A fundamental trade-off in covalent switching and its circumvention by enzyme bifunctionality in glucose homeostasis

The linear framework II: using graph theory to analyse the transient regime of Markov processes

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent chemical species or molecular states, edges represent reactions or transitions and edge labels represent rates that also describe how the system is interacting with its environment. The present paper is a sequel to a recent review of the framework that focussed on how graph-theoretic methods give insight into steady states as rational algebraic functions of the edge labels. Here, we focus on the transient regime for systems that correspond to continuous-time Markov processes. In this case, the graph specifies the infinitesimal generator of the process. We show how the moments of the first-passage time distribution, and related quantities, such as splitting probabilities and conditional first-passage times, can also be expressed as rational algebraic functions of the labels. This capability is timely, as new experimental methods are finally giving access to the transient dynamic regime and revealing the computations and information processing that occur before a steady state is reached. We illustrate the concepts, methods and formulas through examples and show how the results may be used to illuminate previous findings in the literature.

Protein Kinase Signaling Networks Driven by Oncogenic Gq/11 in Uveal Melanoma Identified by Phosphoproteomic and Bioinformatic Analyses

Metastatic uveal melanoma (UM) patients typically survive only 2 to 3 years because effective therapy does not yet exist. Here, to facilitate the discovery of therapeutic targets in UM, we have identified protein kinase signaling mechanisms elicited by the drivers in 90% of UM tumors: mutant constitutively active G protein α-subunits encoded by GNAQ (Gq) or GNA11 (G11). We used the highly specific Gq/11 inhibitor FR900359 (FR) to elucidate signaling networks that drive proliferation, metabolic reprogramming, and dedifferentiation of UM cells. We determined the effects of FR on the proteome and phosphoproteome of UM cells as indicated by bioinformatic analyses with CausalPath and site-specific gene set enrichment analysis. We found that inhibition of oncogenic Gq/11 caused deactivation of PKC, Erk, and the cyclin-dependent kinases CDK1 and CDK2 that drive proliferation. Inhibition of oncogenic Gq/11 in UM cells with low metastatic risk relieved inhibitory phosphorylation of polycomb-repressive complex subunits that regulate melanocytic redifferentiation. Site-specific gene set enrichment analysis, unsupervised analysis, and functional studies indicated that mTORC1 and 6-phosphofructo-2-kinase/fructose-2,6-biphosphatase 2 drive metabolic reprogramming in UM cells. Together, these results identified protein kinase signaling networks driven by oncogenic Gq/11 that regulate critical aspects of UM cell biology and provide targets for therapeutic investigation.

Thermodynamic bounds on ultrasensitivity in covalent switching

Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants and thereby reflect fundamental physical constraints on covalent switching.

The linear framework: using graph theory to reveal the algebra and thermodynamics of biomolecular systems

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent biochemical species or molecular states, edges represent reactions or transitions and labels represent rates. The graph yields a linear dynamics for molecular concentrations or state probabilities, with the graph Laplacian as the operator, and the labels encode the nonlinear interactions between system and environment. The labels can be specified by vertices of other graphs or by conservation laws or, when the environment consists of thermodynamic reservoirs, they may be constants. In the latter case, the graphs correspond to infinitesimal generators of Markov processes. The key advantage of the framework has been that steady states are determined as rational algebraic functions of the labels by the Matrix-Tree theorems of graph theory. When the system is at thermodynamic equilibrium, this prescription recovers equilibrium statistical mechanics but it continues to hold for non-equilibrium steady states. The framework goes beyond other graph-based approaches in treating the graph as a mathematical object, for which general theorems can be formulated that accommodate biomolecular complexity. It has been particularly effective at analysing enzyme-catalysed modification systems and input-output responses.

Emergence and Enhancement of Ultrasensitivity through Posttranslational Modulation of Protein Stability

Signal amplification in biomolecular networks converts a linear input to a steeply sigmoid output and is central to a number of cellular functions including proliferation, differentiation, homeostasis, adaptation, and biological rhythms. One canonical signal amplifying motif is zero-order ultrasensitivity that is mediated through the posttranslational modification (PTM) cycle of signaling proteins. The functionality of this signaling motif has been examined conventionally by supposing that the total amount of the protein substrates remains constant, as by the classical Koshland-Goldbeter model. However, covalent modification of signaling proteins often results in changes in their stability, which affects the abundance of the protein substrates. Here, we use mathematical models to explore the signal amplification properties in such scenarios and report some novel aspects. Our analyses indicate that PTM-induced protein stabilization brings the enzymes closer to saturation. As a result, ultrasensitivity may emerge or is greatly enhanced, with a steeper sigmoidal response, higher magnitude, and generally longer response time. In cases where PTM destabilizes the protein, ultrasensitivity can be regained through changes in the activities of the involved enzymes or from increased protein synthesis. Importantly, ultrasensitivity is not limited to modified or unmodified protein substrates-when protein turnover is considered, the total free protein substrate can also exhibit ultrasensitivity under several conditions. When full enzymatic reactions are used instead of Michaelis-Menten kinetics for the modeling, the total free protein substrate can even exhibit nonmonotonic dose-response patterns. It is conceivable that cells use inducible protein stabilization as a strategy in the signaling network to boost signal amplification while saving energy by keeping the protein substrate levels low at basal conditions.

Allosteric conformational ensembles have unlimited capacity for integrating information

Integration of binding information by macromolecular entities is fundamental to cellular functionality. Recent work has shown that such integration cannot be explained by pairwise cooperativities, in which binding is modulated by binding at another site. Higher-order cooperativities (HOCs), in which binding is collectively modulated by multiple other binding events, appear to be necessary but an appropriate mechanism has been lacking. We show here that HOCs arise through allostery, in which effective cooperativity emerges indirectly from an ensemble of dynamically interchanging conformations. Conformational ensembles play important roles in many cellular processes but their integrative capabilities remain poorly understood. We show that sufficiently complex ensembles can implement any form of information integration achievable without energy expenditure, including all patterns of HOCs. Our results provide a rigorous biophysical foundation for analysing the integration of binding information through allostery. We discuss the implications for eukaryotic gene regulation, where complex conformational dynamics accompanies widespread information integration.

Gene Regulation in and out of Equilibrium

Determining whether and how a gene is transcribed are two of the central processes of life. The conceptual basis for understanding such gene regulation arose from pioneering biophysical studies in eubacteria. However, eukaryotic genomes exhibit vastly greater complexity, which raises questions not addressed by this bacterial paradigm. First, how is information integrated from many widely separated binding sites to determine how a gene is transcribed? Second, does the presence of multiple energy-expending mechanisms, which are absent from eubacterial genomes, indicate that eukaryotes are capable of improved forms of genetic information processing? An updated biophysical foundation is needed to answer such questions. We describe the linear framework, a graph-based approach to Markov processes, and show that it can accommodate many previous studies in the field. Under the assumption of thermodynamic equilibrium, we introduce a language of higher-order cooperativities and show how it can rigorously quantify gene regulatory properties suggested by experiment. We point out that fundamental limits to information processing arise at thermodynamic equilibrium and can only be bypassed through energy expenditure. Finally, we outline some of the mathematical challenges that must be overcome to construct an improved biophysical understanding of gene regulation.

Robustness and parameter geography in post-translational modification systems

Biological systems are acknowledged to be robust to perturbations but a rigorous understanding of this has been elusive. In a mathematical model, perturbations often exert their effect through parameters, so sizes and shapes of parametric regions offer an integrated global estimate of robustness. Here, we explore this “parameter geography” for bistability in post-translational modification (PTM) systems. We use the previously developed “linear framework” for timescale separation to describe the steady-states of a two-site PTM system as the solutions of two polynomial equations in two variables, with eight non-dimensional parameters. Importantly, this approach allows us to accommodate enzyme mechanisms of arbitrary complexity beyond the conventional Michaelis-Menten scheme, which unrealistically forbids product rebinding. We further use the numerical algebraic geometry tools Bertini, Paramotopy, and alphaCertified to statistically assess the solutions to these equations at ∼10⁹ parameter points in total. Subject to sampling limitations, we find no bistability when substrate amount is below a threshold relative to enzyme amounts. As substrate increases, the bistable region acquires 8-dimensional volume which increases in an apparently monotonic and sigmoidal manner towards saturation. The region remains connected but not convex, albeit with a high visibility ratio. Surprisingly, the saturating bistable region occupies a much smaller proportion of the sampling domain under mechanistic assumptions more realistic than the Michaelis-Menten scheme. We find that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity. The apparent monotonic increase in volume of the bistable region remains perplexing because the region itself does not grow monotonically: parameter points can move back and forth between monostability and bistability. We suggest mathematical conjectures and questions arising from these findings. Advances in theory and software now permit insights into parameter geography to be uncovered by high-dimensional, data-centric analysis.

Biological organisms are often said to have robust properties but it is difficult to understand how such robustness arises from molecular interactions. Here, we use a mathematical model to study how the molecular mechanism of protein modification exhibits the property of multiple internal states, which has been suggested to underlie memory and decision making. The robustness of this property is revealed by the size and shape, or “geography,” of the parametric region in which the property holds. We use advances in reducing model complexity and in rapidly solving the underlying equations, to extensively sample parameter points in an 8-dimensional space. We find that under realistic molecular assumptions the size of the region is surprisingly small, suggesting that generating multiple internal states with such a mechanism is much harder than expected. While the shape of the region appears straightforward, we find surprising complexity in how the region grows with increasing amounts of the modified substrate. Our approach uses statistical analysis of data generated from a model, rather than from experiments, but leads to precise mathematical conjectures about parameter geography and biological robustness.

A two-way switch for inositol pyrophosphate signaling: Evolutionary history and biological significance of a unique, bifunctional kinase/phosphatase

The inositol pyrophosphates (PP-InsPs) are a unique subgroup of intracellular signals with diverse functions, many of which can be viewed as reflecting an overarching role in metabolic homeostasis. Thus, considerable attention is paid to the enzymes that synthesize and metabolize the PP-InsPs. One of these enzyme families - the diphosphoinositol pentakisphosphate kinases (PPIP5Ks) - provides an extremely rare example of separate kinase and phosphatase activities being present within the same protein. Herein, we review the current state of structure/function insight into the PPIP5Ks, the separate specialized activities of the two metazoan PPIP5K genes, and we describe a phylogenetic analysis that places PPIP5K evolutionary origin within the Excavata, the very earliest of eukaryotes. These different aspects of PPIP5K biology are placed in the context of a single, overriding question. Why are they bifunctional: i.e., what is the particular significance of the ability to turn PP-InsP signaling on or off from two separate 'switches' in a single protein?

p53-TP53-Induced Glycolysis Regulator Mediated Glycolytic Suppression Attenuates DNA Damage and Genomic Instability in Fanconi Anemia Hematopoietic Stem Cells

Emerging evidence has shown that resting quiescent hematopoietic stem cells (HSCs) prefer to utilize anaerobic glycolysis rather than mitochondrial respiration for energy production. Compelling evidence has also revealed that altered metabolic energetics in HSCs underlies the onset of certain blood diseases; however, the mechanisms responsible for energetic reprogramming remain elusive. We recently found that Fanconi anemia (FA) HSCs in their resting state are more dependent on mitochondrial respiration for energy metabolism than on glycolysis. In the present study, we investigated the role of deficient glycolysis in FA HSC maintenance. We observed significantly reduced glucose consumption, lactate production, and ATP production in HSCs but not in the less primitive multipotent progenitors or restricted hematopoietic progenitors of Fanca^−/− and Fancc^−/− mice compared with that of wild‐type mice, which was associated with an overactivated p53 and TP53‐induced glycolysis regulator, the TIGAR‐mediated metabolic axis. We utilized Fanca^−/− HSCs deficient for p53 to show that the p53‐TIGAR axis suppressed glycolysis in FA HSCs, leading to enhanced pentose phosphate pathway and cellular antioxidant function and, consequently, reduced DNA damage and attenuated HSC exhaustion. Furthermore, by using Fanca^−/− HSCs carrying the separation‐of‐function mutant p53^R172P transgene that selectively impairs the p53 function in apoptosis but not cell‐cycle control, we demonstrated that the cell‐cycle function of p53 was not required for glycolytic suppression in FA HSCs. Finally, ectopic expression of the glycolytic rate‐limiting enzyme PFKFB3 specifically antagonized p53‐TIGAR‐mediated metabolic reprogramming in FA HSCs. Together, our results suggest that p53‐TIGAR metabolic axis‐mediated glycolytic suppression may play a compensatory role in attenuating DNA damage and proliferative exhaustion in FA HSCs. stem cells2019;37:937–947

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

Model discrimination for Ca2+ -dependent regulation of myosin light chain kinase in smooth muscle contraction

Excitation-contraction coupling in smooth muscle is mediated by the Ca²⁺ - and calmodulin-dependent regulation of myosin light chain kinase. The precise mechanism of this regulation remains controversial, and several mathematical models have been proposed for the interaction of the three species. These models have previously been analyzed at steady state primarily by numerical simulation of differential equations, for which parameter values must be estimated from data. Here, we use the linear framework for timescale separation to demonstrate that models of this general kind can be solved analytically for an equilibrium steady state, without having to determine parameter values. This analysis leads to parameter-independent methods for discriminating between the models, for which we propose experiments that could be performed with existing methods.

Energy-speed-accuracy relation in complex networks for biological discrimination

Discriminating between correct and incorrect substrates is a core process in biology, but how is energy apportioned between the conflicting demands of accuracy (μ), speed (σ), and total entropy production rate (P)? Previous studies have focused on biochemical networks with simple structure or relied on simplifying kinetic assumptions. Here, we use the linear framework for timescale separation to analytically examine steady-state probabilities away from thermodynamic equilibrium for networks of arbitrary complexity. We also introduce a method of scaling parameters that is inspired by Hopfield's treatment of kinetic proofreading. Scaling allows asymptotic exploration of high-dimensional parameter spaces. We identify in this way a broad class of complex networks and scalings for which the quantity σln(μ)/P remains asymptotically finite whenever accuracy improves from equilibrium, so that μ_{eq}/μ→0. Scalings exist, however, even for Hopfield's original network, for which σln(μ)/P is asymptotically infinite, illustrating the parametric complexity. Outside the asymptotic regime, numerical calculations suggest that, under more restrictive parametric assumptions, networks satisfy the bound, σln(μ/μ_{eq})/P<1, and we discuss the biological implications for discrimination by ribosomes and DNA polymerase. The methods introduced here may be more broadly useful for analyzing complex networks that implement other forms of cellular information processing.

Inositol Pyrophosphate Synthesis by Diphosphoinositol Pentakisphosphate Kinase-1 is Regulated by Phosphatidylinositol(4,5)bisphosphate

5-diphosphoinositol tetrakisphosphate (5-InsP7) and bisdiphosphoinositol tetrakisphosphate (InsP8) are 'energetic' inositol pyrophosphate signaling molecules that regulate bioenergetic homeostasis. Inositol pyrophosphate levels are regulated by diphosphoinositol pentakisphosphate kinases (PPIP5Ks); these are large modular proteins that host a kinase domain (which phosphorylates 5-InsP7 to InsP8), a phosphatase domain that catalyzes the reverse reaction, and a polyphosphoinositide-binding domain (PBD). Here, we describe new interactions between these three domains in the context of full-length human PPIP5K1. We determine that InsP7 kinase activity is dominant when PPIP5K1 is expressed in intact cells; in contrast, we found that InsP8 phosphatase activity prevails when the enzyme is isolated from its cellular environment. We approach a reconciliation of this disparity by showing that cellular InsP8 phosphatase activity is inhibited by C8-PtdIns(4,5)P2 (IC50 approx. 40 ìM). We recapitulate this phosphatase inhibition with natural PtdIns(4,5)P2 that was incorporated into large unilamellar vesicles. Additionally, PtdIns(4,5)P2 increases net InsP7 kinase activity 5-fold. We oftlinedemonstrate that PtdIns(4,5)P2 is not itself a phosphatase substrate; its inhibition of InsP8 phosphatase activity results from an unusual, functional overlap between the phosphatase domain and the PBD. Finally, we discuss the significance of PtdIns(4,5)P2 as a novel regulator of PPIP5K1, in relation to compartmentalization of InsP7/InsP8 signaling in vivo.

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

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

Intimate connections: Inositol pyrophosphates at the interface of metabolic regulation and cell signaling

Inositol pyrophosphates are small, diffusible signaling molecules that possess the most concentrated three-dimensional array of phosphate groups in Nature; up to eight phosphates are crammed around a six-carbon inositol ring. This review discusses the physico-chemical properties of these unique molecules, and their mechanisms of action. Also provided is information on the enzymes that regulate the levels and hence the signaling properties of these molecules. This review pursues the idea that many of the biological effects of inositol pyrophosphates can be rationalized by their actions at the interface of cell signaling and metabolism that is essential to cellular and organismal homeostasis.

The Significance of the Bifunctional Kinase/Phosphatase Activities of Diphosphoinositol Pentakisphosphate Kinases (PPIP5Ks) for Coupling Inositol Pyrophosphate Cell Signaling to Cellular Phosphate Homeostasis

Proteins responsible for P_i homeostasis are critical for all life. In Saccharomyces cerevisiae, extracellular [P_i] is "sensed" by the inositol-hexakisphosphate kinase (IP6K) that synthesizes the intracellular inositol pyrophosphate 5-diphosphoinositol 1,2,3,4,6-pentakisphosphate (5-InsP₇) as follows: during a period of P_i starvation, there is a decline in cellular [ATP]; the unusually low affinity of IP6Ks for ATP compels 5-InsP₇ levels to fall in parallel (Azevedo, C., and Saiardi, A. (2017) Trends. Biochem. Sci. 42, 219-231. Hitherto, such P_i sensing has not been documented in metazoans. Here, using a human intestinal epithelial cell line (HCT116), we show that levels of both 5-InsP₇ and ATP decrease upon [P_i] starvation and subsequently recover during P_i replenishment. However, a separate inositol pyrophosphate, 1,5-bisdiphosphoinositol 2,3,4,6-tetrakisphosphate (InsP₈), reacts more dramatically (i.e. with a wider dynamic range and greater sensitivity). To understand this novel InsP₈ response, we characterized kinetic properties of the bifunctional 5-InsP₇ kinase/InsP₈ phosphatase activities of full-length diphosphoinositol pentakisphosphate kinases (PPIP5Ks). These data fulfil previously published criteria for any bifunctional kinase/phosphatase to exhibit concentration robustness, permitting levels of the kinase product (InsP₈ in this case) to fluctuate independently of varying precursor (i.e. 5-InsP₇) pool size. Moreover, we report that InsP₈ phosphatase activities of PPIP5Ks are strongly inhibited by P_i (40-90% within the 0-1 mm range). For PPIP5K2, P_i sensing by InsP₈ is amplified by a 2-fold activation of 5-InsP₇ kinase activity by P_i within the 0-5 mm range. Overall, our data reveal mechanisms that can contribute to specificity in inositol pyrophosphate signaling, regulating InsP₈ turnover independently of 5-InsP₇, in response to fluctuations in extracellular supply of a key nutrient.

The significance of the 1-kinase/1-phosphatase activities of the PPIP5K family

The inositol pyrophosphates (diphosphoinositol polyphosphates), which include 1-InsP₇, 5-InsP₇, and InsP₈, are highly 'energetic' signaling molecules that play important roles in many cellular processes, particularly with regards to phosphate and bioenergetic homeostasis. Two classes of kinases synthesize the PP-InsPs: IP6Ks and PPIP5Ks. The significance of the IP6Ks - and their 5-InsP₇ product - has been widely reported. However, relatively little is known about the biological significance of the PPIP5Ks. The purpose of this review is to provide an update on developments in our understanding of key features of the PPIP5Ks, which we believe strengthens the hypothesis that their catalytic activities serve important cellular functions. Central to this discussion is the recent discovery that the PPIP5K is a rare example of a single protein that catalyzes a kinase/phosphatase futile cycle.

Invariants reveal multiple forms of robustness in bifunctional enzyme systems

Experimental and theoretical studies have suggested that bifunctional enzymes catalyzing opposing modification and demodification reactions can confer steady-state concentration robustness to their substrates. However, the types of robustness and the biochemical basis for them have remained elusive. Here we report a systematic study of the most general biochemical reaction network for a bifunctional enzyme acting on a substrate with one modification site, along with eleven sub-networks with more specialized biochemical assumptions. We exploit ideas from computational algebraic geometry, introduced in previous work, to find a polynomial expression (an invariant) between the steady state concentrations of the modified and unmodified substrate for each network. We use these invariants to identify five classes of robust behavior: robust upper bounds on concentration, robust two-sided bounds on concentration ratio, hybrid robustness, absolute concentration robustness (ACR), and robust concentration ratio. This analysis demonstrates that robustness can take a variety of forms and that the type of robustness is sensitive to many biochemical details, with small changes in biochemistry leading to very different steady-state behaviors. In particular, we find that the widely-studied ACR requires highly specialized assumptions in addition to bifunctionality. An unexpected result is that the robust bounds derived from invariants are strictly tighter than those derived by ad hoc manipulation of the underlying differential equations, confirming the value of invariants as a tool to gain insight into biochemical reaction networks. Furthermore, invariants yield multiple experimentally testable predictions and illuminate new strategies for inferring enzymatic mechanisms from steady-state measurements.

Robust network structure of the Sln1-Ypd1-Ssk1 three-component phospho-relay prevents unintended activation of the HOG MAPK pathway in Saccharomyces cerevisiae

BACKGROUND

The yeast Saccharomyces cerevisiae relies on the high-osmolarity glycerol (HOG) signaling pathway to respond to increases in external osmolarity. The HOG pathway is rapidly activated under conditions of elevated osmolarity and regulates transcriptional and metabolic changes within the cell. Under normal growth conditions, however, a three-component phospho-relay consisting of the histidine kinase Sln1, the transfer protein Ypd1, and the response regulator Ssk1 represses HOG pathway activity by phosphorylation of Ssk1. This inhibition of the HOG pathway is essential for cellular fitness in normal osmolarity. Nevertheless, the extent to and mechanisms by which inhibition is robust to fluctuations in the concentrations of the phospho-relay components has received little attention.

RESULTS

We established that the Sln1-Ypd1-Ssk1 phospho-relay is robust-it is able to maintain inhibition of the HOG pathway even after significant changes in the levels of its three components. We then developed a biochemically realistic mathematical model of the phospho-relay, which suggested that robustness is due to buffering by a large excess pool of Ypd1. We confirmed experimentally that depletion of the Ypd1 pool results in inappropriate activation of the HOG pathway.

CONCLUSIONS

We identified buffering by an intermediate component in excess as a novel mechanism through which a phospho-relay can achieve robustness. This buffering requires multiple components and is therefore unavailable to two-component systems, suggesting one important advantage of multi-component relays.

Laplacian Dynamics with Synthesis and Degradation

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first-order degradation and zeroth-order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer an open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of such a matrix. This also completes our previous framework and makes it purely graph theoretical. Lastly, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels in pancreatic β-cells.

MAPK's networks and their capacity for multistationarity due to toric steady states

Mitogen-activated protein kinase (MAPK) signaling pathways play an essential role in the transduction of environmental stimuli to the nucleus, thereby regulating a variety of cellular processes, including cell proliferation, differentiation and programmed cell death. The components of the MAPK extracellular activated protein kinase (ERK) cascade represent attractive targets for cancer therapy as their aberrant activation is a frequent event among highly prevalent human cancers. MAPK networks are a model for computational simulation, mostly using ordinary and partial differential equations. Key results showed that these networks can have switch-like behavior, bistability and oscillations. In this work, we consider three representative ERK networks, one with a negative feedback loop, which present a binomial steady state ideal under mass-action kinetics. We therefore apply the theoretical result present in to find a set of rate constants that allow two significantly different stable steady states in the same stoichiometric compatibility class for each network. Our approach makes it possible to study certain aspects of the system, such as multistationarity, without relying on simulation, since we do not assume a priori any constant but the topology of the network. As the performed analysis is general it could be applied to many other important biochemical networks.

Time-scale separation--Michaelis and Menten's old idea, still bearing fruit

Michaelis and Menten introduced to biochemistry the idea of time-scale separation, in which part of a system is assumed to be operating sufficiently fast compared to the rest so that it may be taken to have reached a steady state. This allows, in principle, the fast components to be eliminated, resulting in a simplified description of the system's behaviour. Similar ideas have been widely used in different areas of biology, including enzyme kinetics, protein allostery, receptor pharmacology, gene regulation and post-translational modification. However, the methods used have been independent and ad hoc. In the present study, we review the use of time-scale separation as a means to simplify the description of molecular complexity and discuss recent work setting out a single framework that unifies these separate calculations. The framework offers new capabilities for mathematical analysis and helps to do justice to Michaelis and Menten's insights about individual enzymes in the context of multi-enzyme biological systems.