Fluctuation sensitivity of a transcriptional signaling cascade

The internal biochemical state of a cell is regulated by a vast transcriptional network that kinetically correlates the concentrations of numerous proteins. Fluctuations in protein concentration that encode crucial information about this changing state must compete with fluctuations caused by the noisy cellular environment in order to successfully transmit information across the network. Oftentimes, one protein must regulate another through a sequence of intermediaries, and conventional wisdom, derived from the data processing inequality of information theory, leads us to expect that longer sequences should lose more information to noise. Using the metric of mutual information to characterize the fluctuation sensitivity of transcriptional signaling cascades, we find, counter to this expectation, that longer chains of regulatory interactions can instead lead to enhanced informational efficiency. We derive an analytic expression for the mutual information from a generalized chemical kinetics model that we reduce to simple, mass-action kinetics by linearizing for small fluctuations about the basal biological steady state, and we find that at long times this expression depends only on a simple ratio of protein production to destruction rates and the length of the cascade. We place bounds on the values of these parameters by requiring that the mutual information be at least one bit—otherwise, any received signal would be indistinguishable from noise—and we find not only that nature has devised a way to circumvent the data processing inequality, but that it must be circumvented to attain this one-bit threshold. We demonstrate how this result places informational and biochemical efficiency at odds with one another by correlating high transcription factor binding affinities with low informational output, and we conclude with an analysis of the validity of our assumptions and propose how they might be tested experimentally.

Figure 1

A schematic representation of a transcriptional regulatory cascade. Each circle represents the population-averaged cellular concentration of the denoted species, and each arrow indicates that the chemical species at the tail of the arrow promotes the transcription of the species at the head of the arrow. The species labeled $S$ is the signal that triggers the cascade, and the subsequent species, labeled $R_1,R_2,...,R_n$, are the first response, second response, etc., to the signal.

Figure 2

The mutual information of an $n$-link transcriptional cascade (solid black curve, in units of bits) plotted versus the dimensionless parameter $\alpha {\sigma }_S/{\sigma }_{R_n|S}$. The gray dashed curves are approximate fits to the curve at small and large parameter values, and the intersection of the red dashed lines marks where the information curve reaches the one-bit threshold.

Figure 3

Plots of the $t\to 0$ and $t\to \infty$ limits of the mutual information of a transcriptional signaling cascade [Eq. (23)] as a function of cascade length. The information is plotted in bits, and all links are assumed to be characterized by the same rate constant ratio, $k/k_D$. (a) The $t\to 0$ limit plotted for, from bottom to top, $k/k_D=3/2$, 2, 5/2, and 3. Solid curves are the exact limit and the dotted curves are the $k\gg k_D$ approximation. (b) The same as panel (a), but for $k/k_D=1/2$, 5/8, 3/4, and 7/8 (also from bottom to top). In this case the dotted lines are the $k\ll k_D$ approximation. (c) Same as panel (a) but for the $t\to \infty$ limit and its $k\gg k_D$ approximation. (d) Same as panel (b) but for the $t\to \infty$ limit and its $k\ll k_D$ approximation.

Figure 4

(a) Stochastic simulation-extracted long-time mutual information $I_{\infty }(n,k)$ (in bits) plotted as a function of cascade length $n$ for $k/k_D=4$. From bottom to top, the different curves are for $\langle N_S\rangle =2$, 4, 8, 16, 32, 64, and 128. (b) The same plot, but for $k/k_D=1/2$. Note that in this panel, the ordinate axis is on a logarithmic scale in order to emphasize the exponential behavior of the curves as $\langle N_S\rangle$ increases.

Figure 5

Four constraints on the kinetic ratio $k/k_D$ are plotted as functions of cascade length. The dashed curve at $k/k_D=1$ and the solid curve sans data points are both lower bounds representing values of $k/k_D$, below which single-bit mutual information cannot be achieved. They can both be derived from Eq. (24). The greatest such lower bound can be derived from Eq. (27) and is plotted as a solid curve with data points. It is valid only for a cascade with equivalent signaling links, unlike the other three constraints, which are always valid if $k$ is replaced with $\overline{k}$. The final dashed curve at $k/k_D=\sqrt{3}$ is a hard upper bound on the possible lower bounds. It can be used in place of the greatest lower bound in the event that the $k_i$ are not equal.

Figure 6

(a) The threshold performance ratio required for one-bit, steady-state information transmission in a single-link transcriptional cascade obeying Hill kinetics, plotted versus the kinetic sensitivity $\nu$. The curves, from bottom to top, are for binding efficiencies $\langle S\rangle /K$ of 1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, and 5. (b) The mutual information of that same system (in bits) plotted as a function of $\nu$ for fixed performance ratio $\langle R\rangle /\langle S\rangle =1$ and the same nine values of the binding efficiency (this time from top to bottom).