Optimizing information transmission in optogenetic Wnt signaling

TL;DR

The results suggest optogenetic Wnt signaling allows for regulatory control beyond a simple binary switch, and provides a framework to apply ideas from information processing to single-cell in vitro experiments.

Abstract

Populations of cells regulate gene expression in response to external signals, but their ability to make reliable collective decisions is limited by both intrinsic noise in molecular signaling and variability between individual cells. In this work, we use optogenetic control of the canonical Wnt pathway as an example to study how reliably information about an external signal is transmitted to a population of cells, and determine an optimal encoding strategy to maximize information transmission from Wnt signals to gene expression. We find that it is possible to reach an information capacity beyond 1 bit only through an appropriate, discrete encoding of signals: using either no Wnt, a short Wnt pulse, or a sustained Wnt signal. By averaging over an increasing number of outputs, we systematically vary the effective noise in the pathway. As the effective noise decreases, the optimal encoding comprises more discrete input signals. These signals do not need to be fine-tuned to achieve near-optimal information transmission. The optimal code transitions into a continuous code in the small-noise limit, which can be shown to be consistent with the Jeffreys prior. We visualize the performance of different signal encodings using decoding maps. Our results suggest optogenetic Wnt signaling allows for regulatory control beyond a simple binary switch, and provides a framework to apply ideas from information processing to single-cell in vitro experiments.

Figures (9)

• Figure 1: (A) Optogenetic control of Wnt-signaling. In the absence of light, there is no Wnt signal and no expression of TopFlash $g$. When the light is activated, Wnt target genes are expressed. We vary the duration $t$ of the Wnt signal and measure the resulting gene expression $g$. (B) The histograms over $g$ are long-tailed, left-skewed, and unimodal; shown here for Wnt signal durations $t = 5$, $10$, $15$, and $20$ hours. Black lines show the gamma distribution from Eq. \ref{['eq:likelihood']}, evaluated at the appropriate $t$. (C) The mean $\mu_g(t)$ of each histogram scales linearly with the standard deviation $\sigma_g(t)$. The black line shows the linear dependence predicted by Eq. \ref{['eq:likelihood']}. (D) Rescaling the histograms (dividing by the standard deviation) shows a collapse of gene expression data. The collapsed data is described well by a gamma distribution with shape parameter $\hat{k} \approx 2.88 \pm 0.01$ and unit variance (black line). (E) The mean gene expression $\mu_g(t)$ grows linearly with time, as captured by Eq. \ref{['eq:likelihood']} (black line). Error bars show the standard deviation. (F) We can view our system analogously to a communication channel where input $t$ is mapped to output $g$ via the noisy transmission probability $p(g \vert t)$.
• Figure 2: (A) The input signal $t^{*}$ leads to a gene output $g$ drawn from the transmission probability $p(g \vert t^{*})$. Based on a measurement of $g$, one can use the posterior distribution $p(t \vert g)$ to infer the input signal. (B) Decoding map $p^{(1)}(t \vert t^{*})$ from Eq. \ref{['eq:decoding_map_1cell']}, showing the average probability assigned to $t$ by the posterior $p(t\vert g)$ given that the true signal is $t^{*}$. Here, the input distribution $p(t)$ is uniform over all possible signals $t \in [0,\infty)$ hours.
• Figure 3: (A) Binary input distribution of optogenetic Wnt signals, containing (i) an "off" state of $t=0$ hours and (ii) an "on" state of $t=\Delta t$ hours. (B) Mutual information $I(g;t)$ as a function of the duration $\Delta t$ of the on state. Results from the data (blue) and predictions from the gamma distribution in Eq. \ref{['eq:likelihood']} (black) are shown. Error bars obtained via subsampling. The upper bound $I \le 1$ (gray line) corresponds to perfect distinguishability of the on and off states
• Figure 4: (A) Optimal encoding of optogenetic Wnt signals for single cells: the Blahut--Arimoto algorithm converges to a discrete solution $p_{\star}^{(1)}(t)$, consisting of three optimally distinguishable Wnt signal durations or "symbols". (B) Decoding map $p_{\star}^{(1)}(t \vert t^\ast)$ obtained using the optimal prior: at the cost of discretizing the space of input signals, we gain distinguishability (cf. Fig. \ref{['fig:decodingmap_1cell']}B).
• Figure 5: (A) We show the information capacity achieved by the optimal encoding of Wnt signals (red line), and show convergence to analytical results in the small-noise and large-noise regimes. (B) The optimal code for Wnt signaling consists of a discrete number of symbols (blue dots). As the effective noise decreases, the optimal number of symbols increases, and approaches a continuous optimal code $p^{(\infty)}_{\star}(t)$ with a heavy tail that decays as $\sim 1/t$. The color of the markers (blue shade) indicates the relative probability mass of the symbols. (C) Decoding maps visualize how encoding strategies affect signal inference. Shown are decoding maps $p^{(N)}(t \vert t^{*})$ for ensembles of $N=2$ (top row) and $N=10$ (bottom row) cells. A uniform prior over opto-Wnt signals (left column) leads to broad posterior distributions, while the optimized discrete choice of signals (right column) yields more distinguishable responses and higher mutual information $I^{(N)}_{\star}(\bar{g} ; t)$, at the cost of discretizing the space of input signals.