Tuesday, 23 October 2018

On Evolution & Random Walk

In one of my most popular papers, I wrote about how the universe is governed by a biased random walk, giving us a Divinely choreographed mathematical skew that can create enough order to facilitate stars, evolution and life. Many of you have asked whether, in that case, biological evolution is also similarly governed by a biased random walk. Actually, nobody has asked that - but it’s one of the most interesting and intelligent questions a reader could have asked if they were on the ball with this, because it’s exactly the right sort of question to be asking.

As a reminder, a random walk describes a path derived from a series of random steps in a mathematical search space - so, for example, whether a drunk man staggers left or right with each step of the walk is entirely random (50% chance of left or right), as is his final destination after N number of steps. Using this model, if you mapped the drunkard at point A, and tried to predict his position after, say, 100 steps, you would not be able to deterministically predict his final location.
So let’s ask the question, then: is biological evolution governed by a random walk process? The answer is yes and no, but mostly yes. If all the proprietary parts in evolution’s mathematical space (which I’ve called ‘morphospace’) are all randomly walking through evolution with their distinct genetic drift and mutations, contingency says that, like a group of drunkards walking independently, we would expect them to have arrived randomly at different evolutionary endpoints. However, evolution is not a purely random walk process - and there are two reasons why: one is fairly simple, and the other is pretty complex. Let’s start with the simple one first, as discussed by Richard Dawkins in his book The Blind Watchmaker.

“I don't know who it was first pointed out that, given enough time, a monkey bashing away at random on a typewriter could produce all the works of Shakespeare. The operative phrase is, of course, given enough time. Let us limit the task facing our monkey somewhat. Suppose that he has to produce, not the complete works of Shakespeare but just the short sentence 'Methinks it is like a weasel', and we shall make it relatively easy by giving him a typewriter with a restricted keyboard, one with just the 26 (capital) letters, and a space bar. How long will he take to write this one little sentence?”

That describes what evolution would be like if it was a random walk process. But of course, it isn’t, as Dawkins is happy to acknowledge:

“We again use our computer monkey, but with a crucial difference in its program. It again begins by choosing a random sequence of 28 letters, just as before ... it duplicates it repeatedly, but with a certain chance of random error – 'mutation' – in the copying. The computer examines the mutant nonsense phrases, the 'progeny' of the original phrase, and chooses the one which, however slightly, most resembles the target phrase METHINKS IT IS LIKE A WEASEL. The sequences progress through each generation:








What this describes is what is called a ratchet effect (cumulative selection) where beneficial traits lock into place, rather like how card players get to keep their favoured cards after each shuffling of the deck. To expound on this, evolution requires four fundamental things to underpin the system.

1) Variation: there is variation in traits.

2) Inheritance: these variations can be passed on to offspring.

3) Differential survival(/reproduction): given the reproductive potential of most organisms, a population should be able to grow (this is not always what happens, of course)

4) Natural selection: those with heritable traits that make them more likely to survive by passing on genetic material.

The above example of METHINKS is not precisely illustrative of how natural selection works; rather it is illustrative of how cumulative selection can lead to rapid change over a relatively short period of time. The analogy was used to answer the criticism that there has not been sufficient time for particular structures to evolve by "random chance.” The analogy shows that random variation can lead to rapid organisation of structure, provided that there is selection for the structure. The analogy defends the 'rapid' part, not the 'selection' part.

Suppose a specific complex sequence, such as just the letters that make up the word 'METHINKS' corresponds to something complicated, like a human eye. The chance of hitting the sequence such as 'METHINKS' by fortuity alone is very small - 1 in 209 billion (That is, 1 in 26 to the power of 8, for this 8 letter sequence, drawn from an alphabet of 26 letters). Similarly, conjuring up a human eye out of nothing also has a vanishingly small probability, it might as well be zero. But, as I said, this is a poor analogy for evolution, because evolution acts as a 'ratchet', so when a correct letter clicks into place, it stays there (as indicated by capital letters), so it can achieve the target phrase in much fewer attempts, say 40:

1) 'sgfcsgo' ...

10) 'fETopcrS' ...

20) 'xETrINsS' ...

30) 'METoINKS' ...


Now the question is, doesn't there have to be an intelligence to compare the target sequence 'METHINKS' against the sequence that evolution is trying out, or in real terms, the comparing of the 'proto-eye' to the target eye that is evolving? Well, in evolution, intermediates give advantages, and when those advantages accumulate, like in poker when you keep the cards you need for a good hand and toss out the bad ones, more sophisticated survival parts are created.
By this model above, the first attempt corresponds to being totally blind. The 10th try might correspond to a patch of photosensitive cells, so the organism can know if it is light or dark. The 20th try might correspond to ridges forming around these cells, so they are in an indentation, and the shadows of the ridges could give some information about which direction the light is coming from. The 30th try could correspond to the ridges starting to close up, so the light comes only through a small hole, so that the organism has much better information about the direction of the light, like a pin-hole camera. The last, 40th try, could correspond to a lens forming over this hole, which focuses light onto the photosensitive cells, resulting in a high quality image.

The point is that 1% of an eye is better than no eye, and 50% of an eye is better than 20% of an eye, and so on. At all stages, this extra light information available to the organism improves its survival value, and so the genes for making 1%, or 20% or 80% or whatever, is preferentially passed on to future generations. So, it's not as if an intelligence compares 20% of an eye to a complete human eye, and said 'ahh, this is better than its cousin, with 15% of an eye, I will let it pass on its genes for making this eye', but simply that when a predator comes along, it will see it before its cousin sees it, so its cousin will get eaten and not pass on its genes for making the 'inferior' 15% of an eye, but the 20% of an eye individual will pass on its genes. Of the offspring, some might have 19% of an eye, others might have 21% of an eye. Then the 21% of an eye will be more likely to survive, and its offspring might have 22% of an eye, and so on, all the way from humble beginnings until a complete, complicated and accurate eye is formed. The target sequence above merely corresponds to something that aids differential reproduction.

Having established all that, here is where we get to the more complex considerations, because underneath all that is a highly complex mathematical picture, which gives us another way to consider random walk. What’s happening in biological evolution underneath that layer is that there is a huge biochemical morphospace that has a connected structure through which evolution’s reducible complexity can traverse. Take, for example, irreducible complexity and reducible complexity - they refer to the arrangement of stable organic structures in evolution’s ‘morphospace’, but they cannot most primarily be understood at the level of the organism, because morphospace is not an adaptive landscape where we visualise the relationship between genotypes (or phenotypes) and reproductive success, and model fitness on the height of the landscape. Morphospace is a mathematical model of the form of connectivity between patterns – so a reducibly complex morphospace means that the biological structures that populate the evolutionary landscape form a connected group.

You may think of the system as being like a gigantic sponge made up of very tiny fibrils that connect the evolutionary structure together. If the connection has no broken off parts then the random walk of evolution can move across the whole structure. In fact, this is a particularly good illustration because sponges are composed entirely of mobile cells which can move about between different layers of tissue and reallocate themselves to take on different tasks. Sponges have totipotency, which as you may know, is the ability of a single cell to divide and produce all the differentiated cells in an organism. This allows any fragment of a sponge to regenerate into a self-sustaining organism.

So biological evolution is random walk, but as I said at the start, it is not entirely random walk. Firstly, because the ratchet effect locks beneficial mutations in place, but secondly because although the biochemical engine of evolution is underwritten by probability envelopes concerning whether a particular genetic trait will be passed on to subsequent organisms - and that, at least in terms of the mutations themselves, does approximate random walk - there are sufficient constraints on the system to bias the model in favour of order.

If we take an evolutionary starting point and then generate a random walk on the organism, then the probability favours random walk statistics (in formal terms,  a Gaussian probability distribution across the search space) - a probability curve in the shape of a distribution graph with no evolutionary biases. In the actual evolutionary landscape, though, what happens is random walk plus incremental variants in the search space; that is, we see a bias in the system that conforms to the ratchet mechanism of natural selection’s operation for fitness.

What you have to remember is that by the time we get to the level of order in the universe that contains our planet’s chemical substrate, the majority of the cosmological groundwork has already been done. It’s a bit like showing a movie at the cinema: your viewing pleasure is the tiny end part that succeeds all the planning: the screenplay, the casting, the rehearsals, the production, and the filming that went into making the movie. In mathematical terms, biological evolution is like showing an ingenious movie at the cinema that God has already written and produced - what we see is the most accessible elements of a complex creation process that involved ingenious twiddling of the mathematical laws to eventuate in a biochemical random walk substrate on which biological life can flourish.

In terms of evolutionary genetics and inheritance, our movie-watching lens of analysis looks like probabilistic search space of numerous configurational possibilities which generates successful survival machines, with genes using bodies as vehicles for propagation, many of them outliving their hosts by millions of years. Evolution, then, has an isotropic random walk directionality, but a relatively constrained search space in terms of the four fundamental underpinnings I mentioned earlier (variation, inheritance, differential survival and natural selection).

So, in simpler terms, going back to our group of drunkards on a random walk - if they all live in the same apartment block, the neighbourhood in which they are walking has enough limitations on the road and path structure, and they meet enough friends along the way to gently nudge them on the right course, and both those things mean that they have more of a chance of all arriving back at the same place.
The kind of biological evolution we see from those cinema seats can only work if the randomness of the mutations plays out within a very small probabilistic search space - and the groundwork for this was already done when the blueprint for the universe was written into the laws of physics. By the time the second law of thermodynamics gets scrambled into action we have an intricately directed form of entropy: where biology is organised under the constraints of the information implicit in its machinery, and at the same time still remains within the ordinances of the second law of thermodynamics.

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