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This analysis is an attempt to show how the additional weight of the sperm carrying Xchromosomes compared to those carrying Ychromosomes results in a lower proportion of conceptions of female babies than male babies.
The forces on a spermatozoa in the direction of forward motion are: 1. the forward force F resulting from its swimming, 2. the backward force cV² due to fluid resistance (where V is the forward velocity and c is coefficient), 3. the force kW due to its weight W and the incline of its path. When the spermatozoa is traveling at a constant speed then
The ratio of the weights W_{f}/W_{m} is about 1.04. The parameters c and k would be the same for the two types of sperm. The swimming force is likely to have a probability distribution which would be the same for the two types.
The sign of k is positive for an uphill journey and negative of a downhill one. The heavier Xladen sperm are at a disadvantage only for an uphill journey. The effect of the different weights is a function of the net amount of uphill journey.
The time T required for a spermatozoa to reach the ovum would be L/V, where L is the length of the path and V is the velocity. The dependence of V on its determiants can be expressed as
Since ln(T)=ln(L)−ln(V)
Since ln(T) = ln(L) − ln(V)
There would be a probability distribution for L as well as for V and the two together generate the probability distribution for T.
The quantity sought is the probability that the spermatozoa with the minimum time is an Xchromosomeladen spermatozoa.
Let p_{f}(T) and p_{m}(T) is the probability distributions for the times required to reach the ovum. These probability distributions generate the probability distributions of the minimum times for a large population of sperm. Let P_{f}(T) and P_{m}(T) be the probability distributions for the minimum times. The probability that a female will be conceived, G, is the sum over all T of the product of the probability that a value T occurs times the probability that minimum time for the Yladen sperm is greater than T. This is
The probability of the conception of a boy, B, is then
If there were no difference in weight of the two types of sperm then G and B would be equal to 0.5. what is needed is an approximation of the form
This would give
For a large sample size the probability distribution of the minimum of a sample tends to a spike at the minimum possible value which for P_{m}(T) would be for the value of T determined by the minimum L and the maximum V, which in turn comes from the maximum F.
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