48 CELL HEREDITY 



As a consequence, because the distribution they predicted was observed, 

 we cannot reject either kind of randomness as a property of mutation. 

 These two components of randomness can be examined separately. For 

 example, when the mutation rate is measured after different periods of 

 growth of bacteria under identical conditions, it is always found to be 

 the same. The existence of the same chance for all, or at least most, 

 of the bacteria has been demonstrated in another way. The method 

 employed uses the Poisson distribution, which must first be described. 

 Consider, as an example, a gentle spring shower in which the rain- 

 drops, affected by so many opposite forces during their formation and 

 fall to earth, are trulv distributed at random. Suppose we were to rule 

 a sheet of paper into 100 squares of equal size and expose it to this 

 shower until exactly 100 hits had been registered; the average number of 

 hits per square would be 1, but it is a matter of experience that each 

 square will not have been hit by one raindrop. Some squares will not 

 have been hit at all, some will have been hit once, some twice, etc. 

 Few squares will have manv hits, of course, and the average number of 

 squares hit not at all, once, twice, etc., is approximated by the Poisson 

 formulation. This is a special case of the binomial theorem which holds 

 when the actual number of events that does occur is insignificantly small 

 compared to the number the system could entertain. Mathematically, 

 it is written 



Ax) = - e-^- (2.1) 



where P{x) is the chance that x events will occur, x! is x factorial (e.g., 

 3! = 1 X 2 X 3), m is the average number of events per unit, and e is 

 2.718, the base of natural logarithms. P^x) is, of course, approximated by 

 the observed frequency with which x events occur per unit, and in this 

 way the equation will pretty well describe the distribution of numbers of 

 raindrops hitting the squares on the paper at random. 



This formulation has a usefulness in biology in general, and especially 

 in genetics. Much of biology, in contrast to classical chemistry, is con- 

 cerned with units, i.e., with discontinuities. Chemists are usually inter- 

 ested in the average behavior of billions upon billions of molecules and 

 can express rates of reaction by differential equations. The biologist, 

 like the physicist studying atomic particles, usually deals with discrete 

 units in small numbers, whether they be organisms, cells, microscopically 

 visible subcellular particles, or operationally defined units. His subject 

 matter is so often particulate that the mathematics he employs is fre- 

 quently the statistics of counting. Furthermore, his chemistry, so often 



