Evaporation-driven convective flows in suspensions of non-motile bacteria

We report a novel form of convection in suspensions of the bioluminiscent marine bacterium $Photobacterium~phosphoreum$. Suspensions of these bacteria placed in a chamber open to the air create persistent luminiscent plumes most easily visible when observed in the dark. These flows are strikingly similar to the classical bioconvection pattern of aerotactic swimming bacteria, which create an unstable stratification by swimming upwards to an air-water interface, but they are a puzzle since the strain of $P.~phosphoreum$ used does not express flagella and therefore cannot swim. Systematic experimentation with suspensions of microspheres reveals that these flow patterns are driven not by the bacteria but by the accumulation of salt at the air-water interface due to evaporation of the culture medium; even at room temperature and humidity, and physiologically relevant salt concentrations, the rate of water evaporation is sufficient to drive convection patterns. A mathematical model is developed to understand the mechanism of plume formation, and linear stability analysis as well as numerical simulations were carried out to support the conclusions. While evaporation-driven convection has not been discussed extensively in the context of biological systems, these results suggest that the phenomenon may be relevant in other systems, particularly those using microorganisms of limited motility.


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In the deep ocean, animals such as certain fish and squid produce light through a symbiotic relationship with 27 bioluminescent bacteria [1]. One well-known luminescent bacterium is Photobacterium phosphoreum, a rod-shaped 28 organism ∼ 3.5 µm long and ∼ 0.5 µm wide, whose bright luminscence can easily be observed when a flask containing air-water meniscus 5 mm plumes FIG. 1. Convective flow observed in a suspension of Photobacterium phosphoreum. The chamber is 6 cm long, 1 mm deep and 1 cm high. Bacteria were cultured to a density of 5.5 × 10 8 cells/cm 3 following the procedure detailed in the text. This image was acquired with Nikon D300s equipped with a 60 mm f/2.8 macro lens, and an exposure time of 10 seconds. Suspensions were studied in chambers of the shape shown in Fig. 2a, constructed using two glass coverslips (Fisher 107 12404070), held together by two layers of tape 300 µm thick (Bio-Rad SLF-3001). This tape has internal dimensions 108 6 × 2 cm, and one long side was cut to create an air-liquid interface. The cuvette, 600 µm in depth, was filled using a 109 plastic syringe connected to a stainless steel needle (Sigma-Aldrich CAD4108) to yield a fluid height of 1 cm, which Imaging, CCS FPR-136-RD). The camera is placed outside the light cones formed by the red LED-ring illuminator. 114 The particles (bacteria or microspheres) suspended in the sample are visible by the light they scatter into the camera. 115 This setup can also be used to capture bacterial bioluminescence and darkfield images quasi-simultaneously by 116 switching on and off the LED ring in synchrony with acquisition of images. This was achieved by adding a second The hypothesis that the chemical reaction leading to light production in P. phosphoreum creates dense components 128 that trigger hydrodynamic plumes was excluded after two control experiments. The first one consisted of placing non-  (Fig. 2), but with a upper boundary sealed except for a single hole (red arrows). Both experiments were performed at optical density OD600nm = 1.5, using the same liquid medium, and the images were taken 75 minutes after the experiment started.
model of plume formation proposed later. As also observed in Fig.3, the center of the plumes is depleted of bacteria 139 or beads in both cases. Videos of these two experiments are provided in the Supplemental Material [27].

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As plume generation evidently does not depend on any life processes of the bacteria, in subsequent experiments 141 described below we focused exclusively on suspensions of microspheres instead of bacteria. Switching to beads also 142 allowed us to explore a wide range of salt concentrations, far beyond that which is physiologically possible using P. 143 phosphoreum, and therefore bacterial medium was substituted by aqueous solutions of salt at various concentrations.

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D. Experiments using microspheres 145 Figure 5 shows convective patterns observed in a suspension of 3 µm diameter microspheres at OD 600nm =1 with a 146 salt concentration of 1% (weight/weight). The bright horizontal line at the top of the image is the air-water meniscus.

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As previously noted, the plumes appear dark in the darkfield image, which implies the absence of beads scattering 148 light in those regions. This situation is very different from conventional bioconvection, in which plumes appear bright    was kept constant and equal to 1% (w/w) and the concentration of beads in both cases is OD 600nm = 1. Using the 177 software ImageJ the plume thickness was measured at three different points, obtaining d 3µm = 0.152 ± 0.008 mm and 178 d 6µm = 0.590 ± 0.025 mm. The ratio between these two is ∼ 3.88, very close to the ratio 4 predicted by the Stokes 179 formula above, consistent with the idea that the plume width is set by the volume of the depleted fluid layer. The 7 volume fraction of beads is ∼ 10 −4 , sufficiently low that we expect no significant corrections to the settling speed nor 181 significant buoyancy effects. It is important to note that while the nature of the experimental setup we used required 182 microspheres for visualization, their concentration is sufficiently low that hydrodynamic interactions between them 183 can be neglected, and are explicitly absent in the mathematical model developed below.

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The dependence on salt concentration. The importance of salt in the plume formation was tested by suspending 185 beads in pure water. In an experiment repeated in triplicate, no plumes appeared after 27 hours of observation, 186 confirming that the presence of salt is necessary to create the convection. Room temperature and humidity conditions 187 were checked in a 24-hours period, finding that the temperature was 21.5 ± 0.5 • C and humidity was 38 ± 2%. These 188 experiments also confirmed that the thermal effect of evaporation, that cools the upper surface and therefore makes 189 the fluid in that region slightly heavier, is not strong enough to produce plumes under our experimental conditions.

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In the model presented in the next sections, thermal effects were not considered, and similar plumes were observed, 191 suggesting that salt in the main driver in the instability presented here.

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The need for evaporation. Finally, experiments of beads suspended in salty water were repeated with a layer of The hypothesis that accumulation of salt due to evaporation is responsible for plume formation is studied in this where β T is the temperature gradient, α the coefficient of thermal expansion, g the acceleration due to gravity, d the 212 depth of the fluid layer, κ the thermal diffusivity and ν the kinematic viscosity.

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The calculation presented here uses a salinity Rayleigh number based on the initial fluid depth h 0 , that compares salt-induced buoyancy salt with the stabilizing effects of diffusion and viscosity, where β is the so- cm in the experiments). For the fluid flow, the Navier-Stokes equations within the Bousinesq approximation are used: where ∆ρ ≡ β(c * − c T ) is the density difference from the uniform salt distribution (see also [20]). The boundary 229 conditions are Equation (3a) is the incompressibility condition, and, in Eq. (3b), p * is the pressure difference from the hydrostatic 231 one, with ρ 0 the fluid density in the initial state with a homogenous distribution of salt, ν the kinematic viscosity, g 232 the acceleration of gravity, andk is the unit vector in the vertical direction. The dynamics of the salt concentration 233 c * follow the advection-diffusion equation with D the salt diffusion constant and u * the velocity field. The boundary conditions are Here, we note that the upper boundary condition is obtained by considering conservation of total salt concentration forces. This results in the dimensionless space and time scales: With these definitions, we obtain a dimensionless height and the Péclet number respectively. Velocity, pressure and salt concentration are also made dimensionless, as In the resulting dimensionless equations of motion we adopt the notation u = (u, w) to identify the horizontal and 245 vertical components of the fluid velocity. The incompressibility condition is and the Navier-Stokes equations are in the horizontal direction, and in the vertical direction, where Sc = ν/D is the Schmidt number and Ra s is the salinity Rayleigh number in (1). The 249 dynamics of the salt concentration is then Finally, the dimensionless height evolves as 251 The typical values of the experimental and material parameters are given in Table II where ε 1, u = (u , w ) is the perturbed velocity field, and c is the perturbed salt concentration at the leading Here, it should be noted that the unsteady effect of the base state and the coefficient H(T ) appears only from the 278 equations at O( δ) because the evaporation speed is so small (i.e. δ 1). Therefore, at the leading order, the base 279 state and H at a given t = T can be considered to be locally 'frozen' over the time scale of instability.  To find the solution for these coupled equations, a normal-mode form is considered, where k gives the modulation of the pattern in the horizontal direction and the sign of the real part of σ indicates 286 the stability of the solution. Using this solution, Eqs. (17)-(18) can be written as a linear system with D = (1/H)d/dz, and the operators A and B are In the linear system, dc 0 /dz is the numerical derivative of c 0 (z, t), which is found by solving Eq. (13). To find the 288 boundary conditions, we recall that the perturbed fluid velocity is u(x, t) = 0 + εu (x, t) at the leading order. At physical parameters used to solve the linear system are shown in Table II. The value of β was taken from Kang et al.

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Considering the salt profile at t 0 = 1 h, the largest eigenvalues were calculated for different wavevectors k and 308 salinity Rayleigh numbers. Figure 10 shows the k − Ra s plane color-coded by the value of the growth rate σ, and the 309 neutral stability curve along which σ = 0. The smallest critical Rayleigh number is Ra * s = 3.8 × 10 4 , with a critical

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Since the linear stability analysis would not be valid in the regime far beyond the critical Rayleigh number, where nonlinearities would become important, we investigated this regime numerically. For simplicity, a two-dimensional geometry was used, shown in Fig. 14. As in the experiments, the initial height of the column of water is h 0 = 1 cm, the length of the cuvette is 6 cm, and the salt concentration is initially homogeneous and equal to c 0 . For the numerical computations, Equations (4a) and (3b)-(3a) are rescaled using the following expressions for length, time, flow speed, pressure and salt concentration: The dimensionless equations to be solved numerically are then The numerical studies were performed using the finite element package Comsol Multiphysics, in which, as in  Table II.

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The Comsol simulation was also verified by imposing periodic boundary conditions (pbc) in the horizontal direction.

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In this case, the given horizontal domain size only allows the simulation to resolve 6 cm/n of horizontal wavelengths 362 (n = 1, 2, 3, ...). For 1% of salt concentration, the distance between initially developing plumes was found to be  The average distance between plumes can be calculated from these images using a similar code to the one described in In the experiments it was clear that the lower the overall salt concentration, the longer it takes for plumes to 379 develop, which can be understood from the fact that a longer time is needed to accumulate sufficient salt at the top 380 boundary. As can be seen in Fig. 16, this feature is also observed in the simulations. Here, the snapshots were taken 381 when the plumes first developed, which is longer as the initial salt concentration decreases. In addition, the flow 382 field magnitude also decreases with lower initial salt concentrations, which is expected since there is less salt being 383 accumulated at the air-water interface. Here we summarize some of the quantitative measurements of plume obtained using the three approaches. We and numerical simulations the system was laterally finite, so geometric confinement effects can occur. Moreover, in 389 the experiments, the cuvette of course had front and back glass walls, so that the dynamics might be considered more 390 akin to Hele-Shaw flow than a truly two-dimensional system. needed for the instability to start is reflected in the value of the largest eigenvalue, which was 7 times larger for 1% with the results obtained using the other two methods.

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The phenomenon of interest here was discovered in a suspension of non-motile marine bacteria, but plumes were 425 also observed using a non-marine bacterium (Serratia) with low salt concentration in the medium. Therefore the 426 convection does not require salt in particular, but merely some component that accumulates due to evaporation. The

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In the ocean, an accumulation of salt can generate "salt fingers", which are formed when cold fresh water surrounds 433 warm salty water. This phenomenon relies on the fact that thermal diffusion is faster than the salt diffusion, which 434 has the consequence that a region high in salt will cool before the salt can diffuse, creating a descending plume of  The boundary condition for c at the free surface is sought here. Using the Leibniz integral rule, let us write the 452 temporal change of the salt concentration over the entire control volume as follows: where ζ(t * , x * ) = h(t * ) + η(t * , x * ) and V = [−∞, ∞] × [0, ζ(t * , x * )]. The first term on the right hand side of (26) is 454 the change rate of the total amount of salt in the system, and the second term represents the rate of change of the 455 salt concentration due to evaporation over the entire free surface. We note that the first term should vanish because 456 the total amount of salt in the system is constant. This implies that the integrand of the second term represents the 457 rate of change of the salt concentration at a given location x * on the free surface. Then, the conservation of the total 458 salt concentration requires the following boundary condition at the free surface: where the constant evaporation rate v e was identified.