Intrinsic fluctuations, robustness, and tunability in signaling cycles

Compartmental exchange regulates steady states and stochastic switching of a phosphorylation network

The phosphoregulation of proteins with multiple phosphorylation sites is governed by biochemical reaction networks that can exhibit multistable behavior. However, the behavior of such networks is typically studied in a single reaction volume, while cells are spatially organized into compartments that can exchange proteins. In this work, we use stochastic simulations to study the impact of compartmentalization on a two-site phosphorylation network. We characterize steady states and fluctuation-driven transitions between them as a function of the rate of protein exchange between two compartments. Surprisingly, the average time spent in a state before stochastically switching to another depends nonmonotonically on the protein exchange rate, with the most frequent switching occurring at intermediate exchange rates. At sufficiently small exchange rates, the state of the system and mean switching time are controlled largely by fluctuations in the balance of enzymes in each compartment. This leads to negatively correlated states in the compartments. For large exchange rates, the two compartments behave as a single effective compartment. However, when the compartmental volumes are unequal, the behavior differs from a single compartment with the same total volume. These results demonstrate that exchange of proteins between distinct compartments can regulate the emergent behavior of a common signaling motif.

Frequency-preference response in covalent modification cycles under substrate sequestration conditions

Covalent modification cycles (CMCs) are basic units of signaling systems and their properties are well understood. However, their behavior has been mostly characterized in situations where the substrate is in excess over the modifying enzymes. Experimental data on protein abundance suggest that the enzymes and their target proteins are present in comparable concentrations, leading to substrate sequestration by the enzymes. In this enzyme-in-excess regime, CMCs have been shown to exhibit signal termination, the ability of the product to return to a stationary value lower than its peak in response to constant stimulation, while this stimulation is still active, with possible implications for the ability of systems to adapt to environmental inputs. We characterize the conditions leading to signal termination in CMCs in the enzyme-in-excess regime. We also demonstrate that this behavior leads to a preferred frequency response (band-pass filters) when the cycle is subjected to periodic stimulation, whereas the literature reports that CMCs investigated so far behave as low-pass filters. We characterize the relationship between signal termination and the preferred frequency response to periodic inputs and we explore the dynamic mechanism underlying these phenomena. Finally, we describe how the behavior of CMCs is reflected in similar types of responses in the cascades of which they are part. Evidence of protein abundance in vivo shows that enzymes and substrates are present in comparable concentrations, thus suggesting that signal termination and frequency-preference response to periodic inputs are also important dynamic features of cell signaling systems, which have been overlooked.

Retroactivity induced operating regime transition in an enzymatic futile cycle

Activated phosphorylation-dephosphorylation biochemical reaction cycles are a class of enzymatic futile cycles. A futile cycle such as a single MAPK cascade governed by two underlying enzymatic reactions permits Hyperbolic (H), Signal transducing (ST), Threshold-hyperbolic (TH) and Ultrasensitive (U) operating regimes that characterize input-output behaviour. Retroactive signalling caused by load due to sequestration of phosphorylated or unphosphorylated form of the substrate in a single enzymatic cascade without explicit feedback can introduce two-way communication, a feature not possible otherwise. We systematically characterize the operating regimes of a futile cycle subject to retroactivity in either of the substrate forms. We demonstrate that increasing retroactivity strength, which quantifies the downstream load, can trigger five possible regime transitions. Retroactivity strength is a reflection of the fraction of the substrate sequestered by its downstream target. Remarkably, the minimum required retroactivity strength to evidence any sequestration triggered regime transition demands 23% of the substrate bound to its downstream target. This minimum retroactivity strength corresponds to the transition of the dose-response curve from ST to H regime. We show that modulation of the saturation and unsaturation levels of the enzymatic reactions by retroactivity is the fundamental mechanism governing operating regime transition.

Molecular switch architecture determines response properties of signaling pathways

Many intracellular signaling pathways are composed of molecular switches, proteins that transition between two states-on and off Typically, signaling is initiated when an external stimulus activates its cognate receptor that, in turn, causes downstream switches to transition from off to on using one of the following mechanisms: activation, in which the transition rate from the off state to the on state increases; derepression, in which the transition rate from the on state to the off state decreases; and concerted, in which activation and derepression operate simultaneously. We use mathematical modeling to compare these signaling mechanisms in terms of their dose-response curves, response times, and abilities to process upstream fluctuations. Our analysis elucidates several operating principles for molecular switches. First, activation increases the sensitivity of the pathway, whereas derepression decreases sensitivity. Second, activation generates response times that decrease with signal strength, whereas derepression causes response times to increase with signal strength. These opposing features allow the concerted mechanism to not only show dose-response alignment, but also to decouple the response time from stimulus strength. However, these potentially beneficial properties come at the expense of increased susceptibility to upstream fluctuations. We demonstrate that these operating principles also hold when the models are extended to include additional features, such as receptor removal, kinetic proofreading, and cascades of switches. In total, we show how the architecture of molecular switches govern their response properties. We also discuss the biological implications of our findings.

On the emergence of cognition: from catalytic closure to neuroglial closure

In an analogous manner as occurred during the development of a connected metabolism that at some point reached characteristics associated with what is called "life"-due mainly to a catalytic closure phenomenon when chemicals started to autocatalyze themselves forming a closed web of chemical reactions-it is here proposed that cognition and consciousness (or features associated with them) arose as a consequence of another type of closure within the nervous system, the brain especially. Proper brain function requires an efficient web of connections and once certain complexity is attained due to the number and coordinated activities of the brain cell networks, the emergent properties of cognition and consciousness take place. Seeking to identify main features of the nervous system organization for optimal function, it is here proposed that while catalytic closure yielded life, neuroglial closure produced cognition/consciousness.

Operating regimes in a single enzymatic cascade at ensemble-level

Single enzymatic cascade, ubiquitously found in cellular signaling networks, is a phosphorylation-dephosphorylation reaction cycle causing a transition between inactive and active states of a protein catalysed by kinase and phosphatase, respectively. Steady-state information processing ability of such a cycle (e.g., MAPK cascade) has been classified into four qualitatively different operating regimes, viz., hyperbolic (H), signal-transducing (ST), threshold-hyperbolic (TH) and ultrasensitive (U). These four regimes represent qualitatively different dose-response curves, that is, relationship between concentrations of input kinase (e.g., pMEK) and response activated protein (e.g., pERK). Regimes were identified using a deterministic model accounting for population-averaged behavior only. Operating regimes can be strongly influenced by the inherently present cell-to-cell variability in an ensemble of cells which is captured in the form of pMEK and pERK distributions using reporter-based single-cell experimentation. In this study, we show that such experimentally acquired snapshot pMEK and pERK distribution data of a single MAPK cascade can be directly used to infer the underlying operating regime even in the absence of a dose-response curve. This deduction is possible primarily due to the presence of a monotonic relationship between experimental observables R_IQR, ratio of the inter-quartile range of the pERK and pMEK distribution pairs and R_M, ratio of the medians of the distribution pair. We demonstrate this relationship by systematic analysis of a quasi-steady state approximated model superimposed with an input gamma distribution constrained by the stimulus strength specific pMEK distribution measured on Jurkat-T cells stimulated with PMA. As a first, we show that introduction of cell-to-cell variability only in the upstream kinase achieved by superimposition of an appropriate input pMEK distribution on the dose-response curve can predict bimodal response pERK distribution in ST regime. Implementation of the proposed method on the input-response distribution pair obtained in stimulated Jurkat-T cells revealed that while low-dosage PMA stimulation preserves the H regime observed in resting cells, high-dosage causes H to ST regime transition.

Intrinsic limits of information transmission in biochemical signalling motifs

All living things have evolved to sense changes in their environment in order to respond in adaptive ways. At the cellular level, these sensing systems generally involve receptor molecules at the cell surface, which detect changes outside the cell and relay those changes to the appropriate response elements downstream. With the advent of experimental technologies that can track signalling at the single-cell level, it has become clear that many signalling systems exhibit significant levels of 'noise,' manifesting as differential responses of otherwise identical cells to the same environment. This noise has a large impact on the capacity of cell signalling networks to transmit information from the environment. Application of information theory to experimental data has found that all systems studied to date encode less than 2.5 bits of information, with the majority transmitting significantly less than 1 bit. Given the growing interest in applying information theory to biological data, it is crucial to understand whether the low values observed to date represent some sort of intrinsic limit on information flow given the inherently stochastic nature of biochemical signalling events. In this work, we used a series of computational models to explore how much information a variety of common 'signalling motifs' can encode. We found that the majority of these motifs, which serve as the basic building blocks of cell signalling networks, can encode far more information (4-6 bits) than has ever been observed experimentally. In addition to providing a consistent framework for estimating information-theoretic quantities from experimental data, our findings suggest that the low levels of information flow observed so far in living system are not necessarily due to intrinsic limitations. Further experimental work will be needed to understand whether certain cell signalling systems actually can approach the intrinsic limits described here, and to understand the sources and purpose of the variation that reduces information flow in living cells.

Exploring the Function of Dynamic Phosphorylation-Dephosphorylation Cycles

Protein phosphorylation is a dynamic post-translational modification critical for biological responses. At the level of individual molecules, phosphorylation dynamics can have important functional implications, but this information is rarely quantified. We discuss how rapid phosphorylation-dephosphorylation cycles could underlie important signaling properties, including the ability to rapidly bind and release proteins.

A Chemical Reaction Network to Generate Random, Power-Law-Distributed Time Intervals

In Lévy walks (LWs), particles move with a fixed speed along straight line segments and turn in new directions after random time intervals that are distributed according to a power law. Such LWs are thought to be an advantageous foraging and search strategy for organisms. While complex nervous systems are certainly capable of producing such behavior, it is not clear at present how single-cell organisms can generate the long-term correlated control signals required for a LW. Here, we construct a biochemical reaction system that generates long-time correlated concentration fluctuations of a signaling substance, with a tunable fractional exponent of the autocorrelation function. The network is based on well-known modules, and its basic function is highly robust with respect to the parameter settings.

Optimal information transfer in enzymatic networks: A field theoretic formulation

Signaling in enzymatic networks is typically triggered by environmental fluctuations, resulting in a series of stochastic chemical reactions, leading to corruption of the signal by noise. For example, information flow is initiated by binding of extracellular ligands to receptors, which is transmitted through a cascade involving kinase-phosphatase stochastic chemical reactions. For a class of such networks, we develop a general field-theoretic approach to calculate the error in signal transmission as a function of an appropriate control variable. Application of the theory to a simple push-pull network, a module in the kinase-phosphatase cascade, recovers the exact results for error in signal transmission previously obtained using umbral calculus [Hinczewski and Thirumalai, Phys. Rev. X 4, 041017 (2014)2160-330810.1103/PhysRevX.4.041017]. We illustrate the generality of the theory by studying the minimal errors in noise reduction in a reaction cascade with two connected push-pull modules. Such a cascade behaves as an effective three-species network with a pseudointermediate. In this case, optimal information transfer, resulting in the smallest square of the error between the input and output, occurs with a time delay, which is given by the inverse of the decay rate of the pseudointermediate. Surprisingly, in these examples the minimum error computed using simulations that take nonlinearities and discrete nature of molecules into account coincides with the predictions of a linear theory. In contrast, there are substantial deviations between simulations and predictions of the linear theory in error in signal propagation in an enzymatic push-pull network for a certain range of parameters. Inclusion of second-order perturbative corrections shows that differences between simulations and theoretical predictions are minimized. Our study establishes that a field theoretic formulation of stochastic biological signaling offers a systematic way to understand error propagation in networks of arbitrary complexity.

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.

Ultrasensitivity and noise amplification in a model of V. harveyi quorum sensing

We analyze ultrasensitivity in a model of Vibrio harveyi quorum sensing. We consider a feedforward model consisting of two biochemical networks per cell. The first represents the interchange of a signaling molecule (autoinducer) between the cell cytoplasm and an extracellular domain and the binding of intracellular autoinducer to cognate receptors. The unbound and bound receptors within each cell act as kinases and phosphotases, respectively, which then drive a second biochemical network consisting of a phosphorylation-dephosphorylation cycle. We ignore subsequent signaling pathways associated with gene regulation and the possible modification in the production rate of an autoinducer (positive feedback). We show how the resulting quorum sensing system exhibits ultrasensitivity with respect to changes in cell density. We also demonstrate how quorum sensing can protect against the noise amplification of fast environmental fluctuations in comparison to a single isolated cell.

Noise and information transmission in promoters with multiple internal States

Based on the measurements of noise in gene expression performed during the past decade, it has become customary to think of gene regulation in terms of a two-state model, where the promoter of a gene can stochastically switch between an ON and an OFF state. As experiments are becoming increasingly precise and the deviations from the two-state model start to be observable, we ask about the experimental signatures of complex multistate promoters, as well as the functional consequences of this additional complexity. In detail, we i), extend the calculations for noise in gene expression to promoters described by state transition diagrams with multiple states, ii), systematically compute the experimentally accessible noise characteristics for these complex promoters, and iii), use information theory to evaluate the channel capacities of complex promoter architectures and compare them with the baseline provided by the two-state model. We find that adding internal states to the promoter generically decreases channel capacity, except in certain cases, three of which (cooperativity, dual-role regulation, promoter cycling) we analyze in detail.

Characterization of tradeoffs in biomolecular signaling

Systems-level tradeoffs are fundamental in engineering, and recent work has highlighted an analogous role for them in biology. However, the extent of validity of these tradeoffs, especially for biomolecular systems, is generally unclear. Here, we address this issue for signaling tradeoffs that can constrain, for a fixed concentration of the signaling protein, a simultaneous enhancement of the gain and range of an amplifier or of the gain and threshold of a switch. We find that these gain-related tradeoffs persist in mathematical models of biomolecular reaction mechanisms that are at the core of large classes of signaling systems. Further, we find that these tradeoffs are also prevalent in the parametric functional forms commonly used to describe input-output curves in experimental analyses. Finally, we find that these tradeoffs can persist even in the presence of transcriptional feedback mechanisms that can change the concentration of the signaling protein. These results present a systematic characterization of these tradeoffs in biomolecular signaling systems.

Time-dependent information transmission in a model regulatory circuit

Many biological regulatory systems respond with a physiological delay when processing signals. A simple model of regulation which respects these features shows how the ability of a delayed output to transmit information is limited: at short times by the time scale of the dynamic input, at long times by that of the dynamic output. We find that topologies of maximally informative networks correspond to commonly occurring biological circuits linked to stress response and that circuits functioning out of steady state may exploit absorbing states to transmit information optimally.

Theoretical aspects of cellular decision-making and information-processing

Microscopic biological processes have extraordinary complexity and variety at the sub-cellular, intra-cellular, and multi-cellular levels. In dealing with such complex phenomena, conceptual and theoretical frameworks are crucial, which enable us to understand seemingly different intra- and inter-cellular phenomena from unified viewpoints. Decision-making is one such concept that has attracted much attention recently. Since a number of cellular behavior can be regarded as processes to make specific actions in response to external stimuli, decision-making can cover and has been used to explain a broad range of different cellular phenomena [Balázsi et al. (Cell 144(6):910, 2011), Zeng et al. (Cell 141(4):682, 2010)]. Decision-making is also closely related to cellular information-processing because appropriate decisions cannot be made without exploiting the information that the external stimuli contain. Efficiency of information transduction and processing by intra-cellular networks determines the amount of information obtained, which in turn limits the efficiency of subsequent decision-making. Furthermore, information-processing itself can serve as another concept that is crucial for understanding of other biological processes than decision-making. In this work, we review recent theoretical developments on cellular decision-making and information-processing by focusing on the relation between these two concepts.

Stochastic approaches in systems biology

The discrete and random occurrence of chemical reactions far from thermodynamic equilibrium, and low copy numbers of chemical species, in systems biology necessitate stochastic approaches. This review is an effort to give the reader a flavor of the most important stochastic approaches relevant to systems biology. Notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy-to-follow presentation of a stochastic framework for modeling subcellular biochemical systems. In particular, we make an effort to show how the notion of propensity, the chemical master equation (CME), and the stochastic simulation algorithm arise as consequences of the Markov property. Most stochastic modeling reviews focus on stochastic simulation approaches--the exact stochastic simulation algorithm and its various improvements and approximations. We complement this with an outline of an analytical approximation. The most common formulation of stochastic models for biochemical networks is the CME. Although stochastic simulations are a practical way to realize the CME, analytical approximations offer more insight into the influence of randomness on system's behavior. Toward that end, we cover the chemical Langevin equation and the related Fokker-Planck equation and the two-moment approximation (2MA). Throughout the text, two pedagogical examples are used to key illustrate ideas. With extensive references to the literature, our goal is to clarify key concepts and thereby prepare the reader for more advanced texts.

Mutual information in time-varying biochemical systems

Cells must continuously sense and respond to time-varying environmental stimuli. These signals are transmitted and processed by biochemical signaling networks. However, the biochemical reactions making up these networks are intrinsically noisy, which limits the reliability of intracellular signaling. Here we use information theory to characterize the reliability of transmission of time-varying signals through elementary biochemical reactions in the presence of noise. We calculate the mutual information for both instantaneous measurements and trajectories of biochemical systems for a Gaussian model. Our results indicate that the same network can have radically different characteristics for the transmission of instantaneous signals and trajectories. For trajectories, the ability of a network to respond to changes in the input signal is determined by the timing of reaction events, and is independent of the correlation time of the output of the network. We also study how reliably signals on different time scales can be transmitted by considering the frequency-dependent coherence and gain-to-noise ratio. We find that a detector that does not consume the ligand molecule upon detection can more reliably transmit slowly varying signals, while an absorbing detector can more reliably transmit rapidly varying signals. Furthermore, we find that while one reaction may more reliably transmit information than another when considered in isolation, when placed within a signaling cascade the relative performance of the two reactions can be reversed. This means that optimizing signal transmission at a single level of a signaling cascade can reduce signaling performance for the cascade as a whole.

Stochastic bimodalities in deterministically monostable reversible chemical networks due to network topology reduction

Recently, stochastic simulations of networks of chemical reactions have shown distributions of steady states that are inconsistent with the steady state solutions of the corresponding deterministic ordinary differential equations. One such class of systems is comprised of networks that have irreversible reactions, and the origin of the anomalous behavior in these cases is understood to be due to the existence of absorbing states. More puzzling is the report of such anomalies in reaction networks without irreversible reactions. One such biologically important example is the futile cycle. Here we show that, in these systems, nonclassical behavior can originate from a stochastic elimination of all the molecules of a key species. This leads to a reduction in the topology of the network and the sampling of steady states corresponding to a truncated network. Surprisingly, we find that, in spite of the purely discrete character of the topology reduction mechanism revealed by "exact" numerical solutions of the master equations, this phenomenon is reproduced by the corresponding Fokker-Planck equations.

The contribution of Notch1 to nephron segmentation in the developing kidney is revealed in a sensitized Notch2 background and can be augmented by reducing Mint dosage

We previously determined that Notch2, and not Notch1, was required for forming proximal nephron segments. The dominance of Notch2 may be conserved in humans, since Notch2 mutations occur in Alagille syndrome (ALGS) 2 patients, which includes renal complications. To test whether mutations in Notch1 could increase the severity of renal complications in ALGS, we inactivated conditional Notch1 and Notch2 alleles in mice using a Six2-GFP::Cre. This BAC transgene is expressed mosaically in renal epithelial progenitors but uniformly in cells exiting the progenitor pool to undergo mesenchymal-to-epithelial transition. Although delaying Notch2 inactivation had a marginal effect on nephron numbers, it created a sensitized background in which the inactivation of Notch1 severely compromised nephron formation, function, and survival. These and additional observations indicate that Notch1 in concert with Notch2 contributes to the morphogenesis of renal vesicles into S-shaped bodies in a RBP-J-dependent manner. A significant implication is that elevating Notch1 activity could improve renal functions in ALGS2 patients. As proof of principle, we determined that conditional inactivation of Mint, an inhibitor of Notch-RBP-J interaction, resulted in a moderate rescue of Notch2 null kidneys, implying that temporal blockage of Notch signaling inhibitors downstream of receptor activation may have therapeutic benefits for ALGS patients.

Amplification and detection of single-molecule conformational fluctuation through a protein interaction network with bimodal distributions

A protein undergoes conformational dynamics with multiple time scales, which results in fluctuating enzyme activities. Recent studies in single-molecule enzymology have observe this "age-old" dynamic disorder phenomenon directly. However, the single-molecule technique has its limitation. To be able to observe this molecular effect with real biochemical functions in situ, we propose to couple the fluctuations in enzymatic activity to noise propagations in small protein interaction networks such as a zeroth-order ultrasensitive phosphorylation-dephosphorylation cycle. We show that enzyme fluctuations can indeed be amplified by orders of magnitude into fluctuations in the level of substrate phosphorylation, a quantity of wide interest in cellular biology. Enzyme conformational fluctuations sufficiently slower than the catalytic reaction turnover rate result in a bimodal concentration distribution of the phosphorylated substrate. In return, this network-amplified single-enzyme fluctuation can be used as a novel biochemical "reporter" for measuring single-enzyme conformational fluctuation rates.

Noise and critical phenomena in biochemical signaling cycles at small molecule numbers

Biochemical reaction networks in living cells usually involve reversible covalent modification of signaling molecules, such as protein phosphorylation. Under conditions of small molecule numbers, as is frequently the case in living cells, mass-action theory fails to describe the dynamics of such systems. Instead, the biochemical reactions must be treated as stochastic processes that intrinsically generate concentration fluctuations of the chemicals. We investigate the stochastic reaction kinetics of covalent modification cycles (CMCs) by analytical modeling and numerically exact Monte Carlo simulation of the temporally fluctuating concentration. Depending on the parameter regime, we find for the probability density of the concentration qualitatively distinct classes of distribution functions including power-law distributions with a fractional and tunable exponent. These findings challenge the traditional view of biochemical control networks as deterministic computational systems and suggest that CMCs in cells can function as versatile and tunable noise generators.

Investigating the two-moment characterisation of subcellular biochemical networks

While ordinary differential equations (ODEs) form the conceptual framework for modelling many cellular processes, specific situations demand stochastic models to capture the influence of noise. The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). While stochastic simulations are a practical way to realise the CME, analytical approximations offer more insight into the influence of noise. Towards that end, the two-moment approximation (2MA) is a promising addition to the established analytical approaches including the chemical Langevin equation (CLE) and the related linear noise approximation (LNA). The 2MA approach directly tracks the mean and (co)variance which are coupled in general. This coupling is not obvious in CME and CLE and ignored by LNA and conventional ODE models. We extend previous derivations of 2MA by allowing (a) non-elementary reactions and (b) relative concentrations. Often, several elementary reactions are approximated by a single step. Furthermore, practical situations often require the use of relative concentrations. We investigate the applicability of the 2MA approach to the well-established fission yeast cell cycle model. Our analytical model reproduces the clustering of cycle times observed in experiments. This is explained through multiple resettings of M-phase promoting factor (MPF), caused by the coupling between mean and (co)variance, near the G2/M transition.

The effects of reversibility and noise on stochastic phosphorylation cycles and cascades

The phosphorylation-dephosphorylation cycle is a common motif in cellular signaling networks. Previous work has revealed that, when driven by a noisy input signal, these cycles may exhibit bistable behavior. Here, a recently introduced theorem on network bistability is applied to prove that the existence of bistability is dependent on the stochastic nature of the system. Furthermore, the thermodynamics of simple cycles and cascades is investigated in the stochastic setting. Because these cycles are driven by the ATP hydrolysis potential, they may operate far from equilibrium. It is shown that sufficient high ATP hydrolysis potential is necessary for the existence of a bistable steady state. For the single-cycle system, the ensemble average behavior follows the ultrasensitive response expected from analysis of the corresponding deterministic system, but with significant fluctuations. For the two-cycle cascade, the average behavior begins to deviate from the expected response of the deterministic system. Examination of a two-cycle cascade reveals that the bistable steady state may be either propagated or abolished along a cascade, depending on the parameters chosen. Likewise, the variance in the response can be maximized or minimized by tuning the number of enzymes in the second cycle.

The Goldbeter-Koshland switch in the first-order region and its response to dynamic disorder

In their classical work (Proc. Natl. Acad. Sci. USA, 1981, 78:6840-6844), Goldbeter and Koshland mathematically analyzed a reversible covalent modification system which is highly sensitive to the concentration of effectors. Its signal-response curve appears sigmoidal, constituting a biochemical switch. However, the switch behavior only emerges in the 'zero-order region', i.e. when the signal molecule concentration is much lower than that of the substrate it modifies. In this work we showed that the switching behavior can also occur under comparable concentrations of signals and substrates, provided that the signal molecules catalyze the modification reaction in cooperation. We also studied the effect of dynamic disorders on the proposed biochemical switch, in which the enzymatic reaction rates, instead of constant, appear as stochastic functions of time. We showed that the system is robust to dynamic disorder at bulk concentration. But if the dynamic disorder is quasi-static, large fluctuations of the switch response behavior may be observed at low concentrations. Such fluctuation is relevant to many biological functions. It can be reduced by either increasing the conformation interconversion rate of the protein, or correlating the enzymatic reaction rates in the network.

Operating regimes of signaling cycles: statics, dynamics, and noise filtering

A ubiquitous building block of signaling pathways is a cycle of covalent modification (e.g., phosphorylation and dephosphorylation in MAPK cascades). Our paper explores the kind of information processing and filtering that can be accomplished by this simple biochemical circuit. Signaling cycles are particularly known for exhibiting a highly sigmoidal (ultrasensitive) input–output characteristic in a certain steady-state regime. Here, we systematically study the cycle's steady-state behavior and its response to time-varying stimuli. We demonstrate that the cycle can actually operate in four different regimes, each with its specific input–output characteristics. These results are obtained using the total quasi–steady-state approximation, which is more generally valid than the typically used Michaelis-Menten approximation for enzymatic reactions. We invoke experimental data that suggest the possibility of signaling cycles operating in one of the new regimes. We then consider the cycle's dynamic behavior, which has so far been relatively neglected. We demonstrate that the intrinsic architecture of the cycles makes them act—in all four regimes—as tunable low-pass filters, filtering out high-frequency fluctuations or noise in signals and environmental cues. Moreover, the cutoff frequency can be adjusted by the cell. Numerical simulations show that our analytical results hold well even for noise of large amplitude. We suggest that noise filtering and tunability make signaling cycles versatile components of more elaborate cell-signaling pathways.

A cell is subjected to constantly changing environments and time-varying stimuli. Signals sensed at the cell surface are transmitted inside the cell by signaling pathways. Such pathways can transform signals in diverse ways and perform some preliminary information processing. A ubiquitous building block of signaling pathways is a simple biochemical cycle involving covalent modification of an enzyme–substrate pair. Our paper is devoted to fully characterizing the static and dynamic behavior of this simple cycle, an essential first step in understanding the behavior of interconnections of such cycles. It is known that a signaling cycle can function as a static switch, with the steady-state output being an “ultrasensitive” function of the input, i.e., changing from a low to high value for only a small change in the input. We show that there are in fact precisely four major regimes of static and dynamic operation (with ultrasensitive being one of the static regimes). Each regime has its own input–output characteristics. Despite the distinctive features of these four regimes, they all respond to time-varying stimuli by filtering out high-frequency fluctuations or noise in their inputs, while passing through the lower-frequency information-bearing variations. A cell can select the regime and tune the noise-filtering characteristics of the individual cycles in a specific signaling pathway. This tunability makes signaling cycles versatile components of elaborate cell-signaling pathways.