Magnetic Field Effects on the Recombination Kinetics of Radical Pairs

464

J. Phys. Chem. B 1998, 102, 464-472

Magnetic Field Effects on the Recombination Kinetics of Radical Pairs B. van Dijk,† J. K. H. Carpenter,‡ A. J. Hoff,† and P. J. Hore*,‡ Huygens Laboratory, UniVersity of Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands, and Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: July 3, 1997; In Final Form: October 28, 1997X

The recombination kinetics of radical pairs able to react only via the singlet channel are calculated within the framework of the radical pair mechanism, treating nuclear spins semiclassically. The form of the decay, which is in general nonexponential, depends on the applied magnetic field strength and the rate of electron spin-lattice relaxation. Analytical expressions are derived for the limiting cases of zero and high field, with very slow or very fast relaxation. The theory is tested and verified by measuring the decay of the secondary radical pair of photosynthetic bacterial reaction centers at low temperatures.

Introduction Despite intense interest in the possible effects of magnetic fields on chemical and biological processes, there are few plausible mechanisms to explain the observed effects. By far the most satisfactory and extensively tested theory is the radical pair mechanism,1-3 which describes how a magnetic field can influence the unpaired electron spins of a radical pair so as to modify reaction product yields, even though the interactions are much smaller than thermal energies. Spin-correlated radical pairssshort-lived intermediates in a variety of chemical reactionssundergo coherent interconversion, or mixing, between their electronic singlet (S) and triplet (T) states. If the S and T pairs react to form distinct chemical products, the yields of those products become sensitive to applied magnetic fields that modify the extent and/or efficiency of ST mixing. For example, if an applied field accelerates mixing in a pair formed in an S state in solution, the yield of the geminate recombination product formed from the S state will be reduced with a corresponding boost in the concentration of the free radicals that escape the solvent cage. Effects of this kind, induced by applied static and/or oscillatory magnetic fields, have been observed both for reactions involving organic radicals in solution and for the electron transfer reactions that form the initial steps of photosynthetic energy conversion.4-6 A sine qua non for the observation of a magnetic field effect (MFE) is that the radical pair can form products via both the singlet and triplet states. The influence of the field is restricted to modifying the proportions of pairs that proceed through the S and the T reaction channels. If only one channel is open, there can be no MFE on the ultimate yield of that product, but there should be a change in the rate of radical pair decay if an applied field modifies the ST mixing step. For example, the amount of triplet product formed from a radical pair initially in a singlet state should rise more rapidly if the applied field facilitates ST mixing, even though the final yield would still be 100%. It is effects such as these that are described * Corresponding author: email, [email protected]. † University of Leiden. ‡ Oxford University. X Abstract published in AdVance ACS Abstracts, December 15, 1997.

theoretically and demonstrated experimentally in this paper. The system we have chosen is the secondary radical pair in the electron transport chain of bacterial photosynthetic reaction centers. This species comprises the reduced electron acceptor Q- (a semiquinone) and the oxidised primary electron donor P+ (a bacteriochlorophyll dimer cation). Both the formation, by photoexcitation, and the recombination of P+Q- proceed solely through electronic singlet states. The origin of the effects described here may be seen from the following argument. Suppose the pure singlet radical pair recombines with a first-order rate constant kS, and that the pure triplet pair is completely unreactive. In a static field, strong compared to the various magnetic interactions in the radical pair, two of the spin eigenstates are pure triplet (T+1 and T-1) and are consequently not initially populated if the pair is formed from a singlet precursor.7 The other two eigenstates are linear combinations of the singlet state S and the remaining triplet state T0; both have 50% singlet character if the two electron spins interact only weakly. Under these circumstances, we can anticipate that the overall recombination rate constant will be 1/ k . In zero or very weak applied field, by contrast, all four 2 S eigenstates are more or less degenerate and, in general, all have some singlet character. On average, we may suppose, each of the eigenstates will have 25% singlet character, so that the overall recombination rate constant would be 1/4kS. Although the latter part of the argument is oversimplified, it is qualitatively correct and predicts a potentially detectable difference in the recombination rates in zero and high field. Further effects arise if the electron spin-lattice relaxation is not slow on the time scale of the recombination. For example, in a strong static field, relaxation can transfer population from the two states that have both singlet and triplet character to the otherwise unpopulated and unreactive T(1 states and thereby modify the radical pair lifetime. The observation of such effects has recently been reported,8,9 together with the use of either pulsed static or static and microwave magnetic fields to pump spins into the T(1 states. Here we focus on the kinetics of recombination in zero field (actually the earth’s magnetic field), in high field, and with a static field jump shortly after creation of the radical pairs by a pulse of light.

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Recombination Kinetics of Radical Pairs Semiclassical Approximation The radicals that constitute the secondary radical pair P+Qin bacterial reaction centers contain many nuclear spins with significant hyperfine couplings10-14sfar too many to countenance an exact numerical calculation of the effect of a magnetic field on the recombination kinetics. We therefore treat the nuclei semiclassically,15,16 an approximation that should be applicable when the electron paramagnetic resonance (EPR) spectra of the two radicals have a more or less Gaussian envelope, as is in fact the case for both P+ and Q-.10-14 The semiclassical approach allows one to develop a simple yet realistic analytical description of the recombination dynamics of P+Q-. Although our theory is tailored to P+Q-, it can easily be generalized, e.g., by allowing kT to be nonzero. As in previous treatments of radical pair dynamics in reaction centers, we assume a completely rigid structure with no relative motion of the radicals. The zero-field spin Hamiltonian of the radical pair, which we assume to have no electron-electron spin-spin coupling (a good approximation for P+Q- where the exchange interaction is at most 0.01 mT and the dipolar coupling is ≈0.1 mT17-19), is written (in angular frequency units):

H ˆ ) gµBB1‚Sˆ 1 + gµBB2‚Sˆ 2

(1)

Sˆ 1 and Sˆ 2 are the spin angular momentum operators of the two electrons, µB is the Bohr magneton, and the two g-values are taken to be isotropic and identical (g1 ) g2 ) g), which should be a good approximation for external magnetic fields less than say 100 mT, the g-value difference in P+Q- being ≈0.001.10-13 The vectors B1 and B2 are the net magnetic fields experienced by the two electrons due to hyperfine interactions. The huge number of possible combinations of nuclear spin orientations in each radical is taken into account by assuming that B1 and B2 can adopt all possible orientations, and that their magnitudes follow a Gaussian distribution law with mean square value15,16

〈B2j 〉 ) 1/3

∑k A2kjIkj (Ikj + 1)

(j ) 1, 2)

(2)

in which Ikj is the nuclear spin quantum number of nucleus k in radical j and Akj is the corresponding hyperfine coupling. As will be seen, the form of this distribution is not important for the magnetic field effects we are about to discuss. Each electron is quantized along its B vector, and the two local fields are entirely uncorrelated. Direct experimental data on the nuclear spin relaxation in P+ or Q- is, to our knowledge, not available. We assume that the nuclear spin relaxation is slow on the time scale of radical pair recombination. In the absence of an external field, only the relatiVe orientation of B1 and B2 matters. Without loss of generality, we therefore choose a coordinate system in which B1 is parallel to the z-axis and B2 is perpendicular to the y-axis. Thus eq 1 becomes

H ˆ ) gµBB1Sˆ 1z + gµBB2[Sˆ 2x sin θ + Sˆ 2z cos θ]

(3)

where B1 and B2 are the magnitudes of the two local fields, θ is the angle between them, and Sˆ jq is the q-component of the spin angular momentum operator of the electron on radical j. In the singlet-triplet basis, |S〉, |T0〉, |T+1〉, |T-1〉, it is easily shown that the matrix representation of H ˆ is

[

J. Phys. Chem. B, Vol. 102, No. 2, 1998 465

1 H ) gµB × 2 0

B1 - B2 cos θ

B1 - B2 cos θ

0

1 B2 sin θ x2 1 B2 sin θ x2



-

]

1 B2 sin θ B1 + B2 cos θ 0 x2 1 B2 sin θ 0 -B1 - B2 cosθ x2 (4)

Diagonalizing this matrix, one finds the eigenstates |j〉 and corresponding energies pωj in the absence of an external magnetic field:

1 1 1 1 1 sin θ |S〉 sin θ |T0〉 + cos θ |T-1〉 |1〉 ) + 2 2 2 x2 x2

( )

( )

( )

1 1 1 1 1 |2〉 ) cos θ |S〉 cos θ |T0〉 + sin θ |T+1〉 2 2 2 x2 x2

( )

( )

( )

1 1 1 1 1 cos θ |S〉 + cos θ |T0〉 + sin θ |T-1〉 |3〉 ) 2 2 2 x2 x2

( )

( )

( )

1 1 1 1 1 sin θ |S〉 + sin θ |T0〉 + cos θ |T+1〉 |4〉 ) + 2 2 2 x2 x2 (5)

( )

( )

( )

1 -ω1 ) ω4 ) gµB(B1 + B2)/p 2 1 ω2 ) -ω3 ) gµB(B1 - B2)/p 2

(6)

In the experiments we wish to discuss, the radical pair is created essentially instantaneously in a spin-correlated singlet state, by means of a light flash. The initial populations p0j of the eigenstates may therefore be obtained from the singlet state density operator, Fˆ 0 ) |S〉〈S|

p0j ) 〈j|Fˆ 0|j〉 ) 〈j|S〉〈S|j〉 ) |〈S|j〉|2 (j ) 1...4)

(7)

whence

p01 ) p04 ) 1/2sin2(1/2θ) ) 1/2s2 p02 ) p03 ) 1/2cos2(1/2θ) ) 1/2c2

(8)

That is, the zero-field states are populated according to their individual singlet character; only when B1 and B2 happen to be perpendicular does one have p01 ) p02 ) p03 ) p04 ) 1/4. Conversely, for those radical pairs in which the local fields are parallel, two of the four states are pure triplet and hence initially empty (p01 ) p04 ) 0) while the other pair, being 50% singlet, 50% triplet are equally populated (p02 ) p03 ) 1/2). However, the spin-correlated radical pair is not formed in a stationary state: there are coherences amongst the four eigenstates which, unlike their populations, are time dependent, oscillating at the frequency corresponding to the energy gap between the pairs of states involved. For example, the coherent superposition of states |1〉 and |3〉 evolves according to

〈1|Fˆ |3〉 ) 〈1|S〉〈S|3〉ei(ω3-ω1)t ) -1/4 sin θ eigµBB2t/p (9)

466 J. Phys. Chem. B, Vol. 102, No. 2, 1998

van Dijk et al.

The lifetime of these zero-quantum coherences (ZQC), whose oscillations are responsible for singlet-triplet interconversion,20 is determined by the irreversible dephasing brought about by spin-spin relaxation (T2) processes. The T2 of single quantum coherences in P+Q- at 15 K, measured by 9 GHz electron spinecho spectroscopy, is of the order of 1 µs,8 and since both zeroand single-quantum relaxation should be sensitive to the spectral density of molecular motions at very low frequencies, it seems likely that the zero-quantum T2 should also be in the microsecond region. As the radical pair recombination has a time constant of ≈25 ms at low temperature,17,21-23 we can ignore the presence of such coherent superpositions and restrict attention to the eigenstate populations, which are independent of the magnitudes of the two hyperfine fields, B1 and B2. Finally in this section, we calculate the rate at which radical pairs recombine from each of the zero-field states, if only singlet pairs are able to recombine. It is perhaps intuitively obvious (see Appendix A) that the recombination rate constant for eigenstate |j〉 is

kj ) kS|〈S|j〉|2 (j ) 1...4)

(10)

i.e. the rate constant appropriate for a pure singlet (kS), scaled by the singlet character of the state in question. Thus, we have

k1 ) k4 ) 1/2 sin2 (1/2θ) kS ) 1/2s2kS k2 ) k3 ) 1/2 cos2 (1/2θ) kS ) 1/2c2kS

(11)

Recombination in Zero Field We are now in a position to determine the recombination kinetics of a radical pair in zero field. For the time being, we ignore the possibility of electron spin-lattice relaxation. From eqs 8 and 11, the fraction of radical pairs remaining at time t is

4

[RP]t,θ )

pj ) 2(p1 + p2) ) A+e-λ t + A-e-λ t ∑ j)1 +

p0j e-k t ) c2 e-(1/2)c k t + s2e-(1/2)s k t ∑ j)1 2

j

S

2

S

λ( ) 1/4kS + 2Wzf ( 1/4Z A( ) 1/2 ( (kS cos2 θ - 8Wzf)/2Z

Averaging this expression over all relative orientations of B1 and B2, one obtains the distinctly nonexponential decay:

[RP]t )

8 - 4(2 + kSt)e

-(1/2)kSt

(kSt)2

(13)

However, this simple expression is of limited utility as electron spin-lattice relaxation cannot always be regarded as much slower than recombination. Relaxation may be included by introducing a first-order rate constant Wzf connecting the following pairs of states: 1T 2, 1 T 3, 2 T 4, 3 T 4. If the initial populations and the decay rates satisfy p01 ) p04, p02 ) p03, k1 ) k4, k2 ) k3 (cf. eqs 8 and 11), and if we neglect the Boltzmann population differences at thermal equilibrium, p1 and p4 will always be equal, as will p2 and p3. With this simplification, one can write the coupled rate equations

dp1/dt ) -(k1 + 2Wzf)p1 + 2Wzf p2 dp2/dt ) -(k2 + 2Wzf)p2 + 2Wzf p1

(14)

The other two relaxation processes, 1 T 4 and 2 T 3, have been omitted because they connect states with populations that are identical at all times and so have no effect. Integration gives

(16)

Z ) xk2S cos2 θ + 64W2zf In the slow relaxation limit (8Wzf , kS), eq 15 reassuringly reduces to eq 12, while in the opposite extreme, (8Wzf . kS), one finds

[RP]t ) e-(1/4)kSt

(17)

which should come as no surprise because, for rapid relaxation, all four states become equally populated and decay with the average rate constant 1/4(k1 + k2 + k3 + k4) ) 1/4kS. To obtain the overall kinetics, eq 15 must be averaged over θ by (numerical) integration. Recombination in High Field Having derived expressions for the recombination kinetics of a relaxing radical pair in zero field, it turns out to be trivial to analyze the equivalent flash photolysis experiment in high field, where “high” means strong compared to the hyperfine interactions. Under these conditions, the two electron spins are quantized along the direction of the applied magnetic field. Only the projections of B1 and B2 onto this axis need be considered: their effect is to cause rapid dephasing of zero-quantum coherences. Therefore, in high field, we can use the formalism developed for zero field, with the single change that θ equals zero. Thus, from eqs 15 and 16, we obtain ′

(12)

(15)

where

4

[RP]t,θ )

-

′

[RP]t ) A′+ e-λ +t + A′- e-λ -t λ′( ) 1/4kS + 2Whf ( 1/4Z′ A′( ) 1/2 ( (kS - 8Whf)/2Z′

(18)

Z′ ) xk2S + 64W2hf where we have used Whf for the relaxation rate to avoid confusion with the zero-field rate Wzf, which may be different. Also note that when θ ) 0, the eigenstates in high field, labeled |5〉, |6〉, |7〉, and |8〉 to distinguish them from the zero-field states |1〉, |2〉, |3〉, and |4〉, are, from eq 5, the familiar high-field spin functions

|5〉 ) |T+1〉 |6〉 ) +

1 1 |S〉 + |T0〉 x2 x2

|7〉 ) -

1 1 |S〉 + |T0〉 x2 x2

(19)

|8〉 ) |T-1〉 The two states |5〉 and |8〉, being pure triplet, are not initially populated, do not subsequently become populated unless the

Recombination Kinetics of Radical Pairs


spins relax before recombining, and are not themselves able to undergo recombination. In the limits of slow and fast relaxation, one finds

(20)

The situation for rapid relaxation is identical to that at zero field provided the relaxation rate Whf, which may be field dependent, is still much faster than kS/8. For slow relaxation only the central pair of states, |6〉 and |7〉, are populated: both are 50% singlet, hence the 1/2kS rate constant. Although it seems reasonable to assume that the four relaxation rates, 1 T 2, 1 T 3, 2 T 4, 3 T 4, in zero field are the same (in each case one state has 1 - 1/2 sin2(1/2θ) triplet character, the other 1 - 1/2 cos2(1/2θ)), this is less evident at high field where each of the four transitions corresponds to a spin flip of either radical 1 or radical 2, which in general will have different relaxation behavior. That this is irrelevant may be seen by solving the rate equations using relaxation rates of, say, W1 for 5 T 7 and 6 T 8 and W2 for 5 T 6 and 7 T 8. It is easily seen that only the sum W1 + W2 is important and that this quantity can be identified with 2Whf in eq 18. The other relaxation pathways (5 T 8 and 6 T 7) can again be omitted because they connect equally populated pairs of states. An alternative way of studying the high-field recombination is to produce the radical pair by flash excitation in zero field and then suddenly switch on a high field after the coherences created by the light flash have had time to dephase, but before significant recombination or relaxation.9 Once again, “high” field means large compared to the hyperfine couplings. Henceforth we will refer to this experiment as “pulsed field” recombination. Projecting the zero-field states onto the highfield states, and omitting the coherences amongst the former, one can obtain the populations of the eigenstates immediately after the field jump:

〈q|j〉〈j|S〉〈S|j〉〈j|q〉

(21)

4

)

high fieldb

pulsed fieldb

/4kS nonexponential eq 13

1 /4kS biexponential 1 /2kS

1 /4kS biexponential 1 /2kS, 2Whf

1

a Relative to the recombination rate constant k . b Recombination is S predicted to be mono- or biexponential with the rate constants shown unless otherwise stated.

after the field step occur with the same first-order rate constants as before, eq 18, but with different amplitudes ′

′

[RP]t ) A′′+ e-λ +t + A′′- e-λ -t A′′( ) 1/2 ( (kS - 24Whf)/6Z′

(24)

The limiting cases, which should be compared with eqs 20, 13, and 17, are

8Whf , kS: [RP]t ) 2/3e-(1/2)kSt + 1/3e-2Whft 8Whf . kS: [RP]t ) e-(1/4)kSt

(25)

In the above we have assumed that no relaxation occurs before the field has reached a value large compared to the hyperfine interactions. If the relaxation occurs on a time scale fast compared to the switching time of the field, all zero-field eigenstates will be equally populated before a substantial field is experienced by the radical pair. Projection of the zero-field states onto the high-field states then yields an equal population (of 25%) of all high-field eigenstates. Equation 25 should then be replaced by

8Whf , kS: [RP]t ) 1/2e-(1/2)kSt + 1/2e-2Whft 8Whf . kS: [RP]t ) e-(1/4)kSt

(26)

Steady-State Illumination

4

∑ j)1

zero fieldb

The predictions of the preceding sections are summarized in Table 1.

p0q ) 〈q|Fˆ 0|q〉

)

relaxation ratea very fast intermediate very slow

8Whf , kS: [RP]t ) e-(1/2)kSt 8Whf . kS: [RP]t ) e-(1/4)kSt

TABLE 1: Predicted Radical Pair Recombination in Zero, High, and Pulsed Field

|〈q|j〉|2|〈S|j〉|2 ∑ j)1

(q ) 5...8)

recombination rate ) kS|〈S|j〉|2pj

Whence, using eqs 5 and 19

(22)

and, after averaging over θ (the angle between the effective hyperfine fields in both radicals in zero field)

〈p5〉 ) 〈p8〉 ) 1/6, 〈p6〉 ) 〈p7〉 ) 1/3

(27)

The rate at which each state is populated is also proportional to its singlet nature

p05 ) p08 ) 1/4sin2 θ p06 ) p07 ) 1/2 - 1/4 cos2 θ

The final type of experiment we wish to consider here involves continuous generation of spin-correlated radical pairs under steady-state conditions. It has already been seen that the recombination rate constant from each radical pair eigenstate is proportional to its singlet character

(23)

i.e., the two “central” energy levels receive twice the population of T+1 and T-1 because of the way the high-field states correlate with the zero-field ones. The recombination/relaxation kinetics

population rate ) k0|〈S|j〉|2

(28)

where k0 is the overall (zeroth order) rate constant for the formation of the singlet radical pairs. k0 is not field dependent, but it is determined by factors such as the rate of absorption of light and the quantum yield of radical pair formation. |〈S|j〉|2 is field dependent, because the eigenstates depend on the applied field strength. Of course, eqs 27 and 28 are only valid for energy levels with nonvanishing singlet character: states |5〉 and |8〉 (eq 19), being pure triplet, would never become populated in the absence of spin-lattice relaxation. In a finite


van Dijk et al. and will recombine more slowly than the average 1/4kS, an effect that is clearly observable in the tail of the decay. When 8Wzf . kS, the relaxation rapidly equilibrates the populations of all levels and the decay is exponential with rate 1/4kS because, on average, the radical pair has 25% singlet character and all energy levels are essentially equally occupied. The high-field decay in the slow-relaxation limit (eq 20) is clearly much faster than the decay in zero field; this arises because only two levels are populated: both have 50% singlet character and therefore recombine with rate 1/2kS. Finally, the pulsed field shows clearly the biexponential decay (eq 25), which in the slow-relaxation limit has rate constants 1/2kS and 2Whf. Experimental Section

Figure 1. Predicted radical pair decay when relaxation is very slow or very fast compared to recombination for zero, high and pulsed field. The limiting values W ) 0 and W ) ∞ are used in eqs 15, 18, and 24.

field, however, these states have a small proportion of singlet character by virtue of the hyperfine coupling. Under steady-state conditions the populating and depopulating rates for each level must be equal, so that (from eqs 27 and 28)

pj ) k0/kS

(29)

That is, all four populations are equal, whatever the field, and there should be no magnetic field effect on the steady-state radical pair concentration. This conclusion is barely changed by the inclusion of spin-lattice relaxation: restoration of thermal equilibrium amongst the radical pair states produces changes in the populations of the order of 1 part in 106 at 20 K in a magnetic field of 50 mT. The time taken for the steady state to be established in the absence of relaxation is of the order of [kS|〈S|j〉|2]-1. Using |5〉-|8〉 to denote here the eigenstates of the semiclassical Hamiltonian in a 50 mT field, the populations of |5〉 and |8〉 would thus rise toward k0/kS much more slowly than those of |6〉 and |7〉. An estimate of the magnitude of this effect can be obtained as follows. In an applied field B0 the angle θ in eqs 3-5 is roughly tan-1 (B2/B0) or ≈ 1°, for B0 ) 50 mT and B2 ) 1 mT (approximately the root-mean-square hyperfine coupling in P+Q-). The populating rates of |5〉 and |8〉 relative to |6〉 and |7〉 are thus in the ratio (eq 8) sin2(1/2θ)/cos2(1/2θ), or ≈ 10-4. In the absence of spin-lattice relaxation, therefore, two of the four radical pair states are so slowly filled that it might be easy experimentally to overlook their contribution. If this were the case, the apparent steady-state concentration of radical pairs at high field would be half that at zero field, where all four energy levels have, in general, significant singlet character. Spin-lattice relaxation, however, will accelerate the formation of a true steady state by transferring spins from |6〉 and |7〉 to |5〉 and |8〉, making it less likely that the contribution of |5〉 and |8〉 would be missed. Radical Pair Decay In Figure 1 we have plotted the calculated radical pair concentration after a flash of light for the different limiting cases of the relaxation rate in zero, high, and pulsed field. Note that when relaxation is very fast all three experiments yield the same monoexponential decay with rate 1/4kS (eqs 17, 20, and 25). The difference between the fast relaxation and slow relaxation limits in zero magnetic field can be understood by noting that some eigenstates have more than 25% singlet character and will therefore recombine faster than the average 1/4kS, and, consequently, some eigenstates have less than 25% singlet character

Photosynthetic reaction centers were used, prepared from the purple bacterium Rhodobacter (Rb.) sphaeroides to investigate the magnetic field effects discussed above. When the primary donor P (a bacteriochlorophyll dimer) is photoexcited, charge separation occurs via the primary acceptor I to the secondary acceptor Q, a quinone molecule.24-27 It takes the electron about 200 ps to reach Q at low temperatures where forward electron transport is blocked and the radical pair recombines nonexponentially to the ground state with a lifetime of about 25 ms.17,21-23 The radical pair decay kinetics are nearly temperature independent from 1 to 80 K.30 The energy of the secondary radical pair is much lower than that of the molecular electronic triplet state of the donor28,29 so that recombination is exclusively to the singlet ground-state PQ even at room temperature. The exchange interaction between P and Q is a few microteslas at most and the dipolar coupling about 0.1 mT.17,18 Native reaction centers contain a non-heme iron center, which considerably enhances spin-lattice relaxation in the radical pair through the coupling of the paramagnetic Fe2+-ion to the electron spins of P+ and, especially, Q-. The relaxation is, however, strongly retarded by removal of the iron or replacement with a diamagnetic metal ion.31 Iron depletion was achieved by chemical modification or genetic manipulation (HisM266Cys mutant). The reaction centers were prepared according to ref 24; chemical Zn-reconstitution was carried out as described in ref 32. Except for the change in spin-lattice relaxation mentioned above, there is no difference in the kinetics when the iron is replaced by zinc.31 When no metal ion is present, the electron transfer time from the primary to the secondary acceptor increases by a factor of about 50.31,33 This increase in charge-transfer time causes the lifetime of the primary radical pair to increase from about 200 ps to 5 ns (a factor of 25) and the yield of the secondary radical pair to drop by 50%. All other kinetics remain unchanged. Of the HisM266Cys reaction centers about 98% do not contain Fe as measured by EPR.34 Of the noniron-containing fraction about 41% contains Zn and 6% contains Mg, so about 50% is metal-free. For the magnetic field effect measurements the sample was mounted between a pair of water-cooled Helmholtz coils fed by two Oltronic B32-20R power supplies operated in series. The power supplies were switched by a field effect transistor, the magnetic field had a 6 ms rise time (1/e), and the maximum field was about 80 mT. The primary electron donor was photoexcited with a xenon flash lamp (≈100 mJ/flash, flash duration less than 10 µs). The transmission of the sample was monitored at the wavelength of maximum absorption of the donor ground state (890 nm at low temperatures) with a simple single-beam spectrometer; the (small) transient part of the signal was selected and amplified by a Tektronix AM 502 differential


Figure 2. Decay of the radical pairs at zero, high, and pulsed magnetic field. The upper panel shows the experimental data from reaction centers of the Rb. sphaeroides HisM266Cys mutant. The data were recorded at 16 K with a field of 50 mT in the high and pulsed field measurements. The rise time of the field in the pulsed field decay was about 6 ms. The lower panel shows the calculated decays using eqs 15, 18, and 24, respectively. The values used for the relaxation rates and kS are indicated in Table 2. In the experimental curves there is a large flash artifact at short times that is not plotted in these figures; for that reason the curves do not start at 1.

amplifier, which also functioned as a high-frequency filter above 3 kHz, and was recorded with a digital averaging oscilloscope. A programmable timer unit was used to trigger the magnetic field power supply, the flash lamp, and the acquisition hardware. The measurements were triggered every 5 s and averaged 1520 times. The time resolution of the equipment was about 1 ms. Results In Figure 2 the zero, high, and pulsed field decays at 16 K of the secondary radical pair of reaction centers of the HisM266Cys mutant of Rb. sphaeroides are plotted (upper panel). The temperature of the sample was low enough for spinlattice relaxation at high field to be slower than recombination.8,9 Indeed, differences between the three curves are observable, indicating that we are not close to the fast-relaxation limit. A convenient way of looking at the data is to take the difference between the decay in high magnetic field (either static or pulsed) and the decay in zero field under the same conditions. Figures 3 and 4 show these difference curves for various temperatures (i.e., various relaxation rates) for the mutant reaction centers (upper panels). In the high-minus-zero field decays (Figure 3) the change in the relaxation rate with temperature is clearly recognized by the change in the shape of the curves (kS hardly changes in the temperature range spanned,


Figure 3. Difference between the decay in high and zero magnetic field. The upper panel shows the experimental data from the reaction centers of the Rb. sphaeroides HisM266Cys mutant. Data were recorded at different temperatures with a field of 50 mT for the high-field measurements. The lower panel shows the simulated curves for different temperatures using eqs 15 and 18. The values used for the relaxation rates and kS are indicated in Table 2. The curves correspond to 16 (a), 25 (b), 40 (c), 60 (d), and 70 K (e).

as may be seen from Figure 5). One sees that at low temperatures (16-25 K) the curves in Figure 3 are initially negative and cross zero at around 20-40 ms. The subsequent positive signal disappears slowly. When the temperature is increased the negative part decreases in intensity until the curves are initially positive around 30 K. Although difficult to see, the 25 K curve crosses zero around 230 ms to become negative again. At 70 K there is almost no difference between the zero and high-field decay. Figure 4 shows the pulsed-minus-zero field decays. All curves initially rise to a maximum, before decaying and eventually changing sign again at t > 300 ms (not shown). At higher temperatures, the difference signal is much smaller and appears to remain negative throughout its decay. Additionally, we show the zero-field decay of the radical pairs recorded at 16 and 90 K in Figure 5. It can be seen that although very small, there is a difference between the two curves. All high/pulsed-field data were measured with a field of about 50 mT. Measurements of the field dependence of the effect (not shown) indicated that it reaches about half of its maximum intensity near 7 mT. The curves for 80 mT looked identical to those at 50 mT, validating our assumption of being in the highfield limit at 50 mT. We have also measured the decay in reaction centers where the iron was removed chemically rather than by genetic modification and replaced with diamagnetic zinc. Similar magnetic field effects were observed. Finally the decay was checked in iron-containing reaction centers which are known


Figure 4. Difference between the decay in pulsed and zero magnetic field. The upper panel shows the experimental data from the reaction centers of the Rb. sphaeroides HisM266Cys mutant. Data were recorded at different temperatures with a field of 50 mT in the pulsed field case, triggered directly after the flash and with a rise time of about 6 ms. The lower panel shows the simulated curves for different temperatures using eqs 15 and 24. The values used for the relaxation rates and kS are indicated in Table 2. The curves correspond to 16 (a), 25 (b), 40 (c), 60 (d), and 70 K (e).

Figure 5. Recombination of the secondary radical pair P+Q- measured in reaction centers of the HisM266Cys mutant at zero magnetic field for different temperatures. In the data there is a large flash artifact at short times that is not plotted in these figures; for that reason the curves do not start at 1.

to have very fast spin-lattice relaxation (on the time scale of recombination). Unsurprisingly, no magnetic field effect was seen. Discussion The Decay at Zero Field. The small difference in the zerofield decays at 16 and 90 K shown in Figure 5 could be due to

van Dijk et al. a change in (the distribution of) kS and/or a change in the zerofield relaxation time. It is known that kS is not completely temperature independent.30 We note that an increase in the zerofield relaxation rate alone cannot be responsible, as can be seen by comparing the data to Figure 1. Temperature Dependence of Spin-Lattice Relaxation Rates. The temperature dependence between 16 and 40 K in the high-minus-zero-field decay is much stronger than the temperature dependence of the zero-field curves in this region (not shown) indicating that this change is mainly caused by a strong temperature dependence of the high-field decay curves. This is also found from high-field measurements at different temperatures (data not shown). The fact that the difference between zero and high field disappears at temperatures around 70 K suggests that both zerofield and high-field relaxation may be considered fast at that temperature (eqs 17 and 20). From this we can directly conclude that the 90 K zero-field decay given in Figure 5 is governed by eq 17. Since we know the lifetime of the radical pair at zero field is about 26 ms, kS must be about 150 s-1 because the zerofield decay rate should be 1/4kS when relaxation is fast. Since the limit of fast relaxation holds when 8W . kS, we know that both Wzf and Whf must be larger than kS/8, thus larger than 20 s-1 above 70 K. In other words spin-lattice relaxation is faster than 50 ms above 70 K at both zero field and 50 mT. Judged by the fact that the difference between the zero-field curves at different temperatures are smaller than that predicted for the zero-field decays with very slow and very fast relaxation (compare Figure 1 with Figure 5), we either do not span the whole range from slow to fast zero-field relaxation (since we know now that zero-field relaxation is fast at higher temperatures we must then conclude that even at the lowest temperatures reached the zero-field relaxation cannot be considered slow compared to the radical pair recombination) or the changes in the zero field recombination are obscured by small changes in the singlet recombination rate kS with temperature. The Nonexponentiality of the Radical Pair Decay. The decay of the radical pair P+Q- in reaction centers of Rb. sphaeroides is known to be nonexponential. Since this observation was also made at higher temperatures, in iron-containing samples,21-23 we can conclude that the nonexponentiality is not (or at least not solely) caused by the spin-selective recombination described above, a conclusion that is further supported by the fact that the measured decay rate depends slightly on the detection wavelength,23 something that can not be explained by our model. The nonexponential decay has been explained in two ways: one is to assume two distinct conformational states of the reaction center, with different decay rates,23 which are supposed to interconvert rapidly at room temperature, explaining the fact that the decay becomes monoexponential around 300 K. At low temperatures the two states pre-exist in the dark.30 The second model assumes a distribution of the distance between the two radicals:22 once again rapid interconversion between different states (distances) is needed to account for the room temperature behavior. The nonexponentiality severely obstructs the analysis of the data to the extent that it does not seem useful to fit the data quantitatively to our model since, although the model also predicts a nonexponential decay for slow relaxation, we assume monoexponential decay when relaxation is fast. For this reason we restrict ourselves to “fitting” the data by eye and to a discussion of differences between the data and the model. Simulated curves are included only to show that we can qualitatively model the type of behavior observed in the data.



TABLE 2: Overview of the Parameters Used for the Simulated Decay Curves in Figures 2, 3, and 4 label

T (K)

kS (s-1)

Whf/kS

Wzf/kS

a b c d e

16 25 40 60 70

150 150 150 150 150

≈5 × 10-2 ≈1 × 10-1 g5 g5 g20

e10-2 e10-2 e10-2 ≈1 × 10-1 g20

It is not our intention to obtain the actual values of kS or the relaxation rates Wzf and Whf. Comparing the Data with the Model. In Figure 2 calculated decay curves are plotted below the experimental data. The values used for the different parameters can be found in Table 2. The general features of the data are reproduced very well. Note that the value for Whf is much smaller than that found in ref 9, using supposedly identical samples with a different origin. (The mutants used in this work were made by A. de Boer in Leiden; the reaction centers used in ref 9 were prepared from a HisM266Cys mutant kindly provided by Dr. J. Allen of the Arizona State University.) The values found here lie more in the range of the relaxation rates found in ref 8; we conclude that, for reasons currently under investigation, the reaction centers studied in ref 9 had an extremely slow spin-lattice relaxation rate. The Decay in a Static Magnetic Field. In Figure 3 we have plotted the expected differences between the high and zerofield decays using eqs 15 and 18. As found experimentally, the curves for slow relaxation (i.e., low temperature) start out negative, cross zero and then go asymptotically to zero. When the relaxation is very slow compared to the recombination (8Whf , kS), the simulated curves (not shown) are negative throughout. Since the lowest temperature data do cross zero, we have to conclude that the limit of 8Whf , kS is not yet reached at 16 K for a 50 mT field. Upon going to higher temperatures the initial negative feature disappears, as in the data. In the limit of fast relaxation the difference is obviously gone, also in agreement with the experiment. Note that the same parameter values were used for the 16 K simulations in Figure 3 and the curves in Figure 2. The Decay in a Pulsed Magnetic Field. The simulated pulsed-minus-zero-field decays (lower panel, Figure 4) lack the initial negative feature observed in the data. We assume that this initial fast negative part is an unwanted effect caused by singlet-triplet mixing in the primary radical pair P+I- resulting in formation of the molecular triplet state 3P. This causes a shift of the baseline upon switching on the field (see also the Conclusions below). The curves have been corrected for this shift so the value at t ) ∞ is zero. The remaining parts of the simulated curves mimic the data quite well at temperatures below about 30 K. However the simulations around 60 K still show a substantial amplitude while the experimental signal is almost zero. It is possible to improve the agreement by using different relaxation rates in the high and pulsed-field cases, an assumption that does not seem justified. We have chosen a set of parameters that agrees reasonably well with the much more structured high-minus-zero field data but is not such a good match to the pulsed-minus-zero field decays. Field Dependence of Spin-Lattice Relaxation Rates. As argued before from the differences between the zero-field recombination curves at different temperatures we either do not span the whole range from fast to slow relaxation (compared to recombination) or the changes in zero-field recombination are obscured by a change in the singlet recombination rate kS. Clearly from Table 2 the latter is the case since we do span a broad range of zero-field relaxation rates.

It may seem odd that the spin-lattice relaxation rate (Table 2) is faster at zero field than at fields that are strong compared to the hyperfine interaction since relaxation processes at zero field do not involve changes in the radical pair energy. However, we have no knowledge of the mechanism(s) that are responsible for the relaxation at the different fields and cannot rule out that relaxation in zero field is indeed slow. Certainly we could not satisfactorily model our data with zero field rates that are faster than the high-field relaxation rates. That the field dependence of the spin-lattice relaxation indeed shows odd behavior was already suggested in ref 9 where no field dependence of the spin-lattice relaxation was found between 40 and 80 mT. Futhermore the spin-lattice relaxation rates we find at low temperature and fields of about 50 mT are in agreement with the rates that were determined using ESE.8 The observation that spin-lattice relaxation is slower in zero field than in high field is at this moment not well understood and warrants further investigation. Conclusions We have shown that the model developed here gives a good picture of the magnetic field effects on the decay of the secondary radical pair in Rb. sphaeroides. The differences between theory and experiment may be explained as follows. First, the radical pair decay is known to be nonexponential17,21-23,30 reflecting either a few discrete values of kS or a distribution of values. The neglect of this deviation from first-order kinetics is probably the largest source of discrepancy between the experimental data and the model described above. Second, since the relaxation rate appears to be sensitive to small changes in conditions, for example, those associated with the sample preparation method, we cannot exclude the possibility that a distribution of relaxation rates exists in the sample. This would cause the high-field decay to deviate from biexponentiality as was indeed found in ref 9. Third, the neglect of the weak magnetic interaction between the two electrons, although a good approximation, is not strictly justified. Fourth, in metal-depleted reaction centers the primary radical pair lives long enough for substantial singlet-triplet mixing to occur before the pair either recombines or proceeds to P+Q-.31,33 This effect can change our results in two ways. First, there can arise direct magnetic field effects on the singlet-triplet mixing process itself or on the (intersystem crossing) decay of the molecular triplet 3P that can be formed from the long-lived primary radical pairs. This can, in turn, change the yield of P+Q- formation in zero and high magnetic field. Furthermore the background illumination used to probe the optical transmission will induce a (small) steady-state P+Q- concentration prior to the light flash; this concentration can change when the field is switched on thus shifting the baseline of the transmission at t ) 0 in the pulsed field measurements. This effect was reported in ref 9 but was left unexplained at the time. Note that it was argued in the theoretical section that this change in equilibrium concentration is not due to a direct magnetic field effect on the P+Q- concentration. As mentioned before, the curves (Figure 4) are corrected for this shift in such a way that the difference signal decays to zero at very long times. The second aspect of the singlet-triplet mixing in P+I- in metal-free reaction centers is that P+Q- will not, as we have assumed, be formed in a pure singlet state.35 Kirmaier et al.31 found that the yield of P+Qin metal-free samples is about 50%, and since our reaction centers have a ≈50% metal content, we expect that roughly


van Dijk et al.

25% of the P+Q- we observe has some probability of being formed in a triplet state. However, this should not greatly affect our conclusions. The difference between the zero-field and high-field recombination kinetics arises principally from the fact that states |5〉 and |8〉 (|T+1〉 and |T-1〉, respectively) are not directly populated in high field. This continues to be true in the presence of P+I- spin mixing, because in high field only the T0 triplet sublevel is mixed with the singlet. Thus, allowing the initial state of P+Q- in the model to be partially triplet would only add to the complexity while not changing the basic features and conclusions. In principle the effects described here can be used to follow spin-lattice relaxation as a function of magnetic field and as a function of temperature, something that is usually difficult with magnetic resonance techniques. The observation that the radical pair decay is in general nonexponential in zero and high external magnetic fields is important to recall when analyzing radical pair data. Only when the spin-lattice relaxation is very fast compared to recombination is the use of a single exponential to describe the data justified. Acknowledgment. We are grateful to A. de Boer for preparing the mutant bacteria and to S. J. Jansen for preparing reaction centers. We thank Dr. S. A. Dzuba for stimulating discussions. This work was supported by the Netherlands Foundation for Chemical Research (SON), financed by the Netherlands Organisation for Scientific Research (NWO). B.v.D. acknowledges a travel grant from the European Science Foundation (ESF). Appendix. Recombination from Partial Singlet States Here we derive an expression for the recombination rate constant of a radical pair in an eigenstate that is only partially singlet in character, in the limit that pure triplet states are unable to recombine. If the recombination rate constant for a pure singlet state is kS, the equation of motion of the density operator is (see, e.g., ref 36)

dFˆ ) -i[H ˆ , F] - 1/2kS[|S〉〈S|Fˆ + Fˆ |S〉〈S|] dt

(30)

where H ˆ is the spin Hamiltonian. In the eigenbasis of H ˆ , |j〉, (j ) 1...4), we have

d 〈j|Fˆ |j〉 ) - 1/2kS〈j|S〉〈S|Fˆ |j〉 -1/2kS〈j|Fˆ |S〉〈S|j〉 dt

(31)

As the eigenstates |k〉 (k ) 1...4) form a complete set, we can write

〈j|S〉〈S|Fˆ |j〉 )

∑k 〈j|S〉〈S|k〉〈k|Fˆ |j〉

(32)

Assuming, as above, that all coherences dephase rapidly, we have

〈j|S〉〈S|Fˆ |j〉 ) |〈S|j〉|2〈j|Fˆ |j〉

(33)

and similarly for 〈j|Fˆ |S〉〈S|j〉. Hence

d 〈j|Fˆ |j〉 ) -kS|〈S|j〉|2〈j|Fˆ |j〉 dt

(34)

〈j|Fˆ (t)|j〉 ) 〈j|Fˆ 0|j〉 e-kjt

(35)

and

where the rate constant for recombination from state j is

kj ) kS|〈S|j〉|2

(36)

References and Notes (1) Adrian, F. J. In Chemically induced magnetic polarization; Muus L. T.; Atkins, P. W.; McLauchlan, K. A.; Pedersen, J. B., Eds; D. Reidel Publishing; Dordrecht-Holland, 1977; p 77 ff. (2) Closs, G. L. J. Am. Chem. Soc. 1969, 91, 4552. (3) Kaptein, R.; Oosterhoff, L. J. Chem. Phys. Lett. 1969, 4, 195. (4) Steiner, U. E.; Ulrich, T. Chem. ReV. 1989, 51, 89. (5) Hoff, A. J. Q. ReV. Biophys. 1981, 14, 599. (6) Boxer, S. G.; Chidsey, C. E. D.; Roelofs, M. G. Annu. ReV. Phys. Chem. 1983, 34, 389. (7) Hore, P. J. In AdVanced EPR, applications in biology and biochemistry; Hoff, A. J., Ed.; Elsevier: Amsterdam, 1989; p 405. (8) Dzuba, S. A.; Proskuryakov, I. I.; Hulsebosch, R. J.; Bosch, M. K.; Gast, P.; Hoff, A. J. Chem. Phys. Lett. 1996, 253, 361-366. (9) van Dijk, B.; Gast, P.; Hoff, A. J. Phys. ReV. Lett. 1996, 18, 4478. (10) McElroy, J. D.; Feher, G.; Mauzerall, D. C. Biochim. Biophys. Acta 1969, 172, 176. (11) McElroy, J. D.; Feher, G.; Mauzerall, D. C. Biochim. Biophys. Acta 1972, 276, 363. (12) Feher, G., Okamura, M. Y.; McElroy, J. D. Biochim. Biophys. Acta 1972, 267, 222. (13) Burghaus, O.; Plato, M.; Rohrer, M.; Mo¨bius, K.; McMillan, F.; Lubitz, W. J. Phys. Chem. 1993, 97, 7639. (14) Feher, G. J. Chem. Soc., Perkin Trans. 2 1992, 11, 1861. (15) Schulten, K; Wolynes P. G., J. Chem. Phys. 1978, 68, 3292. (16) Salikhov K. M.; Molin Yu. N.; Sagdeev R. Z.; Buchachenko A. L. Spin polarization and magnetic effects in radical reactions; Elsevier: Amsterdam, 1984; p 84 ff. (17) van den Brink J. S.; Hulsebosch, R. J.; Gast, P.; Hore, P. J.; Hoff, A. J. Biochemistry 1994, 33, 13668. (18) Bittl, R.; Zech, S. G. J. Phys. Chem. B 1997, 101, 1429. (19) Ermler, A.; Fritzsch G.; Buchanan, S. K.; Michel, H. Structure 1994, 2, 925. (20) Zwanenburg, G.; Hore, P. J. Chem. Phys. Lett. 1993, 203, 65. (21) McElroy, J. D.; Mauzerall, D. C.; Feher, G. Biochim. Biophys. Acta 1974, 333, 261. (22) Kleinfeld, D.; Okamura, M. Y.; Feher, G. Biochemistry 1986, 23, 5780. (23) Parot, P.; Thierry, J.; Verme´glio, A. Biochim. Biophys. Acta 1987, 893, 534. (24) Feher, G.; Okamura, M. Y. In The photosynthetic bacteria; Clayton R. K., Sistrom W. R., Eds.; Plenum Press: New York, 1978, p 349. (25) Parson, W. W. In Chlorophylls; Scheer H., Ed.; CRC Press: Boca Raton, FL, 1991; p 1153. (26) Kirmaier, C.; Holten, D. In The photosynthetic reaction center; Deisenhofer J., Norris J. R., Eds; Academic Press: San Diego, CA, 1993; Vol. 2, p 49. (27) Dressler, K.; Umlauf, E.; Schmidt, S.; Hamm, P.; Zinth, W.; Buchanan, S.; Michel, H. Chem. Phys. Lett. 1991, 183, 270. (28) Arata, A.; Parson, W. W. Biochim. Biophys. Acta 1981, 638, 201. (29) Takiff, L.; Boxer, S. G. J. Am. Chem. Soc. 1988, 110, 4425. (30) Schoepp, B.; Parot, P.; Lavorel, J.; Verme´glio, A. In The Photosynthetic Bacterial Reaction Center 2, structure, spectroscopy and dynamics; Breton, J., Verme´glio, A., Eds.; Nato ASI, series A, Life Sciences, 237; Plenum Press: New York, 1992, p 331. (31) Kirmaier, C.; Holten, D.; Debus, R. J.; Feher, G.; Okamura, M. Y. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 6407. (32) Dzuba, S. A.; Gast, P.; Hoff, A. J. Chem. Phys. Lett. 1995, 236, 595. (33) Liu, B. L.; van Kan, P. J. M.; Hoff, A. J. FEBS Lett. 1992, 289, 23. (34) Williams, J. C.; Paddock, M. L.; Feher, G.; Allen, J. P. Biophys. J. 1991, 59, 142a. (35) Hore, P. J. Mol. Phys. 1996, 89, 1195. (36) Hoff, A. J.; Hore, P. J. Chem. Phys. Lett. 1984, 108, 104.

Magnetic Field Effects on the Recombination Kinetics of Radical Pairs

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