A flight simulator for a bacterium. Down here water behaves like tar, inertia does not exist, and every instinct you have about moving through a fluid is wrong. You cannot coast. You cannot glide. You cannot steer. What you can do is what E. coli does, which is swim in a straight line, get shoved off course by molecules, and decide when to give up and try a new direction.
Every number here is computed from the sliders above using the standard expressions, not looked up. Nothing the sliders can do will get you out of creeping flow: push the body all the way out to fifty microns in plain water and the Reynolds number still only reaches about 0.004. That is rather the point. What does move, by five orders of magnitude, is how long you keep a heading.
Cut the motor on a car and it rolls. Cut it on a bacterium and it stops, essentially instantly, having traveled a distance smaller than an atom.
The velocity decays as exp(-t γ/m) against Stokes drag
γ = 6πμa, so the total coasting distance is just v m/γ.
For a one micron body swimming at thirty microns a second in water at body temperature that
comes out at about 0.1 ångström: roughly a fifth of the Bohr radius, which is
0.529 å.
Purcell put the same figure in his 1977 lecture Life at Low Reynolds Number, and the
instrument panel above recomputes it from whatever you set the sliders to.
This is why there is no glide, no drift and no momentum in this simulator. It is not a simplification. Including inertia would change the trajectory by less than the width of an atom, and pretending otherwise would be the lie.
With inertia gone, the equations of motion lose their time dependence. The consequence is strange and absolute: how fast you move your body makes no difference to where you end up. Only the sequence of shapes matters.
So any stroke that retraces itself, out and back through the same shapes, returns you exactly to where you started. A scallop has one hinge. It can open and it can close, and whatever it does with the timing, it goes nowhere. Purcell called this the scallop theorem, and the simulator reproduces it: the reciprocal stroke gives a net displacement of exactly zero, to the last digit the arithmetic carries.
Escaping needs a stroke that traces a loop in shape space rather than a line, which needs at least two degrees of freedom out of phase. A rotating helix is the cheapest way to do that, which is why bacteria have corkscrews rather than fins.
A bacterium cannot measure a gradient across its own body. It is two microns long, the concentration difference over that distance is minute, and Brownian noise in the count of arriving molecules swamps it. It also cannot hold a heading: rotational Brownian motion turns it through tens of degrees a second, so by the time it has swum a few body lengths it has forgotten which way it was pointing.
What it does instead is measure the concentration in time. If things are getting better, keep going. If not, tumble and pick a new direction at random. That is the entire algorithm, and it is the only control this simulator gives you: a tumble button.
It should not work, and it works extremely well. Of four hundred simulated cells with that single asymmetry, all four hundred reached the source, in a median of fifty two seconds. Four hundred identical cells tumbling at a constant rate managed it three hundred and seventy one times, in a median of a hundred and forty four seconds, with the rest still wandering when the ten minute clock ran out. The race mode runs that experiment in front of you, twenty four against twenty four, and you swim in it.
The cell is modeled as a sphere with Stokes drag. A real E. coli is a rod with a helical bundle, its drag is anisotropic, and the propulsion comes from that anisotropy. The swimming speed here is set rather than derived from motor torque.
The fluid has no memory, no boundaries and no other cells in it. Real swimmers interact hydrodynamically, are attracted to surfaces, and leave wakes that are anything but the clean empty water here.
The cells are drawn several times larger than the scale bar says they are. At a true one micron across they would be two or three pixels and you could not tell a tumble from a run. The positions and the distances are to scale; the bodies are not.
Tumble angles are drawn from a distribution with roughly the right mean. The real distribution is broader and has structure that depends on how many flagella unbundle.
The chemotaxis response is instantaneous. Real receptor adaptation takes a few seconds and has memory, which is precisely what lets a cell compare now against a little while ago.