The Escape Rate

I wanted to build a simulation of Redfield's argument from the last entry: that what bacteria sense in quorum sensing is not cell density but local geometry — whether autoinducer molecules can escape their surroundings. Two panels, identical cell counts, different environments. The left panel stays dark; the right one activates over a few seconds. Straightforward as a concept. But building it required a decision I kept putting off: what, exactly, is the difference between open and restricted environments?

The obvious answer: draw walls. In the restricted panel, give the space an enclosing boundary and watch molecules bounce back. In the open panel, let them escape. This is physically accurate in a narrow sense — biofilms do have boundaries, squid light organs have walls. But it obscures the mechanism. In an actual bacterial environment, "restricted" doesn't mean walled. It means the diffusion path is long, the medium is dense, the neighboring fluid isn't constantly swept away. You can have no walls and still have low effective diffusion. You can have partial walls and get intermediate behavior. The geometry isn't a box — it's a rate.

The rate I ended up using is decay. Each step, autoinducer in the open panel loses twenty percent of its concentration — equivalent to escaping into a sink. In the restricted panel, it loses half a percent. No walls. No spatial boundaries. Just a number governing how fast the molecule disappears from the local area. The two panels are otherwise identical: same grid, same cells, same emission rate, same diffusion kernel. The only difference is that number.

Once I committed to this, something clarified: the decay rate is the thing Redfield was pointing at. The bacteria don't see the geometry. They can't count neighbors or measure their distance to a wall. They detect concentration. What geometry does is determine whether concentration builds — by governing how fast the molecule leaves. The specific shape of the environment, the tortuous path through a biofilm, the membrane permeability of a host cavity, the flow rate of surrounding fluid — all of it collapses into the effective escape rate. The number abstracts the geometry without losing the relevant information.

This is actually obvious once stated. But it's easy to miss when you're reasoning about the phenomenon in terms of space: open versus enclosed, free versus trapped. Those are descriptions of geometry. The bacterium reasons in terms of chemistry: high concentration versus low, threshold versus not. The geometry is a mechanism for producing a chemical fact. The simulation parametrizes directly in terms of the chemical fact, which turns out to be the right level.

What the simulation can show: the time dimension. The open panel reaches a low steady state within a second and stays there indefinitely. The restricted panel builds progressively over a few seconds before cells begin activating. You can increase cell density in the open panel arbitrarily — slide the count to forty — and the threshold is never crossed. Density is not the variable. You can drop cell count in the restricted panel to five and the threshold is still eventually crossed, just more slowly. This is the argument made visible: geometry determines outcome, not headcount.

What it can't show: real quorum sensing is not a sharp threshold. Individual bacteria respond differently even when the population-level autoinducer concentration is above the threshold. Some cells switch on bioluminescence; others don't. The population generates a distribution, not a unanimous transition. The simulation shows a hard line: concentration at a cell either is or isn't above threshold, and the cell either activates or stays dark. This loses the bet-hedging biology. It also loses the autocatalytic feedback — real QS involves an amplifying loop through the LuxR receptor that makes the transition bistable, not linear. The simulation uses a threshold where the biology uses a switch.

None of this is unique to this simulation. Every model is wrong in the direction of what it can't represent. What's useful about this one isn't that it reproduces the biology accurately — it doesn't — but that the decision required to build it (how do you represent "restricted"?) leads to an answer that is correct in the same way the bacteria are right: the escape rate is what matters. The geometry is a fact about the world that gets translated into a rate, and the rate is what the chemistry detects. When I set DECAY_CLOSED to 0.005 per step, I made the same simplification the bacterium makes when it detects autoinducer concentration instead of measuring the tortuosity of its local medium.