Radical pairs may play a role in microtubule reorganization

The exact mechanism behind general anesthesia remains an open question in neuroscience. It has been proposed that anesthetics selectively prevent consciousness and memory via acting on microtubules (MTs). It is known that the magnetic field modulates MT organization. A recent study shows that a radical pair model can explain the isotope effect in xenon-induced anesthesia and predicts magnetic field effects on anesthetic potency. Further, reactive oxygen species are also implicated in MT stability and anesthesia. Based on a simple radical pair mechanism model and a simple mathematical model of MT organization, we show that magnetic fields can modulate spin dynamics of naturally occurring radical pairs in MT. We propose that the spin dynamics influence a rate in the reaction cycle, which translates into a change in the MT density. We can reproduce magnetic field effects on the MT concentration that have been observed. Our model also predicts additional effects at slightly higher fields. Our model further predicts that the effect of zinc on the MT density exhibits isotopic dependence. The findings of this work make a connection between microtubule-based and radical pair-based quantum theories of consciousness.

Quantum physics has been proposed to be part of the solution for the mystery of consciousness. In particular the holistic character of quantum entanglement might provide an answer to the binding problem 14 . In the 1990s, Penrose and Hameroff proposed a theory of consciousness based on quantum computations in MTs [15][16][17][18] . Computational modeling suggested that electron resonance transfer among aromatic amino acid tryptophan (Trp) rings in tubulin (subunits of MTs) in a quantum electronic process could play roles in consciousness 19 . Craddock et al. showed that anesthetic molecules might bind in the same regions and hence result in loss of consciousness 20 . In a recent experiment, Zhang et al. observed a connection between electronic states and vibrational states in tubulin and MTs 21 . However, quantum electronic coherence beyond ultrafast timescales demands more supporting evidence and has been recently challenged experimentally 22 . In contrast, quantum spin coherence could be preserved for much longer timescales 23 . For example, Fisher has proposed that phosphorus nuclear spins could be entangled in networks of Posner molecules, Ca9(PO4) 6 , which could form the basis of a quantum mechanism for neural processing in the brain 24 . However, this particular spin-based model also requires more supporting evidence and recently has faced experimental challenges 25 .
It is known that magnetic fields (MFs) can influence different brain functions [26][27][28][29][30][31][32] . Recently, it has been shown that shielding the geomagnetic field-exposure to hypomagnetic field (HMF)-influences adult hippocampal neurogenesis and hippocampus-dependent cognition in mice, where reactive oxygen species (ROS) are implicated 33 . There exists a considerable amount of evidence showing that MFs affect MTs [34][35][36][37][38][39] . Wang et al. show that exposure to HMF caused tubulin assembly disorder 40 . Moreover, Wu et al. observe that low-frequency sub-millitesla MF modulates the density of MTs in cells 41 . All these observations establish the magnetosensitivity of MTs for wide ranges of MF strengths.
Magnetosensitive reactions often involve radical molecules-transient molecules with an odd number of electrons 42 . A radical pair is a short-lived reaction intermediate comprising two radicals formed in non-equilibrium states whose unpaired electron spins may be in a superposition of singlet (S) and triplet (T) states 43 , depending on the parent molecule's spin configuration 44 . The radical pair mechanism (RPM) is the most promising explanation for weak magnetic field effects on chemical reactivity 45 . Schulten was the first to propose the RPM to explain the magnetoreception of migratory birds 46 , and to date, the RPM is the most well-established model for this phenomenon 47 . Recently, it has also been proposed that RPM may explain xenon induced general anesthesia 13 , lithium effects on hyperactivity 48 and the magnetic field effects on the circadian clock 49 .
Evidence suggests that oxidative stress is vital for regulating actin and MT dynamics 63 . MTs contain specific amino acid residues, including Trp, tyrosine (Tyr), and phenylalanine (Phe), susceptible to oxidation. This, in turn, affects the ability of MT to polymerize and causes the severing of actin microfilaments in neuronal and non-neuronal cells. Contrarily, ROS inhibition causes aberrations in actin polymerization, decreases neurite outgrowth, and affects neurons' normal development and polarization.
To be magnetically sensitive, tubulin must have (para)magnetic agents. With an unpaired electron, radicals are paramagnetic and could serve as magneto-sensing molecules. Trp and O2 .− radicals are well-known radicals, particularly in animal avian magnetoreception, in the form of [ Trp .+ . . . FAD .− ] or [ FADH . . . . O2 .− ], wherein the latter case FADH . is the donor and O2 .− acceptor while in the former case Trp .+ is the donor and FAD .− is the acceptor 45,47,64,65 . It is thought that [ Trp .+ . . . FAD .− ] is formed in a photo-induced process. This case is similar to the NMDA receptor (NMDAR) in Xenon-induced anesthesia 13 . In both tubulin and the NMDAR, Trp is one of the key aromatic molecules 66,67 . It is also known that Trp can absorb environmental photons to reach an excited state, and it is considered in the excitation coherency in tubulins 68 . Furthermore, it is shown that Trp and superoxide can be involved in an electron transfer process 69,70 . For instance, in the context of the bird's magnetic compass, a new study suggests that involvement of other radicals rather than the conventional ones is plausible 71 . Moreover, this model could be adapted for other RP complexes in the microtubule dynamics, e.g. involving flavin and Tyr. Such radical pairs can be formed via flavoenzymes 72,73 .
Various studies have proposed mathematical models for the dynamics and stability of MTs 74,75 . Craddock et al. show that the dynamics of MT can be framed in a simple kinetic model 76 . In the context of the RPM, Player et al. show that quantum effects can be introduced to a kinetic model by considering the quantum effects on the corresponding reaction rates in the chemical equations 77 . Taking the same approach, a new study shows that quantum effects can directly modulate the period of the circadian clock in Drosophila, where the spin dynamics of the RPs are the key elements 49 .
In the present work, we propose that there are naturally occurring RPs in the form of TrpH .+ and O2 .− , which is an important paramagnetic ROS, that play important roles in MT organization, as shown in Fig. 1. Oxygen is abundant, and tryptophan is an essential aromatic ring for tubulin; thus, the formation of such radical pairs can occur broadly. Our model predicts that the applied magnetic fields alter the spin dynamics of such RPs, similar to Ref. 13 , and hence modulate the assembly of MTs in cytoskeleton. It is also known that changes in zinc ion concentration in neurons influence the stability of polymerized MTs 76 . Following the work of Craddock et al. 76 , our model further predicts that the effect of zinc on the MT density exhibits isotopic dependence.
In the following, we review the experimental results for the effects of applied magnetic field 40,41 on the density of MTs. We then describe the quantum spin dynamics of our radical pair model and a simple kinetic model for the MT dynamics. Next, we introduce the quantum effect to the density of the MT, and we show that our model can reproduce the observed magnetic field effects; it further makes new predictions for experiments using lowfrequency MFs. Lastly, the model predicts that the zinc effect on the MT density is isotope dependent. caused disorders in tubulin self-assembly 40 . They show that the absorbance at 350 nm, which is for monitoring tubulin self-assembly, was altered by exposure to 30 min HMF. Average gray volume per cell was reported based on the amount of fluorescence in each cell. About 95% of tubulin were assembled in the GMF, while much less of the tubulin assembled ( ∼ 64% ) in the HMF. In that work, the magnitude of the residual GMF was 10-100 nT.

Radical pair model calculations.
We develop an RP model to reproduce the HMF effects on the MT density observed by Wang et al. 40 . Among the amino acids in MTs, as mentioned above, Trp is redox active 70 , as shown by its involvement in RP formation in the context of cryptochrome, and could feasibly participate in the creation of RPs. We propose that the magnetic field interacts with the spins of RPs on Trp and superoxide. Note, that superoxide is thought to form RPs with flavins in other contexts 78 . The correlated spins of RP are assumed to be in the [ TrpH .+ . . . O2 .− ] form, following Ref. 13 , where the unpaired electron on each molecule couples to the nuclear spins in the corresponding molecule. Oxygen has zero nuclear spin and thus zero coupling with its nucleus.
We consider a simplified system in which the unpaired electron on TrpH .+ is coupled to the Trp's β-proton with the largest isotropic HF coupling constant (HFCC) of 1.6046 mT 79 among all the nuclei in Trp. We consider only Zeeman and HF interactions 45,80 . For the RPs, we assume the g-values of a free electron (which is an excellent approximation in the low-field regime that we are considering here). The Hamiltonian for the RP system reads: where Ŝ A and Ŝ B are the spin operators of radical electron on TrpH .+ and O2 . , respectively, Î A is the nuclear spin operator of TrpH .+ 's β-proton, a A is HFCC, and ω is the Larmor precession frequency of the electrons due to the Zeeman effect. In the model presented here, for zinc effects, a B corresponds to the nuclear spin of zinc. We assumed the initial state of RPs to be singlet states (see the "Discussion" section).
The spin density matrix of RPs can be calculated using the Haberkorn master equation, where the kinetic reaction rates for singlet and triplet products are assumed to be equal 43,81,82 . For simplicity, the kinetic reaction rates were assigned equal first-order rate constants, k. Introducing spin relaxation phenomenologically 83,84 , the fractional triplet yield can be calculated as: where M 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 rate (relaxation rate). In this model, we assumed the reaction rates for singlet and triplet have the same values. Of note, the singlet and triplet product of the RP system in [ TrpH .+ . . . O2 .− ] are H2O2 and O2 −78 , respectively, which are the major ROS in redox regulation of biological activities and signaling 85 .
Here we look at the dependence of the triplet yield on changes in the strength of the external static magnetic field for the [ TrpH .+ . . . O2 .− ] radical complex, as shown in Fig. 2, for k = 10 6 s −1 , and r = 10 5 s −1 with a A = 1.6046 mT. One can notice a fairly strong HMF effect (the triplet yield goes from around 60% to around 40%). Our choices for the rate constants are discussed below and in the "Discussion" section. Quantum effects on microtubule density. Here we explain how quantum effects can play an essential role in tubulin polymerization.
Spin is a magnetic moment and hence obeys the spin conservation. This leads to spin selectivity of chemical reactions. In other words, depending on whether radical pairs are in a singlet or triplet state, the chemical reaction will take different paths. It thus results in modulating the reaction rates. The work of Wang et al. 40 shows that the absence of the GMF decreased the tubulin polymerization. Thus, it is natural to assume that effects of applied magnetic fields and hyperfine interactions can be incorporated into the classical model Eq. (4) by influencing the polymerization rate, k p , similar to other studies for introducing zinc effect into the stability of polymerized MTs 76 . The key assumption in our model is that this rate is influenced by a RP reaction. The effect of the triplet yield change on k p reads: where k ′ p , T , and ′ T are the modified rate constant k p , the triplet yield under natural quantum effects (only GMF and no isotope effects), and the triplet yield resulting from quantum effects due to the external MF effects and hyperfine interactions from isotope effects, respectively.
Magnetic field effects on microtubule density. Here, we look at the explicit effects of an applied magnetic field on the density of microtubules. Using Eq. (4), we explore the parameter space of relaxation rate r and recombination rate k in order to investigate the effects of shielding geomagnetic field on MT's density. Wang et al. report that the ratio MT density of geomagnetic field over hypomagnetic field is about 1.48 40 . Figure 3 show that the ratio of the MT density of GMF over HMF can reach above 1.3, which has the right trend compared to the experimental findings. However the uncertainty of the experiment was not reported. Our model predicts a magnetic dependence of the MT density. Figure 4 show the dependence of the MT density ratio of in GMF compared to applied static magnetic field, for a A = 1.6046 mT based the RP complex of [ TrpH .+ . . . O2 .− ], k = 10 6 s −1 , and r = 10 5 s −1 . Figure 4 indicates that exposure to static magnetic fields stronger than the geomagnetic field could Figure 2. The dependence of the triplet yield of the [ TrpH .+ . . . O2 .− ] complex on applied static magnetic field for r = 10 5 s −1 , k = 10 6 s −1 , a A = 1.6046 mT. The triplet yield goes from around 60% to around 40% by changing the magnetic field strength from GMF to HMF. The strong HMF effect is further emphasized by the inset. www.nature.com/scientificreports/ decrease the the microtubule density ratio. The maximum MT density occurs around 0.05 mT, which is in the range of the GMF.
Zinc isotope effect on microtubule density. The RPM typically leads to isotope effects 13,48 , and thus an isotope effect would be a good test of our proposal. It is known that alterations in zinc ion concentration in neurons influence the stability of polymerized MTs 76 . Zn is a positively charged ion. Thus it is natural to assume that Zn 2+ couples with a molecule with a negative charge. Among all stable isotopes of zinc, only 67 Zn has a nuclear spin of I B = − 5 2 , with a natural abundance of 4 % . Here we explore the isotope effect of zinc on the density of MTs. In this model nuclear spin of zinc modulates the spin dynamics of the RPs via hyperfine interactions. The Hamiltonian for the RP system here reads: where Î 2 is the nuclear spin operator of Zn +2 , and a B = −11.39 mT is the HFCC of Zn +2 .
We look at the effect of 67 Zn on the MT density. We assume that 67 Zn interacts with the superoxide radical. Our model predicts that administration of 67 Zn increases the density of MTs compared to Zn with zero nuclear spin, as shown in Fig. 5. We explored the parameter space of relaxation rate r and recombination rate k in order to investigate the effects of the 67 Zn treatment on MT's density, for a A = 1.6046 mT and a B = −11.39 mT, as shown in Fig. 5. Figure 6 shows the dependence of the MT density ratio of the administration of Zn over 67 Zn on the strength of applied magnetic field based on the RP complex of [ TrpH .+ . . . O2 .− ]. The geomagnetic field is 0.05 mT. The magnetic field modulates rates k d and k p for r = 10 5 s −1 and k = 10 6 s −1 . Our model predicts that administering 67 Zn increases the MT density compared to Zn without nuclear spin.  www.nature.com/scientificreports/ DFT analysis. The ORCA package 86 was used for our Zn 2+ -O2 .− DFT calculations, and the molecular structure was optimized using PBE0/def2-TZVP. The orbitals obtained from the optimization calculations were used to calculate orbital energies as well as the hyperfine coupling constant a B . Using RI-B2GP-PLYP/def2-QZVPP 87 , we obtained a 2 = −11.39 mT. In these calculations, relativistic effects were treated by a scalar relativistic Hamiltonian using the zeroth-order regular approximation (ZORA) 88 . Solvent effects were considered by using the conductor-like polarizable continuum model (CPCM) 89 , with a dielectric constant of 2. The resulting Mulliken charge and spin population of the [ Zn 2+ -O2 .− ] complex indicates that the unpaired electron resides primarily on the O2 molecule but is extended slightly onto the zinc atom, see Table 1. The highest occupied molecular orbital (HOMO) of [ Zn 2+ -O2 .− ] is shown in Fig. 7.

Discussion
In the present work, our main goal was to explore whether an RP model can help explain the magnetic field effects on the MT density. We showed that the quantum effects influence the rates in the chemical kinetics of the MT dynamics, and then this results in a change in the density of MT in cytoskeleton. Our model reproduces the experimental findings of Ref. 40 fairly well, as shown in Fig. 3. Our model further predicts that exposure to static magnetic fields stronger than the geomagnetic field could decrease the microtubule density ratio, with the maximum microtubule density occurs at a magnetic field strength in the range of the geomagnetic field, which might have evolutionary roots. The O2 .− radical is thought to have a fast spin relaxation rate. However, it has also been proposed that this fast spin relaxation can be decreased by reducing the molecular symmetry in the biological environment 64,90-92 , i.e. an asymmetric environment. In this case the degeneracy of the HOMOs ( π g ) of O2 .− will be removed and hence its fast reorientation molecular will be prevented. It is shown experimentally that in frozen solutions 93 and alkali halide crystals 94 such the fast spin relaxation of O2 .− can be hindered. Additionally, Kattnig and Hore show that scavenger species around the superoxide radical can also reduce its fast spin relaxation 95,96 .
The present work predicts that the zinc effects on the MT density exhibits an isotope dependent manner. Isotope effects are generally a good indication for radical pairs. In addition, magnesium (Mg) is required for MT stability and function [97][98][99] . Furthermore, Buchachenko et al. showed that adenosine triphosphate (ATP) production was more than two folds in the presence of 25 Mg compared to 24 Mg in purified pig skeletal muscle PGK 100 . Hence, it would be interesting to perform an experiment to probe Mg isotope effects in MT stability.
The ground state of the oxygen molecule is a triplet state. For this reason, oxygen-based radical pairs are often assumed to start from triplet states. However, if the dioxygen molecule goes to its singlet state, the initial state of the radical pairs will be a singlet state. Singlet oxygen is generated naturally 101,104-106 and therefore there is also the possibility of the pair being created from singlet oxygen. In this case, the initial state of the radical pairs will be singlet states. Furthermore, spin-orbit interactions lead to the non-radiative transition between two electronic states with different spin multiplicity ( e.g. singlet and triplet)-intersystem crossing (ISC), which is ubiquitous and important in fields ranging from chemical physics to chemical biology, including chiral molecules 70,[107][108][109][110][111] .
A well-known indication of the pathogenesis of tauopathy-loss-of-function effects on the microtubules and the gain-of-function effects of the toxic tau species-is oxidative stress, which contributes to tau phosphorylation and the formation of neurofibrillary tangles 112 . However, the mechanisms behind the connection between reactive oxygen species (ROS) generated by oxidative stress and tau hyperphosphorylation are elusive 68 . Further, redox signaling and oxidative stress regulate cytoskeletal dynamics 63 . Proper balances between chemical reduction and oxidation (known as redox balance) are crucial for normal cellular physiology. Imbalances in the production of oxidative species lead to DNA damage, lipid peroxidation, and aberrant post-translational modification of proteins, which could induce injury, cell death, and disease 85 . These findings further emphasize the role of radicals and likely the need for a RPM in the context of the brain. Our proposed RPM model for magnetic field effects and Zn isotope effects on the microtubule organization includes superoxide radical.
It is also worth mentioning that any changes in the natural system could lead to various effects and complexify the entire system. Nonetheless, changes in superoxide concentration would likely affect the polymerization rate. This is in line with other observations in the work of Zhang and colleagues, where they reported that exposure to hypomagnetic fields caused hippocampal neurogenesis in mice to hypomagnetic fields caused hippocampal neurogenesis in mice 33 ; however, this adverse effect was reversible by elevating ROS levels through pharmacological inhibition of ROS removal. Thus, it would be interesting to test whether modulating ROS levels pharmacologically can influence the HMF effects on tubulin polymerization.
Microtubules not only play crucial roles in cell shape, cell transport, cell motility, and cell division, but also are important targets for curing devastating diseases, such as Alzheimer's disease [113][114][115] , Parkinson's diseases 116,117 , and cancer [118][119][120] . The dynamics of MAPs signaling proteins play critical roles in the MT network, hence in the process of synaptic plasticity and brain function. Studies suggest that memory is encoded in MT of neuronal www.nature.com/scientificreports/ dendrites and cell bodies. Anesthetic causes loss of consciousness and memory formation via acting on MTs [121][122][123] . Disintegration and separation of the MT-associated protein tau have been observed in memory neurodegenerative diseases and disorders, e.g., in AD 124 ; MT-stabilizers are currently the target for curing such diseases 62,125,126 . However, the underlying mechanism for such diseases is mostly unknown. Thus this project also paves a potential path to study other functionalities of the body and the brain connected to MTs in the light of the RPM.
In conclusion, our results suggest that quantum effects may underlie the magnetic field effects on the microtubule dynamics. This is likely a similar mechanism to those behind magnetoreception in animals 47 , xenon-induced general anesthesia 13 , lithium treatment for mania 48 , the magnetic field effects on the circadian clock 49 . Our work is thus another piece of evidence that quantum entanglement 14,24,[127][128][129][130][131][132][133][134] may play essential roles in the brain's functions, anesthesia, and consciousness 15,135 . Particularly, the photo-emission of singlet oxygen could serve as quantum messengers to establish long-distance connections 136 that might be essential for consciousness. Our work also provides a potential connection between microtubule-based and spin-based quantum approaches to consciousness.

Data availability
The generated datasets and computational analysis are available from the corresponding author on reasonable request. www.nature.com/scientificreports/