Exact results for noise power spectra in linear biochemical reaction networks

Bye bye, linearity, bye: quantification of the mean for linear CRNs in a random environment

Molecular reactions within a cell are inherently stochastic, and cells often differ in morphological properties or interact with a heterogeneous environment. Consequently, cell populations exhibit heterogeneity both due to these intrinsic and extrinsic causes. Although state-of-the-art studies that focus on dissecting this heterogeneity use single-cell measurements, the bulk data that shows only the mean expression levels is still in routine use. The fingerprint of the heterogeneity is present also in bulk data, despite being hidden from direct measurement. In particular, this heterogeneity can affect the mean expression levels via bimolecular interactions with low-abundant environment species. We make this statement rigorous for the class of linear reaction systems that are embedded in a discrete state Markov environment. The analytic expression that we provide for the stationary mean depends on the reaction rate constants of the linear subsystem, as well as the generator and stationary distribution of the Markov environment. We demonstrate the effect of the environment on the stationary mean. Namely, we show how the heterogeneous case deviates from the quasi-steady state (Q.SS) case when the embedded system is fast compared to the environment.

Frequency spectra and the color of cellular noise

The invention of the Fourier integral in the 19th century laid the foundation for modern spectral analysis methods. This integral decomposes a temporal signal into its frequency components, providing deep insights into its generating process. While this idea has precipitated several scientific and technological advances, its impact has been fairly limited in cell biology, largely due to the difficulties in connecting the underlying noisy intracellular networks to the frequency content of observed single-cell trajectories. Here we develop a spectral theory and computational methodologies tailored specifically to the computation and analysis of frequency spectra of noisy intracellular networks. Specifically, we develop a method to compute the frequency spectrum for general nonlinear networks, and for linear networks we present a decomposition that expresses the frequency spectrum in terms of its sources. Several examples are presented to illustrate how our results provide frequency-based methods for the design and analysis of noisy intracellular networks.

Frequency response in splicing regulation under mRNA auto-depletion control

Nowadays, there exists a huge literature about stochastic model of transcriptional and translational control in gene networks. However, results related to post-transcriptional regulation via splicing and its connection with transcriptional and translational regulation are almost missing in the current literature and only related to the steady state moments investigation. Nowadays, it is becoming of paramount importance the need for modeling post-transcriptional regulation via splicing especially for DNA viruses or retroviruses. However, there exists only few studies in the literature about splicing regulation and none of them investigate its behavior in the frequency domain that can unveil important features of dynamical stochastic systems that cannot be revealed by the sole steady state moment investigation. The aim of this work is to theoretically investigate a simple gene network subject to splicing regulation with negative feedback control, implemented through mRNA auto-depletion under a frequency domain perspective. This study showed the pivotal role of the burst size, enhancing the noise power spectrum, as well as the splicing conversion rates capable to increase and decrease the noise power spectrum in the pre-mRNA and mRNA, respectively, for high values of conversion rates. Importantly, it shows the capability of the mRNA autodepletion control to modulate the noise as a frequency-dependent amplifying control as a function of the negative feedback strengths.

Receptor time integration via discrete sampling

Living cells can measure chemical concentrations with remarkable accuracy, even though these measurements are inherently noisy due to the stochastic binding of the ligand to the receptor. A widely used mechanism for reducing the sensing error is to increase the effective number of measurements via receptor time integration. This mechanism is implemented via the signaling network downstream of the receptor, yet how it is implemented optimally given constraints on cellular resources such as protein copies and time remains unknown. To address this question, we employ our sampling framework [Govern and ten Wolde, Proc. Natl. Acad. Sci. USA 111, 17486 (2014)PNASA60027-842410.1073/pnas.1411524111] and extend it here to time-varying ligand concentrations. This framework starts from the observation that the signaling network implements the mechanism of time integration by discretely sampling the ligand-binding state of the receptor and storing these states into chemical modification states of the readout molecules downstream. It reveals that the sensing error has two distinct contributions: a sampling error, which is determined by the number of samples, their independence, and their accuracy, and a dynamical error, which depends on the timescale that these samples are generated. We test our previously identified design principle, which states that in an optimally designed system the number of receptors and their integration time, which determine the number of independent concentration measurements at the receptor level, equals the number of readout proteins, which store these measurements. We show that this principle is robust to the dynamics of the input and the relative costs of the receptor and readout proteins: these resources are fundamental and cannot compensate each other.

Variance of filtered signals: Characterization for linear reaction networks and application to neurotransmission dynamics

Neurotransmission at chemical synapses relies on the calcium-induced fusion of synaptic vesicles with the presynaptic membrane. The distance of the synaptic vesicle to the calcium channels determines the release probability and consequently the postsynaptic signal. Suitable models of the process need to capture both the mean and the variance observed in electrophysiological measurements of the postsynaptic current. In this work, we propose a method to directly compute the exact first- and second-order moments for signals generated by a linear reaction network under convolution with an impulse response function, rendering computationally expensive numerical simulations of the underlying stochastic counting process obsolete. We show that the autocorrelation of the process is central for the calculation of the filtered signal's second-order moments, and derive a system of PDEs for the cross-correlation functions (including the autocorrelations) of linear reaction networks with time-dependent rates. Finally, we employ our method to efficiently compare different spatial coarse graining approaches for a specific model of synaptic vesicle fusion. Beyond the application to neurotransmission processes, the developed theory can be applied to any linear reaction system that produces a filtered stochastic signal.

Frequency analysis of splicing regulation

In the past decades, mathematical modelers developed a huge literature to model and analyze gene networks under both deterministic and stochastic formalisms. Such literature is predominantly focused on modeling transcriptional and translational regulation, while the development of proper mathematical frameworks to model and study post-transcriptional regulation via splicing and its connection with transcriptional and translational regulation are almost missing. Nowadays, it is becoming of paramount importance the need for modeling post-transcriptional regulation via splicing especially for bacteria or viruses. However, current literature is focused on investigating splicing regulation at steady state and none of them have the purpose to investigate gene networks behavior in the frequency domain, thus providing only a partial investigation about the system dynamical response. The aim of this work is to theoretically investigate a simple gene network subjects to splicing regulation with/without negative feedback control under a frequency domain perspective. This study showed the pivotal role of the burst size, as well as splicing conversion rates to modulate the noise and the power spectrum response. It also shows an interesting behavior under the frequency domain induced by the merging effect of burst size, splicing conversion rates and negative feedback strength.

Information transmission in a two-step cascade: interplay of activation and repression

We present an information-theoretic formalism to study signal transduction in four architectural variants of a model two-step cascade with increasing input population. Our results categorize these four types into two classes depending upon the effect of activation and repression on mutual information, net synergy, and signal-to-noise ratio. Using the Gaussian framework and linear noise approximation, we derive the analytic expressions for these metrics to establish their underlying relationships in terms of the biochemical parameters. We also verify our approximations through stochastic simulations.

A Voxel Model to Decipher the Role of Molecular Communication in the Growth of Glioblastoma Multiforme

Glioblastoma Multiforme (GBM), the most malignant human tumour, can be defined by the evolution of growing bio-nanomachine networks within an interplay between self-renewal (Grow) and invasion (Go) potential of mutually exclusive phenotypes of transmitter and receiver cells. Herein, we present a mathematical model for the growth of GBM tumour driven by molecule-mediated inter-cellular communication between two populations of evolutionary bio-nanomachines representing the Glioma Stem Cells (GSCs) and Glioma Cells (GCs). The contribution of each subpopulation to tumour growth is quantified by a voxel model representing the end to end inter-cellular communication models for GSCs and progressively evolving invasiveness levels of glioma cells within a network of diverse cell configurations. Mutual information, information propagation speed and the impact of cell numbers and phenotypes on the communication output and GBM growth are studied by using analysis from information theory. The numerical simulations show that the progression of GBM is directly related to higher mutual information and higher input information flow of molecules between the GSCs and GCs, resulting in an increased tumour growth rate. These fundamental findings contribute to deciphering the mechanisms of tumour growth and are expected to provide new knowledge towards the development of future bio-nanomachine-based therapeutic approaches for GBM.

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.

Statistics of Nascent and Mature RNA Fluctuations in a Stochastic Model of Transcriptional Initiation, Elongation, Pausing, and Termination

Recent advances in fluorescence microscopy have made it possible to measure the fluctuations of nascent (actively transcribed) RNA. These closely reflect transcription kinetics, as opposed to conventional measurements of mature (cellular) RNA, whose kinetics is affected by additional processes downstream of transcription. Here, we formulate a stochastic model which describes promoter switching, initiation, elongation, premature detachment, pausing, and termination while being analytically tractable. We derive exact closed-form expressions for the mean and variance of nascent RNA fluctuations on gene segments, as well as of total nascent RNA on a gene. We also obtain exact expressions for the first two moments of mature RNA fluctuations and approximate distributions for total numbers of nascent and mature RNA. Our results, which are verified by stochastic simulation, uncover the explicit dependence of the statistics of both types of RNA on transcriptional parameters and potentially provide a means to estimate parameter values from experimental data.

Communication in Plants: Comparison of Multiple Action Potential and Mechanosensitive Signals With Experiments

Both action potentials and mechanosensitive signalling are an important communication mechanisms in plants. Considering an information-theoretic framework, this paper explores the effective range of multiple action potentials for a long chain of cells (i.e., up to 100) in different configurations, and introduces the study of multiple mechanosensitive activation signals (generated due to a mechanical stimulus) in plants. For both these signals, we find that the mutual information per cell and information propagation speed tends to increase up to a certain number of receiver cells. However, as the number of cells increase beyond 10 to 12, the mutual information per cell starts to decrease. To validate our model and results, we include an experimental verification of the theoretical model, using a PhytlSigns biosignal amplifier, allowing us to measure the magnitude of the voltage associated with the multiple AP's and mechanosensitive activation signals induced by different stimulus in plants. Experimental data is used to calculate the mutual information and information propagation speed, which is compared with corresponding numerical results. Since these signals are used for a variety of important tasks within the plant, understanding them may lead to new bioengineering methods for plants.

Molecular Communications With Molecular Circuit-Based Transmitters and Receivers

The performance of a communication link can be improved by maximizing the mutual information between the input and output signals. This paper considers this maximization problem in a molecular communication link where both the transmitter and the receiver are molecular circuit. This general optimization is hard to solve. We simplify the problem by limiting to reactions with linear reaction rates and molecular circuits with a limited number of species. We derive an expression of mutual information and use it for numerical maximization. We show that our parameterized transmitter circuit is able to give mutual information that is close to upper bound obtained in our earlier work.

Communication and Information Theory of Single Action Potential Signals in Plants

Many plants, such as Mimosa pudica (the "sensitive plant"), employ electrochemical signals known as action potentials (APs) for rapid intercellular communication. In this paper, we consider a reaction-diffusion model of individual AP signals to analyze APs from a communication- and information-theoretic perspective. We use concepts from molecular communication to explain the underlying process of information transfer in a plant for a single AP pulse that is shared with one or more receiver cells. We also use the chemical Langevin equation to accommodate the deterministic as well as stochastic component of the system. Finally, we present an information-theoretic analysis of single action potentials, obtaining achievable information rates for these signals. We show that, in general, the presence of an AP signal can increase the mutual information and information propagation speed among neighboring cells with receivers in different settings.

EIciRNA-mediated gene expression: tunability and bimodality

Biological experiments have verified that EIciRNAs (a class of circRNA) produced from pre-mRNA can regulate gene expression, but the effect of regulation remains unexplored. Here, we refine a mechanistic gene model from experimental facts, in which we assume pre-mRNA synthesizes EIciRNAs and mRNAs in a probabilistic manner, with the probability called the pathway strength, and the resulting EIciRNAs positively regulate the pre-mRNA synthesis. We show that there is a critical pathway strength such that the mRNA mean and the mRNA noise reach the highest and lowest levels, respectively. The EIciRNA can induce the unimodal and bimodal mRNA expressions, as well as the transition between them. Our investigation hints that EIciRNA is a non-negligible factor affecting cell-to-cell variability in gene expression.

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.

Robustness of Clocks to Input Noise

To estimate the time, many organisms, ranging from cyanobacteria to animals, employ a circadian clock which is based on a limit-cycle oscillator that can tick autonomously with a nearly 24 h period. Yet, a limit-cycle oscillator is not essential for knowing the time, as exemplified by bacteria that possess an "hourglass": a system that when forced by an oscillatory light input exhibits robust oscillations from which the organism can infer the time, but that in the absence of driving relaxes to a stable fixed point. Here, using models of the Kai system of cyanobacteria, we compare a limit-cycle oscillator with two hourglass models, one that without driving relaxes exponentially and one that does so in an oscillatory fashion. In the limit of low input noise, all three systems are equally informative on time, yet in the regime of high input-noise the limit-cycle oscillator is far superior. The same behavior is found in the Stuart-Landau model, indicating that our result is universal.

Modeling of signaling crosstalk-mediated drug resistance and its implications on drug combination

The efficacy of pharmacological perturbation to the signaling transduction network depends on the network topology. However, whether and how signaling dynamics mediated by crosstalk contributes to the drug resistance are not fully understood and remain to be systematically explored. In this study, motivated by a realistic signaling network linked by crosstalk between EGF/EGFR/Ras/MEK/ERK pathway and HGF/HGFR/PI3K/AKT pathway, we develop kinetic models for several small networks with typical crosstalk modules to investigate the role of the architecture of crosstalk in inducing drug resistance. Our results demonstrate that crosstalk inhibition diminishes the response of signaling output to the external stimuli. Moreover, we show that signaling crosstalk affects the relative sensitivity of drugs, and some types of crosstalk modules that could yield resistance to the targeted drugs were identified. Furthermore, we quantitatively evaluate the relative efficacy and synergism of drug combinations. For the modules that are resistant to the targeted drug, we identify drug targets that can not only increase the relative drug efficacy but also act synergistically. In addition, we analyze the role of the strength of crosstalk in switching a module between drug-sensitive and drug-resistant. Our study provides mechanistic insights into the signaling crosstalk-mediated mechanisms of drug resistance and provides implications for the design of synergistic drug combinations to reduce drug resistance.

Improving the Capacity of Molecular Communication Using Enzymatic Reaction Cycles

This paper considers the capacity of a diffusion-based molecular communication link assuming the receiver uses chemical reactions. The key contribution is we show that enzymatic reaction cycles, which is a class of chemical reactions commonly found in cells consisting of a forward and a backward enzymatic reaction, can improve the capacity of the communication link. The technical difficulty in analyzing enzymatic reaction cycles is that their reaction rates are nonlinear. We deal with this by assuming that the amount of certain chemicals in the enzymatic reaction cycle is large. In order to simplify the problem further, we use singular perturbation to study a particular operating regime of the enzymatic reaction cycles. This allows us to derive a closed-form expression of the channel gain. This expression suggests that we can improve the channel gain by increasing the total amount of substrate in the enzymatic reaction cycle. By using numerical calculations, we show that the effect of the enzymatic reaction cycle is to increase the channel gain and to reduce the noise, which results in a better signal-to-noise ratio and in turn a higher communication capacity. Furthermore, we show that we can increase the capacity by increasing the total amount of substrate in the enzymatic reaction cycle.

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.

On a theory of stability for nonlinear stochastic chemical reaction networks

We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jacobian matrix can be readily computed. Eigenvalues govern the relaxation of fluctuation autocorrelation functions at steady state. Autocorrelation functions reveal the time scales of phenomena underlying the dynamics of nonlinear reaction networks. In accord with the fluctuation-dissipation theorem, these functions are found to be congruent to response functions to small perturbations. Significant differences are observed in the stability of nonlinear reacting systems between deterministic and stochastic modeling formalisms.

Interplay between stochasticity and negative feedback leads to pulsed dynamics and distinct gene activity patterns

Gene expression is an inherently stochastic process that depends on the structure of the biochemical regulatory network in which the gene is embedded. Here we study the dynamical consequences of the interplay between stochastic gene switching and the widespread negative feedback regulatory loop in a simple model of a biochemical regulatory network. Using a simplified hybrid simulation approach, in which only the gene activation is modeled stochastically, we find that stochasticity in gene switching by itself can induce pulses in the system, providing also analytical insights into their origin. Furthermore, we find that this simple network is able to reproduce both exponential and peaked distributions of gene active and inactive times similar to those that have been observed experimentally. This simplified hybrid simulation approach also allows us to link these patterns to the dynamics of the system for each gene state.

Stochastic modular analysis for gene circuits: interplay among retroactivity, nonlinearity, and stochasticity

This chapter introduces a computational analysis method for analyzing gene circuit dynamics in terms of modules while taking into account stochasticity, system nonlinearity, and retroactivity. (1) ANALOG ELECTRICAL CIRCUIT REPRESENTATION FOR GENE CIRCUITS: A connection between two gene circuit components is often mediated by a transcription factor (TF) and the connection signal is described by the TF concentration. The TF is sequestered to its specific binding site (promoter region) and regulates downstream transcription. This sequestration has been known to affect the dynamics of the TF by increasing its response time. The downstream effect-retroactivity-has been shown to be explicitly described in an electrical circuit representation, as an input capacitance increase. We provide a brief review on this topic. (2) MODULAR DESCRIPTION OF NOISE PROPAGATION: Gene circuit signals are noisy due to the random nature of biological reactions. The noisy fluctuations in TF concentrations affect downstream regulation. Thus, noise can propagate throughout the connected system components. This can cause different circuit components to behave in a statistically dependent manner, hampering a modular analysis. Here, we show that the modular analysis is still possible at the linear noise approximation level. (3) NOISE EFFECT ON MODULE INPUT-OUTPUT RESPONSE: We investigate how to deal with a module input-output response and its noise dependency. Noise-induced phenotypes are described as an interplay between system nonlinearity and signal noise. Lastly, we provide the comprehensive approach incorporating the above three analysis methods, which we call "stochastic modular analysis." This method can provide an analysis framework for gene circuit dynamics when the nontrivial effects of retroactivity, stochasticity, and nonlinearity need to be taken into account.

Stochastic oscillations induced by intrinsic fluctuations in a self-repressing gene

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

An effective method for computing the noise in biochemical networks

We present a simple yet effective method, which is based on power series expansion, for computing exact binomial moments that can be in turn used to compute steady-state probability distributions as well as the noise in linear or nonlinear biochemical reaction networks. When the method is applied to representative reaction networks such as the ON-OFF models of gene expression, gene models of promoter progression, gene auto-regulatory models, and common signaling motifs, the exact formulae for computing the intensities of noise in the species of interest or steady-state distributions are analytically given. Interestingly, we find that positive (negative) feedback does not enlarge (reduce) noise as claimed in previous works but has a counter-intuitive effect and that the multi-OFF (or ON) mechanism always attenuates the noise in contrast to the common ON-OFF mechanism and can modulate the noise to the lowest level independently of the mRNA mean. Except for its power in deriving analytical expressions for distributions and noise, our method is programmable and has apparent advantages in reducing computational cost.

Feed-forward loops and diamond motifs lead to tunable transmission of information in the frequency domain

Using a Gaussian model, we study the transmission of time-varying biochemical signals through feed-forward motifs and diamond motifs. To this end, we compute the frequency dependence of the gain, the noise, as well as their ratio, the gain-to-noise ratio, which measures how reliably a network transmits signals at different frequencies. We find that both coherent and incoherent feed-forward motifs can either act as low-pass or high-pass filters for information: The frequency dependence of the gain-to-noise ratio increases or decreases with increasing frequency, respectively. Our analysis of diamond motifs reveals that cooperative activation of the output component can increase the gain-to-noise ratio. This means that from the perspective of information transmission, it can be beneficial to split the input signal in two and recombine the two propagated signals at the output. Cooperative activation can be implemented via the formation of homo- or heteromultimers that then bind and activate the output component or via the binding of individual molecules of the intermediate species to the output component.

Reliability of frequency and amplitude decoding in gene regulation

In biochemical signaling, information is often encoded in oscillatory signals. However, the advantages of such a coding strategy over an amplitude-encoding scheme of constant signals remain unclear. Here we study the dynamics of a simple model gene promoter in response to oscillating and constant transcription factor signals. We find that in biologically relevant parameter regimes an oscillating input can produce a more constant protein level than a constant input. Our results suggest that oscillating signals may be used to minimize noise in gene regulation.

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.

Analytic methods for modeling stochastic regulatory networks

Recent single-cell experiments have revived interest in the unavoidable or intrinsic noise in biochemical and genetic networks arising from the small number of molecules of the participating species. That is, rather than modeling regulatory networks in terms of the deterministic dynamics of concentrations, we model the dynamics of the probability of a given copy number of the reactants in single cells. Most of the modeling activity of the last decade has centered on stochastic simulation, i.e., Monte Carlo methods for generating stochastic time series. Here we review the mathematical description in terms of probability distributions, introducing the relevant derivations and illustrating several cases for which analytic progress can be made either instead of or before turning to numerical computation. Analytic progress can be useful both for suggesting more efficient numerical methods and for obviating the computational expense of, for example, exploring parametric dependence.

Computational study of noise in a large signal transduction network

BACKGROUND

Biochemical systems are inherently noisy due to the discrete reaction events that occur in a random manner. Although noise is often perceived as a disturbing factor, the system might actually benefit from it. In order to understand the role of noise better, its quality must be studied in a quantitative manner. Computational analysis and modeling play an essential role in this demanding endeavor.

RESULTS

We implemented a large nonlinear signal transduction network combining protein kinase C, mitogen-activated protein kinase, phospholipase A2, and β isoform of phospholipase C networks. We simulated the network in 300 different cellular volumes using the exact Gillespie stochastic simulation algorithm and analyzed the results in both the time and frequency domain. In order to perform simulations in a reasonable time, we used modern parallel computing techniques. The analysis revealed that time and frequency domain characteristics depend on the system volume. The simulation results also indicated that there are several kinds of noise processes in the network, all of them representing different kinds of low-frequency fluctuations. In the simulations, the power of noise decreased on all frequencies when the system volume was increased.

CONCLUSIONS

We concluded that basic frequency domain techniques can be applied to the analysis of simulation results produced by the Gillespie stochastic simulation algorithm. This approach is suited not only to the study of fluctuations but also to the study of pure noise processes. Noise seems to have an important role in biochemical systems and its properties can be numerically studied by simulating the reacting system in different cellular volumes. Parallel computing techniques make it possible to run massive simulations in hundreds of volumes and, as a result, accurate statistics can be obtained from computational studies.

Dynamical fluctuations in biochemical reactions and cycles

We develop theory for the dynamics and fluctuations in some cyclic and linear biochemical reactions. We use the approach of maximum caliber, which computes the ensemble of paths taken by the system, given a few experimental observables. This approach may be useful for interpreting single-molecule or few-particle experiments on molecular motors, enzyme reactions, ion-channels, and phosphorylation-driven biological clocks. We consider cycles where all biochemical states are observable. Our method shows how: (1) the noise in cycles increases with cycle size and decreases with the driving force that spins the cycle and (2) provides a recipe for estimating small-number features, such as probability of backward spin in small cycles, from experimental data. The back-spin probability diminishes exponentially with the deviation from equilibrium. We believe this method may also be useful for other few-particle nonequilibrium biochemical reaction systems.

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.

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.

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.

Noise in biological circuits

Noise biology focuses on the sources, processing, and biological consequences of the inherent stochastic fluctuations in molecular transitions or interactions that control cellular behavior. These fluctuations are especially pronounced in small systems where the magnitudes of the fluctuations approach or exceed the mean value of the molecular population. Noise biology is an essential component of nanomedicine where the communication of information is across a boundary that separates small synthetic and biological systems that are bound by their size to reside in environments of large fluctuations. Here we review the fundamentals of the computational, analytical, and experimental approaches to noise biology. We review results that show that the competition between the benefits of low noise and those of low population has resulted in the evolution of genetic system architectures that produce an uneven distribution of stochasticity across the molecular components of cells and, in some cases, use noise to drive biological function. We review the exact and approximate approaches to gene circuit noise analysis and simulation, and review many of the key experimental results obtained using flow cytometry and time-lapse fluorescent microscopy. In addition, we consider the probative value of noise with a discussion of using measured noise properties to elucidate the structure and function of the underlying gene circuit. We conclude with a discussion of the frontiers of and significant future challenges for noise biology.

On the spontaneous stochastic dynamics of a single gene: complexity of the molecular interplay at the promoter

BACKGROUND

Gene promoters can be in various epigenetic states and undergo interactions with many molecules in a highly transient, probabilistic and combinatorial way, resulting in a complex global dynamics as observed experimentally. However, models of stochastic gene expression commonly consider promoter activity as a two-state on/off system. We consider here a model of single-gene stochastic expression that can represent arbitrary prokaryotic or eukaryotic promoters, based on the combinatorial interplay between molecules and epigenetic factors, including energy-dependent remodeling and enzymatic activities.

RESULTS

We show that, considering the mere molecular interplay at the promoter, a single-gene can demonstrate an elaborate spontaneous stochastic activity (eg. multi-periodic multi-relaxation dynamics), similar to what is known to occur at the gene-network level. Characterizing this generic model with indicators of dynamic and steady-state properties (including power spectra and distributions), we reveal the potential activity of any promoter and its influence on gene expression. In particular, we can reproduce, based on biologically relevant mechanisms, the strongly periodic patterns of promoter occupancy by transcription factors (TF) and chromatin remodeling as observed experimentally on eukaryotic promoters. Moreover, we link several of its characteristics to properties of the underlying biochemical system. The model can also be used to identify behaviors of interest (eg. stochasticity induced by high TF concentration) on minimal systems and to test their relevance in larger and more realistic systems. We finally show that TF concentrations can regulate many aspects of the stochastic activity with a considerable flexibility and complexity.

CONCLUSIONS

This tight promoter-mediated control of stochasticity may constitute a powerful asset for the cell. Remarkably, a strongly periodic activity that demonstrates a complex TF concentration-dependent control is obtained when molecular interactions have typical characteristics observed on eukaryotic promoters (high mobility, functional redundancy, many alternate states/pathways). We also show that this regime results in a direct and indirect energetic cost. Finally, this model can constitute a framework for unifying various experimental approaches. Collectively, our results show that a gene - the basic building block of complex regulatory networks - can itself demonstrate a significantly complex behavior.

Hybrid stochastic simplifications for multiscale gene networks

BACKGROUND

Stochastic simulation of gene networks by Markov processes has important applications in molecular biology. The complexity of exact simulation algorithms scales with the number of discrete jumps to be performed. Approximate schemes reduce the computational time by reducing the number of simulated discrete events. Also, answering important questions about the relation between network topology and intrinsic noise generation and propagation should be based on general mathematical results. These general results are difficult to obtain for exact models.

RESULTS

We propose a unified framework for hybrid simplifications of Markov models of multiscale stochastic gene networks dynamics. We discuss several possible hybrid simplifications, and provide algorithms to obtain them from pure jump processes. In hybrid simplifications, some components are discrete and evolve by jumps, while other components are continuous. Hybrid simplifications are obtained by partial Kramers-Moyal expansion [1-3] which is equivalent to the application of the central limit theorem to a sub-model. By averaging and variable aggregation we drastically reduce simulation time and eliminate non-critical reactions. Hybrid and averaged simplifications can be used for more effective simulation algorithms and for obtaining general design principles relating noise to topology and time scales. The simplified models reproduce with good accuracy the stochastic properties of the gene networks, including waiting times in intermittence phenomena, fluctuation amplitudes and stationary distributions. The methods are illustrated on several gene network examples.

CONCLUSION

Hybrid simplifications can be used for onion-like (multi-layered) approaches to multi-scale biochemical systems, in which various descriptions are used at various scales. Sets of discrete and continuous variables are treated with different methods and are coupled together in a physically justified approach.

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

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

Gene-gene cooperativity in small networks

We show how to construct a reduced description of interacting genes in noisy, small regulatory networks using coupled binary spin variables. Treating both the protein number and gene expression state variables stochastically and on equal footing, we propose a mapping that connects the molecular level description of networks to the binary representation. We construct a phase diagram indicating when genes can be considered to be independent and when the coupling between them cannot be neglected, which can lead to synchrony or correlations. We find that an appropriately mapped Boolean description reproduces the probabilities of gene expression states of the full stochastic system very well, and can be transferred to examples of self-regulatory systems with a larger number of gene copies.

Enhanced identification and exploitation of time scales for model reduction in stochastic chemical kinetics

Widely different time scales are common in systems of chemical reactions and can be exploited to obtain reduced models applicable to the time scales of interest. These reduced models enable more efficient computation and simplify analysis. A classic example is the irreversible enzymatic reaction, for which separation of time scales in a deterministic mass action kinetics model results in approximate rate laws for the slow dynamics, such as that of Michaelis-Menten. Recently, several methods have been developed for separation of slow and fast time scales in chemical master equation (CME) descriptions of stochastic chemical kinetics, yielding separate reduced CMEs for the slow variables and the fast variables. The paper begins by systematizing the preliminary step of identifying slow and fast variables in a chemical system from a specification of the slow and fast reactions in the system. The authors then present an enhanced time-scale-separation method that can extend the validity and improve the accuracy of existing methods by better accounting for slow reactions when equilibrating the fast subsystem. The resulting method is particularly accurate in systems such as enzymatic and protein interaction networks, where the rates of the slow reactions that modify the slow variables are not a function of the slow variables. The authors apply their methodology to the case of an irreversible enzymatic reaction and show that the resulting improvements in accuracy and validity are analogous to those obtained in the deterministic case by using the total quasi-steady-state approximation rather than the classical Michaelis-Menten. The other main contribution of this paper is to show how mass fluctuation kinetics models, which give approximate evolution equations for the means, variances, and covariances of the concentrations in a chemical system, can feed into time-scale-separation methods at a variety of stages.

Noise reduction by diffusional dissipation in a minimal quorum sensing motif

Cellular interactions are subject to random fluctuations (noise) in quantities of interacting molecules. Noise presents a major challenge for the robust function of natural and engineered cellular networks. Past studies have analyzed how noise is regulated at the intracellular level. Cell-cell communication, however, may provide a complementary strategy to achieve robust gene expression by enabling the coupling of a cell with its environment and other cells. To gain insight into this issue, we have examined noise regulation by quorum sensing (QS), a mechanism by which many bacteria communicate through production and sensing of small diffusible signals. Using a stochastic model, we analyze a minimal QS motif in Gram-negative bacteria. Our analysis shows that diffusion of the QS signal, together with fast turnover of its transcriptional regulator, attenuates low-frequency components of extrinsic noise. We term this unique mechanism "diffusional dissipation" to emphasize the importance of fast signal turnover (or dissipation) by diffusion. We further show that this noise attenuation is a property of a more generic regulatory motif, of which QS is an implementation. Our results suggest that, in a QS system, an unstable transcriptional regulator may be favored for regulating expression of costly proteins that generate public goods.

Regulation of signal duration and the statistical dynamics of kinase activation by scaffold proteins

Scaffolding proteins that direct the assembly of multiple kinases into a spatially localized signaling complex are often essential for the maintenance of an appropriate biological response. Although scaffolds are widely believed to have dramatic effects on the dynamics of signal propagation, the mechanisms that underlie these consequences are not well understood. Here, Monte Carlo simulations of a model kinase cascade are used to investigate how the temporal characteristics of signaling cascades can be influenced by the presence of scaffold proteins. Specifically, we examine the effects of spatially localizing kinase components on a scaffold on signaling dynamics. The simulations indicate that a major effect that scaffolds exert on the dynamics of cell signaling is to control how the activation of protein kinases is distributed over time. Scaffolds can influence the timing of kinase activation by allowing for kinases to become activated over a broad range of times, thus allowing for signaling at both early and late times. Scaffold concentrations that result in optimal signal amplitude also result in the broadest distributions of times over which kinases are activated. These calculations provide insights into one mechanism that describes how the duration of a signal can potentially be regulated in a scaffold mediated protein kinase cascade. Our results illustrate another complexity in the broad array of control properties that emerge from the physical effects of spatially localizing components of kinase cascades on scaffold proteins.

Signal transduction is the science of cellular communication. Cells detect signals from their environment and use them to make decisions such as whether or when to proliferate. Tight regulation of signal transduction is required for all healthy cells, and aberrant signaling leads to countless diseases such as cancer and diabetes. For example, in higher organisms such as mammals, signal transduction that leads to cell proliferation is often guided by a scaffold protein. Scaffolding proteins direct the assembly of multiple proteins involved in cell signaling by providing a platform for these proteins to carry out efficient signal transmission. Although scaffolds are widely believed to have dramatic effects on how signal transduction is carried out, the mechanisms that underlie these consequences are not well understood. Therefore, we used a computational approach that simulates the behavior of a model signal transduction module comprising a set of proteins in the presence of a scaffold. The simulations reveal mechanisms for how scaffolds can dynamically regulate the timing of cell signaling. Scaffolds allow for controlled levels of signal that are delivered inside the cell at appropriate times. Our findings support the possibility that these signaling dynamics regulated by scaffolds affect cell decision-making in many medically important intracellular processes.

Effects of protein maturation on the noise in gene expression

Fluorescent proteins are frequently used as reporters for gene expression in living cells, either by being expressed in tandem with a protein of interest or through the creation of fusion proteins. The data yielded by the fluorescence output are of considerable interest in efforts to formulate quantitative models of cellular behavior underway in fields such as systems biology and synthetic biology. An often neglected aspect of these proteins, however, is their maturation: Before a fluorescent protein can generate a fluorescent signal, it must mature through a series of steps (folding, cyclization, and oxidation) that may take from many minutes to over a day. The presence of these maturation steps creates a distinction between the observed gene expression profile and the actual profile. We examine this effect through a simplified gene expression model and conclude that fluorescent protein maturation can have significant effects on estimates of both the mean protein levels and the variability in gene expression. The model shows that in many regimes, the observed variability will be increased by the maturation process, but indicates the existence of regimes in which the observed variability will actually be less than the true variability of the target protein. The latter effect arises from a low-pass filtering effect introduced by the chain of maturation reactions. The results suggest that the maturation of fluorescent proteins should be taken into account when using such proteins as quantitative indicators of gene expression levels.