Genetic Redundancies Enhance Information Transfer in Noisy Regulatory Circuits

Guillermo Rodrigo; Juan F. Poyatos

doi:10.1371/journal.pcbi.1005156

Abstract

Cellular decision making is based on regulatory circuits that associate signal thresholds to specific physiological actions. This transmission of information is subjected to molecular noise what can decrease its fidelity. Here, we show instead how such intrinsic noise enhances information transfer in the presence of multiple circuit copies. The result is due to the contribution of noise to the generation of autonomous responses by each copy, which are altogether associated with a common decision. Moreover, factors that correlate the responses of the redundant units (extrinsic noise or regulatory cross-talk) contribute to reduce fidelity, while those that further uncouple them (heterogeneity within the copies) can lead to stronger information gain. Overall, our study emphasizes how the interplay of signal thresholding, redundancy, and noise influences the accuracy of cellular decision making. Understanding this interplay provides a basis to explain collective cell signaling mechanisms, and to engineer robust decisions with noisy genetic circuits.

Author Summary

There is increasing evidence that the presence of molecular noise greatly influences function in biological systems. This could imply, for instance, that genetic circuits adopt particular architectures in order to reduce noise. On the other hand, noise can be beneficial. Here, we show that this could be the case for the functioning of analog to digital genetic devices, which are commonly found in cellular decision making situations. We use the framework of information theory to illustrate first how noise can enhance information transfer in these devices. In those regimes in which noise is detrimental, we discuss how genetic redundancies allow information to be maximized, and how this effect depends on the specifics of the devices, and the interdependence among them. These results provide overall an additional rationale for genetic redundancies in genomic systems.

Citation: Rodrigo G, Poyatos JF (2016) Genetic Redundancies Enhance Information Transfer in Noisy Regulatory Circuits. PLoS Comput Biol 12(10): e1005156. https://doi.org/10.1371/journal.pcbi.1005156

Editor: Alexandre V. Morozov, Rutgers University, UNITED STATES

Received: April 4, 2016; Accepted: September 19, 2016; Published: October 14, 2016

Copyright: © 2016 Rodrigo, Poyatos. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting Information files.

Funding: This work has been supported by BFU2015-66894-P (MINECO/FEDER) and GV/2016/079 (GVA) Grants. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

The biochemistry of cells determines the operation of biological circuits. This biochemistry is inevitably noisy [1–3] what immediately suggests a limitation to the reliable function of these circuits, and thus many early studies examined how the problem of achieving correct operation could nevertheless be solved. Mechanisms such as kinetic proofreading [4], or integral feedback control [5] emerged then as some fundamental solutions. One might ask, on the other hand, to what extent noise could indirectly represent an advantage. An example is found when cell populations, in which noise leads to phenotypic variability, display heterogeneity in stress responses that represent a crucial element for survival [6].

In a more direct situation, noise can turn into a necessary ingredient to facilitate new classes of behaviors not achievable otherwise [7–11]. These valuable behaviors are typically related to cellular decisions, which essentially involve changes in the expression phenotype. Specific biological circuits were therefore shown to employ noise to induce the expression of transient phenotypes [8], or to switch among distinct stable states [9]. That many of these probabilistic dynamics relate to systems whose actions are susceptible to limiting signal values [12] emphasizes the connection between noise, cellular decisions, and threshold response circuits.

The beneficial aspect of noise also forces us to revisit some of the original arguments on the relationship between stochasticity and the structure of biological systems [13, 14]. In particular, the existence of genetic redundancies was frequently interpreted as a mean to enhance reliability of operation (i.e., noise as a disruptive element). This role appeared in consequence as a plausible rationale for the evolutionary maintenance of several copies of a gene or circuit [15]. Instead, we focus here on redundancy as a genetic architecture that, when coupled to the effect of noise in threshold response circuits, enables unique information-processing functions.

We examined this issue within the precise framework of information theory. Biological circuits are in this way interpreted as communication channels, in which an input signal (x) originates –as a result of a cellular decision– an expression output (y), with a given probability (Fig 1A). The uncertainty on the input signal is then reduced by the decision process, whose set of outcomes tells us about the input distribution [16, 17]. An association that is properly quantified by the mutual information (MI), an information-theoretic measure describing the dependence between the input signal and the output phenotype (no matter how they could correlate [18], Fig 1A). Notably, this framework was recently exploited to quantify the functionality of transcriptional regulatory elements [19–21], the accuracy of cell location during developmental processes [22], and the maximal information transmission capacity of noisy signaling pathways [23, 24]. The relevance of redundancies was already manifested in some of these results.

Download:

Fig 1. Intrinsic noise can increase or decrease information transfer in threshold genetic systems.

(A) Any transmission of signals (x) that could lead to an error in the output (y) can be framed as a noisy information channel. The mutual information (MI) quantifies the dependence between input and output distributions, P(x) and P(y), respectively, and can discriminate some associations not detected by the correlation coefficient. For instance, the cartoon illustrates two cases with the same correlation (represented by the eccentricity of the ellipses) but different MI. (B) The channel can describe a gene autoactivating its own expression (y) in a bistable OFF/ON manner, i.e., a simple example of threshold regulatory circuit. Information transfer depends on the relationship between x and the threshold value of activation (x and y in arbitrary units). Three instances of P(x) are shown (uniform distributions with different means; the blue one corresponds to a mean equal to the threshold value). When the signal is always beyond the threshold (red distribution) the circuit exhibit a nonzero MI only when it works stochastically (note the output distributions). Here we considered a binary response (OFF if y < 1, ON otherwise). (C) Resonance in MI as a function of intrinsic noise for the red P(x) in (B). MI values computed with the responses of the device to 10⁴ different signals drawn from the described distribution. Responses are shown explicitly for three noise strengths (inset figures, black dots), together with its averaged stimulus-response profile (red curve). The maximum in MI occurs when the averaged stimulus-response profile is more linear (S1 Fig). (D) Other signal distributions, in which intrinsic noise always reduces MI, can nevertheless exhibit a resonance when the combined response of several units is considered (we show here the case of duplicated threshold devices; N = 2). Colors correspond to those distributions shown in (B). See Materials and Methods for details.

https://doi.org/10.1371/journal.pcbi.1005156.g001

Here, we first illustrate how the stochasticity of biochemical reactions (intrinsic noise) can help to gain information. We then show that information transfer can be amplified, if the combined response of multiple genetic units is considered. The reported amplification is shown to rely on the presence of different factors that contribute to generate variability in the individual response of each unit, like intrinsic noise or genetic heterogeneity (i.e., differences in the biochemical properties). This variability helps to enlarge the capacity of the global output to represent the input distribution. In contrast, we also discuss how factors reducing variability in the responses, like a noise source common to all units (extrinsic noise) or regulatory cross-talk, eventually mitigate the gain.

Results

Intrinsic noise can amplify information transfer

We first analyzed a minimal regulatory circuit implemented by a gene (whose expression we denote as y) autoactivating transcriptionally its own production [25] (Materials and Methods). This is a genetic implementation of a threshold device that, by acting deterministically, becomes activated only if the input signal x crosses a particular limit (Fig 1B). When the signal is stochastic, the response depends of course on the relationship between this threshold and the mean (and variance) of the underlying distribution P(x) (considered for simplicity as a uniform distribution; Fig 1B). A symmetric distribution centered on the threshold would thus originate equally likely the two output values (OFF/ON) (i.e., one bit of information); while the same distribution centered above/below the threshold would produce biased responses (i.e., less than one bit of information).

However, the previous behavior can be affected by the extensive noise sources acting on biological circuits [1–3]. One could ask then to what extent the circuit is reliably representing the signal. With this goal, we computed the response (black dots in subpanels of Fig 1C) to a number of signals drawn from a fixed distribution (red distribution in Fig 1B) and strength of intrinsic noise. To quantify how much information the response conveys about the input, we made use of MI [16–18] (Fig 1A) (Materials and Methods). Values of MI change with noise [main plot in Fig 1C; the mean of P(x) is above the threshold]. For weak noise levels, the circuit works essentially as a deterministic switch, it is always y = ON as x > threshold. For strong noise levels, the device cannot distinguish signal fluctuations, then its behavior is essentially random. In both cases, the gene is processing a limited amount of information (subpanels of Fig 1C, red curves denote the averaged stimulus-response profiles). But the transmission of information presents a maximum for an intermediate noise level. In this regime, the circuit can express its two possible states due to noise (i.e., low values of x can cross the threshold) [26], what precisely contributes to a better representation of the input signal (see also S1 Fig); a characteristic behavior of noisy nonlinear systems known as stochastic resonance (SR) [27, 28].

Moreover, SR disappears when the mean of P(x) is close to the threshold, as stochasticity is now not required to reach the two possible states. Noise always reduces information transfer (Fig 1D, curves for N = 1, N denoting the number of circuits involved). Note here how MI does exhibit an upper limit of one bit when the mean of P(x) exactly matches the threshold, and the circuit is noiseless. MI decreases with noise because signal values above/below the threshold originate in some cases stochastic crossings (e.g., y = ON when x < threshold), and the information content in absence of noise is already high (note in contrast that, in the scenario of SR, MI was very low in absence of noise). Additionally, Fig 1D displays a situation in which a maximum in MI is nevertheless observed (curves for N = 2). This is obtained by increasing the number of devices processing the same input, with y representing the sum of all individual outputs; a phenomenon called suprathreshold SR [29] (see S1 Text, and S2 Fig, for a brief account of suprathreshold SR). Note how the maximum is observed in the noise regime where the information processing of individual units (N = 1) qualitatively declines (from here the presence of several units begins to lose effectiveness). What is apparent is that redundancy boosts information transfer, given a fixed noise level.

Genetic redundancy enhances information transfer

The addition of extra copies of the threshold device, i.e., genetic redundancy, appears then as a potential mechanism to increase the transmission of information in the presence of intrinsic noise. Consider, for instance, a situation in which two devices read in parallel the same input signal, assuming again two possible values of gene expression for each unit. The overall output alphabet [30] consists of three letters: {0 (both copies OFF), 1 (one OFF the other ON), 2 (both ON)}. The new alphabet is linked, of course, to the action of independent (intrinsic) noise sources acting on the two genes, which allows each device to produce an autonomous response (with noise-induced threshold crossings, S2 Fig). The sum of individual responses would give, accordingly, a global output distribution P(y) constituted by three peaks. The extended alphabet helps therefore to enlarge the capacity of the output to represent the input variability; in other words, it contributes to linearize the averaged stimulus-response profile (Fig 2A, see also S3 Fig).

Download:

Fig 2. Genetic redundancy amplifies information transfer in threshold genetic systems.

(A) Input/output distributions depicting information transfer. The input distribution (in yellow) is assumed to be uniform. Output distributions (in gray) illustrate the processing of the signal x, either through a single copy of the threshold device (left) or an array of multiple redundant copies (right). In the latter case, each unit of the array receives the same signal and the output y is the sum of all the individual responses (S2 Fig). Redundancy enlarges the alphabet of the response. This is reflected in the output distribution, and also in the linearization of the averaged stimulus-response profile (black curve). (B) (Left) Array of N threshold devices (circles) whose constituent units correspond to (1) a simple regulated unit, (2) a bistable circuit implemented with a positive feedback, and (3) an excitable circuit constituted by two interlinked positive and negative feedback loops. (Right) Dependence of mutual information (MI) with the number of units (N) for each of these systems relative to the case N = 1. A uniform signal distribution with mean equal to the threshold value was considered. MI does not increase with extra copies for noiseless units (independently of the type of unit; dashed line). See Materials and Methods for details on the modeling of each circuit.

https://doi.org/10.1371/journal.pcbi.1005156.g002

Both the number of units and the type of nonlinearity influence the increment of information transfer. In Fig 2B, we introduced three different threshold devices [25] to show how MI increases with redundancy. For each type, MI relative to the case of no redundancy (i.e., a single unit) was plotted. Specifically, we examined a simple regulated unit, a bistable expression system implemented through a positive feedback, and an excitable device constituted by interlinked positive and negative feedbacks (implemented as the one linked to transient differentiation in Bacillus subtilis [8]) (Materials and Methods). The output of all these devices is given by a continuous variable representing gene expression (note that the response of the bistable unit was regarded as OFF/ON in the previous section, Materials and Methods). This allowed identifying discrepancies in terms of MI among different gene regulatory circuits. In particular, the largest amplification of information content corresponds to those devices whose actions ultimately rely on discontinuous transitions (i.e., the bistable and excitable systems). Out of these two systems, the excitable one presents comparatively larger amplification, although only observed for relatively large arrays. In this system, the response is entirely binary even in presence of noise: either the signal triggers a response or not (S4 Fig). Moreover, the gain in information transfer is much lower for the simple regulated system in which the response profile is continuous (what entails that one unit already has the capacity to reach a relatively large output alphabet). The contribution of redundancy is therefore always much higher in analog-to-digital than in analog-to-analog signaling circuits, and provided they are noisy.

Input signal distribution shapes information transfer

We next studied how the specific distribution of the signal impinging on the genetic circuits (that can encode distinct environmental or genetic conditions [31]) can further modulate the enhancement of information transfer. Knowing this distribution is typically difficult in cellular systems [24]. We examined then to what extent information transfer would be influenced by the shape of P(x). We considered three different signals acting on the array of threshold devices (Materials and Methods). MI increases more strongly with a normally distributed signal (Fig 3A). For this distribution, the mass of x values is closer to the threshold what allows noise to alter more frequently the expected deterministic response of the device. We then analyzed the effect of the relationship between the threshold and the signal mean. When the mean of P(x) is equal to the threshold, a higher increase of MI with genetic redundancy is observed (Fig 3B). Arguably, if the mass of x values is equally distributed above/below the threshold, there exists again more chances for noise-induced threshold crossings. Fine-tuning of the parameters characterizing P(x) contributes thus to a better representation of the input signal by the global output response.

Download:

Fig 3. The distribution of the signal modulates the increase of information transfer due to genetic redundancy.

(A) Effect of the form of the distribution on MI: (1) uniform (covering two orders of magnitude), (2) lognormal (with standard deviation equal to 2/3), and (3) beta in log scale (with the two shape parameters equal to 1/3). In all cases, the mean of the distribution is equal to the threshold value. (B) Effect of the mean of the distribution (here uniform) on MI: (1) equal to the threshold value, (2) and (3) deviated from the threshold value. We considered as threshold device a bistable unit implemented with a positive feedback in all plots (Materials and Methods).

https://doi.org/10.1371/journal.pcbi.1005156.g003

Extrinsic noise and cross-talk limit information transfer

The most important constraint for the gain in information associated to the previous redundant systems is the independence between the noise sources. When these are correlated, P(y) becomes more sharply peaked around a small subset of possible responses (i.e., the output alphabet is more limited; Fig 4A). This applies to biological circuits that, in addition to intrinsic noise, also integrate the effect of extrinsic fluctuations [2, 32]. Extrinsic noise affects all genetic devices in the same manner what eventually correlates individual outputs. For instance, Fig 4B shows how (relative) MI decreases with the strength of extrinsic noise in an array of five bistable units (Materials and Methods). Note however that this redundant architecture still exhibits, for different extrinsic noise levels, a larger MI with respect to the nonredundant case (inset of Fig 4B).

Download:

Fig 4. Extrinsic noise and cross-talk among redundant copies limit information transfer.

(A) Input/output distributions depicting information transfer. Correlation among individual gene responses due to extrinsic noise or cross-talk reduces the response alphabet, and generates a less linear averaged stimulus-response profile (black curve, see Fig 2A for comparison). (B) Dependence of mutual information (MI) with the strength of extrinsic noise (Materials and Methods). Relative MI is with respect to absence of extrinsic noise. For this plot, we considered a system of N = 5 bistable units implemented with positive feedback. The inset shows a direct comparison between N = 1 and N = 5, emphasizing that MI increases with N. (C) Dependence of MI with the degree of cross-talk for the same regulatory system, but now constituted by N = 2 units. Relative MI is with respect to the situation without cross-talk. The inset presents the marginal probability distribution of gene expression of one unit (y₁) in the absence and presence of cross-talk (parameterized by ε = 0 and ε = 0.01, respectively; see Materials and Methods) for the mean value of the input signal (x). Note that when the units are coupled, gene expression becomes unimodal (dashed curve).

https://doi.org/10.1371/journal.pcbi.1005156.g004

Despite the independence of the noise sources, cross-talks between devices can similarly lead to correlations in the individual responses. In a genetic context, one could imagine two independent transcription factors sharing recognition domains [33]. One could also conceive a second unit recently emerged by duplication, and that no process of neofunctionalization yet occurred [34]. Fig 4C indeed shows a decay in (relative) MI for a system of two bistable units when cross-talk between them increases (simulations done without accounting for extrinsic noise, Materials and Methods, see also S5 Fig for a study of asymmetric cross-talk). In this case, the activation of one unit drags the activation of the other, biasing again the output alphabet (inset of Fig 4C). Of note, the decay profile in MI is qualitatively different in the two scenarios. Addition of extrinsic noise contributes to limit information transfer in a progressive manner since it increasingly coordinates responses. In the second situation, outputs are correlated once a certain cross-talk range is reached what is reflected in a sharper decay (similar patterns are expected to be observed when considering other constituent units).

Heterogeneity also contributes to increase information transfer

We proceeded by examining a complementary source of individuality in information processing, which is linked to the heterogeneity within the collection of threshold devices. In the context of genetic circuits, variability in promoter strengths, ribosome-binding sites, proteins half-lives, or protein-DNA binding affinities are all factors that in effect modify threshold values or output responses. Adjusting for each device the values of the biochemical parameters of the model can capture this variation [35]. We specifically explored the implication of threshold heterogeneity in the array of five bistable units in which the threshold values are drawn from a Gaussian distribution (and heterogeneity equates to the associated standard deviation, Materials and Methods).

Notably, we observed again a resonance in information transfer, but this time as a function of the degree of heterogeneity (Fig 5). While moderate levels of heterogeneity allows regulatory circuits to encode complementary aspects of the input signal, hence enhancing information transfer, greater variation becomes detrimental by originating noise-induced threshold crossings over the whole input range. Moreover, since both intrinsic noise and heterogeneity contribute to increase the transmission of information (certainly by enlarging the alphabet of the global output), we also explored to what extent these two sources of individuality act independently [36]. We found that intrinsic noise mitigates the increase in MI due to threshold variability (inset of Fig 5). Intuitively, higher noise levels make indistinguishable those variations associated to the control of gene expression in the threshold unit.

Download:

Fig 5. Genetic heterogeneity among redundant copies amplifies information transfer.

Variation in the biochemical features of the constituent threshold devices (represented by the different colors of the devices, here bistable units, N = 5) leads to a maximum in mutual information (MI). We consider random threshold values drawn from a Gaussian distribution whose standard deviation determines the degree of heterogeneity (see Materials and Methods, no heterogeneity corresponds to a very low, i.e., 10⁻⁴, but nonzero value due the log scale of the x axis). The inset indicates the peak differential MI (i.e., the difference between the largest value of MI with heterogeneity and the value of MI without it) for varying noise levels. This reveals how a situation of stronger intrinsic noise contributes to reduce the improving effect on MI of heterogeneous units (the main plot corresponds to an intrinsic noise amplitude equal to 0.16).

https://doi.org/10.1371/journal.pcbi.1005156.g005

Empirical evidences of the theory

Several analyses support the gain in information transfer established by our theoretical framework. For instance, by explicitly quantifying information transduction, new experimental reports showed how multiple copies of a biochemical system accessing the same signal leads to an increase in MI [23, 24]. This was coupled to multiple gene copies (comparing 1× and 2× diploids [24]) or to a general redundant architecture (with a bush network topology) that could be implemented in an intra- or intercellular manner [23].

Moreover, recent work examined how the output of a LuxR-inducible promoter in Escherichia coli would change with copy number ranging from 1 copy per cell (genome copy) to hundreds (via multicopy plasmids) [37]. We used the available data to confirm the relative MI gain as a function of genetic redundancy (Fig 6A, Materials and Methods). Because the system does not rely on a discontinuous transition (e.g., it is not bistable), the gain saturates with few copies (see the curve labelled as “1” in Fig 2B). We contrasted this analysis with the response to progesterone of a single bistable unit governing the oocyte maturation in Xenopus [38]. The noise-driven differential concentration from oocyte to oocyte (for a particular progesterone level) of the controlling protein makes unreliable the maturation process (i.e., subthreshold levels of progesterone can induce maturation in some oocytes) what limits the amount of information transmission one could achieve in a noiseless situation (Fig 6B). Arguably, the presence of additional copies of the precursor gene would help to reduce such variability and then mitigate the loss of information (see Fig 1D).

Download:

Fig 6. Experimental evidence of the role of genetic redundancies for the transmission of information.

(A) (Top) Regulatory scheme of a synthetic experimental system taken to assess the effect of genetic redundancy on information transfer. An array of N RFPs regulated by LuxR-AHL, implemented through chromosomal integration (N = 1) or plasmids with different copy numbers (N > 1). (Bottom) Dependence of mutual information (MI) with the number of units (N) relative to the system with N = 1 (Materials and Methods). (B) (Left) Regulatory scheme associated to the induction of oocyte maturation in a bistable manner (mediated by GSK3β) by progesterone. (Right) Effect of molecular noise on information transfer. MI for this intrinsically stochastic threshold unit relative to the deterministic case (Materials and Methods). (C) Three examples of genetic regulatory architectures in yeast where redundant genes are coregulated by a common signal. They correspond to pheromone, vitamin B1 and glucose signaling, respectively.

https://doi.org/10.1371/journal.pcbi.1005156.g006

Finally, Fig 6C illustrates other biological contexts in which the signaling architectures, by exhibiting redundancies, could be more effective in terms of information transfer following our predictions [39–41].

Discussion

Binary decisions implemented by means of threshold devices appear in many engineering and physical systems, and have been extensively studied in relation to the detection and transmission of signals. While noise was commonly considered harmful in many of these settings, some work alternatively identified circumstances in which its presence improves performance [27, 28, 42]. In Biology, both the stochastic nature of biochemical reactions and the typical occurrence of thresholds –linked, for instance, to cell fate determination– also anticipates the possibility of beneficial effects. This specifically applies to the case of gene regulatory circuits, in which molecular stochasticity acts in many cases as a core determinant of function [43].

In this work we discussed in detail the benefits of intrinsic molecular noise when multiple threshold regulatory circuits process a common signal. This system exhibits a resonance phenomenon known as suprathreshold SR [29] (S1 Text). The effect establishes the value of the noise-induced uncoupling of the action of each unit. This advantage is manifested as well in a more linear relation between stimulus and response, a type of dose-response alignment that could be important in how precise extracellular conditions determine cell responses, and that was previously associated to negative feedbacks [21]. Our functional analysis therefore reveals redundancies not only as a genetic architecture contributing to robustness [44, 45], or to the adaptation to novel environments through the increase of gene expression levels [46, 47], but also as a mechanism increasing the capacity to transmit reliable information (Fig 7). We suggest that this aspect could contribute to the evolutionary maintenance of genetic redundancy (tradeoffs with the associated genetic load of redundancy could matter, see S6 Fig). That multiple signaling pathways in Saccharomyces cerevisiae overlap supports this hypothesis [48].

Download:

Fig 7. Model of information transfer in gene regulatory circuits.

Intrinsic noise, genetic redundancy, and heterogeneity increase the transmission of information by expanding the capacity of the (summing) global output to represent the input variability. In contrast, extrinsic noise and cross-talk among redundant units become limiting factors by correlating the individual outputs of the units. Threshold genetic units represented here as gray circles; input signal x and output response y characterized by P(x) and P(y) distributions, respectively.

https://doi.org/10.1371/journal.pcbi.1005156.g007

The balance of intrinsic/extrinsic noise also plays an important part to condition the amount of information transferred (Fig 7). Cells implementing regulatory circuits with few representative molecules or living in rich environments would shift this balance towards intrinsic noise [49]. Beyond this genetic/environmental tuning, cellular systems could avoid the loss of information, due to extrinsic noise, when the signal operates dynamically rather than statically [50]. Note that here we considered a static operation. Our results further emphasize how heterogeneity and cross-talk among redundant copies play opposite roles in the maintenance of information content (Fig 7). One could thus interpret the action of several parallel signaling pathways, each conveying approximately one bit of information, as heterogeneous copies of an effective threshold device what enhances information transmission, e.g., this was observed in pathways for the growth factor-mediated gene expression [51].

That a global response –the sum of individual responses, in this case– implemented by parallel processing units could lead to better performance than that of the individual components was proposed in early models of computing, and can indeed be observed at different levels of biological organization: from genes (this work), to living cells [23], to social organisms [52]. In addition, ideas on redundancy and heterogeneity when mounting unreliable components were already present in the initial development of fault-tolerant computation and communication [53, 54], and also permeate to many biological scenarios. Our work substantiates the implications of these notions in cellular decision making by natural [48] and synthetic [55] molecular circuits, and contributes to exemplify how the application of concepts from information theory could lead to a more precise and quantitative understanding of cellular systems.

Materials and Methods

Modeling noisy regulatory systems

As a general regulatory model we examined a redundant system consisting of N different units, each of them activated by the input signal (x). These units correspond to three types, defined by three specific sets of ordinary differential equations that were extended to account for stochastic effects using the Langevin approach [56].

Simple regulated system.

The model for the i-th unit of this type is (1) where expression, and time, are appropriately rescaled to have a dimensionless model. Here, α₀ and α correspond to the basal and maximal expression level, and n governs the steepness of the function. The parameter values used are α₀ = 0.01, α = 2.5, and n = 2. Intrinsic noise is described by an stochastic process ξ_i with 〈ξ_i(t)〉 = 0 and 〈ξ_i(t₀)ξ_i(t₀ + t)〉 = δ(t). Noise amplitude is given by , i.e., the square root of the sum of the propensities [56]. K is proportional to the effective dissociation constant between the transcription factor and the promoter, and determines the number of molecules of the system and then intrinsic noise [57] (K = 100, otherwise specified).

Bistable system.

We considered a gene activating transcriptionally its own expression. This corresponds to a minimal implementation of a bistable system. The model for the i-th unit reads (2) where expression and time are again appropriately rescaled to have a dimensionless model. Parameters definitions and values as before, as well as the statistics of ξ_i. The input signal (x) is introduced in this case as a small perturbation. Noise amplitude is given by , having neglected the effect of x.

Excitable system.

We used a model previously proposed to explain competence in Bacillus subtilis, associated with the capability for DNA uptake from the environment, by which the cell can reach transient (excitable) differentiation [58]. Each of the N units of the system consists of two transcriptional units (y_i and z_i) that implement, in an effective way, interlinked positive and negative feedback loops. The model for the i-th unit reads (3) where expression and time are appropriately rescaled to have a dimensionless model. The parameter values are α₀ = 0.004, α = 0.07, β = 0.826, σ = 5, n = 2, and m = 5 (β and m correspond to maximal expression level and steepness, respectively, while σ describes a ratio of dissociation constants). Here, noise amplitude is constant (, with K = 500), and ξ_i follows the same statistics as before.

Extrinsic noise, cross-talk and heterogeneity

We introduced an additional stochastic process ξ_ex, to account for extrinsic noise common to all units. The correlation time of extrinsic noise is of the order of the cell cycle (the mean is also 0). For simplicity, we assumed systems implemented with short-lived proteins, so that ξ_ex is constant within the time window required for the dynamical unit to reach steady state after reading the signal x (this feature also reduces potential expression dependences on growth rates, e.g., [59]). To examine cross-talk we applied a perturbative approach, with a perturbative parameter ε quantifying the degree of cross-talk. Finally, to study heterogeneity we specifically considered variability in the threshold values of the different units. We modified these values by introducing a Gaussian random number ω (of mean 1), with its standard deviation corresponding to the degree of heterogeneity. Note that only intrinsic noise was considered when accounting for cross-talk or heterogeneity. See full details of these methods in the S1 Text.

Input and output variables

We considered that the threshold regulatory system is initially in a steady state (x = 0) before becoming activated (x ≠ 0 at time t = 0). The signal represents a continuous stimulus with fixed amplitude (x is a step function at t = 0), for the simple and bistable units, or a pulse (for one unit of normalized time) for the excitable one. The amplitude of the signal is given by x = 〈x〉10^u, where u corresponds to a random number uniformly distributed in [−1, +1], unless otherwise specified. Signal stochasticity illustrates fluctuations due to upstream processes, environmental changes or molecular noise. We considered logx as input variable to compute information transfer.

Each threshold unit is able to sense the signal what could alter its expression level as Δy_i = y_i(x) − y_i(x = 0). The output was calculated at steady state, and signal fluctuations occur at a frequency that allows the genetic circuit to respond against the current signal value. In addition, the total differential gene expression of a redundant system can be written as . Since the response of the excitable system is transient, we implemented a Boolean function operating on y_i, setting 1 if the unit was excited or 0 if not. For all main figures, we always treated the threshold units as dynamical systems, i.e., modeled by differential equations. However, in the first section of the paper (Fig 1), the gene expression level (y_i) was treated as a Boolean variable (OFF/ON) after resolving the corresponding differential equation. Expression was treated as a continuous variable in the subsequent sections (Figs 2–5).

Input distributions

We mainly included a uniform distribution P(x) covering two orders of magnitude throughout the manuscript (as described above). However, in Figs 1 and 3B, we analyzed the effect of the mean 〈x〉 of the distribution, with values 0.001 (equal to the threshold value), 0.005 and 0.01. In Fig 2, the mean was fixed to the threshold value, i.e., 〈x〉 = 1 in the simple regulated unit, 〈x〉 = 0.001 in the bistable system, and 〈x〉 = 0.9 in the excitable system. In Figs 4 and 5, concerning to the bistable system, 〈x〉 = 0.005. Moreover, in Fig 3A, we studied the effects of a normal or a beta distribution in log scale, with the mean equal to the threshold value.

Quantification of information transfer

We used mutual information () as a quantitative metric to describe how the global output response of a single cell is sensitive to different concentrations of the input signal [18]. This adds to the quantification by the averaged stimulus-response profile. To calculate , we performed 10⁴ realizations of the pair (x, y) and solved numerically the following integral (4) where we considered logx as input and Δy as output variables. By using the Fokker-Planck equation, we calculated the probability that a unit has a given gene expression level (see more details in S1 Text).

Analysis of experimental data

We considered the dose-response data of a synthetic system composed by a red fluorescent protein (RFP) controlled by the transcription factor LuxR, which is activated by N-acyl homoserine lactone (AHL, the signal) [37]. Indeed, this is a simple regulated unit, which was implemented with different gene copy numbers. Mutual information was calculated between the RFP expression at the population level and the concentration of AHL in log scale (an estimation of the actual values). In addition, we considered the dose-response data of a natural system governing the oocyte maturation in Xenopus [38]. Here, the glycogen synthase kinase 3β (GSK3β) controls the meiotic entry of progesterone (the signal) in the oocytes. This system is bistable and is implemented by an effective positive feedback loop (through two negative regulations). Mutual information was calculated between the phosphorylation state of GSK3β of individual oocytes (considered as a Boolean variable) and the concentration of progesterone in log scale. As a reference, we considered a deterministic scenario with a distribution of progesterone centered in the threshold of the system.

Supporting Information

S1 Text. A detailed description of all the methods used in this study.

https://doi.org/10.1371/journal.pcbi.1005156.s001

(PDF)

S1 Fig. Distance of the average response to the ideal linear response as a function of intrinsic noise.

We computed the change in gene expression Δy (with respect to the initial steady state of the system) caused by the presence of the signal (x) for the bistable unit (N = 1). The distribution of the input signal values is a uniform with a mean of 0.005 and a variance that allows covering two orders of magnitude. Panels correspond to the responses of the device to 10⁴ signal values drawn from the described distribution (black dots) and different intrinsic noise amplitudes. Average response (red curve, mean over all possible responses) and ideal linear response (blue curve) are also shown. The main plot displays the distance between the average and the linear response, which exhibits a minimum for certain amount of noise.

https://doi.org/10.1371/journal.pcbi.1005156.s002

(TIF)

S2 Fig. Suprathreshold stochastic resonance in multilevel threshold systems.

(A) A signal x acting on a summing array of threshold units i (threshold value θ_i), each one experiencing independent noise ξ_i. The output y ranges from 0 to N (the total number of units). (B) Average transmitted information through the array in (A) quantified in terms of the mutual information (). Here, 10⁴ signal values, drawn from a Gaussian distribution with mean 〈x〉 = 1 and σ_x = 1, passed through the threshold device (Heaviside function). All thresholds are set equal to the signal mean, and noise strength is determined by σ_i (of a Gaussian distribution with 〈ξ_i〉 = 0). In this situation, the signal is strongly suprathreshold, yet noise does induce a maximum (stochastic resonance) for all N > 1. (C) Quantification of information transmission using the correlation coefficient cannot distinguish between two situations with low (r₁) and high (r₂) noise strength as compared to the mutual information ( and , respectively). Here, we considered an array of N = 5 threshold units. (D) Response (blue) and averaged response (red) in the regimes of low and high noise in (C); values beyond the threshold highlighted in yellow. While the linear dependence between x and y is relatively similar, the information obtained about x when measuring y is not. This implies similar correlation coefficients but different mutual information. The use of to quantify the amount of information transmission was originally discussed in [29].

https://doi.org/10.1371/journal.pcbi.1005156.s003

(TIF)

S3 Fig. Effect of feedback strength or intrinsic noise on information transmission.

(A) Dependence of mutual information with the feedback strength (or also intrinsic noise amplitude, parameterized by 1/K) of the bistable unit. The inset shows the effective stochastic potential (ϕ) for K = 100 and K = 1000 (with N = 2). Certainly, it shows two potential wells (i.e., two stable steady states), and the threshold of the system is within (note that intrinsic noise is multiplicative, i.e., the amplitude of the stochastic fluctuations depends on the particular gene expression level). In case of higher noise (K = 100), the potential barrier is very low, indicating that it is very easy to have stochastic threshold crossings. However, in case of lower noise (K = 1000), the potential barrier is high, moderating the number of stochastic threshold crossings. The observed trend of information transfer versus intrinsic noise (higher the noise, higher the information transfer) is explained because the stochastic threshold crossings (for a continuous output variable, and to some extent) is the mechanism underlying the linearization of the response and the increase of communication fidelity. (B) Dependence of mutual information with the number of units (N) of the system. Relative mutual information is with respect to N = 1 for different intrinsic noise levels (modulated by K). This shows how the higher the intrinsic noise level, the stronger the amplification of information transfer due to genetic redundancy.

https://doi.org/10.1371/journal.pcbi.1005156.s004

(TIF)

S4 Fig. Influence of the number of units for an excitable system.

(Top) Input/output distributions depicting information transfer. The input distribution (in yellow) is assumed to be uniform (with a mean of 0.9 and a variance that allows covering two orders of magnitude). Output distributions (in gray) illustrate the processing of the signal x, either through a single copy of the threshold device (left) or an array of multiple redundant copies (right). Note that the output is Boolean, setting if the unit is excited or 0 if not. In the latter case, each unit of the array receives the same signal and the output y is the sum of all the individual responses. Redundancy effectively enlarges the alphabet of the response. This is reflected in the output distribution, and also in the linearization of the averaged stimulus-response profile (black curve). (Bottom) Relative mutual information, with respect to N = 1, for a system of N excitable units implemented with interlinked positive and negative feedbacks. According to the same input level, and in the presence of molecular noise, each unit can perform an excursion over the phase space (excitation) or not. Inset shows the phase space where the nullclines (black lines) determine the possible dynamical trajectories (gray lines, deterministic regime). Arrows indicate direction of the dynamics, and the black dot corresponds to a stable steady state. Note that a perturbation (in gene z) induces the excitation of the system provided its magnitude (x) is large enough, otherwise the system falls down to the steady state as it remains within the corresponding basin of attraction.

https://doi.org/10.1371/journal.pcbi.1005156.s005

(TIF)

S5 Fig. Dependence of mutual information with the average degree of cross-talk for the bistable system constituted by 2 units.

We considered a scenario of asymmetric cross-talk (parameterized by ε), i.e., gene 1 is affected by gene 2 with strength ε₂ and gene 2 by gene 1 with strength ε₁, with ε = (ε₁ + ε₂)/2. Note that the average amount of cross-talk is maintained with respect to a scenario of symmetric cross-talk. Three situations are represented (black, ε₁ = ε₂ = ε, symmetric; blue, ε₁ = 1.5ε and ε₂ = 0.5ε; and red, ε₁ = 2ε and ε₂ = 0), all displaying a similar trend.

https://doi.org/10.1371/journal.pcbi.1005156.s006

(TIF)

S6 Fig. Tradeoff between information transfer and genetic load for a bistable genetic system with redundancy.

Simulations with N ranging from 1 to 10 (genetic load). For simplicity, we considered a bacterium of 5000 genes. According to previous work characterizing genetic load in bacteria [60], we can state that the relative growth rate of the cell is 1 − ϕ/0.48, where ϕ is the relative amount of unnecessary protein. In our case, ϕ = S(N − 1)/5000, where S is the number of independent systems that are implemented with redundancy. In case of a specific strategy by one system (S = 1), the genetic load is negligible. However, if redundant systems were more generally observed (e.g., S = 100), the genetic load would be considerable what leads to a Pareto front.

https://doi.org/10.1371/journal.pcbi.1005156.s007

(TIF)

Acknowledgments

We thank M.A. Fares and A. Couce for a critical reading of the manuscript.

Author Contributions

Conceived and designed the experiments: GR JFP.
Performed the experiments: GR JFP.
Analyzed the data: GR JFP.
Contributed reagents/materials/analysis tools: GR JFP.
Wrote the paper: GR JFP.

References

1. McAdams HH, Arkin A. Stochastic mechanisms in gene expression. Proc Natl Acad Sci USA. 1997;94:814–819. pmid:9023339
- View Article
- PubMed/NCBI
- Google Scholar
2. Elowitz MB, Levine AJ, Siggia ED, Swain PS. Stochastic gene expression in a single cell. Science. 2002;297:1183–1186. pmid:12183631
- View Article
- PubMed/NCBI
- Google Scholar
3. Lestas I, Vinnicombe G, Paulsson J. Fundamental limits on the suppression of molecular fluctuations. Nature. 2010;467:174–178. pmid:20829788
- View Article
- PubMed/NCBI
- Google Scholar
4. Hopfield JJ. Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity. Proc Natl Acad Sci USA. 1974;71:4135–4139. pmid:4530290
- View Article
- PubMed/NCBI
- Google Scholar
5. Yi TM, Huang Y, Simon MI, Doyle J. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc Natl Acad Sci USA. 2000;97:4649–4653. pmid:10781070
- View Article
- PubMed/NCBI
- Google Scholar
6. Bishop AL, Ran FA, Sumner ER, Avery SV. Phenotypic heterogeneity can enhance rare-cell survival in ‘stress-sensitive’ yeast populations. Mol Microbiol. 2007;63:507–520. pmid:17176259
- View Article
- PubMed/NCBI
- Google Scholar
7. Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S. Bacterial persistence as a phenotypic switch. Science. 2004;305:1622–1625. pmid:15308767
- View Article
- PubMed/NCBI
- Google Scholar
8. Süel GM, Garcia-Ojalvo J, Liberman LM, Elowitz MB. An excitable gene regulatory circuit induces transient cellular differentiation. Nature. 2006;440:545–550. pmid:16554821
- View Article
- PubMed/NCBI
- Google Scholar
9. Acar M, Mettetal JT, van Oudenaarden A. Stochastic switching as a survival strategy in fluctuating environments. Nat Genet. 2008;40:471–475. pmid:18362885
- View Article
- PubMed/NCBI
- Google Scholar
10. Turcotte M, Garcia-Ojalvo J, Süel GM. A genetic timer through noise-induced stabilization of an unstable state. Proc Natl Acad Sci USA. 2008;105:15732–15737. pmid:18836072
- View Article
- PubMed/NCBI
- Google Scholar
11. Raj A, Rifkin SA, Andersen E, van Oudenaarden A. Variability in gene expression underlies incomplete penetrance. Nature. 2010;463:913–918. pmid:20164922
- View Article
- PubMed/NCBI
- Google Scholar
12. Feinerman O, Veiga J, Dorfman JR, Germain RN, Altan-Bonnet G. Variability and robustness in T cell activation from regulated heterogeneity in protein levels. Science. 2008;321:1081–1084. pmid:18719282
- View Article
- PubMed/NCBI
- Google Scholar
13. Lerner IM. Genetic homeostasis. Oliver and Boyd; 1954.
14. McAdams HH, Arkin A. It’s a noisy business! Genetic regulation at the nanomolar scale. Trends Genet. 1999;15:65–69. pmid:10098409
- View Article
- PubMed/NCBI
- Google Scholar
15. Nowak MA, Boerlijst MC, Cooke J, Smith JM. Evolution of genetic redundancy. Nature. 1997;388:167–171. pmid:9217155
- View Article
- PubMed/NCBI
- Google Scholar
16. Levchenko A, Nemenman I. Cellular noise and information transmission. Curr Opin Biotechnol. 2014;28:156–164. pmid:24922112
- View Article
- PubMed/NCBI
- Google Scholar
17. Bowsher CG, Swain PS. Environmental sensing, information transfer, and cellular decision-making. Curr Opin Biotechnol. 2014;28:149–155.
18. Cover TM, Thomas JA. Elements of Information Theory. Wiley; 2006.
19. Tkacik G, Callan CG, Bialek W. Information flow and optimization in transcriptional regulation. Proc Natl Acad Sci USA. 2008;105:12265–12270. pmid:18719112
- View Article
- PubMed/NCBI
- Google Scholar
20. Libby E, Perkins TJ, Swain PS. Noisy information processing through transcriptional regulation. Proc Natl Acad Sci USA. 2007;104:7151–7156. pmid:17420464
- View Article
- PubMed/NCBI
- Google Scholar
21. Yu RC, Pesce CG, Colman-Lerner A, et al. Negative feedback that improves information transmission in yeast signalling. Nature. 2008;456:755–761. pmid:19079053
- View Article
- PubMed/NCBI
- Google Scholar
22. Dubuis JO, Tkacik G, Wieschaus EF, Gregor T, Bialek W. Positional information, in bits. Proc Natl Acad Sci USA. 2013;110:16301–16308. pmid:24089448
- View Article
- PubMed/NCBI
- Google Scholar
23. Cheong R, Rhee A, Wang CJ, Nemenman I, Levchenko A. Information transduction capacity of noisy biochemical signaling networks. Science. 2011;334:354–358. pmid:21921160
- View Article
- PubMed/NCBI
- Google Scholar
24. Hansen AS, O’Shea EK. Limits on information transduction through amplitude and frequency regulation of transcription factor activity. eLife. 2015;4:e06559. pmid:25985085
- View Article
- PubMed/NCBI
- Google Scholar
25. Wall ME, Hlavacek WS, Savageau MA. Design of gene circuits: lessons from bacteria. Nat Rev Genet. 2004;5:34–42. pmid:14708014
- View Article
- PubMed/NCBI
- Google Scholar
26. Gammaitoni L. Stochastic resonance and the dithering effect in threshold physical systems. Phys Rev E. 1995;52:4691–4698. pmid:9963964
- View Article
- PubMed/NCBI
- Google Scholar
27. Gingl Z, Kiss LB, Moss F. Non-dynamical stochastic resonance: theory and experiments with white and arbitrarily coloured noise. Europhys Lett. 1995;29:191–196.
- View Article
- Google Scholar
28. Gammaitoni L, Hanggi P, Jung P, Marchesoni F. Stochastic resonance. Rev Mod Phys. 1998;70:223–287.
- View Article
- Google Scholar
29. Stocks NG. Suprathreshold stochastic resonance in multilevel threshold systems. Phys Rev Lett. 2000;84:2310–2313. pmid:11018872
- View Article
- PubMed/NCBI
- Google Scholar
30. Shannon CE. A mathematical theory of communication. Bell Syst Tech J. 1948;27:379–423.
- View Article
- Google Scholar
31. Sharpe C, Lawrence N, Martinez-Arias A. Wnt signalling: a theme with nuclear variations. Bioessays. 2001;23:311–318. pmid:11268036
- View Article
- PubMed/NCBI
- Google Scholar
32. Swain PS, Elowitz MB, Siggia ED. Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc Natl Acad Sci USA. 2002;99:12795–12800. pmid:12237400
- View Article
- PubMed/NCBI
- Google Scholar
33. Masquilier D, Sassone-Corsi P. Transcriptional cross-talk: nuclear factors CREM and CREB bind to AP-1 sites and inhibit activation by Jun. J Biol Chem. 1992;267:22460–22466. pmid:1429597
- View Article
- PubMed/NCBI
- Google Scholar
34. Hittinger CT, Carroll SB. Gene duplication and the adaptive evolution of a classic genetic switch. Nature. 2007;449:677–681. pmid:17928853
- View Article
- PubMed/NCBI
- Google Scholar
35. Mayo AE, Setty Y, Shavit S, Zaslaver A, Alon U. Plasticity of the cis-regulatory input function of a gene. PLoS Biol. 2006;4:555. pmid:16602820
- View Article
- PubMed/NCBI
- Google Scholar
36. Hunsberger E, Scott M, Eliasmith C. The competing benefits of noise and heterogeneity in neural coding. Neural Comput. 2014;26:1600–1623. pmid:24877735
- View Article
- PubMed/NCBI
- Google Scholar
37. Zucca S, Pasotti L, Mazzini G, De Angelis MGC, Magni P. Characterization of an inducible promoter in different DNA copy number conditions. BMC bioinformatics. 2012;13:S11. pmid:22536957
- View Article
- PubMed/NCBI
- Google Scholar
38. Justman QA, Serber Z, Ferrell JE, El-Samad H, Shokat KM. Tuning the activation threshold of a kinase network by nested feedback loops. Science. 2009;324:509–512. pmid:19390045
- View Article
- PubMed/NCBI
- Google Scholar
39. Elion EA, Brill JA, Fink GR. FUS3 represses CLN1 and CLN2 and in concert with KSS1 promotes signal transduction. Proc Natl Acad Sci USA. 1991;88:9392–9396. pmid:1946350
- View Article
- PubMed/NCBI
- Google Scholar
40. Wang Y, Pierce M, Schneper L, Güldal CG, Zhang X, Tavazoie S, et al. Ras and Gpa2 mediate one branch of a redundant glucose signaling pathway in yeast. PLoS Biol. 2004;2:e128. pmid:15138498
- View Article
- PubMed/NCBI
- Google Scholar
41. Brion C, Ambroset C, Delobel P, Sanchez I, Blondin B. Deciphering regulatory variation of THI genes in alcoholic fermentation indicate an impact of Thi3p on PDC1 expression. BMC Genomics. 2014;15:1085. pmid:25494835
- View Article
- PubMed/NCBI
- Google Scholar
42. McDonnell MD, Ward LM. The benefits of noise in neural systems: bridging theory and experiment. Nat Rev Neurosci. 2011;12:415–426. pmid:21685932
- View Article
- PubMed/NCBI
- Google Scholar
43. Eldar A, Elowitz MB. Functional roles for noise in genetic circuits. Nature. 2010;467:167–173. pmid:20829787
- View Article
- PubMed/NCBI
- Google Scholar
44. Kafri R, Levy M, Pilpel Y. The regulatory utilization of genetic redundancy through responsive backup circuits. Proc Natl Acad Sci USA. 2006;103:11653–11658. pmid:16861297
- View Article
- PubMed/NCBI
- Google Scholar
45. Keane OM, Toft C, Carretero-Paulet L, Jones GW, Fares MA. Preservation of genetic and regulatory robustness in ancient gene duplicates of Saccharomyces cerevisiae. Genome Res. 2014;24:1830–1841. pmid:25149527
- View Article
- PubMed/NCBI
- Google Scholar
46. Riehle MM, Bennett AF, Long AD. Genetic architecture of thermal adaptation in Escherichia coli. Proc Natl Acad Sci USA. 2001;98:525–530. pmid:11149947
- View Article
- PubMed/NCBI
- Google Scholar
47. Gresham D, Desai MM, Tucker CM, et al. The repertoire and dynamics of evolutionary adaptations to controlled nutrient-limited environments in yeast. PLoS Genet. 2008;4:e1000303. pmid:19079573
- View Article
- PubMed/NCBI
- Google Scholar
48. van Wageningen S, Kemmeren P, Lijnzaad P, et al. Functional overlap and regulatory links shape genetic interactions between signaling pathways. Cell. 2010;143:991–1004. pmid:21145464
- View Article
- PubMed/NCBI
- Google Scholar
49. Volfson D, Marciniak J, Blake WJ, Ostroff N, Tsimring LV, Hasty J. Origins of extrinsic variability in eukaryotic gene expression. Nature. 2006;439:861–864. pmid:16372021
- View Article
- PubMed/NCBI
- Google Scholar
50. Selimkhanov J, Taylor B, Yao J, Pilko A, Albeck J, Hoffmann A, et al. Accurate information transmission through dynamic biochemical signaling networks. Science. 2014;346:1370–1373. pmid:25504722
- View Article
- PubMed/NCBI
- Google Scholar
51. Uda S, Saito TH, Kudo T, Kokaji T, Tsuchiya T, Kubota H, et al. Robustness and compensation of information transmission of signaling pathways. Science. 2013;341:558–561. pmid:23908238
- View Article
- PubMed/NCBI
- Google Scholar
52. Conradt L, Roper TJ. Consensus decision making in animals. Trends Ecol Evol. 2005;20:449–456. pmid:16701416
- View Article
- PubMed/NCBI
- Google Scholar
53. von Neumann J. Probabilistic logics and the synthesis of reliable organisms from unreliable components. Shannon CE, McCarthy J, editors. Princeton University Press; 1956. https://doi.org/10.1515/9781400882618-003
54. Moore EF, Shannon CE. Reliable circuits using less reliable relays. J Franklin I. 1956;262:191–208.
- View Article
- Google Scholar
55. Dueber JE, Mirsky EA, Lim WA. Engineering synthetic signaling proteins with ultrasensitive input/output control. Nat Biotechnol. 2007;25:660–662. pmid:17515908
- View Article
- PubMed/NCBI
- Google Scholar
56. Gillespie D. The chemical Langevin equation. J Chem Phys. 2000;113:297–306.
- View Article
- Google Scholar
57. Rodrigo G, Kirov B, Shen S, Jaramillo A. Theoretical and experimental analysis of the forced LacI-AraC oscillator with a minimal gene regulatory model. Chaos. 2013;23:025109. pmid:23822507
- View Article
- PubMed/NCBI
- Google Scholar
58. Süel G, Garcia-Ojalvo J, Liberman L, MB E. An excitable gene regulatory circuit induces transient cellular differentiation. Nature. 2006;440:545–550. pmid:16554821
- View Article
- PubMed/NCBI
- Google Scholar
59. Nevozhay D, Adams RM, Van Itallie E, Bennett MR, Balázsi G. Mapping the environmental fitness landscape of a synthetic gene circuit. PLoS Comp Biol. 2012;8:e1002480. pmid:22511863
- View Article
- PubMed/NCBI
- Google Scholar
60. Scott M, Gunderson CW, Mateescu E, Zhang Z, Hwa T. Interdependence of cell growth and gene expression: origins and consequences. Science 2010;330:1099–1102. pmid:21097934
- View Article
- PubMed/NCBI
- Google Scholar

[ref1] 1. McAdams HH, Arkin A. Stochastic mechanisms in gene expression. Proc Natl Acad Sci USA. 1997;94:814–819. pmid:9023339
View Article
PubMed/NCBI
Google Scholar

[2] View Article

[3] PubMed/NCBI

[4] Google Scholar

[ref2] 2. Elowitz MB, Levine AJ, Siggia ED, Swain PS. Stochastic gene expression in a single cell. Science. 2002;297:1183–1186. pmid:12183631
View Article
PubMed/NCBI
Google Scholar

[6] View Article

[7] PubMed/NCBI

[8] Google Scholar

[ref3] 3. Lestas I, Vinnicombe G, Paulsson J. Fundamental limits on the suppression of molecular fluctuations. Nature. 2010;467:174–178. pmid:20829788
View Article
PubMed/NCBI
Google Scholar

[10] View Article

[11] PubMed/NCBI

[12] Google Scholar

[ref4] 4. Hopfield JJ. Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity. Proc Natl Acad Sci USA. 1974;71:4135–4139. pmid:4530290
View Article
PubMed/NCBI
Google Scholar

[14] View Article

[15] PubMed/NCBI

[16] Google Scholar

[ref5] 5. Yi TM, Huang Y, Simon MI, Doyle J. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proc Natl Acad Sci USA. 2000;97:4649–4653. pmid:10781070
View Article
PubMed/NCBI
Google Scholar

[18] View Article

[19] PubMed/NCBI

[20] Google Scholar

[ref6] 6. Bishop AL, Ran FA, Sumner ER, Avery SV. Phenotypic heterogeneity can enhance rare-cell survival in ‘stress-sensitive’ yeast populations. Mol Microbiol. 2007;63:507–520. pmid:17176259
View Article
PubMed/NCBI
Google Scholar

[22] View Article

[23] PubMed/NCBI

[24] Google Scholar

[ref7] 7. Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S. Bacterial persistence as a phenotypic switch. Science. 2004;305:1622–1625. pmid:15308767
View Article
PubMed/NCBI
Google Scholar

[26] View Article

[27] PubMed/NCBI

[28] Google Scholar

[ref8] 8. Süel GM, Garcia-Ojalvo J, Liberman LM, Elowitz MB. An excitable gene regulatory circuit induces transient cellular differentiation. Nature. 2006;440:545–550. pmid:16554821
View Article
PubMed/NCBI
Google Scholar

[30] View Article

[31] PubMed/NCBI

[32] Google Scholar

[ref9] 9. Acar M, Mettetal JT, van Oudenaarden A. Stochastic switching as a survival strategy in fluctuating environments. Nat Genet. 2008;40:471–475. pmid:18362885
View Article
PubMed/NCBI
Google Scholar

[34] View Article

[35] PubMed/NCBI

[36] Google Scholar

[ref10] 10. Turcotte M, Garcia-Ojalvo J, Süel GM. A genetic timer through noise-induced stabilization of an unstable state. Proc Natl Acad Sci USA. 2008;105:15732–15737. pmid:18836072
View Article
PubMed/NCBI
Google Scholar

[38] View Article

[39] PubMed/NCBI

[40] Google Scholar

[ref11] 11. Raj A, Rifkin SA, Andersen E, van Oudenaarden A. Variability in gene expression underlies incomplete penetrance. Nature. 2010;463:913–918. pmid:20164922
View Article
PubMed/NCBI
Google Scholar

[42] View Article

[43] PubMed/NCBI

[44] Google Scholar

[ref12] 12. Feinerman O, Veiga J, Dorfman JR, Germain RN, Altan-Bonnet G. Variability and robustness in T cell activation from regulated heterogeneity in protein levels. Science. 2008;321:1081–1084. pmid:18719282
View Article
PubMed/NCBI
Google Scholar

[46] View Article

[47] PubMed/NCBI

[48] Google Scholar

[ref13] 13. Lerner IM. Genetic homeostasis. Oliver and Boyd; 1954.

[ref14] 14. McAdams HH, Arkin A. It’s a noisy business! Genetic regulation at the nanomolar scale. Trends Genet. 1999;15:65–69. pmid:10098409
View Article
PubMed/NCBI
Google Scholar

[51] View Article

[52] PubMed/NCBI

[53] Google Scholar

[ref15] 15. Nowak MA, Boerlijst MC, Cooke J, Smith JM. Evolution of genetic redundancy. Nature. 1997;388:167–171. pmid:9217155
View Article
PubMed/NCBI
Google Scholar

[55] View Article

[56] PubMed/NCBI

[57] Google Scholar

[ref16] 16. Levchenko A, Nemenman I. Cellular noise and information transmission. Curr Opin Biotechnol. 2014;28:156–164. pmid:24922112
View Article
PubMed/NCBI
Google Scholar

[59] View Article

[60] PubMed/NCBI

[61] Google Scholar

[ref17] 17. Bowsher CG, Swain PS. Environmental sensing, information transfer, and cellular decision-making. Curr Opin Biotechnol. 2014;28:149–155.

[ref18] 18. Cover TM, Thomas JA. Elements of Information Theory. Wiley; 2006.

[ref19] 19. Tkacik G, Callan CG, Bialek W. Information flow and optimization in transcriptional regulation. Proc Natl Acad Sci USA. 2008;105:12265–12270. pmid:18719112
View Article
PubMed/NCBI
Google Scholar

[65] View Article

[66] PubMed/NCBI

[67] Google Scholar

[ref20] 20. Libby E, Perkins TJ, Swain PS. Noisy information processing through transcriptional regulation. Proc Natl Acad Sci USA. 2007;104:7151–7156. pmid:17420464
View Article
PubMed/NCBI
Google Scholar

[69] View Article

[70] PubMed/NCBI

[71] Google Scholar

[ref21] 21. Yu RC, Pesce CG, Colman-Lerner A, et al. Negative feedback that improves information transmission in yeast signalling. Nature. 2008;456:755–761. pmid:19079053
View Article
PubMed/NCBI
Google Scholar

[73] View Article

[74] PubMed/NCBI

[75] Google Scholar

[ref22] 22. Dubuis JO, Tkacik G, Wieschaus EF, Gregor T, Bialek W. Positional information, in bits. Proc Natl Acad Sci USA. 2013;110:16301–16308. pmid:24089448
View Article
PubMed/NCBI
Google Scholar

[77] View Article

[78] PubMed/NCBI

[79] Google Scholar

[ref23] 23. Cheong R, Rhee A, Wang CJ, Nemenman I, Levchenko A. Information transduction capacity of noisy biochemical signaling networks. Science. 2011;334:354–358. pmid:21921160
View Article
PubMed/NCBI
Google Scholar

[81] View Article

[82] PubMed/NCBI

[83] Google Scholar

[ref24] 24. Hansen AS, O’Shea EK. Limits on information transduction through amplitude and frequency regulation of transcription factor activity. eLife. 2015;4:e06559. pmid:25985085
View Article
PubMed/NCBI
Google Scholar

[85] View Article

[86] PubMed/NCBI

[87] Google Scholar

[ref25] 25. Wall ME, Hlavacek WS, Savageau MA. Design of gene circuits: lessons from bacteria. Nat Rev Genet. 2004;5:34–42. pmid:14708014
View Article
PubMed/NCBI
Google Scholar

[89] View Article

[90] PubMed/NCBI

[91] Google Scholar

[ref26] 26. Gammaitoni L. Stochastic resonance and the dithering effect in threshold physical systems. Phys Rev E. 1995;52:4691–4698. pmid:9963964
View Article
PubMed/NCBI
Google Scholar

[93] View Article

[94] PubMed/NCBI

[95] Google Scholar

[ref27] 27. Gingl Z, Kiss LB, Moss F. Non-dynamical stochastic resonance: theory and experiments with white and arbitrarily coloured noise. Europhys Lett. 1995;29:191–196.
View Article
Google Scholar

[97] View Article

[98] Google Scholar

[ref28] 28. Gammaitoni L, Hanggi P, Jung P, Marchesoni F. Stochastic resonance. Rev Mod Phys. 1998;70:223–287.
View Article
Google Scholar

[100] View Article

[101] Google Scholar

[ref29] 29. Stocks NG. Suprathreshold stochastic resonance in multilevel threshold systems. Phys Rev Lett. 2000;84:2310–2313. pmid:11018872
View Article
PubMed/NCBI
Google Scholar

[103] View Article

[104] PubMed/NCBI

[105] Google Scholar

[ref30] 30. Shannon CE. A mathematical theory of communication. Bell Syst Tech J. 1948;27:379–423.
View Article
Google Scholar

[107] View Article

[108] Google Scholar

[ref31] 31. Sharpe C, Lawrence N, Martinez-Arias A. Wnt signalling: a theme with nuclear variations. Bioessays. 2001;23:311–318. pmid:11268036
View Article
PubMed/NCBI
Google Scholar

[110] View Article

[111] PubMed/NCBI

[112] Google Scholar

[ref32] 32. Swain PS, Elowitz MB, Siggia ED. Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc Natl Acad Sci USA. 2002;99:12795–12800. pmid:12237400
View Article
PubMed/NCBI
Google Scholar

[114] View Article

[115] PubMed/NCBI

[116] Google Scholar

[ref33] 33. Masquilier D, Sassone-Corsi P. Transcriptional cross-talk: nuclear factors CREM and CREB bind to AP-1 sites and inhibit activation by Jun. J Biol Chem. 1992;267:22460–22466. pmid:1429597
View Article
PubMed/NCBI
Google Scholar

[118] View Article

[119] PubMed/NCBI

[120] Google Scholar

[ref34] 34. Hittinger CT, Carroll SB. Gene duplication and the adaptive evolution of a classic genetic switch. Nature. 2007;449:677–681. pmid:17928853
View Article
PubMed/NCBI
Google Scholar

[122] View Article

[123] PubMed/NCBI

[124] Google Scholar

[ref35] 35. Mayo AE, Setty Y, Shavit S, Zaslaver A, Alon U. Plasticity of the cis-regulatory input function of a gene. PLoS Biol. 2006;4:555. pmid:16602820
View Article
PubMed/NCBI
Google Scholar

[126] View Article

[127] PubMed/NCBI

[128] Google Scholar

[ref36] 36. Hunsberger E, Scott M, Eliasmith C. The competing benefits of noise and heterogeneity in neural coding. Neural Comput. 2014;26:1600–1623. pmid:24877735
View Article
PubMed/NCBI
Google Scholar

[130] View Article

[131] PubMed/NCBI

[132] Google Scholar

[ref37] 37. Zucca S, Pasotti L, Mazzini G, De Angelis MGC, Magni P. Characterization of an inducible promoter in different DNA copy number conditions. BMC bioinformatics. 2012;13:S11. pmid:22536957
View Article
PubMed/NCBI
Google Scholar

[134] View Article

[135] PubMed/NCBI

[136] Google Scholar

[ref38] 38. Justman QA, Serber Z, Ferrell JE, El-Samad H, Shokat KM. Tuning the activation threshold of a kinase network by nested feedback loops. Science. 2009;324:509–512. pmid:19390045
View Article
PubMed/NCBI
Google Scholar

[138] View Article

[139] PubMed/NCBI

[140] Google Scholar

[ref39] 39. Elion EA, Brill JA, Fink GR. FUS3 represses CLN1 and CLN2 and in concert with KSS1 promotes signal transduction. Proc Natl Acad Sci USA. 1991;88:9392–9396. pmid:1946350
View Article
PubMed/NCBI
Google Scholar

[142] View Article

[143] PubMed/NCBI

[144] Google Scholar

[ref40] 40. Wang Y, Pierce M, Schneper L, Güldal CG, Zhang X, Tavazoie S, et al. Ras and Gpa2 mediate one branch of a redundant glucose signaling pathway in yeast. PLoS Biol. 2004;2:e128. pmid:15138498
View Article
PubMed/NCBI
Google Scholar

[146] View Article

[147] PubMed/NCBI

[148] Google Scholar

[ref41] 41. Brion C, Ambroset C, Delobel P, Sanchez I, Blondin B. Deciphering regulatory variation of THI genes in alcoholic fermentation indicate an impact of Thi3p on PDC1 expression. BMC Genomics. 2014;15:1085. pmid:25494835
View Article
PubMed/NCBI
Google Scholar

[150] View Article

[151] PubMed/NCBI

[152] Google Scholar

[ref42] 42. McDonnell MD, Ward LM. The benefits of noise in neural systems: bridging theory and experiment. Nat Rev Neurosci. 2011;12:415–426. pmid:21685932
View Article
PubMed/NCBI
Google Scholar

[154] View Article

[155] PubMed/NCBI

[156] Google Scholar

[ref43] 43. Eldar A, Elowitz MB. Functional roles for noise in genetic circuits. Nature. 2010;467:167–173. pmid:20829787
View Article
PubMed/NCBI
Google Scholar

[158] View Article

[159] PubMed/NCBI

[160] Google Scholar

[ref44] 44. Kafri R, Levy M, Pilpel Y. The regulatory utilization of genetic redundancy through responsive backup circuits. Proc Natl Acad Sci USA. 2006;103:11653–11658. pmid:16861297
View Article
PubMed/NCBI
Google Scholar

[162] View Article

[163] PubMed/NCBI

[164] Google Scholar

[ref45] 45. Keane OM, Toft C, Carretero-Paulet L, Jones GW, Fares MA. Preservation of genetic and regulatory robustness in ancient gene duplicates of Saccharomyces cerevisiae. Genome Res. 2014;24:1830–1841. pmid:25149527
View Article
PubMed/NCBI
Google Scholar

[166] View Article

[167] PubMed/NCBI

[168] Google Scholar

[ref46] 46. Riehle MM, Bennett AF, Long AD. Genetic architecture of thermal adaptation in Escherichia coli. Proc Natl Acad Sci USA. 2001;98:525–530. pmid:11149947
View Article
PubMed/NCBI
Google Scholar

[170] View Article

[171] PubMed/NCBI

[172] Google Scholar

[ref47] 47. Gresham D, Desai MM, Tucker CM, et al. The repertoire and dynamics of evolutionary adaptations to controlled nutrient-limited environments in yeast. PLoS Genet. 2008;4:e1000303. pmid:19079573
View Article
PubMed/NCBI
Google Scholar

[174] View Article

[175] PubMed/NCBI

[176] Google Scholar

[ref48] 48. van Wageningen S, Kemmeren P, Lijnzaad P, et al. Functional overlap and regulatory links shape genetic interactions between signaling pathways. Cell. 2010;143:991–1004. pmid:21145464
View Article
PubMed/NCBI
Google Scholar

[178] View Article

[179] PubMed/NCBI

[180] Google Scholar

[ref49] 49. Volfson D, Marciniak J, Blake WJ, Ostroff N, Tsimring LV, Hasty J. Origins of extrinsic variability in eukaryotic gene expression. Nature. 2006;439:861–864. pmid:16372021
View Article
PubMed/NCBI
Google Scholar

[182] View Article

[183] PubMed/NCBI

[184] Google Scholar

[ref50] 50. Selimkhanov J, Taylor B, Yao J, Pilko A, Albeck J, Hoffmann A, et al. Accurate information transmission through dynamic biochemical signaling networks. Science. 2014;346:1370–1373. pmid:25504722
View Article
PubMed/NCBI
Google Scholar

[186] View Article

[187] PubMed/NCBI

[188] Google Scholar

[ref51] 51. Uda S, Saito TH, Kudo T, Kokaji T, Tsuchiya T, Kubota H, et al. Robustness and compensation of information transmission of signaling pathways. Science. 2013;341:558–561. pmid:23908238
View Article
PubMed/NCBI
Google Scholar

[190] View Article

[191] PubMed/NCBI

[192] Google Scholar

[ref52] 52. Conradt L, Roper TJ. Consensus decision making in animals. Trends Ecol Evol. 2005;20:449–456. pmid:16701416
View Article
PubMed/NCBI
Google Scholar

[194] View Article

[195] PubMed/NCBI

[196] Google Scholar

[ref53] 53. von Neumann J. Probabilistic logics and the synthesis of reliable organisms from unreliable components. Shannon CE, McCarthy J, editors. Princeton University Press; 1956. https://doi.org/10.1515/9781400882618-003

[ref54] 54. Moore EF, Shannon CE. Reliable circuits using less reliable relays. J Franklin I. 1956;262:191–208.
View Article
Google Scholar

[199] View Article

[200] Google Scholar

[ref55] 55. Dueber JE, Mirsky EA, Lim WA. Engineering synthetic signaling proteins with ultrasensitive input/output control. Nat Biotechnol. 2007;25:660–662. pmid:17515908
View Article
PubMed/NCBI
Google Scholar

[202] View Article

[203] PubMed/NCBI

[204] Google Scholar

[ref56] 56. Gillespie D. The chemical Langevin equation. J Chem Phys. 2000;113:297–306.
View Article
Google Scholar

[206] View Article

[207] Google Scholar

[ref57] 57. Rodrigo G, Kirov B, Shen S, Jaramillo A. Theoretical and experimental analysis of the forced LacI-AraC oscillator with a minimal gene regulatory model. Chaos. 2013;23:025109. pmid:23822507
View Article
PubMed/NCBI
Google Scholar

[209] View Article

[210] PubMed/NCBI

[211] Google Scholar

[ref58] 58. Süel G, Garcia-Ojalvo J, Liberman L, MB E. An excitable gene regulatory circuit induces transient cellular differentiation. Nature. 2006;440:545–550. pmid:16554821
View Article
PubMed/NCBI
Google Scholar

[213] View Article

[214] PubMed/NCBI

[215] Google Scholar

[ref59] 59. Nevozhay D, Adams RM, Van Itallie E, Bennett MR, Balázsi G. Mapping the environmental fitness landscape of a synthetic gene circuit. PLoS Comp Biol. 2012;8:e1002480. pmid:22511863
View Article
PubMed/NCBI
Google Scholar

[217] View Article

[218] PubMed/NCBI

[219] Google Scholar

[ref60] 60. Scott M, Gunderson CW, Mateescu E, Zhang Z, Hwa T. Interdependence of cell growth and gene expression: origins and consequences. Science 2010;330:1099–1102. pmid:21097934
View Article
PubMed/NCBI
Google Scholar

[221] View Article

[222] PubMed/NCBI

[223] Google Scholar

Figures

Abstract

Author Summary

Introduction

Results

Intrinsic noise can amplify information transfer

Genetic redundancy enhances information transfer

Input signal distribution shapes information transfer

Extrinsic noise and cross-talk limit information transfer

Heterogeneity also contributes to increase information transfer

Empirical evidences of the theory

Discussion

Materials and Methods

Modeling noisy regulatory systems

Simple regulated system.

Bistable system.

Excitable system.

Extrinsic noise, cross-talk and heterogeneity

Input and output variables

Input distributions

Quantification of information transfer

Analysis of experimental data

Supporting Information

S1 Text. A detailed description of all the methods used in this study.

S1 Fig. Distance of the average response to the ideal linear response as a function of intrinsic noise.

S2 Fig. Suprathreshold stochastic resonance in multilevel threshold systems.

S3 Fig. Effect of feedback strength or intrinsic noise on information transmission.

S4 Fig. Influence of the number of units for an excitable system.

S5 Fig. Dependence of mutual information with the average degree of cross-talk for the bistable system constituted by 2 units.

S6 Fig. Tradeoff between information transfer and genetic load for a bistable genetic system with redundancy.

Acknowledgments

Author Contributions

References