The sensitivity of a radical pair compass magnetoreceptor can be significantly amplified by radical scavengers

Birds have a remarkable ability to obtain navigational information from the Earth’s magnetic field. The primary detection mechanism of this compass sense is uncertain but appears to involve the quantum spin dynamics of radical pairs formed transiently in cryptochrome proteins. We propose here a new version of the current model in which spin-selective recombination of the radical pair is not essential. One of the two radicals is imagined to react with a paramagnetic scavenger via spin-selective electron transfer. By means of simulations of the spin dynamics of cryptochrome-inspired radical pairs, we show that the new scheme offers two clear and important benefits. The sensitivity to a 50 μT magnetic field is greatly enhanced and, unlike the current model, the radicals can be more than 2 nm apart in the magnetoreceptor protein. The latter means that animal cryptochromes that have a tetrad (rather than a triad) of tryptophan electron donors can still be expected to be viable as magnetic compass sensors. Lifting the restriction on the rate of the spin-selective recombination reaction also means that the detrimental effects of inter-radical exchange and dipolar interactions can be minimised by placing the radicals much further apart than in the current model.


Singlet-triplet interconversion in AB
Signalling state anisotropy Supporting Information Figure S1: Anisotropic yields of the signalling state, S, and the scavenging product, X, for a model [FAD  WH + ] radical pair.

Figure S2:
Anisotropic yields of the signalling state, S, and the scavenging product, X, for a model [FAD  WH + ] radical pair.

Figure S3:
Anisotropic yields of the scavenging product, X, for a model [FAD  WH + ] radical pair.

Figure S4:
Anisotropic yields of the scavenging product, X, for various model radical pairs.  Section S1: Derivation of Eq. (3).

Singlet-triplet interconversion in AB
Insight into the origin of singlet-triplet interconversion in AB as a result of a spin-selective AC scavenging reaction may be obtained from the following simple argument. 36 First, we define the usual singlet and triplet states for the AB and AC pairs: where X = B or C. We start with the AB pair in its singlet state, AB S , and the radical C in state  C (  AB C S is considered below). This initial state,    AB C S in the   AB C basis, can be expressed in the ABC product basis using equations (1): and then transformed into the   AC B basis, again using equations (1): Equation (3) shows that the AC pair is 25% singlet and 75% triplet, as would be expected from the absence of correlation between C and either A or B. Now we allow the AC singlets to recombine via a spin-selective scavenging reaction. To see the effect most clearly, we simply remove the first term on the right hand side of equation (3) to give the modified state   :   can be transformed back into the ABC product basis: and then into the original   AB C basis: The proportions of singlet and triplet AB pairs, which were initially 100% and 0% respectively, are now 75% and 25%. If we start with    AB C S instead of  AB C S , the equivalent of equation (7) is: which again gives 75% singlet and 25% triplet. Thus the net effect of the spin-selective AC reaction is to induce singlet-triplet interconversion in the AB pair even though there is initially no spin correlation between C and either A or B.

Signalling state anisotropy
The principal factor behind the unexpectedly large values of  S and  S appears to be the form of the hyperfine interactions of the N5 and N10 nitrogens in FAD  . As the most anisotropic hyperfine interactions in the flavin radical, they seem to reinforce one another and to dominate the spin dynamics of FAD  -containing radical pairs. 34 Both 14 N hyperfine tensors have almost perfect axial symmetry, with parallel symmetry axes, large z-components and near-zero x-and y-components. 11, 34 A consequence is that when the magnetic field is parallel to the symmetry axis, the spin Hamiltonian connects the AB singlet state to AB 0 T but not to AB +1 T or  AB 1 T . By contrast, when the field is perpendicular to the hyperfine symmetry axis, S AB is mixed with all three AB triplet states.
In the parallel configuration, the S AB  T AB interconversion caused by the AC scavenging reaction (see the Appendix in the main text) leads to T AB states which (a) cannot be converted to S AB by the spin Hamiltonian and are therefore unable to return to the ground state, (b) are not scavenged because of the Wigner spin-conservation requirements, and which therefore (c) contribute to a high yield of the non-selectively formed signalling state. It is these states that are responsible for the long-time behaviour shown in Fig. 3b (main text).
In the perpendicular case, the more extensive S AB  T AB mixing means that no T AB states are immune to spin-selective recombination and scavenging. The result is a lower yield of the competing reaction that leads to the signalling state. Intermediate orientations show similar behaviour. It is only when the field is parallel to the dominant hyperfine axis that singlet-triplet mixing in the AB pair becomes restricted and the spike emerges. This qualitative difference between parallel and all other directions of the magnetic field seems to be responsible for the large anisotropies in the yield of the signalling state.

Figure S1
Anisotropic yields of the signalling state, S, and the scavenging product, X, for a model [FAD  WH + ] radical pair. The scavenger is a radical (J = ½) with no hyperfine interactions. (a) and (b) relative anisotropies ( S and  X , respectively), (c) and (d) absolute anisotropies (Δ S and Δ X , respectively), both as a function of the scavenging rate constant, k C , for various values of . The spin system comprises N5 and N10 in FAD  and N1 in WH + . The model is identical to that used for Figures 4(c) and 4(d) except that the scavenger reacted with W •+ instead of FAD •− .  X and Δ X are defined by analogy with  X and Δ X (equations (15) and (14), respectively).

Figure S2
Anisotropic     The anisotropic yields were rescaled to reveal most clearly the increase in spikiness as the lifetime of the radical pair was prolonged. If drawn to scale, the pattern for k f -1 = 1 ms would be 64 times taller than that for k f -1 = 1.23 μs. See Figure S5 for a description of this type of plot.

S1. Derivation of Eq. (3)
Consider a doublet (S A = ½) interacting with a particle with spin S C = J. According to the Clebsch-Gordan series, the tensor product states associated with Â S and Ĉ S can be combined to give eigenstates of the total angular momentum  

S2. Derivation of Eqs (6) and (8)
For the sake of clarity, we focus on the case   C 0 k . The general results given in eqs (6) and (8)  Integrating eq. (4) for   C 0 k , we obtain: which is independent of S and M. In order to evaluate the matrix elements of S AB P , we recouple the angular momenta to yield an eigenbasis of  ˆÂ

S S S S S M S S S S S M S S S S S S S S S S , (S.8)
with the recoupling coefficient given in terms of Wigner 6-j symbol by: