J. theor. Biol. (1975) 55, 95-106
Mechanisms of Conformon Creation in ~t-Helieal Strueturest MAREKZ. Z~im~sra~ Department of Theoretical Chemistry, Jagellonian University, 30-060 Cracow, Poland,§ and Division of Chemistry, National Research Council of Canada, Ottawa KIA OR6, Canada (Received 10 June 1974, and in revised form 17 January 1975) The influence of a quadratic exciton-phonon coupling, (i.e., a change of the vibrational frequency of the interpeptide group with vibrational excitation) on conformon properties is studied. The interpeptide group vibrations are treated classically, which is reasonable at room temperatures. A harmonic model is assumed for the interactions of the peptide groups. A new mechanism of the exciton-phonon coupling leading to the formation of a conformon in the a-helix is proposed. It is shown that even small changes in force constants with peptide group excitation have a marked effect on the conformon size and energy. The decrease of this force constant contributes to the localization of the conformon, and, therefore, to the distortion of the polypeptide chain in the region of the excitation. It also stabilizes the conformon state. The opposite effect occurs in the case, where the force constant of the interpeptide group vibration increases with excitation. The cases of large and small conformons are studied separately. 1. Introduction Volkenstein (1972) introduced the concept of a conformon, i.e. a vibrational or electronic excitation of a macromolecule accompanied by a local deformation of this macromolecule. All biologically active macromolecules show enormous deformability in aqueous solution and their activity depends strongly on their conformation (Volkenstein, 1967). Davydov (I973) developed a quantum theory of contraction of ~-helical proteins under the excitation o f their peptide groups which form three parallel chains of hydrogen bonds along the molecule. He showed that the excitation is localized in the region of several peptide groups, and that the distances between the peptide groups in this region are shortened. Therefore the thickness of an ~-helix increases in this region. This localized excitation which we shall call a conformon, moves along an ~-helix on account of resonance interactions between the peptide groups. Davydov suggested that such a conformon moving along t Issued as N R C C No. 14462.
~: NRCC Postdoctoral Fellow 1974-75. § Permanent address. 95
M. Z .
the myosine filament in the myofibril of the muscle causes relative motion of this filament with respect to the actin filaments, and thus leads to a decrease of the muscle fibre length. Such a change of the muscle fibre length occurs at the expense of energy released in the transformation of an ATP molecule into an ADP molecule in the presence of Ca 2+ ions (Volkenstein, 1971). Davydov suggested that the C = O stretching vibration of the peptide group is excited. This process of excitation occurs at the heads of myosin filaments (Davies, 1963). Davydov's theory which states that in a myofibril the whole myosin filament is the working element, not only its heads as in the theory of Davies (1963), is an example of the important role of conformons in biology. Several experimental facts suggest that the working element in the muscle fibril is the actin filament and that the myosin catalyzes only a conformational change of actin filaments instead of pulling them (see Laki, 1974). If this is true, then the thin filaments propel themselves into the domain of the thick filament. Oplatka et al. (1974) observed that ATP can be hydrolyzed and contraction can occur also in the absence of myosin filaments, but in the presence of actin filaments and the heads of the myosin molecules. In such a case a conformon created with the help of energy released by ATP hydrolysis or by binding the ATP molecules to the protein, propagates along the actin filament. The formalism presented here does not depend on the nature of the filament along which the propagation occurs. The aim of this paper is to analyze some conformon properties (viz. size and energy) in the harmonic approximation. In section 2 a new mechanism of conformon creation is proposed in which hydrogen bonds play a special role. In section 3 the large conformon, i.e. a conformon with the dimension of several interpeptide group distances is studied. This limit arises when the resonance interpeptide group interaction is greater than the deformation energy. The opposite limit, the small conformon case, is studied in section 4. 2. Mechanisms of the Exciton-Phonon Coupling in an at-Helix
Before proceeding to the mathematical formulation of the problem, we shall discuss in this section which possible mechanisms in ~-helical structures can lead to the stabilization of the conformon states. For the creation of a local deformation in a macromolecule accompanying its excitation, changes in the interpeptide group vibrations are necessary. This corresponds to an interaction between the excitation of the ~-helix (exciton) and the interpeptide group vibrations (phonons). We call this interaction the exciton-phonon coupling. A first source of the exciton-phonon coupling term is the dependence of the van der Waals interaction between peptide groups upon their excitation.
C O N F O R M O N S IN ~ - H E L I C A L S T R U C T U R E S
Davydov has shown that the linear exciton-phonon coupling resulting from this dependence leads to a shortening of the interpeptide group distances in the region of the excitation. It is easy to show that inclusion of the quadratic exciton-phonon coupling resulting from the same source leads to fl < 0, i.e. a decrease of the interpeptide group frequency, provided the linear coupling reduces the part of the g-helix embraced by the excitation. However, there may be a second source of the exciton-phonon coupling. In this model the interacting peptide groups are joined by the hydrogen bonds ~ N - - H ' . " O = C ~ . The energy released during the ATP hydrolysis is about 0-48 eV. This energy is high enough to excite the N - - H stretching vibration. This can occur either directly, or, as seems more probable because of the higher reactivity of the C = O group, indirectly via the excitation of the carbonyl group to the second excited vibrational state (an energy of 0.46 eV is needed). This state which lies slightly above the first vibrational excited state of the N - H group, can transfer its energy to the latter state through Fermi resonance. The work of Witkowski (1967) and Mar~chal & Witkowski (1967, 1968) on the theory of infrared spectra of the hydrogen bond, points out that the excitation of the X - - H stretching vibration in a hydrogen bond causes considerable changes in the hydrogen bond vibration. These two modes are strongly coupled and the main coupling term is linear. This linear term gives rise, as can be easily seen from the Reid (1959) curves and from Mar~chal & Witkowski (1968) to a shortening of the O " -O distance for the equilibrium potential energy in the - - O - - H . . . O = C ~ hydrogen bonds. A similar situation should occur for the ~ N - - H ' - . O = C ~ hydrogen bonds which couple peptide groups in the ~-helix. This situation is reminiscent of the case of electronic-vibrational coupling in molecules, and we can expect that a small decrease of the N . - . O vibrational frequency can occur after excitation of the N - - H stretching vibration, just as in excited electronic states of molecules. The same is true for the C = O group excitation, but to a much smaller extent. This mechanism of exciton-phonon coupling in the c~-hetixseems to us to be more important than the mechanism proposed by Davydov, as distortions in the hydrogen bonds which couple peptide groups should be large, namely similar to those occurring in carboxylic acid dimers for which the infrared spectra are well understood. The distortions resulting from changes of van der Waals interaction between peptide groups upon the vibrational excitation of the carbonyl groups, on the other hand, are expected to be small. We therefore suggest that hydrogen bonds play a crucial role in the formation of conformons in the ~-helix structures. J
M. Z . Z G I E R S K I
3. Large Conformon in the Harmonic Approximation Let us assume that the interpeptide group interactions are harmonic, and that the resonance interactions between them do not depend, to a first approximation, on interpeptide group distances. The total wavefunction of a conformon is assumed to be ¢(r, xl, . . . . x.) = Y. am(x1 . . . . . x.)4,m(r-md, xl . . . . . x.) where ~.(r-md,
x l , . . . , x . ) = qb+~(r-md, x l . . . . . x.) ~ ~ ° ( r - l d , x l . . . . . x.)
~b°, ~+ being the wavefunctions for the ruth peptide group in the ground and excited state, respectively, and the functions am(X1, x2 . . . . . x.) are to be determined from the following Schrtdinger equation:
ImP( 2M h2 dXm 022 + 0'5M°92x2m)-
] "a.(xl . . . . .
-J[a.+x(Xl . . . . . x . ) + a . _ l ( X l . . . . ,x.)] = E a . ( x , . . . . . x~).
Here x. denotes the deviation of the nth peptide group from its equilibrium position, M is the reduced mass of the relative vibration of peptide groups which are connected by a hydrogen bond, 090 is the frequency of the relative vibration of peptide groups in their ground states, J is the matrix element of the resonance interaction between nearest-neighbours, ~ is the distortion parameter, which describes the linear exciton-phonon interaction [it is connected with the G parameter in Davydov (1973)], and p = (o~12- o~o)/2~oo.2 2 where o~1 is the frequency of the relative vibrations of peptide groups in the presence of the excitation. The zero of energy in equation (3) is taken to be at the centre of the free-exciton band. The derivation of equation (3) is formally the same as that in Holstein (1959) on the molecular model of a polaron, to which we refer for details. Mechanisms which lead to excitonphonon coupling terms in equation (3) were discussed in the previous section. Now, we shall show that at room temperature, where interpeptide group vibrations (phonons) can be treated classically ( k T >> hcoo) the linear coupling term is necessary to create a conformon state; however, an additional quadratic coupling will be shown to have a marked influence on the conformon properties. Neglecting the kinetic energy term of the phonons in equation (3) (adiabatic approximation for the low energetic phonons), we look for the potential energy of phonons in the presence of an excitation (the C = O or N--I-I stretching vibration of the peptide group), i,e. we look for functions a. 0. The conformon embraces approximately u -1
C O N F O R M O N S IN ~ - H E L I C A L S T R U C T U R E S
peptide groups, and moves along the ~-helix because of the resonance interaction J between peptide groups. The continuum approximation is valid for ~-I ~> 1, i.e. approximately for 0.5? 2 ,~ 2J (fl = 0). The value of 0.5? 2, as we shall see in the next section, is the difference between the bottoms of the eonformon band and the free-exciton band in the limit J = 0, and we call it the peptide group deformation energy. So this approximation is valid for a free-exciton bandwidth larger than the peptide group deformation energy. Now, we can calculate the values of x~°) in the conformon region. We have from equations (6) and (17): x.( ° ) -
where any x(,°) ~ 0 means the shortening of the distance between the nth and (n+ 1)th peptide group in the or-helix. The largest distortion occurs of course at the site n o and equals 73
i.e. again for a constant ? we have greater distortion for fl < 0. Therefore, compared to the fl = 0 limit, the weak quadratic exciton-phonon coupling with fl < 0 gives a greater increase in the ~-helix diameter, and thus a closer contact between the myosin filament in the muscle fibre and an actin filament in the region of excitation. Now, equations (8), (19) and (21) give the following expression for the conformon energy
E = - 2 J - 4-~ + ~
Thus we see that the conformon state (17) has a lower energy than the free-exciton state (12) with k = 0 (J > 0), and that the quadratic excitonphonon coupling with fl < 0 leads to a further decrease in the conformon energy. So the conformon state is further stabilized although greater deformation which is present for fl < 0 requires more energy to be returned to the peptide groups in the form of their vibrational potential energy. However, the influence of the quadratic exciton-phonon coupling on the conformon energy is smaller than its influence on the distortion of the ct-helix. We can conclude that a small decrease of the interpeptide vibrational frequency after excitation of the N - - H or C = O stretching vibration in these groups is highly favourable in order to deform and bend the myosin or actin molecule so as to bring it into closer contact with the surrounding filaments. As a result it yields a more effective movement of the myosin filament relative to the actin filaments and thus a shortening of the muscle fibre.
4. Small Conformon in the Harmonic Approximation
To complete our investigation we now discuss the small conformon. This limit arises when (in analogy to a small polaron) the excitation is localized mainly on a single peptide group so that only this group and the two nearest neighbours are markedly deformed. This localized conformon moves along the peptide group chain through random jumps to neighbouring sites (hopping mechanism). Such a situation occurs when the one-peptide group deformation energy (0.5~ 2 for fi = 0) is greater than the resonance interaction between peptide groups, i.e. for y2 > 2J. The limit J = 0 can be easily investigated, as equation (9) possesses the following asymptotic solution for J > 0: J n a(O) = 6n, n° + "/2(1 --3fl) (6n'n°+ 1 +6"'"°-a)
(1 + 2fl)- ~6,, -o
e 1 = ~2(1 - 3fi)- 2J
El = - 0"5y2(1 - 2fl) + 0(p2),
where the subscript 1 of x(,°), e, and E denotes that these values correspond to the limit of a totally localized conformon on a single peptide group (J = 0). In order to investigate a wider range of the y2/2j ratio, i.e. the range (1, oo), we use the single site approximation, which neglects the coherent scattering of an exciton on two different sites (Velick2~, Kirkpatrick & Ehrenreich, 1968; Sumi & Toyozawa, 1971). Such an approximation can give a correct description of a small conformon moving along the a-helix according to the hopping mechanism which means that it scatters almost independently on each peptide group. In the single site approximation, the Hamiltonian of equation (3), in which only site n is taken into account and the kinetic energy term is neglected, can be written in the following second quantized form (Sumi & Toyozawa, 1971): n n = O"5Mcoox,, 2 2 + A , a + a,, + ~., E ( k ) a ~ a k,
where An = - ~ ' ¢ O o X +
. + M flcOoX.,
a n , a. are the creation annihilation operators of an exciton on the nth site,
CONFORMONS IN 0~-ttELICAL STRUCTURES
a~, ak are the same operators in the quasi-momentum representation, and E(k) is the energy of a free-exciton with the momentum k. Introducing the dimensionless quantity q defined by q = x/~rO2o~- ix n
we get Hn = 0"5~2q2-~2q(1 -flq)a+an + ~, E(k)a~ak.
We treat the x~ vibrations classically (the inequality k T >>no% is assumed to be satisfied). Equation (31) gives the free exciton scattered by a potential -y2q(l -flq) in which the probability of a specific value of q is given by
/ - 7e
W(q) = ~ / 2 ~
exp k - 2--k-T/"
Therefore the Green's function of the scattered exciton has the form (Goldberger & Watson, 1964): +co ( y2q2~ G(E) = -cosdq exp - 2 - ' ~ ] ['G°(E)-I +y2q_fly2q2-]-t (33) and
Go(E) = N - ' ~ Go(k, E) k(34) k Go(k, E) being the Green's function of a flee exciton with momentum k and energy E. The poles of the function (33) that lie outside the free exciton band represent the small conformon states. In order to investigate them, we assume the following model Green's function Go(E): Go(E) -1 = 0"5(E + ~ ) . (35) Such a Green's function was used in work on mixed crystals (Onodera & Toyozawa, 1968). It describes the energy band of width 2J, and correctly reproduces the van Hove singularities of the state density function at band edges. Using this form of the free-exciton function we get the following equation for the q values (distortion) and energies of a small conformon: 1 --2flq 2tq -- 2t(l -- 2flq) + 8tq2(1 -- flq)2 -- O, E/J = tq 2 --
1 + 16t2q2(1 __flq)2 8tq(1 - flq) '
where t = ~2/2j. Equation (36) was solved for several values of t and then the conformon energy E (in units J) was calculated for three values of fl(0, + 0.1 and - 0.1). The results are presented in Figs 1 and 2. The figures show that the effect of
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0'8t~ 0.7 I
FIG. 1. The dependenceof the dimensionlessdeformation in the small conformon region on t for three different values of 13.
the quadratic exciton-phonon coupling on a small conformon is similar to that on a large conformon. The coupling with fl < 0 increases markedly the deformation of a peptide group on which the exciton is located, and reduces its energy. The opposite situation arises for fl > 0. In that case there is a greater tendency to delocalize the small conformon for a given J than for p < 0. For the same J value this delocalization tendency is greater than the corresponding localization tendency for fl < 0, as shown in Fig. 2. All our conclusions apply also to the molecular model of a polaron.
5. Conclusions It is suggested that the coupling of the N - - H stretching vibration with the vibration of a hydrogen bond joining peptide groups in an cvhelix plays a dominant role in the formation of conformons in such structures of proteins. This mechanism parallels that proposed by Davydov but is expected to be more effective. It is shown that the small quadratic exciton-phonon coupling resulting from the reduction of the interpeptide group vibrational frequency leads to stabilization of the conformon state in the ~-helix and to its greater
C O N F O R M O N S IN ~ - H E L I C A L
FIG. 2. The dependence of the small conformon energy on t for three different values of 13. The dashed lines represent the case of a strictly localized conformon.
localization both in the large and the small conformon case. Such an interaction, therefore, can account for the action of the muscle fibre, as it leads to closer contact between the myosin and actin filaments in the region of the excitation. The author is deeply indebted to Professor A. Witkowski for reading the manuscript and helpful comments. The financial support of the PAN-3 research project of the Polish Academy of Sciences is gratefully acknowledged.
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