Billions of birds fly thousands of kilometers every year between their breeding and wintering grounds, helped by an extraordinary ability to detect the direction of the Earth’s magnetic field. The biophysical sensory mechanism at the heart of this compass is thought to rely on magnetically sensitive, light-dependent chemical reactions in cryptochrome proteins in the eye. Thus far, no theoretical model has been able to account for the <5° precision with which migratory birds are able to detect the geomagnetic field vector. Here, using computer simulations, we show that genuinely quantum mechanical, long-lived spin coherences in realistic models of cryptochrome can provide the necessary precision. The crucial structural and dynamical molecular properties are identified.

Migratory birds have a light-dependent magnetic compass, the mechanism of which is thought to involve radical pairs formed photochemically in cryptochrome proteins in the retina. Theoretical descriptions of this compass have thus far been unable to account for the high precision with which birds are able to detect the direction of the Earth's magnetic field. Here we use coherent spin dynamics simulations to explore the behavior of realistic models of cryptochrome-based radical pairs. We show that when the spin coherence persists for longer than a few microseconds, the output of the sensor contains a sharp feature, referred to as a spike. The spike arises from avoided crossings of the quantum mechanical spin energy-levels of radicals formed in cryptochromes. Such a feature could deliver a heading precision sufficient to explain the navigational behavior of migratory birds in the wild. Our results (i) afford new insights into radical pair magnetoreception, (ii) suggest ways in which the performance of the compass could have been optimized by evolution, (iii) may provide the beginnings of an explanation for the magnetic disorientation of migratory birds exposed to anthropogenic electromagnetic noise, and (iv) suggest that radical pair magnetoreception may be more of a quantum biology phenomenon than previously realized.

Migratory birds have a light-dependent magnetic compass (1⇓⇓–4). The primary sensory receptors are located in the eyes (2, 3, 5⇓–7), and directional information is processed bilaterally in a small part of the forebrain accessed via the thalamofugal visual pathway. The evidence currently points to a chemical sensing mechanism based on photo-induced radical pairs in cryptochrome flavoproteins in the retina (8⇓⇓⇓⇓⇓⇓⇓⇓⇓–18). Anisotropic magnetic interactions within the radicals are thought to give rise to intracellular levels of a cryptochrome signaling state that depend on the orientation of the bird’s head in the Earth's magnetic field (8, 9, 19). In support of this proposal, the photochemistry of isolated cryptochromes in vitro has been found to respond to applied magnetic fields in a manner that is quantitatively consistent with the radical pair mechanism (15). Aspects of the radical pair hypothesis have also been explored in a number of theoretical studies, the majority of which have concentrated on the magnitude of the anisotropic magnetic field effect (9, 10, 16, 17, 19⇓⇓⇓⇓⇓⇓⇓–27). Very little attention has been devoted to the matter we address here: the precision of the compass bearing available from a radical pair sensor (28).

To migrate successfully over large distances, it is not sufficient simply to distinguish north from south (or poleward from equatorward) (29). A bar-tailed godwit (Limosa lapponica baueri), for example, was tracked by satellite flying from Alaska to New Zealand in a single 11,000-km nonstop flight across the Pacific Ocean (30). A directional error of more than a few degrees could have been fatal. Because the magnetic compass seems to be the dominant source of directional information (31), and the only compass available at night under an overcast (but not completely dark) sky, migratory birds must be able to determine their flight direction with high precision using their magnetic compass. Studies have shown that migratory songbirds can detect the axis of the magnetic field lines with an accuracy better than 5° (32, 33). Any plausible magnetoreception hypothesis must be able to explain how such a directional precision can be achieved. Previous simulations of radical pair reactions (9, 10, 17, 20, 21) show only a weak dependence on the direction of the geomagnetic field and therefore cannot straightforwardly account for the magnetic orientation of birds in the wild.

Theoretical treatments of radical pair-based magnetoreception typically involve simulations of the quantum spin dynamics of short-lived radicals in Earth strength (∼50 μT) magnetic fields (9, 10, 17). The general aim is to determine how the yield of a reaction product depends on the orientation of the reactants with respect to the magnetic field axis. A crucial element in all such calculations is the presence of nuclear spins whose hyperfine interactions are the source of the magnetic anisotropy (8, 16). Most studies have focused on idealized spin systems comprising the two electron spins, one on each radical, augmented by one or two nuclear spins (9, 21⇓⇓⇓⇓⇓–27, 34). Only a handful has attempted to deal with realistic, multinuclear radical pairs (10, 16, 17, 20). The other critical ingredient in such simulations is the lifetime of the electron spin coherence: if the spins dephase completely before the radicals have a chance to react, there can be no effect of an external magnetic field (35). Several studies have assumed, explicitly or implicitly, that the spin coherence persists for about a microsecond, i.e., the reciprocal of the electron Larmor frequency (1.4 MHz) in a 50-μT field (9, 10, 17, 20). Either because the spin system was grossly oversimplified (9, 21⇓⇓⇓⇓⇓–27, 34), or because of this restriction on the spin coherence time, previous theoretical treatments have generally predicted the reaction yield to be a gently varying (often approximately sinusoidal) function of the orientation of the radical pair in the geomagnetic field. Although capable of delivering information on the direction of the field, such a compass would not provide a precise heading. A more sharply peaked dependence on the field direction would be needed to achieve a compass bearing with an error of 5° or less.

Here, we explore the behavior of cryptochrome-inspired radical pairs with multinuclear spin systems and long-lived (>1 μs) spin coherence. We conclude that there is ample scope for a cryptochrome-based radical pair compass to have evolved with a heading precision sufficient to explain the navigational behavior of migratory birds both in the laboratory and in the wild.

Results

Spin Dynamics Simulations. Product yields of radical pair reactions were calculated as described elsewhere (10, 16, 36⇓–38) by solving a Liouville equation containing (i) the internal magnetic (hyperfine) interactions of the electron spin with the nuclear spins in each radical, (ii) the magnetic (Zeeman) interactions of the two electron spins with the external magnetic field, and (iii) appropriate spin-selective reactions of the singlet and triplet states of the radical pair. As a starting point, we modeled [FAD•− TrpH•+], the radical pair that is responsible for the magnetic sensitivity of isolated cryptochrome molecules in vitro (15). It consists of the radical anion of the noncovalently bound flavin adenine dinucleotide (FAD) cofactor and the radical cation of the terminal residue of the “tryptophan (Trp) triad” electron transfer chain within the protein (39⇓–41). All calculations were performed in a coordinate system aligned with the tricyclic flavin ring system (Fig. 1A): x and y are, respectively, the short and long in-plane axes, and z is normal to the plane. Hyperfine interaction tensors were calculated by density functional theory (SI Appendix, Section S1). Following Lee et al. (16), the 14 largest hyperfine interactions, 7 in FAD•− and 7 in TrpH•+, were included (see SI Appendix, Section S2 for additional simulations including up to 22 nuclear spins.) A magnetic field strength of 50 μT was used throughout. The relative orientation of the two radicals was that of FAD and Trp-342 in Drosophila melanogaster cryptochrome (Protein Data Bank ID code 4GU5) (SI Appendix, Section S1) (42, 43). The initial state of the spin system was a pure singlet. Two approximations (SI Appendix, Sections S3 and S4) were introduced to make simulations of the 16-spin system computationally tractable (9): (i) exchange and dipolar interactions between the radicals were assumed to be negligible, and (ii) the singlet and triplet states were assumed to react to form distinct products with identical first order rate constants, k. The lifetime of the radical pair, τ, is defined as the reciprocal of k. As a measure of the available directional information, we calculated Φ S , the fractional yield of the product formed from the singlet state of the radical pair after the reactions have proceeded to completion (10, 16). Spin relaxation processes were not included in the initial simulations. Further details are given in the SI Appendix, Section S5. Fig. 1. Reaction yields of a [FAD•− TrpH•+] radical pair. (A) The axis system used in the simulations superimposed on the tricyclic flavin ring system. (B) The variation of Φ S with θ for radical pairs with lifetimes between 1 and 100 μs. For clarity, two of the traces have been offset vertically: by −0.001 (light green) and −0.002 (red). θ specifies the direction of the magnetic field in the zx plane of the flavin. (C) The same data as in B (1- to 20-μs lifetimes) presented as 2D polar plots. In each case, only the anisotropic part of Φ S is shown, with red and blue indicating values, respectively, larger and smaller than the isotropic value. The five plots are drawn on the same scale. The blue features at θ = ±90° (labeled ∗ in the 20-μs plot) are the spikes. (D) The anisotropic part of Φ S (10-μs lifetime) presented as a 3D polar plot. A circle in the xy plane (θ = 90°) is included as a guide to the eye. The blue disk in the xy plane (labeled ∗ ) gives rise to the spike. The smaller blue disk, labeled # (also in C), angled at ∼40° to the xy plane, comes principally from the N1 indole nitrogen of TrpH•+. Its tilt reflects the orientation of the indole group of the tryptophan relative to the flavin (42, 43). (E) Visual modulation patterns calculated from Φ S (1- to 20-μs lifetimes) representing the directional information available from an array of cryptochrome-containing magnetoreceptor cells distributed around the retina. The bright spot in the lower half of the pattern arises from the spike. (F) 3D polar plot of Φ S (10-μs lifetime) averaged over a 360° rotation around an axis in the xy plane. This object has been rotated by 90° relative to D and scaled up by a factor of 2.1. The patterns in E were calculated using the same averaging procedure (SI Appendix, Section S6).

Origin of the Spike in Φ S . The approximate axial symmetry of Φ S for τ = 1 μs (Fig. 1C) has been noted before and was attributed principally to the two nitrogens, N5 and N10, in the central ring of the FAD•− radical (10, 16). N5 and N10 are the only nuclei in [FAD•− TrpH•+] with hyperfine tensors that, like Φ S , are approximately axially symmetric around the flavin z axis. It therefore seems probable that they also play a role in creating the spike that arises when τ > 5 μs. This prediction is confirmed by Fig. 2A, which shows Φ S for a very slightly modified version of [FAD•− TrpH•+]. The z components of the hyperfine interactions of N5 and N10 in flavin radicals are large, and the x and y components have small but nonzero absolute values (SI Appendix, Section S7). The calculated principal values of the two interactions are (A xx , A yy , A zz ) = (−0.087, −0.100, 1.757) mT for N5 and (−0.014, −0.024, 0.605) mT for N10 (SI Appendix, Section S1; here 1 mT corresponds to 28 MHz). When A xx and A yy for either N5 or N10 were set to zero, the spike was attenuated by 60–70%; when A xx and A yy for both nitrogens were set to zero, the spike disappeared (Fig. 2A). The rest of Φ S remained essentially unchanged. The strong, sharp component of Φ S for [FAD•− TrpH•+] therefore owes its existence, at least in part, to the form of the hyperfine tensors of N5 and N10 in the flavin radical, i.e., large A zz and small but nonzero | A xx | and | A yy | . Fig. 2. Reaction yields of various radical pairs. (A) Φ S for a [FAD•− TrpH•+] radical pair in which the transverse principal components of selected nitrogen hyperfine interactions (A xx and A yy ) were set to zero: for N5 (blue), N10 (red), and both N5 and N10 (green). Φ S for the unmodified [FAD•− TrpH•+] is shown in black. In all cases, τ = 1 ms. For clarity, three of the traces have been offset vertically by 0.006 (green) and 0.003 (blue and red). (B) Φ S for a [FAD•− Y•] radical pair in which radical Y• contains a single 14N nucleus with an axial hyperfine tensor with principal components (A xx , A yy , A zz ) = (0.0, 0.0, 1.0812) mT (modeled on N1 in TrpH•+). The radical pair lifetimes are as indicated (1–100 μs). The angle between the z axes of Y• and FAD•− was 45°; the intensity of the spike was found to decrease smoothly to zero as this angle was increased from 0° to 90°. For clarity, the five traces for τ < 100 μs have been offset vertically, from top to bottom, by 0.020, 0.016, 0.012, 0.008, and 0.004 respectively. (C) Φ S for toy radical pairs, [X• Y•]. For the red, orange and green traces, X• contains a single 14N hyperfine tensor with principal components (A xx , A yy , A zz ) = (−0.0989, −0.0989, 1.7569) mT. For the blue and black traces, (A xx , A yy , A zz ) = (−0.2, −0.2, 1.7569) and (−0.4, −0.4, 1.7569) mT, respectively. In all five cases, Y• contains a single 14N nucleus with an axial hyperfine interaction: (A xx , A yy , A zz ) = (0.0, 0.0, 1.0812) mT. The two hyperfine tensors have parallel z axes. The radical pair lifetimes are as indicated (10, 33.3, 100 μs); ×2 and ×4 indicate the doubling and quadrupling of A xx and A yy in X•. For clarity, three of the traces have been offset vertically by 0.03 (green) and 0.06 (orange and red). SI Appendix, Section S8 contains an analysis that unambiguously attributes the thin equatorial disk in Fig. 1D to avoided crossings of the quantum mechanical energy levels of the radical pair spin Hamiltonian as a function of the magnetic field direction and predicts that the line shape of a cross-section through the disk (i.e., the spike) will be an upside-down Lorentzian. When A xx and A yy for both nitrogens are set to zero, the avoided crossings become level crossings and the spike vanishes.

Simpler Flavin-Containing Radical Pairs. To obtain further insight into the origin of the spike, simulations were performed for three radical pairs related to [FAD•− TrpH•+] (1). When the TrpH•+ radical was replaced by a hypothetical radical that had no hyperfine interactions, Φ S was found to vary smoothly and approximately sinusoidally with θ and barely changed as τ was increased from 1 to 100 μs (SI Appendix, Section S9) (2). This pattern (gentle, smooth θ dependence, no spike) persisted when a single isotropic hyperfine interaction was present in the second radical (SI Appendix, Section S9) (3). However, when the second radical contained an axially anisotropic hyperfine interaction with A xx = A yy = 0 or A xx = A yy ≠ 0, the spike at θ = 90° reappeared and, as in Fig. 1B, strengthened with increasing lifetime (Fig. 2B). From this and other simulations of flavin-containing radical pairs, it appears that an additional condition for the existence of the spike is that the radical that partners the FAD•− should have at least one nucleus with an anisotropic hyperfine interaction. This condition is amply fulfilled by TrpH•+, in which the indole nitrogen and the aromatic hydrogens all interact anisotropically with the electron spin (16).

A Toy Radical Pair. To confirm and further explore these conclusions, we devised a “toy” radical pair, with a smaller, more manageable spin system, that behaves qualitatively like [FAD•− TrpH•+]. One radical (X•) had a single nitrogen with a hyperfine tensor similar to that of the N5 in FAD•−. The other (Y•) had a single nitrogen with an axial hyperfine tensor modeled on the indole nitrogen in TrpH•+. Like [FAD•− TrpH•+], [X• Y•] shows a spike at θ = 90° superimposed on a rolling background (Fig. 2C). The spike became more pronounced when either the lifetime was prolonged or the amplitudes of the small transverse hyperfine components in X• were increased. For example, doubling A xx and A yy when τ = 10 μs increased the amplitude of the spike by about the same amount as increasing τ from 10 to 33 μs without changing A xx and A yy (Fig. 2C).

Spin Relaxation in the Toy Radical Pair. Of course, the spin coherence does not persist indefinitely but inevitably relaxes toward the equilibrium state in which all spin correlation has vanished. The rate of this process is highly relevant because there can be no magnetic field effect if the spin system equilibrates before the radicals react. The dominant spin relaxation pathways in a cryptochrome-based radical pair probably arise from modulation of hyperfine interactions by low-amplitude stochastic librational motions of the radicals within their binding pockets in the protein. The approach to equilibrium is likely to be highly complex for realistic radicals undergoing realistic motions especially because the external magnetic field is weaker than many of the hyperfine interactions. In general, one can expect a multitude of relaxation pathways, at a variety of rates, not all of which necessarily degrade the performance of the radical pair as a compass sensor (44). To explore the conditions necessary for the spike to survive in the presence of molecular motion, we studied a simple model of the microscopic dynamics of the FAD•− radical in cryptochrome. The tricyclic isoalloxazine moiety was allowed to undergo rotational jumps (+β ↔ −β degrees) around its y axis with a first order rate constant, k r (SI Appendix, Section S10). In the language of magnetic resonance, this rocking motion constitutes a “symmetric two-site exchange” process (45), the effect of which is to modulate the hyperfine field experienced by the electron spin. For a given set of anisotropic hyperfine interactions, the only additional parameters are the rocking angle and the rate constant. To get an initial idea of the expected behavior, we started with the toy radical pair introduced above. Fig. 3A shows Φ S when Y• is stationary and X• undergoes 10° rotational jumps (i.e., β = 5°) around its y axis. The lifetime of the radical pair was fixed at 10 μs, so that any relaxation pathway occurring on this timescale, or faster, could influence Φ S . When the rocking is sufficiently fast (k r ≥ 3 × 109 s−1; Fig. 3A), the differences in the magnetic interactions in the two orientations are averaged by the motion and a single sharp spike is seen at θ = 90°. As k r is reduced, the averaging becomes less efficient, causing attenuation of the spike (without significant broadening) and flattening of the gently varying background (Fig. 3A and SI Appendix, Section S11). Spin relaxation is most efficient when k r is comparable to the strengths of the hyperfine interactions, i.e., ∼108 s−1. Under these conditions, Φ S tends toward 0.25, the statistical singlet fraction expected at thermal equilibrium. Fig. 3. Reaction yields of radical pairs with spin relaxation included. (A) The toy radical pair, [X• Y•]. X• has a single 14N nucleus with hyperfine components (A xx , A yy , A zz ) = (−0.2, −0.2, 1.7569) mT; Y• has a single 14N nucleus with hyperfine components (0.0, 0.0, 1.0812) mT. The two hyperfine tensors have parallel z axes. The radical pair lifetime is 10 μs. X• underwent 10° rotational jumps (i.e., β = 5°) around the y axis with rate constants k r between 3 × 1011 and 108 s−1, as indicated. (B) The [FAD•− Y•] radical pair. FAD•− has seven magnetic nuclei, as in Fig. 1. Y• has single 14N nucleus with hyperfine components (A xx , A yy , A zz ) = (0.0, 0.0, 1.0812) mT. The radical pair lifetime is 10 μs. FAD•− underwent 10° rotational jumps (i.e., β = 5°) around the y axis, with rate constants k r varying between 3 × 1011 and 109 s−1, as indicated. In A and B, the direction of the magnetic field (θ) is varied in the zx plane of the flavin ring system (Fig. 1A). Almost identical results were found for the zy plane. These simulations were performed for a rocking axis (y) perpendicular to the symmetry axis (z) of the hyperfine tensor in X•. Rotation around an axis tilted out of the xy plane results in less extensive modulation of the magnetic interactions, less efficient spin relaxation, and less attenuation of the spike for a given k r . In this respect, Fig. 3A represents the worst case. The behavior of Φ S when k r ≤ 108 s−1 is discussed in SI Appendix, Section S12. In summary, the spike survives if k r ≥ 3 × 109 s−1 (Fig. 3A). This value corresponds to a librational wavenumber of the aromatic ring systems greater than ∼0.1 cm−1.

Spin Relaxation in a Flavin-Containing Radical Pair. We now look at the effects of motion on a more realistic spin system. It proved impractical to repeat the above calculation for the full (16-spin) [FAD•− TrpH•+] radical pair treated above. Instead, we studied [FAD•− Y•] in which FAD•− contained seven nuclear spins (as above) and Y• was the same as in the toy radical pair, [X• Y•]. Fig. 3B shows Φ S for [FAD•− Y•] with the FAD•− radical undergoing 10° rotational jumps (β = 5°) around its y axis with rate constants in the fast exchange regime: 109 s−1 ≤ k r ≤ 3 × 1011 s−1. As was the case for [X• Y•] (Fig. 3A), when τ = 10 μs, the spike at θ = 90° persists for rocking rates down to 3 × 109 s−1 and is even visible when k r = 109 s−1. Similar behavior was found for a [X• TrpH•+] pair in which TrpH•+ underwent 10° jumps (SI Appendix, Section S13). Spin relaxation effects were more pronounced for jumps larger than 10°. Clearly, the dynamics of FAD•− and TrpH•+ in cryptochrome are considerably more complicated than this two-site jump model. However, we can infer from these exploratory studies that the spike in Φ S is not excessively sensitive to reasonably rapid, relatively low-amplitude motions of the type likely to occur for the radicals in their binding sites in cryptochrome. The message we take from these calculations is that radical motions on timescales faster than about 1 ns could allow the spikiness of Φ S to survive.