Biophysical Journal 85:1851-1870 (2003)
© 2003 The Biophysical Society
Charge Recombination and Protein Dynamics in Bacterial Photosynthetic Reaction Centers Entrapped in a Sol-Gel Matrix
Jan M. Kriegl *,
Florian K. Forster * and
G. Ulrich Nienhaus * 
* Department of Biophysics, University of Ulm, Ulm, Germany; and Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois
Correspondence: Address reprint requests to Gerd Ulrich Nienhaus, Dept. of Biophysics, University of Ulm, 89069 Ulm, Germany. Tel.: 49-731-502-3050; Fax: 49-731-502-3059; E-mail: uli{at}uiuc.edu.
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ABSTRACT
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Many proteins can be immobilized in silica hydrogel matrices without compromising their function, making this a suitable technique for biosensor applications. Immobilization will in general affect protein structure and dynamics. To study these effects, we have measured the P+QA- charge recombination kinetics after laser excitation of QB-depleted wild-type photosynthetic reaction centers from Rhodobacter sphaeroides in a tetramethoxysilane (TMOS) sol-gel matrix and, for comparison, also in cryosolvent. The nonexponential electron transfer kinetics observed between 10 and 300 K were analyzed quantitatively using the spin boson model for the intrinsic temperature dependence of the electron transfer and an adiabatic change of the energy gap and electronic coupling caused by protein motions in response to the altered charge distributions. The analysis reveals similarities and differences in the TMOS-matrix and bulk-solvent samples. In both preparations, electron transfer is coupled to the same spectrum of low frequency phonons. As in bulk solvent, charge-solvating protein motions are present in the TMOS matrix. Large-scale conformational changes are arrested in the hydrogel, as evident from the nonexponential kinetics even at room temperature. The altered dynamics is likely responsible for the observed changes in the electronic coupling matrix element.
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INTRODUCTION
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Both natural and engineered proteins have great potential as exquisitely selective, versatile catalysts and recognition elements in many different applications in scientific areas such as synthetic and analytical chemistry, medicine, biotechnology, bioremediation, and environmental monitoring. Often, technical applications demand efficient ways of immobilizing proteins. For instance, industrial bioconversion processes require high-density, heterogeneous catalysts, permitting high activities over extended time periods while tolerating a range of operating conditions. Biosensors and biochips are frequently based on precisely engineered architectures, containing immobilized biomolecules in their fully functional conformation (Gill and Ballesteros, 2000a
,b).
Various ways have been devised to form bioimmobilizates, such as covalent binding to solid surfaces and supported organic films, entrapment in polymer hydrogels, and microencapsulation. A particularly attractive approach promising wide applicability appears to be the formation of biomolecule-polymer composites in which biomolecules are permanently entrapped within covalent polymer networks (Gill and Ballesteros, 2000a,b). Recent years have seen enormous progress in the sol-gel encapsulation of biomolecules in porous silica networks. By modification of the classic sol-gel methods, mild polymerization conditions were developed so that proteins maintain their native structure and characteristic activities when co-polymerized with the silica matrix (Ellerby et al., 1992). The polymer matrices thus produced were shown to be permeable enough to enable diffusion of lower weight substrates while retaining even smaller proteins in the mass range 
10 kD. A large variety of novel biosensors has been developed based on encapsulation of biomolecules in silica sol-gel matrices, with applications ranging from the detection of small amounts of gaseous molecules such as O2, CO, or NO to the determination of glucose content in liquids (Gill and Ballesteros, 1998, 2000a; Tess and Cox, 1999; Livage et al., 2001).
As of yet, little is known about the structural details of the biomolecule-matrix interaction. Protein function is intimately linked to the ability of the protein to perform structural fluctuations among many different conformational substates (Frauenfelder et al., 1991; Nienhaus and Young, 1996). Consequently, the details of the biomolecule-matrix interaction determine if an encapsulated protein will be completely, partly, or not at all functional when embedded in the polymer matrix. Spectroscopic and electrochemical studies on a number of proteins in silica hydrogels and xerogels (Gill and Ballesteros, 1998, 2000a; Livage et al., 2001) have suggested that the proteins are entrapped in their native conformation in tight cages, so that rotations and global conformational changes are restricted, but local motions required for substrate binding and catalysis are still possible. For instance, hemoglobins can be trapped in their different quaternary (R,T) states in the gel, and the polymer matrix suppresses global R-T interconversions entirely (Shibayama and Saigo, 1995; Khan et al., 2000, 2001). These studies showed unambiguously that large-scale dynamics of biomolecules is strongly hindered in the glassy cage. However, the origin of the forces restricting global protein motions is still unknown. Besides the obvious mechanical effects, specific electrostatic interactions between silicate sites and protein surface residues have been proposed to influence the flexibility of the protein (Shibayama and Saigo, 1995; Livage et al., 2001; Gottfried et al., 1999).
In this work, we have examined how encapsulation in a silica sol-gel matrix affects the kinetics of light-induced long-range electron transfer (ET) in reaction centers (RCs) of photosynthetic purple bacteria (Rhodobacter sphaeroides). RCs are membrane-bound pigment-protein complexes that consist of three polypeptide subunits (H, L, and M) and a number of cofactors. ET occurs by sequential electron tunneling between the cofactors, and even slight modifications of their mutual coupling can give rise to substantial variations of ET rates (Marcus and Sutin, 1985; Moser et al., 1992; Dutton and Mosser, 1994; Moser et al., 1995; Gray and Winkler, 1996; Page et al., 1999). This sensitivity enables us to examine the influence of the protein environment on the ET reaction by precise measurements of the ET kinetics. Light absorption initially causes promotion of the electronic system of the special pair (P), a bacteriochlorophyll dimer located close to the periplasmatic side of the membrane, to its first excited singlet state (P*). Subsequently, the electron is transferred within 4 ps via an accessory bacteriochlorophyll to a bacteriopheophytin (HA) and within another 200 ps further to the primary acceptor, a ubiquinone UQ10 (QA), which is located 25 Å away from the special pair, close to the cytoplasmatic side of the protein (Allen et al., 1987; Feher et al., 1989; Ermler et al., 1994). The final, secondary acceptor (QB) takes up two electrons and two protons, leaves its binding pocket and delivers the electrons and protons to the bc1 complex (Feher et al., 1989; Hoff and Deisenhofer, 1997). In the preparations that we have used for this study, the secondary quinone (QB) was removed so that the electron recombines with the hole on the special pair, thus restoring the RC within 
120 ms (room temperature) back to its ground state,
 | (1) |
The last and slowest ET step in the sequence, 
(because the pheophytin is not involved in this particular ET step, we drop the H for simplicity), is particularly attractive for detailed, quantitative studies. It is long-ranged, and the large spatial separation leads to weak coupling of the donor and acceptor electronic states, which ensures nonadiabatic ET from a thermally equilibrated initial manifold of vibrational states.
To assess how protein motions affect the ET reaction, we use the sample temperature as our key control variable. There are two ways in which temperature affects ET:
- The intrinsic temperature dependence arises from the coupling of the ET reaction to the phonon bath. Here we use the spin boson model (Leggett et al., 1987; Warshel et al., 1989; Xu and Schulten, 1992, 1994) to describe the observed, nonexponential kinetics and its temperature dependence. To achieve this, we introduce distributions of energy gaps and electronic coupling elements between donor and acceptor states to account for structural heterogeneity in the sample.
- An extrinsic temperature (and time) dependence arises from light-induced charge separation. The electric field changes substantially perturb the protein and give rise to conformational changes (Noks et al., 1977; Kleinfeld et al., 1984; Woodbury and Parson, 1986; Rubin et al., 1994; Peloquin et al., 1994). They in turn adiabatically influence the key physical parameters and thus the ET kinetics.
The basic strategy with which we exploit the extrinsic temperature dependence to gain insight into the coupling of ET and protein motions can be understood with the schematic in Fig. 1, depicting free energy surfaces of the 
and PQA electronic states as a function of a conformational coordinate q. They are assumed to be overall parabolic but exhibit a certain ruggedness, as can be inferred from the temperature dependence of the observed protein motions. Within the Condon approximation, transitions between the charge-separated and neutral states are vertical in this scheme. The energy gap, 
, the vertical separation between the two surfaces, is the crucial physical parameter governing the ET rate. As it is evident from Fig. 1, varies with q, and therefore, the evolution of the protein ensemble on the lower or upper energy surfaces can be studied by measuring the charge recombination kinetics.
In the experiment, samples are either cooled in the dark, or a strong light source is switched on during cooling at several temperatures TL, keeping the RCs in the charge-separated state. After arrival at 10 K, we measure light-induced ET kinetics while slowly increasing the temperature from 10 to 300 K over a period of several hours. Upon cooling in the dark, the RCs are frozen in the neutral PQA state, and the ensemble of RCs is distributed around the minimum of the lower energy surface (q = 0 in Fig. 1). For simplicity, we shall refer to this as the dark-adapted conformation (D). Light excitation pitchforks the molecules onto the upper surface, where they stay for a few milliseconds before recombination. At 10 K, the ruggedness of the surface is too large, and the RCs will not change conformational coordinate during their brief residence on the surface (Fig. 1). As temperature increases, thermally activated transitions on the upper surface become possible, and the ensemble can progress toward the minimum at q = 1, which we refer to as the light-adapted conformation (L). This motion is accompanied by an energy gap decrease. The theoretical analysis then yields a concomitant decrease of the ET rate.
By exposing the samples to a strong light source while cooling from room temperature to 10 K, samples will essentially be kept in the charge-separated state, and will be arrested in the light-adapted conformation (q = 1). After switching off the light at 10 K, a distribution of molecules has been prepared on the PQA surface at q 1 that cannot relax toward q = 0 because of the ruggedness of the energy landscape. As the temperature is slowly raised over several hours, energy barriers can be overcome, and the ensemble slowly moves to the left. During this relaxation, we can propel the molecules to the upper surface by laser excitation and measure their position along q via the ET kinetics.
In an earlier study, we had measured the kinetics of charge recombination from the primary acceptor in RCs of the carotenoid-deficient strain R26 of Rb. sphaeroides dissolved in glycerol-water mixtures and presented a detailed quantitative picture of the coupling between ET and protein motions solvating the altered charge distribution (McMahon et al., 1998). Here we use this strategy to assess the effect of encapsulation of (carotenoid containing) RCs in a sol-gel matrix on the protein motions accompanying ET in the proteins.
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MATERIALS AND METHODS
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Bacteria growth and RC purification
For this study, we have used RC proteins with engineered poly-His tag (Goldsmith and Boxer, 1996). Bacteria were grown anaerobically under illumination in 2 liters of YCC medium at 34°C for six days, yielding 5 g cells/l of medium (fresh weight) (Clayton and Sistrom, 1978). The cells were harvested at 4°C by centrifugation and kept at -80°C for long-term storage. After thawing, the cells were resuspended in lysis buffer (10 mM Tris, 100 mM NaCl, pH 8.0) and stirred for 30 min at room temperature (Goldsmith and Boxer, 1996). Lysozyme (Sigma-Aldrich Chemie GmbH, Munich, Germany), was added to a final amount of 0.8 mg/g cells (fresh weight) to break the cell membranes. The suspension was stirred for 30 min at 4°C and, after addition of deoxycholic acid (4 mg/g cells), heated to 37°C for 10 min. Subsequently, DNase (20 µg/g cells, Sigma-Aldrich Chemie GmbH, Munich, Germany) was added, and the lysate was stirred for yet another 30 min at room temperature. Lysate and cell debris were separated by centrifugation at 15,000 rpm (rotor SS-34, Sorvall, Newtown, CT) for 15 min. After collecting the supernatant, we essentially followed the purification protocol reported by Goldsmith and Boxer (1996). 0.5% (v/v) LDAO (Fluka, Buchs, Switzerland) and 5 mM imidazole and, thereafter, Ni-NTA resin (Qiagen GmbH, Hilden, Germany), equilibrated in purification buffer (10 mM Tris, 0.1% (v/v) LDAO, pH 8.0), was added to the solution. After gentle stirring at 4°C overnight, the slurry was poured onto an empty column. The poly-His RCs were eluted using 100 mM imidazole in purification buffer. Removal of the imidazole, further purification and concentration was achieved by gel filtration (Sephadex G25 column, Amersham Biosciences Europe GmbH, Freiburg, Germany) and Amicon filtration (molecular weight cutoff 100 kD). For removal of the secondary quinone, QB, RCs were loaded on a Toyopearl DEAE 650S anion exchange column (TosoHaas GmbH, Stuttgart, Germany) and washed with inhibition buffer (0.5 mM 1,10-phenanthroline, 10 mM Tris, 1% (v/v) LDAO; Okamura et al., 1975). The extent of QB removal was checked via the decay of the 
triplet state which appears in RCs that have lost both QB and the more strongly bound primary quinone QA (Volk et al., 1995).
Sample preparation
Purified and QB-extracted RCs were dissolved in a mixture of 75% glycerol and 25% buffer (10 mM Tris, 0.1% (v/v) LDAO, pH 8.0) to a final concentration of roughly 15 µM. The solution was loaded in a 10 x 10 x 2.5 mm3 PMMA cuvette. After injection of the sample, which we will refer to as the bulk solvent sample, the cuvette was sealed to prevent leakage and oxygen access.
The sol-gel matrix was prepared using tetramethoxysilane (TMOS, Sigma-Aldrich Chemie GmbH, Munich, Germany) as the precursor component (Chen et al., 1986; Brinker and Scherer, 1990; Ellerby et al., 1992; Miller et al., 1996). HCl-catalyzed hydrolysis of TMOS was achieved by mixing 0.04 M HCl, millipore water and TMOS (volume ratio 1:15:67). The solution was ultrasonicated for 20 min and kept on ice before subsequent addition of the protein-cryosolvent mixture. The TMOS hydrogel and the RC containing solution (mixture of 75% glycerol and 25% buffer (10 mM Tris, 0.1% (v/v) LDAO, pH 8.0) with a final protein concentration of 15 µM) were mixed at a volume ratio of roughly 3:4 and kept on ice for 10 min. The transparent sample was filled into a 10 x 10 x 2.5 mm3 PMMA cuvette that was also used for the measurements on RCs in bulk solvent. The cuvettes were carefully sealed to prevent evaporation of water and suppress further aging. ET kinetics data were found to be identical at the beginning and end of the investigation, implying that the properties of the sol-gel matrix within the cuvette remained constant during the experiments. The described encapsulation procedure yields a hydrogel with a silica network with pore diameters of <100 Å, in which the RC proteins are embedded (Brinker and Scherer, 1990; Ellerby et al., 1992; Livage et al., 2001).
Time-resolved spectroscopy
The cuvette containing the sample was mounted in a copper sample holder which was attached to the cold finger of a closed-cycle helium refrigerator (model 22, CTI Cryogenics, Mansfield, MA). A digital temperature controller (model 330, Lakeshore Cryotronics, Westerville, OH) was used to adjust the temperature. Light-induced electron transfer was initiated in the samples by a 6-ns (full width at half maximum) pulse (532 nm, 180 mJ) from a frequency doubled, Q-switched Nd:YAG laser (model NY 61, Continuum, Santa Clara, CA). To measure optical absorbance changes in the Soret region of the bacteriochlorophyll molecules, light from a tungsten lamp (model A 1010, PTI, Brunswick, NJ) was passed through a monochromator set at 436 nm, the sample, and a second monochromator onto a photomultiplier tube (model R5600U, Hamamatsu, Middlesex, NJ). The photocurrent was converted into a voltage, amplified and recorded with a digital storage oscilloscope from 10 ns to 50 µs (model TDS 520, Tektronix, Wilsonville, OR) and a home-made logarithmic time-base digitizer (Wondertoy II) from 2 µs to 100 s.
For light-adaptation of the RCs, light from a 250-W tungsten lamp (Oriel, Stratford, CT) was passed through a heat filter and a long pass filter (RG645, Schott GmbH, Mainz, Germany). From the signal amplitudes of transient ET kinetics of RCs cooled in the dark from 280 to 80 K, recorded both under continuous illumination and in the dark, the excitation rate, kL, of the illumination setup can be determined (McMahon et al., 1998; van Brederode, 1999), yielding kL 250 s-1. With an average recombination rate of 10 s-1 for RCs at room temperature, this implies that 95% of the RCs were trapped in the charge-separated state during cooling of the sample. To obtain ensembles with different illumination histories, light was switched on at specific temperatures TL during cooling from 280 to 10 K. After reaching the lowest temperature (10 K), the sample was warmed in the dark at a constant ramp rate of 4.6 mK/s for both dark- and light-adapted samples. ET kinetics after flash excitation were measured every 5 K between 10 and 300 K after 3 min equilibration time at each temperature. At least five transients were averaged at each temperature, with an equilibration time of 1 min.
Numerical computations
Numerical computation of Franck-Condon (FC) factors according to Eq. 42 (see Appendix) was implemented on a SUN UltraSPARC-III architecture available at the computer facility of the University of Ulm, using library functions of the NAG C library (Mark 6, NAG Ltd., Oxford, UK). In our previous work, the spectral density of phonons involved in the ET reaction was truncated at 400 cm-1 to simplify the FC calculation (McMahon et al., 1998). In the new algorithm, we have shifted the cutoff frequency to 3400 cm-1, which is the upper limit of vibrational modes occurring in proteins. The FC calculations were typically performed with a resolution in redox energy and reorganization energy of 20 meV, covering the temperature range from 0 to 305 K at a spacing of 5 K. For spectral densities with a substantial high frequency component, the resolution in redox energy was increased to 
= 1 meV. FC factors within two discrete points of evaluation were obtained via cubic spline interpolation on a logarithmic FC scale. All other data analysis was performed on PC workstations using the PV-WAVE package (Visual Numerics, Boulder, CO).
Experimental results and qualitative discussion
To reliably examine the influence of the sol-gel matrix on the ET properties of the protein, we have performed experiments with bulk solvent and TMOS-encapsulated RCs in parallel, using protein from the same strain and preparation. Fig. 2 shows charge recombination kinetics of RCs dissolved in glycerol/buffer solution (closed symbols) and RCs encapsulated in a sol-gel matrix (open symbols) at 60, 170, and 300 K. For both dark-cooled RCs (Fig. 2 A) and RCs cooled under illumination from 280 K (Fig. 2 B), charge recombination in the TMOS sample is a factor of 2 slower than in solution. All kinetic traces are nonexponential except the solution data near 300 K.
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FIGURE 2 Charge recombination kinetics of RCs in a 75%/25% (v/v) glycerol/buffer mixture (closed symbols) and embedded in TMOS (open symbols) at 60 K (circles), 170 K (triangles), and 300 K (diamonds). (A) Cooled in the dark, and (B) cooled under illumination from 280 K. Lines represent fits according to the model described in Modeling and Quantitative Analysis.
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A phenomenological, model-independent analysis of the ET kinetics provides insight into the subtle differences between both samples. For that purpose, we describe the survival probability of the charge-separated state, N(t), with a distribution of rate coefficients, f(log k),
 | (2) |
A numerical Laplace inversion of Eq. 2 using the maximum entropy method (MEM) yields the distribution of rate coefficients (Steinbach et al., 1992). They are depicted in Fig. 3 as contour plots for RCs in bulk solvent (left column) and TMOS-encapsulated RCs (right column), cooled in the dark (top row), and under illumination from 280 K (bottom row). To illustrate the differences between samples and their illumination histories, the same scale was applied to all four maps.
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FIGURE 3 Contour plots of the distributions of rate coefficients, f(log k), between 10 and 300 K with linear spacing of contour lines. All plots were scaled identically to illustrate the differences between samples and illumination histories. (A) Bulk solvent, cooled in the dark; (B) bulk solvent, cooled under illumination from 280 K; (C) TMOS hydrogel matrix, cooled in the dark; and (D) TMOS hydrogel matrix, cooled under illumination from 280 K.
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To afford a more quantitative comparison, the rate distributions f(log k) in Fig. 3 can be characterized by a logarithmically averaged rate,
 | (3) |
and the standard deviation 
ET to quantify the width of the distribution,
 | (4) |
In Fig. 4 we plot these two quantities for the distributions shown in Fig. 3. We have also included data from an experiment in which the illumination during cooling was switched on at 180 K.
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FIGURE 4 Temperature dependence of parameters characterizing the ET rate distributions. Logarithmically averaged charge recombination rate, log kET, (A) for RCs in bulk solvent and (B) for RCs immobilized in TMOS matrix. Width ET of the rate distributions, (C) for RCs in bulk solvent, and (D) for RCs encapsulated in TMOS. Samples were either cooled in the dark (diamonds) or under illumination from 180 K (crosses) and 280 K (open circles).
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The sample in cryosolvent exhibits the same overall features that were observed in earlier experiments on carotenoid-deficient RCs (McMahon et al., 1998). ET kinetics of dark-cooled RCs speed up by a factor of 6 upon cooling from room temperature to 10 K, from kET 8.7 to 50 s-1. A pronounced step occurs in the temperature range between 160 and 200 K, as seen in Figs. 3 A and 4 A. A slight increase of the average rate coefficient from 7.5 to 8.7 s-1 is observed between at 250 and 300 K. At 10 K, the width of the distribution is ET = 0.2. At 200 K, ET exhibits a broad maximum extending over a temperature interval of 50 K. The distribution narrows toward 300 K, with ET = 0.1 as obtained from the MEM. This width, however, represents exponential (nondistributed) behavior within the noise statistics of our data. A completely different behavior is observed after cooling the cryosolvent sample from 280 K under illumination. At 10 K, kET is about the same as at 300 K, and upon warming, an excursion toward faster rates is evident, extending over temperatures ranging from 120 to 200 K. Above 250 K, the kET data essentially merge with those of the dark-adapted preparation. The width of the rate distribution is about threefold larger at 10 K as compared with the dark-cooled sample. Concomitant with the excursion of kET above 120 K, the width of the distribution narrows and merges with that of the dark-cooled sample above 200 K.
A dip in the rate coefficients occurs universally below 50 K. Its magnitude depends on the illumination conditions and thus reflects light-induced conformational changes. This effect is likely related to observations made by spectral hole burning and will not be considered here (Ganago et al., 1991; Reddy et al., 1992).
The temperature-dependent features in the data (above 50 K) can be understood with the schematic in Fig. 5. Cooling in the dark arrests the dark-adapted state with fast recombination kinetics at 10 K. From 50 to 170 K, the protein ensemble is frozen-in on the timescale of charge separation (Fig. 5, A and B), and the slowing arises solely from the intrinsic temperature dependence of the ET reaction (vide infra). The steep drop from 170 to 200 K and the subsequent step up to 250 K are caused by the protein ensemble relaxing on the energy surface during the (millisecond) lifetime of the charge-separated state (Fig. 5, C and D). The quantitative analysis shows that this leads to a decrease of the energy gap and concomitant slowing of the ET rate. This dynamic model is further supported by the observation that the rate distribution broadens over the temperature interval in which charge recombination occurs while the protein ensemble is diffusing downhill on the excited state surface. Above 250 K, relaxation is complete within the lifetime of the charge-separated state so that recombination occurs from around the minimum of the energy surface at q = 1.
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FIGURE 5 Evolution of the protein ensembles in different temperature ranges due to relaxation on the energy surfaces associated with the neutral and charge-separated states, for samples cooled in the dark (left, A–D) or under illumination from room temperature (right, E–H). A detailed description is given in the text.
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Cooling under illumination arrests the ensemble in the light-adapted state. At 10 K, kET is almost identical to the value at room temperature, reflecting the smaller energy gap at q = 1 (Fig. 5 E). As the temperature is slowly increased, protein dynamics is initially frozen in. But then, at 120 K, an increase of kET signals that the ensemble begins to slowly develop toward q = 0. It migrates over hours on the lower surface (Fig. 5 F), and the relaxation dynamics can be observed over h instead of ms as in the dark-cooled preparation. A maximum occurs at 175 K. At this point, relaxations on the upper surface become important again, leading to a subsequent decay toward 200 K (Fig. 5 G). Above 250 K, the RCs relax completely on the upper energy surface within the lifetime of the charge-separated state (Fig. 5 H). The extent of relaxation can be quantified by the gap between the dark- and light-cooled kET curves. The temperature dependence of the width ET also supports this dynamic scenario. It decreases markedly above 120 K and merges with that of the dark-adapted sample near 200 K. Fig. 4 also includes data measured after switching light on at 180 instead of 280 K during cooling. At 180 K, motions on the rugged energy surface are substantially slowed and partially arrested so that the protein ensemble becomes trapped on part of the way between q = 0 and 1. Consequently, data from this experiment interpolate between the other two data sets.
RCs immobilized in TMOS hydrogels exhibit qualitatively the same behavior as solvent samples. The temperature dependence of the rate distributions is similar, with the step in kET of the dark-cooled sample between 160 and 200 K and the excursion of kET of the light-cooled sample in the intermediate temperature range. It is evident that protein motions solvating the charges are not suppressed in the hydrogel. However, there are also significant differences. For both dark- and light-cooled samples, recombination is slower by a factor of 2 over the entire temperature range in RCs in a sol-gel matrix than in cryosolvent. The overall width ET is larger than that of the bulk sample (Fig. 4, C and D), and the kinetics remain clearly nonexponential even at room temperature. A pronounced difference between solution and hydrogel is apparent from comparing kET above 200 K. Whereas the solvent sample exhibits a discrete step in kET between 200 and 250 K, the TMOS sample shows a smooth and continuous decay, and kET remains constant above 250 K. With a detailed analysis of the temperature dependence of ET in a solution sample we had shown earlier that the step between 200 and 250 K is associated with the presence of global conformational motions (McMahon et al., 1998). The lack of this step suggests, therefore, that global fluctuations between markedly different conformations are suppressed in the sol-gel matrix. This conclusion is also in agreement with the observed widths of the rate distributions. From 200 to 250 K, the rate distribution of the solution sample collapses to essentially exponential behavior, indicating that even the slowest global fluctuations affecting ET become faster than kET. By contrast, the rate distributions of TMOS encapsulated RCs narrow only slightly and stay constant above 250 K. The nonexponential kinetics even at 300 K clearly show that the RCs entrapped in the gel are structurally heterogeneous. While local dynamics is possible in the gel, fluctuations among globally distinct conformations are arrested on the ET timescale.
Finally, there is another significant difference between bulk solvent and encapsulated RC samples. When cooling under illumination from 280 K, the signal amplitude of the solvent sample at 10 K is only 20% of that at 300 K, but 50% for the TMOS sample. This effect is completely reversible, however, and recovery of the signal amplitudes occurs at 250 K in both samples.
From this phenomenological discussion of the kinetics, two main questions arise. First, why are the charge recombination kinetics in RCs that are immobilized in a silica gel markedly slower than in bulk solvent? And second, how are relaxations and fluctuations among different conformational substates influenced by the hydrogel matrix? To address these questions, we present a quantitative analysis of the ET kinetics based on a quantum-mechanical model that allows us to map the distribution of rate coefficients onto a distribution of a physical parameter, the energy gap .
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MODELING AND QUANTITATIVE ANALYSIS
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Microscopic ET rates
In his pioneering work on electron transfer reactions in solution, Marcus derived the well-known formula for the ET rate coefficient (Marcus, 1964; Marcus and Sutin, 1985),
 | (5) |
where V is the electronic coupling between reactant and product state, and 
denote the free energy difference between the two redox states and the reorganization energy, respectively, kB is Boltzmann's constant, 
Planck's constant divided by 2
, and T is the absolute temperature. This expression has been applied successfully to a variety of redox systems in nonbiological as well as in biological systems (Marcus and Sutin, 1985; Gray and Malmström, 1989; Moser et al., 1992).
Marcus's rate law is the classical version of the general expression for nonadiabatic ET transfer, in which the microscopic rate coefficient is given by Fermi's golden rule, using the Condon approximation (Levich and Doganadze, 1959; Jortner, 1976),
 | (6) |
with the thermally averaged FC factor, which is a measure for the overlap of the nuclear wavefunctions between the reactant and product state. Comparison with Eq. 5 shows that, in the classical case, FC depends on the redox energy , the reorganization energy , and the temperature T.
However, Marcus's expression is only applicable at temperatures sufficiently high that equipartition holds for all vibrational modes coupled to the ET reaction. In biological ET, however, quantum effects are important even at room temperature. Therefore, we use the spin-boson model to evaluate the FC factors. This model provides a quantum-mechanical description of ET coupled to a continuous spectrum of harmonic oscillators, which is characterized by its spectral density J(
) (Garg et al., 1985; Onuchic et al., 1986; Warshel and Hwang, 1986; Leggett et al., 1987; Warshel et al., 1989; Xu and Schulten, 1992; Xu and Schulten, 1994). The temperature and free energy dependence of the FC factor is given in terms of the spectral density J(), which is related to the reorganization energy by (Xu and Schulten, 1994)
 | (7) |
We model J()/ as a sum of a Lorentzian, centered at = 0 and characterized by its linewidth s, and a few Gaussians centered at higher frequencies i > 0 cm-1 with variance 
 | (8) |
where Ss and Si denote the coupling strengths. A more elaborate description of the spin-boson model will be given in the Appendix.
In our analysis of the charge recombination kinetics, we describe the structural heterogeneity of the protein ensemble by a distribution of energy gaps, g(), which depends on temperature and the illumination protocol applied before the experiment. The energy gap distribution is related to the distribution of the rate coefficients, f(log k), by
 | (9) |
In contrast to the free energy gap , the spectral density J() (and thus the reorganization energy ) is taken to be temperature independent. Furthermore, the electronic coupling matrix element V is assumed to depend logarithmically on the free energy gap . This is modeled by introducing a pivot point 0 and a stretch factor 
,
 | (10) |
with temperature-independent parameters V0, , and 0. The assumptions underlying Eq. 10 are discussed in detail in McMahon et al. (1998). This approach allows us to recast Eq. 2 in terms of a single physical parameter, the energy gap :
 | (11) |
Determination of temperature-independent ET parameters
From the classical Marcus formula (Eq. 5), which is equivalent to the spin boson model in the high temperature limit (Xu and Schulten, 1992), it is obvious that various combinations of the parameters , V, and characterizing the redox system yield the same rate coefficient at a particular temperature. Whereas the redox energy can be determined by experimental means, for instance redox titration (Moss et al., 1991; Lin et al., 1994; Paschenko et al., 2001), delayed fluorescence measurements (Arata and Parson, 1981), or direct voltammetry (Kong et al., 1998), the reorganization energy is not directly accessible to experiment (Sharp, 1998). To model charge recombination from the primary quinone QA in RCs of Rb. sphaeroides, several authors reported varying combinations of V and , all yielding convenient fits of the ET kinetics (Gunner et al., 1986; Ortega et al., 1996; McMahon et al., 1998; Schmid and Labahn, 2000). Therefore, a self-consistent method is necessary to determine these parameters. In our experimental approach, we measure the charge recombination kinetics as a function of both temperature and illumination protocol. This allows us to extract temperature-independent parameters from the intrinsic temperature dependence of the microscopic ET rates. Moreover, we can also quantify the effects of protein motions on the ET reaction.
To determine the temperature-independent parameters, , V0, , and 0, we exploit a peculiarity of activationless ET. A calculation of rate coefficients k(, T) for fixed reorganization energy (Fig. 6) shows, that, depending on the redox energy , the ET kinetics either speed up or slow down with increasing temperature. Both regions are separated by a region with essentially temperature-independent rate coefficients. In the following, we refer to this point of minimal temperature dependence, {iso, kiso}, as the isokinetic point. This behavior can be examined directly by comparing subsequent kinetics traces N(t, T) with increasing temperature T. The dispersion of the ET kinetics with respect to temperature can be quantified by calculating the sum over all pair differences,
 | (12) |
where nT denotes the total number of kinetic traces. Fig. 7 A shows p(t) for dark-cooled (filled triangles) and light-cooled (open circles) RCs in cryosolvent, calculated for temperatures between 60 and 120 K (5 K intervals). Whereas the kinetics of the dark-cooled sample slow down continuously with increasing temperature, as inferred from the sign of p(t), cooling under illumination leads to a more complex behavior: for t < 0.24 s, the kinetics speed up with temperature; for t > 0.24 s, the opposite is observed. Thus, tiso = 0.24 s indicates an isokinetic point. From the sign of p(t) and the isokinetic point, the location of the distributions of redox energies, g(), can thus be estimated for both samples: the majority of the distribution characterizing dark-cooled RCs in cryosolvent is located at redox energies larger than iso; for bulk RCs cooled under illumination, the distribution is centered near the isokinetic point iso.
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FIGURE 6 Microscopic rate coefficients k(, T) as a function of , calculated with the spin boson model at 10, 50, 100, 150, 200, 250, and 300 K (alternating solid and dotted lines). Also shown are three Gaussian g() distributions, the one that was used to optimize the temperature-independent ET parameters (dotted line), and the fit results for the dark-cooled sample (60 K, solid line) and a sample cooled under illumination from 280 K (60 K, dashed line). For details, see the text.
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FIGURE 7 Sum of pair differences, p(t), from kinetic traces measured between 60 and 120 K. (A) Bulk solvent RCs. (B) RCs embedded in TMOS: dark-cooled (diamonds), sample cooled under illumination from 280 K (open circles); and isokinetic points (arrows).
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Along with the reorganization energy , the electronic coupling matrix element V0 and the factors relating and V0, , and 0, have to be determined. Besides the isokinetic point kiso, we take advantage of two additional features of charge recombination kinetics. First, we assume that the microscopic rate coefficient for nonadiabatic ET, kRT, is well approximated by the Marcus expression, Eq. 5, at the highest temperatures studied. With the redox energy at room temperature (300 K), RT, determined by independent experimental means (Arata and Parson, 1981; Moss et al., 1991; Lin et al., 1994; Paschenko et al., 2001), the interdependence of the electronic coupling, V, and the reorganization energy, , is given by
 | (13) |
where the rate coefficient at room temperature, kRT, is taken from the experimental data. Second, we compare the maximum rate obtained for dark-cooled samples, kmax, and the rate at the isokinetic point, kiso. We define kmax heuristically such that 5% of the distribution covers still faster rates,
 | (14) |
with the rate distribution f(log k) as obtained from a MEM analysis of the charge recombination kinetics of dark cooled RCs. From the spin-boson model, the ratio 
= kmax/kiso can be calculated as a function of and reorganization energy for fixed pivot points 0, whereas different reorganization energies are achieved by an appropriate, concurrent scaling of the coupling constants SS and Si in Eq. 8. At fixed , can be computed numerically as a function of . Both relations lead to a unique interdependence of the temperature-independent parameters , V, and , which is illustrated for RCs in bulk solvent in Fig. 8.
Conformational motions can be neglected below 120 K. Therefore, the protein ensemble cooled under illumination from room temperature remains in the light-adapted conformation, which is characterized by a distribution of energies gaps, gL(), centered at the room temperature value RT. For simplicity, we assume a Gaussian shape,
 | (15) |
With the distribution of energy gaps, gL(), kept fixed between 60 and 120 K and the interdependence of , V0, and as shown in Fig. 8, a parameter triple , V0, and can be iteratively determined that reproduces the observed isokinetic point. Within the optimization procedure, a variation of the pivot point 0 is reflected in a change of the electronic coupling matrix element V0, whereas both the reorganization energy and the scaling factor of the interdependence of the electronic coupling matrix element V0 and redox energy , , remain unaffected. Therefore, we chose 0 = RT in our calculations. The rate at the isokinetic point also depends on the width L of gL(). Thus, we use the width of the distribution of rate coefficients, ET, obtained from the MEM analysis of the ET kinetics (Eq. 4), as an additional criterion in the parameter determination.
As a final step in the determination of the temperature-independent parameters, an appropriate shape of the spectral density J() has to be selected. High frequency modes with short lifetimes give rise to a flattening of the k(, T) curves. The dispersion of the microscopic rate coefficients with respect to temperature, d log k/dT, at fixed redox energy and reorganization energy is reflected in the steepness of the sum of pair differences, p(t). The larger 
the more pronounced are the differences observed between subsequent kinetic traces N(t, Ti) and N(t, Tj) at a fixed distribution of redox energies, and vice versa. This is illustrated in Fig. 9 A, where the sum of pair differences p(t) is shown for different parameterizations of the spectral density (inset, Fig. 9 A) after iterative determination of the temperature-independent ET parameters. The open squares represent the experimental results obtained from bulk RCs cooled under illumination from 280 K. As a quantitative criterion for the choice of parameters governing the spectral density J(), we use the residuals R(t) of the sum of pair differences, shown in Fig. 9 B. They are defined as
 | (16) |
Here, 
is obtained from the parameter optimization, and 
denotes the sum of pair differences calculated from the experimental data. The standard deviations (t) have been determined from the statistics of each kinetic trace N(t, T). Minimization of the residuals R(t) yields a suitable parameterization of the spectral density. Note that we have not applied any fitting routine to analyze each individual kinetic trace so far; the optimization algorithm, nevertheless, reproduces the sum of pair differences very well.
Fig. 6 shows k(, T) curves obtained for bulk solvent RCs from the optimization algorithm described above, together with the static distributions of energy gaps of the light-adapted and dark-adapted conformations. The temperature-independent ET parameters are summarized in Table 1.
The identical parameter optimization procedure has also been applied to RCs embedded in the TMOS matrix. The sums of pair differences p(t) in Fig. 7 B exhibit a similar pattern as their bulk solvent counterparts, suggesting that the analysis applied to bulk solvent RCs should also be valid for TMOS encapsulated RCs. The parameters, which are included in Table 1, are overall similar to those of RCs in bulk solvent. Two differences are noteworthy. The electronic coupling matrix element V0 is reduced compared to bulk solvent. This difference directly affects the position of the isokinetic point, which can be read off from Fig. 7 B as tiso = 0.42 s for TMOS-encapsulated RCs instead of tiso = 0.24 s obtained for bulk RCs. Moreover, the amplitude of p(t) is smaller for TMOS-encapsulated RCs, which can be traced back to a broadening of the distribution of energy gaps. Indeed, the width of the Gaussian characterizing the freeze-trapped, light-adapted protein ensemble is 15 meV larger compared to the value obtained for bulk solvent RCs.
Fit results
After determination of the temperature-independent parameters of the model, the charge recombination kinetics were fitted individually by applying Eq. 11, using a nonlinear least-squares Levenberg-Marquardt algorithm. In the simplest fashion, one can parameterize the distribution of redox energies, g(), by a single Gaussian. However, deviations from a symmetric distribution of energy gaps can be envisioned from different topographies of the potential surfaces representing the charge-separated 
and charge recombined state (PQA), respectively, as depicted in Fig. 1. Flash-induced charge recombination kinetics probe a distribution of redox energies, g(), rather than the initial distribution of conformations, p(q, T = T0). The mapping is given by
 | (17) |
where the redox energy is a function of the conformational coordinate q (Fig. 1).
Only for harmonic potentials with the same first derivative, the mapping is linear, resulting in a conserved shape of p(q, T = T0). In general, this may not be the case. If we assume the initial distribution of conformations to be Gaussian, as expected for evenly distributed random variables, the distribution of redox energies observed in charge recombination kinetics appears as an asymmetric distribution. To account for both symmetric and asymmetric distributions of redox energies, charge recombination kinetics were fitted with a single Gaussian or a sum of two Gaussians,
 | (18) |
As with f(log k), we characterize the g() distribution by its first moment, 
, and its second moment, , given by
 | (19) |
and
 | (20) |
In the fit, the peak and width parameters of the g() were varied. For bulk solvent RCs, a single Gaussian g() distribution described the charge recombination kinetics of both dark-cooled RCs and RCs cooled under illumination from different temperatures appropriately throughout the whole temperature range. Only between 120 and 180 K, the fit quality was slightly reduced. The distributions at 60 K for dark cooled RCs and RCs cooled under illumination from 280 K are depicted as solid and dotted lines in Fig. 6, respectively. Also shown is the static g() distribution that was used in the parameter optimization (dashed line), showing a deviation by only 3% of the peak position from the fit result. The kinetics at 60, 170, and 300 K calculated from the fit are shown as solid lines in Fig. 2, illustrating the ability of our ET model to precisely reproduce the experimental data without introducing fit parameters beyond those mentioned above. The center positions and widths of the Gaussians are depicted as a function of temperature between 60 and 300 K for both dark cooled samples and samples cooled under illumination in Fig. 10, A and C, respectively. They reproduce the behavior deduced from the temperature dependence of rate distributions in Fig. 4, A and C.
For the sol-gel matrix preparation, the kinetics of dark-cooled RCs and RCs cooled under illumination from 180 K were well fitted using a single Gaussian g(). For RCs cooled under light from 280 K, the distribution was clearly asymmetric and, therefore, we used the two-Gaussian distribution. From our data we cannot identify the origin for the asymmetry present in the light-adapted ensemble. Slight distortions of the potential surfaces are conceivable, leading to a nonlinear mapping between and q (Eq. 17). The presence of a significant fraction that was not freeze-trapped in the light-adapted state due to restrictions introduced by the TMOS matrix can be excluded because of the temperature dependence of and (Fig. 10, B and D) which tracks their bulk solvent counterpart perfectly. The individual fits are shown as dashed lines in Fig. 2, A and B. The mean energy gap of the dark-cooled TMOS encapsulated RCs between 60 and 120 K is 4% higher compared to the value obtained for bulk solvent RCs, indicating a slight variation of the energetics governing ET upon sol-gel encapsulation. The parameters characterizing the 60 K distributions of energy gaps for both bulk solvent and TMOS entrapped RCs are compiled in Table 2.
Following our previous work (McMahon et al., 1998), we introduce a phenomenological relaxation function 
(T), which is defined for dark-adapted (D) and light-adapted (L) RCs as
 | (21) |
and
 | (22) |
respectively. Here, the difference D(T0) – D(T1) defines the maximum relaxation amplitude of the conformational coordinate , with D(T1) denoting the fully relaxed protein.
Fig. 11, A and C, shows both D(T) and L(T) and their derivatives with respect to T, |d/dT|, for bulk solvent RCs. We have restricted the relaxation functions to the temperature interval from T0 = 60 K to T1 = 250 K to exclude the additional complications present outside of this temperature window. The relaxation function (T) shows four steps, which represent different relaxation processes activated in subsequent temperature regions. We had earlier assigned these steps to conformational fluctuations in different tiers of the hierarchy of conformational substates, CS3, CS2, CS1, and CS0, in ascending order (McMahon et al., 1998). Relaxation functions and derivatives for TMOS encapsulated RCs are depicted in Fig. 11, B and D. Although they exhibit an overall similar shape compared to their bulk solvent counterparts, there are significant differences. The onset temperatures and amplitudes of the CS2 and CS1 relaxations differ for both preparations. In the hydrogel, these motions occur at somewhat higher temperatures and are not so clearly separated. Most interesting, however, is the behavior of (T) above 200 K. For the dark-cooled bulk solvent preparation, (T) clearly exhibits a well-separated step with a much weaker decay from 200 to 250 K. This feature has been associated with global conformational rearrangements of the RC proteins (McMahon et al., 1998). In contrast, the corresponding relaxation function for the TMOS sample decays smoothly from 180 to 250 K.
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DISCUSSION
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Dependence of ET rates on temperature and illumination protocol
In their pioneering work, Kleinfeld and co-workers observed a drastic decrease of the average rate of charge recombination upon cooling RCs under illumination, which was accompanied by a substantial broadening of the lifetime distribution of the charge separated state, as compared with dark-cooled RCs (Kleinfeld et al., 1984). Their initial analysis invoked a dependence of the donor-acceptor distance on the illumination history, yielding an increase of the mean distance between the special pair P and quinone QA by 1 Å upon cooling under illumination. On a molecular scale, this would imply an unrealistically large shift of the tightly bound cofactors within the protein matrix. By contrast, studies on the variation of the X-band ESP spectra of the 
secondary radical pair in QB-depleted, Zn2+-reconstituted RCs of Rb. sphaeroides R26 when cooled in the dark and under illumination revealed only slight modifications of the cofactor arrangement (van den Brink et al., 1994). Small conformational changes depending on the illumination history were also observed for the special pair (Müh et al., 1997) and the photoactive bacteriopheophytin of the RC of Rb. sphaeroides (Müh et al., 1998) by means of ENDOR/TRIPLE spectroscopy.
A simple mapping of the rate coefficients on the edge-to-edge distance between donor and acceptor cannot neither account for the complex temperature dependence of the recombinatio
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ABSTRACT
INTRODUCTION
