Monitoring ion-channel function in realtime through quantum decoherenceLiam T. Halla,1, Charles D. Hilla, Jared H. Coleb, Brigitte Städlerc, Frank Carusoc, Paul Mulvaneyd,Jörg Wrachtrupe, and Lloyd C. L. Hollenberga
aCentre for Quantum Computer Technology, School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia; bInstitute für TheoretischeFestkörperphysik and Deutsche Forschungsgemeinschaft—Center for Functional Nanostructures, Karlsruher Institut für Technologie, D-76128 Karlsruhe,Germany; cCentre for Nanoscience and Nanotechnology, Department of Chemical and Biomolecular Engineering, University of Melbourne, Parkville,Victoria 3010, Australia; dSchool of Chemistry and Bio21 Institute, University of Melbourne, Parkville, Victoria 3010, Australia; and e3. PhysikalischesInstitut and Stuttgart Research Center of Photonic Engineering, Universität Stuttgart, 70550 Stuttgart, Germany
Edited by Lu Jeu Sham, University of California at San Diego, La Jolla, CA, and approved August 20, 2010 (received for review March 15, 2010)
In drug discovery, there is a clear and urgent need for detection ofcell-membrane ion-channel operation with wide-field capability.Existing techniques are generally invasive or require specializednanostructures. We show that quantum nanotechnology couldprovide a solution. The nitrogen-vacancy (NV) center in nanodia-mond is of great interest as a single-atom quantum probe fornanoscale processes. However, until now nothing was knownabout the quantum behavior of a NV probe in a complex biologicalenvironment. We explore the quantum dynamics of a NV probein proximity to the ion channel, lipid bilayer, and surroundingaqueous environment. Our theoretical results indicate that real-time detection of ion-channel operation at millisecond resolutionis possible by directly monitoring the quantum decoherence ofthe NV probe. With the potential to scan and scale up to an array-based system, this conclusion may have wide-ranging implicationsfor nanoscale biology and drug discovery.
biophysics ∣ magnetometry ∣ nanomagnetometry ∣ open quantum systems
The cell membrane is a critical regulator of life. Its importanceis reflected by the fact that the majority of drugs target mem-
brane interactions (1). Ion channels allow for passive and selec-tive diffusion of ions across the cell membrane (2), whereas ionpumps actively create and maintain the potential gradients acrossthe membranes of living cells (3). To monitor the effect of newdrugs and drug delivery mechanisms, a wide-field ion-channelmonitoring capability is essential (4). However, there are signifi-cant challenges facing existing techniques stemming from thefact that membrane proteins, hosted in a lipid bilayer, require acomplex environment to preserve their structural and functionalintegrity (1, 5–7). Patch clamp techniques are generally invasive,quantitatively inaccurate, and difficult to scale up (8, 9), whereasblack lipid membranes (10, 11) often suffer from stability issuesand can only host a limited number of membrane proteins.
Instead of altering the way ion channels and the lipid mem-brane are presented or even assembled for detection, ourapproach is to consider a unique and inherently noninvasive insitu detection method based on the quantum properties of asingle-atom probe. The atomic probe is a single nitrogen-vacancy(NV) center in a nanodiamond crystal that is highly sensitive tomagnetic fields and shows great promise as a magnetometerfor nanobiosensing (12–19). The NV center in nanodiamondhas already been used as a fluorescence marker in biologicalsystems (20–24). However, up to now there has been no analysisof the effect of the biological environment on the quantumdynamics of the NV center—such considerations are critical tonanobiomagnetometry applications. We explore these issues indetail and, furthermore, show that the rate of quantum decoher-ence of the NV center is sufficiently sensitive to the flow of ionsthrough the channel to allow real-time detection, over and abovethe myriad background effects. In this context, decoherencerefers to the loss of quantum coherence between magneticsublevels of the NV atomic system due to interactions with an
environment. Such superpositions of quantum states are gener-ally fleeting in nature due to interactions with the environment,and the degree and timescale over which such quantum coher-ence is lost can be measured precisely. The immediate conse-quence of the fragility of the quantum coherence phenomenonis that detecting the loss of quantum coherence (decoherence)in such a single-atom probe offers a unique monitor of biologicalfunction at the nanoscale (18, 19).
The NV probe (Fig. 1) consists of a diamond nanocrystal con-taining an NV defect at the end of an atomic force microscopetip, as recently demonstrated (16). For biological applications,a quantum probe must be submersible to be brought withinnanometers of the sample structure, hence the NV system lockedand protected in the ultrastable diamond matrix (Fig. 1A) is thesystem of choice. The NV center alone offers the controllable,robust, and persistent quantum properties such room tempera-ture nanosensing applications will demand, as well as zero toxicityin a biological environment (20–22). Theoretical proposalsfor the use of diamond nanocrystals as sensitive nanoscale mag-netometers (12–14) have been followed by proof-of-principleexperiments (15–17). However, such nanoscale magnetometersemploy only a fraction of the quantum resources at hand anddo not have the sensitivity to detect the minute magnetic momentfluctuations associated with ion-channel operation. In contrast,our results show that measuring the quantum decoherence of theNV induced by the ion flux provides a highly sensitive monitoringcapability for the ion-channel problem, well beyond the limits ofmagnetometer time-averaged field sensitivity (19). To determinethe sensitivity of the NV probe to the ion-channel signal, wedescribe the lipid membrane, embedded ion channels, and theimmediate surroundings as a fluctuating electromagnetic envir-onment and quantitatively assess each effect on the quantumcoherence of the NV center. We consider the diffusion of nuclei,atoms, and molecules in the immediate surroundings of thenanocrystal and the extent to which each source will decoherethe quantum state of the NV. We find that, over and above thesebackground sources, the decoherence of the NV spin levels ishighly sensitive to the ion flux through a single ion channel.Our theoretical findings demonstrate the potential of thisapproach to revolutionize the way ion channels and potentiallyother membrane-bound proteins or interacting species are char-acterized and measured, particularly when scaleup and scanningcapabilities are considered.
Author contributions: L.T.H. and L.C.L.H. designed research; L.T.H., C.D.H., J.H.C., B.S.,F.C., P.M., J.W., and L.C.L.H. performed research; L.T.H., C.D.H., J.H.C., B.S., F.C., P.M.,J.W., and L.C.L.H. contributed new reagents/analytic tools; L.T.H., C.D.H., J.H.C., B.S.,F.C., P.M., J.W., and L.C.L.H. analyzed data; and L.T.H., C.D.H., J.H.C., B.S., F.C., P.M.,J.W., and L.C.L.H. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: [email protected].
www.pnas.org/cgi/doi/10.1073/pnas.1002562107 PNAS ∣ November 2, 2010 ∣ vol. 107 ∣ no. 44 ∣ 18777–18782
APP
LIED
PHYS
ICAL
SCIENCE
S
Dow
nloa
ded
by g
uest
on
Feb
ruar
y 28
, 202
1
This paper is organized as follows. We begin by describing thequantum decoherence imaging system (Fig. 1) implemented usingan NV center in a realistic technology platform. The biologicalsystem is described in detail, and estimates of the sensitivity ofthe NV decoherence to various magnetic field sources are madethat indicate the ability to detect ion-channel switch-on/offevents. Finally, we conduct large-scale numerical simulations ofthe time evolution of the NV spin system, including all magneticfield generating processes, which acts to verify the analytic pictureand provides quantitative results for the monitoring and scanningcapabilities of the system.
ResultsThe energy level scheme of the C3v-symmetric NV system(Fig. 2B) consists of ground (3A), excited (3E), and metastable(1A) states. The ground-state manifold has spin sublevels (m ¼ 0,�1), which in zero field are split by 2.88 GHz. In a backgroundmagnetic field, the lowest two states (m ¼ 0, þ1) are readilyaccessible by microwave control. An important property of the
NV system is that, under optical excitation, the spin levels aredistinguishable by a difference in fluorescence, hence spin-statereadout is achieved by purely optical means (25, 26). These prop-erties are fortuitous in the current context because mammalianion-channel function is known to be insensitive to optical light(27). Because of the simple readout and control, the quantumproperties of the NV system and the interaction with the immedi-ate crystalline environment are well known (28, 29). The coher-ence time of the spin levels is very long even at room temperature:In type 1b nanocrystals T2 ∼ 15 μs (30) and, in isotopically engi-neered diamond, can be as long as 1.8 ms (17) with the use of aspin-echo microwave control sequence (Fig. 2C). More encoura-ging is that these times are predicted to be as high as 75 μs and100 ms, respectively, with the application of optimal microwavepulse sequences (31). Nanocrystals of 5-nm diameter containingstable NV centers have recently been demonstrated (32).
Typical ion-channel species Kþ, Ca2þ, Naþ, and nearby watermolecules are electron spin paired, so any magnetic signal dueto ion-channel operation will be primarily from the motion ofnuclear spins. Ions and water molecules enter the channel inthermal equilibrium with random spin orientations and movethrough the channel over a microsecond timescale. The monitor-ing of ion-channel activity occurs via measurement of the contrastin probe behavior between the on and off states of the ion chan-nel. This protocol requires the dephasing due to ion-channelactivity to be at least comparable to that due to the fluctuatingbackground magnetic signal. We must therefore account forthe decoherence due to the diffusion of water molecules, buffermolecules, saline components, as well as the transversal diffusionof lipid molecules in the cell membrane.
The nth nuclear spin with charge qn, gyromagnetic ratio γn,velocity ~vn, and spin vector ~Sn interacts with the NV spin vector~P and gyromagnetic ratio γp through the time-dependent dipoledominated interaction
H intðtÞ ¼ ∑N
n¼1
κðnÞdip
� ~P· ~Snr3nðtÞ
− 3~P· ~rnðtÞ ~Sn· ~rnðtÞ
r5nðtÞ�; [1]
where κðnÞdip ≡μ04π ℏ
2γpγn are the probe-ion coupling strengths, and~rnðtÞ is the time-dependent ion-probe separation. AdditionalBiot–Savart fields generated by the ion motion, both in thechannel and the extracellular environment, are several orders ofmagnitude smaller than this dipole interaction and are neglectedhere. Any macroscopic fields due to intracellular ion currents areof nanotesla order and are effectively static over T2 timescales.These effects will thus be suppressed by the spin-echo pulsesequence.
In Fig. 3A, we show typical field traces at a probe height of1–10 nm above the ion channel, generated by the ambient envir-onment and the onset of ion-flow as the channel opens. Thecontribution to the net field at the NV probe position from thevarious background diffusion processes dominate the ion-channelsignal in terms of their amplitude. Critically, because the magnet-ometer mode detects the field by acquiring phase over the coher-ence time of the NV center, both the ion-channel signal andbackground are well below the nanotesla Hz−1∕2 sensitivity limitof the magnetometer over the (∼1 ns) self-correlated timescalesof the environment. However, the effect of the various sourceson the decoherence rate of the NV center are distinguishablebecause the amplitude-fluctuation frequency scales are verydifferent, leading to remarkably different dephasing behavior.
To understand this effect, we need to consider the fullquantum evolution of the NV probe. In the midst of this envir-onment, the probe’s quantum state, described by the densitymatrix ρðtÞ, evolves according to the Liouville equation,ddt ρðtÞ ¼ − i
ℏ ½HðtÞρðtÞ − ρðtÞHðtÞ�, where ρðtÞ is the incoherentthermal average over all possible unitary evolutions of the entiresystem, as described by the full Hamiltonian, H ¼ Hnv þH intþ
ADiamond nanocrystal
containing NV center
2.88 GHz microwave
quantum control
BLipid bilayer
Fluorescence
readout
1.0
0.8
0.6
0 05
5-5-5
10
-10 -10
10
x [nm]y [nm]
∆PC
0 50 100 150 200
0.6
0.8
1
Time [ms]Ion channel stateMeasurement record
On
Off
D∆P
Fig. 1. Quantum decoherence imaging of ion-channel operation (simula-tions). (A) A single NV defect in a diamond nanocrystal is placed on an atomicforce microscope tip. The unique properties of the NV atomic level schemeallows for optically induced readout and microwave control of magnetic(spin) sublevels. (B) The nearby cell membrane is host to channels permittingthe flow of ions across the surface. The ion motion results in an effectivefluctuating magnetic field at the NV position which decoheres the quantumstate of the NV system. (C) This decoherence results in a decrease in fluores-cence, which is most pronounced in regions close to the ion-channel opening.(D) Changes in fluorescence also permit the temporal tracking of ion-channeldynamics.
A B C
Fig. 2. Details of the NV center structure, energy levels, and control scheme.(A) NV-center diamond lattice defect. (B) NV spin detection through opticalexcitation and emission cycle. Magnetic sublevels ms ¼ 0 and ms ¼ �1 aresplit by a D ¼ 2.88 GHz crystal field. Degeneracy between the ms ¼ �1
sublevels is lifted by a Zeeman shift, δω. Application of 532 nm green lightinduces a spin-dependent photoluminescence and pumping into the ms ¼ 0
ground state. (C) Microwave and optical pulse sequences for coherent controland readout.
18778 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1002562107 Hall et al.
Dow
nloa
ded
by g
uest
on
Feb
ruar
y 28
, 202
1
Hbg, where Hnv is the Hamiltonian of the NV system, and H intdescribes the interaction of the NV system with the backgroundenvironment (e.g., diffusion of ortho spin water species and ionsin solution) and any intrinsic coupling to the local crystal envir-onment. The self-evolution of the background system is describedby Hbg, which, in the present methodology, is used to obtainthe noise spectra of the various background processes. We notethat the following analysis assumes dephasing to be the dominantdecoherence channel in the system. We ignore relaxation pro-cesses because all magnetic fields considered are at least 4 ordersof magnitude less than 2.88 GHz, and are hence unable to flipthe probe spin over relevant timescales. Phonon excitation inthe diamond crystal may also be neglected (17). Before movingonto the numerical simulations, we consider some importantfeatures of the problem.
The decoherence rate of the NV center is governed by theaccumulated phase variance during the control cycle. Maximaldephasing due to a fluctuating field will occur at the cross-overpoint between the fast and slow fluctuation regimes (19). A mea-sure of this point is the dimensionless ratio Θ≡ f e∕γpσB, whereτe ¼ 1∕f e is the correlation time of the fluctuating signal, withcross-over at Θ ∼ 1.
In what follows, we consider exclusively the behavior of asodium ion channel. Sodium-23 has an effective abundance of100% and a nuclear magnetic moment of μNa ¼ 2.22μN , whereμN ¼ 5.05 × 10−27 JT−1 is the nuclear magneton. Other magneti-cally active ion-channel species include potassium-39, having a93.1% abundance and a nuclear magnetic moment of 0.391μN ,and chlorine-35, having a 75.4% abundance and a nuclear mag-netic moment of 0.821μN , ensuring sodium ions will interact moststrongly with the NV center. We can estimate the field standarddeviation σicB due to the random nuclear spin of ions and boundwater molecules moving through an ion channel (ic) as
σicB ∼ μ04π
1h3p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNNaμ
2Na þ NH2Oμ
2H2O
q, where NNa and NH2O are the
average numbers of sodium ions and water molecules insidethe channel. By modeling the channel as a cylinder with typical
sodium channel influx/outflux rates (33), we may numerically cal-culate the rms fluctuation strength of the ion-channel magneticfield, σicB , as a function of the probe standoff distance hp, as shownin Fig. 3B. The fluctuation rate is defined by the rate at whichions move through the channel and is independent of whetherthe ions are moving into or out of the cell. However, in typicalphysiological processes (neuronal firing for example), the sodiumflux will be inward. Ion flux rates give an effective fluctuation rateof f ice ∼ 3 MHz (33). For probe-channel separations of 2–8 nm,values of Θ range from 0.4 to 40 (Fig. 3C). Thus, the ion-channelflow hovers near the cross-over point, with an induced dephasingrate of Γic ∼ 104–105 Hz.
At these separation distances, the presence of the diamondsurface is expected to have a negligible effect on the ion-channeldynamics. To reduce the decoherence effects of surface spins onthe NV, the surface may be terminated with H or OH moieties.This surface termination essentially replaces the electron spinsassociated with the sp2 hybridized orbitals, with weaker nuclearspins (32). The ions in the channel, also being nuclear spins,couple very weakly to their surrounding environment. We mayestimate their coupling to the diamond surface as f ∼ μ0
4πℏ μ2Nh
−3p ∼
1 Hz, which is negligible compared to the fluctuation rate of thechannel itself. Additionally, we may approximate the ratiobetween the magnetic force on the ions due to the surface spinsand the electric force due to adjacent ions as FB∕FE∼ð3 μ0
4π μ2Nh
−4p Þ∕ð e2
4πε0Δr−2Þ ∼ 10−15, where Δr is the typical distance
between adjacent ions in the channel. Similarly, the ratio betweenthe magnetic force due to the NV spin and the electric forceis FB∕FE ∼ ð3 μ0
4π μBμNh−4p Þ∕ð e2
4πε0Δr−2Þ ∼ 10−12, thus we expect
the presence of the probe to be truly noninvasive.We now consider the dephasing effects of the various sources
of background magnetic fields. The first source of backgroundnoise is the fields arising from the motion of the water moleculesand ions throughout the aqueous solution. Due to the nuclearspins of the hydrogen atoms, liquid water consists of a mixtureof spin neutral (para) and spin-1 (ortho) molecules. The equili-brium ratio of ortho to para molecules (OP ratio) is 3∶1 (34),making 75% of water molecules magnetically active. In biologicalconditions, dissolved ions occur in concentrations 2–3 orders ofmagnitude below this number density and are ignored here (theyare important however for calculations of the induced Starkshift; see below). The rms strength of the field due to the aqueoussolution is σH2O
B ∼ gHμNμ02π
ffiffiffiffiffiffiffiffiffiffiffiffiffiffinH2O
πh3p
q. This magnetic field is there-
fore 1–2 orders of magnitude stronger than the field from the ionchannel (Fig. 3 A and B). The fluctuation rate of the aqueousenvironment is dependent on the self-diffusion rate of the watermolecules. Using DH2O ¼ 3 × 10−9 m2 s−1, the fluctuation rate isfH2Oe ∼DH2O∕ð2hpÞ2. This fluctuation rate places the magneticfield due to the aqueous solution in the fast-fluctuation regime,with ΘH2O ∼ 103–104 (Fig. 3B), giving a comparatively slow de-phasing rate of ΓH2O ∼ fH2O
e Θ−2H2O
∼ 100 Hz and correspondingdephasing envelope DH2O ¼ e−ΓH2O
t.An additional source of background dephasing is the lipid
molecules comprising the cell membrane. Assuming magneticcontributions from hydrogen nuclei in the lipid molecules, lateraldiffusion in the cell membrane gives rise to a fluctuating Bfield, with a characteristic frequency related to the diffusionrate. Atomic hydrogen densities in the membrane are nH ∼ 3×1028 m−3. At room temperature, the populations of the spin statesof hydrogen will be equal, thus the rms field strength is given byσLB ∼ gHμN
μ08π
ffiffiffiffiffiffiffiffiffin 5π
4h3p
q. The strength of the fluctuating field due to
the lipid bilayer is of the order of 10−7 T (Fig. 3A). The diffusionconstant for lateral Brownian motion of lipid molecules in lipidbilayers is DL ¼ 2 × 10−15 m2 s−1 (35), giving a fluctuation
6
2 4 6 8 1010-10
10-8
10-7
10-6
10-5
B
σ B[T
]
Ion channelsLipid bilayerH
2O
Electrolyte
2 4 6 8 10
100
105 CΘ
Ion channelsLipid bilayerH
2O
Electrolyte
FFL
SFL
Time [ms]
H2O signal
Lipid bilayer signal
Ion channel signal
Na ions+
Ion channel
Cell membrane
H2O
A
-10
10
0
0.2 0.4 0.6 0.8 10
Fig. 3. Sources of magnetic field fluctuations and their relative amplitudes.(A) Calculated magnetic field signals from water, ion-channel, and lipidbilayer sources at a probe standoff of 4 nm over a 1 ms timescale. (B) Com-parison of σB for various sources of magnetic fields. (C) Fluctuation regime,Θ ¼ fe∕γpσB, for magnetic field sources vs. probe standoff. Rapidly fluctuat-ing fields (Θ ≫ 1) are said to be in the fast-fluctuating limit (FFL). Slowly fluc-tuating fields (Θ ≪ 1) are in the slow fluctuation limit (SFL). The ion-channelsignal exists in the Θ ∼ 1 regime and therefore has an optimal dephasingeffect on the NV probe.
Hall et al. PNAS ∣ November 2, 2010 ∣ vol. 107 ∣ no. 44 ∣ 18779
APP
LIED
PHYS
ICAL
SCIENCE
S
Dow
nloa
ded
by g
uest
on
Feb
ruar
y 28
, 202
1
frequency of fLe ∼ 125 Hz and ΘL ∼ 10−4 (Fig. 3C). At this fre-quency, any quasi-static field effects will be predominantlysuppressed by the spin-echo refocusing. The leading-order(gradient-channel) dephasing rate is given by (19) ΓL∼
1
2ffiffiffiffiffiffiffi2ffiffi2
pp Θ−1∕2L fLe þ OðΘ−1∕3
L fLe Þ, giving rise to dephasing rates of
the order ΓL ∼ 100 Hz, with corresponding dephasing envelopeDLðtÞ ¼ e−Γ
4L t
4
.The electric fields associated with the dissolved ions also
interact with the NV center via the ground-state Stark effect.The coefficient for the frequency shift as a function of the electricfield applied along the dominant (z) axis is given by R3D ¼ 3.5 ×10−3 HzmV−1 (36). Fluctuations in the electric field may berelated to an effective magnetic field via Beff
z ¼ R3DEz∕γp, whichmay be used in an analysis similar to that above. An analysis usingDebye–Hückel theory (37) shows charge fluctuations of an ionicsolution in a spherical region Λ of radius R behave as hQ2
Λi ¼DEkBTð1þ κRÞe−κR½R coshðκRÞ − sinhðκRÞ
κ �, where DE is the diffu-sion coefficient of the electrolyte, and κ is the inverse Debyelength (lD); lD ¼ 1∕κ ¼ 1.3 nm for biological conditions.Although this analysis applies to a region Λ embedded in aninfinite bulk electrolyte system, simulation results discussedbelow show very good agreement when applied to the systemconsidered here. The electric field variance may be obtained fromhQ2
Λi, giving σE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihE2i − hEi2
p∼ 106 Vm−1, as a function of hp.
Relaxation times for electric field fluctuations are τEe ¼ ϵϵ0ρE(38), where ρE is the resistivity of the electrolyte, giving fEe ∼1∕τEe ¼ 1.4 GHz under biological conditions. Given the relativelylow strength (Fig. 3A) and short relaxation time of the effectiveStark-induced magnetic field fluctuations (Θ ∼ 105) (Fig. 3B), weexpect the charge fluctuations associated with ions in solutionto have little effect on the evolution of the probe.
DiscussionWe now turn to the problem of noninvasively resolving thelocation of a sodium ion channel in a lipid bilayer membrane.When the channel is closed, the dephasing is the result of thebackground activity and is defined by Doff ¼ DH2ODLDED13C
.When the channel is open, the dephasing envelope is definedby Don ¼ DoffDic. Maximum contrast will be achieved by opti-mizing the spin-echo interrogation time, τ, to ensure Doff −Donis maximal. Thus, in the vicinity of an open channel at the point ofoptimal contrast, τ ≈ T2∕2, we expect an ensemble ground-statepopulation of PonðT2
2Þ ¼ 1
2½1þDonðT2
2Þ� ¼ 0.61, and PoffðT2
2Þ ¼
12½1þDoffðT2
2Þ� ¼ 0.93 otherwise. By scanning over an open ion
channel and monitoring the probe via repeated measurementsof the spin state, we may build up a population ensemble for eachlateral point in the sample. The signal-to-noise ratio improveswith the dwell time at each point. Fig. 4 shows simulated scansof a sodium ion channel with corresponding image acquisitiontimes of 4, 40, and 400 s. It should be noted here that the spatialresolution available with this technique is beyond that achievableby magnetic field measurements alone, because for large Θ,ΔP ∝ B2 ∝ h−6p .
We may employ similar techniques to temporally resolve asodium ion-channel switch-on event. By monitoring a singlepoint, we may build up a measurement record, I. In an experi-mental situation, measurement frequency has an upper limit offm ¼ ðτ þ τm þ τ2πÞ−1, where τm ≈ 900 ns is the time requiredfor photon collection, and τ2π is the time required for all threemicrowave pulses. A tradeoff exists between the increaseddephasing due to longer interrogation times and the correspond-ing reduction in measurement frequency. Interrogation times arelimited by the intrinsic T2 time of the crystal. A second tradeoffexists between the variance of a given set of Nτ consecutivemeasurements and the temporal resolution of the probe. Forthe monitoring of a switching event, the spin state may be inferredwith increased confidence by performing a running average over alarger number of data points, Nτ. However, increasing Nτ willlead to a longer time lag before a definitive result is obtained.The uncertainty in the ion-channel state goes as δP ∼ ð ffiffiffiffiffiffi
Nτ
p Þ−1,where Nτ is the number of points included in the dynamicaveraging. We must take sufficient Nτ to ensure thatδP < ΔPðτ;hp;T2Þ ¼ Poff − Pon. The temporal resolution de-pends on the width of the dynamic average and is given byδt ∼Nτðτ þ τmÞ, giving the relationship δt ¼ τþτm
δP2 > τþτm½ΔPðτ;hp;T2Þ�2.
We wish to minimize this function with respect to τ for a givenstandoff (hp) and T2 time.
In reality, not all crystals are manufactured with equal T2
times. An important question is therefore, for a given T2, whatis the best temporal resolution we may hope to achieve? Fig. 5Bshows the optimal temporal resolution as a function of T2. Itcan be seen that δt improves monotonically with T2 until T2
exceeds the dephasing time due the fluctuating backgroundfields (Fig. 5A). Beyond this point, no advantage is found fromextending T2.
A plot of δt as a function of τ is given in Fig. 5C for standoffs of2–6 nm. Solid lines depict the resolution that maybe achievedwith T2 ¼ 300 μs. Dashed lines represent the resolution thatmay be achieved by extending T2 beyond the dephasing timesof background fields. We see that δt diverges as τ → T2 and isoptimal for τ → 1∕Γic.
As an example of monitoring of ion-channel behavior, weconsider a crystal with a T2 time of 300 μs at a standoff of3 nm. Fig. 5C tells us that an optimal temporal resolution of
60 nm 20 nm
|∆P| = |Poff-Pon|
4 s 40 s 400 s
Fig. 4. Simulated spatial scans based on the ion channel as a dephasingsource. Relative population differences are plotted for pixel dwell timesof 10, 100 and 1,000 ms. Corresponding image acquisition times are 4, 40,and 400 s.
2 4 6 8 10
100
105A
Γ [H
z]
10-6 10-4 10-210-4
10-2
100
B
2 nm
3 nm4 nm
5 nm 6 nm
Opt
imal
δt [
s]
10-6 10-5 10-4 10-3 10-210-4
10-3
10-2
10-1
100
Free−evolution time, τ [s]
δt [
s]
C
2 nm
3 nm
4 nm 5 nm
6 nm
T2 = 300µs
Crystal T2 times [s]
Γic
ΓL
ΓH2O
ΓC13
ΓE
101
7 nm
T2 = 15µs
Fig. 5. Temporal characteristics of the measurement protocol. (A) Dephas-ing rates due to the sources of magnetic field plotted as a function ofprobe standoff, hp. (B) Optimum temporal resolution as a function ofcrystal T2 times for hp ¼ 2–6 nm. (C) Temporal resolution as a function ofinterrogation time, τ, for separations of 2–7 nm. The limits correspondingto T2 ¼ 15 μs and T2 ¼ 300 μs are shown as vertical dashed lines.
18780 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1002562107 Hall et al.
Dow
nloa
ded
by g
uest
on
Feb
ruar
y 28
, 202
1
δt ∼ 1.1 ms may be achieved by choosing τ ∼ 100 μs. This inter-rogation time suggests an optimal running average will employNτ ¼ δtðτ þ τmÞ−1 ≈ 11 data points. Fig. 6A shows a simulated de-tection of a sodium ion-channel switch-on event using Nτ ¼ 20,50, and 100 points. The effect of increasing Nτ is shown to givepoorer temporal resolution but also produces a lower variancein the signal, which may be necessary if there is little contrastbetween Poff and Pon. Conversely, decreasing Nτ results in animprovement to the temporal resolution but leads to a largersignal variation.
We now consider an ion-channel switching between states afteran average waiting time of 5 ms (200 Hz) (Fig. 6B). To ensure thecondition δP < ΔP is satisfied, we perform the analysis usingNτ ¼ 20, giving a resolution of δt ≈ 2 ms. The blue curve showsthe response of the NV population to changes in the ion-channelstate. Fourier transforms of the measurement record, FðIÞ,are shown in Fig. 6 C–E). The switching dynamics are clearlyresolvable for heights less than 6 nm. The dominant spectralfrequency is 100 Hz which is half the 200 Hz switching rate asexpected. Beyond 6 nm, the contrast between Poff and Pon istoo small to be resolvable due to the T2 limited temporal resolu-tion, as given in Fig. 5B. This contrast may be improved via themanufacturing of nanocrystals with improved T2 times, allowingfor longer interrogation times (dashed curves, Fig. 5C).
With regard to scaleup to a wide-field imaging capability,beyond the obvious extrinsic scaling of the number of singlechannel detection elements (in conjunction with microconfocalarrays), we consider an intrinsic scaleup strategy using manyNV centers in a bulk diamond probe (39), with photons collected
in a pixel arrangement. Because the activity of adjacent ionchannels is correlated by the micrometer scale activity of themembrane, the fluorescence of adjacent NV centers will likewisebe correlated, thus wide-field detection will occur via a fluores-cence contrast across the pixel. Implementation of this schemeinvolves a random distribution of NV centers in a bulk diamondcrystal. The highest reported NV density is 2.8 × 1024 m−3 (40),giving typical NV–NV couplings of <10 MHz which are strongenough to introduce significant additional decoherence. Weseek a compromise between increased population contrast andincreased decoherence rates due to higher NV densities, nnv,given by Γnv ∼
ffiffiffiffi2π
p3
ℏμ04π γ2pnnv (19).
For ion-channel operation correlated across each pixel, thetotal population contrast ΔΦ between off and on states isobtained by averaging the local NV state population changeΔΦðτÞ ¼ Poffð ~ri; ~rc;τÞ − Ponð ~ri; ~rc;τÞ over all NV positions ~ri andorientations, and ion-channel positions ~rc and species, and max-imizing with respect to τ. As an example, consider a crystal withnnv ¼ 1024 m−3 whose surface is brought within 3 nm of the cellmembrane containing an sodium and potassium ion-channeldensities of ∼2 × 1015 m−2 (41). Higher densities will yield betterresults, however, these have not been realized experimentally asyet, and electron spins in residual nitrogen will begin to induceNV spin flips. We expect ion-channel activity to be correlatedacross pixel areas of 1 × 1 μm, so the population contrast betweenoff and on states is ΔΦ ≈ 15. At these densities, the optimalinterrogation time is τ ∼ 0.8 μs, yielding an improvement inthe temporal resolution by a factor of 10,000, opening up thepotential for single-shot measurements of ion-channel activityacross each pixel.
We have carried out an extensive analysis of the quantumdynamics of a NV diamond probe in the cell-membrane environ-ment and determined the theoretical sensitivity for the detection,monitoring, and imaging of single ion-channel function throughquantum decoherence. Using current demonstrated technology,a temporal resolution in the 1–10 ms range is possible, withspatial resolution at the nanometer level. With the scope forscaleup and unique scanning modes, this fundamentally differentdetection mode has the potential to revolutionize the character-ization of ion-channel action, and possibly other membraneproteins, with important implications for molecular biology anddrug discovery.
Materials and MethodsGeometry. All numerical calculations were performed with Monte Carlomethods using MATLAB, and conducted on a rectangular slab. Reflectiveboundary conditions are imposed at the top and bottom, representingthe diamond crystal and lipid bilayer, respectively. Periodic boundary condi-tions are imposed on the sides of the volume. The lower boundary also hostsion channels with the surface density given in the main text. The NV center isplaced at a variable height above the diamond crystal boundary.
Ion Channel. We assume the waiting time between ion ejections from thechannel to follow a Poissonian distribution about a mean rate of 3 MHz. Eachion will couple to the NV spin via Eq. 1, with rnðtÞ describing the spatialseparation between the NV spin and the dipoles in the ion channel. Uponexiting the channel, the timescales associated with the ion motion willchange to that described by the self-diffusion rate of nuclear spins in the elec-trolyte. Because the addition of ions to the electrolyte at these frequencies isslow compared to the characteristic Brownian motion of the electrolyte, anyions exiting the channel will rapidly diffuse away. Thus for simplicity, ionsare terminated upon exiting the channel, making the nuclear dipole concen-tration near the channel opening equal to that of the bulk.
Magnetic Field of the Electrolyte. For the purpose of modeling the fluctuatingfield due to self-diffusive behavior of the electrolyte, we do not distinguishbetween different species of nuclear dipole, and instead consider aweighted-average nuclear dipole moment as defined by the concentrationsof water molecules and dissolved ions. The coupling of each ion to the NVcenter is again described by Eq. 1, however, couplings between ions, or to the
0 2 4 6 8 10 12 14 16
0.5
0.6
0.7
0.8
0.9
1
Time [ms]
Channel on
Ns = 20
Ns = 50
Ns = 100
A
0 10 20 30 40 50
0.5
0.6
0.7
0.8
0.9
1
Time [ms]
ms=
0 po
pula
tion
B
0 100 2000
0.5
1
fs [Hz]
FT
(I)
A.U
. hp = 4nm C
0 100 2000
0.5
1
fs [Hz]
hp = 5nm D
0 100 2000
0.5
1
fs [Hz]
hp = 6nm E
ms=
0 po
pula
tion
Fig. 6. Theoretical results for the detection of ion-channel operation. (A) Plotillustrating the dependence of temporal resolution (δt) and signal variance(δP) on the number of data points included in the running average (Ns).(B) Simulated reconstruction of a sodium ion-channel signal with a 200 Hzswitching rate using optical readout of an NV center (blue curve). The actualion-channel state (on/off) is depicted by the dashed line, and the green linedepicts the analytic confidence threshold. Fourier transforms ofmeasurementrecords are shown inC–E for standoffsof 4, 5, and6nm, respectively. Switchingdynamics are clearly resolvable for hp < 6 nm, beyond which there is littlecontrast between decoherence due to the ion-channel signal and thebackground.
Hall et al. PNAS ∣ November 2, 2010 ∣ vol. 107 ∣ no. 44 ∣ 18781
APP
LIED
PHYS
ICAL
SCIENCE
S
Dow
nloa
ded
by g
uest
on
Feb
ruar
y 28
, 202
1
mean field, are ignored because the associated timescales are much slowerthan those of their diffusive motion. Because of this weak coupling, each ionis assumed to not flip during its interaction with the NV center. Each of thernðtÞ then describes a random walk of a dipole of random but fixed orienta-tion, as defined by the self-diffusion coefficient of the electrolyte.
Magnetic Field of the Lipid Bilayer. Lipid diffusive motion is modeled in asimilar fashion to the bulk nuclear motion using the diffusion rate givenin the main text, however, the lipids are confined to the two-dimensionalplane comprising the lower boundary of the system geometry.
Electric Field of the Electrolyte. Regions of nonneutral charge density,together with their correlation lengths are found using Monte Carlonumerical techniques outlined in ref. 37. The rms electric field felt by theNV center may then be found by integrating over the charge distributionin the electrolyte.
ACKNOWLEDGMENTS. This work was supported by the Australian ResearchCouncil (ARC). F.C. and P.M. acknowledge support under the ARC FederationFellowship scheme, and L.C.L.H. is the recipient of an ARC Australian Profes-sorial Fellowship. J.W. acknowledges support from the Baden-WuerttembergStiftung, and European Union Projects Nadiatec and SOLID.
1. Reimhult E, Kumar K (2008) Using manifold structure for partially labelled classifica-tion. Trends Biotechnol 26:82–89.
2. Ide T, Takeuchi Y, Aoki T, Yanagida T (2002) Simultaneous optical and electricalrecording of a single ion-channel. Jpn J Physiol 52:429–434.
3. Baaken G, SondermannM, Schlemmer C, Ruhe J, Behrends J-C (2008) Planar microelec-trode-cavity array for high-resolution and parallel electrical recording of membraneionic currents. Lab Chip 8:938–944.
4. Lundstrom K (2006) Structural genomics for membrane proteins. Cell Mol Life Sci63:2597–2607.
5. Fang Y, Frutos A-G, Lahiri J (2002) Membrane protein microarrays. J Am Chem Soc124 2394–2395.
6. Yamazaki V, et al. (2005) Cell membrane array fabrication and assay technology.BMC Biotechnol 5:18.
7. Jelinek R, Silbert L (2009) Biomimetic approaches for studying membrane processes.Biosystems 5:811–818.
8. Damjanovich S (2005) Biophysical Aspects of Transmembrane Signalling (Springer,Berlin), pp 298–302.
9. Quick M (2002) Transmembrane Transporters (Wiley, Hoboken, NJ), pp 194–197.10. Mueller P, Rudin D-O, Tien H-T, Wescott W-C (1962) Reconstitution of cell membrane
structure in vitro and its transformation into an excitable system. Nature 194:979–980.11. Mueller P, Wescott W-C, Rudin D-O, Tien H-T (1963) Methods for formation of single
bimolecular lipid membranes in aqueous solution. J Phys Chem 67:205–209.12. Chernobrod B-M, Berman G-P (2005) Spin microscope based on optically detected
magnetic resonance. J Appl Phys 97:014903.13. Degen C-L (2008) Scanning magnetic field microscope with a diamond single-spin
sensor. Appl Phys Lett 92:243111.14. Taylor J-M, et al. (2008) High-sensitivity diamond magnetometer with nanoscale
resolution. Nat Phys 4:810–816.15. Maze J-R, et al. (2008) Nanoscale magnetic sensing with an individual electronic spin in
diamond. Nature 455:644–647.16. Balasubramanian G, et al. (2008) Nanoscale imaging magnetometry with diamond
spins under ambient conditions. Nature 455:648–651.17. Balasubramanian G, et al. (2009) Ultralong spin coherence time in isotopically
engineered diamond. Nat Mater 8:383–387.18. Cole J-H, Hollenberg L-C-L (2009) Scanning quantum decoherence microscopy.
Nanotechnology 20:495401.19. Hall L-T, Cole J-H, Hill C-D, Hollenberg L-C-L (2009) Sensing of fluctuating nanoscale
magnetic fields using nitrogen-vacancy centers in diamond. Phys Rev Lett 103:220802.20. Yu S-J, Kang M-W, Chang H-C, Chen K-M, Yu Y-C (2005) Bright fluorescent nanodia-
monds: No photobleaching and low cytotoxicity. J Am Chem Soc 127:17604–17605.21. Neugart F, et al. (2007) Dynamics of diamond nanoparticles in solution and cells.
Nano Lett 7:3588–3591.
22. Fu C-C, et al. (2007) Characterization and application of single fluorescent nanodia-monds as cellular biomarkers. Proc Natl Acad Sci USA 104:727–732.
23. Chao J-I, et al. (2007) Nanometer-sized diamond particle as a probe for biolabeling.Biophys J 93:2199–2208.
24. Faklaris O, et al. (2008) Detection of single photoluminescent diamond nanoparticlesin cells and study of the internalization pathway. Small 4:2236–2239.
25. Jelezko F, et al. (2002) Single spin states in a defect center resolved by optical spectro-scopy. Appl Phys Lett 81:2160–2162.
26. Jelezko F, Wrachtrup J (2006) Single defect centres in diamond: A review. Phys StatusSolidi B 203:3207–3225.
27. Kramer R-H, Chambers J-J, Trauner D (2005) Photochemical tools for remote controlof ion channels in excitable cells. Nat Chem Biol 1:360–365.
28. Jelezko F, Gaebel T, Popa I, Gruber A, Wrachtrup J (2004) Observation of coherentoscillations in a single electron spin. Phys Rev Lett 92:076401.
29. Hanson R, Dobrovitski V-V, Feiguin A-E, Gywat O, Awschalom D-D (2008) Coherentdynamics of a single spin interacting with an adjustable spin bath. Science320:352–355.
30. Rabeau J-R, et al. (2007) Single nitrogen vacancy centers in chemical vapor depositeddiamond nanocrystals. Nano Lett 7:3433–3437.
31. Hall L-T, Hill C-D, Cole J-H, Hollenberg L-C-L (2010) Ultra-sensitive diamond magneto-metry using optimal dynamic decoupling. Phys Rev B 82:045208.
32. Bradac C, et al. (2010) Observation and control of blinking nitrogen-vacancy centres indiscrete nanodiamonds. Nat Nanotechnol 5:345–349.
33. LeontiadouH,Mark A-E, Marrink S-J (2007) Ion transport across transmembrane pores.Biophys J 92:4209–4215.
34. Tikhonov V-I, Volkov A-A (2002) Separation of water into its ortho and para isomers.Science 296:2363–2363.
35. Bannai H, Levi S, Schweizer C, Dahan M, Triller A (2006) Imaging the lateral diffusionof membrane molecules with quantum dots. Nat Protoc 1:2628–2634.
36. Vanoort E, Glasbeek M (1990) Electric-field-induced modulation of spin echoes ofn-v centers in diamond. Chem Phys Lett 168:529–532.
37. Kim Y-C, Fisher M-E (2008) Charge fluctuations and correlation lengths in finiteelectrolytes. Phys Rev E 77:051502.
38. Fornes J-A (2000) Dielectric relaxation around a charged colloidal cylinder in anelectrolyte. J Colloid Interf Sci 222:97–102.
39. Steinert S, et al. (2010) High sensitivity magnetic imaging using an array of spins indiamond. Rev Sci Instrum 81:043705.
40. Acosta V-M, et al. (2009) Diamonds with a high density of nitrogen-vacancy centersfor magnetometry applications. Phys Rev B 80:115202.
41. Arhem P, Blomberg C (2007) Ion channel density and threshold dynamics of repetitivefiring in a cortical neuron model. Biosystems 89:117–125.
18782 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1002562107 Hall et al.
Dow
nloa
ded
by g
uest
on
Feb
ruar
y 28
, 202
1
