Analysis of quantum coherence in biology

Journal Pre-proofs Analysis of Quantum Coherence in Biology Igor Khmelinskii, Vladimir I. Makarov PII: DOI: Reference:

S0301-0104(19)31383-7 https://doi.org/10.1016/j.chemphys.2019.110671 CHEMPH 110671

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Please cite this article as: I. Khmelinskii, V.I. Makarov, Analysis of Quantum Coherence in Biology, Chemical Physics (2019), doi: https://doi.org/10.1016/j.chemphys.2019.110671

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Review Paper

Analysis of Quantum Coherence in Biology Igor Khmelinskii (1) and Vladimir I. Makarov (2) 1 University

of the Algarve, FCT, DQF and CEOT, 8005-139, Faro, Portugal

2 University

of Puerto Rico, Rio Piedras Campus, PO Box 23343, San Juan, PR 00931-3343,

USA Corresponding Author: Dr. Vladimir Makarov Department of Physics, UPR, Rio Piedras Campus, PO Box 23343 San Juan PR 00931 USA

Highlights: 1. Theoretic methods for quantum coherency dynamics in biology reviewed; 2. Namely, effective Hamiltonian, Green Function and density matrix methods; 3. Quantum coherency dynamics in photosynthesis, brain and vision discussed; 4. The effective Hamiltonian method recommended for interpreting quantum coherency

effects in biology. Keywords: effective Hamiltonian; Green’s function; density matrix; excitons; neurons; photosynthesis; retinal Müller cells

Abstract We reviewed the tools of quantum physics used in modeling of quantum coherence (QC) effects in different systems, including biological systems, which behave as quantum objects in some of their degrees of freedom. In particular, we considered the usage of the effective Hamiltonian (EH), Green’s function (GF) and density matrix (DM) methods in the analysis of QC, focusing on QC in biological systems. We discussed the two main mechanisms of loss of quantum state coherence: (i) dephasing of the originally prepared coherent wave package and (ii) population relaxation in the same wave package. Dephasing does not affect the quantum state population, e.g. as in spin-spin relaxation, where dephasing is described by the 2 relaxation time. On the other hand, the state population relaxation of the spin wavepackage is attributed to spin-lattice relaxation and is described by the 1 relaxation time. Presently we discussed EH and GF formalisms in terms of the complex energy, dependent on intra- and intersystem interactions that induce state population relaxation. We provided a detailed analysis of these approaches for the exciton relaxation dynamics in a glycine polypeptide chain. The same phenomena were described in the DM formalism using the relaxation matrix. We discussed QC in different biological systems, showing that QC is conserved when the interactions of the coherent wavepackage with other degrees of freedom are weak, as otherwise population relaxation causes loss of QC. We believe that our results will be useful for the researchers in the area of quantum biology.

Graphical abstract

|s>

V

 j   j0   i

s

j

 s   s0   i

2

j 2

j 

First-order interactions between quantum states; the matrix element V describes the strength of the interaction between the state |s> and the set of states {|j>} with the state density ρj. The state energies are characterized by complex values, with the imaginary part describing the respective state width.

Abbreviations used in the text: adenosine triphosphate – ATP; quantum coherence – QC; quantum effect – QE; classical degree of freedom – CDF; quantum degree of freedom – QDF; Müller cell – MC; intermediate filaments – IFs; polypeptide – PP; quantum interference – QI; effective Hamiltonian – EH; Green’s function – GF; density matrix – DM; quantum mechanism – QM; microtubule – MT. I. Introduction A lot of attention was focused over the last 50 years on quantum effects in biological systems [1]. The main focus has been given to the quantum interference (QI; coherence/decoherence) effects in the brain [2], and to the propagation of the ATP hydrolysis energy along a polypeptide chain [3]. The latter effect is also dependent on quantum interference [1]. The signal exchange between neurons is typically described as a classical electrochemical process of ion transport through the cell membranes. However, Penrose [4,5], Stapp [6] and others proposed that quantum effects may play a significant role in the interactions between neurons. Thus, neural networks should be described as quantum communication networks [7-9]. Several authors considered that brain activity may only be understood as a consequence of quantum effects operating in it. Wigner [10] proposed that consciousness was linked to the quantum measurement problem, the same idea was later further developed by Penrose [5]. It was also proposed that consciousness is a macroscopic quantum effect, involving superconductivity [11], superfluidity [12], electromagnetic fields [13], Bose condensation [14], superfluorescence [15] or some other mechanisms [16,17]. However, presently we find no physical evidence for any of these proposed mechanisms. Penrose [5] proposed that QEs occur in brain cell microtubules. It was proposed by Hameroff & Watt [18], Hameroff [19] and Penrose [4,5] that microtubules can process information like a cellular quantum communications network, which operates as a quantum computer. This idea has been later discussed quite extensively [20-26]. All of these quantum modeling approaches are based on defining QDF in the system, which get involved into the quantum communications network. Stapp [27] assumed that QDFs are weakly coupled with the CDF. However, Zeh [28], Zurek [29], Brown [30], Hawking [31] and Hepp [32] believed that environment-induced coherence is rapidly broken in the real brain system; therefore QDFs should be interacting with CDFs strongly. However, in order to obtain quantitative conclusions, we need to correctly

determine the QDFs in the discussed system, and calculate the coupling strength between QDFs and CDFs. The propagation of ATP hydrolysis energy along a polypeptide (PP) chain has been extensively discussed earlier [33-42]. It was proposed that energy propagates in the form of vibrational soliton (Davydov soliton). Davydov soliton model includes vibrational excitation of the C=O group in the polypeptide chain, with the energy of 0.205 eV [43], sourced by ATP hydrolysis, which liberates up to 0.43 eV [44]. This vibrational excitation is transported to the C=O group of a neighboring amino-acid residue by the electric dipole-dipole energy transfer mechanism [45]. Considering decoherence due to population relaxation, the energy propagation length is determined by decoherence time of the originally prepared coherent state. The main parameter defining decoherence time is the relaxation time of the C=O vibrational mode. Typical estimates of this relaxation time at room temperature are in the subpicosecond time scale [43]. Therefore, Davydov soliton model requires strong dipole-dipole interactions between C=O groups in proteins, exceeding 100 cm–1. However, no large shifts of spectral maxima attributable to the C=O vibrational mode were ever observed when passing from individual aminoacids to PP chains [46,47], ruling out such strong interactions. Another difficulty of this model lies in the vibrational energy transfer from the mode to water molecules hydrogen-bonded to the PP chain. Indeed, the bending vibration energy of the water molecule equals 0.204 eV [48], very close to that of the C=O mode, which would induce strong coupling between these two modes and fast dissipation of the C=O vibrational energy. Additionally, this model fails to predict weak interactions between the QDFs and CDFs, which could preclude additional relaxation of the QDFs. To explain high contrast vision of vertebrate eyes, quantum mechanism of light energy propagation along MC IFs was proposed recently [49]. Specifically, it was proposed [49] that MC IFs absorb light in the MC endfeet, and the electronic excitation energy propagates along these IFs from the level of the inner limiting membrane to that of the outer limiting membrane in the form of excitons. Thus, the excitons arrive to the photoreceptor cells, where the energy is transferred to opsin/rhodopsin chromophores by the exchange mechanism. It was proposed that coupling between QDFs and CDFs in the protein IFs is much lower than that expected for Davydov solitons. These proposed properties of the IFs were justified in a model that described quantum confinement in a conductive nanowire representing the IF [47]. The postulated

properties of the IFs [50,51] were later confirmed in the experimental measurements of electric and light energy conductivity of porcine retinal MC IFs, which behave as conductive nanowires and transmit excitons over macroscopic distances without loss [49]. Moreover, it was found [52] that these IFs demonstrate semiconductor properties with the band gap of about 0.64 eV. Detailed ab initio analysis of the zone structure in -helix PP showed that such PPs have three conductive zones in the 0 – 6.5 eV energy range, where the energy gaps between the valence, and the first and second conductive zones are 0.63 to 1.09 eV and 0.87 to 3.92 eV, respectively, depending on the wave vector [53]. This allows the excitons with the energies in the 0.63 to 3.92 eV range to propagate along the -helix PP chain. As the MC IF are formed of numerous PP molecules, the referred zone structure calculations [53] are in good agreement with the experimental band gap measurements [52]. The results obtained allow reconsidering the analysis of quantum interference effects in biological systems, based on experimental measurements, appropriately interpreted theoretically. Thus, presently we shall be taking a fresh look at QI in biological systems. We shall support this approach by the general review of the previously proposed ideas. However, our present attention was focused on coherent effects developing in IFs, which are biological nanowires, quite ubiquitous in living systems. Thus, we discuss in detail the earlier proposed modeling approaches, including the analysis of QI effects in IFs, high-contrast vision of vertebrates, and a hypothetic communication mechanism between brain cells. It has been found that QI effects are not important for ion transport through the brain cell membranes, where the transport is successfully described by the classical-physics electro-diffusion equation [54]. On the other hand, (i) Davydov soliton and (ii) exciton transport along the -helix PP should provide better opportunities for observing QI effects. Presently we report that the criteria for the experimental observation of QI are quite different in these two examples. We believe that the present review will contribute to better understanding the organization and internal workings of living systems, based on quantum coherence effects. II. Possible approaches to the analysis of QI effects Coherent effects were extensively studied earlier in atomic and molecular spectroscopy, where they are well observable [55]. The theory of radiationless dynamics of excited states, based on QI, was discussed by Jortner et al. [56] in a simple model, where discrete states are coupled by an interaction appearing in the first-order perturbation theory. This theory was further developed

by other authors using more comprehensive models [57]. Different approaches were used in the respective analysis, including the EH by Andrews [58], GF by Bixon & Jortner [59] and DM theory [60]. The DM method has been extensively used for the analysis of QI in biological systems [61-63]. Note that while the DM method produces an exact solution of the problem, properly taking into account the excited state relaxation, it is quite cumbersome, as it a requires solving n2 first-order coupled differential equations when the basis set contains n eigenvectors only. Additionally, the DM method becomes even less convenient when applied to the analysis of multi-state systems. However, the EH and the GF methods, the second being a modification of the first, require solving a polynomial equation of degree n only, in order to determine the eigenvalues of the diagonalized Hamiltonian. Thus, both EH and GF approaches result in much simpler calculations compared to those needed in the DM treatment of the same system. We shall illustrate these ideas applying the three methods to the same case studies. a) EH method Let us introduce this method using a simple example (Fig. 1), with one active level |s>, which is initially populated, and coupled by the first-order perturbation V with n inactive {|j>} levels. The zero-order unperturbed states are characterized by the respective complex energy values s(0) = Es(0) - is/2, {j(0) = Ej(0) - ij/2}, where s, j are the energy level widths, determined by the relaxation rates of the respective levels. Note that the state width is created by stochastic interactions between quantum degrees of freedom and classical degrees of freedom or, when the state |s> is optically active, also with the electromagnetic field vector potential. We shall also introduce the density of the |j> levels; assuming that n →  we determine the state density as i = |Ej+1(0) – Ej(0)|–1. Using the system parameters, we shall consider three cases with qualitatively different dynamics of the initially prepared states: 1.  

s Vˆ j  j  1 – resonance limit,

2.   1 – intermediate case, 3.   1 – statistical limit,

s Vˆ j

where

is the average value of the interaction matrix element. The first case describes

the state dynamics in the system of two coupled states. The second case describes several |j> states in resonance with the |s> state, all of which contribute to the interaction. The third case has an infinite number of |j> states interacting with the |s> state. Before considering each of the cases, we need to introduce additional parameters



s Vˆ j

and    j  j ,

j

which characterize the system dynamics, where  j

is the average width of the |j> states. Note

that the  parameter determines the type of system time evolution; namely, QI effects are not observable when  << 1, QI effects may be observable although limited at   1, and QI effects are important when  >> 1. The second parameter  characterizes the form of the |j> state energy spectrum: the energy spectrum is discrete when  <<1, the system dynamics is strongly dependent on the parameter values when   1, and the energy spectrum is quasicontinuum when

 >> 1. Thus, we need to analyze the values of ,  and  parameters in order to predict the system time evolution. a) TheEH method The EH may be represented as follows [62]:

Hˆ eff

  s0  Vs1   10  V   1s ...  ... V 0  ns

... Vsn   ... 0  ... ...  ...  n0  

(1)

Let us first consider  >> 1, which corresponds to numerous |j> states in the interaction resonance with the |s> state. The eigenvalues and eigenvectors of the diagonalized Hamiltonian (1) are given by the solution of the Schrödinger equation:

  s0      V1s  ...   V  ns

Vs1 1   ... 0 0 

... Vsn  Cks    ... 0  Ck1  0 ... ...  ...    ...  n0     Ckn 

(2)

where the eigenvectors are given by: n

 k  Cks s   Ckj j ,

(3)

j 1

and Cks and Ckj comply with:

Ckj Cks



E

Vks

0 

k

 Ej



i  k   j  2

,

(4)

where the k subscript will be described below. The values of Ek and k are determined by diagonalizing the effective Hamiltonian, where the eigenvalues k are obtained by solving the secular equation:

 s0    V1s ... Vns

... Vsn ... 0 0 ... ... ...  n0   

Vs1 1   ... 0 0 

(5)

which may be rewritten as follows: n

s    0 

j 1

VsjV js

 j0   

0 

 Es

n

i  E   s      Vsj 2 j 1

E    E   2i  0

2

j

E    E  0

j

2



s

 

1  s   2 4

0

(6)

Here  =E – i/2 should be found by solving Eq. (5), which may be separated into two coupled polynomial equations, each of the degree n + 1, one obtained as the real part of Eq. (5) and another as its imaginary part: n

Es  E   Vsj 0 

2

 s      Vsj j 1

0

j

0

j 1 n

E    E  E    E   14  2

j

2

 

0

2

s

 s    E j0   E 2  1  s   2

(7)

0

4

Solving this system of equations, we obtain n + 1 values of {Ek} and {k}, and thus the eigenvalues of the effective Hamiltonian (1):

i 2

 k  Ek  k

(8)

Taking into account the relationships (2) and (8), and the normalization of the eigenvectors (3), we obtain the coefficients Cks and Ckj. To simplify the solving of the system (7), we may reduce it to a polynomial equation: n

Es0   E   j 1

2

Vsj

E    E   0

(9)

0

j

Note that the coefficients comply with: Ckj Cks



Vks , Ek  E j0 



(10)



and the s parameter values are calculated from: n

k  Cks  s   Ckj  j 2

2

(11)

j 1

On the other hand, Eqs. (9-11) may be used to analyze the QI properties. To describe time evolution of the originally prepared coherent wavepackage of the quasistationary |k> states, where we present a coherent state as follows: n 1

 0    ak  k

(12)

k 1

Its time evolution is then given by: n 1

 t    ak  k e

Ek k t t  2

i

(13)

k 1

The probability to find the system in its initial state in an arbitrary instant of time t is given by:

Pinit t   Ainit t    0  |  t  2

n 1

  ak e k t  4

k 1

n 1



2

j ,k  j

2

a j ak e



2



n 1

a k 1

 j  k 2

t

2 k

e

i

Ek k 2 t t  2

 E  Ek Cos 2 j  

(14)

 t  

In the same way, we may determine the probability to find the system in any of the basis states |s> or {|j>}. This probability is given by:

Pst t   As t   s |  t  2

n 1

  ak Cks e k t  k 1

2

n 1



j ,k  j

2



n 1

a C k 1

k

a j C js ak Cks e

ks



e

i

Ek k 2 t t  2

 j  k 2

t

 E  Ek Cos 2 j  

 t  

,

(15)

and:

Psj t   A j t   j |  t  2

n 1

  ak Ckj e k t 

n 1

2

k 1



m,k  m

2



n 1

a C k 1

k

kj

am Cmj ak Ckj e

e

i

E k k 2 t t 2 

   m kt 2

 E  Ek  Cos 2 m t   

.

(16)

Eqs. (14-16) describe the system time evolution in general. Coming back to the case of  >> 1, let us consider an approximation k = 0. In this case, the respective probability may be represented as follows:

 n1 P0 t   e k t  Rk   k 1 E  Ek  jk  j 

n 1

S

j ,k  j

kj

 Cos 2 jk t  .

(17)

If  >> 1, then jk >> . In this case, the QI effects are important, and, for example, Eqs. (14) and (15) may get transformed into a biexponential function [64-66]:

P0 t   A f e Af 





 0  k deph t

 As e 0t

n 1

S

j ,k  j

k deph 

jj

2  2 V j h

(18)

n 1

As   Rk k 1

where is the average value of Vsj. In this case, QI describes the dephasing dynamics of the originally prepared wavepackage. This dephasing is described by the fast component with the decay rate 0 + kdeph  kdeph. The slow component with the 0 decay rate describes independent evolution of each of the |k> states. The phasing time of the dephased wavepackage is proportional to I [67]. The QI effects become unimportant in an opposite case, when  << 1, or jk << . The time evolution is then described by a single-exponential function:

P0 t   A0 e  0t .

(19)

Let us now consider the case of  << 1. This corresponds to random resonances that may occur between the two states, whereby the system dynamics is determined by the interaction of only two zero-order states. An approximate secular equation may be written as follows:

E    E E    E   V

2

0

0

s

i

si

0,

(20)

and it has two solutions: E1, 2 

1  0  Es  Ei0    2

E    E    0

s

0

i

2

2  4 Vsi  

(21)

The eigenvectors of these eigenvalues are calculated from the equations:

 Es0   Ek   V is 

Vsi  Ck1   0 Ei  Ek  Ck 2  0 

(22)

Thus, the eigenvectors may be written as follows:

k 

Vsj



2

0 

Vsj  Es  Ek



2

0     s  E s  Ek j    Vsj  

(23)

The initially prepared coherent state may be defined as a superposition of the Eq. (12) states:

 0  a1  1  a2  2

(24)

Provided the |s> state is the only optically active state, the a1 and a2 coefficients may be presented as follows: a1  a2 

Vsj



2

Vsj  Es0   Ek Es0   Ek 2



Vsj  Es0   Ek



2

(25)



2

Therefore, in this case we obtain:

 0   s Thus, the probability to find the system in its initial state may be represented as follows:

(26)

Ps t  



V

2 sj

k 

V

Vsj 2 sj



4

 Es  E1



 Es0   E1 Vsj

e 1t 

 V

2

2

sj



sj



 Es0   E2

2

Vsj  Es0   Ek

V

2



4

0 

 E s  E2



2 2

e 2t

4

2 Vsj

2



2 2

0 

Vsj



2

s 

1  2 2 2 1 2  Es0   E j0   4 Vsj 

 2

e 12t Cos 212t 

E    E   E    E  2

0

s

Vsj

k

2

2

0

s

j

(27)

k

12 





2

Eq. (27) describes QI effects, if 12 << 12. In an opposite case, the time evolution of the initially prepared coherent state is described by a three-exponential function. If Es(0) = Ej(0), Eq. (27) simplifies into: 1  Ps t   e 2

 s  j 2

t

  Vsj 1  Cos 2   

 s  j  t  Vsj  t   e 2 Cos 2    

 t  

(28)

The QI effects were extensively studied in atoms [68] and emission of simple molecules [66]. b) The GF method The GF approach is quite similar to the EH approach. It defines the system effective Hamiltonian in the form: H = H0 – (i/2) + V, where the eigenvalues and the eigenvectors of H0 are determined from the equation: H0|s> = Es(0)|s> or H0|j> = Ei(0)|i>, and the real part of the energies is obtained in  = 3πE/h units. Note that we are considering the general case, thus the matrix elements of the perturbation V are non-zero between all of the system states. Assuming the system was originally prepared in the |s> state coupled by the

perturbation V with the {|i>} states, the time evolution of the |s> state may be described as follows [69]:

s t   e

i

Hˆ t 

s

(29)

The probability amplitude to find the system in its original state is obtained as follows:

As t   s | st   s e

i

Hˆ t 

s

(30)

Applying the Laplace transform, the latter relationship may be represented as follows [67]:





i t 1  As t   e s Hˆ   2i 



G    Hˆ  



1





i t 1  s d  e s G   s d 2i 

(31)

1

where the integration is carried out over complex energy around the poles of the matrix element. Taking into account that G(ε) is defined as the matrix inverse to H – :

 Gss   G1s  ...  G  ns

Gs1 ... G11 ... ... ... Gn1 ...

 0   s Es  i    Gsn  2   G1n   V1s   ... ...   Gnn   Vns  

E10   i

1

...

2

Vn1

    ... V1n   ... ...   ... En0   i n    2  ...

Vs1 

1

Vsn

(32)

the matrix element of interest is given by:

Gss   

M ss   det Hˆ  





(33)

where Mss is the minor of the matrix element. Thus, the final relationship for the probability amplitude of interest may be presented as follows:

As t  

1 M ss   e  it d  2i  det Hˆ  





(34)

Here the integration interval is – <  < +, integrating around each of the poles along a circle with r0  0. The analysis of this relationship requires solving the secular equation:

Es0   i

s 2



Vs1

1

V1s

E10   i

...

...

Vns

2



Vn1

...

Vsn

...

V1n

...

0

(35)

... 0 

... En  i

n 2



and next, the residue theorem is used to calculate the integral (34). Since Eq. (35) is a polynomial equation of the degree n + 1, it may be presented in the form: n

      0

k  s ,1

(36)

k

The solution in its final form may therefore be presented as follows: i t i k t M ss  k  1   As t   e d  e M ss  k Rk  k    n 2i  k     k 





k  s ,1

1

Rk  k  

n

 

k

 m 

(37)

m  s ,1,m  k

where Rk(k) is the residue at the k-th pole. The GF method may be extended to the analysis of the transition probability amplitude between the states |m> and |j>:

Amj t    e

i

k 

t

M mj  k Rk  k  ,

(38)

k

and the respective transition probability is given by |Amj(t)|2. We have to note that both the EH and GF methods require solving a polynomial equation of the degree n + 1. Example For the system of two coupled states, |s> and |1>, and taking into account Eq. (38), we obtain the probability amplitude to find the system in the |s> state as follows:

Ass t  

i 

    i s t i 1 t i 1 t  Vs1 V Vs1  i s t   e  e   1s e   e   s  1 1   s  s   1  

 s0    10 



1 2

        0

0

s

1

2

 4 Vs1  i 2

2   2Vs1 i 0 t  2i t  2v  Ass t   e Cos t   s  1   

0  v

i    0  v  i i 2 2

 s0    10 

(39)

2

       

1 2

0

0

s

1

2

 4 Vs1

2

Therefore, the respective probability may be represented as follows:

Pss t   Ass t 

2

2

V  2v   4 s1 e it Cos 2  t   s  1   

(40)

The latter relationship is the same as that obtained in Eq. (28) using the EH method. Let us now consider the density matrix method, capable of describing the state dynamics exactly. c)

The DM method

The DM method follows from the Heisenberg representation of the time evolution unitary operator [70]: Sˆ t   e

i

Hˆ t 

(41)

where the Hamiltonian is given by:

 E s0   V Hˆ   1s ...  V  ns

Vs1 ... Vsn   E10  ... V1n  ... ... ...   Vn1 ... E n0  

(42)

The eigenvectors of this Hamiltonian are defined by the linear combination of the orthonormal basis states. This set of eigenvectors is given by Eq. (3). Let us now introduce the density matrix operator , with its time evolution described by: i ˆ

i ˆ

Ht  Ht ˆ t   Sˆ 1 t ˆSˆ t   e  ˆe 

The time derivative of (t) is given by:

(43)

dˆ t  i ˆ  Hˆ t   Hˆ t i  Hˆ t   Hˆ t ˆ i ˆ H  Hˆ t   ˆ t Hˆ  He ˆe  e ˆe dt    i

i

i



i



(44)

The latter equation may be transformed into the matrix form as follows: d l ˆ t  j dt







i l Hˆ ˆ t  j  l ˆ t Hˆ j 



i l Hˆ m m ˆ t  j  l ˆ t Hˆ m m Hˆ j  dlj i   H lm  mj  lm H mj dt  m







(45)



On the other hand, our problem is reduced to the analysis of a system of (n + 1)2 first-order differential equations. This system of equations has to be complemented by the diagonal terms describing excitation of the |s> and {|i>} states, and by the matrix describing the relaxation rates of the density matrix elements:

0 W  ss  0 W11  ... ...   0 0 

... 0    ss  ss   ... 0    1s 1s  ... ...   ...   ... Wnn    ns  ns

 s1 s1 ...  sn  sn    1111 ...  1n 1n  ...

 n1 n1

... ...   ...  nn  nn 

(46)

Where:

 lj 

 ll   jj

(47)

2

Thus, the equation system (45) may be rewritten as follows:

dll i  Wll   ll ll   H lm  ml  lm H ml  dt  m dlj i   lj lj   H lm  mj  lm H mj dt  m



Example



(48)

Let us consider the time evolution of a system of two coupled states |s> and |1>, with the same zero-order energy value E0. Initially we assume that ss = 1 and 11 = 0 (Wss = (t); W11 = 0). The resulting equation system may be presented as follows:

d ss   ss  ss  ia1s  ia s1 dt d s1  ia ss   s1 s1  ia11 dt d1s  ia ss   1s 1s  ia11 dt d11   1111  ia s1  ia1s dt V V a  s1  1s  

(49)

To solve this equation system, we have to find the roots of the secular equation:

  ss    ia ia 0

 ia   s1   0 ia

0

ia 0   1s    ia

ia   ss    s1    1s    11     ia   11  

 2 ss    s1   a 2  2 1s    11   a 2   ss    s1    1s    11   





(50)

 2 s1   a 2  ss   11  2    1s     ss    11     4a 2  0 2

The latter equation has four roots, which may be presented as follows:

1  2   s1 3, 4   s1 

1 2

 ss   11 2  16a 2

  s1  i 2a

(51)

Taking into account the initial conditions, the solution for ss(t) is given by:

 ss t  

e   s 1t 1  Cos2at   e s1t Cos 2 at  2

(52)

This is similar to the result obtained above using the EH method, cf. Eq. (28). The difference is significant only if  ss   11   16a 2 . However, the QC effects are only observable if  s1  a , 2

i.e. EH or GF methods are implemented much easier and more conveniently for the description of QC than the DM method. We shall next discuss different biological systems, where QC effects may be observable. III.QC effects in biological systems In this section, we will discuss QC and QM concepts applied to the analysis of (i) photosynthesis, (ii) energy transfer along polypeptide chain and (iii) mechanism of inter-cellular communication in the brain. (i) Photosynthesis The structure of LH2 antenna in Rps. acidophila was reported earlier [71,72]. The photosynthesis involves several different pigments (B850, B875, and P) of the reaction center, except B800. If B800 of the peripheral LH2 antenna is excited, the respective excitation is transferred to the reaction center, with B800 – B850 energy transfer operating at the first step [72-74]. Since the B800-B850 distance is about 18 Å, the energy transfer was attributed to the electric dipole-dipole mechanism, described by the Förster relationship (Volkhard, 2008). The respective energy transfer probability should thus be directly proportional to the spectral overlap between the B800 fluorescence and B850 absorption spectrum [75,76]. The B800 – B850 energy transfer dynamics was investigated by measuring the B800 excited state lifetime at 77 K [72,73,77]. The measured energy transfer times [72,73,77,78] were compared with those calculated using the Förster relationship [79-82], with an acceptable agreement of theoretical predictions with the experimental results. The measured energy transfer times were 2.4 ps at 4 K [83,84], 1.8 ps at 77 K, and 0.7 ps at room temperature [85,86]. The dynamics of the B800 emission was described by a single-exponential function, with no QC observable. The same results were obtained for B800 intramolecular energy transfer, when this complex was excited by 80 fs pulses at 77 K with the emission detected at 790, 800 and 810 nm wavelengths. The measured biexponential emission decay times were 0.35  0.05 and 1.8  0.2 ps [74,87]. The authors interpreted 0.35 ps as the dephasing time; presently, using the phonon density at the excitation energy level of 107/cm–1, we estimated V = 534 cm–1 for the interaction between the states. The fast component of the fluorescence decay was assigned to the dephasing of the originally prepared coherent state, because the coherent width of the exciting laser pulse estimated using the pulse duration ( 1250 cm-1) was significantly larger than the estimated

interaction coupling the system states. On the other hand, the laser pulse excites all of the system states that fall within its spectral width, thereby forming the initial wave package. The measured intramolecular energy transfer in B800 was much faster than the energy transfer between the B800 and B850 complexes. Therefore, the emission decay curves recorded at higher time resolution were biexponential, describing the QC dephasing of the initially prepared coherent state, with subsequent independent emission from the dephased system states, with the energy transferred to the B850 complex. The rate of dephasing, as we already noted in section II, is described by the golden Fermi rule, Eq. (18). The perturbation V is diabatic in the system considered [88]. Note that QC effects in the form of quantum beats were reported for the FennaMatthews-Olson photosynthetic complex extracted from the green sulfur bacterium Chlorobium tepidum [74]. The two-dimensional spectra of this complex show clear beats in the cross-peak between the two excitons in function of the waiting time. According to the authors, the agreement in phase and frequency indicated that those beats were not an experimental artifact [82]. The measured beating period was ca. 100 fs, while the characteristic decay time was 400 fs. Thus, the interaction coupling the two system states was about 333 cm-1, while 12 = 83.3 cm–1. Note that the quantum beat effects reported earlier [82] may be described with good accuracy by Eqs. (28), (40) or (52). In other words, the interacting states are in the interaction resonance, as the energy gap between the zero-order system states is much smaller than the interaction coupling these states. (ii) Energy propagation along the PP chain and protein filaments This issue has been extensively studied earlier [33-43,89-96] with the help of the Davydov soliton theory. The propagation of Davydov soliton along the PP chain may be described by the following Hamiltonian [43]:

 p 2 1 H D    0 Bn Bn  J Bn Bn  Bn Bn    n  w u n  u n1    2M 2   1 u n1  u n1 Bn Bn  H qp  H ph  H int



 













(53)

where B+n(Bn) is the exciton creation (annihilation) operator for the vibrational excitation of the C=O bond (Amide I oscillation) at the site n, un is the displacement operator of the aminoacid residue at the site n, Pn is its conjugate momentum operator, M is the reduced mass of the C=O vibration, w is the elastic constant of the protein molecular chain, χ1 is the nonlinear coupling

parameter, representing the strength of the exciton-phonon interaction, ε0 is the energy of the Amide I exciton (C=O vibration), and J is the dipole–dipole interaction energy between the neighboring aminoacid residues. Taking into account the Hamiltonian of Eq. (53) and using the DM method, a set of N2 (N = n + 1) first-order differential equations may be obtained and used for analyzing the excitation propagation along the PP chain. Here, N is the number of aminoacid residues in the chain. The detailed analysis of this problem is quite complex, therefore we shall use the simpler EH approach instead. Using Davydov’s ideas, we present the Schrödinger equation in the EH formalism as follows:

    J  ...   0 

... 0  Cks     ... 0  Ck1  0 ... ... ...  ...    0 0 ...    Ckn  J

0 J

(54)

Using our previous deductions, we obtain the probability amplitude to find the system in the |i> state, while it was initially prepared in the |s> state, in the form:

Asi t    ak Cki e

i

 k 0  

t

k t 2

(55)

k

and therefore, the respective probability is given by:

Psi t   Asi t    ak 2 Cki e

2  t k

2

k





k ,m k

ak Cki amCmi c



k  m t 2

  k0    m0   Cos t    

(56)

Taking into account Eq. (54), we may determine the Cki coefficients by using the Cks values:

Ck 1 

k

Cks J   k  2  Ck 2     1Cks  J  

(57)

... For simplicity, we shall assume that ak = (n + 1)–1/2. Let us now consider an example of three sequentially coupled states. The respective secular equation and its solutions may be presented as follows:

 J 0

J

0 J     3  2J 2  0 

 J

(58)

10   0;  20,3   2 J Therefore, the eigenvectors of the respective Hamiltonian may be presented as follows:  1  s  2 e 2

1 

 s  2 4

t





J





J

2 

i t  1 s  21  2 e  2

3 

i t  1 s  21  2 e  2

 s  2 1   2 8

 s  2 1   2 8

t

(59)

t

Thus e.g. the probability to find out system in the state |2> at an arbitrary instant in time may be written as:

1  Ps 2 t   e 4

 s   2  2

t

1   e 8

 s  2 1   2 4

t

1   e 4

 s   1   2  3

t

J  Cos t   

       s 1 2 t  2 J  1    s 2 2  t  J  3 Cos t   e Cos t  e       4    2    2 J  1  s 41 2 t   e t  1  Cos 8    

1   e 8

 s  2 1   2 4

t

(60)

The physical nature of the s and i values will be discussed separately. Here, let us for simplicity assume that s = i = 0, and thus the latter relationship may be reduced to:

Ps 2 t  

e  0 t 4

  2 J    J  1  t   1  Cos t   1  Cos  2         

(61)

Fig. 2 shows the predicted values for Ps2(t) at different values of the parameters. Table 1. The plots of Fig. 3 were calculated for the fixed value of the parameter 2J/h = 3.0 cm-1, and different values of 0, listed below.

#

0, cm-1

#

0, cm-1

#

0, cm-1

1

0.3

4

0.6

7

0.9

2

0.4

5

0.7

8

1.0

3

0.5

6

0.8

9

1.5

The results shown in Fig. 2 demonstrate that the role of QC decreases with the 2J/(h0) ratio. Numerical calculations for a PP chain containing 1000 aminoacid residues produced the probability to find the system with the last 999-th residue excited, relative to the probability of having the |s> fragment excited, and the results are plotted in Fig. 3. The results shown in Fig. 3 may be described by a biexponentual function:



Ps ,999  t   A e  0t  e

 kdeph t



(62)

where

k deph 

2 2 J  

(63)

Calculating kdeph directly, and taking into account that J = 3.0 cm-1 and  = 1000/(max(0) - min(0)) (cm-1)-1, produced the result kdeph = 2.8 cm-1. Thus, the calculated dephasing time of the originally prepared wavepackage is in acceptable agreement with the buildup time of the respective probability, with the maximum value of the probability of ca. 8.410–4. Fig. 3 shows a dramatic reduction in the probability for 0 = 1.5 cm-1. Taking into account the previously obtained values of J [97] and the efficiency of the C=O vibrational relaxation in liquid water, we conclude that in the frameworks of Davydov soliton model, the QI effects in a realistic model of a PP remain unexplained, with the peak probability for our system appearing with the C=O vibrational excitation on the last aminoacid residue being extremely low, because the respective probability is only different from zero in the very high n-th order (n = 1000) of the perturbation theory. We therefore must conclude that Davydov soliton theory is not applicable to any real protein molecules or protein complexes. An alternative mechanism for the excitation energy propagation along a PP chain

This approach is based on the calculated zone structure of an -helix PP. Khmelinskii4 & Makarov [51] reported two conductive zones in the near-IR (0.63 – 1.09 eV) and visible (0.87 – 3.92 eV) spectral ranges for an -helix PP. Let us consider ATP hydrolysis energy transfer within the first conductive zone of a PP, initially deposited into a geometrically localized segment of the characteristic length  of the PP chain. Presently we leave aside the mechanism of the energy transfer from the ATP + H2O  ADP + H3PO4 reaction to the electronically excited state of the PP. To simplify calculations, the PP chain was approximated by a cylinder with the diameter d and length L, where the potential for the exciton motion in the first conductive zone along the z axis was approximated by that of a rectangular potential box with infinite walls, with both d and  much smaller than the rectangular box width L [53]. It was shown [52] that the probability of exciton propagation may be represented as follows:

 r ,  , z, t 

2

2



 A  t    E r ,     e e  2   2

 2k 2 2

 2hk    Cos 2k  z  t  dk m   

(64)

The latter relationship describes the exciton propagation along the PP with the average velocity: 

 2h  k e  m 

 2k 2 2

dk

(65)

and the wavepackage dephasing is described by the characteristic time:



2m 2  0.273 2 , s 

(66)

while the dynamics of the energy relaxation is described by the  coefficient. The exciton propagation time may be expressed as  

L



, being dependent on the effective electron mass m.

Considering the ATP hydrolysis energy transfer, and assuming that  ≈ 10 Ǻ (typical size of an aminoacid residue that accepts the ATP hydrolysis energy), we obtain  = 2.7310–14 s. We shall present a detailed consideration of this mechanism in the following section, where the theory of

light energy propagation along Müller cell intermediate filaments is supported by experimental data. Light energy propagation along MC IFs Recently we reported that MC IFs isolated from porcine retina are good conductors of electric current, and of light energy [50-53]. The diameter and length of these nanofibers was ca. 12 nm and 117 m, respectively. It was found [52] that the current – voltage characteristics of these biological fibers correspond to those of a semiconductor with the band gap of 0.63 eV between the valence and the first conductive zone. As we already mentioned, the second conductive zone is located in the 0.87 – 3.92 eV energy range covering the near-IR, visible and near-UV spectral ranges [51]. It was found that PP chains are strongly anisotropic in their light absorption cross-section, with the difference between the longitudinal and perpendicular transitions of ca. 3 orders of magnitude. Thus, assuming a bundle of aligned IFs, the excitons would be created very efficiently at the IF extremity interacting with light, already within a small fraction of the total length of the IFs. The spectrum of light transmission efficiency through aligned MC IFs was also measured earlier [95]. The experimental spectrum was deconvoluted into four subbands with the respective maxima at (1) 453, (2) 493, (3) 546 and (4) 613 nm. Apparently, each of the deconvoluted subbands may be assigned to a different type of MC IFs, as different MC IFs could be present in cone cells sensitive to different spectral ranges of incident light. Recently we also reported [98] that exiton propagation along the MC IFs may be described by biexponential emission decay curves. Thus, these experimental decays were fitted with a biexponential function: t    t 1  2  I t   A0 e  e    

(67)

where A0, 1 and 2 are the respective fitting parameters, listed in Table 2.

Table 2. Values of the A0, 1, 2 and meff/me fitting parameters obtained for the different excitation wavelengths. Here, meff and me are the effective, and the free electron masses, respectively. A0, a.u.

1, ns

2, ns

meff/me

453

0.370.01

41.90.3

261.33.3

0.671

493

0.510.02

52.90.7

239.23.5

0.737

546

0.740.02

46.90.6

253.13.7

0.712

613

0.440.01

39.90.2

272.34.6

0.693

Excitation wavelength, , nm

Using the relation  1 

L



, the effective electron mass for the longitudinal electron motion was

estimated in function of the excitation wavelength [98]. The estimated mass values are also listed in Table 2. The reported results [98] thus provide direct support to the earlier proposed theory [49], stating that high contrast vision of the vertebrate eyes may be attributed to the light energy transmission in the form of excitons. These excitons propagate along the MC IFs from the inner limiting membrane level that receives the image focused by the eye lens, to the outer limiting membrane level, where the photoreceptor cells are located. Each MC is physically connected by its respective IFs to a certain photoreceptor cone cell in the retina [99]. Hypothetic mechanism of communication between neurons Several publications have discussed this issue [62,100-102]. Several mechanisms have been proposed that allegedly could explain the communication between neurons. Of these, we shall only mention the one based on electric signal propagation along MTs existing in neurons [103]. In particular, Hameroff & Penrose [103] proposed that a coherent ionic wavepackage was induced by a jump of the electric potential difference applied to the extremities of the MTs. They assumed that communication between brain cells would thus develop according to laws of quantum physics, and the brain would be working as a quantum computer. The coherent state dynamics of the two subsystems coupled by a MT may be described in the frameworks of the

density matrix method, Eq. (32). However, later it was found [104] that coherence is not maintained in such systems, the main reason being the short relaxation time of the ion linear momentum compared with the time the ion takes to move along the MT. In fact, the 1D ionic transport along an MT should be described as electro-diffusion, where the flux and the master equations for a selected ionic species X may be presented as follows [105]:

J X t   p XMT



z X q t  k BT

z X q t  C X e  C X1 z q t   X k BT 1  e kBT 2 



z X q t  k BT

(68)

d X  S z q t  C X2 e  C X1  p XMT MT X z q t   X dt V2 k BT 1  e kBT

where pXMT is the permeability for the ionic species X transport along the MT, SMT is the effective cross-section of MTs coupling the two communicating subsystems,, V2 is the effective volume, where the ion X is located, zX is the relative ionic charge, q is the absolute value of the electronic charge,  t  is the time-dependent potential difference applied to the MTs, kB is the Boltzmann constant, CX(1) and CX(2) are the concentrations of the ion X in the first and the second subsystems. The processes described by Eq. (68) occur on the millisecond time scale (characteristic time of the (t) variation), being far too slow for the quantum coherence to survive. We believe that the most promising approach to discovering quantum coherence effects in the neuron communication is the ATP hydrolysis energy propagation along a PP chain [43], or the ATP energy propagation along a PP chain by the electronic excitation (exciton) transfer mechanism as proposed by Makarov et al. [49]. As we already noted, the vibrational Davydov soliton is unstable due to strong vibrational relaxation of the vibrationally excited C=O group of the aminoacid residues forming the PP chain to the bending vibrational mode of water molecules hydrogen-bonded to the same PP chain. An alternative mechanism proposed recently [52] is based on theoretical analysis of the zone structure of -helix PPs [52] and the experimentally investigated properties of the MC IFs [51], including longitudinal exciton propagation along these IFs. It was also found [49] that such IFs have semiconductor properties with the band gap of about 0.63 eV between the first conductive and the valence bands. Therefore, we presume that

fast communications between brain cells may be occurring by ATP coherent energy exchange due to the interaction between IFs of different cells. In this case, decoherence of the originally prepared wavepackage in the first conductive zone is created by the wavepackage dephasing without loss of the excitation energy and by the radiationless relaxation of the IF exciton (see above). The radiationless rate constant  in Eq. (64) was evaluated both experimentally and theoretically for the second conductive zone of the MC IFs [50,51], where it was shown that:





 1  e L L 

 2k 2 2



4hk dk   m

(69)

Here, L is the length of the studied system,  is the length of the zone where the nanofiber is excitated [50], k is the wave-vector, m is the effective electron mass and  is the rate constant of the exciton relaxation. The average exciton propagation velocity is defined by the integral in Eq. (69) [50-52]. Presently we calculated  numerically, following the procedure described below. The calculations were carried out for a poly-glycine PP. Using the earlier reported results [52] on ab initio calculations of the zone structure of PP chains, we calculated  using the relationship [106]:

2    VZ Vˆna  FCZ 

2

  EFCZ , ph  EVZ , ph dEVZ , ph  0

2  VZ Vˆna  FCZ h

2

VZ E FCZ 

(70)

where VZ, FCZ are the electronic wavefunctions in the valence and the first conductive zone, Vna is the diabatic perturbation operator, EFCZ,ph and EVZ,ph are selected phonon energies in the first conductive and valence zones, respectively, and VZ(EFCZ,ph) is the phonon density in the valence zone at the energy level of the selected phonon in the first conductive zone. This latter parameter is given by [106]:

VZ EFCZ , ph

15    EFCZ , ph  aN  i  i 1    s

s  1!  k s

s 1

(71)

k 1

where s is number of vibrational degrees of freedom, and I is the fundamental frequency of these vibrational modes. In our estimates we used the vibrational frequencies of the amino-acid

residues. Using glycine, s = 15N, where N is the number of glycine residues in the PP chain. The fundamental frequencies for the modes of interest are shown in Tab. 3. Table 2. Fundamental vibrational frequencies of glycine residues (-NH-CH2-CO-) in a PP chain. Number

Mode

Frequency, cm-1

1

N-H stretch

3319

2

N-H stretch

3311

3

N-H stretch

3224

4

C-H stretch

2927

5

C-H stretch

2926

6

C=O stretch

1666

7

N-H bend

1616

8

N-H bend

1614

9

N-H bend

1493

10

CCN bend

1446

11

CCO bend

1367

12

NCH bend

1284

13

OCH bend

1277

14

NH torsion

1046

15

NH torsion

1025

We excluded the low-frequency modes, which apparently can’t promote the radiationless relaxation discussed. The matrix element of the diabatic perturbation was calculated taking into account the wavefunctions calculated ab initio earlier [53] in the Herzberg-Teller approximation [88], where the perturbation was represented as follows:

 15  Hˆ q, Q  Vˆna    Qi  Qi Q 0  i 1  i  

(72)

where q are the electronic coordinates and Q are the deviations of the nuclear coordinates from their equilibrium positions, for the modes listed in Tab. 3. The calculated value kna = 1.78197 s-1 was obtained for the poly-glycine PP chain. Taking into account the size of the glycine residue (0.7 nm), and assuming that the ATP hydrolysis energy was localized on one of residues at zero time, using Eq. (67), and the relationship k prop 

 L

, we obtained the value of the rate constant

kprop = 3.411011 s-1 m-1 describing the exciton propagation along the PP chain. We therefore conclude that for a typical PP length, kprop >> , or the energy loss by radiationless relaxation of excitons may be neglected, and the ATP hydrolysis energy will be transmitted along the natural PP nanowire without loss. In other words, nature may use this mechanism for fast and efficient short-range communication between neurons in the brain. This transmitted energy should be somehow used at the receiving cell, as we shall discuss in a follow-up publication. IV. Discussion In this section, we shall discuss the methods and approaches considered above. A brief discussion related to quantum biology was already presented, however, no detailed discussion was provided. Here, we will compare the theoretical methods used for the analysis of coherent processes in biological systems, and explore the coherent wave stability problem. a) Comparison of different theoretical methods We presented three different approaches for the analysis of coherent state dynamics: namely EH, GF and DM methods. The EH method has been extensively used to analyze the transition dynamics between coupled quantum states of a physical object [56, 57, 59 64], where the EH was defined as: Heff = H0 + V – i/2. Here, H is the zero-order system Hamiltonian, V is the perturbation coupling the zero-order quantum states and  is the operator describing the relaxation properties of the zero-order states. The latter operator corresponds to the imaginary part of the energy, defining the quantum state width/lifetime, and also transforms the effective Hamiltonian into the non-Hermitian form. Thus,

the eigenvalues of such Hamiltonian are expressed by complex numbers {n(0) – in/2}. As we already noted, the zero-order set of states {n(0)} may be presented with good accuracy as the set of eigenvalues of the Hermitian Hamiltonian H0 + V, provided <|V|> >> <||>, where <|V|> and <||> are the absolute values of the average perturbation coupling the zero-order quantum states and the difference of the state widths, respectively. These results follow from the analysis of a simple model with two coupled resonance states, where the secular equation and its solution in the exact form may be represented as follows:

E0   2   s 1   4 2   s   1  0

2

4

2

 Vs1 

  s   1   0 4

(73)

and: 

 s  1 2

1, 2  E0 

1 4

 s   1 2  16 Vs1 2

 E0  Vs1

(74)

 s   1 2  16 Vs1 2 respectively. The relationship <|V|> >> <||> is typically satisfied in the examples discussed here. Thus, the behavior on the system with N states may be typically analyzed using the EH method, and an N-th degree polynomial equation has to be solved. The GF method is an extension of the EH method into the complex energy space. The problem may be solved by finding the GF poles, or else, once more an N-th degree polynomial equation has to be solved. This method produces time evolution generated by the perturbation V in the zero-order states [69, 107-109]. Both the EH and GF methods define the operator  phenomenologically, therefore the detailed mechanisms defining these parameters have to be considered especially, as we did above for the radiationless relaxation of excitons. The state relaxation dynamics depends on the coherent state stability. Therefore, we will discuss the relaxation mechanisms, first comparing the DM method with EH and GF methods. The DM method uses the Hermitian Hamiltonian in the form H = H0 + V, the Heisenberg evolution operator Sˆ t   e



Hˆ t 

and the density matrix operator ˆ . The diagonal elements of the

latter give the probability to find the system in the respective i-th zero-order state. Using the DM method in the analysis of N coupled zero-order states, we need to solve a system of N2 first-order linear differential equations. Thus, the DM approach is much more complex when we analyze dynamics of a multi-state system. Note, however, that the DM analysis of QC in biological systems considered only two subsystems, coupled by one interaction [77,101,110-111], corresponding to the simple example of two states that we discussed. Thus, the DM method produces the same results for the analysis of two coupled states (see above) as were obtained using EH. Thus, we finally conclude that EH is the most suitable method for the analysis of QC in biological systems. b) The origins of the state width parameter  We already introduced the energy relaxation mechanism described by the golden Fermi rule, see Eq. (70). We shall now consider the energy relaxation due to an external time-dependent perturbation. Such perturbations may be generated by random thermal motion of molecules, with the interaction potential depending on time due to translational and rotational diffusion. In the first limiting case, the lifetime of the complex between the coherently prepared excited species and the surrounding molecules is shorter than the vibration period of the interacting species with respect to each other. We shall also consider the second limiting case, with the complex lifetime much longer than the vibration period. Case 1: Here the relaxation rate constant of the initially prepared state may be represented as follows [113]:

kng 

n Vˆ0 g 

Vˆ t   V0e ng 

 t t   0    

e

ng

2

(75)

En  E g 

where |g> are the other states of the system and V(t) is the time-dependent interaction. Presently we are not considering the relaxation in detail, as it has been discussed earlier [102, 114]. We shall introduce a simple case when the time-dependent perturbation may be described by a

Gaussian function. As follows from Eq. (77), this relaxation mechanism is only important when

ng  1 . Using typical frequencies of 1012 Hz that describe the motion of small molecules (water) at room temperature, we deduce the respective ng  33 cm1 , therefore, this mechanism works only for the coherent states created in the orbital/spin angular momentum degrees of freedom. Thus, it does not apply to translational degrees of freedom, as the notion of coherence does not apply to such states [101]. Case 2: Here the complex lifetime is much longer than the vibration period along the interaction coordinate. Thus, the vibrational excitation energy may get redistributed between different internal degrees of freedom, and the electronic excitation may decay by radiationless transitions within the complex. In either case, the excitation relaxation rate is determined by the interaction energy in the complex. Neutral species may interact by (i) electric dipole-dipole, (ii) dipoleinduced dipole, (iii) quadrupole-dipole, (iv) dispersion, and (v) hydrogen-bond mechanisms. The hydrogen-bond interactions with water molecules are the most important in the biological systems considered. The one-dimensional potential energy surface along the coordinate with the maximum interaction energy may be described by the Morse potential [115]:



U r   Ddis 1  e  r  r0 



2

(76)

where Ddis is the complex dissociation energy,  is the parameter determining the interaction distance and r0 is the equilibrium distance between the interacting species. Thus, the, complex lifetime equals the inverse of the dissociation rate constant, and the latter may be described by:

kdis   vib e



Ddis k BT

(77)

where

 vib  

2 Ddis mH 2 O

(78)

Using the typical value of Ddis ≈ 0.26 eV [13, 116] and  ≈ 3108 cm–1 [115], we estimated the respective frequency of 1.121012 Hz. Therefore, the complex lifetime at room temperature is ca. 3.2110–8 s. On the other hand, the vibrational period is much shorter than the complex lifetime.

Thus, we conclude that the second case may explain the relaxation dynamics of the excited states in a wide range of energies. Therefore, this should be the most probable mechanism for the loss of state coherence in biological systems. V. Conclusions Presently we review the analysis of several aspects addressed of quantum coherence and decoherence dynamics in several biological systems, including the photosynthetic complex, ATP hydrolysis energy transfer along a PP backbone, and light propagation through the inverted retina of vertebrate eye. We analyzed quantum coherence dynamics using different theoretical tools, including the effective Hamiltonian, Green’s function and density matrix methods. The first two approaches are based on the same general ideas, while the DM method is significantly different from the other two. We conclude that EH and GF tools are simpler in practical usage, although DM generates exact solutions. Namely, analyzing quantum coherence dynamics in a system with n quantum states, EH and GF generate a secular equation of the n-th order. For comparison, DM requires a system of n2 linear differential equations of the first order to be solved. We concluded that the EH method is the most convenient for biological applications, and use it in the discussion of coherent state dynamics in the already mentioned biological systems. Detailed analysis of the respective coherent processes produced the following conclusions: -

In photosynthesis, coherent dynamics of energy transfer between the B800 energy donor and the B850 energy acceptor was described by the coherence loss of the originally prepared wavepackage in the system of mixed quantum states that has contribution of both B800 and B850 entities. The respective decoherence time is determined by the golden Fermi rule, and observed as the short-living emission component. At the same time, the long-living emission component is determined by the independent emission of the same mixed quantum states. The overall character of the excited state dynamics depends on the excitation energy, sample temperature, and other factors;

-

The presently discussed exciton mechanism for the ATP hydrolysis energy transfer along the PP chain is a more viable alternative to Davydov’s vibrational soliton hypothesis,

providing a significant progress in understanding the workings of enzymes that enable the functioning of every living cell; -

High-contrast vision of vertebrate eyes may be appropriately described as resulting from exciton transport along retinal MC IFs. These excitons should be propagating in the conductive zone of the MC IFs, which have the properties of a semiconductor. We analyzed the exciton propagation using the EH method. Upon detailed consideration of the physical nature of the phenomenological parameters of our model, we estimated the values of these parameters.

-

Exciton propagation in PP chains, IFs (build of protein molecules) or microtubules, with the excitons produced upon ATP hydrolysis, may play a significant role in communications between the adjacent brain cells.

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Figure 2. Time-dependent probability to find the system in the state |2> in the system of three sequentially coupled levels, and |s> being the initially prepared state. The parameter values used in the calculations are all listed in Table 1.

Figure 3. The calculated probability to find the PP of 1000 aminoacid residues with the excitation localized on the last 999-th residue, assuming the excitation initially prepared at the residue |s>, with 2J/h = 3.0 cm-1 and 1) 0 = 0.3 cm-1; (2) 0 = 1.5 cm-1.

Author statements: Conceptualization: I.K., V.M.; Methodology, Validation, Formal analysis,

Investigation, Writing - original draft I.K. and V.M., Supervision V.M.

Graphical abstract

|s>

V

 j   j0   i

s

j

 s   s0   i

2

j 2

j 

First-order interactions between quantum states; the matrix element V describes the strength of the interaction between the initially created state |s> and the set of states {|j>} with the state density ρj. The state energies are characterized by complex values, with the imaginary part describing the respective state width.

Highlights: 5. Theoretical methods for quantum coherence dynamics in biology reviewed; 6. Namely, effective Hamiltonian (EH), Green’s function and density matrix methods; 7. Quantum coherence dynamics in photosynthesis, brain and vision discussed; 8. The EH method recommended for interpreting quantum coherence effects in biology.