Mathematical and computational analysis of adaptation via feedback inhibition in signal transduction pathways

Effects of saturation and enzyme limitation in feedforward adaptive signal transduction

In this study, the author examines the effects of saturation and enzyme limitation in temporal and spatial signal transduction in a generic feedforward adaptive module. The feedforward module encompasses a range of temporal and spatial signal processing, and this study systematically examines the effect of enzyme limitation/saturating effects in each of the feedforward pathways, and their interplay. It is found that this saturation makes the adaptation inexact, and this effect is more pronounced for higher levels of input signals. Further, it has a very significant role in affecting the temporal dynamics of this module. In examining the role of saturation in the module response to static gradients, the author finds that in certain cases, saturation can completely alter the gradient response. The author examines various aspects of the response systematically using analytical methods and simulations. Overall the author studies a framework and basis for examining and understanding the roles of saturating effects in multiple pathways involved in adaptive responses and sheds light on the relationship and connection between exact and inexact adaptation.

Division of labor by dual feedback regulators controls JAK2/STAT5 signaling over broad ligand range

Cellular signal transduction is governed by multiple feedback mechanisms to elicit robust cellular decisions. The specific contributions of individual feedback regulators, however, remain unclear. Based on extensive time-resolved data sets in primary erythroid progenitor cells, we established a dynamic pathway model to dissect the roles of the two transcriptional negative feedback regulators of the suppressor of cytokine signaling (SOCS) family, CIS and SOCS3, in JAK2/STAT5 signaling. Facilitated by the model, we calculated the STAT5 response for experimentally unobservable Epo concentrations and provide a quantitative link between cell survival and the integrated response of STAT5 in the nucleus. Model predictions show that the two feedbacks CIS and SOCS3 are most effective at different ligand concentration ranges due to their distinct inhibitory mechanisms. This divided function of dual feedback regulation enables control of STAT5 responses for Epo concentrations that can vary 1000-fold in vivo. Our modeling approach reveals dose-dependent feedback control as key property to regulate STAT5-mediated survival decisions over a broad range of ligand concentrations.

Biology using engineering tools: the negative feedback amplifier

Negative feedback is an ubiquitous feature of biological networks. Recent work from Sturm and colleaguespresents experimental evidence that biological negative feedback can serve the same function as it does for engineered systems: robustness to perturbations within the feedback loop. Such behavior has important implications for how to attack deregulated signaling networks containing negative feedback in diseases such as cancer.

Dynamic modeling and analysis of cancer cellular network motifs

With the advent of high-throughput biology, we now routinely scan cells and organisms at practically all levels, from genome to protein, metabolism, signaling and other cellular functions. This methodology allowed biological studies to move from a reductionist approach, such as isolation of specific pathways and mechanisms, to a more integrative approach, where biological systems are seen as a network of interconnected components that provide specific outputs and functions in response to stimuli. Recent literature on biological networks demonstrates two important concepts that we will consider in this review: (i) cellular pathways are highly interconnected and should not be studied separately, but as a network; (ii) simple, recurrent feedback motifs within the network can produce very specific functions that favor their modular use. The first theme differs from the traditional approach in biology because it provides a framework (i.e., the network view) in which large datasets are analyzed with an unbiased view. The second theme (feedback motifs) shows the importance of locally analyzing the dynamic properties of biological networks in order to better understand their functionality. We will review these themes with examples from cell signaling networks, gene regulatory networks and metabolic pathways. The deregulation of cellular networks (metabolism, signaling etc.) is involved in cancer, but the size of the networks and resulting non-linear behavior do not allow for intuitive reasoning. In that context, we argue that the qualitative classification of the 'building blocs' of biological networks (i.e. the motifs) in terms of dynamics and functionality will be critical to improve our understanding of cancer biology and rationalize the wealth of information from high-throughput experiments. From the examples highlighted in this review, it is clear that dynamic feedback motifs can be used to provide a unified view of various cellular processes involved in cancer and this will be critical for future research on personalized and predictive cancer therapies.

Open cascades as simple solutions to providing ultrasensitivity and adaptation in cellular signaling

Cell signaling is achieved predominantly by reversible phosphorylation-dephosphorylation reaction cascades. Up until now, circuits conferring adaptation have all required the presence of a cascade with some type of closed topology: negative-feedback loop with a buffering node, or incoherent feed-forward loop with a proportioner node. In this paper--using Goldbeter and Koshland-type expressions--we propose a differential equation model to describe a generic, open signaling cascade that elicits an adaptation response. This is accomplished by coupling N phosphorylation-dephosphorylation cycles unidirectionally, without any explicit feedback loops. Using this model, we show that as the length of the cascade grows, the steady states of the downstream cycles reach a limiting value. In other words, our model indicates that there are a minimum number of cycles required to achieve a maximum in sensitivity and amplitude in the response of a signaling cascade. We also describe for the first time that the phenomenon of ultrasensitivity can be further subdivided into three sub-regimes, separated by sharp stimulus threshold values: OFF, OFF-ON-OFF, and ON. In the OFF-ON-OFF regime, an interesting property emerges. In the presence of a basal amount of activity, the temporal evolution of early cycles yields damped peak responses. On the other hand, the downstream cycles switch rapidly to a higher activity state for an extended period of time, prior to settling to an OFF state (OFF-ON-OFF). This response arises from the changing dynamics between a feed-forward activation module and dephosphorylation reactions. In conclusion, our model gives the new perspective that open signaling cascades embedded in complex biochemical circuits may possess the ability to show a switch-like adaptation response, without the need for any explicit feedback circuitry.

Adaptive response and enlargement of dynamic range

Many membrane channels and receptors exhibit adaptive, or desensitized, response to a strong sustained input stimulus, often supported by protein activity-dependent inactivation. Adaptive response is thought to be related to various cellular functions such as homeostasis and enlargement of dynamic range by background compensation. Here we study the quantitative relation between adaptive response and background compensation within a modeling framework. We show that any particular type of adaptive response is neither sufficient nor necessary for adaptive enlargement of dynamic range. In particular a precise adaptive response, where system activity is maintained at a constant level at steady state, does not ensure a large dynamic range neither in input signal nor in system output. A general mechanism for input dynamic range enlargement can come about from the activity-dependent modulation of protein responsiveness by multiple biochemical modification, regardless of the type of adaptive response it induces. Therefore hierarchical biochemical processes such as methylation and phosphorylation are natural candidates to induce this property in signaling systems.

Automated ensemble modeling with modelMaGe: analyzing feedback mechanisms in the Sho1 branch of the HOG pathway

In systems biology uncertainty about biological processes translates into alternative mathematical model candidates. Here, the goal is to generate, fit and discriminate several candidate models that represent different hypotheses for feedback mechanisms responsible for downregulating the response of the Sho1 branch of the yeast high osmolarity glycerol (HOG) signaling pathway after initial stimulation. Implementing and testing these candidate models by hand is a tedious and error-prone task. Therefore, we automatically generated a set of candidate models of the Sho1 branch with the tool modelMaGe. These candidate models are automatically documented, can readily be simulated and fitted automatically to data. A ranking of the models with respect to parsimonious data representation is provided, enabling discrimination between candidate models and the biological hypotheses underlying them. We conclude that a previously published model fitted spurious effects in the data. Moreover, the discrimination analysis suggests that the reported data does not support the conclusion that a desensitization mechanism leads to the rapid attenuation of Hog1 signaling in the Sho1 branch of the HOG pathway. The data rather supports a model where an integrator feedback shuts down the pathway. This conclusion is also supported by dedicated experiments that can exclusively be predicted by those models including an integrator feedback.

modelMaGe is an open source project and is distributed under the Gnu General Public License (GPL) and is available from http://modelmage.org.

Raf family kinases: old dogs have learned new tricks

First identified in the early 1980s as retroviral oncogenes, the Raf proteins have been the objects of intense research. The discoveries 10 years later that the Raf family members (Raf-1, B-Raf, and A-Raf) are bona fide Ras effectors and upstream activators of the ubiquitous ERK pathway increased the interest in these proteins primarily because of the central role that this cascade plays in cancer development. The important role of Raf in cancer was corroborated in 2002 with the discovery of B-Raf genetic mutations in a large number of tumors. This led to intensified drug development efforts to target Raf signaling in cancer. This work yielded not only recent clinical successes but also surprising insights into the regulation of Raf proteins by homodimerization and heterodimerization. Surprising insights also came from the hunt for new Raf targets. Although MEK remains the only widely accepted Raf substrate, new kinase-independent roles for Raf proteins have emerged. These include the regulation of apoptosis by suppressing the activity of the proapoptotic kinases, ASK1 and MST2, and the regulation of cell motility and differentiation by controlling the activity of Rok-α. In this review, we discuss the regulation of Raf proteins and their role in cancer, with special focus on the interacting proteins that modulate Raf signaling. We also describe the new pathways controlled by Raf proteins and summarize the successes and failures in the development of efficient anticancer therapies targeting Raf. Finally, we also argue for the necessity of more systemic approaches to obtain a better understanding of how the Ras-Raf signaling network generates biological specificity.

Feedback regulation of EGFR signalling: decision making by early and delayed loops

Human-made information relay systems invariably incorporate central regulatory components, which are mirrored in biological systems by dense feedback and feedforward loops. This type of system control is exemplified by positive and negative feedback loops (for example, receptor endocytosis and dephosphorylation) that enable growth factors and receptor Tyr kinases of the epidermal growth factor receptor (EGFR)/ERBB family to regulate cellular function. Recent studies show that the collection of feedback regulatory loops can perform computational tasks - such as decoding ligand specificity, transforming graded input signals into a digital output and regulating response kinetics. Aberrant signal processing and feedback regulation can lead to defects associated with pathologies such as cancer.

Stochastic adaptation and fold-change detection: from single-cell to population behavior

Background

In cell signaling terminology, adaptation refers to a system's capability of returning to its equilibrium upon a transient response. To achieve this, a network has to be both sensitive and precise. Namely, the system must display a significant output response upon stimulation, and later on return to pre-stimulation levels. If the system settles at the exact same equilibrium, adaptation is said to be 'perfect'. Examples of adaptation mechanisms include temperature regulation, calcium regulation and bacterial chemotaxis.

Results

We present models of the simplest adaptation architecture, a two-state protein system, in a stochastic setting. Furthermore, we consider differences between individual and collective adaptive behavior, and show how our system displays fold-change detection properties. Our analysis and simulations highlight why adaptation needs to be understood in terms of probability, and not in strict numbers of molecules. Most importantly, selection of appropriate parameters in this simple linear setting may yield populations of cells displaying adaptation, while single cells do not.

Conclusions

Single cell behavior cannot be inferred from population measurements and, sometimes, collective behavior cannot be determined from the individuals. By consequence, adaptation can many times be considered a purely emergent property of the collective system. This is a clear example where biological ergodicity cannot be assumed, just as is also the case when cell replication rates are not homogeneous, or depend on the cell state. Our analysis shows, for the first time, how ergodicity cannot be taken for granted in simple linear examples either. The latter holds even when cells are considered isolated and devoid of replication capabilities (cell-cycle arrested). We also show how a simple linear adaptation scheme displays fold-change detection properties, and how rupture of ergodicity prevails in scenarios where transitions between protein states are mediated by other molecular species in the system, such as phosphatases and kinases.

Four-dimensional dynamics of MAPK information processing systems

Mitogen activated protein kinase (MAPK) cascades process a myriad of stimuli received by cell-surface receptors and generate precise spatio-temporal guidance for multiple target proteins, dictating receptor-specific cellular outcomes. Computational modelling reveals that the intrinsic topology of MAPK cascades enables them to amplify signal sensitivity and amplitude, reduce noise and display intricate dynamic properties, which include toggle switches, excitation pulses and oscillations. Specificity of signaling responses can be brought about by signal-induced feedback and feedforward wiring imposed on the MAPK cascade backbone. Intracellular gradients of protein activities arise from the spatial separation of opposing reactions in kinase-phosphatase cycles. The membrane confinement of the initiating kinase in MAPK cascades and cytosolic localization of phosphatases can result in precipitous gradients of phosphorylated signal-transducers if they spread solely by diffusion. Endocytotic trafficking of active kinases driven by molecular motors and traveling waves of protein phosphorylation can propagate phosphorylation signals from the plasma membrane to the nucleus, especially in large cells, such as Xenopus eggs.

Systematic quantification of negative feedback mechanisms in the extracellular signal-regulated kinase (ERK) signaling network

Cell responses are actuated by tightly controlled signal transduction pathways. Although the concept of an integrated signaling network replete with interpathway cross-talk and feedback regulation is broadly appreciated, kinetic data of the type needed to characterize such interactions in conjunction with mathematical models are lacking. In mammalian cells, the Ras/ERK pathway controls cell proliferation and other responses stimulated by growth factors, and several cross-talk and feedback mechanisms affecting its activation have been identified. In this work, we take a systematic approach to parse the magnitudes of multiple regulatory mechanisms that attenuate ERK activation through canonical (Ras-dependent) and non-canonical (PI3K-dependent) pathways. In addition to regulation of receptor and ligand levels, we consider three layers of ERK-dependent feedback: desensitization of Ras activation, negative regulation of MEK kinase (e.g. Raf) activities, and up-regulation of dual-specificity ERK phosphatases. Our results establish the second of these as the dominant mode of ERK self-regulation in mouse fibroblasts. We further demonstrate that kinetic models of signaling networks, trained on a sufficient diversity of quantitative data, can be reasonably comprehensive, accurate, and predictive in the dynamical sense.

Understanding the temporal codes of intra-cellular signals

The health of organisms and cells depends on appropriate responses to diverse internal and external cues, stimuli, or challenges, such as changes in hormone or cytokine levels, or exposure to a pathogen. Cellular responses must be tailored to the identity and intensity of the stimulus and therefore intra-cellular signals must carry information about both. However, signaling mediators often form intricate networks that react to multiple stimuli yet manage to produce stimulus-specific responses. The multi-functionality ('functional pleiotropism') of signaling nodes suggests that biological networks have evolved ways of passing physiologically relevant stimulus information through shared channels. Increasing evidence supports the notion that this is achieved in part through temporal regulation of signaling mediators' activities. The present challenge is to identify the features of temporal activity profile that represent information about a given stimulus and understand how cells read the temporal codes to control their responses.

Signal response sensitivity in the yeast mitogen-activated protein kinase cascade

The yeast pheromone response pathway is a canonical three-step mitogen activated protein kinase (MAPK) cascade which requires a scaffold protein for proper signal transduction. Recent experimental studies into the role the scaffold plays in modulating the character of the transduced signal, show that the presence of the scaffold increases the biphasic nature of the signal response. This runs contrary to prior theoretical investigations into how scaffolds function. We describe a mathematical model of the yeast MAPK cascade specifically designed to capture the experimental conditions and results of these empirical studies. We demonstrate how the system can exhibit either graded or ultrasensitive (biphasic) response dynamics based on the binding kinetics of enzymes to the scaffold. At the basis of our theory is an analytical result that weak interactions make the response biphasic while tight interactions lead to a graded response. We then show via an analysis of the kinetic binding rate constants how the results of experimental manipulations, modeled as changes to certain of these binding constants, lead to predictions of pathway output consistent with experimental observations. We demonstrate how the results of these experimental manipulations are consistent within the framework of our theoretical treatment of this scaffold-dependent MAPK cascades, and how future efforts in this style of systems biology can be used to interpret the results of other signal transduction observations.

Variations in Gnai2 and Rgs1 expression affect chemokine receptor signaling and the organization of secondary lymphoid organs

Ligand bound chemoattractant receptors activate the heterotrimeric G protein Gi to stimulate downstream signaling pathways to properly position lymphocytes in lymphoid organs. Here we show how variations the expression of a chemokine receptor and in two components in the signaling pathway, Gα_i2 and RGS1, affects the output fidelity of the signaling pathway. Examination of B cells from mice with varying numbers of intact alleles of Ccr7, Rgs1, Gnai2, and Gnai3 provided the basis for these results. Loss of a single allele of either Gnai2 or Rgs1 affected CCL19 triggered chemotaxis, while loss of a single allele of Ccr7, which encodes the cognate CCL19 receptor, had little effect. Emphasizing the importance of Gnai2, B cells lacking Gnai3 expression responded to chemokines better than did wild type B cells. At an organismal level, variations in Rgs1 and Gnai2 expression affected marginal zone B cell development, splenic architecture, lymphoid follicle size, and germinal center morphology. Gnai2 expression was also needed for the proper alignment of MOMA-1⁺ macrophages and MAdCAM-1⁺ endothelial cells along marginal zone sinuses in the spleen. These data indicate that chemoattractant receptors, heterotrimeric G-proteins, and RGS protein expression levels have a complex inter-relationship that affects the responses to chemoattractant exposure.

Theoretical analysis of time-to-peak responses in biological reaction networks

Processing of information by signaling networks is characterized by properties of the induced kinetics of the activated pathway components. The maximal extent of pathway activation (maximum amplitude) and the time-to-peak-response (position) are key determinants of biological responses that have been linked to specific outcomes. We investigate how the maximum amplitude of pathway activation and its position depend on the input and wiring of a signaling network. For this purpose, we consider a simple reaction A→B that is regulated by a transient input and extended this to include back-reaction and additional partners. In particular, we show that a unique maximum of B(t) exists. Moreover, we prove that the position of the maximum is independent of the applied input but regulated by degradation reactions of B. Indeed, the time-to-peak-response decreases with increasing degradation rate, which we prove for small models and show in simulations for more complex ones. The identified dependencies provide insights into design principles that facilitate the realization dynamical characteristics like constant position of maximal pathway activation and thereby guide the characterization of unknown kinetics within larger protein networks.

Goldbeter-Koshland model for open signaling cascades: a mathematical study

A modified Goldbeter-Koshland (GK) type model is proposed to study the open signaling cascade with forward activation. Three different regimes related to the ultrasensitivity are clearly defined. When the cascade is sufficiently long, the steady states in the downstream cycles exhibit a limiting behavior. Meanwhile, a temporal switch-like behavior is revealed in the pre-ultrasensitivity regime without any feedback circuitry. A detailed analysis of this phenomenon is provided.

Adaptive response by state-dependent inactivation

Many membrane channels and receptors exhibit adaptive, or desensitized, response to a strong sustained input stimulus. A key mechanism that underlies this response is the slow, activity-dependent removal of responding molecules to a pool which is unavailable to respond immediately to the input. This mechanism is implemented in different ways in various biological systems and has traditionally been studied separately for each. Here we highlight the common aspects of this principle, shared by many biological systems, and suggest a unifying theoretical framework. We study theoretically a class of models which describes the general mechanism and allows us to distinguish its universal from system-specific features. We show that under general conditions, regardless of the details of kinetics, molecule availability encodes an averaging over past activity and feeds back multiplicatively on the system output. The kinetics of recovery from unavailability determines the effective memory kernel inside the feedback branch, giving rise to a variety of system-specific forms of adaptive response-precise or input-dependent, exponential or power-law-as special cases of the same model.

Defining network topologies that can achieve biochemical adaptation

Many signaling systems show adaptation-the ability to reset themselves after responding to a stimulus. We computationally searched all possible three-node enzyme network topologies to identify those that could perform adaptation. Only two major core topologies emerge as robust solutions: a negative feedback loop with a buffering node and an incoherent feedforward loop with a proportioner node. Minimal circuits containing these topologies are, within proper regions of parameter space, sufficient to achieve adaptation. More complex circuits that robustly perform adaptation all contain at least one of these topologies at their core. This analysis yields a design table highlighting a finite set of adaptive circuits. Despite the diversity of possible biochemical networks, it may be common to find that only a finite set of core topologies can execute a particular function. These design rules provide a framework for functionally classifying complex natural networks and a manual for engineering networks. For a video summary of this article, see the PaperFlick file with the Supplemental Data available online.

Specificity and robustness of the mammalian MAPK-IEG network

The mitogen-activated protein kinase cascade is a conserved signal transduction pathway found in organisms of complexity spanning from yeast to humans. In many mammalian tissue types, this pathway can correctly transduce signals from different extracellular messengers, leading to specific and often mutually exclusive cellular responses. The transduced signal is tuned by a complicated set of positive and negative feedback control mechanisms and fed into a downstream gene expression network. This network, based on the immediate early gene system, has two possible, mutually exclusive outcomes. Using a mathematical model, we study how different stimuli lead to different temporal signal structure. Further, we investigate how each of the feedback controls contributes to the overall specificity of the gene expression output, and hypothesize that the complicated nature of the mammalian mitogen-activated protein kinase pathway results in a system able to robustly identify and transduce the proper signal without investing in two completely separate signal cascades. Finally, we quantify the role of the RKIP protein in shaping the signal, and propose a novel mechanism of its involvement in cancer metastasis.

Regulating the total level of a signaling protein can vary its dynamics in a range from switch like ultrasensitivity to adaptive responses

Biological signaling networks can exhibit rich response dynamics including ultrasensitivity, adaptation to persistent stimuli and oscillations. Previous modeling efforts have considered the proteins in these networks as two-state entities and their total levels as fixed quantities. However, inside the cell, most molecules are in constant flux because of various processes such as degradation, synthesis, binding of scaffold proteins and release from vesicles. The resulting freedom in the amount of signaling protein that is available for signaling has not been explored. Here, we analyze the response dynamics of a signaling protein when it enters the signaling pool in one state (modified or unmodified) and exits in both states. When the exit rates of these two states are comparable, a persistent stimulus results in step responses and can produce ultrasensitivity, as shown previously. However, we find that when the exit rates are imbalanced, the signaling protein gives transient responses to persistent stimuli even though the system does not have any explicit feedback. Further, these rates determine the signal range over which the system is responsive. Building small networks from signaling proteins with different exit rates, we show that these systems can exhibit rich behavior. Taken together, these findings indicate that altering the total level of signaling proteins can significantly change their response and provide additional richness in system dynamics. We discuss relevant biological examples in which regulating total protein levels could be exploited to alter signaling behavior.

The effects of cascade length, kinetics and feedback loops on biological signal transduction dynamics in a simplified cascade model

How intracellular signals are propagated with appropriate strength, duration and fidelity over time is poorly understood. To address these issues, intracellular signal transduction was studied both analytically and numerically using a simplified cascade model. The main observations can be summarized as follows: when the response kinetics is of the Michaelis-Menten type, the signal strength will always reach the same magnitude as the cascade length increases, regardless of the type of stimulus applied (i.e. either continuous or unitary pulse). However, when the response kinetics is of the Hill type (Hill coefficient >1), there exists a stimulation threshold. If the stimulus is below the threshold, the signal decays toward zero; in contrast, if the stimulus is above the threshold, the signal amplitude reaches a nonzero steady state. The time taken for the signal to proceed through the cascade increases as the half-maximum point, or Hill coefficient, increases, whereas the duration of the output signal at the end of the cascade decreases as the half-maximum point increases. In the presence of positive feedback, the stimulation threshold increases; under these conditions, the feedback strength necessary for bistability changes (with power-law characteristics) inversely related to the length of the cascade. In the presence of negative feedback, oscillations are induced when the Hill coefficient is greater than 1 and the cascade has more than two steps. Likewise, the feedback strength required to generate oscillations changes (again with power-law characteristics) inversely with the length of the cascade.

Signal duration and the time scale dependence of signal integration in biochemical pathways

Background

Signal duration (e.g. the time over which an active signaling intermediate persists) is a key regulator of biological decisions in myriad contexts such as cell growth, proliferation, and developmental lineage commitments. Accompanying differences in signal duration are numerous downstream biological processes that require multiple steps of biochemical regulation.

Results

Here we present an analysis that investigates how simple biochemical motifs that involve multiple stages of regulation can be constructed to differentially process signals that persist at different time scales. We compute the dynamic, frequency dependent gain within these networks and resulting power spectra to better understand how biochemical networks can integrate signals at different time scales. We identify topological features of these networks that allow for different frequency dependent signal processing properties.

Conclusion

We show that multi-staged cascades are effective in integrating signals of long duration whereas multi-staged cascades that operate in the presence of negative feedback are effective in integrating signals of short duration. Our studies suggest principles for why signal duration in connection with multiple steps of downstream regulation is a ubiquitous motif in biochemical systems.

Dose-to-duration encoding and signaling beyond saturation in intracellular signaling networks

The cellular response elicited by an environmental cue typically varies with the strength of the stimulus. For example, in the yeast Saccharomyces cerevisiae, the concentration of mating pheromone determines whether cells undergo vegetative growth, chemotropic growth, or mating. This implies that the signaling pathways responsible for detecting the stimulus and initiating a response must transmit quantitative information about the intensity of the signal. Our previous experimental results suggest that yeast encode pheromone concentration as the duration of the transmitted signal. Here we use mathematical modeling to analyze possible biochemical mechanisms for performing this “dose-to-duration” conversion. We demonstrate that modulation of signal duration increases the range of stimulus concentrations for which dose-dependent responses are possible; this increased dynamic range produces the counterintuitive result of “signaling beyond saturation” in which dose-dependent responses are still possible after apparent saturation of the receptors. We propose a mechanism for dose-to-duration encoding in the yeast pheromone pathway that is consistent with current experimental observations. Most previous investigations of information processing by signaling pathways have focused on amplitude encoding without considering temporal aspects of signal transduction. Here we demonstrate that dose-to-duration encoding provides cells with an alternative mechanism for processing and transmitting quantitative information about their surrounding environment. The ability of signaling pathways to convert stimulus strength into signal duration results directly from the nonlinear nature of these systems and emphasizes the importance of considering the dynamic properties of signaling pathways when characterizing their behavior. Understanding how signaling pathways encode and transmit quantitative information about the external environment will not only deepen our understanding of these systems but also provide insight into how to reestablish proper function of pathways that have become dysregulated by disease.

Cells must be able to detect and respond to changes in their surroundings. Often environmental cues, such as hormones or growth factors, are received by membrane receptors that in turn activate intracellular signaling pathways. These pathways then transmit information about the stimulus to the cellular components required to elicit an appropriate response. In many cases, the nature of the response depends on the dose of the stimulus. Thus, in addition to relaying qualitative information (e.g., the presence or absence of a stimulus), signaling pathways must also transmit quantitative information about the intensity of the stimulus. Here we introduce “dose-to-duration” encoding as an effective strategy for relaying such information. We demonstrate that by providing a mechanism for overcoming saturation effects, modulation of signal duration increases the range of stimulus concentrations for which dose-dependent responses are possible. This increased dynamic range produces the counterintuitive result of “signaling beyond saturation” in which dose-dependent responses are still possible after apparent saturation of the receptors. Finally, we demonstrate that dose-to-duration encoding is used in the yeast mating response pathway and presents a simple mechanism that can account for current experimental observations.

Adaptive dynamics with a single two-state protein

An important step towards understanding biological systems is to relate simple biochemical elements to dynamics. Here, we present the arguably simplest dynamical element in biochemical networks. It consists of a single protein with two states (active and inactive) and an external signal that catalyses the conversion between these two states. Further, there is steady synthesis and degradation of the inactive and active forms, respectively. As this element captures both structural and dynamical features of biochemical networks at the lowest level, we refer to it as a biochemical network unit (BioNetUnit). Using both simulations and mathematical analysis, we find that BioNetUnit shows perfect adaptation that leads to temporal responses to step changes in the incoming signal. Compared with a well-described adaptive system, which is found in bacterial chemotaxis, BioNetUnit has lower sensitivity and its adaptation time is less robust to the base signal levels. We show that these dynamical limitations lead to 'once-and-only-once' responses for certain signal sequences. These findings demonstrate that BioNetUnit is relevant in adaptive and cyclic processes. In particular, it could be seen as a generic representation for ligand-activated receptors that are desensitized upon continuous activation. The analysis of coupled BioNetUnits will show how the presented dynamics at single unit will change upon increased system complexity and how such systems would mediate biological functions.

Pulsing cells: how fast is too fast?

Signal transduction pathways are complex coupled sets of biochemical reactions evolved to transmit and process information about the state of the immediate cell environment. Can we design experiments that would inform us about the properties and limitations of signal processing? Recent studies suggest that this indeed can be achieved by exciting a cell with carefully designed oscillatory stimuli. Although this analysis has its caveats, complex temporal stimulation of signal transduction networks can serve to rapidly advance our understanding of these information channels and ultimately create intelligent ways of controlling them.

Biological switches and clocks

To introduce this special issue on biological switches and clocks, we review the historical development of mathematical models of bistability and oscillations in chemical reaction networks. In the 1960s and 1970s, these models were limited to well-studied biochemical examples, such as glycolytic oscillations and cyclic AMP signalling. After the molecular genetics revolution of the 1980s, the field of molecular cell biology was thrown wide open to mathematical modellers. We review recent advances in modelling the gene-protein interaction networks that control circadian rhythms, cell cycle progression, signal processing and the design of synthetic gene networks.

The impact of the regulatory design on the response of epidermal growth factor receptor-mediated signal transduction towards oncogenic mutations

Epidermal growth factor receptor (EGFR)-mediated signal transduction is often hyperactivated in tumour cells and therefore considered a promising target for cancer therapy. A number of computational models have been developed which describe the pathway in great detail. These models are similar in their description of the activation events. The deactivation of the EGFR signalling seems to be cell type-specific and is less understood. Deactivation via receptor internalization, feedback inhibition of son of sevenless (SOS) by double phosphorylated, extracellular signal-regulated kinase (ERKPP) or transiently activated Ras-GTPase activating protein (Ras-GAP) proteins is discussed to play a role. In this study we address the question of to what extent the effect of oncogenic perturbations on EGFR signalling depend on the specific regulation structure. This is investigated using a detailed pathway model under two regulatory modes: the negative feedback via ERKPP to SOS and feed-forward deactivation via transiently activated Ras-GAP proteins. We show that the effect of receptor overexpression differs qualitatively under both regulations. In the system with transiently activated Ras-GAP it may result in an attenuation of the ERK activation. Such a nonintuitive effect was also observed experimentally. In general we find the model with transiently activated Ras-GAP to have a higher robustness towards receptor overexpression and Ras mutations. In particular, we demonstrate that this model can compensate for these oncogenic perturbations if the regulation is strong. The negative feedback can not protect the system against Ras mutations. A general sensitivity analysis, however, shows a higher robustness of the model under negative feedback, indicating the limited significance of such analyses for the prediction of specific oncogenic perturbations.

Kinetic insulation as an effective mechanism for achieving pathway specificity in intracellular signaling networks

Intracellular signaling pathways that share common components often elicit distinct physiological responses. In most cases, the biochemical mechanisms responsible for this signal specificity remain poorly understood. Protein scaffolds and cross-inhibition have been proposed as strategies to prevent unwanted cross-talk. Here, we report a mechanism for signal specificity termed "kinetic insulation." In this approach signals are selectively transmitted through the appropriate pathway based on their temporal profile. In particular, we demonstrate how pathway architectures downstream of a common component can be designed to efficiently separate transient signals from signals that increase slowly over time. Furthermore, we demonstrate that upstream signaling proteins can generate the appropriate input to the common pathway component regardless of the temporal profile of the external stimulus. Our results suggest that multilevel signaling cascades may have evolved to modulate the temporal profile of pathway activity so that stimulus information can be efficiently encoded and transmitted while ensuring signal specificity.