… Data aequatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa …

## Category: mathematical physics

### Why do Volvox spin?

For today’s High-Energy-Physics/General-Relativity Colloquium, we had a speaker whose research is rather far from the usual topics. Raymond Goldstein of DAMTP gave a talk on the physics of multicellular organisms, with particular focus (since the field is so broad and so new for most of the audience members) on the example of Volvox, a kind of green algae composed of spherical colonies of about 50,000 cells.

One of the very interesting things about them is that, if you look under a microscope (or even a magnifying glass! Each colony is about half a millimeter across, so you can even see them with the naked eye), they spin. (Yes, the Goldstein lab has its own YouTube channel.)

(The video also shows how their motion can be constrained by hydrodynamical bound states formed due to their individual spinning motion.)

Now, we have a pretty good idea of the very basic locomotive mechanism of these organisms. Each colony is formed with an exterior ball of “swimming” cells and some interior balls of “reproducing” cells. The swimming cells each have two flagella pointed outwards into the surrounding fluid. Their beating will give rise to the motion for the whole colony. But the strange thing is that they do not swim straight: the cells colonies tend to travel in one direction, will spinning with the axes aligned with the direction of travel. Why? Isn’t is inefficient to expand extra energy to spin all the time? This was a central question around which the presentation today was built.

Two main results were described in the talk today. First is a result about how the two flagella of each cell interacts. It was observed (some time ago) that, by direct observation under a microscope, the two flagella can exhibit three types of interaction. First is complete synchronisation: the two flagella beats in unison, like how a swimmer’s arms move when pulling the breaststroke. This is observed 85% of the time. Then there is “slippage”, where for some reason one flagellum is slips out-of-phase from the other briefly, and recovers after a while. This happens about 10% of the time. And lastly there is a completely lack of synchronisation when the two flagella beats with different frequencies for about 5% of the time. The original report on this surmised that this difference represents three different “types” of cells: since each observation is short in time, they didn’t observe much in terms of transitions from one type to the other. What was discovered more recently is that, in fact, the three behaviour all belong to the one single type of cells making up Volvox, and the transition is stochastic!

Now, why this may be surprising is the following: each flagellum is a mechanical beater and has some innate characteristic frequency at which it beats. So in an ideal, linear situation, the two independent flagella should not interact. And so there cannot be reinforcements of any type. Now, one may guess that since the two flagella are swimming in water, the hydrodynamics may serve as a medium for the interaction. However, a pure hydrodynamic interaction should lead to something like sympathy, a phenomenon first observed by Huygens. Basically, Huygens put two pendulum clocks on the same wall, and set their pendulums to be of arbitrary phase relative to each other. After some time, however, he discovers that invariably the clocks settle down to a state where the their pendulums are completely out of phase. This “tuning” is attributed to vibrations being passed along the supporting beams of the wall.

(One can do a similar experiment at home with a board, two metronomes, and two soda cans. A dramatic example is shown below.)

But the problem with the synchronisation theory is that it can only explain the 85% of the time occurrence of completely in phase swimming, but not the other 15%. The solution to this problem requires real consideration of the chaos on a molecular level. As it turns out, one of the force we have so far neglected is the force driving the flagella. This is dependent on the biochemical processes inside the cells. By considering the biochemical noise which contributes a stochastic forcing on the entire system, one can recover the other 15% of out-of-phase behaviour. (The noise is not thermal, as thermal noise should have much lower amplitude than required to cause the phenomena.)

The second beautiful result described in the talk was on the spinning of the colonies, and its relation to phototaxis, the attraction of the green algae to light. How are to two related? It is a quite magnificent feat of evolution. Now, in this colony of 50,000 cells, there is no central nervous system, so how do the cells coordinate their motion to swim toward the light? You cannot rely on chemical signals, or even hydrodynamical synchronisation, since the physical distance between the cells are typically larger than the cells themselves. The effects of this signalling would be too weak and too slow. It is more reasonable to expect the behaviour to be “crowd sourced” (for viewers of the Ghost in the Shell anime series, a cellular level of “stand alone complex”): each cell is programmed to behave in a certain way, and when taken as a whole, their joint behaviour gives rise to the desired response of the colony as a whole.

Well, at the level of the cell, what can they do? Each cell is equipped with a photosensing organelle. And like the classic Gary Larsen cartoon, each cell is really only capable of stimulus-response. Experimentally it was confirmed that each individual cell reacts to light. When a cell is initially “facing away” (the light-sensor is direction sensitive) from the light source, and turns to “see” the light, the stimulus would shock the cell into slowing down its flagella’s beating. After a very short while the cell gets used to the light, and the beating resumes in earnest. The reverse change from light to darkness, however, does not cause changes in the beating of the flagella.

And this explains the spinning of the Volvox! Imagine the colony swimming along, minding its own business, when suddenly light hits one side of the colony. The cells on the lit side slows down its flagella beating, and gradually recovers its beating as it rotates out of view of the light source. So the net effect of the spinning of the colony is that “new” cells kept being brought into view of the light, receive the shock, slows its flagella, and recovers as it “retreats into the night”, only to be shocked again “as sun rises the next day”. So the flagella beats more fervently on the dark side of the colony compared to the bright side, so, as anyone who has tried swimming one-armed would know, the colony will slowly turn toward the light source.

The best part about this process is that it is self-correcting. As the axis of rotation gets more and more aligned with the light source, more and more of the cells experience an “Alaskan summer” with the “sun” perpetually overhead. These cells that are not brought back into darkness no-longer receive the periodic shock that slows their flagella, and so swim equally as hard through the entire “day”, and therefore no longer contributes to turning. When the spin axis is perfectly aligned with the light source, the entire “northern hemisphere” is perpetually illuminated, while the “southern” is not, so until the light-source changes directions, the colony will cease to change directions and move straight toward the light source.

For this all to work, it requires that the spin rate of the colony be exactly the same as the rate at which the cells recover from the shock of seeing the light. And this is experimentally confirmed. (An interesting question brought up at the end is whether we can use this as a laboratory test for evolution: if we add some syrup or something to the water to make it more viscous, the spin rate will necessarily slow down. Then the original strand of Volvox will not be as effective at swimming toward the light. It would be interesting to see whether after a few hundred generations, a mutant strand evolves with slower recovery time from illumination.)

### The “Hoop Conjecture” of Kip Thorne and Spherically Symmetric Space-times

Abstract. (This being a rather long post, I feel the need to write one.) In the post I first gather some miscellaneous thoughts on what the hoop conjecture is and why it is difficult to prove in general. After this motivation, I show also how the statement becomes much easier to state and prove in spherical symmetry: the entire argument collapses to an exercise in ordinary differential equations. In particular, I demonstrate a theorem that is analogous, yet slightly different, from a recent result of Markus Khuri, using much simpler machinery.

The Hoop conjecture is a proposed criterion for when a black-hole will form under gravitational collapse. Kip Thorne, in 1972 [see Thorne, Nonspherical Gravitational Collapse: a Short Review in Magic without Magic] made the conjecture that (I paraphrase here)

Horizons form when and only when a mass $M$ gets compacted into a region whose circumference $C$ in EVERY direction is bounded by $C \lesssim M$.

This conjecture, now widely under the name of “Hoop conjecture”, is deliberately vague. (This seemed to have been the trend in physics, especially in general relativity. Conjectures are often stated in such a way that half the effort spent in proving said conjectures are used to find the correct formulation of the statement itself.) Read the rest of this entry »

### Aharonov-Bohm effect’s analogue in water waves

Rather indirectly through Claude (a not-so-short story there) I learned of a paper by Michael Berry and collaborators titled “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue” (a copy can be found on Berry’s website at number 96). It is among the most pleasurable papers I have read in physics to date.

To understand why this Chandrasekhar-esque paper tickles me so, I need to explain a little bit of the physics.

The Aharonov-Bohm effect
The Aharonov-Bohm effect is one of the classic illustrations of the difference between classical and quantum mechanics. More importantly though, it also demonstrated that the use of the vector magnetic potential is not a mere computational convenience, but that the potential also manifests itself in physical effects. Read the rest of this entry »