Dynamical principles in biological processes

Transcription

1 Proc. Natl. Acad. Sci. USA Vol. 95, pp , February 998 Chemistry Dynamical principles in biological processes E. W. SCHLAG, S.H.LIN, R.WEINKAUF, AND P. M. RENTZEPIS Institute for Physical Theoretical Chemistry, Technical University of Munich, D Garching, Germany; Institute of Atomic Molecular Sciences, Academia Sinica, Taipei, Taiwan, Republic of China; Department of Chemistry, University of California, Irvine, CA Contributed by P. M. Rentzepis, October 27, 997 ABSTRACT The purpose of this paper is to propose certain dynamical principles in biological systems, which can be used to explain the effectiveness of charge transfer or excitation transfer in biological systems. Some of these systems are accessible experimentally.. INTRODUCTION In this paper we shall discuss some dynamical principles that govern biological processes. For convenience of discussion, we use the biological electron transfer as an example. Of course, the same principles can be applied to other biological processes. The states involved in the process under consideration may be detected characterized by experimental means such as ultrafast spectroscopy. Electron transfer reactions play ey roles in a great many biological processes, including collagen synthesis, steroid metabolism, immune response, drug activation, neurotransmitter metabolism, nitrogen fixation, respiration, photosynthesis ( 4). The latter two processes are of fundamental significance because they provide most of the energy that is required for the maintenance of life. From the viewpoint of global bioenergetics, aerobic respiration photosynthesis are complementary processes. The oxygen that is evolved by photosynthetic organisms is consumed by aerobic microbes animals. Similarly, the end products of aerobic respiratory metabolism, (CO 2 H 2 O), are the major nutritional requirements of photosynthetic organisms. The global C, H, O cycles thus are largely caused by aerobic respiration photosynthesis. In other words, most biological processes consist of a series of dynamical events with different time scales. In this paper we attempt to find some general principles that strongly influence the biological processes. 2. THEORY A typical model for biological electron transfer (ET) can be expressed as ( 4) PA A 2 A 3...A n A O hv P*A A 2...A n A O P A A 2...A n A. [2-] where the first step denotes photoexcitation, the second step represents ET. This charge separation is of paramount importance. However, charge separation must be maintained, therefore the energy-wasting bac reaction must be minimized. Although in Eq. 2- we use the photoinduced ET as illustration, the theoretical treatment that follows can be easily generalized to other cases. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby mared advertisement in accordance with 8 U.S.C. 734 solely to indicate this fact. 998 by The National Academy of Sciences $2.000 PNAS is available online at From Eq. 2- we can see that the ET can be accomplished either in one step T P*A A 2...A n A O P A A 2...A n A, [2-2] or in several steps (i.e., sequential ET), P*A A 2...A n A O P A A 2...A n A O n P A A 2 A 3...A n A...O P A A 2...A n A. [2-3] The ET described by Eq. 2-2 usually is called the ET by super-exchange interaction (or ET through bonds). We first shall treat the model described by Eq For convenience we shall rewrite Eq. 2-3 as P O 2 n A O A 2 O A 3...O A O K Products, [2-4] where K represents the rate constant for any reaction associated with P A A 2... A n A n A after ET. For example, P P * A...A n A, A P A A 2...A n A, A P A A 2...A n A, etc. Notice that dp P [2-5] dt da dt A P [2-6] da n dt n A n n A n [2-7] da dt na n KA. [2-8] Here we have ignored the bac transfers; the condition for its validity will be discussed later. To solve Eqs. 2-5 through 2-8 we shall use Laplace transformation; A m p 0 dte pt A m t. [2-9] Abbreviation: ET, electron transfer. To whom reprint requests should be addressed. pmrentze@ uci.edu. 358

2 Chemistry: Schlag et al. Proc. Natl. Acad. Sci. USA 95 (998) 359 P p P0 p, [2-0] A p P0 p p, [2-] A 2 p P0 p 2 p p, [2-2] A n p n n2 P p n p n p 2 p p, [2-3] A p n p K n P p n p 2 p p, [2-4] where P(0) denotes the initial concentration of P(t). Carrying out the inverse Laplace transformation of Eqs. 2-0 through 2-4 it yields: Pt P0e t, [2-5] A t P0 et e t, [2-6] At P0 n n K n n K... K ekt, n n... e nt... [2-7] K n n n n If K n, n..., then in the time region of Kt, Eq. 2-7 reduces to A t P0e Kt [2-8] Eq. 2-8 shows that for the case in which the ET is very effective, the intermediate steps involving ET do not have any effect on the chemical reaction (or other processes) described by K. Next, we discuss the condition under which the reverse processes in Eq. 2-4 can be ignored. For example, if we let the rate constant for the reverse process A 2 3 A be, then At room temperature, we find exp G 2 RT 0. [2-9] G 2 500cm 69mev. [2-20] It is well nown that in photosynthesis charge separation is of paramount importance. The ey problem is to maintain the charge separation, which involves minimizing the energywasting bac reaction. Reaction centers contain an ordered array of secondary electron acceptors, (A, A 2, A 3... ), that optimize the G 0 that occurs at each step described by Eq Thus the bac reaction is circumvented by optimizing forward electron transfer that rapidly removes electrons from A (see Eq. 2-9). As the acceptors are separated by greater greater distances from P, the probability of the bac electron transfer to P decreases. Now we consider the one-step model described by Eq In this case, the quantum mechanical rate constant for T can be expressed as (5-9) T 2 T fi 2 P iv fv iv v v 2 E fv E iv, [2-2] where the electronic matrix element T fi is given by T fi V fi m V fm V mi...v E i E fi T 2 fi, m [2-22] V fi, etc. denote the ET matrix elements i.e., V fi f V i. The first term in Eq V fi describes the so-called direct ET or ET through space, whereas the second term T (2) fi in Eq describes the super-exchange ET or ET through bonds (or bridge groups). The summation over m in Eq covers all the possible intermediate states. In other words, all the possible paths are to be included in the calculation of T (2) fi (see Appendix). We consider the calculation of T (2) fi. In this case, we have N m C mn n, [2-23] n where, ' P A A 2...A N A, 2 ' P A A 2 A 3...A N A, etc. If we consider only the most effective path, it follows that C H E m C 2 H 2... C N H N 0, C H 2 C 2 H 22 E m... C N H 2N 0, C H N C 2 H N2... C N H NN E m 0. [2-24] In particular if m, i.e., ' P A A, then where T fi V fi V fmv mi E i E m, [2-25] V fi f V i P A A VP*A A, [2-26] V fm f V m P A A VP A A, [2-27] V mi m V i P A AVP*A A. [2-28] Other examples are discussed in the Appendix. In most cases, no general expressions can be obtained except for the case where all the intermediate groups are equivalent. In this case, the theory of molecular exciton can be applied we find 2 C mn N E m 2cos 2 sin mn N [2-29] m N, [2-30] where m,2,...n, H H 22...H NN, V nn V nn.

3 360 Chemistry: Schlag et al. Proc. Natl. Acad. Sci. USA 95 (998) T fi V fi 2 i f N N m sin 2 m N m E i 2cos m N, [2-3] where i V i f f V N. Here we have i P*A A 2...A N A. For N 2, i f T fi V fi E i 2 2, [2-32] for N 3, i f 2 T fi V fi () E i E i [2-33] For the case of (E i ) 2 2, Eq. 2-3 reduces to T fi V fi () N i f N E i N. [2-34] It is important to note that the ET rate constant (see Eq. 2-2) can be separated into the electronic part T fi 2 the nuclear part, v v P av fv iv 2 (E fv E iv ). Although T fi 2 will depend on the intermediate groups, the nuclear part that determines the temperature free energy dependence of ET is relatively insensitive to the intermediate groups is determined mainly by the donor acceptor groups. Proteins that function as electron transfer entities typically place their prosthetic groups in a hydrocarbon environment may provide hydrogen bonds (in addition to ligs) to assist in stabilizing both the oxidized the reduced forms of the cofactor. Metal-lig bonds remain intact upon electron transfer to minimize inner-sphere reorganization (0, ). Many of the complex multisite metalloenzymes (e.g., cytochrome c oxidase, xanthine oxidase, nitrogenase FeMo protein) contain redox centers that function as intramolecular electron transferases, moving electrons tofrom other metal centers that bind exogenous ligs during enzymatic turnover. 3. DISCUSSION We show, in the following example, that the existence of one or more fast reverse processes in the same reaction does not favor ET, P*A A 2 A O 2 P A A 2 A - 0 P A A 2 A O P A A 2 A O K Products. [3-] For convenience, we let Therefore P P*A A 2 A, A P A A 2 A, A 2 P A A 2 A; A P A A 2 A. [3-2] dp P, [3-3] dt da P dt A A 2, [3-4] da 2 dt A 2 A 2, [3-5] da dt 2A 2 KA. [3-6] For the case in which are fast enough that the equilibrium is established in the ET processes, we obtain where A 2 P P 0 2 P et e Kt et2p e Kt K K P, [3-7] 2 P. [3-8] Under the conditions P 2 K, t, P 2 t, Eq. 3-7 reduces to A P 0 e Kt, [3-9] which suggests that only in this case will the ET still be very effective. Next we consider the branching effect. A m n 2 A 2 O 2 A 2 O n A 3...O A n n A 3...O A n. [3-0] Using Laplace transformation, we find A n n p n2 A [3-] p n p A n n p n2 A [3-2] p n p A n t A 0... [3-3] A n t A 0... [3-4] That is, the branching effect is determined by the relative magnitudes of. 4. CONCLUSION In this paper we have used the ET process as means for discussing some dynamical principles associated with biological processes. We have compared the sequential ET process with the through bond (i.e., super-exchange) ET process, discussed the conditions under which the sequential process is most effective. We also have shown that various paths associated with the super-exchange ET process can be represented diagrammatically. The effect of branching reversible processes also has been presented.

4 Chemistry: Schlag et al. Proc. Natl. Acad. Sci. USA 95 (998) 36 APPENDIX In this appendix we shall consider the N 2 case N 3 case. For the N 2 case, we have C m H E m C m2 H 2 0 C m H 2 C m2 H 22 E m 0. [A-] [A-2] The results for the case of equivalent intermediate groups are given by Eq Here we shall consider an interesting case in which the A 2 group has a much higher energy than that the A group, i.e., H 22 H. In this case, we find 2 E 2 H 22 H 22 H [A-3] Suppose that C m H E m C m2 H 2 0 C m H 2 C m2 H 22 E m C m3 H 23 0 C m2 H 32 C m3 H 33 E 0. H H 33. [A-0] [A-] [A-2] [A-3] That is, we have PA A 2 A 3 A. We shall consider only the case in which A 2 has the highest energy, i.e., H 22 H H 33. The approximate solution of Eqs. A-0 A-2 is given by E H [A-4] 2 2 H 22 H [A-4] 2 3 [A-5] 2 E H H 22 H [A-5] E 2 H 22 2H 2 2 H 22 H [A-6] where H 2. H H 2 22 [A-6] H H 22 H H 22 H 3 E 3 H 2H 2 2 H 22 H H 2 [A-7] [A-8] f i T fi V fi E i E E i E 2 fv E i E H 2 H 22 H 2. [A-9] Notice that i f 2 V i E i E 2. [A-7] f V P A A 2 A VP A A 2 A 2 V i P A A 2 AVP*A A 2 A. Eq. A-7 can graphically be expressed as follows: V fi P*A A 2 A 3 P A A 2 A f i E i E E i E 2 P*A A 2 A 3 P A A 2 A 3 P A A 2 A 3 P A A 2 A i f V E i E P*A A 2 A 3 P A A 2 A 3 P A A 2 A f 2 V i E i E 2 P*A A 2 A 3 P A A 2 A 3 P A A 2 A. [A-8] [A-9] In other words, each term in Eq. A-7 can be represented by a reaction path. To determine which path is most important, one has to examine the magnitudes of E i E m the ET matrix element, f V V i. Next we consider the N 3 case. In this case we have T fi V fi fv 2 2 V i E i E 2 i f 3 V i. E i E f V Again Eq. A-20 can be graphically represented as follows V fi P*A A 2 A 3 A 3 P A A 2 A 3 A f V 2 2 V i E i E 2 P*A A 2 A 3 A 3 P A A 2 A 3 A 3 P A A 2 A 3 A f V i E i E P*A A 2 A 3 A 3 P A A 2 A 3 A 3 P A A 2 A 3 A f 3 V i E i E P*A A 2 A 3 P A A 2 A 3 A 3 P A A 2 A 3 A. [A-20] S.H.L. wishes to than the National Science Council of the Republic of China for supporting this wor.. Clayton, R. K. (980) Photosynthesis: Physical Mechanism Chemical Pattern (Cambridge Univ. Press, Cambridge, U.K.). 2. Mathews, B. W. & Fenna, R. E. (980) Acc. Chem. Res. 3, Deisenhofer, J., Epp, O., Mii, K., Huber, R. & Michel, M. (984) J. Mol. Biol. 80, Marcus, R. A. (956) J. Chem. Phys. 24,

5 362 Chemistry: Schlag et al. Proc. Natl. Acad. Sci. USA 95 (998) 5. Levich, V. G. (966) Adv. Electrochem. Eng. 4, Kestner, N. R., Logan, J. & Jortner, J. (974) J. Phys. Chem. 78, Lin, S. H. (989) J. Chem. Phys. 90, Hopfield, J. J. (977) Biophys. J. 8, Lin, S. H., Hayashi, M., Alden, R. G., Suzui, S., Gu, X. Z. & Lin, Y. Y. (995) in Femtosecond Chemistry, eds. Manz, J. & Wöste, L. (VCH, New Yor), Chapter 22, pp Gray, H. B. & Malmström, B. G. (983) Comm. Inorg. Chem. 2, Meyer, T. E. & Cusanovich, M. A. (989) Biochem. Biophys. Acta 975, 28.