Radical Scavenging Could Answer the Challenge Posed by Electron–Electron Dipolar Interactions in the Cryptochrome Compass Model

Many birds are endowed with a visual magnetic sense that may exploit magnetosensitive radical recombination processes in the protein cryptochrome. In this widely accepted but unproven model, geomagnetic sensitivity is suggested to arise from variations in the recombination rate of a pair of radicals, whose unpaired electron spins undergo coherent singlet–triplet interconversion in the geomagnetic field by coupling to nuclear spins via hyperfine interactions. However, simulations of this conventional radical pair mechanism (RPM) predicted only tiny magnetosensitivities for realistic conditions because the RPM’s directional sensitivity is strongly suppressed by the intrinsic electron–electron dipolar (EED) interactions, casting doubt on its viability as a magnetic sensor. We show how this RPM-suppression problem is overcome in a three-radical system in which a third “scavenger” radical reacts with one member of the primary pair. We use this finding to predict substantial magnetic field effects that exceed those of the RPM in the presence of EED interactions in animal cryptochromes.


Spin dynamics simulations
Dynamics of the spin densityρ(t) are described by the Liouville-von Neumann equation, where the time-independent Hamiltonian may be written as sum of energy operators: including magnetic Zeeman effectsĤ Zee , electron-electron dipolar (EED) interactionsĤ dip , electron-nuclear "hyperfine" couplingĤ hf , and electron-electron exchangeĤ ex . We neglect decoherence processes, assumed slow relative to the lifetime of the radical pair, as reflected in the rate constants for scavenging k X and escape k f . As usual, square brackets [ , ] denote commutation, whereas brace brackets { , } indicate anti-commutation of the enclosed terms.
In detail, the Zeeman HamiltonianĤ Zee = i g i µ Bˆ S i · B describes interaction of the magnetic field denoted B with each radical i's spin angular momentum (expressed as unitless vector operatorsˆ S i ), where µ B is the Bohr magneton. The dipolar HamiltonianĤ dip defines the energy of the EED interactions, where D ij (r ij ) = µ 0 g i g j µ 2 B /(4π| r ij | 3 ), r ij = r j − r i gives the displacement from radical i to j where u ij = r ij /| r ij | is the corresponding unit vector, g i and g j designate electronic gfactors, and µ 0 denotes the magnetic permeability of free space. The hyperfine Hamiltonian H hf defines the energy of interaction of the ith electron spin with the n i magnetic nuclei (enumerated as i ) within the radical i: The anisotropic hyperfine coupling tensors A i define the electron-nuclear magnetic couplings where they are present. The exchange Hamiltonian is of the formĤ ex = i>j J ij 2ˆ S i ·ˆ S j + 1 2 . The projectorP ab S defines the singlet state of any two radicals a and b. Recombination is recovered in the R3M case where (a, b) = (1, 2), whereas scavenging of either of either member of the radical pair is modeled otherwise, i.e., either (a, b) = (1, 3) or (a, b) = (2, 3).
We modeled hyperfine (HF) coupling interactions up to a total of four magnetic nuclei. For Cl Cry4, the flavin hyperfine tensors were as given above, whereas the tryptophan tensors were rotated to reflect its altered relative orientation in this protein: For the flavin semiquinone, the following three tensors were used: The position of the scavenger radical was varied in three dimensions by assuming a virtual "shell" of possible locations on the vertices of a sphere-like geodesic polyhedron of radius R, while varying R. For each type of scavenger reaction (scavenging of the anion or cation of the primary radical pair), the polyhedron was centred around the scavenger's target radical. We initially sampled scavenger coordinates at ∆R = 1 • A radial increments, recursively adjusting the radius increment and number of vertices to generate a smooth MFE function Γ for each tunneling decay constant β tested. The "R3M" 1 recombinationbased systems were approximated by simulating primary-pair recombination with a third, unreactive "bystander" radical nearby. Figure S1: Graphical representation of DmCry4 (PDB ID: 4GU5) 2 as a ribbon diagram. 3 Labels indicate the FAD prosthetic group and Trp terad residues, superimposed on the protein ribbon to illustrate their positions in the molecule. Figure S2: Pertinent residues in the crystal structure of Cl Cry4 (red) and the homology model of Er Cry4 from Ref. [4] (blue). The root-mean-square displacement (RMSD) of the relevant part of both of these structures amounts to only 1.2 • A and the differences in inter-radical distances are negligible (compared to their fluctuations as assessed in molecular dynamics simulation). Note in particular that DmCry and Cl Cry4 too differ by a RMSD of 1.2 • A, while the simulated MFEs of these systems are practically identical (cf. Fig. 2 and S8). No intrinsic enhancement of magnetosensitvity is expected for Er Cry4 within the constraints of the suggested model. The superiority recently established for Er Cry4 in vitro appears to be the result of tuned rate constants. 5 However, the rate constants established there do not permit substantial magnetosensitivity in the geomagnetic field. Changes in the relative radical orientation are likewise not expected to dramatically boost the magnetosensitivity. 6   Table S1: Summary of three-radical reactions. The table shows the initial spin configuration of the geminate radical pair, the charge recombination mechanism, and the reaction scheme as given in the main manuscript. Bystander-enhanced schemes (in the top two rows) recover the RPM in the large-distance limit of the distance from the primary pair to bystander B • .

Radical Trio Notation Geminate Pair
Mechanism Scheme

Unconstrained models
We began by carrying out a systematic exploration of the reaction phase space, setting   A and rates k opt X . For the FAD •− / W •+ radical pairs, we have studied the relative orientations as found in Cl Cry4 and DmCry. The FADH • / Z •− models are applicable to both species. ET rates are given in GHz. Brackets indicate which radical is being scavenged by S • (cf Table S1). For comparison, see also Figures S2 -S5.

DmCry
Cl Subsequently, we developed a rough three-parameter approximation of the optimal k 13 , k opt 13 , intended to optimize the resulting Γ over all space for the reaction ( S • / FAD •− ) / W •+ in the DmCry system: Although the expression for k opt 13 given in eq. (S5) cannot be characterized as an exactly optimized function for all Rs, we found that it nevertheless provided a sense of asymptotically optimal behavior in the MFE, while delivering large MFEs at modest values of R.
When eq. (S5) was applied as the ET rate constant in the other scavenger-reaction systems listed in Table S2, it uniformly predicted large "optimal" values of the resulting MFEs in the asymptotic limit R → ∞, where substantial scavenger-mediated ET rates could not be considered plausible indicators of MFE-mediating scavenging rates in realistic biological systems. Ad hoc preliminary findings obtained from applying eq. (S5) in the other model systems indicated that unconstrained optimizations of the scavenging rate in the asymptotic distance limit would not lead to realistic MFE predictions-a hypothesis which was later borne out in the results of the studies we performed as a consequence using ET rates bounded by Marcus theory (viz. Figs. S3, S4, S5).
. Plots show the maximal Γ as a function of the inter-radical distance R obtained for DmCry using eq. (S5) (green), superimposed on the maximal Γ realizable for activationless ET for four tunneling media: covalently-bound (blue), typical protein (red), "soft" vacuum (yellow), and "hard" vacuum (purple), shown up close in each inset plot.  Brackets are used in the subfigure labels to indicate the radicals involved in the scavenging process. Note that the anisotropic MFE becomes negligible in the large scavenging-radius limit as ET with the scavenger approaches zero and the primary radical recombination has been neglected. . The legend specifies the identity of the scavenged radical, where brackets indicate the pair involved in the scavenging process. Note that the anisotropic MFE becomes negligible in the large scavenging-distance limit as the ET with the scavenger approaches zero and the primary radical recombination has been neglected.

Scavenging-mediated Absolute Anisotropies in Cl Cry4
Figure S10: Maximum absolute MFE (∆Φ f ) by scavenger distance R from the scavenged radical S • , for models of FAD •− /W •+ primary-pair types in Cl Cry4 based on activationless ET through four tunneling media: covalently-bound (blue), typical protein (red), "soft" vacuum (yellow), and "hard" vacuum (purple). Subfigures a and c show MFEs from simulations wherein W •+ was scavenged by S • , whereas subfigures b and d show results for FAD •− scavenged by S • . Tunneling decay parameters are indicated by colour, in figure. Brackets in the subfigure labels indicate which radical is involved in the scavenging process. Note that the anisotropic MFE becomes negligible in the large scavenging-radius limit as ET with the scavenger approaches zero, and that the primary radical recombination has been neglected.

Scavenging-mediated Absolute Anisotropies in DmCry
Figure S12: Maximum absolute MFE (∆Φ f ) by scavenger distance R from the scavenged radical S • , for models of FAD •− /W •+ primary-pair types in DmCry, based on activationless ET through four tunneling media: covalently-bound (blue), typical protein (red), "soft" vacuum (yellow), and "hard" vacuum (purple). Subfigures a and c show MFEs from simulations wherein W •+ was scavenged by S • , whereas subfigures b and d show results for FAD •− scavenged by S • . Tunneling decay parameters are indicated by colour, in figure. Brackets in the subfigure labels indicate which radical is involved in the scavenging process. Note that the anisotropic MFE becomes negligible in the large scavenging-radius limit as ET with the scavenger approaches zero, and that the primary radical recombination has been neglected. The maximum MFEs predicted using the FADH • / ( Z •− far / S •− ) reactions showed a monotonic dependence on the type of tunneling assumed, predicting smaller MFEs for weak couplings, and larger MFEs for stronger couplings. The hard vacuum-mediated model predicted optimal MFEs of just 5% for scavengers located 6 • A from the Z far radical, inside the protein cavity (but outside the protein surface) close to Trp536 in DmCry. The "soft" vacuum-mediated model predicted optimal MFEs of 14% for locations 7 • A away from the Z far radical, in the protein cavity (but outside the protein surface) in close proximity to residues Arg237 and Met266. The model of typical through-protein tunneling predicted optimal MFEs approaching 37% for locations within 10   For the RPM-model k −1 b = 3 µs while the corresponding rate was neglected in the triad model, as we have discussed in the main manuscript. The scavenger radical S • has been assumed at the location of maximal sensitivity at a distance R = 9.9

RF magnetic field effects
• A from the W •+ D for which Γ = 24.1 % is realized in the absence of RF fields. The RPM model only provides Γ = 0.13 % The RF perturbation was assumed to have amplitude 5 µT and be oriented parallel to the molecular x-axis. The MFE is here assessed in terms of 2(Y max − Y min )/(Y max + Y min ), where Y max,min is the yield evaluated for the static magnetic field direction of maximal/minimal reaction yield in the absence of the RF perturbation, i.e. in terms of the relative change of the yield at these two selected orientations of the geomagnetic field. To elicit a RF magnetic field sensitivity like that observed in behaviour experiments, a significantly longer lifetime or amplified RF field amplitude needed to be assumed for both the RPM-model and the model suggested here. Horizontal dashed lines indicate the MFE at zero RF frequency.