Alternative radical pairs for cryptochrome-based magnetoreception
Alpha Lee, Jason C. S. Lau, Hannah J. Hogben, Till Biskup, Daniel R. Kattnig and P. J. Hore
Electronic Supplementary Material
1.
Simulation methods
The simulations of the anisotropic reaction yields of radical pair reactions were performed as
described elsewhere [1-3]. The singlet and triplet states of the radical pair were assumed to
undergo spin-selective reactions to give different products with the same first order rate
constant, k  106 s 1 . The following commonly used assumptions were made: the inter-radical
exchange and dipolar interactions are negligible; both radicals have isotropic g-values equal
to g e (2.0023), the free electron g-value; g-anisotropy, nuclear Zeeman interactions, nuclear
quadrupolar interactions and spin relaxation are negligible. The geomagnetic field intensity
was taken to be 50 T. With the exception of figure 3, the singlet product yields were
calculated for 450 directions of the magnetic field covering the hemisphere: 0    12  ,
0    2 . For Figure 3, 1250 field directions were used.
The singlet yield was calculated as [2]:

S  k  pS (t ) e kt dt
0
in which the fraction of radical pairs in the singlet state at time t is [3]:
pS (t ) 
1
(A)
(B)
   R pq
(t ) R pq
(t )
4 p x, y ,z q x , y ,z
with
(m)
R pq
(t ) 
1
ˆ
ˆ
Tr  Sˆmp e  iH mt Sˆmq eiH mt 


Zm
Nm
Z m    2 I mj  1
j 1
Sˆmp ( p  x, y, z ) are the electron spin operators and Hˆ m is the spin Hamiltonian of radical m
(m = A, B). I mj is the spin quantum number of nucleus j in radical m and N m is the number
of nuclei in radical m.
2.
Analytical solutions
For simple spin systems, algebraic expressions for the singlet yield can sometimes be
obtained by integrating the Liouville von-Neumann equation (in which ˆ (t ) is the spin
density operator and
superoperator):
ˆ
Hˆ  Hˆ 1ˆ  1ˆ  Hˆ T
is the spin Hamiltonian commutator
dˆ (t )
ˆ
ˆ
ˆ
   iHˆ  k1ˆ  ˆ (t )   Lˆ ˆ (t ) .
dt


ˆ
The Liouvillian, L̂ , is not time-dependent and the initial state of the radical pair is pure
singlet so that:
 ˆ
ˆ ˆ (0) 
ˆ (t )  exp  Lt
 
1
ˆˆ Pˆ S ,
exp  Lt
ZAZB
and the singlet product yield is:

 S  k  pS (t ) e  kt dt
0

 k  Tr  Pˆ S ˆ (t )  e  kt dt
0
 
k
ˆˆ Pˆ S  e  kt dt

Pˆ S exp  Lt



0
ZAZB


k  ˆ S ˆˆ1 ˆ S 

 P L P .
ZAZB 

 
where P̂S is the singlet projection operator.
As an example, we consider a radical pair [A B] containing a single nitrogen nucleus with a
hyperfine interaction that satisfies Axx  Ayy  0 . The spin Hamiltonian is:
Hˆ  Hˆ A  Hˆ B

Hˆ A  3aSˆAz Iˆz   SˆAz cos  SˆAx sin



Hˆ B   SˆBz cos  SˆBx sin .
where a  aiso is the isotropic hyperfine interaction and  is the electron Larmor frequency
   e B, B  50 T  . In the basis BAmI , BAmI , BAmI , BAmI , where
mI  1, 0,  1, the matrix representations of Ĥ and P̂S are block diagonal with blocks of the
form:
c b
 1
 2s
 12 s

 0


b 0
s 
1

0 b
2s

1
1
c  b 
2s
2s
1
2
s
1
2
s
0
1
2
and
2
0 0

1
0 2
 0  12

0 0
0
 12
1
2
0
0

0
0

0
ˆ
respectively, where s   sin , c   cos , and b  32 mIa . As a result, L̂ is also block
ˆ
diagonal, containing nine 16  16 blocks making it fairly easy to obtain L̂1 and hence  S .
The result, in the limit of very slow radical recombination ( k  0 ), is:
S (a,   k ) 
567a 4  9a 2 2  8 4  27a 2 (3a 2  5 2 )cos 2
12(9a 2   2  6a cos )(9a 2   2  6a cos )
so that when a   :
S (a    k ) 
1
1
 7  cos 2  and  S  .
12
6
3.
Representation of hyperfine tensors
The hyperfine tensors in figure 1 are represented as follows. The distance from the nucleus in
question to the plotted three-dimensional surface in the direction  ,  is proportional to
 Axx

r T  Ayx
A
 zx
Axy
Ayy
Azy
Axz 

Ayz  r
Azz 
where the matrix is the full hyperfine tensor (with Apq  Aqp ) and
rT   sin cos , sin sin  , cos  .
4.
Atom numbering scheme
Figure S1. Atom numbering scheme used for the FAD and Trp radicals.
3
5.
Hyperfine interactions
Table S1. Hyperfine tensors used to simulate the reaction yield anisotropy of radical pairs
containing the FAD radical. Calculated using density functional theory in Gaussian‐03 [4]
at the UB3LYP/EPR‐III level.
Nucleus
A
aiso 
N5
0.0 
 0.0989 0.0039


0.0 
 0.0039 0.0881
 0.0
0.0
1.7569 

N10
0.0 
 0.0190 0.0048


0.0 
 0.0048 0.0196
 0.0
0.0
0.6046 

H6
0.0 
 0.2569 0.1273


0.0 
 0.1273 0.4711
 0.0
0.0
0.4336 

H8 (3)
0.0
0.0 
 0.4399


0.4399
0.0 
 0.0
 0.0
0.0
0.4399 

H
0.0
0.0 
 0.4070


0.4070
0.0 
 0.0
 0.0
0.0
0.4070 

Tqq

0.5233
1.2336
0.6101
0.6234
0.1887
0.4159
0.2031
0.2128
0.3872
0.1896
0.0464
0.1432
0.4399
0.0
0.0
0.0
0.4070
0.0
0.0
0.0

Calculated by Dr Ilya Kuprov, Department of Chemistry, University of Southampton. The calculation was
done for the radical anion of 7,8,10-trimethyl isoalloxazine (lumiflavin) in vacuo. All hyperfine parameters
are in mT. The atom numbering scheme is as shown in figure S1.

Full hyperfine tensors in the FAD axis system shown in figure 1a.

Isotropic hyperfine interactions.

Principal anisotropic components of the hyperfine tensors (arranged in descending order of magnitude).
Notes:
H8 methyl group: the anisotropic components are small (< 0.08 mT) and were not included in the spin dynamics
simulations. The average of the three isotropic couplings, (0.6493 + 0.6493 + 0.0212)/3 mT, was used for all
three methyl protons on the assumption that methyl group rotation is fast enough to average the three
interactions.
H: the anisotropic components are small (< 0.09 mT) and were not included. One of the  protons was
included in the simulations, with an isotropic hyperfine coupling equal to the largest of those calculated for the
CH3 group (0.4070, 0.4070, 0.0189 mT).
4
Choice of nuclei: the spin dynamics calculations were performed using the 7 nuclei with the largest isotropic
couplings
5
Table S2. Hyperfine tensors used to simulate the reaction yield anisotropy of radical pairs
containing the TrpH  radical. Calculated using density functional theory in Gaussian‐03 [4]
at the UB3LYP/EPR‐III level.
Nucleus
A
aiso 
N1
 0.0336 0.0924 0.1354 


 0.0924 0.3303 0.5318 
 0.1354 0.5318 0.6680 


H1
 0.9920 0.2091 0.2003 


 0.2091 0.2631 0.2803 
 0.2003 0.2803 0.5398 


H2
 0.2843 0.1757 0.1525 


 0.1757 0.2798 0.0975 
 0.1525 0.0975 0.2699 


H4
 0.5596 0.1956 0.1657 


 0.1956 0.4020 0.0762 
 0.1657 0.0762 0.5021 


H6
 0.0506 0.0622 0.0889 


 0.0622 0.3100 0.0297 
 0.0889 0.0297 0.2642 


H7
 0.4355 0.1541 0.1239 


 0.1541 0.2777 0.0864 
 0.1239 0.0864 0.377 


H
 1.5808 0.0453 0.0506 


 0.0453 1.5575 0.0988 
 0.0506 0.0988 1.6752 



Tqq

0.3215
0.7596
0.3745
0.3851
0.5983
0.5914
0.1071
0.4843
0.2780
0.2855
0.0919
0.1936
0.4880
0.3001
0.0480
0.2520
0.2083
0.1979
0.0494
0.1485
0.3636
0.2540
0.0594
0.1945
1.6046
0.1521
0.0456
0.1065
Calculated by Dr Ilya Kuprov, Department of Chemistry, University of Southampton. The calculation was
done for the radical cation of tryptophan in vacuo. All hyperfine parameters are in mT. The atom numbering
scheme is as shown in figure S1.

Full hyperfine tensors in the same axis system as FAD (table S1 and figure 1a).

Isotropic hyperfine interactions.

Principal anisotropic components of the hyperfine tensors (arranged in descending order of magnitude).
6
Notes:
H: one of the  proton with the larger isotropic hyperfine interaction (1.6046 mT) was included in the spin
dynamics simulations. The other had a much smaller aiso (0.0457 mT).
Choice of nuclei: the spin dynamics calculations were performed using the 7 nuclei with the largest isotropic
couplings.
7
Table S3. Hyperfine tensors used to simulate the reaction yield anisotropy of radical pairs
containing the FADH radical. Calculated using density functional theory in Gaussian‐03 [4]
at the UB3LYP/EPR‐III level.
Nucleus
A
aiso 
N5
 0.0906 0.0028 0.0036 


 0.0028 0.0799 0.1383 
 0.0036 0.1383 1.4645 


N10
 0.0032 0.0034 0.0013 


 0.0034 0.0203 0.0936 
 0.0013 0.0936 0.7346 


H5
 1.3794 0.0387 0.0046 


 0.0387 0.0822 0.0833 
 0.0046 0.0833 0.9472 


H8  3
0.0
0.0 
 0.2554


0.2554
0.0 
 0.0
 0.0
0.0
0.2554 

H
 0.1652 0.0377 0.0120 


 0.0377 0.2568 0.0423 
 0.0120 0.0423 0.1501 


Tqq

0.4313
1.0454
0.5195
0.5259
0.2506
0.4961
0.2413
0.2548
0.8029
0.7298
0.1522
0.5776
0.2554
0.0
0.0
0.0
0.1908
0.0939
0.0386
0.0554

Calculated by Dr Ilya Kuprov, Department of Chemistry, University of Southampton. The calculation was
done for the radical anion of 7,8-dimethyl 10-ethyl isoalloxazine in vacuo. All hyperfine parameters are in
mT. The atom numbering scheme is as shown in figure S1. The protonation site in FADH is N5.

Full hyperfine tensors.

Isotropic hyperfine interactions.

Principal anisotropic components of the hyperfine tensors (arranged in descending order of magnitude).
Notes:
H8 methyl group: the anisotropic components are small (< 0.05 mT) and were not included in the spin dynamics
simulations. The average of the three isotropic couplings, (0.3737 + 0.3845 + 0.0080)/3 mT, was used for all
three methyl protons on the assumption that methyl group rotation is fast enough to average the three
interactions.
H: The  proton with the larger isotropic hyperfine coupling was included in the spin dynamics simulations.
Choice of nuclei: the spin dynamics calculations were performed using the 7 nuclei with the largest isotropic
couplings.
8
6.
Model radical pairs: one
14N
hyperfine interaction
Figure S2. Calculated reaction yield anisotropy, S , of a one-nitrogen radical pair as a
( N)
function of Txx and Tyy , with aiso
= +0.5 mT. The principal values of the hyperfine tensor are
Aqq  aiso  Tqq ( q  x, y, z ), where Txx  Tyy  Tzz  0 . The maxima in S are found for axial
(N)
( N)
hyperfine interactions, when two of the Tqq are equal to  aiso
and the third equals 2aiso
, i.e.
( N)
when two of the Aqq are zero and the third equals 3aiso .
Figure S3. Calculated reaction yield anisotropy, S , of a one-nitrogen radical pair as a
( N)
function of aiso
, the isotropic hyperfine coupling, with Axx  Ayy  0 . Radical pair lifetime =
( N)
1 s. Magnetic field strength = 50 T. The asymptotic value of S at large aiso
is close to
the limiting value of 1/6 (see main paper, section 2.5).
9
7.
References
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of weak magnetic fields on free radical recombination reactions. Mol. Phys. 95, 7189.
Till, U., Timmel, C.R., Brocklehurst, B. & Hore, P.J. 1998 The influence of very
small magnetic fields on radical recombination reactions in the limit of slow
recombination. Chem. Phys. Lett. 298, 7-14.
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