in order to describe the regulation of gene expression in prokaryotes. It is shown that the derived models have all the qualitative properties of the activation and inhibition regulatory mechanisms observed in experiments. The basic construction considers genes as templates for protein production, where regulation processes result from activators or repressors connecting to DNA binding sites. All the parameters in the models have a straightforward biological meaning. After describing the general properties of the basic mechanisms of positive and negative gene regulation, we apply this framework to the self-regulation of the trp operon and to the genetic switch involved in the regulation of the lac operon. One of the consequences of this approach is the existence of conserved quantities depending on the initial conditions that tune bifurcations of fixed points. This leads naturally to a simple explanation of threshold effects as observed in some experiments.

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

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

Topological units of environmental signal processing in the transcriptional regulatory network of Escherichia coli

Recent evidence indicates that potential interactions within metabolic, protein-protein interaction, and transcriptional regulatory networks are used differentially according to the environmental conditions in which a cell exists. However, the topological units underlying such differential utilization are not understood. Here we use the transcriptional regulatory network of Escherichia coli to identify such units, called origons, representing regulatory subnetworks that originate at a distinct class of sensor transcription factors. Using microarray data, we find that specific environmental signals affect mRNA expression levels significantly only within the origons responsible for their detection and processing. We also show that small regulatory interaction patterns, called subgraphs and motifs, occupy distinct positions in and between origons, offering insights into their dynamical role in information processing. The identified features are likely to represent a general framework for environmental signal processing in prokaryotes.

On the use of qualitative reasoning to simulate and identify metabolic pathways

MOTIVATION

Perhaps the greatest challenge of modern biology is to develop accurate in silico models of cells. To do this we require computational formalisms for both simulation (how according to the model the state of the cell evolves over time) and identification (learning a model cell from observation of states). We propose the use of qualitative reasoning (QR) as a unified formalism for both tasks. The two most commonly used alternative methods of modelling biochemical pathways are ordinary differential equations (ODEs), and logical/graph-based (LG) models.

RESULTS

The QR formalism we use is an abstraction of ODEs. It enables the behaviour of many ODEs, with different functional forms and parameters, to be captured in a single QR model. QR has the advantage over LG models of explicitly including dynamics. To simulate biochemical pathways we have developed 'enzyme' and 'metabolite' QR building blocks that fit together to form models. These models are finite, directly executable, easy to interpret and robust. To identify QR models we have developed heuristic chemoinformatics graph analysis and machine learning procedures. The graph analysis procedure is a series of constraints and heuristics that limit the number of ways metabolites can combine to form pathways. The machine learning procedure is generate-and-test inductive logic programming. We illustrate the use of QR for modelling and simulation using the example of glycolysis.

AVAILABILITY

All data and programs used are available on request.

Dynamic influence of feedback enzyme inhibition and transcription attenuation on the tryptophan operon response to nutritional shifts

A mathematical model of the tryptophan operon is developed. This model considers all of the system known regulatory mechanisms: repression, transcription attenuation, and feedback enzyme inhibition. Special attention is paid to the estimation of all the model parameters from reported experimental data. The model equations are numerically solved. An analysis of these solutions reveals that transcription attenuation helps to speed up the operon response to nutritional shifts, while enzyme inhibition increases the operon stability.

Near-critical behavior of aminoacyl-tRNA pools in E. coli at rate-limiting supply of amino acids

The rates of consumption of different amino acids in protein synthesis are in general stoichiometrically coupled with coefficients determined by codon usage frequencies on translating ribosomes. We show that when the rates of synthesis of two or more amino acids are limiting for protein synthesis and exactly matching their coupled rates of consumption on translating ribosomes, the pools of aminoacyl-tRNAs in ternary complex with elongation factor Tu and GTP are hypersensitive to a variation in the rate of amino acid supply. This high sensitivity makes a macroscopic analysis inconclusive, because it is accompanied by almost free and anticorrelated diffusion in copy numbers of ternary complexes. This near-critical behavior is relevant for balanced growth of Escherichia coli cells in media that lack amino acids and for adaptation of E. coli cells after downshifts from amino-acid-containing to amino-acid-lacking growth media. The theoretical results are used to discuss transcriptional control of amino acid synthesis during multiple amino acid limitation, the recovery of E. coli cells after nutritional downshifts and to propose a robust mechanism for the regulation of RelA-dependent synthesis of the global effector molecule ppGpp.

A steady-state modeling approach to validate an in vivo mechanism of the GAL regulatory network in Saccharomyces cerevisiae

Cellular regulation is a result of complex interactions arising from DNA-protein and protein-protein binding, autoregulation, and compartmentalization and shuttling of regulatory proteins. Experiments in molecular biology have identified these mechanisms recruited by a regulatory network. Mathematical models may be used to complement the knowledge-base provided by in vitro experimental methods. Interactions identified by in vitro experiments can lead to the hypothesis of multiple candidate models explaining the in vivo mechanism. The equilibrium dissociation constants for the various interactions and the total component concentration constitute constraints on the candidate models. In this work, we identify the most plausible in vivo network by comparing the output response to the experimental data. We demonstrate the methodology using the GAL system of Saccharomyces cerevisiae for which the steady-state analysis reveals that Gal3p neither dimerizes nor shuttles between the cytoplasm and the nucleus.

Metabolic design based on a coupled gene expression—metabolic network model of tryptophan production in Escherichia coli

The presumably high potential of a holistic design approach for complex biochemical reaction networks is exemplified here for the network of tryptophan biosynthesis from glucose, a system whose components have been investigated thoroughly before. A dynamic model that combines the behavior of the trp operon gene expression with the metabolic network of central carbon metabolism and tryptophan biosynthesis is investigated. This model is analyzed in terms of metabolic fluxes, metabolic control, and nonlinear optimization. We compare two models for a wild-type strain and another model for a tryptophan producer. An integrated optimization of the whole network leads to a significant increase in tryptophan production rate for all systems under study. This enhancement is well above the increase that can be achieved by an optimization of subsystems. A constant ratio of control coefficients on tryptophan synthesis rate has been identified for the models regarding or disregarding trp operon expression. Although we found some examples where flux control coefficients even contradict the trends of enzyme activity changes in an optimized profile, flux control can be used as an indication for enzymes that have to be taken into account in optimization.

Autoregulation of regulatory proteins is key for dynamic operation ofGALswitch inSaccharomyces cerevisiae

Autoregulation and nucleocytoplasmic shuttling play important roles in the operation of the GAL regulatory system. However, the significance of these mechanisms in the overall operation of the switch is unclear. In this work, we develop a dynamic model for the GAL system and further validate the same using steady-state and dynamic experimental expression data. Next, the model is used to delineate the relevance of shuttling and autoregulation in response to inducing, repressing, and non-inducing-non-repressing media. The analysis indicates that autoregulation of the repressor, Gal80p, is key in obtaining three distinct steady states in response to the three media. In particular, the analysis rationalizes the intuitively paradoxical observation that the concentration of repressor, Gal80p, actually increases in response to an increase in the inducer concentration. On the other hand, although nucleocytoplasmic shuttling does not affect the dynamics of the system, it plays a dominant role in obtaining a sensitive response to galactose. The dynamic model was also used to obtain insights on the preculturing effect on the system behavior.

Modeling operon dynamics: the tryptophan and lactose operons as paradigms

Understanding the regulation of gene control networks and their ensuing dynamics will be a critical component in the understanding of the mountain of genomic data being currently collected. This paper reviews recent mathematical modeling work on the tryptophan and lactose operons which are, respectively, the classical paradigms for repressible and inducible operons.