GENERAL DISCUSSION 347 



represented by S. In other words, among a large number of these chromosomes subjected 

 to umt dose the proportion S will undergo a breakage which is not restituted and which 

 therefore leads to the loss of the chromosome (resulting either in hypoploid descendant 

 cells or lethal bridge-formation between them). Then the proportion of these chromosomes 

 not affected in this way, i.e. the potential chromosome-survivors among them, is (1 — S). 

 If now the dose is d instead of unity the proportion of these clrromosomes in which there 

 is no effective breakage is (1 — SyK Moreover, if we take, instead of chromosome S, one 

 whose effective length or breakabUity is LS, the proportion unaffected by unit dose is 

 (1 — S)^ and the proportion unaffected by dose (Z is (I — >S')^''. Following the customary 

 procedure, Ave may represent {1 — S) by e~°, where — a is the natural logarithm of 

 (1 — S). Then the proportion of chromosome-survivors, for a chromosome of effective 

 length LS at dose d, is simply e~"'^^ , and the proportion of these chromosomes lost is 

 1 — e~"^'^. When the chromosome dealt with is the single X-chromosome of man or 

 Drosophila tliis same expression also represents the proportion of cells lost in the critical 

 tissue and stage under consideration. 



When Drosophila cliromosomes that occur in pairs are dealt with, cell death occurs 

 only if both members of the same pair are lost from the same cell. The probability or 

 proportion of these killed cells may be taken as approximately {SLd)^, for any cliromo- 

 some type of effective length SL, at dose d, so long as the product SLd, that is, the total 

 chance of a given chromosome being lost, is not more than about 0-1. Thus the proportion 

 of ceUs surviving from tliis cause of death will m that case be about [1 — (SLd)"^]. For 

 greater accuracy one may reckon as follows. Since the frequency of loss of any given 

 cliromosome is (1 — e~*^^'), the frequency of lethal loss involving both members of that 

 pair is (1 — e""^'')^. Hence the frequency of avoidance of this cause of cell death is 

 1 — (1 — e-«^'')2. This can be reduced to 2e-«if' — e'^'^^'K 



To get the total frequency of cell survival from all cliromosome losses one multiphes 

 together the survival frequencies found for each chromosome-tj^e, using the formula 

 Q-aLd fQj. ^i^g single X of males and one of the formulae just given for all cliromosome 

 pairs wluch have two separately viable liomologues present. If, however, one or both 

 members of a pair contains a deficiency or other recessive cell-lethal, the same expression 

 is used as for the X of males (with suitable change of the value of L), provided that only 

 one member of the pair is thus affected. If both members are affected the frequency of 

 survival from damage to the cliromosomes of this pair is the square of the frequency 

 obtaining when only one member is affected, i.e. it is q-^(^^^K 



Where two or more chromosome types are sensibly alike in their effective chromo- 

 some length — as is the case for the second and third chromosomes of Drosophila and 

 probably in some cases for different cliromosomes belonging to a given size-group in 

 man — the calculation is of course simpUfied by grouping the like ones together. One then 

 raises the figure representing the frequency of survival based on one pair to a power 

 representing the number of these pairs. 



By these means one arrives at an algebraic expression representing the frequency of 

 surviving cells for any given dose. In this expression, only the value of the constant a 

 (or S) is unknoA\Ti, when an organism such as Drosophila is dealt with — one in wliicli the 

 relations between the effective breakage frequencies, or "lengths", of the different cliromo- 

 some types are known. Of course it has been assumed in all these calculations that the 

 total dose and dose-rate have been low enough to allow the effective breakages in any two 

 chromosomes to have been independent of one another in somatic cells of the given kinds. 



It is possible to get empirical evidence regarding the validity of these expressions and, 

 if they prove not invalid, to solve for a and for the frequencies of cells killed by (or 

 surviving) given doses, if one has observations on the frequencies of individuals surviving 



