Dynamical principles in biological processes - Semantic Scholar

Proc. Natl. Acad. Sci. USA Vol. 95, pp. 1358–1362, February 1998 Chemistry

Dynamical principles in biological processes E. W. SCHLAG†, S. H. LIN‡, R. WEINKAUF†,

AND

P. M. RENTZEPIS§¶

†Institute for Physical and Theoretical Chemistry, Technical University of Munich, D-86748 Garching, Germany; ‡Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei, Taiwan, Republic of China; and §Department of Chemistry, University of California, Irvine, CA 92619-2025

Contributed by P. M. Rentzepis, October 27, 1997

From Eq. 2-1 we can see that the ET can be accomplished either in one step

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.

kT P*A 1A 2 . . . A nA O ¡ P 1A 1A 2 . . . A nA 2, or in several steps (i.e., sequential ET),

1. INTRODUCTION

k ¡ P 1A 2 P*A 1A 2 . . . A nA O 1 A 2 . . . A nA

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 and characterized by experimental means such as ultrafast spectroscopy. Electron transfer reactions play key roles in a great many biological processes, including collagen synthesis, steroid metabolism, immune response, drug activation, neurotransmitter metabolism, nitrogen fixation, respiration, and photosynthesis (1–4). 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 and photosynthesis are complementary processes. The oxygen that is evolved by photosynthetic organisms is consumed by aerobic microbes and animals. Similarly, the end products of aerobic respiratory metabolism, (CO2 and H2O), are the major nutritional requirements of photosynthetic organisms. The global C, H, and O cycles thus are largely caused by aerobic respiration and 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.

k1 kn O ¡ P 1A 1A 2 ¡ P 1A 1A 2 . . . A nA 2. 2 A 3 . . . A nA . . . O [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. 2-3. For convenience we shall rewrite Eq. 2-3 as k1 k2 kn k K PO ¡ A1 O ¡ A2 O ¡ A3 . . . O ¡AO ¡ Products,

[2-4]

where K represents the rate constant for any reaction associated with P1A1A2 . . . AnAnA2 after ET. For example, P [ 1 2 P*A1 . . . AnA, A1 [ P1A2 1 A2 . . . AnA, A [ P A1A2 . . . AnA , etc. Notice that dP 5 2kP dt

[2-5]

dA 1 5 2k 1A 1 1 kP dt

2. THEORY

[2-6]

------

A typical model for biological electron transfer (ET) can be expressed as (1–4)

------

hv ¡ P*A 1A 2 . . . A nA O ¡ PA 1A 2A 3 . . . A nA O P 1A 1A 2 . . . A nA 2.

[2-2]

dA n 5 k n21A n21 2 k nA n dt

[2-1]

dA 5 k nA n 2 KA. dt

where the first step denotes photoexcitation, and the second step represents ET. This charge separation is of paramount importance. However, charge separation must be maintained, and therefore the energy-wasting back reaction must be minimized. Although in Eq. 2-1 we use the photoinduced ET as illustration, the theoretical treatment that follows can be easily generalized to other cases.

[2-7] [2-8]

Here we have ignored the back transfers; the condition for its validity will be discussed later. To solve Eqs. 2-5 through 2-8 we shall use Laplace transformation; # m~p! 5 A

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. §1734 solely to indicate this fact.

E

`

dte 2pt A m~t!.

[2-9]

0

Abbreviation: ET, electron transfer. ¶To whom reprint requests should be addressed. e-mail: pmrentze@ uci.edu.

© 1998 by The National Academy of Sciences 0027-8424y98y951358-5$2.00y0 PNAS is available online at http:yywww.pnas.org.

1358

Chemistry: Schlag et al.

Proc. Natl. Acad. Sci. USA 95 (1998)

It follows that P# ~p! 5

P~0! , p1k

[2-10]

Now we consider the one-step model described by Eq. 2-2. In this case, the quantum mechanical rate constant for kT can be expressed as (5-9) kT 5

k P~0! # 1~p! 5 A , z p 1 k1 p 1 k

[2-11]

k1 k P~0! # 2~p! 5 , A z z p 1 k2 p 1 k1 p 1 k

[2-12]

2p uT u2 É fi

kn21 kn22 k1 k P~0! , z ... ... z z p 1 kn p 1 kn21 p 1 k2 p 1 k1 p 1 k

[2-13]

kn21 kn k1 k P~0! z , ... ... z z p 1 K p 1 kn p 1 k2 p 1 k1 p 1 k

[2-14]

and # ~p! 5 A

where P(0) denotes the initial concentration of P(t). Carrying out the inverse Laplace transformation of Eqs. 2-10 through 2-14 it yields: P~t! 5 P~0!e 2kt, A 1~t! 5 P~0!

S

OO v

v9

Pivu^Qfv9uQiv&u2 d~Efv9 2 Eiv!,

[2-21]

where the electronic matrix element Tfi is given by Tfi 5 Vfi 1

-----# n~p! 5 A

1359

[2-15]

D

k k e 2k1t , e 2kt 1 k1 2 k k 2 k1

O m

VfmVmi 1 . . . 5 Vfi 1 Tfi~2!, Ei 2 Em

[2-22]

and Vfi, etc. denote the ET matrix elements i.e., Vfi 5 ^FfuVuFi&. The first term in Eq. 2-22 Vfi describes the so-called direct ET or ET through space, whereas the second term T(2) fi in Eq. 2-22 describes the super-exchange ET or ET through bonds (or bridge groups). The summation over m in Eq. 2-22 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

O N

Fm 5

C mnf n,

n

[2-23]

2 1 where, f1 ' P1A2 1 A2 . . . ANA, f2 ' P A1A2 A3 . . . ANA, etc. If we consider only the most effective path, it follows that

[2-16]

C 1~H 11 2 E m! 1 C 2H 12 1 . . . 1 C NH 1N 5 0, C 1H 21 1 C 2~H 22 2 E m! 1 . . . 1 C NH 2N 5 0,

------

F

A~t! 5 P~0!

kn21 k kn z ... e2Kt, kn 2 K kn21 2 K k2K

------

G

kn21 k kn z ... e2knt 1 . . . . 1 K 2 kn kn21 2 kn k 2 kn

-----[2-17]

If K ,, kn, kn21 . . . k1, k then in the time region of K t ; 1, Eq. 2-17 reduces to A ~t! 5 P~0!e 2Kt

S

D

[2-19]

At room temperature, we find 2 DG812 . 500cm 21~69mev!.

In particular if Fm 5 f1, i.e., f1 ' T fi 5 V fi 1

[2-18]

Eq. 2-18 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 A2 3 A1 be k91, then DG812 k1 5 exp 2 . 10. k91 RT

C1HN1 1 C2HN2 1 . . . 1 CN~HNN 2 Em! 5 0.

[2-20]

It is well known that in photosynthesis charge separation is of paramount importance. The key problem is to maintain the charge separation, which involves minimizing the energywasting back reaction. Reaction centers contain an ordered array of secondary electron acceptors, (A1, A2, A3 . . . ), that optimize the DG0 that occurs at each step described by Eq. 2-3. Thus the back reaction is circumvented by optimizing forward electron transfer that rapidly removes electrons from A2 1 (see Eq. 2-19). As the acceptors are separated by greater and greater distances from P1, the probability of the back electron transfer to P1 decreases.

P 1A 2 1 A,

[2-24]

then

V fmV mi , Ei 2 Em

[2-25]

where V fi 5 ^F fuVuF i& 5 ^P 1A 1A 2uVuP*A 1A&, V fm 5 ^F fuVuF m& 5

^P 1A 1A 2uVuP 1A 2 1 A&,

[2-26] [2-27]

and V mi 5 ^F muVuF i& 5 ^P 1A 2 1 AuVuP*A 1A&.

[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 and we find

S D

2 C mn 5 N11

1 2

sin

mn p N11

[2-29]

and E m 5 a 1 2 b cos

mp , N11

[2-30]

where m 5 1, 2, . . . N, a 5 H11H22 . . . 5HNN, and b 5 Vnn11 5 Vnn21. It follows that

1360


O

2bibf N Tfi 5 Vfi 1 N 1 1 m51


mp N11 , mp a 2 Ei 1 2bcos N11

[2-31]

b ibb f , ~E i 2 a ! 2 2 b 2

[3-5]

dA 5 k 2A 2 2 KA. dt

where bi 5 ^f1uVuFi& and bf 5 ^FfuVufN&. Here we have fi 5 P*A1A2 . . . ANA. For N 5 2, T fi 5 V fi 1

dA 2 5 k 1A 1 2 ~k91 1 k 2!A 2, dt

~ 2 1!msin2

[2-32]

and for N 5 3,

[3-6]

For the case in which k1 and k91 are fast enough that the equilibrium is established in the ET processes, we obtain A 5 ~k2P# 1!

S

#

D

k P0 e2kt 2 e2Kt e2tk2P1 2 e2Kt 2 , K2k ~k2P# 1 2 k! K 2 P# 1k2

[3-7]

k1 . k 1 1 k91

[3-8]

where

bibfb2 Tfi 5 Vfi 1 (21) . ~a 2 Ei!@~a 2 Ei!2 2 2b2#

[2-33]

For the case of (a2Ei)2 .. b2, Eq. 2-31 reduces to T fi 5 V fi 1 (21) N

b ib fb N21 . ~ a 2 E i! N

[2-34]

It is important to note that the ET rate constant (see Eq. 2-21) can be separated into the electronic part uTfiu2 and the nuclear part, Sv Sv9 Pavu^Ufv9uUiv&u2d(Efv92Eiv). Although uTfiu2 will depend on the intermediate groups, the nuclear part that determines the temperature and free energy dependence of ET is relatively insensitive to the intermediate groups and is determined mainly by the donor and acceptor groups. Proteins that function as electron transfer entities typically place their prosthetic groups in a hydrocarbon environment and may provide hydrogen bonds (in addition to ligands) to assist in stabilizing both the oxidized and the reduced forms of the cofactor. Metal-ligand bonds remain intact upon electron transfer to minimize inner-sphere reorganization (10, 11). Many of the complex multisite metalloenzymes (e.g., cytochrome c oxidase, xanthine oxidase, and nitrogenase FeMo protein) contain redox centers that function as intramolecular electron transferases, moving electrons toyfrom other metal centers that bind exogenous ligands during enzymatic turnover.

P# 1 5

Under the conditions P# 1k2 .. K, and kt .. 1, P# 1k2 .. t, Eq. 3-7 reduces to A 5 P 0e 2Kt,

which suggests that only in this case will the ET still be very effective. Next we consider the branching effect. k1

A1

K P 1A 1A 2A 2 O ¡ Products.

[3-1]

Using Laplace transformation, we find [3-11]

k9n22 k9n21 A 1~0! z ... ... . p p 1 k9n21 p 1 k 1 1 k91

[3-12]

and

It follows that A n~t! 5

k1 A ~0! 1 . . . k 1 1 k91 1

[3-13]

A9n~t! 5

k91 A ~0! 1 . . . k 1 1 k91 1

[3-14]

4. CONCLUSION [3-2]

Therefore

dA 1 5 k P 2 k 1A 1 1 k91A 2, dt

k n22 k n21 A 1~0! z ... ... p p 1 k n21 p 1 k 1 1 k91

#n5 A

That is, the branching effect is determined by the relative magnitudes of k1 and k91.

2 1 P ; P*A 1A 2A, A 1 ; P 1A 2 1 A 2A, A 2 ; P A 1A 2 A;

dP 5 2kP, dt

k92 k9n21 A92 O ¡ A93 . . . O ¡ A9n.

and

For convenience, we let

A ; P 1A 1A 2A 2.

k2 k n21 A2 O ¡ A3 . . . O ¡ An [3-10]

n

# 9n 5 A

k1 k2 k ¡ P 1A 2 0 P 1A 1A 2 ¡ P*A 1A 2A O 1 A 2A | 2A O k91

m k91

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,

[3-9]

[3-3] [3-4]

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, and 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 and reversible processes also has been presented.



1361

APPENDIX

C m1~H 11 2 E m! 1 C m2H 12 5 0

[A-10]

In this appendix we shall consider the N 5 2 case and N 5 3 case. For the N 5 2 case, we have

C m1H 21 1 C m2~H 22 2 E m! 1 C m3H 23 5 0

[A-11]

C m2H 32 1 C m3~H 33 2 E! 5 0.

[A-12]

H 11 5 H 33.

[A-13]

C m1~H 11 2 E m! 1 C m2H 12 5 0

[A-1]

C m1H 21 1 C m2~H 22 2 E m! 5 0.

[A-2]

The results for the case of equivalent intermediate groups are given by Eq. 2-32. Here we shall consider an interesting case in which the A2 group has a much higher energy than that the A1 group, i.e., H22 .. H11. In this case, we find

That is, we have PA91A02A93A. We shall consider only the case in which A02 has the highest energy, i.e., H22 .. H11 5 H33. The approximate solution of Eqs. A-10–A-12 is given by

b2 H 22 2 H 11

[A-3]

E 1 5 H 11

b f H 22 2 H 11 1

[A-4]

F1 5

b2 H 22 2 H 11

[A-5]

E 2 5 H 22 1

E 2 5 H 22 1 F2 5 f2 1

Suppose that

E 1 5 H 11 2 and

1

Î2 ~ f 1 2 f 3!

F2 5 f2 1 F1 5 f1 1

b f H 11 2 H 22 2

[A-6]

where b 5 H12. It follows that

b fbb i ^F fuVu f 1& 1 ~E i 2 E 1!~E i 2 E 2! Ei 2 E1

z bi 1 bf

^ f 2uVuF i& . Ei 2 E2

F3 5 [A-7]

2uH 12u 2 H 22 2 H 11

[A-16]

2uH 12u 2 H 22 2 H 11

S

f1 1 f3 2

[A-17] [A-18]

D

2H 12 f . H 22 2 H 11 2

[A-19]

It follows that T fi 5 V fi 1

Notice that ^F fuVu f 1& 5 ^P 1A 1A 2A 2uVuP 1A 2 1 A 2A&

1

Î2

[A-15]

H 12 H 12 f1 1 f H 22 2 H 11 H 22 2 H 11 3

E 3 5 H 11 2 and

T fi 5 V fi 1

[A-14]

^F fuVu f 2&^ f 2uVuF i& 1 1 @^F fuVu f 1& Ei 2 E2 Ei 2 E1

z b i 1 b f^ f 3uVuF i&#.

[A-8]

[A-20]

Again Eq. A-20 can be graphically represented as follows

and ^ f 2uVuF i& 5 ^P 1A 1A 2 2 AuVuP*A 1A 2A&.

V fi

[A-9]

P*A91A 02A93A 3 P 1A91A 02A93A 2

Eq. A-7 can graphically be expressed as follows:

^F fuVu f 2&^ f 2uVuF i& Ei 2 E2

V fi P*A 1A 2A 3 P 1A 1A 2A 2

P*A91A 02A93A 3 P 1A91A 022A93A 3 P 1A91A 02A93A 2

b fbb i ~E i 2 E 1!~E i 2 E 2!

^F fuVu f 1& b i Ei 2 E1

2 2 1 1 2 P*A 1A 2A 3 P 1A 2 1 A 2A 3 P A 1A 2 A 3 P A 1A 2A

P*A91A 02A93A 3 P 1A912A 02A93A 3 P 1A91A 02A93A 2

b i^F fuVu f 1& Ei 2 E1

b f^ f 3uVuF i& Ei 2 E1

1 2 P*A 1A 2A 3 P 1A 2 1 A 2A 3 P A 1A 2A

b f^ f 2uVuF i& Ei 2 E2 1 2 P*A 1A 2A 3 P 1A 1A 2 2 A 3 P A 1A 2A .

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 Ei 2 Em and the ET matrix element b, ^FfuVuf1& and ^fuVuFi&. Next we consider the N 5 3 case. In this case we have

P*A91A 02A 3 P 1A91A 02A932A 3 P 1A91A 02A93A 2. S.H.L. wishes to thank the National Science Council of the Republic of China for supporting this work. 1. 2. 3. 4.

Clayton, R. K. (1980) Photosynthesis: Physical Mechanism and Chemical Pattern (Cambridge Univ. Press, Cambridge, U.K.). Mathews, B. W. & Fenna, R. E. (1980) Acc. Chem. Res. 13, 309–317. Deisenhofer, J., Epp, O., Miki, K., Huber, R. & Michel, M. (1984) J. Mol. Biol. 180, 385–398. Marcus, R. A. (1956) J. Chem. Phys. 24, 966–978.

1362


5. Levich, V. G. (1966) Adv. Electrochem. Eng. 4, 249. 6. Kestner, N. R., Logan, J. & Jortner, J. (1974) J. Phys. Chem. 78, 2148–2166. 7. Lin, S. H. (1989) J. Chem. Phys. 90, 7103–7113. 8. Hopfield, J. J. (1977) Biophys. J. 18, 311–321. 9. Lin, S. H., Hayashi, M., Alden, R. G., Suzuki, S., Gu, X. Z. & Lin,

Proc. Natl. Acad. Sci. USA 95 (1998) Y. Y. (1995) in Femtosecond Chemistry, eds. Manz, J. & Wo ¨ste, L. (VCH, New York), Chapter 22, pp. 633–667. 10. Gray, H. B. & Malmstro ¨m, B. G. (1983) Comm. Inorg. Chem. 2, 203–205. 11. Meyer, T. E. & Cusanovich, M. A. (1989) Biochem. Biophys. Acta 975, 1–28.