Distance determination in spin-correlated radical pairs in photosynthetic reaction centres by electron spin echo envelope modulation

16 April 1999

Chemical Physics Letters 303 Ž1999. 593–600

Distance determination in spin-correlated radical pairs in photosynthetic reaction centres by electron spin echo envelope modulation C.E. Fursman, P.J. Hore

)

Physical and Theoretical Chemistry Laboratory, Oxford UniÕersity, South Parks Road, Oxford OX1 3QZ, UK Received 21 January 1999

Abstract Distances between radicals in photosynthetic reaction centres can be determined from the strong ‘out-of-phase’ electron spin echo envelope modulation ŽESEEM. caused by the electron spin–spin dipolar interaction. The precision with which the dipolar coupling can be measured in such experiments and the optimum strategy for sampling the time-domain spin echo signal are explored using the method of Cramer–Rao lower bounds. Consideration of the contributions to the ESEEM from ´ anisotropic hyperfine interactions suggests that the precision is much higher than hitherto estimated and that the accuracy may not be affected if nuclear modulations are ignored. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction The ‘out-of-phase’ electron spin echo envelope modulation ŽESEEM. of spin-correlated radical pairs has been exploited recently to obtain the distances between radicals involved in solar energy conversion in photosynthetic systems w1–19x. Such measurements are possible because both radicals can be excited simultaneously by short, intense microwave pulses with the result that the ESEEM is dominated by the magnetic dipolar coupling of the two electron spins which, for point dipoles, is proportional to the inverse cube of the separation. In terms of the Žaxial. dipolar interaction parameter D and the Žisotropic.

) Corresponding author. E-mail: [email protected]; fax: q44 1865 275410

exchange interaction J, the echo modulation frequency is w2x

ve s 2 J y 2 D cos 2u y 13 ,

Ž 1.

where u is the angle between the direction of the magnetic field of the ESE spectrometer and the principal axis of the dipolar interaction. Fig. 1 shows the Fourier transform of a simulated two-pulse ESEEM signal for an orientationally disordered sample. Spectra of this type have been observed for various q photosynthetic radical pairs: P865 Qy A in bacterial req action centres w3,5,6,10,11,16x; P700 Ay 1 in photosysq w x w x tem I w5–7,9,13x; and P680 Qy 8,15 and YZox Qy A A 8 in photosystem II ŽP denotes the primary electron donor, Q the quinone acceptor, and YZ the redox-active tyrosine residue.. The spectrum has prominent

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 1 8 5 - 2

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C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

photosynthetic radical pairs? The purpose of this Letter is to discuss these issues, which need to be resolved if ESEEM is to be used reliably to extract distance information from noisy data, to characterise small structural differences between closely similar samples w16x, and to avoid the systematic errors that may come from the use of an inexact model w18x.

2. Cramer–Rao lower bounds ´

Fig. 1. Sine Fourier transform of the ESEEM of a spin-correlated radical pair, calculated using Eq. Ž5. with Dsy112 mT, J s 0, T s 2.5 ms. The frequency axis has been converted to magnetic field units. The frequencies f < and f H , defined in Eq. Ž2., are indicated: they do not exactly coincide with the turning points of the spectrum because of the homogeneous line-broadening. The inset shows the two-pulse ESE sequence: the interval D Žtypically 1 ms. after the light flash which creates the radical pairs, allows zero quantum coherences to relax.

features corresponding to the canonical orientations, u s 0 Žparallel. and u s pr2 Žperpendicular. at frequencies w3x: f 5 s " Ž 2 J y 43 D . , f H s " Ž 2 J q 32 D . .

Ž 2.

Although values of the coupling parameters can be obtained directly from the spectrum via f 5 and f H w3x, this approach has its drawbacks. Several factors conspire to obscure or distort the positions of the relatively weak parallel features: these include noise, homogeneous linebroadening, artefacts due to the imperfect extrapolation of signals corrupted by the instrumental ‘dead-time’, and peaks arising from the nuclear modulations caused by anisotropic hyperfine interactions w12,18,19x. Such problems may lead to systematic errors in the determination of D and J. Some of these pitfalls can be avoided by least squares model-fitting in the time-domain, using an adjustable echo dephasing time, and by ignoring data affected by the dead-time. Several questions then arise. With what precision can the coupling parameters, and in particular D, be determined? What are the experimental conditions that optimise this precision? What is the appropriate model function for

An obvious way of assessing the likely experimental error in the value of a parameter obtained by model-fitting is to repeat the quantitation procedure many times on the same simulated data, perturbed by different sets of random noise. The standard deviations of the observed parameters from their true values give an estimate of the experimental uncertainty. Although such ‘Monte-Carlo’ methods w20x have the advantages of directness and generality, they tend to be time consuming. A more efficient approach uses the theory of Cramer–Rao lower ´ bounds ŽCRLB. w21–25x. For an unbiased estimation method the error in a parameter, as measured by its standard deviation, must be greater than or equal to the CRLB. In many cases, model-fitting methods result in parameters with standard deviations that equal the lower bounds. The Cramer–Rao lower bound for a parameter u k ´ is its standard deviation sŽ u k ., calculated from the Fisher information matrix, F:

(

s Ž u k . s Ž Fy1 . k k ,

Ž 3.

i.e. the square root of the k th diagonal element of the inverse of F. If the noise is Gaussian, with standard deviation s , the elements of F are given by Fjk s

1

s2

Ý Re n

ž

E yˆn) E yˆn Eu j Eu k

/

,

Ž 4.

in which yˆn is the model function at the nth data point Ž n s 1,2, . . . , N .. The natural model function for ESEEM of radical pairs is the powder average of the pure electron– electron modulation with exponential damping: yˆn s h eyt n r T

2p

p

H0 H0

sin Ž vetn . sin u d u d f ,

Ž 5.

C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

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where tn is the nth value of the delay t in the two-pulse spin echo sequence shown in Fig. 1. This expression is appropriate for ‘hard pulses’ that excite the whole spectrum uniformly, and when the coupling between the electron spins is weak, in the sense that the difference in their EPR Želectron paramagnetic resonance. frequencies greatly exceeds the strength of their mutual coupling w18x. The parameters u k of the model are thus D, J, the relaxation time T and an amplitude h. To determine the CRLBs for a given experimental measurement, one needs estimates of the parameters and the noise level s . We have chosen, as a representative ŽX-band. ESEEM signal, that of the radical pair Pq Qy in bacterial reaction centres of A Rhodobacter Ž Rb.. sphaeroides R-26 in which the Fe 2q was replaced Zn2q, at a temperature of 100 K ŽFig. 2. w11x. The delay time t has values between 200 and 2760 ns in steps of 20 ns. Very similar signals have been reported by other authors w5,6,10x. The standard deviation of the noise was estimated from the final 1 ms of the signal. Non-linear least-squares optimisation w26x gave the following parameters: D s y116.6 mT; T s 1.15 ms;

J s q0.51 mT;

hrs s 134 ,

Ž 6.

and the simulation shown in Fig. 2A. The values of D and J may be compared with those found by Dzuba et al. w11x Ž D s y115 mT, J s q0.7 mT. by fitting to a slightly different model function, in the frequency domain. The absolute value of h is arbitrary, reflecting the scaling of the experimental data. The Cramer–Rao lower bounds are ´ s Ž D . s 0.55 mT;

s Ž J . s 0.19 mT;

s Ž T . s 0.043 ms;

s Ž h . rs s 2.52 .

Ž 7.

Extremely similar values for the standard deviations were obtained by the Monte-Carlo procedure referred to above, confirming that the uncertainty in each parameter is indeed equal to its CRLB. This calculation was roughly 10 4 times slower than the Cramer–Rao computation. The lower bounds are ´ somewhat smaller than the generous errors estimated in Ref. w11x: "5 mT for D and q1.5 mTry 0.3 mT for J. Using DrmT s y2780 Ž Rrnm.y3 , the values of D and sŽ D . in Eqs. Ž6. and Ž7. give a Pq–Qy A

Fig. 2. Model-fitting of the experimental ESEEM signal Ždots. of the radical pair Pq Qy A in bacterial reaction centres of Rb. sphaeroides R-26. ŽA. Without hyperfine interactions, using Eq. Ž5. with parameters as in Eq. Ž6.. ŽB. With anisotropic hyperfine interactions to two spin-1r2 nuclei, using Eq. Ž10. with parameters as in Eq. Ž14. and AA s119 mT, A B s155 mT, vA I s 43.2 mT, v B I s61.0 mT, cA s 2.308, c B s 39.28.

separation of R s 2.878 nm with an error of 0.005 nm. In agreement with previous analyses of the ‘outof-phase’ ESEEM signal from both bacteria and plants, the exchange interaction is seen to be non-zero and positiÕe. This appears to be the first objective calculation of the experimental error in J. Independent evidence for J ) 0 comes from simulation of the spin-polarized X-band EPR spectra of PqQy A in Rb. sphaeroides w27x which are highly sensitive to small isotropic radical–radical interactions. Despite these results, it might be thought that at a separation of ; 3 nm, J should be negligible. If the value of J

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C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

is fixed at zero and the model-fitting repeated, one obtains the following parameters with a marginally Ž1%. higher x 2 Žsum of squared residuals.: D s y115.3 mT; T s 1.14 ms;

hrs s 133

Ž 8.

with CRLBs: s Ž D . s 0.16 mT;

s Ž T . s 0.043 ms;

s Ž h . rs s 2.50 .

Ž 9.

The change in D can easily be understood. The coupling parameters extracted from the data are principally controlled by the strong modulation at the perpendicular frequency f H , so that if DX is the value of D obtained assuming J s 0, we have f H s 2 J q 2 Dr3 f 2 DXr3. Taking D and J from Eq. Ž6., one finds DX s y115.1 mT in good agreement with Eq. Ž8.. The Cramer–Rao method also allows one to pre´ dict how the uncertainty in D depends on the signalto-noise ratio and the dephasing time. Specifically, sŽ D . is directly proportional to srh, and increases as T is reduced Ž sŽ D . s 1.05 mT for T s 0.5 ms, and 0.40 mT for T s 2.0 ms.. Neither result is qualitatively surprising: the lower the quality of the data and the more rapidly the modulations disappear into the noise, the poorer the precision of the parameters that control the frequency of the modulations. In this context one might ask what signal-to-noise ratio would be necessary to achieve a given precision in the radical–radical distance, R. Using sŽ D . A srh and D s y116.6 mT, one finds that an uncertainty of "0.01 nm corresponds to hrs s 60, i.e. a noise level 2.2 times higher than the data in Fig. 2.

3. Experimental optimisation In an experiment in which a continuous signal is sampled at regular time intervals, one is faced with choosing a number of experimental variables, e.g. the interval between samples Dt , the number of time points N, and the extent of signal averaging M. Of course, the choice is strongly influenced by the form of the signal to be digitised, but it is rare that this decision is made totally objectively, that is in such a way as to maximise the precision with which the parameterŽs. of interest can be quantified for a given

measurement time. For example, there is often a trade-off between N and M. If, as in the ESEEM experiment discussed here, the overall experiment time is essentially proportional to the product N = M, we may ask whether it is better to sample just once per time point Ž M s 1. and maximise the number of points N, or to perform more extensive signal averaging with fewer time points. Clearly, signal averaging improves the signal-to-noise ratio, but this may be counterproductive if it is achieved by reducing the maximum value of the time variable Žs NDt . to the point that significant information in the tail of the signal is missed. The answer is conveniently provided by the Cramer–Rao lower bounds, whose val´ ues are minimised by the optimum sampling strategy w23–25,28x. Suppose the overall duration of the ESEEM measurement is fixed, for example by the number of laser flashes to which the sample can be subjected before it starts seriously to degrade or, for flowed samples, by the amount of material available. If the measurement is repeated M times at each value of t , the signal-to-noise ratio will improve by a factor of 'M . The lower bounds, which are inversely proportional to the signal-to-noise ratio, therefore scale as 'N , for fixed N = M. The optimum compromise between signal averaging and number of t-values may be found by choosing N so as to minimise sŽ u k .'N . Fig. 3 shows the dependence of sŽ D .'N and sŽ J .'N on tmax Žs NDt . using Dt s 20 ns as in Ž . the Pq Qy A data. The optimum tmax minimum CRLB for both parameters is close to 660 ns, which gives a sŽ D .'N smaller by a factor of about 2 than for the 2760 ns used above. That is, the expected error in D could have been halved by cutting the acquisition time tmax by three-quarters and quadrupling the extent of signal averaging. At first sight it is surprising that a relatively small value of tmax , near the second maximum of the experimental data Žsee Fig. 2. and well before the signal disappears into the noise, should be the optimum. Certainly such a high degree of truncation of the experimental signal would be inappropriate if D and J were to be estimated from the Fourier transform, because of the ‘sinc-wiggles’ that would arise from the discontinuity at t s tmax in the time-domain signal. However, for time-domain model-fitting

C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

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Fig. 3. Scaled Cramer–Rao lower bounds for D Žsquares. and J Ždiamonds., calculated using Eqs. Ž5. and Ž6., as a function of tma x ´ Žs NDt . for Dt s 20 ns. The optimum value of tma x corresponds to the minimum in sŽ D .'N , i.e. 660 ns.

it is clear that the earlier, more intense part of the ESEEM signal contains more information than does the tail. Less obviously, it seems that it is only necessary to sample about one cycle of the principal oscillation Žfrequency ; 2 Dr3. in order to determine reasonably precise values of D and J. One may also ask whether there is any advantage in changing the sampling interval Dt , and hence N, keeping tmax fixed at 660 ns. The scaled Cramer–Rao ´ lower bounds for D and J were found to be essentially independent of N if it is greater than or equal to 5 Ž Dt - 115 ns.. Although the digitised signal appears smoother for sampling intervals smaller than ; 115 ns, it contains no extra information. For smaller N Žlarger Dt . the uncertainties in both parameters increase sharply.

4. Nuclear modulations The Cramer–Rao lower bounds only provide a ´ good guide to the errors in the values of parameters determined by model-fitting if the model provides an adequate description of the experimental data. Although the ESEEM simulation in Fig. 2A broadly agrees with the data, the fit is not very impressive. In

particular, there is a shoulder just after the second zero-crossing, close to 500 ns, that is not, and cannot, be reproduced by the model function of Eq. Ž5.. A possible source of mismatch is the echo modulation that arises from anisotropic hyperfine interactions w12,18,19x. If the electron spins are weakly coupled, the modulation function for a radical pair should be simply the product of the pure electron– electron modulation and the pure electron–nuclear modulation that would be observed for the two radicals in the absence of radical–radical interactions w18x. A modified model function may be obtained by extending the earlier treatment w18x from one to two spin-1r2 nuclei, one in each radical Žlabelled A and B.. In reality, the nuclear ESEEM of the photosynthetic radical pair is a highly complicated and imperfectly known function of the many hyperfine and quadrupolar interactions in the two radicals. Nevertheless, a two-nucleus approximation is quite revealing. Thus, Eq. Ž5. becomes: yˆn s h eyt n r T

2p

p

H0 H0

sin Ž vetn . 12  fA Ž t .

qf B Ž t . 4 sin u d u d f ,

Ž 10 .

C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

598

where the nuclear modulation functions fAŽt . and f B Žt . are, as usual w29x, given by fQ Ž t . s 1 y 12 k Q 1 y cos v Q a t y cos v Q b t q 12 cos Ž v Q a q v Q b . t q 12 cos Ž v Q a y v Q b . t .

Ž 11 .

v Q a and v Q b are the NMR frequencies of nucleus Q Žs A or B. coupled to an m s q1r2 or m s y1r2 electron spin, and k Q is the depth of the modulation:

(Ž a y v s (Ž a q v

vQ a s

1 2

Q

QI

.

vQ b

1 2

Q

QI

.

kQ s

ž

2

q 14 bQ2 ,

2

q 14 bQ2 ,

2

bQ v Q I

vQ a vQ b

/

.

Ž 12 .

aQ , bQ and v Q I are the secular and pseudo-secular hyperfine coupling constants, and the nuclear Zeeman frequency of nucleus Q. The anisotropic hyperfine interactions are assumed to be axially symmetric, with the form aQ s A Q Ž 3 cos 2 ´ Q y 1 . , bQ s 3 A Q sin ´ Q cos ´ Q ,

Ž 13 .

where ´ Q is the angle between the axis of the hyperfine interaction and the magnetic field, and A Q is the coupling constant. The axes of the two hyperfine interactions make angles cA and c B with respect to the electron–electron dipolar axis; all three vectors are assumed to lie in a plane. To see whether Eq. Ž10. provides a more satisfactory description of the ESEEM data, we need to repeat the model-fitting procedure, now with 10 parameters: D, J, T, and h, and, for each radical, A Q , v Q I , and c Q . Although the 10-dimensional x 2 surface is highly structured, the global minimum, located by an extensive search, is substantially deeper than the many local minima. The result, shown in Fig. 2B, is seen to be a significant improvement on Eq. Ž5. ŽFig. 2A.: x 2 fell by a factor of 4 and the best fit parameters were: D s y116.6 mT; J s q0.90 mT; T s 0.93 ms;

hrs s 247.2.

Ž 14 .

Comparing these with Eq. Ž6., one sees that D has not changed, that J has roughly doubled but is

still tiny, and that T is a little smaller. The value of hrs has approximately doubled because the nuclear modulations have the effect of reducing the amplitude of the calculated signal, so that a larger h is needed to get agreement between simulated and experimental data. Although the anisotropic hyperfine interactions change the shape of the ESEEM spectra Žsee below., they evidently do not markedly shift the frequencies of the parallel and perpendicular features which principally determine D. The revised model can only be held to be meaningful if the values of the new variables are judged to be sensible. Given the somewhat unrealistic nature of the two-nucleus approximation, it seems unprofitable to inspect the hyperfine parameters directly. A more satisfactory approach is to use them to simulate the echo modulation arising solely from the hyperfine interactions, free from the dominant modulation caused by the electron spin–spin coupling. This is achieved by evaluating eyt r T

2p

p

H0 H0

1 2

 gA Ž t . q g B Ž t . 4 sin u d u d f , Ž 15 .

where g Q Žt . is the time-dependent part of fQ Žt ., i.e. Ž fQ Žt . q 12 k Q y 1. using the parameter values in Eq. Ž14. and the caption to Fig. 2B. The Fourier transform of this signal, shown in Fig. 4, has principal peaks at 2.8 and 3.9 MHz and a smaller feature near 1 MHz. This spectrum shows striking similarities to the two-pulse X-band ESEEM signals of the individual radicals: Pq Žin chromatophores of Rhodospirillum rubrum w30x. has a major peak at 2.64 MHz and 2q Ž a minor one at 0.90 MHz; Qy A in Zn -substituted Rb. sphaeroides R-26 reaction centres w31x. has a strong peak at 4.1 MHz, and three less well-resolved lines below 2 MHz. This correspondence provides encouragement that it is physically realistic to extend the model to embrace anisotropic hyperfine interactions. The CRLBs for D, J, T and h using the 10parameter model are: s Ž D . s 0.47 mT;

s Ž J . s 0.17 mT;

s Ž T . s 0.029 ms;

s Ž h . rs s 5.18 .

Ž 16 .

Surprisingly, the first three are smaller than for the simpler model without nuclear modulations. Although for a fixed amplitude h, the lower bounds for

C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

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Fig. 4. The Fourier transform Žabsolute value mode. of the nuclear ESEEM ŽEq. Ž15.. derived from the experimental data for Pq Qy A in Rb. sphaeroides R-26 by deconvolution of the dominant electron–electron modulation.

D, J and T are as expected larger Žin general, the more parameters in a model, the higher their standard deviations., this effect is outweighed by the increase in hrs noted above.

An impression of the improvement afforded by the inclusion of nuclear modulations is provided by the frequency domain spectra in Fig. 5. To avoid problems associated with extrapolating the experi-

Fig. 5. Sine Fourier transforms of the three ESEEM signals in Fig. 2. The agreement with the experimental measurement Ždots. is more impressive with Žsolid line. than without Ždashed line. the anisotropic hyperfine interactions. The frequency axis has been converted to magnetic field units.

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C.E. Fursman, P.J. Hore r Chemical Physics Letters 303 (1999) 593–600

mental signal back to t s 0, the first 10 points Žt s 0–180 ns. of the two simulations were set to zero prior to Fourier transformation. Despite the resulting baseline undulation in all three spectra, the differences in the quality of the two fits are striking.

5. Conclusions Cramer–Rao lower bounds provide a rapid and ´ simple route to the uncertainties in the values of parameters obtained by model-fitting. The method suggests that the precision of the dipolar coupling in photosynthetic radical pairs obtained from two-pulse ESEEM may be better than previously estimated Žtypical errors quoted are "4 mT w3,5–8x.. For a given measurement time, the precision may be optimised by recording a truncated echo modulation signal and using the time saved to perform more extensive signal averaging. Although a simple model neglecting the modulations arising from anisotropic hyperfine interactions, does not give very impressive agreement with the experimental data, the indications are, for bacterial reaction centres at least, that the value of D is not seriously in error. A more elaborate model, including nuclear modulations, gives an excellent fit to the data, yielding parameters consistent with previously measured spectra of the individual radicals.

Acknowledgements We are grateful to Professor A.J. Hoff ŽLeiden. and Dr. S.A. Dzuba ŽNovosibirsk. for providing ESEEM data of PqQy A in Rb. sphaeroides R-26, and to Professor Hoff for his continuing support and helpful comments on the manuscript.

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