We present the development of an analytical model that can be used for the rational design of a biosensor based on shifts in the local surface plasmon resonance (LSPR) of individual gold nanoparticles. The model relates the peak wavelength of light scattered by an individual plasmonic nanoparticle to the number of bound analyte molecules and provides an analytical formulation that predicts relevant figures-of-merit of the sensor such as the molecular detection limit (MDL) and dynamic range as a function of nanoparticle geometry and detection system parameters. The model calculates LSPR shifts for individual molecules bound by a nanorod, so that the MDL is defined as the smallest number of bound molecules that is measurable by the system, and the dynamic range is defined as the maximum number of molecules that can be detected by a single nanorod. This model is useful because it will allow

The unique optical properties of plasmonic nanoparticles have led to the development of a novel class of label-free biomolecular sensors. These biosensors exploit the phenomenon of localized surface plasmon resonance (LSPR) which endows unique light scattering and absorption properties to noble metal nanoparticles that are a function of the local environment. This behavior transduces binding events at the nanoparticle surface into a macroscopically measurable optical signal^{1−6} and forms the basis of LSPR biosensors. Herein, we report the derivation and experimental verification of an analytical model that predicts the figures-of-merit of nanoparticle LSPR biosensors operating at the single particle limit and provides guidance for the rational design of LSPR biosensors.

LSPR arises from the resonant oscillation of conduction electrons on the surface of metal nanoparticles. The energy associated with this resonance is a function of the nanoparticle composition, size, shape, and the surrounding dielectric environment.^{7−12} Noble metal nanoparticles with features in the 10−100 nm size scale exhibit plasmonic resonances at optical frequencies. Consequently, such nanoparticles exhibit a characteristic LSPR spectrum, with one or more peaks in the visible light range that correspond to specific electron oscillation resonances. The LSPR peak location and intensity are sensitive to the local refractive index (RI) surrounding the nanoparticle and are the basis of their utility as biosensors. Binding of analyte molecules to nanoparticles that are decorated with a receptor specific for that analyte alters the local RI in the vicinity of the plasmonic nanostructure, resulting in a shift of the LSPR spectrum. This shift can be measured either as a change in the peak intensity of the scattered light or as a shift in the LSPR peak wavelength (λ*).

This approach to biomolecular detection was initially proposed for plasmonic particles in suspension,(^{14,15} and was subsequently validated by other groups^{16−18} and extended to the limit of single nanoparticles.^{5,19−25} By miniaturizing the sensor down to a single nanoparticle, the detection system is reduced to the size-scale that is commensurate with the size of biomolecular analytes. To further understand the mechanics of single-nanoparticle detection, investigators have also been studying the specific details of plasmonic nanostructures that are responsible for their use in sensors such as the distance dependence of sensitivity^{26−28} and the resonant electric field enhancement distribution.^{29,30} However, to date, these developments have not been integrated into a coherent quantitative framework that allows the effect of these parameters on relevant figures-of-merit of biosensors, such as their limit of detection and dynamic range, to be predicted.

Motivated by the goal of rationally designing LSPR sensors, in this paper, we integrate recent progress in understanding the structural details of metal nanoparticles that control their plasmonic behavior with various measurement system parameters that impact measurement uncertainty into a quantitative model that is capable of predicting the response of a single, receptor-functionalized nanoparticle to discrete analyte binding events. The result is an analytical model that quantifies the LSPR shift of a gold nanorod caused by the local refractive index increase from the presence of a target biomolecule. The principle of this work is similar to that of Stemberg

The utility of this model is two-fold. First, it provides an analytical model that allows

As a first approximation, the MDL of nanoparticle sensors is determined by their composition, size, and shape of the NPs. Recognizing that the development of a generalized model that could account for the dependence of the MDL of an LSPR sensor on all of these parameters was likely to prove computationally intensive and possibly intractable, we focused instead on developing a model that was applicable to gold nanorods as plasmonic transducers of binding events for the following reasons. First, although silver particles are more sensitive than gold particles of the same shape and size,(^{34−36} Third, gold nanorods can be conveniently synthesized to exhibit plasmon resonances with peak wavelengths ranging from 700 to 900 nm and beyond simply by tuning their aspect ratio and size.^{34,37−40} This spectral region is particularly useful for biosensing because the background absorption and scattering of endogeneous chromophores from biological mixtures (^{41,42} We note that LSPR of several other geometries of gold nanoparticles such as nanoshells, nanodiscs,(^{43,44,46} which allows more accurate determination of peak shifts as described later in the model details. Additionally, the line widths of gold nanorod scattering spectra have been shown to be a function of the length and aspect ratio of the gold nanorods(

We sought to develop an analytical model that estimates the MDL of an LSPR sensor for a specific analyte−receptor pair based on the geometric dimensions of the gold nanorod using a specified spectral detection system. Thus, the model is an equation that predicts the minimum number of detectable analyte molecules based on input parameters that consist of the nanorod dimensions and optical system parameters. Building the model involved several steps; first, the spectral detection system and data analysis algorithms were analyzed to determine the measurement uncertainty in detecting LSPR peak wavelengths. Then, the average number of analyte molecules that must bind to induce an LSPR shift equal to the measurement uncertainty for a given nanorod geometry was determined. This is the smallest number of bound molecules that can be reliably be detected by the proposed system and is defined as its MDL. It is important to note that we define the MDL by the amount of material bound to the surface of the nanoparticle, and not the concentration of the analyte in the surrounding medium. This choice was made so that the focus of the model is the interaction of the bound analyte with the plasmonic nanoparticles and the subsequent LSPR signal generated, and not the mass transport kinetics of the sensor system. Therefore, the optimal nanoparticle can be determined for a proposed analyte strictly by using the model to predict which geometry will offer the lowest MDL.

The first step in the derivation of the model was estimation of the minimum LSPR shift that can be reliably measured for a particular detection system. We assumed that the total uncertainty is the sum of uncertainties induced by two factors as represented in _{system}, and uncertainty introduced by the peak fitting to the gathered nanoparticle scattering spectrum, _{fit}.

A thorough discussion of the _{system} of the microspectroscopy system utilized in this work has been reported elsewhere.(_{system} was ∼0.3 nm for the optical detection system described in that work. The same experimental setup was used in this study; hence we assume that _{system} is 0.3 nm.

Next, uncertainty due to the data analysis method must be considered. The measurement noise was modeled as a Gaussian distribution. Although the physical model of noise in an optical measurement is a Poisson distribution due to shot noise, it can be accurately and conveniently modeled by a Gaussian distribution for a large number of photons because it is merely a counting problem. This yields the following relationship for _{fit}_{d}, and read noise, _{r}, according to the following:

For a shot noise-limited measurement (SNR = √^{−1}·s^{−1} and read noise of less than 4 electrons RMS such that a shot noise-limited measurement (defined here as a measurement for which shot noise exceeds dark and read noise sources by an order of magnitude) will have an SNR > 13 for a typical integration time of 20 s. The role of the nanoparticle geometry in affecting measurement uncertainty becomes apparent as the signal level; that is, the amount of light scattered is proportional to the nanoparticle’s scattering cross section, _{sca}. Thus in the shot noise-limited regime, the following relationship holds for SNR:
_{sca}.(_{sca} predicted by Mie theory of the sphere, a value of 0.385 is determined for _{sca} may be calculated by rewriting

Now, with _{sca} of the particle, even if the measurement is not shot noise-limited. _{sca} can be determined analytically as a function of nanorod geometry by applying the model developed by Kuwata

The next step in the development of the model was to analytically translate this uncertainty limit from units of wavelength shift to number of bound molecules as a function of nanorod geometry. In order to formulate this relationship in a manner that is analytically simple, several approximations and assumptions were made. First, we assumed that the nanorods are cylindrical in shape with a length (^{48,52} We also assumed that the LSPR peak shift, Δλ_{LSPR}, resulting from a bound analyte is proportional to the total LSPR shift that is expected if the entire surrounding medium increased to the RI of the analyte. The proportionality constant is the ratio of the analyte volume to the total sensing volume of the nanorod. This assumption yields the following relationship:
_{S} is the total sensing volume of the nanorod, _{D} is the volume of the detected analyte, _{D}·ΔRI, where _{D} is the volume of analyte bound to the nanorod and ΔRI is the difference between the RI of the analyte and that of the surrounding medium. In the case where the approximate size of the analyte is known, the detection volume _{D} can be replaced with the product _{A} where _{A} is the volume of the analyte molecule and _{LSPR} yields an expression relating the measured peak shift of the LSPR spectrum to the optical detection mass:

Solving

In order to determine the MDL of the system, we replace Δλ_{LSPR} in _{M}, the MDL of the system in terms of the minimum detectable number of bound biomolecules.
_{M} is a function of the length _{M} is determined by several parameters on the right-hand side. Careful examination of these parameters is required to fully understand the dependence of each on the nanorod geometry and the effect each has on the overall MDL of the system. We note that _{M} is dependent upon the detection system employed for measuring the LSPR shifts of single nanoparticles because of the dependence on the peak measurement uncertainty _{M} is also a function of the spatially dependent RI sensitivity

In order to determine the function _{M} analytically, we next sought to define the parameters _{sca}) of gold nanorods as a function of nanorod geometry. This model provides the complete LSPR scattering spectrum of the nanorod with a simple analytical formula based on nanorod length and diameter. The model is based on fits to finite-difference time-domain with the depolarization factor calculated by electrostatic approximation.(_{sca} at the peak wavelength for nanorods of arbitrary length and diameter. A second analytical model, based on Gans’s extension of Mie theory that simulates LSPR spectra as a function of the nanorod aspect ratio,^{53−55} was also used to simulate spectra as a means of validation. These two models provided consistent results within the geometric range of nanorods studied (length 50−100 nm and diameter 15−50 nm). These predictions were then further verified experimentally by comparison to scattering spectra of individual, chemically synthesized nanorods as shown in

Comparison of simulated

It has been shown that LSPR shifts induced by local RI have a strong distance dependence as a result of the exponential decrease in field enhancement further from the nanoparticle surface.(^{29,30} These observations suggest that the location at which a target analyte binds to the nanorod (

With these assumptions clarified, we sought to generate an analytical function that describes _{S} and the decay length of electric field enhancement _{d} were also determined as a function of nanorod geometry. _{S} is defined as the fixed distance from the nanorod surface containing 95% of _{M}, of an arbitrary nanorod can be analytically estimated from the length and diameter of the nanorod.
_{S} is the sensing volume, _{A} is the analyte volume, ΔRI is the RI difference between the analyte and the surrounding medium, _{system} is the peak measurement uncertainty resulting from uncertainty in the physical detection of the LSPR peak, _{fit} is peak determination uncertainty due to data fitting, _{0} is the bulk RI sensitivity, _{d} is the decay length of the resonant electric field.

Right axis: experimental wavelength shift of a nanorod (63.3 ± 8.2 nm × 24.9 ± 4.9 nm) as a function of deposited polyelectrolyte thickness (red dots) and shifts predicted by

_{0}. We emphasize this point because much work in the field has been devoted to synthesizing nanoparticles with different geometries and compositions in order to optimize the RI sensitivity of the nanoparticle.^{45,58,59} Clearly, the nanoparticle bulk sensitivity is an important contributor to the overall MDL; however, it is not necessarily the most important. In fact, Miller ^{60,61} The parameter _{sca} can be measured more accurately due to their increased scattering signal as seen in

_{sca} that cannot produce enough scattered light to be visualized in the dark-field microscopy setup. For this reason, no data are shown for nanorods with LSPR peaks outside of this range or with SNR below the cutoff of 30. We assumed the streptavidin molecules have a volume(^{3} and RI of 1.57.(

(A) Calculated molecular limits of detection for gold nanorods of arbitrary dimension based on

In addition to the absolute MDL, the dynamic range (DR) is an important figure-of-merit (FOM) of a biomolecular sensor. For the purposes of this model, the DR is defined as the theoretical maximum number of analyte molecules that are detectable by a single nanorod. This definition was chosen because it is consistent with the model output, which is quantified as the number of bound molecules. The DR was determined by calculating the total surface area available for binding and dividing it by the footprint of a bound analyte molecule. This value was then scaled by a factor of 0.9, which assumes a hexagonal packing density of hard spheres yielding the highest possible coverage that could be achieved in practice. By this definition, a larger nanorod will obviously exhibit a higher DR because of its larger surface area. However, larger nanorods also tend to have higher MDLs because of the increased sensing volume. To balance these considerations, we calculate a composite FOM for nanorods that includes both the DR and MDL of these nanorod plasmonic sensors operating in the single nanorod mode. The composite FOM is calculated simply as the ratio of the DR

To experimentally test the results provided by the model, gold nanorods were synthesized with dimensions that were as close to the geometry predicted by

Left: TEM of rods used in this study. Nanorod length is measured to be 63.3 ± 8.2 nm and diameter 24.9 ± 4.9 nm (

A dose−response curve was determined by incubating identical samples of biotin-functionalized gold nanorods that were chemisorbed on glass slides in streptavidin solutions that spanned a range of protein concentration.

To investigate the relationship between the experimental detection limit measured by analyte concentration and the MDL predicted by

Fits to scattering spectra of a single nanorod in water (blue), after biotin conjugation (green), and subsequently in 100 pM streptavidin (red). The 0.52 nm shift corresponds to approximately 27 streptavidin molecules by

A 0.52 nm shift was observed upon incubation of the biotin-functionalized gold nanorods in 100 pM streptavidin, which translates to ∼27 streptavidin molecules bound by a single nanorod, by use of

As a further check of the relevance of the MDLs predicted by our model, we observed that the mean LSPR shift at saturation is 5.4 nm (^{2} of biotin-activated surface is available for the binding of streptavidin molecules. Each streptavidin molecule is a tetramer of four identical biotin binding subunits, and the entire tetramer is roughly ellipsoidal with axes of 5.4 nm × 5.8 nm × 4.8 nm.^{65,66} Thus, the footprint that a streptavidin molecule would occupy on the binding surface is approximately 25 nm^{2}. In this orientation, a maximum of only 320 molecules would be expected to fit on the nanorod surface. Approximating the streptavidin molecules as hard spheres, the model of random sequential adsorption(

Additionally, we compare the mass of streptavidin presumed bound at saturation by ^{2} have been experimentally determined for streptavidin layers formed over biotin layers on flat substrates.^{68,69} These binding densities yield an estimated ∼220 bound streptavidin molecules to the available nanorod binding area of 8000 nm^{2}. Again, these estimates are slightly lower than the 280 molecules estimated from

We systematically explored the parameters in

The product _{A}

The total sensing volume _{S} of the nanoparticle is determined by the nanoparticle shape and decay length _{d}. Smaller _{S} and _{d} are indicative of nanostructures with small, intense electric field enhancements. In general, the sensing volume will vary proportionally with the cube of the decay length simply because it is a three-dimensional volume subtended by the decay length. In the case of rods, smaller rods have shorter decay lengths. So ideally, one would want to use the smallest nanorods possible. However, smaller rods also exhibit smaller scattering cross sections _{sca}, which means they scatter less light under dark-field illumination. For rods in particular, _{sca} varies approximately as the square of the rod length and proportionately to rod diameter.(_{sca} would be reduced by about a factor of 8. On the other hand, to maintain an adequate SNR of the collected spectra, the intensity of the illumination source must be increased. An upper limit to this intensity exists because at some point the nanorods will melt due to photothermal effects.^{70,71} The use of a broadband illumination source, such as a supercontinuum fiber laser, would allow the complete scattering spectrum to be collected and would provide a light output that is 6−10-fold more intense than the tungsten filament used in the experimental setup used here. Hence,

The decay length of the electric field enhancement _{d} is determined by the geometry of the nanoparticles. ^{30,55,72} Thus, analyte molecules that bind near the end of the nanorod will cause a greater LSPR shift than those binding on the lateral portion. For this reason,

The binding distance _{d}^{−1}). For the case of nanorods with decay length _{d} generally on the order of 10−20 nm, this results in a factor of 1.1 improvement (

The bulk RI sensitivity of a nanostructure _{0} is dependent on its size, shape, and material composition. However, Miller and Lazarides have observed that _{0} is correlated with the LSPR peak wavelength, regardless of nanoparticle shape.^{60,61} Thus, for a nanostructure of known material composition, its bulk RI sensitivity can be predicted simply by characterizing its LSPR peak. Because longer wavelength resonances exhibit higher bulk RI sensitivities, the highest sensitivities will be from particles with LSPR peaks in the red end of the spectrum. Assuming the detection is to remain in the visible light spectrum, LSPR peaks at wavelengths up to 800 nm could be measured. Thus, bulk RI sensitivities as high as 600 nm/RIU could be expected. This is more than a two-fold improvement over the 260 nm/RIU sensitivity of nanorods used in this study. However, it is worth noting that, for the case of nanorods, longer resonances correspond to larger rods with larger sensing volumes and longer decay lengths.

The peak measurement uncertainties _{fit} and _{system} represent the most direct way at improving sensor detection limits. Obviously, the smaller the wavelength shifts that can be reliably detected, the greater the accuracy and lower the overall MDL will be. For the system used in these experiments, _{fit} was found to be approximately 0.02 nm while _{system} was found to be 0.3 nm. Thus, the contribution to MDL is dominated by _{system}, whereas _{fit} acts more as an absolute limit for noise levels as discussed above. We performed an in-depth analysis of the contributing factors to _{system} and found that a large portion of the uncertainty is the result of physical system stability.(_{system} can be reduced to the order of 0.005 nm.(_{fit} resulting in a 15-fold decrease in MDL.

These possible enhancement factors are displayed in

technique summary | variable from eq | potential enhancement factor |
---|---|---|

drying | ΔRI | 2.4 |

brighter illumination source | _{sca}, _{S}, and _{d} | 7 |

shorter binding moiety | 1.1 | |

LSPR peaks in IR | _{0} | 2 |

silver nanoparticles | _{0} | 1.5 |

high spectral resolution detection system | _{system} | 15 |

total | ∼800 |

In conclusion, we have presented a simple mathematical model that analytically relates physical detection parameters of a single nanoparticle LSPR sensor to the minimum number of detectable analyte molecules. The utility of this model is two-fold. It can be used to select the optimum nanoparticle geometry for a desired detection system completely analytically, forgoing otherwise necessary comprehensive, experimental characterization. The minimum number of detectable molecules can be estimated as well as the number of molecules detected at saturation, which provides insight into the dynamic range of the system. Additionally, the model provides a framework through which the effects of potential system improvements can be assessed. Analysis of the theoretical limits of the dependent variables of

Hydrogen tetrachloroaurate trihydrate (HAuCl_{4}), sodium borohydride, ascorbic acid, phosphate buffered saline (PBS) tablets, poly(allylamine hydrochloride) (PAH), poly(4-styrenesulfonate, sodium salt) (PSS), silver nitrate, and mercaptohexadecanoic acid (MHA) were purchased from Sigma. Glass coverslips, ethanol, methanol, sodium chloride (NaCl), and hydrochloric acid (HCl) were purchased from VWR. Cetyltrimethylammonium bromide (CTAB) was purchased from Fluka. (+)-Biotinyl-3,6,9-trioxaundecanediamine (biotin−amine), 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC), _{3}SH) was purchased from Prochimia. Distilled water purified by a reverse-osmosis filtration system (18 MΩ·cm, PureFlow Inc.) was used for all experiments.

Gold nanorods were chemically synthesized by a seed-mediated growth procedure^{34,76} similar to that described previously.^{19,36} Spherical gold seed particles were prepared as follows: to a mixture of 7.5 mL of 0.1 M CTAB in water and 0.250 mL of 0.01 M HAuCl_{4} was added under vigorous stirring 0.6 mL of chilled 0.01 M NaBH_{4}. The brown suspension was then gently heated and stirred for a few minutes. Gold nanorods were synthesized in a water bath at 29 °C as follows. To 95 mL of 0.1 M CTAB in water were added 0.6 mL of silver nitrate and 0.64 mL of 0.1 M ascorbic acid. The mixture was swirled after the addition of each reagent to ensure mixing. Sixty microliters of gold seed particles was added, and the mixture was inverted and allowed to sit overnight, resulting in a purple-colored suspension of gold nanorods. Excess CTAB was removed from the gold rod suspension by centrifugation twice at 4500 rpm for 30 min. The gold nanorods were resuspended in water to a total volume of 10 mL and stored at room temperature until further use.

Number 1 thickness rectangular 9 × 26 mm coverglasses were cleaned ^{14,77,78} Between each polymer incubation step, an extinction spectrum was collected with a Varian Cary 300Bio UV−visible spectrophotometer. In this manner, plots were generated of λ

Scattering spectra were collected from individual nanorods using the Zeiss Axiovert 200 (Plan Neofluor 100X objective) dark-field illumination microspectroscopy system described previously.^{19,50} Scattering spectra of individual nanorods were collected by imaging the microscope field of view to a line-imaging spectrometer (Acton Research SpectraPro 2150i). Streptavidin detection experiments with single gold nanorods were also performed as previously reported.(_{3}SH and 0.5 mM MHA. The EG_{3}SH molecules of the mixed monolayer serve two purposes. They prevent nonspecific adsorption as well as modulate the surface density of the MHA molecules and hence of biotin that is subsequently chemically conjugated to the −COOH groups. Biotin was conjugated to the nanorods by incubating the SAM-functionalized nanorods in an aqueous solution of 0.4 M EDC and 0.1 M NHS for 7 min to convert the COOH groups to NHS esters. The chips were rinsed with water and were then immediately incubated in biotin−amine for 2 h and rinsed with water again. Streptavidin binding was performed by incubating the biotin-functionalized nanorods with a continuous flow of a streptavidin solution in PBS for 2 h. LSPR shifts were measured as the difference between the λ* of fits to the measured scattering spectra. To ensure the observed LSPR shifts resulted from specific interaction between biotin and streptavidin, controls for the binding experiments were performed by incubating biotin-functionalized gold nanorods in streptavidin solution that had been presaturated with excess (1 mM) free biotin.(

This work was supported by a grant from the Centers for Disease Control (NCID; R01 CI-00097) to A.C. G.J.N. acknowledges the support of a graduate fellowship from the NIH through a Biotechnology training grant (GM8555) awarded to the Center of Biomolecular and Tissue Engineering at Duke University. The authors also wish to thank the anonymous reviewers for their constructive suggestions that, we believe, helped improve this paper.

An in-depth derivation of the distance dependence of nanoparticle sensitivity to local refractive index is available. This material is available free of charge

nn8006465_si_001.pdf