PAME: plasmonic assay modeling environment

Substrate types

PAME supports two substrates: optical fibers and chips. Substrates mediate the interaction between light and the multilayer stack through a weighting function, $\sum _i^{N}f\left({\theta }_i\right)$, where θ_i corresponds to the angle of the ith incident light ray onto the substrate. The chip is meant to describe simple configurations, for example a gold film deposited on a glass slide and illuminated from below at a single angle, θ_o. In this case, $\sum _i^{N}f\left({\theta }_i\right)=\delta \left({\theta }_o\right)$. For optical fibers, the propagation modes are determined by properties of the fiber itself, such as its numerical aperture, core and cladding materials, and its ability to maintain polarization states. Furthermore, the placement of the multilayer on different regions of the fiber has a significant effect on f(θ_i), and hence on the optical response of the sensor. The two most common orientations, either transversally³ along the propagation direction on the fiber, or axially on the cleaved fiber endface, are shown in Fig. 2. Both of these orientations have been realized as biosensors (Lipoprotein et al., 2012; Shrivastav, Mishra & Gupta, 2015), with the axial configuration, often referred to as a “dip sensor” because the endface is dipped into the sample, appearing more often in recent years (Mitsui, Handa & Kajikawa, 2004; Wan et al., 2010; Jeong et al., 2012; Sciacca & Monro, 2014). PAME does not presently support multilayers along bent regions of the fiber, or along sharpened optical tips (Issa & Guckenberger, 2006; Library et al., 2013) and assumes all rays to be plane waves. For advanced waveguide design and modal analysis, we recommend Lumerical™ MODE solutions.

PAME’s Substrate interface queries users to configure chip and optical fiber parameters, rather than working directly with f(θ_i). Users can choose between multilayer orientation, polarization state (P, S or unpolarized), and range of angles, all from which PAME builds f(θ). The interface is user-friendly, and attempts to obviate incompatible or unphysical settings. For instance, the ellipsometric amplitude (Ψ) and phase (δ) depend on ratios of P-polarized to S-polarized light reflectance, but users may opt to only compute S-waves, resulting in errant calculations downstream. Anticipating this, PAME provides an ellipsometry mode, which when enabled, prevents the polarization state from being changed. By combining substrate types with context modes, PAME provides a simple interface for modeling a number of common optical setups.

Multilayer stack

Figure 2 depicts the multilayer stack, in which each dielectric slab is assumed to be homogenous, and of uniform thickness; heterogeneous materials must be homogenized through an effective medium theory (EMT). Furthermore, the multilayer model presumes that layers are connected by smooth and abrupt boundaries to satisfy Fresnel’s equations. The first and last layers, conventionally referred to as “substrate” and “solvent,” are assumed to be semi-infinite, with incident light originating in the substrate. The treatment of anisotropic layers without effective medium approximations is discussed in the Future Improvement section.

A light ray incident on the stack at angle θ, as set by f(θ), will reflect, refract and absorb, in accordance with Fresnel’s equations. For example, in a simple 2-layer system, the light reflectance, Γ(λ), at the boundary between n₁, n₂ is (1)${\displaystyle R=\frac{1}{2}\left(r_s+r_p\right)}$ (2)${\displaystyle R=\frac{1}{2}\left({\left|\frac{n_{1}cos{\theta }_i-n_{2}cos{\theta }_t}{n_{1}cos{\theta }_i+n_{2}cos{\theta }_t}\right|}^2+{\left|\frac{n_{1}cos{\theta }_t-n_{2}cos{\theta }_i}{n_{1}cos{\theta }_t+n_{2}cos{\theta }_i}\right|}^2\right),}$ where θ_i, θ_t are the angles of incidence upon, and transmission into n₂ from n₁, and r_s and r_p are the complex reflection coefficients of the S and P-polarized light. For N-layers, Fresnel’s equations are solved recursively using the transfer matrix method(TMM), also referred to as the recursive Rouard method (Rouard, 1937; Lecaruyer et al., 2006). In addition to the reflectance, transmittance and absorbance, a variety of optical quantities are computed in the multilayer, including the Poynting vector, the complex wave vector angle, ellipsometric parameters, and film color. For a thorough treatment of light propagation in multilayer structures, see Orfanidis (2008) and Steed (2013). PAME offers a simple tabular interface for adding, removing, and editing materials in arbitrarily many layers, as shown in Fig. 2C. PAME delegates the actual TMM calculation to an adapted version of the Python package, tmm (Byrnes, 2012).

PAME material classes

PAME includes three material categories: bulk materials, composite materials and nanomaterials. A bulk material such as a gold film is sufficiently characterized by its index of refraction. The optical properties of a gold nanoparticle, however, depend on the index of gold, particle size, the surrounding medium, a particle-medium mixing model, and other parameters. A nanoparticle with a shell is even more intricate. PAME encapsulates a rich hierarchy of materials in an object-oriented framework to ensure compatibility with the multilayer stack and interactive plotting interface.

Bulk material

In PAME, a “bulk” material refers to a single, homogeneous substance, fully characterized by its complex index of refraction, $\tilde{n}=n+i\kappa$, or dielectric function, $\tilde{e}=e+iϵ$, which are related through a complex root $\left(\tilde{e}=\sqrt{\tilde{n}}\right)$, which gives the relations ${\displaystyle e=n^2-{\kappa }^{2}n=\sqrt{\frac{\sqrt{e^2+{ϵ}^2}+e}{2}}}$ ${\displaystyle ϵ=2n\kappa \kappa =\sqrt{\frac{\sqrt{e^2+{ϵ}^2}-e}{2}}.}$ Here n and e, and optical quantities derived from them, are understood to be dispersive functions of wavelengths, n(λ), e(λ). The refractive index n is assumed to be independent of temperature and non-magnetic at optical frequencies⁴. The index of refraction of bulk materials is obtained through experimental measurements, modeling, or through a combination of the two; for example, measuring n at several wavelengths, and fitting to a dispersion model such as the Sellmeier equation, (3)${\displaystyle n\left(\lambda \right)=\sqrt{1+\frac{A_1{\lambda }^2}{{\lambda }^2-B_1^2}+\frac{A_2{\lambda }^2}{{\lambda }^2-B_2^2}+\frac{A_3{\lambda }^2}{{\lambda }^2-B_3^2}}+\cdots .}$ PAME is bundled with several dispersion models, including the Cauchy, Drude and Sellmeier relations, as well as two freely-available⁵ refractive index catalogs: Sopra (2008) and RefractiveIndex.INFO (Polyanskiy, 2015), comprising over 1,600 refractive index files. PAME includes a materials adapter to browse and upload materials as shown in Fig. 3. Selected materials are automatically converted, interpolated, and expressed in the working spectral unit (nm, eV, cm, …) and range. PAME’s plots respond to changes in material parameters in real time.

Nanomaterials

In PAME, nanomaterials are treated as a special instance of a composite material⁶ whose properties depend on a core material, a medium material, possibly an intermediate shell material, and particle size. A key distinction between nanomaterials and their bulk counterparts is that the optical properties of nanoparticles are highly sensitive to both the particle size and the permitivity of the surrounding medium. The implicit optical properties of spheroidal nanoparticles, such a extinction cross section, σ_ext, are solved analytically through Mie Theory (Bohren, 1983; Jain et al., 2006). This is a fundamentally important quantity, as the position and shift in the extinction cross section maximum, known as the localized plasmon resonance (Willets & Van Duyne, 2007; Anker et al., 2008), is often the best indicator of the state of the nanoparticles comprising the system. While the full solution to the extinction cross section is described by the sum of an infinite series of Ricatti–Bessel functions, described in full in Lopatynskyi et al. (2011), for brevity consider the approximate expression (Jeong et al., 2012; Van de Hulst, 1981): (4)${\displaystyle {\sigma }_{\text{ext}}\approx \underset{\approx {\sigma }_{\text{scatt}}}{\underbrace{\frac{128{\pi }^5}{3{\lambda }^4}{R}^{6}Im{\left[\frac{m^2-1}{m^2+2}\right]}^2}}\underset{\approx {\sigma }_{\text{abs}}}{\underbrace{-\frac{8{\pi }^2}{\lambda }{R}^3\left[\frac{m^2-1}{m^2+2}\right]}},}$ where m is the ratio of the refractive index of the core particle material to that of the suspension medium (e.g., gold to water). The polynomial dependence of R, λ and m demonstrate the high variability in the optical properties of nanoparticles, and by extension, the biosensors that utilize them. Typical cross sections from silver and gold nanospheres are depicted in Fig. 4.

While optical constants derived from Mie Theory are computed analytically, it is important to recognize that in a dielectric slab, nanomaterials are represented by an effective dielectric function; thus, optical constants like transmittance or reflectance will be computed using an effective dielectric function representing the nanoparticle layer. PAME currently supports nanospheres and nanospheres with shells; planned support for exotic particle morphologies is described in the Future Improvements section. Similar treatments of nanoparticle layers with effective media approximations have been successful (Li et al., 2006; Liu et al., 2011), even for non-spherical particles, and for ensembles of different sized particles (Battie et al., 2014). This is a salient difference between PAME, and numerical approaches like the boundary element method(BEM), discrete dipole approximation(DDA) and finite-difference time-domain(FDTD): PAME relies on mixing theories, and hence is constrained by any underlying assumptions of the mixing model. For a more in-depth discussion on nanoparticle modeling, see Myroshnychenko et al. (2008) and Trügler (2011).

Simulation and data analysis

PAME’s interactivity makes it ideal for exploring the relationships between system variables, while the simulation environment provides the means to systematically increment a parameter and record the correspond response; for example, incrementing the fill fraction of inclusions in a nanoparticle shell to simulate protein binding, or incrementing the refractive index of the solvent to simulate a refractometer. Because most updates in PAME are automatically triggered, simulations amount to incrementing parameters in a loop and storing and plotting the results.

PAME’s simulation interface simplifies the process of setting simulation variables and storing results. It is comprised of three tabs: a Selection tab (Fig. 5) for setting simulation parameters and value ranges. Variable names like layer1.d refers to the thickness of the first layer in the multilayer stack, and material2.shell_thickness is the size of the nanoparticle shell in nanometers of material2. PAME provides suggestions, documentation and a tree viewer to choose simulation variables, and alerts users to errant inputs, or invalid ranges; for example, if users try to simulate a volume fraction beyond its valid range of 0.0 to 1.0. The Notes/IO tab provides a place to record notes on the simulation, and configure the output directory. PAME can store all of its state variables in every cycle of the simulation, including the entire multilayer structure, and all computed optical quantities, but this can lead to storing large quantities of redundant data. The Storage tab lets users pick and choose their storage preference, and even specify the quantities that should be regarded as “primary” for easy access when parsing.

PAME provides a SimParser object to interact with saved simulations, which while not required, is intended to be used inside an IPython Notebook (Perez & Granger, 2007) environment. The SimParser stores primary results in Pandas (McKinney & Millman, 2010) and scikit-spectra (Hughes & Liu, 2014) objects for easy interaction and visualization, and the remaining results are stored in JSON. This allows for immediate analysis of the most important simulation results, with the remaining data easily accessed later through a tree viewer and other SimParser utilities.

Case 1: refractometer

Plasmonic sensors respond to changes in their surrounding dielectric environments, and are commonly utilized as refractometers (Mitsui, Handa & Kajikawa, 2004; Punjabi & Mukherji, 2014), even going so far as to measure the refractive index of a single fibroblast cell (Lee et al., 2008). Refractive index measurements can also be used to measure sensitivity and linear operating ranges. A common approach is to immerse the sensor in a medium such as water, and incrementally change the index of refraction of the medium by mixing in glycerine or sucrose. Because the index of refraction as a function of glycerine concentration is well-known (Glycerine Producers’ Association , 1963), sensor response can be expressed in refractive index units (RIU). This is usually taken a step further in biosensor designs, where the RIUs are calibrated to underlying biophysical processes (e.g., protein absorption), either through modeling as PAME does, or through orthogonal experimental techniques such as Fourier Transform Infrared Spectroscopy (Tsai et al., 2011). This calibration process has been described previously (Jeong & Lee, 2011; Richard & Anna, 2008), and is usually carried through in commercial plasmonic sensors. This quantifies the analyte binding capacity of the sensor, an important parameter for assessing binding models⁷, non-equilibrium sensing, and performing one-step measurements, for example estimating the glucose levels in a blood sample. As a first use case, PAME is used to calibrate sensor response to increasing concentrations of glycerine for an axial fiber comprised of a 24 nm layer of gold nanoparticles.

A dip sensor was constructed using a protocol and optical setup similar to that of Mitsui, Handa & Kajikawa (2004). In brief, optical fiber probes were cleaved, submerged in boiling piranha (3:1 H₂SO₄ : H₂O₂), functionalized with 0.001% (3-Aminopropyl) trimethoxysilane for 60 min in anhydrous ethanol under sonication, dryed in an oven at 120 °C, and coated with 24 nm AuNPs to a coverage of about 40 ± 5%, as verified by SEM imaging (Hughes et al., 2015). The fiber was submerged in 2 mL of distilled water under constant stirring, and glycerine droplets were added incrementally until the final glycerine concentration was 32%, with each drop resulting in a stepwise increase in the reflectance as shown in Figs. 6A and 6C). This system was simulated using the stack described in Fig. 1, where the organosilanes were modeled as a 2 nm-thick layer of a Sellmeier material (Eq. (4)), with coefficients: A₁ = 6.9, A₂ = 3.2, A₃ = 0.89, B₁ = 1.6, B₂ = 0.0, B₃ = 50.0. These coefficients led to excellent agreement between experiment and simulation during the self-assembly process of the AuNP film.

Figures 6C and 6D shows the strong agreement between measured and simulated response to increasing glycerine, and PAME is able to show the reflectance spectrum free of the influence of the LED light source in the dataset(b,a). It is clear that while the nanoparticle’s reflectance is prominent around λ_max ≈ 560 nm, the combination of both an increase in reflectance, and a blue-shift of spectral weight yield a 485 nm peak in the normalized reflectance spectrum. Neither is indicative of the free-solution plasmon resonance peak at λ_max ≈ 528 nm, and maintaining a correspondence between the reflectance centroid and plasmon resonance can lead to misinterpretation. Furthermore, the shape of this glycerine response profile is very sensitive to parameters like organosilane layer thickness and nanoparticle size and coverage, and by fitting to the simulated response, one may then estimate these parameters which are otherwise difficult to measure. This simple example provides valuable insights into the relationship between glycerine concentration and reflectance on a dip sensor.

Case 2: multiplexed Ag–Au sensor

Sciacca & Monro (2014) recently published a multiplexed biosensor in which both gold and silver nanoparticles were deposited on the endface of a dip sensor and their reflectance was monitored simultaneously. In their experiment, the gold colloids were functionalized with anti-apoE, the antibody to an overexpressed gastric cancer biomarker, apoE. The silver colloids were functionalized with a non-specific antibody. The authors showed that the plasmon resonance peak of the gold particles shifted appreciably in response to apoE, while the silver did not. Furthermore, the gold peak did not respond to CLU, an underexpressed gastric cancer biomarker while the uncoated silver particles did, presumably due to non-specific binding. In effect, the multiplexed sensor provides a built-in negative control and can identify specific events more robustly. The ability to multiplex two or more colloids to a single sensor has great potential. To gain insights into this system, the sensor’s response to a mid-sized protein like NeutrAvidin (60 kDa) was simulated⁸. To our knowledge, this is the first attempt at modeling a multiplexed dip sensor containing two nanoparticle species.

A dip sensor was configured in PAME, composed of a 2.5 nm thick layer of organosilanes⁹ and a 92 nm layer of mixed protein-coated nanoparticles in water. A 3-layer composite material model was used to represent the mixed nanoparticles. Materials 1 and 2 were set to 80 nm AuNPs and 60 nm AgNPs, respectively, using the dielectric functions described in Fig. 4; material 3 was set to water. Au–Au effects and Ag–Ag effects are taken into account, but PAME’s 2-phase EMTs cannot account for Au–Ag effects (3-phase and N-phase EMTs (Luo, 1997; Zhdanov, 2008) will be implemented in upcoming releases). Therefore, the combined layer, ${\tilde{n}}_{\mathrm{AuAg}}$, is weighted in proportion to the fill fraction as, ${\tilde{n}}_{\mathrm{AuAg}}={\varphi }_1{\tilde{n}}_{\mathrm{Au}}+{\varphi }_2{\tilde{n}}_{\mathrm{Ag}}$. Nanoparticle coverage was chosen so as to produce a large reflectance, with approximately equal contributions from gold and silver; the actual coverage used in Sciacca & Monro (2014) is not stated. Ultimately, 45.56% of the surface sites were covered in gold, and 18.64% in silver. NeutrAvidin was modeled as a 6 nm sphere (Tsortos et al., 2008) of dispersive refractive index, n ≈ 1.5 (Sarid & Challener, 2010), filling a 6 nm-wide shell on the nanoparticles from 0 to 95% coverage (617 proteins per AuNP and 363 per AgNP), as shown in Fig. 7.

Despite using several approximations, the simulation provides many insights into multiplexed sensors. First, the 60 nm AgNPs reflect much more efficiently, despite AgNPs and AuNPs having nearly identical extinction cross sections (Fig. 4). This is because silver particles are more efficient scatterers (Lee & El-Sayed, 2006), and reflectance depends exponentially on scattering cross section (Quinten, 2011). Therefore, reflectance sensors composed of highly-scattering particles can utilize sparse nanoparticle films, which are less susceptible to aggregation (Scarpettini & Bragas, 2010) and electrostatic and avidity effects. Secondly, the normalized response to protein binding is about 0.08 units for silver, and 0.06 for gold; however, there are 2.33 more proteins on gold than silver. Therefore, considering the response per molecule, 60 nm silver spheres are 3.12 time more sensitive to protein binding than 80 nm gold spheres. Experiments have confirmed similar three-fold enhancement to protein-induced plasmon resonance shifts in aqueous solutions of AuNPs and AgNPs (Sun & Xia, 2002; Mayer, Hafner & Antigen, 2011). This suggests a correspondence between shifts measured in free solution and the reflectance in optical fibers, despite little similarity in the qualitative profile of the response. Nusz et al. (2009) has suggested a figure of merit to objectively compare shifts vs. intensity responses.

Finally, if response is partitioned into two spectral regions, such that 385 nm < λ ≤ 500 nm corresponds to silver, and λ ≥ 500 nm to gold, then Fig. 7 illuminates an important result: despite a clear separation in the peaks of the reflectance spectra, the response to NeutrAvidin spans both partitions. For example, in Fig. 7C the gold region (λ ≥ 550 nm) clearly responds to proteins binding to silver nanoparticles. This could lead to misinterpretations; for example, the response at λ ≈ 570 nm could be misattributed to non-specific binding onto gold, when in fact there is only binding to silver. In Sciacca & Monro (2014), both the gold and silver spectral regions responded to apoE, when only gold is coated with anti-apoE. While the signal in the silver region could be due to non-specific interactions between the anti-apoE and AgNPs, these simulations show that it could simply be due to spectral overlap in the gold and silver response, the extent of which depends on the dielectric function of the protein, the Au–Ag coupling and other factors.