Quantum Cognition: The possibility of processing with nuclear spins in the brain
QQuantum Cognition: The possibility of processing with nuclear spins in the brain
Matthew P. A. Fisher Department of Physics, University of California, Santa Barbara, CA 93106 (Dated: September 1, 2015)The possibility that quantum processing with nuclear spins might be operative in the brain is explored. Phos-phorus is identified as the unique biological element with a nuclear spin that can serve as a qubit for suchputative quantum processing - a neural qubit - while the phosphate ion is the only possible qubit-transporter .We identify the “Posner molecule”, Ca ( PO ) , as the unique molecule that can protect the neural qubits on verylong times and thereby serve as a (working) quantum-memory . A central requirement for quantum-processingis quantum entanglement . It is argued that the enzyme catalyzed chemical reaction which breaks a pyrophos-phate ion into two phosphate ions can quantum entangle pairs of qubits. Posner molecules, formed by bindingsuch phosphate pairs with extracellular calcium ions, will inherit the nuclear spin entanglement. A mechanismfor transporting Posner molecules into presynaptic neurons during vesicle endocytosis is proposed. Quantummeasurements can occur when a pair of Posner molecules chemically bind and subsequently melt, releasing ashower of intra-cellular calcium ions that can trigger further neurotransmitter release and enhance the probabilityof post-synaptic neuron firing. Multiple entangled Posner molecules, triggering non-local quantum correlationsof neuron firing rates, would provide the key mechanism for neural quantum processing. Implications, both invitro and in vivo, are briefly mentioned.
PACS numbers: “I can calculate the motion of heavenly bodies, but not themadness of men” - Isaac Newton
A. Introduction
It has long been presumed that quantum mechanics cannotplay an important (functional) role in the brain, since main-taining quantum coherence on macroscopic time scales (sec-onds, minutes, hours,...) is exceedingly unlikely in a wetenvironment (although see [3,4] and references therein).Small molecules, or even individual ions, while described inprinciple by quantum mechanics, rapidly entangle with thesurrounding environment, which causes de-phasing of anyputative quantum coherent phenomena. However, there isone exception: Nuclear spins are so weakly coupled to theenvironmental degrees of freedom that, under some circum-stances, phase coherence times of five minutes or perhapslonger are possible.
Putative quantum processing with nuclear spins in the wetenvironment of the brain - as proposed by Hu and Wu in Ref.[3] - would seemingly require fulfillment of many unrealiz-able conditions: for example, a common biological elementwith a long nuclear-spin coherence time to serve as a qubit,a mechanism for transporting this qubit throughout the brainand into neurons, a molecular scale quantum memory for stor-ing the qubits, a mechanism for quantum entangling multiplequbits, a chemical reaction that induces quantum measure-ments on the qubits which dictates subsequent neuron firingrates, among others.Our strategy, guided by these requirements and detailed be-low, is one of “reverse engineering” - seeking to identify thebio-chemical “substrate” and mechanisms hosting such puta-tive quantum processing. Remarkably, a specific neural qubitand a unique collection of ions, molecules, enzymes and neu-rotransmitters is identified, illuminating an apparently single path towards nuclear spin quantum processing in the brain.
B. The Neural qubit - phosphorus nuclear-spin
The nucleus of every element is characterized by a half-integer spin-magnitude (I = 0 , , · · · ) and for I (cid:54) = 0 anassociated magnetic dipole moment which precesses aroundmagnetic fields at the nucleus. These magnetic fields arisefrom nuclear magnetic moments of nearby atoms/ions. Nucleiwith I > also have an electric quadrupole moment whichcouples to electric field gradients generated by charges ofnearby electrons/nuclei. Magnetic and electric field perturba-tions cause quantum decoherence of the nuclear spin - anath-ema to quantum processing - so that the “coherence time”, t coh , must be maximized when seeking a possible biologicalarena for nuclear spin processing.In the biochemical setting electric fields are the primarysource of decoherence for nuclei with I > , while I = spins are more weakly decohered only by magnetic fields. Forexample, a solvated Li + isotope with I = has t coh ∼ s,while the Li + isotope (with very small electric quadrupolemoment) is an “honorary I = ” with t coh as long as 5minutes! Thus, the element hosting the putative neural qubitmust have nuclear-spin I = .Among the most common biochemical elements, carbon,hydrogen, nitrogen, oxygen, phosphorus and sulphur, and thecommon ions Na + , K + , Cl − , Mg and Ca , besides hydro-gen, only phosphorus has a nucleus with spin I = . Thisidentifies the phosphorus nucleus as our putative neural qubit . C. Qubit transporter - the phosphate ion
Phosphorus is bound into the inorganic phosphate ion PO − (abbreviated as Pi) in biochemistry, present in energy trans- a r X i v : . [ q - b i o . N C ] A ug port molecules such as ATP and in poly-phosphate chains in-cluding the pyrophosphate ion, P O − (abbreviated as PPi). The tetrahedrally coordinated oxygen cage surrounding phos-phorus in Pi resembles the cage of oxygens in the hydrationshell of solvated Li + . However, the phosphorus spin coher-ence time for solvated Pi, t coh ∼ s, is significantly shorterthan the coherence time of solvated Li + , ∼ min. Thisdifference can be attributed to the proton that binds to Pi atphysiological pH - the proton and phosphorus nuclear spinsin HPO − (HPi) interact via the electrons and significantlyreduce the phosphorus spin coherence time to ∼ s.The solvated phosphate ion can nevertheless serve as aneffective qubit transporter diffusing ∼ µ m (the cellularscale) in roughly − s, while maintaining spin coherence.Qubit memory-storage (and processing) on times of secondsor longer will, however, require another molecule, which wenext discuss. D. Qubit memory - the Posner molecule
If another biological cation (with I = 0 ) can displace theproton in binding to the phosphate ions, longer spin coher-ence times might be possible. The presence of bone mineral,calcium-phosphate, indicates that under some physiologicalconditions calcium ions can out-compete the proton in bind-ing to phosphates. Indeed, several recent in vitro studies havefound evidence for a stable calcium-phosphate molecule, Ca ( PO ) (see Figure 1), in simulated body fluids (SBF)appropriate for the extracellular fluid of mammals. Thesenanometer diameter “Posner molecules” are likely present inreal extracellular body fluid as well.The phosphorus spins in a Posner molecule are expected tohave very long coherence times, which we estimate as fol-lows. The magnetic dipole fields from protons in nearbywater molecules will cause the phosphorus spins to precessat frequencies of order f dip ∼ Hz, naively suggestingmilli-second coherence times. But due to the rapid tumblingof Posner molecules in water (with rotation frequencies oforder f rot ∼ Hz), the magnitude and direction of thedipole magnetic field at a given phosphorus nucleus will varyrapidly with time ( f − rot ∼ psec), averaging to zero. Resid-ual magnetic field fluctuations will lead to “directional diffu-sion” of the phosphorus spins with very long coherence times, t coh (cid:39) f rot /f dip ∼ s ∼ day. Actual coherence timescould well be even longer, since 5 of the 64 spin states in thePosner molecule, with zero (total) spin, I tot = 0 , will be virtu-ally blind to decoherence. E. Enzyme catalyzed qubit entanglement “Quantum entanglement” between qubits is necessary forquantum processing.
While a single qubit state can be ex-pressed in terms of “up” and “down” basis states, a pair ofqubits will have four basis states, | ↑↑(cid:105) , | ↑↓(cid:105) and so on. Thetwo-qubit “spin-singlet” state, | s (cid:105) = [ | ↑↓(cid:105) − | ↓↑(cid:105) ] / √ , em-bodies a form of entanglement which lies at the heart of quan- 𝜑 ’ a a ’ FIG. 1: Two Posner molecules, Ca ( PO ) , with calcium ions(blue) in a bcc arrangement (eight at the corners and one at the cen-ter of a cube) as viewed along the (111) axis. The six phosphateions - a tetrahedron of oxygens (red) surrounding a central phospho-rus (purple) - are on the cube faces and reduce the symmetry to S ,with one remaining 3-fold symmetry axis. As shown, the two Posnermolecules are oriented with the 3-fold axis out of the page (molecule a ) and into the page (molecule a (cid:48) ), with ϕ and ϕ (cid:48) denoting the ro-tation angles around their respective symmetry axis. In this orien-tation quantum chemistry calculations indicate that the two Posnermolecules can chemically bind to one another, releasing of order an eV of energy. tum mechanics and serves as the “unit of currency” for labo-ratory quantum computing efforts. If the first spin is measuredto be “up”, then the second spin will be found “down” - andvice versa - independent of the spatial separation of the twospins - a non-local entanglement referred to by Einstein as“spooky action at a distance”. Two-qubit states describe the phosphorus nuclear spins inthe important biochemical ion pyrophosphate (PPi), a linearphosphate-dimer created in the hydrolysis reaction ATP → AMP + PPi.
The four time-independent (stationary) statesof the two interacting phosphorus spins are the (para) spin-singlet and three (ortho) spin-triplet states, | t + (cid:105) = | ↑↑(cid:105) , | t − (cid:105) = | ↓↓(cid:105) and | t (cid:105) = [ | ↑↓(cid:105) + | ↓↑(cid:105) ] / √ . A general spinstate of PPi can be written as a linear combination of thesefour stationary states.Quantum mechanics is usually presumed irrelevant in de-scribing the translational, vibrational and rotational motion ofmolecules or ions diffusing in water. However, an adequatedescription of the hydrolysis reaction of interest to us, PPi → Pi + Pi, requires a full quantum treatment of the molec-ular rotations . With rotations included the quantum state ofPPi can be presented as a sum of singlet and triplet terms ofthe form, Ψ PPi = c s ψ s (ˆ r ) | s (cid:105) + c t ψ t (ˆ r ) | t (cid:105) , (1)where | t (cid:105) = (cid:80) m a m | t m (cid:105) ( m = 0 , ± ) with normalizations, (cid:104) t | t (cid:105) = 1 and | c s | + | c t | = 1 . Here ˆ r is a unit vectorspecifying the ions orientation. Being identical fermions, Ψ PPi must change sign under the interchange of the two phospho-rus atoms - which corresponds to an exchange of the spins anda 180 degree body rotation, ˆ r → − ˆ r . Since the spin-singletchanges sign under exchange, | s (cid:105) → −| s (cid:105) , while the spin-triplets do not, the corresponding orbital wavefunctions mustsatisfy, ψ s/t (ˆ r ) = ± ψ s/t ( − ˆ r ) , so that both terms in Eq. (1)change sign under full exchange, as required. The first andsecond terms in Ψ PPi are direct analogs of the para and orthostates of molecular hydrogen - perhaps we might call thempara and ortho pyrophosphate - but, quite generally, the ap-propriate state of both H and PPi should be a quantum linearsuperposition of the para and ortho states.The stationary states of freely rotating PPi (“spherical har-monics”) are labelled by an integer angular momentum L ≥ (in units of (cid:126) ). Under a 180 degree rotation the spherical har-monics are multiplied by ( − L , so that ψ s and ψ t can beexpressed in terms of even and odd angular momentum wave-functions, respectively. For PPi tumbling in water, L ∼ ,so the distinction between even and odd values of L is unim-portant.However, the enzyme catalyzed hydrolysis reaction PPi → Pi + Pi to which we next turn, requires first slowing ( L ∼ O (1) ) and then stopping the PPi rotation, a process which willpresumably depend on the sign of ( − L . This suggests areaction rate dependent on the nuclear spin state, different forthe singlet and triplet states .
1. The enzyme pyrophosphatase
We illustrate this within a simple model of the enzymepyrophosphatase. The four magnesium ions inside the en-zyme pocket, each with charge +1 appropriate when singlybonded to an enzyme oxygen, will attract pyrophosphateP O − into the pocket (see Figure 2). When rapidly rotatingPPi will “look” like a spherical shell of charge, held in placeby the magnesium ions, symmetrically arranged for maximumstability, as depicted in Figure 2a. But in the aspherical elec-trostatic environment in the pocket, PPi will lose angular mo-mentum and, once slowed, will tend to align along preferredorientations, held by the “pinning” potentials attracting thenegatively charged PPi oxygens to the positive magnesiumions, of strength v x and v y (see Figure 2b and 2c).Consider shape deformations of the enzyme pocket whicheither increase the distance between the two magnesium ionsalong the y-axis (decreasing v y ) or bring them closer together(increasing v y ). When v y is greatly increased and the PPi ionis oriented along the y-axis, weak chemical bonds should formbetween the end oxygens of PPi and the two magnesium ions,as depicted in Figure 2c. By thereby “pulling” on the electronsof PPi, the magnesium ions will weaken the chemical bondsbetween the phosphorus ions and the central oxygen, whichwill then be susceptible to hydrolysis driving the transition,P O − + H O → × HPO − . The liberated phosphate ions(Pi) can then leave the pocket, completing the reaction (seeFigure 2d).The effect of the spins on this reaction can be re-vealed by retaining only four orbital configurations, with PPialigned along the preferred orientations, | x ± (cid:105) , | y ± (cid:105) . The sin-glet/triplet wavefunctions have only two states each, | x s/t (cid:105) = PPi -2-2
Mg+ Mg+Mg+Mg+ v x v x t rot t rot +- v y v y H O PiPi (a) (b)(c) (d)
FIG. 2: The pyrophosphate ion P O − (PPi), shown rotating insidethe pocket of the enzyme pyrophosphatase in (a), is attracted by fourenzyme bound Mg + ions. Once the rotation slows, PPi will orientalong preferred directions to align with the Mg + ions, as in (b) and(c), with v x , v y denoting respective “pinning” strengths. When PPibinds (chelation) to two Mg + ions, as depicted in (c), the weakenedinternal covalent bonds of PPi will facilitate the hydrolysis reaction,PPi + H O → Pi + Pi. The phosphorus nuclear spins in the two lib-erated phosphate ions (Pi) will be singlet entangled - dashed purpleline in (d) - even after they leave the enzyme pocket. ( | x + (cid:105) ± | x − (cid:105) ) / √ , with the same for x → y . The orbitaldynamics can be described by a simple Hamiltonian, H s = − t s ( | x s (cid:105)(cid:104) y s | + c.c. ) − v x | x s (cid:105)(cid:104) x s | − v y | y s (cid:105)(cid:104) y s | , (2)with s → t for the triplet sector. The first term describes ro-tational motion while the other terms are the “pinning” poten-tials from the magnesium ions. Crucially, due to their differentsymmetries the singlet and triplet rotation rates are very differ-ent t s/t = t + rot ± t − rot , where t + rot , t − rot are the rates for the ion torotate clockwise or counterclockwise, respectively (see Figure2b). Crucially, when t + rot = t − rot as we now assume, the tripletwavefunction cannot rotate, t t = 0 , due to a destructive quan-tum interference between the clockwise and counterclockwise“trajectories”.Consider now a shape deformation of the enzyme pocket todrive the chemical reaction, by varying v y at fixed v x , startingwith v y << v x , where both the singlet and triplet groundstates will have PPi oriented along the x-axis (Figure 2b).Now gradually increase v y until v y >> v x . Under this evo-lution the singlet wavefunction will rotate and align with they-axis, where it can bind to the magnesium ions driving thechemical reaction (Figure 2c). However, the triplet wavefunc-tion cannot rotate, and will get “stuck” along the x-axis, un-able to take part in the chemical reaction. Strikingly, after the reaction the phosphorus nuclear spins in the two separatedphosphate ions will be entangled in a singlet , as depicted bythe purple dashed line in Figure 2d. Relaxing the spatial sym-metry that gave t t = 0 will lead to a non-zero, but small,probability of triplet entangled phosphate ions being released.If these (singlet) entangled phosphate pairs are released intothe extracellular fluid, they can combine with calcium ions toform Posner molecules, where the phosphorus nuclear spinscan be “held” in memory. Moreover, if two such Posnermolecules share an entangled phosphate pair, their spins willbe entangled , as depicted in Figure 3a. Generally, one can en-visage complex clusters of highly entangled Posner molecules(see Figure 3b) providing an ideal setting for quantum pro-cessing, as we next discuss. F. Quantum processing with Posner molecules
Consider first the spin and rotational states of a single Pos-ner molecule. Quantum chemistry calculations find a cubic ar-rangement for the calcium ions (eight at the cube corners andone at the cube center) but the phosphate ions on the six facesreduce the cubic symmetry to S , with one 3-fold symmetryaxis along a cube diagonal (see Figure 1). The quantumstates of the six phosphorus spins can be labelled by the totalspin, I tot = 0 , , , , and by a “pseudo-spin”, σ = 0 , ± , en-coding the transformation properties under a 3-fold rotation, | σ (cid:105) → ω σ | σ (cid:105) with ω = e i π/ . The quantum state with bothspin and rotations can be expressed as, | Ψ Pos (cid:105) = (cid:88) σ c σ | ψ σ (cid:105)| σ (cid:105) , (3)with the choice of normalizations (cid:104) σ | σ (cid:105) = (cid:104) ψ σ | ψ σ (cid:105) = 1 and (cid:80) σ | c σ | = 1 . The orbital wavefunction ψ σ ( ϕ ) depends onthe angle ϕ of rotation about the 3-fold symmetry axis. Fermistatistics requires | Ψ pos (cid:105) be invariant under a 120 degree rota-tion that interchanges the positions and spins of the phospho-rus ions, implying, ψ σ ( ϕ +2 π/
3) = ¯ ω σ ψ σ ( ϕ ) with ¯ ω = ω ∗ . The nuclear spin and rotational states are thus quantum en-tanglement in the Posner molecule .The quantum state for two Posner molecules ( a and a (cid:48) ) canbe expressed as, | Ψ aa (cid:48) (cid:105) = (cid:88) σσ (cid:48) C aa (cid:48) σσ (cid:48) | ψ aσ (cid:105)| ψ a (cid:48) σ (cid:48) (cid:105)| σσ (cid:48) (cid:105) aa (cid:48) , (4)with (normalized) orbital states depending on the rotation an-gles ϕ and ϕ (cid:48) , multiplying a common (normalized) spin state | σσ (cid:48) (cid:105) aa (cid:48) which encodes (possible) entanglement between thespins in the two Posner molecules. Under a 120 degree rota-tion about the respective 3-fold symmetry axes, the spin stateis multiplied by ω σ and ω σ (cid:48) , with the orbital states multi-plied by compensating factors, ¯ ω σ and ¯ ω σ (cid:48) . The wavefunction C aa (cid:48) σσ (cid:48) , a × complex matrix satisfying (cid:80) σσ (cid:48) | C aa (cid:48) σσ (cid:48) | = 1 ,encodes pseudo-spin and rotational entanglement between thetwo Posner molecules , inherited from the spin entanglementprovided, C aa (cid:48) σσ (cid:48) (cid:54) = c aσ c a (cid:48) σ (cid:48) . Quantum processing with spins in the brain will require“projective measurements”, induced by chemical reactionswhich can stimulate subsequent biochemical activity. Thechemical binding of Posner molecules, present in vitro inexperiment, might provide such a mechanism. Considertwo Posner molecules that approach one another oriented withanti-parallel 3-fold symmetry axis (see Figure 1). Quantumchemistry calculations reveal that a chemical binding is thenpossible, and will lower their energy by roughly an electronvolt. This reaction can be described in terms of the angles ofrotation, ϕ and ϕ (cid:48) , about the 3-fold symmetry axis of the twoPosner molecules. Attracted by Van der Waals forces the twoPosner molecules might stick and (rapidly) rotate on one an-other, putting ϕ = ϕ ≡ φ . But chemical binding will requirestopping their rotations.The dynamics of the angle φ can be described in terms ofa common wavefunction, χ σσ (cid:48) ( φ ) = ψ aσ ( φ ) ψ a (cid:48) σ (cid:48) ( φ ) , whichtransforms as, χ ( φ + 2 π/
3) = ¯ ω σ + σ (cid:48) χ ( φ ) , and satisfies aSchrodinger equation, Hχ = Eχ , with Hamiltonian, ˆ H = ˆ (cid:96) / I pair + V ( ˆ φ ) . (5)Here ˆ (cid:96) = − i (cid:126) ∂ φ and I pair is the moment of inertia of the twoco-rotating Posner molecules. Due to the S symmetry, thepotential of interaction will satisfy, V ( φ + 2 π/
6) = V ( φ ) .To induce chemical binding we take a very strong (delta-function) interaction, V = − V (cid:80) n =1 δ ( φ − πn/ , andseek bound state solutions with E < . Such bound states,which correspond to chemically bonded Posner molecules,exist only if σ + σ (cid:48) = 0 . Thus the binding reaction of twoPosner molecules induces a “projective measurement” onto astate with zero total pseudo-spin , releasing an electron volt ofenergy! The probability that a Posner pair binds (after stick-ing) is P aa (cid:48) react = (cid:80) σσ (cid:48) | C aa (cid:48) σσ (cid:48) | δ σ + σ (cid:48) , . Once chemically boundthe Posner molecules can no longer rapidly rotate, and are pre-sumably easier to melt via hydrolysis, as discussed in the nextsection.
1. Quantum Entangled Chemical Reactions
The chemical binding of multiple Posner molecules withentangled nuclear spins might allow for complex quantumprocessing. Consider a simple example of two entangledpairs, | Ψ aa (cid:48) (cid:105) ⊗ | Ψ bb (cid:48) (cid:105) , created and situated as in Figure 3c.We introduce a variable r = 0 , with r = 1 when a reactionbinding the Posner pair { ab } proceeds and r = 0 when it doesnot, and another variable r (cid:48) = 0 , for the Posner pair { a (cid:48) b (cid:48) } .The joint probability distribution function, P rr (cid:48) , for these tworeactions ( P the probability that both reactions proceed, forexample) can be expressed in terms of their common wave-functions as, P rr (cid:48) = (cid:88) σ a σ a (cid:48) (cid:88) σ b σ b (cid:48) | C aa (cid:48) σ a σ a (cid:48) | | C bb (cid:48) σ b σ b (cid:48) | g r ( σ a , σ b ) g r (cid:48) ( σ a (cid:48) , σ b (cid:48) ) (6)with g ( σ, σ (cid:48) ) = δ σ + σ (cid:48) , and g = 1 − g .Being interested in quantum entanglement between thesetwo reactions, we define an “entanglement measure”, E = FIG. 3: A pair of entangled Posner molecules in (a). The purple dashed lines represent singlet entangled phosphorus nuclear spins. Acomplex of highly entangled Posner molecules in (b). With two pairs of entangled Posner molecules, labelled ( a, a (cid:48) ) and ( b, b (cid:48) ) as in panel (c),a chemical binding between one member in each pair - the black box connecting ( a, b ) - can change the probability of a subsequent bindingof the other members of the pair, ( a (cid:48) , b (cid:48) ) . If the Posner molecules chemically bind after being transported into two presynaptic neurons asdepicted in (d), they will be susceptible to melting, releasing their calcium into the cytoplasm enhancing neurotransmitter release, therebystimulating (quantum) entangled postsynaptic neuron firing. [ δrδr (cid:48) ] , where δr = r − [ r ] and δr (cid:48) = r (cid:48) − [ r (cid:48) ] . Thesquare brackets denote an average with respect to P rr (cid:48) , with [ f rr (cid:48) ] = (cid:80) rr (cid:48) P rr (cid:48) f rr (cid:48) , for an arbitrary function f rr (cid:48) . Thequantity E will depend on the quantum state of the four Pos-ner molecules.With no entanglement between the four Posner molecules,the wavefunctions take a product form, C aa (cid:48) σσ (cid:48) = c aσ c a (cid:48) σ (cid:48) and C bb (cid:48) σσ (cid:48) = c bσ c b (cid:48) σ (cid:48) , as does the distribution function, P rr (cid:48) = p r p (cid:48) r (cid:48) .One can readily verify from Eq.(6) that this corresponds to E = 0 . A positive value, E > , implies an enhancementin the tendency for both reactions to proceed together, while E < reflects an anti-correlation - when one reaction pro-ceeds the other is less likely to, and vice versa. For genericentanglement between the spins in the Posner molecule pairs,one will have E (cid:54) = 0 , indicating that the chemical reactionsthemselves have become quantum entangled, even if spatiallyseparated . Clouds of multiple entangled Posner moleculescan induce correlated, non-local binding reactions, a powerful setting for quantum processing.
G. Quantum processing with neurons
To be functionally relevant in the brain, the dynamics andquantum entanglement of the phosphorus nuclear spins mustbe capable of modulating the excitability and signaling ofneurons - which we take as a working definition of “quan-tum cognition”. Phosphate uptake by neurons might providethe critical link. In presynaptic glutamatergic neurons thevesicular transmembrane protein VGLUT brings glutamateinto the vesicles driven by proton gradients (the vesicleis acidic with pH = 5.5). In the original discovery paper in 1994, VGLUT was reported to have a sequence homologyto a (rabbit) kidney phosphate transporter, which brings phos-phate into cells driven by a sodium concentration gradient. Moreover, VGLUT (which was, at the time, named BNPI forbrains sodium-phosphate transporter ) was found to uptakephosphate when expressed in Xenopus oocytes, in a sodium-concentration dependent manner. We propose that VGLUTplays a dual physiological role , both transporting glutamateinto presynaptic vesicles and transporting phosphate ions intothe presynaptic neurons during vesicle endocytosis - andthat this enables neuron uptake of Posner molecules, as de-tailed below.A rapid influx of calcium following an incoming ac-tion potential triggers the presynaptic vesicles to fuse withthe cell wall and release glutamate into the synaptic cleft(exocytosis). During subsequent endocytosis these vesiclesare retrieved (from the cell wall or in a “kiss-and-run” mode )and brought back into the presynaptic neuron. In this processthe sodium-rich extracellular fluid (with pH=7.4) will enterthe vesicle, perhaps engulfing Posner molecules floating in thesynaptic cleft. After pinch off and retreat the vesicle interiorwill become acidic due to proton pumps. Once the pH dropsbelow 6, we anticipate that any enveloped Posner moleculeswill melt via hydrolysis (“proton attack”) releasing phosphateand calcium ions into the vesicle interior.Due to the high Na + concentration in the vesicle interior af-ter endocytosis, the transmembrane protein VGLUT, now ex-posed to a large sodium concentration gradient, might trans-port the phosphate ions out of the vesicle into the cytoplasm.With local cytoplasmic calcium levels elevated during exo-cytosis, these phosphate ions could recombine with calcium,forming Posner molecules inside the neuron. In effect, gluta-mate release has triggered the influx of Posner molecules intothe presynaptic neurons .If a chemical bond subsequently forms between two Posnermolecules in the lower pH=7 environment of the cytoplasm,the stationary (non-rotating) dimer will be susceptible to melt-ing via hydrolysis (“proton attack”) - liberating 18 calciumions which could stimulate further glutamate release, therebyenhancing the firing of the postsynaptic neuron.During cellular uptake, nuclear spin entanglement betweentwo different Posner molecules will be retained, even if trans-ported into two different neurons. The uptake of many Posnermolecules could induce nuclear spin entanglement betweenmultiple presynaptic neurons. The chemical binding and sub-sequent melting of two Posner molecules inside a given neu-ron would then influence the probability of Posner moleculesbinding and melting in other neurons. This could lead to non-local quantum correlations in the glutamate release and post-synaptic firing across multiple neurons .A simple example with two neurons illustrating this criti-cal link between nuclear spin entanglement and neuron firingrates is depicted in Figure 3d. Compound and more elaborateprocesses involving multiple Posner molecules and multipleneurons are possible, and might enable complex nuclear-spinquantum processing in the brain.
H. Prospects
In this paper an apparently unique mechanism for quan-tum processing in the brain has been explored. The phos- phorus nuclear spins in phosphate ions serve as qubits, pair-wise entangled during hydrolysis of pyrophosphate, engulfedand protected inside Posner molecules, inducing entangle-ment of the nuclear spins and rotational states of multiplePosner molecules, which can be transported into presynap-tic glutamatergic neurons during vesicle endocytosis, withintra-cellular calcium being released by subsequent bindingand melting of the Posner molecules, stimulating further glu-tamate release, thereby enhancing, and quantum-entangling,postsynaptic neuron excitability and activity!An intricate story, with multiple links in the chain of re-quired processes. We briefly mention some experiments thatmight serve to refute, or perhaps strengthen, the hypothesis ofnuclear spin quantum processing in the brain.Dynamic light scattering and cryoTEM could be employedto explore the concentration of Posner molecules in simulatedbody fluids, upon varying the ion concentrations, pH and othercontrol variables. (Attempting to establish whether Pos-ner molecules are present in real body fluids, while challeng-ing, would also be critical.) Liquid state NMR methods couldbe used to measure the spin dynamics (e.g. spin coherencetimes) of the phosphorus nuclei inside Posner molecules. Calcium and oxygen isotopes (with non-zero nuclear spin) ifincorporated into the Posner molecules would presumably de-cohere the phosphorus nuclear spins, which might be acces-sible with NMR. Determining the prospects of replacing thecentral calcium ions in the Posner molecule with “impurity”elements - for example lithium and mercury ions, energeti-cally favorable in quantum chemistry calculations - wouldalso be instructive. If replacement is possible, varying thelithium and mercury isotopes and examining the effects onphosphorus spin coherence inside the Posner molecules couldalso be interesting.Many aspects of the mechanisms proposed in this pa-per could be explored in vitro. Establishing controlof pyrophosphate hydrolysis catalyzed by the enzymepyrophosphatase, in vitro would be a first step. Withcalcium present the released phosphate ions should bind intoPosner molecules. Probing possible phosphorus nuclear spinentanglement between multiple Posner molecules might bepossible by separating the solution into two separate contain-ers, lowering their pH to induce melting of chemically bondedPosner molecules and measuring the calcium release with cal-cium fluorescence molecules. Quantum entanglement wouldbe revealed by coincidences and correlations between the flu-orescence emitted from the two containers. If present, onecould envisage performing quantum processing, and, conceiv-ably, designing and implementing a liquid state nuclear-spinquantum computer. The mechanism for neuron uptake of Posner molecules, ar-guably required for in vivo quantum processing with phos-phorus nuclear spins, relies on the transport of phosphateby VGLUT from the presynaptic vesicle interior into thecytoplasm. In vitro experiments further establishing andcharacterizing the potential (sodium-concentration driven)phosphate transport by VGLUT would be essential. If the phosphorus nuclear spins inside Posner molecules areplaying a functional role in the brains of mammals (or, possi-bly, other vertebrates), then perturbations of the nuclear spinsmight have behavioral manifestations. Strong time and spa-tially dependent magnetic fields would be expected to modifythe phosphorus spin dynamics inside Posner molecules, andcould be characterized with NMR. Might this inform trans-cranial magneto stimulation protocols, modifying their ef-ficacy in treating mental illness? If two lithium ions can beincorporated inside the Posner molecules during molecule for-mation (replacing the central divalent calcium cation) theywould tend to decohere the phosphorus nuclear spins, offeringa possible explanation for the remarkable efficacy of lithiumin tempering mania in patients with bipolar disorder. If this isindeed the mechanism, one might expect a lithium isotope de-pendence on the behavioral response. Remarkably, a lithiumisotope dependence on the mothering behavior of rats chroni-cally fed either Li or Li - having elevated or depressed alert-ness levels, respectively - has indeed been reported. Repro-ducing this striking experiment would be paramount. Chronicingestion of the calcium-43 isotope, which has a large I = 7 / nuclear spin, might also possibly have deleterious effects onmice and rats. Might an exploration of the effects of shockwaves on the mechanical stability and nuclear spin dynam-ics (induced via excitation of vibrational modes) of Posnermolecules free floating in water have some relevance to braintrauma? It is hoped that the various experiments suggested abovemight be informative, in and of themselves - and possibly in refuting, or supporting, the hypothesis of nuclear-spin quan-tum processing in the brain.
Acknowledgments
I am grateful to many, many physicists, chemists and neu-roscientists for invaluable input and suggestions, including,Ehud Altman, Leon Balents, Bill Bialek, David Cory, RobertEdwards, Marla Feller, Daniel Fisher, Michael Fisher, YvetteFisher, Craig Garner, James Garrison, Mike Gazzaniga, SteveGirvin, Tarun Grover, Songi Han, Paul Hansma, Matt Helge-son, Alexej Jerschow, Ilia Kaminker, Charlie Kane, DavidKleinfeld, Ken Kosik, Allan MacDonald, Michael Miller,Thomas Mueggler, Ryan Mishmash, Lesik Motrunich, NickRead, Richard Reimer, Jeff Reimer, Steve Shenker, BorisShraiman, T. Senthil, Chuck Stevens, Ashvin Vishwanath,Shamon Walker, David Weld, Xiao-Gang Wen and XuemeiZhang. I would like to especially acknowledge Michael Swiftfor his quantum chemistry calculations of Posner molecules.This research was supported in part by the National ScienceFoundation under Grant No. DMR-14-04230, and by the Cal-tech Institute of Quantum Information and Matter, an NSFPhysics Frontiers Center with support of the Gordon and BettyMoore Foundation (M.P.A.F.). Thanks to the Aspen Center ofPhysics where some parts of this work were completed. M. Tegmark, “Importance of quantum decoherence in brain pro-cesses,” Physical Review E , 4194 (2000). C. Seife, “Cold Numbers Unmake the Quantum Mind,” Science , 5454 (2000). H. Hu and M. Wu, “Spin-mediated consciousness theory: possi-ble roles of neural membrane nuclear spin ensembles and param-agnetic oxygen,” Medical Hypothesis , 633 (2004). R. Penrose and S. Hameroff, “Consciousness in the Universe:Neuroscience, Quantum Space-Time Geometry and Orch ORTheory,” Journal of Cosmology (2011). P. J. Hore, “Nuclear Magnetic Resonance,” Oxford Science Pub-lications (2011). F. W. Wehrli, “Temperature dependent Spin-lattice relaxation ofLi-6 in Aqueous Lithium chloride,” Journal of Magnetic Reso-nance , 527 (1976). K. Gottfried and T. M. Yan, “Quantum Mechanics: Fundamen-tals,” Springer Verlag Press (2003). B. Alberts, D. Bray, A. Johnson, J. Lewis, M. Raff, K. Roberts,and P. Walter, “Essential Cell Biology: An introduction to theMolecular Biology of the Cell,” Garland Publishing (1998). H. Lodish, D. Baltimore, S. Zipursky, P. Matsudaira, andJ. Downell, “Molecular Cell Biology,” WH Freeman and Co(1999). C. Nicholson and E. Sykov, “Extracellular space structure re-vealed by diffusion analysis,” Trends Neurosci. , 207 (1998). X. Su, K. Sun, F. F. Z. Cui, and W. Landisc, “Organization ofapatite crystals in human woven bone,” Bone , 150 (2003). K. Onuma and I. Atsuo, “Cluster Growth Model for Hydroxyap-atite,” Chem. Mater. , 3346 (1998). K. Onuma and I. Atsuo, “Clustering of Calcium Phosphate
CaCl H P O KClH ,” J. Phys. Chem. B , 8230 (1999). A. Dey, P. Bomans, F. Muller, J. Will, G. Frederik, P.M.andde With, and N. Sommerdijk, “The role of prenucleation clus-ters in surface-induced calcium phosphate crystallization,” NatureMaterials , 1010 (2010). L. Wang, S. Li, E. Ruiz-Agudo, C. Putnis, and A. Putnis, “Pos-ner’s cluster revisited: direct imaging of nucleation and growth ofnanoscale phosphate clusters at the calcite-water interface,” Crys-tEngComm , 6252 (2012). M. Swift, C. Van de Walle, and M. Fisher, to be published . M. A. Nielsen and I. L. Chuang, “Quantum Computation andQuantum Information,” Cambridge University Press (2010). R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,“Quantum entanglement,” Rev. Mod. Phys. , 865 (Jun 2009). F. M. Harold, “Inorganic polyphosphates in biology: structure,metabolism and function,” Bacteriol Rev , 772 (1966). E. Wilson, “The Statistical Weights of the Rotational Levels ofPolyatomic Molecules, Including Methane, Ammonia, Benzene,Cyclopropane and Ethylene,” Journal of Chemical Physics , 276(1935). L. Yang, R. Liao, J. Yu, and R. Liu, “DFT Study on the Mech-anism of Escherichia coli Inorganic Pyrophosphatase,” J. Phys.Chem. B , 6505 (2009). G. Treboux, P. Layrolle, N. Kanzaki, K. Onuma, and I. Atsuo,“Existence of Posner’s Cluster in Vacuum,” J. Phys. Chem. A ,5111 (2000). N. Kanzaki, G. Treboux, S. Onuma, K. Tsutsumi, and I. Atsuo,“Existence of Posner’s Cluster in Vacuum,” Biomaterials , 2921(2001). X. Yin and M. Stott, “Biological Calcium Phosphates and Pos- ner’s Cluster,” Journal of Chemical Physics , 3717 (2003). F. Grases, M. Zelenkova, and O. Sohnel, “Structure and formationmechanism of calcium phosphate concretions formed in simulatedbody fluid,” Urolithiasis , 9 (2014). L. Du, S. Bian, B. Gou, Y. Jiang, J. Huang, Y. Gao, and Y. Zhao,“Structure of Clusters and Formation of Amorphous CalciumPhosphate and Hydroxyapatite: From the perspective of coordi-nation chemistry,” Cryst. Growth Des. , 3103 (2013). R. Ahdut-Hacohen, D. Duridanova, H. Meiri, and R. Rahamimoff,“Hydrogen ions control synaptic vesicle ion channel activity inTorpedo electromotor neurons,” The Journal of Physiology ,347 (2004). B. Ni, X. Wu, J. Wang, and S. M. Paul, “Regional Expressionand Cellular Localization of the Na + -dependent Inorganic Phos-phate Co-transporter of Rat Brain,” The Journal of Neuroscience , 5789 (1995). E. Bellocchio, H. Hu, A. Pohorille, J. Chan, V. M. Pickel, andR. H. Edwards, “The Localization of the Brain-Specific InorganicPhosphate Transporter Suggests a Specific Presynaptic Role inGlutamatergic Transmission,” The Journal of Neuroscience ,8648 (1998). E. Bellocchio, R. J. Reimer, R. T. Fremeau, and R. H. Edwards,“Uptake of Glutamate into Synaptic Vesicles by an InorganicPhosphate Transporter,” Science , 957 (2000). B. Ni, P. R. Rostek, N. S. Nadi, and S. M. Paul, “Cloning and ex-pression of a cDNA encoding a brain-specific Na + -dependent in- organic phosphate co-transporter,” Proc. Nod. Acad. Sci. , 5607(1994). J. E. Lever, “Active Phosphate Ion Transport in Plasma MembraneVesicles Isolated from Mouse Fibroblasts,” The Journal of Biolog-ical Chemistry , 2081 (1978). W. Boron and E. Boulpaep, “Medical Physiology: a cellular andmolecular approach (2 ed.),” Philadelphia: Saunders, (2012) . E. R. Kandel, J. H. Schwawrtz, T. M. Jessell, S. A. Sieegelbaum,and A. J. Hudspeth, “Principles of Neuroscience,” McGraw Hill(2013). S. Gandhi and C. Stevens, “Three modes of synaptic vesicularrecycling revealed by single-vesicle imaging,” Nature , 607(2003). M. Glinn, B. Ni, and S. M. Paul, “Characterization of Na + -dependent Phosphate Uptake in Cultured Fetal Rat Cortical Neu-rons,” J. Neurochem. , 2358 (1995). A. T. Barker, R. Jalinous, and I. L. Freeston, “Non-Invasive Mag-netic Stimulation of Human Motor Cortex,” The Lancet , 1106(1985). J. A. Sechzer, K. W. Lieberman, G. J. Alexander, D. Weidman,and P. E. Stokes, “Aberrant Parenting and Delayed Offspring De-velopment in Rats Exposed to Lithium,” Biol Psychiatry , 1258(1986). A. Maas, N. Stocchetti, and R. Bullock, “Moderate and severetraumatic brain injury in adults,” Lancet Neurology7