Fluid flow enhances the effectiveness of toxin export by aquatic microorganisms: A first-passage perspective on microvilli and the concentration boundary layer

A central challenge for organisms during development is determining a means to efficiently export toxic molecules from inside the developing embryo. For aquatic microorganisms, the strategies employed should be robust with respect to the variable ocean environment and limit the chances that exported toxins are reabsorbed. As a result, the problem of toxin export is closely related to the physics of mass transport in a fluid. In this paper, we consider a model first-passage problem for the uptake of exported toxins by a spherical embryo. By considering how macroscale fluid turbulence manifests itself on the microscale of the embryo, we determine that fluid flow enhances the effectiveness of toxin export as compared to the case of diffusion-limited transport. In the regime of a large Péclet number, a perturbative solution of the advection-diffusion equation reveals that a concentration boundary layer forms at the surface of the embryo. The model results suggest a functional role for cell surface roughness in the export process, with the thickness of the concentration boundary layer setting the length scale for cell membrane protrusions known as microvilli. We highlight connections between the model results and experiments on the development of sea urchin embryos.

Figure 1

The first-passage probability ${\mathrm{\Pi }}_D$ as a function of the microvilli tip location ${\xi }^{\prime }=1+\frac{h}{R}$ for the case of pure diffusion. The microvilli have length $h$ and the embryo has radius $R$.

Figure 2

The dimensionless Péclet number $\text{Pe}$ (blue line) and Reynolds number $\text{Re}$ (red line) as a function of the turbulent kinetic energy dissipation rate $\varepsilon \left({\text{m}}^2{\text{s}}^{-3}\right)$.

Figure 3

The time-averaged velocity field $\langle \stackrel{*}{\stackrel{⃗}{u}}\rangle$ in the $(x_2,x_3)$ plane (red arrows). Velocity streamlines starting at the tips of microvilli $\left({\xi }^{\prime }=1.05\right)$ are shown as blue lines.

Figure 4

The concentration boundary layer thickness $\ell =R{\text{Pe}}^{-\frac{1}{3}}$ as a function of $\text{Pe}$. At large $\text{Pe}$, the length of embryonic microvilli $h\approx \ell$.

Figure 5

The steady-state concentration profile $c/c_{\infty }$, normalized by the far-field concentration $c_{\infty }$, as a function of the dimensionless radial variable $\xi$. The case of pure diffusion $\left(\text{Pe}=0\right)$, Eq. (69), is shown as a blue line. The concentration profile (along the line $\theta =\pi /4)$ in the advection-dominated regime $\left(\text{Pe}=100\right)$, Eq. (47), is shown as a red line.

Figure 6

Steady-state concentration contours of $c/c_{\infty }$ in the $(x_2,x_3)$ plane for the case of strong advection $\left(\text{Pe}=100\right)$. Note the thickness of the concentration-boundary layer. The concentration rapidly approaches its far-field value $\left(c/c_{\infty }=1\right)$ in a thin layer surrounding the embryo.

Figure 7

Steady-state concentration contours of $c/c_{\infty }$ in the $(x_2,x_3)$ plane for the case of pure diffusion $\left(\text{Pe}=0\right)$. Note the thickness of the concentration-boundary layer. At twice the embryo radius, the concentration has approached roughly half of its far-field value $\left(c/c_{\infty }=1\right)$.

Figure 8

The first-passage probability $\mathrm{\Pi }$ as a function of the microvilli tip location ${\xi }^{\prime }$. The result for the case of pure diffusion $\left(\text{Pe}=0\right)$, ${\mathrm{\Pi }}_D$, is shown as a blue line. The zeroth-order result for the case of strong advection $\left(\text{Pe}=2062\right)$, ${\mathrm{\Pi }}_0$, is shown as a red line. Note that in the advection-dominated case, the values of ${\xi }^{\prime }$ over which there is a rapid decrease in absorption probability agree quite well with the length of embryonic microvilli. Microvilli of height $h=2$, 5, and $10\mu \text{m}$ correspond to ${\xi }^{\prime }=1.05$, 1.125, and 1.25, respectively.

Figure 9

The average number of ATP molecules $\langle N_{\text{ATP}}\rangle$ required to efflux a single toxin molecule as a function of the microvilli tip location ${\xi }^{\prime }$. The result for the case of pure diffusion $\left(\text{Pe}=0\right)$ is shown as a blue line. The zeroth-order result for the case of strong-advection $\left(\text{Pe}=2062\right)$ is shown as a red line.

Figure 10

(a) Hundreds of thousands of microvilli (short lines) roughen the surface of a spherical embryo of radius $R\approx 40\mu \text{m}$ in a microscale velocity gradient (blue arrows). (b) Schematic of an individual microvillus, a cylindrical cell membrane protrusion of radius $\varrho \approx 0.1\mu \text{m}$ and length $h\approx 2\mu \text{m}$. (c) In the multiple scattering calculation, the effect of surface roughness is to displace the smooth surface by a distance $\lambda$.

Figure 11

The first-passage probability $\mathrm{\Pi }$ as a function of the microvilli tip location ${\xi }^{\prime }$. The effect of surface roughness is to increase the first-passage probability. Heuristic estimates for the magnitude of the effect are provided by the dashed lines.