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Genetic alterations such as point mutations, chromosomal rearrangements, modification of DNA methylation and chromosome aberrations accumulate during the lifetime of an organism. They can be caused by intrinsic errors in the DNA replication and repair as well as by external factors such as exposure to mutagenic substances or radiation. The main purpose of the present work is to begin an exploration of the stochastic nature of non-equilibrium DNA alteration caused by events such as tautomeric shifts. This is done by modeling the genetic DNA code chain as a sequence of DNA-bit values ('1' for normal bases and '-1' for abnormal bases). We observe the number of DNA-bit changes resulting from the random point mutation process which, in the model, is being induced by a stochastic Brownian mutagen (BM) as it diffuses through the DNA-bit systems. Using both an analytical and Monte Carlo (MC) simulation techniques, we observe the local and global number of DNA-bit changes. It is found that in 1D, the local DNA-bit density behaves like 1/t, the global total number of the switched (abnormal) DNA-bit increases as t. The probability distribution P(b, 0, t) of b(0, t) is log-normal. It is also found that when the number of mutagens is increased, the number of the total abnormal DNA-bits does not grow linearly with the number of mutagens. All analytic results are in good agreement with the simulation results.