The arrangement of anisotropic spin couplings can optimize sensitivity of the cryptochrome radical pair to the direction of geomagnetic field

Abstract

Sensing of the geomagnetic field direction by many living organisms is commonly thought to involve radical pairs, such as those formed photochemically between the flavin and tryptophan radicals in the cryptochrome proteins. Previous theoretical studies have shown that strongly axial hyperfine couplings in the cryptochrome radicals greatly enhance the formation of a signaling state of the protein when the magnetic field is directed perpendicular to the hyperfine axis of either of the radicals. However, further analysis led to the conclusion that sharpness of detecting those magnetic directions is strongly suppressed by the inter-radical electron spin coupling. Here, we perform theoretical simulations of the compass function for a set of arrangements of the intra- and inter-radical spin couplings in the idealized cryptochrome radical pair, and find certain arrangements that preserve the sharpness in detecting the direction of the geomagnetic field. One particular arrangement, with the hyperfine axes of the radicals orthogonal to the symmetry axis of inter-radical coupling, provides even sharper field-direction sensitivity than that contributed solely by the anisotropy of the hyperfine coupling.

Figures (10)

• Figure 1: The arrangements of the FAD$^{\bullet{-}}$ and Trp$^{\bullet{+}}$ radicals and the spin couplings in the considered models of the cryptochrome radical pair. The surfaces display the shapes of the couplings (see the text). (1) RP1: the radical pair with the single HFC at each radical, N5 for FAD$^{\bullet{-}}$ and N1 for Trp$^{\bullet{+}}$. (2) RP2: the same structure and HFC interaction as in RP1, but with inclusion of the EED spin coupling. (3) RP3: the same structure as in RP1 and RP2, but with the electron exchange spin coupling added to the couplings of RP2. (4) RP4: the arrangement of the radicals and couplings derived from these arrangements and couplings in RP3 such that the HFC axis of each radical becomes orthogonal to the EED axis.
• Figure 2: The response of RP1 to the direction of the geomagnetic field. The symbols $\CIRCLE$ and $\blacktriangle$ indicate the spikes at the field directions orthogonal to the HFC axes FAD N5 and Trp N1, respectively. (a) The direction dependence of the interconversion triplet yield as a 3D polar graph. The direction of the magnetic field is defined in the coordinate frame shown in the plot. For the magnetic field pointing from the origin to the 3D surface, the distance from the origin to the surface gives the yield value according to the color scale. The vertical axis of the coordinate frame (the $z$-axis) is orthogonal to the HFC axes of the radicals, cf. Figure 1(1). (b) The dependence of the triplet yield (solid lines) and the yield of the signaling state (dashed lines) on the magnetic field direction as 2D graphs. Left plot: the yields varying with the azimuthal angle $\varphi$ for the selected values (indicated near the curves) of the polar angle $\vartheta$, with $\varphi$ and $\vartheta$ defining the magnetic field direction in the coordinate frame of the polar graph in Figure 2(a). Right plot: the yields varying with $\vartheta$ for the $\varphi$ values indicated near the curves.
• Figure 3: The energies and singlet weights of the spin states responsible for the sharp magnetic anisotropy of the RP1 model. (a) The dependence of the spin-state energies of the model radical pair on the polar angle $\vartheta$ at the azimuthal angle $\varphi=71.26^\circ$. The solid lines show the bunches of the energy levels responsible for the spikes in the yield profiles at $\vartheta=0$ and $\vartheta=180^\circ$, and the dashed lines show the other levels of in a total of 36 spin states. The numbers enumerate the levels in order of increasing energy. (b) The avoided crossing of the levels 1 and 2 at the positions of spikes. (c) The singlet weights of the levels 1 and 2. Note sharp variation of the weights at the spike positions.
• Figure 4: The field-direction response of RP2. (a) The 3D polar graph for the interconversion triplet yield. (b) The triplet and signaling yields varying with one of the angles, $\vartheta$ or $\varphi$, at the selected values (indicated near the curves) of the other angle.
• Figure 5: The spin-state properties of the RP2 model. (a) The spin-energy levels varying with the angle $\theta$ between the magnetic field and EED axis. The solid lines show the levels 19 and 22, and the dashed lines show the remaining 34 out of 36 levels. (b) The RP2 levels 19 and 22 (solid lines) compared to the singlet S and triplet T$_0$ levels of the electron spin pair (dashed lines). The notation T$_0$ reflects the definition of the triplet states with respect to the EED axis (see text). (c) The singlet weights of the spin states 19 and 22.