The tree of life in more mathematical detail

Dear Tom,

So today I learnt from Wikipedia article following an interview with Leonard Susskind that the string theory landscape was inspired by the biological fitness landscape, itself derived from free energy landscapes. I have no clue about the first and some clue about the third. However, I have a pretty decent clue about the second.

There are 3 dimensions that define a fitness landscape. The X and Z dimensions represent an area of all hypothetical individual genotypes - each genotype is a particular combination of genes that could give rise to a hypothetical individual. It can represent a specific variant (A, T, C or G) at a particular genomic site, or in my utilization later, the precise sequence of the entire genome (both chromosomes if diploid) of an individual organism. The Y dimension represents the fitness of each genotype under a fixed set of environmental conditions. Fitness means the capacity of the theoretical genotype under those conditions to survive and reproduce relative to all other genotypes.

Fitness itself is thus a theoretical construct. However, for a given species of organism, if the value of a certain trait (phenotype) correlates with the fitness of a genotype, then can said trait value be a component of fitness. Furthermore, one can even view the vector of all possible phenotypes for all possible traits as the better proxy for fitness than one single trait. Anyhow, this has led to more mathematical names for such landscapes - conditional genotype-fitness or genotype-phenotype maps.

These landscapes and maps have been incredibly useful to address the specific problem of how populations evolve through natural selection of genotypes where in principle, the fittest genotypes in a given environment are those that are selected for under a specific set of conditions. If the conditions change over time so that the landscape changes, the correspondence between the new landscape and the old is of great interest because it answers the specific subquestions of which genotypes are fittest in all conditions, some conditions, or no conditions. That is of course bounded by Sewall Wright's mathematical ability, which was freakishly high for someone who considered themselves a biologist born in the early 20th century.

However, there are two fundamental truths about biological populations that are not fully addressed by fitness landscapes.

The first is the much larger role that 'drift' plays in genotype evolution. Drift means any change in fitness of the population not caused by a determinable/persistent/recurring change in condition and was indeed recognized and coined by Wright. Reduction in population size, even if an unbiased set of genotypes were eliminated, has also called a precondition for drift to 'increase in strength' because rare genotypes have a higher shot at reproduction. The thing is, to my knowledge, it was deemed as minor compared to selection due to the lack of DNA sequences available.

Stronger support for drift came years later in the form of selectively neutral mutations first discovered by Motoo Kimura, one of the first people to study real DNA sequences (rather than hypothetical genotypes). Analysing the same stretch of genome across a panel of individuals available for him to study, he found that most mutations occur in sites that either do not code for a protein or code for the same amino acid in a protein, as opposed to those that would change amino acid identities. This is consistent with selection tending to weed out 'functionally deleterious' mutations (probably though not necessarily through death of such mutants) while drift maintains the other 'neutral' mutations at some equilibrium frequency in the population. This has since been found to be broadly true across populations of many organisms, though the relative ratios of the type of mutation depends on the taxa and taxonomic range one considers for comparison. Drift has been further invoked for the evolution of revolutionary transitions in life, though there is rich debate on these sort of topics. Nonetheless, there is this dynamic interplay between both selection and drift that determine the resulting range of genotypes in a population in a given time.

The second is that actual living populations of individuals are not a continuous X by Z area of genotypes. They are discrete dots that fall within this area. Not every theoretical genotype exists (or has existed) at any moment of time on this planet, even though there has been as many genotypes as individuals (arguably individual cells) at said moments of time. That is to say, there are and has been a lot genotypes on this planet.

Thus, nature of evolution in biological populations is insufficiently explained by shifts in conditional landscapes over time.

I propose here a more informative model of evolution in biological populations. An individual with a unique genotype can be represented as a dot in the X-Z space. Reproduction occurs by an existing dot in the X - Z area giving rise to new real dots with distinct genotypes that constitute the next generation. The reason behind this is because of Darwin's law of common descent of all life. Each and every individual must come from at least one individual before it (2 if sexually-reproducing - can't think of 3-parent children but it probably does exist somewhere). The only Y axis that makes sense to me is real time, where fitness (survival from birth until exact age where reproduction occurs) would be recoded as the 'length' of the connection between new dots spawned by old dots.

Now that I think of it, I have stumbled upon a more complex version of the Wright-Fisher model, invented by the aforementioned Wright and another father of population genetics (and modern frequentist statistics), Ronald Fisher. This model summarises evolution in a hypothetical population with a constant number of individuals (dots) per generation and non-overlapping generation - each new dot must be derived from exactly one of the old dots, 'picked' at random, and the old dots 'die'.

How would environmental change and phenotypic change be represented? In other words, how would drift and natural selection be represented? My model nor the Wright Fisher give no explicit information on these two factors (just genotype and the time from birth to reproduction). Maybe these can collectively be viewed as shifts in the relative lengths of old dot-new dot connections considering a local population of dots under a defined window of time. Environmental conditions and phenotypic means and variabilities for all individuals alive at the start and at the end of the time window can either be empirically measured/estimated or represented as parameters that are a function of the distributions of dots at the start and the end of this window. These two forces underlying evolution would be consistent with some cutoff for a large change in phenotypic mean, phenotypic variability, difference in the number of dots before and after, and for the case of natural selection, change in environmental conditions. Perhaps considering them as distinct force components explaining evolution maybe better than calling them separate.

Perhaps there will be some weirdness too if we consider the type of dots that can appear across different forms of life. Species that can do horizontal gene transfer means one individual can have two dots, even though it is the same cell. Furthermore, an individual might have >2 parents. Having more than one set of chromosomes and meiosis means the genotype spaces is much wider for some populations of multicellular organisms. That being said, in principle we could also model each individual cell, even in a multicellular organism, as a dot. Each of our cells probably has one or two mutations that affect our collective functioning as a multicellular individual. Sadly, although I might be wrong at the moment of writing, I don't believe we have the technology yet to do single cell DNA sequencing of the whole genome.

I'm sure I am missing some more other biological complexities in this complex model of the evolution of genotype-based life. However, I can't help but wonder if quantum information theory can help model this unwieldy, discrete, noisy system that is the evolution of biological genotype-populations. In particular, I hypothesize that the number and identities of dots that appear in time and the average length that they exist for, though bearing a stochastic component, might be predictable to a certain degree of accuracy with some knowledge of the changes in environment, mean phenotype, phenotype variance, selection, and drift parameters. Maybe it's not and we'll have to stick with simplified fitness landscapes for now. But if you think it was possible somehow, I would loveeee to here your thoughts on how this would be done.

Best,
James