Information bounds the robustness of self-organized systems

Abstract

Self-assembled systems, from synthetic nanostructures to developing organisms, are composed of fluctuating units capable of forming robust functional structures despite noise. In this Letter we ask: are there fundamental bounds on the robustness of self-organized nano-systems? By viewing self-organization as noisy encoding, we prove that the positional information capacity of short-range classical systems with discrete states obeys a bound reminiscent of area laws for quantum information. This universal bound can be saturated by fine-tuning transport coefficients. When long-range correlations are present, global constraints reduce the need for fine-tuning by providing effective integral feedback. Our work identifies bio-mimetic principles for the self-assembly of synthetic nanosystems and rationalizes, on purely information-theoretic grounds, why scale separation and hierarchical structures are common motifs in biology.

Figures (7)

• Figure 1: Reproducible self-assembly from the balance between transport and interactions. a. Synthetic self-assembled systems with prescribed boundaries are subject to noise, which can limit yield and functionality. b. Viable biological self-organization requires precise establishment of chemical gradients in response to pre-existing or external symmetry breaking. c. The self-organizing process can be understood as a noisy encoding process. Are there limits on the quality of the encoder depending on the nature of the spatial process, represented here by a nonlinear differential operator $\mathcal{L}$? d. We focus on a minimal model of cellular decision-making, where two interacting species of particles diffuse around and carry opposing signals generated at diametrically opposed signaling sources. At a specified time $T$, the sign of the local difference in signal concentration sets the cell fates. e. This system is mapped onto a diffusive Ising model, where particles diffuse at a set rate and change type according to local concentrations. f. At fixed particle numbers, we find that cell fates are most reproducible at intermediate diffusion. g. This reproducibility is quantified by information-theoretic quantities. While the outcomes are most reproducible at large $D$, the most informative patterns require intermediate diffusion. Simulation in g are done with $\Delta x = L/8$, $a=L/48$, $T=500$, $\gamma=1$, $\beta = 2\beta_c$, $h_0=2\beta$, where $\beta_c = \ln(1+\sqrt{2})$; distributions are estimated from $500$ replicates.
• Figure 2: A universal bound on positional information in short-ranged correlated systems. a. Variance in domain position is minimal when domains walls have a width $\ell \approx L$. If $\ell$ is too large, then domain wall solutions are unstable. Signal bias is here normalized by average total density. Full black line indicates mean signal, dashed lines one standard deviation, lighter lines are replicates. b. This optimum in variance can be understood through the energetic cost of shifting the domain wall: if $\ell \ll L$, then translating the domain wall changes very little the value of the signal at the boundary $\Delta m\approx 0$. If $\ell > L$ then the front solution is unstable. These effects materialize as a sharply peaked confinement stiffness. c. Positional information is maximal at intermediate diffusion, and saturate as the particle number density $N_{\Delta x}$ increases. Here, we have $N=16$ cells. d. The maximal value of PI at constant particle density $a = L/96$ depends on the number of cells $N=L/\Delta x$. e. This saturation can be explained by considering the relationship between positional information and the marginal probability $p(\theta)$ of finding a red region at $\theta$: in short-ranged correlated systems, PI is maximal when $p(\theta)$ is piecewise linear; in diffusive systems, a different optimum exists. f. This constraint leads to a system-size dependent bound on PI for systems with short-ranged correlations. All simulations have $h_0=3\beta, \beta=3\beta_c/2$, $\gamma=1$ and are averaged over $500$ replicates.
• Figure 3: Long-ranged interactions can stabilize patterns, allowing for increased robustness. a. A conserved number of molecules of two types bind to a membrane with a rate proportional to their concentration in the reservoir. Once on the membrane, the molecules diffuse with constant $D$ and unbind at a rate $r$ in the absence of molecules of the other type in the vicinity. When the other molecule type is present, the unbinding rate increase nonlinearly, leading to bistable surface concentration dynamics. b. The dependence of the binding rate on the reservoir particle number leads to global coupling which stabilizes the front position: if there are more $A$ (blue) than $B$ (red) bound, then binding of $B$ is favored, and vice versa. c. The positional information as a function of $D$ for variable noise is now much higher, and stays at its maximal value for a wider range of $D$. d. The marginal probability of finding $B$ at position $\theta$ can now be non-convex. Here we keep $a=L/96$. e. The maximal PI can now reach the theoretical maximum, beating the bound for short-ranged systems, with increasing particle numbers $L/a$ (lower noise) leading to better PI. In all simulations $r=1, k=4/3, N_0^{A,B} = 6(L/a), T=5\cdot 10^3/D, L=4, \alpha=30$, and results are averaged over $1000$ replicates.
• Figure E1: Understanding the emergence of collective behavior in finite systems. a. Starting from diffusive microscopic dynamics described by a master equation, we use exact coarse-graining methods to obtain a fluctuating hydrodynamics description in terms of a SPDE. Spatio-temporal averaging of fluctuating dynamics gives an effective deterministic hydrodynamic model. b. This process allows us to understand the emergent collective dynamics and use continuum modeling tools even in small discrete systems. c-d. We find that the collective response from microscopic simulations rapidly converge to their continuum predictions (black lines) with increasing particle numbers (c) and diffusivity (d). Dashed lines indicates expected absolute signal bias for a finite coarse-grained domain size $\Delta x$. e. The linear response in Fourier domain from simulations is well-described by the renormalization predictions, validating that the fluctuating hydrodynamic picture is accurate even in small systems with less than tens of particles per coarse-graining domain.
• Figure E2: The Ising model with external sources displays saturating positional information a. We consider a 1D Ising chain with broken translational symmetry by having non-zero opposing magnetic field $\pm h$ at two diametrically-opposed points. $J$ is related to the correlation length in this model by $\ell \sim 1/\ln J$. Here states are represented as white and black dots. b. As we vary the coupling strength for a given magnetic field strength $h=3$ for variable temperature $T$, we find a similar optimal coupling $\tilde{J}^*$ as in the DIM. Interestingly, as we reduce temperature (noise amplitude), the optimal range of $\tilde{J}$ expands but the PI stays bounded. Here, $N=8$. Dashed line indicates theoretical prediction for the bound. c. Marginal probability profiles are similar to the ones in the DIM case: Small $J$ lead to exponentially-peaked probabilities around source points, while large $J$ homogenize the system. Optimal values of $J$ lead to a sawtooth profile. d. Parameter search of $J$ and $h$ for variable $N$ show that the optimal region of $J,h$ and optimal value of PI both become smaller with increasing $N$.

Theorems & Definitions (1)

• Theorem