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9. Epigenetic evolution within theory of open quantum systems

In paper (Asano et al., 2012b), a general model of the epigenetic evolution unifying neo-Darwinian with neo-Lamarckian approaches was created in the framework of theory of open quantum systems. The process of evolution is represented in the form of adaptive dynamics given by the quantum(-like) master equation describing the dynamics of the information state of epigenome in the process of interaction with surrounding environment. This model of the epigenetic evolution expresses the probabilities for observations which can be done on epigenomes of cells; this (quantum-like) model does not give a detailed description of cellular processes. The quantum operational approach provides a possibility to describe by one model all known types of cellular epigenetic inheritance.

To give some hint about the model, we consider one gene, say $g$. This is the system $S$ in Section 8.1. It interacts with the surrounding environment  $\varepsilon$ a cell containing this gene and other cells that send signals to this concrete cell and through it to the gene $g$. As a consequence of this interaction some epigenetic mutation $\mu$ in the gene $g$ can happen. It would change the level of the $g$-expression.

For the moment, we ignore that there are other genes. In this oversimplified model, the mutation can be described within the two dimensional state space, complex Hilbert space ${\mathcal {H}}_{epi}$ (qubit space). States of $g$ without and with mutation are represented by the orthogonal basis $|0\rangle$,$|1\rangle$; these vectors express possible epigenetic changes of the fixed type $\mu$.

A pure quantum information state has the form of superposition$|\psi \rangle _{epi}=c_{0}|0\rangle +c_{1}|1\rangle$.

Now, we turn to the general scheme of Section 8.2 with the biological function $F$  expressing $\mu$-epimutation in one fixed gene. The quantum Markov dynamics (24) resolves uncertainty encoded in superposition $|\psi \rangle _{epi}$ (“modeling epimutations as decoherence”). The classical statistical mixture , ${\rho }_{steady}$see (30), is approached. Its diagonal elements $p_{0},p_{1}$give the probabilities of the events: “no $\mu$-epimutation” and “$\mu$-epimutation”. These probabilities are interpreted statistically: in a large population of cells,  $M$ cells,$M\gg 1$ , the number of cells with $\mu$-epimutation is $N_{m}\approx p_{1}M$. This $\mu$-epimutation in a cell population would stabilize completely to the steady state only in the infinite time. Therefore in reality there are fluctuations (of decreasing amplitude) in any finite interval of time.

Finally, we point to the advantage of the quantum-like dynamics of interaction of genes with environment — dynamics’ linearity implying exponential speed up of the process of epigenetic evolution (Section 8.4).