Ready, set, mutate... and may the best microbe win

There are two issues here that need to be separated. They are linearity and multiplicity of variables.

As I'm sure you know, any system of purely linear differential equations in multiple variables can be solved, no matter how many variables there are. The solution is simply an eigenvalue problem; the time evolution of the system is a sum of exponentials. So, if we regard the evolution problem as the operator equation Fx = y , where F is the selective pressure, x are the genomic variables, and y the relative survival of those genes in the next generation, the fact that there are multiple selective pressures and each may affect the survival of several genes independently is irrelevant. Your 'trade-off' issue (and I must say I'd prefer to avoid Paley-esque comparison of living systems with cars) just involves two pressures having opposite signed coefficients. No problem; while a trade-off may naturally slow evolution of one gene (by countering the pressure), it won't prevent the system reaching the optimum.

The real issue is whether the system is truly linear - i.e. whether there are terms of the form Fx₁x₂. In that case, evolution, proceeding by slow steps, might genuinely not be able to find the global optimum, but might get stuck. I suspect much of speciation is in fact a result of getting stuck in local optima. However, if the 'fitness landscape', in the form of F, changes enough, you can get moved out of the local optimum. This is in essence why catastrophic events cause rapid evolution.

OK, Sorry, this is going to be another rambling post. :-(

So, if we regard the evolution problem as the operator equation Fx = y, where F is the selective pressure, x are the genomic variables, and y the relative survival of those genes in the next generation, the fact that there are multiple selective pressures and each may affect the survival of several genes independently is irrelevant.

Don't know how to answer here without giving the wrong impression...

I understand completely about time evolution and linear differential equations being sums of exponentials. And I agree that the use of such a formulation is applicable to the survival problem. I merely do not know how applicable it is to a problem with multiple survival pressures: if you can combine the survival pressures into a single F, or if you have multiple F's which commute, then OK.

But I don't know enough about modeling evolutionary survival in this way to know if my question is even germane to the issue. If you have any references I'll put them on my "to read" pile. [Warning, I'm currently getting to reading put on my list 5 months ago. It does make me look like a jerk to admit that, but I'd rather do that than give a misleading impression.]

Note: What I'm trying to get at here is the modeling of mutation / survival rates in the face of something like Malaria and sickle cell anemia. You have multiple competing causes of morbidity / mortality; and either one can kill at a young enough age to prevent passing on of genes. Whichever survival pressure is 'stronger' for a given population will (I suspect) depend sensitively on the particular environmental conditions (# of mosquitos, # of predators you have to run away from so sickle cell means you're out of breath and get eaten, etc., etc.)

Your 'trade-off' issue (and I must say I'd prefer to avoid Paley-esque comparison of living systems with cars) just involves two pressures having opposite signed coefficients.

I hate to admit it, but I never read Paley. I chose cars for two reasons: 1) textbook description of classes and inheritance in Java, 2) it's a great way to illustrate 'design' tradeoffs in a way non-experts can appreciate. No, I don't necessarily mean "Design" in the ID sense, but in the "what is the way to minimize some global error given a number of competing constraints"? I was thinking of several talks I attended within the Dept. of Defence, concerning design tradeoffs for various armored vehicles. What enhances survivability in a classic Warsaw Pact mass attack might get your rear handed to you in a tight urban environment. (*)

No problem; while a trade-off may naturally slow evolution of one gene (by countering the pressure), it won't prevent the system reaching the optimum.

Well said, indeed. The first sentence is EXACTLY my point. By using an artificially simplified system such as the rising-temperature bacterial tank, you obtain estimates of the effective mutation rate for a given environmental pressure. That's fantastic, that is real-life empirical data. My only point was, outside the laboratory, there are more likely to be other selection pressures, so your estimate of the rate from *this* experiment is likely inexact. What *are* the error bars? Not to 'disprove' selection, but just to make sure you are aware of any approximations. The breakdown of the ideal gas law does not invalidate the atomic / molecular theory of matter: but you'd still better keep in mind its limitations.

In that case, evolution, proceeding by slow steps, might genuinely not be able to find the global optimum, but might get stuck.

One of the military symposia I attended had a fascinating discussion of just these issues with regards to simulated evolutionary methods for target acquisition...and of course (IIRC I first read about it in a paper by Monty Pettit) you have a similar problem in protein folding when you cannot sample all of configuration space, a "steepest descents" method may get caught in a local minimum. Matching the sampling step size to the size over which the topography of the potential energy varies is important (**). Think 'broad shallow minium vs. deep narrow minimum'. With a broad minimum, you are likely to find it but end up in a configuration which is a physically uninteresting local minimum; with a deep narrow minimum, it is probably very important but your steps are likely to pass right by it. The use of a simulated evolutionary method avoids this by randomly sampling 'all over' the configuration space.

The steepest descent method can be thought of as taking a golf ball on the golf course, setting it down (assume the greens are fast!), and watching where it rolls.

The simulated evolution method is akin to going out on the course in the rain, and looking where the water has puddled.

Real life evolution seems to be somewhere between these two: you can take arbitrary jumps from your starting position (like the nylon bug) but you're not *truly* sampling over the "whole" of the configuration space.

This topic has made me very happy. I'm going to wander off somewhere to purr for awhile ;-)

Cheers!

(*) Think of modeling a function, using a couple of parameters. Is it better to minimize the RMS error throughout the entire function, or is it better to get "almost exact" everywhere except in a limited region, where your fit *sucks* ? The answer, as usual, is "it depends."

(**) THIS is what I think is what is missing in evolutionary theory. There must be a sensitive interplay between raw mutation rates, birthrate, environmental changes, and other populations (say of food or predators). The selection pressures will to keep the target population "on the edge" (so they need to evolve), but not be so strong that they just drive everybody extinct.

But even in the case of mass extinctions, the mass extinctions are not the end of the story. In fact, the dying out of other species within a habitat or region merely becomes just one more "selection pressure" for the folks left behind...

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