Soliton dynamics in a thermalized molecular chain with transversal degree of freedom

26 September 1994 PHYSICS LETTERS A

ELSEVIER

PhysicsLetters A 193 (1994) 148-153

Soliton dynamics in a thermalized molecular chain with transversal degree of freedom N. Flytzanis a,.,~, A.V. Savin b,2, y. Zolotaryuk c,3 a Physics Department, University of Crete, 71409 Heraklion, Greece b Institute for Physical and Technical Problems, Prechistenka Street 13/7, Moscow 119034, Russian Federation c Taras Shevchenko University of Kiev, Physics Department, Academician Glushkov Avenue 6, 252127 Kiev 127, Ukraine

Received 14 June 1994;acceptedfor publication 7 July 1994 Communicatedby A.R. Bishop

Abstract

A numerical investigation of the dynamics of acoustic solitons in an anharmonic molecular chain with thermalized longitudinal and transversal vibrations is presented. It is shown that acoustic solitons can serve as an effective energy carrier only if the frequency spectrum of the transversal vibrations of the molecules (optical phonons) of the chain is located higher than the spectrum of longitudinal vibrations (acoustic phonons). Soliton interaction with the low-frequencytransversal vibrations causes its quick degradation.

1. I n t r o d u c t i o n

Strong anharmonicity in the intermolecular interaction, which can be observed in some quasi-one-dimensional molecular systems, favors the appearance of acoustic solitons. Excitations of this type represent local regions of compression of intermolecular bonds, which propagate with ultrasonic velocity. Nowadays dynamical properties of solitons are well-studied [ 1,2]. Particle-like behavior and stability with respect to the interaction with longitudinal thermal vibrations (acoustic phonons) of the chain allows us to invoke solitons for the explanation of a number of physical phenomena. So in Refs. [ 3,4 ] the idea of solitons was used in the explanation of the mechanism

of the effective transmission of energy in the hydrolysis of the ATP molecule in a-helix protein macromolecules. In a thermalized molecular chain not only acoustic phonons are present, but also optical ones, which are caused by transversal vibrations of the backbone constrained by the environment. In the present paper a numerical study of the interaction of the acoustic soliton with the transversal thermal vibrations of the molecules at physiological temperatures T = 300 K is carried out. In Section 2 we present the model along with a choice for the parameters involved. In Section 3 we discuss the chain thermalization and in Section 4 the simulation method. The numerical results are discussed in Section 5 along with the conclusions and their implication for energy localisation.

* Correspondingauthor. ' E-mail:[email protected]. 2 E-mail:[email protected]. 3 E-mail:[email protected]. 0375-9601/94/$07.00 © 1994Elsevier ScienceB.V. All rights reserved SSDI 0375-9601 ( 94 )00576-1

N. Flytzanis et al. / Physics Letters A 193 (1994) 148-153

When a > ao the chain will be in the strained state. In the linear approximation Eqs. (2) and (3) will take the form

2. Two-dimensional model of the quasi-onedimensional molecular chain We consider a chain of molecules (atoms) of equal mass M along the x axis a distance a apart, which can move in the x, y plane. The Hamiltonian of the chain can be written in the form H= ~

[I2 M ( x"2 "2 n + yn)

+ V(p.) + Vo(y.) ] ,

(1)

n

where a dot denotes differentiation with respect to time t; x. and y. are the displacement components for the nth molecule from its equilibrium position; V(p) is the intermolecular interaction potential; p . = d . - a o is the dilatation of the nth bond of the chain, with d. = [ ( a + x . + l - x . ) 2 + (y.+, _ y . ) 2 ] 1/2 being the distance between the nth and ( n + 1 )th molecules, and ao ~
(2)

Myn=F(pn)(Yn+I-y.)/d. --F(pn-1 ) (Yn --Yn-I ) /dn-I -KoYn ,

(3)

corresponds to Hamiltonian (1), where F ( p ) = d V(p)/dp. For small strains a polynomial expansion can be written as F (p) = K p - 7P2.~_..., with K = d E1I/ d2p Ip=0 the linear elastic constant of the intermolecular interaction, and 7 = - ½daV/d3plp=o the cubic anharmonicity constant.

z = Kou2/2

149

/

n-1 n n+l Fig. 1. Schematic representation of the two-dimensional model of the molecularchain.

M.~. = K ( x . + 1 - 2x. + x . _ 1) , M p . = K ( a - a o ) ( Y.+ I - 2y. + Y . - 1 ) / a-- Koy. . The velocity of sound for longitudinal waves is cL=a~, and for transversal ones cx= x / K a ( a - a o ) / M . The dispersion relation for longitudinal vibration has the form 092 (k) = 4 ( K / M ) X sin2( ½ka), and for transversal vibrations it is 092(k) = K o / M + [ 4 K ( a - a o ) / a M ]

sin2(½ka),

where k is the wave number. The velocity of sound for transversal vibrations differs from zero only in the strained state (a > ao). In Ref. [ 5 ] it was shown that in this case even in the chain with harmonic intermolecular interaction (i.e. for V ( p ) = ~ g p 2) envelope solitons exist (since the equations for x and y are nonlinear), which describe localised vibrations with transversal motion. Solitons of this type have the velocity spectrum 0~V a0. We shall consider only the unstrained chain ( a = a 0 ) when the dispersion of optical phonons equals zero and due to anharmonicity of the intermolecular interaction only longitudinal acoustic solitons exist, which have velocity spectrum v> CL. The advantage of using the Toda potential is that we can isolate the effect of the transverse vibrations without combining it with the influence of nonintegrability even in the absence of transverse vibrations. We choose the values of the parameters of our model corresponding to the hydrogen-bonded chain of peptide groups ( P G ) in the or-helix protein. The effective mass of PG equals M = 190.7 × 10-27 kg [ 6 ], with spacing a = ao = 5.05/k, the elasticity constant of the hydrogen bond K = 9.64 N / m , and energy of the bond % = 0.17 eV. It is often convenient to describe the intermolecular interaction by the Morse potential V(p) = % [ e x p ( - t i p ) - 1] 2 ,

(4)

where the phenomenological potential parameter fl= ~ . The anharmonicity parameter for this case is 7= 3Eofl 3. The Morse chain is not a fully integrable system so that it is not possible to obtain exact soliton solutions as in the case of the Toda chain

N. Flytzaniset aL / PhysicsLettersA 193 (1994) 148-153

150

which is completely integrable [ 1 ]. In the region of compression the Toda potential

V(p) =Kb -~ [b -~ exp( - b p ) + p - b -1 ] = ½Kp2- ~Kbp3 +...

(5)

approximates well the Morse potential (4). Therefore we shall use the Toda potential (5) with the value of the phenomenological parameter b=2~/K= 3fl=0.399 A -~. Ko will be treated as a variable parameter in this paper.

3. Thermalization of the molecular chain

Following Ref. [ 7 ] we describe the dynamics of the thermalized molecular chain by the system of the Langevin equations

- M F 2 , +~n,

(6)

M~n = F(p,) (Yn+1 - Yn) /dn - F(p,_l ) ×(Yn-Yn-,)/dn-1-Koy.-MFyn+q.,

(7)

where F = 1/tr with t~ the vibrational relaxation time of the chain; ~.(t) and ~/n(t) are normally distributed d-correlated random forces which describe the interaction of molecules in the chain with the bath at the temperature T. The random forces obey the following correlation relations, (~n(t)~m(t') ) = ( q , ( t ) q m ( t ' ) )

where G(p) = b -~ [exp( - b p ) + 1 ], ~n(z) =~n(t)/K, and On( 3) = tin( t) / K with ~r= tr/to the dimensionless relaxation time, so that the prime in (9) and (10) indicates differentiation with dimensionless time. We will integrate the system of equations (9), (10) by the standard Runge-Kutta method of fourth-order precision with constant step Az. Then we change from continuous time • to discrete time, IAT, and from random functions ~ (x), On(~) to a sequence of random d-correlated values ~n(/Az), 0n(/Ar). The correlation functions in (8) will take the form

(~.(laT)~.,(kaT)) = (On(lAZ)Om(kZXO) =a2,~n,.,~,

where a 2 = 2kBT/rrArK. For the full thermalization of vibrations in the chain it is necessary that the upper boundary of the frequency spectrum for the random forces (Dr=Tr/Arto satisfies the inequality (.Or > COL ( ~ / a ) = 2 ~ , and (Dr > (DT = ~

(in this case the molecular system

will apprehend random forces as purely white noise). If we take the step of integration Az=0.01, and the elasticity parameter Ko ~<5K, then the above inequality will always be satisfied. Thermalization of the chain can be characterized by the instantaneous temperature T/(z) = K Z (Xn,2 +Yn,2 )/2NkB,

(8)

where kB is the Boltzmann constant. We introduce the dimensionless time z= t ~ for convenience in the numerical integration (one unit of the dimensionless time corresponds to the time of traversal for one cell of the chain at the longitudinal sound speed to=x/rM/K= 1.40649× 10-13). Then, taking into account the form of the Toda potential (5), we can rewrite the system of equations (6), (7) in the form

x"=G(pn)(x,+, -xn + a ) l / d n - G ( p , _ l ) × (xn-xn_~ +a)/dn_, - x ' J z ~ + ( , ,

(10)

N

= 2MFkB Td( t-t')dnm , (~n(t)q,,(t')) =0,

X (Yn-Y,_~)/dn_~-Koy~-y'/~r=O~,

(~,(IAz)O,,(kAz)) = 0 ,

Mgn = F(pn) (xn+t - x n +a)/dn - F ( p n _ , ) X ( X n - - X n _ 1 +a)/dn_l

y~ = G(p.) (y.+, - y n ) / d . - G ( p . _ ~ )

(9)

n=l

where N is the number of molecules in the chain. If at the initial moment of time z= 0 the chain is in the equilibrium state, then the average value of its temperature at a later time T is ~r(r)= (T~(Q)=T[1-exp(-z/zr)]. Therefore for full thermalization of the chain it is sufficient to integrate the system (9), (10) during the time v= 10r,, after which the average temperature of the chain ]P(z) will practically coincide with the temperature of the bath T. In the following we will take the value of the relaxation time T,= 100, which allows us to obtain sufficiently quickly a thermalized state of the chain, and the temperature T = 300.

N. Flytza nis et aL / Physics Letters A 193 (1994) 148-153

151

/

4. Simulation of the propagation through the thermalized area of the chain

pn(Z) = -b-'

Let us consider the propagation of the acoustic soliton through the thermalized area of the chain. In the numerical simulation of the dynamics we use fixed boundary conditions x l -- O, Yl = O, XN =-- O, YN--- O. We take the number of molecules N = 300-800, depending on the length of the thermalized area. From those a total number of N1 molecules is in contact with the bath with indices n = 11, 12 ..... 10 + NI. The dynamics will be given by the system of Langevin equations (9), (10), while the rest of the system satisfies the same equation without random forces and relaxation terms. At the initial time z = 0 we put the soliton at one end of the chain ( n = 1, 2, ..., 10) just before the thermalized region. An appropriate initial condition is the solution of the Toda soliton [ 1 ]

ln~l +

sinh2(q)

cosh2[at--~ST5) -s~]T

y,(z) = 0 , where s = V/CL is a dimensionless velocity, and the dimensionless parameter q can be determined from the equation s = sinh ( q ) / q . In the second region (n = 11, .... 10+N, ) we shall use a thermalized initial condition obtained under fixed boundary conditions at the ends of the thermalized region. The rest of the chain ( n = 11 + N,,..., N) is initially undisturbed with x , = O, y , = 0 , x;, =0, y;, =0. Thermal vibrations of molecules, random external forces and damping forces influence the dynamics while propagating through the thermalized area. To estimate only the influence of the thermal vibrations we consider the dynamics of the soliton with zr = oo. In that case the energy of the chain N

E= ~ [½K(x~ +y~)+V(p,,)+ '~Koy.]2 n=l t=300.O

t:O.O

0.4

0.4

(a)

(b)

0.3

0.3

0.2

0.2

0.I

0.i

o) v

0

0

-0.1

-0.i 0

i00

200

300

400

i00

500

200

t=300,.O

500

0.25

(c)

(d)

0.4

~

400

t=300.0

0.6

~

300 n

n

0.15

0.2

E o

0.05

o -0.05 -o.2 -0.15

-0.4

-0.25

-0.6 0

i00

200

300 n

400

500

0

i00

200

300

400

500

n

Fig. 2. Propagation of the acoustic soliton through the thermalized area of the chain of NI = 50 molecules, Ko/K= 5, zr= 100, kp- 0.600. (a) Initial and (b) final (z= 300 ) energy distribution along the chain; (c) final relative displacement and (d) y-displacement

152

N. Flytzanis et al. / Physics Letters A 193 (1994) 148-153

is an integral of motion for the system. The precision of the numerical integration can be checked by monitoring energy conservation. The numerical experiments which were carried out have shown that the energy of the chain is conserved with a precision of up to six digits for the whole integration time z= 300 using a step of integration Az= 0.01. Let us mention that thermal vibrations can create by thermal activation acoustic solitons. Thus, according to Refs. [ 8,9 ] in the thermalized Toda chain with values of the parameters which correspond to the double-DNA helix at T-- 310 the ratio of the average number of thermally activated solitons Ns to the number of cells Nequals 0.31. Here, however, we only consider the dynamics of the injected acoustic soliton with energy Es = 0.5 eV (s = 1.6, q = 1.76 ). For this energy of the soliton it is easy to separate it from thermally activated solitons, whose velocities are slightly larger than the velocity of sound and whose energy is of the order of kBT. After propagating through the thermalized area the input soliton quickly outruns the radiated one by the bath acoustic (longitudinal) phonons and the thermally activated small-amplitude solitons (Fig. 2), which allows as easy identification of it. Optical (transversal) phonons have zero dispersion, so that the soliton interacts with them only in the thermalized area, in contact with the bath. The transmission factor of the soliton kp is the ratio Ep/ Es, where Ep is the energy of the soliton after passing the thermalized area.

5. Simulations and discussion The value of the transmission ratio kp depends on the thermalized state of the chain at the initial moment of time and on the random forces during propagation. In that sense kp has a random value. Physically important is not its value, obtained with a certain realization of the thermalized state, but its average value/~p and square-root standard deviation ap from that value. To estimate these for different values of K0 and "['r we calculate the values of {~,}~o for 50 independent realizations of the initial thermalized state and calculate the average ~ and a v. In Table 1 we show the results obtained for a length ofthermalized chain N~ = 27.

Table 1 Average energy transmission ratio (/~) and square root standard deviation (ap) as a function of the ratio Ko/Kfor two dissipation times.

zoo

Ko/ K

kp

~p

100

0.1 0.5 1.0 5.0 oo 0.1 0.5 1.0 5.0 <~

0.157 0.474 0.630 0.853 0.848 0.220 0.533 0.784 0.996 1.0

0.057 0.139 O. 164 0.140 0.123 0.083 0.161 0.133 0.027 0.0

ov

In the limit Ko = ~ in our two-dimensional model, the dynamics of the soliton is essentially the same as in the one-dimensional Toda model with thermal fluctuations. In the absence of the damping (Zr = ~ ) and external random forces the transmission ratio kp = 1 for all independent realizations of the initial thermalized state (/¢p = 1, a p = 0 ) . Damping ( Z r < ~ ) causes the decreasing of the energy of the soliton. So for Zr=100, and T = 0 K the transmission ratio /%o=0.830<1, and for T = 3 0 0 K we have /~p= 0.848 =/q,o with ap = 0.123. This allows us to draw the conclusion that in the one-dimensional model of the chain the acoustic soliton is stable to the interaction with longitudinal thermal vibrations. The energy of the soliton is decreased only due to the overcoming of the damping force. In the limit Ko = ~ random external forces cause only weak fluctuations of the energy, which will not be seen when averaging over time. For finite values of Ko/K in the two-dimensional model the dynamics of the soliton depends on the ratio Ko/K. High-frequency transversal vibrations do not have a significant influence on the dynamics of the soliton. So for Ko/K=5 and zr=oo the average value/q, =0.996 -,~ 1, and ap=0.027. It is interesting to mention that for some realizations of the thermalized state of the chain, interaction with the transversal vibrations causes even the growth of the energy of the soliton (kp> 1 ). On the other hand, low-frequency transversal vibrations cause the gradual destruction of the soliton. As it is seen from the data of

N. Flytzanis et al. / Physics Letters A 193 (1994) 148-153

Table 1, with the decrease of the ratio Ko/K the value of the transmission ratio/q, monotonously decreases. Thus, at relatively high temperatures the inequality Ko/K> 1 is a necessary condition for the possibility of the stable motion of the acoustic solitons through the thermalized molecular chain. We have also investigated the transmission of energy as a function of the length of the thermalized region. In Fig. 3 we present some preliminary results for Ko/K= 0.5 and a relaxation time To = 100. We see that the soliton transfers a significant part of its energy over distances which are of biological significance. The points are fitted by an exponential function as a guide to the eye, since other polynomial expressions could fit with the same rms error. At lower temperatures preliminary results show that a single exponent is not sufficient. This problem will be investigated in detail in the future, especially in the region of small disorder where asymptotic behavior can be more simple. In Table 2 we give the length Lr over which the soliton energy is decreased by one half. We see that for low Ko/K ratio the value of Lr is equally low for low or high relaxation times. When the frequency of transversal vibrations is much higher than the longitudinal band the value of L~ has increased by a factor

Ko/K=0.5,

To=I00.0,

error=0.0011

0.8 0.6 0.4 0.2 0 i0

20

30

40

%

100 Ko/K Lr

0.1 6

oo 0.5 22

5.0 120

0.1 6

0.5 30

5.0 large

of 20 for To= 100 and for extremely large relaxation time it is much larger, indicating that for large Ko/K values the Langevin damping is the only loss mechanism and the soliton can attain long range propagation. The numerical investigation shows that in the therrealized anharmonic chain acoustic solitons can serve as effective energy carriers, only if the frequency spectrum of the transversal vibrations (optical phonons) is located higher then the spectrum of the longitudinal vibrations (acoustical phonons). Interaction of the solitons with the low-frequency transversal vibrations leads to their quick degradation. On the other hand, interaction with the high-frequency vibrations leads to long range propagation. The length of the disordered region over which the energy is decreased to one half also depends strongly on the force constant ratio.

--

1

0

Table 2

References

"

1.2 0.2311+0.7535*exp(-0.045*x)

153

50

60

N1

Fig. 3. Transmission of energy coefficient as a function of the thermalized length N1.

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