Statistics of optimal information flow in ensembles of regulatory motifs

Genetic regulatory circuits universally cope with different sources of noise that limit their ability to coordinate input and output signals. In many cases, optimal regulatory performance can be thought to correspond to configurations of variables and parameters that maximize the mutual information between inputs and outputs. Since the mid-2000s, such optima have been well characterized in several biologically relevant cases. Here we use methods of statistical field theory to calculate the statistics of the maximal mutual information (the “capacity”) achievable by tuning the input variable only in an ensemble of regulatory motifs, such that a single controller regulates $N$ targets. Assuming (i) sufficiently large $N$, (ii) quenched random kinetic parameters, and (iii) small noise affecting the input-output channels, we can accurately reproduce numerical simulations both for the mean capacity and for the whole distribution. Our results provide insight into the inherent variability in effectiveness occurring in regulatory systems with heterogeneous kinetic parameters.

Figure 1

Scheme of one controller (e.g., a transcription factor) regulating $N$ independent targets. $P_{\mathrm{x}}\left(x\right)$ is the probability distribution of the controller level $x$, recalled by the dashed black lines in middle panels. The channel is represented by the mean target levels ${\left\{{\overline{g}}_i\left(x\right)\right\}}_{i=1}^N$ (solid lines in middle panels), modeled as Hill functions of $x$ (the controller can either activate or repress each of the $N$ targets). The error bars represent $\pm {\sigma }_i\left(x\right)$, where ${\sigma }_i^2\left(x\right)$ is the intrinsic noise variance of the $i\mathrm{th}$ target. The presence of noise induces a probabilistic relationship between the levels of the controller and the targets: the outputs of the regulatory network are the probability distributions ${\left\{P_{\mathrm{g}}\left(g_i\right)\right\}}_{i=1}^N$, whose shape depends on the matching between the controller distribution and the noisy channel.

Figure 2

Blue markers: probability distributions $P_I\left(I^{★}\right)$ obtained numerically for ensembles of ${10}^4$ regulatory motifs with different $N$ constructed by choosing ${\overline{g}}_i$ and ${\sigma }_i^2$ as in (2) and (3), by sampling each $K_i$ independently from a uniform distribution in $[x_{min},x_{max}]$, with $x_{min}={10}^{-2}$, $x_{max}=1$, and by setting $h_i=2$ for each $i=1,\cdots ,N$. The red line represents the best fit to a GEV distribution [30]. Blue and red vertical lines mark, respectively, the positions of the numerical mean and of the mean of the GEV fitted to the data. The purple star indicates $I_{\mathrm{opt}}^{★}$, Eq. (11). The maximum number of target molecules was set to $M_{max}=100$.

Figure 3

Probability distributions $P_{\lambda }(\lambda ;x)$ at four different values of the controller level $x$ for $N=5$ targets. The vector $\mathit{K}$ was sampled from a uniform distribution in $[x_{min},x_{max}]$, $x_{min}={10}^{-2}$, $x_{max}=0.1,1,10$ (respectively top, middle, and bottom rows). Hill indices were fixed to $h_i=2$, $\forall i=1,\cdots ,N$. Red markers represent the numerical results, while solid lines display analytical results corresponding to the Gaussian [Eq. (35), cyan] and cubic [Eqs. (52, 53, 54), magenta] approximations.

Figure 4

Comparison between numerical results for the average capacity $\langle {log}_2\tilde{Z}\rangle$ (markers) and analytical results for the annealed average ${log}_2\langle \tilde{Z}\rangle$ obtained within the Gaussian [Eq. (35), cyan] and cubic (Eqs. (52, 53, 54), magenta) approximations. Vectors $\mathit{K}$ in the top panels and in the bottom panels are sampled respectively from a uniform and a uniform in log-scale distributions in $[x_{min},x_{max}]$, $x_{min}={10}^{-2}$, while Hill indices are fixed to $h_i=2$, $\forall i=1,\cdots ,N$. The dashed line gives the value of ${\tilde{I}}_{\mathrm{opt}}^{★}$ for each $x_{max}$. The absolute difference between quenched and annealed averages estimated at $x_{max}=40$ through the argument given in Appendix pp2 in the uniform (log-uniform) case is $\simeq 4\times {10}^{-3}$ ($2\times {10}^{-2}$) for $N=5$ and $\simeq {10}^{-3}$ ($3\times {10}^{-3}$) for $N=20$. Note that in the log-uniform case, for $N=20$ and large $x_{max}$, the difference between the cubic approximation and the quenched numerical average is comparable to the difference between quenched and annealed numerical averages.

Figure 5

Probability distributions $P_I\left({\tilde{I}}^{★}\right)$ for different $x_{max}$ (rows) and $N$ (columns). Markers represent the distribution obtained numerically from ${10}^4$ samplings of $\mathit{K}$ from a uniform distribution in $[x_{min},x_{max}]$ ($x_{min}={10}^{-2}$) and Hill indices fixed at $h_i=2\forall i=1,\cdots ,N$ (left two columns), and from ${10}^4$ samplings of $\mathit{h}$ from a uniform distribution in $[1,5]$ and fixed $\mathit{K}$ (chosen randomly in $[0.01,0.1]$ for the case $x_{max}=0.1$ and properly rescaled for the cases $x_{max}=1$ and $x_{max}=10$). The red and green lines represent the analytical form (71) with fitted and estimated parameters, respectively (see text for details). In all panels, purple stars denote the value of ${\tilde{I}}_{\mathrm{opt}}^{★}$.

Figure 6

Same as Fig. 5 but with $\mathit{K}$ sampled from a uniform distribution in log scale in $[x_{min},x_{max}]$, with fixed $x_{min}={10}^{-2}$ and $h_i=2\forall i=1,\cdots ,N$.