Functional window of the avian compass

The functional window is an experimentally observed property of the avian compass that refers to its selectivity around the geomagnetic-field strength. We show that the simple radical-pair model, using biologically feasible hyperfine parameters, can qualitatively explain the salient features of the avian compass as observed in behavioral experiments: its functional window, as well as disruption of the compass action by radio-frequency fields of specific frequencies. Further, we show that adjustment of the hyperfine parameters can tune the functional window, suggesting a possible mechanism for its observed adaptation to field variation. While these lend support to the radical-pair model, we find that in its simplest form—or even with minor augmentations—it cannot quantitatively explain the observed width of the functional window. This suggests deeper generalization of the model, possibly in terms of more nuclei or more subtle environmental interaction than has been considered hitherto. Finally, we examine a possible biological purpose for the functional window; even assuming evolutionary benefit from radical-pair magnetoreception, it seems likely that the functional window could be just a corollary thereof, imparting no additional advantage.

Figure 1

(a) Compass sensitivity as a function of the Zeeman field for ($a_x,a_y,a_z,k_s,k_t)=(0.345\mathrm{G},0.345\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1},2\times {10}^4{\mathrm{s}}^{-1}$). The plot clearly reveals a “functional window” centered around 47 $\mu \mathrm{T}$. Inset: The functional window, for the linear scale of the magnetic field. (b) Singlet yield as a function of the geomagnetic field inclination for ($a_x,a_y,a_z,k_s,k_t)=(0.345\mathrm{G},0.345\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1},2\times {10}^4{\mathrm{s}}^{-1}$) with (red circles) and without (black squares) a 150-nT RF field of 1.315-MHz frequency. The figure distinctly shows the degradation of compass sensitivity in the presence of the RF field, a defining characteristic of the avian compass. (c) Variation of the center of the functional window (denoted $H_C$) as a function of the axial hyperfine strength ($a_z$)—indicative of functional window adaptation.

Figure 2

The general trend of the functional window with hyperfine and recombination rate parameters is shown here. The functional window is plotted for the parameter values of ($a_x,a_y,a_z,k_s,k_t)=(0.345\mathrm{G},0.345\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1},2\times {10}^4{\mathrm{s}}^{-1}$), ($0.690\mathrm{G},0.690\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1},2\times {10}^4{\mathrm{s}}^{-1}$), and ($0.690\mathrm{G},0.690\mathrm{G},9\mathrm{G},{10}^5{\mathrm{s}}^{-1},{10}^5{\mathrm{s}}^{-1}$). $a_z$ is kept constant in these plots. The plots show that upon increasing (decreasing) the values of $a_x$ and $a_y$, the height of the functional window decreases (increases). An increment in the recombination rate shifts the lower portion of the functional window to higher values of the magnetic field.

Figure 3

(a) Change in sensitivity of the compass as a function of the RP recombination rate. (b) Sensitivity as a function of the Zeeman-field strength when singlet and triplet recombination rates are different. We take $k_s>k_t$, which is implied by RP theory [29]. The curve is plotted for ($k_s,k_t)=(2\times {10}^4{\mathrm{s}}^{-1},{10}^4{\mathrm{s}}^{-1}$), ($3\times {10}^4{\mathrm{s}}^{-1}, 2\times {10}^4{\mathrm{s}}^{-1}$), ($4\times {10}^4{\mathrm{s}}^{-1}, 2\times {10}^4{\mathrm{s}}^{-1}$), and ($4\times {10}^4{\mathrm{s}}^{-1}, 3\times {10}^4{\mathrm{s}}^{-1}$).

Figure 4

Compass sensitivity as a function of the Zeeman-field strength in the presence of environmental noise [30]. The sensitivity is analyzed for $(a_x,a_y,a_z,k_s,k_t)=(0.345\mathrm{G},0.345\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1},2\times {10}^4{\mathrm{s}}^{-1})$ along with environmental noise rates ($\mathrm{\Gamma }$) of ${10}^4, 2\times {10}^4, {10}^5$, and ${10}^6{\mathrm{s}}^{-1}$. As is clear, the functional window property vanishes when the noise rate is higher than the RP recombination rate.

Figure 5

Compass sensitivity as a function of the Zeeman field for various hyperfine and recombination rate parameters. The sensitivity is plotted for ($a_x,a_y,a_z,k$(=$k_s,k_t\left)\right)=(0.345\mathrm{G},0.345\mathrm{G},1\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1}$), ($0\mathrm{G},0\mathrm{G},1\mathrm{G},5\times {10}^4{\mathrm{s}}^{-1}$), ($9\mathrm{G},9\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1}$), and ($18.80\mathrm{G},37.60\mathrm{G},0\mathrm{G},5\times {10}^5{\mathrm{s}}^{-1}$). The plot intends to show the sensitivity behavior for parameters other than the ones mentioned in the text.

Figure 6

Compass sensitivity as a function of the Zeeman field for ($a_x,a_y,a_z)=(0.345\mathrm{G},0.345\mathrm{G},9\mathrm{G}$) and recombination rates of $k=({10}^5{\mathrm{s}}^{-1},{10}^6{\mathrm{s}}^{-1},{10}^7{\mathrm{s}}^{-1},{10}^8{\mathrm{s}}^{-1}$). The plot shows the change in the functional window property of the avian compass when the recombination rate is increased. We observe that the functional window property starts to become distorted when the recombination rate is increased beyond ${10}^6{\mathrm{s}}^{-1}$. The functional window becomes nonexistent for $k={10}^7{\mathrm{s}}^{-1}$ and $k={10}^8{\mathrm{s}}^{-1}$.

Figure 7

Compass sensitivity as a function of the Zeeman-field strength in the presence of environmental noise [14]. The sensitivity is plotted for $(a_x,a_y,a_z,k_s,k_t)=(0.345\mathrm{G},0.345\mathrm{G},9\mathrm{G},2\times {10}^4{\mathrm{s}}^{-1},2\times {10}^4{\mathrm{s}}^{-1})$ along with environmental noise rates ($\mathrm{\Gamma }$) of ${10}^4, 2\times {10}^4, {10}^5$, and ${10}^6{\mathrm{s}}^{-1}$. The sensitivity in this case is much lower than that is for the noise model used in the text (cf. Fig. 4).