Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 92, Issue 8, 2711-2726, 15 April 2007
doi:10.1529/biophysj.106.097139
Biophysical Theory and Modeling
Magnetic Field Effects in Arabidopsis thaliana Cryptochrome-1
Ilia A. Solov’yov*, Danielle E. Chandler† and Klaus Schulten†, 
, 
* Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Frankfurt am Main, Germany
† Department of Physics, and Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois
Address reprint requests to Klaus Schulten, Dept. of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801.
Abstract
The ability of some animals, most notably migratory birds, to sense magnetic fields is still poorly understood. It has been suggested that this “magnetic sense” may be mediated by the blue light receptor protein cryptochrome, which is known to be localized in the retinas of migratory birds. Cryptochromes are a class of photoreceptor signaling proteins that are found in a wide variety of organisms and that primarily perform regulatory functions, such as the entrainment of circadian rhythm in mammals and the inhibition of hypocotyl growth in plants. Recent experiments have shown that the activity of cryptochrome-1 in
Arabidopsis thaliana is enhanced by the presence of a weak external magnetic field, confirming the ability of cryptochrome to mediate magnetic field responses. Cryptochrome’s signaling is tied to the photoreduction of an internally bound chromophore, flavin adenine dinucleotide. The spin chemistry of this photoreduction process, which involves electron transfer from a chain of three tryptophans, can be modulated by the presence of a magnetic field in an effect known as the radical-pair mechanism. Here we present and analyze a model of the flavin-adenine-dinucleotide-tryptophan chain system that incorporates realistic hyperfine coupling constants and reaction rate constants. Our calculations show that the radical-pair mechanism in cryptochrome can produce an increase in the protein’s signaling activity of ∼10% for magnetic fields on the order of 5G, which is consistent with experimental results. These calculations, in view of the similarity between bird and plant cryptochromes, provide further support for a cryptochrome-based model of avian magnetoreception.
Introduction
The ability of some animals to perceive the Earth’s magnetic field has been known since the 19th century 1,2 and has been studied scientifically since the 1960s 3. The best-studied example is the use of the geomagnetic field by migratory birds for orientation and navigation. Reviews of these studies can be found in Wiltschko and Wiltschko 4,5,8, Beason 6, and Mouritsen and Ritz 7. Despite decades of research, the mechanism of avian magnetoreception remains elusive. The two candidates discussed most often are a magnetite-based mechanism 9,10,11,12,13,14,15,16,17,18 and a chemical reaction mechanism called the radical-pair model 19,20,21,22. Evidence suggests that birds use both types of magnetoreception simultaneously, using small magnetite particles to form a magnetic “map” while using a radical-pair mechanism as the basis of the orientational compass 11.
There are several reasons to prefer a radical-pair-based compass over one based on magnetite. The avian compass is an inclination compass, sensitive only to the inclination of the Earth’s magnetic field lines and not to their polarity 4,5. The avian compass is also known to be highly sensitive to the strength of the ambient magnetic field, requiring a period of acclimation before orientation can occur at intensities differing from that of the natural geomagnetic field 23. Further evidence favoring a radical-pair-based compass is offered by recent experiments probing the effects of low-intensity radiofrequency radiation on bird orientational behavior 24,25,26,27. Furthermore, the avian compass is light-dependent, as first suggested by theory 19,21, normally requiring light in the blue-green range to function properly 28,29 and is known to be localized in the right eye of migratory birds 30. A radical-pair model in which a light-driven, magnetic-field-dependent chemical reaction in the eye of the bird modulates the visual sense indeed predicts these properties 19,20,21,22,31,32,33,34. Finally, a protein harboring blue-light-dependent radical-pair formation, cryptochrome, is found localized in the retinas of migratory birds 35,36, where its effects could intercept the visual pathway.
The radical-pair mechanism, in general, involves a process by which a pair of spin-1/2 radicals leads to distinct reaction products for the spins in either an overall singlet or triplet state. The mechanism has been explored for a variety of model systems 19,20,22,32,37,38,39. In such instances, hyperfine coupling, exchange, dipole-dipole, and Zeeman interactions acting on the electron spins can induce magnetic field effects in the reaction yields.
The radical-pair mechanism supposedly linked to the avian compass arises in the protein cryptochrome 22. Cryptochrome is a signaling protein found in a wide variety of plants and animals 40,41,42, and is highly homologous to DNA photolyase 43,44. The role of cryptochrome varies widely among organisms, from the entrainment of circadian rhythms in vertebrates to the regulation of hypocotyl elongation and anthocyanin production in plants 45,46,47. The role of cryptochrome as a magnetic compass, as suggested in Ritz et al. 22 and Ahmad et al. 48, is still hypothetical.
Photolyase and cryptochrome both internally bind the chromophore flavin adenine dinucleotide (FAD). In photolyase, the presence of FAD in its fully reduced FADH− state is necessary for its DNA repair activity. The FAD cofactor, which typically exists in photolyase in its semireduced FADH form, is brought to the FADH− state by a series of light-induced electron transfers involving a chain of three tryptophans that bridge the space between FAD and the protein surface 49,50,51,52,53.
Although little is presently known about the activity of cryptochromes, it has been suggested 54,55 that a light-induced autophosphorylation reaction is involved in the early stage of cryptochrome’s signaling activity. Recent experiments 56 have shown that light-induced electron transfer from a tryptophan chain conserved from photolyase is the dominant FAD reduction pathway in Arabidopsis thaliana (mouse-eared cress) cryptochrome, and that disruption of this photoreduction pathway impedes the protein’s autophosphorylation activity. However, although photolyase seems to be activated when the semireduced FADH form is converted to the fully reduced FADH− form, cryptochrome seems to be activated when the fully oxidized FAD form is converted to the semireduced FADH form 57.
The tryptophan chain in Arabidopsis cryptochrome consists of Trp-324, Trp-377, and Trp-400, as shown in Fig. 1. Trp-324 is located near the periphery of the protein body, and Trp-400 is proximal to the flavin cofactor, with Trp-377 located in between. Before light activation of cryptochrome, the flavin cofactor is present in its fully oxidized FAD state. FAD absorbs blue light photons, being promoted thereby to an excited state, FAD*. FAD* is then protonated, likely from a nearby aspartic acid 58, producing FADH+. Once the electronically excited flavin is in the FADH+ state, light-induced electron transfer is initiated. An electron first jumps from the nearby Trp-400 into the hole left by the excited electron in FADH+, forming FADH+Trp-400+. An electron then jumps from Trp-377 to Trp-400, forming FADH+Trp-377+, and subsequently from Trp-324 to Trp-377, forming FADH+Trp-324+. Finally, Trp-324+ becomes deprotonated to Trp-324dep, i.e., forming FADH+Trp-324dep50, fixing the electron on the FADH cofactor. This scenario is summarized in Fig. 2.
However, before the final deprotonation takes place, it is possible for the electron on FADH to back-transfer to one of the tryptophans, which quenches the signaling state. This back-transfer, leading to the formation of FADH+, as shown in Fig. 2, can only occur if the spins of the two unpaired electrons are in an overall singlet state. An external magnetic field can influence the overall electron spin state through the Zeeman interaction acting jointly with hyperfine coupling to the nuclear spins associated with the hydrogen and nitrogen atoms 37. If the overall spin state is triplet, electron back-transfer and formation of FADH+ cannot occur, extending the time cryptochrome stays in its signaling state. This, in turn, could affect the visual perception of a bird, as described in Ritz et al. 22, permitting the bird to visually discern the magnetic field.
In this article, we seek to investigate computationally the electron transfer and spin dynamics in cryptochrome as depicted in Fig. 2. This requires an atomic-level structure of the protein. Unfortunately, no structures of avian cryptochromes are available yet. The only available structure at this time is that of Arabidopsis thaliana cryptochrome-1 43. However, the cryptochromes of birds and plants are very similar. A BLAST 59 comparison of Erithacus rubecula (European robin) cryptochrome-1a and cryptochrome-1b with Arabidopsis thaliana cryptochrome-1 gives expect values of 3×10−38 and 2×10−37, respectively, with 28% sequence identity for each (see Fig. 3). Therefore, we will base our computational analysis on the electron transfer and spin dynamics of Arabidopsis cryptochrome-1.
In regard to the similarity of avian and plant cryptochromes, a recent experiment on the effect of an external magnetic field on Arabidopsis thaliana seedlings 48 is encouraging. It was found that signaling from cryptochrome-1, measured through a hypocotyl inhibition and anthocyanin production assay, is enhanced when seedlings are placed in a magnetic field of 5G, compared with an assay at an Earth-strength (0.5G) magnetic field. Mutant seedlings lacking cryptochromes showed no change under different magnetic-field strengths. This observation suggests that the plant cryptochrome spends a longer time in its signaling state when placed in an external magnetic field of 5G than it spends under Earth-strength magnetic field conditions.
In this article, a model of the FADH-tryptophan chain system is developed and analyzed. The model incorporates realistic electron-transfer rate constants and magnetic interactions for electron spins. Our goal is to show that a weak magnetic field can have a measurable effect on cryptochrome signaling.
Theory
In this section a calculation of cryptochrome activation and its magnetic-field dependence is outlined. This calculation expands upon previous work 22 by creating a relatively realistic model of the radical-pair system in cryptochrome-1. This is achieved by incorporating realistic hyperfine coupling tensors for FADH and tryptophan, by including multiple tryptophans in the photoreduction pathway, and by using realistic reaction rate constants for electron forward transfer, electron back-transfer, and tryptophan deprotonation.
Radical-pair Hamiltonian
The Hamiltonian for the intermediate radical-pair systems, FADH+Trp-400+, FADH+Trp-377+, or FADH+Trp-324+, as shown in Figure 2 and Figure 4, is the sum of two Hamiltonians for each radical pair, e.g., a Hamiltonian for FADH and a Hamiltonian for Trp-400+. In addition, a Hamiltonian 
arises that accounts for the exchange and dipolar interactions within the radical pair.
The Hamiltonian for one specific pair is denoted generically
The Hamiltonians
and
as explained in
20, are composed of a Zeeman interaction term and a hyperfine coupling interaction term and are written
where
is the spin operator of nucleus
i,
is the electron spin operator,
is the hyperfine coupling tensor for nucleus
i,
μB=5.78843×10
−9eV/G is the Bohr magneton, and
B0 cos
θ) is the external magnetic field. The nuclear spins, electron spins, and external magnetic field are depicted in a so-called semiclassical manner in
Fig. 4. As explained in detail by Schulten and co-workers
20,
32, in the semiclassical picture, the electrons precess in the local magnetic field corresponding to the term
in Eq.
(2), with contributions from the external field
and from the nuclear spins
The sum over
i in Eq.
(2) is performed over all nuclei of one radical;
j denotes the FADH or tryptophan radical. The operator
is the so-called
g-tensor, which can be brought to the following diagonal form in an appropriate coordinate system
The diagonal values are called
g-factors. In this article, an isotropic
g-tensor is assumed, with
gxx=
gyy=
gzz=
g=2.
The dimension of the Hamiltonian in Eq. (2) is determined by the dimensions of the spin spaces of the nuclei. The spin operator in Eq. (2) can be written
where (
σx,
σy,
σz) are the Pauli spin matrices
60 and
En is the identity matrix of dimension
n. The dimension of the identity matrices is determined by the dimension of the spin spaces of the corresponding nuclei, shown in Eqs.
(4) as a subscript. Each nuclear spin is coupled to the electron spin by a hyperfine coupling tensor
mThis hyperfine coupling tensor is split into an isotropic part and an anisotropic part:
The isotropic part can be written (for the sake of simplicity, the index
j is left out)
where
are hyperfine coupling constants.
The isotropic part of the hyperfine tensor is diagonal in the same basis as the g-tensor, but the anisotropic part, in general, is not. The hyperfine axes define the orthonormal basis in which the anisotropic part of the hyperfine tensor is diagonal. To compute the inner product 
the electron and nuclear spins must be rotated into the same basis. If the coordinate frames of the g-tensor and of the anisotropic hyperfine tensor are denoted as (x, y, z) and (x′, y′, z′), corresponding to the unit vectors 
and 
respectively, then the anisotropic part of the tensor can be written
Here the rotated spin matrices are
In our model, we consider each of the three tryptophans to be identical, and we neglect orientational differences between them, so each will have an identical Hamiltonian.
In addition to the Zeeman and hyperfine coupling interaction terms one needs to account, in general, also for the electron-electron exchange and dipolar interactions in the radical pair. This is done through the term in Eq. (1). These interactions play an important role when the distances between the radicals are small. The part of the Hamiltonian describing the electron-electron exchange and dipolar interactions is
where
and
are the unpaired electron spins on the FADH and Trp- radicals, respectively. The functions
J(
R) and
D(
R) describe the strength of the exchange and dipolar couplings and are assumed, as is often done, to take the simple functional form
In Eqs. (13), R is the edge-to-edge distance between the radicals, J0 is the exchange coupling constant, 
is the unit vector in the direction of 
and β is a range parameter. The exchange and dipolar coupling parameters rapidly decrease with the distance between the radicals and can be neglected if the distance between the radicals is sufficiently large. It is possible to estimate the values of the coupling parameters for given distances between FADH and Trp- radicals using Eqs. (13). The characteristic distances RFADH–Trp-400, RFADH–Trp-377, and RFADH–Trp-324 are 6.0, 8.9, and 13.3Å, respectively. The values for J0 and β are taken, from a study of acyl-ketyl biradicals 61,62, to be J0=7×109 G and β=2.14Å−1; these values are typical for radical pairs in solution. With these values for J0, β, and R, one makes the following estimates for the exchange coupling parameters: J(RFADH–Trp-400)=18,568G, J(RFADH–Trp-377)=37G, and J(RFADH–Trp-324)=0.006G. The estimated values for the dipolar coupling parameters are D(RFADH–Trp-400)=43G, D(RFADH–Trp-377)=13G, and D(RFADH–Trp-324)=4G.
The estimated exchange interaction in the FADH+Trp-400+ radical pair is significantly larger than the hyperfine interaction, which is characterized by a coupling constant ( in Eq. (8)) of ∼10G per nucleus (see below). In the FADH+Trp-377+ radical pair, the exchange interaction is significantly smaller than for the FADH+Trp-400+ pair, but is comparable with the typical hyperfine interaction. In the FADH+Trp-324+ radical pair, the exchange interaction is much smaller than both the typical hyperfine interaction and the external magnetic field. It must be stressed that the given estimates are qualitative and the real exchange interaction in cryptochrome may be significantly different from the values given above. An accurate calculation of the exchange interaction depends on knowledge of the constants J0 and β, and the coupling parameter is especially sensitive to the constant β. For example, a value of β=4.28Å−161 produces J(RFADH–Trp-400)=0.05G, J(RFADH–Trp-377)=2×10−7G, and J(RFADH–Trp-324)=4.8×10−15G, all of which are negligible in comparison with the hyperfine interaction. The estimated dipole-dipole interaction appears to be of the same order of magnitude as the hyperfine interaction term for the FADH+Trp-400+ and FADH+Trp-377+ radical pairs, but is notably smaller than the typical hyperfine interaction for the FADH+Trp-324+ radical pair.
Large values of the exchange or dipolar coupling parameters in the Hamiltonian of a radical pair mean that the singlet-triplet interconversion process in the radical pair will be suppressed 61. The large estimates for the exchange and dipolar couplings for the FADH+Trp-400+ and FADH+Trp-377+ pairs would then seem problematic for the production of a magnetic field effect. However, because the characteristic rate for electron transfer from Trp-377 to Trp-400 and from Trp-324 to Trp-377 is of the same order of magnitude as the singlet-triplet interconversion rate (see rate constants below), neglecting the exchange and dipolar interaction terms for these pairs will not significantly affect the spin dynamics. As is further illustrated in Fig. 4, the main contribution to the spin dynamics of the system comes from the FADH+Trp-324+ radical pair due to the disparity in the lifetimes, τ(FADH+Trp-400+) ≈ τ(FADH+Trp-377+) ≈ 10ns and τ(FADH+Trp-324+) ≈ 100ns, so that the neglect of the exchange and dipolar interaction in the FADH+Trp-400+ and FADH+Trp-377+ pairs is acceptable. For the FADH+Trp-324+ radical pair, the estimates for both the exchange and dipolar couplings are smaller than the typical hyperfine interaction, so the exchange and dipolar interactions may be neglected for this pair as well. For these reasons, we have chosen to neglect the term Hint in Eq. (1) and consider only the effects of the Zeeman and hyperfine interaction terms.
Stochastic Liouville equation
To describe FAD photoreduction and a radical-pair-based magnetic-field effect in cryptochrome, we extend the description in Ritz et al. 22 and include three intermediate radical pairs, i.e., FADH+Trp-400+, FADH+Trp-377+, and FADH+Trp-324+, as shown in Figure 2 and Figure 4. The time evolution of the corresponding spin system is described through a modified stochastic Liouville equation 63. For this purpose, three density matrices ρi are defined for the states 1≤i≤3, corresponding to FADH+Trp-400+, FADH+Trp-377+, and FADH+Trp-324+. Each density matrix follows a stochastic Liouville equation that describes the spin motion and also takes into account the transitions into and out of a particular state from or into other states, as illustrated in Fig. 2. The equations that govern the evolution of the density matrices ρi are generalizations of Eq. (3) in Ritz et al. 22 and read
Here,
and
are the projection operators onto the singlet and triplet states of the electron spin pair, which are defined as
where
and
denote the unpaired electrons on FADH and Trp, respectively.
in Eqs.
(16) is the Hamiltonian associated with the radical pair that consists of FADH and the
ith tryptophan. Since all tryptophans are assumed to be identical, we set
The rate constants associated with the process of electron jumping from one tryptophan to the next are denoted by
k1 and
k2. The rate constants for electron back-transfer from each of the three tryptophans are denoted
and
and
kd is the rate constant associated with tryptophan deprotonation
50. [
A,
B]
±=
AB±
BA denotes the commutator and anticommutator, respectively. We will adopt the following assumptions and notational conventions about the rate constants:
These assumptions will be rationalized below in the “Rate constants” section.
To illustrate the derivation of Eqs. (16), we explain the right-hand side of Eq. (17). The first term describes the electron spin motion; the second and third terms describe the loss of density due to the electron hole transition FADH+Trp-377+→FADH+Trp-324+; the fourth term describes the electron back-transfer FADH+Trp-377+→FADH++Trp-377; the last two terms account for the electron hole transition FADH+Trp-400+→FADH+Trp-377+. The second, third, fifth, and sixth terms correspond to spin-independent reactions, but the fourth term describes a manifestly spin-dependent reaction, as electron back-transfer is only permitted when the FADH+Trp-377+ radical pair is in an overall singlet electron spin pair state.
By using the relationship 
and collecting terms, Eqs. (16) can be rewritten
Simplifying once more, the final set of coupled differential equations for our model is obtained:
We assume that the system begins with the hole (left from electron transfer) on the first tryptophan and with the electron spin pair in the singlet (rather than triplet) state, so that the initial conditions are
This assumption is inspired by experimental data from photolyase
53. The actual initial state of cryptochrome is not known and might be a triplet state; however, the results of our calculation would be qualitatively similar had we chosen a triplet state for the initial condition. The system of differential equations (Eqs.
(26)) was solved numerically.
We do not include in our model the possibility of electrons transferring backward in the tryptophan chain, i.e., electrons undergo the transfers Trp-377→Trp-400 or Trp-324→Trp-377, but never the transfers Trp-400→Trp-377, or Trp-377→ Trp-324. Although the latter transfers are feasible, calculations in the literature 52 and our own estimates presented below suggest that the rate constants for electrons transferring backward in the chain are 2–3 orders of magnitude smaller than the rate constants for forward transfer. This implies that the probability for such behavior is small and, therefore, we neglect this reverse electron transfer in our model.
Hyperfine coupling
The cryptochrome activation yield is very sensitive to the hyperfine coupling tensors chosen for the FADH and tryptophan radicals. For the yield to acquire a dependence on the angle between the magnetic field vector and the radical-pair axis, the hyperfine tensor of at least one radical must have significant anisotropy. One of the improvements made in our model of the FADH-tryptophan radical pair is to use realistic hyperfine coupling tensors for the two radicals, rather than relying on an order-of-magnitude guess as was done in Ritz et al. 22. Information regarding the hyperfine tensors of nuclei in FADH and tryptophan in photolyase and other molecules has been published 64,65,66. We assume that the hyperfine tensors for FADH and tryptophan in cryptochrome are similar to those exhibited by related systems. Indeed, the possibility for magnetic field effects in photolyase has previously been examined using similar hyperfine tensors 67. Our model differs from those previously considered in that it allows for a more complex reaction mechanism in which electron transfer and back-transfer rate constants are considered.
The hyperfine coupling constants and principal hyperfine axes used in the calculation are presented below (see Table 2). Because of the computational cost of calculating the activation yield for systems with a high-dimensional Hamiltonian, we include only up to four nuclei in each of our models of the FADH-tryptophan radical pair. Several combinations of nuclei in each radical were considered, and the activation yield for each configuration was calculated. However, the dependence of the activation yield on the magnetic field is sensitive to the choice of nuclei and associated hyperfine coupling constants. Fig. 5 shows the labeling used for the nuclei in FADH and tryptophan.
In this article, we take into consideration two representative choices of nuclei. The first choice includes the nuclei N5 in FADH and H5 and Hβ1 in tryptophan; the second choice includes N5 and H5 in FADH and H5 and Hβ1 in tryptophan. The two radical-pair models are listed in Table 1, and the corresponding hyperfine coupling constants are given in Table 2. We included in our choices the nuclei with the strongest hyperfine coupling, according to the literature, as the calculated magnetic field dependence of cryptochrome activation proved to be most sensitive to the influence of these nuclei. We then modified the coupling constants from the values reported in the literature and chose values that gave the largest change in activation upon increase of the magnetic field to 5G (Table 2). Our goal was to demonstrate the possibility of obtaining a large (on the order of 10%) variation (either an increase or decrease) in activation yield when the magnetic field is varied according to the experiments reported in Ahmad et al. 48.
| | |
| Radical-pair model | Nuclei in FADH | Nuclei in tryptophan | |
|---|
| 1 | N5 | H5,  | |
| 2 | N5, H5 | H5, | |
| | |
| | |
| Hyperfine constants and axes chosen for FADH | |
|---|
| Nucleus | aiso (G) | Tii (G) | Hyperfine axes | |
| N5 | 3.93 | −4.98 | 0.4380 | 0.8655 | −0.2432 | |
| | | −4.92 | 0.8981 | −0.4097 | 0.1595 | |
| | | 0, 9.89* | −0.0384 | 0.2883 | 0.9568 | |
| H5 | −7.69 | −6.16 | 0.9819 | 0.1883 | −0.0203 | |
| | | −1.68 | −0.0348 | 0.2850 | 0.9579 | |
| | | 7.84 | −0.1861 | 0.9398 | −0.2864 | |
| Hyperfine constants and axes chosen for tryptophan | |
| | |
| Nucleus | aiso (G) | Tii (G) | Hyperfine axes | |
|---|
| | 16 | 0.00 | 1.000 | 0.000 | 0.000 | |
| | | 0.00 | 0.000 | 1.000 | 0.000 | |
| | | 0.00 | 0.000 | 0.000 | 1.000 | |
| H5 | 5 | 0.00 | 1.000 | 0.000 | 0.000 | |
| | | 0.00 | 0.000 | 1.000 | 0.000 | |
| | | 0.00 | 0.000 | 0.000 | 1.000 | |
| | |
* The value of 9.89G was used for radical-pair model 1 and 0.00G for radical-pair model 2.
|
To determine the magnetic field effect on cryptochrome activation more precisely, one needs to obtain more accurate values for the hyperfine coupling constants for the relevant nuclei in FADH and tryptophan. The results presented below can only show the feasibility of obtaining a significant magnetic field effect in cryptochrome based on estimates for the hyperfine coupling within the radical pair.
Rate constants
For realistic estimates of the reaction rate constants for electron forward transfer, electron back-transfer, and tryptophan deprotonation, we used a combination of experimental values from the literature 50,53 and our own theoretical estimates.
As indicated in Fig. 2, we denote the rate constants for forward electron transfer Trp-377→Trp-400 and Trp-324→Trp-377 by k1 and k2. These transfers correspond to an electron jumping between tryptophans in the direction opposite to that of the arrows shown in Fig. 2, as the arrows actually indicate hole transfer. We denote the rate constants for reverse electron transfer by (k1)′ and (k2)′. The electron forward transfer rate constants were experimentally determined for DNA photolyase and estimated to be ∼108s−150,53.
The rate constant for electron transfer can be estimated if one considers the tunneling process of an electron through protein. The rate constant is commonly expressed as the product of two factors 68. The first factor is an electronic term arising from the strength of the coupling of the electron donor/acceptor wavefunctions, leading to a roughly exponential fall-off in the electron tunneling rate with distance through the insulating barrier and, accordingly, is proportional to exp(−βR), where R is the edge-to-edge distance and β is proportional to the square root of the barrier height; the second factor depends on the energy, λ, required to repolarize the protein matrix upon electron transfer, and the driving force, ΔG, for the electron transfer. These quantities are depicted in the Marcus diagram 68,69 shown in Fig. 6. Both classical 69 and quantum mechanical 70,71,72 versions of the Marcus theory of electron transfer suggest a roughly parabolic dependence of log rate on ΔG.
Electron tunneling between covalently bridged redox centers in synthetic systems (β≈0.9Å−1) 73 is clearly much faster than tunneling through vacuum (β≈2.8–3.5Å−1) 74,75. Earlier experimental examination of tunneling in proteins suggested an intermediate value (β≈1.4Å−1) corresponding to a weighted average of the two extreme β values 74,76. A simple empirical expression that incorporates an exponential decay of the tunneling rate constant k (in s−1) with edge-to-edge distance R (in Å) and a parabolic dependence of the rate on ΔG and λ (in eV) is 77
The coefficient 0.6 corresponds to
β=1.4Å
−1 on a common log scale, whereas the coefficient 3.1 collects the room-temperature constants for the quantized nuclear term
71, as suggested by extensive studies of photosynthetic reaction centers
74,
78,
79. Equation
(32) has proven to be a useful approximation for electron-transfer rate constants in the absence of a detailed protein structure; however, it does not explicitly address the variations in polypeptide structure or whether those variations have been naturally selected to influence tunneling rate constants to physiological advantage.
The use of the edge-to-edge distance R in Eq. (32) provides only a rough estimate of the electron tunneling rate constant. The edge-to-edge distance is suitable in the case when the molecules are static, but in a protein at thermal equilibrium, the tryptophans move and rotate, and the average distance between donor and acceptor groups offers a better variable for the spatial dependence of the electron transfer rate. Accordingly, we substitute in Eq. (32) the average distance between tryptophans,
for R, where
i and
j denote the atoms in the first and second tryptophan, respectively, and
Npairs is the total number of atomic pairs. The average distance between Trp-377 and Trp-400 calculated from Eq.
(33) is 7.21Å, whereas the average distance between Trp-324 and Trp-377 is 8.37Å. With Δ
G=−0.2eV (see
Figure 2 and
Figure 6) and the generic value
λ=1.0eV for the reorganization energy of electron-tunneling processes in proteins
76, we estimate that
k1=4.9×10
8s−1 and
k2=9.9×10
7s−1 for electron transfer from Trp-377 to Trp-400 and from Trp-324 to Trp-377, respectively.
The value ΔG is estimated to be negative (see Figure 2 and Figure 6), despite differences in the polarities of the tryptophan environments 50. From inspection of the crystal structure, it was suggested 50 that the polarities increase and, hence, the potentials decrease in the order Trp-400, Trp-377, Trp-324. The value for ΔG in DNA photolyase is calculated and discussed in Popovic et al. 52.
The estimates above for k1 and k2 are in good agreement with experimentally determined values and correctly reproduce the order of magnitude of the electron-transfer rate constants. For a more accurate evaluation of the rate constants, it is necessary to employ a more detailed model that accounts explicitly for the structure and vibrations of the protein; such a model 80 is far beyond the scope of this study. Since the estimated rate constants are of the same order of magnitude as the experimentally measured values, we will use the experimentally measured rate constants in our calculations. The estimated rate constants k1 and k2 are of about the same order of magnitude, which supports our assumption, k1=k2, used in the system of coupled stochastic Liouville equations, Eqs. (16).
The rate constants for electron transfers Trp-400→Trp-377 and Trp-377→Trp-324 can also be estimated through Eq. (32). In this case, we employ ΔG=0.2eV for both processes. Thus, one estimates 
and 
for Trp-400→Trp-377 and Trp-377→Trp-324 transitions, respectively. These rate constants are significantly smaller than k1 and k2 and, accordingly, electron transfer in the reverse direction of the Trp-400, Trp-377, Trp-324 chain can be neglected.
The rate constants for electron back-transfer from FADH to a tryptophan, 
and 
(see Fig. 2) can also be estimated through Eq. (32). For this purpose, one needs to know the distances between the fragments, the reorganization energies, and the driving forces. The characteristic distances RFADH–Trp-400, RFADH–Trp-377, and RFADH–Trp-324 are 6.0, 8.9, and 13.3Å, respectively. The reorganization energies are expected to increase with increased distance between the two fragments and, thus, we choose them as 0.85, 1.0, and 1.4eV for the pairs FADH+Trp-400, FADH+Trp-377, and FADH+Trp-324, respectively. The driving forces for these processes can be estimated from the energy diagram in Fig. 2. Since cryptochrome is excited by a blue light photon, the energy difference between the ground and excited states should be ∼2.6eV. The initial electron transfer step, from Trp-400 to FADH, proceeds downhill with a driving force of ∼0.5eV 50. The next two electron-transfer steps proceed with a decrease in energy of 0.2V 50,52. Accordingly, the driving energies are ΔGFADH–Trp-400=−2.1eV, ΔGFADH–Trp-377=−1.9eV, and 
With these driving energies, one obtains 
and 
for the electron back-transfers FADH→Trp-400, FADH→Trp-377, and FADH→Trp-324, respectively. The rate constants compare well with each other. For the sake of simplicity, we will consider the three rate constants to be equal, assuming a value of 107s−1 (see Table 3).
| | |
| Process | Rate constant | Estimate (s−1) | Measured value (s−1) | |
|---|
| Electron forward transfer | k1=k2=ket | 1×108 | 1×108, 50,53 | |
| Electron reverse transfer | (k1)′ | 1.6×106 | — | |
| Electron reverse transfer | (k2)′ | 3.3×105 | — | |
| Electron back-transfer |  | 1×107 | — | |
| Tryptophan deprotonation | kd | — | 3.3×106, 50,53 | |
| Singlet-triplet interconversion | Ω | 8.3×107 | — | |
| | |
The measured deprotonation rate of Trp-324 at pH 7.4 is kd=3.3×106s−150,53. This rate constant can also be estimated, but it depends on the temperature and on the concentration of the external agent, which induces deprotonation.
To test the feasibility of singlet-triplet interconversion to facilitate magnetic-field-dependent cryptochrome activation, it is necessary to estimate the characteristic time for the process, which should be of the same order of magnitude as (or shorter than) the time needed for forward electron transfer. In this subsection, an estimate for the singlet-triplet interconversion time is derived for a model radical pair with one nucleus and two electrons. In this case, the spin Hamiltonian, given by Eq. (1) with Hint neglected, is
where
and
are the
g-tensors of the electrons in radicals
A and
B, which comprise the radical pair;
D denotes the spin-1/2 nuclei involved in hyperfine coupling to one of the electron spins.
To describe the spin motion, one needs to choose the basis states of the wavefunction. For three spin-1/2 particles, eight basis states are required, which will be denoted as ψi, where i=1, 2, …, 8. The determination of the basis states is an exercise in basic quantum mechanics, and the reader is referred to an introductory textbook 60. If the two electrons of the radical pair are found in the singlet state, then the corresponding basis states are
where
αi and
βi are the eigenfunctions of the operator
Siz with eigenvalues 1/2 and −1/2, respectively;
αD and
βD denote the analogous states for the nuclear spin.
Another six states are connected with the triplet states of the radical pair, which will be denoted as 
and 
:
If the radical pair is found in the triplet state, the total spin of the system can be 1/2 or 3/2. The basis states that describe the state of the system with total spin 1/2 are
whereas the basis states that describe the four states of the system with total spin 3/2 are
One can now calculate, for example, the transition probability from a singlet state, e.g., the one described by wavefunction
ψ1, to a triplet state, e.g., the one described by wavefunction
ψ3. This transition is possible if the conditions
are met, where
and
are the energy expectation values of the system in states corresponding to
ψ1 and
ψ3, respectively. The matrix element for the
transition can be evaluated in terms of the parameters specifying the Hamiltonian
34. One obtains
and the energies of states
ψ1 and
ψ3 calculated likewise are
With
gA=
gB=2,
A33=16G,
B=0.5G,
μB=5.78843×10
−9eV/G, one obtains
and
E3 =−1.158×10
−8eV, and conditions defined in Eqs.
(47) are satisfied.
If the system is initially in state ψ1, the probability to find it in state ψ3 at a later time t is
as long as the other six states (
ψ2 and
ψ4–
ψ8) are neglected. Thus, the radical pair oscillates from singlet to triplet state with characteristic frequency
With the values above, one obtains Ω≈8.3×10
7s−1, which is indeed of the same order of magnitude as the electron forward transfer rate constant,
ket=1×10
8s−1, discussed above.
The estimated values of the rate constants for various processes considered in the calculation are compiled in Table 3. It should be noted that the measured rate constants referenced in this article refer to photolyase, not cryptochrome, and could easily be off by an order of magnitude for cryptochrome. However, since accurate data for the rate constants in cryptochrome are not available, we used our estimated values in conjunction with the measured rate constants from photolyase. Although the values presented here must be considered approximate, the fact that magnetic field effects are observed in Arabidopsis (which would not be possible for unsuitable cryptochrome rate constants, as explained below in the Discussion) suggests that our values are likely accurate to within an order of magnitude.
Cryptochrome activation yield
Once the density matrix has been obtained as a solution of the coupled stochastic Liouville equations (Eqs. (26)), observables of interest can be evaluated. The main quantity of interest is the activation yield of cryptochrome. This yield corresponds to the formation of the product FADH+Trp-324dep. The yield depends on the strength and orientation of the magnetic field, described through (B0, θ, ϕ) and is given by the expression
In case the cryptochrome is oriented randomly relative to the external field, the total yield is averaged over
θ and
ϕ,
The magnetic-field dependence of Φ(
B0,
θ,
ϕ) and
develops due to the electron back-transfer reaction FADH+Trp
+→FADH
++Trp (see
Fig. 2) and, in particular, due to the reaction FADH+Trp-324
+→FADH
++Trp-324. This reaction is possible only in the singlet state of the FADH+Trp-324
+ radical pair, and its yield is given by
One can recognize that the activation yield is determined by the function
Consequently, we refer to
and its complement
as the singlet and triplet state populations, respectively.
Results and discussion
The theory and methods described above have been used to study spin dynamics in cryptochrome. In the following sections, the magnetic field dependence of the formation of FADH stabilized by deprotonation of Trp-324+ to Trp-324dep, averaged over the orientation of the magnetic field, is analyzed by means of the observable 
defined in Eq. (55). We found that the suggested radical-pair mechanism is consistent with a cryptochrome-mediated magnetic-field response in Arabidopsis thaliana. The dependence of the activation yield Φ(B0, θ, ϕ), defined in Eq. (54), on the orientation (θ, ϕ) of an external magnetic field is also discussed and it is shown that cryptochrome activation might serve as an inclination compass. Results are presented on the time evolution of the singlet population and discussed in detail.
Magnetic-field dependence of activation yield
The dynamics of electron spins is gove
and 
from the nuclear spins on the two radicals. The spin precession continuously alters the relative spin orientation, causing the singlet (antiparallel) ↔ triplet (parallel) interconversion underlying the magnetic field effect. The nuclei which are actually included in our calculations (radical-pair model 2, see text) are labeled.
