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.
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.
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).
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).
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).
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.
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.