## Abstract

Stochasticity inherent to biochemical reactions (intrinsic noise) and variability in cellular states (extrinsic noise) degrade information transmitted through signaling networks. We analyzed the ability of temporal signal modulation—that is, dynamics—to reduce noise-induced information loss. In the extracellular signal–regulated kinase (ERK), calcium (Ca^{2+}), and nuclear factor kappa-B (NF-κB) pathways, response dynamics resulted in significantly greater information transmission capacities compared to nondynamic responses. Theoretical analysis demonstrated that signaling dynamics has a key role in overcoming extrinsic noise. Experimental measurements of information transmission in the ERK network under varying signal-to-noise levels confirmed our predictions and showed that signaling dynamics mitigate, and can potentially eliminate, extrinsic noise–induced information loss. By curbing the information-degrading effects of cell-to-cell variability, dynamic responses substantially increase the accuracy of biochemical signaling networks.

## Dynamic signals enhance information transfer

Cells need to process information about their external environment reliably to survive. However, variation, or noise, in biochemical reactions, or in the states of individual cells, make it hard for a cell to detect concentration, rather than just the presence or absence of an activating ligand. Selimkhanov *et al.* show that cellular signaling circuits get around this problem by continually monitoring signals over time. Such dynamic responses in cultured human cells more effectively distinguish signals from noise and thus avoid loss of information transmitted to the cell from external signals.

*Science*, this issue p. 1370

The role of biological signaling networks is to reliably transmit specific information about the extracellular environment to downstream effectors, allowing the cell to adjust its physiological state to changing conditions. The stochasticity of molecular interactions that underlies various forms of “noise” in biological systems can interfere with signal transduction and degrade the transmitted information. How signaling networks perform their core functions in the presence of noise is a fundamental question. Information-theoretic approaches allow estimation of the maximal possible information transmission capacity of noisy biochemical networks (*1*–*11*). Previous applications of such methods to the analysis of signaling networks suggested that as a result of noise, cells lose most of the information about the concentration of ligands (*12*–*14*). Thus far, the information-theoretic analyses of signaling networks have been based on scalar measurements performed at a single time point. However, the information on activating ligands is often encoded using a dynamic signal represented by a multivariate vector that contains a single cell’s response at multiple time points (*15*–*18*).

To test the hypothesis that dynamic responses contain more information than static responses, we performed single live-cell measurements of three key signaling pathways (Fig. 1): extracellular signal–regulated kinase (ERK), calcium (Ca^{2+}), and nuclear factor kappa-B (NF-κB) [supplementary materials (SM) section 1.1]. Fully automated computational image analysis (SM section 1.2) allowed us to measure the response of 910,121 individual live cells (figs. S1 to S6 and tables S1 to S3). The large sample size was instrumental for analyzing high-dimensional multivariate responses. In all three pathways, there was substantial variability within the dynamic (Fig. 1, C to E) and nondynamic (Fig. 1F) single-cell responses across multiple concentrations of activating ligands.

To analyze the implications of noise on information loss, we used an information-theoretic approach to calculate the information transmission capacity of a dynamic signaling network. The information transmission capacity [also referred to as channel capacity (*19*)] is measured as the maximal possible mutual information between the measured response and the activating ligand concentration. To calculate the mutual information between a dynamic response (a vector) and the ligand concentration (a scalar), we expanded on a previously described algorithm (*14*). The algorithm uses continuous multidimensional response data and a k–nearest-neighbor approach to estimate the conditional probability density for each cell’s response (SM section 2). We thus estimated the information transmission capacity of the dynamic response and of several types of static responses. For all single–time point static scalar responses, we found transmission capacity (<1 bit) (*12*, *13*) (Fig. 2A). However, across all three signaling pathways, the dynamic response had significantly higher information transmission capacity than several scalar responses previously described (*20*, *21*) (Fig. 2, B and C, Student’s *t* test, *P* < 0.05 for all comparisons, table S6). These estimates should be considered as lower bounds because they do not exclude variability resulting from experimental imperfections.

To elucidate the origins of the enhanced information transmission capacity of dynamic signaling responses, we developed a mathematical theory using information-theoretic formalism (SM section 3). The theory explicitly accounts for the information-degrading effects of intrinsic and extrinsic noise sources in the context of multivariate responses. Intrinsic noise adds to uncertainty in all dimensions (i.e., time points) independently from one another. In contrast, the extrinsic variability in cellular states produces fluctuations that are constrained by the signaling network that generates the dynamics. Therefore, the fluctuating components generated by extrinsic noise at different time points are deterministically dependent on one another. As a result, intrinsic and extrinsic noise sources have different effects on the information transmission capacity of multivariate responses. In the case of purely intrinsic noise, additional measurements increase the information logarithmically because of simple ensemble averaging (*12*). In the case of purely extrinsic noise, a sufficient number of dynamical measurements can provide complete information about the a priori uncertain internal state of the cell and therefore lead to a substantial gain in the information about the activating ligand (Fig. 3A).

To test our analytical prediction that the multivariate dynamic response can completely eliminate the information loss that results from introduction of extrinsic noise (SM section 3), we used computer simulations of ERK responses based on a published kinetic model (SM section 4.1) (*22*) (fig. S15). We generated sets of simulated ERK activity trajectories in response to an increasing number of ligand concentrations. We varied model values for ERK and mitogen-activated protein kinase kinase (MEK) according to a uniform distribution (±20% mean value) to mimic extrinsic noise and measured the information transmission capacity. Our analysis supported the analytical prediction and showed that whereas the univariate response, based on maximal ERK dynamics, had limited information transmission capacity, the dynamic multivariate response can transmit complete information about ligand concentration (Fig. 3B). An intuitive demonstration for the limitation of univariate response and the ability of multivariate response to overcome extrinsic noise is shown in Fig. 3, C and D. Superficially, the trajectories of two populations of simulated responses of ERK activity to two input concentrations of epidermal growth factor (EGF) appear overlapping (Fig. 3C), but in fact, they are completely separable when considering joint distributions (*23*). Plotting the distribution of ERK activity at *t* = 9 and *t* = 24 min on a two-dimensional (2D) plane (Fig. 3D) shows that the responses to a single varied parameter input lies on a one-dimensional (1D) curve within a 2D space. The two 1D manifolds for different inputs are completely separated from each other (inset), but overlap considerably in any 1D projection. This simple example demonstrates how the extrinsic variability of a single parameter can in principle be completely eliminated with measurements from only two time points.

The accuracy of a response can be characterized by its signal-to-noise ratio (SNR). The mutual information and the system’s SNR are related; however, this relationship is strongly affected by the noise properties (intrinsic versus extrinsic) and the type of the response. Our analytical theory predicts a different relationship between mutual information and SNR for three different types of responses: (i) scalar responses that do not distinguish between intrinsic and extrinsic noise; (ii) multivariate responses without any dynamic component (redundant measurements) that can only reduce intrinsic noise; and (iii) dynamic responses that combine the benefits of redundant measurements with efficient mitigation of extrinsic variability. We varied the SNR in the ERK network by partial inhibition of the ERK kinase MEK with six different dosages of the inhibitor U0126. At each MEK inhibition level, we measured ERK response to eight EGF levels. A total of 48 conditions were measured in four biological replicates (fig. S23). At each MEK inhibitor level, we calculated the mutual information and the SNR from single-cell responses (SM section 4.3.1). In total, Fig. 4 contains 535,107 cell responses (tables S4 and S5). As expected, for a scalar response, the formula relating the mutual information and the overall SNR is in very good agreement with our experimental measurements. The theoretical prediction of mutual information for the redundant measurement case requires knowledge of intrinsic–to–extrinsic noise ratio (IER). IER was estimated in two ways: by (i) quantifying the fluctuations in the later (quasistationary) portion of the response time series of our ERK data (fig. S21A) (ii) using data for repeated measurements of single-cell responses (*24*) (fig. S21B). The predicted mutual information based on redundant responses required IER values that are two to four orders of magnitude higher than experimentally estimated IER values (SM section 4.3.3) (fig S17). In contrast, the measured mutual information values were in good agreement with the theoretical prediction for a dynamic response based on a computational ERK model (SM section 4.1) (fig. S18). Overall, this analysis demonstrates that the substantial information gain from multivariate measurements is indeed the direct result of the dynamic nature of ERK response.

The robustness of biological systems is epitomized by their ability to function in the presence of a large variability in cellular states (*25*, *26*). Signaling dynamics allow biochemical networks to mitigate variability in the cellular state and thereby maximize the information transmission capacity of signaling networks. Although the theory and observations presented here focus on the information transmission capacity of the dynamics of a single signaling molecule, the extension of our analysis to the case of multiple signaling molecules responding to one ligand is straightforward. Not all of the information contained in the dynamical responses may actually be used by cells. Yet, because reliable information transmission is a fundamental function of cellular signaling networks, it is plausible that evolutionary pressures shaped the cellular machinery to maximize the reliable decoding of multivariate dynamic signals.