Electron spin relaxation can enhance the performance of a cryptochrome-based magnetic compass sensor

The rates and yields of free radical reactions are often sensitive to applied magnetic fields, a phenomenon known as the radical pair mechanism [1, 2]. It has been suggested [3] that migratory birds sense the direction of the Earth's magnetic field [4–7] by means of radical pairs formed photochemically in cryptochrome proteins located in their retinas [8]. Experimental and theoretical support for a radical pair mechanism of compass magnetoreception is accumulating (reviewed in [9–17]) although it is not yet clear whether cryptochrome is actually the sensor.

In vitro photo-excitation of the flavin adenine dinucleotide (FAD) cofactor in cryptochromes leads to the formation of radical pairs via sequential electron transfers along the 'Trp-triad', a chain of three tryptophan residues within the protein [18–21]. Donation of an electron to the excited singlet state of the FAD leads to the flavin radical, FAD^•−, and the radical form of the terminal tryptophan, ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ (supplementary figure S1). This [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}]$ radical pair in both Arabidopsis thaliana cryptochrome-1 (AtCry1) and a closely related DNA photolyase has been shown to respond to weak magnetic fields (∼1 mT) and has at least some of the properties required of a magnetic compass sensor [22, 23]. The origin of the magnetic field effect lies in a combination of the spin dynamics of the radical pair and its spin-selective recombination reactions, figures 1(a) and (b) and supplementary figure S1 [9].

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Various spin interactions determine the magnetic field effect. The hyperfine interactions [24] of the electron spins with magnetic nuclei (mainly ¹H and ¹⁴N) result in coherent, oscillatory interconversion of the electronic singlet and triplet states of the radical pair. The electronic Zeeman interaction with the external magnetic field modifies this time-dependence, altering the probability of spin-dependent radical recombination and changing the yields of the reaction products (figures 1(a) and (b)). The anisotropy of the hyperfine interactions causes the effect of the external field to depend on its direction, as required for a compass sensor. Two other important interactions—usually neglected in this context—are the electron–electron exchange and dipolar couplings [25, 26], both of which depend on r, the distance between the centres of electron spin density in the two radicals. The exchange interaction is usually assumed to have the exponential form:

$$\begin{eqnarray}&&J(r)={J}_{0}\quad \mathrm{exp}(-\beta r),\end{eqnarray} \tag{ 1 }$$

with a range parameter β = 14 nm⁻¹ for radicals embedded in a protein [25, 27]. The strength of the interaction, J₀, has a complex dependence on the electronic properties of the radicals and is difficult to calculate with any accuracy. The dipolar interaction, however, can reliably be obtained by assuming the electron spins to be point dipoles (normally an excellent approximation for radical pairs [25]):

$$\begin{eqnarray}&&D(r)=-\displaystyle \frac{2.785\;{\rm{mT}}}{{(r/{\rm{nm}})}^{3}}=-\displaystyle \frac{77.80\;{\rm{MHz}}}{{(r/{\rm{nm}})}^{3}}.\end{eqnarray} \tag{ 2 }$$

In the absence of other spin interactions, J(r) and D(r) remove the degeneracy of the singlet (S) and the three triplet (T₊₁, T₀, T₋₁) states:

$$\begin{eqnarray}&&E({\rm{S}})=+J(r),\quad E({{\rm{T}}}_{m})=-J(r)+D(r)\left({m}^{2}-\displaystyle \frac{2}{3}\right);\end{eqnarray} \tag{ 3 }$$

(m, the magnetic quantum number of the triplet, equals 0 or ±1). These interactions hinder efficient singlet–triplet interconversion and so degrade the sensitivity of the radical pair to the geomagnetic field [25].

The magnetically sensitive [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}]$ radical pair in AtCry1 [23] has r = 1.90 nm [28], giving D(r) = −11 MHz and $|J(r)|=6.3\,{\rm MHz}$ (assuming $|{J}_{0}|$ ≈ 8.0 × 10⁷ T and β = 14 nm⁻¹ [25]). Experimental measurements of J(r) for [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ ] in two DASH-type cryptochromes are in rough agreement with this estimate: +2.0 MHz for Xenopus laevis [29] and +4.2 MHz for Synechocystis [30]. A more exact calculation of D(r), avoiding the point-dipole approximation, confirmed the −11 MHz value to within 0.6 MHz [25]. It is therefore clear that both $|J(r)|$ and $|D(r)|$ are likely to be larger than the electron Zeeman interaction (∼1.4 MHz) with the Earth's magnetic field (∼50 μT), and cannot therefore be dismissed.

Although values of $|J(r)|$ or $|D(r)|$ larger than a few MHz would be sufficient to mar the performance of a radical pair as a geomagnetic sensor, the effects of the two interactions can partially cancel under certain circumstances, specifically when D(r)/J(r) ≈ +6 or −3 [25]. Using estimates of $|{J}_{0}|$ derived from data on photosynthetic reaction centre proteins [31–33], it has been argued that such ratios can be expected when r ≈ 2.0 ± 0.2 nm, i.e. a separation close to that of FAD^•− and ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ in cryptochromes (∼1.9 nm) [25]. However, the values of J(r) and D(r) are not known for any of the four avian cryptochromes, and it has not been established how effective the cancellation would be for radicals as complex as flavin and tryptophan. It is therefore appropriate to ask whether the damaging effects of J(r) and D(r) could be circumvented in some other way. One possibility, discussed later, is that 'the quantum Zeno effect immunises the avian compass against the deleterious effects of exchange and dipolar interactions' [34]. Here, we examine whether electron spin relaxation could come to the rescue.

Spin relaxation, the process that drives the electron–nuclear spin system of the radical pair towards thermal equilibrium [24], is normally expected to attenuate the sensitivity of the compass by destroying the spin coherence that is essential for its operation [35]. It seems plausible that the optimum performance would require negligible loss of coherence during the lifetime of the radical pair. There have been several theoretical treatments of spin relaxation in the context of magnetoreception, almost all of which have dealt with generic relaxation mechanisms furnished with phenomenological rate constants [36–41]. Intriguingly, a few of these studies have indicated that the compass sensitivity could be enhanced by the inclusion of certain abstract spin relaxation pathways [37, 38, 40, 42, 43]. In the following, we discuss the spin relaxation that arises from realistic spin interactions modulated by realistic protein dynamics. Specifically, we show that the time dependence of J(r) and D(r) arising from stochastic fluctuations in the separation of the FAD^•− and ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ radicals in cryptochrome has the potential to boost the magnetic field effect and so counteract the detrimental effects these interactions would have in the absence of molecular motion.

We start by establishing the principle that the harmful effect of an exchange interaction on a radical pair compass can be alleviated by fluctuations in the strength of that interaction. To do so, we chose the simplest possible radical pair comprising two electron spins, one of which is coupled to a single spin-$\tfrac{1}{2}$ nucleus via an anisotropic hyperfine interaction. The eigenvalues of the anisotropic hyperfine tensor [24], ${A}_{xx}={A}_{yy}=0,$ ${A}_{zz}=42MHz,$ were chosen to resemble the hyperfine tensor of the N5 interaction in FAD^•− [44]. Singlet and triplet pairs were allowed to react to form distinct products (figure 1(a)) with identical rate constants, k = k_S = k_T = 10⁶ s⁻¹, such that the radical pair lifetime was 1 μs. At time t = 0, the radical pair was in a pure singlet state with no nuclear polarisation. The performance of this toy compass was judged by the anisotropy of the singlet yield:

$$\begin{eqnarray}&&{\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}={{\rm{\Phi }}}_{{\rm{S}}}(0)-{{\rm{\Phi }}}_{{\rm{S}}}(90^\circ ),\end{eqnarray} \tag{ 4 }$$

where ${{\rm{\Phi }}}_{{\rm{S}}}(\theta )$ is the final yield of the singlet product (P_S in figure 1(a)) and θ is the angle between the external magnetic field (50 μT) and the symmetry axis of the hyperfine tensor.

${{\rm{\Phi }}}_{{\rm{S}}}(\theta )$ was determined using the equation of motion for the radical pair density operator, $\hat{\rho }(t)$:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t}=-{\rm{i}}[{\hat{H}}_{0},\hat{\rho }(t)]-k\hat{\rho }(t)-{k}_{{\rm{STD}}}({\hat{P}}^{{\rm{S}}}\hat{\rho }(t){\hat{P}}^{{\rm{T}}}+{\hat{P}}^{{\rm{T}}}\hat{\rho }(t){\hat{P}}^{{\rm{S}}})=-\hat{\hat{{\rm{\Omega }}}}\hat{\rho }(t).\end{eqnarray} \tag{ 5 }$$

${\hat{H}}_{0}$ is the time-independent part of the spin Hamiltonian:

$$\begin{eqnarray}&&{\hat{H}}_{0}={\gamma }_{{\rm{e}}}{B}_{0}[({\hat{S}}_{{\rm{A}}x}+{\hat{S}}_{{\rm{B}}x})\mathrm{sin}\;\theta +({\hat{S}}_{{\rm{A}}z}+{\hat{S}}_{{\rm{B}}z})\mathrm{cos}\;\theta ]+{A}_{zz}{\hat{S}}_{{\rm{A}}z}{\hat{I}}_{z}-{J}_{{\rm{av}}}\left(2{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}}+\displaystyle \frac{1}{2}\hat{E}\right),\end{eqnarray} \tag{ 6 }$$

in which the three terms are, respectively, the electron Zeeman, electron–nuclear hyperfine, and average exchange interactions. ${\hat{{\boldsymbol{S}}}}_{{\rm{A}}}$ and ${\hat{{\boldsymbol{S}}}}_{{\rm{B}}}$ are the electron spin angular momentum operators of the two radicals, and $\hat{{\boldsymbol{I}}}$ is the spin operator of the nucleus in radical A. ${B}_{0}$ = 50 μT is the strength of the external magnetic field. The other terms in equation (5) account for the radical recombination reactions and the effects of fluctuations in J(r). The latter is derived in appendix A. ${k}_{{\rm{STD}}}$ is the rate constant of the spin relaxation caused by J-modulation, and ${\hat{P}}^{{\rm{S}}}$ and ${\hat{P}}^{{\rm{T}}}$ are the singlet and triplet projection operators. If the fluctuations in the exchange interaction occur on a faster timescale than the coherent spin dynamics, ${k}_{{\rm{STD}}}$ is given by (appendix A)

$$\begin{eqnarray}&&{k}_{{\rm{STD}}}=4\langle {[J(r(t))-{J}_{{\rm{av}}}]}^{2}\rangle {\tau }_{{\rm{c}}},\end{eqnarray} \tag{ 7 }$$

where $\langle \cdots \rangle$ indicates a time average, ${J}_{{\rm{av}}}$ = $\langle J(r(t))\rangle ,$ and the correlation time ${\tau }_{{\rm{c}}}$ characterises the timescale of the fluctuations in r. The rate of relaxation is thus proportional to the mean square deviation of the exchange parameter from its mean value and is faster the slower the fluctuations. It is clear from equation (5) that J-modulation relaxes density matrix elements of the type $\langle {\rm{S}}|\hat{\rho }(t)|{{\rm{T}}}_{m}\rangle ;$ it is therefore referred to as a 'singlet–triplet dephasing' (STD) mechanism [45]. We note that the relaxation term in equation (5) can also be obtained phenomenologically using Lindblad noise operators, ${\hat{L}}_{j}$ [36, 46]:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t}+{\rm{i}}[{\hat{H}}_{0},\hat{\rho }(t)]+k\hat{\rho }(t)={k}_{{\rm{STD}}}\displaystyle \sum _{j}\left[{\hat{L}}_{j}\hat{\rho }(t){\hat{L}}_{j}^{\quad \dagger }-\displaystyle \frac{1}{2}({\hat{L}}_{j}^{\;\dagger }{\hat{L}}_{j}\hat{\rho }(t)+\hat{\rho }(t){\hat{L}}_{j}^{\;\dagger }{\hat{L}}_{j})\right]\end{eqnarray} \tag{ 8 }$$

with ${\hat{L}}_{j}=\{{\hat{P}}^{{\rm{S}}},{\hat{P}}^{{\rm{T}}}\}$ or ${2}^{-\mathrm{1/2}}({\hat{P}}^{{\rm{S}}}-{\hat{P}}^{{\rm{T}}})$.

Finally, the singlet yield once all radicals have recombined is:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{S}}}(\theta )=k\displaystyle {\int }_{0}^{\infty }{\rm{Tr}}[{\hat{P}}^{{\rm{S}}}\hat{\rho }(t)]{\rm{d}}t=k\displaystyle {\int }_{0}^{\infty }{\rm{Tr}}[{\hat{P}}^{{\rm{S}}}\mathrm{exp}(-\hat{\hat{{\rm{\Omega }}}}t)\hat{\rho }(0)]{\rm{d}}t,\end{eqnarray} \tag{ 9 }$$

where the superoperator $\hat{\hat{{\rm{\Omega }}}}$ is defined in equation (5) and $\hat{\rho }(0)=\tfrac{1}{2}{\hat{P}}^{{\rm{S}}}$.

Figure 2(a) plots the anisotropy ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}$ (equation (4)) versus ${k}_{{\rm{STD}}}$ for four values of ${J}_{{\rm{av}}}.$ When ${k}_{{\rm{STD}}}$ is small (e.g. 10³ s⁻¹), there is essentially no relaxation during the 1 μs lifetime of the radicals. In this limit, ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}$ is dramatically reduced when the average exchange interaction is comparable to or stronger than the geomagnetic field (50 μT). However, when ${J}_{{\rm{av}}}\ne 0,$ values of ${k}_{{\rm{STD}}}$ greater than ∼10⁶ s⁻¹ cause ΔΦ_S to rise towards the values appropriate for ${J}_{{\rm{av}}}=0.$ When the relaxation is faster still (${k}_{{\rm{STD}}}$ > 10⁸ s⁻¹), ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}$ drops towards zero. Fluctuations in J(r) resulting in intermediate relaxation rates (10⁶ s⁻¹ < ${k}_{{\rm{STD}}}$ < 10⁸ s⁻¹) therefore boost the compass sensitivity well above the level expected for a time-independent exchange interaction (dashed lines in figure 2(a)). Similar behaviour was found for modulation of the dipolar interaction (appendix B and supplementary figure S2) and for the reaction scheme in figure 1(b) (supplementary figure S3).

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To put figure 2(a) in context, J-modulation was compared with a 'random fields' relaxation (RFR) mechanism in which unspecified, uncorrelated molecular motions cause the expectation values of all three spin angular momentum operators (${\hat{S}}_{x},$ ${\hat{S}}_{y},$ and ${\hat{S}}_{z})$ of both electrons to relax independently with the same rate constant, ${k}_{{\rm{RFR}}}$ (appendix C). As shown in figure 2(b), this relaxation mechanism does not enhance ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}$ but causes it to fall towards zero at much smaller values of the rate constant than was the case in figure 2(a). Clearly, not all relaxation mechanisms are able to rescue the sensitivity of a radical pair compass.

To shed light on the differences between figures 2(a) and (b), figure 2(c) plots ${{\rm{\Phi }}}_{{\rm{S}}}(\theta )$ for $\theta =0$ and $\theta =90^\circ$ as a function of the relaxation rate constant for the two mechanisms (with ${J}_{{\rm{av}}}$ = 100 μT). It can be seen that very fast relaxation (>10¹⁰ s⁻¹) destroys the anisotropy either by locking the radical pair into its initial singlet state (STD) or by relaxing the spins to the equilibrium state in which the singlet probability is 0.25 (RFR). For rate constants between 10⁶ s⁻¹ and 10⁸ s⁻¹, STD is much less effective at relaxing ${{\rm{\Phi }}}_{{\rm{S}}}(\theta )$ towards 0.25 when the magnetic field is parallel to the hyperfine axis ($\theta =0)$ than when it is perpendicular ($\theta =90^\circ ).$ For the STD mechanism, therefore, the anisotropy in the reaction yield is considerably larger when 10⁶ s⁻¹ $\lesssim$ ${k}_{{\rm{STD}}}$ $\lesssim$ 10⁸ s⁻¹ than in the absence of relaxation (${k}_{{\rm{STD}}}$ < 10⁴ s⁻¹). The RFR mechanism, by contrast, relaxes the spins isotropically and so drives ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}$ monotonically to zero as ${k}_{{\rm{RFR}}}$ is increased.

STD has been invoked to account for the observed magnetic-field dependence of the flavin–tryptophan radical pair reaction in AtCry1 [23]. J-modulation was considered to arise from reversible electron hopping between the distal and medial members of the Trp-triad, i.e. [FAD^•− TrpH_B ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$] $\rightleftarrows$ [FAD^•− ${{{\rm{TrpH}}}_{{\rm{B}}}}^{\cdot +}$ TrpH_C] (figure 1(c)). As described above, the former (denoted RP₁) has r₁ = 1.90 nm and exchange and dipolar interactions of a few MHz; the latter (denoted RP₀) has a centre-to-centre distance r₀ = 1.32 nm in AtCry1 and estimated interactions: $|J({r}_{0})|\approx 21\;{\rm{GHz}},$ D(r₀) = −34 MHz [25]. With such a large exchange interaction in RP₀, reversible electron hopping results in strong modulation of $J(r)$ and potentially rapid relaxation of singlet–triplet coherences in RP₁. Provided k₀₁, the electron hopping rate constant for RP₀ → RP₁, is not much larger than $|J({r}_{0})|,$ then every time RP₁ is converted into RP₀ the coherence experiences a large phase shift before the electron jumps back. Under these conditions, the dephasing rate constant k_STD in equation (5) is expected to be equal to k₁₀, the rate constant for RP₀ ← RP₁ [23].

This prediction can be verified with a toy model similar to that used above. Figure 2(d) shows the singlet yield anisotropy determined (i) as a function of k_STD using equation (5) and (ii) as a function of k₁₀ by solving a Liouville equation containing explicit reversible electron hopping between RP₀ and RP₁, coherent spin dynamics in both radical pairs, and spin-selective recombination for RP₁ (figure 1(c) and appendix D). The parameters for method (ii) were chosen to approximate those determined from the observed magnetic field effects on [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}]$ in AtCry1 [23]. The two calculations are in excellent agreement confirming that the dephasing rate constant is indeed given by k₁₀. Note also that the value of k₁₀ (= 1.1 × 10⁷ s^–1) determined from the magnetic field effects on AtCry1 [23] corresponds to a position in figure 2(d) close to the maximum enhancement of ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}.$ However, this should be treated with caution because of our use of a grossly simplified spin system for figure 2(d) (one nuclear spin instead of ∼10; eight spin states instead of ∼2¹²).

We now turn to STD spin relaxation for the [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}]$ radical pair in cryptochrome. Fluctuations in the separation and relative orientation of the two radicals were characterised by means of all-atom molecular dynamics (MD) simulations [47]. The calculations were performed using a modified version of the crystal structure of AtCry1 (PDB ID 1U3C [28]) containing the flavin radical, FAD^•−, and the radical form of the terminal residue of the Trp-triad, ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ (W324) with force-field parameters and atomic charges as in [48]. The protein was solvated in an aqueous solution of 50 mM NaCl in a periodic box of dimensions 103 × 111 × 99 ų containing a total of 113 455 atoms. After extensive equilibration, seven statistically independent MD trajectories, three spanning 300 ns and four covering 100 ns, were accumulated for the microcanonical ensemble at 300 K. The initial configuration of the system was identical in the seven simulations; stochastic behaviour emerged from the Langevin thermostat [49], as implemented in NAMD [50]. The protein dynamics were sampled at intervals of 0.5 or 5 ps. The accumulated trajectories cover 1.3 μs. Because the motions of the radicals are considerably faster than both spin relaxation and radical recombination, it is not necessary for the simulations to include the reactions or to span the lifetime of the radicals. Electron hopping between TrpH_B and TrpH_C [23] was not included. Further details of the simulations can be found in [47].

Motional correlation functions, calculated from the MD trajectories, were used to obtain spectral densities and hence a Redfield relaxation superoperator [51, 52] which, when combined with (i) electron Zeeman and hyperfine interactions, (ii) average exchange and dipolar interactions, and (iii) spin-selective recombination reactions, allowed calculation of the singlet yield ${{\rm{\Phi }}}_{{\rm{S}}}(\theta ,\phi ),$ where the polar angles θ and ϕ specify the direction of the external 50 μT magnetic field (appendix A, equation (20)). The slow relaxation arising from cross-correlation of the exchange and the dipolar interactions was neglected. The reaction yield anisotropy, ${\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}},$ was calculated as the difference between the maximum and minimum values of ${{\rm{\Phi }}}_{{\rm{S}}}(\theta ,\phi ).$ Spin relaxation induced by fluctuations in the hyperfine interactions was the subject of [47] and is not treated here. The forward electron transfer steps along the Trp-triad are complete within a nanosecond which is too fast for singlet–triplet coherence to be generated in the intermediate radical pairs [FAD^•− ${{{\rm{TrpH}}}_{{\rm{A}}}}^{\bullet +}]$ and [FAD^•− ${{{\rm{TrpH}}}_{{\rm{B}}}}^{\bullet +}].$ Consequently, STD relaxation is expected to occur only in [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$].

Figures 3(a)–(c) show the variation in r, J(r) and D(r) over the course of MD trajectories spanning 1.3 μs in total. J(r) and D(r) were calculated using equations (1) and (2), with $|{J}_{0}|$ = 8 × 10⁷ T and β = 14 nm⁻¹. The mean distance between the centres of spin density (± standard deviation; see supplementary figure S4) is r_av = 1.89 ± 0.05 nm. The mean interaction strengths (± standard deviation) are $|{J}_{{\rm{av}}}|=8.42\quad \pm \quad 5.64\,{\rm MHz}$ and ${D}_{{\rm{av}}}=-11.5\quad \pm \quad 0.9MHz.$ The maximum and minimum separations are r_max = 2.43 nm and r_min = 1.69 nm, so that $|J({r}_{\mathrm{min}})|=115MHz,$ $|J({r}_{\mathrm{max}})|=3.85kHz,$ D(r_min) = −16.1 MHz, and $D({r}_{\mathrm{max}})$ = −5.5 MHz.

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Figure 4 gives an impression of the positions of the two radicals in AtCry1 in the average structure and in the two extreme conformations, r = r_min and r = r_max. The variation in the radical separation is mainly attributable to the greater mobility of the ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ radical compared to the more snugly bound FAD^•− radical located in the centre of the protein. Representations of the semi-occupied molecular orbitals and the spin densities in the two radicals are shown in supplementary figure S4.

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Given the uncertainty in the value of J₀ (equation (1)), we henceforth treat $|{J}_{{\rm{av}}}|$ as an adjustable parameter and model the exchange interaction as $J(r)={J}_{{\rm{av}}}\mathrm{exp}(-\beta (r-{r}_{{\rm{av}}})),$ with β = 14 nm⁻¹ and r_av = 1.89 nm as above. Except for very small values of $|{J}_{{\rm{av}}}|,$ the fluctuations in $J(r)$ are much larger than those in $D(r),$ meaning that J-modulation should be the dominant source of spin relaxation.

The auto-correlation functions

$$\begin{eqnarray}&&{g}_{K}(\tau )=\langle [K(t)-{K}_{{\rm{av}}}][K(t+\tau )-{K}_{{\rm{av}}}]\rangle \end{eqnarray} \tag{ 10 }$$

(K = r, J, or D, and ${K}_{{\rm{av}}}=\langle K(t)\rangle )$ that characterise the relevant motions of the radical pair are highly non-exponential as a result of motions on timescales ranging from picoseconds to nanoseconds (figure 3(d)). The average time constant for the decay of ${g}_{r}(\tau )$ (8.5 ns) is determined mainly by low-amplitude components in the range τ ≥ 10 ns. In combination with the large fluctuations in $J(r(t))$ noted above, these comparatively slow motions tend to dominate the spin relaxation. By contrast, the rapidly decaying components of ${g}_{r}(\tau )$ (τ < 500 fs) are of minor importance. The average time constants for ${g}_{J}(\tau )$ and ${g}_{D}(\tau )$ are 8.5 ns and 7.6 ns, respectively, reflecting the different r-dependence of the two interactions.

Figures 5(a) and (b) show, respectively, the singlet yield averaged over all directions of the magnetic field, ${\bar{{\rm{\Phi }}}}_{{\rm{S}}},$ and the relative anisotropy, ${\rm{\Gamma }}={\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{S}}}/{\bar{{\rm{\Phi }}}}_{{\rm{S}}},$ both as functions of ${J}_{{\rm{av}}}.$ The lifetime of the radical pair ($\tau ={k}_{{\rm{S}}}^{-1}={k}_{{\rm{T}}}^{-1})$ was 1 μs. When $|{J}_{{\rm{av}}}|\lt 5MHz,$ the protein dynamics have a minimal effect on ${\bar{{\rm{\Phi }}}}_{{\rm{S}}}$ (figure 5(a)). For larger $|{J}_{{\rm{av}}}|,$ relaxation tends to reduce ${\bar{{\rm{\Phi }}}}_{{\rm{S}}}$ towards the statistical value of 0.25. The effect of spin relaxation on the anisotropy (figure 5(b)) is small when $|{J}_{{\rm{av}}}|$ is outside the range 2 MHz < $|{J}_{{\rm{av}}}|\lt 25\,{\rm MHz}$ but shows a complex behaviour within this range. For most of these intermediate values of $|{J}_{{\rm{av}}}|,$ STD boosts the anisotropy above that found when relaxation is excluded. Γ is particularly strongly enhanced for positive values of the exchange interaction (7 MHz < ${J}_{{\rm{av}}}$ < 18 MHz).

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Figure 5(c) offers additional insight by plotting the relative singlet yield anisotropy Γ against the radical pair lifetime for a few representative values of J_av. In the absence of relaxation (orange lines), Γ is small for very short lifetimes (τ < 10 ns), grows as τ approaches the reciprocal of the Larmor period (∼700 ns), and eventually reaches a plateau when τ > ∼10 μs. When relaxation is included (green lines) and $|{J}_{{\rm{av}}}|$ is small (e.g. 0.0 and −0.75 MHz), the anisotropy is always less than or equal to that in the absence of relaxation. For larger J_av however (e.g. +16.8 MHz and +9.0 MHz), the relaxation causes a substantial increase in Γ for radical pairs with intermediate lifetimes (10 ns < τ < 1 μs). For example, when J_av = +16.8 MHz, and τ = 100 ns, the STD mechanism boosts Γ by a factor of 34. No enhancement of any magnitude was found in our earlier study of the relaxation resulting from fluctuations in the hyperfine interactions in FAD^•− and ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ [47].

In summary, unless the mean exchange interaction is too large or too small, the thermal fluctuations in the separation of the FAD^•− and ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}$ radicals in cryptochrome can induce spin relaxation of such a rate as to enhance the magnetic sensitivity, sometimes quite substantially.

In the following paragraphs we discuss other ways in which Nature might have avoided the problems posed by exchange and dipolar interactions in a [FAD^•− ${{{\rm{TrpH}}}_{{\rm{C}}}}^{\bullet +}]$ cryptochrome magnetosensor. We then comment on possible reasons why the sensitivity enhancements afforded by fluctuations in the radical separation might not be manifested in cryptochrome.

5.1. Asymmetric recombination rates

5.2. Alternative radical pairs

5.3. Relaxation-enhanced compass sensitivity in cryptochrome

Previous work on radical pair magnetoreceptors has sought to identify, and in some cases quantify, the factors that control their sensitivity to the direction of the geomagnetic field. These properties include the identity of the radicals [44, 63, 64], their separation [25] and relative orientation [65], their hyperfine interactions [39, 44, 60, 66, 67], and their chemical kinetics [9, 67, 68]. Clearly, the molecular dynamics of the radicals must be added to this list given the profound effects electron spin relaxation can have on the spin-coherence. As argued here, modulation of spin interactions by molecular motions does not always simply degrade the performance of the sensor by destroying coherence. Under some circumstances, protein dynamics and appropriate magnetic interactions combine to produce anisotropic spin relaxation pathways that enhance the response of the compass to the direction of the geomagnetic field. Intermediate spin relaxation rates were found to be most effective. Evolution, acting through random mutations in the sequence of an avian cryptochrome, could have tuned not only its structural, magnetic and kinetic properties but also its dynamics for the purpose of magnetic sensing. In this regard, magnetoreception is no different from many other protein functions in which dynamics as well as structure play an essential role [69]. As in other areas of 'quantum biology' [70–72], interactions of the quantum system with a fluctuating environment appear to be critical for the quantum behaviour [58, 71]. An example, with a clear parallel to the present work, is 'environment assisted quantum transport' in which intermediate dephasing rates enhance the efficiency of photosynthetic energy transfer [73, 74].

We are grateful to Ye Jin and Jonathan Manouch for preliminary calculations. We thank the University of Oxford Advanced Research Computing (ARC) facility (http://dx.doi.org/10.5281/zenodo.22558) and the DeiC National HPC Center, University of Southern Denmark for generous allocation of CPU time. IAS is grateful for financial support from the Lundbeck Foundation and the Russian Science Foundation (Grant No. 14-12-00342). PJH thanks the European Research Council (under the European Union's 7th Framework Programme, FP7/2007-2013/ERC Grant Agreement No. 340451) and the Air Force Office of Scientific Research (Air Force Materiel Command, USAF Award No. FA9550-14-1-0095) for financial support.

Equation (7) and the STD term in equation (5) can be derived as follows. We start with the equation of motion of the radical pair density operator, $\hat{\rho }(t)$:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t}=-{\rm{i}}[{\hat{H}}_{0}+{\hat{H}}_{1}(t),\hat{\rho }(t)]-k\hat{\rho }(t).\end{eqnarray} \tag{ 11 }$$

${\hat{H}}_{0}$ is the time-independent part of the spin Hamiltonian containing the Zeeman and hyperfine interactions and the average exchange interaction. k (= k_S = k_T) is the rate constant for the formation of reaction products from the singlet and triplet states of the radical pair (figure 1(a)). ${\hat{H}}_{1}(t)$ contains the time-dependence of the exchange interaction arising from fluctuations in the radical separation:

$$\begin{eqnarray}&&{\hat{H}}_{1}(t)=F(t){\hat{h}}_{1},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&F(t)=J(r(t))-{J}_{{\rm{av}}}=J(r(t))-\langle J(r(t))\rangle ,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\hat{h}}_{1}=-\left(2{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}}+\displaystyle \frac{1}{2}\hat{E}\right)={\hat{P}}^{{\rm{S}}}-{\hat{P}}^{{\rm{T}}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\hat{P}}^{{\rm{S}}}=\displaystyle \frac{1}{4}\hat{E}-{\hat{{\boldsymbol{S}}}}_{A}\cdot {\hat{{\boldsymbol{S}}}}_{B}\;{\rm{and}}\;{\hat{P}}^{{\rm{T}}}=\displaystyle \frac{3}{4}\hat{E}+{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}}.\end{eqnarray} \tag{ 15 }$$

${\hat{P}}^{{\rm{S}}}$ and ${\hat{P}}^{{\rm{T}}}$ are the singlet and triplet projection operators, $\hat{E}$ is the identity operator, and ${\hat{{\boldsymbol{S}}}}_{{\rm{A}}}$ and ${\hat{{\boldsymbol{S}}}}_{{\rm{B}}}$ are the electron spin angular momentum operators of the two radicals. We assume that the correlation function for $J(r(t))$ decays exponentially with a correlation time ${\tau }_{{\rm{c}}}$,

$$\begin{eqnarray}&&\langle F(t)F(t+\tau )\rangle =\langle F{(t)}^{2}\rangle \mathrm{exp}(-\tau /{\tau }_{{\rm{c}}})=\langle {[J(r(t))-{J}_{{\rm{av}}}]}^{2}\rangle \mathrm{exp}(-\tau /{\tau }_{{\rm{c}}})\end{eqnarray} \tag{ 16 }$$

and that the fluctuations in $F(t)$ are fast enough that

$$\begin{eqnarray}&&\langle {[J(r(t))-{J}_{{\rm{av}}}]}^{2}\rangle {\tau }_{{\rm{c}}}^{2}\ll 1.\end{eqnarray} \tag{ 17 }$$

Then, in the extreme narrowing limit , ${\omega }_{\alpha \beta }^{2}{\tau }_{{\rm{c}}}^{2}\quad \ll \quad 1$ ($\forall \alpha ,\beta ),$ where ${\omega }_{\alpha \beta }$ are ${\hat{H}}_{0}$ eigenvalue-differences, equation (11) becomes:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t}+{\rm{i}}[{\hat{H}}_{0},\hat{\rho }(t)]+k\hat{\rho }(t)=-{\tau }_{{\rm{c}}}\langle F{(t)}^{2}\rangle [{\hat{h}}_{1},[{\hat{h}}_{1},\hat{\rho }(t)]]=-{k}_{{\rm{STD}}}({\hat{P}}^{{\rm{S}}}\hat{\rho }(t){\hat{P}}^{{\rm{T}}}+{\hat{P}}^{{\rm{T}}}\hat{\rho }(t){\hat{P}}^{{\rm{S}}}),\end{eqnarray} \tag{ 18 }$$

with ${k}_{{\rm{STD}}}$ given by equation (7). The operator part of equation (18) can be obtained using equations (14) and (19) and

$$\begin{eqnarray}&&{({\hat{P}}^{{\rm{S}}})}^{2}={\hat{P}}^{{\rm{S}}};\quad {({\hat{P}}^{{\rm{T}}})}^{2}={\hat{P}}^{{\rm{T}}};\quad {\hat{P}}^{{\rm{S}}}{\hat{P}}^{{\rm{T}}}={\hat{P}}^{{\rm{T}}}{\hat{P}}^{{\rm{S}}}=0;\quad {\hat{P}}^{{\rm{S}}}+{\hat{P}}^{{\rm{T}}}=\hat{E}.\end{eqnarray} \tag{ 19 }$$

We note that this approach could be extended to somewhat slower fluctuations, i.e. outside the extreme narrowing limit, using standard Redfield relaxation theory.

For the toy radical pair discussed in sections 2 and 3, the time-independent spin Hamiltonian is given by equation (6). More generally, for a radical pair with multiple, non-diagonal, rhombic hyperfine interactions, ${\hat{H}}_{0}$ has the form

$$\begin{eqnarray}&&{\hat{H}}_{0}={\hat{H}}_{0{\rm{A}}}+{\hat{H}}_{0{\rm{B}}}-{J}_{{\rm{av}}}\left(2{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}}+\displaystyle \frac{1}{2}\hat{E}\right),\\ &&{\hat{H}}_{0K}={\gamma }_{{\rm{e}}}{B}_{0}[{\hat{S}}_{Kx}\;\mathrm{sin}\;\theta \;\mathrm{cos}\;\phi +{\hat{S}}_{Ky}\;\mathrm{sin}\;\theta \;\mathrm{sin}\;\phi +{\hat{S}}_{Kz}\;\mathrm{cos}\;\theta ]+\displaystyle \sum _{j=1}^{{N}_{K}}[{\hat{{\boldsymbol{S}}}}_{K}\cdot {{\bf{A}}}_{Kj}\cdot {\hat{{\boldsymbol{I}}}}_{Kj}],\end{eqnarray} \tag{ 20 }$$

where K = $\{{\rm{A}},{\rm{B}}\},$ and ${\hat{{\boldsymbol{I}}}}_{Kj}$ and ${{\bf{A}}}_{Kj}$ are the nuclear spin angular momentum operator and hyperfine tensor of nucleus j in radical K. N_K is the number of nuclear spins in radical K. Note that the spatial degrees of freedom are included in the spin Hamiltonian by means of effective parameters such as ${{\bf{A}}}_{Kj}$ and ${J}_{{\rm{av}}}$.

A rough estimate of k_STD may be obtained by assuming a Gaussian distribution of separations, centred at $r={r}_{{\rm{av}}},$ with standard deviation, σ:

$$\begin{eqnarray}&&f(r)=\displaystyle \frac{1}{\sqrt{2\pi }}\displaystyle \frac{1}{\sigma }\mathrm{exp}\left(-{\displaystyle \frac{(r-{r}_{{\rm{av}}})}{2{\sigma }^{2}}}^{2}\right).\end{eqnarray} \tag{ 21 }$$

Then, using equations (1) and (7):

$$\begin{eqnarray}&&{k}_{{\rm{STD}}}=4{\tau }_{{\rm{c}}}\displaystyle {\int }_{0}^{\infty }f(r){[J(r(t))-{J}_{{\rm{av}}}]}^{2}{\rm{d}}r=4{\tau }_{{\rm{c}}}{J}_{0}^{2}\mathrm{exp}({\beta }^{2}{\sigma }^{2})\mathrm{exp}(-2\beta {r}_{{\rm{av}}})(\mathrm{exp}[{\beta }^{2}{\sigma }^{2}]-1).\end{eqnarray} \tag{ 22 }$$

When J₀ = 8 × 10⁷ T, β = 14 nm⁻¹, r_av = 1.9 nm, σ = 0.1 nm, and τ_c = 1 ns, equation (22) gives k_STD = 2.7 × 10⁸ s⁻¹.

The relaxation arising from fluctuations in the dipolar interaction may be treated in the same way as J-modulation. The analogous expressions to equations (13) and (14) are:

$$\begin{eqnarray}&&F(t)=D(r(t))-{D}_{{\rm{av}}}=D(r(t))-\langle D(r(t))\rangle ,\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\hat{h}}_{1}=\displaystyle \frac{2}{3}(3{\hat{S}}_{{\rm{A}}z}{\hat{S}}_{{\rm{B}}z}-{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}})=\displaystyle \frac{1}{3}{\hat{P}}^{{{\rm{T}}}_{+}}+\displaystyle \frac{1}{3}{\hat{P}}^{{{\rm{T}}}_{-}}-\displaystyle \frac{2}{3}{\hat{P}}^{{{\rm{T}}}_{0}},\end{eqnarray} \tag{ 24 }$$

where ${\hat{P}}^{{{\rm{T}}}_{m}}$ is the projection operator for the ${{\rm{T}}}_{m}$ triplet state (m = 0, ±1). The evolution of the density operator is:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t}+{\rm{i}}[{\hat{H}}_{0},\hat{\rho }(t)]+k\hat{\rho }(t)=-{\tau }_{{\rm{c}}}\langle F{(t)}^{2}\rangle [{\hat{h}}_{1},[{\hat{h}}_{1},\hat{\rho }(t)]]\\ &&\quad \quad =-{\tau }_{{\rm{c}}}\langle {[D(r(t))-{D}_{{\rm{av}}}]}^{2}\rangle \left[\displaystyle \frac{1}{9}{\hat{P}}^{{\rm{S}}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{+}}+\displaystyle \frac{1}{9}{\hat{P}}^{{{\rm{T}}}_{+}}\hat{\rho }(t){\hat{P}}^{{\rm{S}}}+\displaystyle \frac{1}{9}{\hat{P}}^{{\rm{S}}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{-}}+\displaystyle \frac{1}{9}{\hat{P}}^{{{\rm{T}}}_{-}}\hat{\rho }(t){\hat{P}}^{{\rm{S}}}\right.\\ &&\quad \quad \;\;\;\;\;+\displaystyle \frac{4}{9}{\hat{P}}^{{\rm{S}}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{0}}+\displaystyle \frac{4}{9}{\hat{P}}^{{{\rm{T}}}_{0}}\hat{\rho }(t){\hat{P}}^{{\rm{S}}}+{\hat{P}}^{{{\rm{T}}}_{+}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{0}}+{\hat{P}}^{{{\rm{T}}}_{0}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{+}}\\ &&\quad \quad \;\;\;\;\;+{\hat{P}}^{{{\rm{T}}}_{-}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{0}}+{\hat{P}}^{{{\rm{T}}}_{0}}\hat{\rho }(t){\hat{P}}^{{{\rm{T}}}_{-}}\phantom{\rule{0ex}{2.95ex}}].\end{eqnarray} \tag{ 25 }$$

It can be seen from equation (25) that D-modulation leads not only to STD but also to triplet–triplet dephasing; density matrix elements such as $\langle {{\rm{T}}}_{+1}|\hat{\rho }(t)|{{\rm{T}}}_{0}\rangle$ and $\langle {{\rm{T}}}_{-1}|\hat{\rho }(t)|{{\rm{T}}}_{0}\rangle$ (but not $\langle {{\rm{T}}}_{+1}|\hat{\rho }(t)|{{\rm{T}}}_{-1}\rangle )$ are relaxed. ${\hat{H}}_{0}$ in equation (25) is given by equations (6) or (20) with the exchange term replaced by $\tfrac{2}{3}{D}_{{\rm{av}}}(3{\hat{S}}_{{\rm{A}}z}{\hat{S}}_{{\rm{B}}z}-{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}})$.

Using r_av = 1.9 nm, σ = 0.1 nm, and τ_c = 1 ns (as above, equation (21)), the dipolar relaxation rate constant is:

$$\begin{eqnarray}&&{k}_{{\rm{D}}}={\tau }_{{\rm{c}}}\langle {[D(r(t))-{D}_{{\rm{av}}}]}^{2}\rangle ={\tau }_{{\rm{c}}}\displaystyle {\int }_{0}^{\infty }f(r){[D(r(t))-{D}_{{\rm{av}}}]}^{2}{\rm{d}}r=\;1.4\times {10}^{5}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 26 }$$

This value is substantially smaller than the corresponding estimate of k_STD for J-modulation (2.7 × 10⁸ s⁻¹) because the dipolar interaction has a much weaker dependence on r.

A corresponding treatment of the 'random fields' relaxation mechanism proceeds as follows. The time-dependent part of the spin Hamiltonian is

$$\begin{eqnarray}&&{\hat{H}}_{1}(t)=\displaystyle \sum _{K}\displaystyle \sum _{j}{F}_{Kj}(t){\hat{S}}_{Kj},\end{eqnarray} \tag{ 27 }$$

with

$$\begin{eqnarray*}&&\langle {F}_{Kj}(t)\rangle =0\;{\rm{and}}\;\langle {F}_{Kj}(t){F}_{K^{\prime} j^{\prime} }(t+\tau )\rangle ={\delta }_{KK^{\prime} }{\delta }_{jj^{\prime} }\langle F{(t)}^{2}\rangle \mathrm{exp}(-\tau /{\tau }_{{\rm{R}}}),\end{eqnarray*}$$

where ${\hat{S}}_{Kj}$ is the j-component (j = x, y, z) of the electron spin operator for radical K (K = A, B). Hence:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t}+{\rm{i}}[{\hat{H}}_{0},\hat{\rho }(t)]+k\hat{\rho }(t)=-{\tau }_{{\rm{R}}}\langle F{(t)}^{2}\rangle \displaystyle \sum _{K}\displaystyle \sum _{j}[{\hat{S}}_{Kj},[{\hat{S}}_{Kj},\hat{\rho }(t)]]\\ &&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\;\;\;=-{k}_{{\rm{RFR}}}\left\{\displaystyle \frac{3}{2}\hat{\rho }(t)-\displaystyle \sum _{K}\displaystyle \sum _{j}{\hat{S}}_{Kj}\hat{\rho }(t){\hat{S}}_{Kj}\right\},\end{eqnarray} \tag{ 29 }$$

with

$$\begin{eqnarray}&&{k}_{{\rm{RFR}}}=2{\tau }_{{\rm{R}}}\langle F{(t)}^{2}\rangle .\end{eqnarray} \tag{ 30 }$$

The effects of the J-modulation caused by electron hopping between RP₀ and RP₁ (figure 1(c)) were modelled by means of coupled Liouville equations for the two density operators:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\sigma }_{0}(t)}{{\rm{d}}t}=-{L}_{0}{\sigma }_{0}(t)+{k}_{10}{\sigma }_{1}(t)-{k}_{01}{\sigma }_{0}(t),\\ &&\displaystyle \frac{{\rm{d}}{\sigma }_{1}(t)}{{\rm{d}}t}=-{L}_{1}{\sigma }_{1}(t)-{k}_{10}{\sigma }_{1}(t)+{k}_{01}{\sigma }_{0}(t),\end{eqnarray} \tag{ 31 }$$

in which k₁₀ and k₀₁ are the first order rate constants for RP₁ → RP₀ and RP₀ → RP₁ respectively and ${\sigma }_{0}(t)$ and ${\sigma }_{1}(t)$ are the vector representations of the two density operators, ${\hat{\rho }}_{0}(t)$ and ${\hat{\rho }}_{1}(t).$ The matrix representations of the two Liouvillian superoperators are

$$\begin{eqnarray}&&{L}_{0}={\rm{i}}({\hat{H}}_{0}^{(0)}\otimes \hat{E}-\hat{E}\otimes {\hat{H}}_{0}^{(0){\rm{T}}}),\\ &&{L}_{1}={\rm{i}}({\hat{H}}_{0}^{(1)}\otimes \hat{E}-\hat{E}\otimes {\hat{H}}_{0}^{(1){\rm{T}}})+\displaystyle \frac{{k}_{{\rm{b}}}}{2}({\hat{P}}^{{\rm{S}}}\otimes \hat{E}+\hat{E}\otimes {\hat{P}}^{{\rm{S}}})+{k}_{{\rm{f}}}\hat{E}\otimes \hat{E}.\end{eqnarray} \tag{ 32 }$$

The rate constants ${k}_{{\rm{b}}}$ and ${k}_{{\rm{f}}}$ are defined in figure 1(c). Superscript T denotes transpose. The two spin Hamiltonians are

$$\begin{eqnarray}&&{\hat{H}}_{0}^{(0)}=-{J}_{{\rm{RP}}0}\left(2{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}}+\displaystyle \frac{1}{2}\hat{E}\right),\\ &&{\hat{H}}_{0}^{(1)}=-{J}_{{\rm{RP1}}}\left(2{\hat{{\boldsymbol{S}}}}_{{\rm{A}}}\cdot {\hat{{\boldsymbol{S}}}}_{{\rm{B}}}+\displaystyle \frac{1}{2}\hat{E}\right)+{\hat{H}}_{{\rm{Z}}}^{(1)}+{\hat{H}}_{{\rm{HFC}}}^{(1)}.\end{eqnarray} \tag{ 33 }$$

${\hat{H}}_{{\rm{Z}}}^{(1)}$ and ${\hat{H}}_{{\rm{HFC}}}^{(1)}$ are the Zeeman and hyperfine interactions in RP₁ (see equation (6)). It was assumed that ${J}_{{{\rm{RP}}}_{{\rm{0}}}},$ the exchange interaction in RP₀, is so large that the Zeeman and hyperfine interactions in RP₀ have a negligible effect on its spin dynamics. The singlet product yield was calculated by assembling a joint Liouville space for the two sites:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\left(\begin{array}{c}{\sigma }_{0}(t)\\ {\sigma }_{1}(t)\end{array}\right)=-\left(\begin{array}{cc}{L}_{0}+{k}_{01}E & -{k}_{10}E\\ -{k}_{01}E & {L}_{1}+{k}_{10}E\end{array}\right)\left(\begin{array}{c}{\sigma }_{0}(t)\\ {\sigma }_{1}(t)\end{array}\right)=-{\boldsymbol{\Lambda }}\left(\begin{array}{c}{\sigma }_{0}(t)\\ {\sigma }_{1}(t)\end{array}\right)\end{eqnarray} \tag{ 34 }$$

so that

$$\begin{eqnarray}&&\left(\begin{array}{c}{\sigma }_{0}(t)\\ {\sigma }_{1}(t)\end{array}\right)=\mathrm{exp}(-{\boldsymbol{\Lambda }}t)\left(\begin{array}{c}{\sigma }_{0}(0)\\ {\sigma }_{1}(0)\end{array}\right).\end{eqnarray} \tag{ 35 }$$

The singlet yield is then

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{S}}}={k}_{{\rm{b}}}\displaystyle {\int }_{0}^{\infty }{\rm{Tr}}[{P}^{{\rm{S}}}{\sigma }_{1}(t)]\;{\rm{d}}t\\ \quad \;\;={k}_{{\rm{b}}}\displaystyle {\int }_{0}^{\infty }\left(\begin{array}{cc}0 & {P}^{{\rm{S}}}\end{array}\right)\mathrm{exp}(-{\boldsymbol{\Lambda }}t)\left(\begin{array}{c}{\sigma }_{0}(0)\\ {\sigma }_{1}(0)\end{array}\right)\;{\rm{d}}t\\ \quad \;\;=\displaystyle \frac{{k}_{{\rm{b}}}}{Z}\left(\begin{array}{cc}0 & {P}^{{\rm{S}}}\end{array}\right){{\boldsymbol{\Lambda }}}^{-1}\left(\begin{array}{c}{P}^{{\rm{S}}}\\ 0\end{array}\right)\end{eqnarray} \tag{ 36 }$$

using the initial condition ${\sigma }_{0}(0)\quad =\quad {P}^{{\rm{S}}}/Z$ where Z is the total number of nuclear spin configurations.