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 The Statistical Dynamics of the Sex Ratio

This analysis is an attempt to show how the additional weight of the sperm carrying X-chromosomes compared to those carrying Y-chromosomes 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

#### F − cV² − kW and thus V = [(F − kW)/c]½

The ratio of the weights Wf/Wm 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 X-laden 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

#### V = [F(1 − (k/F)W)/c]½and hence ln(V) = ½[ln(F) + ln(1−(k/F)W) − ln(c)] and since ln(1-x) is approximately equal −x ln(V) = ½[ln(F) −(k/F)W − ln(c)] and thus ∂ln(V)/∂W = −½(k/F)

Since ln(T)=ln(L)−ln(V)

#### ∂ln(T)/∂W = ½(k/F)and W∂ln(T)/∂W = ∂ln(T)/∂ln(W) = ½(kW/F)

Since ln(T) = ln(L) − ln(V)

#### ln(T) = ln(L) − ½ln[(F−kW)/c]

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 X-chromosome-laden spermatozoa.

Let pf(T) and pm(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 Pf(T) and Pm(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 Y-laden sperm is greater than T. This is

#### G = ∫0∞Pf(T)[∫T∞Pm(S)dS]dT which can be expressed as G = ∫0∞Pf(T)[1 −∫0TPm(S)dS]dT or, equivalently G = 1 − ∫0∞Pf(T)[∫0TPm(S)dS]dT

The probability of the conception of a boy, B, is then

#### B = ∫0∞Pf(T)[∫0TPm(S)dS]dT

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

#### B = 0.5 + ΔW∫0∞(∂Pm(T)/∂W)[∫0TPm(S)dS]dT

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 Pm(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.