Robust signal processing in living cells

Cellular signaling networks have evolved an astonishing ability to function reliably and with high fidelity in uncertain environments. A crucial prerequisite for the high precision exhibited by many signaling circuits is their ability to keep the concentrations of active signaling compounds within tightly defined bounds, despite strong stochastic fluctuations in copy numbers and other detrimental influences. Based on a simple mathematical formalism, we identify topological organizing principles that facilitate such robust control of intracellular concentrations in the face of multifarious perturbations. Our framework allows us to judge whether a multiple-input-multiple-output reaction network is robust against large perturbations of network parameters and enables the predictive design of perfectly robust synthetic network architectures. Utilizing the Escherichia coli chemotaxis pathway as a hallmark example, we provide experimental evidence that our framework indeed allows us to unravel the topological organization of robust signaling. We demonstrate that the specific organization of the pathway allows the system to maintain global concentration robustness of the diffusible response regulator CheY with respect to several dominant perturbations. Our framework provides a counterpoint to the hypothesis that cellular function relies on an extensive machinery to fine-tune or control intracellular parameters. Rather, we suggest that for a large class of perturbations, there exists an appropriate topology that renders the network output invariant to the respective perturbations.     

Feedback control architecture and the bacterial chemotaxis network

Bacteria move towards favourable and away from toxic environments by changing their swimming pattern. This response is regulated by the chemotaxis signalling pathway, which has an important feature: it uses feedback to 'reset' (adapt) the bacterial sensing ability, which allows the bacteria to sense a range of background environmental changes. The role of this feedback has been studied extensively in the simple chemotaxis pathway of Escherichia coli. However it has been recently found that the majority of bacteria have multiple chemotaxis homologues of the E. coli proteins, resulting in more complex pathways. In this paper we investigate the configuration and role of feedback in Rhodobacter sphaeroides, a bacterium containing multiple homologues of the chemotaxis proteins found in E. coli. Multiple proteins could produce different possible feedback configurations, each having different chemotactic performance qualities and levels of robustness to variations and uncertainties in biological parameters and to intracellular noise. We develop four models corresponding to different feedback configurations. Using a series of carefully designed experiments we discriminate between these models and invalidate three of them. When these models are examined in terms of robustness to noise and parametric uncertainties, we find that the non-invalidated model is superior to the others. Moreover, it has a 'cascade control' feedback architecture which is used extensively in engineering to improve system performance, including robustness. Given that the majority of bacteria are known to have multiple chemotaxis pathways, in this paper we show that some feedback architectures allow them to have better performance than others. In particular, cascade control may be an important feature in achieving robust functionality in more complex signalling pathways and in improving their performance.     

Robustness and fragility in the yeast high osmolarity glycerol (HOG) signal‐transduction pathway

Cellular signalling networks integrate environmental stimuli with the information on cellular status. These networks must be robust against stochastic fluctuations in stimuli as well as in the amounts of signalling components. Here, we challenge the yeast HOG signal‐transduction pathway with systematic perturbations in components’ expression levels under various external conditions in search for nodes of fragility. We observe a substantially higher frequency of fragile nodes in this signal‐transduction pathway than that has been observed for other cellular processes. These fragilities disperse without any clear pattern over biochemical functions or location in pathway topology and they are largely independent of pathway activation by external stimuli. However, the strongest toxicities are caused by pathway hyperactivation. In silico analysis highlights the impact of model structure on in silico robustness, and suggests complex formation and scaffolding as important contributors to the observed fragility patterns. Thus, in vivo robustness data can be used to discriminate and improve mathematical models.     

Thermal robustness of signaling in bacterial chemotaxis

Temperature is a global factor that affects the performance of all intracellular networks. Robustness against temperature variations is thus expected to be an essential network property, particularly in organisms without inherent temperature control. Here, we combine experimental analyses with computational modeling to investigate thermal robustness of signaling in chemotaxis of Escherichia coli, a relatively simple and well-established model for systems biology. We show that steady-state and kinetic pathway parameters that are essential for chemotactic performance are indeed temperature-compensated in the entire physiological range. Thermal robustness of steady-state pathway output is ensured at several levels by mutual compensation of temperature effects on activities of individual pathway components. Moreover, the effect of temperature on adaptation kinetics is counterbalanced by preprogrammed temperature dependence of enzyme synthesis and stability to achieve nearly optimal performance at the growth temperature. Similar compensatory mechanisms are expected to ensure thermal robustness in other systems.     

Strong negative feedback from Erk to Raf confers robustness to MAPK signalling

Protein levels within signal transduction pathways vary strongly from cell to cell. Here, we analysed how signalling pathways can still process information quantitatively despite strong heterogeneity in protein levels. We systematically perturbed the protein levels of Erk, the terminal kinase in the MAPK signalling pathway in a panel of human cell lines. We found that the steady‐state phosphorylation of Erk is very robust against perturbations of Erk protein level. Although a multitude of mechanisms exist that may provide robustness against fluctuating protein levels, we found that one single feedback from Erk to Raf‐1 accounts for the observed robustness. Surprisingly, robustness is provided through a fast post‐translational mechanism although variation of Erk levels occurs on a timescale of days.     

Coherent coupling of feedback loops: a design principle of cell signaling networks

MOTIVATION: It is widely accepted that cell signaling networks have been evolved to be robust against perturbations. To investigate the topological characteristics resulting in such robustness, we have examined large-scale signaling networks and found that a number of feedback loops are present mostly in coupled structures. In particular, the coupling was made in a coherent way implying that same types of feedback loops are interlinked together. RESULTS: We have investigated the role of such coherently coupled feedback loops through extensive Boolean network simulations and found that a high proportion of coherent couplings can enhance the robustness of a network against its state perturbations. Moreover, we found that the robustness achieved by coherently coupled feedback loops can be kept evolutionarily stable. All these results imply that the coherent coupling of feedback loops might be a design principle of cell signaling networks devised to achieve the robustness.     

The role of multiple chemotactic mechanisms in a model of chemotaxis in C. elegans: different mechanisms are specialised for different environments

Unlike simpler organisms, C. elegans possesses several distinct chemosensory pathways and chemotactic mechanisms. These mechanisms and pathways are individually capable of driving chemotaxis in a chemical concentration gradient. However, it is not understood if they are redundant or co-operate in more sophisticated ways. Here we examine the specialisation of different chemotactic mechanisms in a model of chemotaxis to NaCl. We explore the performance of different chemotactic mechanisms in a range of chemical gradients and show that, in the model, far from being redundant, the mechanisms are specialised both for different environments and for distinct features within those environments. We also show that the chemotactic drive mediated by the ASE pathway is not robust to the presence of noise in the chemical gradient. This problem cannot be solved along the ASE pathway without destroying its ability to drive chemotaxis. Instead, we show that robustness to noise can be achieved by introducing a second, much slower NaCl-sensing pathway. This secondary pathway is simpler than the ASE pathway, in the sense that it can respond to either up-steps or down-steps in NaCl but not both, and could correspond to one of several candidates in the literature which we identify and evaluate. This work provides one possible explanation of why there are multiple NaCl sensing pathways and chemotactic mechanisms in C. elegans: rather than being redundant the different pathways and mechanism are specialised both for the characteristics of different environments and for distinct features within a single environment.     

Adaptation Dynamics in Densely Clustered Chemoreceptors

In many sensory systems, transmembrane receptors are spatially organized in large clusters. Such arrangement may facilitate signal amplification and the integration of multiple stimuli. However, this organization likely also affects the kinetics of signaling since the cytoplasmic enzymes that modulate the activity of the receptors must localize to the cluster prior to receptor modification. Here we examine how these spatial considerations shape signaling dynamics at rest and in response to stimuli. As a model system, we use the chemotaxis pathway of Escherichia coli, a canonical system for the study of how organisms sense, respond, and adapt to environmental stimuli. In bacterial chemotaxis, adaptation is mediated by two enzymes that localize to the clustered receptors and modulate their activity through methylation-demethylation. Using a novel stochastic simulation, we show that distributive receptor methylation is necessary for successful adaptation to stimulus and also leads to large fluctuations in receptor activity in the steady state. These fluctuations arise from noise in the number of localized enzymes combined with saturated modification kinetics between the localized enzymes and the receptor substrate. An analytical model explains how saturated enzyme kinetics and large fluctuations can coexist with an adapted state robust to variation in the expression levels of the pathway constituents, a key requirement to ensure the functionality of individual cells within a population. This contrasts with the well-mixed covalent modification system studied by Goldbeter and Koshland in which mean activity becomes ultrasensitive to protein abundances when the enzymes operate at saturation. Large fluctuations in receptor activity have been quantified experimentally and may benefit the cell by enhancing its ability to explore empty environments and track shallow nutrient gradients. Here we clarify the mechanistic relationship of these large fluctuations to well-studied aspects of the chemotaxis system, precise adaptation and functional robustness.     

Funneled landscape leads to robustness of cell networks: yeast cell cycle

We uncovered the underlying energy landscape for a cellular network. We discovered that the energy landscape of the yeast cell-cycle network is funneled towards the global minimum (G0/G1 phase) from the experimentally measured or inferred inherent chemical reaction rates. The funneled landscape is quite robust against random perturbations. This naturally explains robustness from a physical point of view. The ratio of slope versus roughness of the landscape becomes a quantitative measure of robustness of the network. The funneled landscape can be seen as a possible realization of the Darwinian principle of natural selection at the cellular network level. It provides an optimal criterion for network connections and design. Our approach is general and can be applied to other cellular networks.     

Evolution of robustness to noise and mutation in gene expression dynamics

Phenotype of biological systems needs to be robust against mutation in order to sustain themselves between generations. On the other hand, phenotype of an individual also needs to be robust against fluctuations of both internal and external origins that are encountered during growth and development. Is there a relationship between these two types of robustness, one during a single generation and the other during evolution? Could stochasticity in gene expression have any relevance to the evolution of these types of robustness? Robustness can be defined by the sharpness of the distribution of phenotype; the variance of phenotype distribution due to genetic variation gives a measure of 'genetic robustness', while that of isogenic individuals gives a measure of 'developmental robustness'. Through simulations of a simple stochastic gene expression network that undergoes mutation and selection, we show that in order for the network to acquire both types of robustness, the phenotypic variance induced by mutations must be smaller than that observed in an isogenic population. As the latter originates from noise in gene expression, this signifies that the genetic robustness evolves only when the noise strength in gene expression is larger than some threshold. In such a case, the two variances decrease throughout the evolutionary time course, indicating increase in robustness. The results reveal how noise that cells encounter during growth and development shapes networks' robustness to stochasticity in gene expression, which in turn shapes networks' robustness to mutation. The necessary condition for evolution of robustness, as well as the relationship between genetic and developmental robustness, is derived quantitatively through the variance of phenotypic fluctuations, which are directly measurable experimentally.     

Robustness to noise in gene expression evolves despite epistatic constraints in a model of gene networks

Stochastic noise in gene expression causes variation in the development of phenotypes, making such noise a potential target of stabilizing selection. Here, we develop a new simulation model of gene networks to study the adaptive landscape underlying the evolution of robustness to noise. We find that epistatic interactions between the determinants of the expression of a gene and its downstream effect impose significant constraints on evolution, but these interactions do allow the gradual evolution of increased robustness. Despite strong sign epistasis, adaptation rarely proceeds via deleterious intermediate steps, but instead occurs primarily through small beneficial mutations. A simple mathematical model captures the relevant features of the single‐gene fitness landscape and explains counterintuitive patterns, such as a correlation between the mean and standard deviation of phenotypes. In more complex networks, mutations in regulatory regions provide evolutionary pathways to increased robustness. These results chart the constraints and possibilities of adaptation to reduce expression noise and demonstrate the potential of a novel modeling framework for gene networks.     

Accounting for robustness in modeling signal transduction responses

Abstract

A classical question in systems biology is to find a Boolean model which is able to predict the observed responses of a signaling network. It has been previously shown that such models can be tailored based on experimental data. While fitting a minimum-size network to the experimentally observed data is a natural assumption, it can potentially result in a network which is not so robust against the noises in the training dataset. Indeed, it is widely accepted now that biological systems are generally evolved to be very robust. Therefore, in the present work, we extended the classical formulation of Boolean network construction in order to put weight on the robustness of the created network. We show that our method results generally in more relevant networks. Consequently, considering robustness as a design principle of biological networks can result in more realistic models.     

Mutual repression enhances the steepness and precision of gene expression boundaries

Embryonic development is driven by spatial patterns of gene expression that determine the fate of each cell in the embryo. While gene expression is often highly erratic, embryonic development is usually exceedingly precise. In particular, gene expression boundaries are robust not only against intra-embryonic fluctuations such as noise in gene expression and protein diffusion, but also against embryo-to-embryo variations in the morphogen gradients, which provide positional information to the differentiating cells. How development is robust against intra- and inter-embryonic variations is not understood. A common motif in the gene regulation networks that control embryonic development is mutual repression between pairs of genes. To assess the role of mutual repression in the robust formation of gene expression patterns, we have performed large-scale stochastic simulations of a minimal model of two mutually repressing gap genes in Drosophila, hunchback (hb) and knirps (kni). Our model includes not only mutual repression between hb and kni, but also the stochastic and cooperative activation of hb by the anterior morphogen Bicoid (Bcd) and of kni by the posterior morphogen Caudal (Cad), as well as the diffusion of Hb and Kni between neighboring nuclei. Our analysis reveals that mutual repression can markedly increase the steepness and precision of the gap gene expression boundaries. In contrast to other mechanisms such as spatial averaging and cooperative gene activation, mutual repression thus allows for gene-expression boundaries that are both steep and precise. Moreover, mutual repression dramatically enhances their robustness against embryo-to-embryo variations in the morphogen levels. Finally, our simulations reveal that diffusion of the gap proteins plays a critical role not only in reducing the width of the gap gene expression boundaries via the mechanism of spatial averaging, but also in repairing patterning errors that could arise because of the bistability induced by mutual repression.     

Phenotypic variability promotes diversity and stability in competitive communities

Intraspecific variation is at the core of evolutionary theory, and yet, from an ecological perspective, we have few robust expectations for how this variation should affect the dynamics of large communities. Here, by adapting an approach from evolutionary game theory, we show that the incorporation of phenotypic variability into competitive networks dramatically alters the dynamics across ecological timescales, stabilising the systems and buffering the communities against demographic perturbations. The beneficial effects of phenotypic variability are strongest when there are substantial differences among phenotypes and when phenotypes are inherited with moderately high fidelity; yet even low levels of variation lead to significant increases in diversity, stability, and robustness. By identifying a simple and ubiquitous stabilising force in competitive communities, this work contributes to our core understanding of how biological diversity is maintained in natural systems.     

The Relationship between Gene Network Structure and Expression Variation among Individuals and Species

Variation among individuals is a prerequisite of evolution by natural selection. As such, identifying the origins of variation is a fundamental goal of biology. We investigated the link between gene interactions and variation in gene expression among individuals and species using the mammalian limb as a model system. We first built interaction networks for key genes regulating early (outgrowth; E9.5–11) and late (expansion and elongation; E11-13) limb development in mouse. This resulted in an Early (ESN) and Late (LSN) Stage Network. Computational perturbations of these networks suggest that the ESN is more robust. We then quantified levels of the same key genes among mouse individuals and found that they vary less at earlier limb stages and that variation in gene expression is heritable. Finally, we quantified variation in gene expression levels among four mammals with divergent limbs (bat, opossum, mouse and pig) and found that levels vary less among species at earlier limb stages. We also found that variation in gene expression levels among individuals and species are correlated for earlier and later limb development. In conclusion, results are consistent with the robustness of the ESN buffering among-individual variation in gene expression levels early in mammalian limb development, and constraining the evolution of early limb development among mammalian species.