Where Have All the Bad Mutations Gone?

By Wesley R. Elsberry

A feedback item on the talk.origins archive asserted that any probability analysis of evolution has to account for the problem that harmful mutations happen more commonly than beneficial ones, and yet one is not apparently reminded of this when looking at the fossil record. Why is it that we don't see all the discarded mistakes, since those should have happened more often?

There is an approach using probability analysis that shows why we may expect that for any particular fossil, the odds that it shows one or more beneficial traits are higher than that it shows one or more harmful traits.

We can set up the probability problem such that it is parameterized, allowing us to treat the beneficial mutation case and the harmful mutation case in the same fashion.

Each species has a (large but) finite number of organisms that live and die over its time of residence. We'll call this number N.

Each species will tend to have a characteristic probability that any particular individual's remains become a part of the fossil record and available for discovery later. We'll call this probability the "Per Individual Preservation Potential" or PIPP.

For each class of mutations that we are interested in, there are a set of parameters that we can apply to characterize the effect of these mutations. First, there is the per-individual probability that such a mutation occurs at all. Let's call this probability C. A mutation may be retained in the population at some frequency, or it can be lost shortly after appearing. The probability that any particular mutation of the type in question is retained in the population can be called R. (Thus the probability of loss is (1-R).) When a mutation is lost, it will have affected one or some small number of individuals. The average number of individuals affected by a mutation lost shortly after appearance is L. Mutations can occur at various times in the history of the species, and thus those individuals that lived before retetion of a mutation cannot possibly have the trait. The average time at which a mutation enters the population can be expressed as a number between 0 (time of speciation event producing the species) and 1 (time of extinction of the species). If a mutation is retained, then it will tend to achieve some equilibrium freqeuncy in the population after introduction. We'll call this proportion A.

The number of individuals which might be seen as fossils bearing a trait of a particular mutation type can be calculated as
Mut. type fossils = ((Individuals having mutation type that were lost)
+ (Individuals having mutation type that were retained))

We aren't quite at the point of plugging in our terms discussed above. Each individual organism might carry more than one mutation of the type being considered. What we need is to calculate the probability that an individual does not carry any such trait. First, we need the probability that an individual does not carry a particular trait of the type in question.

Probability that a trait that occurs is not carried by an individual =
1 - ((probability of a retained mutation) + (prob. of a lost mutation)) 1 - ((((N-(N*K))*R*A)/N) + ((1-R)*L/N))

Given the probability that an individual doesn't carry a particular trait of the type being examined and an assumption that multiple such traits are independent of each other, we can work out the probability that an individual does not carry any such trait.

Probability that an individual carries no trait of a type =
(Probability of an individual not carrying a particular trait) ^ (Number of instances of the type of trait) =
(1 - ((((N-(N*K))*R*A)/N) + ((1-R)*L/N))) ^ (C*N)

We have in hand a set of parameters to apply. How do the numbers that get installed in these parameters differ between the beneficial mutation case and the harmful mutation case? Certain of them won't. The number of individuals is the same. The timing of retention or loss should not be assumed to differ. It is likely that the number of individuals affected by a trait that is lost will be very small whether the trait was beneficial or harmful. This leaves three parameters, C, R, and A. The probability of occurrence C can be stipulated to be larger for the harmful mutation case than for the beneficial mutation case. However, both R and A should be very much larger in the beneficial mutation case than in the harmful mutation case. A harmful mutation will only be retained in a population rarely, and then it will be confined to a very small proportion of the population. A beneficial mutation, on the other hand, will tend to be retained far more commonly, and its representation in the population will tend to be very high.

What one finds when these equations are applied is that there are values for the parameters which yield an expectation of many more beneficial traits than harmful traits to be seen in fossilized specimens. Further, these values are arguably reasonable values. Let's assume that any sort of mutation susceptible to discovery from fossil remains is rare, perhaps one in a billion (1E-9). Further, let's postulate that a harmful mutation is only slightly rarer, at 1E-10. Let's make beneficial mutations a nice round million times rarer than harmful ones, at 1E-16. For probability of loss of a beneficial mutation, let's be pessimistic and say that half of them are lost outright. Take 3 as our number of individuals affected by a mutation before loss for either beneficial or harmful mutations. For beneficial mutations, take the proportion for retained traits as one half. For harmful, this proportion is more like one in ten thousand or 1E-4. Likewise, harmful mutations are far more likely to be eliminated, again let's take one in ten thousand as our figure. I'll summarize...

Parameter    Beneficial   Harmful
C             1E-16        1E-10
R             0.5          1E-4
A             0.5          1E-4
K             0.5          0.5
L             3            3

The selection of numbers favors, if anything, the anti-evolutionist position. Are beneficial mutations a million times rarer than harmful ones? No. Is the average proportion of beneficial traits at equilibrium 0.5? Usually, it would be very close to 1.0. Are beneficial mutations retained only half of the time? The real number probably lies closer to 0.75.

Plugging those in, one will find that there is a broad range of numbers of individuals for which many more beneficial traits will be evident than harmful ones. The range covers several orders of magnitude. At very small total N, one would find that the PIPP will make it unlikely that any fossils at all can be found. At very large total N, the numbers of both beneficial and harmful mutations available make it unlikely that any individual carries none of each. At the intermediate values, beneficial mutations are much more common than harmful ones.

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