Combined influence of hydrodynamics and chemotaxis in the distribution of microorganisms around spherical nutrient sources

We study how the interaction between hydrodynamics and chemotaxis affects the colonization of nutrient sources by microorganisms. We use an individual-based model and perform probabilistic simulations to ascertain the impact of important environmental and motility characteristics on the spatial distribution of microorganisms around a spherical nutrient source. In general, we unveil four distinct regimes based on the distribution of the microorganisms: (i) strong surface colonization, (ii) rotary-diffusion-induced “off-surface” accumulation, (iii) a depletion zone in the spatial distribution, and (iv) no appreciable aggregation, with their occurrence being contingent on the relative strengths of hydrodynamic and chemotactic effects. More specifically, we show that the extent of surface colonization first increases, then reaches a plateau, and finally decreases as the nutrient availability is increased. We also show that surface colonization reduces monotonically as the mean run length of the chemotactic microorganisms increases. Our study provides insight into the interplay of two important mechanisms governing microorganism behavior near nutrient sources, isolates each of their effects, and thus offers greater predictability of this nontrivial phenomenon.

Figure 1

A schematic of the problem being solved, showing a spherical nutrient source of radius $R$, a spherical swimming microorganism of radius $b$ oriented along the unit vector $\mathbf{p}$, and the spherically symmetric chemoattractant distribution around it $C\left(r\right)$. The origin of a fixed coordinate system $XYZ$ lies at the center of the source. The coordinate system defined by the unit vectors $\widehat{\mathbf{r}}$, ${\widehat{\mathbf{r}}}^{\perp }$ and ${\widehat{\mathbf{r}}}^{\perp }\times \widehat{\mathbf{r}}$ can rotate and translate with respect to the fixed coordinate system, as the microorganism moves through the fluid. In a quiescent, unbounded fluid ($h\to \infty$), the microorganism will swim along the direction $\mathbf{p}$. The hydrodynamic-interaction-induced translational velocity, ${\mathbf{u}}_{HI}$, and rotational velocity, ${\mathbf{\Omega }}_{HI}$, of the microorganism are expressed as functions of its separation from the surface $h$, and its in-plane orientation $\theta$ [see Eqs. (4) and (5)]. Note that $h$ is the dimensionless separation of the microorganism from the source.

Figure 2

(a) The concept of the critical trapping radius [25]: The swimmer trajectory around the smaller sphere escapes, while that around the larger sphere (whose radius is greater than a critical trapping radius) gets trapped. The swimmers' initial orientation, $\mathbf{p}\left(0\right)={\mathbf{e}}_Y$. (b) Alternatively, for a fixed radius, only the swimmer with ${\alpha }_D$ larger than a critical dipole strength will get trapped around the sphere. (c) The concept of the basin of attraction [25]: The swimmer whose initial location is marked by a circle (resp. square) and whose trajectory is shown by a solid line (resp. by a dashed line), starts inside (resp. outside) the basin of hydrodynamic attraction, and thus it gets trapped onto (resp. escapes) the surface. The swimmers' initial orientation, $\mathbf{p}\left(0\right)={\mathbf{e}}_X$. It is important to note that the basin of attraction is defined only in cases when hydrodynamic trapping is ensured.

Figure 3

An illustration of the effect of hydrodynamics on the motion of the microorganism as it gets trapped onto the surface of the nutrient source. The thin blue arrows are the microorganism's intrinsic motility $V_s\mathbf{p}$, the thick orange arrows are the hydrodynamic component of microorganism motion toward the center of the nutrient $\left({\mathbf{u}}_{HI}·\widehat{\mathbf{r}}\right)$, and the black arrows are the instantaneous velocity, $d{\mathbf{x}}_2/dt$ [Eq. (17)]. [(i), (ii)] Hydrodynamics—if strong enough—rotates the microorganism such that it always maintains a constant separation $h_t(\approx 1)$ and in-plane angle ${\theta }_t$, and such that $\left({\mathbf{u}}_{HI}+V_s\mathbf{p}\right)·\widehat{\mathbf{r}}\le 0$. As a result, the microorganism moves tangentially along the surface and stays trapped. (iii) Rotary diffusion—if significant—can cause the microorganism to rotate to an in-plane angle greater than ${\theta }_t$ which reduces the hydrodynamic attraction, causes $\left({\mathbf{u}}_{HI}+V_s\mathbf{p}\right)·\widehat{\mathbf{r}}>0$, and thus leads to escape.

Figure 4

A schematic of the effect of chemotaxis strength on the accumulation around the nutrient source. The left, central, and right columns show the $x\text{−}y$, $y\text{−}z$, and $x\text{−}z$ projections, respectively, of the microorganisms' trajectories. The microorganisms are located initially at $\left[x\left(0\right),y\left(0\right),z\left(0\right)\right]=\left[0,-40,0\right]$, and oriented arbitrarily. It is important to note that in the absence of chemotaxis, most of the microorganisms would just swim away from the source without appreciably changing their orientations. The upper (resp. lower) row represents strong (resp. weak) chemotaxis, which could either be due to $C_0/K_D=1.0$ (resp. $C_0/K_D\ll 1.0$), or a small (resp. large) value of ${\tau }^{*}$. Clearly, strong (resp. weak) chemotaxis leads to the microorganisms being, in general, closer to (resp. further from) the nutrient.

Figure 5

(a) Visualization of the different behaviors elicited by the mechanisms discussed in Table 1. The starting positions are shown by black dots. Red: this microorganism is unable to locate the source in the time for which the simulations were run. Blue: this microorganism chemotaxes close enough to the source, but does not enter the basin of hydrodynamic attraction. Magenta: in this case, the microorganism does make contact with the source, but the hydrodynamic attraction is not strong enough for trapping to occur. Green: an example of a successful trapping wherein chemotaxis and hydrodynamics work in conjunction to bring and trap a microorganism onto the source. See main text for details about the regimes in which such behaviors occur. (b) The time evolution of the distance from the source, $h\left(t\right)$, of trajectories in panel (a).

Figure 6

(a) Variation of the surface concentration, $C_s$, with the dipole strength ${\alpha }_D$ for $D\approx 0$ (negligible rotary diffusion) and $D=7.5\times {10}^{-4}$ (moderate rotary diffusion). (b) Main figure: The distribution $f\left(r\right)$ for $D\approx 0$, and ${\alpha }_D=0.1$ (green dashed line), ${\alpha }_D=0.6$ (orange dash-dotted line), ${\alpha }_D=0.7$ (blue solid line), ${\alpha }_D=1.0$ (red dotted line). Inset: The distribution $f\left(r\right)$ for ${\alpha }_D=0.7$ for $D\approx 0$ and $7.5\times {10}^{-4}$ [corresponding surface concentrations are shown in panel (a) by filled symbols]. Notice the drastic difference in the values of $C_s$ and $f\left(r\right)$ for the two different values of rotary diffusivities.

Figure 7

Variation of the surface concentration $C_s$ with (a) $C_0/K_D$, and, (b) ${\tau }^{*}$. In each case, $C_s$ is highest when ${\alpha }_D>0.65$ and $D$ is negligible, as expected based on the discussion in Sec. 3c. Also, the results are independent of $D$ for ${\alpha }_D\approx 0$. For small ${\tau }^{*}$, $C_s$ varies almost linearly with ${\tau }^{*}$.

Figure 8

The bulk distribution $f\left(r\right)$ as a function of [(a), (b)] $C_0/K_D$ and [(c), (d)] ${\tau }^{*}$. Note the almost similar distributions for $C_0/K_D=0.1,1.0,10.0$, just like the corresponding $C_s$ values in Fig. 7. In conjunction with Fig. 7, it is evident how rotary diffusion causes more microorganisms to stay in the bulk. For weak chemotaxis, there is no appreciable accumulation anywhere in the bulk. $f\left(r\right)$ increases to a maximum and then decays to zero for weak chemotaxis in the panels (b) and (d). See main text for details.

Figure 9

The four qualitatively different behaviors, or spatial distributions $f\left(r\right)$, that can be realized due to the combined influence of hydrodynamics (abbreviated in the legend as HI) and chemotaxis (abbreviated in the legend as Ch) on the locomotion of microorganisms around a spherical nutrient source. $\uparrow$ (resp. $\downarrow$) denotes a strong (resp. weak) influence. The inset shows the surface colonization $C_s$ for each of the four behaviors, with correspondence based on marker type.

Figure 10

The bulk distribution $f\left(r\right)$ as a function of (a) $C_0/K_D$ and (b) ${\tau }^{*}$ for negligibly small hydrodynamic attraction (${\alpha }_D\approx 0$) and $D\approx 0$, $D=7.5\times {10}^{-4}$.

Figure 11

The bulk distribution $f\left(r\right)$ for two different cases: (i) when the value of $D_r$ in Eq. (8) is taken to be half of the rotary diffusivity in an unbounded fluid, $D_{r0}$, in the entire domain (solid line) and (ii) when Eq. (A1) is used to assign bacterial rotary diffusivities based on separation of the microorganism from the source (dashed line marked with circles). The surface colonization values are within $1.25\%$ of each other. The value of the dimensionless rotary diffusivity in unbounded fluid is $7.5\times {10}^{-4}$, i.e., $D_{r0}b/V_s=7.5\times {10}^{-4}$.

Figure 12

Comparison of the bulk distribution $f\left(r\right)$ for combined chemotactic and hydrodynamic attraction to (i) a rigid sphere (asterisks) and (ii) a clean drop with viscosity ratio $\lambda =10$ corresponding to crude oil (circles), for the baseline simulation parameters given in Sec. 3a. The difference between the two cases is not very significant. The surface colonization for the rigid sphere ($C_{s,\mathrm{rigid}}=0.3589$) is 4% larger than that for the drop ($C_{s,\mathrm{drop}}=0.3446$). For motion around the drop, the hydrodynamics-induced linear and angular velocities are taken from Ref. [48].