Quarterly Reviews of Biophysics g, I (1976), pp. 21-33 Printed in Great Britain
Energy transfer and dynamical structuref JOSEF EISINGER Bell Laboratories, Murray Hill, New Jersey 07974 U.S.A.
INTRODUCTION
21
F 6 R S T E R ENERGY TRANSFER THEORY REVIEWED
22
A SIMPLE MODEL FOR INTRAMOLECULAR ENERGY TRANSFER
23
AVERAGING REGIMES FOR DONOR AND ACCEPTOR MOTION
25
EXPERIMENTAL CONSIDERATIONS
27
TRANSFER DEPOLARIZATION
29
CONCLUSION
32
REFERENCES
33
INTRODUCTION
It is only some 50 years ago that biophysicists obtained reliable experimental methods for estimating the molecular weights of biological macromolecules, chiefly as a result of Svedberg's work in developing the ultracentrifuge as an analytical instrument (Svedberg & Pedersen, 1940). Having gained some understanding of the size of proteins, interest soon thereafter turned to the shape and rotational relaxation times of these molecules, and Perrin's work on fluorescence depolarization helped to lay the foundations there (Perrin, 1929). Biophysics had to wait for the development of X-ray spectroscopy of proteins and nucleic acids to provide a picture of the interior structure of biological macromolecules. This work, beginning in the 1940s andflourishingtoday, gave us detailed t This paper was presented at the symposium on Dynamics of Macromolecules in solution at the 5th International Biophysics Congress in Copenhagen, Denmark, 4-9 August 1975. [21]
22
J. EISINGER
atomic structures which are, however, of necessity static: the X-ray structures of molecules may not be identical to solution structures, which depend in turn on solution conditions (e.g. ionicity, pH and the presence of substrates). Crystal structures are moreover incapable of providing information on the rate and extent of motion of particular parts of a macromolecule. To complement such static models biophysicists have sought to develop diverse techniques which allow the study of the dynamical structure of biomolecules, and several of these techniques will be discussed in the course of the present symposium. The subject I will address myself to in this lecture is how energy transfer can be used to provide information on distances, orientations and motion within biomolecules in solution.
FORSTER ENERGY TRANSFER THEORY REVIEWED
Let me begin by reviewing the basic theory of Forster (or dipole-dipole) energy transfer.^ Forster (1948 a, b, 1965) pointed out that if a donor molecule (D) has an emission spectrum which overlaps the absorption spectrum of an acceptor molecule (A) the electric dipole interaction between these chromophores provides a mechanism for long-range non-radiative energy transfer which can readily be distinguished from the trivial radiative energy transfer (i.e. one involving the emission and subsequent absorption of a photon). If we use an asterisk to denote a molecule in an electronically excited state this process can be written as D* + A^A
+ D*.
(1)
The rate of energy transfer can be shown (Forster, 1948 b) to depend on the following factors: the radiative lifetime of D*; the extent to which the donor emission spectrum overlaps the acceptor absorption spectrum (this being required for energy conservation in equation (1)); the polarizability of the medium intervening between D and A; and,finally,two factors which are of paramount importance to the present considerations: R, the distance between D and A, and K2, the so-called orientation factor which depends on the relative orientations of the transition dipoles f This is the only kind of energy transfer I shall discuss. The only other kind of energy transfer of practical importance in biochemistry is one mitigated by the exchange interaction which requires an overlap of the donor and acceptor wavefunctions and is observable only if the donor and acceptor chromophores are nearest neighbours. For aromatic molecules the range of this energy transfer has been estimated to be less than ~ 6 A (Eisinger, Feuer & Lamola, 1969).
Energy transfer and dynamical structure 23
directions of the donor and acceptor. If these are characterized by the unit vectors D and A, and R is the unit vector along the separation vector joining D and A, #c* = D.A-3(D.R)(A.R). (2) Since the dipole-dipole interaction energy has an inverse third power dependence on R, the rate of energy transfer which in the 'very weak' interaction limit (Eisinger & Lamola, 1971) is proportional to the square of the interaction energy may conveniently be written as kT = CkDK*Br*t
(3)
"where kD is the rate of donor de-excitation (radiative and non-radiative) in the absence of A and where C is a constant which depends only on the nature of D and A and the intervening medium but is independent of their relative orientation or separation. A related useful parameter is the efficiency of energy transfer, given by T_
h
T
A SIMPLE MODEL FOR INTRAMOLECULAR ENERGY TRANSFER
While Forster's theory was originally developed for energy transfer between molecules in solution, it appealed to biophysicists because it appeared to provide a means for estimating intramolecular distances from the rate of efficiency of energy transfer between donor and acceptor chromophores attached to the same macromolecule. The inverse sixthpower dependence of kT on R made this approach a powerful one, since even a rough value of kT (or T) would lead to reasonably precise values of R. As a result intramolecular energy transfer was hailed as being capable of providing a' spectroscopic ruler' for molecules in solution. The fact that a knowledge of/c2 was required before R could be evaluated with confidence (cf. equation (3)) was of course appreciated but this difficulty was in general surmounted by assuming isotropic distributions for the transition moments of D and A and assigning an average value (generally | ) to /c2.f This procedure has been criticized on several counts in that it t For a single D, A pair K* can have any value between o and 4. Since R evaluated from an intramolecular energy transfer experiment has a sixth root dependence on the assumed value of /ca it has sometimes been argued that the = # assumption can lead to only minor errors in R unless is actually nearly zero, which is deemed unlikely. This argument is fallacious since relative orientations of D and A which correspond to vanishing *ca values are statistically predominant (Eisinger & Dale, 1974).
24
J. EISINGER
Fig. i. A schematic view of the model discussed in the text. The donor and acceptor transition moments are assumed to be oriented with equal probability anywhere within the volume or over the surface of two cones, which are characterized by their axis vectors (ru, re) and their half angles (x!rA, i/fj)).
does not properly take into account either (i) the rate of D and A motion relative to kT and kD (Perrin, 1929) or (2) the magnitude of this orientational motion. Recently a fairly simple model which attempts to take these two factors into account, was developed (Dale & Eisinger, 1974, 1975) and I will review it here since it illustrates the intimate connexion between energy transfer and dynamical structure. Consider first a single macromolecular substrate carrying a D, A pair separated by a fixed distance R and suppose that transition moments, D and A, are free to orient themselves with equal probability anywhere within the volume or over the surface of two cones each of which may be characterized by its half angle and by the orientation of its axis. If the cone half angle is \n this corresponds to the isotropic assumption over all space or a plane respectively and if it is zero the transition moments have fixed orientation relative to the macromolecular substrate. If the D (or A) chromophore is attached to the substrate by a single bond, constraining the transition moment to the surface of the cone can be expected to provide a valid model while the constraint to the volume of the cone is a better model if there are several bonds between the chromophore and the substrate. While this model can in any case only provide
Energy transfer and dynamical structure 25
an approximation of the orientational freedom of D and A, it is clearly a great improvement over the ' isotropic' assumption since steric considerations make an isotropic distribution of D and A quite unlikely. The model described above is illustrated in Fig. 1. AVERAGING REGIMES FOR DONOR AND ACCEPTOR MOTION
Having defined a model for the orientations and the orientational freedom of a donor acceptor pair we must now consider the rate of chromophore motion relative to the energy transfer rate (kT) and the donor de-excitation rate (kD). From a practical point of view only two limiting cases need to be considered: the so-called dynamic and static averaging corresponding to the motion of the chromophores being fast and slow, respectively, compared to kT and kD. To illustrate the dramatic difference between these two averaging regimes two ensembles of macromolecules endowed with identical A, D pairs separated by a distance R are shown in Fig. 2. Both A and D are, for simplicity's sake, supposed to have isotropic orientational distributions with respect to the separation vector. In Fig. 2 (a) all orientations are sampled before transfer occurs (dynamic averaging limit) so that the transfer rate for each molecule of the ensemble is identical and since the average value of K2 is §-, for any molecule, this is also the appropriate value of for the whole ensemble. This situation is entirely different if the D and A groups do not move appreciably before transfer occurs, which is the static averaging regime pictured in Fig. 2 (b). The transfer rate for each D, A pair is now different and depends on the orientations of D and A as well as on R. Consequently, for instance, a smaller separation can be counteracted by a less favourable orientation in each molecule and one cannot express the statically averaged kT as a function of an average value of K2 independent of R. This important conclusion (Eisinger & Dale, 1974; Dale & Eisinger, 1974, 1975) can be arrived at more generally by noting that the dynamically averaged transfer is according to equation (4).
where (K\) is the average over all allowed orientations of Dt and At on molecule i. The statically averaged transfer efficiency, on the other hand, is given by
26
J. EISINGER
Fig. 2. Two ensembles of macromolecules each of which is endowed with the same donor acceptor pair. The distributions of D and A orientations are assumed to be isotropic. (a) The dynamic averaging regime in which both transition moments adopt all orientations before transfer occurs. The average orientation factor for the ensemble is $, the same as K2 for each molecule. (6) The static averaging regime in which transfer takes place before D or A change their orientations. An average orientation factor independent of R cannot be defined.
and a knowledge of (/c|> does not permit one to evaluate R from
Energy transfer and dynamical structuref JOSEF EISINGER Bell Laboratories, Murray Hill, New Jersey 07974 U.S.A.
INTRODUCTION
21
F 6 R S T E R ENERGY TRANSFER THEORY REVIEWED
22
A SIMPLE MODEL FOR INTRAMOLECULAR ENERGY TRANSFER
23
AVERAGING REGIMES FOR DONOR AND ACCEPTOR MOTION
25
EXPERIMENTAL CONSIDERATIONS
27
TRANSFER DEPOLARIZATION
29
CONCLUSION
32
REFERENCES
33
INTRODUCTION
It is only some 50 years ago that biophysicists obtained reliable experimental methods for estimating the molecular weights of biological macromolecules, chiefly as a result of Svedberg's work in developing the ultracentrifuge as an analytical instrument (Svedberg & Pedersen, 1940). Having gained some understanding of the size of proteins, interest soon thereafter turned to the shape and rotational relaxation times of these molecules, and Perrin's work on fluorescence depolarization helped to lay the foundations there (Perrin, 1929). Biophysics had to wait for the development of X-ray spectroscopy of proteins and nucleic acids to provide a picture of the interior structure of biological macromolecules. This work, beginning in the 1940s andflourishingtoday, gave us detailed t This paper was presented at the symposium on Dynamics of Macromolecules in solution at the 5th International Biophysics Congress in Copenhagen, Denmark, 4-9 August 1975. [21]
22
J. EISINGER
atomic structures which are, however, of necessity static: the X-ray structures of molecules may not be identical to solution structures, which depend in turn on solution conditions (e.g. ionicity, pH and the presence of substrates). Crystal structures are moreover incapable of providing information on the rate and extent of motion of particular parts of a macromolecule. To complement such static models biophysicists have sought to develop diverse techniques which allow the study of the dynamical structure of biomolecules, and several of these techniques will be discussed in the course of the present symposium. The subject I will address myself to in this lecture is how energy transfer can be used to provide information on distances, orientations and motion within biomolecules in solution.
FORSTER ENERGY TRANSFER THEORY REVIEWED
Let me begin by reviewing the basic theory of Forster (or dipole-dipole) energy transfer.^ Forster (1948 a, b, 1965) pointed out that if a donor molecule (D) has an emission spectrum which overlaps the absorption spectrum of an acceptor molecule (A) the electric dipole interaction between these chromophores provides a mechanism for long-range non-radiative energy transfer which can readily be distinguished from the trivial radiative energy transfer (i.e. one involving the emission and subsequent absorption of a photon). If we use an asterisk to denote a molecule in an electronically excited state this process can be written as D* + A^A
+ D*.
(1)
The rate of energy transfer can be shown (Forster, 1948 b) to depend on the following factors: the radiative lifetime of D*; the extent to which the donor emission spectrum overlaps the acceptor absorption spectrum (this being required for energy conservation in equation (1)); the polarizability of the medium intervening between D and A; and,finally,two factors which are of paramount importance to the present considerations: R, the distance between D and A, and K2, the so-called orientation factor which depends on the relative orientations of the transition dipoles f This is the only kind of energy transfer I shall discuss. The only other kind of energy transfer of practical importance in biochemistry is one mitigated by the exchange interaction which requires an overlap of the donor and acceptor wavefunctions and is observable only if the donor and acceptor chromophores are nearest neighbours. For aromatic molecules the range of this energy transfer has been estimated to be less than ~ 6 A (Eisinger, Feuer & Lamola, 1969).
Energy transfer and dynamical structure 23
directions of the donor and acceptor. If these are characterized by the unit vectors D and A, and R is the unit vector along the separation vector joining D and A, #c* = D.A-3(D.R)(A.R). (2) Since the dipole-dipole interaction energy has an inverse third power dependence on R, the rate of energy transfer which in the 'very weak' interaction limit (Eisinger & Lamola, 1971) is proportional to the square of the interaction energy may conveniently be written as kT = CkDK*Br*t
(3)
"where kD is the rate of donor de-excitation (radiative and non-radiative) in the absence of A and where C is a constant which depends only on the nature of D and A and the intervening medium but is independent of their relative orientation or separation. A related useful parameter is the efficiency of energy transfer, given by T_
h
T
A SIMPLE MODEL FOR INTRAMOLECULAR ENERGY TRANSFER
While Forster's theory was originally developed for energy transfer between molecules in solution, it appealed to biophysicists because it appeared to provide a means for estimating intramolecular distances from the rate of efficiency of energy transfer between donor and acceptor chromophores attached to the same macromolecule. The inverse sixthpower dependence of kT on R made this approach a powerful one, since even a rough value of kT (or T) would lead to reasonably precise values of R. As a result intramolecular energy transfer was hailed as being capable of providing a' spectroscopic ruler' for molecules in solution. The fact that a knowledge of/c2 was required before R could be evaluated with confidence (cf. equation (3)) was of course appreciated but this difficulty was in general surmounted by assuming isotropic distributions for the transition moments of D and A and assigning an average value (generally | ) to /c2.f This procedure has been criticized on several counts in that it t For a single D, A pair K* can have any value between o and 4. Since R evaluated from an intramolecular energy transfer experiment has a sixth root dependence on the assumed value of /ca it has sometimes been argued that the = # assumption can lead to only minor errors in R unless is actually nearly zero, which is deemed unlikely. This argument is fallacious since relative orientations of D and A which correspond to vanishing *ca values are statistically predominant (Eisinger & Dale, 1974).
24
J. EISINGER
Fig. i. A schematic view of the model discussed in the text. The donor and acceptor transition moments are assumed to be oriented with equal probability anywhere within the volume or over the surface of two cones, which are characterized by their axis vectors (ru, re) and their half angles (x!rA, i/fj)).
does not properly take into account either (i) the rate of D and A motion relative to kT and kD (Perrin, 1929) or (2) the magnitude of this orientational motion. Recently a fairly simple model which attempts to take these two factors into account, was developed (Dale & Eisinger, 1974, 1975) and I will review it here since it illustrates the intimate connexion between energy transfer and dynamical structure. Consider first a single macromolecular substrate carrying a D, A pair separated by a fixed distance R and suppose that transition moments, D and A, are free to orient themselves with equal probability anywhere within the volume or over the surface of two cones each of which may be characterized by its half angle and by the orientation of its axis. If the cone half angle is \n this corresponds to the isotropic assumption over all space or a plane respectively and if it is zero the transition moments have fixed orientation relative to the macromolecular substrate. If the D (or A) chromophore is attached to the substrate by a single bond, constraining the transition moment to the surface of the cone can be expected to provide a valid model while the constraint to the volume of the cone is a better model if there are several bonds between the chromophore and the substrate. While this model can in any case only provide
Energy transfer and dynamical structure 25
an approximation of the orientational freedom of D and A, it is clearly a great improvement over the ' isotropic' assumption since steric considerations make an isotropic distribution of D and A quite unlikely. The model described above is illustrated in Fig. 1. AVERAGING REGIMES FOR DONOR AND ACCEPTOR MOTION
Having defined a model for the orientations and the orientational freedom of a donor acceptor pair we must now consider the rate of chromophore motion relative to the energy transfer rate (kT) and the donor de-excitation rate (kD). From a practical point of view only two limiting cases need to be considered: the so-called dynamic and static averaging corresponding to the motion of the chromophores being fast and slow, respectively, compared to kT and kD. To illustrate the dramatic difference between these two averaging regimes two ensembles of macromolecules endowed with identical A, D pairs separated by a distance R are shown in Fig. 2. Both A and D are, for simplicity's sake, supposed to have isotropic orientational distributions with respect to the separation vector. In Fig. 2 (a) all orientations are sampled before transfer occurs (dynamic averaging limit) so that the transfer rate for each molecule of the ensemble is identical and since the average value of K2 is §-, for any molecule, this is also the appropriate value of for the whole ensemble. This situation is entirely different if the D and A groups do not move appreciably before transfer occurs, which is the static averaging regime pictured in Fig. 2 (b). The transfer rate for each D, A pair is now different and depends on the orientations of D and A as well as on R. Consequently, for instance, a smaller separation can be counteracted by a less favourable orientation in each molecule and one cannot express the statically averaged kT as a function of an average value of K2 independent of R. This important conclusion (Eisinger & Dale, 1974; Dale & Eisinger, 1974, 1975) can be arrived at more generally by noting that the dynamically averaged transfer is according to equation (4).
where (K\) is the average over all allowed orientations of Dt and At on molecule i. The statically averaged transfer efficiency, on the other hand, is given by
26
J. EISINGER
Fig. 2. Two ensembles of macromolecules each of which is endowed with the same donor acceptor pair. The distributions of D and A orientations are assumed to be isotropic. (a) The dynamic averaging regime in which both transition moments adopt all orientations before transfer occurs. The average orientation factor for the ensemble is $, the same as K2 for each molecule. (6) The static averaging regime in which transfer takes place before D or A change their orientations. An average orientation factor independent of R cannot be defined.
and a knowledge of (/c|> does not permit one to evaluate R from
