Find complexation rates

[prismofeverything]

Per the discussion on this PR: https://github.com/CovertLab/wcEcoli/pull/429 I am creating an issue to encode the idea that the current rates for complexation are placeholders and need thought/care/updating at some point in the future.

[jmason42]

Based on this discussion it looks like realistically it is somewhere between 1 per second and 1000 per second, it looks like many of the reactions are assisted in some way and translation proximity seems to play a role as well.

What's the source on 1000 association reactions per second? Some of the fastest enzyme catalyze at a bit over the diffusion limit, putting them at an association rate constant just over 10^9 /M-s, or equivalently 1 reaction per second assuming two reactive molecules. And that's for small molecules (which are likely less hindered by the cytoplasmic environment). Everything I've seen for protein-protein association suggests 10^5-10^6 /M-s as an average.

Translation proximity is plausible but it's only one of a litany of effects that could modulate association rates (and most of them are hindrances rather than enhancements).

[prismofeverything]

What's the source on 1000 association reactions per second?

We estimated it from a figure in Neidhardt, though I admit it was rough. I just needed a value at the time, so I don't think we should put too much stock in it. The other basis for this number was that it is high enough to reproduce existing behavior, though I believe we could go down to as low as 10 and still succeed with that (but I would have to test that).

I can add in a rate of 1 reaction a second and see what happens, I'll report back. One concern from Travis was that complexation is a basis for the creation of functional ribosomes so missing those reactions could reduce our overall growth rate, but it doesn't hurt to check.

We should also examine my assertion that this value plays the role of the rate for a single reaction if they were the only two molecules in the system. Here is my reasoning for that:

If there are only two molecules in the system, say A and B, which combine to make X, then the propensity calculation will be:

RATE * (count of A) * (count of B) = RATE * 1 * 1 = RATE

Which then is used directly as the lambda parameter to sample the exponential distribution to find out how much time passes. Since the mean of an exponential distribution is 1 / lambda that means the expected time passing for this single reaction is 1 / RATE. So if RATE = 1000, we expect 1 / 1000 reactions to occur in our time step (1 second), or 1000 reactions a second. Am I missing something here or is there a flaw in my logic? It seems straightforward to me but I know things can get subtle here.

[jmason42]

It's a bit of a hassle so I won't ask for it, but I would be happy to take a look at the figure in question if you attach an image.

The other basis for this number was that it is high enough to reproduce existing behavior, though I believe we could go down to as low as 10 and still succeed with that (but I would have to test that).

As the person responsible for the existing behavior, I can assure you that there is no reason to believe the existing behavior is correct.

One concern from Travis was that complexation is a basis for the creation of functional ribosomes so missing those reactions could reduce our overall growth rate, but it doesn't hurt to check.

It will almost certainly affect growth. The fact that ribosome assembly takes some time should be modeled, and if necessary, should be accounted for in other places (the expression rate of ribosomes, polymerization rates, etc.). If it takes minutes for a ribosome to assemble, that's a very profound delay in response, particularly under environmental shifts.

It seems straightforward to me but I know things can get subtle here.

No, that's all correct. I'll clarify a few things, though. Your RATE is actually a rate constant; specifically, a stochastic rate constant. The 'rate' is the overall expression. Regardless, if your overall rate is 1000 per second, then that certainly implies that you'd expect 1000 reactions in a second, although that gets a little confusing since we only have only possible pair of reactants to begin with. In the stochastic domain it's easier to say that the expected (average) time to reaction is 1 millisecond.

One last thing that I've neglected to mention is that stochastic rate constants are not intrinsic properties (at least, not for second-order reactions) and should be recomputed from a kinetic rate constant (which is intrinsic) and the cell volume.