Cumulative vs. Simultaneous Mutations

It may be that many of the amino acids do not affect the functionality of the target protein. For example, one could argue that two specific mutations are needed, but only 212 of the other amino acids are necessary for functionality. In this case, we just change the number of total amino acids from the actual 424 to a ‘functional’ 212, as the others are irrelevant.

The mathematical intuition for this result is the following. The number of paths to the top row increases by a factor of 4 as one moves through time. In the chloroquine example, the top nodes represent all the combinations with two more up movements than down movements: up-up, up-down-up-up, etc. The probability of these successful paths declines by a factor of 0.001 because each path requires the extra p(u); a downward move implies an additional implausible upward move to get back to its earlier closeness to the target. The net probability–the number of paths times the probability, 4 × 0.001—therefore decreases by orders of magnitude at each step. Adding numbers that decline by an order of magnitude or more leaves the first digit unaffected, as when you add the numbers 0.01+0.0001+0.000001; the first term, 0.01, approximates the sum. In this way, the probability of a direct path number approximates the cumulative probability.

If we consider that mutations are like Poisson probabilities, happening randomly through time, the likelihood of two sequential specific mutations is equivalent to the probability of two simultaneous specific mutations. The direct sequence of mutations is precisely like the probability of a simultaneous mutation.

In the infinite limit, the odds of reaching the target are 1.0 regardless of how close one is to a target. The standard way of modeling probabilities ignores the pathways and looks at limits, obscuring this result.

While many paths lead to the target, as a practical matter, they are irrelevant. One can imagine all sorts of intermediate configurations of amino acids that are useful for something else, but that is not science unless one has a reasonable model or an empirical estimate of the ratio of these viable configurations to those possible. We all know that to turn a bicycle into a motorcycle you need several parts to work together. One can imagine each part being added without customers noticing, but one can also imagine a point where the thing does not work at all. To change an ion channel from one that lets only sodium across a membrane to one that allows only potassium, it too needs many specific mutations before it would work at either task. It could be there are many intermediate configurations that do something entirely different, but without any actual data, it takes a lot of faith to imagine these unseen functional intermediates are all buried in our past.

This process helps understand why looking at evolution in real-time on fruit flies, and E. coli shows nothing that would extrapolate to novel functionality. Instead, the more common involves loss of function. For example, losing melanin is advantageous for humans in northern latitudes because it allows humans to get more vitamin D. At the same time, the risk of skin cancer is lower with less sunlight, so the benefit of melanin is less. The sort of evolution we see all over within families like cats, dogs, and bears, however, does not explain how we got cats, dogs, and bears from a common ancestor.