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October 28, 2012

C 2012 American Chemical Society

Stochasticity in Single-MoleculeNanoelectrochemistry: Origins,Consequences, and SolutionsPradyumna S. Singh,† Enno Katelhon,‡,§ Klaus Mathwig,† Bernhard Wolfrum,‡,§ and Serge G. Lemay†,*

†MESAþ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands and ‡Institute of Bioelectronics (ICS-8/PGI-8) and§JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany

The advent of techniques for detectingand manipulating individual moleculesin solution counts among the most

impactful developments of the last twodecades.1 Currently, the most mature andestablished single-molecule techniques relyon either force transduction2 or opticalmethods.3 For example, a particularly power-ful approach is fluorescence: the detection ofsingle fluorophores4,5;even within livingcells6�8

;has become a widespread tool inmany branches of science. Similarly sensitiveprobes based on purely electrical or electro-chemical detection, whether through electro-static coupling or via the transfer of electronsbetween the molecule and an electrode, arebeing actively pursued, both as enablers of newtypesof fundamentalexperimentsandassensorelements in integrated lab-on-a-chip devices.The main strategy for detecting single mol-

ecules electrochemically is so-called redox cy-cling,9,10 the repeated oxidation and reduction

of a target analyte, which allows each mole-cule to contribute multiple electrons to theoverall current. Such an approachwas used byFan and Bard11�13 in a scanning electrochem-ical microscope (SECM) configuration: nano-meter-high cavitieswere formedby impinginga wax-coated metal tip onto a conductingsubstrate. A related approach was employedby Sun and Mirkin by immersing recessednanoelectrodes in a mercury drop.14 Mostrecently, our group demonstrated single-molecule sensitivity in microfabricated nano-electrochemical devices, which consist of twinelectrodes embedded in a nanochannel withinterelectrode spacings of 70 nm.15 Currentfluctuations with amplitudes of 20 fA wereshown to be consistent with the entry andegress of single individual molecules throughthe detection volume of the device. Singlenanoparticles functionalized by redox-activemoieties16 can also, in principle, be detectedusing the same strategy.17

* Address correspondence tos.g.lemay@utwente.nl.

Received for review July 11, 2012and accepted October 27, 2012.

Published online10.1021/nn3031029

ABSTRACT Electrochemical detection of single molecules is being actively

pursued as an enabler of new fundamental experiments and sensitive analytical

capabilities. Most attempts to date have relied on redox cycling in a nanogap,

which consists of two parallel electrodes separated by a nanoscale distance.

While these initial experiments have demonstrated single-molecule detection at

the proof-of-concept level, several fundamental obstacles need to be overcome

to transform the technique into a realistic detection tool suitable for use inmore

complex settings (e.g., studying enzyme dynamics at single catalytic event level,

probing neuronal exocytosis, etc.). In particular, it has become clearer that stochasticity;the hallmark of most single-molecule measurements;can become the

key limiting factor on the quality of the information that can be obtained from single-molecule electrochemical assays. Here we employ random-walk simulations

to show that this stochasticity is a universal feature of all single-molecule experiments in the diffusively coupled regime and emerges due to the inherent properties

of Brownian motion. We further investigate the intrinsic coupling between stochasticity and detection capability, paying particular attention to the role of the

geometry of the detection device and the finite time resolution of measurement systems. We suggest concrete, realizable experimental modifications and

approaches tomitigate these limitations. Overall, our theoretical analyses offer a roadmap for optimizing single-molecule electrochemical experiments, which is not

only desirable but also indispensable for their wider employment as experimental tools for electrochemical research and as realistic sensing or detection systems.

KEYWORDS: nanoelectrochemistry . single-molecule electrochemistry . redox cycling . nanofluidics . nanochannel .stochastic electrochemistry . random walks

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Can these experiments progress from (initiallyimportant) one-off proof-of-concept demonstrationsto robust, reliable, and routine detection tools in morecomplex experiments? We believe that the use ofmicrofabrication methods for the development ofnanogap transducers embedded in fluidic nanochan-nels offers a viable route due to several advantagesover traditional tip-based methods: devices with iden-tical geometries within a high degree of precision canbe fashioned,18 a variety of geometries can be generatedwhich can be tailored to the intended application,19 andintegration with microfluidic systems is facilitated. Con-sequently, the approach is scalable to more complexexperiments under a diverse set of conditions. None-theless, significant conceptual and experimental hurdlesremain to be overcome. Most importantly on the con-ceptual side, most realistic experimental situations re-quire coupling of thenanoscale detection region to sometype of reservoir from which a sample containing theanalyte of interest is introduced. This however allowssingle molecules undergoing Brownian motion to enterand leave the detection region at random, creating aninherently stochastic response. That is, the system isnondeterministic and its description is necessarily prob-abilistic. This was manifest in single-molecule effortsreported thus far, whether via inadvertent cracks in thewax insulationarounda tip11 or thepresenceof apurpose-made access channel15 (a notable exception being theexperiments of Sun and Mirkin14). This particular kind ofstochasticity in single-molecule signals raises importantquestions about the widespread applicability of this ap-proach as a probe in “real” analytical systems, whereusually sensitive, accurate measurements are required tobe carried out in rapid time in a reproducible and deter-ministic manner.Here, we explore the fundamental limitations of

redox-cycling-based electrochemical detection of singlemolecules using random-walk simulations.20�23 Thesesimulations capture the essential behavior of freely diffus-ing molecules in solution and reveal that, while mostexperimental approaches todatehave focusedonenhanc-ing the current per molecule by reducing the interelec-trode separation, the free diffusion of molecules imposesits own inherent limitation on the quality of informationthat can be derived in such experiments, even with an“ideal” measurement system. We further investigate theinfluence of various factors such as the geometry of thedetection region, the mechanism of coupling to the bulkreservoir, and the effect of finite time resolution of themeasurement electronics. We conclude by discussingpossible experimental alternatives to overcome the limita-tions imposed by the stochastic nature of Brownianmotion, with a view to enable more widespread deploy-ment of this detection strategy in sensitive analyticalsystems or complex fundamental experiments.The paper is organized as follows: We first discuss a

three-dimensional (3D) diffusion model of a molecule

undergoing redox cycling between two electrodes. Wethen show that the behavior on the time scales ofinterest can equally be well-described by a muchsimpler one-dimensional (1D) model that capturesthe behavior of molecules diffusing in and out of adetection region. We then assess the role of the finitebandwidth of the measurement electronics since,coupled with the stochastic nature of Brownian mo-tion, these two factors impose the most importantlimitation upon the quality of information one canobtain in a single-molecule experiment. Finally, wediscuss ways to overcome these limitations either bytrapping the molecule or through the implementationof directed flow (advection) in the system.

THEORY

In order to facilitate interpretation and relevantcomparison with experiments, the ensuing discussionwill invoke a prototypical detection device with realisticgeometric and experimental parameters, as sketched inFigure 1a. Thedetection regionof this nanogapdevice isthe volume enclosed by the two plane-parallel electro-des constituting the floor and roof of a nanochannel.

Figure 1. (a) Schematic geometry of a prototypical deviceinvoked for the simulations. (b) Individual electron-transferevents corresponding to oxidation (blue) and reduction(red) of a redox-active molecule confined within the detec-tion region of the device. The area enclosed by each spikecorresponds to the charge on one electron, �e. (c) Oxida-tion and reduction currents obtained after averaging andconvolving the currents in (b) with an instrumental impulseresponse with a time constant of 100 ms.

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The electrodes are separated by a distance z = 50 nm,and the length of the electrodes is L = 50 μm. Thedetection region is assumed to extend across the wholewidth of a nanochannel. We further assume that thisdevice is connected to bulk reservoirs via nanochannelson both ends through which analyte molecules areintroduced in the system. These molecules are free todiffuse in and out of the detection region of the device.The molecules are assigned a diffusion coefficient, D =1� 10�9 m2/s; this value is typical since it lies within theexperimentally determined diffusion coefficients of fer-rocenyl species in aqueous and non-aqueous solutions.Microfabrication methods allow such geometries to beachieved.24,25 Recently, SECM configurations have alsobeen demonstrated that achieve spacings on the orderof 100 nm.26 In all of these cases, because the electrodedimensions themselves are much larger than the verysmall interelectrode separation, the resulting geometryhas a very highly skewed aspect ratio (z, L). As we willarguebelow, this canbeexploited to greatly simplify thetheoretical modeling of the system.When molecules are present in the detection region

of the device, they undergo redox cycling, which leadsto a current through the electrodes. The behavior of ananalyte molecule as it traverses the device can beanalyzed at various scales of both length and time,and because of the skewed geometry, these timescales can usually be separated. On the ultrafast timescales, one can, in principle, observe thediscreteness ofeach individual shuttling event as the molecule im-pinges on one electrode or the other.27 For the geom-etry considered here, this time scale would be on theorder of τs ∼ z2/D = 2.5 μs. Figure 1b shows currentresulting from a full 3D Brownian dynamics simulationof such a particle in the prototypical geometry (fordetails of the simulation and the geometry used, seeSupporting Information). Each spike in Figure 1b cor-responds to a single encounter with the electrode andthe subsequent transfer of a single electron. Note thatthe spikes alternate sequentially between the oxidizingand the reducing electrode. This is expected: after anoxidation event, the molecule must of necessity bereduced before it can be oxidized again.These simulations assumed an extremely high time

resolution, Δt = 1 ns. In practice, however, sensitiveelectrochemical instrumentation systems typicallyhave a measurement bandwidth that is much slowerthan this time scale in order to achieve sufficient sensi-tivity. The effect of a (linear) measurement circuit on atime-dependent signal can easily be accounted for bycomputing the convolution of the “real” signal with theso-called impulse response of the electronics. This con-volution is effectively a weighted average of the signal,with the weight being determined by the impulseresponse. For a simple first-order low-pass filter with acutoff frequency of f 0 = 1/2πτ, the impulse response issimply a decaying exponential, (1/τ)exp(�t/τ), with time

constant τ. Figure 1c shows simulated data like those inFigure 1b after convolution with an instrumental timeresponse of τ = 100 ms. The effect is dramatic: theindividual electron-transfer events are no longer re-solved, and the observed current is instead essentiallyconstant. The magnitude of this current, ip ≈ 64 fA,corresponds to the value expected for one freely diffus-ing molecule, ip = neD/z2, where n is the number ofelectrons transferred per encounter with the electrodeand�e is the chargeon the electron.11 Similar argumentsregarding the limiting role of the instrument responsehave been made previously by Amatore et al. regardingthe observation of individual electron-transfer events inPAMAMdendrimers cappedwith ruthenium(II) bisterpyr-idine redox moieties adsorbed on Pt nanoelectrodes.28

Consequently, although the molecules undergo dif-fusion in three dimensions, the dimension perpendi-cular to the electrodes can be safely ignored since theexperimentalist is essentially “blind” to the motion inthese directions. Furthermore, in the channel-like geom-etry assumed here, motion parallel to the electrodesbut perpendicular to the axis of the channel is irrele-vant since the walls of the channel confine moleculesto the channel in this direction. Motion along the axisof the channel, on the other hand, cannot be ignored:the typical time scale for diffusive transport along thisdimension is τ∼ L2/D= 2.5 s, which is readily accessiblein experiments. Therefore, while a comprehensivedescription of microscopic trajectories must necessa-rily incorporate diffusion in all dimensions, the use of2D or 3D simulations adds little in practice for under-standing experimentally relevant channel-like nano-gap systems since the main observable experimentallyis the entry and egress of individualmolecules from thedetection region of the device. For this reason, wefocus below on random-walk simulations in only onedimension, namely, along the length of the detectionregion. This allows focusing on the features of thesystem that are most relevant for experiments.We consider one-dimensional random walks under-

taken by a discrete particle. The detection region of thedevice of length L (μm) is described by a lattice pointson which the particle undertakes its random walk. Themolecules canmovediscretely on this latticewith stepsof (1. In real units, the steps therefore have lengthΔL = L/a. The corresponding time interval betweeneach step is Δt = (ΔL)2/2D. At each step, the particlescan take a unit step to the right with a probability 1/2 orto the left with a probability 1/2. While here we onlyconsider discrete steps of fixed size, one could also,in principle, represent each step with a continuousdistribution. However, the probability distribution ob-tained for the random walk on length scales longerthan ΔL is the same in both cases.29 We therefore usethe simpler, discrete model which is more amenable toanalytical analysis. The boundaries of the detectionregion and the presence of the reservoir were simulated

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in different ways depending on the purpose of eachspecific simulation, as described in theMethods section.

RESULTS

Time Evolution. Figure 2 shows a trajectory gener-ated for a single particle as it undergoes a randomwalk.The shaded gray represents the detection region. Ateach step, we assess whether the particle is in thedetection area of the device or outside it. If it is insidethe detection area of the device, we define the occu-pancy in the device to be 1, else to be 0 (in anexperiment, a molecule in the detection region of thedevice contributes ip amount of current and 0 currentotherwise). As the particle undergoes its random walk,it can repeatedly enter and leave the detection region,leading to rapid telegraph-like oscillations in the current�time response with varying amount of time spent in thedetection region for eachevent, as seen inFigure2b.Mostoften, the molecule spends only a very short time in thedetection region before exiting again, leading to veryrapid oscillations of the occupancy.

Distribution of Residence Times. What is the typical timespent by the particle in the detection region? Ourintuition about Brownian motion suggests that thisshould be on the order of L2/2D. This reasoning ishowever misleading. The duration of an event is bydefinition the time elapsed between the moment thata molecule enters the detection region and the mo-ment when it leaves the detection region for the first

time, a quantity known as the time of first passage.Much is known about first passage times since they areof great interest in diverse fields such as neuroscience,diffusion-controlled chemical reactions, ecology, andfinance.30,31 For the present case, after a particle hasentered the device, there is, in principle, an infinite setof trajectories that the particle can follow until it leavesfor the first time. The resulting distribution of residencetimes is given by the solution to the classic Gambler'sRuin problem in random-walk theory.32 The probabilityof the particle lasting n steps in the device before itencounters either end of the device for the first timehas the analytic form33

Ptot(n) ¼ 1a ∑

a � 1

v¼ 1cosn�1

πv

asin

πv

asin

πv

a

þ ∑a � 1

v¼ 1cosn�1

πv

asin

πv

asin

πv(a � 1)a

!

(1)

Here, a is once again the number of lattice points thatmakeup the lengthof thedevice, andn canbe convertedinto residence time, t, according to t = n(ΔL2/2D). In eq 1,the first term within brackets corresponds to the prob-ability of exiting from the same side that the particleenters, while the second term corresponds to the prob-ability of traversing the detection region and exiting atthe other end of the device.

Figure 3 compares the analytically computed prob-ability distribution and the simulated distribution ofresidence times. There is excellent agreement, natu-rally, since both simulation and analytical calculationrepresent the samemodel. The plot further shows thatthe most probable residence times are very short. Thisis a consequence of the fact that it is more likely for themolecule to exit very quickly from the same side as ithas entered than to propagate toward the center of thedevice. At short time scales, the probability distributiondiverges as t�3/2, where t is the residence time. Thus,even if themotion of the particleswas probed on amuchfaster time scale, the same behavior would be observed.On longer time scales, a “shoulder” is seen in the prob-ability distribution, which corresponds to the alternativescenario where the particle traverses the entire length ofthe device and exits from the other end. The mean timefor first passage, considering only these latter events, isgiven by L2/6D, consistent with intuition.15,30

Figure 2. (a) Trajectory of a particle as it undergoes arandomwalk. The gray regiondenotes the detection region.(b) Corresponding occupancy plot of the particle within thedefined detection region.

Figure 3. Distribution of residence times of the molecule inthe detection region. The black line denotes the distributioncalculated according to eq 1, while the red line is thecorresponding distribution from a random-walk simulation.

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How does this rapid entry and exit of moleculescompare with typical experimental time scales? Theinverse relationship between the gain and the mea-surement bandwidth of amplifiers in electronic mea-surement circuits ensures that the more sensitive acurrent measurement system, the lower its time reso-lution. In particular, the impulse response of electro-meters capableofmeasuring femtoampere level currentsis on the order of a few hundredmilliseconds.15 Compar-ing this to the distribution of residence times (Figure 3), itis clear that most events have residence times that aremuch shorter than this experimentally accessible timescale, and only the relatively rare, atypically long eventscan be resolved.

The consequence of finite time resolution is furtherillustrated in Figure 4a, which shows a typical occupancy�time trace obtained for the case of ÆNæ= 0.2. As expected,it reveals rapid telegraph-like oscillations correspondingto the presence of 0, 1, or 2 molecules over time. Toreproduce the effect of the instrument, this trace wasthen convolved with a first-order impulse response asdescribed earlier (τ = 100 ms). This filtering results inconsiderable smoothing of the occupancy�time trace(Figure 4b) such that only the events of durations muchlonger than τ are able to attain the full magnitude ofoccupancy as in the unfiltered case. On the other hand,the very rapid events are heavily averaged and notindividually resolved in the convolved traces. This leadsto noisy-looking curves with apparent plateaus at valuesof the current that do not correspond to integer numberof particles.

It is, inprinciple, possible todeconvolve the smoothedcurve to recover the original data. In practice, this is notpossible, however, due to the presence of additionalinstrumental noise superimposedon the single-moleculesignals inexperimental data. This noise is amplifiedby thedeconvolution procedure, leading to no improvement inthe quality of the signals.

Quantifying the Occupancy. In order to extract quanti-tative information from stochastic signals, a statisticaldescription is required. The average occupancy in thedetection region of the device, ÆNæ, can be controlledvia the bulk concentration of the reservoir, C, since thetwo are in diffusing equilibrium. Neglecting double-layer effects, which is appropriate under the high saltconditions considered here, this is simply ÆNæ = CVNA,where NA is Avogadro's constant and V is the volumeof the detection region of the device (more generally,ÆNæ is set by the chemical potential of the reservoir). Aswas made manifest by Figure 4a, however, the instan-taneous number of molecules present, N(t), cannot becontrolled and stochastic fluctuations between differ-ent values of N occur. In general, the probability ofhaving exactlyN particles at any given time, PN, is givenby the Gibbs distribution, PN � ZN exp(μN/kBT), whereZN is the partition function for N particles, μ is thechemical potential of the redox species, kB is Boltzmann's

constant, and T is the absolute temperature.34 For thecase of non-interacting particles, which is appropriate fora dilute solution, this is analogous to an ideal gas; in thiscase, the probability distribution reduces to the well-knownPoissondistribution,PN= ÆNæNexp(�ÆNæ)/N!.11�13,15

Figure 5a,d,g shows the probability distributionsobtained by histogramming occupancy�time tracesgenerated by random-walk simulations for three dif-ferent values of ÆNæ. In order to get sufficient statistics,simulations of 105 time steps were repeated 200 times.The resulting probability amplitudes agree to within1% with the Poisson distribution for PN.

Implicit in this analysis is the assumption of infinitetime resolution. To evaluate the experimentally feasibledistribution, we convolved the simulated occupancy�time traces with the assumed impulse response andthen histogrammed the resulting traces. Figure 5e,fshows histograms obtained for two different time con-stants of the impulse response, τ = 0.1 and 1 s, respec-tively, for ÆNæ = 1. In comparison to the unfiltered traces,the histograms for the filtered traces are considerablysmeared, such that the weight in the probability dis-tribution shifts from the peaks to values intermediatebetween integer multiples of ip. This effect becomesmore pronounced as τ increases.

The basis of this smearing of histograms can beunderstood by considering the broad distribution ofresidence times togetherwith the finite time resolutioninherent to measurement systems. Because the vastmajority of events are of very short durations, they areheavily smoothed in experiments, resulting in values ofcurrent that are much smaller than ip. The longerevents which do result in integer multiples of ip forthe current are rare and make only a minor contribu-tion to the histograms. Taken together, these twofactors preempt the observation of distinct peaks in

Figure 4. (a) Occupancy�time trace for an average occu-pancy, ÆNæ = 0.2. (b) Corresponding current�time plot afterconvolving the trace in (a) with a model impulse responsewith a time constant τ = 100 ms. This shows how the finitetime response of the measurement electronics precludesthe observation of rapid rise and fall in current as themolecule enters and leaves the detection region.

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the histograms corresponding to the presence ofinteger number of molecules. We emphasize that thislimitation arises from the fundamental properties ofBrownian motion of particles since we have not in-cluded any extrinsic noise (e.g., instrumental noise) tothe traces. Needless to say, the unavoidable presenceof extrinsic noise in any measurement will only exacer-bate the smearing of histograms.

OVERCOMING STOCHASTICITY

Wehave seen how the intrinsic properties of randomwalks combined with the finite time resolution ofmeasurement systems act in concert to limit the abilityto resolve single molecules. We now explore possibleapproaches to mitigate this effect.

Role of Access Channel. In the geometry of the deviceconsidered thus far, the detection region is connectedto 1D access channels on either side through whichmolecules can enter. The sharp boundary between thechannel and the detection region, coupled with thespace-filling property of random walks, increases thelikelihood that most trajectories of particles follow aback-and-forth path at the edge. However, what if thedetection region of the device was directly connectedto a 3D reservoir? One would expect qualitativelydifferent behavior because, unlike in 1D, there is onlya small chance in a 3D random walk that the particle

will return to its point of origin. In other words, thereis high probability that, once a particle diffuses tothe edge of the detection region and enters the 3Dreservoir, it escapes forever. While not influencing thedistribution of dwell times, this should eliminate theclustering of short events and could thus potentiallyreduce the smearing out of histograms.

Figure 6a shows occupancy�time traces obtainedin this way for ÆNæ = 0.2. Comparing with Figure 4ashows that, unlike the case of the 1D reservoir wherelong portions of the trace have 0 occupancy, in thepresent case, the short events are distributed fairlyuniformly. As a result, the long events in the convolveddata (Figure 6b) appear less noisy. Comparing thecorresponding histogram (Figure 6c) to those of Figure 5,the only difference is that, in the present case, theamplitude corresponding to 0 occupancy is muchreduced. However, for the peaks of interest, there isno significant difference between the two cases. Wetherefore conclude that coupling to a 3D reservoirinstead of a 1D channel represents only aminor improve-ment to the system's ability to resolve single-moleculeevents.

Trapping of Molecules. We now consider another pos-siblemechanismbywhich to overcome stochasticity;trapping of molecules in the detection region. Whileconceptually straightforward, this approach has the

Figure 5. Histograms of simulated traces for different average occupancies and measurement time constants. Rows (top tobottom): Average occupancies of ÆNæ = 0.2, 1, and 2, respectively. Columns (left to right): Traces filtered with varying timeconstants of the impulse response, τ = 0, 0.1, and 1 s, respectively.

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disadvantage that a reference electrode can no longerbe employed. The earliest examples of electrochemicaldetection of single molecules adopted this strategy:11

an electrode that was shrouded by insulating wax waspressed against a conducting substrate, thereby seal-ing off the detection volume from the external solu-tion. However, fluctuations in the current were stillobserved which were attributed to intermittent trap-ping. More recently, Sun and Mirkin were able toensure trapping of molecules in nanoscale cavities.14

Although they did not perform amperometric mea-surements, the number of trapped molecules wasinferred from the (essentially time-independent) mag-nitude of the limiting currents in cyclic voltammetryexperiments.

In the context of an integrated nanofluidic system,any analytical method that relies on the mechanicaltrapping of molecules within the detection regionwould require some manner of valves35 to automatethe opening and closing of access to the device. Suchvalves are likely to result in some dead volume oneither extremity of the detection region. Here weexplore the effect of this dead volume on stochasticityof the resultant current. We assume that the averageoccupancy in the detection region is ÆNæ = 1. Further,we define the total volume of the system Vtotal toinclude the volume of the detection region (V) andthe dead volume (Vdead). We investigate the effect of

keeping ÆNæ constant and increasing Vtotal. For the caseVdead = 0, the occupancy is equal to 1. Thus, amper-ometrically, only a constant current ip is expected, asshown in Figure 7a. If, however, there is some deadvolume present on either side of the detection region,into which molecules can diffuse, then fluctuations inthe occupancy (and current) will be seen. The prob-ability of finding Nmolecules in the detection region issimply given by the binomial distribution,

PN ¼ ÆNæ=pN

!pN(1 � p)(ÆNæ=p) � N

where p=V/Vtotal is the individual probability of findingeach particle in the detection region. Figure 7b showsthe occupancy traces obtained for Vtotal = 2V. Asexpected, the occupancy has a binomial distribution,with P1 = 0.5 and P0 = P2 = 0.25 (see SupportingInformation). As the dead volume increases(Figure 7c), however, the occupancy traces becomeincreasingly indistinguishable from the case wherethere is no trapping and the detection region iscoupled to an infinite reservoir (see histograms inSupporting Information). This is not surprising sincethe binomial distribution asymptotically converges tothe Poisson distribution for small p and large N.Experimentally, this implies that, for any improvementsfrom confinement to be noticeable, the dead volume

Figure 6. (a) Occupancy�time plot for a device configura-tion inwhich the detection region is directly coupled to a 3Dreservoir, with an average occupancy of ÆNæ = 0.2. (b)Corresponding current�time plot after convolving thetrace in (a) with a model impulse response with a timeconstant, τ = 100 ms. (c) Histogram of the convolved tracesfor an average occupancy of ÆNæ = 1.

Figure 7. Occupancy traces for particles trapped in thedevice with sealed boundaries for (a) Vdead = 0; (b) Vdead =V; and (c) Vdead = 2V. The inset in each plot shows theassociated geometry. In all cases, the assumed averageoccupancy in the detection region is ÆNæ = 1.

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must be significantly smaller than V. Alternatively, themeasurement must integrate over such a long periodof time (.Ltotal

2 /D, where Ltotal is the total lengthincluding the length along the dead volume) that thefluctuations are averaged over. In this sense, trappingof molecules benefits from small device dimensions.

Hydrodynamic Flow. A way to overcome stochasticitythat is particularly suitable for fluidic devices is throughthe introduction of advective flow in the system. We usethe term advection to denote a flow that is parallel to thesurface of the two electrodes so as to distinguish it fromconvection, which is typically used to describe perpendi-cular flow. The solution containing the redox-activeanalyte could be advected through the device usingeither pressure or electrokinetic methods. This is particu-larly relevant for cases where sample exchange is desir-able. Since advective flow disrupts the diffusive equilib-rium between the detection region and the outside,how does it impact the stochasticity of the signals? Themain effect is to directly alter the residence time of themolecule in the detection region, with the average resi-dence time given by tflow = L/v, where v is the velocity ofthe fluid and L is the length of the detection region asabove.Because the instantaneous rmsvelocityof themole-cules is much larger than v, diffusion (and therefore ip)remains entirely unaffected; the flowmerely introduces apersistent bias. In order for the flow to be effective increating a well-defined residence time, however, the rmsfluctuations in the displacement due to diffusion, Δx,

should be much less than the length of the device: Δx =(2DΔtflow)

1/2 , L. Rearranging, this leads to a conditionfor the required velocity, v. 2D/L. For a device of lengthL = 200 μm, this corresponds to v. 10 μm/s.

Figure 8 shows occupancy traces for three differentfluid velocities. For the case where v is less than thecritical value (Figure 8a), the occupancy trace showsshort events similar to the case where there is no flowin the system. Conversely, when v is much greater thanthe critical value (Figure 8b,c), the short events are nolonger seen, and the occupancy traces show uniform,well-defined residence times in the detection region.Figure 9 shows the simulated distribution of residencetimes under conditions of fluid flow. A prominent, well-defined peak develops at values of the residence timeclose to the advection time (near 2 s in Figure 9) thatcorrespond to particles that enter from one side of thedevice and exit from the other. As can be seen,increasing the fluid velocity shifts this peak to shorterresidence times, consistent with intuition. It also in-creases the absolute probability of such events occur-ring, especially compared to the case when there is noflow in the system. The simulated curves also showsmall, very rapid oscillations; these are an artifact of thesimulation that arise from having two discrete stepsizes of varying lengths corresponding to diffusion andflow (ΔL and Δxflow, respectively).

DISCUSSION AND CONCLUSIONS

Our focus has been to delineate the fundamentalfeatures of Brownian motion that have a direct impacton single-molecule electrochemical detection. In realexperiments, there are at least two additional factorsthat can significantly impact the quality of the informa-tion that can be extracted from amperometric data.The first is dynamic adsorption of analyte molecules tothe electrode surface: intermittent adsorption results

Figure 8. Current�time traces in the presence of flow withdifferent fluid flow velocities in a device with L = 200 μm: (a)1 μm/s, (b) 50 μm/s, (c) 100 μm/s. In all cases, the traces havebeen convolved with an impulse response with time con-stant τ = 100 ms.

Figure 9. Distribution of residence times as a function offluid flow velocity obtained from simulations for a devicewith L = 200 μm. Red, 200 μm/s; blue, 100 μm/s; black,analytical solution (in the absence of flow) using eq 1. Thesmall, noise-like oscillations arise from interference be-tween the two discrete step sizes present in the simulation.

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in a decrease of the effective diffusion coefficient of theanalyte in the detection region,36 causing the currentper molecule to have a value lower than the purelydiffusive case ip = eD/z2. Furthermore, individual ad-sorption�desorption events add an additional com-ponent to the noise, although they are likely to occuron time scales much faster than the instrumentalresponse and therefore likely to be averaged. Variousapproaches including the functionalization of electro-des with self-assembled monolayers with tailored endgroups can be used to minimize adsorption.37 A sec-ond aspect we have not considered so far is the back-ground noise and contributions to it from instrumentalnoise, leakage currents, etc. Here we focused on an idealscenario, where the only source of stochasticity is Brow-nian motion itself. Real experiments will have consider-able extrinsic noise superimposed on the signal andthereby further obscure single-molecule features in thesignals. As in most single-molecule techniques, elimina-tionof thebackgroundnoise is often asbig a challenge asthe selective amplification of the desired signal.In summary, we have used random-walk simulations

to elucidate the properties of redox-cycling-basednanogap sensors suitable for electrochemically detect-ing individual molecules. We argued that, while full 3Dsimulations can give insight into individual electron-transfer events, 1D random-walk simulations suffice tocapture the dynamics that are experimentally accessi-ble in channel-like nanogap detectors. These 1D simu-lations show that, in regimes where the detection

region is diffusively coupled to a bulk reservoir, theinherently stochastic nature of Brownian motioncoupled with the finite time resolution of measure-ment electronics causes significant deviations from theideally expected Poisson distribution for the faradaiccurrent. We have further explored three specific stra-tegies by which this significant limitation can beovercome.Our results show that the mechanical trapping of

molecules in the detection region can ameliorate thesmearing of the current distribution, as long as deadvolumes can be neglected or that the signals areintegrated for a sufficiently long time. Direct couplingof the detection volume to a 3D reservoir, on the otherhand, leads only to minor enhancement of the qualityof the signals. Finally, we showed that advective flow,wherein the solution containing the redox-active ana-lyte is actively driven through the detection region,leads to essentially deterministic signals if the flow issufficiently rapid. We propose that this latter approachoffers the most promising route for overcoming sto-chasticity and creating reliable single-molecule detec-tors. The limitations imposed by the inherent nature ofBrownian motion, as delineated in this paper, repre-sent the key bottleneck in the wider adoption andutilization of single-molecule electrochemical sensitiv-ity in more complex, realistic assays. However, themethods proposed herein to circumvent these hin-drances can, in our view, enable a more widespreademployment of this attractive capability.

METHODSTo compute the distribution of residence times of particles in

the device, a single particle was injected at one end of thedetection region (using a= 20) and the random-walk simulationwas run until the particle exited either end of this region. Thesimulation was run repeatedly, and the total number of stepstaken before exiting was histogrammed.To simulate the behavior of molecules trapped in a closed

volume, we considered the two edges of the 1D lattice (a = 100)to be perfectly reflecting boundaries. If a random walker took astep that wouldmake it exit the simulated region, it was insteadmoved in the opposite direction.To simulate the effect of coupling the ends of the detection

volume to a 3D reservoir, we treated the detection region as a 1Dlattice (a = 100) with perfectly absorbing boundaries. As particlesundergoing the randomwalk touched the edges of the detectionregion, they were removed from the system. Simultaneously, newparticles were introduced into the system at each time step and ateach end of the detection regionwith a finite probability, p (wherep, 1). Particles were thus introduced into the detection region ata rateΓin = p/Δt, while they exitedwith rateΓout = ÆNæ/τ0, where τ0is the mean lifetime of the particle in the detection region and ÆNæis the average number of particles in the detection region (oraverage occupancy); τ0 can be evaluated numerically using τ0 =(ΔL2/2D)∑nnPtot (n). At steady state, Γin = Γout, so that ÆNæ = pτ0/Δt.The average number of particles present in the detection region isthus proportional to the parameter p; for our prototypical geo-metry, p = 0.005 corresponds to ÆNæ = 1.To simulate the coupling of the detection region to a reservoir

via a nanochannel, in which transport is also effectively 1D, weemployed simulations on a large 1D lattice of 104 points. The

detection region of the device was represented by a smallsubset of this large simulation domain (a = 100, or 1% of thetotal). The part of the simulation domain outside the detec-tion region thus played the role of the nanochannel and wasalso sufficiently large to represent the reservoir. Periodicboundary conditions were enforced; particles exiting at oneend of the simulation domain were reintroduced at the otherend, ensuring that the “bulk” concentration remained con-stant throughout the simulation. The positions of a numberof random walkers appropriate for the simulated concentra-tion were randomly initialized; the walkers were movedindependently at each time step, and the total number ofmolecules in the detection region was calculated after eachstep.To simulate an advective flow of the solvent, a translation of

magnitudeΔxflow = vΔtwas superimposed on the random-walkmotion for each walker at each time step.

Conflict of Interest: The authors declare no competingfinancial interest.

Acknowledgment. We thank M. Zevenbergen, F. Pedaci, andS. Feldberg for insightful discussions. We gratefully acknowledgefinancial support from The Netherlands Organization for ScientificResearch (NWO) and the European Research Council (ERC).

Supporting Information Available: Details of the full 3Drandom-walk simulation and the geometry used for particleconfined in the detection region of a prototypical device. Histo-grams from simulated traces for particles trapped in the devicewith sealed boundaries and differing dead volumes. This materialis available free of charge via the Internet at http://pubs.acs.org.

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Supporting information for: Stochasticity in

Single-Molecule Nanoelectrochemistry: Origins,

Consequences and Solutions

Pradyumna S. Singh,† Enno Kätelhön,‡,¶ Klaus Mathwig,† Bernhard Wolfrum,‡,¶

and Serge G. Lemay∗,†

MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands,

Institute of Bioelectronics (ICS-8/PGI-8), Forschungszentrum Jülich GmbH, 52425 Jülich,

Germany, and JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich

GmbH, 52425 Jülich, Germany

E-mail: s.g.lemay@utwente.nl

3D Random Walk

A three-dimensional simulation is used to model the current response of a single molecule inside

the active area of a nanofluidic sensor. It is assumed that the molecule can only adopt two oxidation

states and that it reacts immediately whenever it touches an appropriately biased electrode. The

molecule’s pathway is calculated via a diffusive random walk. The space available for diffusion

is limited by two infinite plane-parallel plates that are separated by a gap of 50 nm and represent

∗To whom correspondence should be addressed†MESA+ Institute for Nanotechnology, University of Twente‡Institute of Bioelectronics (ICS-8/PGI-8), Forschungszentrum Jülich¶JARA-Fundamentals of Future Information Technology, Forschungszentrum Jülich

S1

the top- and bottom electrodes (see Figure S1). Initially, the molecule is released in between the

electrodes and can then diffuse freely within the given boundaries.

Figure S1: Schematic representation of the geometry used in the three-dimensional random-walksimulation. A single redox-active molecule can diffuse freely in between two infinite plane-parallelelectrodes.

The software is written in C++ and compiled via the GNU Compiler Collection (GCC). It

utilizes the pseudo random number generator Mersenne Twister (mt19937), which is a part of

GNU Scientific Library (GSL). During each iteration of the random walk, which corresponds to

a time interval dt, individual pseudo random numbers are generated for the movement in all three

Cartesian dimensions. The molecule’s coordinate is then randomly either increased or decreased

by the fixed one-dimensional spatial step size dr1d . The lateral and temporal step sizes dr1d and

dt are linked via the following equation that can be derived from the one-dimensional diffusion

equation:

dr1d =√

2 ·D ·dt (1)

The simulation data presented in the article utilize a diffusion coefficient of D = 1×10−9 m2/s and

a temporal step width of dt = 1×10−9 s.

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Current Distribution for Trapped Molecules

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(a) (b)

(c) (d)

Figure S2: Histograms from simulated traces for particles trapped in the device with sealed bound-aries and differing dead volumes (see Fig. 7 of main text): (a) Vdead = V and (c) Vdead = 2V . (b)and (d) are histograms for traces corresponding to (a) and (c), respectively, that have been filteredwith the model impulse response (τ = 100 ms). In all cases the average occupancy 〈N〉= 1.

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