Sensitivity and entanglement in the avian chemical compass

The radical pair mechanism can help to explain avian orientation and navigation. Some evidence indicates that the intensity of external magnetic fields plays an important role in avian navigation. In this paper, using a two-stage model, we demonstrate that birds could reasonably detect the directions of geomagnetic fields and gradients of these fields using a yield-based chemical compass that is sensitive enough for navigation. Also, we find that the lifetime of entanglement in this proposed compass is angle dependent and long enough to allow adequate electron transfer between molecules.

Figure 1

The reaction scheme of the radical pair mechanism in cryptochrome. $k_b={\tau }_b$ and $k_f={\tau }_f$ are the first-order rate constants for recombination of the initial radical pair and formation of the secondary pair from the initial one, respectively. $k_s={\tau }_s$ is the rate constant for the decay of the secondary pair. The green two-headed arrows indicate the interconversion of the singlet and triplet states of the radical pairs.

Figure 2

Left: A coupled radical pair with neighboring effective nuclear spins (green arrows, $\to )$ in an external magnetic field $\vec{B}$ (blue arrows, $\to )$. Right: The direction of $\vec{B}$ depicted in the molecular coordinate frame, where the $z$ axis is the $z$ axis of the radical pair [23].

Figure 3

Angle dependence of the magnetic field effect as a function of the external field. The MFE is the ratio of the difference between the triplet yields of the secondary pair at nonzero external fields and zero field $\left(\Delta {\Phi }_T={\Phi }_T-{\Phi }_{0T}\right)$ and the triplet yields at zero field, which is $\Delta {\Phi }_T/{\Phi }_{0T}$. This graph depicts the MFE as a function of external fields $\vec{B}$ for five different polar angles $\theta$ between the electron spin and the magnetic field. $\theta =0^{\circ }\left(\text{black,}\text{–}\square \text{–}\right),{30}^{\circ }$ (red,) $,{60}^{\circ }$ (blue,) $,{85}^{\circ }$ (pink,) $$, and ${90}^{\circ }$ (green,) $$. The decay rates are shown in Fig. 1.

Figure 4

The magnetic sensitivity of the chemical compass as a function of angle. The sensitivity is defined as $\partial {\Phi }_T/\partial B$ in ${\mathrm{T}}^{-1}$. There is a rapid increase in this sensitivity between ${80}^{\circ }$ and ${90}^{\circ }$.The sensitivity under geomagnetic field is of the order of ${10}^{-3}$, requiring a strong magnification mechanism in birds to utilize it.

Figure 5

Intensity dependence of the magnetic sensitivity. This graph shows the sensitivity as a function of external magnetic field $\vec{B}$ as a function of polar angle $\theta$ between the $z$ axis of the radical pair and the magnetic field, i.e., $\theta =0^{\circ }\left(\text{black,}\text{–}\square \text{–}\right),{30}^{\circ }$ (red,) $,{60}^{\circ }$ (blue,) $,{85}^{\circ }$ (pink,) $$, and ${90}^{\circ }$ (green,) $$. The interior graph magnifies the data for angles $\theta =0^{\circ }\left(\text{black,}\text{–}\square \text{–}\right),{30}^{\circ }$ (red,) $$, and ${60}^{\circ }$ (blue,) $$.

Figure 6

Entanglement of the initial radical pair $\left[\mathrm{FAD}{•}^{-}\mathrm{TrpH}{•}^{+}\right]$ as a function of external fields for four polar angles, $\theta =0^{\circ }\left(\text{black}\right),{30}^{\circ }\left(\text{red}\right),{60}^{\circ }\left(\text{blue}\right),\text{and}{90}^{\circ }\left(\text{green}\right)$. Since the entanglement of he initial pair $\left[\mathrm{FAD}{•}^{-}\mathrm{TrpH}{•}^{+}\right]$ is compressed within $0.1\mu \mathrm{s}$, the time scale of the graph for that is from 0 to 0.1 $\mu \mathrm{s}$. The other graphs range from 0 to 0.8 $\mu \mathrm{s}$, and the entanglement of the initial pair at $0^{\circ },{30}^{\circ },\text{and}{60}^{\circ }$ differ after 0.3 $\mu \mathrm{s}$.

Figure 7

The magnetic sensitivity as a function of angles under different hyperfine coupling tensors. The black curve (–$\square$–) describes the situation under the hyperfine tensor mentioned in the previous context, while the red one (–$\circ$–) shows the situation under a different set of hyperfine tensors, diag$\left\{{\widehat{A}}_{11}\right\}=\{-0.650\mathrm{G}$, −0.566 G, 17.071 $\mathrm{G}\}$, diag$\left\{{\widehat{A}}_{12}\right\}=\{5.71\mathrm{G}$, 5.81 G, 27.07 $\mathrm{G}\}$, diag$\left\{{\widehat{A}}_{21}\right\}=\{$−0.792 G, −0.692 G, 14.060 $\mathrm{G}\}$, diag$\left\{{\widehat{A}}_{22}\right\}=\{$−1.32 G,−1.15 G, 27.64 $\mathrm{G}\}$.