Universal entrainment mechanism controls contact times with motile cells

Contact between particles and motile cells underpins a wide variety of biological processes, from nutrient capture and ligand binding to grazing, viral infection, and cell-cell communication. The window of opportunity for these interactions depends on the basic mechanism determining contact time, which is currently unknown. By combining experiments on three different species—Chlamydomonas reinhardtii, Tetraselmis subcordiforms, and Oxyrrhis marina—with simulations and analytical modeling, we show that the fundamental physical process regulating proximity to a swimming microorganism is hydrodynamic particle entrainment. The resulting distribution of contact times is derived within the framework of Taylor dispersion as a competition between advection by the cell surface and microparticle diffusion, and predicts the existence of an optimal tracer size that is also observed experimentally. Spatial organization of flagella, swimming speed, and swimmer and tracer size influence entrainment features and provide tradeoffs that may be tuned to optimize the estimated probabilities for microbial interactions like predation and infection.

Figure 1

(a) Snapshot of typical particle entrainment events for OM. The colloid $\left(r_{\mathrm{P}}=0.5\mu \mathrm{m}\right)$ is shown with a white arrow. Scale bar: $10\mu \mathrm{m}$. (b) Diagram and approximated streamlines $r_{\mathrm{SL}}\left(\theta \right)=r_{\mathrm{S}}+b\sqrt{2/3}/sin\left(\theta \right)$ of the swimmer-generated flow in its comoving frame. (c) [resp. (d)] Typical experimental [resp. numerical] tracer trajectories in the frame of CR. Scale bar: $10\mu \mathrm{m}$. (e) PDF of entrainment length $L$ obtained with tracers of radius $r_{\mathrm{P}}=0.5\mu \mathrm{m}$ for three different organisms: red triangles: OM; grey circles: CR; black squares: TS. Above a peak at value $L_{\mathrm{M}}^{\left(\mathrm{S}\right)}$, those are well fitted by an exponential distribution (phenomenological) with characteristic length scale $L_{\mathrm{J}}^{\left(\mathrm{S}\right)}$. Inset: semi-log plot of the same data. (f) Comparison of experimental (filled grey circles) and numerical (empty grey circles) jump length distributions using our outboard CR, for small impact parameters and normalized by $L_{\mathrm{M}}$, the length at the maximum: $L_{\mathrm{M}}^{\left(\mathrm{sim}\right)}\approx 14.5\mu \mathrm{m}$ and $L_{\mathrm{M}}^{\left(\exp \right)}\approx 9.4\mu \mathrm{m}$. The best fit using Eq. (19) (solid line) agrees well with the experimental data. Inset: semi-log plot of the same data. (g) Conditional PDF of the rescaled impact parameter $\tilde{b}=2b/w$ given an entrainment event $\mathrm{PDF}\left(\tilde{b}\right|\mathrm{jump})$. Curves are fitted by exponential distributions with characteristic lengths ${\tilde{b}}^{\left(\mathrm{TS}\right)}=1.0\pm 0.3,{\tilde{b}}^{\left(\mathrm{CR}\right)}=0.30\pm 0.05$, and ${\tilde{b}}^{\left(\mathrm{OM}\right)}=0.70\pm 0.25$. The color code is the same as in (e). (h) The same data in a semi-log plot with the curve for OM (red triangles) shifted to highlight the two regimes in the distribution for CR (grey circles).

Figure 2

Entrainment simulations by the outboard swimmer of Brownian tracer particles of various sizes. Shown are spatial PDFs of the beads, azimuthally and time averaged, as seen in the co moving frame of a CR alga, obtained by averaging over an ensemble of ${10}^3$ particles released in front of the body at $b=0\mu \mathrm{m}$ with a constant surface-to-surface distance of $1.5\mu \mathrm{m}$. (a) Small tracers with $r_{\mathrm{P}}=0.01\mu \mathrm{m}$ diffuse away quickly and are not entrained for a long time. (b) and (c) Intermediate-sized tracers with $r_{\mathrm{P}}=0.072,0.52\mu \mathrm{m}$ primarily flow along streamlines close to the swimmer body, in its no-slip layer, and are the furthest entrained. (d) Large tracers with $r_{\mathrm{P}}=1.38\mu \mathrm{m}$ flow along paths far from the swimmer body, and are entrained less.

Figure 3

Average entrainment length $\langle L\rangle$ as a function of tracer size. (a) Simulated as in Fig. 2 for OM, CR, and EC (open red triangles, grey circles, and blue diamonds respectively). The initial surface-to-surface distance is $11\mu \mathrm{m}$ for OM, $1.5\mu \mathrm{m}$ for CR, and $0.5\mu \mathrm{m}$ for EC. Solid lines show the corresponding analytical prediction [Eq. (15)]; dashed lines without noise [Eq. (5)]. The same data on a log-log scale are in Fig. S8(b). (b) Average entrainment length obtained experimentally with CR for different tracer sizes. The error bars represent the standard error on the mean (s.e.m.). Inset: The same data obtained in simulations with tracers initially located in front of the swimmer at impact parameters $b$ uniformly distributed in $[0,r_{\mathrm{S}}]$. Dashed lines are guides to the eye. (c) Analytical $\langle L\rangle$ as a function of swimming speed at fixed $r_{\mathrm{P}}=1\mu \mathrm{m}$. Solid lines [Eq. (15)]: OM red, CR grey, EC blue. Dashed lines [Eq. (5)]: without noise. (d) The optimal bead size $r_{\mathrm{P}}^{*}$ as a function of swimmer size and speed, obtained from Eq. (13) with constant value $\lambda =4$. Markers represent a few typical model organisms—grey circle: CR; black square: TS; red pyramid: OM; blue triangle: Euglena gracilis; orange asterisk: Bdellovibrio bacteriovorus; five-pointed green star: Peranema trichophorum.

Figure 4

Schematics of the analytical model: (a) A Brownian particle is advected over a solid surface (green axis) in a linear shear flow along the $x$ direction characterized by a strain rate $U$. The tracer of radius $r_{\mathrm{P}}$, initially located at $x=0$ and $\epsilon =r_{\mathrm{P}}$, is free to diffuse in any direction, but cannot cross the line $\epsilon =r_{\mathrm{P}}$ due to steric interactions. (b) This situation is equivalent to a tracer free to diffuse through the surface $y=0$, but with a modified flow that is nonzero at the wall and always positive. (c) Parameters $\langle \epsilon \rangle$ [black circles, Eq. (23)] and $w$ [red circles, Eq. (26)] obtained from the fitting procedure of $\mathrm{PDF}\left(L\right)$, Eq. (19), for the four tracer sizes, probed with CR. The analytical estimates of these parameters (full lines) are in good semiquantitative agreement. In particular the minimum in $\langle \epsilon \rangle$ is captured at the right location.

Figure 5

Contact time fluctuations can affect the probability of a successful entrainment interaction. (a) Map of the relative width $\sqrt{\mathrm{var}\left(T\right)}/\langle T\rangle$ of the contact time distribution as a function of tracer size $r_{\mathrm{P}}$ and swimming speed $v_{\mathrm{S}}$. Slow swimmers and small tracers show larger relative fluctuations in contact time. The solid line is the optimal tracer size for a given swimming velocity, Eq. (13). (b) and (c) Success probabilities of cell-object interactions for three different particle sizes $(r_{\mathrm{P}}=0.1\mu \mathrm{m}$: pink curve; $r_{\mathrm{P}}=0.7\mu \mathrm{m}$: blue curve; $r_{\mathrm{P}}=1.6\mu \mathrm{m}$; green curve) in two simple cases: In (b) the interaction requires a finite time $T_i$ to happen, and in (c) the interaction is described by a constant rate of success $\mathrm{\Omega }$ per unit time. In the first case, fluctuations are important in setting the relative probabilities of success, while in the latter case the average contact time $\langle T\rangle$ (vertical dashed lines) mainly determines the chances of successful interaction.