Noise propagation and signaling sensitivity in biological networks: a role for positive feedback

Interactions between genes and proteins are crucial for efficient processing of internal or external signals, but this connectivity also amplifies stochastic fluctuations by propagating noise between components. Linear (unbranched) cascades were shown to exhibit an interplay between the sensitivity to changes in input signals and the ability to buffer noise. We searched for biological circuits that can maintain signaling sensitivity while minimizing noise propagation, focusing on cases where the noise is characterized by rapid fluctuations. Negative feedback can buffer this type of noise, but this buffering comes at the expense of an even greater reduction in signaling sensitivity. By systematically analyzing three-component circuits, we identify positive feedback as a central motif allowing for the buffering of propagated noise while maintaining sensitivity to long-term changes in input signals. We show analytically that noise reduction in the presence of positive feedback results from improved averaging of rapid fluctuations over time, and discuss in detail a particular implementation in the control of nutrient homeostasis in yeast. As the design of biological networks optimizes for multiple constraints, positive feedback can be used to improve sensitivity without a compromise in the ability to buffer propagated noise.     

Noise propagation in two-step series MAPK cascade

Series MAPK enzymatic cascades, ubiquitously found in signaling networks, act as signal amplifiers and play a key role in processing information during signal transduction in cells. In activated cascades, cell-to-cell variability or noise is bound to occur and thereby strongly affects the cellular response. Commonly used linearization method (LM) applied to Langevin type stochastic model of the MAPK cascade fails to accurately predict intrinsic noise propagation in the cascade. We prove this by using extensive stochastic simulations for various ranges of biochemical parameters. This failure is due to the fact that the LM ignores the nonlinear effects on the noise. However, LM provides a good estimate of the extrinsic noise propagation. We show that the correct estimate of intrinsic noise propagation in signaling networks that contain at least one enzymatic step can be obtained only through stochastic simulations. Noise propagation in the cascade depends on the underlying biochemical parameters which are often unavailable. Based on a combination of global sensitivity analysis (GSA) and stochastic simulations, we developed a systematic methodology to characterize noise propagation in the cascade. GSA predicts that noise propagation in MAPK cascade is sensitive to the total number of upstream enzyme molecules and the total number of molecules of the two substrates involved in the cascade. We argue that the general systematic approach proposed and demonstrated on MAPK cascade must accompany noise propagation studies in biological networks.     

Engineering aspects of enzymatic signal transduction: photoreceptors in the retina

Identifying the basic module of enzymatic amplification as an irreversible cycle of messenger activation/deactivation by a "push-pull" pair of opposing enzymes, we analyze it in terms of gain, bandwidth, noise, and power consumption. The enzymatic signal transduction cascade is viewed as an information channel, the design of which is governed by the statistical properties of the input and the noise and dynamic range constraints of the output. With the example of vertebrate phototransduction cascade we demonstrate that all of the relevant engineering parameters are controlled by enzyme concentrations and, from functional considerations, derive bounds on the required protein numbers. Conversely, the ability of enzymatic networks to change their response characteristics by varying only the abundance of different enzymes illustrates how functional diversity may be built from nearly conserved molecular components.     

Stochastic radial basis functions

Stochastic signal processing can implement gaussian activation functions for radial basis function networks, using stochastic counters. The statistics of neural inputs which control the increment and decrement operations of the counter are governed by Bernoulli distributions. The transfer functions relating the input and output pulse probabilities can closely approximate gaussian activation functions which improve with the number of states in the counter. The means and variances of these gaussian approximations can be controlled by varying the output combinational logic function of the binary counter variables.     

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

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

Noise Management by Molecular Networks

Fluctuations in the copy number of key regulatory macromolecules (“noise”) may cause physiological heterogeneity in populations of (isogenic) cells. The kinetics of processes and their wiring in molecular networks can modulate this molecular noise. Here we present a theoretical framework to study the principles of noise management by the molecular networks in living cells. The theory makes use of the natural, hierarchical organization of those networks and makes their noise management more understandable in terms of network structure. Principles governing noise management by ultrasensitive systems, signaling cascades, gene networks and feedback circuitry are discovered using this approach. For a few frequently occurring network motifs we show how they manage noise. We derive simple and intuitive equations for noise in molecule copy numbers as a determinant of physiological heterogeneity. We show how noise levels and signal sensitivity can be set independently in molecular networks, but often changes in signal sensitivity affect noise propagation. Using theory and simulations, we show that negative feedback can both enhance and reduce noise. We identify a trade-off; noise reduction in one molecular intermediate by negative feedback is at the expense of increased noise in the levels of other molecules along the feedback loop. The reactants of the processes that are strongly (cooperatively) regulated, so as to allow for negative feedback with a high strength, will display enhanced noise.     

On sensitivity amplification in intracellular signaling cascades

Sensitivity amplification has long been regarded as a virtually universal property of signal transduction cascades, yet a comprehensive parameter analysis remains a challenge even for relatively simple networks. We use a fast and accurate method to compute properties of multilevel cascades of activation-inactivation cycles and show that the monocyclic cascades amplify sensitivity only under specific conditions. In particular, it is found that efficient sensitivity amplification in a cascade, relative to the sensitivities of individual cycles, requires asymmetry in saturation of converter enzymes, with inhibitors much more saturated than activators.     

The potential for signal integration and processing in interacting MAP kinase cascades

The cellular response to environmental stimuli requires biochemical information processing through which sensory inputs and cellular status are integrated and translated into appropriate responses by way of interacting networks of enzymes. One such network, the mitogen-activated protein (MAP) kinase cascade is a highly conserved signal transduction module that propagates signals from cell surface receptors to various cytosolic and nuclear targets by way of a phosphorylation cascade. We have investigated the potential for signal processing within a network of interacting feed-forward kinase cascades typified by the MAP kinase cascade. A genetic algorithm was used to search for sets of kinetic parameters demonstrating representative key input-output patterns of interest. We discuss two of the networks identified in our study, one implementing the exclusive-or function (XOR) and another implementing what we refer to as an in-band detector (IBD) or two-sided threshold. These examples confirm the potential for logic and amplitude-dependent signal processing in interacting MAP kinase cascades demonstrating limited cross-talk. Specifically, the XOR function allows the network to respond to either one, but not both signals simultaneously, while the IBD permits the network to respond exclusively to signals within a given range of strength, and to suppress signals below as well as above this range. The solution to the XOR problem is interesting in that it requires only two interacting pathways, crosstalk at only one layer, and no feedback or explicit inhibition. These types of responses are not only biologically relevant but constitute signal processing modules that can be combined to create other logical functions and that, in contrast to amplification, cannot be achieved with a single cascade or with two non-interacting cascades. Our computational results revealed surprising similarities between experimental data describing the JNK/MKK4/MKK7 pathway and the solution for the IBD that evolved from the genetic algorithm. The evolved IBD not only exhibited the required non-monotonic signal strength-response, but also demonstrated transient and sustained responses that properly reflected the input signal strength, dependence on both of the MAPKKs for signaling, phosphorylation site preferences by each of the MAPKKs, and both activation and inhibition resulting from the overexpression of one of the MAPKKs.     

From spiking neuron models to linear-nonlinear models

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.     

G-protein-coupled enzyme cascades have intrinsic properties that improve signal localization and fidelity

G-protein-coupled enzyme cascades are used by eukaryotic cells to detect external signals and transduce them into intracellular messages that contain biological information relevant to the cell's function. Since G-protein-coupled receptors that are designed to detect different kinds of external signals can generate the same kind of intracellular response, effective signaling requires that there are mechanisms to increase signal specificity and fidelity. Here we examine the kinetic equations for the initial three stages in a generic G-protein-coupled cascade and show that the physical properties of the transduction pathway result in two intrinsic features that benefit signaling. 1), The response to a single activated receptor is naturally confined to a localized spatial domain, which could improve signal specificity by reducing cross talk. 2), The peak of the response generated by such a signaling domain is limited. This saturation effect reduces trial-to-trial variability and increases signaling fidelity by limiting the response to receptors that remain active for longer than average. We suggest that this mechanism for reducing response fluctuations may be a contributing factor in making the single photon responses of vertebrate retinal rods so remarkably reproducible.     

Effect of the MAPK cascade structure, nuclear translocation and regulation of transcription factors on gene expression

The mitogen activated protein kinase (MAP kinase) cascade system represents a highly conserved prototype of signal transduction by enzyme cascades. One of the best-studied properties of the MAPK system is its ability to convert graded input stimulus to switch-like all-or-none responses. Previous theoretical studies have centered on quantifying dual phosphorylated MAPK as a final output response and have not incorporated its influence on the regulation of gene expression. The main objective of the current work is to understand the regulatory effect of positive feedback loop embedded in the MAPK cascade, nuclear translocation of active MAPK, phosphorylation and activation of nuclear target proteins on the regulation of specific gene expression. To achieve this objective, we have simulated the MAPK cascade system, which resembles Hog1p activation pathway in yeast, at steady state. Thus, the input signal to the MAPK system is correlated with gene expression as a final system-level output response. The steady state simulation results suggest that other than regulating the signal propagation through cascades, the nuclear translocation of activated MAPK and subsequent regulation of gene expression represent one of the key modes to control the threshold level of response. This work proposes that, it is essential to consider the compartmental distributions of signaling species and the corresponding regulatory mechanisms of gene expression to study the system-level performance of signaling modules such as the MAPK cascade. Such an analysis will relate the extracellular cues to the final phenotypic response by capturing the mechanistic details of the signaling pathway.     

Integration of cellular signals in chattering environments

Cells are constantly exposed to fluctuating environmental conditions. External signals are sensed, processed and integrated by cellular signal transduction networks, which translate input signals into specific cellular responses by means of biochemical reactions. These networks have a complex nature, and we are still far from having a complete characterization of the process through which they integrate information, specially given the noisy environment in which that information is embedded. Guided by the many instances of constructive influences of noise that have been reported in the physical sciences in the last decades, here we explore how multiple signals are integrated in an eukaryotic cell in the presence of background noise, or chatter. To that end, we use a Boolean model of a typical human signal transduction network. Despite its complexity, we find that the network is able to display simple patterns of signal integration. Furthermore, our computational analysis shows that these integration patterns depend on the levels of fluctuating background activity carried by other cell inputs. Taken together, our results indicate that signal integration is sensitive to environmental fluctuations, and that this background noise effectively determines the information integration capabilities of the cell.     

Allosteric interactions and bifunctionality make the response of glutamine synthetase cascade system of Escherichia coli robust and ultrasensitive

Glutamine synthetase (GS) regulation in Escherichia coli by reversible covalent modification cycles is a prototype of signal transduction by enzyme cascades. Such enzyme cascades are known to exhibit ultrasensitive response to primary stimuli and act as signal integration systems. Here, we have quantified GS bicyclic cascade based on steady state analysis by evaluating Hill coefficient. We demonstrate that adenylylation of GS with glutamine as input is insensitive to total enzyme concentrations of GS, uridylyltransferase/uridylyl-removing enzyme, regulatory protein PII, and adenylyltransferase/adenylyl-removing enzyme. This robust response of GS adenylylation is also observed for change in system parameters. From numerical analyses, we show that the robust ultrasensitive response of bicyclic cascade is because of allosteric interactions of glutamine and 2-ketoglutarate, bifunctionality of converter enzymes, and closed loop bicyclic cascade structure. By system level quantification of the GS bicyclic cascade, we conclude that such a robust response may help the cell in adapting to different carbon and nitrogen status.     

Biology using engineering tools: The negative feedback amplifier

Negative feedback is a ubiquitous feature of biological networks. Recent work from Sturm and colleagues¹ presents 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.Key words: negative feedback, signal transduction, quantitative modeling, mitogen activated protein kinase, spatiotemporal, dynamicsHow does a cell in an organism differentiate signals from noise while being immersed in bath of growth factors and hormones? This issue is one of the major challenges in understanding how the high-fidelity and specificity of biological responses are generated. Engineers have considered similar problems for a long time. Can we borrow concepts from engineering to understand biology?In the early 20^th century, the reliable transmission of telephone signals became a growing problem as lines became longer. To transmit signals over longer distances, amplification was needed, but noise and distortion were added every time the signal was amplified. A solution to this problem was the negative feedback amplifier (NFA), an invention made in 1927 by Harold Stephen Black, an employee of Bell Laboratories, who proposed that the output of the amplifier be used to attenuate its input, creating a negative feedback loop.² This proposal seemed so counterproductive that it took nine years for a patent to be issued. Why would one want to decrease the magnitude of a signal that needs to be amplified? There are actually two good reasons for doing this. First, consider the effects of signal amplification on the dynamic range of the transduction system. With higher amplification (gain), the output will be saturated at lower input magnitudes, reducing the dynamic range. However, with negative feedback in place, the amplification is diminished, thus allowing the output to respond to a greater range of input magnitudes, in a more linear fashion. Second, consider that during signal transmission, there is a perturbation to the amplifier, such that its output magnitude is now suddenly too high. With a negative feedback loop in place, this increased magnitude is passed back to the amplifier input as increased negative feedback strength, consequently attenuating the input signal and mitigating the effects of the perturbation. Thus, although the amplifier gain is reduced by the negative feedback, the NFA affords increased resistance to perturbations within the amplifier and a greater range of responsiveness to different input strengths. These properties seem very useful for biological signaling networks that constantly have to deal with intrinsic and extrinsic noise while simultaneously responding to a sea of variedconcentration growth factors.Recent work demonstrates that the mammalian Extracellular-signal Regulated Kinase 1/2 (ERK1/2) cascade has properties similar to the NFA found in electronic systems, namely, it amplifies input signals, makes input-output response curves more linear and confers robustness to perturbations within the feedback loop.¹ But how exactly does an enzymatic kinase cascade function as a NFA? And what are the implications for biology and medicine?     

Role of Relaxation Time Scale in Noisy Signal Transduction

Intra-cellular fluctuations, mainly triggered by gene expression, are an inevitable phenomenon observed in living cells. It influences generation of phenotypic diversity in genetically identical cells. Such variation of cellular components is beneficial in some contexts but detrimental in others. To quantify the fluctuations in a gene product, we undertake an analytical scheme for studying few naturally abundant linear as well as branched chain network motifs. We solve the Langevin equations associated with each motif under the purview of linear noise approximation and derive the expressions for Fano factor and mutual information in close analytical form. Both quantifiable expressions exclusively depend on the relaxation time (decay rate constant) and steady state population of the network components. We investigate the effect of relaxation time constraints on Fano factor and mutual information to indentify a time scale domain where a network can recognize the fluctuations associated with the input signal more reliably. We also show how input population affects both quantities. We extend our calculation to long chain linear motif and show that with increasing chain length, the Fano factor value increases but the mutual information processing capability decreases. In this type of motif, the intermediate components act as a noise filter that tune up input fluctuations and maintain optimum fluctuations in the output. For branched chain motifs, both quantities vary within a large scale due to their network architecture and facilitate survival of living system in diverse environmental conditions.     

How robust are switches in intracellular signaling cascades?

Since all-or-none decisions of the cell are controlled by extracellular signals, cells have biochemical switches within their intracellular signaling networks. Central elements of these switches are multisite phosphorylation, enzymic saturation, and amplification by cascades. Moreover, positive feedback can contribute to switch-like behavior termed also ultrasensitivity. Here we analyse the robustness of these mechanisms exemplified by models of the three-molecule MAPK-cascade and the single-molecule Goldbeter-Koshland switch. We show that the ultrasensitivity in the MAPK-cascades is more robust against changes of the kinetic parameters than the Goldbeter-Koshland switch. If multiple parameters are changed randomly, the effects of parameter changes can compensate each other in the cascade leading to a remarkable robustness of the switch-like behavior. The different degrees of robustness can be traced back to the different mechanisms of generating ultrasensitivity. While in the Goldbeter-Koshland switch the saturation of the enzymes are crucial, in the MAPK-cascade the adjustment of working ranges determines the ultrasensitivity. Our results indicate that amplification of ultrasensitivity in cascades and multisite phosphorylation might be a design principle to achieve robust switches.     

Tunable signal processing in synthetic MAP kinase cascades

The flexibility of MAPK cascade responses enables regulation of a vast array of cell fate decisions, but elucidating the mechanisms underlying this plasticity is difficult in endogenous signaling networks. We constructed insulated mammalian MAPK cascades in yeast to explore how intrinsic and extrinsic perturbations affect the flexibility of these synthetic signaling modules. Contrary to biphasic dependence on scaffold concentration, we observe monotonic decreases in signal strength as scaffold concentration increases. We find that augmenting the concentration of sequential kinases can enhance ultrasensitivity and lower the activation threshold. Further, integrating negative regulation and concentration variation can decouple ultrasensitivity and threshold from the strength of the response. Computational analyses show that cascading can generate ultrasensitivity and that natural cascades with different kinase concentrations are innately biased toward their distinct activation profiles. This work demonstrates that tunable signal processing is inherent to minimal MAPK modules and elucidates principles for rational design of synthetic signaling systems.