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About the coupling between librational motions and conformational transitions in chain molecules
Volume 173,number $6
CHEMICALPHYSICSLETTERS
19October 1990
About the coupling between librational motions and conformational transitions in chain molecules Giorgio J. Moro Istituto di Chimica Fisica, Universitridi Parma, via/e delle Scienze, 43100 Parma. Italy
Received23 July 1990
The conformationaltransitions lead to reorientationsof a smallfraction only of a chain molecule.This followsfrom the analysis of the barrier crossing problem according to the Framers theory. Therefore, the librational motion operating after a transition is responsible for the attainment of the new equilibrium distribution. A simplified treatment is presented for the model system of an infinite chain of rotors. It is shown that the reorientations of the chain units induced by a conformational transition, propagate along the chain with increasing time delays. T’hisconstitutes an effective mechanism of coupling between the librational motion and the conformational dynamics in long chain molecules.
1. Introduction
In short chain molecules the librational motions (i.e. small amplitude displacements of the torsional angles about the locations of the minima of the intramolecular potential) and the conformational transitions are uncoupled in the presence of sizable energy barriers separating the conformers. Thus,. given the potential and the friction components, the Kramers theory [ 1 ] and its generalization to multidimensional diffusion operators [ 2 ] can be used to calculate the transition rates w(j+k) determining the time evolution of the conformational populations [ 3-61,
afxu= g [Pk(t) w(k+A -P,(O +e.M) I. at
be fully specified by the dynamics of the conformer population equation ( 1). The rotational relaxation of a chain unit with respect to a laboratory frame is treated in a similar fashion, by considering also the motions of the molecule as a whole and the recoil rotations defined as the reorientations of two chain segments after a rotation around their connecting bond [lo]. This method, however, is not adequate to describe the rotational relaxation in very long (infinite) chains. In fact, eq. (2) implies that the chain units reorient all together after a conformational transition. In other words, the rotational displacements induced by a conformational transition propagate instantaneously along the chain, no matter how far
(1)
Furthermore, the rotational dynamics of the system can be described in terms of the rotational isomeric state (RIS) theory [ 71 generalized to the time domain [ 5,6,8,9], under the condition that the tor-
sional variables x relax very quickly (i.e. in a time much shorter than the lifetimes of the conformational states) to the new equilibrium position xj (given by the potential minimum) after a transition: pj(x; t) X&X-xi) P/(t) .
(2)
Therefore, the torsional angles relaxation results to ElsevierScience Publishers B.V. (North-Holland)
Fig. I. The chain of rotors represented as an ensemble of rods constrained to rotate about a common axis.
503
CHEMICAL PHYSICS LETTERS
Volume 173, number 5,6
away. To avoid this unphysical behaviour, one should describe in detail how the system approaches the new equilibrium position after a transition. Obviously this requires an analysis of the coupling between the librational motion and the conformational transitions. A simplified treatment is presented here for the model system of an infinite chain of identical rotors sketched in fig. 1.
2. Chain reorientation during a confomational transition
The chain of rotors was already considered in the past to study cooperative motions [ 11,121 or as model system for polymers [ 13,141. In the overdamped regime, the following diffusion equation describes the time evolution of the system:
ap(a;
~
t)
=D$(&
at
+
&~P(a;i),
(3)
where the array (Ycollects the orientations oi of the rotors with respect to a laboratory frame (see fig. 1) and Do is the diffusion coefficient of each rotor. The potential V is assumed to be factorized with respect to the torsional angles x, = LY, - CY,_, , W)=
(4)
c KG;> > i
with the elementary torsional potential V, having a fixed number of equivalent minima. In order to apply the Kramers theory, the following normal mode analysis should be performed near the saddle point (S) of the internal potential connecting two stable states j and k [ 5,6]: ( 1 /kB T) DV~*)U~+“U~
,
19 October 1990
letting the initial and the final positions according to the constraint of the least potential energy along u ‘, one calculates the displacements A(j+k) of the torsional coordinates caused by a conformational transition. The only free parameter is the curvature ratio p of the elementary torsional potential V,(x), defined as:
where xmin (xmax) is one of the equivalent minima (maxima) [ 151. The details of the calculations will be presented in a forthcoming paper [ 161. In fig. 2 the orientational displacements Aarjof the rotors due to the conformational transition with a unitary change of the torsional angle x0, are represented for some values of the parameter p. It is evident that, independently of the particular features of the potential, the rotational displacements of few rotors only are required in a conformational transition. This illustrates the localized nature of the conformational transitions in polymers, as discovered by Skolnick and Helfand [ 3,4], which explains the independence of the interconversion rates on the chain length. But one can look at this result from another perspective: a conformational transition generates a non-equilibrium configuration and the librational motion operating after the transition must be responsible for the attainment of the new equi-
(5)
where D is the diffusion matrix in the torsional angles representation and V.$” is the curvature matrix of the potential at the saddle point. The unique negative eigenvalue J_,identifies the reactive mode, and it allows the calculation of the transition rate w(i~k)=(IA,I/27C)exp(-~slkeT)
-10
given the free energy difference A& between the saddle point and the starting configuration [ 121. The corresponding eigenvector u’ defines the most probable direction of crossing the saddle point. By se504
-8
-6
-4
-2
0
2
4
6
8 il0
(6) Fig. 2. Reorientations AIY,of the rotors caused by a torsional transition of unitary length between II_, and a, for p=O.5 (triangles), p= 1.0 (circles) and p=2.0 (squares). The dashed line indicates the orientational displacements required to reach the new equilibrium position.
librium orientations (dashed line in fig. 2) at large distances.
with the drift of its centery( t) given by the equation
[I61 y{t)&+exp(
3. Rotational relaxation after a amfonnational transition A simplified pictureof the effects of the librational motion after a given &k) conformational transition, can be derived according to the following approximations: (i) The state of the system before the transition (t=O- ) is approximated by the Gaussian distribution obtained from the expansion of the equilibrium distribution about the starting conformation. The distribution function on the set of torsional angles x at t=O- is written as
,
(8)
where x3 indicates the jth minimum of the internal potential and G(z) is the normalized Gaussian distribution, G(Z)= IDet(VC2’/2JckaT) iIf2 xexp( -z%%/2kgT)
,
(9)
determined by the curvature matrix Vc2) of the potential minima in the torsional angle representation. Because of the factorized form of the potential eq. (4)) V(‘) has only diagonal elements given by k,T/ 2, with 2 indicating the mean square displacement of the torsional angles within a potential well. (ii) The conformational transition at t= 0 shifts the center of the Gaussian distribution by dCj-+k) without modifying its shape. In the following, y(t) will indicate the center of the Gaussian distribution as function of the time, therefore y(O+)=p(O-)+dO’-Jc)
)
1,
[y(O+) -d].
(12)
(13)
Wt)=Y(t)-Y(O-)
the corresponding changes Atii(t) of the rotors orientation can be calculated as function of the time. In fig. 3 they are represented for some value of the time T scaled with respect to the elementary frequency of the librational motion _
(14)
The curvature ratio p= 1 was taken in the calculations_Fig. 3 clearly shows that the librational motion drives the reorientations of the rotors after the conformational transition with different time scales depending on the distance from the location of the transition. The time delay &, defined as the time required by the ith rotor to reduce to half the angular distance from the new equilibrium position, )=fAai(~),
(15)
(10)
with y(O-) =xj. (iii ) The librational relaxation at t > 0 is described by the diffusion equation (3) with the harmonic approximation for the potential of the kth conformation. Under this condition, the Gaussian shape of the distribution is preserved also fur t> 0, P(x; t) =(7x-y(t)
-DV%/kJ)
(iv) The average change of the torsional angles is calculated from the displacement of center of the Gaussian distribution. From the average displacement of the torsional coordinates
z=2D,t/x’! P(x;O-)=G(x-1’)
19October 1990
CHEMICAL.PHYSICS LETTERS
Volume 173, number$6
(11)
~20
-15
-18
-5
0
5
10
15
i
20
Fig. 3. Reorientations of the rotors for different times after a transition of unitary length between a_, and cx,,:t=O (circles), 7=8 (qmes), r-64 (triangles) and7=512 (crosses).Tbeorientational displacements less than 10e3 are not represent. The dashed line indicates the new equilibrium position for r=m. 505
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
1000
%I
0 0
750 0 0
0
500
0 0 0 0
250
0
n
5
0
10
15
i
20
Fig. 4. Time delay as function of the position.
is the convenient parameter to characterize the position dependence of the rotational relaxation. They are represented in fig. 4 for the same physical situation of fig. 3. Note that the angular distance between the potential minima xi and .B?does not affect the values of the time delays. The results of fig. 4 can be compared to a residence time rR= 801 for the conformational states, as calculated from a twofold cosine torsional potential with a 5kBT barrier.
19 October 1990
even greater than the residence time of the local conformational states. (B) The extensions of the RIS theory [ 71 to the time domain [ 5,6,8,9], ultimately based on eq. (2) that is on the assumption of a fast equilibration of the system after a transition, describe correctly the rotational dynamics of short chains only, as long as they are characterized by negligible time delays. (C) A fully microscopic theory of the rotational relaxation in long chain molecules (polymers) requires the inclusion of both the conformational states and the librational degrees of freedom, together with an explicit treatment of their dynamical coupling
[161. Acknowledgement This work has been supported by the Italian Ministry of the University and the Scientific and Technological Research. The author is indebted to A. Ferrarini and P.L. Nordio for suggestions and for many stimulating discussions.
References 4. Conclusions Theoretical tools suitable to describe in detail and self-consistently the coupling between the librational motion and the conformational transitions are not available. The analysis presented here is rather crude since it is essentially based on the deterministic behaviour of the system (see eq. ( 12) ) without taking into account the spreading of the distribution which is likely to occur because of transitions at different times. Furthermore, no attempt is made to describe the interference effects coming from transitions at different chain positions. In spite of these simplifications, some general conclusion can be drawn. (A) The reorientations determined by a conformational transition at a given position, propagate along the chain with increasing time delay. With long enough chains, the time delays can be comparable or
506
[ 1 ] H.A. Kramers, Physica 7 ( I 940) 284. [2] J.S. Larger, Ann. Phys. 54 (1969) 258. [3] J. Skolnickand E. Helfand, J. Chem. Phys. 72 (1980) 5489. [ 41 E. Helfandand J. Skolnick, J. Chem. Phys. 77 (1982) 1295. [5] A. Ferrarini, G. More and P.L. Nordio, Mol. Phys. 63 (1988) 225. [6] G.J. Moro, k Ferrarini, A. Polimeno and P.L. Nordio, in: Reactive and flexible molecules in liquids, ed. Th. Dorfmuller (Kluwer, Dordrecht, 1989) p. 107. [7] P.J. Flory, Statistical mechanics of chain molecules (Interscience, New York, 1969). [ 81 I. Bahar and B. Erman, J. Chem. Phys. 88 (1988) 1228. [ 91 I. Bahar, B. Erman and L. Monnerie, Macromolecules 22 (1989) 431. [lO]G.Moro,Chem.Phys. 118(1987) 167,181. [ 111I.E. Shore and R. Zwanzig, J. Chem. Phys. 63 (1975) 5445. [ 121 G.P. Zientara and J.H. Freed, J. Chem. Phys. 15 (1983) 3077. [ 131 R. Cook and L.L. Livomese Jr., Macromolecules 16 (1983) 920. [ 141R. Cookand E. Helfand, J. Chem. Phys. 82 (1985) 1599. f 15] A. Ferrarini, G. Moro, P.L. Nordio and A. Polimeno, Chem. Phys.Lztters 151 (1988) 531. [ 161G.J. Moro, manuscript in preparation.
CHEMICALPHYSICSLETTERS
19October 1990
About the coupling between librational motions and conformational transitions in chain molecules Giorgio J. Moro Istituto di Chimica Fisica, Universitridi Parma, via/e delle Scienze, 43100 Parma. Italy
Received23 July 1990
The conformationaltransitions lead to reorientationsof a smallfraction only of a chain molecule.This followsfrom the analysis of the barrier crossing problem according to the Framers theory. Therefore, the librational motion operating after a transition is responsible for the attainment of the new equilibrium distribution. A simplified treatment is presented for the model system of an infinite chain of rotors. It is shown that the reorientations of the chain units induced by a conformational transition, propagate along the chain with increasing time delays. T’hisconstitutes an effective mechanism of coupling between the librational motion and the conformational dynamics in long chain molecules.
1. Introduction
In short chain molecules the librational motions (i.e. small amplitude displacements of the torsional angles about the locations of the minima of the intramolecular potential) and the conformational transitions are uncoupled in the presence of sizable energy barriers separating the conformers. Thus,. given the potential and the friction components, the Kramers theory [ 1 ] and its generalization to multidimensional diffusion operators [ 2 ] can be used to calculate the transition rates w(j+k) determining the time evolution of the conformational populations [ 3-61,
afxu= g [Pk(t) w(k+A -P,(O +e.M) I. at
be fully specified by the dynamics of the conformer population equation ( 1). The rotational relaxation of a chain unit with respect to a laboratory frame is treated in a similar fashion, by considering also the motions of the molecule as a whole and the recoil rotations defined as the reorientations of two chain segments after a rotation around their connecting bond [lo]. This method, however, is not adequate to describe the rotational relaxation in very long (infinite) chains. In fact, eq. (2) implies that the chain units reorient all together after a conformational transition. In other words, the rotational displacements induced by a conformational transition propagate instantaneously along the chain, no matter how far
(1)
Furthermore, the rotational dynamics of the system can be described in terms of the rotational isomeric state (RIS) theory [ 71 generalized to the time domain [ 5,6,8,9], under the condition that the tor-
sional variables x relax very quickly (i.e. in a time much shorter than the lifetimes of the conformational states) to the new equilibrium position xj (given by the potential minimum) after a transition: pj(x; t) X&X-xi) P/(t) .
(2)
Therefore, the torsional angles relaxation results to ElsevierScience Publishers B.V. (North-Holland)
Fig. I. The chain of rotors represented as an ensemble of rods constrained to rotate about a common axis.
503
CHEMICAL PHYSICS LETTERS
Volume 173, number 5,6
away. To avoid this unphysical behaviour, one should describe in detail how the system approaches the new equilibrium position after a transition. Obviously this requires an analysis of the coupling between the librational motion and the conformational transitions. A simplified treatment is presented here for the model system of an infinite chain of identical rotors sketched in fig. 1.
2. Chain reorientation during a confomational transition
The chain of rotors was already considered in the past to study cooperative motions [ 11,121 or as model system for polymers [ 13,141. In the overdamped regime, the following diffusion equation describes the time evolution of the system:
ap(a;
~
t)
=D$(&
at
+
&~P(a;i),
(3)
where the array (Ycollects the orientations oi of the rotors with respect to a laboratory frame (see fig. 1) and Do is the diffusion coefficient of each rotor. The potential V is assumed to be factorized with respect to the torsional angles x, = LY, - CY,_, , W)=
(4)
c KG;> > i
with the elementary torsional potential V, having a fixed number of equivalent minima. In order to apply the Kramers theory, the following normal mode analysis should be performed near the saddle point (S) of the internal potential connecting two stable states j and k [ 5,6]: ( 1 /kB T) DV~*)U~+“U~
,
19 October 1990
letting the initial and the final positions according to the constraint of the least potential energy along u ‘, one calculates the displacements A(j+k) of the torsional coordinates caused by a conformational transition. The only free parameter is the curvature ratio p of the elementary torsional potential V,(x), defined as:
where xmin (xmax) is one of the equivalent minima (maxima) [ 151. The details of the calculations will be presented in a forthcoming paper [ 161. In fig. 2 the orientational displacements Aarjof the rotors due to the conformational transition with a unitary change of the torsional angle x0, are represented for some values of the parameter p. It is evident that, independently of the particular features of the potential, the rotational displacements of few rotors only are required in a conformational transition. This illustrates the localized nature of the conformational transitions in polymers, as discovered by Skolnick and Helfand [ 3,4], which explains the independence of the interconversion rates on the chain length. But one can look at this result from another perspective: a conformational transition generates a non-equilibrium configuration and the librational motion operating after the transition must be responsible for the attainment of the new equi-
(5)
where D is the diffusion matrix in the torsional angles representation and V.$” is the curvature matrix of the potential at the saddle point. The unique negative eigenvalue J_,identifies the reactive mode, and it allows the calculation of the transition rate w(i~k)=(IA,I/27C)exp(-~slkeT)
-10
given the free energy difference A& between the saddle point and the starting configuration [ 121. The corresponding eigenvector u’ defines the most probable direction of crossing the saddle point. By se504
-8
-6
-4
-2
0
2
4
6
8 il0
(6) Fig. 2. Reorientations AIY,of the rotors caused by a torsional transition of unitary length between II_, and a, for p=O.5 (triangles), p= 1.0 (circles) and p=2.0 (squares). The dashed line indicates the orientational displacements required to reach the new equilibrium position.
librium orientations (dashed line in fig. 2) at large distances.
with the drift of its centery( t) given by the equation
[I61 y{t)&+exp(
3. Rotational relaxation after a amfonnational transition A simplified pictureof the effects of the librational motion after a given &k) conformational transition, can be derived according to the following approximations: (i) The state of the system before the transition (t=O- ) is approximated by the Gaussian distribution obtained from the expansion of the equilibrium distribution about the starting conformation. The distribution function on the set of torsional angles x at t=O- is written as
,
(8)
where x3 indicates the jth minimum of the internal potential and G(z) is the normalized Gaussian distribution, G(Z)= IDet(VC2’/2JckaT) iIf2 xexp( -z%%/2kgT)
,
(9)
determined by the curvature matrix Vc2) of the potential minima in the torsional angle representation. Because of the factorized form of the potential eq. (4)) V(‘) has only diagonal elements given by k,T/ 2, with 2 indicating the mean square displacement of the torsional angles within a potential well. (ii) The conformational transition at t= 0 shifts the center of the Gaussian distribution by dCj-+k) without modifying its shape. In the following, y(t) will indicate the center of the Gaussian distribution as function of the time, therefore y(O+)=p(O-)+dO’-Jc)
)
1,
[y(O+) -d].
(12)
(13)
Wt)=Y(t)-Y(O-)
the corresponding changes Atii(t) of the rotors orientation can be calculated as function of the time. In fig. 3 they are represented for some value of the time T scaled with respect to the elementary frequency of the librational motion _
(14)
The curvature ratio p= 1 was taken in the calculations_Fig. 3 clearly shows that the librational motion drives the reorientations of the rotors after the conformational transition with different time scales depending on the distance from the location of the transition. The time delay &, defined as the time required by the ith rotor to reduce to half the angular distance from the new equilibrium position, )=fAai(~),
(15)
(10)
with y(O-) =xj. (iii ) The librational relaxation at t > 0 is described by the diffusion equation (3) with the harmonic approximation for the potential of the kth conformation. Under this condition, the Gaussian shape of the distribution is preserved also fur t> 0, P(x; t) =(7x-y(t)
-DV%/kJ)
(iv) The average change of the torsional angles is calculated from the displacement of center of the Gaussian distribution. From the average displacement of the torsional coordinates
z=2D,t/x’! P(x;O-)=G(x-1’)
19October 1990
CHEMICAL.PHYSICS LETTERS
Volume 173, number$6
(11)
~20
-15
-18
-5
0
5
10
15
i
20
Fig. 3. Reorientations of the rotors for different times after a transition of unitary length between a_, and cx,,:t=O (circles), 7=8 (qmes), r-64 (triangles) and7=512 (crosses).Tbeorientational displacements less than 10e3 are not represent. The dashed line indicates the new equilibrium position for r=m. 505
Volume 173, number 5,6
CHEMICAL PHYSICS LETTERS
1000
%I
0 0
750 0 0
0
500
0 0 0 0
250
0
n
5
0
10
15
i
20
Fig. 4. Time delay as function of the position.
is the convenient parameter to characterize the position dependence of the rotational relaxation. They are represented in fig. 4 for the same physical situation of fig. 3. Note that the angular distance between the potential minima xi and .B?does not affect the values of the time delays. The results of fig. 4 can be compared to a residence time rR= 801 for the conformational states, as calculated from a twofold cosine torsional potential with a 5kBT barrier.
19 October 1990
even greater than the residence time of the local conformational states. (B) The extensions of the RIS theory [ 71 to the time domain [ 5,6,8,9], ultimately based on eq. (2) that is on the assumption of a fast equilibration of the system after a transition, describe correctly the rotational dynamics of short chains only, as long as they are characterized by negligible time delays. (C) A fully microscopic theory of the rotational relaxation in long chain molecules (polymers) requires the inclusion of both the conformational states and the librational degrees of freedom, together with an explicit treatment of their dynamical coupling
[161. Acknowledgement This work has been supported by the Italian Ministry of the University and the Scientific and Technological Research. The author is indebted to A. Ferrarini and P.L. Nordio for suggestions and for many stimulating discussions.
References 4. Conclusions Theoretical tools suitable to describe in detail and self-consistently the coupling between the librational motion and the conformational transitions are not available. The analysis presented here is rather crude since it is essentially based on the deterministic behaviour of the system (see eq. ( 12) ) without taking into account the spreading of the distribution which is likely to occur because of transitions at different times. Furthermore, no attempt is made to describe the interference effects coming from transitions at different chain positions. In spite of these simplifications, some general conclusion can be drawn. (A) The reorientations determined by a conformational transition at a given position, propagate along the chain with increasing time delay. With long enough chains, the time delays can be comparable or
506
[ 1 ] H.A. Kramers, Physica 7 ( I 940) 284. [2] J.S. Larger, Ann. Phys. 54 (1969) 258. [3] J. Skolnickand E. Helfand, J. Chem. Phys. 72 (1980) 5489. [ 41 E. Helfandand J. Skolnick, J. Chem. Phys. 77 (1982) 1295. [5] A. Ferrarini, G. More and P.L. Nordio, Mol. Phys. 63 (1988) 225. [6] G.J. Moro, k Ferrarini, A. Polimeno and P.L. Nordio, in: Reactive and flexible molecules in liquids, ed. Th. Dorfmuller (Kluwer, Dordrecht, 1989) p. 107. [7] P.J. Flory, Statistical mechanics of chain molecules (Interscience, New York, 1969). [ 81 I. Bahar and B. Erman, J. Chem. Phys. 88 (1988) 1228. [ 91 I. Bahar, B. Erman and L. Monnerie, Macromolecules 22 (1989) 431. [lO]G.Moro,Chem.Phys. 118(1987) 167,181. [ 111I.E. Shore and R. Zwanzig, J. Chem. Phys. 63 (1975) 5445. [ 121 G.P. Zientara and J.H. Freed, J. Chem. Phys. 15 (1983) 3077. [ 131 R. Cook and L.L. Livomese Jr., Macromolecules 16 (1983) 920. [ 141R. Cookand E. Helfand, J. Chem. Phys. 82 (1985) 1599. f 15] A. Ferrarini, G. Moro, P.L. Nordio and A. Polimeno, Chem. Phys.Lztters 151 (1988) 531. [ 161G.J. Moro, manuscript in preparation.