Some Introductory Ideas Concerning the Application of Information Theory in Biology 57 



Given (dldX) p^d) as some function of A, equation (3) may be regarded as a 

 differential equation for //(A). This equation has a simple form if the y,v,(A) 

 and the c,,(/l) may be replaced by their averages y(/l) and iJ{?>.). 



{dHldX) = log2 e 2 {p{i)J{X)[ p^ij) + I] -{-p{i)m [-^p,{j) + I] loge/;,(;) 



-\-p,{j)\og,pij){dicrA)p{i)] (11) 



{dHldXy= -J{X) log2 e lp{i)p,{j) loge/^(7) 

 +i J(A) log2 e 2 p{i) loge PiCj) 

 +log2 e 2 Piij) HePiij) {dldX) pii) (12) 



Substituting equation (2) in equation (12) and rearranging we have 

 {dHldX) + J{X)H = J{X)H, + ya) 2 pii) \o^z Piij) 



i.j 



+ lPi(j)iog,p,(j)(dld?i)p{i) (13) 



i.j 



The third term on the right of equation (13) is negligible for biological 

 systems. To show this we must discuss first the method of calculating the 

 {dldX) pU). By definition (3) the following relation holds: 



p{i)=lpii)p^ii)- (14) 



i 



Form the derivative with respect to A and substitute equation (4) : 



{dldX) p{i) = llpij) {dldX) p,{i) + pAi) (dIdX) p(j)] ( 1 5) 



j 



{djdX) pii) = - 2 y,, /.,(/•) pij) + 2 q. pij) + 2 pM) i^m p(j) ( 1 6) 



j i J 



The equations (16) are a set of differential equations for the p{i). They may 

 be rearranged in the usual form: 



{dldX) pii) - 2 PjO) idldX) pij) = - 2 Jji PiU) pij) + 1 c,i pij) i 1 7) 

 j j J 



We are interested in the conditions when the id/dX) pii) vanish. The condition 

 is of course that the terms on the right of the equations (17) are all equal and 

 that the detenninant of the coefficients of the idjdX) pii) be different from zero. 

 Among the circumstances in which this will occur are those where all p^ii) = 

 q and all Pi{k) = p ii j^ k). That is, all letters are equally probable and one kind 

 of error is as likely as the other. In my paper in Part V the behavior of dH/dX 

 under the much stronger conditions that the J^j and c^j vanish at A = will be 

 needed. Then, of course, providing that the determinant of the coefficients of 

 the idldX) pii) be different from zero, all id I dX) pii) = 0. It may therefore be 

 expected that except under most exceptional and special conditions the idfdX) pii) 

 will be very small or will vanish. 



It can be further shown that for a nearly perfect system the coefficients of 

 the idldX)pii) in equation (13) are small compared to one. Dancoff and 



