Single-Particle Plasmon Sensing of Discrete Molecular Events

Article pubs.acs.org/JPCC

Single-Particle Plasmon Sensing of Discrete Molecular Events: Binding Position versus Signal Variations for Different Sensor Geometries Virginia Claudio,† Andreas B. Dahlin,† and Tomasz J. Antosiewicz*,†,‡ †

Department of Applied Physics, Chalmers University of Technology, SE-41296 Göteborg, Sweden Centre of New Technologies, University of Warsaw, Ż wirki i Wigury 93, 02-089 Warsaw, Poland

‡

ABSTRACT: The sensitivity of a surface plasmon to the dielectric environment makes it a viable tool in detecting single molecules. To be able to precisely determine sensed molecular concentrations and carry out precise analyses of single-molecule binding/ unbinding events in real time it is necessary to quantify rigorously the relation between the number of bound molecules and the spectral response of the plasmonic sensor. However, this is challenging as this relation is subject to an uncertainty which is highly dependent on the spatially varying response of the plasmonic nanosensor of choice. The origin of this uncertainty is little understood, and its effect is often disregarded in quantitative sensing experiments. Here, we employ stochastic diffusion-reaction simulations of biomolecular interactions on a sensor’s surface combined with electromagnetic calculations of the plasmon resonance peak shift of three metal nanosensors (disk, cone, dimer) to clarify the interplay between position-dependent binding probability and inhomogeneous sensitivity distribution in determining the statistical characteristics of the total signal upon molecular binding. This approach is generally applicable regardless of the specific transduction mechanism at the basis of sensing. Here we identify how this interplay affects the feasibility of using certain plasmonic sensors for sensing low concentrations or real-time monitoring of individual binding reactions and how illumination conditions may affect the level of uncertainty of the measured signal upon molecular binding.

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an electric signal,11 or an electromagnetic resonance as a means of transduction. In the latter case an important class of nanoresonators is represented by metal nanoparticles. When exposed to light they respond with a localized surface plasmon resonance (LSPR) whose line shape and spectral position are very sensitive to changes of the refractive index in the nearby environment. Very high average sensitivity down to the singlemolecule detection12,13 has recently been matched by the realtime detection of individual binding events to the nanosensor,14,15 thus offering proof-of-concept of true singlemolecule observations by LSPR. To make quantitative use of such measurements, it is important to give an estimate of the confidence level offered by a specific nanoparticle, experimental setup, and data analysis method. Naturally, for real-time observations, not only the base noise level but also the acquisition speed are of major importance. However, a significant contribution to the uncertainty is often disregarded, namely, the influence of the geometrical features of the nanosensor itself. In particular, a nonhomogeneous field distribution leads to a position-dependent peak shift, hence to a nonhomogeneous sensitivity.15,16 This is as well the case for other types of sensors (not based on plasmons or light, such as force cantilevers). Moreover, the positions on the sensor

INTRODUCTION The details of biochemical reactions are crucial to understanding life’s secrets1 and therefore to implement diagnostic tools and drug screening platforms for the tailored cure of complex diseases. One of the main burdens to overcome in the life sciences is represented by the limitations associated with ensemble measurements where details on the molecular properties are lost in the averaging and reaction dynamics are simply hard to determine, especially when the concentration of interacting molecular species is low and sensitivity is poor. In comparison, a large amount of chemical, biological, and medical information can be uncovered by single-molecule investigations.1−4 In recent years considerable efforts have led to interesting advances in the detection of single molecules. Many methods are based on fluorescence microscopy, in which fluorescent labels are attached to the molecules of interest and monitored with advanced optical techniques.5,6 Direct observation of single-molecule optical absorption has also been achieved.7−9 However, on one hand the use of labels can heavily affect the true kinetic behavior of the molecules under observation, while on the other hand extremely low noise levels are required for the detection of pure optical absorption, raising issues on the complexity and operational stability of the setups utilized in these experiments. Usually the presence of the molecule is instead amplified by a transducer. The most common ways of avoiding fluorescent labeling then are to utilize a force signal,10 © 2014 American Chemical Society

Received: December 13, 2013 Revised: March 6, 2014 Published: March 11, 2014 6980

dx.doi.org/10.1021/jp412219v | J. Phys. Chem. C 2014, 118, 6980−6988

The Journal of Physical Chemistry C

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Figure 1. Scheme of electromagnetic calculations and stochastic diffusion-reaction simulations. (a) We consider three plasmonic sensors illuminated by linearly polarized light: nanodisk, nanocone, and nanodimer. The colormaps indicate 3-D distributions of relative resonance peak shifts induced by molecule binding at a particular position. The color scale is different for each sensor (we also used unpolarized light for the disk, which resulted in an axially symmetric peak shift map, not shown here). The blue dots mark discretized receptor positions at which we place molecules (represented by dielectric spheres) to calculate the induced peak shift. The receptor positions on the surface of a disk-shaped sensor are obtained by first discretizing the cross section (white line) to find radii of circles (cyan lines) which are further divided. (b) Scheme of the stochastic diffusion-reaction simulation volume V for N molecules diffusing with time steps Δtc, Δtm, and Δtfwhere Δtc > Δtm > Δtfdepending on their distance from a sensor. (c) Drawing of the reaction volume (black circle) and the distance-dependent binding probability (shadow) for one molecule (yellow star) in the vicinity of one receptor (green). The four different cases are representative of the effect of volume exclusion due to geometrical constraints on the accessible volume. (d) The total binding probability for one molecule depends on the number of free receptors in its vicinity; e.g., the green molecule can only attempt to bind to one receptor. The yellow is in the range of three receptors; however, one of them is occupied, and the binding probability is calculated as a sum over only two receptors.

where molecules diffuse to and bind are not random as commonly assumed. It is a well-known fact that the probability distribution of such positions depends greatly on the shape of the object itself.17 Moreover, experimental efforts have been reported to selectively functionalize the edges of plasmonic structures to take advantage of the strongly enhanced electric field there.18,19 However, even in those cases the electric field, and consequently the measured peak shift upon molecule binding, is not completely uniform, and to the best of our knowledge the question of how this aspect influences the result of a bioassay in the majority of nanosensing platforms has so far been disregarded. In a previous work,20 a model was developed that relates the sensitivity of a nanoplasmonic resonator to the local field in which the analyte is placed. Thus, one can obtain maps of the local shifts. Here we simulate single molecular diffusion and reaction on the sensor surface with high spatial and temporal resolution. We elucidate the role of nonhomogeneous sensitivity (which here we define as sensor-specif ic uncertainty) and preferential binding positions determined by diffusion, and we propose a way to compare various sensor geometries for the optimization of single-molecule sensing. This work combines electromagnetic simulations with stochastic simulations of molecule diffusion and binding. The electromagnetic part aims at analyzing the optical response of plasmonic metal resonators to the binding of molecules (modeled as dielectric spheres) and assigning an appropriate numerical value to the induced response (peak shift value). The stochastic part serves to elucidate the quasi-random nature of molecule binding to receptors attached to the surface of considered plasmonic nanosensors. This paper is organized as follows. We begin by describing the methodology behind this work, where we focus on the stochastic simulations, but also give a brief introduction to the electromagnetic part. Next, we

present results of diffusion and reaction studies and combine them with plasmonic sensing. Finally, we discuss the stochastic nature of nanoplasmonic sensing and conclude.

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METHODS We set up a sensing experiment in silico by modeling free diffusion of noninteracting molecules in a continuous volume and their binding/unbinding to a nanosensor. Our model takes inspiration from the field of stochastic simulations of diffusioncontrolled reactions,21−28 and it is designed ad hoc to model the binding of single entities to the 3-D surface of various nanostructures with high spatial and temporal accuracy. The sensor surface is discretized in a way representing different molecular receptor locations. When a molecule is close enough to such receptors and if at least one of the receptors is free, the probability of binding to one of them becomes higher than zero (but lower than one), thus the only interaction between molecules is through the occupancy state of receptors. Further, if a receptor is occupied, under reversible binding conditions the probability of unbinding is taken into consideration. We distinguish this case of reversible binding from the simpler case of perfect absorption in which molecules binding to the sensor are removed from the volume and the receptor still remains free afterward (no tracking of receptors’ occupation). Each receptor location is associated with a certain sensitivity, and the output of the experiment is a time trace of the total signal related to all stochastic events happening on the sensor. This approach is valid for any sensor geometry and transduction principle as long as a spatial-dependent sensitivity is known or can be calculated. Here we briefly present the scheme we have employed to obtain the spatial sensitivity distribution of molecule binding to a metal nanoparticle, i.e., a 3-D peak shift map. We shall now briefly describe how such 3D peak shift maps are obtained using a disk as an example (see 6981



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then used together with stochastic simulation results to obtain temporal peak shift traces that are subsequently further analyzed. They are also interpolated in 3-D to allow easy visualization of the calculated sensitivities (see Figure 1a). The peak shift maps plotted in Figure 1a are for linear polarization; however, for the disk we have also used unpolarized light, and this illumination condition results in an axially symmetric peak shift distribution. The sensors are placed on the surface of a substrate above which target molecules diffuse freely, bound only by reflecting boundary conditions of a large cubic box limiting the simulation volume. Here we assume that the liquid medium in which diffusion occurs has the same refractive index as the substrate (e.g., MgF2 is very close to water) to neglect optical substrate effects. Diffusion is modeled through a Wiener process.29 Given the diffusion constant D, the mean one-dimensional diffusion length in a time interval dt is equal to s = (2Ddt)1/2. At each time step the spatial shift in x, y, and z directions is calculated by multiplying s by a random number from the normal (0,1) distribution. The mean diffusion step in all directions is equal to zero, while the mean absolute value and the standard deviation are both equal to (6Ddt)1/2.17 In our simulations we use an algorithm based on three time discretization steps (Figure 1b). First, a coarse time step is used for molecules far away from the sensor Δtc, and it gives a large mean diffusion length compared to the sensor size. This allows a rather quick simulation of diffusion in the large volume. However, if one would use only a large time step, the sensor would be effectively transparent to the molecules, as they would jump through or past the sensor. Thus, for molecules in a smaller volume close to the sensor we use a fine time step Δtf that gives a mean 1-D diffusion about sf = 1.4 nm for D = 10 μm2/s. If any molecule is close to a binding receptor we evaluate the binding probability within the relevant time step, in this case Δtf. Concurrently, in each time step the unbinding probability of already bound molecules is evaluated, and if an unbinding attempt is successful, the molecule begins to diffuse from the next time step. Between the coarse and fine volumes a transition region (Δtm) is present to mediate diffusion of molecules from near the sensor to the “bulk solution”. Its radius is eight times larger than the mean 1-D coarse time step ensuring that a molecule from the coarse region cannot reach the inner fine region in one time step. Similarly, the fine region radius is 13 times larger than the mean transition region step. In the simulations presented in this work we have used Δtc = 1 ms, Δtm = 10 μs, and Δtf = 0.1 μs. In more detail, the probability of binding Pb to any receptor is given by Pb = 1 − exp(−∑receptors(kon/Vr)dt), where kon is the binding rate and Vr the spherical reaction volume of radius given by the distance between the molecule and the receptor,23 which is nonzero only if the receptor is free (Figure 1c). The sum runs over all receptors whose distance from the molecule is less than a critical distance (reaction radius) (Figure 1d). The choice of this distance within a certain range is in our case not very critical, although too small of a distance leads to underestimation of the equilibrium surface coverage. Once the binding probability has been calculated, a first random number r1 is generated and compared to Pb. If r1 > Pb, the molecule is still free to diffuse. If r1 ≤ Pb then binding occurs, and a second random number r2 is used to evaluate to which receptor within the reaction radius the molecule will bind. This is done by evaluating the cumulative probability and comparing

Figure 1a). Metal nanoparticles with sizes on the order of a few tens of nanometers are capable of supporting a localized surface plasmon resonance. The resonance position, amplitude, and shape depend on the geometrical arrangement of metal as well as its dielectric environment. Thus, a change of the surroundings (e.g., binding of a molecule with a different refractive index than that of the surrounding medium) affects the resonance in a measurable way, usually quantified by a shift of resonance position (peak shift). To connect stochastic diffusion/binding in silico experiments with the response of a plasmonic sensor we need to calculate such 3-D peak shift maps using the same spatial discretization employed for localizing receptors on the surface of the metal nanoparticle. We begin by choosing a representation for the considered moleculeit is assumed to be a dielectric sphere of 3 nm in radius and have a refractive index of 2. The choice of this value is arbitrary and serves only to make the peak shifts easier to calculate in regions with a small field enhancement; using a different value would simply rescale the peak shifts without qualitatively changing their distribution. We place receptors no closer to each other than 5 nm, so that the binding of a molecule at one will not prohibit an adjacent receptor from binding due to insufficient space for another molecule to fit. The receptor positions (x,y,z)Δr, where Δr ≈ 6 nm is the approximate linear discretization step, are calculated in the following way. For instance, a disk is a cylindrically symmetric object so in this case we first discretize the exterior outline of the (ρ,z) cross section marked by the thick white line (Figure 1a). The blue spheres placed along the white line mark equally spaced (∼Δr) radii, which define circles which are subsequently further discretized in the azimuthal direction. Each circle is divided into equidistant segments (again approximately Δr long along the thin cyan lines) meaning that subsequent rings have an increasing number of points. The marked positions on the particles show places at which we place the dielectric sphere to calculate the peak shifts induced by molecular binding, where these peak shifts form the basis of surface plasmon resonance sensing. An efficient way to calculate and estimate the induced peak shifts is by calculating the energy density displacement caused by the sensed particle.20 The most straightforward approach, the one employed here (finite-difference time-domain, nonuniform spatial discretization of 5 Å around the resonators), is to use a numerical method, preferably a time domain one, to obtain spectra quickly. Naturally, frequency domain schemes are also feasible, although they will require several runs to determine the peak shift. Here, we investigate three distinct resonators with different characteristics: a disk with relatively weak and exposed plasmonic hot spots (radius 25 nm, height 25 nm, edge rounding of 5 nm), a cone with a single exposed hot spot (radius 23 nm, taper angle 20°, edge and apex rounding of 2 nm), and a dimer with a strong but relatively inaccessible hot spot (two identical disks as described above separated by 6 nm). In Figure 1a we present all three resonators with marked binding sites and sensitivity maps. At each of the positions marked by a sphere in Figure 1a we place, one at a time, a target molecule (3 nm in radius, n = 2) that we want to sense and calculate the resulting peak shift. We did not take into account the refractive index of the binding receptors, as this would merely result in a red shift of the entire spectrum; however, the binding positions were situated 31.5 Å from the surface (3 nm radius plus a 1.5 Å separation between the surface of the dielectric sphere and the metal surface). These peak shifts are 6982



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with r2. The first receptor for which the cumulative probability is higher than r2 is the one where the molecule binds. Simulation parameters are properly taken to obtain a good compromise between overcoming the computational limits (e.g., slow speed and handling of many digits for very small probabilities) and choosing realistic quantities. In particular, our aim was to ensure the observation of reversible binding events at a low surface coverage ( 1000 s). The dashed lines are set ±0.64 below (green) and above (orange) the average signal. The polarized light time trace is characterized by a 20% larger signal variance, a result of a larger variance of the calculated 3-D peak shift map for polarized light than for unpolarized. The variance is enhanced by preferential binding at the edge where the inhomogeneity of the peak shift is the largest for polarized light.

differs. Namely, for unpolarized light it is circularly symmetric and mirror symmetric in the polarized case. By utilizing the same geometry and the same time trace (same receptor occupation over time) for both polarizations we highlight the contribution of the different sensitivity distributions alone. This is qualitatively seen in the time traces in Figure 4a,c. The dashed horizontal lines set ±0.64 about the average (standard deviation of the peak shift for polarized light) demonstrate that for unpolarized light far fewer occurrences of exceptionally large/small peak shifts are observed (standard deviation for unpolarized light is 0.53). This larger peak shift spread for polarized light is evident in histograms (Figure 4b,d, average over 30 time traces). Hence, the uncertainty of the average peak shift induced by the same average number of molecules is larger for the polarized case than for the unpolarized one. This reflects the greater inhomogeneity of the sensitivity distribution for polarized light. Particularly, this difference in the spread of the signal is enhanced by the effect of geometry in “directing” the molecules preferentially at the edges of the disk, where the degree of inhomogeneity is the highest. Indeed, since fully covered disks are assumed to have the same total peak shift, this difference will diminish as the concentration increases, while it will be most apparent for single binding events per sensor. Finally, we compare the three types of nanostructures discussed in this study by plotting the probability distribution of the total peak shift under the same concentration and reaction parameters (Figure 5). The disk (red, blue) provides a much narrower distribution compared to the dimer (black) and the cone (green) but also a much lower average signal. The cone offers the highest signal for the same concentration, although the spread is too big for estimating it with accuracy. A closer

Figure 5. Probability distribution of the total signal produced by a disk (blue for unpolarized and red for polarized illumination), a dimer (black), and a cone (green) upon binding/unbinding of molecules for a concentration of 92 pM. The plot is obtained as a histogram of the time traces at equilibrium (t > 2600 s) adding up to more than 105 events, with a bin size of 0.1 nm. The probability is normalized to the average number of detected molecules for each sensor (ca. 11 on the disk, 26 on the dimer, 15 on the cone). The inset shows a magnification of the main plot. Due to the small spread of possible peak shifts the disk gives a better response for estimating the concentration of molecules in solution. The cone and dimer give a clearer on/off detection signal indicating the (mere) presence of a target molecule but poorer quantification of concentration.

look at the distribution for the cone reveals the presence of an interesting pattern across the possible peak shifts. The pattern repeats itself at regular intervals corresponding approximately to multiples of the highest peak shift (12 nm) produced by a 6985



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molecule binding on one of five available receptors on the first ring around the tip of the cone. The probability peaks right above zero and right above a multiple of the highest peak shift, suggesting the presence of many molecules away from the top of the cone where the signal is the lowest. This seems in contradiction with our previous considerations on the positiondependent binding probability for the cone. However, it should be noted that the conclusions based on Figures 2b and 3b refer to a probability per single receptor location. The probability distribution in Figure 5 is the sum of all occurring binding events, and the ratio between the number of receptors located at the bottom of the cone and those at the tip is ca. 5. The details of the pattern originate from the discretized nature of the data and a high gradient of the sensitivity distribution for this structure. In a real-time single-particle experiment the possible shifts are also discrete due to the fixed position of receptors, although their location is not as structured as in the current analysis but is rather random and they are more densely packed than assumed here. Hence, while the fine and sharp structure, shown in Figure 5, observed in our simulations will disappear, we still expect to observe the presence of primary peaks corresponding to one or multiple bindings at or close to the hot spot. However, due to a larger peak shift variability the structure is expected to be smeared out. We expect this to be most noticeable for the cone, where the subsequent maxima might overlap. In a measurement on an ensemble of cones the slight difference between the sensitivities of each cone and the precise receptor locations is expected to further smear out the pattern in the total peak shift. The same applies to the dimer; however, the largest peak shifts occur when a molecule binds in the gap, and as a consequence, the probability of observing large shifts is small when compared to the cone. Any other sensor offering a hot spot or a steep gradient in the sensitivity map is subject to this consideration (e.g., a rod- or star-shaped nanoresonator but not a disk or a ring). The consequence of these phenomena on actual sensing experiments is a greater uncertainty than what has been taken into account so far when estimating the number of molecules on the surface of a sensor, concentration in solution, and reaction parameters. The sensor-specif ic uncertainty, combined with the geometry-dependent diffusion and binding, affects considerably the quantitative analysis in cases where low to medium surface coverage is reached. For instance, we can use the traces in Figure 4 as the basis for discussing a realistic experimental situation in which we want to detect a few molecules. If our detection apparatus has a noise level lower (see green dashed line) than our signal, then a disk resonator under unpolarized light gives a better measurable signal (less uncertainty). On the other hand, if the noise level in our experiment is slightly higher than the average signal (see orange dashed line), a larger spread in the signal might actually be desirable since the upper tail is more easily detectable yielding more discernible counts. Following this line of thought, in the case of a single-molecule detection experiment we may choose to use a cone resonator since the exposed hot spot is the best geometry available to capture molecules in the region with high sensitivity (the tip) and therefore detect a clear signal above the experimental noise level. However, it is important to recognize that such inhomogeneous sensitivity distribution is arguably useful in a quantitative analysis, and its use should rather be limited to yes/no type of detections. This is especially true when the time resolution is lower than the typical reaction time constants.

Assuming that our acquisition time allows us to resolve individual binding and unbinding events on the surface and that we have a good signal-to-noise for each position on the sensor, we could simply count molecules over time one by one and even use the time intervals between individual events to extract directly the information on the reaction constants. The flaw in this reasoning is that we cannot be certain that the single steps that would appear in the time trace of the signal are coming from individual molecules and that all the binding events are detectable (above the noise level). Except in cases where a single receptor is present on preferably the same location on each sensor,28,32 the observation of clear steps rising from the noisy signal is not equivalent to a real detection of all molecules binding and unbinding. We have to keep in mind that a molecule bound to a low sensitivity region simply will not be seen, and we will not know of its presence, while a couple of molecules binding to the same low sensitivity region in a relatively short time interval may appear to produce one single binding step similar to one induced by a molecule binding at a high sensitivity spot. We suggest that at best one can combine calculations of sensitivity distribution with measurements to assess what percentage of events will likely be missed in a given setup and accompany the results of a quantitative analysis with the corresponding confidence level comprehensive of all sources of uncertainty, i.e., those coming from the setup as well as those intrinsic to the system being measured as discussed herein.

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SUMMARY AND CONCLUSIONS The interpretation of a label-free transduction signal originating from few or single molecules is far from being obvious. Even the nicest signal with the lowest background contains sources of uncertainty that are intrinsic to the inhomogeneous nature of sensors. Moreover, diffusion in three-dimensional space causes an inhomogeneous and nonrandom distribution of molecules on the sensor’s surface. This inhomogeneity can persist even after equilibrium establishment. We have shown how both these effects add extra uncertainty to the conventionally assumed purely Poissonian behavior of molecular binding to nanoparticles. Therefore, the determination of average values in ensemble measurements as well as the direct counting of individual molecules are both subject to a lower confidence interval than commonly assumed. It is then clear that also the determination of reaction rates is far from accurate not only when using simple parameters fitting to binding curves but also when more advanced methods are utilized such as parameters fitting to event histograms,33 fluctuation analysis,34 or step finding algorithms.35 In fact, on one hand the information looked for is just hidden by the noise, and on the other hand the validity of the signal above the noise level is limited by the temporal resolution, i.e., the acquisition speed compared to the commonly faster kinetics of biomolecular reactions. We propose that all surface-based bioassays involving an inhomogeneous feature (any topographic variation from a flat surface) should be accompanied by an assessment of the uncertainty to be labeled as ultrasensitive. Our model aims in this direction, though other phenomena influencing the kinetics of the process could be taken into account, such as temperature variations and electrostatic and optical forces. Moreover, our approach can be expanded to infer hidden information. For instance, based on the knowledge of where molecules are more likely to bind and on the spatial dependence of the magnitude 6986



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(11) Zheng, G.; Patolsky, F.; Cui, Y.; Wang, W. U.; Lieber, C. M. Multiplexed Electrical Detection of Cancer Markers with Nanowire Sensor Arrays. Nat. Biotechnol. 2005, 23, 1294−1301. (12) Mayer, K. M.; Hao, F.; Lee, S.; Nordlander, P.; Hafner, J. H. A Single Molecule Immunoassay by Localized Surface Plasmon Resonance. Nanotechnology 2010, 21, 255503. (13) Chen, S.; Svedendahl, M.; Antosiewicz, T. J.; Käll, M. PlasmonEnhanced Enzyme-Linked Immunosorbent Assay on Large Arrays of Individual Particles Made by Electron Beam Lithography. ACS Nano 2013, 7, 8824−8832. (14) Zijlstra, P.; Paulo, P. M. R.; Orrit, M. Optical Detection of Single Non-Absorbing Molecules Using the Surface Plasmon Resonance of a Gold Nanorod. Nat. Nanotechnol. 2012, 7, 379−382. (15) Ament, I.; Prasad, J.; Henkel, A.; Schmachtel, S.; Sönnichsen, C. Single Unlabeled Protein Detection on Individual Plasmonic Nanoparticles. Nano Lett. 2012, 12, 1092−1095. (16) Sannomiya, T.; Hafner, C.; Vörös, J. In Situ Sensing of Single Binding Events by Localized Surface Plasmon Resonance. Nano Lett. 2008, 8, 3450−3455. (17) Berg, H. C. Random Walks in Biology; Princeton University Press: Princeton, 1983. (18) Beeram, S. R.; Zamborini, F. P. Selective Attachment of Antibodies to the Edges of Gold Nanostructures for Enhanced Localized Surface Plasmon Resonance Biosensing. J. Am. Chem. Soc. 2009, 131, 11689−11691. (19) Zijlstra, P.; Paulo, P. M. R.; Yu, K.; Xu, Q.-H.; Orrit, M. Chemical Interface Damping in Single Gold Nanorods and Its Near Elimination by Tip-Specific Functionalization. Angew. Chem. 2012, 124, 8477−8480. (20) Antosiewicz, T. J.; Apell, S. P.; Claudio, V.; Käll, M. A Simple Model for the Resonance Shift of Localized Plasmons Due to Dielectric Particle Adhesion. Opt. Express 2012, 20, 524−533. (21) Smoluchowski, M. V. Versuch Einer Mathematischen Theorie der Koagulationskinetik Kolloider Lösungen. Z. Phys. Chem. 1917, 92, 129−168. (22) Collins, F. C.; Kimball, G. E. Diffusion-Controlled Reaction Rates. J. Colloid. Sci. 1949, 4, 425−437. (23) Gillespie, D. T. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys. 1976, 22, 403−434. (24) Gillespie, D. T. Stochastic Simulation of Chemical Kinetics. Annu. Rev. Phys. Chem. 2007, 58, 35−55. (25) Erban, R.; Chapman, S. J. Stochastic Modelling of ReactionDiffusion Processes: Algorithms for Bimolecular Reactions. Phys. Biol. 2009, 6, 046001. (26) Andrews, S. S.; Bray, D. Stochastic Simulation of Chemical Reactions with Spatial Resolution and Single Molecule Detail. Phys. Biol. 2004, 1, 3−4. (27) Mahmutovic, A.; Fange, D.; Berg, O. G.; Elf, J. Lost in Presumption: Stochastic Reactions in Spatial Models. Nat. Methods 2012, 9, 1163−1166. (28) Boghossian, A. A.; Zhang, J.; Le Floch-Yin, F. T.; Ulissi, Z. W.; Bojo, P.; Han, J.-H.; Kim, J.-H.; Arkalgud, J. R.; Reuel, N. F.; Braatz, R. D.; et al. The Chemical Dynamics of Nanosensors Capable of SingleMolecule Detection. J. Chem. Phys. 2011, 135, 084124. (29) Box, G. E. P.; Muller, M. E. A Note on the Generation of Random Normal Deviates. Ann. Math. Stat. 1958, 29, 610−611. (30) Squires, T. M.; Messinger, R. J.; Manalis, S. R. Making It Stick: Convection, Reaction and Diffusion in Surface-Based Biosensors. Nat. Biotechnol. 2008, 26, 417−426. (31) Sheehan, P. E.; Whitman, L. J. Detection Limits for Nanoscale Biosensors. Nano Lett. 2005, 5, 803−807. (32) Suzuki, K.; Hosokawa, K.; Maeda, M. Controlling the Number and Positions of Oligonucleotides on Gold Nanoparticle Surfaces. J. Am. Chem. Soc. 2009, 131, 7518−7519. (33) Wei, R.; Gatterdam, V.; Wieneke, R.; Tampe, R.; Rant, U. Stochastic Sensing of Proteins with Receptor-Modified Solid-State Nanopores. Nature Nanotechnol. 2012, 7, 257−263.

of the signal one could use an extreme sensitivity gradient (such as that offered by a cone) to make an informed guess on the total number of molecules bound to it by observing the distribution of the signal as in Figure 5, thus avoiding the need for passivation of low sensitivity regions, but rather exploiting the hot spot as well as the low sensitivity regions. Various technological advances have already been made that if combined together could finally provide a platform for multiplexed label-free single-molecule studies with ultrahigh temporal resolution.36−39 The optimization of sensors and setups will only be fully exploited by the adoption of a more informed statistical analysis of the signal40 regardless of the type of transduction mechanism at hand. Finally, we note that stochastic simulations of biomolecular interactions with ultrahigh spatial and temporal resolution are a versatile and expandable tool,41 which can provide insight and support in the design and data analysis of single-molecule sensing.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Mikael Käll for stimulating discussions and for initializing this project. We would like to acknowledge Alexander M. Berezhkovskii at NIH/CIT, Bethesda, Maryland, for lessons and insights on stochastic modeling of diffusioncontrolled reactions. In addition we thank S. Peter Apell for his valuable comments. This project was supported by the Swedish Foundation for Strategic Research through the project RMA11 Functional Electromagnetic Metamaterials for Optical Sensing. T.J.A. acknowledges support from the Polish National Science Center via the project 2012/07/D/ST3/02152.

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