Noise management by molecular networks

Extrinsic noise of the target gene governs abundance pattern of feed-forward loop motifs

Feed-forward loop (FFL) is found to be a recurrent structure in bacterial and yeast gene transcription regulatory networks. In a generic FFL, transcription factor (TF) S regulates production of another TF X while both of these TFs regulate production of final gene-product Y. Depending upon the regulatory programs (activation or repression), FFLs are grouped into two broad classes: coherent (C) and incoherent (I), each class containing four distinct types (C1-C4 and I1-I4). These FFL types are experimentally observed to occur with varied frequencies, C1 and I1 being the abundant ones. Here we present a stochastic framework singling out the absolute value of the normalized covariance of X and Y to be the determining factor behind the abundance of FFLs while considering differential promoter activities of X and Y. Our theoretical construct employs two possible signal integration mechanisms (additive and multiplicative) to synthesize Y while steady-state population level of S remains fixed or becomes tunable reflecting two possible environmental signaling scenarios. Our model categorically points out that abundant FFLs exhibit higher amount of the designated metric which has a biophysical connotation of extrinsic noise for the target gene Y. Our predictions emanating from an overarching analytical expression utilizing biologically plausible parametric conditions are substantiated by stochastic simulation.

Qualitative and quantitative nature of mutual interactions dictate chemical noise in a democratic reaction network

The functions of a living cell rely on a complex network of biochemical reactions that allow it to respond against various internal and external cues. The outcomes of these chemical reactions are often stochastic due to intrinsic and extrinsic noise leading to population heterogeneity. The majority of calculations of stochasticity in reaction networks have focused on small regulatory networks addressing the role of timescales, feedback regulations, and network topology in propagation of noise. Here we computationally investigated chemical noise in a network with democratic architecture where each node is regulated by all other nodes in the network. We studied the effects of the qualitative and quantitative nature of mutual interactions on the propagation of both intrinsic and extrinsic noise in the network. We show that an increased number of inhibitory signals lead to ultrasensitive switching of average and that leads to sharp transition of intrinsic noise. The intrinsic noise exhibits a biphasic power-law scaling with the average, and the scaling coefficients strongly correlate with the strength of inhibitory signal. The noise strength critically depends on the strength of the interactions, where negative interactions attenuate both intrinsic and extrinsic noise.

Investigation of chemical noise in multisite phosphorylation chain using linear noise approximation

Quantitative and qualitative nature of chemical noise propagation in biochemical reaction networks depend crucially on the topology of the networks. Multisite reversible phosphorylation-dephosphorylation of target proteins is one such recurrently found topology that regulates host of key functions in living cells. Here we analytically calculated the stochasticity in multistep reversible chemical reactions by determining variance of phosphorylated species at the steady state using linear noise approximation to investigate the effect of mass action and Michaelis-Menten kinetics on the noise of phosphorylated species. We probed the dependence of noise on the number of phosphorylation sites and the equilibrium constants of the reaction equilibria to investigate the chemical noise propagation in the multisite phosphorylation chain.

Identification of a bet-hedging network motif generating noise in hormone concentrations and germination propensity in Arabidopsis

Plants have evolved to exploit stochasticity to hedge bets and ensure robustness to varying environments between generations. In agriculture, environments are more controlled, and this evolved variability decreases potential yields, posing agronomic and food security challenges. Understanding how plant cells generate and harness noise thus presents options for engineering more uniform crop performance. Here, we use stochastic chemical kinetic modelling to analyse a hormone feedback signalling motif in Arabidopsis thaliana seeds that can generate tunable levels of noise in the hormone ABA, governing germination propensity. The key feature of the motif is simultaneous positive feedback regulation of both ABA production and degradation pathways, allowing tunable noise while retaining a constant mean level. We uncover surprisingly rich behaviour underlying the control of levels of, and noise in, ABA abundance. We obtain approximate analytic solutions for steady-state hormone level means and variances under general conditions, showing that antagonistic self-promoting and self-repressing interactions can together be tuned to induce noise while preserving mean hormone levels. We compare different potential architectures for this 'random output generator' with the motif found in Arabidopsis, and report the requirements for tunable control of noise in each case. We identify interventions that may facilitate large decreases in variability in germination propensity, in particular, the turnover of signalling intermediates and the sensitivity of synthesis and degradation machinery, as potentially valuable crop engineering targets.

Role of noise and parametric variation in the dynamics of gene regulatory circuits

Stochasticity in gene expression impacts the dynamics and functions of gene regulatory circuits. Intrinsic noises, including those that are caused by low copy number of molecules and transcriptional bursting, are usually studied by stochastic simulations. However, the role of extrinsic factors, such as cell-to-cell variability and heterogeneity in the microenvironment, is still elusive. To evaluate the effects of both the intrinsic and extrinsic noises, we develop a method, named sRACIPE, by integrating stochastic analysis with random circuit perturbation (RACIPE) method. RACIPE uniquely generates and analyzes an ensemble of models with random kinetic parameters. Previously, we have shown that the gene expression from random models form robust and functionally related clusters. In sRACIPE we further develop two stochastic simulation schemes, aiming to reduce the computational cost without sacrificing the convergence of statistics. One scheme uses constant noise to capture the basins of attraction, and the other one uses simulated annealing to detect the stability of states. By testing the methods on several synthetic gene regulatory circuits and an epithelial-mesenchymal transition network in squamous cell carcinoma, we demonstrate that sRACIPE can interpret the experimental observations from single-cell gene expression data. We observe that parametric variation (the spread of parameters around a median value) increases the spread of the gene expression clusters, whereas high noise merges the states. Our approach quantifies the robustness of a gene circuit in the presence of noise and sheds light on a new mechanism of noise-induced hybrid states. We have implemented sRACIPE as an R package.

Noise propagation in an integrated model of bacterial gene expression and growth

In bacterial cells, gene expression, metabolism, and growth are highly interdependent and tightly coordinated. As a result, stochastic fluctuations in expression levels and instantaneous growth rate show intricate cross-correlations. These correlations are shaped by feedback loops, trade-offs and constraints acting at the cellular level; therefore a quantitative understanding requires an integrated approach. To that end, we here present a mathematical model describing a cell that contains multiple proteins that are each expressed stochastically and jointly limit the growth rate. Conversely, metabolism and growth affect protein synthesis and dilution. Thus, expression noise originating in one gene propagates to metabolism, growth, and the expression of all other genes. Nevertheless, under a small-noise approximation many statistical quantities can be calculated analytically. We identify several routes of noise propagation, illustrate their origins and scaling, and establish important connections between noise propagation and the field of metabolic control analysis. We then present a many-protein model containing >1000 proteins parameterized by previously measured abundance data and demonstrate that the predicted cross-correlations between gene expression and growth rate are in broad agreement with published measurements.

Interplay of synergy and redundancy in diamond motif

The formalism of partial information decomposition provides a number of independent components which altogether constitute the total information provided by the source variable(s) about the target variable(s). These non-overlapping terms are recognized as unique information, synergistic information, and redundant information. The metric of net synergy conceived as the difference between synergistic and redundant information is capable of detecting effective synergy, effective redundancy, and information independence among stochastic variables. The net synergy can be quantified using appropriate combinations of different Shannon mutual information terms. The utilization of the net synergy in network motifs with the nodes representing different biochemical species, involved in information sharing, uncovers rich store for exciting results. In the current study, we use this formalism to obtain a comprehensive understanding of the relative information processing mechanism in a diamond motif and two of its sub-motifs, namely, bifurcation and integration motif embedded within the diamond motif. The emerging patterns of effective synergy and effective redundancy and their contribution toward ensuring high fidelity information transmission are duly compared in the sub-motifs. Investigation on the metric of net synergy in independent bifurcation and integration motifs are also executed. In all of these computations, the crucial roles played by various systemic time scales, activation coefficients, and signal integration mechanisms at the output of the network topologies are especially emphasized. Following this plan of action, we become confident that the origin of effective synergy and effective redundancy can be architecturally justified by decomposing a diamond motif into bifurcation and integration motif. According to our conjecture, the presence of a common source of fluctuations creates effective redundancy. Our calculations reveal that effective redundancy empowers signal fidelity. Moreover, to achieve this, input signaling species avoids strong interaction with downstream intermediates. This strategy is capable of making the diamond motif noise-tolerant. Apart from the topological features, our study also puts forward the active contribution of additive and multiplicative signal integration mechanisms to nurture effective redundancy and effective synergy.

Parallel arrangements of positive feedback loops limit cell-to-cell variability in differentiation

Cellular differentiations are often regulated by bistable switches resulting from specific arrangements of multiple positive feedback loops (PFL) fused to one another. Although bistability generates digital responses at the cellular level, stochasticity in chemical reactions causes population heterogeneity in terms of its differentiated states. We hypothesized that the specific arrangements of PFLs may have evolved to minimize the cellular heterogeneity in differentiation. In order to test this we investigated variability in cellular differentiation controlled either by parallel or serial arrangements of multiple PFLs having similar average properties under extrinsic and intrinsic noises. We find that motifs with PFLs fused in parallel to one another around a central regulator are less susceptible to noise as compared to the motifs with PFLs arranged serially. Our calculations suggest that the increased resistance to noise in parallel motifs originate from the less sensitivity of bifurcation points to the extrinsic noise. Whereas estimation of mean residence times indicate that stable branches of bifurcations are robust to intrinsic noise in parallel motifs as compared to serial motifs. Model conclusions are consistent both in AND- and OR-gate input signal configurations and also with two different modeling strategies. Our investigations provide some insight into recent findings that differentiation of preadipocyte to mature adipocyte is controlled by network of parallel PFLs.

Statistics and simulation of growth of single bacterial cells: illustrations with B. subtilis and E. coli

The inherent stochasticity of molecular reactions prevents us from predicting the exact state of single-cells in a population. However, when a population grows at steady-state, the probability to observe a cell with particular combinations of properties is fixed. Here we validate and exploit existing theory on the statistics of single-cell growth in order to predict the probability of phenotypic characteristics such as cell-cycle times, volumes, accuracy of division and cell-age distributions, using real-time imaging data for Bacillus subtilis and Escherichia coli. Our results show that single-cell growth-statistics can accurately be predicted from a few basic measurements. These equations relate different phenotypic characteristics, and can therefore be used in consistency tests of experimental single-cell growth data and prediction of single-cell statistics. We also exploit these statistical relations in the development of a fast stochastic-simulation algorithm of single-cell growth and protein expression. This algorithm greatly reduces computational burden, by recovering the statistics of growing cell-populations from the simulation of only one of its lineages. Our approach is validated by comparison of simulations and experimental data. This work illustrates a methodology for the prediction, analysis and tests of consistency of single-cell growth and protein expression data from a few basic statistical principles.

Taking chances and making mistakes: non-genetic phenotypic heterogeneity and its consequences for surviving in dynamic environments

Natural selection has shaped the strategies for survival and growth of microorganisms. The success of microorganisms depends not only on slow evolutionary tuning but also on the ability to adapt to unpredictable changes in their environment. In principle, adaptive strategies range from purely deterministic mechanisms to those that exploit the randomness intrinsic to many cellular and molecular processes. Depending on the environment and selective pressures, particular strategies can lie somewhere along this continuum. In recent years, non-genetic cell-to-cell differences have received a lot of attention, not least because of their potential impact on the ability of microbial populations to survive in dynamic environments. Using several examples, we describe the origins of spontaneous and induced mechanisms of phenotypic adaptation. We identify some of the commonalities of these examples and consider the potential role of chance and constraints in microbial phenotypic adaptation.

Evolution of Modular Networks Under Selection for Non-Linearly Denoising

Many biological and artifact networks often represent modular structures in which the network can be decomposed into several subnetworks. Here, we propose a simple model for the modular network evolution based on the nonlinear denoising in node activities. This model suggests that modular networks can evolve under certain conditions — if the stipulated goals for the networks or the input and target output pairs involve modular features, or if the signal transfer in a node is carried out in a nonlinear manner with respect to the saturation at the upper and lower bounds. Our model highlights the positive role played by noise in modular network evolution.

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.

Redundancy in information transmission in a two-step cascade

We present a stochastic framework to study signal transmission in a generic two-step cascade S→X→Y. Starting from a set of Langevin equations obeying Gaussian noise processes we calculate the variance and covariance while considering both linear and nonlinear production terms for different biochemical species of the cascade. These quantities are then used to calculate the net synergy within the purview of partial information decomposition. We show that redundancy in information transmission is essentially an important consequence of Markovian property of the two-step cascade motif. We also show that redundancy increases fidelity of the signaling pathway.

Effect of Intrinsic Noise on the Phenotype of Cell Populations Featuring Solution Multiplicity: An Artificial lac Operon Network Paradigm

Heterogeneity in cell populations originates from two fundamentally different sources: the uneven distribution of intracellular content during cell division, and the stochastic fluctuations of regulatory molecules existing in small amounts. Discrete stochastic models can incorporate both sources of cell heterogeneity with sufficient accuracy in the description of an isogenic cell population; however, they lack efficiency when a systems level analysis is required, due to substantial computational requirements. In this work, we study the effect of cell heterogeneity in the behaviour of isogenic cell populations carrying the genetic network of lac operon, which exhibits solution multiplicity over a wide range of extracellular conditions. For such systems, the strategy of performing solely direct temporal solutions is a prohibitive task, since a large ensemble of initial states needs to be tested in order to drive the system--through long time simulations--to possible co-existing steady state solutions. We implement a multiscale computational framework, the so-called "equation-free" methodology, which enables the performance of numerical tasks, such as the computation of coarse steady state solutions and coarse bifurcation analysis. Dynamically stable and unstable solutions are computed and the effect of intrinsic noise on the range of bistability is efficiently investigated. The results are compared with the homogeneous model, which neglects all sources of heterogeneity, with the deterministic cell population balance model, as well as with a stochastic model neglecting the heterogeneity originating from intrinsic noise effects. We show that when the effect of intrinsic source of heterogeneity is intensified, the bistability range shifts towards higher extracellular inducer concentration values.

G Protein-Coupled Receptor Signaling Networks from a Systems Perspective

The signal-transduction network of a mammalian cell integrates internal and external cues to initiate adaptive responses. Among the cell-surface receptors are the G protein-coupled receptors (GPCRs), which have remarkable signal-integrating capabilities. Binding of extracellular signals stabilizes intracellular-domain conformations that selectively activate intracellular proteins. Hereby, multiple signaling routes are activated simultaneously to degrees that are signal-combination dependent. Systems-biology studies indicate that signaling networks have emergent processing capabilities that go far beyond those of single proteins. Such networks are spatiotemporally organized and capable of gradual, oscillatory, all-or-none, and subpopulation-generating responses. Protein-protein interactions, generating feedback and feedforward circuitry, are generally required for these spatiotemporal phenomena. Understanding of information processing by signaling networks therefore requires network theories in addition to biochemical and biophysical concepts. Here we review some of the key signaling systems behaviors that have been discovered recurrently across signaling networks. We emphasize the role of GPCRs, so far underappreciated receptors in systems-biology research.

Regulation of burstiness by network-driven activation

We prove that complex networks of interactions have the capacity to regulate and buffer unpredictable fluctuations in production events. We show that non-bursty network-driven activation dynamics can effectively regulate the level of burstiness in the production of nodes, which can be enhanced or reduced. Burstiness can be induced even when the endogenous inter-event time distribution of nodes' production is non-bursty. We find that hubs tend to be less susceptible to the networked regulatory effects than low degree nodes. Our results have important implications for the analysis and engineering of bursty activity in a range of systems, from communication networks to transcription and translation of genes into proteins in cells.

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.

Noise propagation in synthetic gene circuits for metabolic control

Dynamic control of enzyme expression can be an effective strategy to engineer robust metabolic pathways. It allows a synthetic pathway to self-regulate in response to changes in bioreactor conditions or the metabolic state of the host. The implementation of this regulatory strategy requires gene circuits that couple metabolic signals with the genetic machinery, which is known to be noisy and one of the main sources of cell-to-cell variability. One of the unexplored design aspects of these circuits is the propagation of biochemical noise between enzyme expression and pathway activity. In this article, we quantify the impact of a synthetic feedback circuit on the noise in a metabolic product in order to propose design criteria to reduce cell-to-cell variability. We consider a stochastic model of a catalytic reaction under negative feedback from the product to enzyme expression. On the basis of stochastic simulations and analysis, we show that, depending on the repression strength and promoter strength, transcriptional repression of enzyme expression can amplify or attenuate the noise in the number of product molecules. We obtain analytic estimates for the metabolic noise as a function of the model parameters and show that noise amplification/attenuation is a structural property of the model. We derive an analytic condition on the parameters that lead to attenuation of metabolic noise, suggesting that a higher promoter sensitivity enlarges the parameter design space. In the theoretical case of a switch-like promoter, our analysis reveals that the ability of the circuit to attenuate noise is subject to a trade-off between the repression strength and promoter strength.

Adaptability of non-genetic diversity in bacterial chemotaxis

Bacterial chemotaxis systems are as diverse as the environments that bacteria inhabit, but how much environmental variation can cells tolerate with a single system? Diversification of a single chemotaxis system could serve as an alternative, or even evolutionary stepping-stone, to switching between multiple systems. We hypothesized that mutations in gene regulation could lead to heritable control of chemotactic diversity. By simulating foraging and colonization of E. coli using a single-cell chemotaxis model, we found that different environments selected for different behaviors. The resulting trade-offs show that populations facing diverse environments would ideally diversify behaviors when time for navigation is limited. We show that advantageous diversity can arise from changes in the distribution of protein levels among individuals, which could occur through mutations in gene regulation. We propose experiments to test our prediction that chemotactic diversity in a clonal population could be a selectable trait that enables adaptation to environmental variability.

DOI:http://dx.doi.org/10.7554/eLife.03526.001

Bacterial colonies are generally made up of genetically identical cells. Despite this, a closer look at the members of a bacterial colony shows that these cells can have very different behaviors. For example, some cells may grow more quickly than others, or be more resistant to antibiotics. The mechanisms driving this diversity are only beginning to be identified and understood.

Escherichia coli bacteria can move towards, or away from, certain chemicals in their surrounding environment to help them navigate toward favorable conditions. This behavior is known as chemotaxis. The signals from all of these chemicals are processed in E. coli by just one set of proteins, which control the different behaviors that are needed for the bacteria to follow them. Different numbers of these proteins are found in different—but genetically identical—bacteria, and the number of proteins is linked to how the bacteria perform these behaviors.

It has been suggested that diversity can be beneficial to the overall bacterial population, as it helps the population survive environmental changes. This suggests that the level of diversity in the population should adapt to the level of diversity in the environment. However, it remains unknown how this adaptation occurs.

Frankel et al. developed and combined several models and simulations to investigate whether differences in chemotaxis protein production help an E. coli colony to survive. The models show that in different environments, it can be beneficial for the population as a whole if different cells have different responses to the chemicals present. For example, if a lot of a useful chemical is present, bacteria are more likely to survive by heading straight to the source. If not much chemical is detected, the bacteria may need to move in a more exploratory manner.

Frankel et al. find that different amounts of chemotaxis proteins produce these different behaviors. To survive in a changing environment, it is therefore best for the E. coli colony to contain cells that have different amounts of these proteins. Frankel et al. propose that the variability of chemotaxis protein levels between genetically identical cells can change through mutations in the genes that control how many of the proteins are produced, and predict that such mutations allow populations to adapt to environmental changes.

The environments simulated in the model were much simpler than would be found in the real world, and Frankel et al. describe experiments that are now being performed to confirm and expand on their results. The model could be used in the future to shed light on the behavior of other cells that are genetically identical but exhibit diverse behaviors, from other bacterial species to more complex cancer cells.

DOI:http://dx.doi.org/10.7554/eLife.03526.002

Nonlinear biochemical signal processing via noise propagation

Single-cell studies often show significant phenotypic variability due to the stochastic nature of intra-cellular biochemical reactions. When the numbers of molecules, e.g., transcription factors and regulatory enzymes, are in low abundance, fluctuations in biochemical activities become significant and such "noise" can propagate through regulatory cascades in terms of biochemical reaction networks. Here we develop an intuitive, yet fully quantitative method for analyzing how noise affects cellular phenotypes based on identifying a system's nonlinearities and noise propagations. We observe that such noise can simultaneously enhance sensitivities in one behavioral region while reducing sensitivities in another. Employing this novel phenomenon we designed three biochemical signal processing modules: (a) A gene regulatory network that acts as a concentration detector with both enhanced amplitude and sensitivity. (b) A non-cooperative positive feedback system, with a graded dose-response in the deterministic case, that serves as a bistable switch due to noise-induced ultra-sensitivity. (c) A noise-induced linear amplifier for gene regulation that requires no feedback. The methods developed in the present work allow one to understand and engineer nonlinear biochemical signal processors based on fluctuation-induced phenotypes.

Modular network evolution under selection for robustness to noise

Real networks often exhibit modularity, which is defined as the degree to which a network can be decomposed into several subnetworks. The question of how a modular network arises is still open to discussion. The leading hypothesis is that high modularity evolves under multiple goals, which are decomposable to subproblems, as well as under the evolutionary constraint that selection prefers sparse links in a network. In the present study, we investigate an alternative evolutionary constraint entailing increased robustness to noise. To examine this, we present noise-interfused network models involving an analytically solvable linear system and biologically inspired nonlinear systems. The models demonstrate that it is possible to evolve a modular network under both modularly changing goal orientations and enhancing robustness to noise, thereby reducing sensitivity to noise. By performing theoretical analyses of linear systems, it is shown that the evolutionary constraint enforces the establishment of well-balanced noise sensitivities of multiple noise sources and leads to a modular network underlying a modular structure in goals. Moreover, computer simulations confirm that the presented mechanisms of modular network evolution are robust to variations of nonlinearity in network functions. Our findings suggest a positive role for the presence of noise in network evolution.

Noise attenuation in the ON and OFF states of biological switches

Biological switches must sense changes in signal concentration and at the same time buffer against signal noise. While many studies have focused on the response of switching systems to noise in the ON state, how systems buffer noise at both ON and OFF states is poorly understood. Through analytical and computational approaches, we find that switching systems require different dynamics at the OFF state than at the ON state in order to have good noise buffering capability. Specifically, we introduce a quantity called the input-associated Signed Activation Time (iSAT) that concisely captures an intrinsic temporal property at either the ON or OFF state. We discover a trade-off between achieving good noise buffering in the ON versus the OFF states: a large iSAT corresponds to noise amplification in the OFF state in contrast to noise buffering in the ON state. To search for biological circuits that can buffer noise in both ON and OFF states, we systematically analyze all three-node circuits and identify mutual activation as a central motif. We also study connections among signal sensitivity, iSAT, and noise amplification. We find that a large iSAT at the ON state maintains signaling sensitivity while minimizing noise propagation. Taken together, the analysis of iSATs helps reveal the noise properties of biological networks and should aid in the design of robust switches that can both repress noise at the OFF state and maintain a reliable ON state.

Robustness of signal transduction pathways

Signal transduction pathways transduce information about the outside of the cell to the nucleus, regulating gene expression and cell fate. To reliably inform the cell about its surroundings, information transfer has to be robust against typical perturbation that a cell experiences. Robustness of several mammalian signaling pathways has been studied recently by quantitative experimentation and using mathematical modeling. Here, we review these studies, and describe the emerging concepts of robustness and the underlying mechanisms.

Synthetic biology: advancing the design of diverse genetic systems

A major objective of synthetic biology is to make the process of designing genetically encoded biological systems more systematic, predictable, robust, scalable, and efficient. Examples of genetic systems in the field vary widely in terms of operating hosts, compositional approaches, and network complexity, ranging from simple genetic switches to search-and-destroy systems. While significant advances in DNA synthesis capabilities support the construction of pathway- and genome-scale programs, several design challenges currently restrict the scale of systems that can be reasonably designed and implemented. Thus, while synthetic biology offers much promise in developing systems to address challenges faced in the fields of manufacturing, environment and sustainability, and health and medicine, the realization of this potential is currently limited by the diversity of available parts and effective design frameworks. As researchers make progress in bridging this design gap, advances in the field hint at ever more diverse applications for biological systems.

Gene network requirements for regulation of metabolic gene expression to a desired state

Gene circuits that control metabolism should restore metabolic functions upon environmental changes. Whether gene networks are capable of steering metabolism to optimal states is an open question. Here we present a method to identify such optimal gene networks. We show that metabolic network optimisation over a range of environments results in an input-output relationship for the gene network that guarantees optimal metabolic states. Optimal control is possible if the gene network can achieve this input-output relationship. We illustrate our approach with the best-studied regulatory network in yeast, the galactose network. We find that over the entire range of external galactose concentrations, the regulatory network is able to optimally steer galactose metabolism. Only a few gene network parameters affect this optimal regulation. The other parameters can be tuned independently for optimisation of other functions, such as fast and low-noise gene expression. This study highlights gene network plasticity, evolvability, and modular functionality.

Understanding complexity in neurodegenerative diseases: in silico reconstruction of emergence

Healthy functioning is an emergent property of the network of interacting biomolecules that comprise an organism. It follows that disease (a network shift that causes malfunction) is also an emergent property, emerging from a perturbation of the network. On the one hand, the biomolecular network of every individual is unique and this is evident when similar disease-producing agents cause different individual pathologies. Consequently, a personalized model and approach for every patient may be required for therapies to become effective across mankind. On the other hand, diverse combinations of internal and external perturbation factors may cause a similar shift in network functioning. We offer this as an explanation for the multi-factorial nature of most diseases: they are "systems biology diseases," or "network diseases." Here we use neurodegenerative diseases, like Parkinson's disease (PD), as an example to show that due to the inherent complexity of these networks, it is difficult to understand multi-factorial diseases with simply our "naked brain." When describing interactions between biomolecules through mathematical equations and integrating those equations into a mathematical model, we try to reconstruct the emergent properties of the system in silico. The reconstruction of emergence from interactions between huge numbers of macromolecules is one of the aims of systems biology. Systems biology approaches enable us to break through the limitation of the human brain to perceive the extraordinarily large number of interactions, but this also means that we delegate the understanding of reality to the computer. We no longer recognize all those essences in the system's design crucial for important physiological behavior (the so-called "design principles" of the system). In this paper we review evidence that by using more abstract approaches and by experimenting in silico, one may still be able to discover and understand the design principles that govern behavioral emergence.

Duel of the fates: the role of transcriptional circuits and noise in CD4+ cells

CD4+ T cells play key roles in orchestrating adaptive immune responses, and are a popular model for mammalian cell differentiation. While immune regulation would seem to require exactly adjusted mRNA and protein expression levels of key factors, there is little evidence that this is strictly the case. Stochastic gene expression and plasticity of cell types contrast the apparent need for precision. Recent work has provided insight into the magnitude of molecular noise, as well as the relationship between noise, transcriptional circuits and epigenetic modifications in a variety of cell types. These processes and their interplay will also govern gene expression patterns in the different CD4+ cell types, and the determination of their cellular fates.

Post-transcriptional regulatory networks play a key role in noise reduction that is conserved from micro-organisms to mammals

RNA-binding proteins (RBPs) are core regulators of mRNA transcript stability and translation in prokaryotes and eukaryotes alike. Genome-wide studies in yeast have shown intriguing relationships between the expression dynamics of RBPs, the structure of post-transcriptional regulatory networks of RBP-mRNA binding interactions and noise reduction in post-transcriptionally regulated expression profiles. In the present study, we assembled and compared the genomic properties of RBPs and integrated transcriptional and post-transcriptional regulatory networks in four species: Escherichia coli, yeast, mouse and human. We found that RBPs are consistently regulated to have minimal levels of protein noise, that known noise-buffering network motifs are enriched in the integrated networks and that post-transcriptional feedback loops act as regulators of other regulators. These results support a general model where RBPs are the key regulators of stochastic noise-buffering in numerous downstream cellular processes. The currently available datasets do not allow clarification of whether post-transcriptional regulation by RBPs and by noncoding RNAs plays a similar or distinct role, although we found evidence that specific combinations of transcription factors, RBPs and micro-RNAs jointly regulate known disease pathways in humans, suggesting complementarity rather than redundancy between both modes of post-transcriptional regulation.

Discreteness-induced concentration inversion in mesoscopic chemical systems

Molecular discreteness is apparent in small-volume chemical systems, such as biological cells, leading to stochastic kinetics. Here we present a theoretical framework to understand the effects of discreteness on the steady state of a monostable chemical reaction network. We consider independent realizations of the same chemical system in compartments of different volumes. Rate equations ignore molecular discreteness and predict the same average steady-state concentrations in all compartments. However, our theory predicts that the average steady state of the system varies with volume: if a species is more abundant than another for large volumes, then the reverse occurs for volumes below a critical value, leading to a concentration inversion effect. The addition of extrinsic noise increases the size of the critical volume. We theoretically predict the critical volumes and verify, by exact stochastic simulations, that rate equations are qualitatively incorrect in sub-critical volumes.

Mitochondrial variability as a source of extrinsic cellular noise

We present a study investigating the role of mitochondrial variability in generating noise in eukaryotic cells. Noise in cellular physiology plays an important role in many fundamental cellular processes, including transcription, translation, stem cell differentiation and response to medication, but the specific random influences that affect these processes have yet to be clearly elucidated. Here we present a mechanism by which variability in mitochondrial volume and functionality, along with cell cycle dynamics, is linked to variability in transcription rate and hence has a profound effect on downstream cellular processes. Our model mechanism is supported by an appreciable volume of recent experimental evidence, and we present the results of several new experiments with which our model is also consistent. We find that noise due to mitochondrial variability can sometimes dominate over other extrinsic noise sources (such as cell cycle asynchronicity) and can significantly affect large-scale observable properties such as cell cycle length and gene expression levels. We also explore two recent regulatory network-based models for stem cell differentiation, and find that extrinsic noise in transcription rate causes appreciable variability in the behaviour of these model systems. These results suggest that mitochondrial and transcriptional variability may be an important mechanism influencing a large variety of cellular processes and properties.

Interplay between gene expression noise and regulatory network architecture

Complex regulatory networks orchestrate most cellular processes in biological systems. Genes in such networks are subject to expression noise, resulting in isogenic cell populations exhibiting cell-to-cell variation in protein levels. Increasing evidence suggests that cells have evolved regulatory strategies to limit, tolerate or amplify expression noise. In this context, fundamental questions arise: how can the architecture of gene regulatory networks generate, make use of or be constrained by expression noise? Here, we discuss the interplay between expression noise and gene regulatory network at different levels of organization, ranging from a single regulatory interaction to entire regulatory networks. We then consider how this interplay impacts a variety of phenomena, such as pathogenicity, disease, adaptation to changing environments, differential cell-fate outcome and incomplete or partial penetrance effects. Finally, we highlight recent technological developments that permit measurements at the single-cell level, and discuss directions for future research.

Adjusting phenotypes by noise control

Genetically identical cells can show phenotypic variability. This is often caused by stochastic events that originate from randomness in biochemical processes involving in gene expression and other extrinsic cellular processes. From an engineering perspective, there have been efforts focused on theory and experiments to control noise levels by perturbing and replacing gene network components. However, systematic methods for noise control are lacking mainly due to the intractable mathematical structure of noise propagation through reaction networks. Here, we provide a numerical analysis method by quantifying the parametric sensitivity of noise characteristics at the level of the linear noise approximation. Our analysis is readily applicable to various types of noise control and to different types of system; for example, we can orthogonally control the mean and noise levels and can control system dynamics such as noisy oscillations. As an illustration we applied our method to HIV and yeast gene expression systems and metabolic networks. The oscillatory signal control was applied to p53 oscillations from DNA damage. Furthermore, we showed that the efficiency of orthogonal control can be enhanced by applying extrinsic noise and feedback. Our noise control analysis can be applied to any stochastic model belonging to continuous time Markovian systems such as biological and chemical reaction systems, and even computer and social networks. We anticipate the proposed analysis to be a useful tool for designing and controlling synthetic gene networks.

Stochastic gene expression at the single cell level can lead to significant phenotypic variation at the population level. To obtain a desired phenotype, the noise levels of intracellular protein concentrations may need to be tuned and controlled. Noise levels often decrease in relative amount as the mean values increase. This implies that the noise levels can be passively controlled by changing the mean values. In an engineering perspective, the noise levels can be further controlled while the mean values can be simultaneously adjusted to desired values. Here, systematic schemes for such simultaneous control are described by identifying where and by how much the system needs to be perturbed. The schemes can be applied to the design process of a potential therapeutic HIV-drug that targets a certain set of reactions that are identified by the proposed analysis, to prevent stochastic transition to the lytic state. In some cases, the simultaneous control cannot be performed efficiently, when the noise levels strongly change with the mean values. This problem is shown to be resolved by applying extra noise and feedback.

Unraveling the design principle for motif organization in signaling networks

Cellular signaling networks display complex architecture. Defining the design principle of this architecture is crucial for our understanding of various biological processes. Using a mathematical model for three-node feed-forward loops, we identify that the organization of motifs in specific manner within the network serves as an important regulator of signal processing. Further, incorporating a systemic stochastic perturbation to the model we could propose a possible design principle, for higher-order organization of motifs into larger networks in order to achieve specific biological output. The design principle was then verified in a large, complex human cancer signaling network. Further analysis permitted us to classify signaling nodes of the network into robust and vulnerable nodes as a result of higher order motif organization. We show that distribution of these nodes within the network at strategic locations then provides for the range of features displayed by the signaling network.

Strong negative feedback from Erk to Raf confers robustness to MAPK signalling

This study shows that MAPK signalling is robust against protein level changes due to a strong negative feedback from Erk to Raf. Surprisingly, robustness is provided through a fast post-translational mechanism although variation of Erk levels occurs on a timescale of days.

MAPK signalling is robust against variation in protein level.

Robustness is mediated by a negative feedback to Raf.

Loss of negative feedback due to mutation in B-Raf opens the door for targeted intervention.

Protein levels within signal transduction pathways vary strongly from cell to cell. For example, it has been reported that concentrations of the last kinase within the MAPK signalling module, Erk, varies about four-fold between clonal cells under the same conditions. In the present study, we analysed how signalling pathways can still process information quantitatively despite strong heterogeneity in protein levels. Mathematical analysis of isolated phosphorylation–dephosphorylation cycles predicts that phosphorylation of a signalling molecule is proportional to the protein concentration. We systematically perturbed the protein levels of Erk in human cell lines by siRNA. We found that the steady-state phosphorylation of Erk is very robust against perturbations of Erk protein level, suggesting that there are mechanisms that provide robustness to the pathway against protein fluctuations. Using mathematical modelling, we identified three potential mechanisms that may provide robustness against fluctuating protein levels:

1. Kinetic effects (saturation of the activating kinase Mek),

2. Transcriptional negative feedbacks,

3. Negative feedbacks on the post-translational level.

By experimental analysis of the systems, which included analysis of Erk phosphorylation under Mek overexpression, measuring transcript levels of negative feedback regulators, and application of generic inhibitors of transcription and translation, we could exclude kinetic effects and transcriptional negative feedback as mechanisms of robustness.

By analysing a panel of cell lines, we found that cells are robust as long as the signal passes through Raf-1. In contrast, cells where the pathway is activated by a mutation in B-Raf lose robustness. Detailed molecular analysis of the system shows that a single post-translational feedback to Raf mediates robustness. Thus, robustness is provided through a fast post-translational mechanism although variation of Erk levels occurs on a timescale of days.

Protein levels within signal transduction pathways vary strongly from cell to cell. Here, we analysed how signalling pathways can still process information quantitatively despite strong heterogeneity in protein levels. We systematically perturbed the protein levels of Erk, the terminal kinase in the MAPK signalling pathway in a panel of human cell lines. We found that the steady-state phosphorylation of Erk is very robust against perturbations of Erk protein level. Although a multitude of mechanisms exist that may provide robustness against fluctuating protein levels, we found that one single feedback from Erk to Raf-1 accounts for the observed robustness. Surprisingly, robustness is provided through a fast post-translational mechanism although variation of Erk levels occurs on a timescale of days.

Structural and functional organization of RNA regulons in the post-transcriptional regulatory network of yeast

Post-transcriptional control of mRNA transcript processing by RNA binding proteins (RBPs) is an important step in the regulation of gene expression and protein production. The post-transcriptional regulatory network is similar in complexity to the transcriptional regulatory network and is thought to be organized in RNA regulons, coherent sets of functionally related mRNAs combinatorially regulated by common RBPs. We integrated genome-wide transcriptional and translational expression data in yeast with large-scale regulatory networks of transcription factor and RBP binding interactions to analyze the functional organization of post-transcriptional regulation and RNA regulons at a system level. We found that post-transcriptional feedback loops and mixed bifan motifs are overrepresented in the integrated regulatory network and control the coordinated translation of RNA regulons, manifested as clusters of functionally related mRNAs which are strongly coexpressed in the translatome data. These translatome clusters are more functionally coherent than transcriptome clusters and are expressed with higher mRNA and protein levels and less noise. Our results show how the post-transcriptional network is intertwined with the transcriptional network to regulate gene expression in a coordinated way and that the integration of heterogeneous genome-wide datasets allows to relate structure to function in regulatory networks at a system level.

Negative feedback and physical limits of genes

This paper compares the auto-repressed gene to a simple one (a gene without auto-regulation) in terms of response time and output noise under the assumption of fixed metabolic cost. The analysis shows that, in the case of non-vanishing leak expression rate, the negative feedback reduces both the switching on and switching off times of a gene. The noise of the auto-repressed gene will be lower than the one of the simple gene only for low leak expression rates. Summing up, for low, but non-vanishing leak expression rates, the auto-repressed gene is both faster and less noisier compared to the simple one.

How structure determines correlations in neuronal networks

Networks are becoming a ubiquitous metaphor for the understanding of complex biological systems, spanning the range between molecular signalling pathways, neural networks in the brain, and interacting species in a food web. In many models, we face an intricate interplay between the topology of the network and the dynamics of the system, which is generally very hard to disentangle. A dynamical feature that has been subject of intense research in various fields are correlations between the noisy activity of nodes in a network. We consider a class of systems, where discrete signals are sent along the links of the network. Such systems are of particular relevance in neuroscience, because they provide models for networks of neurons that use action potentials for communication. We study correlations in dynamic networks with arbitrary topology, assuming linear pulse coupling. With our novel approach, we are able to understand in detail how specific structural motifs affect pairwise correlations. Based on a power series decomposition of the covariance matrix, we describe the conditions under which very indirect interactions will have a pronounced effect on correlations and population dynamics. In random networks, we find that indirect interactions may lead to a broad distribution of activation levels with low average but highly variable correlations. This phenomenon is even more pronounced in networks with distance dependent connectivity. In contrast, networks with highly connected hubs or patchy connections often exhibit strong average correlations. Our results are particularly relevant in view of new experimental techniques that enable the parallel recording of spiking activity from a large number of neurons, an appropriate interpretation of which is hampered by the currently limited understanding of structure-dynamics relations in complex networks.

Evolution of bet-hedging mechanisms in cell cycle and embryo development stimulated by weak linkage of stochastic processes

Our current understanding of the origin and evolution of the cell cycle is largely filled with gaps and unresolved questions. Numerous similarities between the processes comprising the cell cycle in distant organisms from the Pro- and Eukaryota kingdoms provide some clues about the course that evolution has taken. Contemporary Prokaryotes and Eukaryotes regulate their cell cycles in a quite similar way, using a master oscillator that regulates cell division. Despite this striking similarity, they use entirely different molecules for this purpose. The necessity to keep the master oscillator intact for the survival of every cell/organism allows evolutionary changes in only the secondary mechanisms and processes of the cell cycle. This is especially clear in oocytes and embryos, which have a direct impact on the reproductive success of an adult organism. Here, we present examples of cues driving such mild evolutionary changes of certain aspects of cell cycle progression in oocytes and early embryos. We suggest that weak linkages between core processes that rely on randomness (stochasticity) have led to the evolution of strategies increasing fitness similar to bet-hedging, a stochastic-based survival strategy of risk minimization widely implemented by populations of bacteria, yeast, arthropods, and birds. Stochastic diversification of phenotypes by isogenic cells increases their fitness in unpredictable environments and improves their survival rate upon exposure to stress, a trait beneficial in evading antibiotic treatment by bacteria or withstanding chemotherapy by cancer cells. The evolution of bet-hedging has been observed experimentally for bacteria and attributed to specific molecular mechanisms involved in this strategy. In this chapter, we set out to answer whether similar strategies could have evolved at the level of oocytes and embryos. We indicate possible evolutionary cues capable of realizing bet-hedging-like mechanisms.

Origins of stochastic intracellular processes and consequences for cell-to-cell variability and cellular survival strategies

Quantitative analyses of the dynamics of single cells have become a powerful approach in current cell biology. They give us an unprecedented opportunity to study dynamics of molecular networks at a high level of accuracy in living single cells. Genetically identical cells, growing in the same environment and sharing the same growth history, can differ remarkably in their molecular makeup and physiological behaviors. The origins of this cell-to-cell variability have in many cases been traced to the inevitable stochasticity of molecular reactions. Those mechanisms can cause isogenic cells to have qualitatively different life histories. Many studies indicate that molecular noise can be exploited by cell populations to enhance survival prospects in uncertain environments. On the other hand, cells have evolved noise-suppression mechanisms to cope with the inevitable noise in their functioning so as to reduce the hazardous effects of noise. In this chapter, we discuss key experiments, theoretical results, and physiological consequences of molecular stochasticity to introduce this exciting field to a broader community of (systems) biologists.

What it takes to understand and cure a living system: computational systems biology and a systems biology-driven pharmacokinetics-pharmacodynamics platform

The utility of model repositories is discussed in the context of systems biology (SB). It is shown how such repositories, and in particular their live versions, can be used for computational SB: we calculate the robustness of the yeast glycolytic network with respect to perturbations of one of its enzyme activities and one transport system. The robustness with respect to perturbations in the key enzyme phosphofructokinase is surprisingly large. We then note the upcoming convergence of pharmacokinetics-pharmacodynamics (PK-PD) and bottom-up SB. In PK alone, quite a few one-, two- or more compartment models provide valuable initial guesses and insights into the absorption, distribution, metabolism and excretion of particular drugs. These models are phenomenological however, forbidding implementation of molecule-based tools and network information. In order to facilitate a fruitful synergy between SB and PK-PD, and between PK and PD, we present a new platform that combines standard phenomenological models used in the PK/PD field with mechanism-based SB models and approaches.

Effect of feedback on the fidelity of information transmission of time-varying signals

Living cells are continually exposed to environmental signals that vary in time. These signals are detected and processed by biochemical networks, which are often highly stochastic. To understand how cells cope with a fluctuating environment, we therefore have to understand how reliably biochemical networks can transmit time-varying signals. To this end, we must understand both the noise characteristics and the amplification properties of networks. In this paper, we use information theory to study how reliably signaling cascades employing autoregulation and feedback can transmit time-varying signals. We calculate the frequency dependence of the gain-to-noise ratio, which reflects how reliably a network transmits signals at different frequencies. We find that the gain-to-noise ratio may differ qualitatively from the power spectrum of the output, showing that the latter does not directly reflect signaling performance. Moreover, we find that autoactivation and autorepression increase and decrease the gain-to-noise ratio for all of frequencies, respectively. Positive feedback specifically enhances information transmission at low frequencies, while negative feedback increases signal fidelity at high frequencies. Our analysis not only elucidates the role of autoregulation and feedback in naturally occurring biological networks, but also reveals design principles that can be used for the reliable transmission of time-varying signals in synthetic gene circuits.

Trade-offs and noise tolerance in signal detection by genetic circuits

Genetic circuits can implement elaborated tasks of amplitude or frequency signal detection. What type of constraints could circuits experience in the performance of these tasks, and how are they affected by molecular noise? Here, we consider a simple detection process–a signal acting on a two-component module–to analyze these issues. We show that the presence of a feedback interaction in the detection module imposes a trade-off on amplitude and frequency detection, whose intensity depends on feedback strength. A direct interaction between the signal and the output species, in a type of feed-forward loop architecture, greatly modifies these trade-offs. Indeed, we observe that coherent feed-forward loops can act simultaneously as good frequency and amplitude noise-tolerant detectors. Alternatively, incoherent feed-forward loop structures can work as high-pass filters improving high frequency detection, and reaching noise tolerance by means of noise filtering. Analysis of experimental data from several specific coherent and incoherent feed-forward loops shows that these properties can be realized in a natural context. Overall, our results emphasize the limits imposed by circuit structure on its characteristic stimulus response, the functional plasticity of coherent feed-forward loops, and the seemingly paradoxical advantage of improving signal detection with noisy circuit components.

Mutual information in time-varying biochemical systems

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

Transcriptional feedbacks in mammalian signal transduction pathways facilitate rapid and reliable protein induction

Signal transduction pathways translate the cellular context into context-dependent expression of genes. In response to extracellular stimuli, proteins have to be up-regulated quickly and reliably. However, rapid and reliable control of target genes by a signalling pathway faces two major challenges. Firstly, swift changes in protein levels require short-lived proteins, which is not resource-optimal. Secondly, gene expression is an intrinsically noisy process, and fluctuations in the protein numbers are likely to reduce the functionality of the proteins. Mammalian signalling pathways frequently induce the transcription of their own inhibitors, resulting in negative feedback regulation. However, the functional role of these transcriptional feedbacks in mammalian signal transduction is unclear. Here, we analyse a mathematical model of a prototypical signalling pathway, the MAPK cascade, in order to investigate how transcriptional negative feedbacks may help to overcome the challenge of fast and reliable gene induction. It is shown that a transcriptional negative feedback helps to decouple protein stability and response times, thus allowing for swift up-regulation even of long-lived proteins. Furthermore, transcriptional negative feedbacks filter out the extrinsic component of gene expression noise, which dominates the uncertainty in gene expression in mammalian cells, thus making gene expression more reliable. Model analysis predicts that both goals can be achieved if (i) proteins and mRNAs of the feedback regulators and (ii) mRNAs of all targets are short-lived. These predictions are confirmed by large-scale measurements of mRNA and protein half-lives. Therefore, the design of the mammalian signal transduction network with its rapid feedback inhibition allows for swift and reliable target gene expression.

Sensitivity summation theorems for stochastic biochemical reaction systems

We investigate how stochastic reaction processes are affected by external perturbations. We describe an extension of the deterministic metabolic control analysis (MCA) to the stochastic regime. We introduce stochastic sensitivities for mean and covariance values of reactant concentrations and reaction fluxes and show that there exist MCA-like summation theorems among these sensitivities. The summation theorems for flux variances is shown to depend on the size of the measurement time window () within which reaction events are counted for measuring a single flux. It is found that the degree of the -dependency can become significant for processes involving multi-time-scale dynamics and is estimated by introducing a new measure of time-scale separation. This -dependency is shown to be closely related to the power-law scaling observed in flux fluctuations in various complex networks.