Figures and Tables
Fig. 1: Theoretical hypothesis and experimental setup.(A ) Three typical patterns of movement behaviors of microorganisms, showing the ‘run-and-stop’, ‘run-reverse-flick’, ‘run-and-tumble’, and the ‘circular run-and-reverse’ pattern of marine diatoms. (B ) The characteristics of an optimization model by adjusting movement behavioral plasticity. The dash line shows peaks predicted fitness and therefore what would be expected in nature. When the environment changes, its optimal value would change accordingly. (C ) The schematic of the experimental setup (not to scale). (D ) An example of the observed movement trajectories. (E ) Scanning electron microscope image of speciesNavicula arenaria var. rostellata shows an boat-shape cell, where the two raphes can spray the extracellular polymeric substances (EPS) to obtain self-propulsion.
Fig. 2: Experimental observations and theoretical predictions of the circular run-and-reversal behaviors of diatom Navicula arenaria var. rostellata. (A ) A typical cell trajectory containing circular run and reversal behaviors captured with a microscopy at 4 frames per second (see Movie S1 for more trajectories) for 5 min. (B ) Cropping of the partial trajectory depicts a reversal behavior with zoom in on the panel (A), where the running from CCW switches to CW through a reversal behavior, and vice versa. The arrows indicate the moving direction of the diatom cells. (C ) Experimental data showing the movement velocity before and after a reversal occurrence; for clarity, not all speeds of the time series are shown here. (D ) and (E ) Predictions of spatial trajectory and reversal event obtained from model (1) with parameters value \(V_{0}\)= 17 \(\mu m/s\), \(D_{\theta}=0.0054\)\(\text{rad}^{2}\)/s, \(\nu=0.02\ s^{-1}\), and \(\omega=\pi/3\)6 rad/s. Colorbars in panel (A, D) depict the time (see Movie S2 for theoretical simulations).
Fig. 3: Comparing the laboratory measurements and simulation results with theoretical (analytical) predictions on diffusion behaviors of diatom cells. (A ) Statistical distribution of 1704 reversal interval time \(t\) from the 29 experimental individuals trajectories, which can be well fitted by an exponential distribution\(f\propto\ e^{-0.016t}\) with the slope of -0.016. (B ) The measured probability density functions of cells’ displacements as a function of displacement normalized by its standard deviation (\(\sigma=\sqrt{\left\langle{x}^{2}\right\rangle}\)) along the\(\ x\)-axis direction for different times. A fit to the data with Gumbel law (solid black lines) and Gaussian model (dashed green lines) are shown for two different time scales, where the Gumbel law of the distribution imply slower diffusion at a long-time scale. (C ) Mean squared displacement (MSD) for three different values of the rotational diffusion coefficient \(D_{\theta}\) obtained by performing the numerical simulations of model (1) and comparison with the experiments (circles symbols), respectively. By decreasing the strength of rotational diffusion in the model, the scaling behaviors of the MSD vs. time becomes consistent with confined diffusivity from ballistic behaviors similarly to cage-effect emergence after the characteristic times (\(\sim 25\ s\)). Parameters are \(\omega=\pi/36\) rad/s,\(\nu=0.02\ s^{-1}\), \(D_{\theta}=0.0054\) \(\text{rad}^{2}\)/s and \(V_{0}=\)17 \(\mu\)m/s. The dashed lines are a guide to the eye to mark the change of the scaling law with 2.0 and 1.0 respectively, the solid line corresponds to the trend predicted by theory Eq. (4). (D ) Correlation of measured and predicted changes in the direction of cells moving. Experimental data ( symbols) have error bars representing lower and upper SD. Corresponding analytical predictions (solid line and dashed line with triangle symbols) are given by theory Eq. (3) and numerical simulations of model (1) respectively. The dashed line indicates 0 to guide the eye in (B).
Fig. 4: Theoretical prediction of optimal foraging strategies with spatially randomized nutrient targets. (A) Schematic representation (not to scale) of diatom cells blindly searching for randomly distributed nutrient resources (dots). The cells placed in a two-dimensional space move with constant speed\(\mathbf{V}_{\mathbf{0}}\) and variable orientation described in model (1). The capture radius \(\mathbf{r}_{\mathbf{c}}\) is about 20\(\mathbf{\mu}\)m size (dashed circle area). (B) The distinctive exponential function,\(\mathbf{n}\left(\mathbf{t}\right)\mathbf{=A}\mathbf{e}^{\mathbf{-\tau t}}\)with the decay rate τ, was used to describes the foraging efficiency of diatom movement strategy with respect to various value of\(\mathbf{D}_{\mathbf{\theta}}\) and \(\mathbf{\nu}\). (C, D) The efficiency of captured nutrients as a function of\(\mathbf{D}_{\mathbf{\theta}}\) and \(\mathbf{\nu}\), respectively. Foraging efficiency is calculated by averaging over 1000 trajectories with various \(\mathbf{\omega}\), where the plot is scaled to the maximum value at \(\mathbf{D}_{\mathbf{\theta}}\mathbf{=0.3}\) and\(\mathbf{\nu=0.0001\ }\mathbf{s}^{\mathbf{-1}}\) respectively (see fig. S6 for without scaling). (E, F) The analytical prediction of effective diffusivity from theory Eq. (5), coinciding with directly numerical simulations of model (1). The dashed lines and gray shaded area represent mean \(\mathbf{\pm}\mathbf{2\ }\)SD from the experimentally measured values of \(\mathbf{D}_{\mathbf{\theta}}\)and ν for Navicula arenaria var.rostellata .
Fig. 5: Theoretical and experimental results implicate the emergence of the foraging efficiency for various behavioral strategies.(A ) Heatmap of foraging efficiency (colorbar) with respect to (\(D_{\theta},\nu\))-parameter space obtained from randomly distributed nutrient targets and constant movement speed for \(\omega=\pi/36\)rad/s and \(V_{0}=17\) \(\mu\)m/s. Optimal foraging occurs over a window of behavioral parameters of \(\nu\ \)and \(D_{\theta}\), and is indicated by the yellow areas. The boundaries of the optimal regions change sharply with increasing reversal rate (white dashed lines with intervals \(\tau=0.1\)). In the low reversal rate limit, there are nonlinear effects of the rotational diffusion on diatom foraging. The colored-solid dots correspond to the experimentally measured rotational diffusion coefficients versus reversal rate on diatom Navicula arenaria var. rostellata and the colorscale indicates the scaled foraging efficiency, \(\tau\) from 0 to 1.0. (B ) Theoretical prediction of Eq. (5) on the effective diffusivity as functions of the rotational diffusivity and reversal rate. It shows a similar spatial profile comparison with directly numerical simulations.
Fig. 6: Pairwise invasibility plot (PIP) of behavioral strategy . The PIP indicates that the movement behavioral strategy of rotational diffusivity evolves toward a stable point 0.2 (vertical dashed line). For a range of resident (x -axis) and mutant (y -axis) movement strategies, the PIP describes whether a mutant has a higher (green) or a lower (blue) fitness than the resident. Plus and minus symbols indicate combinations resulting in positive and negative invasion fitness, respectively. Here, the PIP shows that the rotational diffusivity with 0.2 is the sole evolutionarily stable strategy (ESS). Simulation parameters with\(\nu=0.02\ s^{-1}\), and \(\omega=\pi/36\ rad/s\).
Fig. 7: Reversal behaviors depend on the ambient dSi concentration. (A ) The diffusivity of diatom cells maximizes at an ambient dSi concentration of about 30 mg/L and declines at low and high dSi concentrations. The reversal events show a sharply increase when dSi goes beyond 60 mg/L, but it maintains a plateau at low dSi. The grayscale rectangle indicates typical dSi concentrations in coastal ecosystems. (B ) Efficiency diffusion coefficient, showing a monotonic decline with increased reversal events, which have a maximized dispersal coefficient about \(\nu=0.02\ s^{-1}\) coincident with model predictions.
Table 1. Statistical properties of measured experimental parameters on diatom movement behaviors. Experimental statistics of behavioral parameters on diatom cells at dSi concentrations of 15 mg/L.n , number of individuals.