Evolution of design principles in biochemical networks

Synchronization of stochastic expressions drives the clustering of functionally related genes

Functionally related genes tend to be chromosomally clustered in eukaryotic genomes even after the exclusion of tandem duplicates, but the biological significance of this widespread phenomenon is unclear. We propose that stochastic expression fluctuations of neighboring genes resulting from chromatin dynamics are more or less synchronized such that their expression ratio is more stable than that for unlinked genes. Consequently, chromosomal clustering could be advantageous when the expression ratio of the clustered genes needs to stay constant, for example, because of the accumulation of toxic compounds when this ratio is altered. Evidence from manipulative experiments on the yeast GAL cluster, comprising three chromosomally adjacent genes encoding enzymes catalyzing consecutive reactions in galactose catabolism, unequivocally supports this hypothesis and elucidates how disorder in one biological phenomenon-gene expression noise-could prompt the emergence of order in another-genome organization.

Hybrid CME-ODE method for efficient simulation of the galactose switch in yeast

It is well known that stochasticity in gene expression is an important source of noise that can have profound effects on the fate of a living cell. In the galactose genetic switch in yeast, the unbinding of a transcription repressor is induced by high concentrations of sugar particles activating gene expression of sugar transporters. This response results in high propensity for all reactions involving interactions with the metabolite. The reactions for gene expression, feedback loops and transport are typically described by chemical master equations (CME). Sampling the CME using the stochastic simulation algorithm (SSA) results in large computational costs as each reaction event is evaluated explicitly. To improve the computational efficiency of cell simulations involving high particle number systems, the authors have implemented a hybrid stochastic–deterministic (CME–ODE) method into the publically available, GPU‐based lattice microbes (LM) software suite and its python interface pyLM. LM and pyLM provide a convenient way to simulate complex cellular systems and interface with high‐performance RDME/CME/ODE solvers. As a test of the implementation, the authors apply the hybrid CME‐ODE method to the galactose switch in Saccharomyces cerevisiae, gaining a 10–50× speedup while yielding protein distributions and species traces similar to the pure SSA CME.

Modular bond-graph modelling and analysis of biomolecular systems

Bond graphs can be used to build thermodynamically-compliant hierarchical models of biomolecular systems. As bond graphs have been widely used to model, analyse and synthesise engineering systems, this study suggests that they can play the same rôle in the modelling, analysis and synthesis of biomolecular systems. The particular structure of bond graphs arising from biomolecular systems is established and used to elucidate the relation between thermodynamically closed and open systems. Block diagram representations of the dynamics implied by these bond graphs are used to reveal implicit feedback structures and are linearised to allow the application of control-theoretical methods. Two concepts of modularity are examined: computational modularity where physical correctness is retained and behavioural modularity where module behaviour (such as ultrasensitivity) is retained. As well as providing computational modularity, bond graphs provide a natural formulation of behavioural modularity and reveal the sources of retroactivity. A bond graph approach to reducing retroactivity, and thus inter-module interaction, is shown to require a power supply such as that provided by the ATP ⇌ ADP + Pi reaction. The mitogen-activated protein kinase cascade (Raf-MEK-ERK pathway) is used as an illustrative example.

The yeast galactose network as a quantitative model for cellular memory

Recent experiments have revealed surprising behavior in the yeast galactose (GAL) pathway, one of the preeminent systems for studying gene regulation. Under certain circumstances, yeast cells display memory of their prior nutrient environments. We distinguish two kinds of cellular memory discovered by quantitative investigations of the GAL network and present a conceptual framework for interpreting new experiments and current ideas on GAL memory. Reinduction memory occurs when cells respond transcriptionally to one environment, shut down the response during several generations in a second environment, then respond faster and with less cell-to-cell variation when returned to the first environment. Persistent memory describes a long-term, arguably stable response in which cells adopt a bimodal or unimodal distribution of induction levels depending on their preceding environment. Deep knowledge of how the yeast GAL pathway responds to different sugar environments has enabled rapid progress in uncovering the mechanisms behind GAL memory, which include cytoplasmic inheritance of inducer proteins and positive feedback loops among regulatory genes. This network of genes, long used to study gene regulation, is now emerging as a model system for cellular memory.

Validation of a model of the GAL regulatory system via robustness analysis of its bistability characteristics

Background

In Saccharomyces cerevisiæ, structural bistability generates a bimodal expression of the galactose uptake genes (GAL) when exposed to low and high glucose concentrations. This indicates that yeast cells can decide between using either the limited amount of glucose or growing on galactose under changing environmental conditions. A crucial requirement for any plausible mechanistic model of this system is that it reproduces the robustness of the bistable response observed in vivo against inter-individual parametric variability and fluctuating environmental conditions.

Results

We show how a control-theoretic analysis of the robustness of a model of the GAL regulatory network may be used to establish the model’s plausibility in characterizing the persistent memory of different carbon sources, without the need for extensive simulations. Chemical Reaction Network Theory is used to establish that the proposed network model is compatible with structural bistability. The robustness of each of the two operative conditions against fluctuations of the species concentrations is demonstrated by studying the Domains of Attraction of the corresponding equilibrium points. Finally, we use a global robustness analysis method based on Semi-Definite Programming to evaluate the modification of the bistable steady states induced by multiple parametric variations throughout bounded regions of the parameter space.

Conclusions

Our analysis provides convincing evidence for the robustness, and hence plausibility, of the GAL regulatory network model. The proposed workflow also demonstrates the power of analytical methods from control theory to provide a direct quantitative characterization of the dynamics of multistable biomolecular regulatory systems without recourse to extensive computer simulations.

Gene network requirements for regulation of metabolic gene expression to a desired state

Gene circuits that control metabolism should restore metabolic functions upon environmental changes. Whether gene networks are capable of steering metabolism to optimal states is an open question. Here we present a method to identify such optimal gene networks. We show that metabolic network optimisation over a range of environments results in an input-output relationship for the gene network that guarantees optimal metabolic states. Optimal control is possible if the gene network can achieve this input-output relationship. We illustrate our approach with the best-studied regulatory network in yeast, the galactose network. We find that over the entire range of external galactose concentrations, the regulatory network is able to optimally steer galactose metabolism. Only a few gene network parameters affect this optimal regulation. The other parameters can be tuned independently for optimisation of other functions, such as fast and low-noise gene expression. This study highlights gene network plasticity, evolvability, and modular functionality.

Structural bistability of the GAL regulatory network and characterization of its domains of attraction

Bistability is a system-level property, exploited by many biomolecular interaction networks as a key mechanism to accomplish different cellular functions (e.g., differentiation, cell cycle, switch-like response to external stimuli). Bistability has also been experimentally found to occur in the regulatory network of the galactose metabolic pathway in the model organism Saccharomyces cerevisiae. In this yeast, bistability generates a persistent memory of the type of carbon source available in the extracellular medium: under the same experimental conditions, cells previously grown with different nutrients generate different responses and get stably locked into two distinct steady states. The molecular interactions of the GAL regulatory network have been thoroughly dissected through wet-lab experiments; thus, this system provides a formidable benchmark to our ability to characterize and reproduce in silico the behavior of bistable biological systems. To this aim, a number of models have been proposed in the literature; however, we found that they are not able to replicate the persistent memory behavior observed in (Acar et al., 2005 ). The present study proposes a novel model of the GAL regulatory network, which, in addition to reproducing in silico the experimental findings, can be formally analyzed for structural multistability of the network, using chemical reaction network theory (CRNT), and allows the characterization of the domains of attraction (DA). This work provides further insights into the GAL system and proposes an easily generalizable approach to the study of bistability-associated behaviors in biological systems.

Mathematical model of GAL regulon dynamics in Saccharomyces cerevisiae

Genetic switches are prevalent in nature and provide cells with a strategy to adapt to changing environments. The GAL switch is an intriguing example which is not understood in all detail. The GAL switch allows organisms to metabolize galactose, and controls whether the machinery responsible for the galactose metabolism is turned on or off. Currently, it is not known exactly how the galactose signal is sensed by the transcriptional machinery. Here we utilize quantitative tools to understand the S. cerevisiae cell response to galactose challenge, and to analyze the plausible molecular mechanisms underlying its operation. We work at a population level to develop a dynamic model based on the interplay of the key regulatory proteins Gal4p, Gal80p, and Gal3p. To our knowledge, the model presented here is the first to reproduce qualitatively the bistable network behavior found experimentally. Given the current understanding of the GAL circuit induction (Wightman et al., 2008; Jiang et al., 2009), we propose that the most likely in vivo mechanism leading to the transcriptional activation of the GAL genes is the physical interaction between galactose-activated Gal3p and Gal80p, with the complex Gal3p-Gal80p remaining bound at the GAL promoters. Our mathematical model is in agreement with the flow cytometry profiles of wild type, gal3Δ and gal80Δ mutant strains from Acar et al. (2005), and involves a fraction of actively transcribing cells with the same qualitative features as in the data set collected by Acar et al. (2010). Furthermore, the computational modeling provides an explanation for the contradictory results obtained by independent laboratories when tackling experimentally the issue of binary versus graded response to galactose induction.

Understanding the dynamic behavior of genetic regulatory networks by functional decomposition

A number of mechanistic and predictive genetic regulatory networks (GRNs) comprising dozens of genes have already been characterized at the level of cis-regulatory interactions. Reconstructions of networks of 100's to 1000's of genes and their interactions are currently underway. Understanding the organizational and functional principles underlying these networks is probably the single greatest challenge facing genomics today. We review the current approaches to deciphering large-scale GRNs and discuss some of their limitations. We then propose a bottom-up approach in which large-scale GRNs are first organized in terms of functionally distinct GRN building blocks of one or a few genes. Biological processes may then be viewed as the outcome of functional interactions among these simple, well-characterized functional building blocks. We describe several putative GRN functional building blocks and show that they can be located within GRNs on the basis of their interaction topology and additional, simple and experimentally testable constraints.

External control of the GAL network in S. cerevisiae: a view from control theory

While there is a vast literature on the control systems that cells utilize to regulate their own state, there is little published work on the formal application of control theory to the external regulation of cellular functions. This paper chooses the GAL network in S. cerevisiae as a well understood benchmark example to demonstrate how control theory can be employed to regulate intracellular mRNA levels via extracellular galactose. Based on a mathematical model reduced from the GAL network, we have demonstrated that a galactose dose necessary to drive and maintain the desired GAL genes' mRNA levels can be calculated in an analytic form. And thus, a proportional feedback control can be designed to precisely regulate the level of mRNA. The benefits of the proposed feedback control are extensively investigated in terms of stability and parameter sensitivity. This paper demonstrates that feedback control can both significantly accelerate the process to precisely regulate mRNA levels and enhance the robustness of the overall cellular control system.

Mathematical modeling of biomolecular network dynamics

Mathematical and computational models have become indispensable tools for integrating and interpreting heterogeneous biological data, understanding fundamental principles of biological system functions, genera-ting reliable testable hypotheses, and identifying potential diagnostic markers and therapeutic targets. Thus, such tools are now routinely used in the theoretical and experimental systematic investigation of biological system dynamics. Here, we discuss model building as an essential part of the theoretical and experimental analysis of biomolecular network dynamics. Specifically, we describe a procedure for defining kinetic equations and parameters of biomolecular processes, and we illustrate the use of fractional activity functions for modeling gene expression regulation by single and multiple regulators. We further discuss the evaluation of model complexity and the selection of an optimal model based on information criteria. Finally, we discuss the critical roles of sensitivity, robustness analysis, and optimal experiment design in the model building cycle.

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.

Metabolic gene regulation in a dynamically changing environment

Natural selection dictates that cells constantly adapt to dynamically changing environments in a context-dependent manner. Gene-regulatory networks often mediate the cellular response to perturbation, and an understanding of cellular adaptation will require experimental approaches aimed at subjecting cells to a dynamic environment that mimics their natural habitat. Here we monitor the response of Saccharomyces cerevisiae metabolic gene regulation to periodic changes in the external carbon source by using a microfluidic platform that allows precise, dynamic control over environmental conditions. We show that the metabolic system acts as a low-pass filter that reliably responds to a slowly changing environment, while effectively ignoring fast fluctuations. The sensitive low-frequency response was significantly faster than in predictions arising from our computational modelling, and this discrepancy was resolved by the discovery that two key galactose transcripts possess half-lives that depend on the carbon source. Finally, to explore how induction characteristics affect frequency response, we compare two S. cerevisiae strains and show that they have the same frequency response despite having markedly different induction properties. This suggests that although certain characteristics of the complex networks may differ when probed in a static environment, the system has been optimized for a robust response to a dynamically changing environment.

Quantification of intracellular phosphorylated carbohydrates in HT29 human colon adenocarcinoma cell line using liquid chromatography-electrospray ionization tandem mass spectrometry

The quantitative understanding of the role of sugar phosphates in regulating tumor energetic metabolism at the proteomic and genomic level is a prerequisite for an efficient rational design in combined drug chemotherapy. Therefore, it is necessary to determine accurately the concentration of the main sugar phosphate pools at the lower concentrations present in the often-limited volume of tumor cell samples. Taking as an example the human adenocarcinoma cell line HT29, we here report a fast and reliable quantitative method based on the use of liquid nitrogen, a weak acid extraction, and liquid chromatography-electrospray ionization tandem mass spectrometry to quantify simultaneously the intracellular concentration of sugar phosphate pools. The method was set up using standard addition curves. Thus, it is possible to identify and quantify hexose phosphate, pentose phosphate, and triose phosphate pools up to 0.02-0.10 ng x microL(-1), depending on the analyte. The method developed was here used for the quantitative study of changes in phosphorylated carbohydrates of central carbon metabolism when high or low glucose concentration conditions are induced in vitro in the HT29 human colon adenocarcinoma cell line.

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

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

Transcriptional noise and cellular heterogeneity in mammalian macrophages

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

Multiscale modeling of cardiac cellular energetics

Multiscale modeling is essential to integrating knowledge of human physiology starting from genomics, molecular biology, and the environment through the levels of cells, tissues, and organs all the way to integrated systems behavior. The lowest levels concern biophysical and biochemical events. The higher levels of organization in tissues, organs, and organism are complex, representing the dynamically varying behavior of billions of cells interacting together. Models integrating cellular events into tissue and organ behavior are forced to resort to simplifications to minimize computational complexity, thus reducing the model's ability to respond correctly to dynamic changes in external conditions. Adjustments at protein and gene regulatory levels shortchange the simplified higher-level representations. Our cell primitive is composed of a set of subcellular modules, each defining an intracellular function (action potential, tricarboxylic acid cycle, oxidative phosphorylation, glycolysis, calcium cycling, contraction, etc.), composing what we call the "eternal cell," which assumes that there is neither proteolysis nor protein synthesis. Within the modules are elements describing each particular component (i.e., enzymatic reactions of assorted types, transporters, ionic channels, binding sites, etc.). Cell subregions are stirred tanks, linked by diffusional or transporter-mediated exchange. The modeling uses ordinary differential equations rather than stochastic or partial differential equations. This basic model is regarded as a primitive upon which to build models encompassing gene regulation, signaling, and long-term adaptations in structure and function. During simulation, simpler forms of the model are used, when possible, to reduce computation. However, when this results in error, the more complex and detailed modules and elements need to be employed to improve model realism. The processes of error recognition and of mapping between different levels of model form complexity are challenging but are essential for successful modeling of large-scale systems in reasonable time. Currently there is to this end no established methodology from computational sciences.

Dizzy: stochastic simulation of large-scale genetic regulatory networks

We describe Dizzy, a software tool for stochastically and deterministically modeling the spatially homogeneous kinetics of integrated large-scale genetic, metabolic, and signaling networks. Notable features include a modular simulation framework, reusable modeling elements, complex kinetic rate laws, multi-step reaction processes, steady-state noise estimation, and spatial compartmentalization.

Replacement of a conserved tyrosine by tryptophan in Gal3p of Saccharomyces cerevisiae reduces constitutive activity: implications for signal transduction in the GAL regulon

The ability of Saccharomyces cerevisiae to utilize galactose is regulated by the nucleo-cytoplasmic shuttling of a transcriptional repressor, the Gal80 protein. Gal80 interacts with the transcriptional activator Gal4 in the nucleus and inhibits its function, preventing induction of the GAL genes. In response to galactose, the relative amounts of Gal80 in the cytoplasm and the nucleus are modulated by the action of a signal transducer, Gal3. Although it has been speculated that Gal3 binds galactose, this has not been experimentally demonstrated. In this study, we show that replacement of a conserved tyrosine in Gal3 by tryptophan leads to a reduction of its constitutive activity in the absence of galactose. In addition, this mutant protein was fully functional in vivo only when high concentrations of galactose were present in the medium. When overexpressed, the mutant was found to activate the genes GAL1 and GAL7/10 differentially. The implications of these findings for the fine regulation of GAL genes, and its physiological significance, are discussed.

Is the regulation of galactose 1-phosphate tuned against gene expression noise?

The average number of mRNA molecules per active gene in yeast can be remarkably low. Consequently, the relative number of copies of each transcript per cell can vary greatly from moment to moment. When these transcripts are encoding metabolic enzymes, how do the resulting variations in enzyme concentrations affect the regulation of metabolic intermediates? Using a kinetic model of galactose utilization in yeast, we analysed the transmission of noise from transcription and translation on metabolic intermediate regulation. In particular, the effect of the kinetic properties of the galactose-1-phosphate uridylyltransferase reaction on the transmission of noise was analysed.

Stochastic variation in the concentration of a repressor activates GAL genetic switch: implications in evolution of regulatory network

In Saccharomyces cerevisiae, a recessive mutation in the signal transducer encoded by GAL3 leads to a significant lag in the induction of GAL genes, referred to as long term adaptation phenotype (LTA). Further, gal3 mutation in combination with other genetic defects leads to the non-inducibility of GAL genes. It was shown that the expression of GAL1 encoded galactokinase, a redundant GAL3 like signal transducer, eventually substitutes for the lack of GAL3 signal transduction function. However, how GAL1 gets induced in the absence of GAL3 is not clear. We hypothesize that GAL1 induction in gal3 cells exposed to galactose is due to a stochastic decrease in the repressor, Gal80p concentration, leading to heterogeneity in the population. This observation explains not only LTA observed in gal3 cells but also explains the non-inducibility of gal3 mutants in combination with other genetic defects. By recruiting a dedicated signal transducer, GAL3, S. cerevisiae GAL switch has evolved to overcome the fortuitous induction, which occurs due to low signal to noise ratio in certain mutants of Escherichia coli and Kluveromyces lactis.

Control of internal and external noise in genetic regulatory networks

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

A method for estimating stochastic noise in large genetic regulatory networks

MOTIVATION

Genetic regulatory networks are often affected by stochastic noise, due to the low number of molecules taking part in certain reactions. The networks can be simulated using stochastic techniques that model each reaction as a stochastic event. As models become increasingly large and sophisticated, however, the solution time can become excessive; particularly if one wishes to determine the effect on noise of changes to a series of parameters, or the model structure. Methods are therefore required to rapidly estimate stochastic noise.

RESULTS

This paper presents an algorithm, based on error growth techniques from non-linear dynamics, to rapidly estimate the noise characteristics of genetic networks of arbitrary size. The method can also be used to determine analytical solutions for simple sub-systems. It is demonstrated on a number of cases, including a prototype model of the galactose regulatory pathway in yeast.

AVAILABILITY

A software tool which incorporates the algorithm is available for use as part of the stochastic simulation package Dizzy. It is available for download at http://labs.systemsbiology.net/bolouri/software/Dizzy/

CONTACT

dorrell@systemsbiology.org

SUPPLEMENTARY INFORMATION

A conceptual model of the regulatory part of the galactose utilization pathway in yeast, used as an example in the paper, is available at http://labs.systemsbiology.net/bolouri/models/galconcept.dizzy