Investigating microscale patchiness of motile microbes under turbulence in a simulated convective mixed layer

Abstract

Microbes play a primary role in aquatic ecosystems and biogeochemical cycles. Spatial patchiness is a critical factor underlying these activities, influencing biological productivity, nutrient cycling and dynamics across trophic levels. Incorporating spatial dynamics into microbial models is a long-standing challenge, particularly where small-scale turbulence is involved. Here, we combine a fully 3D direct numerical simulation of convective mixed layer turbulence, with an individual-based microbial model to test the key hypothesis that the coupling of gyrotactic motility and turbulence drives intense microscale patchiness. The fluid model simulates turbulent convection caused by heat loss through the fluid surface, for example during the night, during autumnal or winter cooling or during a cold-air outbreak. We find that under such conditions, turbulence-driven patchiness is depth-structured and requires high motility: Near the fluid surface, intense convective turbulence overpowers motility, homogenising motile and non-motile microbes approximately equally. At greater depth, in conditions analogous to a thermocline, highly motile microbes can be over twice as patch-concentrated as non-motile microbes, and can substantially amplify their swimming velocity by efficiently exploiting fast-moving packets of fluid. Our results substantiate the predictions of earlier studies, and demonstrate that turbulence-driven patchiness is not a ubiquitous consequence of motility but rather a delicate balance of motility and turbulent intensity.

Fig 1. Depth structure of the simulated flow. The white-blue gradient represents the density gradient of the fluid (Δρ/ρ₀ = (ρ − ρ₀)/ρ₀), where ρ₀ is the reference density ρ₀ at z = 0).

(a) Turbulent kinetic energy (k) vs depth in the fluid DNS. The red line denotes the mean value of k at all simulated depths. (b) Turbulent dissipation rate (ϵ) vs depth in the fluid DNS. The red line denotes the mean value of ϵ at each simulated depth, while the golden ribbon shows the variance of ϵ at each depth. Variances are not known for panel (a) since k is itself computed from the variances of the fluid velocities. The Shallow, Mid, and Deep depth regions are labelled in red text and delimited by black dashed lines. Turbulence was strongest near the fluid surface and declined with depth, with essentially quiescent waters below the density interface.

Fig 2. Dimensionless stability (Ψ) and swimming (Φ) number values for microbes at different depths within our IBM. Expected patchiness is greater when Ψ ≈ 1 and for large Φ.

(a) The stability number Ψ of simulated microbes varied through the Deep (light blue), Mid (yellow) and Shallow (red) depth regions of the flow. To contextualise our simulations with respect to real world flows, grey boxes illustrate the range of the stability number Ψ at expected values of ϵ and ν within a convective oceanic mixed layer (see S2 Text). The stability number Ψ of our microbes is broadly similar to that expected in such real world conditions. (b) The swimming number Φ of simulated microbes also varied through the Deep (light blue), Mid (yellow) and Shallow (red) depth regions of the flow. Here again, grey boxes illustrate the range of the swimming number Φ at expected values of ϵ and ν within a convective oceanic mixed layer convective turbulence (see again S2 Text). The swimming number of microbes in our IBM overlaps with realistic values, but was generally lower (see Discussion).

Fig 3. Q-statistic over time (solid lines) in different depth regions.

Subplots in the left-hand column, (a), (c), (e), are from two simulated microbe populations representative of non-agile microbe behaviour. Subplots in the right-hand column, (b), (d), (f), are from two simulated microbe populations representative of agile microbe behaviour. Within each subplot, the dashed gray line represents the mean value $\overline{Q}$ (w.r.t. time) of the Q-statistic for the two simulations and depth region plotted therein. Non-agile microbes were not more concentrated in patches than non-motile microbes, whereas agile microbes in the deep region formed patches over twice as concentrated as non-motile microbes (Q > 1). Note that agile microbes in the shallower regions exhibited weak but negative mean patch enhancement. Full results for every combination of motility parameters (B, v_swim) and each depth region are plotted in Figs E, F and G in S3 Text.

Fig 4. Violin plot comparison of the distribution of Q-values at all times for agile and non-agile microbes in the combined Shallow-Mid regions and the Deep region.

Mean values are marked in white. In the Shallow-Mid regions, motile microbes were not significantly more concentrated in patches than non-motile microbes (Q ≈ 0). In the Deep region, motile microbes formed patches over twice as concentrated as non-motile microbes (Q ≥ 1), but non-agile microbes could do so only transiently ($\overline{Q}\approx 0$) whereas agile microbes were consistently more patch concentrated than non-motile microbes ($\overline{Q}=0.27$).

Fig 5. Normalised distributions of microbe polar angles between 20 − 60s across all depths in three motile simulations with v_swim = 100 μm s⁻¹ and B = 1s, B = 3s and B = 5s respectively.

A polar angle of 0°C would represent orientation directly “upwards” towards the fluid surface. Mean polar angles for each simulation are marked and annotated in black. Microbes were subject to a constant balancing between their inherent tendency to orient towards the vertical (captured by the reorientation timescale parameter B) and the disorienting effect of turbulence. Faster reorientation (low B) resulted in a more vertical orientation than in the case of slower reorientation (high B), and slower reorientation also produced more homogeneous orientations. Distributions for all combinations of motility parameters are plotted in Fig H in S3 Text.

Fig 6. Normalised distributions of the magnitude and polar angle of effective velocity in each depth region of two simulations, respectively characteristic of non-agile (a, c, e) and agile (b, d, f) microbes.

Microbes in the deep region had near-horizontal effective velocity, which acted to restrict their movement in the vertical direction. Also in the deep region, the difference in the magnitude of effective velocity (“effective speed”) was many times larger than the difference in microbial swimming speed. Elsewhere, effective speeds were very similar between all simulations, and the effective velocity was less vertically constrained.

Fig 7. Vertical fluid velocities (in ms⁻¹) at t = 60 s during the SPARKLE DNS.

Axes are labelled in units of DNS cell side-length. (a) shows the velocities at the surface and sides of the simulation. (b) is a top-down view of the fluid surface demonstrating rising regions of less dense fluid (red) pushing aside denser, falling fluid (blue). (c) is a side-on cross-section view through the center of the simulation, demonstrating the depth profile of these cooler (blue) and warmer (red) regions.

Fig 8. Microbial motion within the simulated flow.

(a) A snapshot of microbe positions at t = 25 s of the (B, v_swim) = (5 s, 10 μm s⁻¹) motile simulation. (b) Sample of six 3D microbe trajectories from t = 0–60s in the (B, v_swim) = (5 s, 10 μm s⁻¹) motile simulation. Each uniquely-coloured sequence of dots represents a single microbe’s trajectory. Owing to the periodic boundaries in the longitudinal and latitudinal directions, trajectories may appear discontinuous when a microbe moves through such a boundary (e.g. green trajectory). Microbes spent time in each of the three depth regions considered in our analysis, mostly due to advection by the surrounding fluid, but also through individual locomotion in less fast-moving regions of the fluid. In particular, long sojourns were noticeable at greater depths where turbulent fluid motion is less intense.