Life has been a mystery for most physicists since the question of Maxwell’s demon. In the present paper, the self-reproduction, which is characteristic of the organism, is shown to be essential in resolving the paradox of Maxwell’s demon. For this purpose, a new thermodynamic quantity of biological activity is introduced to represent the molecular events in an organism recently revealed by molecular biology. This quantity gives a measure that the entropy production arising from the difference between acquired energy and stored energy compensates for the negative entropy of systematizing the organism, and is considered to be proportional to the self-reproducing rate of the organism as the first approximation. The equation of replicator dynamics consisting of self-reproducing rate, death rate and mutation terms contains all known types of evolution of unicellular organisms. When the mutation term is restricted to the point mutation mainly due to the nucleotide base changes in genes, this equation automatically leads to Darwinian evolution that the mutant with the higher increase rate is selected to become prevailing in the population. Throughout this evolution, the nucleotide bases in genes are converged to the special arrangement exhibiting the optimal increase rate of the organism. Moreover, the mutants having experienced gene duplication first decline to the minor members in the population but some of the descendants recover as a new style of organisms by generating new gene(s) from the counterpart of duplicated genes. This evolutionary process to expand the repertoire of genes is mathematically formulated by solving the equation of replicator dynamics up to the higher order of mutation terms. The present theoretical approach can be not only extended to the multicellular diploid eukaryotes but also applied to explain the origin of genes in the self-reproducing proto-cells formed anciently.
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