Emergent Loewner Dynamics in Slime Mold Growth

Abstract

Growth fronts of slime molds are characterized through a direct geometric analysis based on Loewner evolutions, using experimentally acquired time-resolved images. The associated Loewner driving functions reconstructed from expanding pseudopod boundaries display statistical properties consistent with Gaussian-like behavior. A geometric estimate of the diffusivity parameter~$κ$ is inferred from fractal scaling, while Brownian diagnostics are assessed on the reconstructed driving signal. These findings show that the boundaries of a growing living organism display statistical and geometric properties consistent with emergent Loewner dynamics over experimentally accessible scales. This study establishes a quantitative framework for analyzing biological growth interfaces and suggests new connections between morphogenesis, stochastic geometry, and network reorganization under varying environmental conditions. We provide, to our knowledge, the first explicit reconstruction of a Loewner driving function from a living growth interface, revealing an emergent Brownian-like conformal growth regime at expanding fronts.

Figures (21)

• Figure 1: Time-compressed growth sequence of Physarum polycephalum. Snapshots extracted from a 36-s video representing a 24-h experiment (1 frame every 2 min). Times indicate the corresponding experimental time.
• Figure 2: Multilevel Loewner reconstruction. Time-resolved imaging yields growth contours $C_{t_k}$. The analysis is performed on three geometries: pseudopods, the extracted network, and intensity-based regions (brightening/dimming). Each configuration is mapped to a Loewner driving signal $U_t$ for statistical diagnostics.
• Figure 3: Global statistical diagnostics of the reconstructed Loewner driving function. Q-Q plot, power spectral density, Hill-type tail analysis, and distributions of fractal dimension and diffusivity parameter $\kappa$ computed on the largest connected component of the pseudopod network.
• Figure 4: Local Q-Q plots of the driving function increments $\Delta U_t$ (pseudopods). Representative outer spatial windows (2, 3, 4, 7, 8, 12). The empirical quantiles closely follow the theoretical Gaussian quantiles, with mild deviations at extreme tails.
• Figure 5: Local power spectral density (PSD) with regression (pseudopods). Log--log representation of the PSD of the reconstructed driving signal for representative outer windows (2, 3, 4, 7, 8, 12). A linear regression is performed over the selected frequency range, providing an estimate of the local scaling exponent $\beta$ in $S(\omega) \propto \omega^{-\beta}$. The slopes remain broadly compatible with $\beta \approx 2$, while exhibiting inter-window variability attributable to finite-size effects at low frequencies and noise-dominated behavior at high frequencies.

Theorems & Definitions (9)

• Definition 8.1: Lévy Process
• Proposition 8.1: Brownian Motion as a Wiener Process
• Definition 8.2
• Proposition 8.2: Power Spectrum of a Stationary Signal
• Corollary 8.3: Brownian Power Spectrum
• Remark 8.1
• Definition 8.3
• Theorem 8.4
• Proposition 8.5