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Proceedings of the National Academy of Sciences Vol. 66, No. 2, pp. 441-444, June 1970 The Dynamics of , II. The Steady-State Theory of Mutation Rates Henry Eyring and Betsy J. Stover* DEPARTMENT OF CHEMISTRY AND DIVISION OF RADIOBIOLOGY, DEPARTMENT OF , UNIVERSITY OF UTAH, SALT LAKE CITY Communicated February 24, 1970 Abstract. The statistical effect of internal irradiation on the survival of beagles is similar to that of aging, with the exception that occurs earlier. Since biological processes are close to- equilibrium most of the time, the above observation suggests that death ensues when the rate of insult has exceeded the rate of recovery to a sufficient extent that the reserves are depleted and the steady state can no longer be maintained. The steady-state theory of mutation rates is derived from first principles, and, through absolute rate theory, is com- pletely general and is not limited to a specific kind of cellular alteration. How- ever, the nature of the cellular alterations that lead to nonsurvival are considered. The young growing system is also considered in the context of this theory.

Introduction. Survival data from an extensive program of observing the pro- tracted biological effects of and other pertinent radionuclides have been treated successfully by the theory to be developed in this and a subsequent paper.1 The statistical effect of irradiation, in this case, was found to be similar to that of aging. The statistical death curves have substantially the same shape for the irradiated dogs as for the controls, except that death comes at an earlier age. The specific or the degenerative of old age thus simply take their toll earlier as a result of the weakening by irradiation. The effect of , i.e., disinfectants, has been interestingly treated as an irreversible attack on r separate sites with death occurring when all these sites are incapacitated. Thus, if p is the probability that a site is operative and q that it is incapacitated, then p + q = 1, and, further, the term qr in the binomial expansion of (p + q)r is the probability of death. If p is decreasing as the result of a chemical reaction with the rate constant k, then p = e-kt and q = (1 -e-a) so that qr = 1 - (1 - e-kt)r and the fraction surviving at time.t is f = 1 - (1 _e-kt)r. (1) In those cases in which the rate constant, k, varies with time, e-kt should be replaced by e - fkdt wherever e -'t appears in the above discussion. t = 0 Curtis has discussed radiation and aging in an interesting way in the context of the mutation hypothesis of aging and of the action of irradiation to accelerate the aging process.3 441 Downloaded by guest on September 27, 2021 442 BIOCHEMISTRY: EYRIVG AND STOVER PRoc. N. A. S.

Another quite successful way of considering the onset of or death is to plot on a log-log scale the rate of onset N, of death or a particular disorder against the age of the subject, t. The result is frequently a fairly straight line. Thus N = ct - 1 (2) and N = ctr/r. (3) In this case r is interpreted as the number of centers which must be changed to bring about the onset of death or diseases, as the case may be. These centers have frequently been interpreted to be which are being modified with a consequent change in the nature of the affected . Burch has interpreted the onset of autoimmune diseases, such as inflammatory polyarthritis, rheumatoid arthritis, and that of aging in terms of such mutations in a parent cell or a stem cell.4 The quantity cdr in equation (2) is a scale and must be pro- portional to a rate constant and thus reflect changes in the environment effecting the rate of mutation. Here the nature of the mutation process is unspecified. A steady-state theory of mutation rates: Typically, biological processes pro- ceed at near equilibrium so that, for example, a change in the concentration of some product may set in motion compensatory or reverse reactions which bring the system back into biological balance. In developing the steady-state theory of mutation rates, we suppose that there are r sites in r or less than r cells, i,e., one or more per cell, subject to a critical change of which n of these have been changed at the time t. Then, if vj is the rate at which a single unchanged site is changing, and vj is the rate at which the change in one of the altered sites is disappearing, we can write vi(r - n) = vjn (4) or vir n= ~~~~~~~~~~(5) Vj + Vj and n 1 r (vj/v,) + 1 (6) is the fraction of changed sites and the fraction of unchanged sites p = 1 -q is of+rjofrj 1 ~~~~~~~~~(7) Vi + + V-V j Vi + V 1 + (V(/Vj) Now, according to absolute rate theory KkT In wherehCit2e..ciKviT -AGToi/RThf= e (AG~oi/RT-Z ci) (8) where cj is the concentration of the ith constituent, and Downloaded by guest on September 27, 2021 VOL. 66, 1970 BIOCHEMISTRY: EYRINiG AND STOVER 443

KkT hj e -(AG+oj/RT -: Incj) (9) Some of these constituents, ci, may be or other reactants that are being used up by irradiation, aging, poisons, or other stresses. Thus, we can write dci -k1c. (10) dt - Hence in ci = -kit + in c0,. (11) In a steady state such as this, the process of modifying the site need not be the reverse of the process by which this site is mended. (For example, if a critical molecule is destroyed, the steady state is maintained by synthesis of another molecule of the same kind.) Changing an or other entity will alter the value of q = n/r. By combining equations (7)-(9) and (11) we ob- tain for the fraction of altered sites, n/r

q = n/r= (1 + ea-bt)-1 (12) where a- bt [(AGt~o- AG10)/RT- In coi + ZInco] + i j (Z k -j kj)t. (13) Then the probability of a site's being unaltered, p, is p = (1 - q) = (1 + e-(a-bt))>l (14) The probable nature of mutations causing : There are 100-200 known types of depending on the method of classification. The fact that radia- tion, the process of aging, and many types of carcinogens cause similar types of cancer suggest that they act nonspecifically through a common mechanism. They thus presumably act by decreasing (a - bt); that is, they increase the dif- ference between breaking as compared with the rate of repair bringing about a genetic change resulting in cancer.5-8 The first possibility is that the critical chromosome break leads to the elimination of the for re- pressing growth and thus leads to uncontrolled growth of the modified cell. The second possibility is that a gene is lost which controls the surface adhesion between like cells and thus promotes growth through a looser structure which allows the escape of inhibitor and/or the diffusion into the cell of nutrients, which in turn leads to uncontrolled growth as well as to the promotion of me- tastasis from the poorly fastened cells. In the process of , perfectly healthy unattached cells must have acquired genes which led to their attach- ment to and from organs, and with this clumping came inhibition of growth due to the slow escape of specific inhibitors, also genetically controlled, as well as poorer access of nourishment. The above types of mutation probably represent gene Downloaded by guest on September 27, 2021 444 BIOCHEMISTRY: EYRING AND STOVER Puoc. N. A. S.

or suppression. The third possibility is that cancer caused by a may also arise from the addition of genes, promoting uncontrolled growth. However, a virus may also cause gene deletion. Since mutations involving both the gain and loss of genes occur, it will not be surprising to find there are malig- nant conditions arising from both causes. Our formal theory does not distin- guish between deletion or addition of genetic material but only requires that a chromosome be modified to cause nonsurvival. The steady state of the growing system: The biological steady state of the growing system differs from that of the mature system. In maturity the re- serves are maintained by steady-state processes, but during growth the reserves must not only be maintained but continuously increased by these processes. Thus the rates of making genetic material must exceed the rates of loss, but the net difference in rates must be close to constant and must change slowly as the system progresses from one growth phase to the next. The manifestation of critical cellular changes that lead to mutations, and which are induced by ir- radiation, poisons, infections, and other damaging agents, will occur sooner if these critical changes occur during periods of rapid growth such as gestation, infancy, and, perhaps to a lesser extent, adolescence. This is consistent with experimental observation. Further, a very large dose of irradiation markedly alters the values of ci or cj, or both, to decrease the net difference in rates that is necessary for growth, and the result is retardation of growth and development. Another exan ple of the effect on growing systems of large changes in the values of ci and/or c; is malnutrition. Conclusion. The steady-state theory of mutation rates has been derived in this paper, and it will be extended to include a number of mechanisms of non- survival in a subsequent paper. An application has already been reported.' * One of us (H. E.) would like to thank the National Institutes of Health for its support and the other (B. J. S.), the U.S. Atomic Energy Commission, Contract At. (11-1)-119. 1 Stover, B. J., and H. Eyring, these PROCEEDINGS, 66, 132 (1970). 2 Johnson, F. H., H. Eyring, and M. J. Polissar, in The Kinetic Basis of (New York: John Wiley & Sons, Inc., 1954), pp. 453-463. 3 Curtis, H. J., Symp. Soc. Exptl. Biol., 21, 51 (1967). 4 Burch, P. R. J., Lancet, 1, 1253; and 2, 299 (1963). 5 Pettijohn, D., and P. Hanawalt, J. Mol. Biol., 9, 395 (1964). 6 Olivera, B. M., and I. R. Lehman, these PROCEEDINGS, 57, 1426 (1967). 7 Olivera, B. M., and 1. R. Lehman, these PROCEEDINGS, 57, 1700 (1967). 8 Olivera, B. M., and I. R. Lehman, J. Mol. Biol., 36. 261 (1968). Downloaded by guest on September 27, 2021