Mexican jumping beans exhibit diffusive motion

Organisms across many lengthscales generate specific strategies for motion that are crucial to their survival. Here, we detail the motion of a nontraditional organism, the Mexican jumping bean, where a larva encapsulated within a seed blindly moves the seed in search of shade. Using image analysis techniques, we quantitatively describe the motion of these objects as active particles. From this experimental data, we build a computational simulation that quantitatively captures the motion of these beans. And we further evaluate the effectiveness of using the observed diffusive strategy to find shade, suggesting that the random walk is an advantageous strategy for survival.

Figure 1

Bean size distribution. (a) Photograph of six beans of various sizes resting on a two-dimensional (2D) surface on their different faces: two on their curved edge and four on a flat face. A line is displayed to define where the pole-to-pole length $L$ is measured for a bean. (b) Probability distribution of bean lengths $P\left(L\right)$ for $N=142$ beans.

Figure 2

Effective diffusion constant is size dependent. (a) Example of experimental data with photograph of bean overlaid with bean trajectory over time. (b) Mean squared displacement MSD as a function of lag time $\tau$ for $N=32$ beans at a temperature of $37.2\pm 1.5{}^{\circ }\mathrm{C}$. Each set of markers corresponds to experimental data for a given bean, and lines are best fits. (c) MSD as a function of $\tau$ for $N=5$ beans at $26\pm 3{}^{\circ }\mathrm{C}$. (d) Scaling parameter $\alpha$ for beans at 26 ${}^{\circ }\mathrm{C}$ and 37.2 ${}^{\circ }\mathrm{C}$. Two-sided Student's $t$ test gives $p>0.05$. (e) Effective diffusion constant $D$ as a function of bean size $L$ for beans at $37.2\pm 1.5{}^{\circ }\mathrm{C}$ (red symbols) and beans at $26\pm 3{}^{\circ }\mathrm{C}$ (blue symbols), with line of best fit ($p<0.01$) calculated only for beans at $37.2\pm 1.5{}^{\circ }\mathrm{C}$. (f) Effective diffusion constant for beans with $7.5<L<9.5\mathrm{mm}$ at 26 ${}^{\circ }\mathrm{C}$ and 37.2 ${}^{\circ }\mathrm{C}$. Two-sided Student's $t$ test gives $p<0.05$. All data points used in generating box plots in (d) and (f) are overlaid atop respective box plots.

Figure 3

Step details of bean motion. (a) Example of displacement $\delta$ normalized by maximum displacement ${\delta }_m$ and (b) delay $t_D$ normalized by maximum delay $t_{D,m}$ for a single bean over 10 minutes. (c) Probability distribution of delay $t_D$ with fitted inverse gamma distribution. (d) Probability distribution of displacement $\delta$ with fitted exponential distribution. (e) Probability distribution of relative angular displacement where $\theta =0$ degrees are forward jumps. Data shown in (c)–(e) are aggregated over all $N=32$ beans at $37.2\pm 1.5{}^{\circ }\mathrm{C}$ and each contain $\sim 9300$ total steps.

Figure 4

Simulated bean motion. Mean squared displacement MSD as a function of lag time $\tau$ of simulation trajectories with an isotropic (black) angular distribution and a distribution directly sampled from experimental data in Fig. 3 (gray). Dashed lines are lines of best fit for MSD $\sim {\tau }^{\alpha }$ with $\alpha =1.01$ and $\alpha =1.02$ for the isotropic and experimental angular distributions, respectively.

Figure 5

First passage to shade. (a) Schematic of circle with radius $R$ denoting shade and an object at initial position $r>R$ away from the shade. (b) Eventual hitting probability $\varepsilon$ as a function of object initial position $r/R$ for diffusive objects (black) and ballistic objects (red).