Store:Khrennikov13

8. Open quantum systems: interaction of a biosystem with its environment

As was already emphasized, any biosystem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}  is fundamentally open. Hence, dynamics of its state has to be modeled via an interaction with surrounding environment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} . The states of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} are represented in the Hilbert spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}}  and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} . The compound system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon}  is represented in the tensor product Hilbert spaces . This system is treated as an isolated system and in accordance with quantum theory, dynamics of its pure state can be described by the Schrödinger equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\tfrac{d}{dt}\Psi(t)=\widehat{H}\Psi(t)(t), \Psi(0)=\Psi_0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (21)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)}  is the pure state of the system $S+\varepsilon$ and ${\hat {\mathcal {H}}}$ is its Hamiltonian. This equation implies that the pure state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)}  evolves unitarily :Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(t)=\hat{U}(t)\psi_0} . Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U}(t)=e^{-it\hat{\mathcal{H}}}} . Hamiltonian (evolution-generator) describing information interactions has the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}=\hat{\mathcal{H}}_s+\hat{\mathcal{H}}_\varepsilon+{\mathcal{\hat H_{S,\varepsilon}}}} , where  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}_s} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{H}}_\varepsilon} are Hamiltonians of the systems and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal{\hat H_{S,\varepsilon}}}} is the interaction Hamiltonian.12 This equation implies that evolution of the density operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{R}}(t)}  of the system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon}  is described by von Neumann equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{d\widehat{R}}{dt}(t)=-i[\widehat{H},\widehat{R},(t)], \widehat{R}(0)=\widehat{R}_0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (22)}

However, the state  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathcal{R}}(t)} is too complex for any mathematical analysis: the environment includes too many degrees of freedom. Therefore, we are interested only the state of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} ; its dynamics is obtained via tracing of the state of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S+\varepsilon} w.r.t. the degrees of freedom of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon}  :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\rho}(t)=Tr_\mathcal{H}\widehat{R}(t)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (23)}

Generally this equation, the quantum master equation, is mathematically very complicated. A variety of approximations is used in applications.

8.1. Quantum Markov model: Gorini–Kossakowski–Sudarshan–Lindbladequation

The simplest approximation of quantum master equation (23) is the quantum Markov dynamics given by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation (Ingarden et al., 1997) (in physics, it is commonly called simply the Lindblad equation; this is the simplest quantum master equation):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{d\widehat{\rho}}{dt}(t)=-i[\widehat{H},\widehat{\rho},(t)]+ \widehat{L}[\widehat{\rho}(t),\widehat{\rho}(0)=\widehat{\rho}_0}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (24)}

where Hermitian operator (Hamiltonian) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\mathcal{H}}}  describes the internal dynamics of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}  and the superoperator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{{L}}} , acting in the space of density operators, describes an interaction with environment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} . This superoperator is often called Lindbladian. The GKSL-equation is a quantum master equation for Markovian dynamics. In this paper, we have no possibility to explain the notion of quantum Markovianity in more detail. Quantum master equation (23) describes generally non-Markovean dynamics.

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8.2. Biological functions in the quantum Markov framework

We turn to the open system dynamics with the GKSL-equation. In our modeling, Hamiltonian  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{\mathcal{H}}} and Lindbladian  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{{L}}} represent some special biological function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}  (see Khrennikov et al., 2018) for details. Its functioning results from interaction of internal and external information flows. In Sections 10, 11.3,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is some psychological function; in the simplest case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} represents a question asked to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} (say  is a human being). In Section 7, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}   is the gene regulation of glucose/lactose metabolism in Escherichia coli bacterium. In Sections 9, 11.2,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} represents the process of epigenetic mutation. Symbolically biological function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}  is represented as a quantum observable: Hermitian operator  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{F}} with the spectral decomposition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widehat{F}=\sum_xx\widehat{E}^F(x)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} labels outputs of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . Theory of quantum Markov state-dynamics describes the process of generation of these outputs.

In the mathematical model (Asano et al., 2015b, Asano et al., 2017b, Asano et al., 2017a, Asano et al., 2015a, Asano et al., 2012b, Asano et al., 2011, Asano et al., 2012a), the outputs of biological function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}   are generated via approaching a steady state of the GKSL-dynamics:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t \to \infty}\widehat{\rho}(t)=\widehat{\rho}_{steady}}

$(25)$

such that it matches the spectral decomposition of ${\widehat {F}}$, i.e.,

${\widehat {\rho }}_{steady}=\sum _{x}p_{x}{\widehat {E}}^{F}(x)$

$(26)$

where

$p_{x}\geq \sum _{x}p_{x}=1$

$(26)$

This means that ${\widehat {\rho }}_{steady}$ is diagonal in an orthonormal basis consisting of eigenvectors of ${\widehat {F}}$. This state, or more precisely, this decomposition of density operator ${\widehat {\rho }}_{steady}$, 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 ${\widehat {\rho }}_{0}$ interacting with environment $\varepsilon$. (We recall that mathematically the interaction is encoded in the Lindbladian ${\widehat {L}}$) Resulting from this interaction, biological function $F$ produces output $x$ with probability $p_{x}$. We remark that in the operator terms the probability is expressed as $p_{x}=Tr{\widehat {\rho }}_{steady}{\widehat {E}}^{F}(x)$

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

It should be noted that limiting state  ${\widehat {\rho }}_{steady}$ expresses the stability with respect to the influence of concrete environment $\varepsilon$. 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:

$i[{\widehat {H}},{\widehat {\rho }}_{steady}]={\widehat {L}}[{\widehat {\rho }}_{steady}]$

$(27)$

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.

8.3. Operation of biological functions through decoherence

To make the previous considerations concrete, let us consider a pure quantum state as the initial state. Suppose that a biological function  $F$ is dichotomous, $F=0,1$, and it is symbolically represented by the Hermitian operator that is diagonal in orthonormal basis $|0\rangle$,$|1\rangle$ . (We consider the two dimensional state space — the qubit space.) Let the initial state has the form of superposition

$\psi \rangle =c_{0}|0\rangle +c_{1}|1\rangle$

$(28)$

where $c_{j}\in C,|c_{0}|^{2}+||c_{1}|^{2}=1$. The quantum master dynamics is not a pure state dynamics: sooner or later (in fact, very soon), this superposition representing a pure state will be transferred into a density matrix representing a mixed state. Therefore, from the very beginning it is useful to represent superposition (28) in terms of a density matrix:

${\widehat {\rho }}_{0}={\begin{vmatrix}|c_{0}|^{2}&c_{0}{\bar {c}}_{1}\\{\bar {c}}_{0}c_{1}&|c_{1}|^{2}\end{vmatrix}}$

$(29)$

State’s purity, superposition, is characterized by the presence ofnonzero off-diagonal terms.

Superposition encodes uncertainty with respect to the concrete state basis, in our case $|0\rangle$,$|1\rangle$. Initially biological function $F$  was in the state of uncertainty between two choices $x=0,1$. This is genuine quantum(-like) uncertainty. Uncertainty, about possible actions in future. For example, for psychological function (Section 10)  $F$ representing answering to some question, say “to buy property” ( $F=1$) and its negation ( $F=0$) , a person whose state is described by superposition (28) is uncertain to act with ( $F=1$)  or with ( $F=0$) . Thus, a superposition-type state describes individual uncertainty, i.e., uncertainty associated with the individual biosystem and not with an ensemble of biosystems; with the single act of functioning of $F$  and not with a large series of such acts.

Resolution of uncertainty with respect to ${\widehat {F}}-basis$ is characterized by washing off the off-diagonal terms in (29) The quantum dynamics (24) suppresses the off-diagonal terms and, finally, a diagonal density matrix representing a steady state of this dynamical systems is generated:

${\widehat {\rho }}_{0}={\begin{vmatrix}p_{0}&0\\0&p_{1}\end{vmatrix}}$

$(30)$

This is a classical statistical mixture. It describes an ensemble of biosystems; statistically they generate outputs $F=\alpha$ with probabilities $p_{\alpha }$. In the same way, the statistical interpretation can be used for a single system that performs $F$-functioning at different instances of time (for a long time series).

In quantum physics, the process of washing off the off-diagonal elements in a density matrix is known as the process of decoherence. Thus, the described model of can be called operation of biological function through decoherence.

8.4. Linearity of quantum representation: exponential speed up for biological functioning

The quantum-like modeling does not claim that biosystems are fundamentally quantum. A more natural picture is that they are a complex classical biophysical systems and the quantum-like model provides the information representation of classical biophysical processes, in genes, proteins, cells, brains. One of the advantages of this representation is its linearity. The quantum state space is a complex Hilbert space and dynamical equations are linear differential equations. For finite dimensional state spaces, these are just ordinary differential equations with complex coefficients (so, the reader should not be afraid of such pathetic names as Schrödinger, von Neumann, or Gorini–Kossakowski–Sudarshan–Lindblad equations). The classical biophysical dynamics beyond the quantum information representation is typically nonlinear and very complicated. The use of the linear space representation simplifies the processing structure. There are two viewpoints on this simplification, external and internal. The first one is simplification of mathematical modeling, i.e., simplification of study of bioprocesses (by us, external observers). The second one is more delicate and interesting. We have already pointed to one important specialty of applications of the quantum theory to biology. Here, systems can perform self-observations. So, in the process of evolution say a cell can “learn” via such self-observations that it is computationally profitable to use the linear quantum-like representation. And now, we come to the main advantage of linearity.

The linear dynamics exponentially speeds up information processing. Solutions of the GKSL-equation can be represented in the form ${\widehat {\rho }}(t)=e^{t{\widehat {\Gamma }}}{\widehat {\rho }}$, where ${\widehat {\Gamma }}$ is the superoperator given by the right-hand side of the GKSL-equation. In the finite dimensional case, decoherence dynamics is expressed via factors of the form $e^{t{(ia-b)}}$, where $b>0$. Such factors are exponentially decreasing. Quantum-like linear realization of biological functions is exponentially rapid comparing with nonlinear classical dynamics.

The use of the quantum information representation means that generally large clusters of classical biophysical states are encoded by a few quantum states. It means huge information compressing. It also implies increasing of stability in state-processing. Noisy nonlinear classical dynamics is mapped to dynamics driven by linear quantum(-like) equation of say GKSL-type.

The latter has essentially simpler structure and via selection of the operator coefficients encoding symbolically interaction within the system  $S$ and with its surrounding environment $\varepsilon$,  $S$ can establish dynamics with stabilization regimes leading to steady states.