Constructional Selection on Pleiotropy

Suppose a gene newly added to the genome has pleiotropy vector $\mbox{\boldmath\(p\)\unboldmath }_{n+1}$, and affects $k_{n+1} = \sum_{j=1}^{f} m_{n+1\,j}$ fitness components, which become resampled from the interval [0,1]. Let y be the sum, before the new gene is added, of the fitness components the new gene is going to alter. The probability that the new sum will be less than y is:

 

$$\Pr[S_k < y]$$ F_k(y) = (2)

$$\frac{1}{k!} \sum_{i=0}^{k} (-1)^i {k \choose i} \left(\frac{y-i + \vert y-i\vert}{2}\right)^k,$$ =

where S_k is the sum of k independent uniform random variables on [0,1], from [10].

Then, from equation ( (1), the probability that the new gene will produce a fitness increase is 1-F_k<<1152>>n+1(y). When the average of the fitness components to be altered by the new gene is above 1/2, the greater k_n+1 is, the less the chance that the new gene will produce a fitness increase, precipitously less so for highly adapted fitness components. Since the new gene is kept only if it produces a fitness increase, constructional selection will filter out genes with high k.

Suppose that there is an underlying probability density s(k) of pleiotropy values k for genes newly added to the genome. Then the density s^*(k) of pleiotropy values among genes that are kept by the genome (i.e. which improve fitness) will be

$$s^*(k) = s(k) \sum_{\mbox{\boldmath\(p\)\unboldmath }\in\{0,... ...ldmath }^{\rm T}\mbox{\boldmath\(\phi\)\unboldmath })\right]/N$$ (3)

where $\mbox{\boldmath\(\phi\)\unboldmath }$ is the vector of fitness components before the gene was added, $\Pr[\mbox{\boldmath\(p\)\unboldmath }\vert k]$ is the probability of sampling pleiotropy vector $\mbox{\boldmath\(p\)\unboldmath }$ given that the new gene's pleiotropy value is k, and N is the normalizer so that $\sum_k s^*(k) = 1$.