Some Introductory Ideas Concerning the Application of Information Theory in Biology 53 



some fault cither in the duplication process in the germ line or the somatic 

 line or from incorrect rcad-o(T of any kind. //„ may be expressed in terms of 

 the read-off or transition probabilities (3) of a letter of kind / to a letter of 

 kindy, Piij). The probability of letter / is p{i). 



H=H,-\-y p{i) p^ij) log2 p,{j) (2) 



Consider the case where these probabilities are a function of some variable 1. 

 In the application of these considerations A is the measure of some deleterious 

 influence such as dose of ionizing radiation. Form the derivative dHjd?.: 



ciHldX = log2 e 2 (MO ic¥>^) P.ij) + Pii) loge pSi) {dIdX) p,{j) 



+ P.(j)ioi,p,{J){dldX)p{i)] (3) 



The absolute value of dHfdX will become indefinitely large because of the 

 second term in equation (3) as any p^{j) approaches zero if p{i) ^ and 

 (dldX) pi{j) 7^ 0. This may happen, in particular, if any p/ij) approaches one 

 for then SL\lpi{k), (j ^ k) approach zero. This situation {p,{j) = 1) corresponds 

 to the assumption that there is always a correct reproduction in the DNA 

 duplication or in the RNA read-off. Under these circumstances the first term 

 is finite and the third term is zero. 



Watson and Crick regard a mutation as being reflected by a change in 

 order of the nucleotide bases in DNA. This is apparently always possible; 

 they have suggested a biochemical scheme by which this can be affected. This 

 means that in a real biological system p{i) ^ and {djdX) p/ij) 7^ 0. A real 

 ensemble of organisms will be represented by an ensemble of genetic messages. 

 This will be true even if the ensemble is isogenic. Some noise must exist in the 

 genetical information; if the noise is less than equilibrium it is quickly intro- 

 duced. 



There is some experimental evidence in support of this conclusion. Burdette 

 (4) prepared populations of isogenic Drosophila. One strain had the same low 

 incidence of tumors in both sexes (about 4 per cent) and the other had a high 

 incidence (about 60 to 80 per cent) even greater in males than in females. The 

 tumor incidence of the isogenic strains was initially much lower in each case 

 than the stock from which it originated. But in each case, by the twelfth genera- 

 tion, the tumor incidence of the isogenic strain had returned to about the same 

 rate as that of the original stock. Tumor incidence is a morphological mal- 

 function and, as shown in this and other experiments, is under genetic control. 



The fact that all flies were not tumor bearing and the gradual return of the 

 isogenic strains to the tumor incidence of the strains from which they were 

 selected, reflects the accumulation of errors in the genome. The results of the 

 experiment are in accord with the proposition proved above. 



Representation of the Ensemble 0/ Organisms by a Probability Distribution in H: 

 piH, A) 



If we grant that perfect systems do not exist, the other side of the coin is, 

 how imperfect may they be? This question was first discussed by Dancoff and 

 QuASTLER (5) and their conclusion, which is known as Dancoff's principle, 

 states that the amount of redundance is just that required to reduce the error 



