Emergent memory in cell-like active systems

Abstract

Active systems across scales, ranging from molecular machines to human crowds, are usually modeled as assemblies of self-propelled particles driven by internally generated forces. However, these models often assume memoryless dynamics and no coupling of internal active forces to the environment. Here, guided by the example of living cells, which have recently been shown to display multi-timescale memory effects, we introduce a general theoretical framework that goes beyond this paradigm by incorporating internal state dynamics and environmental sensing into active particle models. We show that when the self-propulsion of an agent depends on internal variables with their own complex dynamics - modulated by local environmental cues - environmental memory spontaneously emerges and gives rise to new classes of behaviours. These include memory-induced responses, adaptable localization in complex landscapes, suppression of motility-induced phase separation, and enhanced jamming transitions. Our results demonstrate how minimal information processing capabilities, intrinsic to non-equilibrium agents with internal states like living cells, can profoundly influence both individual and collective behaviours. This framework bridges cell-scale activity and large-scale intelligent motion in cell assemblies, and opens the way to the quantitative analysis and design of systems ranging from synthetic colloids to biological collectives and robotic swarms.

Figures (5)

• Figure 1: $\mathbf{(a)}$ Minimal model of crawling cell (blue) interacting with an external environment $U$. Its internal degrees of freedom $\mathbf{s}$ (orange beads) evolve in response to the local environment $\bm{\mathcal{E}}_x$ (green). To linear order, the response function $\mathbf{R}$ can be modelled by a network of (non reciprocal) harmonic springs. The self-propulsion force $p(\mathbf{s})$ drives the motion of the cell through the landscape. $\mathbf{(b)}$ Trap-and-release protocol applied to examples of linear spring networks: shaken Rouse chain (see SM IV for definition), which displays environmental memory (top) and equilibrium Rouse chain, with environment-insensitive dynamics (bottom). For $t<0$, the particle (blue monomer of position $x$) is confined by an harmonic trap, then released at $t=0$. $\mathbf{(c)}$ Memory-induced transient dynamics of the polarity correlator $c(t,T)=\langle p(t+T)p(T) \rangle$ for persistent MPs. Colours from red to blue denote increasing observation times after release $T/\tau_m = 0, 0.1, 0.5, 1$. Blue also corresponds to stationary state. Symbols show numerics, lines analytics (SM VI). Black symbols: AOUPs, insensitive to the protocols. $\mathbf{(d)}$ Same analysis for antipersistent MPs. Black symbols: equilibrium dimers, also protocol-insensitive. $\mathbf{(e)}$ Mean squared increments $\Delta(t,T)=\langle \left( x(t+T)-x(T) \right)^2 \rangle$ for persistent MPs, highlighting again protocol-dependent transients (analytics in SM VI). Curves for the control protocol collapse on the stationary (blue) line. Inset: crossover from short-time diffusion $D$ to long-time diffusion $D_\star$. $\mathbf{(f)}$ Same as in $\mathbf{(e)}$ for antipersistent MPs. Simulation details in SM VIII.
• Figure 2: $\mathbf{(a)}$ Stationary distribution $p(x)$ in a hard box of size $L=1$ for persistent MPs (orange) and AOUPs (red). Green dashed line shows uniform Boltzmann measure. Inset: perturbative predictions from SM VII (same colours) compared with simulations. $\mathbf{(b)}$ Same as $\mathbf{(a)}$ for antipersistent particles: antipersistent AOUPs (blue), MPs (magenta), equilibrium dimers (green). $\mathbf{(c)}$ Rescaled profiles $Lp(x/L)$ for persistent MPs for different box sizes $L$ (in units $\ell_\star$). $\mathbf{(d)}$ Same as $\mathbf{(c)}$ for antipersistent MPs. $\mathbf{(e)}$ Polarity correlator $c_s(t) = \langle p(t)p(0) \rangle$ for persistent AOUPs (red) and MPs (orange $\to$ green as $L$ decreases). The black curve shows predictions for AOUPs (SM VII). Inset: correlation amplitude $c_s(0)$ vs. $L/\ell_\star$. Orange-to-green and red curves respectively show the exact harmonic results for MPs (SM VII) and the AOUP prediction. $\mathbf{(f)}$ Same as $\mathbf{(e)}$ for antipersistent AOUPs (blue) and MPs (magenta $\to$ green). Simulation details in SM VIII.
• Figure 3: $\mathbf{(a,b)}$ Localization of MPs in a $1$d isolated trap $U$ (black). $\mathbf{(a)}$ Persistent case: Averaged densities inside/outside the trap for persistent MPs (bold orange) and AOUPs (bold red). Thin lines: nonaveraged profiles. Inset: ratios of trapping probabilities for persistent MPs vs. AOUPs at different ratios $\ell_D/\ell_{tr}$. $\mathbf{(b)}$ Antipersistent case: Same as in $\mathbf{(a)}$ for antipersistent MPs (magenta) and equilibrium dimers (green, following Boltzmann measure). $\mathbf{(c,d,e)}$ Localization of MPs in a $2$d corrugated landscape of Gaussian pillars. $\mathbf{(c)}$$2d$ heatmap of the landscape. Inset: 3D view. $\mathbf{(d)}$ Ratio of stationary distributions of persistent MPs vs. AOUPs in the landscape. $\mathbf{(e)}$ Same as $\mathbf{(d)}$ for antipersistent MPs vs. equilibrium dimers. Numerical details in SM VIII.
• Figure 4: $\mathbf{(a)}$ MIPS in assemblies of interacting persistent AOUPs ($\overline{\phi}=0.6, \, \tau=200$) colored by local packing fraction. Colorbar applies to all panels. $\mathbf{(b,c,d)}$ Persistent MPs under the same conditions at $\overline{\phi}=0.3,0.6,0.8$, showing no phase separation. $\mathbf{(e)}$ Probability distribution of local packing fraction $P(\phi)$ of configurations $\mathbf{(a-d)}$: black bimodal for AOUPs, unimodals (purple, salmon, gold) for persistent MPs ($\overline{\phi}=0.3,0.6,0.8$). $\mathbf{(f)}$ AOUP phase diagram: gas/liquid binodals (blue/red squares) delimit MIPS region. Overlaid: most probable local packing fractions of persistent MPs at different $\overline{\phi}=0.3,0.6,0.8$ (purple, salmon, gold); round dots show no phase separation. Dashed line $\tau=200$ marks conditions of $\mathbf{(a-e)}$. Numerical details in SM VIII.
• Figure 5: $\mathbf{(a,b)}$ Selected trajectories of antipersistent MPs and matched equilibrium dimers at $\overline{\phi}=0.72$. MPs are jammed while equilibrium dimers remains mobile. $\mathbf{(c)}$ Mean-squared displacement for dense assemblies of antipersistent MPs. Colors to increasing $\overline{\phi}$ from $0.6$ to $0.7$ (cyan to magenta). All curves show short- and long-time diffusive scaling and a plateau develops at higher densities, indicating transient caging and the onset of jamming. $\mathbf{(d)}$ Effective diffusion coefficient $D_{\rm eff}$ vs. $\overline{\phi}$ for Brownian particles (bold black line/squares), equilibrium dimers (dotted lines/triangles) and antipersistent MPs (bold lines/dots). Colors indicates $\ell_D$. Colored ticks below the $x$-axis mark the jamming densities for antipersistent MPs (colored) and equilibrium brownian or dimers (black). Inset: effective particle radius $r_0+\delta r$ vs. $\ell_D$ for antipersistent MPs (dots) and equilibrium dimers (triangles); $\delta r$ is defined so that their jamming densities matches the one of Brownian particles. More details in SM VIII.