ON -THE ENERGY TRANSFER IN BIOLOGICAL SYSTEMS* By Jonn Avery, Zouran Bay,{ AND ALBERT SzenT-GYORGYI INSTITUTE FOR MUSCLE RESEARCH AT THE MARINE BIOLOGICAL LABORATORY, WOODS HOLE, MASSACHUSETTS Communicated September 16, 1961 It seems likely that in various biological systems energy consuming chemical reactions are coupled to delocalized states of energy. Let us suppose that the circles in Figure 1 represent an aggregate of, say, 100 molecules to which we com- municate an energy quant. We suppose one of the molecules, labeled m, to be close to another molecule or molecular aggregate S, which will eventually use the energy, and thusactasasink. The question is how the energy quant can reach that sink. Two different mechanisms could be proposed, an individual and a, collective one. (a) The individual picture: At any given instant, the energy of excitation is considered to belong to one individual molecule of the system. Through a dipole- dipole interaction between two neighboring molecules in resonance, the energy “Jumps” from one molecule to another. The energy in this picture ‘migrates’ in a way similar to the Brownian motion of a particle, until it reaches 8. This pic- ture has been treated by J. and F. Perrin,’ and T. Férster.? (b) The collective picture: In this picture, which has been initiated by Frenkel? in solid-state physics and worked out by others,‘ the energy is delocalized, behaves more like a wave and can be directly transmitted to the sink. We feel that this mode of transmission, which, with certain modifications, holds also for an electron, has not been sufficiently appreciated in biology. Such delocalized states are repre- sented in quantum mechanics as a superposition of localized states. Thus, for example, we might let the symbol 4, represent the absorbing molecules with the excitation localized on, say, the Kth molecule. Then the delocalized state would be represented by a superposition of the localized wave functions: a;P, + A2Pe + soe + anPy. One describes the entire system (molecules plus sink) by the product of the wave function above and another wave function (call it £) which represents the sink. We can let & represent the sink before it has trapped the energy, &, afterwards. The probability per unit time that the energy will be trapped is proportional to the square of the quantum mechanical matrix element of the coupling energy H’ which connects the initial state W; = (ah + aed, + ... + ayby)bo to the final state Vy = Dok, where % is the wave function of the aggregate of molecules in the ground state. Because of the short range of the forces which couple the sink to the collection of the - molecules, the matrix element will be zero, unless a, * 0. The matrix element for the transition of the energy to S will be equal to the coefficient a,, multiplied by the coupling energy: 1742 Vou. 47, 1961 BIOCHEMISTRY: AVERY ET AL. 1743 SV PAH'W dr = S (Bok) *M (a9, + AgPs + ore + ay®y) Kodr = Om S (Pees) *H OpEd. If the sink is strongly coupled to the adjacent molecule, that is, if S (Gobi) *H1b,,Lod is very large, then the probability per unit time for the energy to be trapped will be large, even though at the time of trapping a, (i.e., the probability of the excitation being near to the sink) is small. In the individual picture, S could absorb the energy only if it happened to migrate tom. A random migration of this kind would be slow if the number of the mole- cules in the collection were large, for example, 100 as in Figure 1. The excitation would have to make on the order of (100)? jumps before reaching S. O O ne XxX OQ O Q 000 Vf gomS Fia. 1. Speaking roughly, the individual picture is valid when the time needed for the thermal motions to destroy phase relationships is smaller than the time needed for the excitation to make a single jump. Let us try to make this criterion more pre- cise. In the individual picture, one makes the perturbation expansion assuming that the coupling energy which causes excitation to migrate can be treated as a small quantity. Since the range of coupling is very short, a jump to the next nearest (from A to C in Fig. 2) neighbor is a second-order effect, involving exci- tation of the nearest neighbors (B) as intermediate states. The number of these intermediate neighbor molecules will be called n. In order for the picture of indi- vidual jumps to be valid, the perturbation series must converge—i.e., it must be easier for the excitation to make a short jump from A to B than a long one from AtoC. If we assume that the effect of thermal motions is to broaden the ex- cited level into a Lorentzian shape of width AT, then the condition for convergence of the perturbation series turns out to be Tt,/n? > 1. Here t, is the time needed for a jump to an adjacent molecule. We can try to apply these considerations to biological systems, say, to a granum of a chloroplast. 1f one accepts the idea of the photosynthetic unit of Emerson and Arnold,’ then the situation is similar to that in Figure 1. A photon is absorbed somewhere in a group of about 200 chlorophyll molecules and is transmitted to an enzyme system (S) where it is utilized. Franck and Livingston® have pointed out that since the fluorescence yield of chlorophyll in vivo is only 10~%, the absorption must take place in a thousandth of the fluorescence decay time (10~* sec), thus in 10—!! sec. If the transfer proceeded by individual random jumps, it would have to make 10‘ of these in 10-1! seconds, that is, 4, would 1744 BIOCHEMISTRY: DELUCA AND ENGSTROM Proc. N. A. 8. have to be 10-% seconds. A reasonable figure for the collision broadening of the excited level would be given by ! ~ 101% sec~!. Since the chlorophyl molecules are thought to be arranged in a monomolecular layer. we let n = 2, then Tt — ~ 2X 10-3, n Therefore the condition for convergence of the perturbation series is by no means fulfilled, and in order to get a valid approximation, one would have to solve the secular equation, which would lead to nonlocalized states. Further work in this direction is in progress. The above considerations hold also for more extensive systems. It seems possible that they can be applied to biological processes other than photosynthesis. For example, excitation processes’ in the retina could be mentioned. The light wave focused on one visual rod is coherent within the area of this rod. It szems probable that this collective character of the excitation is important for the forma- tion of the signal in the adjoining nervous system. It also seems possible that collective activities play a role in the function of the central nervous system and relations may be found, say, between conscience and nonlocalized electronic states, or S and storage of memory, * This research was supported by grants from The Commonwealth Fund, the National Science Foundation (Grant B-10805) and The National Institutes of Health, (Grant H-2042, C5-7, BBC). T On leave from the National Bureau of Standards, Washington, D.C. 1Perrin, F., Ann. Chem. et Phys., 17, 283 (1932). * Forster, T., Ann. Phys., 2, 55 (1948); Radiation Res., suppl. 2, 326 (1960). 3 Frenkel, J., Phys. Rev., 37, 17, 1276 (1931). 4 See the review of M. Kasha, Revs. Mod. Phys., 31, 162 (1959). 5 Emerson, R., and W. Arnold, J. Gen. Physiol., 15, 391 (1932); 16, 191 (1932). 6 Franck, J., and R. Livingston, Revs. Mod. Phys., 21, 505 (1949).