Theoretical calculations of RYDMR effects in photosynthetic bacteria

13 February 1987

CHEMICAL PHYSICS LETTERS

Volume 134, number 1

THEORETICAL CALCULATIONS OF RYDMR EFFECTS IN PHOTOSYNTHETIC BACTERIA D.A. HUNTER a, A.J. HOFF b and P.J. HORE a a Physical Chemistry Laboratory, Oxford University, Oxford OXI 3QZ, UK b Department

of Biophysics, Huygens Laboratory,

State University, 2300 RA Leyden,

The Netherlands

Received 29 November 1986

A detailed quantitative analysis of low-field RYDMR (reaction yield detected magnetic resonance) spectra of the photosynthetic bacterium Rhodobacter sphaeroides is presented. Reaction centres, in which the first stable electron acceptor (a quinone) had been pre-reduced, show pronounced temperature dependence around 280 K. This is interpreted in terms of a change in the separation of the constituents of the primary radical pair formed by photoinduced electron transfer.

1. Introduction

‘PI

In photosynthesis, the energy of light is harnessed by a series of chemical reactions, PIX 3

lPIX -+ P+I-x

+ P+Ix-

k-

3-

P+I--

P+I-

b

+ . .. . \

lP is an excited singlet state of the primary donor, P; I is an early transient electron acceptor and X is the first stable acceptor. In photosynthetic bacteria the primary radical pair, P+I -, is formed in less than 5 ps and has a lifetime of 200 ps. The latter is increased to about 1.5 ns when X (a ubi- or mena-quinone) is previously reduced (to X-) or removed. Under these conditions P+I- , which is formed in a singlet state, may revert directly to the ground state P. Alternatively, it may convert to a triplet radical pair whence it can form 3P, an excited triplet state of P. These reactions and some of the symbols to be used later are summarized in fig. 1. Interconversion of the singlet and triplet states of P+I- is mediated principally by the hyperfine interactions in the two radicals but may also be influenced by a static magnetic field via the electron Zeeman interaction. This process is at the heart of most observations of chemically induced dynamic nuclear and electron polarization (CIDNP and CIDEP) and of magnetic field effects on chemical reactions [l-4] . 6

3PI PI Fig. 1. Simplified reaction scheme for blocked photosynthetic reaction centres.

Another way to alter the efficiency of singlettriplet interconversion and hence the quantum yield of 3P (for example) is to apply electromagnetic radiation of the correct frequency to induce EPR (electron paramagnetic resonance) transitions within the radical pair. Such resonances increase or decrease the triplet character of the radical pair wavefunction, and so assist or hinder the production of 3P. Any change in reactant or product concentrations so produced may be detected very sensitively by optical absorption. Typically one would monitor such changes while varying the strength of a dc magnetic field to bring the 0 009-2614/87/$ (North-Holland

03.50 0 Elsevier Science Publishers Physics Publishing Division)

B.V.

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EPR transitions into resonance with a microwave field of fixed frequency. This technique is known as reaction yield detected magnetic resonance or RYDMR [5-71. Recently Moehl et al. [8] reported low-field RYDMR experiments on the photosynthetic bacterium Rhodobacter (formerly Rhodopseudomonas) sphaeroides R-26. Using continuous photolysis, spectra of P+I- were measured by monitoring the absorption of P at 860 nm. By using a low microwave frequency (300 MHz) it was possible to avoid inducing EPR transitions in 3P which are shifted out of resonance by the zero-field splitting. The spectra of quinone-depleted samples were essentially independent of temperature over the range 160 to 295 K. By contrast those of pre-reduced reaction centres showed a pronounced temperature variation around 280 K. In this communication we present a detailed quantitative analysis of these results which suggests that there is a temperature-dependent change in the reaction centre of pre-reduced Rb. sphaeroides affecting both the separation of P+ and I- and the rates of their reactions to give P and 3P.

LETTERS

13 February

1987

tions is deferred till later. These conditions simplify the problem to that tackled by Lersch et al. [9,10] and by Tang and Norris [l l] . Following these authors the triplet quantum yield, $T, may be written

+T = k,

7

dt

TrI~Tdt)l>

(1)

0

where the projection operator PT picks out the triplet components of the radical pair density matrix, p, whose time evolution is given by the Liouville equation dp/dt = -i[H,

p] .

(2)

The representation of the spin Hamiltonian (H) of P+I- (in angular frequency units) written to include the two reaction channels, is [lo] IS)

1%)

IT,)

IT-r)

J- fiks

0

Q

0

0

4 : 0

-J+

Aw

-

2-1’2 WI 0

$kT

2-112 -I-

a1

ii&T

2-112

Wl

0 2-112

-J -

Wl

Aw

-

;ikT

2. Theory is the exchange interaction; Q is half the difference in the hyperfine fields experienced by P+ and I- (the difference in Zeeman interactions also causes singlettriplet interconversion but is negligible at the low fields considered here); w1 is the microwave field strength and Aw the offset from resonance. J is defined such that, in the absence of hyperfine interactions (Q = 0) and the microwave field (~1 = 0), the IS) state has an energy 2Jabove IT,) when J is positive. Though eq. (2) may easily be solved numerically, even this calculation is inconveniently time consuming. Consequently we have adopted and developed a suggestion of Lersch and Michel-Beyerle [ 10,121 which is to treat the hyperfine term Q as a perturbation, an approximation that allows one to solve eq. (2) analytically. Rather than proceed with the density matrix, it is simpler to deal with the probability amplitudes that comprise it. Writing the radical pair wavefunction (\k) as a linear combination of the eigenstates in the absence of hyperfine interactions one has:

J

A complete theoretical description of low-field RYDMR of pre-reduced photosynthetic reaction centres would need to include the unpaired electron spins on I’+, I- and X- ; nuclear spins on all three radicals; pairwise electron-electron exchange and dipolar interactions as well as the microwave field effects. Moreover, for experiments involving continuous (rather than flash) photolysis, all the reactions shown in fig. 1 (and more besides) would have to be included in the calculation. Such a task, though relatively straightforward, would require a formidable amount of computer time. Instead, we make some simplifying assumptions that render feasible least-squares fitting of a theoretical lineshape to the experimental spectra. Specifically we assume: that the presence of X- may be ignored; that the high-field approximation is valid; that only the steps labelled k,, kT and ISC in fig. 1 need be considered; that the electron-electron dipolar interaction between P+ and I- has negligible effect and finally that the electron Zeeman and hyperfine interactions are isotropic. A discussion of these approxima-

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0

;(k,

u(t) = exp (-iHt)

a(0).

(5)

The states 1~~)are linear combinations of IT/); u(t) is the column vector (as, a+ 1, ao, CZ_~)and u(0) specifies the initial (singlet) radical pair state (1, 0, 0,O). Correcting the basis states IS>, 17j>to first order in Q:

IS’)= IS) +

’ c j=_l


Ej

I

bj'

-

1,

(rjlTo)Q =

ES -ej

j=-l,O,

[exp (-ie,

t) - exp (-iej

t)] ,

1,

(8)

and hence ’ c i--l

@,=ik,

l(Tol~j~12Q2 le, - ei12

1 1 e*s - e s +---*--, ei* - ej

(

X-

1 ej - es

1 es* - ej 1

(9)

where ES and ej (i = -1, 0, 1) are the complex eigenvalues of the Hamiltonian in the absence of singlettriplet interconversion. Evaluating eq. (9) explicitly gives the difference in triplet yields with and without the microwave field, @&+

k, + k, Q”w; - @T(O) = 2ks -$-1

X

(2J + s2)2 + i(ks + kT)2 1

t (2J-52)2ti(k

4s

2 tk T )2-4J2t’(k

4s

+k T )2 00)

8

(11)

Eq. (10) differs from the analogous result derived from ref. [IO] in having ks t kT instead of kT - ks in the denominators and ks t kT instead of kT in front of the large parentheses. The discrepancy arises from the approximation kT > ks made in ref. [IO] which amounts to neglecting the last three terms in the large parentheses in eq. (9).

(7)

J

gives

ai

- ks)2 + 4J2 B Q2.

3. Results

IS), j=-l,O,

es -e.

1987

(6)

'j),

(Tolrj)Q I$,=

13 February

with a2 = Aw2 t wf. Evidently when singlet-triplet interconversion is slow the shape of the RYDMR spectrum is determined solely by J, ks t kT and the microwave field strength w 1. This expression is valid when [lo]

s dt jzglai(rbj(t)*,

@p,= k,

LETTERS

The experimental RYDMR spectra [g] of prereduced reaction centres of Rb. sphaeroides R-26 at five temperatures between 236 and 293 K are shown in fig. 2 (dots). The signal is centred at = 110 G, as expected for radicals with g = 2 and a microwave frequency of about 300 MHz. The spectral intensity, in arbitrary units, is proportional to the triplet concentration in the presence of the microwave field minus that in its absence. Although the absolute intensity of the RYDMR effect is not known, the relative intensities are as indicated by the axis labelling. Lineshapes calculated from eq. (10) were matched to these measurements by a non-linear least-squares procedure [ 13 ] . All five spectra were fitted simultaneously by varying the values of twelve parameters, namely J and ks t kT at each of the five temperatures, the microwave field strength (unknown but calculated to be less than 2 G [ 8 ] ) and a scaling factor. The results are shown in figs. 2 and 3. Attempts to repeat this analysis keeping either J or ks + kT independent of temperature gave much less satisfactory agreement between theory and experiment. A similar treatment of spectra recorded at 211 K gave: ]2JI = 10.4 + 0.3 G, ks t kT = 3.8 f 0.1 X IO8 s-1 for pre-reduced reaction centres and ]2JI = 10.1 = 2.6 rt 0.1 X lo8 s-l for reaction +0.5G,ks+k, centres from which X had been removed. The errors quoted here and drawn in fig. 3 represent 95% confidence limits. The sign of J cannot be determined using eq. (10).

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CHEMICAL PHYSICS LETTERS

13 February 1987

5 4 2

T (K) Fig. 3. Temperature dependence of l2A (dots and left axis) and ks + kT (squares and right axis) determined by leastsquares fitting of the experimental spectra shown in fig. 2. 1

279K

4. Discussion

281K

-a

*.

0

40

w

120

wo

Magnetic Field (Gauss) Fig. 2. Experimental (dots) and calculated (lines) RYDMR spectra of pre-reduced reaction centres of Rb. sphaeroides R26.

An approximate theory has been used to analyse RYDMR spectra of Rb. sphaeroides reaction centres. Before discussing the implications of the temperature dependence summarized in fig. 3, we examine the errors introduced by the many approximations set out above. Although iterative curve fitting of the kind just outlined becomes prohibitively time consuming once any of the above approximations is relaxed, individual spectra may be calculated in a reasonable time. Accordingly, the Liouville equation (2) was expanded [14] to include the reaction steps in fig. 1 together with production of the singlet radical pair by continuous photolysis, thermally activated repopulation of the radical pair triplet state from 3P, and spinlattice relaxation within 3P and decay of 3P to P. Literature values were used for the extra rate constants [15-l 71, with the exception of the spin lattice relaxation time, 3 T, , which was chosen by trial and error. Numerical solution of these coupled differential equations in the steady-state limit gave RYDMR spectra which were essentially identical to those of fig. 2 provided 3T, was less than 10m7 s and ks less than lo8 s-l. Hoff and Proskuryakov [18] found 10B5 > 3T1 > 10m6 s, where the higher limit is the triplet lifetime (about 5 ps at 300 K) and the 9

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CHEMICAL PHYSICS LETTERS

lower limit followsfrom the observation that, at 300 K, the EPR spectrum of 3P taken 1 ~.lsafter a flash has the same electron spin polarization pattern as found with cw EPR at cryogenic temperatures (where spin-lattice relaxation is very slow). With 3T, of 10-5-10-6 s, however, the calculated spectra were substantially broadened and bore little resemblance to the experimental lineshapes. The origin of this effect is unclear. In these calculations J and ks + k, were fixed at the values summarized in fig. 3. To test the validity of the perturbation theory approach, these calculations were repeated with realistic hyperfine couplings. Following Hoff and Rademaker [ 191, we averaged the triplet yield over Gaussian distributions of hyperfine interactions with widths of 9 G (P+) and 16 G (I-). This caused no significant change in the lineshapes. Although eq. (11) is not satisfied for the larger values of (SlaT,) that arise from such widths, they have low weighting in the average and do not much affect the lineshape. To investigate the influence of the dipolar interaction between P+ and I- we returned to the flash photolysis approximation in which only the reaction Step ks, kT and singlet-triplet interconversion are considered. Terms appropriate for an axial dipolar interaction were added to Hand the triplet yield averaged over a random distribution of reaction centre orientations. No significant change in the lineshape occurred until the dipolar parameter IDI reached 20 G. Though values of ID1 of this size and larger have been suggested on several occasions [20] , calculations based on the crystallographic coordinates of Rb. viridis [21] put IDI at about 11 G [22]. Furthermore the degree of symmetry of the experimental spectra suggests that low-field effects are not very important, and therefore that D cannot be very large. Like Lersch and Michel-Beyerle [lo], we found that the lineshape for oriented reaction centres is very sensitive to D and to the orientation with respect to the magnetic field. Extensive but not exhaustive calculations including the electron spin on X- indicate that the lineshapes of fig. 2 are not much affected by the exchange interaction between I- and X- if it is less than about 3 G and that the calculated lineshapes cannot be made to match the experimental spectra if it is much larger than 3 G. We now turn to a consideration of the temperature dependence observed for pre-reduced reaction 10

13 February 1987

centres. A possible source of these effects can be seen in the “two-site” model of Haberkorn et al. [23] . This was introduced to account for the discrepancy between the value of J for P+I- determined from magnetic field effects on the yield of 3P [24,25] and the much larger value deduced from the rate of appearance of P+I- measured by optical spectroscopy [26]. The idea is that P+I- consists of two forms in equilibrium, one in which the unpaired electrons are close (C) and the other where they are more distant (D). Assuming that JC is large and JD zero, Haberkorn et al. obtained the following approximate expression for the effective exchange interaction between P+ and I- : J eff x

k$,IJs

7

(12)

where kC is the rate for the reaction C + D and k, that for D + C. Equating the equilibrium constant k,/k, to the Boltzmann factor, exp (-AE,,/kT) where AE DC = E, - EC is the energy gap between close and distant sites one finds J eff =

O$/Jc) exp(-AE&T).

(13)

Assuming that E, = E(I-) and E C = E(lP), AE, is 0.17 eV [27] and eq. (13) predicts a substantial decrease in Jeff as the temperature is lowered over the range covered by fig. 3. Such behaviour is certainly not seen in reaction centres devoid of quinone, while pre-reduced samples show an increase in the exchange parameter as the temperature is reduced. This seems to exclude the two-site mechanism. Returning to the “one-site” mechanism which we have implicitly assumed throughout, another source of temperature dependence may be devined. According to Anderson [28] J should be given by J = 2e2AETlAE;,,

(14)

in which e is the electronic overlap integral between p and I-, AERp is the energy difference between lP and PI-, and AET that between IP and SP. If one postulates a contraction of the reaction centre as the temperature is decreased or a conformational change that brings P+ and I- closer together then the consequent increase in E would explain the observed increase in J. Such a change in the reaction centre cannot, however, account for the temperature dependence of ks + kT which, through the Fermi golden rule,

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CHEMICAL PHYSICS LETTERS

(1% should also increase with increasing E. In this expression (\kfl\ki> is the overlap integral between the initial and final vibrational states (energies E, and Ef) and pi is the probability of finding the reactants in initial vibrational state i. Conceivably the temperature dependence Of pi counteracts the effect of e2, so causing kS + k, to rise between 236 and 279 K. Finally we note a discrepancy between the RYDMR signals observed by Moehl et al. [8] and by Bowman et al. [6] . The widths (at half maximum height) of Moehl’s spectra were very similar for quinone-depleted and reduced samples. This contrasts with the significant change in width reported by Bowman: 30 G for quinone-depleted and 135 G for quinone-reduced samples. A possible explanation for this difference lies in the sample preparations. In ref. [8] the reduction was done with ascorbate and illumination during freezing to cryogenic temperature, a procedure known to yield singly reduced quinone [29-311. No significant changes in linewidth were observed on going from 80 to 240 K [8,32] . In ref. [6], dithionite was used as the reducing agent. Prolonged illumination or sampling in flashing light at room temperature may then lead to double reduction [29]. It is possible that the extra charge on the quinone accelerates one or more of the decay routes of the radical pair, leading to additional lifetime broadening.

Acknowledgement We are indebted to Dr. J.R. Norris for suggesting solving eq. (2) under the conditions of continuous photolysis and to Dr. W. Lersch for providing a copy of his “Diplomarbeit”.

[l] L.T. Muus, P.W. Atkins, K.A. McLauchlan and J.B. Pedersen, eds., Chemically induced magnetic polarization (Reidel, Dordrecht, 1977). [2] A.J. Hoff, Quart. Rev. Biophys. 14 (1982) 599. [3] A.J. Hoff, Quart. Rev. Biophys. 17 (1984) 153. [4] S.G. Boxer, C.E.D. Chidsey and M.G. Roelofs, Ann. Rev. Phys. Chem. 34 (1983) 417. [S] E.L. Frankevich and S.I. Kubarev, in: Triplet state ODMR spectroscopy, ed. R.H. Clarke (Wiley, New York, 1982) p. 137.

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161 M.K. Bowman,D.E. Budil, G.L. Closs,A.G. Kostka, C.A. Wraight and J.R. Norris, Proc. Natl. Acad. Sci. US 78 (1981) 3305. I71 J.R. Norris, M.K. Bowman, D.E. Budil, J. Tang, C.A. Wraight and G.L. Closs, Proc. Natl. Acad. Sci. US 79 (1982) 5532. 181 K.W. Moehl, E.J. Lous and A.J. Hoff, Chem. Phys. Letters 121 (1985) 22. I91 W. Lersch, A. Ogrodnik and M.E. Michel-Beyerle, Z. Naturforsch. 37a (1982) 1454. [lOI W. Lersch and M.E. Michel-Beyerle, Chem. Phys. 78 (1983) 115. [Ill J. Tang and J.R. Norris, Chem. Phys. Letters 94 (1983) 77. I121 W. Lersch, Diplomarbeit, Technische Universitat, Munich (1982). [I31 A. Jones, Computer J. 13 (1970) 301. [I41 D.A. Hunter, Part II Thesis, Oxford University, Oxford (1986). I151 H.J. den Blanken and A.J. Hoff, Biochim. Biophys. Acta 724 (1983) 52. t161 C.E.D. Chidsey , L. Takiff, R.A. Goldstein and S.G. Boxer, Proc. Natl. Acad. Sci. US 82 (1985) 6850. [I71 C.C. Schenck, R.E. Blankenship and W.W. Parson, Biochim. Biophys. Acta 680 (1982) 44. [I81 A.J. Hoff and 1.1. Proskuryakov, Chem. Phys. Letters 115 (1985) 303. [I91 A.J. Hoff and H. Rademaker, in: Chemically induced magnetic polarisation, eds. L.T. Muus, P.W. Atkins, K.A. McLaachlan and J.B. Pedersen (Reidel, Dordrechr, 1977) ch. 24, p. 397. PO1 A.J. Hoff, Photochem. Photobiol. 43 (1986) 727. I211 J. Deisenhofer, 0. Epp, K. Miki, R. Huber and H. Michel, J. Mol. Biol. 180 (1984) 385. WI A. Ogrodnik, W. Lersch, M.E. Michel-Beyerle, J. Deisenhofer and H. Michel, in: Antennas and reaction centers of photosynthetic bacteria. Structure, interactions, and dynamics, ed. M.E. Michel-Beyerle (Springer, Berlin, 1986) p. 198. 1231 R. Haberkorn, M.E. Michel-Beyerle and R.A. Marcus, Proc. Acad. Sci. US 76 (1979) 4185. 1241 A.J. Hoff, H. Rademaker, R. van Grondelle and L.N.M. Duysens, Biochim. Biophys. Acta 460 (1977) 547. 1251 R.E. Blankenship, T.J. Schaafsma and W.W. Parson, Biochhn. Biophys. Acta 461 (1977) 297. [ 261 J.L. Martin, J. Breton, A.J. Hoff, A. Migus and A. Antonelli, Proc. Natl. Acad. Sci. US 83 (1985) 957. [ 271 N.W. Woodbury and W.W. Parson, Biochhn. Biophys. Acta 767 (1984) 345. [28] P.W. Anderson, Phys. Rev. 115 (1959) 2. [29] M.Y. Okamura, R.A. Isaacson and G. Feher, Biochim. Biophys. Acta 546 (1979) 304. [30] A. de Groot, E.J. Lous and A.J. Hoff, Biochim. Biophys. Acta 808 (1985) 13. [31] P. Gast, R.A. Mushlin and A.J. Hoff, J. Phys. Chem. 86 (1982) 2886. [32] E.J. Lous, private communication.

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