Adaptation to time-varying environments in a reaction-diffusion model

TL;DR

The paper addresses how adaptation to time-varying environments can arise from basic physical processes. It develops a minimal reaction-diffusion model with many locally stable states and diffusion-enabled reproduction to show that environmental sequences are encoded in final states and can confer a competitive advantage in space. Adaptation emerges from environmental exposure alone, from spatial reproduction, and from teacher–student interactions, with learning, generalization, and collective learning demonstrated. The results provide a physically grounded framework for adaptation, connecting memory-like behavior in disordered and driven systems to social-like learning phenomena in coupled reactors.

Abstract

We present a spatially-extended system of chemical reactions exhibiting adaptation to time-dependent influxes of reactants. Here adaptation is defined as improved reproductive success, namely the ability of one of the many locally stable states available to the system to expand in space at the expense of other states. We find that adaptation can arise simply by environmental exposure to sequences of varying influxes. This adaptation is specific to the temporal sequence yet flexible enough to generalize to related sequences. It is enhanced through repeated exposure to the same environmental sequence, representing a form of learning, and through spatial interactions, enabling natural selection to act and representing a form of collective learning. Finally, adaptation benefits from a nearby adapted state, representing a form of teacher-guided learning. By combining environmental drives and reproduction within a stochastic reaction-diffusion dynamics framework, our model lays a foundation for a theory of adaptation grounded in physical principles.

Figures (7)

• Figure 1: Systems, states and environmental sequences. A system is specified by $N$ nodes, with mutual inhibitions between some pairs of nodes, represented by $\vdash\!\dashv$. The environment $\Lambda$ and the state of the system $\sigma$ are $N$-dimensional vectors with elements in $\{0,1\}$. Dynamically, a node becomes active, $\sigma_i=1$, when its influx is turned on, $\Lambda_i=1$, and no other node that inhibits it is active; a node becomes inactive as soon as its influx turns off, to $\Lambda_i=0$. A. A system with $N=2$ nodes, with an inhibitory interaction. When all influxes are on, i.e., $\Lambda=(1,1)$, the system can be in two states, $\sigma=(1,0)$ or $\sigma=(0,1)$. Top graph: continuous dynamics, showing stochastic switches between the two states; bottom: the binary model used here. As indicated in the table, turning off one of the fluxes eliminates the possibility of one of the states. B. The same principles apply to larger networks combining multiple nodes and interactions between them, where more states are present and where the relation between the influxes and the states is more intricate, as illustrated here with a network of $N=4$ nodes. C. An environmental sequence is defined by a series of $2m$ nodes $(i_1,\dots,i_{2m})$ where each node appears an even number of times. It represents a series of events where fluxes $\Lambda_i$ are successively turned either on or off, starting and ending with all fluxes on. For example, in the environmental sequence $(1,2,1,2)$, $\Lambda_1$ is turned off, followed by $\Lambda_2$, before $\Lambda_1$ is turned on, followed by $\Lambda_2$. Here, the dynamics of the system with $N=2$ is represented when experiencing two particular environmental sequences, $(1,2,1,2)$ and $(1,2,2,1)$, starting each time from the two possible initial conditions, $\sigma=(1,0)$ and $\sigma=(0,1)$. In this example, the final state depends on the environmental sequence while the initial state is forgotten.
• Figure 2: Encoding of information from environmental sequences in final states. A. Mean distance between states reached after experiencing the same environmental sequence for varying values of $m$. Two trajectories are run under the same environmental sequence and the distance (normalized Hamming distance) between the two states at the end is reported. For $m$ sufficiently large ($m/N\gg 1$) states obtained after the same environmental sequence are identical. Inset: mean number of active nodes (productivity) as a function of the number of steps with $m=500$, starting from 3 different initial conditions, showing convergence. B. Number of different states reached after a long environmental sequence ($m=1000$) when sampling uniformly environmental sequences, measured as the Shannon diversity, with an exponential fit, shown as a function of the number $N$ of reactions. C. Distance between states obtained from two environmental sequences differing by a given number of swaps, with $m=1000$. D. Mean number of different state-cycles for a given environmental sequence (from exhaustive search).
• Figure 3: Competition and adaptation. A. Two states obtained in independent reactors under cycles of two distinct environmental sequences $\epsilon_1$ and $\epsilon_2$ are competed in two coupled reactors under cycles of environmental sequence $\epsilon_1$. Signed success is the fraction of times the state $\sigma_3$ resulting from the competition is more similar to $\sigma_1$, the state evolved under the same environment, minus the fraction of cases where it is more similar to $\sigma_2$. B. The two environmental sequences $\epsilon_1$ and $\epsilon_2$ share a number of last steps in common. Given that there are $2m$ steps, they are identical when steps$/m=2$. C. The environmental sequences of the two environments differ by $n$ swaps, showing ability to discriminate changes in the order of steps within an environmental sequence. D. Two competing states are obtained from the same trajectory at different times, an young state $\sigma_1$ and a old state $\sigma_2$. E. Here the old state is taken after $64$ cycles, the young one from a varying number of cycles.
• Figure 4: Generalization. A. Same scheme as Fig. \ref{['fig:adapt']}A but B. the environmental sequences applied during the competition differs from the two used in the previous step. The environmental sequences $\epsilon_1$ and $\epsilon_3$ are independently derived by $n$ swaps from a reference $\epsilon_0$, while the environmental sequence $\epsilon_2$ is an independent random permutation of $\epsilon_0$. C. Generalization success as a function of $n/m$.
• Figure 5: Coupling to an adapted state. A. A teacher is defined as a state that went through cycles of an environmental sequence. This state seeds a reactor that is coupled to another reactor. After a varying number of cycles in the same environmental sequence (x-axis in B), the final state of the second reactor defines an instructed student. Similarly, an untaught student is defined in the absence of a teacher. Finally, the instructed and uninstructed students compete in two coupled reactors. B. Advantage in competition of being instructed as a function of the number of cycles in the second phase of the protocol. As expected, when learning over sufficiently many cycles, there is no longer an advantage to having a teacher present.