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shear

Figure 4.7 The profile of velocity of a fluid flowing inside a tube. a: Velocity in a tube assumes a parabolic velocity profile, ranging from v = 0 at the tube walls and the maximum velocity (indicated by the longest arrow) at the center. b: Distribution of shear in a tube with a flowing fluid. Shear is greatest where the velocity gradient, dv/dr, is steepest (at the walls) and falls to null at the center.

shear

Figure 4.7 The profile of velocity of a fluid flowing inside a tube. a: Velocity in a tube assumes a parabolic velocity profile, ranging from v = 0 at the tube walls and the maximum velocity (indicated by the longest arrow) at the center. b: Distribution of shear in a tube with a flowing fluid. Shear is greatest where the velocity gradient, dv/dr, is steepest (at the walls) and falls to null at the center.

ity profile tilts all the cells in the column toward the walls. The cells now congregate at the walls of the tube rather than at the center. This phenomenon is known as gyrotaxis—literally, "arranged by a vortex."

Of course, this explanation relies on the occult process that is the cause of the trouble in the first place. Unmasking the real answer requires us to break the problem down into three steps:

1. Large-scale gradients in oxygen concentration arise in the culture as a consequence of the organisms' collective metabolism.

2. In response to these gradients, the organisms redistribute themselves in the culture according to the oxygen concentration gradients. This imposes a new large-scale gradient in potential energy in the form of a spatial redistribution of mass within the culture.

3. Gravity acts on this new PE gradient to oppose the redistribution of mass, generating in the process large-scale flows within the culture.

Let us now turn to just how these steps work together to generate bioconvection.

How Do Bioconvection Plumes Arise? We can now return to the remarkable appearance of bioconvection plumes in a petri dish of Chla-mydomonas. Like all interesting phenomena, this one has two explanations: a short simple one that masks a more complicated, but more interesting, explanation. The short simple explanation is that the order is imposed by the energy flowing through the organisms. In this case, the work being done comes not from an external agency, like a mixing spoon or a baseboard heater, but from the chemical potential energy in the culture broth being channeled through the cells:

STEP 1: ESTABLISHING THE GRADIENTS IN OXYGEN CONCENTRATION

First let us imagine a culture of randomly distributed microorganisms, contained in a petri dish that we can either open to, or close off from, the air as we wish. Let us suppose initially that the dish is closed to the air.

We will first focus on a single microorganism and the small parcel of fluid (culture broth) that surrounds it.4 For various reasons, I will treat each microorgan

4 ,um, and since it is roughly spherical it will occupy a volume of ism and its associated parcel of fluid as if they were an inseparable unit.5 As the microorganism consumes oxygen, it necessarily takes it from its associated parcel of fluid. The concentration of oxygen in this parcel, which we designate as the oxygen partial pressure, or pO2, therefore declines by some small amount proportional to its depletion. If we measure the oxygen concentration in our parcel at several times thereafter, we will see the pO2 declining with respect to time, the rate of decline being exactly equal to the rate at which the microorganism is consuming oxygen.

The same thing will happen everywhere in the culture: wherever there is a microorganism, the pO2 in its surrounding parcel of liquid will be declining. If we assume, just for convenience, that every microorganism is consuming oxygen at the same rate, the pO2 is pretty much the same no matter where in the culture it is measured. In other words, oxygen is uniformly distributed and uniformly depleted through the culture. To put it more formally, there will be no gradient in oxygen partial pressure in any spatial dimension: no matter which way a microorganism faces, whether it be up, down, north, south, east, or west, it will confront the same oxygen partial pressure everywhere. Consequently, the movements of oxygen in the culture are dominated by small-scale variations of oxygen partial pressure between an individual microorganism and its associated parcel of fluid.

When we open the top of the culture to the air, however, there is a remarkable change. No longer do all the organisms in the culture confront the same oxygen concentration no matter which direction they face. The unfortunate ones deep in the fluid still are roughly 2.7 x 10-10 cm3. If the culture density is about 106 per cm3, each Chlamydomonas will have about 10-6 cm3 of culture fluid to occupy, or roughly 3,700 times the volume of the cell itself, or a cube roughly 100 ,um on a side, 25 times longer than the length of the cell itself.

5. At very small scales, objects and the fluids they are embedded in are held together to a great extent by the fluid's viscosity, which contributes a sort of "added mass" to the object. Conse quently, very small objects in fluids always tend to travel with the fluids surrounding them.

surrounded, initially at least, by parcels of liquid with the same low pO2. The lucky ones at the surface, however, are adjacent to the rich oxygen source of the air. There is now an oxygen partial pressure difference between the air and the surface parcel of liquid. As dictated by the Second Law, oxygen will move from the air into the topmost parcel, increasing its oxygen concentration.

This increase at the top level will set in motion a sort of "bucket brigade" for the transport of oxygen deeper into the culture. The bucket brigade works like this. The movement of oxygen from the air into a surface parcel makes its pO2 higher than that of the fluid parcel just below it. An oxygen partial pressure difference between the parcels is established that will drive oxygen downward at a rate proportional to the partial pressure difference between the parcels. The same thing happens for each pair of parcels we encounter from the top to the bottom of the culture, that is between parcels just below the surface and parcels below them, and so on. The end result is a large-scale gradient in oxygen partial pressure that drives a large-scale flow of oxygen downward into the culture.

This downward movement of oxygen is inefficient, for two reasons. First, the oxygen moves downward by diffusion, which is very fast over short distances, such as across the radius of a cell, but is very slow over "long" distances, such as the few millimeters from the top to the bottom of the culture. Second, each parcel of liquid contains an oxygen-consuming microorganism. Because the microorganism in a surface parcel will consume some of the oxygen passing through that level, less oxygen will be available to move to the next parcel below. It is as if the microorganisms were each exacting an "oxygen tax" against the movement of oxygen molecules past them. The end result will be a vertical gradient of oxygen partial pressure in the culture, with high partial pressures at the top and lower and lower partial pressures toward the bottom, as each microorganism extracts its "oxygen tax." Unlike the closed culture, where oxygen in all the parcels was de pleted uniformly, each parcel's pO2 in the open culture is now steady with respect to time, reflecting the replacement of oxygen in the parcel as it is consumed by the contained microorganism. Also, the cells at the top levels enjoy higher, and presumably more equable, partial pressures, because their oxygen consumption rates can be met without the intervention of lots of "middlemen" each taking their piece of the flux as oxygen flows down to the bottom. The poor cells at the bottom, on the other hand, must rely on the trickle-down from the top, and so must operate at lower partial pressures.

Let us now put this scenario into a more general context of energy. Oxygen in the culture can move from one place to another only if there is a source of potential energy to move it: oxygen molecules have mass, and it therefore requires work to move them around. The potential energy gradient moving the oxygen comes from the difference in oxygen concentrations between two points in the culture. Indeed, the way we express oxygen concentration as its partial pressure is a direct measure of the potential energy that does the work of moving oxygen.

In the closed culture, the distribution of potential energy is diffuse: gradients in oxygen partial pressure extend no further than that between a microorganism and its associated parcel of fluid. If the microorganisms are distributed diffusely through the culture, so too will be the small-scale gradients in potential energy in the culture. When we open the culture to the air, however, we have now imposed a large-scale potential energy field, the top-to-bottom gradient in oxygen partial pressure. Part of this new potential energy field comes from having one surface of the culture exposed to a rich source of oxygen (to use our jargon, the environmental potential energy), and part comes from energy being channeled through living things (manifest in the "oxygen tax" extracted as oxygen moves past the microorganisms). As a result, a gradient of potential energy is imposed on the culture on a much larger scale than the gradients affecting the living things in the culture.

STEP 2: REDISTRIBUTING THE MICROORGANISMS

The next step depends on the locomotory activities of the microorganisms. Remember that the negative gravitaxis in Chlamydomonas is partially "dumb"—that is, it relies on the cell's distribution of mass always pointing it up. However, swimming in these organisms is also activated when the cell finds itself in poor conditions. Chlamydomonas in the top layers will power locomotion only to the extent they need to stay there— life is good, so why move? Cells in the bottom layers, however, will divert more energy through their locomotory organs, powering the higher levels of locomotion needed to climb to the top. Thus, the large-scale gradient of oxygen partial pressures also elicits a large-scale gradient in the magnitude of the metabolic energy stream flowing through the culture. Work is done on the culture as a whole, but its distribution reflects the distribution of oxygen partial pressure. The farther the cell must travel, the more work is done. As a result of this work, cells will accumulate in a thin layer at the surface levels of the culture.

Because these cells have a net density that is slightly greater than the water they live in, their accumulation at the surface of the culture makes the top layers of culture fluid heavier than the lower layers. This concentration of higher density is obviously unstable, in the same way that a brick balanced on the top of a broomstick would be. Unlike the brick, which would fall immediately if it were not held up somehow, the mass at the top of the culture can be stabilized to a degree. When unstable distributions of mass in fluids, called inversions, do collapse, they do so in a fairly controlled way that generates the orderly bio-convection cells.

STEP 3: GENERATING THE BIOCONVECTION CELLS

The factors that stabilize and destabilize inversions can be easily appreciated with an analogy. Imagine a pan of water being heated on a hot plate. The heat imposes an unstable inversion on the water: the water at the bottom, being hotter than the water at the top, is more buoyant. The warmer water at the bottom will therefore "want" to rise to the top, but its rise will be blocked by the layers of cooler, denser water above it, which sit on it like a pot lid.

Inversions are stabilized by the fluid's viscosity, which quantifies how resistant a fluid is to flow. In the inversion that develops in a pan of heated water, the buoyant water at the bottom pushes upward against the heavier water on top with some force, let us call it Fb. Ordinarily, that buoyant force would drive an upward flow. For this to happen, however, the buoyant water at the bottom must flow through the heavier layer of water on top, and this will be resisted by the water's viscosity; let us call it Fv. As long as the buoyant forces pushing the water up are smaller than the viscous forces resisting it (that is, Fb << Fv), the inversion will be stable. If the buoyant forces match or exceed the viscous forces resisting it, however, the inversion will break up. The events just as the inversion begins to break up are of particular interest to us, so let us consider them in some detail.

It is very hard to heat a surface like the bottom of a pan absolutely uniformly. There will inevitably be some parts of the pan bottom that are heated more than others. There, the water will be warmer, and a bit more buoyant, than it will be in other, less well heated areas. The horizontal boundary between the layers of the inversion will therefore not be perfectly flat but will exhibit transient humps, where "bubbles" of locally more buoyant water rise slightly higher than adjacent parcels of cooler and slightly less buoyant water.

If one continues to heat the pan, eventually enough energy will be imparted to one of these bubbles so that its buoyant force overcomes the viscous forces resisting its upward flow. It will then "punch through" the layer of heavier water keeping it down. The hole punched through the dense upper layer now opens up a pathway for the bottom layer of "suppressed" warmer fluid everywhere else in the pan. The more buoyant water on the bottom of the pan can now rise through the hole opened for it, and the denser top water helps the flow along by pressing down on it with its greater weight. The result is the sudden appearance of a vigorous upward-flowing plume of warm and buoy ant water, leading ultimately to the turnover of the inversion.

When Chlamydomonas migrate to the uppermost layers of the culture, an inversion of sorts also occurs, even though the culture's temperature is uniform. The accumulation of organisms at the surface forms a dense layer of fluid, which gravity should make sink were it not spread out on top of a less dense layer below. The stability of this inversion is governed by the same physical rules that govern the temperature inversion in a heated pan. In this case, however, it is random fluctuations in density of microorganisms that lead to the inversion's breakdown. Locally dense collections of cells form "bubbles" (anti-bubbles, really) of dense culture that will tend to sink into the less densely populated and more buoyant culture medium below it. Usually, these anti-bubbles are buoyed up until they are dispersed by Brownian motion.6 At some point, however, one of the anti-bubbles will become dense enough to overcome the buoyant force holding it up, and it will begin to sink.

The mere presence of a sufficiently dense anti-bubble is not enough to generate a bioconvection plume, however. The maximum increase of density of the fluid is limited by the density of the microorganisms themselves. Even if the organisms were packed as tightly as possible, the top layers will be only about 5 percent denser than the water: in reality, the density differences will be less. What is needed in addition is that the anti-bubble be composed of living microorganisms: an anti-bubble of dead cells is never dense enough to generate a plume. Bioconvection plumes, however, appear reliably and rapidly (within minutes) in living cultures.

To understand what happens next, remember the mechanism of hydrodynamic focusing: a downward-

6. Brownian motion is named after the Scottish botanist Robert Brown. In 1858, Brown reported a peculiar random motion of microscopic particles suspended in fluids. The motion itself is caused by random variations in the forces exerted on these particles by the atoms of the fluids in which the particles are suspended. This force is significant only on the small scale of cells and molecules, for which it is a randomizing force.

pointing velocity profile will tilt cells toward its center, with the degree of tilt being directly proportional to the shear. So armed, let us now follow the development of a bioconvection plume in detail.

Suppose first an anti-bubble of densely packed microorganisms is sufficiently dense to start it sinking slowly in the culture fluid (Fig. 4.8). If these cells were dead, the increased density would be short-lived, because Brownian motion would quickly disperse them. In an anti-bubble of living cells, however, a very different dynamic occurs. When the anti-bubble begins to sink, it drags adjacent parcels of fluid along with it, and these drag adjacent parcels along with them, and so on and so on. The result is a velocity profile that extends some distance away from the sinking anti-bubble: the velocity is highest at the center of the sinking anti-bubble and falls off gradually away from the center (Fig. 4.8). Any living microorganism caught in this flow field will be rotated toward its center, just as it is when the culture experiences hydrodynamic focusing. The surrounding Chlamydomonas, because they swim in whatever direction they are pointed, will therefore congregate at the center of the anti-bubble. This will, in turn, increase the number and density of cells in the anti-bubble, which will increase its sinking speed, which will increase the steepness of the velocity gradients, which will tilt more cells toward the center of the field, which will increase the anti-bubble's density still further and . . .

You get the idea, I hope. The initial sinking of the bubble sets up the conditions for promoting its own sinking rate, a condition we call positive feedback. The positive feedback only works with living cells. The macroscopic result is the generation of robust convection plumes through the culture, within minutes, each plume centered on a transiently formed anti-bubble of organisms dense enough to start sinking. Eventually, these plumes will compete with one another for microorganisms to feed it, and it will come as no surprise that the initially stronger plumes will eventually incorporate the initially weaker plumes into them, forming the curtains of bioconvection that develop (Fig. 4.3).