Collective is different: Information exchange and speed-accuracy trade-offs in self-organized patterning

TL;DR

This work develops a minimal, analytically tractable model of self-organized patterning via local cell–cell communication and lateral inhibition. By framing the dynamics as a stochastic reaction network, it enables exact computation of trajectory-based mutual information and transfer entropy, revealing how information flows between cells during pattern formation. The key findings show a speed–accuracy trade-off, that globally optimal patterning does not maximize intercellular information transfer, and that instantaneous information can be non-monotonic in time, with experimental data from Drosophila SOP formation qualitatively supporting these predictions. The results illuminate fundamental principles of decentralized self-organization and provide a quantitative bridge between theory and live-cell imaging of Delta–Notch signaling.

Abstract

During development, highly ordered structures emerge as cells collectively coordinate with each other. While recent advances have clarified how individual cells process and respond to external signals, understanding collective cellular decision making remains a major challenge. Here, we introduce a minimal, analytically tractable, model of cell patterning via local cell-cell communication. Using this framework, we identify a trade-off between the speed and accuracy of collective pattern formation and, by adapting techniques from stochastic chemical kinetics, quantify how information flows between cells during patterning. Our analysis reveals counterintuitive features of collective patterning: globally optimized solutions do not necessarily maximize intercellular information transfer and individual cells may appear suboptimal in isolation. Moreover, the model predicts that instantaneous information shared between cells can be non-monotonic in time as patterning occurs. An analysis of recent experimental data from lateral inhibition in Drosophila pupal abdomen finds a qualitatively similar effect.

Figures (9)

• Figure 1: Self-organization through lateral inhibition. (A) In our model, cells have an internal state abstractly ranging from an inhibitor-like state to an inhibited-like state, along with a receiver state which can be activated by neighboring cells. This allows cells to self-organize into a target pattern, here 3 cells organize so that there is one inhibitor cell and two inhibited cells. Throughout, we number cells counterclockwise starting from the leftmost cell; here cell 3 is the inhibitor. (B) Possible transitions for a single cell in the model and corresponding transition rates. Only the rate of receiving the signal, $k^+$, is a function of the neighboring cell states. (C) Delta-Notch signaling as a motivating biological example of lateral inhibition. When Delta (orange) on one cell binds to Notch (blue) on a neighboring cell, Notch is cleaved, releasing its intracellular domain and activating downstream transcriptional responses.
• Figure 2: Self-organizing model is constrained by a speed-accuracy trade-off. (A) Kymographs showing example trajectories from optimized system with error rate $\epsilon = 0.02$ (top) and $\epsilon = 0.1$ (bottom), illustrating how one cell advances stochastically towards the inhibitor state and signals to its neighbors. Color represents internal state, blue line represents a cell receiving a signal. The corresponding multicellular state for select time points is illustrated above for the first kymograph. Throughout this figure, a model with $M=3$ cells with $N=6$ internal states is shown. (B) Average internal state for each cell conditioned on a successful trajectory where the inhibitor state was reached by cell 3, for $\epsilon=0.02$ (top) and $\epsilon = 0.1$ (bottom). Solid lines show the exact average, faint lines show $100$ stochastic realizations. (C) Optimized model parameters for $\epsilon=0.02$ (left) and $\epsilon = 0.1$ (right). (D) Optimizing the mean first passage time over the set of model parameters $p$, while constraining the error, finds a speed-accuracy trade-off curve that all models are bounded by (black points). The same trade-off curve obtained using our sampling-based approach (teal points) was computed using $10,000$ Gillespie simulations, with error bars showing $1.96 \times$ standard error. Additionally, $5000$ randomly sampled parameter values (gray points) are shown, demonstrating that the entire region above the trade-off curve is accessible. Blue and red dotted lines show the error rates $\epsilon=0.02$ and $\epsilon = 0.1$ respectively. (E) Effective "energy landscape" for internal state dynamics are shown for a cell that is not receiving a signal ($s=0$, top) and a cell that is receiving a signal ($s=1$, bottom). Points represent the "energy" level of each state and lines represent the height of the effective energy barrier between states (SM Sec. II).
• Figure 3: Quantifying information shared between cell trajectories finds a non-monotonic rate of information transfer as well as a greater amount of information shared between trajectories than final states alone. (A) For any pair of cells, we can compute the transfer entropy (TE) rates in both directions $\dot{\mathcal{T}}^{u_1\to u_2}$, $\dot{\mathcal{T}}^{u_2\to u_1}$, the mutual information (MI) rate $\dot{I}= \dot{\mathcal{T}}^{u_1\to u_2} + \dot{\mathcal{T}}^{u_2\to u_1}$ (left), as well as the corresponding integrals (right), shown here for the optimized model with $\epsilon = 0.02$. (B) Total information transferred between any pair of cells for optimized models as the error rate is varied. Each value represents the limiting mutual information or transfer entropy for a given model at large time. Instantaneous mutual information shared between the final states of both cells is also shown. (C) Transfer entropy rates between cell 3 and the remaining cells, $\dot{\mathcal{T}}^{u_3\to [u_1,u_2]}$, the converse $\dot{\mathcal{T}}^{[u_1,u_2]\to u_3}$, as well as the mutual information rate (left) along with the corresponding integrals (right), shown here for the optimized model with $\epsilon = 0.02$. (D) Total transfer entropy from one cell to the remaining cells and the converse, as well as the mutual information for optimized models as the allowed error rate is varied. Instantaneous mutual information shared between the final state of cell 3 and the remaining cells is also shown. Throughout, each computation is averaged over $n=10,000$ Monte-Carlo samples and shaded regions show $\pm 1.96\times$ standard error. Integral values are expressed in units of nats, while rates are given in nats per unit time.
• Figure 4: Conditioning on final cell fates reveals persistent and asymmetric information transfer between cells. Here, all information theoretic quantities are computed for a process conditioned on the terminal state of cell $3$ ending up as an inhibitor $u_3=N$, while cells 1 and 2 are inhibited, $u_1=u_2=0$. (A-B) Schematic where arrow thickness is proportional to the long time conditional transfer entropy between pairs of cells (A) and the inhibitor cell and the remaining pair (B). (C) Conditioned mutual information and transfer entropy rates between cell $2$ and cell $3$ (top) as well as their corresponding integrals (bottom). Interestingly, the conditional information transferred from the inhibited cells is small but positive. (D) Conditional mutual information and transfer entropy rates between cell $3$ and and its neighbors (top), alongside their corresponding integrals (bottom). Throughout, each computation is averaged over $n=10,000$ Monte-Carlo samples, and the shaded regions show $1.96 \times$ standard error. Integral values are expressed in units of nats, while rates are given in nats per unit time.
• Figure 5: Instantaneous information quantities can be non-monotonic in time during self-organization. The information quantities, $CI$, $I(u_1(t);u_2(t))$, and $I([u_1(t),u_2(t)];u_3(t))$, are computed from the instantaneous state of the system at time $t$, shown here for $\epsilon=0.02$. All quantities are expressed in units of nats.

Theorems & Definitions (2)

• Lemma 1
• Proposition 1