POLYSPERMY IO9 



where a = radius of the egg, n = no. of sperm/ml., and c = mean 

 speed of the spermatozoa. When the appropriate values are sub- 

 stituted in this equation, Z, the number of sperm-egg colUsions 

 per second, is found to be o-i6, i-6, and 16 for sperm densities of 

 10^, 10^, and lo'^ per ml., respectively. These values for Z are 

 probably too high because no account is taken of dead spermatozoa 

 in the suspensions. Although the percentage of dead sperm in 

 mammalian suspensions can be estimated by the live-dead stain- 

 ing technique (Lasley et al., 1942), and is often about 10%, no 

 method is at present available for doing this in suspensions of sea- 

 urchin spermatozoa. As the cortical change takes twenty seconds 

 to pass over the egg surface, there will in that time be respectively 

 3-2, 32 and 320 collisions at the three sperm densities in question. 

 About half of these will collide with parts of the egg surface already 

 covered by the cortical change, so that at a sperm density of 10'^ /ml., 

 only I /i 60th of the spermatozoa colliding with the egg surface are 

 capable of fertilizing it, if the cortical change is the block to poly- 

 spermy. How can the probability, p, of a sperm-egg collision 

 being successful, be estimated? Elementary probability theory 

 shows,* and experiments confirm, that if unfertilized eggs and 

 spermatozoa are left in contact with each other for a series of known 

 sperm-egg interaction times t^, ta, . . . tn sees, (n > 45), and the 

 number of fertilized and unfertilized eggs are later counted in each 

 case, the proportion of unfertilized eggs, u, is given by the equation 



log u = — at . . . (2) 



where a is the sperm-egg interaction rate or fertilization parameter 

 with dimensions [T]~^. 



Assuming that the spermatozoa do not change their fertilizing 

 capacity during the experimental period, a is a measure of the 

 receptivity of the egg surface to spermatozoa and its value may 

 reflect the fact that, on a submicroscopic scale, an egg surface is 



* Divide the interval (o, t) during which the eggs and spermatozoa are in 

 contact into r sub-intervals, each of duration t. Let the probability of an egg be- 

 ing fertilized during t be p, and of not being fertilized, q, where p + q = i • 

 There being r sub-intervals of duration r during (o, t), the probability of an egg 

 not being fertilized throughout (o, t) is, by the Product Theorem 



q'' = exp(r log q) = exp(t/T. log q) 

 = exp(^at) 



where a = — i jr. log q 



The theoretical proportions of unfertilized and fertilized eggs are, therefore, 

 u = exp(— at) and f = i — exp(— at). 



