Difference between revisions of "Store:QLMen14"

===8.2. Biological functions in the quantum Markov framework===

We turn to the open system dynamics with the GKSL-equation. In our modeling, Hamiltonian  <math>\widehat{\mathcal{H}}</math> and Lindbladian  <math>\widehat{{L}}</math> represent some special ''biological function'' <math>F</math> (see Khrennikov et al., 2018)<ref>Khrennikov A., Basieva I., PothosE.M., Yamato I.

Quantum Probability in Decision Making from Quantum Information Representation of Neuronal States, Sci. Rep., 8 (2018), Article 16225</ref> for details. Its functioning results from interaction of internal and external information flows. In Sections 10, 11.3,  <math>F</math> is some ''psychological function''; in the simplest case <math>F</math> represents a question asked to  <math>S</math> (say  is a human being). In Section 7, <math>F</math>  is the ''gene regulation'' of glucose/lactose metabolism in Escherichia coli bacterium. In Sections 9, 11.2,  <math>F</math> represents the process of ''epigenetic mutation''. Symbolically biological function <math>F</math> is represented as a quantum observable: Hermitian operator  <math>\widehat{F}</math> with the spectral decomposition <math>\widehat{F}=\sum_xx\widehat{E}^F(x)</math>, where <math>x</math> labels outputs of <math>F</math>. Theory of quantum Markov state-dynamics describes the process of generation of these outputs.

In the mathematical model (Asano et al., 2015b,<ref>Asano M., Khrennikov A., Ohya M., Tanaka Y., Yamato I.

Springer, Heidelberg-Berlin-New York(2015)</ref> Asano et al., 2017b,<ref>Asano M., Basieva I., Khrennikov A., Yamato I.

Prog. Biophys. Mol. Biol., 130 (Part A)(2017), pp. 88-98</ref> Asano et al., 2017a,<ref>Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y.

J. Math. Psychol., 78 (2017), pp. 2-12</ref> Asano et al., 2015a,<ref>Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., Yamato I.

Quantum information biology: from information interpretation of quantum mechanics to applications in molecular biology and cognitive psychology. Found. Phys., 45 (10) (2015), pp. 1362-1378</ref> Asano et al., 2012b,<ref>Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., Yamato I.

AIP Conf. Proc., 1508 (2012), p. 75</ref> Asano et al., 2011,<ref>Asano M., Ohya M., Tanaka Y., BasievaI., Khrennikov A.

J. Theor. Biol., 281 (1) (2011), pp. 56-64</ref> Asano et al., 2012a<ref>Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., I

Syst. Synth. Biol., 6 (2012), pp. 1-7</ref>), the outputs of biological function <math>F</math>  are generated via approaching a ''steady state'' of the GKSL-dynamics:  

such that it matches the spectral decomposition of <math>\widehat{F}</math>, i.e.,   

This means that <math>\widehat{\rho}_{steady}</math> is diagonal in an orthonormal basis consisting of eigenvectors of <math>\widehat{F}</math>. This state, or more precisely, this decomposition of density operator <math>\widehat{\rho}_{steady}</math>, is the classical statistical mixture of the basic information states determining this biological function. The probabilities in state’s decomposition (26) are interpreted statistically.

Consider a large ensemble of biosystems with the state <math>\widehat{\rho}_0</math> interacting with environment <math>\varepsilon</math>. (We recall that mathematically the interaction is encoded in the Lindbladian <math>\widehat{{L}}</math>) Resulting from this interaction, biological function <math>F</math> produces output <math>x</math> with probability <math>p_x</math>. We remark that in the operator terms the probability is expressed as <math>p_x=Tr\widehat{\rho}_{steady}\widehat{E}^F(x)</math>

This interpretation can be applied even to a single biosystem that meets the same environment many times.

It should be noted that limiting state  <math>\widehat{\rho}_{steady}</math> expresses the stability with respect to the influence of concrete environment <math>\varepsilon</math>. Of course, in the real world the limit-state would be never approached. The mathematical formula (25) describes the process of stabilization, damping of fluctuations. But, they would be never disappear completely with time.

We note that a steady state satisfies the stationary GKSL-equation:  

It is also important to point that generally a steady state of the quantum master equation is not unique, it depends on the class of initial conditions.