 |
INTRODUCTION |
Several studies of the electron spin-lattice
relaxation times (T1) of radicals in
metalloproteins have been reported (Bowman et al., 1979
; Norris et al.,
1980; Calvo et al., 1982; Sahlin et al., 1987; Styring and Rutherford,
1988; Innes and Brudvig, 1989; Hirsh et al., 1992a, b, 1993;
Koulougliotis et al., 1995, 1997; Galli et al., 1995, 1996;
Deligiannakis and Rutherford, 1996; Waldeck et al., 1997; Hung et al.,
2000; Telser et al., 2000; Bar et al., 2001). T1
is the time needed to reach thermodynamic equilibrium between the spin
system and the molecular (lattice) vibrations.
T1 is shortened when the radical interact with
the spin of a fast relaxing paramagnetic ion in its vicinity. Using an
appropriate model, one can evaluate the spin-spin interaction from the
experimentally determined value of the spin-lattice relaxation time.
The spin-spin interaction contains information about the electronic and
spatial structure of the spin-dimer, which in many cases are important
in understanding electron transfer processes.
The spin-spin interactions between the metal ion and the radicals
contains exchange and dipole-dipole contributions. Exchange interactions are related to two-electron exchange integrals and to
overlap integrals of the magnetic orbitals that provide the superexchange path (Anderson, 1959). Their evaluation requires a
detailed knowledge of the electronic structure of the
molecular bridge connecting the spins. This has been done only for
unpaired spins connected by simple chemical paths (Kahn, 1993).
Analyses based on experimental values for a large number of compounds
indicate that the exchange contribution Jo
dominates over the dipolar contribution for short distances
R between the spins (R < 
), while
dipolar contributions dominate for longer distances (R > ) (Coffman and Buettner, 1979; Hoffmann et al., 1994). The
value of has been determined empirically and has changed with time
as the experimental database expanded. The most recent value of proposed by Hoffmann et al. (1994) is ~35 Å. However, it should be
kept in mind that the value of depends on the electronic structure
of the pathway and, therefore, will vary for different classes of
compounds. The RC, which has been optimized for efficient electron
transfer, may not necessarily be representative of an average protein.
Relatively little is known about exchange interactions between unpaired
spins within a protein. The connecting paths involve covalent and
non-covalent bonds, H-bonds, and space jumps. Exchange interactions
have been related to electron transfer rates when the unpaired spins
are components of an electron transfer reaction (Hopfield, 1974;
Okamura et al., 1979a, b; DeVault, 1984; Hendrickson, 1985;
Michel-Beyerle et al., 1988; Calvo et al., 2000). Thus, a determination
of the exchange interaction can contribute to the understanding of the
electron transfer processes.
Dipolar interactions are related to the distance between the
interacting spins and to the orientation of the applied magnetic field
with respect to the molecular axes (Slichter, 1990). Thus, dipolar
interactions provide direct information about the three-dimensional molecular structure (e.g., Calvo et al., 2000). This provides important
information for proteins whose x-ray structure has not been determined.
Spin-spin interactions often modify the electron paramagnetic resonance
(EPR) signal. Thus, EPR spectroscopy may be used in these cases to
evaluate spin-spin interactions (Bencini and Gatteschi, 1990). [In
this paper we use Kelvin as the energy (E) units
(i.e., E/kB, where
kB is the Boltzmann constant). The conversion
factors are: 1 K = 1.3805 × 10
23 Joule = 8.617 × 105 eV = 0.6950 cm1 = 2.0837 × 1010 Hz = 7443.7 Gauss (for
g = 2)]. In some cases, in particular when the
interaction is very small, no observable effect on the EPR spectrum is
observed. In these cases, the spin-spin interaction may express itself
in a change of the spin-lattice relaxation time. This situation is
analyzed in detail in this paper. Thus, spin-lattice relaxation
measurements complement standard EPR spectroscopy for the
characterization of spin-spin interactions.
The system studied in this work is the reaction center (RC) of
the photosynthetic bacterium Rhodobacter sphaeroides. The RC is a membrane-bound pigment protein complex that performs the primary
photochemistry by coupling light-induced electron transfer to vectorial
proton transfer across the bacterial membrane (reviewed by Cramer and
Knaff (1991)). Light-induced electron transfer proceeds from a primary
donor (a bacteriochlorophyll dimer, D), through a series of electron
donor and acceptor molecules (a bacteriopheophytin 
and a quinone
molecule QA) to a loosely bound secondary quinone (QB), which serves as a mobile electron and proton carrier.
There is an Fe2+ ion located between QA and
QB whose properties have been studied by several techniques
(reviewed by Feher and Okamura, 1999).
The electronic structures of the ionized pigments
(Q
, Q
,
, and D+) have been studied in
detail by EPR and ENDOR (reviewed by Feher (1992) and Lubitz and Feher
(1999)). The x-ray structure of the RC of Rb. sphaeroides is
well known (Allen et al., 1986, 1987a, b, 1988; Chang et al., 1986;
Yeates et al., 1987, 1988; Ermler et al., 1994; Stowell et al., 1997;
Abresch et al., 1999). Fig. 1 displays
the structure of the cofactors and the Fe2+ ion within the
RC when Q is reduced (Stowell et al.,
1997). The distances between the Fe2+ ion and the
cofactors, and between some pairs of cofactors, are indicated. For
Q, Q, and
they were taken between the centers of the
cofactor rings. For D+, the distance was taken from the
midpoint between the Mg2+ ions of the two
bacteriochlorophyll molecules.
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FIGURE 1
Structure of the cofactors in the photosynthetic
reaction center from Rb. sphaeroides, with the protein
polypeptide chains in the background (Stowell et al., 1997). Distances
are determined between the Fe2+ ion and the centers of the
rings of , Q, and
Q. For D+, the distance was taken
from the center of the line connecting the two Mg2+
atoms.
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The temperature dependence of the spin-lattice relaxation time of
Q in Rb. sphaeroides had been previously studied by us (Calvo et al., 1982). In that work we showed
that the relaxation times, which were of the order of microseconds in
the temperature range between 1.3 and 4.2 K, are a consequence of the
coupling of the Q spin with the fast relaxing
Fe2+ spin. The data were well fitted by a two-step process
allowing simultaneous transitions of the iron and quinone spins. The
analysis gave a zero field splitting between the two lowest levels of
the Fe2+ spin, in good agreement with the value obtained
from magnetic susceptibility and EPR data (Butler et al., 1980, 1984).
In this work we extend the previous investigation to measure the
spin-lattice relaxation times of three cofactor radicals
(Q, Q, and
) within the photosynthetic reaction center as a
function of temperature (T), in the range between 1.4 and
4.2 K (in this temperature range the contribution of higher excited
states of the Fe2+ can be neglected). The contribution of
the Fe2+ to the relaxation times of the cofactor spins was
obtained by comparing T1 in native RCs, with
T1 in RCs in which Fe2+ was replaced
by diamagnetic Zn2+ (Debus et al., 1986; Utschig et al.,
1997). The results are analyzed using a theoretical model, which
explicitly takes into account the crystal field splitting and
rhombicity of the Fe2+. This is an important point because
the more plentiful phonons at the energy of the zero field splitting,
which is considerably larger than the Zeeman energy, are more effective
in the two-phonon relaxation process of the cofactor spins. The
relaxation process for the cofactors is similar to that proposed by
Orbach (1961) for rare earth ions having low excited energy states, and
to that observed for Fe3+ in heme proteins (Scholes et al.,
1971; Herrick and Stapleton, 1976). Other authors addressing the
problem of spin-lattice relaxation of cofactors in the RC (Bowman et
al., 1979; Norris et al., 1980; Hirsh and Brudvig, 1993; reviewed by
Lakshmi and Brudvig, 2000) have neglected to take the crystal field
splitting and rhombicity into account, which can result in an error in
the predicted value of T1 of several orders of
magnitude. These authors follow essentially the theoretical treatments
of Bloembergen et al. (1948, 1949, 1961) and Abragam (1955, 1961),
which were developed for the relaxation of nuclear spins. These
theories are applicable to electron-electron interaction only when the
fast relaxing magnetic ion does not have a crystal field splitting
(e.g., a low-spin Fe3+ ion).
There are two different aspects of the relaxation problem. One deals
with the evaluation of the spin-spin interaction from measured
relaxation times. This is accomplished by using the spin-dimer model
developed in this work. The other aspect deals with the determination
of the source of the interaction (exchange or dipolar) and its relation
to the structure (e.g., distances between cofactors) which
for the RC in Rb. sphaeroides is well known. Our analysis of
the data provides a check of the validity of our model so that it can
be used with confidence to investigate less well-characterized metaloproteins. An analysis of the distance dependence of the exchange
interaction between unpaired spins within the RC can shed light on
electron transfer matrix elements. Preliminary results of this work
have been reported (Calvo et al., 1982, 1999).
| |
MATERIALS AND METHODS |
Preparation of reaction center samples
Reaction centers were isolated from Rb. sphaeroides
R26 and purified as described previously (Isaacson et al., 1995).
Zinc-containing RCs were obtained by replacing the native iron with
zinc using the procedure developed by Debus et al. (1986) and modified
by Utschig et al. (1997). The detergent LDAO in the buffer was
exchanged with maltoside by binding the RCs to a DEAE column, washing
with 10 mM Tris-Cl, 0.04%
n-dodecyl-
-D-maltoside, eluting with 0.2 M
NaCl in the same buffer, and dialyzing against 10 mM Tris-Cl pH 8, 0.04% n-dodecyl--D-maltoside.
The value of T1 was found to depend on the
amount of oxygen in solution. Consequently, great care was taken to
deoxygenate the solution by adding, under an argon atmosphere, 0.9%
glucose, 7.5 units/ml glucose oxidase, and 7.5 units/ml catalase to the buffer. Furthermore, the freezing protocol and buffer composition, which could affect T1, were kept the same for
the Fe2+ and Zn2+ RCs. This was particularly
important for the samples in which the values
of T1 of Fe2+
and Zn2+ were of the same order of magnitude.
The free radical states of the cofactors were prepared as follows.
Q
The Q RC samples were diluted to an
optical absorbance A
= 20 in the
deoxygenated buffer described above to which 100 µM stigmatellin was
added to displace QB. Thirty-five microliters of the sample
was placed in a 2 mm I.D. quartz tube filled with argon and
equilibrated for 30 min. After adding 1.2 mM 3,6-diaminodurene (Aldrich, Milwaukee, WI) to reduce D+, the sample
was given a 1 µs saturating laser flash (500 mJ) at 590 nm from a dye
laser (PhaseR Corp., New Durhan, NH) to form DQ, and immediately plunged into liquid nitrogen.
Q
For the Q sample, a five times excess
of ubiquinone (Sigma, St. Louis, Mo)/RC in ethanol solution was dried
onto a vial. RCs were added to the vial and stirred for ~4 h at
23°C, resulting in a QB occupancy of 
80%.
Q was then made in the same manner as
Q, but without the addition of stigmatellin.
The RCs used to make Fe2+ and
Zn2+ were treated with the same
metal replacement procedure, i.e., to metal-depleted RCs
either Fe2+ or Zn2+ was added (Debus et al.,
1986; Utschig et al., 1997); was made as
described by Okamura et al. (1979b) with the following modifications.
RCs were diluted into buffer containing 50 mM Tris pH 8, 0.1% Triton
X-100, 0.2 mM cytochrome c2, 0.9% glucose, 7.5 units/ml glucose oxidase, and 7.5 units/ml catalase to a final concentration of A = 20. A 35 µl
aliquot of the sample was placed in a 2 mm I.D. quartz tube filled with
argon and equilibrated for 30 min; ~100 mM sodium dithionite and 100 mM Tris base was added and was generated by
illuminating with a tungsten light source (P = 0.4
W/cm2) after being filtered with 2 cm of water and a 660 nm
cutoff filter (Corning 2-64). The accumulation of
was monitored by the optical absorption at 645 nm on a Cary 50 spectrophotometer (Varian, Inc., Palo Alto, CA). When
the absorbance at 645 nm reached a plateau after 1-2 min, indicating a
maximal concentration of , the sample was
plunged into liquid nitrogen.
EPR and relaxation time measurements
Spin-lattice relaxation measurements were performed at 9 GHz by
measuring the recovery of the EPR signal after a saturating microwave
pulse, using a superheterodyne EPR spectrometer of local design (Feher,
1957; McElroy et al., 1974). To switch between saturating and measuring
microwave power levels, a HP8735A Pin diode modulator (Hewlett-Packard,
Palo Alto, CA) was used. The modulator was bypassed by two directional
couplers having a total attenuation of 30 db. Thus, when the modulator
was in the off state (~50 db attenuation) the level of the measuring
microwave power was governed by the by-pass arm, and not by the pin
diode modulator. This scheme resulted in an improved reproducibility and baseline stability required for the signal averaging of the EPR
recoveries. To minimize the 30 dB overload during the saturating microwave pulse a second HP 8735A pin diode modulator and by-pass arm
in front of the 9 GHz preamp was used. The two modulators were adjusted
to keep the overall gain of the system constant. The time response of
the EPR signal was recorded with a LeCroy 9310M digital oscilloscope
(LeCroy Corp, Chestnut Ridge, NY). The recovery signals were averaged
on the oscilloscope and then transferred to a 450 MHz PC (Windows 2000)
equipped with an AT-GPIB adapter card (National Instruments (NI),
Austin, TX) running LabView 5.1 (NI). A Lab PC+ (NI) analog-to-digital
acquisition card was used to control the magnetic field for EPR, and to
supply various control signals. A solid metal TE102 cavity,
constructed of high purity Al-Mg alloy (Al 95%-Mg 5%), which has a
low paramagnetic background at liquid helium temperatures, was used.
The electrical conductivity of this alloy does not increase on cooling
to these temperatures, keeping a constant cavity Q of 4000 and allowing the use of field modulation up to 2 KHz.
Two ranges of values of T1 were measured. For
QFe2+ and
QFe2+ T1
was in the range of 1-20 µs. In this time domain, the recovery of the
EPR signal was observed using field modulation with a boxcar integrator
(Isaacson, 1968) connected to a lock-in amplifier (EG&G 7260 DSP, now
Amertek, Inc., Oak Ridge, TN). In all other samples T1 was longer than 100 ms and only the lock-in
amplifier was used. In all cases many recoveries were averaged to
improve the signal-to-noise (S/N) ratio. The relaxation time
T1 was defined as the 1/e time constant of the
exponential recovery of the signal; it was calculated from the data
using a commercial fitting program (Origin 6.1, OriginLab Corp.,
Northampton, MA). The possibility of multi-exponential recoveries was
considered and is discussed later.
The sample of Fe2+ and all
Zn2+-containing samples have long relaxation times
T1 and are, therefore, easily saturated. The low microwave power required for no-saturation reduces the S/N ratio. To
improve the S/N ratio we worked under slightly saturating conditions using higher microwave powers. To obtain T1
under these conditions we used the relation (Slichter, 1990):
|
(1)
|
where W is the microwave power and 
is a constant that
depends on experimental conditions. T1 was
obtained by measuring the EPR recovery,
(1/T1)meas, as a function of W over
a 9 dB range, and extrapolated to zero power at each temperature. This
extrapolation is valid for single exponential decays, as was observed
in all cases.
| |
EXPERIMENTAL RESULTS |
The temperature dependence of the relaxation time
T1 was measured at 9 GHz (X-band) between 1.4 and 4.2 K. For QFe2+ and
QFe2+, the relaxation measurements
were performed at a magnetic field H corresponding to the
peak of the absorption 
" signal (g = 1.8). In all
other samples it was measured at the peaks of d"/dH. The recovery of
the signal following a saturating microwave pulse at 2.15 K is shown
for QFe2+ and
QZn2+ in Fig.
2, and for
Fe2+ and
Zn2+ in Fig.
3. The data are well fitted with a single
exponential as seen from the small difference (residuals) between the
observed and fitted curves. The results
for QFe2+ samples are similar
to those for QFe2+ (results not
shown).
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FIGURE 2
Recovery of the EPR signal (solid lines)
after a saturating microwave pulse, observed at  = 9 GHz, and
T = 2.15 K from (a)
QFe2+ and (b)
QZn2+. Data are fitted with a
single exponential function (gray dots). The difference
between the experimental curves and the fit (i.e.,
residuals), are also shown. Note the difference in time scales of ~5
orders of magnitude between (a) and (b). The
value of T1 indicated in (b) is
slightly shorter than the value extrapolated to zero power (see Fig.
4).
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FIGURE 3
Recovery of the EPR signal (solid lines)
after a saturating microwave pulse, observed at = 9 GHz, and
T = 2.15 K from (a)
Fe2+, (b)
Zn2+. Data are fitted with a single
exponential function (gray dots). The difference between the
experimental curves and the fit (i.e., residuals) are also
shown. The contribution of the Fe2+ ion to the relaxation
of the spin is obtained by subtracting the rate
determined in (b) from that in (a).
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An example of the dependence of the measured recovery
(1/T1)meas on the microwave power
W at 2.15 K is shown for
QZn2+ in Fig.
4. The value of
(1/T1) is obtained by a linear extrapolation of
the data to zero power (see Eq. 1).
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FIGURE 4
The observed relaxation rates
1/T1 for
QZn2+ at 2.15 K, at different
microwave powers applied during the recovery phase. The solid line
represents a fit to Eq. 1. Extrapolation to zero microwave power yields
the inherent spin-lattice relaxation rate.
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The values of the relaxation rates (1/T1)
as a function of temperature T measured between 1.4 and 4.2 K, for QFe2+ and
QFe2+ are shown in Fig. 5
a, for
QZn2+ and for
QZn2+ in Fig. 5 b, and
for Fe2+ and
Zn2+ in Fig.
6. Each data point in Figs. 5 and 6 was
obtained as described by Figs. 2-4.
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FIGURE 5
Temperature dependence of the observed relaxation rate
1/T1 at = 9 GHz for (a)
QFe2+ and
QFe2+, (b)
QZn2+ and
QZn2+.
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The value of T1 of
QFe2+ is ~5 orders of magnitude
shorter than that observed in
QZn2+ (Fig. 5, a and
b). Similar results were obtained for
QFe2+. The value of
T1 of Fe2+ is
shorter than that observed in Zn2+
(Fig. 6), but the difference is much smaller than for the quinones. These results confirm the role of the Fe2+ ion in the
relaxation of the cofactors. The effect of the Fe2+ is much
more pronounced for the quinone acceptors Q and Q, which are closer to the
Fe2+ ion than (see Fig. 1).
| |
THEORY |
Spin-Hamiltonian description of the spin-dimer
The spins of each oxidized or reduced cofactor s
(s = 1/2) that participate in the electron transfer chain of the
photosynthetic reaction center and the Fe2+ spin
S (S = 2) give rise to a coupled spin-dimer. The spin-spin interaction between s and S is
described by the Hamiltonian:
|
(2)
|
where J is the interaction tensor. In
simple cases J is isotropic, and Eq. 2 gives the Heisenberg
exchange (
Jos · S) described
by a single, scalar, quantity Jo. Anisotropic contributions to J in Eq. 2 can arise from dipolar interactions or from higher order exchange terms (Bencini and
Gatteschi, 1990). Antisymmetric exchange can arise from higher order
contributions of the spin-orbit interaction (Moriya, 1960). Both of
these exchange contributions are usually smaller than the isotropic
value Jo. Furthermore, we will show that
only one term in Eq. 2 contributes to the relaxation of the cofactors
in the RC, and thus anisotropies are not relevant for this work.
The spin-Hamiltonian Hs describing the
properties of the Fe2+-Cof spin-dimer (Cof stands for the
cofactors Q, Q,
and of the photosynthetic RC) in an external
magnetic field H, can be written as
(Abragam and Bleaney, 1970; Butler et al., 1980, 1984):
![H<SUB><UP>s</UP></SUB>=D[<UP>S</UP><SUP><UP>2</UP></SUP><SUB><UP>z</UP></SUB>−<UP>S</UP>(<UP>S</UP>+1)/3]+E(<UP>S</UP><SUP><UP>2</UP></SUP><SUB><UP>x</UP></SUB>−<UP>S</UP><SUP><UP>2</UP></SUP><SUB><UP>y</UP></SUB>)](/content/vol83/issue5/fulltext/2440/img013.gif)
|
(3)
|
where D and E are zero field splitting
parameters of the Fe2+; gFe is
the g-tensor of the Fe2+ spin,
gCof is the g-factor of s.
These g-values are close to being isotropic, which justifies
taking their average values, gCof 
2.0046 for Q and Q (Isaacson et al., 1995), and gCof 2.0036 for (Okamura et al., 1979b).
µB is the Bohr magneton, and HsS
is given by Eq. 2. Equation 3 is written in the coordinate axes in which the zero field splitting is diagonal, and characterized by
D and E. When the Fe2+ is replaced by
diamagnetic Zn2+, the terms of Eq. 3 involving S
are zero. The best characterized RC spin-dimer is
QFe2+ in Rb.
sphaeroides (Butler et al., 1980, 1984), for which the values of
the parameters are:
|
(4)
|
where it was assumed that the principal axes of the exchange
interaction and the zero field splitting tensor are the same. It is
expected that the values of D, E, and the components of the
g-tensor of Fe2+ do not change for
QFe2+ and
Fe2+. However, the values of
J will be different for the different cofactors.
The levels scheme predicted by Eq. 3 with the parameters for the dimer
QFe2+ given in Eq. 4 is shown in
Fig. 7 a (Butler et al.,
1984). The fivefold (2S + 1) degeneracy of the energy levels of
the S = 2 spin of the Fe2+ ion is split by the terms
D and E in Eq. 3. The first and second excited
levels are 3.2 K and 15 K above the ground state. Each of the five
levels has a twofold degeneracy due to the spin of the cofactor
radical. This degeneracy is split by the external field H,
as shown in Fig. 7 a for a magnetic field applied along the
y-direction, which is defined by the zero field splitting terms of the spin Hamiltonian in Eq. 3. The EPR signal centered at
g 1.8 observed at helium temperatures in randomly
oriented frozen QFe2+ dimers is a
superposition of the EPR signals arising from the ground state doublet
(the high field side), and from the excited state doublet (the low
field side), as shown in Fig. 7 b (Butler et al., 1984). The
main feature of both contributions to the spectrum of
QFe2+ is a strong anisotropy of
the g-tensor, with the principal g-value along
the y-direction (gy) considerably
displaced from gx and gz
(gx and gz are close to
the center of the line at g ~ 1.8). This is a
consequence of the large magnetic moment induced in the two lowest
states of the Fe2+ ion when the external magnetic field is
applied along the y-direction (Butler et al., 1984). The
other three doublets are not populated in the temperature range at
which the EPR spectra were observed, and therefore do not contribute.
The spectrum in Fig. 7 b corresponds to
QFe2+; similar results were
obtained for QFe2+. For the
Fe2+ dimer, the spin-spin
interaction is much smaller, and the effect of the zero field splitting
is buried in the width of the signal, which is
mainly due to hyperfine interactions with the proton nuclei.
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FIGURE 7
(a) Energy levels of the spin-dimer
QFe2+, obtained from Eq. 3 with
the magnetic field H applied along the y-axis of the crystal
field. The spin S = 2 of the iron ion gives rise to five energy
levels, each being twofold degenerate due to the interaction with the
spin s = 1/2 of the cofactor. (b) Contributions
of the two lowest doublets to the EPR spectrum observed at = 9 GHz and 2.1 K. Modified from Butler et al. (1984).
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Spin-lattice relaxation of a coupled dimer
We present now a model that explains our spin-lattice relaxation
data. Details of the calculations are given in Appendix A.
The spin-lattice interaction
To calculate the spin-lattice relaxation of the coupled dimer we
add to Eq. 3 the spin-lattice interactions
Hsl(s), acting on the spin of the
cofactor, and HSl(S), acting on the
Fe2+ spin. They couple s and S,
respectively, to the electric field produced by the local distortions
associated with the thermal vibrations (see Appendix A). Adding to Eq. 3 the interaction of the spin-dimer with the thermal vibrations we
obtain:
|
(5)
|
HSl(S) induces transitions
between the five levels of S, with the absorption and
emission of phonons. Since Fe2+ is a non-Kramers' ion
(even number of electrons), its energy levels are split by the
crystalline electric field.
HSl(S), therefore, couples directly
to the lattice vibrations, making Fe2+ a fast
relaxer (Shiren, 1962, 1963; Watkins and Feher, 1962). The
cofactors, however, having a spin of 1/2, are purely
magnetic entities (Kramers' doublet) and relax only in the
presence of H through admixtures of higher levels through
spin-orbit interactions (Van Vleck, 1940; Abragam and Bleaney, 1970;
Orbach and Stapleton, 1972). Consequently,
Hsl(s) produces a weak interaction
making the cofactors slow relaxers.
Interlevel Transition Rates
Fig. 8, a-e
focuses on the two lower doublet states displayed in Fig. 7
a. Only these two levels are populated below 4.2 K and are,
therefore, responsible for the low temperature EPR spectrum and
relaxation processes of
QFe2+ (Fig. 7 b). Fig.
8 a shows the two lowest energy levels of an isolated
Fe2+ ion that are separated by the zero field splitting 
(dotted line). The solid arrows indicate transitions between
the energy states |1
and |2 produced by
HSl(S) with rates
k
and
k
, with the absorption and emission of
phonons, respectively. No EPR transition between these two levels are
induced at the employed microwave frequencies and magnetic fields and,
therefore, the Fe2+ ion is EPR silent in unreduced RCs.
Fig. 8 b includes the splittings 
1 and
2 of the Fe2+ levels produced by its
coupling with the s = 1/2 spin of the cofactor. For simplicity, we
assume that 1 2 = .
Dashed arrows labeled with rates k
and
k
indicate spin-lattice transitions
produced by Hsl(s). These transitions
change the spin state of s (from |
to | or
vice-versa), with the absorption or emission of phonons, without changing the spin state of S (|1 and |2). Fig. 8
c considers transitions labeled
k
and k
that change the spin states of both s and S. A
quantum mechanical calculation of kM, produced
by HSl(S) in the presence of
HsS, is given in Appendix A. The thick solid arrows in Fig. 8, d and e indicate the
transitions induced by the microwave field, which give rise to the
observed EPR signal (Fig. 7 b). Fig. 8, d and
e also show the pairs of transitions contributing to the two
phonon relaxation processes for each of the two doublets with solid and
dashed arrows. Each doublet relaxes via two phonons characterized by
rates k and
k, or k and k (Fig. 8, d and
e). These processes are much more effective than those
produced by kCof (Fig. 8 b).
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FIGURE 8
The two lowest doublet states of the
QFe2+ spin-dimer and the transitions
between them. (a) The Fe2+ levels with a
splitting (dotted double arrow) at zero magnetic field.
The spin-lattice transitions proceed with rates
k and k
(solid arrows). (b) Application of a magnetic
field splits the Fe2+ levels due to the spin of
Q by an amount (dotted double
arrow). The direct relaxation of Q proceeds
with rates k and
k. (c) Spin-lattice
transitions k and
k (dashed arrows) that change
the states of both spins (s and S).
(d) Two-phonon transitions with rates
kM and kFe, respectively,
produce a net relaxation of the spins in the lowest doublet.
(e) Same as (d) for the higher doublet. The EPR
transitions are indicated by heavy arrows.
|
|
The values of the transition rates are (Abragam and Bleaney, 1970) (see
Appendix A):
|
(6a)
|
|
(6b)
|
|
(6c)
|
where n
, n
, and
n± are the equilibrium populations of the
phonons with energies x = , and ± ,
respectively. They are given at temperature T by the phonon occupancy numbers (Bose factors), nx = (exp(x/kBT) 1)1.
ACof, AFe, and
AM represent the strength of the interactions giving rise to each of the processes displayed in Fig. 8, a
and c. The magnitudes of AFe and
ACof involve elastic constants (or sound
velocities), phonon densities appropriate to the protein at the site of
the spin-dimer, and matrix elements of
HSl(S) and
Hsl(s) (see Appendix A), respectively
(Van Vleck, 1940; Orbach, 1961; Abragam and Bleaney, 1970). We show in
Appendix A that AM is proportional to
AFe, the proportionality constant involving
components of J (Eq. 2) and the crystal field splitting
parameters of the Fe2+ ion (Eqs. 3 and 4).
Relaxation rate (1/
)Fe of the Fe2+ ion
Let N1 and N2 be the
populations of levels |1 and |2 in Fig. 8 a. The
rate equations for N1 and
N2 arising from
k and k
are:
|
(7)
|
Using Eqs. 6b and 7 and the condition
(N1)eq
k = (N2)eq k
for the equilibrium populations
(N1)eq and
(N2)eq, we obtain (Abragam and
Bleaney, 1970):
![<UP>d</UP>(N<SUB>1</SUB>−N<SUB>2</SUB>)/<UP>d</UP>t=<UP>−</UP>(1/&tgr;)<SUB><UP>Fe</UP></SUB>[(N<SUB>1</SUB>−N<SUB>2</SUB>)−(N<SUB>1</SUB>−N<SUB>2</SUB>)<SUB><UP>eq</UP></SUB>]](/content/vol83/issue5/fulltext/2440/img030.gif)
|
(8)
|
for the relaxation of the two lowest levels of the
Fe2+ ion. Equation 8 shows that as t 
,
(N1 N2) approaches exponentially its thermal equilibrium value (N1 N2)eq = (N1 + N2) tanh(/2kBT) with a characteristic time Fe defined by:
|
(9)
|
The temperature dependence of Fe enters into Eq. 9
through the phonon occupancy factor n. This
equation is valid at low temperatures where only phonon transitions
involving the two lowest levels of the Fe2+ ion contribute
to the relaxation (Van Vleck, 1940; Abragam and Bleaney, 1970). At
higher temperatures, when other excited levels of the Fe2+
ion are populated, Eq. 8 needs to be replaced by a set of coupled differential equations that give rise to (m 1)
relaxation times, where m is the number of populated
Fe2+ levels (see Appendix B).
Relaxation rate of the cofactors in the presence of spin-spin
coupling
Let N1
, N1
,
N2, and N2 be the
populations of the states |1, |1, |2 and
|2 shown in Fig. 8. At low temperatures, when only these levels
are populated, the total number N of dimers is,
|
(10)
|
From Fig. 8 and Eqs. 6, the rate equation for
N1 is:
Similar rate equations may be written for
N1, N2, and
N2. The differences in populations of the
spin up states | and spin down states | that give rise
to the EPR transitions (Fig. 8 b) are:
|
(11)
|
and,
|
(12)
|
The recovery of the EPR signal was measured at the pea
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ABSTRACT
INTRODUCTION
