Geometry and dynamical morphology of growing bacterial colonies

TL;DR

The paper develops a geometry-first, time-resolved framework to analyze non-equilibrium growth of bacterial colonies by tracking time-dependent area $A(t)$, perimeter $P(t)$, and boundary/internal shape descriptors. Across five Bacillus subtilis strains, many show a robust area--perimeter relation $A \sim P^{\alpha}$ with $\alpha \approx 2$, indicating growth controlled by a single geometric length scale; however, transient local reorganizations cause breakdowns in this scaling, accompanied by excursions in boundary descriptors such as the boundary fractal dimension $D_b$. The study demonstrates that visual diversity in colony morphologies does not imply different growth dynamics and introduces time-resolved geometry as a coarse-grained diagnostic for departures from single-scale geometric constraints in living systems. It further suggests extending the framework to internal morphology and three-dimensional growth to explore how interior structure couples to boundary dynamics.

Abstract

We study non-equilibrium bacterial colony growth using a geometry-first, time-resolved analysis of morphology. From time-lapse microscopy data, we track the coupled evolution of area, perimeter, and boundary-sensitive shape descriptors along the full growth history. We find that non-equilibrium growth can exhibit extended intervals of compact area--perimeter scaling with exponent $α\approx 2$, consistent with growth governed by a single effective geometric length scale, as well as time-localized breakdowns of this scaling during ongoing growth. These breakdowns coincide with transient boundary reorganization while bulk area growth remains sustained. Our results demonstrate that visually distinct morphologies can arise within the same geometric growth regime, and that departures from single-scale behavior reflect intrinsic dynamical restructuring rather than growth arrest. More broadly, this work establishes time-resolved geometry as a coarse-grained framework for identifying when non-equilibrium growth departs from single-scale geometric constraints in living systems.

Figures (8)

• Figure 1: Geometric analysis pipeline for extracting time-resolved observables from colony images. (a) Representative microscopy image at a single time point. (b) Binary mask obtained through preprocessing and adaptive thresholding; the enclosed area $A(t)$ and perimeter $P(t)$ are computed from the segmented footprint and its contour, respectively. (c) Box-counting construction applied to the extracted colony boundary to estimate the effective boundary fractal dimension $D_b$, using box sizes $\epsilon = 5, 10, 20,$ and $40$ pixels, based on the scaling relation $N(\epsilon)\sim \epsilon^{-D_b}$. (d) Box-counting applied to the filled colony mask to estimate the space-filling (mass) fractal dimension $D_f$, using the same box sizes. (e) Log--log box-counting plots for the boundary and filled colony interior, yielding estimates of $D_b$ and $D_f$. All fractal-dimension analyses are performed using a boundary thickness of one pixel.
• Figure 2: Time-resolved snapshots of colony growth morphology for four Bacillus subtilis strains: (a) strain 100, (b) strain 108, (c) strain 106, and (d) strain 102. Five representative time points (I)--(V) are shown for each strain, spanning early to late stages of growth. The colony footprints shown here correspond to those used in the geometric analysis. These snapshots provide a qualitative reference for the temporal evolution of colony shape, boundary organization, and internal patterns across strains.
• Figure 3: Time evolution of colony area $A(t)$ (red, dashed) and perimeter $P(t)$ (green, solid) for multiple bacterial strains, shown in separate panels. Time is reported in hours, with $t_{0}$ denoting the first time point at which the colony boundary is reliably identified. Area and perimeter are plotted on separate vertical axes to emphasize the concurrent evolution of bulk growth and boundary morphology. For all strains, both quantities increase overall with time, indicating sustained growth without fragmentation or arrest, while strain-dependent temporal features are most pronounced in the perimeter. All strains are analyzed over comparable spatial extents, with each panel spanning the full observation window for the corresponding strain. Non-monotonic features in the perimeter signal transient boundary reorganizations that are not apparent from area growth alone.
• Figure 4: Temporal evolution of shape order parameters for all strains. Shown are the circularity $C(t)$ (top row), compactness $\xi(t)=P^{2}/A(t)$ (second row), effective boundary fractal dimension $D_{b}(t)$ (third row), and mass fractal dimension $D_{f}(t)$ (bottom row) for strains 100, 108, 106, and 102 (columns left to right). Boundary-sensitive descriptors ($C$, $\xi$, $D_{b}$) exhibit strain-dependent temporal structure, while the mass fractal dimension $D_{f}$ evolves more gradually toward values characteristic of compact two-dimensional growth. For strain 106, vertical dashed lines mark characteristic times associated with extrema in $C(t)$, coinciding with correlated responses across boundary-sensitive descriptors and with the morphological configurations shown in Fig. \ref{['fig:morphology']}(c). For strain 102, a vertical dashed line indicates the onset of stabilization in boundary-sensitive descriptors following a rapid initial reorganization, beyond which all observables evolve smoothly.
• Figure 5: Time-ordered area--perimeter scaling of colony growth across strains. Log--log plots of colony area $A(t)$ versus perimeter $P(t)$, colored by time, for four bacterial strains (top row: strains 100 and 108; bottom row: strains 106 and 102). Strains 100 and 108 exhibit a robust single power-law relation $A \sim P^{\alpha}$ over the full growth trajectory, consistent with compact two-dimensional growth governed by a single geometric length scale. Strain 106 displays a clear crossover between two distinct scaling regimes (open circles), with the inset highlighting the intermediate region connecting these regimes. Strain 102 shows an early-time curved trajectory associated with rapid morphological reorganization, followed by the emergence of a stabilized power-law scaling regime at later times (open circle). For strain 106, the crossover region coincides with extrema in boundary-sensitive shape descriptors (Fig. \ref{['fig:shape']}) and with the intermediate-time morphologies shown in Fig. \ref{['fig:morphology']}(c).