A hidden feedback in signaling cascades is revealed

Mathematical basis and toolchain for hierarchical optimization of biochemical networks

Biological signalling systems are complex, and efforts to build mechanistic models must confront a huge parameter space, indirect and sparse data, and frequently encounter multiscale and multiphysics phenomena. We present HOSS, a framework for Hierarchical Optimization of Systems Simulations, to address such problems. HOSS operates by breaking down extensive systems models into individual pathway blocks organized in a nested hierarchy. At the first level, dependencies are solely on signalling inputs, and subsequent levels rely only on the preceding ones. We demonstrate that each independent pathway in every level can be efficiently optimized. Once optimized, its parameters are held constant while the pathway serves as input for succeeding levels. We develop an algorithmic approach to identify the necessary nested hierarchies for the application of HOSS in any given biochemical network. Furthermore, we devise two parallelizable variants that generate numerous model instances using stochastic scrambling of parameters during initial and intermediate stages of optimization. Our results indicate that these variants produce superior models and offer an estimate of solution degeneracy. Additionally, we showcase the effectiveness of the optimization methods for both abstracted, event-based simulations and ODE-based models.

Transient frequency preference responses in cell signaling systems

Ligand-receptor systems, covalent modification cycles, and transcriptional networks are the fundamental components of cell signaling and gene expression systems. While their behavior in reaching a steady-state regime under step-like stimulation is well understood, their response under repetitive stimulation, particularly at early time stages is poorly characterized. Yet, early-stage responses to external inputs are arguably as informative as late-stage ones. In simple systems, a periodic stimulation elicits an initial transient response, followed by periodic behavior. Transient responses are relevant when the stimulation has a limited time span, or when the stimulated component's timescale is slow as compared to the timescales of the downstream processes, in which case the latter processes may be capturing only those transients. In this study, we analyze the frequency response of simple motifs at different time stages. We use dose-conserved pulsatile input signals and consider different metrics versus frequency curves. We show that in ligand-receptor systems, there is a frequency preference response in some specific metrics during the transient stages, which is not present in the periodic regime. We suggest this is a general system-level mechanism that cells may use to filter input signals that have consequences for higher order circuits. In addition, we evaluate how the described behavior in isolated motifs is reflected in similar types of responses in cascades and pathways of which they are a part. Our studies suggest that transient frequency preferences are important dynamic features of cell signaling and gene expression systems, which have been overlooked.

Design patterns of biological cells

Design patterns are generalized solutions to frequently recurring problems. They were initially developed by architects and computer scientists to create a higher level of abstraction for their designs. Here, we extend these concepts to cell biology to lend a new perspective on the evolved designs of cells' underlying reaction networks. We present a catalog of 21 design patterns divided into three categories: creational patterns describe processes that build the cell, structural patterns describe the layouts of reaction networks, and behavioral patterns describe reaction network function. Applying this pattern language to the E. coli central metabolic reaction network, the yeast pheromone response signaling network, and other examples lends new insights into these systems.

Targeting Human Proteins for Antiviral Drug Discovery and Repurposing Efforts: A Focus on Protein Kinases

Despite the great technological and medical advances in fighting viral diseases, new therapies for most of them are still lacking, and existing antivirals suffer from major limitations regarding drug resistance and a limited spectrum of activity. In fact, most approved antivirals are directly acting antiviral (DAA) drugs, which interfere with viral proteins and confer great selectivity towards their viral targets but suffer from resistance and limited spectrum. Nowadays, host-targeted antivirals (HTAs) are on the rise, in the drug discovery and development pipelines, in academia and in the pharmaceutical industry. These drugs target host proteins involved in the virus life cycle and are considered promising alternatives to DAAs due to their broader spectrum and lower potential for resistance. Herein, we discuss an important class of HTAs that modulate signal transduction pathways by targeting host kinases. Kinases are considered key enzymes that control virus-host interactions. We also provide a synopsis of the antiviral drug discovery and development pipeline detailing antiviral kinase targets, drug types, therapeutic classes for repurposed drugs, and top developing organizations. Furthermore, we detail the drug design and repurposing considerations, as well as the limitations and challenges, for kinase-targeted antivirals, including the choice of the binding sites, physicochemical properties, and drug combinations.

A nested bistable module within a negative feedback loop ensures different types of oscillations in signaling systems

In this article, we consider a double phosphorylation cycle, a ubiquitous signaling component, having the ability to display bistability, a behavior strongly related to the existence of positive feedback loops. If this component is connected to other signaling elements, it very likely undergoes some sort of protein-protein interaction. In several cases, these interactions result in a non-explicit negative feedback effect, leading to interlinked positive and negative feedbacks. This combination was studied in the literature as a way to generate relaxation-type oscillations. Here, we show that the two feedbacks together ensure two types of oscillations, the relaxation-type ones and a smoother type of oscillations functioning in a very narrow range of frequencies, in such a way that outside that range, the amplitude of the oscillations is severely compromised. Even more, we show that the two feedbacks are essential for both oscillatory types to emerge, and it is their hierarchy what determines the type of oscillation at work. We used bifurcation analyses and amplitude vs. frequency curves to characterize and classify the oscillations. We also applied the same ideas to another simple model, with the goal of generalizing what we learned from signaling models. The results obtained display the wealth of oscillatory dynamics that exists in a system with a bistable module nested within a negative feedback loop, showing how to transition between different types of oscillations and other dynamical behaviors such as excitability. Our work provides a framework for the study of other oscillatory systems based on bistable modules, from simple two-component models to more complex examples like the MAPK cascade and experimental cases like cell cycle oscillators.

Dynamics and Sensitivity of Signaling Pathways

Purpose of Review

Signaling pathways serve to communicate information about extracellular conditions into the cell, to both the nucleus and cytoplasmic processes to control cell responses. Genetic mutations in signaling network components are frequently associated with cancer and can result in cells acquiring an ability to divide and grow uncontrollably. Because signaling pathways play such a significant role in cancer initiation and advancement, their constituent proteins are attractive therapeutic targets. In this review, we discuss how signaling pathway modeling can assist with identifying effective drugs for treating diseases, such as cancer. An achievement that would facilitate the use of such models is their ability to identify controlling biochemical parameters in signaling pathways, such as molecular abundances and chemical reaction rates, because this would help determine effective points of attack by therapeutics.

Recent Findings

We summarize the current state of understanding the sensitivity of phosphorylation cycles with and without sequestration. We also describe some basic properties of regulatory motifs including feedback and feedforward regulation.

Summary

Although much recent work has focused on understanding the dynamics and particularly the sensitivity of signaling networks in eukaryotic systems, there is still an urgent need to build more scalable models of signaling networks that can appropriately represent their complexity across different cell types and tumors.

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.

Optimizing enzymatic catalysts for rapid turnover of substrates with low enzyme sequestration

Enzymes are central to both metabolism and information processing in cells. In both cases, an enzyme's ability to accelerate a reaction without being consumed in the reaction is crucial. Nevertheless, enzymes are transiently sequestered when they bind to their substrates; this sequestration limits activity and potentially compromises information processing and signal transduction. In this article, we analyse the mechanism of enzyme-substrate catalysis from the perspective of minimizing the load on the enzymes through sequestration, while maintaining at least a minimum reaction flux. In particular, we ask: which binding free energies of the enzyme-substrate and enzyme-product reaction intermediates minimize the fraction of enzymes sequestered in complexes, while sustaining a certain minimal flux? Under reasonable biophysical assumptions, we find that the optimal design will saturate the bound on the minimal flux and reflects a basic trade-off in catalytic operation. If both binding free energies are too high, there is low sequestration, but the effective progress of the reaction is hampered. If both binding free energies are too low, there is high sequestration, and the reaction flux may also be suppressed in extreme cases. The optimal binding free energies are therefore neither too high nor too low, but in fact moderate. Moreover, the optimal difference in substrate and product binding free energies, which contributes to the thermodynamic driving force of the reaction, is in general strongly constrained by the intrinsic free-energy difference between products and reactants. Both the strategies of using a negative binding free-energy difference to drive the catalyst-bound reaction forward and of using a positive binding free-energy difference to enhance detachment of the product are limited in their efficacy.

Monostationarity and Multistationarity in Tree Networks of Goldbeter-Koshland Loops

A major challenge in systems biology is to elicit general properties in the face of molecular complexity. Here, we introduce a class of enzyme-catalysed biochemical networks and examine how the existence of a single positive steady state (monostationarity) depends on the network structure, enzyme mechanisms, kinetic rate laws and parameter values. We consider Goldbeter-Koshland (GK) covalent modification loops arranged in a tree network, so that a substrate form in one loop can be an enzyme in another loop. GK loops are a canonical motif in cell signalling and trees offer a generalisation of linear cascades which accommodate network complexity while remaining mathematically tractable. In particular, they permit a modular, recursive proof strategy which may be more widely applicable. We show that if each enzyme follows its own complex reaction mechanism under mass action kinetics, then any network is monostationary for all appropriate parameter values. If the kinetics is non-mass action with a plausible monotonicity requirement, and each enzyme follows the Michaelis-Menten mechanism, then monostationarity is preserved. Surprisingly, a single GK loop with a complex enzyme mechanism under non-mass action monotone kinetics can have more than one positive steady state (multistationarity). The broader interplay between network structure, enzyme mechanism and kinetics remains an intriguing open problem.

The role of stochastic sequestration dynamics for intrinsic noise filtering in signaling network motifs

The relation between design principles of signaling network motifs and their robustness against intrinsic noise still remains illusive. In this work we investigate the role of cascading for coping with intrinsic noise due to stochasticity in molecular reactions. We use stochastic approaches to quantify fluctuations in the terminal kinase of phosphorylation-dephosphorylation cascade motifs and demonstrate that cascading highly affects these fluctuations. We show that this purely stochastic effect can be explained by time-varying sequestration of upstream kinase molecules. In particular, we discuss conditions on time scales and parameter regimes which lead to a reduction of output fluctuations. Our results are put into biological context by adapting rate parameters of our modeling approach to biologically feasible ranges for general binding-unbinding and phosphorylation-dephosphorylation mechanisms. Overall, this study reveals a novel role of stochastic sequestration for dynamic noise filtering in signaling cascade motifs.

Graphical requirements for multistationarity in reaction networks and their verification in BioModels

Thomas' necessary conditions for the existence of multiple steady states in gene networks have been proved by Soulé with high generality for dynamical systems defined by differential equations. When applied to (protein) reaction networks however, those conditions do not provide information since they are trivially satisfied as soon as there is a bimolecular or a reversible reaction. Refined graphical requirements have been proposed to deal with such cases. In this paper, we present for the first time a graph rewriting algorithm for checking the refined conditions given by Soliman, and evaluate its practical performance by applying it systematically to the curated branch of the BioModels repository. This algorithm analyzes all reaction networks (of size up to 430 species) in less than 0.05 second per network, and permits to conclude to the absence of multistationarity in 160 networks over 506. The short computation times obtained in this graphical approach are in sharp contrast to the Jacobian-based symbolic computation approach. We also discuss the case of one extra graphical condition by arc rewiring that allows us to conclude on 20 more networks of this benchmark but with a high computational cost. Finally, we study with some details the case of phosphorylation cycles and MAPK signalling models which show the importance of modelling the intermediate complexations with the enzymes in order to correctly analyze the multistationarity capabilities of such biochemical reaction networks.

Emergent memory in cell signaling: Persistent adaptive dynamics in cascades can arise from the diversity of relaxation time-scales

The mitogen-activated protein kinase (MAPK) signaling cascade, an evolutionarily conserved motif present in all eukaryotic cells, is involved in coordinating crucial cellular functions. While the asymptotic dynamical behavior of the pathway stimulated by a time-invariant signal is relatively well-understood, we show using a computational model that it exhibits a rich repertoire of transient adaptive responses to changes in stimuli. When the signal is switched on, the response is characterized by long-lived modulations in frequency as well as amplitude. On withdrawing the stimulus, the activity decays over long timescales, exhibiting reverberations characterized by repeated spiking in the activated MAPK concentration. The long-term persistence of such post-stimulus activity suggests that the cascade retains memory of the signal for a significant duration following its removal. The molecular mechanism underlying the reverberatory activity is related to the existence of distinct relaxation rates for the different cascade components. This results in the imbalance of fluxes between different layers of the cascade, with the reuse of activated kinases as enzymes when they are released from sequestration in complexes. The persistent adaptive response, indicative of a cellular “short-term” memory, suggests that this ubiquitous signaling pathway plays an even more central role in information processing by eukaryotic cells.

Core signalling motif displaying multistability through multi-state enzymes

Bistability, and more generally multistability, is a key system dynamics feature enabling decision-making and memory in cells. Deciphering the molecular determinants of multistability is thus crucial for a better understanding of cellular pathways and their (re)engineering in synthetic biology. Here, we show that a key motif found predominantly in eukaryotic signalling systems, namely a futile signalling cycle, can display bistability when featuring a two-state kinase. We provide necessary and sufficient mathematical conditions on the kinetic parameters of this motif that guarantee the existence of multiple steady states. These conditions foster the intuition that bistability arises as a consequence of competition between the two states of the kinase. Extending from this result, we find that increasing the number of kinase states linearly translates into an increase in the number of steady states in the system. These findings reveal, to our knowledge, a new mechanism for the generation of bistability and multistability in cellular signalling systems. Further the futile cycle featuring a two-state kinase is among the smallest bistable signalling motifs. We show that multi-state kinases and the described competition-based motif are part of several natural signalling systems and thereby could enable them to implement complex information processing through multistability. These results indicate that multi-state kinases in signalling systems are readily exploited by natural evolution and could equally be used by synthetic approaches for the generation of multistable information processing systems at the cellular level.

Modelling compartmentalization towards elucidation and engineering of spatial organization in biochemical pathways

Compartmentalization is a fundamental ingredient, central to the functioning of biological systems at multiple levels. At the cellular level, compartmentalization is a key aspect of the functioning of biochemical pathways and an important element used in evolution. It is also being exploited in multiple contexts in synthetic biology. Accurate understanding of the role of compartments and designing compartmentalized systems needs reliable modelling/systems frameworks. We examine a series of building blocks of signalling and metabolic pathways with compartmental organization. We systematically analyze when compartmental ODE models can be used in these contexts, by comparing these models with detailed reaction-transport models, and establishing a correspondence between the two. We build on this to examine additional complexities associated with these pathways, and also examine sample problems in the engineering of these pathways. Our results indicate under which conditions compartmental models can and cannot be used, why this is the case, and what augmentations are needed to make them reliable and predictive. We also uncover other hidden consequences of employing compartmental models in these contexts. Or results contribute a number of insights relevant to the modelling, elucidation, and engineering of biochemical pathways with compartmentalization, at the core of systems and synthetic biology.

Signaling Architectures that Transmit Unidirectional Information Despite Retroactivity

A signaling pathway transmits information from an upstream system to downstream systems, ideally in a unidirectional fashion. A key obstacle to unidirectional transmission is retroactivity, the additional reaction flux that affects a system once its species interact with those of downstream systems. This raises the fundamental question of whether signaling pathways have developed specialized architectures that overcome retroactivity and transmit unidirectional signals. Here, we propose a general procedure based on mathematical analysis that provides an answer to this question. Using this procedure, we analyze the ability of a variety of signaling architectures to transmit one-way (from upstream to downstream) signals, as key biological parameters are tuned. We find that single stage phosphorylation and phosphotransfer systems that transmit signals from a kinase show a stringent design tradeoff that hampers their ability to overcome retroactivity. Interestingly, cascades of these architectures, which are highly represented in nature, can overcome this tradeoff and thus enable unidirectional transmission. By contrast, phosphotransfer systems, and single and double phosphorylation cycles that transmit signals from a substrate, are unable to mitigate retroactivity effects, even when cascaded, and hence are not well suited for unidirectional information transmission. These results are largely independent of the specific reaction-rate constant values, and depend on the topology of the architectures. Our results therefore identify signaling architectures that, allowing unidirectional transmission of signals, embody modular processes that conserve their input/output behavior across multiple contexts. These findings can be used to decompose natural signal transduction networks into modules, and at the same time, they establish a library of devices that can be used in synthetic biology to facilitate modular circuit design.

A Discrete Dynamical System Approach to Pathway Activation Profiles of Signaling Cascades

In living organisms, cascades of covalent modification cycles are one of the major intracellular signaling mechanisms, allowing to transduce physical or chemical stimuli of the external world into variations of activated biochemical species within the cell. In this paper, we develop a novel method to study the stimulus-response of signaling cascades and overall the concept of pathway activation profile which is, for a given stimulus, the sequence of activated proteins at each tier of the cascade. Our approach is based on a correspondence that we establish between the stationary states of a cascade and pieces of orbits of a 2D discrete dynamical system. The study of its possible phase portraits in function of the biochemical parameters, and in particular of the contraction/expansion properties around the fixed points of this discrete map, as well as their bifurcations, yields a classification of the cascade tiers into three main types, whose biological impact within a signaling network is examined. In particular, our approach enables to discuss quantitatively the notion of cascade amplification/attenuation from this new perspective. The method allows also to study the interplay between forward and "retroactive" signaling, i.e., the upstream influence of an inhibiting drug bound to the last tier of the cascade.

Ultrasensitivity in signaling cascades revisited: Linking local and global ultrasensitivity estimations

Ultrasensitive response motifs, capable of converting graded stimuli into binary responses, are well-conserved in signal transduction networks. Although it has been shown that a cascade arrangement of multiple ultrasensitive modules can enhance the system's ultrasensitivity, how a given combination of layers affects a cascade's ultrasensitivity remains an open question for the general case. Here, we introduce a methodology that allows us to determine the presence of sequestration effects and to quantify the relative contribution of each module to the overall cascade's ultrasensitivity. The proposed analysis framework provides a natural link between global and local ultrasensitivity descriptors and it is particularly well-suited to characterize and understand mathematical models used to study real biological systems. As a case study, we have considered three mathematical models introduced by O'Shaughnessy et al. to study a tunable synthetic MAPK cascade, and we show how our methodology can help modelers better understand alternative models.

Minimal oscillating subnetwork in the Huang-Ferrell model of the MAPK cascade

Prompted by the recent growing evidence of oscillatory behavior involving MAPK cascades we present a systematic approach of analyzing models and elucidating the nature of biochemical oscillations based on reaction network theory. In particular, we formulate a minimal biochemically consistent mass action subnetwork of the Huang-Ferrell model of the MAPK signalling that provides an oscillatory response when a parameter controlling the activation of the top-tier kinase is varied. Such dynamics are either intertwined with or separated from the earlier found bistable/hysteretic behavior in this model. Using the theory of stability of stoichiometric networks, we reduce the original MAPK model, convert kinetic to convex parameters and examine those properties of the minimal subnetwork that underlie the oscillatory dynamics. We also use the methods of classification of chemical oscillatory networks to explain the rhythmic behavior in physicochemical terms, i.e., we identify of the role of individual biochemical species in positive and negative feedback loops and describe their coordinated action leading to oscillations. Our approach provides an insight into dynamics without the necessity of knowing rate coefficients and thus is useful prior the statistical evaluation of parameters.

Bridging the gap between modules in isolation and as part of networks: A systems framework for elucidating interaction and regulation of signalling modules

While signalling and biochemical modules have been the focus of numerous studies, they are typically studied in isolation, with no examination of the effects of the ambient network. In this paper we formulate and develop a systems framework, rooted in dynamical systems, to understand such effects, by studying the interaction of signalling modules. The modules we consider are (i) basic covalent modification, (ii) monostable switches, (iii) bistable switches, (iv) adaptive modules, and (v) oscillatory modules. We systematically examine the interaction of these modules by analyzing (a) sequential interaction without shared components, (b) sequential interaction with shared components, and (c) oblique interactions. Our studies reveal that the behaviour of a module in isolation may be substantially different from that in a network, and explicitly demonstrate how the behaviour of a given module, the characteristics of the ambient network, and the possibility of shared components can result in new effects. Our global approach illuminates different aspects of the structure and functioning of modules, revealing the importance of dynamical characteristics as well as biochemical features; this provides a methodological platform for investigating the complexity of natural modules shaped by evolution, elucidating the effects of ambient networks on a module in multiple cellular contexts, and highlighting the capabilities and constraints for engineering robust synthetic modules. Overall, such a systems framework provides a platform for bridging the gap between non-linear information processing modules, in isolation and as parts of networks, and a basis for understanding new aspects of natural and engineered cellular networks.

Sustained oscillations in the MAP kinase cascade

The MAP kinase cascade is a network of enzymatic reactions arranged in layers. In each layer occurs a multiple futile cycle of phosphorylations. The fully phosphorylated substrate then serves as an enzyme for the layer below. This paper focusses on the existence of parameters for which Hopf bifurcations occur and generate periodic orbits. Furthermore it is explained how geometric singular perturbation theory allows to generalize results from simple models to more complex ones.

Push-Pull and Feedback Mechanisms Can Align Signaling System Outputs with Inputs

Many cell signaling systems, including the yeast pheromone response system, exhibit "dose-response alignment" (DoRA), in which output of one or more downstream steps closely matches the fraction of occupied receptors. DoRA can improve the fidelity of transmitted dose information. Here, we searched systematically for biochemical network topologies that produced DoRA. Most networks, including many containing feedback and feedforward loops, could not produce DoRA. However, networks including "push-pull" mechanisms, in which the active form of a signaling species stimulates downstream activity and the nominally inactive form reduces downstream activity, enabled perfect DoRA. Networks containing feedbacks enabled DoRA, but only if they also compared feedback to input and adjusted output to match. Our results establish push-pull as a non-feedback mechanism to align output with variable input and maximize information transfer in signaling systems. They also suggest genetic approaches to determine whether particular signaling systems use feedback or push-pull control.

Signaling cascades transmit information downstream and upstream but unlikely simultaneously

Background

Signal transduction is the process through which cells communicate with the external environment, interpret stimuli and respond to them. This mechanism is controlled by signaling cascades, which play the role of intracellular transmitter, being able to transmit biochemical information between cell membrane and nucleus. In theory as well as in practice, it has been shown that a perturbation can propagate upstream (and not only downstream) a cascade, by a mechanism known as retroactivity. This study aims to compare the conditions on biochemical parameters which favor one or the other direction of signaling in such a cascade.

Results

From a mathematical point of view, we show that the steady states of a cascade of arbitrary length n are described by an iterative map of second order, meaning that the cascade tiers are actually coupled three-by-three. We study the influence of the biochemical parameters in the control of the direction of transmission – upstream and/or downstream – along a signaling cascade. A numerical and statistical approach, based on the random scan of parameters describing a 3-tier signaling cascade, provides complementary findings to the analytical study. In particular, computing the likelihood of parameters with respect to various signaling regimes, we identify conditions on biochemical parameters which enhance a specific direction of propagation corresponding to forward or retro-signaling regimes. A compact graphical representation is designed to relay the gist of these conditions.

Conclusions

The values of biochemical parameters such as kinetic rates, Michaelis-Menten constants, total concentrations of kinases and of phosphatases, determine the propensity of a cascade to favor or impede downstream or upstream signal transmission. We found that generally there is an opposition between parameter sets favoring forward and retro-signaling regimes. Therefore, on one hand our study supports the idea that in most cases, retroactive effects can be neglected when a cascade which is efficient in forward signaling, is perturbed by an external ligand inhibiting the activation at some tier of the cascade. This result is relevant for therapeutic methodologies based on kinase inhibition. On the other hand, our study highlights a less-known part of the parameter space where, although the forward signaling is inefficient, the cascade can interestingly act as a retro-signaling device.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0303-2) contains supplementary material, which is available to authorized users.

Ultrasensitive Negative Feedback Control: A Natural Approach for the Design of Synthetic Controllers

Many of the most important potential applications of Synthetic Biology will require the ability to design and implement high performance feedback control systems that can accurately regulate the dynamics of multiple molecular species within the cell. Here, we argue that the use of design strategies based on combining ultrasensitive response dynamics with negative feedback represents a natural approach to this problem that fully exploits the strongly nonlinear nature of cellular information processing. We propose that such feedback mechanisms can explain the adaptive responses observed in one of the most widely studied biomolecular feedback systems—the yeast osmoregulatory response network. Based on our analysis of such system, we identify strong links with a well-known branch of mathematical systems theory from the field of Control Engineering, known as Sliding Mode Control. These insights allow us to develop design guidelines that can inform the construction of feedback controllers for synthetic biological systems.

Spatial Control of Biochemical Modification Cascades and Pathways

Information transmission in cells occurs through complex networks of proteins and genes and is relayed through cascades of biochemical modifications, which are typically studied through ordinary differential equations. However, it is becoming increasingly clear that spatial factors can strongly influence chemical information transmission in cells. In this article, we systematically disentangle the effects of space in signaling cascades. This is done by examining the effects of localization/compartmentalization and diffusion of enzymes and substrates in multiple variants of chemical modification cascades. This includes situations where the modified form of species at one stage 1) acts as an enzyme for the next stage; 2) acts as a substrate for the next stage; and 3) is involved in phosphotransfer. Our analysis reveals the multiple effects of space in signal transduction cascades. Although in some cases space plays a modulatory effect (itself of interest), in other cases, spatial regulation and control can profoundly affect the nature of information processing as a result of the subtle interplay between the patterns of localization of species, diffusion, and the nature of the modification cascades. Our results provide a platform for disentangling the role of space and spatial control in multiple cellular contexts and a basis for engineering spatial control in signaling cascades through localization/compartmentalization.

Retroactivity in the Context of Modularly Structured Biomolecular Systems

Synthetic biology has intensively promoted the technical implementation of modular strategies in the fabrication of biological devices. Modules are considered as networks of reactions. The behavior displayed by biomolecular systems results from the information processes carried out by the interconnection of the involved modules. However, in natural systems, module wiring is not a free-of-charge process; as a consequence of interconnection, a reactive phenomenon called retroactivity emerges. This phenomenon is characterized by signals that propagate from downstream modules (the modules that receive the incoming signals upon interconnection) to upstream ones (the modules that send the signals upon interconnection). Such retroactivity signals, depending of their strength, may change and sometimes even disrupt the behavior of modular biomolecular systems. Thus, analysis of retroactivity effects in natural biological and biosynthetic systems is crucial to achieve a deeper understanding of how this interconnection between functionally characterized modules takes place and how it impacts the overall behavior of the involved cell. By discussing the modules interconnection in natural and synthetic biomolecular systems, we propose that such systems should be considered as quasi-modular.

Paradoxical results in perturbation-based signaling network reconstruction

Mathematical models are extensively employed to understand physicochemical processes in biological systems. In the absence of detailed mechanistic knowledge, models are often based on network inference methods, which in turn rely upon perturbations to nodes by biochemical means. We have discovered a potential pitfall of the approach underpinning such methods when applied to signaling networks. We first show experimentally, and then explain mathematically, how even in the simplest signaling systems, perturbation methods may lead to paradoxical conclusions: for any given pair of two components X and Y, and depending upon the specific intervention on Y, either an activation or a repression of X could be inferred. This effect is of a different nature from incomplete network identification due to underdetermined data and is a phenomenon intrinsic to perturbations. Our experiments are performed in an in vitro minimal system, thus isolating the effect and showing that it cannot be explained by feedbacks due to unknown intermediates. Moreover, our in vitro system utilizes proteins from a pathway in mammalian (and other eukaryotic) cells that play a central role in proliferation, gene expression, differentiation, mitosis, cell survival, and apoptosis. This pathway is the perturbation target of contemporary therapies for various types of cancers. The results presented here show that the simplistic view of intracellular signaling networks being made up of activation and repression links is seriously misleading, and call for a fundamental rethinking of signaling network analysis and inference methods.

Impact of upstream and downstream constraints on a signaling module's ultrasensitivity

Much work has been done on the study of the biochemical mechanisms that result in ultrasensitive behavior of simple biochemical modules. However, in a living cell, such modules are embedded in a bigger network that constrains the range of inputs that the module will receive as well as the range of the module's outputs that network will be able to detect. Here, we studied how the effective ultrasensitivity of a modular system is affected by these restrictions. We use a simple setup to explore to what extent the dynamic range spanned by upstream and downstream components of an ultrasensitive module impact on the effective sensitivity of the system. Interestingly, we found for some ultrasensitive motifs that dynamic range limitations imposed by downstream components can produce effective sensitivities much larger than that of the original module when considered in isolation.

Determinants of cell-to-cell variability in protein kinase signaling

Cells reliably sense environmental changes despite internal and external fluctuations, but the mechanisms underlying robustness remain unclear. We analyzed how fluctuations in signaling protein concentrations give rise to cell-to-cell variability in protein kinase signaling using analytical theory and numerical simulations. We characterized the dose-response behavior of signaling cascades by calculating the stimulus level at which a pathway responds (‘pathway sensitivity’) and the maximal activation level upon strong stimulation. Minimal kinase cascades with gradual dose-response behavior show strong variability, because the pathway sensitivity and the maximal activation level cannot be simultaneously invariant. Negative feedback regulation resolves this trade-off and coordinately reduces fluctuations in the pathway sensitivity and maximal activation. Feedbacks acting at different levels in the cascade control different aspects of the dose-response curve, thereby synergistically reducing the variability. We also investigated more complex, ultrasensitive signaling cascades capable of switch-like decision making, and found that these can be inherently robust to protein concentration fluctuations. We describe how the cell-to-cell variability of ultrasensitive signaling systems can be actively regulated, e.g., by altering the expression of phosphatase(s) or by feedback/feedforward loops. Our calculations reveal that slow transcriptional negative feedback loops allow for variability suppression while maintaining switch-like decision making. Taken together, we describe design principles of signaling cascades that promote robustness. Our results may explain why certain signaling cascades like the yeast pheromone pathway show switch-like decision making with little cell-to-cell variability.

Cells sense their surroundings and respond to soluble factors in the extracellular space. Extracellular factors frequently induce heterogeneous responses, thereby restricting the biological outcome to a fraction of the cell population. However, the question arises how such cell-to-cell variability can be controlled, because some cellular systems show a very homogenous response at a defined level of an extracellular stimulus. We derived an analytical framework to systematically characterize the cell-to-cell variability of intracellular signaling pathways which transduce external signals. We analyzed how heterogeneity arises from fluctuations in the total concentrations of signaling proteins because this is the main source of variability in eukaryotic systems. We find that signaling pathways can be highly variable or inherently invariant, depending on the kinetic parameters and the structural features of the cascade. Our results indicate that the cell-to-cell variability can be reduced by negative feedback in the cascade or by signaling crosstalk between parallel pathways. We precisely define the role of negative feedback loops in variability suppression, and show that different aspects of the dose-response curve can be controlled, depending on the feedback kinetics and site of action in the cascade. This work constitutes a first step towards a systematic understanding of cell-to-cell variability in signal transduction.

Simplifying biochemical models with intermediate species

Mathematical models are increasingly being used to understand complex biochemical systems, to analyse experimental data and make predictions about unobserved quantities. However, we rarely know how robust our conclusions are with respect to the choice and uncertainties of the model. Using algebraic techniques, we study systematically the effects of intermediate, or transient, species in biochemical systems and provide a simple, yet rigorous mathematical classification of all models obtained from a core model by including intermediates. Main examples include enzymatic and post-translational modification systems, where intermediates often are considered insignificant and neglected in a model, or they are not included because we are unaware of their existence. All possible models obtained from the core model are classified into a finite number of classes. Each class is defined by a mathematically simple canonical model that characterizes crucial dynamical properties, such as mono- and multistationarity and stability of steady states, of all models in the class. We show that if the core model does not have conservation laws, then the introduction of intermediates does not change the steady-state concentrations of the species in the core model, after suitable matching of parameters. Importantly, our results provide guidelines to the modeller in choosing between models and in distinguishing their properties. Further, our work provides a formal way of comparing models that share a common skeleton.

Branched motifs enable long-range interactions in signaling networks through retrograde propagation

Branched structures arise in the intra-cellular signaling network when a molecule is involved in multiple enzyme-substrate reaction cascades. Such branched motifs are involved in key biological processes, e.g., immune response activated by T-cell and B-cell receptors. In this paper, we demonstrate long-range communication through retrograde propagation between branches of signaling pathways whose molecules do not directly interact. Our numerical simulations and experiments on a system comprising branches with JNK and p38MAPK as terminal molecules respectively that share a common MAP3K enzyme MEKK3/4 show that perturbing an enzyme in one branch can result in a series of changes in the activity levels of molecules “upstream” to the enzyme that eventually reaches the branch-point and affects other branches. In the absence of any evidence for explicit feedback regulation between the functionally distinct JNK and p38MAPK pathways, the experimentally observed modulation of phosphorylation amplitudes in the two pathways when a terminal kinase is inhibited implies the existence of long-range coordination through retrograde information propagation previously demonstrated in single linear reaction pathways. An important aspect of retrograde propagation in branched pathways that is distinct from previous work on retroactivity focusing exclusively on single chains is that varying the type of perturbation, e.g., between pharmaceutical agent mediated inhibition of phosphorylation or suppression of protein expression, can result in opposing responses in the other branches. This can have potential significance in designing drugs targeting key molecules which regulate multiple pathways implicated in systems-level diseases such as cancer and diabetes.

Logic-based models in systems biology: a predictive and parameter-free network analysis method

Highly complex molecular networks, which play fundamental roles in almost all cellular processes, are known to be dysregulated in a number of diseases, most notably in cancer. As a consequence, there is a critical need to develop practical methodologies for constructing and analysing molecular networks at a systems level. Mathematical models built with continuous differential equations are an ideal methodology because they can provide a detailed picture of a network's dynamics. To be predictive, however, differential equation models require that numerous parameters be known a priori and this information is almost never available. An alternative dynamical approach is the use of discrete logic-based models that can provide a good approximation of the qualitative behaviour of a biochemical system without the burden of a large parameter space. Despite their advantages, there remains significant resistance to the use of logic-based models in biology. Here, we address some common concerns and provide a brief tutorial on the use of logic-based models, which we motivate with biological examples.

Minimal models for cell-cycle control based on competitive inhibition and multisite phosphorylations of Cdk substrates

The eukaryotic cell cycle is characterized by alternating oscillations in the activities of cyclin-dependent kinase (Cdk) and the anaphase-promoting complex (APC). Successful completion of the cell cycle is dependent on the precise, temporally ordered appearance of these activities. A modest level of Cdk activity is sufficient to initiate DNA replication, but mitosis and APC activation require an elevated Cdk activity. In present-day eukaryotes, this temporal order is provided by a complex network of regulatory proteins that control both Cdk and APC activities via sharp thresholds, bistability, and time delays. Using simple computational models, we show here that these dynamical features of cell-cycle organization could emerge in a control system driven by a single Cdk/cyclin complex and APC wired in a negative-feedback loop. We show that ordered phosphorylation of cellular proteins could be explained by multisite phosphorylation/dephosphorylation and competition of substrates for interconverting kinase (Cdk) and phosphatase. In addition, the competition of APC substrates for ubiquitylation can create and maintain sustained oscillations in cyclin levels. We propose a sequence of models that gets closer and closer to a realistic model of cell-cycle control in yeast. Since these models lack the elaborate control mechanisms characteristic of modern eukaryotes, they suggest that bistability and time delay may have characterized eukaryotic cell divisions before the current cell-cycle control network evolved in all its complexity.

Intrinsic feedbacks in MAPK signaling cascades lead to bistability and oscillations

Previous studies have demonstrated that double phosphorylation of a protein can lead to bistability if some conditions are fulfilled. It was also shown that the signaling behavior of a covalent modification cycle can be quantitatively and, more importantly, qualitatively modified when this cycle is coupled to a signaling pathway as opposed to being isolated. This property was named retroactivity. These two results are studied together in this paper showing the existence of interesting phenomena--oscillations and bistability--in signaling cascades possessing at least one stage with a double-phosphorylation cycle as in MAPK cascades.

Dynamics and stability of a three-dimensional model of cell signal transduction

In this paper, we consider a three-dimensional model of cell signal transduction. In this model, the deactivation of signalling proteins occur throughout the cytosol and activation is localized to specific sites in the cell. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of ordinary differential equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo sustained oscillations, hysteresis and other complex behaviors. The simulations of the full three-dimensional system agree with simulations of the reduced ordinary differential equations.

Exploration of trade-offs between steady-state and dynamic properties in signaling cycles

In the intracellular signaling networks that regulate important cell processes, the base pattern comprises the cycle of reversible phosphorylation of a protein, catalyzed by kinases and opposing phosphatases. Mathematical modeling and analysis have been used for gaining a better understanding of their functions and to capture the rules governing system behavior. Since biochemical parameters in signaling pathways are not easily accessible experimentally, it is necessary to explore possibilities for both steady-state and dynamic responses in these systems. While a number of studies have focused on analyzing these properties separately, it is necessary to take into account both of these responses simultaneously in order to be able to interpret a broader range of phenotypes. This paper investigates the trade-offs between optimal characteristics of both steady-state and dynamic responses. Following an inverse sensitivity analysis approach, we use systematic optimization methods to find the biochemical and biophysical parameters that simultaneously achieve optimal steady-state and dynamic performance. Remarkably, we find that even a single covalent modification cycle can simultaneously and robustly achieve high ultrasensitivity, high amplification and rapid signal transduction. We also find that the response rise and decay times can be modulated independently by varying the activating- and deactivating-enzyme-to-interconvertible-protein ratios.

Retroactive signaling in short signaling pathways

In biochemical signaling pathways without explicit feedback connections, the core signal transduction is usually described as a one-way communication, going from upstream to downstream in a feedforward chain or network of covalent modification cycles. In this paper we explore the possibility of a new type of signaling called retroactive signaling, offered by the recently demonstrated property of retroactivity in signaling cascades. The possibility of retroactive signaling is analysed in the simplest case of the stationary states of a bicyclic cascade of signaling cycles. In this case, we work out the conditions for which variables of the upstream cycle are affected by a change of the total amount of protein in the downstream cycle, or by a variation of the phosphatase deactivating the same protein. Particularly, we predict the characteristic ranges of the downstream protein, or of the downstream phosphatase, for which a retroactive effect can be observed on the upstream cycle variables. Next, we extend the possibility of retroactive signaling in short but nonlinear signaling pathways involving a few covalent modification cycles.

Different designs of kinase-phosphatase interactions and phosphatase sequestration shapes the robustness and signal flow in the MAPK cascade

Background

The three layer mitogen activated protein kinase (MAPK) signaling cascade exhibits different designs of interactions between its kinases and phosphatases. While the sequential interactions between the three kinases of the cascade are tightly preserved, the phosphatases of the cascade, such as MKP3 and PP2A, exhibit relatively diverse interactions with their substrate kinases. Additionally, the kinases of the MAPK cascade can also sequester their phosphatases. Thus, each topologically distinct interaction design of kinases and phosphatases could exhibit unique signal processing characteristics, and the presence of phosphatase sequestration may lead to further fine tuning of the propagated signal.

Results

We have built four architecturally distinct types of models of the MAPK cascade, each model with identical kinase-kinase interactions but unique kinases-phosphatases interactions. Our simulations unravelled that MAPK cascade’s robustness to external perturbations is a function of nature of interaction between its kinases and phosphatases. The cascade’s output robustness was enhanced when phosphatases were sequestrated by their target kinases. We uncovered a novel implicit/hidden negative feedback loop from the phosphatase MKP3 to its upstream kinase Raf-1, in a cascade resembling the B cell MAPK cascade. Notably, strength of the feedback loop was reciprocal to the strength of phosphatases’ sequestration and stronger sequestration abolished the feedback loop completely. An experimental method to verify the presence of the feedback loop is also proposed. We further showed, when the models were activated by transient signal, memory (total time taken by the cascade output to reach its unstimulated level after removal of signal) of a cascade was determined by the specific designs of interaction among its kinases and phosphatases.

Conclusions

Differences in interaction designs among the kinases and phosphatases can differentially shape the robustness and signal response behaviour of the MAPK cascade and phosphatase sequestration dramatically enhances the robustness to perturbations in each of the cascade. An implicit negative feedback loop was uncovered from our analysis and we found that strength of the negative feedback loop is reciprocally related to the strength of phosphatase sequestration. Duration of output phosphorylation in response to a transient signal was also found to be determined by the individual cascade’s kinase-phosphatase interaction design.

Mathematical modeling reveals the functional implications of the different nuclear shuttling rates of Erk1 and Erk2

The mitogen-activated protein kinase (MAPK) family of proteins is involved in regulating cellular fates such as proliferation, differentiation and apoptosis. In particular, the dynamics of the Erk/Mek system, which has become the canonical example for MAPK signaling systems, have attracted considerable attention. Erk is encoded by two genes, Erk1 and Erk2, that until recently had been considered equivalent as they differ only subtly at the sequence level. However, these proteins exhibit radically different trafficking between cytoplasm and nucleus and this fact may have functional implications. Here we use spatially resolved data on Erk1/2 to develop and analyze spatio-temporal models of these cascades, and we discuss how sensitivity analysis can be used to discriminate between mechanisms. Our models elucidate some of the factors governing the interplay between signaling processes and the Erk1/2 localization in different cellular compartments, including competition between Erk1 and Erk2. Our approach is applicable to a wide range of signaling systems, such as activation cascades, where translocation of molecules occurs. Our study provides a first model of Erk1 and Erk2 activation and their nuclear shuttling dynamics, revealing a role in the regulation of the efficiency of nuclear signaling.

Variable elimination in post-translational modification reaction networks with mass-action kinetics

We define a subclass of chemical reaction networks called post-translational modification systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop a mathematical framework based on the notion of a cut (a particular subset of species in the system), which provides a linear elimination procedure to reduce the number of variables in the system to a set of core variables. The steady states are parameterized algebraically by the core variables, and graphical conditions for when steady states with positive core variables imply positivity of all variables are given. Further, minimal cuts are the connected components of the species graph and provide conservation laws. A criterion for when a (maximal) set of independent conservation laws can be derived from cuts is given.

Unraveling the complex regulatory relationships between metabolism and signal transduction in cancer

Cancer cells exhibit an altered metabolic phenotype, known as the Warburg effect, which is characterized by high rates of glucose uptake and glycolysis, even under aerobic conditions. The Warburg effect appears to be an intrinsic component of most cancers and there is evidence linking cancer progression to mutations, translocations, and alternative splicing of genes that directly code for or have downstream effects on key metabolic enzymes. Many of the same signaling pathways are routinely dysregulated in cancer and a number of important oncogenic signaling pathways play important regulatory roles in central carbon metabolism. Unraveling the complex regulatory relationship between cancer metabolism and signaling requires the application of systems biology approaches. Here we discuss computational approaches for modeling protein signal transduction and metabolism as well as how the regulatory relationship between these two important cellular processes can be combined into hybrid models.

CROSSTALK FACILITATES SPATIAL SIGNAL PROPAGATION THROUGH MAPK CASCADES

In intracellular mitogen-activated protein kinase (MAPK) cascades, it has been shown that signals can be propagated across the cell cytosol in the form of phosphoprotein waves arising from the bistable response of MAPK to active MAPK kinase. Without such a bistable response, however, they can not propagate into distant cell compartments, although a long positive feedback endows a mechanistically-distinct bistable response of MAPK to extracellular signal. Here we provide a compensate means that uses crosstalk between parallel identical pathways of MAPK cascades. For a spherical cell, we find that both unidirectional and bidirectional crosstalk with enhancement of phosphorylation can facilitate phosphoprotein signal propagation from the plasma membrane to the periphery of cell nucleus. Moreover, different shallow spatial gradients of biphosphorylated MAPK occur in the cytosol under different strengths of pathway interactions. These results suggest that crosstalk would be utilized by living organisms for spatial information transfer and cellular decision-making processing.

A generic model for open signaling cascades with forward activation

In this work, cellular signal transduction in an open cascade with forward activation was studied. By proposing a generic model which captures the common features of major existing models in the literature, it is showed how signaling profile changes during the propagation along the cascade. In particular, a typical OFF-ON-OFF switch-like transient behavior with prolonged temporary ON state is revealed, where OFF and ON represent the states of low level and high level concentrations, respectively. Analytically this phenomenon is closely related to uniform convergence of the active protein concentration of downstream cycles in the finite time range and its failure in the entire time domain. Consequently a classification of open signaling cascade which can sustain OFF-ON-OFF behavior in the far downstream cycles is accessible. Relevant biological issues, such as delayed activation of downstream reaction cycles, signal amplification and prolonged signal duration, to the generic model is discussed.

Load-induced modulation of signal transduction networks

Biological signal transduction networks are commonly viewed as circuits that pass along information--in the process amplifying signals, enhancing sensitivity, or performing other signal-processing tasks--to transcriptional and other components. Here, we report on a "reverse-causality" phenomenon, which we call load-induced modulation. Through a combination of analytical and experimental tools, we discovered that signaling was modulated, in a surprising way, by downstream targets that receive the signal and, in doing so, apply what in physics is called a load. Specifically, we found that non-intuitive changes in response dynamics occurred for a covalent modification cycle when load was present. Loading altered the response time of a system, depending on whether the activity of one of the enzymes was maximal and the other was operating at its minimal rate or whether both enzymes were operating at submaximal rates. These two conditions, which we call "limit regime" and "intermediate regime," were associated with increased or decreased response times, respectively. The bandwidth, the range of frequency in which the system can process information, decreased in the presence of load, suggesting that downstream targets participate in establishing a balance between noise-filtering capabilities and a circuit's ability to process high-frequency stimulation. Nodes in a signaling network are not independent relay devices, but rather are modulated by their downstream targets.

Kinase inhibitors can produce off-target effects and activate linked pathways by retroactivity

BACKGROUND

It has been shown in experimental and theoretical work that covalently modified signaling cascades naturally exhibit bidirectional signal propagation via a phenomenon known as retroactivity. An important consequence of retroactivity, which arises due to enzyme sequestration in covalently modified signaling cascades, is that a downstream perturbation can produce a response in a component upstream of the perturbation without the need for explicit feedback connections. Retroactivity may, therefore, play an important role in the cellular response to a targeted therapy. Kinase inhibitors are a class of targeted therapies designed to interfere with a specific kinase molecule in a dysregulated signaling pathway. While extremely promising as anti-cancer agents, kinase inhibitors may produce undesirable off-target effects by non-specific interactions or pathway cross-talk. We hypothesize that targeted therapies such as kinase inhibitors can produce off-target effects as a consequence of retroactivity alone.

RESULTS

We used a computational model and a series of simple signaling motifs to test the hypothesis. Our results indicate that within physiologically and therapeutically relevant ranges for all parameters, a targeted inhibitor can naturally induce an off-target effect via retroactivity. The kinetics governing covalent modification cycles in a signaling network were more important for propagating an upstream off-target effect in our models than the kinetics governing the targeted therapy itself. Our results also reveal the surprising and crucial result that kinase inhibitors have the capacity to turn "on" an otherwise "off" parallel cascade when two cascades share an upstream activator.

CONCLUSIONS

A proper and detailed characterization of a pathway's structure is important for identifying the optimal protein to target as well as what concentration of the targeted therapy is required to modulate the pathway in a safe and effective manner. We believe our results support the position that such characterizations should consider retroactivity as a robust potential source of off-target effects induced by kinase inhibitors and other targeted therapies.

Long signaling cascades tend to attenuate retroactivity

Signaling pathways consisting of phosphorylation/dephosphorylation cycles with no explicit feedback allow signals to propagate not only from upstream to downstream but also from downstream to upstream due to retroactivity at the interconnection between phosphorylation/dephosphorylation cycles. However, the extent to which a downstream perturbation can propagate upstream in a signaling cascade and the parameters that affect this propagation are presently unknown. Here, we determine the downstream-to-upstream steady-state gain at each stage of the signaling cascade as a function of the cascade parameters. This gain can be made smaller than 1 (attenuation) by sufficiently fast kinase rates compared to the phosphatase rates and/or by sufficiently large Michaelis-Menten constants and sufficiently low amounts of total stage protein. Numerical studies performed on sets of biologically relevant parameters indicated that ∼50% of these parameters could give rise to amplification of the downstream perturbation at some stage in a three-stage cascade. In an n-stage cascade, the percentage of parameters that lead to an overall attenuation from the last stage to the first stage monotonically increases with the cascade length n and reaches 100% for cascades of length at least 6.