Is this model of understanding of what's natural selection and what's not, correct? [closed]

Question

Here in this account I just want to make sure, that I've grasped the concept of natural selection as is usually spoken by evolutionary biologists, truly the wording here are non standard and in some sense sloppy, yet I want to make sure that I've got this concept right.

My account on what constitutes natural selection and what's not:
We say that there is a causal relationship from [A] to [B], to mean that there is a process in which A is a part of, that results in changes in B.

Also we can paraphrase the above by saying that there is a causal relationship that [A] has on [B]; or by saying that [B] is the consequence of a causal relationship from [A].

Now Natural selection seems to require those as prerequisites:

1. There is a causal relationship from some [hereditary material g] to the [survival and reproductive status of individuals harboring g].

More precisely and incorporating some standard terminology this is:

There is a causal relationship from a [genotype g] to [absolute fitness of g].

We call such hereditary material as fitness related hereditary material.

2. A variation exists between some fitness related hereditary material as regards their final causal effect on the degree of their fitness in a common environment $E$. Mathematically speaking, this is:

$\exists E, g_1, g_2( E \text { is an environment } \wedge g_1,g_2 \text { are fitness related hereditary material } \wedge g_1 \neq g_2 \wedge fit_E(g_1) \neq fit_E(g_2))$

In English: There exists an environment $E$ and $g_1, g_2$ where both are fitness related hereditary material such that $g_1$ is different from $g_2$ and fitness of $g_1$ in enviornment $E$ is different from fitness of $g_2$ in environment $E$.

Two fitness related hereditary material that have the same final effect on the degree of their fitness (even if through different causal mechanisms) in a common environment $E$, are said to be isofit$_E$, while those that differ are said to be anisofit$_E$. Formally:

$g_1 \ isofit_E \ g_2 \iff fit_E(g_1)=fit_E(g_2)$

$g_1 \ anisofit_E \ g_2 \iff fit_E(g_1) \neq fit_E(g_2)$

Of course anisofit fitness related hereditary material might even have a difference in the direction of their effect on fitness, so a positive direction means that "the causal relationship from the hereditary material to its fitness, is towards increasing its fitness"; while the opposite is for negative direction.

Now for every hereditary material $g$ the population of all individuals in environment $E$ that harbour $g$, is to be called the "$g$ population in $E$".

3. There is an environment $E$ that has two anisofit$_E$ fitness related hereditary material populations living in $E$.

If we have 1 and 2 and 3, then we can have Natural selection!

The reason is because Natural selection is the differential (i.e., non-equal) size of populations of anisofit fitness related hereditary material, living under a common environment, that is caused by the differential causal relationship those hereditary materials have on survival and reproduction of individuals in their populations under that common environment.

The above serve as a definition of natural selection, albeit a long one, but I think it captures the intended meaning given to that term by evolutionary biologists.

Now from the above definition we get to infer two properties that natural selection has:

1. Natural selection is always not neutral. Note "non-equal" part in the definition. The reason is because we have two anisofit fitness related hereditary material populations living under a common environment, and the difference in their size is attributable to the effect of those hereditary materials on their fitness in that environment, so since those are aniso-fit, then clearly the size of populations of them would be different.

2. For the same reason outlined in 1, we expect natural selection to be "co-directional", i.e. the difference in sizes of the populations of those anisofit hereditary materials, must parallel the difference in the effect of those hereditary materials on their fitness in that common environment, so the population with the hereditary material causing "higher" fitness would have the "bigger" size! In other words natural selection increase the size of the population of the hereditary material that causes higher fitness on the expense of the size of the population of the hereditary material that causes lower fitness.

On the other hand suppose we have some environment in which there are two anisofit fitness related hereditary material populations. Now if some environmental, or recombinative genetic change inflicts those two populations, that works either in a neutral manner, i.e. causes equal population sizes of those anisofit fitness related hereditary material, or works in an opposite-directional manner, i.e. in a direction that is opposite of the expected direction mentioned above, better be termed as "contra-directional". In this situation even if the size of the populations of those hereditary materials is different (imparting the appearance of a selection) still that difference is not explained by the effect of those anisofit fitness related hereditary material on their fitness in that environment! So this would not be an example of natural selection! It would be an example of an environmental factor that caused a "genetic drift", or of a genetic recombination process that caused a "genetic drift" also.

So we in effect have a struggle between "natural selection" which works in the direction of increasing adaptation with the environment, on one hand, and "random selection" (or sometimes called neutral selection or non-selection) which doesn't necessarily work in the direction of increasing adaptation with the environment.

So in some sense "evolution" is determined by the struggle of those two kinds of mechanism of change.

If random change prevails, then evolution would not necessarily move in the direction of increasing adaptation of living organisms with their environment. While if "natural selection" prevails, then evolution would proceed in the direction of increasing adaptation to the environment.

Answer 1

I've mostly skimmed the formalism you introduced, but getting to the 2nd half of your post, the answer is yes, you understood correctly the distinction between natural selection (aka adaptive evolution) and drift (aka neutral evolution), as well as the fact that it's not a given that natural selection would be the predominant effect in arbitrary circumstances. Small population sizes, high rates of mutation, weak genetic repair mechanisms, etc. can all lead to chance being the predominant effect.

The conditions needed for natural selection to be predominant have been investigated in lots and lots of publications. A quick overview and brief list of such publications is found in

• Duret, L. (2008) Neutral theory: The null hypothesis of molecular evolution. Nature Education

One final point I'll make here is that we need to be careful to define what we're talking about being selected. In general, for phenotypes there's not much debate that adaptive (rather neutral) evolution dominates in nature. The research and debates mostly concerned what happens to genotypes. This is somewhat obvious since there are numerous mechanisms that mediate the translation from genotype to phenotype.

A departing teaser, there's still a lot debate concerning how the mechanisms that enable low-error genetic transmission (the surpassing the famous error threshold) came about in the primordial soup. One interesting adaptation we can observe today is that bacteria can alter their rate of mutation in response to environmental changes; they function a bit like the Hulk superhero does: if you stress them, bacteria mutate faster (aka stress-induced mutagenesis).

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And to save you from rediscovering a certain wheel (and posting it here), in the biologically-inspired (but not biologically faithful) realm of "genetic algorithms", there's "fundamental theorem" (of Holland) that states under what conditions (in that simplified world) [natural] selection is likely to be the source of high-fitness schemata (an equivalent of well-adapted genotypes). But the theorem doesn't exclude that such schemata can simply appear by mere chance mutation.

More generally, there's the Price equation; it partitions total evolutionary change in that due to natural selection and that due to other causes (including transmission changes, like mutation). However, without additional assumptions, this is as good as it gets. To quote from SA Frank:

No general conclusion about total evolutionary change is possible, because the complete range of forces that can perturb populations remains unpredictable. However, we can express an elegant equilibrium condition. At equilibrium, the gain in information by [natural] selection must be exactly balanced by the decay in information caused by other evolutionary forces.

In the simplified world of genetic algorithms, such additional assumptions are made (e.g. there's a specific transmission model) a point that is missed (IMHO) by people who diss Holland's theorem as merely restating the Price equation.

And I should at least name-drop the Wright–Fisher model which seems to be the most studied by those interested in a closer approximation of biology (as opposed to solving optimization problems, which is the goal of genetic algorithms). The Wright–Fisher model also makes specific assumptions about transmission; it seems a lot variants of it have been investigated. On this model and a lot more, see chapters 1, 6 & 7 in Durrett's book Probability Models for DNA Sequence Evolution.

Anyway, my main point in this second half of my answer is direct you to well-known formal treatments of evolution by natural selection.