Chiral-induced spin selectivity in the formation and recombination of radical pairs: cryptochrome magnetoreception and EPR detection

We start by briefly reviewing the conventional description of the spin dynamics and spin-selective reactivity of radical pairs before outlining the changes required to incorporate CISS effects. Consider the formation of a radical pair by the transfer of an electron from a closed shell donor to a photoexcited singlet acceptor:

$$\begin{equation}\mathrm{D}+{}^{1}\mathrm{A}^{{\ast}}\to \left[{\mathrm{D}}^{{\bullet}+}{\mathrm{A}}^{{\bullet}-}\right].\end{equation} \tag{ 1 }$$

For organic molecules, in which the spin–orbit coupling is small, such reactions generally conserve spin angular momentum and produce the radical pair predominantly in an electronic singlet state:

$$\begin{equation}\left\vert \mathrm{S}\right\rangle =\frac{1}{\sqrt{2}}\left(\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle -\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle \right),\end{equation} \tag{ 2 }$$

in which $\left\vert \alpha \right\rangle$ and $\left\vert \beta \right\rangle$ are the $m={\pm}\frac{1}{2}$ electron spin states. Various subsequent reactions are possible, including spin-selective back electron transfer to regenerate D and A in their (singlet) ground states,

$$\begin{equation}{}^{1}\left[{\mathrm{D}}^{{\bullet}+}\enspace {\mathrm{A}}^{{\bullet}-}\right]\enspace \enspace \enspace {k}_{\mathrm{S}}\enspace {\to }\enspace \enspace \mathrm{D}+\mathrm{A},\end{equation} \tag{ 3 }$$

and/or formation of the triplet state of one of the reactants, e.g.

$$\begin{equation}{}^{3}\left[{\mathrm{D}}^{{\bullet}+}\enspace {\mathrm{A}}^{{\bullet}-}\right]\enspace \enspace \enspace {k}_{\mathrm{T}}\enspace {\to }\enspace \enspace {}^{3}\mathrm{D}+\mathrm{A}.\end{equation} \tag{ 4 }$$

Reactions (3) and (4) are both spin-conserving. Additionally there may be reactions that involve just one of the radicals (e.g. a change in protonation state), are therefore not subject to spin-selection rules and occur at equal rates for singlet and triplet pairs, e.g.

$$\begin{equation}{}^{1,3}\left[{\mathrm{D}}^{{\bullet}+}{\mathrm{A}}^{{\bullet}-}\right]\enspace {k}_{\mathrm{F}}\enspace {\to }\enspace {\mathrm{D}}^{{\bullet}+}+\mathrm{A}{\mathrm{H}}^{{\bullet}}.\end{equation} \tag{ 5 }$$

In equations (3)–(5), k_S, k_T, and k_F are first order rate constants.

In almost all treatments of radical pair spin dynamics over the last 40 years, such reactions have generally (and very successfully) been modelled by means of 'Haberkorn' operators [54] in a master equation of the form,

$$\begin{equation}\frac{\mathrm{d}\hat{\rho }\left(t\right)}{\mathrm{d}t}=-i{\left[\hat{H},\hat{\rho }\left(t\right)\right]}_{-}-\frac{1}{2}{k}_{\mathrm{S}}{\left[{\hat{P}}^{\mathrm{S}},\hat{\rho }\left(t\right)\right]}_{+}-\frac{1}{2}{k}_{\mathrm{T}}{\left[{\hat{P}}^{\mathrm{T}},\hat{\rho }\left(t\right)\right]}_{+}-{k}_{\mathrm{F}}\hat{\rho }\left(t\right),\end{equation} \tag{ 6 }$$

where $\hat{\rho }\left(t\right)$ and $\hat{H}$ are the spin density operator and spin Hamiltonian of the radical pair comprised of the two unpaired electrons and all hyperfine-coupled nuclei. ${\left[B,C\right]}_{{\pm}}=BC{\pm}CB$, and ${\hat{P}}^{\mathrm{S}}$ and ${\hat{P}}^{\mathrm{T}}$ are projection operators onto the singlet and triplet electronic subspaces. For a radical pair formed in a singlet state with no nuclear spin polarization, $\hat{\rho }\left(0\right)={\hat{P}}^{\mathrm{S}}/N$, where N is the total number of nuclear spin states. At least one of k_S and k_T must be non-zero for the yields of the reaction products to be affected by a weak applied magnetic field. The anti-commutator terms in equation (6), originally proposed phenomenologically [54–56], have recently been derived explicitly for electron transfer reactions [57, 58].

As usual in simulations of the anisotropic effects of weak magnetic fields on radical pairs [59–62], $\hat{H}$ has the general form:

$$\begin{equation}\hat{H}=\boldsymbol{\omega }\cdot \left({\hat{\mathbf{S}}}_{\mathrm{D}}+{\hat{\mathbf{S}}}_{\mathrm{A}}\right)+\sum\limits _{i\in \left\{\mathrm{D},\mathrm{A}\right\}}\sum\limits _{k}{\hat{\mathbf{S}}}_{i}\cdot {\mathbf{A}}_{ik}\cdot {\hat{\mathbf{I}}}_{ik}+{\hat{\mathbf{S}}}_{\mathrm{D}}\cdot \mathbf{D}\cdot {\hat{\mathbf{S}}}_{\mathrm{A}}-J\left(2{\hat{\mathbf{S}}}_{\mathrm{D}}\cdot {\hat{\mathbf{S}}}_{\mathrm{A}}+\frac{1}{2}\right).\end{equation} \tag{ 7 }$$

${\hat{\mathbf{S}}}_{\mathrm{D}}$ and ${\hat{\mathbf{S}}}_{\mathrm{A}}$ are the electron spin operators of the two radicals and ${\hat{\mathbf{I}}}_{ik}$ is the spin operator of nucleus k in radical i. $\boldsymbol{\omega }=-{\gamma }_{\mathrm{e}}B\left(\mathrm{sin}\enspace \theta \enspace \mathrm{cos}\enspace \phi ,\enspace \mathrm{sin}\enspace \theta \enspace \mathrm{sin}\enspace \phi ,\enspace \mathrm{cos}\enspace \theta \right)$ is the external magnetic field vector (in angular frequency units) with magnitude $\left\vert {\gamma }_{\mathrm{e}}\right\vert B$ and direction specified by angles θ and ϕ. A_ik is the hyperfine tensor that couples electron i with nucleus k. D and J are, respectively, the dipolar coupling tensor and the exchange interaction of the two electron spins. D has eigenvalues 4D/3, −2D/3, −2D/3, where D varies as the inverse cube of the separation of the radicals.

How should equation (6) be modified to account for CISS-type spin polarization effects when the radical pair is formed by electron transfer in a chiral medium such as a protein? We start with the limiting case of 100% CISS polarization such that only an electron with spin β can move from D to ¹A^* in reaction (1), leaving behind an electron with spin α in D^•+ [26, 51]. The result is a radical pair initially in the state $\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle$ with the electron spins quantized along the direction of the electron transfer, i.e. the vector connecting the centres of D and A (figure 1). The following can easily be adapted for the initial state $\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle$, as would be the case for the opposite chirality of the intervening medium.

Zoom In Zoom Out Reset image size

In the general case, where the polarization is less than 100%, we propose that the initial state of the radical pair can be written ${\hat{P}}^{\mathrm{I}}=\left\vert {\psi }_{\mathrm{I}}\right\rangle \left\langle {\psi }_{\mathrm{I}}\right\vert$ where

$$\begin{align}\hfill \left\vert {\psi }_{\mathrm{I}}\right\rangle & =\mathrm{cos}\left(\frac{1}{2}\chi \right)\left\vert \mathrm{S}\right\rangle +\mathrm{sin}\left(\frac{1}{2}\chi \right)\left\vert {\mathrm{T}}_{0}\right\rangle \hfill \\ \hfill & =\frac{1}{\sqrt{2}}\left[\mathrm{sin}\left(\frac{1}{2}\chi \right)+\mathrm{cos}\left(\frac{1}{2}\chi \right)\right]\enspace \enspace \left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle +\frac{1}{\sqrt{2}}\left[\mathrm{sin}\left(\frac{1}{2}\chi \right)-\mathrm{cos}\left(\frac{1}{2}\chi \right)\right]\enspace \left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle ,\hfill \end{align} \tag{ 8 }$$

with $\left\vert {\mathrm{T}}_{0}\right\rangle$ the m = 0 triplet state,

$$\begin{equation}\left\vert {\mathrm{T}}_{0}\right\rangle =\frac{1}{\sqrt{2}}\left(\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle +\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle \right).\end{equation} \tag{ 9 }$$

Hence

$$\begin{equation}\begin{aligned}\hfill {\hat{P}}^{\mathrm{I}}& =\frac{1}{2}\left(1+\mathrm{sin}\left(\chi \right)\right)\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle \left\langle {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\vert +\frac{1}{2}\left(1-\mathrm{sin}\left(\chi \right)\right)\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle \left\langle {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\vert \hfill \\ \hfill & \quad -\frac{1}{2}\enspace \mathrm{cos}\left(\chi \right)\left(\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle \left\langle {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\vert +\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle \left\langle {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\vert \right)\hfill \\ \hfill & =\frac{1}{4}\hat{E}-{\hat{S}}_{\mathrm{D}Z}{\hat{S}}_{\mathrm{A}Z}-\left({\hat{S}}_{\mathrm{D}X}{\hat{S}}_{\mathrm{A}X}+{\hat{S}}_{\mathrm{D}Y}{\hat{S}}_{\mathrm{A}Y}\right)\mathrm{cos}\left(\chi \right)+\frac{1}{2}\left({\hat{S}}_{\mathrm{D}Z}-{\hat{S}}_{\mathrm{A}Z}\right)\mathrm{sin}\left(\chi \right),\hfill \end{aligned}\end{equation} \tag{ 10 }$$

in which ${\hat{S}}_{\mathrm{D}Q}$ and ${\hat{S}}_{\mathrm{A}Q}$ $\left(Q\in X,Y,Z\right)$ are components of the electron spin angular momentum operators of the two radicals. The Z-axis here is the CISS quantization axis. In equations (8) and (10), the angle χ determines the extent of the CISS polarization of the initial state, the limiting cases being χ = 0, ${\hat{P}}^{\mathrm{I}}=\left\vert \mathrm{S}\right\rangle \left\langle \mathrm{S}\right\vert$ (pure singlet, no CISS) and χ = π/2, ${\hat{P}}^{\mathrm{I}}=\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle \left\langle {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\vert$ (100% CISS, no RPM). The four operator terms in the final line of equation (10) are as follows: the identity operator, two-spin order, zero quantum coherence (ZQC), and single-spin polarizations, all defined with respect to the CISS Z-axis. It is clear that a CISS contribution to the initial state modifies the amplitude of the ZQC and introduces equal and opposite spin polarization in the two radicals. It also causes the initial state to be anisotropic.

The CISS effect couples the linear momentum of an electron to its spin angular momentum such that electron transport is more efficient when they are either parallel or antiparallel, depending on the chirality of the medium [17]. This correlation implies that if only an electron with spin α can move from D to ¹A^* then only an electron with spin β can move in the opposite direction when the electron returns from A^•− to D^•+ (figure 1). We therefore propose that the corresponding projection operator for back electron transfer should be ${\hat{P}}^{\mathrm{R}}=\left\vert {\psi }_{\mathrm{R}}\right\rangle \left\langle {\psi }_{\mathrm{R}}\right\vert$ where

$$\begin{equation}\begin{aligned}\hfill \left\vert {\psi }_{\mathrm{R}}\right\rangle & =\mathrm{cos}\left(\frac{1}{2}\chi \right)\left\vert \mathrm{S}\right\rangle -\mathrm{sin}\left(\frac{1}{2}\chi \right)\left\vert {\mathrm{T}}_{0}\right\rangle \hfill \\ \hfill & =-\frac{1}{\sqrt{2}}\left[\mathrm{sin}\left(\frac{1}{2}\chi \right)-\mathrm{cos}\left(\frac{1}{2}\chi \right)\right]\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle -\frac{1}{\sqrt{2}}\left[\mathrm{sin}\left(\frac{1}{2}\chi \right)+\mathrm{cos}\left(\frac{1}{2}\chi \right)\right]\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle \hfill \end{aligned}\end{equation} \tag{ 11 }$$

and

$$\begin{equation}{\hat{P}}^{\mathrm{R}}=\frac{1}{4}\hat{E}-{\hat{S}}_{\mathrm{D}Z}{\hat{S}}_{\mathrm{A}Z}-\left({\hat{S}}_{\mathrm{D}X}{\hat{S}}_{\mathrm{A}X}+{\hat{S}}_{\mathrm{D}Y}{\hat{S}}_{\mathrm{A}Y}\right)\mathrm{cos}\left(\chi \right)-\frac{1}{2}\left({\hat{S}}_{\mathrm{D}Z}-{\hat{S}}_{\mathrm{A}Z}\right)\mathrm{sin}\left(\chi \right).\end{equation} \tag{ 12 }$$

The only difference between ${\hat{P}}^{\mathrm{I}}$ and ${\hat{P}}^{\mathrm{R}}$ is the sign of the spin-polarization term, $\left({\hat{S}}_{\mathrm{D}Z}-{\hat{S}}_{\mathrm{A}Z}\right)$. χ plays the same role here as in equations (8) and (10), with limiting cases χ = 0, ${\hat{P}}^{\mathrm{R}}=\left\vert \mathrm{S}\right\rangle \left\langle \mathrm{S}\right\vert$ (pure singlet, no CISS) and χ = π/2, ${\hat{P}}^{\mathrm{R}}=\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle \left\langle {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\vert$ (100% CISS, no RPM). If $\left\vert {\beta }_{\mathrm{D}}{\alpha }_{\mathrm{A}}\right\rangle$ is favoured over $\left\vert {\alpha }_{\mathrm{D}}{\beta }_{\mathrm{A}}\right\rangle$ in the forward electron transfer, and vice-versa in the recombination step, χ will be in the range $\left[0,\enspace -\pi /2\right]$.

Assuming the rate of the triplet reaction in equation (4) is negligible (as is the case in cryptochromes [3, 63]), the master equation for the density operator with CISS effects included becomes

$$\begin{equation}\frac{\mathrm{d}\hat{\rho }\left(t\right)}{\mathrm{d}t}=-i{\left[\hat{H},\hat{\rho }\left(t\right)\right]}_{-}-\frac{1}{2}{k}_{\mathrm{R}}{\left[{\hat{P}}^{\mathrm{R}},\hat{\rho }\left(t\right)\right]}_{+}-{k}_{\mathrm{F}}\hat{\rho }\left(t\right),\end{equation} \tag{ 13 }$$

where k_R is the rate constant for back electron transfer, [D^•+ A^•−] → D + A, and the initial condition is $\hat{\rho }\left(0\right)={\hat{P}}^{\mathrm{I}}/N$. We propose equations (10) and (12) here in the same spirit as the Haberkorn approach [54] in an attempt to merge Carmeli's observations of electron spin polarization in photosynthetic reaction centres (outlined above) [39] with the conventional RPM. Other formulations are possible. For example, the singlet and triplet parts of $\left\vert {\psi }_{\mathrm{I}}\right\rangle$ and $\left\vert {\psi }_{\mathrm{R}}\right\rangle$ could have different phases, and ${\hat{P}}^{\mathrm{I}}$ and ${\hat{P}}^{\mathrm{R}}$ could be mixed states instead of pure states [41]. An advantage of the forms we have chosen is that they involve the minimum number of parameters.

The new equation of motion, equation (13), and the associated initial condition differ from the conventional forms (equation (6) and $\hat{\rho }\left(0\right)={\hat{P}}^{\mathrm{S}}/N$) in that they break time-reversal symmetry. This can be seen to be a consequence of the antisymmetry of angular momentum operators under the time-reversal operator, $\hat{T}$, for example $\hat{T}\enspace {\hat{S}}_{\mathrm{D}z}\enspace {\hat{T}}^{-1}=-{\hat{S}}_{\mathrm{D}z}$ [64, 65]. The term for the Zeeman interaction in the spin Hamiltonian, equation (7), $\boldsymbol{\omega }\cdot \left({\hat{\mathbf{S}}}_{\mathrm{D}}+{\hat{\mathbf{S}}}_{\mathrm{A}}\right)$, is therefore antisymmetric. All other terms in $\hat{H}$ (hyperfine, dipolar, and exchange interactions) and the projection operators ${\hat{P}}^{\mathrm{S}}$ and ${\hat{P}}^{\mathrm{T}}$ in equation (6) contain only bi-linear products of spin operators and are therefore symmetric under time-reversal. For example:

$$\begin{equation}\hat{T}\enspace {\hat{P}}^{\mathrm{S}}\enspace {\hat{T}}^{-1}=\hat{T}\left[\frac{1}{4}\hat{E}\enspace \enspace -\sum\limits _{j=x,y,z}{\hat{S}}_{\mathrm{D}j}{\hat{S}}_{\mathrm{A}j}\right]{\hat{T}}^{-1}=\frac{1}{4}\hat{E}-\sum\limits _{j=x,y,z}\left(\hat{T}\enspace {\hat{S}}_{\mathrm{D}j}{\hat{T}}^{-1}\right)\left(\hat{T}\enspace {\hat{S}}_{\mathrm{A}j}{\hat{T}}^{-1}\right)={\hat{P}}^{\mathrm{S}}.\end{equation} \tag{ 14 }$$

These properties result in the reaction yields being even functions of the magnetic field, ω . By contrast, neither of the CISS spin projection operators, ${\hat{P}}^{\mathrm{I}}$ and ${\hat{P}}^{\mathrm{R}}$ in equations (10) and (12), has time-reversal symmetry because of the $\left({\hat{S}}_{\mathrm{D}z}-{\hat{S}}_{\mathrm{A}z}\right)$ terms, and are in fact interconverted by $\hat{T}$:

$$\begin{equation}\hat{T}\enspace {\hat{P}}^{\mathrm{I}}\enspace {\hat{T}}^{-1}={\hat{P}}^{\mathrm{R}}\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad \hat{T}\enspace {\hat{P}}^{\mathrm{R}}\enspace {\hat{T}}^{-1}={\hat{P}}^{\mathrm{I}}.\end{equation} \tag{ 15 }$$

In passing, we note an apparent parallel between the CISS effects discussed here and the ALTADENA experiment [66–68] (see appendix).

We use the reaction scheme in figure 2 to assess the effects of including CISS in the radical pair model of magnetoreception in migratory birds. In cryptochrome, the donor and acceptor radicals are TrpH^•+ and FAD^•−, respectively, formed by intra-protein electron transfer from a tryptophan residue (TrpH) to the photo-excited singlet state of the non-covalently bound chromophore, flavin adenine dinucleotide (FAD) [63]. This reaction is spin-selective ($\hat{\rho }\left(0\right)={\hat{P}}^{\mathrm{I}}/N$) and is assumed to be much faster than subsequent processes. The radical pair undergoes coherent spin dynamics as a result of the Zeeman, hyperfine, dipolar and exchange interactions and either reverts to the ground state with rate constant k_R and spin-selectivity given by ${\hat{P}}^{\mathrm{R}}$, or reacts non-spin-selectively with rate constant k_F to give a signalling state of the protein. The latter is thought to occur via deprotonation and subsequent reduction of the TrpH^•+ radical, and protonation of the FAD^•− radical, to give a state of the cryptochrome with altered binding affinity to a signalling partner [69]. The ultimate yield of the signalling state once all radical pairs have reacted is

$$\begin{equation}{{\Phi}}_{\mathrm{F}}\enspace \enspace =\enspace \enspace {k}_{\mathrm{F}}\underset{0}{\overset{\infty }{\int }}\mathrm{Tr}\left[\hat{\rho }\left(t\right)\right]\enspace \mathrm{d}t,\end{equation} \tag{ 16 }$$

in which $\hat{\rho }\left(t\right)$ is obtained from equation (13) and Tr is the trace over the electron and nuclear spin spaces. Φ_F depends on the direction of the external magnetic field through the anisotropy of the hyperfine and dipolar interactions (and CISS when χ ≠ 0) and is the source of the directional information from which the bird is assumed to derive a compass bearing. In what follows, we present this information as Φ_R = 1 − Φ_F, the yield of the recombination reaction. Φ_R was calculated for a range of magnetic field directions $\left(\theta ,\enspace \phi \right)$ (equation (7)) to obtain the following measure of the anisotropy of the reaction yield:

$$\begin{equation}{\Delta}{{\Phi}}_{\mathrm{R}}={\Delta}{{\Phi}}_{\mathrm{F}}={\mathrm{max}}_{\theta ,\phi }\left({{\Phi}}_{\mathrm{R}}\right)-{\mathrm{min}}_{\theta ,\phi }\left({{\Phi}}_{\mathrm{R}}\right).\end{equation} \tag{ 17 }$$

Zoom In Zoom Out Reset image size

Before presenting calculations of magnetic field effects, we discuss how CISS could manifest in the EPR spectra of radical pairs. We consider the simple case of a weakly coupled pair of electrons, with different g-values, an isotropic exchange interaction, and no dipolar or hyperfine interactions. At thermal equilibrium, its EPR spectrum would comprise four equally intense lines arranged as two well-separated doublets (figure 3(A)). If the same radical pair were created by electron transfer, with the magnetic field of the EPR spectrometer (z) aligned parallel to the CISS quantization axis (Z), the initial spin state, $\hat{\rho }\left(0\right)={P}^{\mathrm{I}}/N$, would be given by equation (10) with X, Y, Z replaced by x, y, z, respectively:

$$\begin{equation}{\hat{P}}^{\mathrm{I}}=\frac{1}{4}\hat{E}-{\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}-\left({\hat{S}}_{\mathrm{D}x}{\hat{S}}_{\mathrm{A}x}+{\hat{S}}_{\mathrm{D}y}{\hat{S}}_{\mathrm{A}y}\right)\mathrm{cos}\left(\chi \right)+\frac{1}{2}\left({\hat{S}}_{\mathrm{D}z}-{\hat{S}}_{\mathrm{A}z}\right)\mathrm{sin}\left(\chi \right).\end{equation} \tag{ 18 }$$

Zoom In Zoom Out Reset image size

When there is no CISS contribution (χ = 0), $\hat{\rho }\left(0\right)$ comprises correlated two-spin polarization, ${\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}$, and ZQC, $\left({\hat{S}}_{\mathrm{D}x}{\hat{S}}_{\mathrm{A}x}+{\hat{S}}_{\mathrm{D}y}{\hat{S}}_{\mathrm{A}y}\right)$ [70, 71]. In a continuous wave, time-resolved EPR experiment, in which the magnetic field is swept in the presence of a constant microwave field, the oscillatory ZQ term would average to zero, leaving only the two-spin order [72]. The latter gives rise to the antiphase doublets that are characteristic of spin-correlated radical pairs (figure 3(B)) [72–76]. In the same experiment, the final term in equation (18), which would most clearly reveal a CISS effect, corresponds to the spectrum in figure 3(C) in which the doublet signals of the two radicals are equally and oppositely polarized. In general, when 0 < χ < π/2, the EPR spectrum (figure 3(D)) would be a linear combination of figures 3(B) and (C) with the inner pair of lines in the four-line spectrum stronger or weaker than the outer pair.

Now contrast this with the situation in which the spectrometer field is parallel to the molecular X-axis. The initial state is now given by:

$$\begin{equation}{\hat{P}}^{\mathrm{I}}=\frac{1}{4}\hat{E}-{\hat{S}}_{\mathrm{D}x}{\hat{S}}_{\mathrm{A}x}-\left({\hat{S}}_{\mathrm{D}y}{\hat{S}}_{\mathrm{A}y}+{\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}\right)\mathrm{cos}\left(\chi \right)+\frac{1}{2}\left({\hat{S}}_{\mathrm{D}x}-{\hat{S}}_{\mathrm{A}x}\right)\mathrm{sin}\left(\chi \right).\end{equation} \tag{ 19 }$$

Again assuming that the coherences (i.e. x and y operator terms in equation (19)) would be averaged out in a continuous wave, time-resolved experiment, we are left with:

$$\begin{equation}{\hat{P}}^{\mathrm{I}}=\frac{1}{4}\hat{E}-{\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}\enspace \mathrm{cos}\left(\chi \right),\end{equation} \tag{ 20 }$$

so that the spectrum would resemble figure 3(B) rather than figure 3(D). Thus, one could expect a clear difference in the spectra when the magnetic field is parallel and perpendicular to the CISS axis.

The schematic spectra in figure 3 are for an ensemble of idealised, aligned radical pairs. In reality, one would need to consider also g-anisotropy, dipolar and hyperfine interactions [72, 73, 76]. Nevertheless, it might be possible to identify the contribution from the $\left({\hat{S}}_{\mathrm{D}z}-{\hat{S}}_{\mathrm{A}z}\right)$term which would be diagnostic of a CISS component in the spin selectivity.

For a randomly oriented ensemble of radical pairs one would have to average over all directions of the magnetic field with respect to the CISS quantization axis. Rotating ${\hat{P}}^{\mathrm{I}}$ in equation (10) from the molecular frame (X, Y, Z) to the laboratory frame (x, y, z) gives:

$$\begin{equation}\begin{aligned}\hfill \hat{\rho }\left(0\right)\propto \mathrm{exp}\left(-i{\hat{S}}_{\mathrm{D}\mathrm{A}y}{\theta }_{\mathrm{C}}\right){\hat{P}}^{\mathrm{I}}\enspace \mathrm{exp}\left(i{\hat{S}}_{\mathrm{D}\mathrm{A}y}{\theta }_{\mathrm{C}}\right)& =\frac{1}{4}E-{\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}\left[{\mathrm{cos}}^{2}\left({\theta }_{\mathrm{C}}\right)+{\mathrm{sin}}^{2}\left({\theta }_{\mathrm{C}}\right)\mathrm{cos}\left(\chi \right)\right]\hfill \\ \hfill & \quad +\frac{1}{2}\enspace \mathrm{cos}\left({\theta }_{\mathrm{C}}\right)\mathrm{sin}\left(\chi \right)\left({\hat{S}}_{\mathrm{D}z}-{\hat{S}}_{\mathrm{A}z}\right)+\dots \hfill \end{aligned}\end{equation} \tag{ 21 }$$

where θ_C is the angle between the magnetic field and the CISS axis, ${\hat{S}}_{\mathrm{D}\mathrm{A}q}={\hat{S}}_{\mathrm{D}q}+{\hat{S}}_{\mathrm{A}q}$, and only the non-oscillating polarization terms, involving exclusively z-operators, have been retained. Spherical averaging causes the $\left({\hat{S}}_{\mathrm{D}z}-{\hat{S}}_{\mathrm{A}z}\right)$ term to vanish, leaving the average initial state:

$$\begin{equation}\left\langle \hat{\rho }\left(0\right)\right\rangle \propto \frac{1}{4}E-{\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}\left[\frac{1}{3}+\frac{2}{3}\enspace \mathrm{cos}\left(\chi \right)\right].\end{equation} \tag{ 22 }$$

The spectrum would therefore comprise antiphase doublets (figure 3(B)), as expected in the absence of CISS. Any CISS effects would therefore be difficult to distinguish from the conventional RPM polarization.

Finally, we consider the electron spin echo envelope modulation (ESEEM) experiment which has been used extensively to determine the distance between radicals in radical pairs via the echo modulation arising from the electron dipolar coupling [71, 77–79]. Applying the pulse sequence (45°)_x − τ − (180°)_x − τ − echo [71, 80] to $\hat{\rho }\left(0\right)$ in equation (21) gives the expected out-of-phase component from the ${\hat{S}}_{\mathrm{D}z}{\hat{S}}_{\mathrm{A}z}$ term,

$$\begin{equation}-\frac{1}{4}\enspace \mathrm{sin}\left(2J\tau \right)\left({\hat{S}}_{\mathrm{D}x}+{\hat{S}}_{\mathrm{A}x}\right)\left[{\mathrm{cos}}^{2}\left({\theta }_{\mathrm{C}}\right)+{\mathrm{sin}}^{2}\left({\theta }_{\mathrm{C}}\right)\mathrm{cos}\left(\chi \right)\right],\end{equation} \tag{ 23 }$$

as well as a signal in the in-phase channel originating in the $\left({\hat{S}}_{\mathrm{D}z}-{\hat{S}}_{\mathrm{A}z}\right)$ term,

$$\begin{equation}\frac{1}{2\sqrt{2}}\enspace \mathrm{cos}\left(2J\tau \right)\left({\hat{S}}_{\mathrm{D}y}-{\hat{S}}_{\mathrm{A}y}\right)\mathrm{cos}\left({\theta }_{\mathrm{C}}\right)\mathrm{sin}\left(\chi \right).\end{equation} \tag{ 24 }$$

As before, the latter should be observable for an oriented sample but would average to zero for a randomly oriented collection of radical pairs.

We have used the equations set out in section 2 to simulate the anisotropic reaction yields of radical pair reactions taking place in the geomagnetic field. The aim is to see how CISS might affect the signal that is thought to form the basis of magnetic compass sensing in migratory birds. We start with a basic 'toy' model and then proceed to a semi-realistic model of the [FAD^•− TrpH^•+] radical pair that gives rise to the magnetic field effects observed for purified cryptochromes [63, 81, 82] and which could be the magnetic sensor in vivo [3].

To get a clear initial impression of the changes that can be expected when CISS effects are included, we start with a radical pair with the simplest possible internal magnetic interactions. One radical contains a single spin-½ nucleus with an axial hyperfine interaction and principal components a_xx = a_yy = 0, a_zz = 1.5 mT. The other radical has no nuclear spins. The electron–electron dipolar interaction, D = −0.4 mT, corresponds to the ∼1.9 nm separation of FAD^•− and Trp_CH^•+ [83] where Trp_CH is the third of the four tryptophan residues that make up the electron transfer chain in avian Cry4 [84–88]. The CISS quantization axis is taken to be parallel to the dipolar axis (the line of centres of the two radicals) which makes an angle of 45° with the hyperfine z-axis. The exchange interaction of the two electrons in [FAD^•− Trp_CH^•+] is much smaller than D [86, 87]; we have taken it to be zero here. The external magnetic field strength is B = 50 μT, approximately the mean geomagnetic field. Spin relaxation was not included.

Figure 4 shows the reaction yield anisotropy (ΔΦ_R, equation (17)) plotted as a function of the rate constants, k_R and k_F for six values of the parameter χ. Without CISS (χ = 0), the maximum anisotropy (∼0.007) is small and occurs when k_R ≈ k_F ⩽ 10⁷ s⁻¹ [89]. As the contribution of CISS polarization in the formation and recombination steps is increased, the maximum value of ΔΦ_R increases (by a factor of ∼20 when χ = π/2) and now occurs when k_R ≫ k_F.

Zoom In Zoom Out Reset image size

Figures 5(A) and (B) show the shape of the anisotropy of the reaction yield, Φ_R, for two sets of rate constants chosen to be close to the conditions that maximise ΔΦ_R when χ = 0 and χ = π/2, respectively: $\left({k}_{\mathrm{R}},\enspace {k}_{\mathrm{F}}\right)=\left({10}^{6},\enspace {10}^{6}\right){\mathrm{s}}^{-1}$ in figure 5(A) and $\left({k}_{\mathrm{R}},\enspace {k}_{\mathrm{F}}\right)=\left({10}^{8},\enspace {10}^{6}\right){\mathrm{s}}^{-1}$ in figure 5(B). In this representation, the spherical average of the reaction yield, $\left\langle {{\Phi}}_{\mathrm{R}}\right\rangle$, i.e. the part of Φ_R that is independent of the magnetic field direction, has been subtracted to reveal the anisotropic component, ${\bar{{\Phi}}}_{\mathrm{R}}$, which contains the directional information. Red/blue regions correspond to positive/negative ${\bar{{\Phi}}}_{\mathrm{R}}$, i.e. reaction yields larger/smaller than $\left\langle {{\Phi}}_{\mathrm{R}}\right\rangle$. It is clear from figures 5(A) and (B) that the shape of the anisotropy plots is strongly dependent on χ, that exact inversion symmetry (see below) is only found for a pure RPM initial state (i.e. when χ = 0), and that there is a significant antisymmetric component even for small CISS contributions (e.g. χ = 0.05π). The latter can be seen more clearly in figures 5(C) and (D) where the symmetric (${\Delta}{\bar{{\Phi}}}_{\mathrm{R}}^{+}$) and antisymmetric (${\Delta}{\bar{{\Phi}}}_{\mathrm{R}}^{-}$) parts of ${\bar{{\Phi}}}_{\mathrm{R}}$,

$$\begin{equation}{\Delta}{\bar{{\Phi}}}_{\mathrm{R}}^{{\pm}}=\frac{1}{2}\left[{\mathrm{max}}_{\theta ,\phi }\left({\bar{{\Phi}}}_{\mathrm{R}}\left(\theta ,\phi \right){\pm}{\bar{{\Phi}}}_{\mathrm{R}}\left(\theta +\pi ,\phi \right)\right)-{\mathrm{min}}_{\theta ,\phi }\left({\bar{{\Phi}}}_{\mathrm{R}}\left(\theta ,\phi \right){\pm}{\bar{{\Phi}}}_{\mathrm{R}}\left(\theta +\pi ,\phi \right)\right)\right],\end{equation} \tag{ 25 }$$

are plotted as a function of χ.

Zoom In Zoom Out Reset image size

Although toy models have the potential to reveal some aspects of the spin dynamics of radical pairs, they can also be misleading. We have therefore performed similar calculations for a semi-realistic model of the [FAD^•− Trp_CH^•+] radical pair in Cry4. The two nuclei with the largest anisotropic hyperfine interactions in FAD^•− (nitrogens N5 and N10) and in Trp_CH^•+ (N1 and its directly bonded proton H1) were included [60, 61]. The relative orientation of the isoalloxazine ring system of the FAD and the indole ring of Trp_CH was taken from the x-ray structure of pigeon Cry4 [83]. As in the case of the toy model, the electron–electron interactions were D = −0.4 mT and J = 0, and the external magnetic field was again 50 μT. The dipolar axis, again assumed parallel to the CISS quantization axis, was also taken from the pigeon Cry4 structure [83]. See the supplementary information (https://stacks.iop.org/NJP/23/043032/mmedia) for atom labelling schemes and hyperfine tensors.

Figure 6 shows some results of these simulations. In figure 6(A) we see that the reaction yield anisotropy grows as χ is increased from 0 to π/2, and its maximum shifts from k_R ≈ 10⁶ s⁻¹ to k_R ≈ 10⁷ s⁻¹ when k_F ≈ 10⁶ s⁻¹. The latter contrasts with the toy model for which the largest anisotropy was found when k_R ≈ 10⁸ s⁻¹ (for χ = π/2) (figure 4). We attribute this difference to the presence of hyperfine interactions in both radicals which facilitate the double spin-flip, α_D β_A ↔ β_D α_A, that must take place before recombination can occur (figure 1). Another difference between the two models is that ΔΦ_R is larger when one radical has no hyperfine interactions (see, e.g. references [60, 90–92]). Figures 6(B) and (C) show the symmetric and antisymmetric parts of ${\bar{{\Phi}}}_{\mathrm{R}}$ for k_R = 10⁶ s⁻¹, k_F = 10⁶ s⁻¹ and k_R = 10⁸ s⁻¹, k_F = 10⁶ s⁻¹, respectively. In both cases there is a significant antisymmetric component even for small values of χ.

Zoom In Zoom Out Reset image size