Figure 1

A schematic diagram of a self-regulating gene. The gene $\Gamma$ is depicted by a thick black line and a promoter start signal. Gene products $g$ denoted as blue circles can bind to the regulatory sites (one in this example) that control the expression of $\Gamma$. Direct control over the expression of $\Gamma$ is exerted by molecules of the transcription factor $c$ (green diamonds, two binding sites).Reuse & Permissions

Figure 2

Monostable, critical, and bistable (green, red, and blue lines, left to right, respectively) behavior of the self-activating gene for $n_g=2$. (a) The phase diagram as a function of $K_g$ and the input-dependent term. In the region between the black solid lines two solutions ${\overline{g}}_{1,2}$ exist for every value of the input $c$ ($y$ axis). The corresponding critical value is at $K_g^{*}=1/8$ (cusp of the black solid lines). Circles and crosses represent analytical approximations to the exact boundary of the bistable region for $n_g=2$ and large $1/K_g$ (see text). For three choices of $K_g$ denoted by vertical dashed lines, the input-output relations $\overline{g}\left(c\right)$ are plotted in (b). (b) The critical solution (red) has an infinite total derivative ${\overline{g}}^{\prime }\left(c\right)$ at $\theta -n_c\log (1+c/K_c)=3.29$, $\overline{g}=0.25$. The bistable system (blue) has three solutions, two stable and one unstable, for a range of inputs that can be read out from the plot in (a).Reuse & Permissions

Figure 3

Computing information transmission close to the critical point. (a) An input-output relation $\overline{g}\left(c\right)$ for $n_g=2$, $C=1$, and $K_g=K_g^{*}\left(n_g\right)+0.004$, showing a self-activating gene with an almost critical value of $K_g$. (b) Noise in the input from Eq. (14), which is the integrand in the expression for information in Eq. (36), shows an incipient divergence between red vertical bars. (Inset) Zoom-in of the peak shows that it can be sampled well by increasing the number of bins. Different plot symbols indicate domain discretization into 500 (black dots) and 5000 (red circles) bins. At the critical point the divergence is hard to control numerically. (c) An alternative way of computing the same information, by integrating in the output domain as in Eq. (69). Shown is the integrand in the Gaussian noise approximation [black, $S_2\left(\overline{g}\right)$ from Eq. (66)] and with quartic corrections [red (light gray), $S_4\left(\overline{g}\right)$ from Eq. (67)]. At the critical point higher-order corrections regularize the integrand, while away from the critical point the integrand smoothly joins with the Gaussian approximation. This approach is stable numerically both away from and in the critical regime. (d) Information $I={\log }_2\tilde{Z}$ as a function of $K_g$ for $n_g=2$. Critical $K_g^{*}=1/8$ is denoted by a dashed red line. Integration across $\overline{g}$ in the output domain with quartic corrections (squares) agrees well with the integration across $c$ in the input domain (crosses) away from $K_g^{*}$, but also smoothly extends to the critical $K_g^{*}$. This is a cut across the capacity plane in Fig. 4a (denoted by a dashed yellow line) for $C=1$.Reuse & Permissions

Figure 4

Information transmission as a function of self-activating parameters $\{n_g,K_g\}$ for three values of $C$; $C=0.1$ (left column), $C=1$ (middle column), $C=10$ (right column). For each value of $\{n_g,K_g\}$ the three remaining parameters $\{c_{1/2},n_c,K_c\}$ are optimized to maximize information. (a) The capacity planes showing the decrease in capacity in bits (colormap) from the maximal value achieved for an optimal choice of all parameters; the optimal set of parameters is denoted by a colored square on a yellow circle. The white upper left region of each capacity plane corresponds to the bistable region. For low $C=0.1$ the maximum is achieved in the interior of the domain (cyan, line (1)), while for $C=1,10$ $n_g$ is driven to 0 (magenta, line (2)), corresponding to the non-self-interacting solution. Red line (3) represents an example critical system at $n_g=2$ and the green line (4) represents a self-interacting system with a high value for $n_g$. The yellow dashed line in $C=1$ plane represents a cut shown in detail in Fig. 3d. The yellow solid line in $C=0.1$ plane represents a cut shown in detail in Fig. 5a. The red dashed boundary of the critical region in $C=1$ plane is analyzed in detail in Figs. 5b and 5c. (b) Input-output relations for four example systems denoted by colored lines in (a). Despite substantially different values for $\{n_g,K_g\}$, the optimization of remaining parameters makes the input-output curves look very similar to the optimal solutions, except for the critical [red (solid)] curves. (c) The effective input noise ${\sigma }_c\left(c\right)$ for the selected systems.Reuse & Permissions

Figure 5

Information transmission close to the critical line. (a) A detailed scan of information for $C=0.1$ as the function of $K_g$ for $n_g=2$ and optimal choice for ${\theta }_c$ for every $K_g$ exhibits a peak for a nontrivial value of $K_g$. (b) The information transmission along the critical line for $C=1$ as a function of $n_g$ indicates that while including quartic corrections is important (upper line), the information at the critical line does not exhibit any large jumps (not shown). (c) A detailed scan of capacity close to the critical line for $C=1$ as a function of $n_g$ (vertical axis). Shown in color is the increase in information in bits compared to the value very close to the critical $I^{*}\left(n_g\right)$, as a function of the distance in $K_g$ from the critical value (horizontal axis). Across the range of $n_g$, going from the critical axis into the monostable domain increases the information.Reuse & Permissions

Figure 6

Information transmission in a system where all parameters $\theta$ are optimized as a function of the maximal input concentration $C$. The self-interacting system [red (crosses and circles)] allows for an arbitrary MWC-like regulatory function Eq. (38) with parameters $\theta =\{c_{1/2},n_c,K_c,n_g,K_g\}$. The noninteracting system [black (solid dots)] only has the MWC parameters $n_c,K_c$ and the leak $L$ (see Ref. [7]) which can be reexpressed in terms of $c_{1/2}$. The (red) line with circles shows self-activating solutions which are optimal for $C<1$, while the (red) line with crosses shows self-repressing solutions, optimal for $C>1$. Plotted on the secondary vertical axis in green (solid line) is the ratio between the self-interacting contribution to $F$, and the input contribution to $F$ in the expression for the MWC regulatory function Eq. (38). For $C\sim 1$ where the interacting and noninteracting solution join, this term falls to 0, as expected.Reuse & Permissions

Figure 7

Information transmission as a function of self-repressing parameters $\{n_g,K_g\}$ for $C=1$ (left column) and $C=10$ (right column); plot conventions are the same as in Fig. 4. (a) The capacity decrease from the maximum value (achieved at the parameter choice indicated by a yellow circle) as a function of $\{n_g,K_g\}$. The maximum information transmission is achieved for a noninteracting case (magenta, line (2)), for $C=1$. In contrast, for $C=10$, there is a nontrivial optimum for small values of $K_g$ and $n_g\approx -2.5$ (red, line (3)). (b) The mean input-output solutions for three example systems from A (cyan, line (1); magenta, line (2); red, line (3)). (c) The effective noise in the input, ${\sigma }_c\left(c\right)$, for example solutions in (a).Reuse & Permissions

Figure 8

The dependence of information transmission in the self-activating case on the ratio ${\gamma }_{\max }/C$. (a) For various choices of $C$ indicated on the plot, the information transmission with the optimal choice with respect to all parameters $\theta$ is shown as a function of ${\gamma }_{\max }/C$. Two special systems of interest [arrows, blue (dark gray) and green (light gray)] are chosen for the lowest value of $C$. (b) The mean input-output relation for the blue and green system. The green system has a higher transmission, a steeper activation curve but a smaller dynamic range. (c) The effective noise in the input, ${\sigma }_c\left(c\right)$, for the blue and green systems. The green system is closer to critical at the point where the mean input-output curve has the highest curvature and the noise exhibits a dip.Reuse & Permissions

Figure 9

Crossing the critical line (dashed red) into the bistable region. (a) Capacity as a function of $n_g$ at fixed $K_g=0.05$, with the optimal choice of $\{n_c,K_c,c_{1/2}\}$ parameters, for three values of $C$ [$C=0.1,1,10$ dark (gray) to bright red (gray), respectively]. Dots show capacity calculation using the bistable code that can handle multiple branches using Eq. (85), solid line uses the monostable integration as in Eq. (36). (b) Optimal capacity at very high ratios ${\gamma }_{\max }/C={10}^3$ for different value of $C$ ($C={10}^{-3},{10}^{-2},{10}^{-1},1$: circles, crosses, squares, and stars, respectively). The optimum is pushed toward the critical line from the monostable side for ${\gamma }_{\max }/C$ large and $C$ small. In all cases, information in the bistable regime is smaller than in the monostable regime.Reuse & Permissions