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Edited by: Dmytro Hovorun, National Academy of Sciences of Ukraine, Ukraine

Reviewed by: Josep M. Ribó, University of Barcelona, Spain; JunLi Ren, South China University of Technology, China; José Pedro Cerón-Carrasco, Catholic University San Antonio of Murcia, Spain

This article was submitted to Theoretical and Computational Chemistry, a section of the journal Frontiers in Chemistry

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In general, for chemical reactions occurring in systems, where fluctuations are not negligibly small, it is necessary to introduce a master equation for distribution of probability of fluctuations. It has been established that the monomolecular reactions of a type as _{0}. However, the consideration of the Löwdin mechanism as autocatalytic non-linear chemical reactions such as

The role of mutations in DNA is crucial for human aging, metabolic and degenerative disorders and cancer, as well as for biological evolution of living systems (Löwdin, ^{−7}–10^{−9}) per base pair per cell division (Drake et al.,

G-C → G* −

Based on the rare tautomeric hypothesis by Watson-Crick, Löwdin (Löwdin, ^{*}-T^{*}, G^{*}-C^{*}) have been extensively studied in terms of their lifetime, the probability of occurrence and the energy by using different theoretical approaches (Florian et al.,

In the present work, we will show that the Löwdin's mechanism of spontaneous mutations results in a non-Poissonian distribution function of mutagenic tautomeric forms of DNA by using first-order autocatalysis. It is well-known that the first- and higher order autocatalysis was applied to different biological processes, including reproduction (Eigen,

In our model, the autocatalytic reaction of first order is applied to the process of double proton transfer in DNA, which gives a non-Poissonian distribution of tautomeric states of hydrogens along the chain which as a result of replication leads to spontaneous mutations.

The search of experiments on the distribution function of spontaneous mutations to verify our model leads to the Luria-Delbruck's experiments and distribution of mutants (Luria and Delbrück, _{0}.

In general case, for chemical reactions occurring in systems, where fluctuations are not considered negligible small, it is necessary to introduce a master-equation for distribution of probability (Gardiner, _{0} can be derived. It is well-known that at considering the reactions of

Here A denotes protons along the DNA strand which are in their regular stable position, X is the protons in the tautomeric state (_{−} are the rate constants of forward and reverse chemical reactions, respectively. The first reaction corresponds to the processes of the transfer of a proton from the regular position into the tautomeric state and vice versa, while the second reaction corresponds to the generation of tautomeric forms of nucleotide bases due to the interaction of the single proton in the tautomeric state with the regular proton and its relaxation into the single proton tautomeric state.

The double-well potential for a single proton tunneling.

Denoting the number of protons in the tautomeric state by

Here w is the transition probability per unit time. The crux of the master equation is to determine the transition rates explicitly for each chemical reaction. We investigate all transitions leading to N or going away from it. For spontaneous mutations in DNA we consider two types of transition—the tautomeric proton generation (

1. In the direction _{1} (“birth” of a tautomeric proton X). The number of the transitions per second is equal to the occupational probability, _{A} along the DNA strand and the reaction rate _{1}. So, for transition → _{1}_{A}_{1}_{A}

For this reaction the total transition rate is received in the following form:

2. In a similar way, we can find the reverse process of the first reaction. Here the number of tautomeric protons is decreased by 1 (“death” of a metastable proton X). Considering the “death” process for transitions _{ − 1} direction (“death” of X) the total transition rate is equal to:

3. For the reaction (2), in the _{2} direction (“birth” of a tautomeric proton X), the total transition rate can be written:

4. For the reaction (2) in the _{−2} direction (“death” of a tautomeric proton X), the total transition rate can be written:

Thus, by taking into account (Equations 4–7) the transition rates for the reactions (1) and (2) are given:

The stationary solution of the master equation (3) is determined as Eigen (

Let us denote the probabilities of two transitions by the following expressions:

The condition of extremes of stationary

We can find the extreme solutions _{0} of

The solution of Equation (15) for _{0} can be received:

The positive value of _{0} can be obtained if _{ − 2} + _{ − 1}_{2}_{A}_{2}/_{−2}, we can receive the probability distribution function with one maximum (

It is seen from Equation (17) that N_{o} is positive when the transfer of proton from the regular position into the tautomeric state proceeds slower than its reverse reaction (1), while the second proton is transferred almost instantaneously (2) compared to the reverse process of the first reaction (1).

Non-poissonian distribution function P(N) of spontaneous mutations in DNA by Löwdin's mechanism.

If the principle of detailed balance is satisfied for both chemical reactions (1) and (2), then the Poisson distribution function for fluctuations can be deduced (Haken,

Dividing Equation (18) by Equation (19) and using the explicit expressions (Equations 8 and 11), we find the relation:

Here μ is a constant. By using Equation (20) we can rewrite the detailed balance equation:

By inserting Equation (21) into Equation (12) at normalization condition of

In general, however, for non-equilibrium processes, where the detailed balance principle is not valid, the character of distribution function

In this work, we applied first-order autocatalysis to the Löwdin's mechanism of spontaneous mutation formation, in which concerted double proton transfers in DNA lead to the formation of tautomeric forms nucleotide bases during the DNA replication. The stochastic model results in a master-equation for the distribution function of tautomeric nucleotide bases, the stationary solution of which is a non-Poissonian function with the assumption that the processes of double proton transfers during DNA replication are far from equilibrium conditions. We suppose that these peculiarities of Löwdin's mechanism of spontaneous mutation formation should be taken into account in the discussion of the origin of spontaneous mutation formation. It is interesting to note the possibility of accumulation of point mutations locally on DNA due to the cooperation effect between tautomeric hydrogens. The cooperation effect should increase the order of autocatalytic reactions. Such fluctuations can be possible due to the U-negative effect (Anderson,

All datasets generated for this study are included in the article/Supplementary material.

NT was the author of the idea about application of stochastic theory to spontaneous mutations in DNA and the main contributor of the manuscript. BO was the author of the idea about the possibility of soliton generation in DNA based on accumulation of metastable proton fluctuations.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.