Nonlocal decoding of positional and correlational information during development

TL;DR

The paper addresses how nonlocal information from cell-cell communication can sharpen positional readouts in morphogen-driven patterning by leveraging spatial correlations. It formalizes Bayes-optimal decoding with correlational information (CI) and structural priors, deriving upper bounds on positional information gain under Relative Locations Prior (RLP) and Absolute Locations Prior (ALP). It then offers algorithmic approximations—primarily spatial convolution (pooling) and divisive normalization—that realign local readouts into effective signals amenable to local decoding, and demonstrates that a minimal reaction-diffusion network implementing these steps reproduces the predicted gains. The findings show that CI can significantly boost patterning precision, with the direction and magnitude of gains depending on the chosen prior, and reveal a plausible, mechanistic route for nonlocal decoding via simple biochemical networks. Overall, the work highlights the lattice geometry and intercellular communication as central determinants of gradient readout efficiency and provides a framework linking optimal information processing to concrete developmental circuitry.

Abstract

In many developmental systems, cells differentiate into a tissue by reading out morphogen concentration fields, a process fundamentally limited by noise. How much can the precision of this process be improved by nonlocal information, e.g., via cell-cell communication? Using a Bayes-optimal framework, we show that positional inference depends crucially on morphogen spatial correlations and on the ``structural prior'' that encodes the geometry of the cellular lattice performing the readout. We derive upper bounds on positional information gain due to nonlocal readout and identify signal processing algorithms that approximate optimal positional inference, as well as simple chemical reaction schemes which implement such algorithms. Our theory suggests that correlational information can be exploited to significantly enhance developmental precision.

Figures (8)

• Figure 1: (a) Cells arranged into an approximate 1D lattice (squares; here and throughout the paper, we consider $N=100$ cells) read out a local, noisy morphogen signal (circles with different shades of green) to infer their positions and commit to corresponding discrete cell fates (here, blue/white/red square colors). Signal noise can result in imprecise fate assignments (left); its impact could be mitigated by information exchange between cells, e.g., via diffusion-mediated averaging or other cell-cell coupling mechanisms (right). (b) Three example morphogen ensembles (first column; color lines = individual profiles; black = mean) carrying the same PI and thus having identical local decoding maps (second column) petkova_Optimal_2019, but differing in the spatial correlation structure (third column--see Section S2.A in SMnote for details on the covariance matrix parametrization) and therefore in the amount of correlational information, CI (fourth column; dashed vertical line at $I=\log_2(N)\approx 6.6$ bits = perfect specification of every cell in the lattice.)
• Figure 2: A focal cell $i$ (pink square) improves its positional inference with nonlocal morphogen information $g_j$ read out by neighboring cells $j\in\mathcal{N}_i$ (squares left and right of the pink focal cell). (a) For RLP, the nonlocal signal is interpreted in the context of perfectly known relative location, $d_{ij}$, of each "sender cell" that conveys the nonlocal signal. In reality, the relative distances would be imperfectly known (illustrated by springs). (b) For ALP, absolute locations, $x_j$, of "sender cells" are assumed known. In reality, the absolute locations would be imperfectly known (illustrated by springs). (c) Correlational information (CI) in the morphogen ensemble increases with the correlation length of the Gaussian fluctuations, $L_{\rm corr}$, and their magnitude $\sigma_E$ relative to the intrinsic (uncorrelated) noise with magnitude $\sigma_I$ (colorbar). (d) Maximal PI increase with nonlocal decoding using RLP (circles) or ALP (triangles). Color indicates the neighborhood size $|\mathcal{N}_i|$ (dark blue: $\pm 1$ nearest neighbor; dark red: entire system of $N$ cells). Here and for the rest of the paper, we consider $\sigma_{E} /\sigma_{I} = 15$ (violet line in (c)). Dashed grey line: $\Delta \mathrm{PI}$ corresponding to perfect cellular specification.
• Figure 3: (a) Algorithmic transformations map morphogen profiles $\mathbf{g}$ into effective profiles $\mathbf{g}^{\rm eff}$ via spatial convolution and/or divisive normalization; these effective profiles can be subsequently decoded locally via the optimal rule of Eq. \ref{['stddec']}. (b) Information gain $\Delta PI$ for convolution and divisive normalization in the $L_{{\rm corr}}=0$ and $L_{{\rm corr}} \gg N$ limits (blue and green), respectively. $\Delta \mathrm{PI}$ achieved by: (c) convolution with exponential kernels of varying radii (colorbar; green for the optimal radius), across a range of CI values; (d) divisive normalization with a normalization constant obtained via convolution with kernels of varying radii; (e) sequential application of convolution (variable radius) and global normalization (sum over all $N$ cells). (f) For each correlation length $L_\mathrm{corr}$, the optimal kernel radius differs across the three algorithmic strategies. (g) A minimal reaction-diffusion circuit with two diffusible normalizer species ($N_1$, $N_2$) implements algorithmic operations of (a-f). (h-k) Same analyses as in (b-e), but using the reaction--diffusion implementation. (i)$\Delta \mathrm{PI}$ associated with the steady-state profile $N_1$ while varying its diffusion length $\sqrt{D_1}/N$. (j)$\Delta \mathrm{PI}$ associated with $G/N_1$ while varying $\sqrt{D_1}/N$. (k)$\Delta \mathrm{PI}$ associated with $N_1/N_1$ while varying $\sqrt{D_1}/N$, with $\sqrt{D_2}/N$ fixed at $\approx 2.23 \gg \sqrt{D_1}/N$. (l) For each correlation length, the reaction--diffusion model exhibits optimal diffusion lengths analogous to the optimal kernel radii in (f).
• Figure S1: $CI$, $\Delta PI_{{\rm RLP}}$, and $\Delta PI_{{\rm ALP}}$ as a function of the correlation length of the fluctuations and the ratio between the magnitude of the " extrinsic" and " intrinsic" parts of the fluctuations. The results shown here are for the case in which the neighborhood corresponds to the full embryo size i.e., $|\mathcal{N}_i|=N$.
• Figure S2: Algorithmic $\Delta PI$ for convolution, divisive normalization, and their sequential application. The three rows correspond to different shapes of the convolution kernels. The color map represents the size of the convolution kernel. Right column: optimal kernel sizes.