Cryptochrome magnetoreception: four tryptophans could be better than three

The biophysical mechanism of the magnetic compass sensor in migratory songbirds is thought to involve photo-induced radical pairs formed in cryptochrome (Cry) flavoproteins located in photoreceptor cells in the eyes. In Cry4a—the most likely of the six known avian Crys to have a magnetic sensing function—four radical pair states are formed sequentially by the stepwise transfer of an electron along a chain of four tryptophan residues to the photo-excited flavin. In purified Cry4a from the migratory European robin, the third of these flavin–tryptophan radical pairs is more magnetically sensitive than the fourth, consistent with the smaller separation of the radicals in the former. Here, we explore the idea that these two radical pair states of Cry4a could exist in rapid dynamic equilibrium such that the key magnetic and kinetic properties are weighted averages. Spin dynamics simulations suggest that the third radical pair is largely responsible for magnetic sensing while the fourth may be better placed to initiate magnetic signalling particularly if the terminal tryptophan radical can be reduced by a nearby tyrosine. Such an arrangement could have allowed independent optimization of the essential sensing and signalling functions of the protein. It might also rationalize why avian Cry4a has four tryptophans while Crys from plants have only three.


Introduction
The remarkable magnetic compass sense that helps night-migratory songbirds navigate thousands of kilometres [1,2] is thought to have a photochemical mechanism [3][4][5][6][7][8]. The axial nature [1,9] and the light-dependence [10] of the birds' responses to the geomagnetic field, together with the involvement of the birds' visual system in processing magnetic compass information [11,12], are consistent with the formation of transient, magnetically sensitive radical pairs in photoreceptor cells in the retina [5]. The molecule that plays host to this photochemistry seems likely to be a member of the cryptochrome (Cry) family of proteins [13][14][15], a possibility first suggested more than 20 years ago [3]. Of the six known avian Crys [14,[16][17][18][19][20][21][22][23][24][25], Cry1a and Cry4a are the main contenders (reviewed in [6,7]). There is also debate about whether the magnetically sensitive radical pairs are formed directly by photo-excitation of the protein or indirectly as intermediates during 'dark' back-reactions [26][27][28][29][30][31][32]. The identity of any 'dark' radical pair is unknown and vertebrate Cry1a does not seem to bind the crucial flavin adenine dinucleotide (FAD) chromophore at all strongly in vitro [33]. We focus here on Cry4a in which flavin-tryptophan radical pairs [34][35][36][37][38] arise from a series of electron transfers along a chain of aromatic amino acid residues that stretches approximately 25 Å from the FAD in the interior of the protein out to its surface [23,39]. In contrast to plant Crys, in which three tryptophans (Trp), or two tryptophans and a tyrosine (Tyr), constitute the electron transfer pathway [13], animal and animal-like Crys possess a tetrad of tryptophans (e.g. avian Cry4s [23,39]) or three tryptophans plus a terminal tyrosine (e.g. Chlamydomonas reinhardtii Cry [40]). Figure 1 shows the structure of the flavin component of the FAD and the four tryptophans (W) in pigeon (Columba livia, Cl) Cry4a, labelled: A (W395), B (W372), C (W318) and D (W369) [37]. The sequence numbers are the same for Cry4a from the night-migratory European robin (Erithacus rubecula, Er) [21]. Also shown is the sidechain of Tyr319, positioned at the far end of the Trp-tetrad, in contact with solvent. Photo-excitation of FAD in ErCry4a is followed by four consecutive electron transfers between adjacent donors/ acceptors, producing four sequential radical pairs: [FAD •− Trp X H •+ ] (abbreviated RP1 X , X = A, B, C or D) [39]. The separations of the flavin and tryptophan radicals in these four states of ClCry4a are approximately 8, 13, 18 and 21 Å, respectively [37]. Judging by molecular dynamics simulations, the electron transfer chain in ErCry4a has a very similar structure with very similar distances between the key components [39].
A recent study of purified robin Cry4a by Xu et al. [39], the first of its kind for any migratory animal Cry, has shed considerable light on the performance of the protein as a potential magnetoreceptor. Spectroscopic measurements were made on the wild-type (WT) protein, and four mutants, W X F (X = A, B, C or D), in which each of the four tryptophans, in turn, had been replaced with phenylalanine (F) to block electron transfer at different points along the chain. The main findings are as follows. (i) Unlike some other avian Crys [33], ErCry4a can be purified with the essential FAD chromophore stoichiometrically bound. (ii) Blue-light irradiation of W A F, W B F and W C F mutants either yielded no detectable radicals (W A F) or produced FAD and Trp radicals that are too short-lived to be magnetically sensitive (W B F and W C F). In the W D F mutant and the WT protein, however, light-induced radicals with lifetimes in excess of 100 ns were identified. (iii) From measurements of the radical-radical separations, it is clear that RP1 D is the dominant transient charge-separated state in the WT protein. (iv) Smaller magnetic field effects were found for purified WT ErCry4a than for two proteins with only three tryptophans: the W D F mutant of ErCry4a and Cry1 from the model plant, Arabidopsis thaliana (AtCry1) [39].
If Cry4a is the magnetic sensory molecule in migratory songbirds and if, in vivo, the RP1 C state is more magnetically sensitive than RP1 D (as is the case in vitro), then one might wonder why Cry4a has a Trp-tetrad instead of a Trp-triad. This is the question we address here. We explore the proposal [39] that, under the right conditions, a Trp-tetrad would be consistent with high detection sensitivity and might have allowed independent evolutionary optimization of the two essential functions of the protein-sensing and signalling.
The key assumption underlying this idea is that RP1 C and RP1 D interconvert by fast reversible electron hopping, i.e. FAD †À Trp C H †þ Trp D H $ FAD †À Trp C H Trp D H †þ [39]. Figure 2a shows part of the conventional Cry reaction scheme in the case that the magnetic field effect stems from a single radical pair (RP1 = RP1 C or RP1 D ) [41]. RP1 is formed by electron transfer along the tryptophan triad or tetrad to the photo-excited FAD (not shown) and interconverts coherently between its singlet ( S RP1) and triplet ( T RP1) states. At the same time, S RP1 returns to the ground state (GS) by spinselective back electron transfer (rate constant k r ; r = recombination reaction) while both S RP1 and T RP1 can proceed to a stabilized radical pair state, RP2 (rate constant k f ; f = forward reaction) [41]. In the latter step, a proton is lost from the indole nitrogen of the tryptophan radical,

Radical pair reaction schemes
(RP2 D ). The magnetic field effect manifests as a change in the yield of RP2 and hence that of the signalling state (SS), a more stable form of the protein in which we assume the tryptophan radical has been reduced (Trp • → TrpH) and the flavin radical protonated (FAD •− → FADH • ). SS then returns to the GS of the protein on a much longer timescale. We assume RP2 lives long enough in vivo (more than 10 µs) that its electron spins are fully relaxed before conversion to SS so that it generates no additional magnetic field effects.
In the simulations described below, we calculate the dependence of the quantum yield of SS (Φ SS , assumed to equal royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210601 that of RP2) on the direction of the external magnetic field with respect to an array of mutually aligned Cry molecules. This quantity represents the 'signal' from which a bird could derive a magnetic compass bearing. Conversion of GS to SS is thought to lead to a conformational change that alters the protein's binding affinity to signalling partners and thereby initiates a biochemical signalling cascade [42]. The notion that RP1 C and RP1 D might jointly be responsible for the magnetic sensitivity of ErCry4a came from estimates of electron transfer rate constants (figure 3) derived from molecular dynamics simulations [39]. The first two steps along the chain of four tryptophans (RP1 A → RP1 B and RP1 B → RP1 C ) were found to be rapid, exergonic and essentially irreversible. At each stage, forward electron transfer is two orders of magnitude faster than direct return to the GS, such that the RP1 C state would be formed in high yield. By contrast, RP1 C and RP1 D were found to have free energies differing by approximately k B T at physiological temperatures, with similar forward (k CD ) and backward (k DC ) electron transfer rate constants for the interconversion of the two states. The estimates of k CD and k DC (approx. 10 10 s −1 [39]), are considerably faster than both the singlet-triplet interconversion and the subsequent reactions of RP1 C and RP1 D , implying that both radical pairs may contribute to magnetic sensing. We, therefore, explore a modified reaction scheme, figure 2b, involving the two interconverting radical pairs in which one electron spin is on the flavin and the other resides on either Trp C H or Trp D H [39]. The singlet states of both pairs can return to the GS (rate constants k Cr and k Dr ) and the TrpH •+ radicals can be deprotonated to form the RP2 C and RP2 D states (rate constants k Cf and k Df ), which then proceed to the SS, again assumed to contain FADH • as the only radical. In this modified reaction scheme, the yield of the SS, Φ SS , remains the quantity of interest and is defined as the sum of the yields of RP2 C and RP2 D .

Methods
The reactions shown in figure 2b were modelled by means of coupled stochastic Liouville equations, one for each of the two states, RP1 C and RP1 D (see electronic supplementary material, section S1 for details). The spin Hamiltonians of the two radical pairs comprised electron Zeeman, electron-nuclear hyperfine and electron-electron dipolar interactions. Haberkorn operators were used for the recombination (k Cr and k Dr ) and forward (k Cf and k Df ) reaction steps [43]. Dipolar tensors [44] were calculated using the centre-to-centre vectors for FAD-Trp C and FAD-Trp D in ClCry4a [37]. The intensity of the geomagnetic field was 50 µT in all simulations. The anisotropy of the quantum yield of the SS, was calculated as a measure of the magnetic compass sensitivity, where Φ SS is the sum of the yields of RP2 C and RP2 D . The maximum and minimum values of Φ SS were determined by sampling, respectively, 1601 (figure 4) and 98 (figure 5) spherically distributed magnetic field directions. Note that this ΔΦ SS differs from the quantity plotted in figure 4c of Xu et al. [39] which is the change in the isotropic yield of the SS induced by a 50 µT magnetic field, calculated using the reaction scheme in figure 2a.
Electron spin relaxation, with rate constant k relax , was included by modelling the effects of isotropic, randomly fluctuating local magnetic fields [45] (electronic supplementary material, equation S7). The rate of spin relaxation of radical pairs in Cry has not been determined experimentally. The best estimates of k relax come from a study of Cry1 from the plant A. thaliana, using a combination of all-atom molecular dynamics simulations and Bloch-Redfield relaxation theory [45,46]. Librational motions of the FAD •− and TrpH •+ radicals and fluctuations in their positions and dihedral angles modulate hyperfine and dipolar interactions and thereby induce spin relaxation at rates in the range 10 6 -10 7 s −1 . The simulations described below were performed with k relax = 10 6 s −1 (figure 4) and k relax = 10 5 , 10 6 or 10 7 s −1 (figure 5). Figure 2. (a) Conventional reaction scheme in which magnetic field effects on the yield of the signalling state (SS) come from a single radical pair, RP1 (=RP1 C or RP1 D ) [41]. Singlet and triplet states are labelled S and T, respectively. (b) Modified reaction scheme in which RP1 C and RP1 D both contribute to the magnetic field effect [39]. The vertical black arrows indicate the formation of singlet radical pairs by spin-conserving electron transfer along the Trp-triad or tetrad to the photoexcited singlet state of FAD. Orange arrows: electron transfer reactions. Green arrows: TrpH •+ deprotonation reactions. Curved red/blue arrows: coherent singlettriplet interconversion. The symbols beside the arrows are rate constants. GS denotes the ground state of the protein.   [39]. All other rate constants, and the free energies of the various states come from molecular dynamics simulations [39]. FAD* is the photo-excited singlet state of FAD. The RP2 and SS states are not shown.
royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210601 Values of k relax ≈ 10 6 s −1 allow time for electron Larmor precession (frequency = 1.4 MHz in a 50 µT field) to affect the spin dynamics before the spin coherence is irreversibly lost.
If RP1 C and RP1 D interconvert sufficiently rapidly, we anticipate that they can be treated as a single 'composite' radical pair described by figure 2a with a single stochastic Liouville equation (see electronic supplementary material, section S1 for details). The hyperfine and dipolar interactions of this composite species are averages, weighted by the fractional equilibrium populations of RP1 C and RP1 D : Weighted-average rate constants were obtained similarly: Hyperfine tensors, calculated by density functional methods [47], were rotated to match the relative orientations of FAD, Trp C and Trp D in the crystal structure of ClCry4a [37,48]. As the computational resources required for the simulations scale steeply with the number of spins, only a subset of the hyperfine interactions in the flavin and tryptophan radicals could be included. From among the nuclei with the largest anisotropic hyperfine interactions, three were selected (see electronic supplementary material, figure S1 for atom labelling schemes): N5 in FAD •− (FN5) and N1 and H1 in each of Trp C H •+ and Trp D H •+ (WN1, WH1). FN5 and WN1 were used for the calculations shown in figure 4. All three nuclear spins were used for figure 5. The Liouvillian matrices for the two-site and composite models had dimensions 32Z 2 and 16Z 2 , respectively, where Z = 27 or 108 for the two-and three nuclei calculations, respectively.

Results
We start by comparing the two-site RP1 C ↔ RP1 D approach (figure 2b) with the composite model (figure 2a) in which the two rapidly interconverting radical pairs act as a single entity with weighted-average properties. Figure 4 shows the dependence of the reaction yield anisotropy, ΔΦ SS , on the rate constants (k CD and k DC ) for interconversion of RP1 C and RP1 D for two sets of recombination and forward reaction rate constants (k Cr , k Cf , k Dr and k Df ). The first set, used for figure 4a,c, corresponds to the extreme case in which recombination is exclusively from RP1 C and the forward reaction is exclusively that of RP1 D : k Cr = k Df = 1.0 × 10 6 s −1 , k Dr = k Cf = 0. In the second set, used for figure 4b,d, k Cr = 1.2 × 10 7 s −1 and k Dr = 3.4 × 10 5 s −1 (estimates from Xu et al. [39]) and k Cf = k Df = 1.0 × 10 6 s −1 . For both sets, the 1.0 × 10 6 s −1 values were chosen (i) to allow time for the 50 µT magnetic field to significantly affect the spin dynamics and (ii) so that the forward reaction can compete with recombination.
In figure 4a,b, ΔΦ SS is plotted (in colour) as a function of f C , the fraction of radical pairs in the RP1 C state, for five values of k CD , with k DC given by f C k CD /(1 − f C ) (equation (3.2)). Also shown are the equivalent calculations for the composite radical pair (in black). As anticipated, the  (FN5 and WN1), there is no reason to think that the composite model would be significantly less valid for radical pairs with a more realistic number of nuclear spins (see electronic supplementary material, section S1.4 for details). Figure 4 confirms that if the RP1 C ↔ RP1 D interchange is fast enough, the composite model provides a reliable picture of the overall magnetic sensitivity of the system. This is a considerable simplification both conceptually and computationally and has allowed figure 5 to be calculated with three instead of two nuclear spins. This difference explains the less structured appearance of figure 5b compared to figure 4a,b (see electronic supplementary material, section S4).
With its validity confirmed, the composite model was then used to explore the dependence of the signal on the different degrees of freedom available to the system: the reaction rate constants, the spin-relaxation rate and the position of the equilibrium. Figure 5a shows contour plots of ΔΦ SS calculated for weighted-average rate constants k r h i and k f h i in the range 10 4 −10 9 s −1 ( y-and x-axes, respectively; see electronic supplementary material, table S4) for seven values of f C and three spin-relaxation rate constants. Note that these data are not presented in the same way as in figure 4 in which specific values of k Cr , k Cf , k Dr and k Df were used. By plotting ΔΦ SS as a function of k r h i and k f h i in figure 5, two contour plots with the same value of k relax (i.e. in the same row) and different values of f C only differ in the weighted-average parameters of the TrpH •+ hyperfine and FAD •− -TrpH •+ dipolar interactions.
Within each contour plot, the maximum signal occurs for values of k r h iand k f h i near the centre of the 10 4 −10 9 s −1 range, with k r h i % 3 k f h i. This can be rationalized as follows [49]. If the recombination and forward reactions are too slow, the magnetic field effects are attenuated by spin relaxation. If they are too fast, there is insufficient time for the 50 µT magnetic field to affect the spin dynamics. If k r h i and k f h i royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210601 are too different, the competition between the two reactions is ineffective.
Each column in figure 5a shows the effect of spin relaxation for a given value of f C . When the spins relax more rapidly, the signal strength drops and its maximum occurs for larger values of k r h i and k f h i. The variations along the rows of figure 5a reflect the changes in the average dipolar and hyperfine interactions for different proportions of RP1 C and RP1 D . Generally speaking, the signal is largest when 0.0 ≤ f C ≤ 0.1 and drops as f C increases. These variations can be seen more clearly in figure 5b for selected values of k r h i and k f h i. They appear to result mainly from the dependence of the mean dipolar interaction on f C : the larger the dipolar interaction, the more it inhibits the singlet-triplet mixing caused by the magnetic field [44] (electronic supplementary material, section S4.3). Using centre-to-centre distances from FAD •− to the two TrpH •+ radicals (electronic supplementary material, table S1), the average dipolar interaction rises from D h i ¼ À8 The overall conclusion that can be drawn from figure 5 is that if there are no constraints on the values of the averaged rate constants k r h i and k f h i, the largest signal available from the composite radical pair should occur for 0.0 ≤ f C ≤ 0.1, i.e. ≥90% RP1 D rapidly interconverting with ≤ 10% RP1 C . h i % 6 Â 10 6 s À1 (when k relax = 10 6 s −1 ) and k f h i % 9 Â 10 6 s À1 (when k relax = 10 7 s −1 ).

Discussion
The immediate question raised by (ii) and (iii) is whether the values of k r h i and k f h i required to achieve large ΔΦ SS are (i) realistic and (ii) compatible with small values of f C . The answer depends on the rate of spin relaxation (with which we start the discussion below). Before doing so, we note that Xu et al. [39] determined the strength of the dipolar interaction, D, in the FAD •− -TrpH •+ radical pair formed photochemically in WT ErCry4a and hence the centre-tocentre separation of the radicals. Based on the crystal structure of the highly homologous pigeon protein, ClCry4a [37], the difference between the values of 〈D〉 expected for f C = 0.0 and f C = 0.1 was within the experimental error in the measurement of D. These experiments are, therefore, consistent with a small fraction of RP1 C ( f C ≤ 0.1) in rapid exchange with RP1 D .

Spin-relaxation rates
As described in §3, there being no experimental measurements of spin-relaxation rates of radicals in Crys, the best estimates of k relax come from molecular dynamics simulations combined with Bloch-Redfield relaxation theory which suggest values in excess of 10 6 s −1 [45]. For the electrons to relax as slowly as 10 5 s −1 the protein would either have to be almost rigid or the radicals within it would have to undergo very rapid, very low amplitude librational and torsional motions. Neither extreme is plausible.
A number of authors have used relaxation rates much slower than 10 6 s −1 in computer simulations of magnetic field effects purporting to be relevant to magnetoreception. More commonly spin relaxation has been completely ignored. In our view, it is unrealistic to assume, in effect, that flavin and tryptophan radicals in a large protein behave in the same way as small radicals undergoing picosecond rotational diffusion in a non-viscous solvent, the only situation in which one could expect relaxation rates slower than about 10 6 s −1 at physiological temperatures.
To summarize, the discussion below is based on the premise that k relax ≥ 10 6 s −1 . This implies (figure 5a) that k r h i and k f h i must lie in the approximate range 10 6 -10 8 s −1 .

Recombination rates
We look first at the case of f C = 0 in which RP1 D is solely responsible for the magnetic field effects. In this limit, the condition that k r h i % k f h i ≥ 10 6 s −1 is simply k Dr ≈ k Df ≥ 10 6 s −1 . A rough upper limit on the rate constant for direct back electron transfer from FAD •− to Trp D H •+ (in s −1 ), assuming zero activation energy, can be obtained from [50]: Such small values of k r h i are not compatible with a large ΔΦ SS when k relax ≥ 10 6 s −1 . It is, therefore, difficult to see how RP1 D acting alone in ErCry4a could form the basis of a sensitive magnetic compass. Müller et al. [51] reached the same conclusion based on measurements of electron transfer rates in Xenopus laevis (6-4) photolyase which also has a Trp-tetrad.
We now look at f C = 0.1 to see whether a 1 : 9 combination of RP1 C and RP1 D in rapid equilibrium (figure 5) could make for a more satisfactory sensor. Applying equation (5.1) to RP1 C , with an edge-to-edge separation of 13.6 Å (electronic supplementary material, table S1), one obtains an approximate upper limit on k Cr of 1.1 × 10 7 s −1 . Xu et al. [39] arrived at the same number, k Cr ≈ (1.2 ± 0.5) × 10 7 s −1 , using the slightly smaller separation of 13.3 Å and by including an activation energy term in equation (5.1). Combining k Cr = 1.2 × 10 7 s −1 with k Dr = 1.1 × 10 5 s −1 (from above) gives k r h i ¼ 0:1k Cr þ 0:9k Dr % 1:3 Â 10 6 s À1 which satisfies one of the conditions for ΔΦ ss to be relatively large, namely k r h i ! 10 6 s À1 .

Tryptophan deprotonation rates
For a composite radical pair with k r h i % 1:3 Â 10 6 s À1 and f C = 0.1, to give a large value of ΔΦ SS , the mean rate constant for the forward reaction, k f h i, would (using the k r h i % 3 k f h i condition) need to be ≈ 4.3 × 10 5 s −1 . This reaction, in which the TrpH •+ radicals are stabilized by loss of the indole proton (WH1) to form neutral Trp • radicals, has been studied royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210601 for several members of the Cry-photolyase superfamily. Deprotonation time constants span four orders of magnitude, from 100-400 ps [52][53][54], to 200-400 ns [51,[55][56][57][58], to 1-4 µs [41,51,59,60]. The very short, sub-nanosecond, lifetimes are for proteins that have an internal H + acceptor and/or water molecules close to the indole nitrogen atom [52][53][54]; there is no evidence for either feature in the crystal structure of ClCry4a or in the molecular dynamics simulations of ErCry4a for either Trp C H •+ or Trp D H •+ [39]. Slower deprotonation (greater than 100 ns) occurs when the solvent acts as the H + acceptor. In only two cases have deprotonation rates been measured for Trp C H •+ and Trp D H •+ in the same protein (the former by replacing Trp D by phenylalanine). Müller et al. [51] found time constants of 400 ns for Trp C H •+ and 2.5 µs for Trp D H •+ in X. laevis (6-4) photolyase, while Xu et al. [39] reported a 100 ns component in the decays of both RP1 C and RP1 D in ErCry4a. The latter was interpreted in terms of a composite radical pair with k Cf h i % k Df h i % ð5 À 10Þ Â 10 6 s À1 . While most of these measurements on purified proteins are inconsistent with k f h i % 4:3 Â 10 5 s À1 , there is no reason why release of the indole proton from either Trp C H •+ or Trp D H •+ in vivo necessarily occurs at the same rate as for the purified protein. Interactions of Crys with other proteins, required either for molecular alignment or signal transduction [42,61], could reduce the solvent accessibility and hence the deprotonation rate. Additionally, reasonably large values of ΔΦ SS can still be expected (figure 5a) even though the optimum condition, k r h i ¼ 3 k f h i, may not be satisfied exactly. In summary, it seems possible that k f h i, like k r h i, could, in vivo, fall in the range required for a large ΔΦ SS .

Tyr319 reduction
Potentially, there is an alternative reaction pathway from the composite radical pair to the SS. Tyr319 (figure 1) has an edge-to-edge distance to Trp D of 3.9 Å (electronic supplementary material, table S1), and appears well placed to be oxidized by Trp D H •+ . In several members of the photolyase-Cry superfamily there is a tyrosine at the far end of the Trp-triad that donates an electron to the terminal Trp C H •+ radical, thus extending the electron transfer chain and stabilizing the FAD •− radical against back electron transfer [40,53,54,[62][63][64][65][66]. The tyrosine radical (TyrO • ) so formed is solvent-exposed and therefore able to be reduced by exogenous electron donors, potentially allowing the efficient formation of a SS containing FADH • as the only radical.
If Tyr319 oxidation rather than Trp C,D H •+ deprotonation is the major pathway to the SS, the reaction scheme in figure 2b changes to that shown in figure 6. Recombination may be taken to occur exclusively from RP1 C (smaller donor-acceptor separation than RP1 D ), while the SS is formed from RP1 D via spin-independent electron transfer from Tyr319 to Trp D H •+ . Reduction of Trp C H •+ by Tyr319 is likely to be an order of magnitude slower given the approximately 3 Å larger donoracceptor distance. Recombination of both RP1 D and [FAD •− TyrO • ] to the GS is assumed to be negligibly slow due to the large edge-to-edge distances, 16.8 and 20.3 Å, respectively (electronic supplementary material, table S1).
The conditions required for this modified reaction scheme -(figure 6) to deliver large values of ΔΦ SS are exactly the same as for the reactions in figure 2b, namely 10 6 s −1 < k r h i % 3 k f h i < 10 8 s −1 where the subscript in k f h i now refers to electron transfer from Tyr319 to Trp D H •+ . If, as assumed in figure 6, k Cr ≫ k Dr and k Cf ≪ k Df , then k r h i % f C k Cr and k f h i % ð1 À f C Þk Df . As long as k Df is in the approximate range 10 6 −10 8 s −1 , the scheme in figure 6 could be just as suitable as that in figure 2b for efficient magnetic compass sensing.
There is some evidence that Tyr319 does indeed donate an electron to Trp D H •+ in avian Cry4a. Otsuka et al. [38] have reported long-lived TyrO • radicals in chicken (Gallus gallus) Cry4a while Zoltowski et al. [37] found that mutating Tyr319 to aspartic acid in ClCry4a decreased the quantum yield of FAD photoreduction and modified the photoreduction kinetics. Xu et al. [39], however, did not detect lightinduced TyrO • radicals in robin Cry4a either by transient absorption or electron paramagnetic resonance, although a possible explanation in the former case is that the UV-visible absorption band of TyrO • is narrow and heavily overlapped by the bands of the various FAD and Trp species. Nevertheless, it seems possible that Tyr319 could be the terminal electron donor in ErCry4a in vivo where the rate of electron transfer to Trp D H •+ might be tuned by protein-protein interactions with signalling partners. If so, then reduction of Trp D H •+ by Tyr319 could, in conjunction with spin-selective recombination of RP1 C , give rise to a significant ΔΦ SS .
An interesting feature of the scheme in figure 6 is that the two rate constants k r h i ¼ f C k Cr and k f h i ¼ ð1 À f C Þk Df depend on the properties of different tryptophan radicals (C and D, respectively), providing scope for independent optimization of k r h i and k f h i. Amino acid mutations in the neighbourhood of Trp C H •+ , for example, could tune k r h i without affectingk f h i, and vice versa. This situation could also occur for the scheme in figure 2b if k Cf ≪ k Df and k Cr ≫ k Dr (see electronic supplementary material, section S2). By contrast, if only one radical pair is involved (figure 2a), a mutation that led to a favourable change in k r might well have the opposite effect on k f .

Conclusion
By means of spin dynamics simulations, we have explored the potential advantages of simultaneously involving the third and fourth sequentially formed flavin-tryptophan radical pairs in magnetic sensing and signalling in avian Cry4a. A composite radical pair with weighted-average properties of its two components, could, at least in theory, offer 'the best of S RP1 D T RP1 D S RP1 C T RP1 C k CD k DC FAD • -TyrO • k Df k Df SS GS k Cr Figure 6. Modified version of figure 2b in which the major route to the SS is via reduction of Trp D H •+ by Tyr319 instead of deprotonation of Trp C H •+ and Trp D H •+ . Contrary to a recent speculation [38], the edge-to-edge separation of FAD •− and TyrO • (20.3 Å, electronic supplementary material, table S1) is such that this radical pair is even less likely than RP1 D to recombine rapidly enough to give a significant magnetic field effect.
royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 18: 20210601 both worlds'. That is, the stronger magnetic sensitivity afforded by [FAD •− Trp C H •+ ] and the superior potential of [FAD •− Trp D H •+ ] to form a SS via oxidation of Tyr319 ( figure 6). Plants, whose Crys contain only three tryptophans, have no known biological requirement to respond to the direction of the Earth's magnetic field, and so might not need to separate the magnetic sensing and signalling functions in the same way as a migratory bird. The Cry from Drosophila melanogaster has four tryptophans, like avian Cry4a, but lacks the terminal tyrosine. This could be relevant if the various magnetic behaviours reported for fruit flies turn out to offer no biological advantage to these non-migratory animals [67][68][69][70][71][72][73][74][75].
Clearly, experiments are needed to test this idea. One possibility would be to mutate amino acid residues in the neighbourhood of the two tryptophans in such a way as to shift the position of the putative equilibrium. For example, introducing a negative charge or removing a positive charge in the vicinity of Trp C could be expected to stabilize [FAD •− Trp C H •+ ] and so change the magnetic sensitivity. Another option would be to seek conditions for the in vitro experiments that more closely resemble those in vivo. For example, it could be that electron transfer from Tyr319 to the fourth tryptophan radical is favoured by protein-protein interactions and could be revealed by studying Cry4a in association with one of the potential signalling partners identified by Wu et al. [42] Thus, it may be possible to get further insight into whether four tryptophans (or four tryptophans and a tyrosine) are better than three.
Data accessibility. The data are provided in the electronic supplementary material [76].