Radical pairs can explain magnetic field and lithium effects on the circadian clock

Drosophila’s circadian clock can be perturbed by magnetic fields, as well as by lithium administration. Cryptochromes are critical for the circadian clock. Further, the radical pairs in cryptochrome also can explain magnetoreception in animals. Based on a simple radical pair mechanism model of the animal magnetic compass, we show that both magnetic fields and lithium can influence the spin dynamics of the naturally occurring radical pairs and hence modulate the circadian clock’s rhythms. Using a simple chemical oscillator model for the circadian clock, we show that the spin dynamics influence a rate in the chemical oscillator model, which translates into a change in the circadian period. Our model can reproduce the results of two independent experiments, magnetic field and lithium effects on the circadian clock. Our model predicts that stronger magnetic fields would shorten the clock’s period. We also predict that lithium influences the clock in an isotope-dependent manner. Furthermore, our model also predicts that magnetic fields and hyperfine interactions modulate oxidative stress. The findings of this work suggest that the quantum nature of radical pairs might play roles in the brain, as another piece of evidence in addition to recent results on xenon anesthesia and lithium effects on hyperactivity.

. (a) Simplified models of the circadian clock feedback loop in Drosophila. CLOCK (CLK) and CYCLE (CYC) promote the tim and per genes. PER and TIM first accumulate in the cytoplasm and then enter into the nucleus to block their gene transcription. Upon light, absorption CRY binds to TIM, and this results in the degradation of TIM 67,68 . (b) Flavinsemiquinone, FADH · , and superoxide O ·− 2 radical pair in CRY, considered in the RPM model in the present work. The radical pair undergoes interconversion between singlet and triplet states.
The circadian oscillations in Drosophila can be modeled by incorporating the formation of a complex between the PER and TIM proteins and introducing negative feedback loops 69 , which are the key to the rhythmicity of PER and TIM and their mRNA transcription. The models can be described by a set of a few kinetic equations 70 . However, modeling Drosophila CC 70 can be further simplified into two nonlinear equations 67 . Furthermore, Player et al. 71 show that quantum effects such as magnetic field effects and hyperfine interaction of radical pairs can be introduced to the chemical oscillator by considering the quantum effects on the corresponding reaction rates.
Here, we propose that the RPM could be the underlying mechanism behind the lithium treatment effects and MF effects on Drosophila's CC. MF via the Zeeman interaction and lithium nucleus via HFIs modulate the recombination dynamics of singlet-triplet interconversion in the naturally occurring RPs in the [ FADH · ... O ·− 2 ] complex, shown in Fig. 1b, and hence influence the period of the CC.
In the following, we review the quantitative experimental results for the effects of applied magnetic field 25 and lithium 40 on the period of Drosophila's CC. Next, we briefly describe the quantum spin dynamics for the radical pair model where the magnetic field effects and the HFIs are relevant. Moving on, we present our singlet yield calculation for the RP system, inspired by the CRY-based model of birds' avian magnetoreception 72 . Later we use a simple model for the mathematical presentation of Drosophila's CC, following the work on Tyson et al. 67 . Then we introduce the quantum effect to the period of the CC model, and we show the consistency of our model's predictions and the experimental findings on the magnetic field and lithium treatment effects. Finally, we discuss new predictions for experiments.

Results
Magnetic field and lithium treatment effects on circadian clock and RPM. Results from prior experiments. Here, we focus on the effects of static MF on Drosophila's CC observed by Yoshii et al. 25 . The authors conducted experiments to observe the effects of static magnetic fields with different intensities, [0, 150, 300, 500] µT , on changes in the period of Drosophila's CC under blue light illumination, shown in Table 1. These magnetic fields are, excepting the control of 0 µT , approximately 3, 6, and 10 times stronger than natural magnetic fields, respectively. That observation revealed that the period alterations significantly depended on the strength of the magnetic field such that the period change reached a maximum of 0.522 ± 0.072 h at 300 µT . In this experiment, the geomagnetic field was shielded, and the arrhythmic flies were excluded from the analysis. We also consider the results of the experiment conducted by Dokucu et al. 40 observing the effects of chronic lithium administration on Drosophila's CC for a range of doses [0, 300] mM. It was shown that lithium treatment lengthens the CC with a maximum prolongation of 0.7 ± 0.217 h at 30 mM of lithium compared to zero lithium intake, see Table 2. In that work, the lethality of lithium up to 30 mM was relatively low until the end of the experiments. Here, we consider that 30 mM is the optimal concentration of lithium where all RPs interact with lithium atoms. We assume that the lithium administered in that work was in its natural abundance, 92.5% and 7.5% of 7 Li and 6 Li , respectively. Here we will refer to the natural lithium as Li. In our model here, 0 µT of MF and 0 mM of lithium are our control sets for MF and lithium effects on the CC.

RPM model.
We develop an RP model to reproduce static MFs and lithium administration effects on the rhythmicity of Drosophila's CC observed in Ref. 25 and Ref. 40 , respectively. Taking into account the facts that the CC is associated with oxidative stress levels under light exposure 45,46,64,73,74 and applied MF 75,76 , and the CC is affected by lithium intake, we propose that the applied magnetic field interacts with the spins of RPs on FADH and superoxide, and the nuclear spin of lithium modulates the spin state of the radical on superoxide. The correlated spins . We consider a simplified system in which the unpaired electron is coupled to the flavin's nitrogen nucleus with an isotropic HF coupling constant (HFCCs) of 431.3 µT 77 . In this model, for simplicity, we consider only Zeeman and HF interactions 48,78 . Following the work of Hore 72 , the anisotropic components of the hyperfine interactions are excluded, which are only relevant when the radicals are aligned and immobilized 79 . The RPs are assumed to have the g-values of a free electron. The Hamiltonian for the RP system reads as follows: where Ŝ A and Ŝ B are the spin operators of radical electron A and B, respectively, Î A is the nuclear spin operator of the isoalloxazine nitrogen of FADH · , similar to Refs. 39,72 , Î B is the nuclear spin operator of the Li nucleus, a A and a B are HFCCs, taken from 39,77 , and ω is the Larmor precession frequency of the electrons due to the Zeeman effect. Ref 72 uses two nitrogen atoms, while for the purpose of the computational cost we use only the largest one without loss of generality. Of note, oxygen has a zero nuclear spin and thus its HFCC equals zero, ( a B = 0 ), however in the model for lithium effects a B corresponds to the nuclear spin of lithium. We assumed that the RPs start off from singlet states (see the Discussion section).
Singlet yield calculation. The singlet yield resulting from the radical pair mechanism can be obtained by solving the Liouville-von Neumann equation for the spin state of the radical pair throughout the reaction. Using the eigenvalues and eigenvectors of the Hamiltonian, the ultimate singlet yield, S , for periods much greater than the RP lifetime 72 has the following form: , is the nuclear spin multiplicity, P S is the singlet projection operator, |m� and |n� are eigenstates of Ĥ with corresponding eigenenergies of ω m and ω n , respectively, k is the RP reaction rate, and r is the RP spin-coherence lifetime rate (relaxation rate).
Here we look at the sensitivity of the singlet yield to changes in the strength of the external magnetic field for the [ FADH · ... O ·− 2 ] radical complex. Figure 2 illustrates the dependence of the singlet yield of the [ FADH · ... O ·− 2 ] complex on external magnetic field B with a maximum yield in [280-360] µ T for k = 4 × 10 7 s −1 and r = 3 × 10 7 s −1 with a 1A = 431.3 µT . In our model, the magnetic dependence of singlet yield is the foundation of the magnetic sensitivity of the circadian clock. Using the singlet yield, we can reproduce the experimental finding on the effects of applied MF 25 and lithium administration 40 on the period of the circadian clock of Drosophila, as we discuss below. It is worth mentioning that the singlet-product of the RP system in [ 80 , which is the major ROS in redox regulation of biological activities and signaling 81 .
Circadian clock model. We use a simple mathematical model for the circadian clock of Drosophila, following the work of Tyson et al. 67 . Despite its simplicity, the model is very well known 69,82-84 and captures the most important part of the clock's function. In this model, PER monomers are rapidly phosphorylated and degraded, whereas PER/TIM dimers are less susceptible to proteolysis, shown in Fig. 1a. In this context, it is also assumed www.nature.com/scientificreports/ that the cytoplasmic and nuclear pools of dimeric protein are in rapid equilibrium. With these considerations, it is possible to write the mathematical model in two coupled equations as follows: where , P t (t) and M(t) are the total protein and the mRNA concentrations, respectively.
For the descriptions and values of the parameters, see Table 3. In this simple model, k p3 represents the role of CRY's light activation and hence proteolysis of protein. By solving Eqs. 3 and 4, we obtain the oscillation of protein and mRNA concentrations. Figure 3 shows the explicit time-dependence of protein and mRNA concentrations and the parametric representation of the chemical oscillator limit cycle for Drosophila's CC. To obtain the period of the clock, we take the average differences between successive peaks and likewise troughs of either P t (t) or M(t) by keeping track of when the derivative is zero.
Effects of singlet yield change on circadian clock. The effects of applied magnetic fields and hyperfine interactions can be introduced to the chemical oscillator of the circadian clock by modifying the rate k f 71 , following the work of Player et al., see Methods. In the CC Eqs. (3) and (4) the corresponding rate is k p3 , which represents the role of CRY's light activation and hence proteolysis of protein and is 0.1 h −1 for the natural cycle of the clock. Hence for the occasions with no singlet yields effects, this value must be retained. The singlet yield effects on k p3 can be written as follows: Table 3. Parameter values for the circadian clock of Drosophila, taken from the work of Tyson et al. 67 . C m and C p are characteristic concentrations for mRNA and protein, respectively.   www.nature.com/scientificreports/ where k ′ p3 , S , and ′ S are the modified rate constant k p3 , the singlet yield with no quantum effects, and the singlet yield resulted from quantum effects due to the Zeeman and/or hyperfine interactions, respectively.

Name Value Units Description
Based on the above considerations, here, we calculate the explicit effects of an applied magnetic field and the hyperfine interactions on the period of the CC. Using Eqs. (3), (4), and (5), we explored the parameter space of relaxation rate r and recombination rate k in order to find allowed regions for which our model can reproduce both experimental findings of static MF of 300 µT 25 and 30 mM of lithium 40 effects on Drosophila's CC, which respectively lengthen the clock's period by 0.224 ± 0.068 h and 0.567 ± 0.11 h . The results are shown in Fig. 4. We find an allowed region where the model reproduces both experiments, see Fig. 4. The parameters for calculating the period of the circadian clock are taken from Table 3. As discussed above, k p3 corresponds to the degradation of TIM due to blue light exposure. For the MF effects under blue light illumination, we set k p3 = 0.085 h −1 to obtain the control period of the circadian clock 25.8 ± 0.14 h under blue light illumination observed in Ref. 25 . Figure 5 shows the effects of lithium on the rhythmicity of CC, such that 7 Li lengthens the period of the clock longer than 6 Li . For the effects of lithium on the circadian clock, the geomagnetic field of 50 µT is taken into account. Figure 6 shows the effects of 300 µT MF on the CC.
The model here reproduces the dependence of the CC's period on the applied MF's strength and Li administration, shown in Fig. 7. The model predicts that further increases in the intensity of the MF would shorten the period of the clock significantly. For the cases considering MF effects solely, for both the experimental data and the RPM model, the period reaches a maximum between 0 µT and 500 µT and exhibits reduced effects at both lower and slightly higher field strengths, shown in Fig. 7a. For the cases of 6 Li or without lithium intake, the largest prolongation of the period occurs in the same range of magnetic field as well, shown in Fig. 7b. Another prediction of the model is that 7 Li prolongs the clock's period stronger than 6 Li , which has a smaller spin compared to 7 Li , see Fig. 7b. In this model, in the cases where lithium effects are considered, the geomagnetic effects of 50 µT are also considered. For the comparison between our model and the experimental data on the lithium effects, we assume that natural lithium was administered in the experiment 40 . Following Ref. 39 , we considered a7 Li = −224.4 µT . Note that moderate changes to the HFCC of lithium do not impact our results. Figure 7 shows that the dependence of the period on applied MFs and lithium effects calculated by the RPM model used in the present work is consistent with the experimental observations. We compare the maximum lengthening of the period in both the RPM model and experimental data 25

Discussion
In this project, we aimed to probe whether a RP model can explain the experimental findings for both the effects of static magnetic field 25 and lithium 40 on the circadian clock in Drosophila. We showed how the quantum effects affect the rates, which then yields a change in the period of the clock. This is a significant step forward compared to the previous studies on xenon anesthesia 66 and the lithium effects on hyperactivity 39 , where the quantum effects were correlated to experimental findings without explicitly modeling the related chemical reaction networks. With a set of reasonable parameters, our model reproduces the experimental findings, as shown in Figs. 4, and 7. In addition, this strengthens the previously proposed explanation for the effects of lithium on hyperactivity 39 via the circadian clock. We proposed that applied magnetic fields and nuclear spins of lithium influence the spin state of naturally occurring [ FADH · ... O ·− 2 ] radical pairs in the circadian clock. This is inspired by the observations that the Drosophila circadian clock is altered by external magnetic fields 24,25 , which is accompanied by modulations in the ROS level 64,75 , and by lithium administration 40 . Let us note that it has also been suggested that lithium exerts its effects by inhibiting Glycogen synthase kinase-3 (GSK-3) 85,86 . However, while the presence of RPs is a natural explanation for magnetic field effects, their existence in GSK-3 requires experimental support.
The suggested [ FADH · ...O ·− 2 ] radical pairs depend on a form of FAD in the Drosophila CRY photocycle that is thought to be unusual. However, Baik et al. showed the possibility that light-activated CRY in Drosophila neurons express a FADH · neutral semiquinone state 87 . They conclude that further investigation would shed more light  Table 3. The black, red, blue and purple colors indicate zero-lithium, 6 Li , 7 Li , and Li, respectively. Lithium administration prolongs the period of the clock, such that 7 Li has more potency than 6 Li. Of note, there is a large body of evidence that ROS are involved in the context of magnetosensing and the circadian clock modulations 63,64,73,74,76,90 . it has been shown that oscillating magnetic fields at Zeeman resonance can influence the biological production of ROS in vivo, indicating coherent S-T mixing in the ROS formation 80 . Additionally, it has been observed that extremely low frequency pulsed electromagnetic fields cause defense mechanisms in human osteoblasts via induction of O ·− 2 and H 2 O 2 75 . Sherrard et al. 63 observed that weak pulsed electromagnetic fields (EMFs) stimulate the rapid accumulation of ROS, where the presence of CRY was required 63 . The authors of that work concluded that modulation of intracellular ROS via CR represents a general response to weak EMFs. Further, Sheppard et al. 90 demonstrated that MFs of a few millitesla can indeed influence transfer reactions in Drosophila CRY. It has also been shown that illumination of Drosophila CRY results in the enzymatic conversion of molecular oxygen to transient formation of superoxide O ·− 2 and accumulation of hydrogen peroxide H 2 O 2 in the nucleus of insect cell cultures 73 . These findings indicate the light-driven electron transfer to the flavin in CRY signaling 74 .
The feasibility for the O ·− 2 radical to be involved in the RPM is a matter of debate in this scenario due to its likely fast spin relaxation rate r. Because of fast molecular rotation, the spin relaxation lifetime of O ·− 2 is thought to be on the orders of 1 ns 91,92 . Nonetheless, it has also been pointed out that this fast spin relaxation can be decreased on account of its biological environment. Additionally, Kattnig et al. 93,94 proposed that scavenger species around O ·− 2 can also reduce its fast spin relaxation. Moreover, in such a model, the effects of exchange and dipolar interactions can also be minimized.
It is often assumed that in the RP complexes involving superoxide are formed in triplet states, as opposed to the case considered here. This is because the ground state of the oxygen molecule is a triplet state. The initial state for RP formation could also be its excited singlet state, which is a biologically relevant ROS [95][96][97] . Further, the transition of the initial RP state from triplet to singlet could also take place due to spin-orbit coupling 98,99 .
Our model predicts that increasing the intensity of the applied magnetic field will shorten the period of the clock. This is a significant new prediction of our model that would be very interesting to check. The isotopicdependence of the period is another prediction of our present model, such that 7 Li lengthens the period of the clock longer than 6 Li.
The circadian clock not only controls the rhythms of the biological processes, but it also has intimate connections to other vital processes in the body 100 and particularly in the brain 101 . It has been suggested that environmental perturbations in the circadian period could increase the risk of selected cancers and hence the circadian clock could be a therapeutic target for cancer risks 102 . It also appears that the way drugs function depends on the circadian clock 103,104 . Notably, it has been shown that the circadian clock is vital for maintaining the antioxidative defense 105 . Moreover, it has been suggested that the circadian clock could be a new potential target for  www.nature.com/scientificreports/ anti-aging 106,107 and neurodegenerative disorders therapeutics 108 . Thus this project also paves a potential path to study other functionalities of the body and the brain connected to the circadian clock in the light of the RPM. To sum up, our results suggest that quantum effects may underlie the magnetic field and lithium effects on the circadian clock. A similar mechanism is likely to be at the heart of magnetoreception in animals 109 , xenoninduced anesthesia 66 , and lithium treatment for mania 39 . Our work is thus another piece of evidence that quantum effects may play essential roles in the brain's functionalities [110][111][112][113][114][115][116][117][118][119][120][121] .

Methods
Quantum effects and chemical oscillator. The effects of applied magnetic fields and hyperfine interactions can be introduced, following the work of Player et al. 71 , by assuming that the FAD signaling of CRY to TIM and hence the TIM degradation in the CC process proceed by RPs, shown in Eq. 6: where rate constants k 1 and k −1 are assumed to conserve electron spin. External magnetic fields and the hyperfine interactions can influence the overall rate of production of FADH · by altering the extent and timing of coherent singlet/triplet interconversion in RP and so changing the probability that it reacts to give FADH · rather than returning to FAD * . Based on a spin dynamics calculation, one can describe the effect of applied magnetic fields and the HFIs on the kinetics of the CC simply by modifying the rate constant k p3 122-125 which corresponds to the degradation of TIM in Eqs. (3) and (4). The spin dynamics of the RP in Eq. (6) can be written as follows: where L is the Liouvillian, [..., ...] and {..., ...} are the commutator and anti-commutator operators, ρ(t) is the spin density operator of the RP system; its trace, Tr[ρ(t)] , equals the concentration of RPs divided by the fixed concentration of FAD * in Eq. 6. As RPs are short-lived intermediates, their concentrations are very low, and hence one can obtain the steady-state solutions as follows (see Ref. 71 ): where is the singlet yield of the RPM. It is, therefore, possible to introduce the singlet yield of the RPM to the chemical reaction by modifying the rate k f . In the CC Eqs. (3) and (4) the corresponding rate is k p3 , which is 0.1 h −1 for the natural cycle of the clock without blue light illumination and 0.085 h −1 for blue light illumination. The singlet yield effects on k p3 can be written as follows: where k ′ p3 , S , and ′ S are the modified rate constant k p3 , the singlet yield with no quantum effects, and the singlet yield resulted from quantum effects due to the Zeeman and/or hyperfine interactions, respectively.

Data availability
The generated datasets and computational analysis are available from the corresponding author on reasonable request.