J. theor. Biol. (1977) 69, 239-263

Conformational Dynamics in Biological Electron and Atom Transfer Reactions R. R. DOGONADZE AND A. M. KUZNETSOV Institute of Electrochemistry, USSR Academy of Sciences, Leninskij Prospect 3 1, Moscow V-l I, USSR AND

J. UI,STRUP? Fritz-Haber-Institut der Max-Plan&-Gesellschqft. 1 Berlin 33 Dahlem, Faradalweg 46 (Received 8 November 1976, and in revisedform 8 June 1977) On the basisof a generaltheory for multiphonon radiationlesselectron and atom transfer reactionsa formalismwhich incorporatesmany features of biological reaction mediaand active centres,is outlined. The formalism is basedon an effective Hamiltonian description for the media and the concept of quantum dynamic processes.Within this conceptual framework the reaction probability can be calculatedin generalterms,and such features as conformational adiabaticity, intermediate states with no relaxation of nuclear modes,selectivity of enzyme reactions, and energy recuperationcan be rationalized.

1. Introduction Several attempts have recently been made towards the formulation of a theory of elementary electron transfer (ET) processesin biological systems. These approaches have centred around the following mechanisms. (a) Thermal ionization and subsequent free electron band mobility to the acceptor site (Cope, 1963). (b) Thermally activated “hopping” of an electron along a chain of trapping sites from the donor to the acceptor (Winfield, 1965; Gutmann & Lyons, 1967; Tanako, Swanson, Kaltai & Dickerson, 1973). (c) Modification of (a) and (b) by electron-lattice interactions leading to polaron motion in the protein structure (Cope & Straub, 1969; Kemeny -f Permanent address: Chemistry Department A, Building 307, The Technical University ol” Denmark, 2800 Lyngby, Denmark. T.B. 239 16

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& Rosenberg, 1970; Vol’kenstein, 1972; Kemeny & Goklany, 1973; Kemeny, 1974). (d) Quantum mechanical electron tunnelling between donor and acceptor centres of fixed nuclear configurations (DeVault & Chance, 1966; Grigorov, 1969; Grigorov & Chernavskij, 1972; Blumenfel’d & Chernavskij, 1973). (e) Multiphonon radiationless transitions induced by coupling to the intramolecular and medium modes of the system (Hopfield, 1974; Jortner, 1976). (f) Multiphonon electronic transitions analogous to (e) via an inner sphere mechanism in which a bridge atomic group ensures an efficient overlap between donor and acceptor centres (Dogonadze, Ulstrup & Kharkats, 1973a,b). The various physical mechanisms were recently discussed (Jortner, 1976), and it was concluded that ET and atom group transfer (AT) in biological systems generally must proceed by mechanisms (e) and (f). However, the application of the theoretical formalism extensively developed for both molecular and solid-state radiationless transitions (Englman & Jortner, 1970; Kubo & Toyozawa, 1955) and for thermal ET and AT reactions in low-molecular weight solvents (Marcus, 1956; Levich, 1966; Dogonadze, 1971; Vorotyntsev, Dogonadze & Kuznetsov, 1972; Dogonadze & Kuznetsov, 1973; Kestner, Logan & Jortner, 1974) must then account for the following effects characteristic for biological reactions in contrast to reactions between mobile ions in low-molecular solvents as follows. (i) The ET and AT centres are not freely mobile but fixed in membrane structures. (ii) As a conseqnece of(i) the reactions may proceed in interphase regions between a polar and a less polar medium. (iii) In contrast to ET reactions in low-molecular weight solvents, longdistance ET may be favoured in the globular protein media (Krishtalik, 1974). (iv) Biological processes are frequently accompanied by conformational changes, and there is evidence that the overall rates of certain reactions are determined by such changes (Vol’kenstein, 1969; Blumenfel’d, 1972; Blumenfel’d, 1974). Conformational fluctuations and the coupling between electronic and conformational states leading to the formation of conformons in analogy to polarons have been discussed by several authors (Vol’kenstein, 1969; Kemeny & Rosenberg, 1970; Vol’kenstein, 1971, 1972; Blumenfel’d, 1972; Kemeny & Goklany, 1973; Kemeny, 1974). However, generally these approaches are qualitative, and no validity criteria for the models invoked are given. On the other hand, the quantum theory of chemical reaction rates in condensed media based on (e) and (f) above can be adapted to

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account, in principle quantitatively, also for many features of conformational relaxation in biological processes. This adaption will be the purpose of the present work. 2. Conformational

Properties of Biological Reaction Centres

Biological macromolecules possess a very large number of quasiequilibrium configurations, commonly named conformational states, which are stable towards small atomic displacements and which determine the secondary and tertiary structures of the molecules (Kitaigorodskij, 1973). Since the corresponding equilibrium conformational energies may only differ slightly, the macromolecules are expected to fluctuate between different conformational states. This is borne out by experimental evidence (Lindersterm-Lang & Schellman, 1959; Vol’kenstein, 1969; Blumenfel’d, 1972) and the characteristic time for such transitions, induced by thermal motions in the surroundings, is typically longer by many orders of magnitude than that associated with intramolecular modes (Blumenfel’d, 1972, 1974). However, conformational frequencies cannot straightaway be identified with the low-frequency and long-wave modes which are always present in systems, where a sufficiently large number of atoms perform coupled oscillations of small amplitude, e.g. acoustical frequencies. The nature of conformational transitions is qualitatively different from usual low-frequency modes and more similar to some kind of chemical conversion. Various conformational states usually have different free energies, although degenerate states also may prevail. However, biological macromolecules in vivo have very specific functions which is related to the fact that the energy of one particular configuration of the macromolecule is lower than the energy of all others, e.g. the helix conformation. The molecules therefore stay in this configuration for most of the time and only occasionally move into other states. In this sense the low-energy conformation corresponds to a global energy minimum or absolute equilibrium. The conformational states may be distorted by external forces, such as pressure or an electric field of either an external charge distribution, or a charge redistribution resulting from a chemical reaction. If the perturbation is sufficiently small the distortion only implies a shift in the equilibrium position of the atoms but not a transition to a new conformational state. Such a transition would imply a more drastic rearrangement of the secondary and tertiary structures. In view of the interaction between the conformational states and the electronic charge distribution of the molecule, changes in the latter distribution induce conformational reorganization which in turn affects the electronic structure around the reaction centre. Tn this way a

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transferring electron or atomic group may not only induce distortions in a particular conformational state but also shift the relative energies of different conformational states in such a way that a new conformational state has the lowest energy after the process. We shall name reactions which are not accompanied by changes in the equilibrium conformational state, “conformationally adiabatic”, whereas reactions in which such changes do occur will be called “conformationally non-adiabatic”. 3. The Effective Hamiltonian

for the Conformational

States

We shall account for the role of conformational dynamics in biological processes by introducing effective Hamiltonians for the collective conformational motion. This approach is valid for all kinds of collective excitations and was recently evaluated particularly for the theory of the kinetics of lowmolecular weight reactants in condensed media (Dogonadze & Kuznetsov, 1971, 1973; Dogonadze & Kornyshev, 1972). One of the fundamental concepts in this method, as in the quantum theory of condensed phase chemical reactions in general, is the potential energy surfaces of the different electronic states. For a particular electronic state of a reaction which involves collective motion of either solvent molecules or macromolecules the potential energy surface is represented by a complicated surface in a many-dimensional space, spanned by generalized solvent or conformational co-ordinates. In line with the conclusions above, the surfaces have a number of local minima for particular co-ordinate values in the configurational co-ordinate space, each of which are determined by a large number of atomic co-ordinates. As seen from Fig. 1, each local minimum is separated from the neighbouring minima by local potential energy barriers.

FIG. 1. Potential ponds to absolute

energy V(Q) equilibrium.

as a function

of a configurational

co-ordinate.

Q0 corres-

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The transition of the system from one local minimum to another, e.g. from the value of Q = Q, to Q = Q2 in Fig. 1, proceeds via the interfering potential barrier, and depending on the parameter values of the system, this transition may be either of sub-barrier quantum nature, or a classical transition over the barrier. In both cases the motion along the generalized conformational co-ordinate is fundamentally different from motion along a “normal” co-ordinate of a low-frequency oscillator, for which the corresponding potential energy “surface” is shown in Fig. 2. Thus, the motion along the conformational or solvent co-ordinate to a certain extent possesses a random character; if the system at a given moment has a particular local equilibrium co-ordinate value, then the time evolution of the population of

l-------

__-..--.-. .---..-..

FIG. 2. Potential energy U(q) as a function of the co-ordinate dimensional oscillator.

q for a “normal”

one-

the potential well is not described by simple harmonic motion in configuration space but as a relaxational process. A detailed description of such kind of motion is at present not possible but in a number of cases it is not essential for the understanding of the conformational dynamics. As noted above, the Franck-Condon principle is of crucial importance for non-adiabatic ET and AT reactions. This implies that the reaction only proceeds with a finite probability when the electronic levels in the initial and final states are equalized due to fluctuations in the nuclear sub-system. If the chemical reaction does not induce changes in the conformational states this means that the interaction between the electronic sub-system a.nd the nuclear sub-system which determines the conformational changes is weak. In such cases the role of the conformational motion is restricted to a possible orientation of the reaction centres favorably relative to each other. The corresponding potential energy surfaces in the initial and final electronic

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states are illustrated in Fig. 3, and the Franck-Condon principle is fulfilled in this case by fluctuations only in the intramolecular modes of the active centre and the possible solvent modes. For such reactions the theoretical formalism is identical to that for chemical reactions in low-molecular weight solvents. On the other hand, the potential energy surfaces for a reaction which is accompanied by conformational changes are shown in Fig. 4. The electronic reorganization is here preceded by a complicated motion from Q,i to Q* on the potential energy surface. After the electronic reorganization in the

FIG. 3. Distortion of a conformational potential energy surface when the equilibrium co-ordinate value is shifted from Q,, to QO, (conformational adiabaticity).

FIG. 4. Change of conformational state when the equilibrium from QO, to Q,O (conformational non-adiabaticity).

co-ordinate value is shifted

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intersection region of the surfaces the molecule is formed in a conformational state which deviates strongly from equilibrium and subsequently has to relax into its new equilibrium state from Q* to QJO. The kinetics of such reactions depend on the interaction between the electrons and the nuclei, inducing the conformational changes. If the interactions strongly depend on the detailed topology of the surfaces, then the motion along the conformational coordinate must be considered in corresponding detail. However, the interaction may be determined with sufficient accuracy by a substantially smaller number of parameters, rr” (which usually are electric or other field components and therefore functions of space co-ordinates), the magnitudes of which depend on the population in a group of local potential wells rather than in a single well. For example, the interaction may be determined by the effective electric field from the whole nuclear system outside the reaction centre. In such cases it would not depend strongly on the details of the surface and change continuously when the population of the wells is varied. The theoretical formalism for the kinetics of reactions of the latter kind is essentially a description of the time dependence of the parameters, n,,, and in particular their fluctuation probabilities (Dogonadze & Kuznetsov, 1971, 1!>73; Dogonadze & Komyshev, 1972). This is most simply achieved for linear systems for which the deviations in internal parameters (under the influence of sufficiently small perturbations) are proportional to the perturbations, and where the fluctuations are consequently also small. In the following we shall consider such media and postpone an analysis of nonlinear media to a subsequent section. In a linear medium a change in the state of the active centre induces a change in the equilibrium values of the parameters, n,, proportional to the shift of the corresponding parameter in the active centre. For example, a change in the charge, or a deformation in the structure of the active centre under the influence of mechanical forces will lead to a change in the electric field (or dipole moment), or the deformation field, respectively, proportional to the active centre parameter changes. The fluctuations of the linear medium can now be described by introducing an effective Hamiltonian (Dogonadze & Kuznetsov, 1971, 1973; Dogonadze & Komyshev, 1972). By this method a set of harmonic oscillators of frequencies cc; and normal coordinates Qi are introduced in order to describe the fluctuations of the parameters rr”, i.e.

where Ai is a set of coefficients to be determined subsequently. In practice p takes nearly continuous values, and summations over this variable are in fact integrations over the appropriate frequencies. If n”, being a field component, is a function of the atomic co-ordinates, equation (1) is essentially

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a relation between the Fourier components of the corresponding quantities. The effective Hamiltonians of the conformational subsystem in the initial (i) and final (f) states then take the form : E3F”” = ~+9Chw~(Q::-Q~,i)*+Zi, n, P

(2)

@”

(3)

= rT+3Chw~(Q::-Q~,,)*+Zf, n. P

where p is the nuclear kinetic energy, QiOoiand Qiol the equilibrium coordinates of the effective oscillators in the initial and final state, respectively, and I, and If the corresponding electronic energies. The frequencies and oscillator strengths of the effective oscillators are determined by the fluctuation spectrum and intensity, respectively, for the particular frequency. These quantities can be related to the proportionality coefficients of the linear response of the system to an external perturbation, i.e. to the retarded temperature Green’s function, GR(~) (Abrikosov, Gorkov & Dzyaloshinskij, 1963). Apart from Hy and Hy, the Hamiltonians of the total system in the initial, Hi, and final state, H,, also contain the Hamiltonians of the active centres, HyCand Hy, and of the external solvent, Hi and H;, i.e.

Hi = HPO"+H;'+H:,

(4)

H/= Hy+HF+H;.

(5)

Applying the solvent model developed previously, which essentially implies the introduction of effective Hamiltonians for this subsystem, the total potential energies in the initial and final states take the form:

where qky is a set of solvent co-ordinates related to the Fourier components of the solvent polarization by a boson quantization and summed over all momenta (k) and elementary excitations (v), qLuo and q{“, the corresponding equilibrium values of this co-ordinate in the initial and final state, respectively, okV the corresponding frequencies, ri and rf the set of normal co-ordinates of the active centre in the initial and final state, respectively, and Urc and Uy the intramolecular potentials in the initial and final states. No restrictions have so far been imposed on the latter potentials, and both frequency and co-ordinate shifts, anharmonicity, mixing of modes, etc. can therefore be

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incorporated. It should also be noted that the modes represented by equations (6) and (7) take into account the quantum or classical nature of the conformational, solvent, and intramolecular fluctuations, since no conditions have been imposed on the frequencies of any of these subsystems.

4. The Transition Probability The Hamiltonians of equations (4) and (5) and the potential energies of equations (6) and (7) are formally identical to the Hamiltonians applied in several works on the theory of multiphonon radiationless transitions (Englman & Jortner, 1970) and ET reactions (Levich, 1966; Dogonadze, 1971; Vorotyntsev, Dogonadze & Kuznetsov, 1972; Kestner, Logan & Jortner, 1974). The transition probability per unit time between two potential energy surfaces in equations (6) and (7) can therefore be written straightaway in first order perturbation theory. Instead of presenting the most general results (Vorotyntsev, Dogonadze & Kuznetsov, 1972; Dogonadze & Kuznetsov, 1973), we shall restrict ourselves to a formalism which corresponds to physically realistic parameter values for the systems considered and to a discussion of some implications of the model. Most characteristic relaxation times for transition between different conformational states vary between 1 s and low6 s (Blumenfel’d, 1972, 1974). The oscillators corresponding to the conformational relaxation therefore meet the condition: hw; 4 k,T,

(8)

where k, is Boltzmann’s constant and T the absolute temperature, and they represent a purely classical subsystem. The same condition is fulfilled by the majority of the solvent oscillators for which the highest absorption maximum of electromagnetic radiation is located in the Debye region. A minor part (CZ20 %) of the absorption occurs in a frequency range where the oscillators are of quantum nature (Vorotyntsev, Dogonadze & Kuznetsov, 1970) but for the sake of simplicity this will be neglected in the following. On the other hand, most intramolecular modes of the active centres have high frequencies (500-3000 cm-‘), and at room temperature these modes are therefore adequately classified as a quantum subsystem. Furthermore, these modes do not necessarily have the same frequencies or equilibrium co-ordinates in the initial and final states, nor do they need to be harmonic. We only assume for the sake of simplicity, that interconversion between classical and quantum modes during the reaction can be neglected. The expression for the transition probability per unit time, Wfi, is then (Vorotyntsev, Dogonadze & Kuznetsov,

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1972; Dogonadze & Kuznetsov, 1973; Kestner, Logan & Jortner, 1974) Wfi = w$$cZ~: c exp ( - sV/kBT)S,, w exp [ - (Ey +E, u,w

+ E, - E, + AI)‘/4(Es”” + ES)k,T-J

(9)

where AI is the free energy of reaction,

is the reorganization

energy of the conformational

E, = (87C)-‘C]&i-J&\ k

subsystem, and

24k~rl”Ims(k,

2n

J o ole(k

o) do N21

is the solvent reorganization energy (Dogonadze, 1971; Dogonadze & Kuznetsov, 1971, 1973). e(k, o) is here the dielectric permittivity function of the wave vector k, and frequency o, and Dki and D,, the Fourier components of the electric induction vectors of the field generated by the charge distribution in the initial and final state, respectively. The physical meaning of the reorganization energies is clear from Fig. 5. Furthermore, the transmission coefficient, rc, has the form: K = 2(AE/2)~/[h2&(E~

+E,)k,Ta-

3]*

(11)

i

FIG. 5. Potential energy surfaces determined by the classical co-ordinates of the system, {Ret}. {&} and {R,,} are the equilibrium co-ordinates of the initial and final states, respectively, EA the activation energy, E, the reorganization energy of the classical modes, and AZ the heat of reaction.

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where AE is the resonance splitting of the electronic levels in the intersection region of the potential energy surfaces. The effective frequency of motion on the many-dimensional potential energy surface, ~,~r, is given by the relation (Dogonadze & Urushadze, 1971, 1973):

4kBT%Ime(k, w) x(Ef”^+EJ-’ d I@, ~41’

1

(12)

Finally, u and w are quantum numbers of the intramolecular high-frequency oscillators in the initial and final state, respectively, E, and E, the corresponding energies, and Zr,,, the statistical sum of these oscillators. The Frank-Condon factors, SO,w, take various forms depending on the nature of the intramolecular potentials. Their explicit form can be found elsewhere (Dogonadze, 1971; Dogonadze & Kuznetsov, 1973; Ulstrup & Jortner, 1975), and it is sufficient to notice here that they increase rapidly with increasing u and w due to the increasing overlap of the intramolecular nuclear wave functions. Equation (9) presents the reaction probability in a convenient, physically transparent form. For each set of quantum numbers, u and W, potential energy surfaces with respect to classical co-ordinates can be defined. The system is found on the initial state surface corresponding to the quantum number u with a probability given by the Gibbs factor exp (- EJkJ). It moves along the set of classical co-ordinates and reaches the intersection region with a probability given by the last factor in equation (9) which has the form of an Arrhenius activational barrier for the particular pair of surfaces characterized by u and W. The time spent by the system in this region is determined by ~,rr and the reorganization energies of the classical subsystems, E’,” and Es. During this time the electron is transferred from the initial to the final state with a probability given by K, and the nuclear quantum subsystem undergoes a corresponding transition with a probability given by the Franck Condon nuclear overlap factors, S,+,.

5. Implications

of the Linear Fluctuation Model

Several conclusions of relevance to experimentally investigated biological reactions can be drawn from the formalism presented above: (i) The interaction between the active centres and the conformational subsystem are directly reflected in the reaction probability through the conformational reorganization energy, E,‘““, and the “heat effect”, AI. Since it

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also determines the configuration of the reaction complex, it indirectly affects the tunnelling distance of the nuclear quantum subsystem as well. In conformationally adiabatic reactions only negligible shifts occur in the equilibrium values of the normal co-ordinates, Q&, and for such reactions ,Fn = 0. Conformational distortions may occur, but they are only weakly associated with changes in the electronic states of the active centre. On the other hand, in conformationally non-adiabatic reactions there is a shift in the conformational equilibrium co-ordinates, and Ef”” # 0. In such reactions the population of the various local minima affects the electronic structure of the active centre, and changes in the latter in turn induce changes in the conformational states (Qg,i -+ QzO, # Q$J* The stronger the coupling between the active centre and the conformational subsystem, the higher is also Ef”“. (ii) The conformational reorganization energy, E,EO”, is related to the temperature Green’s function which is generally a finite temperature measure of the linear response of the system to an external perturbation (Abrikosov, Gorkov & Dzyaloshinskij, 1963). The equilibrium values of the generalized conformational co-ordinates, Qi,, are thus related to the force field components of the active centre, fi, by the linear equation (Abrikosov, Gorkov & Dzyaloshinskij, 1963; Dogonadze & Kuznetsov, 1971);

This gives for the conformational

reorganization

energy: (14)

where f~i and fif are the force field components of the active centre in the initial and final state, respectively. The response function, Im G,(w,), can be related to various physical properties of the macromolecule depending on the nature of the interaction between the active centre and the surroundings. For example, if the interaction is electrostatic, Im G,(o,) is related to the dielectric permittivity function see equation (lo)]; if the interaction arises from elastic deformations, Im G,(w,) contains the elasticity module, and if f” represents a pressure field, then Im G,(w,) depends on the compressibility of the macromolecular medium. The quantity AZ (Fig. 5) is determined by the sum of the electronic energy differences of the active centre in the initial and final states, and the corresponding differences in equilibrium conformational and solvation energies associated with the interaction between the active centre and the surroundings. Since the latter energies depend on the medium response to the force field

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from the active centre, they can also be expressed by Im G,(o,), viz., EF”“= xEii; ET= CE;, 4

solv -_ g Eli;

EF”

= z E;,

(15) (16)

where EC”” and Es”” are the inertial parts of the conformational and salvation energy, respectively (the inertialess parts are included in the electronic energy difference below). Since : Eii = 2(m0,,)-~ Im G,,(o$(J$~; Eli = (~IT’w,)-

EE, = 2(nw,)-’

Im GL(O,)(Dki)‘;

El,

Im G,,(c~,)(j~~)~,

= (47r20,)-’

Im Gk(q)(DkfJ,

(17) (181

and AI = AZ, +Ey

- ,y” + ,;,l” - EqO’“,

(19)

it is noted that also AI depends on Im G,(o,) and Im GL(ul,). In general, large values of Im G,,(o,) and Af," = f;,--fpni, i.e. strong coupling between the active centre and the conformational modes, favour large conformational reorganization energies. Since the activational factor in equation (9) strongly decreases with increasing EC”“, we expect conformationally adiabatic reactions to be faster than conformationally non-adiabatic reactions, provided that other conditions are identical. (iii) The effective frequency, o,rr, determines the time spent by the system in the intersection region of the potential energy surfaces of the classical modes. It has the form of a weighted average of all the classical frequencies [equation (12)], and the contribution of a particular mode is larger the higher its frequency and reorganization energy. ~,~r is therefore only significantly influenced by frequencies corresponding to transitions between conformational states, in conformationally strongly non-adiabatic reactions. From equation (12) it is noted that ceeffis largely determined by the conformational parameters when the condition :

i 4k]T” o Im a(k, w)/je(k, o)12 dw, 0

(20)

is fulfilled. The value of m,tf is furthermore only important in electronically adiabatic reactions. For electronically non-adiabatic reactions K 4 1 and must be included in the rate expression of equation (9). However, since K is inversely proportional to w,rl, the overall dependence of Wfi on o,tl cancels, and c),tt only determines the non-adiabaticity condition. The important conclusion can therefore be drawn that the rate of electronically non-adiabatic reactions does not depend on the relaxation time for conformational trans-

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itions, provided that they proceed under stationary conditions. Literature reports (Blumenfel’d, 1972, 1974) that the conformational relaxation times determine the rate of the overall reactions must therefore concern either electronically adiabatic reactions or non-stationary reactions. For electronically adiabatic reactions rcS,, w --, 1. For sufficiently exothermic reactions and in the absence of high-frequency quantum modes equation (9) shows that Wfi then approaches w,rr/2n, i.e. the overall rate may be determined by the conformational relaxation time. (iv) Equation (9) formally contains a sum of Arrhenius factors weighted by the Franck Condon overlap factors. However, even in the absence of co-ordinate shifts in intramolecular quantum modes, the temperature dependence of W,i may be more involved than apparent from equation (9). This is due to the special nature of the globular macromolecular structures, and in terms of the present model it is reflected in a temperature dependence of the parameters of the effective Hamiltonian. The quantities Qg which characterize the interaction between the active centre and the protein matrix, do not depend on the detailed conformational structure but rather on certain bulk parameters which determine the occupation of particular conformational states. These parameters depend on temperature, and by variation of the latter, changes occur both in the occupation of the various conformational states and in the probability of transitions between them. For a linear medium this is formally reflected as a temperature variation in the response of the molecular properties to external forces, i.e. in a temperature variation of ImG,,(o,) [cf. the temperature variation of the dielectric permittivity, s(k, o)]. The temperature dependence of the medium response properties can be illustrated by a system in which intramolecular :quantum ‘modes are absent, and the interaction between the active centre and the medium is dominated by a single conformational mode. Es, E,, and E, can then be ignored in comparison to E,‘““, and the temperature dependence of Wfi can be written: -d In Wri/[d (l/k,T)]

= (ES”” +AZJ2/(4E3 +(l/k,T) d [(E$” +AZJ2/4E9”“]/[d

(l/k,T)].

(21)

Inserting equations (14), (17) and (19) into equation (21) and assuming that the reaction is not strongly exothermic or endothermic, i.e. (AZ,)” Q 4EykBT, this equation can be rewritten :

-d In Yd

(l/W)

= K(fr -Aj2 +f: -., ?/4(ff

-0’ >

x [Im G’“” +fl/k,T) dImG’““/d (ljk,7’)] +([~f~-~)*+ff~~~]AZ~}/[2tf~-fi)2]~

(22)

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where G”“” is the retarded temperature Green’s function of the conformationa mode considered. From equation (22) it is obvious that if the temperature dependence of Im G’“” is sufficiently strong, W,, may depend in a complicated fashion on T. For example, if Im G’“” is proportional to T, then the second factor of the first term of equation (22) vanishes, and In Wfi decreases in a linear fashion with increasing T-'. On the other hand, if Im G’“” is inversely proportional to T, then the same factor of equation (22) equals 21m G”“” CCT-', and the “effective” activation energy decreases with increasing temperature. Physically the temperature dependence of Im G,(o,) leads to a temperature dependence of the coupling between the active centre and the conformational s-ubsystem, i.e. Ey and AI. Im G,(w,) may display a particularly strong dependence on T in certain temperature regions corresponding to the occurrence of denaturation, phase transitions, and other co-operative processes resulting from substantial changes in the secondary, tertiary, and quaternary structures. Even the relatively crude model based on the effective Hamiltonian is therefore in principle capable of accounting for the complicated temperature dependence observed in certain biological processes.?

6. Extension to Non-linear Interactions The theory presented above is adequate for macromolecular and solvent media which respond linearly to external perturbations. If the fluctuations of the parameters Z, (which determine the interaction between the active centre and the surroundings) are non-linear the reaction probability can still be estimated by a different, semiclassical approach developed for ET and AT reactions involving small mobile ions (Dogonadze & Urushadze, 1971; Dogonadze & Kuznetsov, 1973). In this procedure the intramolecular quantum modes are deconvoluted as before, and the parametric dependence of the free energies on the generalized classical co-ordinated, n,, in the initial and final states, Ui(n,) and UJrc,) assumed to be known. A set of free energy surfaces, corresponding to different values of the nuclear quantum numbers, is thus defined, and the reorganization of the total quantum subsystem occurs at the intersection hypersurface of the classical modes, where the quantum energies of the initial and final states equalize. The probability of reaching this region is determined generally by the MaxwellBoltzmann distribution function with respect to 7c, and 5,, but the domt If the frequenciesof some modes have values in the region k,T/fi, they may change from being largely quantum to largely classical over a sufficiently wide temperature interval. This will also give rise to a temperature dependence of (d In WJ[d (l/kJ)].

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inating contribution to the reaction probability is provided by trajectories passing the saddle point of the hypersurface. It should be emphasized that the formulation of the reaction probability for non-linear media requires a more detailed information about the rc,dependence of the conformational free energy. In return, a more involved functional dependence can also describe more features of the reaction, compared to the effective Hamiltonians. In particular, a change of the functions tJ,(n,) and U/(n,,) with temperature may account for the abrupchanges of the activational parameters observed in certain systems (Blumenfel’d, 1974). 7. The Dynamics of Composite Reactions So far we have considered processes in which the discrete electronic structure of an active centre is changed by the occurrence of a single elementary reaction. However, biological reaction centres are usually arranged in ordered membrane structures in which the centres successively accept and donate electrons. This is so in particular for the respiratory chain in mitochondrial membranes and for the primary elementary ET steps in photosynthesis. Several ET centres are thus typically involved in a strictly ordered sequence of ET reactions which are highly selective towards ultimate donors and acceptors. The presence of several intermediate electronic and/or conformational states in the overall reactions implies that a quantum mechanical theory must operate with several potential energy surfaces each corresponding to a specific intermediate state. The transition of the total system from the initial to the final state is then analogous to its motion between the corresponding surfaces via the many-dimensional intermediate state potential energy surfaces. The transitions between each pair of surfaces may proceed essentially in two different ways (Dogonadze, Ulstrup & Kharkats, 1974; Dogonadze & Kuznetsov, 1977): (i) If the relaxation times, r,, of all the modes of the system in the intermediate state are small compared to the characteristic time, t,, required for the system to reach the region of intersection with the potential energy surface of the immediately following state, i.e. if z,


Ui(Qzl). After the process the conformational s,tate deviates strongly from equilibrium, and the system begins to relax along the co-ordinate QP. However, provided that U,,(Qt,) > U,,(QzZ), where U,,(Q,) corresponds to the first intermediate potential energy surface, the system may not succeed in dissipating its energy but proceeds without relaxation to the point Qp* where a subsequent ET or AT reaction may occur. This is particularly obvious if the relative position of the surfaces is that of Fig. 6(b); in such cases the system does not even pass through the equilibrium configuration of the intermediate state. The overall activation t This concept and its theoretical implications is a more stringent tool for the analysis of co-operative elementary chemical reactions than the usually applied terminology of “concerted” or “synchronous” reactions. T.B. 17

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energy for the dynamical process is determined by the intersection point Q& of highest absolute energy. However, the potential energy surfaces corresponding to real systems are many-dimensional. Relaxation of a particular parameter into a new equilibrium value therefore does not necessarily imply relaxation of all energy space. The relaxation of the given parameter may be the result of a complicated motion along many modes without change of the total energy (Kharkats, 1972). In the general case, for many-dimensional surfaces expressions

FIG. 6. Potential energy surfaces for a reaction passing through two intermediate states with no relaxation in the classical modes. In (b) the system passes from the first to the second intermediate state surface before reaching the minimum of the former.

for the probability of the dynamical transition can be given, provided that one particular saddle point, e.g. between the surfaces U,,,(Q,) and U,,,,,(Q,) is considerably higher than all the others. The quantitative criteria for this condition can be formulated by noting that a classical trajectory of the system beginning at this point {Q$), may proceed with thermal velocity to the equilibrium co-ordinate values of either the reagents or the products only through classically available regions of configuration space. If the conformational fluctuations are linear, the classical trajectory method then gives for the overall transition probability per unit time (Kuznetsov &

CONFORMATIONAL

K.harkats, 1976; Kharkats, m,m+l Wfi = %Lk.

2iT

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1972):

n,m+Jj~~-~.~~~s,s+~exp C-(Zm-ZJ/k,T)l

x exp [-(EIC,“i,,+l

+E,m’m’1+AZ,,m+1)2/4(ES~,,+1

+E::9’n+‘)k,T], (1 < m, s > m) (24)

where the indices m, m+ 1 refer to elementary transitions from the mth to the (m+ I)th intermediate state. If high-frequency intramolecular modes are also present, averaging over these states can be included as well, in line with equation (9) and previous work (Dogonadze, Ulstrup & Kharkats. 1074). However, for the sake of simplicity this will be omitted here. The transition probability for dynamical reactions is obviously extremely sensitive to the properties of the potential energy surfaces. This can be ihustrated further by considering a transition via one intermediate state. If we further assume that the frequencies of all the conformational modes are equal, a proper turning of the co-ordinate axis shows that the motion on the many-dimensional surfaces is equivalent to motion on a three-dimensional surface determined by two of the new co-ordinates (Kharkats, 1972). Crosssections of such surfaces are shown in Fig. 7 for different relative positions of the surfaces.

RG. 7. Sections of constant energy of potential energy surfaces for motion in a manydimensional co-ordinate space through one intermediate state. (a) Only one equilibrium co-ordinate value is shifted. (b) Both equilibrium co-ordinate values are shifted.

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AND J. ULSTRUP

For the relative positions shown in Fig. 7(a) only one equilibrium coordinate value is shifted. The trajectory then follows the q1 axis almost completely, and the activation energy is determined by the saddle point of the highest energy, as seen from the reaction profile shown in Fig. 8(a). On the other hand, if both modes undergo equilibrium co-ordinate shifts [Fig. 7(b)] this same picture only arises if the energy of one saddle point,

L--.--

-___

-

.-.

.-

.-

FIG. 8. Potential energy profiles for motion along the co-ordinate q1in Fig. 8(a) corresponds to Fig. 7(a); (b) to Fig. 7(b).

e.g. the point (qrr, q&) is considerably higher than that of the other one, in which case the activation is approximately determined by the former. If the energies of both saddle points are similar, then the potential energy profile along q1 takes the form shown in Fig. 8(b). Such a trajectory would be too energy-consuming, and it can be shown (Kharkats, 1972) that a curvi-linear trajectory is energetically more advantageous. However, this also requires additional energy over the energy of the highest saddle point, and more the higher the curvature of the trajectory. Physically this activation energy increase corresponds to deformation in additional chemical bonds and is the basis of the highly selective nature of enzyme catalysis and other biological reactions. Thus, if the reaction can proceed along two routes corresponding to approximately equal saddle point energies (Fig. 7) then the system will follow the trajectory of the smallest curvature. We can therefore conclude that the key problem in the general formulation of the reaction probability for many-dimensional systems is not the accumulation of energy in a particular mode but the most favourable distribution of the energy among the various modes, for the system to pass along a curvilinear classical trajectory in phase space from the initial to the final

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DYNAMICS

259

state. If the potential energy surfaces are located favourably relative to each other, then the system will reach the second saddle point [(&, L&) in Fig. 71 within a time of the order of the characteristic vibrational time for the classical modes in the intermediate state after it has gained sufficient energy to pass the first saddle point. On the other hand, if the relative position of the surfaces is unfavourable, more energy is required than corresponding to the highest saddle point. If this additional energy is not acquired the system will either not reach the second saddle point at all, or only do so after a much longer time than the relaxation time in the intermediate state, due to a. much more involved trajectory in the intermediate state. We therefore conclude that a mediator effectively transmitting the electron by modifying the relative position of the potential energy surfaces may fundamentally change the rate and direction of the chemical reaction. Finally, the concept of quantum dynamical reactions also conveniently ctovers energy recuperation in biological processes (Blumenfel’d, 1974). This can be illustrated by reference to Fig. 9 which shows the potential energy surfaces of a three-level reaction. The first step is strongly exothermic, corresponding to the formation of some chemical bond (Q,), whereas the second is strongly endothermic implying that some other chemical bond

FIG. 9. Potential energy surfaces for a reaction consisting of an exothermic and subsequent endothermic step. The surfaces are suitably situated for recuperation of the energy liberated in the first step.

260

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(Q,) is ruptured. If the intermediate state surface is located in such a way that the system does not reach the second saddle point before all modes have relaxed, e.g. if the second step is too endothermic, then all energy liberated by the first step is dissipated. On the other hand, by a position of the intermediate state surface suitable for reaction, such as shown in Fig. 9, the system will reach the second intersection region with practically no loss of energy after having passed the first one, and the rupture of the second chemical bond may occur by means of the recuperated energy from the first transition. Only a minor part of the total energy may be dissipated as vibrational energy, as indicated in Fig. 9. For reactions which involve a strongly exothermic first step the energy recuperation may be described in an analogous fashion. This seems to be the essence of several coupled ET reactions in the photosynthetic and respiratory chains, where the recuperated energy of an ET step can be used in the synthesis of energy-rich chemical bonds such as the synthesis of ATP from ADP (Chance, De Vault, Legalais, Mela & Yonetani, 1967; Blumenfel’d Koltover, 1972). 8. Discussion We have presented a general quantum mechanical formalism for the theoretical description of important features of elementary and co-operative chemical reactions in biological systems. The formalism is based on the effective Hamiltonian model for media (solvent and protein) which respond linearly to external perturbations, and a semi-classical approach for nonlinearly responding media. Within these models the formalism accounts in quantitative detail for the behaviour of the various subsystems of the reacting macromolecules, i.e. the electrons, the high-frequency intramolecular modes, the conformational modes, and the solvent. The fundamental conclusion is that reactions involving biological macromolecules and reactions in lowmolecular weight systems can both be described within the same formalism but in estimates of kinetic parameters and phenomenological relationships, such as the Briinsted and Arrhenius laws, account must of course be taken of the different nature of the media in which these reactions occur. In most general terms this is done by means of the imaginary part of the appropriate temperature Green’s functions, Im G,(c+,) which determine the field response functions, or more concretely, the solvation and reorganization energies. The formalism presented conveniently accounts for the effects (i)-(iv) listed above. The most pronounced distinctions between reactions in macromolecular and low-molecular weight systems are the presence of conformational fluctuations (Vol’kenstein, 1969; Blumenfel’d, 1972) and of intermediate states with no conformational relaxation in the former systems. It

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DYNAMICS

261

was noted that the conformational fluctuations enter as an additional classical subsystem, which contributes to the effective frequency, reorganization energy, etc., as any other classical modes. However, the actual frequency values entering these expressions are not identical to the real conformational vibrational frequencies but related to them via the boson quantization of the field of interaction with the active centre, and the corresponding response functions. For these reasons there is no justification for defining elementary biological processes solely in terms of conformational relaxation, such as suggested earlier (Blumenfel’d, 1972) even though the conformational subsystem may give the dominating contribution to the activation energy and effective frequency. In fact, it was noted that for a certain class of reactions, i.e. electronically non-adiabatic ET reactions, the reaction probability only depends on the conformational fluctuations via ErO, and the heat of reaction. Furthermore, the complicated dependence of the reaction probability on temperature and other parameters does not in itself invalidate the conceptual framework usually applied in chemical kinetics, e.g. the Arrhenius law. Provided that elementary reactions can be def?ned, these laws are also valid for biological reactions, but they require quantitative modification in view of the dependence of Im G”(o,) on these parameters. In quantum dynamical reactions intermediate states may provide special reaction channels in two ways. Firstly, since the estimated ET distance in the bacteria-chlorophyll~ytochrome c reaction is considerable (2 12-l 3A) (Jortner, 1976) the electronic exchange matrix elements are small. This raises the possibility that a superexchange coupling mechanism via highenergy virtual, states may operate (Anderson, 1950; Halpern & Orgel, 1960; Dogonadze, Ulstrup & Kharkats, 1973a,b). The overall matrix element would then be:

where VR is the matrix element for coupling the initial and final states directly, V>,r and V~i the corresponding matrix elements for coupling to an intermediate state, d, and E represents the total energy for the state given by the subscript (Kharkats, Madumarov & Ulstrup, 1974). Even if (Ed-&) z 1 eV, values of V&rand I$, of about 3 x lo-‘[eV are sufficientIt provide a dominating superexchange contribution to the integral (Jortner, 1976). However, in quantum dynamical enzyme reactions the second way of operation of the intermediate state is more likely. The energy of these states is here sufficiently low to ensure that the activation energy for transition to the intermediate state is lower than the activation energy for the direct transition from the initial to the final state. This was discussed extensively above and in previous reports. As also noted above, the incorporation of

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intermediate states with no conformational relaxation covers effects expressed in several theories of enzyme reactions: entropy trapping (Koshland, 1962), lowering of activation energy (Eyring, Lumry & Spikes, 1954), induced conformational changes (Perutz, 1967; Vol’kenstein, 1971), and energy recuperation (Blumenfel’d, 1974). Finally, we notice that the conformational relaxation times of isolated membrane components of the respiratory chain has been estimated to be much longer than the turn-over rates of these components in ET reactions in the membranes (Pecht & Faraggi, 1972). This suggests that the coupled membrane ET reactions are plausibly described as quantum dynamic processes in line with Fig. 6. Moreover, it can be shown that the mechanism suggested recently for the primary processes of the ATP synthesis, associated with this chain of ET steps (Blumenfel’d & Koltover, 1972; Chance, 1972) can straightaway be reformulated in terms of the present theory of quantum dynamic processes and energy recuperation. This work was initiated while A.M.K. was a visitor at Chemistry Department A of the Technical University of Denmark and completed during the stay of J.U. at the Fritz-Haber-Institut. We wish to express gratitude to the two institutes for hospitality. The Fritz-Haber-Institut and the Ministry of Education of Denmark are also thanked for financial support. REFERENCES Methods of Quantum Field Theory in Statistical ANDERSON, P. W. (1950). Phys. Rev. 79, 350. BLUMENFEL’D, L. A. ( 1972). Biofizika 17,954. BLUMENFEL’D, L. A. (1974). Problems of Biological Physics. Moscow: Nauka. BLUMENFEL’D, L. A. & CHERNAVSKIJ, D. S. (1973). J. theor. Biol. 39, 1. BLUMENFEL’D, L. A. & KOLTOVER, V. K. (1972). Molekulyarnuya Biologiyu 6, 161. CHANCE, B. (1972). FEBS Lett. 23, 3. CHANCE, B., DEVAIJLT, D., LEGALAIS, V., MELA, L. & YONETANI, T. (1967). In Fusl Reactions and Primary Processes in Chemical Kinetics, 5th Nobel Symposium (S. Claesson,

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