Natural Selection

I should probably confess that at the time of writing this post, I did not have any clue of what natural selection is (or for that matter, evolution). Looking back, I just see how silly I was. Six years of experience in biological and medical institutes has taught me to understand and appreciate science in its broad and pristine perspective. Let me explain the idea behind natural selection.

Natural selection is one of the mechanisms through which evolution can happen. Evolution can happen if any of these five key factors are present in a population system:

1. Mutation

2. Natural selection

3. Non-random mating

4. Genetic drift &

5. Gene flow

If none of these are present, then at least for diploids, the allelic and genotypic frequencies will remain constant (the famous Hardy-Weinberg equilibrium) through generations and no evolution will take place. At the end of the day, evolution is just a gradual change in the allele frequencies across generations and it is not synonymous with natural selection. Although selection need not be the only factor in evolution, it is a key mechanism when the population is sufficiently large. Selection requires at least three necessary and sufficient conditions in order to maintain evolution:

1. Variation of traits within a population
2. Heritability of traits
3. The traits should confer fitness (reproductive) or survival advantage.

First, selection requires variation in phenotypes which are amply provided by mutations or polymorphisms. Technically, mutations provide variations in genotypes that account for phenotypes through the principles of molecular biology. They provide the variety (or phenotypic traits) so that nature can pick and retain the traits that are beneficial for survival/reproduction. 

The second key factor for selection to take place is the heritability of traits. Selection can act only on phenotypes and not on genotypes. Therefore, for a trait to evolve in a population, it should be heritable in the first place. If a trait is not heritable, it will not be passed into future generations and the trait will go into an evolutionary dead-end.

Third, the traits should benefit survival or reproduction, otherwise known as fitness advantage. The traits that are not beneficial are weeded out through environmental or genetic machineries.

Natural selection is simple yet a profound concept. Selection works based on a subset of population rather than individual(s). It is a statistical property, like color for instance. Individual atoms/molecules may not have a specific color, but color can emerge due to aggregation. Similarly, when a group of individuals in a population develop a certain trait that is more favorable to survival/reproduction, assuming the trait is heritable, there is a good chance of passing it on to the next generation by sheer probability.

One of the Best Non-constructive Proofs

This is one of the finest proofs in mathematics as it completely exposes the non-constructive philosophy of mathematics as propounded by David Hilbert.

Theorem: There exists irrational numbers p and q such that p^q is a rational number.

Proof: Consider the number a = √2 ^√2 .

Case I:

a is a rational number. We are done. Just take p = √2 and q = √2. Therefore, p^q is a rational number.

Case II:

a is an irrational number. In this case, take p = a and q = √2.

Thus, by the laws of arithmetic, p^q = (√2 ^√2)^√2 = (√2 ) ²  = 2, a rational number. 

QED.

The beauty of this proof is that at the end of the day (or the proof) we really don't know if a is a rational number or an irrational number. Note that this is technically a valid proof without any fallacies. 

Mathematicians who harp on constructivism will frown at this proof and the law of excluded middle is the real culprit. For more details about constructivism (and Brower-Hilbert controversy) look into this link.

Note: It turns out that a is indeed an irrational number as a consequence of Gelfand-Schneider Theorem since √2 is an algebraic number (as it is the solution of the polynomial equation x²-2 = 0).

On Highly Recurrent Cancer Mutations

I hit upon this insight which I thought I should share it with you. This is purely from analytical perspective.

One would wonder why in some cancers, certain residues/mutations (like R132H, V600E etc) are almost always found at high frequencies. There could be a simple game theoretical explanation to it.

In game theory, there is something called mixed and pure strategies. For instance, let us say Charlie and Ruth are playing Rock-Paper-Scissors game. Charlie/Ruth can choose to call “Rock" all the time (or for that matter either “Paper" or “Scissors" all the time). This is called pure strategy – that is, choosing a particular option all the time. On the other hand, Charlie/Ruth can alternate between rock,  paper and scissors in certain frequencies. For instance, calling rock 5/10 times, paper 2/10 times and scissors 3/10 times. This is called mixed strategy. 

There is a nice theorem that says: if Charlie “knows” what Ruth is going to play, then the optimal counter strategy for Charlie is always a pure strategy and not a mixed strategy. The same result holds for Ruth as well. This is a very interesting result because regardless of what the other person plays (could be pure or mixed with any frequency), if a player “knows” what is going to be played, the optimal play for him is to choose a pure strategy. And pure strategy is choosing one option all the time.

Therefore, applying this in a cancer setting: if a gene like IDH or BRAF somehow “senses” the frequency of other events, the optimal strategy for it would be a pure strategy or in other words, mutating at a particular residue all the time. That could just be R132H or V600E. The gene just needs to sense the pattern rather than the frequency of actual events in order to get mutating at a particular residue almost always.

I know this argument devoid of any biology does not say why that particular residue, but it could be a small rationale for why we see what we see.