Stochastic Dependence of Mutation and Killing

On the basis of the definition of mutation frequency given in equation (6), nonlinearities in frequency curves would be expected to arise only from the existence of multilesion mutation-induction processes. However, nonlinearities in mutation-frequency curves can arise if mutation and killing are not stochastically independent processes as assumed in Section 3 and in writing equation (6). We can generalize the formalism of Section 3 and 4 to allow for stochastic dependence of mutation and killing in the following way  

If the probability of macrocolony formation by a cell in which a mutation has been fixed at the assayed locus differs slightly from that in which no such mutation has occurred, then we can write the surviving fraction of mutant cells after a mutagen dose x in the form 

Stochastic independence of mutation and killing implies that 5(a ) = 1. For linear hit functions in equation (52), 8(x) = kim/ki = 6, a constant. Even for nonlinear hit functions, we would expect, on general radiobiological grounds, that 8 x) would be at most a slowly varying function of x that could be approximated adequately by a constant 8. 

It is clear from equation (52) that both positive (upward-bending) and negative (downward-bending) departures from the basic mutant-frequency pattern determined by Hm x) could occur depending on whether 6(a ) was less or greater than unity, respectively. For the purely linear case Lk,Lm), we can write 

This equation explicitly shows how a quadratic component in mutation frequency can arise either if m2 0 or 6 1, or if both conditions obtain. 

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FIGURE 8. Effect of stochastic dependence of mutation and killing (6 1) superimposed on a basically linear response pattern Lk,Lm). Mutation-frequency curves calculated from equation (55) for mi = 10 and Ai = 0.1 (ergs/mm ) for various values of h. The curves for 6 > 1 lie below those for 6 < 1 merely because a single, fixed value of mi was used to calculate all curves. 

If the relevant physical lesions are formed in direct proportion to dose, and if no dose-dependent processes are involved in the conversion of initial lesions to biological hits, then the hit function will be linear. More generally, H x) could be some more complex function that, nonetheless, can always be represented by an infinite power series in x with no constant term (since there can be no induced hits for zero dose). We denote the expected number of lethal and mutational hits at dose x by Hk x) and Hm(x), respectively. Thus, on the basis of single-event Poisson statistics, population homogeneity, and stochastic independence of mutation and killing, we can write... 

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