Feynman-Enderlein Path Integral for Single-Molecule Nanofluidics
FFeynman-Enderlein Path Integral for Single-Molecule Nanofluidics
Siddharth Ghosh †∗ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK.Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, UK.Maxwell Centre, Cavendish Laboratory, University of Cambridge, Cambridge, UK.St John’s College, University of Cambridge, Cambridge, UK.
I present a photon statistics method for quasi-one dimensional sub-diffraction limited nanofluidicmotions of single molecules using Feynman-Enderlein path integral approach. The theory is vali-dated in Monte Carlo simulation platform to provide fundamental understandings of Knudsen typeflow and diffusion of single molecule fluorescence in liquid. Distribution of single molecule burst sizecan be precise enough to detect molecular interaction. Realisation of this theoretical study considersseveral fundamental aspects of single-molecule nanofluidics, such as electrodynamics, photophysics,and multi-molecular events/molecular shot noise. I study two different sizes of molecules, one with 2nm and another with 20 nm hydrodynamic radii driven by a wide range of flow velocities. The studyreports distinctly different velocity dependent nanofluidic regimes, which have not been theoreticallyas well as experimentally reported earlier. Experimental single-molecule fluorescence bursts insideall-silica nanofluidic channels are used to validate the robustness of the method. It is not restrictedto single molecule environment of uniform electrodynamic interactions and can be used to investigatecomplex refractive index mismatch related non-uniform single-molecule electrodynamic interactionsas well. This fundamental investigation of single-molecule nanofluidics has a potential to acceleratethe progress of dynamic and complex single-molecule experiments, such as dynamic heterogeneity,biomolecular interactions of misfolded proteins, and nanometric cavity electrodynamics.
I. INTRODUCTION
In dynamic single-molecule experiments [1–4], for ex-ample in single molecule nanofluidics [5, 6], one of thehardest problems is to identify if the signals are purelysingle-molecule events due to the convoluted complex-ity associated with electrodynamic interactions/Purcelleffect [7–10], photophysics [11, 12], electrostatic effect[13, 14], multi-molecular interactions [15], and con-fined/Knudsen diffusion [16, 17]. Feynman described thisproblem as, “ ...as we go down in size, there are a num-ber of interesting problems that arise. All things do notsimply scale down in proportion... There will be severalproblems of this nature that we will have to be ready todesign for. ” [18]. Statistical mechanics at nanometriclength scale and molecular self assembly can potentiallyanswer many questions related to life on earth as wellas reveal unfamiliar physics [19–25]. Tracking of singlemolecules [26] has been a centre of this field [2, 27] whereincreasing accuracy of spatial position and temporal in-formation for dynamic behaviour of a molecule in twodimensions were two key problems. The state-of-the-artsolid-state and soft nanoengineering provides sufficientprecision to handle individual single molecules by over-coming diffusion induced slip from detection volume andnon-uniform electrodynamics induced artefacts in single-molecule fluorescence signals. In this paper, by nanoflu- ∗ [email protected]; also affiliated to Single-Molecule OpticsGroup, Huygens Laboratory, Leiden Institute of Physics, LeidenUniversity, The Netherlands. Open Academic Research UK CIC,Cambridge, UK. Open Academic Research Council, Hyderabad,India. idic, I am referring to Knudsen flow/diffusion environ-ment where two spatial dimensions are close to or smallerthan the mean free path of molecule’s diffusion as well assmaller spatial confinement than the wavelength of lightbeing used for single-molecule detection [6]. Examples ofsuch environment are molecular motion inside tunnellingnanotube or membrane nanotube [28], nanopores, sin-gle molecule sequencing - linearise DNA in nanochan-nel array [29], zeolite-catalysis [30], and mass transfer incarbon nanotubes [31, 32]. Diffraction limited detectionof these single-molecule nanofluidic motions possess twoproblems – (a) how do we know that a single moleculeis flowing or crawling in the detection volume? (b) Howdo we know that more than one molecule are presentin the detection volume? Photon antibunching [33] andstepwise photobleaching [34] are two widely used meth-ods to resolve this spatio-temporal problem. Those twostrategies are not useful for dynamic systems since pho-ton statistics is limited by the number of photons. Single-molecule methods have gained a significant popularity tobecoming a standard in analytical chemistry and struc-tural biology as predicted by Keller [35]. However, thelimited understanding of this spatio-temporal problem isrestricting single-molecule spectroscopy at its fullest ad-vantage within the mathematical physics and physical-chemistry communities. Single-molecule spectroscopistsshould have the ability to model the transient behaviourof the molecules through a confocal laser beam consider-ing the molecule cycles through excitation and emission,which produces photon emission and significantly influ-enced by the aforementioned parameters.I demonstrate a method to overcome this barrierand to predict and model the characteristics of single-molecule bursts and their spatio-temporal distributions a r X i v : . [ phy s i c s . a t m - c l u s ] F e b inside nanofluidic environment. The transiting singlemolecules through the focus do not have enough timeto provide the reasonable number of photons to performthe antibunching or photobleaching. In contrast, burstsize distribution [15] or burst variant analysis with dy-namic heterogeneity and static heterogeneity [11, 36] pro-vides useful information to distinguish between molec-ular shot noise [37], multi-molecule events, and single-molecule events.The Feynman-Enderlein path integral approach inMonte Carlo (FEPI MC) is presented for two differentsizes of single molecules – 2 nm and 20 nm. I have per-formed the simulation for two overlapping cofocal focias used in two-foci correlation spectroscopy[38, 39]. Theadvantage of having two foci over a single focus is that itcan extract the velocity of the flowing molecules, whichis difficult to obtain in a single focus setup where themethod depends strongly on the complex molecular de-tection function [40, 41]. If we can solve the problem fora two foci system, information for transiting moleculesthrough a single focus can be extracted by turning offanother focus in the simulation. II. COMPUTING BURST SIZE DISTRIBUTION
I use the Feynman path integral [42] to model singlemolecule burst size distribution using the burst searchingtheory developed by Enderlein et al. [15, 43]. Hence, Iterm the modified path integral for single-molecule fluo-rescence as Feynman-Enderlein path integral. I look intothe temporal probability of existence of an unbleachedsingle molecule while passing through a confocal laserbeam (supplementary Figure 1). The existence proba-bility decay exponentially after detecting a photon whenno further photon is detected. Once the next photongets detected instantaneously the probability goes to a larger value due to a possibility of a photon being emittedfrom the molecule instead of the background. The non-vanishing probability of background photon keeps the ex-istence probability always less than unity. This time evo-lution of the existence probability is an important modelfor single molecule burst identification. It is unavoidableto ignore the photobleaching effect from the expressionof the existence probability within a given time intervalin case of single-molecule nanofluidics. Derivation for thephotobeaching contribution can be found in [43].For pure single-molecule transits, let us consider thatat τ = 0 when the molecule with trajectory r ( τ ) startedflowing outside the confocal volume, and τ = T is largeenough to consider that the molecule have crossed thedetection volume. The existence probability of singlemolecule BSD is given by P ( N ) where 1 stands for‘single-molecule crossing’. In the model, I have consid-ered one-step photobleaching and negligible triplet-state(longer triplet state can be handled as well). A simplePoisson distribution can describe the photon detectionstatistics of these single molecules’ sub-subensemble as X f = (cid:90) τ X f [ r ( τ )]d τ (1)where within the time interval δτ the molecule at posi-tion r gives rise to the probability of detecting a photonas X f ( r ) δτ . The probability of detecting N photons dur-ing the time interval { , T } will be superposition of thesePoisson distributions as (equation 2), P ( N ) = (cid:90) ∞ X Nf N ! exp( − X f ) · P [ X f ]d X f (2)where P [ X f ] is the distribution function in a form of apath integral, which runs over all possible photobleachingtimes and molecular trajectories. P [ X f ] = (cid:90) T (cid:90) D r ( τ ) δ (cid:34) X f − (cid:90) τ bl X f [ r ( τ )] (cid:35) X bl [ r ( τ bl )]d τ bl × exp (cid:34) − (cid:90) τ bl (cid:32) [ r ( τ ) − v ] D + X bl [ r ( τ )])d τ (cid:33)(cid:35) p [ r ]+ (cid:90) D r ( τ ) δ (cid:34) X f − (cid:90) T X f [ r ( τ )]d τ (cid:35) × exp (cid:34) − (cid:90) T (cid:32) [ r ( τ ) − v ] D + X bl [ r ( τ )])d τ (cid:33)(cid:35) p [ r ] (3)The δ [ f ( r )] is δ -function functional, p ( r ) δ r is theprobability of finding a molecule in the small volume δ r at position r at time t = 0, δτ X bl r is the probability ofphotobleaching the molecule within δτ at position r , D isthe diffusion constant of the molecule, and v is the flowvelocity. The path integration (cid:82) D r ( τ ) in equation 3,runs over all possible paths starting from a random point r and random end point. The first and second terms in equation 3 represent the contribution of molecules thatphotobleach and do not photobleach, respectively wherethe X f ( r ) = η ( r )Φ f σI ( r ) and X bl ( r ) = Φ bl σI ( r ). Here, I ( r ) is the spatial dependence of intensity of the confo-cal laser beam (photons/area/time), η ( r ) is the collec-tion and detection efficiency of optics and electronics, Φ f and Φ bl are the fluorescence and photobleaching quan-tum yields, and σ is the absorption cross-section. Since FIG. 1.
Single-molecule and multi-molecule events inside a channel that is narrower than the diffraction limit. (a) The black dots represent photoactive single molecules, when they appear in the detection volume photon emission isrepresented with a blurry circle that is spatially larger than the diameter of the foci. (b) Multi-molecule (MM) events aresituations when more than one photoactive molecule is present in a single focus or two foci, and they are emitting photonsat the same time. Such signals are not easily distinguishable from molecules being dragged or crawling within the detectionvolume, which may result into photobleached/photo-inactive molecules – white dot represent them. (c) FEPI MC simulated100 ms binned SM and MM events of 2 nm molecules. i. Short single-molecule bursts – two similar bursts at the same timerefer to emission being detected by two detectors – the arrow represents such events. ii. Short single-molecule bursts with highintensity/photon numbers. iii. Reappearing bursts within millisecond timescale. iv. Long bursts suggest crawling of a singlemolecule. the intensity of the excitation beam is low, no opticalsaturation effect was present. The all-silica nanochan-nels show negligible background and a steady backgroundwith respect to observation time [6]. So, I neglect thebackground effects. The analytical expression of P ( N )is not possible; since the equation 3 runs over an infinitenumber of path, so it is fundamentally difficult to find asolution. Monte-Carlo sampling is done to calculate the P ( N ) where random paths are chosen and remainingintegration is performed numerically. A large number ofrandom sampling over different paths leads to sufficientlyprecise approximation of P ( N ).Multi-molecule events i.e. close succession of morethan one molecules flowing and leading to a single pho-ton burst always have a nonzero probability. This willcause additional peaks with extra temporal broadeningof the photon burst at higher burst size. I use a weight-ing factor w k i.e. proportional to the probability that n molecules pass through the confocal volume, which areseparated by less than the mean transit time leading to asingle unresolved burst. Due to these n -molecule bursts,the BSD is the convolution of BSD for all ( n − P ( N ) = (cid:88) N P ( N ) P ( N
2) (4)where N = N + N , and BSD for n > P n ( N ) = (cid:88) N ∗ P ( N ) P ( N ...P ( N n )= (cid:88) N P ( N ) P n − ( N ) (5)where N ∗ = N + N + ... + N n .If τ s is the minimum time between molecules leadingto an unresolved prolonged burst, τ m is the initial inter-val time during delivery of the molecules from the reser-voir, w n is proportional to the probability of n successivemolecules, which are separated by less than τ s time. Noother molecules are present before and after this ‘ molec-ular train ’ within τ S . In case of random arrival of themolecules, the probability density of τ before the arrivalof the next molecule is τ − m exp( τ /τ m ). For the prob-ability of τ > τ s is exp( − τ s /τ m ), which results to theweighting factor as w n = exp (cid:32) − τ s τ m (cid:33)(cid:34) n − (cid:89) j =1 (cid:90) τ s exp (cid:32) − τ j τ m (cid:33) d τ j τ m (cid:35) exp (cid:32) − τ s τ m (cid:33) = exp (cid:32) − τ s τ m (cid:33)(cid:34) − exp (cid:32) − τ s τ m (cid:33)(cid:35) n − (6)Here, I consider a constant value of τ s , which is modi-fied in the algorithm with a thresholding loop. The initialcondition of the numerical simulation was fed with spatialdependence of the intensity of the excitation laser I ( r ),collection efficiency η ( r ), and initial distribution of themolecule p ( r ). The intensity distribution of a diffractionlimited confocal laser beam is Gaussian. The nanochan-nels width and height are much smaller than the FWHMof the focus but the length is not (Figure 1). For thesimplicity, I could neglect the Gaussian distribution intwo axes for the quasi-1D nature of molecules’ presenceinside nanochannel but I haven’t in the actual algorithm. I have considered it as: I ( x, y ) = 2 Wπr ω exp (cid:34) − x − y r ω (cid:35) (7)where W denotes the power of the laser and r ω is thewaist radius. The spatial dependence (molecules areflowing along the x axis) of optical collection efficiencyis given by: η ( x, y ) = η π (1 − cos ψ ) × (cid:34) arcsin sin θ cos ψ − cos ψ arctan (cid:32) cos ψ sin θ (cid:112) sin ψ − sin θ (cid:33)(cid:35) θ max θ min (8)where η is the maximum value of the col-lection efficiency, ψ = arcsin(NA /n ), θ max =max ( − arctan(( d − x ) / | y | ) , − ψ ) and θ min =max ( − arctan(( d − x ) / | y | ) , − ψ ), NA is the value ofthe numerical aperture, n is the refractive index, and d is the pinhole diameter in the object space. Here, θ min < θ max ; if θ min > θ max , then η ( x, y ) will be zero.Employing a simple hydrodynamic model, I have cal-culated the distribution of the molecules at a position x = x outside the slice of the confocal volume with diffrac-tion limited focus. I have neglected the third dimensiondue to the nano-confinement. Although the moleculesare confined within a diffraction limited space and onlyfree to move in one direction, the multi-molecule eventsare still plausible since the molecules are smaller thanthe width and height of the nanochannels. Molecules areconsidered to be uniformly distributed over x + c = y ; c is a spatial constant. From this starting plane, moleculesaccelerate to the laminar flow velocity and undergo aconfined diffusion. The starting point x = x of theMonte-Carlo simulation is considered at − ω where thelight intensity is negligible. Then, initial probability dis-tribution p is given by: p ( x, y ) = x − x π yDτ (cid:90) y d r (cid:90) π d φ × exp (cid:34) − ( y − r sin φ ) Dτ (cid:35) (9) τ is determined with respect to the pulsed frequencyand the following equation: x − x inj v = τ − k − [1 − exp( kτ )] (10) k is an empirical flow acceleration constant. III. SINGLE-MOLECULE ANDMULTI-MOLECULE BURSTS IN CONFINEDFLOW
The laminar flow profile is approximated to be negli-gible. However, further modelling suggests that I can-not neglect this despite the flow cross-sectional diame-ter is smaller than the confocal detection volume. Thenanochannels are placed at the centre of the confocal vol-ume. The diffusion coefficients of 2 nm and 20 nm sizemolecules are 218 µ m / s and 21.8 µ m / s, respectively.The time step used in the Monte-Carlo simulation was∆ τ = 50 µ s, T was set to 3 ms, and the simulationssampled for 10 paths.I use an experimental configuration as shown in Fig-ure 1 (and supplementary Figure 2) single molecules areflowing inside a nanochannel ( d nc = 50 nm), and thenanochannel is placed inside two 640 nm foci are placed.The foci are shown in two different colours to emphasisthey have orthogonal polarisation from each other – greenand violet represent vertical and horizontal polarisation,respectively from the optical axis. They are spatiallyseparated but slightly ( ≈
50 nm) overlapping two eachother. A free flowing or diffusing single molecule passingby these two foci produces two bursts as schematicallyshown in Figure 1a with green and violet time trace sig-nals. Ideally, single molecule bursts of a single speciesfrom two foci should have similar time duration and in-tensities. If a molecule interacts with the wall, whichis highly likely when the mean free path is compara-ble to d nc . Another possibility for longer time durationwith higher intensity of bursts is in the case of multiplemolecules being present in the foci with close successionsas shown in Figure 1(b). Among these mutlimoleculesthere could be some photobleached molecules. In the sim-ulation, I have considered an existence probability of asingle molecule while undergoing photobleaching as men-tioned in the theory section and supplementary Figure1. A photobleached molecule also interacts with othermolecules, and if it is adsorbed to the surface of the FIG. 2.
Electrodynamics of single-dipole and single-molecule bursts. (a) Left figures – horizontal dipole and rightfigures – vertical (right) dipoles of λ nm are two ideal cases – electrodynamic interaction of single dipoles with heterogeneousinterface strongly depends on refractive index mismatch; the electrical component of the emission pattern for dipoles creates areuniform doughnut shape for n / n = 1.00/1.00 (top panel), n / n = 1.00/1.33 has a larger distribution of electrical componenttowards n for horizontal as well as vertical dipoles (mid panel), the trend continues for n / n = 1.00/1.44 (bottom panel) – theemission intensity is larger for vertical dipole than horizontal dipole. (b) The FEPI MC simulation calculated single-moleculebursts of 20 nm molecule by considering the random orientation of single-dipole and its electrodynamic interactions at 1 µ m/s,10 µ m/s, and 100 µ m/s flow velocities; characteristic burst sizes are ranging from less than 20 to 200 ms with low and highphoton counts. Two signals at same time represent photons being emitted from two foci – non-identical bursts are due to twolaser beams are orthogonally polarised to each other resulting to non-identical photon absorption by the single molecule. nanochannel then it is a constant source of interactionwith the passing by molecules. Another instance thatshows similar bursts characteristics with multi-moleculebursts is crawling of single molecules along the wall ofnanochannels. In the FEPI MC simulations, I have triedto understand these multi-molecule events and single-molecule events. In Figure 1(c), I show FEPI MC simu-lated 100 ms binned single-molecule bursts of 2 nm sizedmolecules. The FEPI MC simulated bursts are hard todistinguish from an experimental data, like in the supple-mentary Figure 4 or earlier report [6], which evidence therobustness of the algorithm. The arrow in the time-traceshow some exemplary single molecule bursts. The firstarrow in Figure 1(c)i represent a single-molecule burstswith minimal interaction in both the foci. In Figure1(c)ii, I found a single-molecule burst with high inten-sity/photon numbers, which also shows a crawling effectas shown by the first arrow. significantly long reappear-ing bursts within 250 ms is shown in Figure 1(c)iii, whichseemed to be showing photo-inactive behaviour due tomulti-molecule interactions. Another multi-molecule in-teractions for nearly 200 ms is shown in multi-moleculeinteractions.To distinguish between crawling of single moleculeson the nanochannel wall due to Knudsen type flow andmulti-molecule interactions, we should investigate theelectrodynamic interactions of single-molecule fluores-cence with its surrounding, and later compare this withthe temporal distributions. Figure 2a shows the electro-dynamic interaction of single dipole with respect to ori-entation and refractive index mismatch of the interface.I plot the electrical field component at 640 nm wave-length for single horizontal and vertical dipoles’ emissionwith respect to optical axis along of excitation. At a uni- form interface with refractive indices n / n = 1.00/1.00,uniform emission pattern is observed with a doughnutshape. Single-molecule experiments are often performedin water. Our single-molecule nanofluidic experimentswere also taken place in water; n / n = 1.00/1.33 rep-resent an air-water interface where the dipole emissionwill have larger distribution of electrical component to-wards n . For larger difference in refractive indices mis-match n / n = 1.00/1.44, which is for air-silica/quartzinterface as we used in the experiment. A noticeabledifference is larger emission pattern for vertical dipolethan horizontal dipole since absorption cross-section of adipole is a function of its orientation/polarisibility withrespect to the field of excitation. The fluctuation in thebursts and its non-identical shapes in the Figure 1c andFigure 2b are justified considering the stochastic electro-dynamic interactions due to the randomness of confinedinteractions inside the nanofluidic channels. In Figure2c, we have shown the bursts for different flow velocitiesranging from 1 µ m/s to 100 µ m/s of 20 nm sized fluo-rophores. The two foci are orthogonally polarised fromeach other. Hence, differences in fluorescence intensitiesof single molecule bursts are not unexpected. In somecases, we see clearly anti-correlated intensities, for exam-ple, the high intensity bursts of 1 µ m/s and 10 µ m/s.The single-molecule bursts inside the box of 100 µ m/s ishighly anti-correlated. IV. TEMPORAL DISTRIBUTION OF BURSTS
The temporal burst size distribution of non-interactingsingle molecules should be a δ -function. The FEPI MCconsiders a δ -function functional considering the complex FIG. 3.
Single-molecule burst size distribution. (a) LHS: Histogram of 1000 paths in focus 1 and 2 for 20 nm sized singlemolecules flowing at the velocities 0 . µ m/s (230 ms with highest occurrence), 0 . µ m/s (220 ms with highest occurrence),1 µ m/s (210 ms with highest occurrence), and 10 µ m/s. Multiple peaks are present in 0 . µ m/s and 0 . µ m/s, a flatteningfeature (200 to 250 ms with highest occurrence) appears due to the flow velocity contribution in 1 µ m/s and 10 µ m/s. RHS:Histogram of time distribution between focus 1 and focus 2, the spread narrows down from top to bottom due to increasingflow velocity. Molecular interactions are visible in 0 . µ m/s to 1 µ m/s in positive and negative sides. (b) LHS: Burst sizedistribution in focus 1 and 2 of 2 nm single molecules flowing at velocities 0 . µ m/s, 0 . µ m/s, 1 µ m/s, and 10 µ m/s – highestoccurrence within 50 ms. The short timescale bursts from 0 to 60 ms are due to single molecule transitions. Larger burst sizesup to 400 ms are due to strong interactions with the nanofluidic environment. At 1 µ m/s and 10 µ m/s, the single moleculebursts and other interactions are smeared together cannot be distinguished separately. RHS: Time distribution of moleculesbetween focus 1 and focus 2 – narrower distribution at 10 µ m/s compare to low flow velocities. interactions of single-molecule fluorescence experimentsinside the nanofluidic channel as shown in the equation3. The histogram of temporal response of single-moleculebursts will show the statistical nature of the complex in-teractions. As shown in Figure 3a, a histogram of single-molecule burst size (defined with time unit) for 20 nmsized single molecules paths through focus 1 and 2 differ-ent velocities ranging from 0 . µ m/s to 10 µ m/s. Here,I overlap the histograms of 1000 path through focus 1(blue) and focus 2 (red). All the histograms have a broadtime distribution of approximately 400 ms. If we lookcarefully, at slower velocities for example at 0 . µ m/sand 0 . µ m/s, we can find discreet peaks within the broadtime distribution. These are due to favourable interac-tions, which can be temporal signatures of nanoconfinedphotophysical interactions of single molecules. Amongthem at this time range 210 ms to 230 ms single-molecule bursts are most frequently occurring, such as 230 ms for0 . µ m/s has, 220 ms for 0 . µ m/s, 210 ms for 1 µ m/s,and 200 ms to 250 ms for 10 µ m/s. At 10 µ m/s, a 50 msflat peak is observed, which suggests equal probability ofseveral kinds of interaction. Since we have two sources ofsignal for a single molecule, let us mathematically com-pare the time distribution between focus 1 and focus 2.The spread between two foci reduces from 400 ms to 200ms with increasing velocities as shown in the right panelof Figure 3a.A contrasting difference is observed in the burst sizedistribution if we reduce an order of magnitude in the sizeof the single molecule. For 2 nm size, the peak the his-togram shifts to faster timescale of 25 – 50 ms and shapeof the histogram is positive skewed as shown in Figure3b. The histogram for 0 . µ m/s shows high occurrencesat fast timescale 25 ms, 60 ms, and 80 ms along with sev- FIG. 4.
Burst size distribution of fast nanofluidics regime . (a) At flow velocity 100 µ m/s, 20 nm (D = 21.8 µ m /s)has highest occurrence of 33 ms bursts and 2 nm (D = 21.8 µ m /s) with 20 ms with a global Poissonian distribution. (b) At1000 µ m/s flow velocity, 20 nm molecule shows highest occurrence at 1.5 ms, and 2 nm molecule has flat distribution of highoccurrence from 1 to 2 ms. (c) Single-molecule nanofluidic regimes – in FEMC simulation, the effect of nanofluidic regimes onsingle-molecule bursts of 20 nm single molecules is considerably visible compare to 2 nm single molecules. eral low occurrences of long bursts at slow timescale upto 400 ms. The distribution at faster timescale are forsingle-molecule bursts, which had less interactions andpassed through the foci with short intervals. Specifically,the sub-25 ms timescale bursts are non-interacting single-molecule crossings, and the 50 to 100 ms timescale burstsare responsible for molecular shot noise i.e. molecularinteractions among multiple single molecules. Molecu-lar shot noise are temporary events when molecule tomolecule interacts, such events can be spectrally resolvedby studying the shift in emission peak as well as withphononic spectroscopy if high SNR is available. Theslower occurrences are due to single-molecule interactionswith the walls, earlier we referred them as crawling eventsof single molecule or multi-molecules. At slow velocities,the single-molecule crossings, molecular shot-noise, andcrawling events can be well identified, which get smearedaway at fast velocities as shown in Figure 3b. The timedistribution between focus 1 and focus 2 is narrower at10 µ m/s compare to slow flow velocities as shown in theright panel of Figure 3b. The time distributions also showfine temporal following the analysis timescale of differentsingle-molecule bursts characteristics.From the findings shown in Figure 3, I classify the flowvelocities ranging from 0 . µ m/s to 10 µ m/s as ‘slownanofluidic regime’ because the diffusion behaviour andstrong interactions of confinement effect are dominatingcharacteristic feature of this regime. Irrespective of theorder of magnitude differences in flow velocities, we canidentify reproducible slow nanofluidic regime characteris-tics. Following the scaling law, the 20 nm molecules showsingle-molecule bursts at around 220 ms while sub-25 mstimescale events are for 2 nm single molecule bursts. Thebroad spread of occurrence in 20 nm single molecules atthis nanofludic regime show strong interactions/crawlingeffect and molecular shot noise.Unlike the slow nanofluidic regime, the characteris- tics of histogram from 100 µ m/s is so different from oneflow velocity to another that I use different timescales toshow the details of their histograms. Thus, I define ’fastnanofluidic regime’ from 100 µ m/s velocity as shown inFigure 4. Figure 4a shows histograms of 20 nm and 2 nmsingle molecules at 100 µ m/s with highest occurrence at33 ms and 20 ms, respectively. At this flow regime, theaforementioned interactions of single molecules are notstrong, specifically for the 20 nm molecules. The molec-ular shot noise and crawling events are limited. However,for the 2 nm molecules, the molecular shot noise scalesdown to faster time scale as we can see a distribution ofthis peak at 38 ms along with a small contribution dueto crawling events. It is noticeable that the sub-20 msbursts for 2 nm molecules is consistent at fast nanofluidicregime. The time distributions between two foci also sug-gest the same in the right panel. The trend continues to1000 µ m/s in Figure 4b. At this flow velocity, it is hardto distinguish between the histograms of 20 nm and 2nm single molecules, except the fine features. At the fastnanofluidic regime, the driving force for flow large enoughto overcome the crawling events. For an overall under-standing, Figure 4c shows single-molecule burst sizes ofthe two identified nabofluidic regimes with respect to theflow velicities. V. SUMMARY
The Feynman-Enderlein path integral is a powerfulmethod to model the complex single-molecule nanoflu-idics and show two distinct single-molecule nanofluidicregimes. I have showed how to resolve the complexityof single-molecule nanofluidics by integrating electrody-namics. In future, the method will be integrated with theelectrostatic effects to deal with the Debye-length relatedissues. The FEMC method has not been used widelyin the single-molecule experiments. This method opensup several avenues in biophysics as well as quantum hy-drodynamics. Single fluorophores to misfolded proteins,nanobodies, and quantum dots in solid state nanochan-nels/tunnelling nanotubes or within lipid bilayers are allrelevant for this work.
VI. ACKNOWLEDGEMENT
The research was funded by the German ResearchFoundation/Deutsche Forschungsgemeinschaft (DFG) –Project number 405479535 (DFG Research Fellowship – PI Siddharth Ghosh). I am most thankful to ProfessorJoerg Enderlein for many important discussions during2012-2016 from where this work is initiated and for lay-ing the foundation of this work. I am also thankful toProfessor Jeremy Baumberg’s comments during the S3IC2020 Conference in Munich and later encouragement towork in this field. The work utilised computational re-sources provided by Professor Tuomas Knowles at theMaxwell Centre and Yusuf Hamied Department of Chem-istry, University of Cambridge. My sincere gratitude toProfessor Allard Mosk for his valuable comments in themanuscript. [1] H. P. Lu, L. Xun, and X. S. Xie, Science , 1877(1998).[2] A. E. Cohen and W. Moerner, Proceedings of the Na-tional Academy of Sciences , 4362 (2006).[3] P. Zijlstra, P. M. Paulo, and M. Orrit, Nature nanotech-nology , 379 (2012).[4] M. D. Baaske, M. R. Foreman, and F. Vollmer, Naturenanotechnology , 933 (2014).[5] J. F. Lesoine, P. A. Venkataraman, P. C. Maloney, M. E.Dumont, and L. Novotny, Nano letters , 3273 (2012).[6] S. Ghosh, N. Karedla, and I. Gregor, Lab on a Chip ,3249 (2020).[7] E. M. Purcell, in Confined Electrons and Photons (Springer, 1995) pp. 839–839.[8] J. Enderlein, Biophysical Journal , 2151 (2000).[9] N. Karedla, S. C. Stein, D. H¨ahnel, I. Gregor, A. Chizhik,and J. Enderlein, Physical review letters , 173002(2015).[10] F. Westerlund, F. Persson, A. Kristensen, and J. O.Tegenfeldt, Lab on a Chip , 2049 (2010).[11] C. Eggeling, S. Berger, L. Brand, J. Fries, J. Schaffer,A. Volkmer, and C. Seidel, Journal of biotechnology ,163 (2001).[12] M. Orrit, Colloids and Surfaces B: Biointerfaces , 396(2009).[13] P. v. Debye and E. H ”u ckel, phys. Z , 185 (1923).[14] R. H. French, V. A. Parsegian, R. Podgornik, R. F. Ra-jter, A. Jagota, J. Luo, D. Asthagiri, M. K. Chaudhury,Y.-m. Chiang, S. Granick, et al. , Reviews of ModernPhysics , 1900 (2010).[15] J. Enderlein, D. L. Robbins, W. P. Ambrose, and R. A.Keller, The Journal of Physical Chemistry A , 6089(1998).[16] K. Malek and M.-O. Coppens, The Journal of chemicalphysics , 2801 (2003).[17] S. Li, Y. Wang, K. Zhang, and C. Qiao, Industrial &Engineering Chemistry Research , 21772 (2019).[18] R. P. Feynman, California Institute of Technology, Engi-neering and Science magazine (1960).[19] E. Schrodinger, What is life?: With mind and matter andautobiographical sketches (Cambridge University Press,2012).[20] M. Loose, E. Fischer-Friedrich, J. Ries, K. Kruse, andP. Schwille, Science , 789 (2008). [21] T. Litschel, K. A. Ganzinger, T. Movinkel, M. Heymann,T. Robinson, H. Mutschler, and P. Schwille, New Journalof Physics , 055008 (2018).[22] R. Golestanian, Physical review letters , 108102(2015).[23] P. Illien, X. Zhao, K. K. Dey, P. J. Butler, A. Sen, andR. Golestanian, Nano letters , 4415 (2017).[24] M. Ghosh, S. Ghosh, M. Seibt, K. Y. Rao, P. Peretzki,and G. M. Rao, CrystEngComm , 622 (2016).[25] M. Ghosh, S. Ghosh, H. Attariani, K. Momeni, M. Seibt,and G. Mohan Rao, Nano letters , 5969 (2016).[26] J. Enderlein, Single Molecules , 225 (2000).[27] A. E. Cohen, Trapping and manipulating single moleculesin solution , Ph.D. thesis, Stanford University (2007).[28] J. Ranzinger, A. Rustom, M. Abel, J. Leyh, L. Kihm,M. Witkowski, P. Scheurich, M. Zeier, and V. Schwenger,PLoS One , e29537 (2011).[29] J. Jeffet, A. Kobo, T. Su, A. Grunwald, O. Green, A. N.Nilsson, E. Eisenberg, T. Ambjornsson, F. Westerlund,E. Weinhold, et al. , ACS nano , 9823 (2016).[30] Z. Ristanovic, A. D. Chowdhury, R. Y. Brogaard,K. Houben, M. Baldus, J. Hofkens, M. B. Roeffaers, andB. M. Weckhuysen, Journal of the American ChemicalSociety , 14195 (2018).[31] J. K. Holt, H. G. Park, Y. Wang, M. Stadermann, A. B.Artyukhin, C. P. Grigoropoulos, A. Noy, and O. Bakajin,Science , 1034 (2006).[32] R. H. Tunuguntla, R. Y. Henley, Y.-C. Yao, T. A. Pham,M. Wanunu, and A. Noy, Science , 792 (2017).[33] T. Basch´e, W. Moerner, M. Orrit, and H. Talon, Physicalreview letters , 1516 (1992).[34] S. Ghosh, A. M. Chizhik, N. Karedla, M. O. Dekaliuk,I. Gregor, H. Schuhmann, M. Seibt, K. Bodensiek, I. A.Schaap, O. Schulz, et al. , Nano letters , 5656 (2014).[35] R. A. Keller, W. P. Ambrose, P. M. Goodwin, J. H. Jett,J. C. Martin, and M. Wu, Applied Spectroscopy , 12A(1996).[36] J. P. Torella, S. J. Holden, Y. Santoso, J. Hohlbein, andA. N. Kapanidis, Biophysical journal , 1568 (2011).[37] D. Chen and N. J. Dovichi, Analytical Chemistry , 690(1996).[38] T. Dertinger, V. Pacheco, I. von der Hocht, R. Hart-mann, I. Gregor, and J. Enderlein, ChemPhysChem ,433 (2007). [39] S. Chiantia, J. Ries, N. Kahya, and P. Schwille,ChemPhysChem , 2409 (2006).[40] P. S. Dittrich and P. Schwille, Analytical chemistry ,4472 (2002).[41] G. U. Nienhaus, P. Maffre, and K. Nienhaus, in Methodsin enzymology , Vol. 519 (Elsevier, 2013) pp. 115–137. [42] R. P. Feynman and L. M. Brown,
Feynman’s thesis: anew approach to quantum theory (World Scientific, 2005).[43] J. Enderlein, D. L. Robbins, W. P. Ambrose, P. M. Good-win, and R. A. Keller, Bioimaging5