corresponding to the case of $\Gamma_{\downarrow\uparrow}=0$. We also observe
this trend in a systematic comparison of the fidelity at the Bell-state time
for a range of $\Gamma_{\downarrow\uparrow}/\Gamma_{\uparrow\downarrow}$
(Figs. 14(e) and 14(f)), where the light-shift gate fidelity appears
relatively insensitive to the imbalance of absorption/decay processes, while
the performance of the Mølmer-Sørenson gate can be drastically altered by the
value of this quantity.

Figure 14: Comparison of the two-qubit gate fidelity for the light-shift gate
(left) and Mølmer-Sørenson gate (right). (a, b) Sample dynamics for
$\Gamma_{+}=0.05J$ (red), $0.25J$ (green), with solid and dotted lines
corresponding to the case of $\Gamma_{-}=0$ and $\Gamma_{-}=\Gamma_{+}$,
respectively ($\Gamma_{el}=0$). Dashed lines correspond to $\Gamma_{el}=0.05J$
(red), $0.25J$ (green), with $\Gamma_{+}=0$. Vertical lines denote Bell-state
time $tJ=\pi/2$. (c, d) Fidelity at the Bell-state time for a range of
$\Gamma_{el}$ and various values of $\Gamma_{+}$, with solid and dotted lines
corresponding to $\Gamma_{-}=0$ and $\Gamma_{-}=\Gamma_{+}$, respectively. (e,
f) Fidelity at the Bell-state time for a range of $\Gamma_{+}$ and
$\Gamma_{\downarrow\uparrow}/\Gamma_{\uparrow\downarrow}$, with
$\Gamma_{el}=0$.

## References

  * Bruzewicz  _et al._ [2019] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Applied Physics Reviews 6, 021314 (2019).
  * Monroe  _et al._ [2021] C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Reviews of Modern Physics 93, 025001 (2021).
  * Brewer  _et al._ [2019] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, 27Al+ quantum-logic clock with a systematic uncertainty below $10^{-18}$, Physical Review Letters 123, 033201 (2019).
  * Gilmore  _et al._ [2021] K. A. Gilmore, M. Affolter, R. J. Lewis-Swan, D. Barberena, E. Jordan, A. M. Rey, and J. J. Bollinger, Quantum-enhanced sensing of displacements and electric fields with two-dimensional trapped-ion crystals, Science 373, 673 (2021).
  * Roos  _et al._ [2006] C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, ‘Designer atoms’ for quantum metrology, Nature (London) 443, 316 (2006).
  * Wang  _et al._ [2021] P. Wang, C.-Y. Luan, M. Qiao, M. Um, J. Zhang, Y. Wang, X. Yuan, M. Gu, J. Zhang, and K. Kim, Single ion qubit with estimated coherence time exceeding one hour, Nature Communications 12, 233 (2021).
  * Ballance  _et al._ [2016] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, High-fidelity quantum logic gates using trapped-ion hyperfine qubits, Physical Review Letters 117, 060504 (2016).
  * Gaebler  _et al._ [2016] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, High-fidelity universal gate set for 9Be+ ion qubits, Physical Review Letters 117, 060505 (2016).
  * Sutherland  _et al._ [2020] R. T. Sutherland, R. Srinivas, S. C. Burd, H. M. Knaack, A. C. Wilson, D. J. Wineland, D. Leibfried, D. T. C. Allcock, D. H. Slichter, and S. B. Libby, Laser-free trapped-ion entangling gates with simultaneous insensitivity to qubit and motional decoherence, Physical Review A 101, 042334 (2020).
  * Tinkey  _et al._ [2022] H. N. Tinkey, C. R. Clark, B. C. Sawyer, and K. R. Brown, Transport-enabled entangling gate for trapped ions, Physical Review Letters 128, 050502 (2022).
  * Clark  _et al._ [2021] C. R. Clark, H. N. Tinkey, B. C. Sawyer, A. M. Meier, K. A. Burkhardt, C. M. Seck, C. M. Shappert, N. D. Guise, C. E. Volin, S. D. Fallek, H. T. Hayden, W. G. Rellergert, and K. R. Brown, High-fidelity Bell-state preparation with 40Ca+ optical qubits, Physical Review Letters 127, 130505 (2021).
  * Pagano  _et al._ [2018] G. Pagano, P. W. Hess, H. B. Kaplan, W. L. Tan, P. Richerme, P. Becker, A. Kyprianidis, J. Zhang, E. Birckelbaw, M. R. Hernandez, Y. Wu, and C. Monroe, Cryogenic trapped-ion system for large scale quantum simulation, Quantum Science and Technology 4, 014004 (2018).
  * Zhang  _et al._ [2017] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature (London) 551, 601 (2017).
  * Kranzl  _et al._ [2022] F. Kranzl, M. K. Joshi, C. Maier, T. Brydges, J. Franke, R. Blatt, and C. F. Roos, Controlling long ion strings for quantum simulation and precision measurements, Physical Review A 105, 052426 (2022).
  * Joshi  _et al._ [2022] M. K. Joshi, F. Kranzl, A. Schuckert, I. Lovas, C. Maier, R. Blatt, M. Knap, and C. F. Roos, Observing emergent hydrodynamics in a long-range quantum magnet, Science 376, 720 (2022).
  * Richerme [2016] P. Richerme, Two-dimensional ion crystals in radio-frequency traps for quantum simulation, Physical Review A 94, 032320 (2016).
  * Bohnet  _et al._ [2016] J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Quantum spin dynamics and entanglement generation with hundreds of trapped ions, Science 352, 1297 (2016).
  * Gärttner  _et al._ [2017] M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nature Physics 13, 781 (2017).
  * Itano  _et al._ [1998] W. M. Itano, J. J. Bollinger, J. N. Tan, B. Jelenković, X.-P. Huang, and D. J. Wineland, Bragg diffraction from crystallized ion plasmas, Science 279, 686 (1998).
  * D’Onofrio  _et al._ [2021] M. D’Onofrio, Y. Xie, A. J. Rasmusson, E. Wolanski, J. Cui, and P. Richerme, Radial two-dimensional ion crystals in a linear Paul trap, Physical Review Letters 127, 020503 (2021).
  * Qiao  _et al._ [2022] M. Qiao, Z. Cai, Y. Wang, B. Du, N. Jin, W. Chen, P. Wang, C. Luan, E. Gao, X. Sun, H. Tian, J. Zhang, and K. Kim, Observing frustrated quantum magnetism in two-dimensional ion crystals, arXiv:2204.07283 (2022).
  * Mielenz  _et al._ [2016] M. Mielenz, H. Kalis, M. Wittemer, F. Hakelberg, U. Warring, R. Schmied, M. Blain, P. Maunz, D. L. Moehring, D. Leibfried, and T. Schaetz, Arrays of individually controlled ions suitable for two-dimensional quantum simulations, Nature Communications 7, ncomms11839 (2016).
  * Wu  _et al._ [2021] Y.-K. Wu, Z.-D. Liu, W.-D. Zhao, and L.-M. Duan, High-fidelity entangling gates in a three-dimensional ion crystal under micromotion, Physical Review A 103, 022419 (2021).
  * Duan [2022] L.-M. Duan, Robust gate design for large ion crystals through excitation of local phonon modes, arXiv:2207.04583 (2022).
  * Kiesenhofer  _et al._ [2023] D. Kiesenhofer, H. Hainzer, A. Zhdanov, P. C. Holz, M. Bock, T. Ollikainen, and C. F. Roos, Controlling two-dimensional Coulomb crystals of more than 100 ions in a monolithic radio-frequency trap, arXiv:2302.00565 (2023).
  * Korenblit  _et al._ [2012] S. Korenblit, D. Kafri, W. C. Campbell, R. Islam, E. E. Edwards, Z.-X. Gong, G.-D. Lin, L.-M. Duan, J. Kim, K. Kim, and C. Monroe, Quantum simulation of spin models on an arbitrary lattice with trapped ions, New Journal of Physics 14, 095024 (2012).
  * Jurcevic  _et al._ [2014] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body system, Nature (London) 511, 202 (2014).
  * Figgatt  _et al._ [2017] C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Debnath, and C. Monroe, Complete 3-qubit Grover search on a programmable quantum computer, Nature Communications 8, 1918 (2017).
  * Joshi  _et al._ [2020] M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions, Physical Review Letters 124, 240505 (2020).
  * Leibfried  _et al._ [2003] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate, Nature (London) 422, 412 (2003).
  * Ge  _et al._ [2019a] W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, M. Foss-Feig, and J. J. Bollinger, Stroboscopic approach to trapped-ion quantum information processing with squeezed phonons, Physical Review A 100, 043417 (2019a).
  * Mølmer and Sørensen [1999] K. Mølmer and A. Sørensen, Multiparticle entanglement of hot trapped ions, Physical Review Letters 82, 1835 (1999).
  * Sørensen and Mølmer [1999] A. Sørensen and K. Mølmer, Quantum computation with ions in thermal motion, Physical Review Letters 82, 1971 (1999).
  * Itano and Wineland [1981] W. M. Itano and D. J. Wineland, Precision measurement of the ground-state hyperfine constant of 25Mg+, Physical Review A 24, 1364 (1981).
  * Britton  _et al._ [2012] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins, Nature (London) 484, 489 (2012).
  * Affolter  _et al._ [2020] M. Affolter, K. A. Gilmore, J. E. Jordan, and J. J. Bollinger, Phase-coherent sensing of the center-of-mass motion of trapped-ion crystals, Physical Review A 102, 052609 (2020).
  * Haljan  _et al._ [2005] P. C. Haljan, K.-A. Brickman, L. Deslauriers, P. J. Lee, and C. Monroe, Spin-dependent forces on trapped ions for phase-stable quantum gates and entangled states of spin and motion, Physical Review Letters 94, 153602 (2005).
  * Cui  _et al._ [2022] K. Cui, J. Valencia, K. T. Boyce, D. R. Leibrandt, and D. B. Hume, Scalable quantum logic spectroscopy, arxiv:2207.11768 10.48550/arxiv.2207.11768 (2022).
  * Uys  _et al._ [2010] H. Uys, M. J. Biercuk, A. P. VanDevender, C. Ospelkaus, D. Meiser, R. Ozeri, and J. J. Bollinger, Decoherence due to elastic Rayleigh scattering, Physical Review Letters 105, 200401 (2010).
  * Ozeri  _et al._ [2007] R. Ozeri, W. M. Itano, R. B. Blakestad, J. Britton, J. Chiaverini, J. D. Jost, C. Langer, D. Leibfried, R. Reichle, S. Seidelin, J. H. Wesenberg, and D. J. Wineland, Errors in trapped-ion quantum gates due to spontaneous photon scattering, Physical Review A 75, 042329 (2007).
  * Moore  _et al._ [2023] I. D. Moore, W. C. Campbell, E. R. Hudson, M. J. Boguslawski, D. J. Wineland, and D. T. C. Allcock, Photon scattering errors during stimulated Raman transitions in trapped-ion qubits, Physical Review A 107, 032413 (2023).
  * Kim  _et al._ [2009] K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, and C. Monroe, Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes, Physical Review Letters 103, 120502 (2009).
  * Tan  _et al._ [2021] W. L. Tan, P. Becker, F. Liu, G. Pagano, K. S. Collins, A. De, L. Feng, H. B. Kaplan, A. Kyprianidis, R. Lundgren, W. Morong, S. Whitsitt, A. V. Gorshkov, and C. Monroe, Domain-wall confinement and dynamics in a quantum simulator, Nature Physics 17 (2021).
  * Blatt and Roos [2012] R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nature Physics 8 (2012).
  * Lin  _et al._ [2011] G.-D. Lin, C. Monroe, and L.-M. Duan, Sharp phase transitions in a small frustrated network of trapped ion spins, Physical Review Letters 106, 230402 (2011).
  * Ge  _et al._ [2019b] W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, J. J. Bollinger, and M. Foss-Feig, Trapped ion quantum information processing with squeezed phonons, Physical Review Letters 122, 030501 (2019b).
  * Ball  _et al._ [2019] H. Ball, C. D. Marciniak, R. N. Wolf, A. T.-H. Hung, K. Pyka, and M. J. Biercuk, Site-resolved imaging of beryllium ion crystals in a high-optical-access Penning trap with inbore optomechanics, Review of Scientific Instruments 90, 053103 (2019).
  * McMahon  _et al._ [2020] B. J. McMahon, C. Volin, W. G. Rellergert, and B. C. Sawyer, Doppler-cooled ions in a compact reconfigurable Penning trap, Physical Review A 101, 013408 (2020), publisher: American Physical Society.
  * Landsman  _et al._ [2019] K. A. Landsman, Y. Wu, P. H. Leung, D. Zhu, N. M. Linke, K. R. Brown, L. Duan, and C. Monroe, Two-qubit entangling gates within arbitrarily long chains of trapped ions, Physical Review A 100, 022332 (2019).
  * Debnath  _et al._ [2016] S. Debnath, N. M. Linke, C. Figgatt, K. A. Landsman, K. Wright, and C. Monroe, Demonstration of a small programmable quantum computer with atomic qubits, Nature (London) 536, 63 (2016).
  * Hayes  _et al._ [2012] D. Hayes, S. M. Clark, S. Debnath, D. Hucul, I. V. Inlek, K. W. Lee, Q. Quraishi, and C. Monroe, Coherent error suppression in multiqubit entangling gates, Physical Review Letters 109, 020503 (2012).
  * Choi  _et al._ [2014] T. Choi, S. Debnath, T. A. Manning, C. Figgatt, Z.-X. Gong, L.-M. Duan, and C. Monroe, Optimal quantum control of multimode couplings between trapped ion qubits for scalable entanglement, Physical Review Letters 112, 190502 (2014).
  * Sawyer and Brown [2021] B. C. Sawyer and K. R. Brown, Wavelength-insensitive, multispecies entangling gate for group-2 atomic ions, Physical Review A 103, 022427 (2021).
  * Thom  _et al._ [2018] J. Thom, B. Yuen, G. Wilpers, E. Riis, and A. G. Sinclair, Intensity stabilisation of optical pulse sequences for coherent control of laser-driven qubits, Applied Physics B 124, 90 (2018).
  * Bethe and Salpeter [1957] H. A. Bethe and E. E. Salpeter,  _Quantum mechanics of one- and two-electron atoms_ (Springer-Verlag, Heidelberg, 1957).
  * Foss-Feig  _et al._ [2013] M. Foss-Feig, K. R. A. Hazzard, J. J. Bollinger, and A. M. Rey, Nonequilibrium dynamics of arbitrary-range Ising models with decoherence: An exact analytic solution, Physical Review A 87, 042101 (2013).


[table]capposition=top

# Radical pairs may play a role in microtubule reorganization

Hadi Zadeh-Haghighi  Department of Physics and Astronomy, University of
Calgary, Calgary, AB, T2N 1N4, Canada  Institute for Quantum Science and
Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada  Hotchkiss
Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada
Christoph Simon  Department of Physics and Astronomy, University of Calgary,
Calgary, AB, T2N 1N4, Canada  Institute for Quantum Science and Technology,
University of Calgary, Calgary, AB, T2N 1N4, Canada  Hotchkiss Brain
Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada

###### Abstract

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 show 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.

## Introduction

The question of how we perceive and experience the world we live in is a
fascinating, long-standing open question in neuroscience, philosophy and
psychology. This conscious experience vanishes during dreamless sleep or under
general anesthesia [1, 2]. Every day millions of surgeries all over the world
would not be possible without anesthetics. However, despite a century of
research, the mechanisms by which anesthetics cause a reversible loss of
consciousness remain obscure [3]. During anesthesia, old memories are
preserved, but no new memory is formed. Furthermore, anesthetics act
relatively selectively on consciousness, as many non-conscious brain
activities, including sensory-evoked potentials, persist during anesthesia.
Understanding anesthesia, apart from its benefits in designing and developing
novel anesthetics, could help explain the mystery of consciousness.

The connections between theories of anesthesia, related to neural correlates
of consciousness, and the Meyer Overton correlation—solubility of anesthetics
in a non-polar, ‘hydrophobic’ medium—has not been completely understood, and
direct anesthetic effects on synaptic receptors are variable and inconsistent.
Further, single-cell organisms perform cognitive activities predominantly by
cytoskeletal microtubules (MTs) and are inhibited by anesthetic gases even
without synapses or networks [4]. Bernard showed that anesthetics act directly
on cytoplasm, depending on cytoskeletal proteins’ dynamics comprising actin
filaments and MTs [5]. Franks and Lieb found that anesthetics act directly
within proteins in non-polar hydrophobic regions [6]. This led some anesthetic
researchers to search for new target protein/molecule for anesthetics.
Moreover, Eckenhoff et al. found that anesthetics bind to 23 membrane proteins
and 34 cytoplasmic proteins including actin, and tubulin [7, 8]. Studies
suggest anesthetics exert their effects via acting on protein reaction
networks involved in neuronal growth, proliferation, division, and
communication, which depend on MTs [9]. Although the affinity of anesthetics
binding to tubulin is a thousand times weaker than to membrane protein, the
abundance of tubulin is thousand to ten thousand times more than membrane
protein sites.

It has also been proposed that anesthetics act on quantum electronic activity
in neuronal hydrophobic regions rather than binding to specific receptors.
Consistently, Turin et al. showed that specific electron spin resonance (ESR)
signals, consistent with free electrons, can be observed during anesthesia
[10]. The same authors proposed that the anesthetic action may involve some
form of electron transfer. Moreover, Li et al. showed experimentally that
isotopes of xenon with non-zero nuclear spin had reduced anesthetic potency in
mice compared with isotopes with no nuclear spin [11]. These findings are
consistent with the idea that different nuclear spins of anesthetics can
modulate the electron transferring process differently. Motivated by this line
of thought, a recent study shows that radical pairs may explain the mechanism
behind xenon-induced anesthesia [12].

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 [13]. In the 1990s, Penrose and
Hameroff proposed a theory of consciousness based on quantum computations in
MTs [14, 15, 16, 17]. 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
[18]. More recently, Craddock et al. showed that anesthetic molecules might
bind in the same regions and hence result in anesthesia [19]. However, quantum
electronic coherence beyond ultrafast timescales demands more supporting
evidence and has been recently challenged experimentally [20]. In contrast,
quantum spin coherence could be preserved for much longer timescales [21]. 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 [22]. However, this
particular spin-based model model also requires more supporting evidence and
recently has faced experimental challenges [23].

It is know that magnetic fields (MFs) can influence different brain functions
[24, 25, 26, 27, 28, 29, 30]. 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
ROS are implicated [31]. There exists a considerable amount of evidence
showing that MFs affect MTs [32, 33, 34, 35, 36, 37]. Wang et al. show that
exposure to HMF caused tubulin assembly disorder [38]. Moreover, Wu et al.
observe that low-frequency sub-millitesla MF modulates the density of MTs in
cells [39]. 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 [40]. 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 [41], depending on the parent molecule’s spin configuration [42].
The radical pair mechanism (RPM) is the most promising explanation for weak
magnetic field effects on chemical reactivity [43]. Schulten was the first to
propose the RPM to explain the magnetoreception of migratory birds [44], and
to date, the RPM is the most well-established model for this phenomenon [45].
Recently, it has also been proposed that RPM may explain xenon induced general
anesthesia [12], lithium effects on hyperactivity [46] and the magnetic field
effects on the circadian clock [47].

MTs are made of $\alpha-\beta$ tubulin heterodimers. The distribution and
organization of MTs in cells are governed by a large number of MT-associated
proteins (MAPs) [48]. MTs play crucial roles in cell shape, cell transport,
cell motility, cell division [49, 50, 51, 52, 53, 54, 55], neuronal polarity
[56], information processing of living systems [57], synaptic activity [58],
regulating the precise timing of nerve spikes [59], and Alzheimer’s disease
(AD) [60].

Evidence suggests that oxidative stress is vital for regulating actin and MT
dynamics [61]. 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.

Various studies have proposed mathematical models for the dynamics and
stability of MTs [62, 63]. Craddock et al. show that the dynamics of MT can be
framed in a simple kinetic model [64]. In the context of the RPM, Player et
al. show that quantum effects can be introduced to the chemical oscillator
model by considering the quantum effects on the corresponding reaction rates
in the chemical equations [65]. 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 PRs are the key elements [47].

Figure 1: Schematic presentation of TrpH^.+ [the aromatic molecule] and O2^.-
[the red molecule] radical pair, considered in the RPM model in the present
work, similar to Ref. [12]. The radical pair undergoes interconversion between
singlet and triplet states. Image rendered using Avogadro
(https://avogadro.cc).

In the present work, we propose that there are naturally occurring RPs in the
form of TrpH^.+ and O2^.-, which is an important ROS, that play important
roles in MT organization, as shown in Fig. 1. Our model predicts that the
applied magnetic fields alter the spin dynamics of such RPs, similar to Ref.
[12], and hence modulate the assembly of MTs in cytoskeleton. It 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 [39, 38] on the density of MTs. We then describe the
quantum spin dynamics of our radical pair model by 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 low-frequency
MFs. Lastly, the model predicts that the zinc effect on the MT density is
isotope dependent.

## Results

### Prior experiment

Wang et al. report that shielding Earth’s geomagnetic field (GMF) (0.025-0.065
mT) caused disorders in tubulin self-assembly [38]. 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 ($\sim$ 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. [38]. Among the amino acids in MTs, as mentioned above, Trp is
redox active [66], 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 [67]. The correlated spins of RP are assumed to be in the [TrpH^.+ …
O2^.-] form, following Ref. [12], 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 $\beta$-proton with the largest isotropic HF coupling
constant (HFCC) of 1.6046 mT [68] among all the nuclei in Trp. We consider
only Zeeman and HF interactions [69, 43]. 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:

$\hat{H}=\omega\hat{S}_{A_{z}}+a_{A}\mathbf{\hat{S}}_{A}.\mathbf{\hat{I}}_{1}+\omega\hat{S}_{B_{z}},$
(1)

where $\mathbf{\hat{S}}_{A}$ and $\mathbf{\hat{S}}_{B}$ are the spin operators
of radical electron on TrpH^.+ and O2^., respectively, $\mathbf{\hat{I}}_{A}$
is the nuclear spin operator of TrpH^.+’s $\beta$-proton, $a_{A}$ is HFCC, and
$\omega$ 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).

Using the Liouville-von Neumann equation for the spin state of the radical
pair, we calculate the singlet yield resulting from the radical pair mechanism
throughout the reaction. The ultimate triplet yield, $\Phi_{T}$, for periods
much greater than the RP lifetime [70] has the following form:

$\Phi_{T}=\frac{3}{4}+\frac{k}{4(k+r)}-\frac{1}{M}\sum_{m=1}^{4M}\sum_{n=1}^{4M}|\bra{m}\hat{P}^{S}\ket{n}|^{2}\frac{k(k+r)}{(k+r)^{2}+(\omega_{m}-\omega_{n})^{2}},$
(2)

where $M$ is the nuclear spin multiplicity, $\hat{P}^{S}$ is the singlet
projection operator, $\ket{m}$ and $\ket{n}$ are eigenstates of $\hat{H}$ with
corresponding eigenenergies of $\omega_{m}$ and $\omega_{n}$, 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^- [67], respectively, which are
the major ROS in redox regulation of biological activities and signaling [71].

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.

### Chemical kinetics model for MT dynamics

We use a simple mathematical model for the dynamics of MT, following the work
of Craddock et al. [64]. This model is based on the interconversion of the
free tubulins, Tu, and the MT in cytoskeleton:

$\ch{Tu<>[k_{\mathrm{p}}][k_{\mathrm{d}}]MT},$ (3)

where $k_{p}$ and $k_{d}$ are the polymerization rate and the depolymerization
rate, respectively. The chemical equation reads as follows:

$\frac{[dMT(t)]}{dt}=k_{p}[Tu(t)]-k_{d}[MT(t)],$ (4)

which yields $[MT(t)]=\frac{k_{p}[P]}{k_{p}+k_{d}}(1-e^{-(k_{p}+k_{d})t})$.

Craddock et al. suggest that changes in the concentration of zinc can be
incorporated by modulating the polymerization rate [64]. In the present work,
it is assumed that the total tubulin protein concentration [P] is 400 $\mu$M,
such that [MT]+[Tu]=[P], $k_{p}$ = 90 s-1 and $k_{d}$ = 150 s-1. These values
are chosen such that we could be able to reproduce the experimental findings,
which are comparable to those in Ref. [64].

### Quantum effects on microtubule density

The effects of triplet yield change can be introduced to the chemical equation
of the MTs by modifying the kinetic rates, following the work of Player et al.
[65, 47]. In the chemical kinetics equations for MT oscillations, Eq. 4, the
corresponding rate is $k_{p}$, which is the MT polymerization rate [64]. 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:

$k^{\prime}_{p}\propto k_{p}\frac{\Phi_{T}^{\prime}}{\Phi_{T}},$ (5)

where $k^{\prime}_{p}$, $\Phi_{T}$, and $\Phi^{\prime}_{T}$ 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 Eqs. 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 [38]. Fig. 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. Fig. 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. Fig. 4 indicates that exposure to
static magnetic fields stronger than the geomagnetic field could 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 [12, 46], 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
[64]. 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 ^67Zn has a nuclear spin of $I_{B}=-\frac{5}{2}$, 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:

$\hat{H}=\omega\hat{S}_{A_{z}}+a_{A}\mathbf{\hat{S}}_{A}.\mathbf{\hat{I}}_{1}+\omega\hat{S}_{B_{z}}+a_{B}\mathbf{\hat{S}}_{B}.\mathbf{\hat{I}}_{2},$
(6)

where $\mathbf{\hat{I}}_{2}$ 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 ^67Zn on the MT density. We assume that ^67Zn
interacts with the superoxide radical. Our model predicts that administration
of ^67Zn 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 ^67Zn
treatment on MT’s density, for $a_{A}=1.6046$ mT and $a_{B}=-11.39$ mT, as
shown in Fig. 5. Fig. 6 shows the dependence of the MT density ratio of the
administration of Zn over ^67Zn 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 ^67Zn increases the
MT density compared to Zn without nuclear spin.

## 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. [38]
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
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 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. It thus would be interesting to perform an experiment to probe
such isotope effects.

The O2^.- radical is thought to have a fast spin relaxation rate [72, 73].
However, it has also been proposed that this fast spin relaxation can be
decreased by the biological environment. Additionally, Kattnig and Hore show
that scavenger species around the superoxide radical can also reduce its fast
spin relaxation [74, 75].

The ground state of the oxygen molecule is singlet, and due to this, it is
often assumed that in the RP complexes superoxide are formed in triplet
states, as opposed to the case considered in the present work, where it is
assumed to be formed as a singlet. Studies have shown the formation of
superoxide from singlet oxygen [66]. Thus it seems reasonable that the initial
state for RP formation could also be its excited singlet state, which is also
which is its excited state (and is also a biologically relevant ROS) [76, 77,
78]. Further, the transition of the initial RP state from triplet to singlet
could also take place due to spin-orbit coupling [79, 80].

Let us also note that 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 [81, 82].

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 [83]. However, the mechanisms behind the
connection between reactive oxygen species (ROS) generated by oxidative stress
and tau hyperphosphorylation are elusive [84]. Further, redox signaling and
oxidative stress regulate cytoskeletal dynamics [61]. 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
[71]. 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.

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 [85, 86, 87], Parkinson’s
diseases [88, 89], and cancer [90, 91, 92]. 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 dendrites and cell bodies. Anesthetic causes loss of
consciousness and memory formation via acting on MTs [93, 94, 95].
Disintegration and separation of the MT-associated protein tau have been
observed in memory neurodegenerative diseases and disorders, e.g., in AD [96];
MT-stabilizers is currently the target for during such diseases [97, 60, 98].
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 [45], xenon-induced
general anesthesia [12], lithium treatment for mania [46], the magnetic field
effects on the circadian clock [47]. Our work is thus another piece of
evidence that quantum entanglement [22, 13, 99, 100, 101, 102, 103, 104, 105,
106] may play essential roles in the brain’s functions, anesthesia, and
consciousness [107, 14]. Particularly, the photo-emission of singlet oxygen
could serve as quantum messengers to establish long-distance connections [108]
that might be essential for consciousness. Our work also provides a potential
connection between microtubule-based and spin-based quantum approaches to
consciousness.

## Methods

### DFT Analysis

The ORCA package [109] 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[110], 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) [111]. Solvent effects were
considered by using the conductor-like polarizable continuum model (CPCM)
[112], 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.

Table 1: Mulliken charge and spin population of [Zn^2+… O2^.-]. Atom | Charge Population | Spin Population
---|---|---
O | -0.203235 | 0.554431
O | -0.203243 | 0.554413
Zn | 1.406478 | -0.108844
Sum | 1 | 1

## Data Availability

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

## References

  * [1] Koch, C., Massimini, M., Boly, M. & Tononi, G.  Neural correlates of consciousness: progress and problems.  _Nature Reviews Neuroscience_ 17, 307–321 (2016).
  * [2] Mashour, G. A.  Integrating the science of consciousness and anesthesia.  _Anesthesia & Analgesia_ 103, 975–982 (2006).
  * [3] Brown, E. N., Purdon, P. L. & Van Dort, C. J.  General anesthesia and altered states of arousal: a systems neuroscience analysis.  _Annual review of neuroscience_ 34, 601–628 (2011).
  * [4] JA Craddock, T., R Hameroff, S., T Ayoub, A., Klobukowski, M. & A Tuszynski, J.  Anesthetics act in quantum channels in brain microtubules to prevent consciousness.  _Current topics in medicinal chemistry_ 15, 523–533 (2015).
  * [5] Perouansky, M.  The quest for a unified model of anesthetic action: a century in claude bernard’s shadow.  _The Journal of the American Society of Anesthesiologists_ 117, 465–474 (2012).
  * [6] Franks, N. & Lieb, W.  Do general anaesthetics act by competitive binding to specific receptors?  _Nature_ 310, 599–601 (1984).
  * [7] Xi, J., Liu, R., Asbury, G. R., Eckenhoff, M. F. & Eckenhoff, R. G.  Inhalational anesthetic-binding proteins in rat neuronal membranes.  _Journal of Biological Chemistry_ 279, 19628–19633 (2004).
  * [8] Pan, J. Z., Xi, J., Tobias, J. W., Eckenhoff, M. F. & Eckenhoff, R. G.  Halothane binding proteome in human brain cortex.  _Journal of proteome research_ 6, 582–592 (2007).
  * [9] Pan, J. Z., Xi, J., Eckenhoff, M. F. & Eckenhoff, R. G.  Inhaled anesthetics elicit region-specific changes in protein expression in mammalian brain.  _Proteomics_ 8, 2983–2992 (2008).
  * [10] Turin, L., Skoulakis, E. M. & Horsfield, A. P.  Electron spin changes during general anesthesia in drosophila.  _Proceedings of the National Academy of Sciences_ 111, E3524–E3533 (2014).
  * [11] Li, N. _et al._ Nuclear spin attenuates the anesthetic potency of xenon isotopes in mice: implications for the mechanisms of anesthesia and consciousness.  _Anesthesiology_ 129, 271–277 (2018).
  * [12] Smith, J., Zadeh-Haghighi, H., Salahub, D. & Simon, C.  Radical pairs may play a role in xenon-induced general anesthesia.  _Scientific Reports_ 11, 6287 (2021).
  * [13] Simon, C.  Can quantum physics help solve the hard problem of consciousness?  _Journal of Consciousness Studies_ 26, 204–218 (2019).
  * [14] Hameroff, S. & Penrose, R.  Consciousness in the universe: A review of the ‘orch or’theory.  _Physics of life reviews_ 11, 39–78 (2014).
  * [15] Stuart, H.  Quantum computation in brain microtubules? the penrose–hameroff ‘orch or ‘model of consciousness.  _Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences_ 356, 1869–1896 (1998).
  * [16] Matsuno, K.  The internalist enterprise on constructing cell motility in a bottom-up manner.  _BioSystems_ 61, 115–124 (2001).
  * [17] Hagan, S., Hameroff, S. R. & Tuszyński, J. A.  Quantum computation in brain microtubules: Decoherence and biological feasibility.  _Physical Review E_ 65, 061901 (2002).
  * [18] Hameroff, S., Nip, A., Porter, M. & Tuszynski, J.  Conduction pathways in microtubules, biological quantum computation, and consciousness.  _Biosystems_ 64, 149–168 (2002).
  * [19] Craddock, T. J. _et al._ Computational predictions of volatile anesthetic interactions with the microtubule cytoskeleton: Implications for side effects of general anesthesia.  _PLoS One_ 7, e37251 (2012).
  * [20] Cao, J. _et al._ Quantum biology revisited.  _Science advances_ 6, eaaz4888 (2020).
  * [21] Hu, H. & Wu, M.  Spin-mediated consciousness theory: possible roles of neural membrane nuclear spin ensembles and paramagnetic oxygen.  _Medical Hypotheses_ 63, 633–646 (2004).
  * [22] Fisher, M. P.  Quantum cognition: The possibility of processing with nuclear spins in the brain.  _Annals of Physics_ 362, 593–602 (2015).
  * [23] Chen, R., Li, N., Qian, H., Zhao, R.-H. & Zhang, S.-H.  Experimental evidence refuting the assumption of phosphorus-31 nuclear-spin entanglement-mediated consciousness.  _Journal of Integrative Neuroscience_ 19, 595–600 (2020).
  * [24] Romero, M. C., Davare, M., Armendariz, M. & Janssen, P.  Neural effects of transcranial magnetic stimulation at the single-cell level.  _Nature communications_ 10, 1–11 (2019).
  * [25] Wang, R., Fan, Y. & Wu, Y.  Spontaneous electromagnetic induction promotes the formation of economical neuronal network structure via self-organization process.  _Scientific reports_ 9, 1–13 (2019).
  * [26] Lenz, M. _et al._ Repetitive magnetic stimulation induces plasticity of inhibitory synapses.  _Nature communications_ 7, 1–13 (2016).
  * [27] Rauš Balind, S., Selaković, V., Radenović, L., Prolić, Z. & Janać, B.  Extremely low frequency magnetic field (50 hz, 0.5 mt) reduces oxidative stress in the brain of gerbils submitted to global cerebral ischemia.  _PLoS One_ 9, e88921 (2014).
  * [28] Pacini, S. _et al._ Effect of 0.2 t static magnetic field on human neurons: remodeling and inhibition of signal transduction without genome instability.  _Neuroscience letters_ 267, 185–188 (1999).
  * [29] Grehl, S. _et al._ Cellular and molecular changes to cortical neurons following low intensity repetitive magnetic stimulation at different frequencies.  _Brain stimulation_ 8, 114–123 (2015).
  * [30] Manikonda, P. K. _et al._ Influence of extremely low frequency magnetic fields on ca2+ signaling and nmda receptor functions in rat hippocampus.  _Neuroscience letters_ 413, 145–149 (2007).
  * [31] Zhang, B. _et al._ Long-term exposure to a hypomagnetic field attenuates adult hippocampal neurogenesis and cognition.  _Nature communications_ 12, 1–17 (2021).
  * [32] Vassilev, P. M., Dronzine, R. T., Vassileva, M. P. & Georgiev, G. A.  Parallel arrays of microtubles formed in electric and magnetic fields.  _Bioscience reports_ 2, 1025–1029 (1982).
  * [33] Glade, N. & Tabony, J.  Brief exposure to high magnetic fields determines microtubule self-organisation by reaction–diffusion processes.  _Biophysical chemistry_ 115, 29–35 (2005).
  * [34] Bras, W. _et al._ The susceptibility of pure tubulin to high magnetic fields: a magnetic birefringence and x-ray fiber diffraction study.  _Biophysical journal_ 74, 1509–1521 (1998).
  * [35] Zhang, L. _et al._ 27 t ultra-high static magnetic field changes orientation and morphology of mitotic spindles in human cells.  _Elife_ 6, e22911 (2017).
  * [36] Qian, A.-R. _et al._ Large gradient high magnetic field affects the association of macf1 with actin and microtubule cytoskeleton.  _Bioelectromagnetics: Journal of the Bioelectromagnetics Society, The Society for Physical Regulation in Biology and Medicine, The European Bioelectromagnetics Association_ 30, 545–555 (2009).
  * [37] Luo, Y. _et al._ Moderate intensity static magnetic fields affect mitotic spindles and increase the antitumor efficacy of 5-fu and taxol.  _Bioelectrochemistry_ 109, 31–40 (2016).
  * [38] Wang, D. L., Wang, X. S., Xiao, R., Liu, Y. & He, R. Q.  Tubulin assembly is disordered in a hypogeomagnetic field.  _Biochemical and Biophysical Research Communications_ 376, 363–368 (2008).
  * [39] Wu, X. _et al._ Weak power frequency magnetic fields induce microtubule cytoskeleton reorganization depending on the epidermal growth factor receptor and the calcium related signaling.  _PloS one_ 13, e0205569 (2018).
  * [40] Rodgers, C. T. & Hore, P. J.  Chemical magnetoreception in birds: the radical pair mechanism.  _Proceedings of the National Academy of Sciences_ 106, 353–360 (2009).
  * [41] Steiner, U. E. & Ulrich, T.  Magnetic field effects in chemical kinetics and related phenomena.  _Chemical Reviews_ 89, 51–147 (1989).
  * [42] Timmel, C. R., Till, U., Brocklehurst, B., Mclauchlan, K. A. & Hore, P. J.  Effects of weak magnetic fields on free radical recombination reactions.  _Molecular Physics_ 95, 71–89 (1998).
  * [43] Hore, P. J. & Mouritsen, H.  The radical-pair mechanism of magnetoreception.  _Annual review of biophysics_ 45, 299–344 (2016).
  * [44] Schulten, K., Swenberg, C. E. & Weller, A.  A biomagnetic sensory mechanism based on magnetic field modulated coherent electron spin motion.  _Zeitschrift für Physikalische Chemie_ 111, 1–5 (1978).
  * [45] Xu, J. _et al._ Magnetic sensitivity of cryptochrome 4 from a migratory songbird.  _Nature_ 594, 535–540 (2021).
  * [46] Zadeh-Haghighi, H. & Simon, C.  Entangled radicals may explain lithium effects on hyperactivity.  _Scientific Reports_ 11, 12121 (2021).
  * [47] Zadeh-Haghighi, H. & Simon, C.  Radical pairs can explain magnetic field and lithium effects on the circadian clock.  _arXiv preprint arXiv:2107.10677_ (2021).
  * [48] Brouhard, G. J. & Rice, L. M.  Microtubule dynamics: an interplay of biochemistry and mechanics.  _Nature reviews Molecular cell biology_ 19, 451–463 (2018).
  * [49] Nogales, E.  Structural insights into microtubule function.  _Annual review of biophysics and biomolecular structure_ 30, 397–420 (2001).
  * [50] Wu, J. & Akhmanova, A.  Microtubule-organizing centers.  _Annual review of cell and developmental biology_ 33, 51–75 (2017).
  * [51] Akhmanova, A. & Steinmetz, M. O.  Tracking the ends: a dynamic protein network controls the fate of microtubule tips.  _Nature reviews Molecular cell biology_ 9, 309–322 (2008).
  * [52] Akhmanova, A. & Steinmetz, M. O.  Control of microtubule organization and dynamics: two ends in the limelight.  _Nature reviews Molecular cell biology_ 16, 711–726 (2015).
  * [53] Redwine, W. B. _et al._ Structural basis for microtubule binding and release by dynein.  _Science_ 337, 1532–1536 (2012).
  * [54] Monroy, B. Y. _et al._ A combinatorial map code dictates polarized microtubule transport.  _Developmental cell_ 53, 60–72 (2020).
  * [55] Monroy, B. Y. _et al._ Competition between microtubule-associated proteins directs motor transport.  _Nature communications_ 9, 1–12 (2018).
  * [56] Burute, M., Jansen, K. I., Mihajlovic, M., Vermonden, T. & Kapitein, L.  Matrix elasticity gradients guide neuronal polarity by controlling microtubule network mobility.  _bioRxiv_ (2021).
  * [57] Sanchez-Castro, N. _et al._ Microtubules as one-dimensional crystals: Is crystal-like structure the key to the information processing of living systems?  _Crystals_ 11, 318 (2021).
  * [58] Dixit, R., Ross, J. L., Goldman, Y. E. & Holzbaur, E. L.  Differential regulation of dynein and kinesin motor proteins by tau.  _Science_ 319, 1086–1089 (2008).
  * [59] Singh, P. _et al._ Cytoskeletal filaments deep inside a neuron are not silent: They regulate the precise timing of nerve spikes using a pair of vortices.  _Symmetry_ 13, 821 (2021).
  * [60] Congdon, E. E. & Sigurdsson, E. M.  Tau-targeting therapies for alzheimer disease.  _Nature Reviews Neurology_ 14, 399–415 (2018).
  * [61] Wilson, C. & González-Billault, C.  Regulation of cytoskeletal dynamics by redox signaling and oxidative stress: implications for neuronal development and trafficking.  _Frontiers in cellular neuroscience_ 9, 381 (2015).
  * [62] White, D., de Vries, G., Martin, J. & Dawes, A.  Microtubule patterning in the presence of moving motor proteins.  _Journal of theoretical biology_ 382, 81–90 (2015).
  * [63] Bowne-Anderson, H., Zanic, M., Kauer, M. & Howard, J.  Microtubule dynamic instability: a new model with coupled gtp hydrolysis and multistep catastrophe.  _BioEssays_ 35, 452–461 (2013).
  * [64] Craddock, T. J. _et al._ The zinc dyshomeostasis hypothesis of alzheimer’s disease.  _PloS one_ 7, e33552 (2012).
  * [65] Player, T. C., Baxter, E. D., Allatt, S. & Hore, P.  Amplification of weak magnetic field effects on oscillating reactions.  _Scientific Reports_ 11, 1–9 (2021).
  * [66] Saito, I., Matsuura, T. & Inoue, K.  Formation of superoxide ion from singlet oxygen. use of a water-soluble singlet oxygen source.  _Journal of the American Chemical Society_ 103, 188–190 (1981).
  * [67] Usselman, R. J. _et al._ The quantum biology of reactive oxygen species partitioning impacts cellular bioenergetics.  _Scientific reports_ 6, 1–6 (2016).
  * [68] Lee, A. A. _et al._ Alternative radical pairs for cryptochrome-based magnetoreception.  _Journal of The Royal Society Interface_ 11, 20131063 (2014).
  * [69] Efimova, O. & Hore, P.  Role of exchange and dipolar interactions in the radical pair model of the avian magnetic compass.  _Biophysical Journal_ 94, 1565–1574 (2008).
  * [70] Hore, P. J.  Upper bound on the biological effects of 50/60 hz magnetic fields mediated by radical pairs.  _Elife_ 8, e44179 (2019).
  * [71] Sies, H. & Jones, D. P.  Reactive oxygen species (ros) as pleiotropic physiological signalling agents.  _Nature reviews Molecular cell biology_ 21, 363–383 (2020).
  * [72] Player, T. C. & Hore, P.  Viability of superoxide-containing radical pairs as magnetoreceptors.  _The Journal of chemical physics_ 151, 225101 (2019).
  * [73] Hogben, H. J., Efimova, O., Wagner-Rundell, N., Timmel, C. R. & Hore, P.  Possible involvement of superoxide and dioxygen with cryptochrome in avian magnetoreception: origin of zeeman resonances observed by in vivo epr spectroscopy.  _Chemical Physics Letters_ 480, 118–122 (2009).
  * [74] Kattnig, D. R.  Radical-pair-based magnetoreception amplified by radical scavenging: resilience to spin relaxation.  _The Journal of Physical Chemistry B_ 121, 10215–10227 (2017).
  * [75] Kattnig, D. R. & Hore, P.  The sensitivity of a radical pair compass magnetoreceptor can be significantly amplified by radical scavengers.  _Scientific reports_ 7, 1–12 (2017).
  * [76] Kerver, E. D. _et al._ In situ detection of spontaneous superoxide anion and singlet oxygen production by mitochondria in rat liver and small intestine.  _The histochemical journal_ 29, 229–237 (1997).
  * [77] Miyamoto, S., Martinez, G. R., Medeiros, M. H. & Di Mascio, P.  Singlet molecular oxygen generated by biological hydroperoxides.  _Journal of photochemistry and photobiology B: Biology_ 139, 24–33 (2014).
  * [78] Kanofsky, J. R.  Singlet oxygen production by biological systems.  _Chemico-biological interactions_ 70, 1–28 (1989).
  * [79] Goushi, K., Yoshida, K., Sato, K. & Adachi, C.  Organic light-emitting diodes employing efficient reverse intersystem crossing for triplet-to-singlet state conversion.  _Nature Photonics_ 6, 253–258 (2012).
  * [80] Fay, T. P. & Manolopoulos, D. E.  Radical pair intersystem crossing: Quantum dynamics or incoherent kinetics?  _The Journal of chemical physics_ 150, 151102 (2019).
  * [81] Liu, Q. _et al._ Mical3 flavoprotein monooxygenase forms a complex with centralspindlin and regulates cytokinesis.  _Journal of Biological Chemistry_ 291, 20617–20629 (2016).
  * [82] Yamauchi, T. & Fujisawa, H.  Disassembly of microtubules by the action of calmodulin-dependent protein kinase (kinase ii) which occurs only in the brain tissues.  _Biochemical and biophysical research communications_ 110, 287–291 (1983).
  * [83] Ballatore, C., Lee, V. M.-Y. & Trojanowski, J. Q.  Tau-mediated neurodegeneration in alzheimer’s disease and related disorders.  _Nature reviews neuroscience_ 8, 663–672 (2007).
  * [84] Kurian, P., Obisesan, T. & Craddock, T. J.  Oxidative species-induced excitonic transport in tubulin aromatic networks: Potential implications for neurodegenerative disease.  _Journal of Photochemistry and Photobiology B: Biology_ 175, 109–124 (2017).
  * [85] Gong, C.-X. & Iqbal, K.  Hyperphosphorylation of microtubule-associated protein tau: a promising therapeutic target for alzheimer disease.  _Current medicinal chemistry_ 15, 2321–2328 (2008).
  * [86] Alonso, A. d. C., Grundke-Iqbal, I. & Iqbal, K.  Alzheimer’s disease hyperphosphorylated tau sequesters normal tau into tangles of filaments and disassembles microtubules.  _Nature medicine_ 2, 783–787 (1996).
  * [87] Iqbal, K., Liu, F., Gong, C.-X. & Grundke-Iqbal, I.  Tau in alzheimer disease and related tauopathies.  _Current Alzheimer Research_ 7, 656–664 (2010).
  * [88] Feng, J.  Microtubule: a common target for parkin and parkinson’s disease toxins.  _The Neuroscientist_ 12, 469–476 (2006).
  * [89] Kett, L. R. _et al._ Lrrk2 parkinson disease mutations enhance its microtubule association.  _Human molecular genetics_ 21, 890–899 (2012).
  * [90] Parker, A. L., Kavallaris, M. & McCarroll, J. A.  Microtubules and their role in cellular stress in cancer.  _Frontiers in oncology_ 4, 153 (2014).
  * [91] Mukhtar, E., Adhami, V. M. & Mukhtar, H.  Targeting microtubules by natural agents for cancer therapy.  _Molecular cancer therapeutics_ 13, 275–284 (2014).
  * [92] Jordan, M. A. & Wilson, L.  Microtubules as a target for anticancer drugs.  _Nature Reviews Cancer_ 4, 253–265 (2004).
  * [93] Craddock, T. J., Tuszynski, J. A. & Hameroff, S.  Cytoskeletal signaling: is memory encoded in microtubule lattices by camkii phosphorylation?  _PLoS computational biology_ 8, e1002421 (2012).
  * [94] Janke, C. & Kneussel, M.  Tubulin post-translational modifications: encoding functions on the neuronal microtubule cytoskeleton.  _Trends in neurosciences_ 33, 362–372 (2010).
  * [95] Janke, C. & Bulinski, J. C.  Post-translational regulation of the microtubule cytoskeleton: mechanisms and functions.  _Nature reviews Molecular cell biology_ 12, 773–786 (2011).
  * [96] Goldstein, L. E. _et al._ Chronic traumatic encephalopathy in blast-exposed military veterans and a blast neurotrauma mouse model.  _Science translational medicine_ 4, 134ra60–134ra60 (2012).
  * [97] Brunden, K. R. _et al._ Epothilone d improves microtubule density, axonal integrity, and cognition in a transgenic mouse model of tauopathy.  _Journal of Neuroscience_ 30, 13861–13866 (2010).
  * [98] Li, C. & Götz, J.  Tau-based therapies in neurodegeneration: opportunities and challenges.  _Nature Reviews Drug Discovery_ 16, 863–883 (2017).
  * [99] Adams, B. & Petruccione, F.  Quantum effects in the brain: A review.  _AVS Quantum Science_ 2, 022901 (2020).
  * [100] Gauger, E. M., Rieper, E., Morton, J. J., Benjamin, S. C. & Vedral, V.  Sustained quantum coherence and entanglement in the avian compass.  _Physical review letters_ 106, 040503 (2011).
  * [101] Bandyopadhyay, J. N., Paterek, T. & Kaszlikowski, D.  Quantum coherence and sensitivity of avian magnetoreception.  _Physical review letters_ 109, 110502 (2012).
  * [102] Cai, J., Guerreschi, G. G. & Briegel, H. J.  Quantum control and entanglement in a chemical compass.  _Physical review letters_ 104, 220502 (2010).
  * [103] Kominis, I.  Magnetic sensitivity and entanglement dynamics of the chemical compass.  _Chemical Physics Letters_ 542, 143–146 (2012).
  * [104] Pauls, J. A., Zhang, Y., Berman, G. P. & Kais, S.  Quantum coherence and entanglement in the avian compass.  _Physical review E_ 87, 062704 (2013).
  * [105] Tiersch, M., Guerreschi, G., Clausen, J. & Briegel, H.  Approaches to measuring entanglement in chemical magnetometers.  _The Journal of Physical Chemistry A_ 118, 13–20 (2014).
  * [106] Zhang, Y., Berman, G. P. & Kais, S.  Sensitivity and entanglement in the avian chemical compass.  _Physical Review E_ 90, 042707 (2014).
  * [107] Hameroff, S. R., Craddock, T. J. & Tuszynski, J. A.  Quantum effects in the understanding of consciousness.  _Journal of integrative neuroscience_ 13, 229–252 (2014).
  * [108] Kumar, S., Boone, K., Tuszyński, J., Barclay, P. & Simon, C.  Possible existence of optical communication channels in the brain.  _Scientific reports_ 6, 36508 (2016).
  * [109] Neese, F.  The orca program system.  _Wiley Interdisciplinary Reviews: Computational Molecular Science_ 2, 73–78 (2012).
  * [110] Goerigk, L. & Grimme, S.  A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions.  _Physical Chemistry Chemical Physics_ 13, 6670–6688 (2011).
  * [111] Van Lenthe, E. v., Snijders, J. & Baerends, E.  The zero-order regular approximation for relativistic effects: The effect of spin–orbit coupling in closed shell molecules.  _The Journal of chemical physics_ 105, 6505–6516 (1996).
  * [112] Marenich, A. V., Cramer, C. J. & Truhlar, D. G.  Universal solvation model based on solute electron density and on a continuum model of the solvent defined by the bulk dielectric constant and atomic surface tensions.  _The Journal of Physical Chemistry B_ 113, 6378–6396 (2009).

## Acknowledgements

The authors would like to thank Jack Tuszyński for valuable discussions. This
work was supported by the Natural Sciences and Engineering Research Council of
Canada.

## Author contributions statement

H.ZH. and C.S. conceived the project; H.ZH. performed the calculations; H.ZH.
and C.S. wrote the paper; C.S. supervised the project.

## Competing Interests

The authors declare no competing interests.

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. Figure 3: The RPM model prediction of the
microtubule density ratio in geomagnetic field compared to hypomagnetic field
based on the RP complex of [TrpH^.+ … O2^.-]. The geomagnetic field and
hypomagnetic field are 0.05 mT and 10 nT, respectively. The magnetic field
modulates rate $k_{p}$ for $a_{A}=1.6046$ mT. The maximum HMF effect is
obtained for $k\in[7\times 10^{5},11\times 10^{5}]$ s-1 and
$r\in[10^{5},2\times 10^{5}]$ s-1. Figure 4: The dependence of the microtubule
density ratio in geomagnetic field compared to applied static magnetic field
based on the RP complex of [TrpH^.+ … O2^.-]. The geomagnetic field is 0.05
mT. The magnetic field modulates rate $k_{p}$ for $a_{A}=1.6046$ mT,
$r=10^{5}$ s-1, and $k=10^{6}$ s-1. HMF causes strong decrease on the
microtubule density. The maximum microtubule density occurs around GMF. Figure
5: The RPM model prediction of the MT density ratio of the administration of
Zn (with zero nuclear spin) over ^67Zn (with nuclear spin of
$I_{B}=-\frac{5}{2}$) based on the RP complex of [TrpH^.+ … O2^.-]. ^67Zn The
geomagnetic field has a strength of 0.05 mT. ^67Zn’s nuclear spin modulates
rate $k_{p}$ for $a_{A}=1.6046$ mT and $a_{B}=-11.39$ mT. Figure 6: The
dependence of the MT density of the administration of Zn (with zero nuclear
spin) [Red] and ^67Zn (with nuclear spin of $I_{B}=-\frac{5}{2}$) [Blue] 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 rate
$k_{p}$ for $a_{A}=1.6046$ mT, $a_{B}=-11.39$ mT, $r=10^{5}$ s-1, and
$k=10^{6}$ s-1. Figure 7: The highest occupied molecular orbital of
[Zn^2+-O2^.-]. Imaged rendered using Avogadro (https://avogadro.cc).


# Early Prediction of Geomagnetic Storms by
Machine Learning Algorithms

Iris Yan
Amity Regional High School
Woodbridge, CT 06525
USA

###### Abstract

Geomagnetic storms (GS) occur when solar winds disrupt Earth’s magnetosphere.
GS can cause severe damages to satellites, power grids, and communication
infrastructures. Estimate of direct economic impacts of a large scale GS
exceeds $40 billion a day in the US. Early prediction is critical in
preventing and minimizing the hazards. However, current methods either predict
several hours ahead but fail to identify all types of GS, or make predictions
within short time, e.g., one hour ahead of the occurrence. This work aims to
predict all types of geomagnetic storms reliably and as early as possible
using big data and machine learning algorithms. By fusing big data collected
from multiple ground stations in the world on different aspects of solar
measurements and using Random Forests regression with feature selection and
downsampling on minor geomagnetic storm instances (which carry majority of the
data), we are able to achieve an accuracy of 82.55% on data collected in 2021
when making early predictions three hours in advance. Given that important
predictive features such as historic Kp indices are measured every 3 hours and
their importance decay quickly with the amount of time in advance, an early
prediction of 3 hours ahead of time is believed to be close to the practical
limit.

###### Contents

  1. 1 Introduction
  2. 2 The method
    1. 2.1 Random Forests regression
  3. 3 Experiments
  4. 4 Conclusions

##  1 Introduction

Geomagnetic storms (GS) are major disturbances to the Earth’s magnetosphere
caused by solar winds [5]. In Earth’s atmosphere, there is a surrounding layer
formed by Earth’s magnetic field called magnetosphere. The solar winds are
created by high-temperature outgoing streams of plasma (i.e., charged
particles) emitted from the solar flares, which continuously buffet and
compress the Earth’s magnetosphere. Geomagnetic storms occur when there is a
sudden change of plasma movements, electric currents and hence magnetic forces
in the Earth’s magnetosphere. It is interesting to note that geomagnetic
storms can also produce the beautiful aurora borealis one may observe in the
sky. An illustration of the solar winds, Earth’s magnetosphere and GS can be
seen in Figure 1.

Figure 1: Illustration of solar winds, Earth’s magnetosphere and geomagnetic
storms. The dark disc indicates the Earth. The dashed line indicates the
surface of the Earth’s magnetosphere, where bow shock occurs when the solar
winds interact with the Earth’s magnetosphere.

During the occurrence of geomagnetic storms, there is a buildup and discharge
of electrons which resembles lightning. GS can cause severe damages to
satellites, power grids, and communication infrastructures. Estimate of direct
economic impacts of a large scale GS exceeds $40 billion a day in the US alone
[13]. GS can cause electrical blackouts and internet outages on massive scales
that may stay unrepaired for months. A noted recent occurrence of GS was on
February 3–4, 2022, and it was reported that 38 out of 49 Starlink satellites
were destroyed by an intense geomagnetic storm. Early prediction is critical
in preventing and minimizing the potential hazards that would be caused by GS.
However, current methods either predict GS several hours ahead but fail to
identify all the major types [14], or make predictions not early enough (i.e.,
only within one hour of the occurrence) [17]. This work aims to predict all
types of geomagnetic storms reliably and as early as possible using machine
learning algorithms.

Early predictions of geomagnetic storms is very challenging. It is nearly
impossible to approach the problem from the physics of geomagnetic storms, due
to its high complexity [10]. One has to use current (or historic) solar
measurements to predict the geomagnetic disturbance in the future. However,
current measurements of solar activity entail very little information about
the magnitude of future geomagnetic storms [8]. The intensity of geomagnetic
storms is commonly measured by the Kp index which takes value in the range of
0 – 9. For instance, the largest recorded geomagnetic storm, which occurred in
1859, had an estimated Kp index in the range of 8.2 – 8.7 by Wagner and his
coauthors [16]. The Feb 3-4, 2022 geomagnetic storm had an estimated Kp index
around 5.0 – 5.3 [1]. Figure 2 shows that a given level of future (e.g., 3
hours later) Kp index can correspond to a wide range of values for current
solar measurements.

Figure 2: Wide range of solar field magnitudes for a given future (e.g., 3
hours later) Kp-index.

Another challenge lies in the extreme sparsity of the signal and the highly
unbalanced nature of the data in the sense that major storms (Kp indices $>4$)
occur rarely [7]. A quick inspection of historic data shows that major storms
carry less than 3% of the data. To detect such extremely sparse signals is
like finding a needle in haystacks. This can lead to the masking phenomenon in
machine learning, where classes with a few data instances may be ignored by
algorithm. Additionally, as the data eventually collected at the ground
stations are transmitted from satellites (which have sensors on board to
collect measurements about various solar activities, geomagnetic disturbances
etc), factors such as unfavorable weather conditions or interferences from
electromagnetic signals can introduce substantial noises and distortion to
data collected at the ground stations.

This work aims to overcome the above challenges by learning highly complex
patterns using Random Forests (RF) regression [2], and using techniques
including downsampling and feature selection to further overcome technical
challenges present in the data. In the remaining of this paper, we describe
our approach and implementation in Section 2. This is followed by a report of
experimental results in Section 3. Finally we conclude in Section 4.

##  2 The method

The early prediction of geomagnetic storms is formulated as a prediction of
future (i.e., 3 hours after the time of prediction) Kp index of the storm
using past measurements of solar activities, past density and Kp indices. Our
approach consists of data collection, data fusion, feature selection,
downsampling and the training of Random Forests regression model for
prediction. Here, instead of trying very hard on improving the algorithm, we
adopt techniques like feature selection and downsampling to “improve” the
datasets for the purpose of sparse pattern detection. Figure 3 is a
illustration of our approach. For the rest of this section, we will describe
individual steps in our approach.

Figure 3: Overall architecture of the proposed approach.

One important feature of this work is the use of big data collected from
multiple sources in the analysis. We have measurements of solar activities or
intensities of geomagnetic disturbances from NASA OMNIWeb [5, 6, 12], the
Kyoto World Data Center in Tokyo, Japan [9], the German Center for Geosciences
[11], respecively. The OMNIWeb data includes solar wind speed, proton density,
wind temperature, field magnitude averages, measured every 5 minutes, for up
to 9 hours before the time of the prediction. The Kyoto data included past Dst
(i.e., density) indices, measured every hour, for up to 3 hours before the
time of the prediction. The GFZ data included past Kp indices, measured every
3 hours, up to 24 hours before the time of the prediction.

Together the data translates to 2.1 million data points, 2679 time events and
780 features (variables). For each point, there are 780 features associated
with it, including Kp indices, field magnitude average, solar wind direction
vector components Bx, By, Bz, solar wind speed, electron and particle density,
and solar wind temperature for time intervals with feature-specific interval
lengths during the last 12 hours till the current time, and the response
variable is the Kp index three hours later from the time of prediction. Note
that here the feature-specific time interval lengths are 5 minutes for
features generated from the OMNIWeb data, one hour for those from the Kyoto
data, and three hours for past Kp indices from the GFZ data.

The next step is feature selection. As there are 780 features after the
initial data fusion, and some features may be more important than others and
many features may not be as informative (or otherwise would introduce noises)
so it might be better to discard them. This would help reduce potential wrong
generalizations deduced from data. So we selectively include important
features (out of 780) using the built-in feature selection tool provided by
RF. RF produces a rank of individual features according to their importance,
and we use the top 50 or 100 features according to this rank. A further step
is downsampling to discard data from larger classes (so that the minority
classes or those data points associated with major storms carry a higher
percentage of the data). This creates a more balanced distribution of data
across different classes, which can often improve accuracy in prediction.

###  2.1 Random Forests regression

Random Forests [2] regression is the algorithm used for model fitting in our
analysis. RF is an ensemble of decision trees with each tree constructed on a
bootstrap sample of the data. Each tree is built by recursively partitioning
the data. At each tree node (the root node corresponds to the bootstrap
sample), RF randomly samples a number of features and then select one that
would lead to an “optimal” partition of that node. Each tree recursively
narrows down which Kp index an instance should fall into. This process is
continued recursively until the tree is fully grown. There are two modes in
RF, classification and regression, and we use regression in this work. That
is, we grow each tree until each leaf node has at most 5 data points or
consists of data points of the same value. The fitted value for a data point
falling into a node is taken as the average of values of all points in the
same leaf node. The final decision of a forest is taken from the average of
decisions from individual trees. The superior empirical performance of RF has
been demonstrated by a number of studies [2, 4, 3]. RF is easy to use (e.g.,
very few tuning parameters), has a remarkable built-in ability for feature
selection. We use the R package randomForest in this work.

For prediction, we try to learn the relationship between the future Kp index
and the current (or historic) solar measurements. This is a regression
problem, and a simple choice would be to use linear regression. Will linear
regression be a sensible choice to model the relationship of interest? We
conduct an exploratory analysis of the data by visualizing the data—to see how
the data looks like. As the raw data has a dimension of 780, we reduce the
data dimension by principle component analysis and visualize the data by its
top two principal directions.

Figure 4: Principal component visualization of data. The points are drawn by
their value along the top 2 principal directions, and labeled by Kp indices
(rounded for clarity) and marked by different colors and shapes.

Figure 4 is a principal component visualization of the data. Together the top
two principal directions captures 73.62% of the variation in the data so the
visualization should be a fairly faithful representation of the original data.
The figure shows that the data (or the underlying pattern) is highly nonlinear
and complicated: data points with same Kp indices are scattering around while
those with different Kp indices are highly mixed. This motivates us to use RF
regression in this work, as it can capture nonlinear patterns in the data. As
our task is to build a model to separate data points corresponding to
different Kp indices, the figure also shows that this task is challenging.

##  3 Experiments

All the data used in our experiments are from 2021, and the first 9 months
data are used for training and the next three months data for test. The number
of trees in RF is fixed at 100 for all experiments, and the number of
variables to select at each node split is the default $p/3$ where $p$ is the
number of features in the data. In this section, we report experimental
results.

Figure 5: Variable importance. Kp indices, FMAs, and other solar activity
measurements taken at different time are arranged by the time they were
measured, with more recent ones displayed first. The numbers in the label of
points indicate the number of minutes till the current moment at which time we
are to predicate the geomagnetic disturbance 3 hours later. For clarity,
features other than FMAs or dst’s are not labeled.

To select important features, we use the built-in RF feature importance tools
to rank the features (which rank the features by their contribution in the
reduction of mean square errors in RF regression). We find that Kp indices
before the time of prediction and field magnitude averages (FMA) are most
important features. The importance of different features are shown in Figure
5. We can see that variable importances decrease quickly with their associated
time (till the current moment), meaning that the further a variable is into
the past, the less informative it would be for predictions of the future
(e.g., magnitude of geomagnetic storms or Kp index 3 hours later). This
increases the difficulty of earlier predictions, as the amount of historic
data that can be used is limited to the last few hours. Due to limited
availability of Kp index measurements in current data, 3 hours in advance
seems to have achieved the practical limit in time for early prediction.
Currently the Kp index is measured every 3 hours, an even earlier prediction
(i.e., 6 hours ahead) with much less informative variables will lead to
unsatisfactory results (not reported here due to limit in space). Compared to
3-hour early prediction, one can only use features 6-hour ahead with those
corresponding to the last 3-6 hours will not be used, as illustrated in Figure
6. However, as shown in Figure 5, features associated with the last 3-6 hours
are much more important than those 6-hour earlier for a given target time to
predict.

Figure 6: Illustration of the difficulty in early prediction. For a given
target time to predict, a 3-hour early prediction will use all features up to
3-hour ahead and a 6-hour early prediction will use all features up to 6-hour
ahead, and the difference is the features associated with time between 3-6
hours ahead.

We assess the performance by the accuracy of prediction defined as the
percentage of instances for which the predicted Kp index is within 1 of the
actual Kp index. Figure 7 shows the relative performance of different
algorithmic choices. RF regression outperforms linear regression, which is
consistent with the nonlinear pattern observed in Figure 4. Feature selection
of top 100 features improved the accuracy of RF regression; reducing to top 50
features further improves the accuracy. This implies that the data, in
particular historic solar measurements in the very distant past, are less
informative. Additional downsampling with top 50 features was able to achieve
an accuracy of 82.55% (we observe that upsampling of the minor classes can
slightly improve accuracy, but results not reported here as not substantial).
For the first time, all types of geomagnetic storms can be predicted 3-hour
ahead, two hours earlier than previous work [17].

Figure 7: Accuracy by different algorithms in predicting Kp index 3 hours in
advance. When L=2, only 1/2 of data with lower Kp indices was used in RF
regression training (downsampling).

##  4 Conclusions

For the first time, all types of geomagnetic storms can be successfully
predicted 3 hours ahead using machine learning algorithms. This was enabled by
the use of big data from multiple sources from the world, coupled with
techniques to ”improve” the dataset for sparse pattern detection, including
feature selection and downsampling. The algorithm achieves a prediction
accuracy at 82.55% by keeping only the most informative features (top 50
features), and downsampling data instances that occur more often (i.e., lower
Kp indices) by half.

The importance of variables from solar measurements decreases very quickly as
the amount of advance in prediction time increases. Measurements taken 6 hours
in advance becomes barely informative, indicating that the achieved 3 hours in
advance prediction has likely reached the practical limit, unless the Kp
indices can be measured more frequently than the current every 3 hours.

Predicting geomagnetic storms accurately in advance will provide a sufficient
warning time window for preparation. This will greatly prevent or decrease
damages to satellites, power grids, and communication infrastructures. Our
future work include the use of an ensemble model combining multiple predictive
algorithms, or to provide a probabilistic confidence [15] for specific
predictions for example those for a very intense geomagnetic storm. Another
possibility is to incorporate the physics of geomagnetic storms [10] in the
prediction model to potentially improve accuracy or robustness of the
prediction.

## Acknowledgements

I would like to thank my science research teacher, Mrs. Piscitelli, and my
mentor Dr. Chen of Google Inc. for their advice in project planning and
suggestions in literature. I am also grateful to the editor, the associate
editor, and the anonymous reviewers for their constructive comments and
suggestions.

## References

  * [1] R. Bojilova and P. Mukharov.  Comparative Analysis of Global and Regional Ionospheric Responses during Two Geomagnetic Storms on 3 and 4 February 2022.  Remote Sensing, 15(7):1739, 2023.
  * [2] L. Breiman.  Random Forests.  Machine Learning, 45(1):5–32, 2001.
  * [3] R. Caruana, N. Karampatziakis, and A. Yessenalina.  An empirical evaluation of supervised learning in high dimensions.  In Proceedings of the Twenty-Fifth International Conference on Machine Learning (ICML), pages 96–103, 2008.
  * [4] R. Caruana and A. Niculescu-Mizil.  An empirical comparison of supervised learning algorithms.  In Proceedings of the 23rd International Conference on Machine Learning (ICML), 2006.
  * [5] NOAA/NWS Space Wether Prediction Center.  Geomagnetic Storms.  https://www.swpc.noaa.gov/phenomena/geomagnetic-storms.
  * [6] NOAA/NWS Space Wether Prediction Center.  Real Time Solar Wind.  https://www.swpc.noaa.gov/products/real-time-solar-wind.
  * [7] S. C. Chapman, R. B. Horne, and N. W. Watkins.  Using the aa index over the last 14 solar cycles to characterize extreme geomagnetic activity.  Geophysical Research Letters, 47:e2019GL086524, 2019.
  * [8] H. A. Elliott, J. M. Jahn, and D. J. McComas.  The Kp index and solar wind speed relationship: insights for improving space weather forecasts.  Space Weather, 11(6):339–349, 2013.
  * [9] World Data Center for Geomagnetism (Kyoto).  Final Dst index.  https://wdc.kugi.kyoto-u.ac.jp.
  * [10] W. D. Gonzalez, J. A., Joselyn, Y. Kamide, H. W. Kroehl, G. Rostoker, B. T. Tsurutani, and V. M. Vasyliunas.  What is a geomagnetic storm?  Journal of Geophysical Research, 99(A4):5771–5792, 1994.
  * [11] GFZ German Research Center for Geosciences.  Kp index.  https://www.gfz-potsdam.de/en/kp-index/.
  * [12] NASA.  SPDF - Omniweb Service.  https://www.swpc.noaa.gov/products/planetary-k-index.
  * [13] E. J. Oughton, A. Skelton, R. B. Horne, A. W. Thomson, and C. T. Gaunt.  Quantifying the daily economic impact of extreme space weather due to failure in electricity transmission infrastructure.  Space Weather, 15(1):65–83, 2017.
  * [14] T. V. Podladchikova and A. A. Petrukovich.  Extended geomagnetic storm forecast ahead of available solar wind measurements.  Space Weather, 10:7001, 2012.
  * [15] V. Vovk and G. Shafer.  A tutorial on conformal prediction.  Journal of Machine Learning Research, 9:371–421, 2008.
  * [16] .D. Wagner, R. Neuhauser, and R. Arit.  Auroral oval reconstruction for historical geomagnetic storms in the 18th and 19th century.  Astronomical Notes, 344(3):e22078, 2023.
  * [17] D. Zhu, S. A. Billings, M. Balikhin, S. Wing, and D. Coca.  Data derived continuous time model for the dst dynamics.  Geophysical Research Letters, 33(4):L04101, 2006.