Optimal signal processing in small stochastic biochemical networks

Abstract

We quantify the influence of the topology of a transcriptional regulatory network on its ability to process environmental signals. By posing the problem in terms of information theory, we do this without specifying the function performed by the network. Specifically, we study the maximum mutual information between the input (chemical) signal and the output (genetic) response attainable by the network in the context of an analytic model of particle number fluctuations. We perform this analysis for all biochemical circuits, including various feedback loops, that can be built out of 3 chemical species, each under the control of one regulator. We find that a generic network, constrained to low molecule numbers and reasonable response times, can transduce more information than a simple binary switch and, in fact, manages to achieve close to the optimal information transmission fidelity. These high-information solutions are robust to tenfold changes in most of the networks' biochemical parameters; moreover they are easier to achieve in networks containing cycles with an odd number of negative regulators (overall negative feedback) due to their decreased molecular noise (a result which we derive analytically). Finally, we demonstrate that a single circuit can support multiple high-information solutions. These findings suggest a potential resolution of the "cross-talk" phenomenon as well as the previously unexplained observation that transcription factors that undergo proteolysis are more likely to be auto-repressive.

Figure 1. Table of circuits (top 12 by the optimality statistic).

Extrapolated average mutual information over range of 25 to 120 molecules at γ = 0.001 and γ = 0.01.

Figure 2. Table of circuits (bottom 12 by the optimality statistic).

Extrapolated average mutual information over range of 25 to 120 molecules at γ = 0.001 and γ = 0.01.

Figure 3

(a) Circuit 19 with an odd number of negative regulators in cycle and (b) Circuit 11 with an even number of negative regulators in cycle. (c) and (d) We ran multiple optimizations ϑ_* = argmax_ϑ L. For each optimization run, we plot the mutual information I _* = I(C,G|ϑ_*) vs. the mean number of molecules of the reporter protein 〈N_G〉. Below 10 copies we saw poor LNA performance (cf. Text S1). Input distribution p(c) = 1/||C|| and ||C|| = 8 so that I(C,G)≤H(C) = 3 bits. Blue squares and red triangles are for γ = 0.001 and γ = 0.01, respectively. The blue and red linearly interpolated lines correspond to the convex hull for each respective γ value. The black solid curve gives the numerically evaluated optimal bound (cf. Results: Determining Optimal Bounds) and dashed curve gives analytic bound for any input distribution (cf. Materials and Methods : Maximum Mutual Information for a Fixed Copy Number). Inset: 〈I〉 as a function of the inverse fraction of data included m [cf. Results: (Almost) Optimal Circuits] in the analysis. Blue and red correspond to two different γ values. Linear regression extrapolated to case of infinite data (y-intercept). The results represent two typical circuits with 1-cycles. Note that here, as in Fig. 4 and Fig. 5, circuits on the left have higher 〈I〉 values as well as narrower gaps between the two γ values than circuits on the right.

Figure 4

(a) Circuit 23 with an odd number of negative regulators in cycle and (b) Circuit 5 with an even number of negative regulators in cycle. (c) and (d) Same as in Fig. 3 for these two circuits with 2 cycles.

Figure 5

(a) Circuit 13 with an odd number of negative regulators in cycle and (b) Circuit 17 with an even number of negative regulators in cycle. (c) and (d) Same as in Fig. 3 for these two circuits with 3 cycles.

Figure 6

Bar graphs for 〈I〉 values for the two classes of circuits: odd (blue) includes circuits with cycles containing an odd number of repressors and even (green) includes circuits with cycles containing an even number of repressors. Top γ₁ = 0.001, middle γ₂ = 0.01, and bottom 〈I(γ₁)〉−〈I(γ₂)〉. For all 3 measures, there is a statistically significant difference between the two classes of circuits as calculated by the U Test (top p = 0.0002, middle p = 0.0003 and bottom p = 0.01).

Figure 7

(a) The objective function L and (b) the mutual information I as a function of the input parameters s_X and s_Y corresponding to the small molecules “strength” on transcription factors X and Y (cf. Materials and Methods : Model and Parameters) for circuit 2. The rest of the parameters are held constant for this figure. The five labeled peaks correspond to 5 distinct behaviors or unique signal encodings (cf. Fig. 8 and Table 1).

Figure 8. The conditional p(G|C) is plotted for each of the 5 maxima of the constrained information shown in Fig. 7.

Colors denote each individual conditional p(G|C = c) where C takes 8 possible and equally likely, states. Since these are all high information solutions, the individual conditionals are all separated well. Note that at, each maximum, the colors are arranged differently, highlighting the fact that the conditionals are different, and therefore the network behaves differently at each of these high information solutions. The arrangement of these individual conditionals is summarized in Table 1.

Figure 9

Top-left: Spectra for the numerically calculated Hessian at each of the corresponding 5 peaks labeled in Fig. 7. Soft modes (→0) are directions in which L has small curvature; hard modes (→−∞) are directions in which L has large curvature. Many eigendirections exhibit small curvature (magnitude of eigenvalue less than 10⁻² for peaks 2–4 and 10⁻¹ for peaks 1 and 5), demonstrating that the maxima are robust to large deviations in parameter space. Colored panels: Magnitude of contribution from each parameter to each eigenvector for each of the five Hessians. Mode index is sorted as in top-left figure (from least curvature to greatest curvature). Row labeled leak corresponds to parameter a ₀. Paired rows labeled X, Y, Z, and G correspond to the two parameters, K and a, describing the gene regulation function for each transcription factor (X, Y, Z) and reporter protein (G). Rows labeled r correspond to the decay rates of each of the 3 transcription factors. Rows labeled s correspond to the input parameters modulating the three transcription factors. For all five peaks, the two most soft modes correspond to a ₀ and a mixture of K_Y, a_Y, respectively. s_X and s_Z contribute mostly to the hard modes.