Resource Limitations induce Phase Transitions in Biological Information Processing

TL;DR

It is found that resource limitations such as sensing reliability, intrinsic stochasticity, and energy cost induce discontinuous phase transitions between memoryless and memory-based strategies, even though the minimal models fall within the standard linear-quadratic-Gaussian class.

Abstract

Biological information processing manifests a huge variety in its complexity and capability among different organisms, which presumably stems from the evolutionary optimization under limited computational resources. Starting from the simplest memory-less responsive behaviors, more complicated information processing using internal memory may have developed in the evolution as more resources become available. In this letter, we report that optimal information processing strategy can show discontinuous transitions along with the available resources, i.e., reliability of sensing and intrinsic dynamics, or the cost of memory control. In addition, we show that transition is not always progressive but can be regressed. Our result obtained under a minimal setup suggests that the capability and complexity of information processing would be an evolvable trait that can switch back and forth between different strategies and architectures in a punctuated manner.

Figures (4)

• Figure 1: Schematic diagram of a minimal model of cellular information processing. (a) Stochastic dynamics of environmental state $x_{t}\in\mathbb{R}$, e.g., the fluctuation of environmental ligands, that the cell is sensing for adaptation. (b) Noisy receptor dynamics $y_{t}\in\mathbb{R}$, which transmits the environmental information into the cell. (c) Intracellular memory dynamics $z_{t}\in\mathbb{R}$, which encodes and accumulates the observed receptor signals for further information processing. (d) Downstream molecules exploit both instantaneous receptor observation $y_{t}$ and the memory state $z_{t}$ to generate optimal responses $\hat{x}^{*}_{t}$ (the orange curve) to the current environmental state $x_{t}$ (the black curve). The parameters are $D=100$, $E=500$, $F=1$, $Q=10$, and $M=1$.
• Figure 2: Phase transitions between optimal estimation strategies for control gain $\Pi_{zx}$ [Eq. (\ref{['eq: ABIP-optimal memory control']})] and objective function $J[v, \hat{x}]$ as functions of $E$ (a,b) and $F$ (c,d). Blue, brown, and orange dots correspond to the values for the strategies satisfying the stationarity condition of $J[v, \hat{x}]$. The responsive strategy is optimal in the blue region whereas the memory-dependent strategy is optimal in the orange region. The memory-dependent strategy (orange dots) appears discontinuously in the blue region where it is suboptimal (the responsive one is still optimal) and the optimality switches at the boundary of the blue and orange regions. The rest of the parameters are $D=100$, $Q=10$, and $M=1$.
• Figure 3: (a) A phase diagram of memory effect $\chi$ for observation noise $E$ and intrinsic noise $F$. Memory effect $\chi$, defined by $(J_{\rm without\ memory}^{*}-J_{\rm with\ memory}^{*})/J_{\rm without\ memory}^{*}$, quantifies the relative contribution of memory. $J_{\rm with/without\ memory}^{*}$ is the minimum value of the objective function with/without memory. (b) Phase boundaries as a function of $E$ and $Q/MF$. Blue, orange, and green curves correspond to the cases where $F$, $M$, and $Q$ are varied, respectively. (c,d) Control gain $\Pi_{zx}$ (c) and objective function $J$ (d) as a function of $E$ for $F=10$, demonstrating the forward and reverse transitions between responsive and memory-dependent strategies. The color codes and the rest of the parameters are the same as in Fig. \ref{['fig:transition']}.
• Figure 4: (a) Schematic diagram of a target tracking problem. An agent, e.g., a cell, estimates the position of the target $x_{t}^{\rm target}$ from noisy observation $y_{t}$ and memory $z_{t}$ and tracks it by controlling its own position $x_{t}^{\rm agent}$ through state control $u_{t}$. The memory control $v_{t}$ is jointly optimized with $u_{t}$ to compress past observation history $y_{t-dt},...,y_{0}$ into memory $z_{t}$ so that it provides the relevant information for state control. (b) The trajectories of the target $x_{t}^{\rm target}$ and the agent $x_{t}^{\rm agent}$ for $E=500$ and $F=1$. (c) Phase diagram of memory effect $\chi$ as function of observation noise $E$ and intrinsic noise $F$. Memory effect $\chi$ is defined as in Fig. \ref{['fig:diagram']}. (d,e) Phase transitions for control gain $\Pi_{zx}$ (d) and the objective function (e) as functions of $E$. The color codes are the same as in Fig. \ref{['fig:transition']}. The rest of the parameters are $D=100$, $Q=10$, $R=1$, and $M=1$.