Cryptochrome magnetoreception: four tryptophans could be better than three

Abstract

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.

1. Introduction

The remarkable magnetic compass sense that helps night-migratory songbirds navigate thousands of kilometres [1,2] is thought to have a photochemical mechanism [3–8]. 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 [13–15], a possibility first suggested more than 20 years ago [3]. Of the six known avian Crys [14,16–25], 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 [26–32]. 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 [34–38] 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_XH^•+] (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].

Figure 1.

Structure of the electron transfer chain in ClCry4a (PDB: 6PU0 [37]) showing the FAD chromophore, the tryptophan tetrad and Tyr319. The numbers between adjacent groups are centre-to-centre separations in Å. The numbers at the bottom of the figure are centre-to-centre distances from the FAD. The orange arrows indicate the four sequential electron transfers. Only the isoalloxazine part of the FAD is shown.

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_XF (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_AF, W_BF and W_CF mutants either yielded no detectable radicals (W_AF) or produced FAD and Trp radicals that are too short-lived to be magnetically sensitive (W_BF and W_CF). In the W_DF 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_DF 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. ${\mathrm{FAD}}^{\bullet -}{\mathrm{Trp}}_{\mathrm{C}}{\mathrm{H}}^{\bullet +}{\mathrm{Trp}}_{\mathrm{D}}\mathrm{H}\leftrightarrow {\mathrm{FAD}}^{\bullet -}{\mathrm{Trp}}_{\mathrm{C}}{\mathrm{H}\hspace{0.17em}\mathrm{Trp}}_{\mathrm{D}}{\mathrm{H}}^{\bullet +}$ [39].

2. Radical pair reaction schemes

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 (^SRP1) and triplet (^TRP1) states. At the same time, ^SRP1 returns to the ground state (GS) by spin-selective back electron transfer (rate constant k_r; r = recombination reaction) while both ^SRP1 and ^TRP1 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, TrpH^•+ → Trp^•, to produce either [FAD^•− Trp^•_C] (RP2_C) or [FAD^•− Trp^•_D] (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 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].

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 photo-excited singlet state of FAD. Orange arrows: electron transfer reactions. Green arrows: TrpH^•+ deprotonation reactions. Curved red/blue arrows: coherent singlet–triplet interconversion. The symbols beside the arrows are rate constants. GS denotes the ground state of the protein.

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_BT 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. 10¹⁰ 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_CH or Trp_DH [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.

Figure 3.

Approximate rate constants (in s⁻¹) for electron transfer reactions in ErCry4a. Values for RP1_A → GS and RP1_B → GS come from transient absorption experiments [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.

3. 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,

$$\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{SS}}=max({\mathit{\Phi }}_{\mathrm{SS}})-min({\mathit{\Phi }}_{\mathrm{SS}}),$$ 3.1

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.

Figure 4.

Comparisons of the anisotropy of the SS quantum yield, ΔΦ_SS, for the two-site and composite radical pair models. (a,b) dependence of ΔΦ_SS on f_C with k_CD values in s⁻¹ as indicated. (c,d) Anisotropy surface plots for f_C = 0.1, 0.5 and 0.9 (corresponding to the vertical dashed lines in (a) and (b)) for the composite and two-site models with k_CD = 3 × 10¹⁰ s⁻¹. (a,c) k_Cr = k_Df = 10⁶ s⁻¹, k_Dr = k_Cf = 0. (b,d) k_Cr = 1.2 × 10⁷ s⁻¹, k_Dr = 3.4 × 10⁵ s⁻¹, k_Cf = k_Df = 10⁶ s⁻¹. The spin-relaxation rate was k_relax = 10⁶ s⁻¹ in all cases. In (a) and (b) the black trace, obtained using the composite model, corresponds to the two-site model in the limit of infinitely fast electron hopping. The anisotropy surface plots (c) and (d) were obtained by calculating Φ_SS(Ω) for 1601 spherically distributed magnetic field directions, Ω. The distance from the centre of the plot to the surface in a direction Ω is proportional to the difference between Φ_SS(Ω) and the spherical average, $\overline{\mathit{\Phi }}$_SS. Red and blue indicate magnetic field directions in which Φ_SS(Ω) is larger and smaller, respectively, than $\overline{\mathit{\Phi }}$_SS. See electronic supplementary material, section S3 for additional anisotropy surface plots.

Figure 5.

(a) Dependence of ΔΦ_SS on log₁₀$⟨k_{\mathrm{f}}⟩$and log₁₀$⟨k_{\mathrm{r}}⟩$, calculated using the composite model for seven values of f_C (columns) and three values of k_relax (10⁵, 10⁶, 10⁷ s⁻¹, rows). The nuclear spins included in the calculation were FN5, WN1 and WH1. (b) Variation of ΔΦ_SS with f_C, for four combinations of $⟨k_{\mathrm{r}}⟩$ and $⟨k_{\mathrm{f}}⟩$, with k_relax = 10⁶ s⁻¹. The data shown in (b) were taken from the middle row of (a). The values of $⟨k_{\mathrm{r}}⟩$ and $⟨k_{\mathrm{f}}⟩$ for (b) were chosen to satisfy the condition for ΔΦ_SS to be large, i.e. $⟨k_{\mathrm{r}}⟩\approx 3⟨k_{\mathrm{f}}⟩$ and 3 × 10⁵ s⁻¹ ≤ k_f ≤ 10⁷ s⁻¹.

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 10⁶–10⁷ s⁻¹. The simulations described below were performed with k_relax = 10⁶ s⁻¹ (figure 4) and k_relax = 10⁵, 10⁶ or 10⁷ s⁻¹ (figure 5). Values of k_relax ≈ 10⁶ 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:

$$f_{\mathrm{C}}=\frac{k_{\mathrm{DC}}}{k_{\mathrm{CD}}+k_{\mathrm{DC}}}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{f}_{\mathrm{D}}=\frac{k_{\mathrm{CD}}}{k_{\mathrm{CD}}+k_{\mathrm{DC}}}.$$ 3.2

Weighted-average rate constants were obtained similarly:

$$⟨k_{\mathrm{r}}⟩=f_{\mathrm{C}}{k}_{\mathrm{Cr}}+f_{\mathrm{D}}{k}_{\mathrm{Dr}}\mathrm{and}⟨k_{\mathrm{f}}⟩=f_{\mathrm{C}}{k}_{\mathrm{Cf}}+f_{\mathrm{D}}{k}_{\mathrm{Df}}.$$ 3.3

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_CH^•+ and Trp_DH^•+ (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 32Z² and 16Z², respectively, where Z = 27 or 108 for the two- and three nuclei calculations, respectively.

4. 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 × 10⁶ s⁻¹, k_Dr = k_Cf = 0. In the second set, used for figure 4b,d, k_Cr = 1.2 × 10⁷ s⁻¹ and k_Dr = 3.4 × 10⁵ s⁻¹ (estimates from Xu et al. [39]) and k_Cf = k_Df = 1.0 × 10⁶ s⁻¹. For both sets, the 1.0 × 10⁶ 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_Ck_CD/(1 − f_C) (equation (3.2)). Also shown are the equivalent calculations for the composite radical pair (in black). As anticipated, the correspondence between the two-site and composite models improves as k_CD and k_DC are increased, with respectable, albeit not perfect, agreement when k_CD > 10¹⁰ s⁻¹, a condition satisfied by the k_CD and k_DC values estimated by Xu et al.: (1.3 ± 0.4) × 10¹⁰ s⁻¹ and (1.5 ± 0.4) × 10¹⁰ s⁻¹, respectively. The similarity of the predictions of the two models can also be seen from the three-dimensional representations of the anisotropic component of Φ_SS shown in figure 4c,d for three values of f_C with k_CD = 3 × 10¹⁰ s⁻¹. Although the calculations shown in figure 4 included only two hyperfine interactions (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 $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}\hspace{0.17em}⟨k_{\mathrm{f}}⟩$ in the range 10⁴−10⁹ 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 $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}\hspace{0.17em}⟨k_{\mathrm{f}}⟩$ 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 $⟨k_{\mathrm{r}}⟩\mathrm{and}⟨k_{\mathrm{f}}⟩$ near the centre of the 10⁴−10⁹ s⁻¹ range, with $⟨k_{\mathrm{r}}⟩\approx 3⟨k_{\mathrm{f}}⟩$. 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 $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}\hspace{0.17em}⟨k_{\mathrm{f}}⟩$ 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 $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}\hspace{0.17em}⟨k_{\mathrm{f}}⟩$. 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 $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}⟨k_{\mathrm{f}}⟩$. 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 $⟨D⟩=-8.1\hspace{0.17em}\mathrm{MHz}$ (f_C = 0) to −14.3 MHz (f_C = 1).

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 $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}⟨k_{\mathrm{f}}⟩$, 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.

5. Discussion

Three main conclusions come from the simulations presented in figures 4 and 5. (i) Provided their interconversion is fast enough (k_CD and k_DC > 10¹⁰ s⁻¹), the third (RP1_C) and fourth (RP1_D) radical pairs formed by sequential electron transfers along the Trp-tetrad in ErCry4a should behave as a single entity with weighted-average magnetic and kinetic properties (figure 4). (ii) If there are no restrictions on the values of the mean rate constants, $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}⟨k_{\mathrm{f}}⟩$, the largest anisotropic signals (ΔΦ_SS) can be expected when the equilibrium proportion of RP1_D is 90–100% (i.e. f_C = 0.0–0.1, figure 5). (iii) The largest values of ΔΦ_SS (figure 5a) occur when $⟨k_{\mathrm{r}}⟩\approx 3⟨k_{\mathrm{f}}⟩$ and $⟨k_{\mathrm{f}}⟩\approx 2\hspace{0.17em}\times \hspace{0.17em}{10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$ (when k_relax = 10⁵ s⁻¹), $⟨k_{\mathrm{f}}⟩\approx 6\hspace{0.17em}\times \hspace{0.17em}{10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$ (when k_relax = 10⁶ s⁻¹) and $⟨k_{\mathrm{f}}⟩\approx 9\hspace{0.17em}\times \hspace{0.17em}{10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$ (when k_relax = 10⁷ s⁻¹).

The immediate question raised by (ii) and (iii) is whether the values of $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}⟨k_{\mathrm{f}}⟩$ 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-to-centre 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.

5.1. 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 10⁶ s⁻¹ [45]. For the electrons to relax as slowly as 10⁵ 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 10⁶ 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 10⁶ s⁻¹ at physiological temperatures.

To summarize, the discussion below is based on the premise that k_relax ≥ 10⁶ s⁻¹. This implies (figure 5a) that $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\mathrm{and}⟨k_{\mathrm{f}}⟩$ must lie in the approximate range 10⁶–10⁸ s⁻¹.

5.2. 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 $⟨k_{\mathrm{r}}⟩\approx ⟨k_{\mathrm{f}}⟩$ ≥ 10⁶ s⁻¹ is simply k_Dr ≈ k_Df ≥ 10⁶ s⁻¹. A rough upper limit on the rate constant for direct back electron transfer from FAD^•− to Trp_DH^•+ (in s⁻¹), assuming zero activation energy, can be obtained from [50]:

$${\log }_{10}{k}_{\mathrm{r}}\le 13-0.6\hspace{0.17em}(R-3.6)$$ 5.1

where R (in Å) is the edge-to-edge separation of the electron donor and acceptor. With R = 16.8 Å for RP1_D (electronic supplementary material, table S1), equation (5.1) gives k_Dr ≤ 1.1 × 10⁵ s⁻¹. Using R = 16.0 Å and an approximate activation energy, Xu et al. [39] obtained a similar estimate: k_Dr ≈ (3.4 ± 1.5) × 10⁵ s⁻¹. Such small values of $⟨k_{\mathrm{r}}⟩$ are not compatible with a large ΔΦ_SS when k_relax ≥ 10⁶ 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 × 10⁷ s⁻¹. Xu et al. [39] arrived at the same number, k_Cr ≈ (1.2 ± 0.5) × 10⁷ 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 × 10⁷ s⁻¹ with k_Dr = 1.1 × 10⁵ s⁻¹ (from above) gives $⟨k_{\mathrm{r}}⟩\hspace{0.17em}=\hspace{0.17em}0.1k_{\mathrm{Cr}}\hspace{0.17em}+\hspace{0.17em}0.9k_{\mathrm{Dr}}\hspace{0.17em}\approx \hspace{0.17em}1.3\hspace{0.17em}\times \hspace{0.17em}{10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$ which satisfies one of the conditions for ΔΦ_ss to be relatively large, namely $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\ge \hspace{0.17em}{10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$.

5.3. Tryptophan deprotonation rates

For a composite radical pair with $⟨k_{\mathrm{r}}⟩\hspace{0.17em}\approx \hspace{0.17em}1.3\hspace{0.17em}\times \hspace{0.17em}{10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$ and f_C = 0.1, to give a large value of ΔΦ_SS, the mean rate constant for the forward reaction, $⟨k_{\mathrm{f}}⟩$, would (using the $⟨k_{\mathrm{r}}⟩\approx 3⟨k_{\mathrm{f}}⟩$ condition) need to be ≈ 4.3 × 10⁵ 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 for several members of the Cry-photolyase superfamily. Deprotonation time constants span four orders of magnitude, from 100–400 ps [52–54], to 200–400 ns [51,55–58], 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 [52–54]; there is no evidence for either feature in the crystal structure of ClCry4a or in the molecular dynamics simulations of ErCry4a for either Trp_CH^•+ or Trp_DH^•+ [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_CH^•+ and Trp_DH^•+ in the same protein (the former by replacing Trp_D by phenylalanine). Müller et al. [51] found time constants of 400 ns for Trp_CH^•+ and 2.5 µs for Trp_DH^•+ 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 $⟨k_{\mathrm{Cf}}⟩\hspace{0.17em}\approx \hspace{0.17em}⟨k_{\mathrm{Df}}⟩\hspace{0.17em}\approx \hspace{0.17em}(5-10)\times {10}^6\hspace{0.17em}{\mathrm{s}}^{-1}$.

While most of these measurements on purified proteins are inconsistent with $⟨k_{\mathrm{f}}⟩\hspace{0.17em}\approx \hspace{0.17em}4.3\hspace{0.17em}\times \hspace{0.17em}{10}^5\hspace{0.17em}{\mathrm{s}}^{-1}$, there is no reason why release of the indole proton from either Trp_CH^•+ or Trp_DH^•+ 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, $⟨k_{\mathrm{r}}⟩=3⟨k_{\mathrm{f}}⟩$, may not be satisfied exactly. In summary, it seems possible that $⟨k_{\mathrm{f}}⟩$, like $⟨k_{\mathrm{r}}⟩$, could, in vivo, fall in the range required for a large ΔΦ_SS.

5.4. 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_DH^•+. 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_CH^•+ radical, thus extending the electron transfer chain and stabilizing the FAD^•− radical against back electron transfer [40,53,54,62–66]. 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,DH^•+ 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_DH^•+. Reduction of Trp_CH^•+ by Tyr319 is likely to be an order of magnitude slower given the approximately 3 Å larger donor–acceptor 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).

Figure 6.

Modified version of figure 2b in which the major route to the SS is via reduction of Trp_DH^•+ by Tyr319 instead of deprotonation of Trp_CH^•+ and Trp_DH^•+. 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.

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 10⁶ s⁻¹ < $⟨k_{\mathrm{r}}⟩\approx 3⟨k_{\mathrm{f}}⟩$ < 10⁸ s⁻¹ where the subscript in $⟨k_{\mathrm{f}}⟩$ now refers to electron transfer from Tyr319 to Trp_DH^•+. If, as assumed in figure 6, k_Cr ≫ k_Dr and k_Cf ≪ k_Df, then $⟨k_{\mathrm{r}}⟩\approx f_{\mathrm{C}}{k}_{\mathrm{Cr}}\hspace{0.17em}\mathrm{and}⟨k_{\mathrm{f}}⟩\approx (1-f_{\mathrm{C}})k_{\mathrm{Df}}$. As long as k_Df is in the approximate range 10⁶−10⁸ 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_DH^•+ 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 light-induced 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_DH^•+ might be tuned by protein–protein interactions with signalling partners. If so, then reduction of Trp_DH^•+ 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 $⟨k_{\mathrm{r}}⟩=f_{\mathrm{C}}{k}_{\mathrm{Cr}}$ and $⟨k_{\mathrm{f}}⟩=(1-f_{\mathrm{C}})k_{\mathrm{Df}}$ depend on the properties of different tryptophan radicals (C and D, respectively), providing scope for independent optimization of $⟨k_{\mathrm{r}}⟩$ and $⟨k_{\mathrm{f}}⟩$. Amino acid mutations in the neighbourhood of Trp_CH^•+, for example, could tune $⟨k_{\mathrm{r}}⟩$ without affecting$⟨k_{\mathrm{f}}⟩$, 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.

6. 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 both worlds'. That is, the stronger magnetic sensitivity afforded by [FAD^•− Trp_CH^•+] and the superior potential of [FAD^•− Trp_DH^•+] 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 [67–75].

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_CH^•+] 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].

Authors' contributions

S.Y.W. and Y.W. performed the spin dynamics simulations. S.Y.W. calculated the solvent accessibilities with assistance from I.A.S. All authors discussed the results. Y.W., H.M. and I.A.S. commented on the manuscript which was written by S.Y.W. and P.J.H.

Competing interests

The authors declare no competing interests.

Funding

We acknowledge the financial support of the Deutsche Forschungsgemeinschaft, project nos. 395940726 (SFB 1372, Magnetoreception and Navigation in Vertebrates) and GRK1885 and the European Research Council (under the European Union's Horizon 2020 research and innovation programme, grant agreement no. 810002, Synergy Grant, QuantumBirds). I.A.S. thanks the Lundbeck Foundation, the Volkswagen Foundation (Lichtenberg Professorship) and the Ministry for Science and Culture of Lower Saxony (‘Simulations meet experiments on the nanoscale: opening up the quantum world to artificial intelligence’, SMART). Computational resources for the simulations were provided by the DeiC National HPC Center, SDU and the CARL Cluster at the Carl-von-Ossietzky University of Oldenburg, which is supported by the DFG under INST grant no. 184/157-1 FUGG and the Ministry of Science and Culture (MWK) of the Lower Saxony State. The work was also supported by the North German Supercomputing Alliance (HLRN).