The lifetime of molecular (Davydov's) solitons

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The lifetime of molecular (Davydov’s) solitons

A. Davydov

To cite this version:

A. Davydov. The lifetime of molecular (Davydov’s) solitons. Journal de Physique I, EDP Sciences,

1991, 1 (11), pp.1649-1660. �10.1051/jp1:1991232�. �jpa-00246442�

Classification

Physics

Abstracts

87.10 71.50 71.38

Proofs not corrected

by

the author

The lifetime of molecular (Davydov's) sofitons

A. S.

Davydov

Institute for Theoretical

Physics,

Acad. of Science of the Ukr. SSR, Kiev, U-S-S-R-

(Received 4

February

1991,

accepted

in

final form

12

July 1991)

Abstract. The lifetime of

Davydov's

solitons in _a one-dimensional system ^is studied

theoretically.

The _process of thermalization and the

properties

of solitons at finite temperature

are

investigated.

The process of soliton creation and

disintegration

of soliton _are discussed.

1. Introduction.

There is _{a very}

important problem

in the science of

bioenergetics

how to store and

transport biological

_energy, ^which

appeared

at about _some time in the

protein

structure. An

answer to this

problem

_was

suggested

in 1973

by Davydov [1-3]

who

proposed

a model for the

energy transport ⁱⁿ

quasi-one

dimensional

biological

_systems. The basic idea for this model is that

transport

energy is done due to

separated formations,

so-called

Davydov's solitons,

that

freely

travel

through

the

system.

Theoretical

investigations

of the lifetime of

Davydov's

solitons in one-dimensional

systems

prove ^to be the most difficult

problem

in the

theory

of solitons. In the present paper we

discuss _some

questions

which _are to be solved to estimate

correctly

the lifetime of solitons in molecular systems.

2.

Stationary

and

nonstationary

states.

Translational invariance is necessary for

freely moving

solitons. These

stationary

states must also be

eigenstates

of _a translational

operator.

^{This is} true for _a

theory

of

Davydov's

solitons

where

velocity

is a

good

quantum ^number.

I do not agree with _an assertion stated in the _paper

[4] by

Bolterauer that _any

stationary

state must be

only

nonlocalized

eigenstate

of translational operator. ^He considered localized soliton states _as

nonstationary

and tried to estimate the time of their transfer into the states of

nonlocal

plane

_waves. So he obtained for lifetime _{r a very} small value _r m10~ ^~~ _s. These values _are smaller than the lifetime of _one isolated intramolecular vibration

(~10~

^l~

s)

in

condensed medium. In

reality

autholocalized state

(soliton)

is _an exact

eigenstate

of _a translational

operator,

because the localized site in _an infinite chain may be distributed _{on any} part ^{of the chain.}

Both nonlocalized

(exciton)

and localized

(soliton)

states are

stationary

states of the _same

Schr6dinger

nonlinear

equation

with total model Hamiltonian H. Soliton is lower in energy than the

plane

_wave

by

_a finite _{energy gap.} Translations between them _are forbidden.

Exact

eigenstates

of any

Hamiltonian, H,

are

stationary

states.

They

have _a definite _energy and _an infinite lifetime.

Nonstationary

states have _no

exactly

definite energy but have finite

lifetime for

passing

into _a stable final state.

Nonstationary

states _are revealed _as states described

by

the

part, Ho,

of _a total

Hamiltonian,

H. The

remaining

part, V

= H

Ho,

of it will be the _reason for transition to the other states. The division of Hamiltonian H into parts

Ho

and V is

nonsingle-valued,

_so the lifetime obtained

by

these calculations is also non-one-valued.

The

problem

of

calculating

the lifetime of solitons has arisen in _recent years with _a view to

clearing

_up whether the lifetime of

Davydov's

soliton at _nonzero temperature ^is

long enough

for it to be used in

biology.

I think that many of the

previous

estimates of

Davydov's

soliton

stability

at finite temperature should be revised.

An

unsatisfactory

theoretical calculation of the lifetime of

Davydov's

solitons is

given

_now

in the

following

_papers Lomdahl and Kerr

[5]

and Lawrence et al.

[6]. They

modelled the temperature effect

by adding arbitrary damping

and noise terms to

dynamic equations.

The numerical studies _were classical

and, thus,

the

stabilizing

_quantum effects

reflecting special properties

of solitons _were

neglected.

In the _paper written

by Cottingham

and Schweitzer

[7]

there _was obtained the estimate for lifetime of

soliton,

_r,

equal

to 10~ ~~-10~^~~ _s. This time is very short to be real.

They perform partial diagonalization

of the total

Hamiltonian, H,

and

using

the rest of its terms, V

= H-

Ho,

in the first-order

perturbation theory,

estimated the

probability

for transitions to delocalized states. As _we indicated

earlier,

these estimates _are not

single-valued.

In the _paper

[8] Wang,

Brown and

Lindenberg

have done simulations for collisions of

phonon

_wave

packets

with

Davydov

solitons

by

the

quantum

Monte-Carlo method. These calculations show that solitons cannot exist in _an

equilibrium

_system above 10K. The

quantum dynamical

calculations reveal that if

they

_are formed

by

_some

nonequilibrium mechanism, they

last at most two

picoseconds.

So it is asserted in the _{paper :} the _« crisis of

bioenergetics

» is

[9]

^still with _us.

On the other hand Bolterauer

[10]

and other authors

(see

references in

[I I])

found solitons to be stable at T~ 30 K. Cruzeiro _et al.

[I I]

derived _a

thermally averaged

Hamiltonian and

found stable solitons at 300 K.

3. Thennalizafion and sofitons.

In nonlinear

systems

^with

dispersion,

that is in the

medium,

where the

phase velocity

at monochromatic values is the function of _wave

length

^{and its}

amplitude,

the

perfect

_way of the energy transport ^{is realized}

by

the nonlinear

solitary

_waves. These _waves transfer _energy without loss and preserve their form. These unusual

properties

of

bell-shaped moving

local

excitation enabled

Zabusky

and Kruskal

[12]

to call them solitons.

In _{contrast to} the monochromatic _waves, which describe

periodic repetition

in space such _as the elevation and

deepening

on the surface of water, or contraction and rare-action of _a

density,

_or deviation from average values of other

physical properties,

solitons _are

characterized

by single excitations, spread

as unit with constant

velocity

without

damping.

For the first time

solitary

_waves were observed

by

Scott Rassel _more than 150 years ago.

Many

times he observed the movements of the

barges along

the channel

Edinburgh-Glasgow.

He

published

these observations in _{a paper «}

Report

_on Waves

[13]

^{in 1844. He discovered} that under _a

sharp

stop of a

barge

from it _a

part

^{of the} waves is

separated

and with

big velocity rolled, receiving

the form of _a

single elevation, continuing

its _way

along

the channel without any noticeable

change

in its

shape

and does not decrease its

velocity.

This _{wave was} called the

wave of translation _or

solitary

_wave.

Only

after _one hundred _years the interest in the

solitary

_{wave was} renewed. Particular interest induced the paper which _was

published

in 1955

by

collaborators of the Los Alamos Scientific

Laboratory Fermi,

Pasta and Ulam

[14].

In this _paper the condition of

thermalizing

energy in nonlinear vibrational

systems

was

investigated.

Using

the _new

computer they attempted

to clear _up the conditions of the thermalization of the vibrations in _a chain of

periodically

situated

particles

between which linear and

quadratic

forces operate.

It _was well known that in _a condensed medium the vibrations of atoms _can be

represented

in the form of _a

superposition

of monochromatic vibrations. In _a linear medium these vibrations _are

independent.

Under weak

nonlinearity

between monochromatic vibrations

interactions arise which reduce to the thermalization. The

precise

calculations

by Fermi,

Pasta and Ulam showed that in _a

system

^with

quadratic nonlinearity

the thermalization does not

occur. This result

appeared

to be

paradoxical

for _a

long

time. It _was resolved

by Zabusky

and

Kruskal

[12] only

ten years after.

They

showed that

long-wave

excitations in _a discrete chain

are described

by

nonlinear

equations.

Their solutions _were stable

bell-shaped

excitations. It

was found that the nonlinear interaction does not _cause the

exchange

of energy between them. This _was the _reason for the absence of thermalization in the nonlinear

system

^which was

investigated.

The

exceptional stability

of solitons is

stipulated by

the mutual influence of two

phenomena

_: the

dispersion

and the

nonlinearity.

The

dispersion

induces diffusion of the localized

excitation, organized

from the monochromatic _waves. In the linear system this diffusion does not

compensate

^{because of the}

independence

of

plane

_waves. In the nonlinear

system

^{there takes}

place

the rearrangement of the energy between them. The _energy is taken

away from fast

going

_waves and _passes to the _waves

lagging.

As _a result the excitation

extending

_{as a} unit is made up.

4. The

properties

of sofitons at different

temperatures.

At

present

^while

investigating

the thermal

stability

of _a soliton

nobody really

^considers the lifetime of _a soliton created at _some moment

(t

₌

o),

but considers its

properties

in _a system which is in contact with the thermostat at different temperatures.

When

investigating

the

properties

of solitons at different

temperatures

^it is very

important

to take into account that there _are two

types

^of

phonons

in the

theory

of solitons _:

I)

the virtual

phonons

which describe the

displacements

of the

equilibrium positions

of molecules of

a chain under creation of the soliton _; and

2)

the real

phonons

which describe the vibrations of

molecules around the _new

equilibrium positions. Only

real

phonons

_go into thermal

equilibrium.

The virtual

phonons

do not

depend

_on the temperature.

The method of

separating

real and virtual

phonons

_was

proposed by Davydov

in ₁₁

5]

where the influence of

temperature

on the

properties

of solitons _was

investigated.

A.

Davydov

_was the first to consider the

quantum

^{mechanical effect of finite}

temperature

on the

properties

of solitons. He used _a model of the

quasi-one-dimensional

chain of N

periodically repeated

neutral molecules at sites _na

maintaining

contact with _a thermostat _at temperature T # o.

Stationary

states of _one intramolecular vibration _{or one extra} electron in this chain in the

short-range approximation

_are described

by

the Hamiltonian

H

=

Ho

₊

_H~~

₊

Hi~~, (4,1)

where

N

Ho

₌ J

£ (2 At A~ (At A~~

j + h-c-

)) (4.2)

n=1

is the operator of _energy counted off from the _energy band bottom of _a free

quasi-particle (an electron,

_or intramolecular

vibration)

with effective _mass

m ₌

h~/2 ^a~

J.

(4.3)

For _a

quantum description,

it is convenient to express of the

displacements,

_u~ at the n-th molecule from its

equilibrium position,

_na,

through

the operator ^of

creation, b(,

and

annihilation, _b~,

of

phonons by

the formula

~J~ 1/2

"~

2 MN Vo

I ~

^~~~ _{~~~ ~} ~

~~

~~~

~~~~

'

~

_~~

'

~~'~~

where M is the _mass of _a

molecule,

_a is the

equilibrium

distance between

molecules,

Vo

^{is the}

velocity

of _a

long-wave

sound. The _wave

number,

_{q, runs} N discrete values.

The energy

operator

^of

short-range

deformational interaction of _a

quasi-particle

with the

displacements has,

in linear

approximation,

the form

Hj~~ =

N~ ^~/~

£ F(q) A( A~(b~

+

b±~) exp(inf ), (4.5)

nq

F(q)

=

F *

(- q)

=

iaf ^~~~~

~

~~~

f

sin

f (4.6)

2

MVO

Here

af

is the energy of the deformational

potential.

The

operator

of the

longitudinal

deformation _energy of the chain has the form

H~~

=

£ e(q) b( _b~

,

(4.7)

q

where

E

(q)

= h q

Vo (4.8)

is the _energy of _a

phonon

with _{a wave} number _q.

Stationary

states of the chain _are described

by

the _average of the energy

3C

=

(1l'(H( 1l'), (1l'(1l')

=1.

(4.9)

In this

expression,

the _wave function

(1l')

is defined

by (1l')

₌

£ fl w~(t) S~(t) At (..

v

~..

) (4,10)

n q

where

S~~(t)

_m exp

($~~(t) b~ $](t) _{b/ (4.ll)}

is the

unitary displacement

_operator. The functions

fi~~(t)

are modulated

plane

_waves

Sqn ( t)

"

p

qn

(t )

_~XP

(~ l~i ) (4. 12)

The action of the

unitary

operator _S~~ upon the

operators b~

^and

b(

leads _to their

displacement by complex

numbers

fi~~

^and

$(~

^because

~q

_" ~nq ~q ~nq

~q $qn (~. ~~)

These

equations

show that the interaction of a

quasi-particle

with the chain results in the vibration of the molecules about the _new

equilibrium positions _fi~~.

These vibrations _are characterized

by

the _new creation and annihilation

operators (b(, _b~)

of real

phonons.

The real

phonons

describe the vibrations of molecules around the _new

equilibrium positions. Only

real

phonons

_go into thermal

equilibrium.

The function

(4.12)

which describes the

displacements

of the

equilibrium positions

of molecules of _a chain under creation of soliton is

temperature-independent.

In

quantum theory

these

displacements correspond

to the virtual

phonons.

Take into account these

properties

of real and virtual

phonons,

_very

important

in the

theory

of

solitons,

which circumscribe

thermodynamically equilibrium

states.

In the chain in thermal

equilibrium

with the thermostat at temperature

T,

the statistical

average of model Hamiltonian

(4.9)

is reduced _to

replacing

the quantum ^number

v~ ^of real

phonons by

their averages

«v~»= [exp(@) -lj~ _(4.14)

After

calculation,

the

averaged

value of

(4.9)

is transformed to the energy functional

«

_3C

»

"

z _11121Wqn

_i~

i~~P(~ K) ~S _~g~

_~

~'~'ii1

)

N~

~~~F(q) _#qn j~ (ban

+

p tan)

+ +

E(q)i« _"q »

+

~qni~ iPqni~i) ^~~~~~

where the

Debye-Waller

factor

exp(- W~)

is defined

by

ll~

=

£ _p~~

_j~

^ill

₊ 2

II v~ )). _(4.16)

~

q

The energy functional

(4,15)

describes both nonlocalized

(banda)

and autolocalized states.

The autolocalized states _are

stationary

states in which the distances _or the orientations of the molecules

change

in _some finite

region

of the

chain,

I-e- there is _a local violation of translational

symmetry.

^This localization

region

_{may occupy any part} of the chain.

Consequently, general

translational symmetry is

preserved. Therefore,

these autolocalized

states _are characterized

by

the energy and total momentum related to the movement of the

localization

region along

the chain with _a constant

velocity

that

depends

_on the value of the

wave number k.

The

stationary

nonlocalized states _are also characterized

by

certain value of _wave number

k,

but in these _states the

probability

^of distribution of _a

quasi-particle

is

independent

of

place (n)

and _so the motion of _a

quasi-particle

is absent.

By superposing stationary

nonlocalized

states we can form

nonstationary

states

(having

_no

strictly

defined

energy)

in which the

probability

of

finding

the

particle

is _nonzero in finite

region

that _moves

along

the chain with _a group

velocity.

The size of the localized

region

in this state,

however,

_grows

continuously

with

time,

I-e- the _wave

packet

« smears ».

The autolocalized states

arising

under the

short-range

interaction _are described

by

nonlinear differential

equations. They

_are

usually

called solitons to

distinguish

them from the autolocalized states first introduced in 1933

by

Landau

[16]

and elaborated

by

Pekar

ii?]

when

describing

the electron motion in ionic

crystals.

The latter states _were called

polarons,

because

they

_are caused

by

^the

long-range (Coulomb)

interaction of electrons with the field of

electric

polarization

of _a

crystal

which is described

by

the

longitudinal optical phonons.

The

properties

of

polarons

are defined

by integro-differential equations.

SMOOTH SINGLE SOLITON. Now _we

investigate

the _case when the autolocalised

quasi-

particle

_occurs in finite

region,

of _a

long

aN

(N

»

I)

molecular chain

[3].

Assume that the state of

quasi-particle

ⁱⁿ such _a chain is described

by

the function

w~(t)

=

W(n) expii(kan wt)j (4,17~

with _a fixed _wave

number, k,

and the real _nonzero

amplitude ~li(n) only

in _some finite

(not

too

small) region

of _a

chain,

I-e- at values of _n that

satisfy

the condition

(fl-n( wNo«N. (4,18)

In this _case, the function

(p~~

(~ in the modulated _wave

(4.12)

also

equals

_zero

beyond

the

region (4,18)

and

depends weakly

_{on n} inside this

region.

Since inside the

region (4.18)

the function

p~~

(~ is

weakly dependent

_{on n we} equate it to _a constant value

p~~(~.

Near the fl in the

region

in which

quasi-particles

_are

mainly distributed,

the energy functional

(4,15)

takes the form

«3c»= J~2jw~j~- iwzw~~j+c,c.jexp(-W~)+

+

£ ^e(q)(« _v~ ^»

+

pan

~)

~N~~~~£F(q)jwnl~(Pqn+P?q,n)I, E(q)"haq, ^(4.19)

where _n

= fl. Since the site fl _can be in _any

part

^{of the}

chain,

henceforth _we shall not indicate the tilde

sign explicitly.

The

amplitudes _p~~

in

(4.19)

_are

weakly dependent

_{on n} in the excitation

region,

_{so we can}

use an

approximate equality

PqnPjn+I "PIn Pq,n+I

_"

lPqnl~ (~'~°)

We take

expressions (4,16)

and

(4.19)

into account to obtain the differential difference

equations

3#~

ih

=

J~2 #~ (#~~

j +

#~_j) exp(- W~)]

at

N~ ^~~~

z F(g) _~bn(pan

₊

p tan)

,

(4.21)

~~

~fl~~

=

e(q) p

~

N~

~/~F*(q) _#n

_{(~ +}

at

+

J[ («

_Vq

^»

+

f~jwnl~ _Pqn eXp(- wn)j ^(4.22)

The real

phonons

with _wave numbers q

correspond

to molecular vibrations about _new

equilibrium positions

with

frequencies Vo(q _(.

The time

change

in the

amplitudes

of

displacements

from

equilibrium positions _p~~

_are defined

by

the

velocity

of the movement of

a

quasi-particle along

the chain

ih

3p~ ^fat

₌

(q Vp~~ (4.23)

Using (4.23),

_we find from

(4.22)

and its

complex conjugate

F *

(q ) #n

^~

(4.24)

~~~ N~~°(f( E0(1+ V~~ 3n G(f)

+

~)

^~

where

s

m

vi v0,

a

n "

JEi~

"

ibn

^~ eXp

(- wn (4'25)

Here so =

hvola

^is the maximum

phonon

_energy.

In states in which the

quasi-particle

location

region

exceeds much the distance between the molecules

#~

_(~ «

l,

and _at low

temperatures (&

_~

8~) equations (4.21)

and

(4.22)

become

16( _{=Ji2wn~ (wn+I} +wn-I)~XP(~ ~oi +G(n)jwni~wn, (4.26)

where _a nonlinear parameter

G(n)

is G

(n

m G

=

~

~, D

m a ~

x

~/MV/, (4.27)

s

and

Debye-Waller

factor

exp(- Hj

is defined

by f

"

Bn('

_{+ 2 ~XP}

("~0/8) ~XP(~ Wn)] (4.28)

Here

B~

_m 7.13 _x lo

(

_so

J~)~

^~

~

~

« l

(4.29)

s

In the continuum

approximation, (4.20)

takes the form

ih

_~

2

J[

I exp

(- Hj]

₊ ^~~

~~~

~~ ~ ~

+ G

# )

#

=

0

(4.30)

at _m

3x~

Its solutions _on the infinite chain _can be written in the form

# (x, t)

=

W

(z)

_exp

ji(kz wt)j (4.31)

Where W

(z)

is _a smooth real function in the

system

of

coordinates,

_z,

moving

with constant

velocity V, thus,

z=x-xo-Vt, V=hk/m«Vo. (4.32)

It

obeys

the

equation

A

+

~~

~~

+ GW

(z)j

^W

^(z)

^{= 0}

^(4.33)

2 fir dz with

d

= m exp

(- w) (4.34)

The localized solution of

(4.33),

normalized _on the infinite chain

by

the condition

lm

a~ W

(z)

dz

= ,

(4.35)

m

is defined

by

the function

W(z)

=

(aQ12)~/~sech (zQ), (4.36)

with parameters

Q

=

G ~XP

(W)

~

a~X~exp(w)

4 aJ ^' 16 J

(~.37)

The energy of the chain deformation in the

region

of n-th molecule is

~

~

D

l~p

^{y~~~ ~~} ~

D ² exp

(

_w

)

~~ ~~~

~~~ 2

a(I s~)

2Y

J(I s~)

The energy of the

quasi-particle

in the

potential

field of the deformation well

U(z)

=

GW

~(z)

=

~~~~P

~~~ sech~(zQ), _(4.39)

8

J(I

_s

)

that _moves with with

velocity V,

is defined

by

Am

= 2

Jj

I exp

(-

w

)j

+

a~(k~ Q

_{~) J exp}

(-

w

(4.40)

The first term in

(4.40)

indicates _a decrease in _resonance interaction caused

by

the

fluctuations at intermolecular distances. The second term characterizes the _energy

gain by binding

at the

quasi-particle

in the field

(4.39).

To calculate the total _energy

E( V),

transferred

by

_a

moving soliton,

_we must add to

(4.40)

the _energy

spent

to form the deformation. So _we obtain

~~ ~~

~

*" _~

~def

=

2

Jj

i exp

(-

_w

~j

D~

exp

(w)

j

~

48

J(

I

s~)

^~

i'~~

~~P (W

(4.41)

At temperature ^{& that is} smaller than maximum energy of

phonons,

_so, and small velocities

(s~« _I),

the soliton total _energy

(4.4I)

_can be written _as

E(V)

=

E(o)

_{+ M~~,}

V~,

_&

~ so,

(4.42)

where

E(°)

=

2

J[I exp(- w)]

^{~ ~}

~~P(w)

48 J

(4.43)

is the energy of _a soliton at rest, and

1~2~

⁴

M~,

= m exp

(w )

₊

~ ~ ~

(4.44)

12 h M

Vo

is its effective _mass. The momentum transferred

by

soliton is P

( V)

= m V exp

(w [I

+

a~

x

~/12

h~

M~ VI] (4.45)

At _zero

temperature

^{the function W} =

0. With

rising temperature

the energy

gain

when the rest

quasi-particles

_are bound with

deformation,

is defined

by

AE

=

~ ^~

+

W(2 J~

_{D ~)}

/48 J,

_w «

(4.46)

If the

inequality 2J~~ D~

_is

fulfilled,

the

increasing temperature (under

&~

so)

causes

increase in Wand stabilizes the soliton. Its

binding

_energy and effective _mass

increase,

but in

agreement

^with

(4.36)

the effective size

(~

l

IQ)

decreases.

At

temperatures exceeding

the maximum

phonon

_energy

(&~ so)

and small velocities

(s~«

I

),

the nonlinear

parameter

G that _enters in

(4.30)

takes the form

G = D [1 ²

&/qr so] (4.47)

It decreases

linearly

with

increasing temperature.

In the _same

approximation

the

Debye-

Waller factor

W

~

f ₍₁

2

8/area) (4.48)

qrso increases.

So, increasing temperature (at

2 &

~

so)

is

accompanied by

the

increasing

soliton size and the

decreasing

of its maximum

amplitude.

5.

Investigations

of the

properties

of sofitons

by

numerical methods.

The

properties

of

Davydov's

solitons at different

temperatures

were

investigated by

Cruzeiro and others

ii Ii. They

used the

thermally averaged

Hamiltonian in the form _:

HT

"

£ llE14nl~ J(ibl ibn-

I

e~~'~~'

₊

wl ibn+

I

e~'~~')j

n

ibn1~

^{N~ ~~~}

£ F(q) e'~~~(p

qn +

Pig,

n

)

+ q

+

Z*nq

₊

l~bnl~ Z*nq(Pq

₊

lPqnl~)1 (5.i)

q q

where

wn,n±I

"

£ I(Pq

+

i) pj Pqn±I

+

q

+

?Pqn PS±i (Pq+ ^{ilPqnl~+ lPq,n±i l~i (5.2)}

This Hamiltonian differs from

(4,19)

in _an

insignificant specification.

A

dynamical equation

derived from the Hamiltonian

(5.I)

_was

investigated by

numerical solutions _at different temperatures

provided

that the _norm is

conserved,

I-e-

~

N

~ £ #~ ~j

=

0

(5.3)

n=i

This condition is

equivalent

to the

assumption

that the soliton

always

exists.

Therefore,

such _an

investigation

of the

properties

of solitons _cannot

give

the _answer relative _to its lifetime.

The authors of the _paper

[11]

state that

Davydov's predictions

of the thermal effect _on soliton

propagations

are

confirmed,

I-e- _an increase in the effective _mass of

soliton, corresponding

to _a decrease in soliton

velocity

_as

temperature

^{increases. Also} as

temperature

increases the soliton becomes _more

expanded.

Because

Davydov's approximate equations

_were not valid for soliton velocities around the sound

velocity, Davydov

could not

predict

whether the transition from the soliton to exciton

state would be continuous _or discontinuous. It _was shown in the paper

[11]

that the transition from _a soliton state to an exciton state is continuous _».

This statement has _a conventional sense because any exciton _state

(wave packets)

has the

spatial

extension in the

time,

but

spatial

size of _a soliton is constant.

A

general

conclusion of the authors of reference

[11]

is «that _an

analysis

based _on

Davydov's assumption

does indeed

imply

that this soliton is robust at

physiological

temperature

(310 K)

_».

6. The

toJ1ologjcal stability

^of sofitons.

When

investigating

the lifetime of solitons _we need to know the initial time when it _was created. The time evolution of soliton state will

depend essentially

_on the time when the

thermodynamic equilibrium

with _a thermostat is established.

The soliton _can pass a

long

distance before its parameters

(W

and

others)

will be

thermalized. As _we know while

studying

the lifetime of solitons due to thermal motion

nobody

has taken into account these very

important

circumstances.

The main

disadvantage

of current theoretical research of the lifetime of excitons is also the

neglect

of the

topological stability

of solitons.

The soliton is

organized

_{as a}

quasi-particle coupled

with local deformation _on the chain.

The _space distribution of _a

quasi-particle (an

exciton for _a vibrational excitation _{or an} electron of the conduction

band)

in _a system

f

= x Vt

moving

with

velocity

V is defined

by

the bell-

shaped

function

8~(I

=

(2 Q )~ ^Sech~ (Qi )

The decrease in the intermolecular distances in the

region

of _a bisoliton is described

by

the function _p

(f

W

~(f _).

This decrease is caused

by displacements

of

equilibrium positions

of molecules and is described

by

the function

u(f)

= A

[I

tanh

&f

₌

(°' _2A,

^~~

^f

^{~ °}

at

f~o.

So,

when _a soliton _moves with

velocity

V all

equilibrium positions

of molecules behind it

are

displaced by

the value 2

A,

but in front of

it,

the

positions

of molecules _are not

changed.

For the soliton to

disappear,

_one needs to waste _an energy ^to transfer _a

quasi-particle

into _a free non-local state

(exciton

_or electron in the conduction

band)

and retum all molecules

which _were

displaced

to their initial states. This circumstance prevents the destruction of _a soliton and guarantees ^its

topological stability.

On account of

topological stability

solitons can be created and

disappear only

at the ends of molecular chain. This very

important property

was not taken into account in works devoted to the calculations of the lifetime of _a soliton. Therefore the estimate of the lifetime of solitons

made

previously require

_a total revision. We remind _{once more} that

usually

_one considers not the

lifetime,

that is the time of the existance of _a soliton from the moment of its appearance, but

only

the

properties

of _an

existing

^soliton at different

temperatures.

7. Sofiton

generation

in molecular chain.

Brizhik,

and the authors of

[18],

have made an attempt to

study

the evolution of the excitation distributed in _an infinite molecular chain

given

at the initial time _moment in different forms for different values of exciton

phonon coupling

constant.

However,

the

question

of creation of such excitation in _an initial time _was not considered.

As has been shown

by

Scott

[19]

for

particular

initial condition _one finds _a threshold

requirement

_on the anharmonic

parameter (x~/«JJ

below which solitons will not appear in molecular chain. For

single

channel model

alpha-heli

of

protein

^{molecules the value}

«J/x~

is about 0.52. Here _« is the

hydrogen

bond

spring

constant, x the

exciton-phonon interaction,

J is the nearest

neighbour dipole-dipole coupling

_energy.

The author and Brizhik

[18]

have shown that the value of this threshold

depends strongly

upon the

shape

of the initial conditions. The threshold value

approaches

_{zero as} the initial conditions

approach

the

hyperbolic

secant

shape

of soliton.

As _was said in section

6,

solitons _can be created and

destroyed

at the ends of the chain

only.

The

possible

mechanism of soliton creation _may be the

following.

An electron

beam, light quanta,

^local

hydrolyses

molecule ATP etc. excited the

impurity

molecule at the end of the chain

principal

molecule. The excitation transfer into the

neighbouring

molecule of the chain with which the

impurity

molecule is connected.

This process has been studied

by

Brizhik and others

[20].

This _process is characterized

by

two

parameters

:

parameter

^defines a nonreasonance excitation transfer from _an

exciting impurity

molecule onto _a

neighbouring

one of the basic chain and Jis the _resonance excitation

transfer _constant between the basic molecules of _a chain. If the condition r » J is

satisfied,

the excitation of

impurity

molecule will be localized at the molecule of chain nearest to the

impurity

at the moment to

satisfying

the

inequality

hr~ _« to « hJ~ In this _case the inverse transfer of the excitation to

impurity

is

impossible.

The excitation will transfer

along

the chain

by

resonance mechanism.

8. Process of

disintegration

sofitons.

Lifetimes of solitons _are determined

by

the

velocity

of

disintegration

when the localized

quasi- particle

_goes to the nonlocalized exciton states. There is _a low and _a fast _process of

disintegration

of solitons. At low process the

complete

annihilation of soliton takes

place

that

is,

the transfer of _a

quasi-particle

from _a localized _{state into a} nonlocalized state

(exciton)

and removal of _a local deformation in the chain. At fast processes the transfer of

quasi-particle

from _a localized state into nonlocalized takes

place

_so fast that _a local deformation has _no time to

disappear.

As is

known,

at the moment of

fight absorption by

^molecular

systems

^the coordinates of

heavy particles

_are unable _to

displace (the

Frank-Condon

principle).

Since the formation of _a soliton is connected with the

displacement

of

equilibrium positions

of

heavy particles (the peptide

_groups in the muscle

molecules,

for

example).

So under the influence of elec-

tromagnetic radiation,

solitons

disintegrate

into

rapidly relaxating

excitons and local deformation of the chain. After a local deformation

spreading along

the chain.

This process will be called fast annihilation of the soliton. The _process of fast annihilation

can occur at any

place

of the chain. At fast annihilation the _energy

expenditure

is

required,

U,

which

equals

the

binding

_energy

quasi-particle

with local deformation of the chain.

For rest of the solitons _an energy deformation of chain is

equal

to

U/3. So,

for slow _process

one needs to

spend

the energy

equal

to 2

U/3.

Using

the idea of solitons

Davydov proposed

in 1973 _{a new}

hypothesis

of the mechanism of the

shortening

of _sarcomere

length

that evolved contraction of striated muscle

[2, 21].

According

to this

hypothesis

under _{a nerve}

impulse

the calcium ions reach the first series of

myosin

molecule heads at the ends of thick filament initiate the

hydrolysis

of the ATP molecules attached _to them. The energy released generates solitons in _a

long

helical section of

myosin

molecules.

They

_move from the ends to the center and the

displacement

of the

surrounding

action

proteins.

Spending

their kinetic energy for the works necessary to contract the muscle

fiber,

the solitons _are slowed down

and, stepping

_near the centers of the thick filaments

(H-band),

_are

annihilated, giving

_up the rest of their energy ^to thermal motion. This is the _reason for the

heating

of the muscle

during

their work.

Thus, only

the kinetic energy of solitons is used in the contraction of the muscle fibers.

Thus,

the

disintegration (so

and

lifetime)

of soliton _are

always stipulated by

extemal action

on the soliton.

References

[1] ^{DAVYDOV A.} S., J. Theor. Biol. 38

(1973)

559;

Phys.

Scr. 20

(1979)

387 _;

Usp.

Fiz. Nauk. 138

(1982)

603 Sov.

Phys.

JETP 78

(1980)

789.

[2] DAVYDOV A. S.,

Biology

and Quantum Mechanics

(Pergamon,

New York, 1982).

[3] ^DAVYDOV A. S., Solitons in Molecular Sastem (D. Reidel Publish. Corp. Dordrecht, 1985) Second Edition (1991).

[4] BOLTERAUER H., Quantum effect _on

Davydov

soliton, In _: Davydov's Soliton Revisited, P. L.

Christiansen and A. C. Scott Eds., NATO ASI Series,

Physics

243

(1991)

99-107.

[5] LOMDAHL P. S. and KERR M. C.,

Phys.

Rev. Lett. 55

(1985)

1239.

[6] LAWRENCE A. F., MCDANiEL J. C., CHANG D. B., PIERCE B. M. and BiRGE R. R.,

Phys.

Rev. A 33

(1986)

l18.

[7] ^{GOTTiNDHAM J.} P., SCHWEiTzER J. W., Perturbation estimate of the lifetime of

Davydov

soliton at 300 K, Rev. Lett. 62

(1985)

1752.

[8] WANG X., BROWN D. W., LiNDERBERG K.,

Phys.

Rev. Lett. 62

(1989)

1796.

[9] See in _: Ann. N. Y. Acad. Sci. 227

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108.

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(Manchester University Press, 1989), p. 619-624.

[I I] ^CRUzEiRO L., HALDING J., CHRiSTiANSEN P. L., SCOVGAARD O. and SCOTT A. C., Temperature effects _on Davydov soliton,

Phys.

Rev. A 37 (1988) 880.

[12] ZABUSKY N. J. and KRUSKAL M. D., Phys. Rev. Lett. Is (1965) 240.

[13] SKOTT-RUSSEL J.,

Report

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Roy.

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(Edinburgh)

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Los Alamos, Report LA

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theory

of motion of

quasi-particle

in molecular chain with thermal vibration taken into account,

Phys.

Stat. Sol.

(b)

138

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559.

[16]

LANDAU L., Uber die

Bewegung

der Electronen in kristalloetter,

Phys.

Z. Sowjetunion 8 (1933) 664.

[17J PEKAR C. I., Autolocalization of _an electron in

crystals,

Zh.

Eksp.

Tear. Fiz, 16

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335.

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BRizHiK L. S. and DAVYDOV A. S., Soliton excitation in one-dimensional molecular system,

Phys.

Stat. Solidi (b) lls

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615.

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