Optimally designed nanolayered metal-dielectricparticles as probes for massively multiplexedand ultrasensitive molecular assaysAnil K. Kodalia,b,c, Xavier Llorad, and Rohit Bhargavab,c,e,1

aDepartment of Mechanical Science and Engineering, bBeckman Institute for Advanced Science and Technology, cMicro and Nanotechnology Laboratory,dNational Center for Supercomputing Applications, and eDepartment of Bioengineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Edited* by Karl Hess, Beckman Institute, Urbana, IL, and approved June 14, 2010 (received for review March 31, 2010)

An outstanding challenge in biomedical sciences is to devise apalette of molecular probes that can enable simultaneous andquantitative imaging of tens to hundreds of species down to ultra-low concentrations. Addressing this need using surface-enhancedRaman scattering-based probes is potentially possible. Here, wetheorize a rational design and optimization strategy to obtain re-producible probes using nanospheres with alternating metal andreporter-filled dielectric layers. The isolation of reporter moleculesfrom metal surfaces suppresses chemical enhancement, and conse-quently signal enhancements are determined by electromagneticeffects alone. This strategy synergistically couples interstitialsurface plasmons and permits the use of almost any molecule asa reporter by eliminating the need for surface attachment. Geneticalgorithms are employed to optimize the layer dimensions to pro-vide controllable enhancements exceeding 11 orders of magnitudeand of single molecule sensitivity for nonresonant and resonantreporters, respectively. The strategy also provides several other op-portunities, including a facile route to tuning the response of thesestructures to be spectrally flat and localization of the enhancementwithin a specific volume inside or outside the probe. The spectrallyuniform enhancement for multiple excitation wavelengths and fordifferent shifts enables generalized probes, wheras enhancementtuning permits a large dynamic range by suppression of enhance-ments from outside the probe. Combined, these theoretical calcu-lations open the door for a set of reproducible and robust probeswith controlled sensitivity for molecular sensing over a concentra-tion range of over 20 orders of magnitude.

Raman spectroscopy ∣ surface enhanced ∣ chemical imaging ∣vibrational spectroscopy

Surface-enhanced Raman scattering (1) (SERS)-based probes,consisting of nanostructured particles, are strongly emerging

for biomedical applications. SERS-based probes (2, 3) are excep-tionally attractive as they offer quantitative enhancement of sig-nal with facile readout (4, 5), extensively multiplexed imaging (6),and ultrasensitive assays (7, 8)—but not all at the same time. TheSERS effect is typically prominent in nanoscale metal-dielectricenvironments in which the signal of a proximal organic moleculecan be rationally tailored (9) and enhanced to the extent thatsingle molecules may be detected (10, 11). Hence, SERS probestypically contain nanoscale metallic structures and organic mole-cules (12) that act as a quantitative reporter for the presence ofthe probe. The signal of this reporter is greatly enhanced to trans-duce biochemical species of low concentration at the molecular(13), cellular (14), and tissue levels (15, 16) to measurable signals.The achieved enhancement depends on the reporters’ molecularcharacteristics as well as nanoscale size, shape, geometry, localaggregation state, and surface characteristics of the metal. Theseparameters can potentially be controlled to tune the reporters’signal, especially to maximize sensitivity of detection. Controllingand tuning the enhancement of the reporter’s signal, however, is amajor ongoing challenge. Given the large number of factorsinfluencing enhancement, designing particles for specific en-

hancement levels remains an active theoretical challenge whilesimultaneously controlling variability in their response remainsa practical hurdle. Both aspects are actually closely linked. Varia-bility in SERS signal arises due to the synergistic effects of themetal’s atomic mobility (17), surface reorganization (18), and thereporter’s surface mobility. The net result is an unpredictable var-iation in enhancement, including blinking (19) or “hot spots”(20). Indeed, the intractability of controlling enhancement hasled to theorization of a “SERS-uncertainty principle” (21) anda practical choice between (a) unstructured colloids providingan exceptionally large (22, 23) but uncontrolled enhancement[up to single molecule level (24)] and uncertain spatial localiza-tion or (b) well-defined and controlled probes using self-assem-bling monolayer reporters (25) but of lower enhancement andlimited reporter diversity.

Variability in signal intensities (19) of a nanoparticle popula-tion (26) can be reduced, for example, by using self-assembledmonolayers, whereas resonant dyes (27, 28) can be employedfor ultrasensitive applications. These specialized remedies, how-ever, are not conducive to designing a palette of reliable probesfor high throughput and ultrasensitive multiplexing. Although theutility of controlled nanostructures on making the SERS effectusable (29) is now undisputed, a rational framework to designgeneralized SERS probes for a desired enhancement level,spectral selectivity, and size is lacking. In turn, this limits ourcapability to design assays in terms of both multiplexing and sen-sitivity based on theoretical models. In modeling probes to tailorSERS enhancement, two mechanisms—electromagnetic andchemical—are generally invoked. Electromagnetic enhancement(EE) is now well understood (30) and is usually larger comparedto the chemical enhancement (CE) (31). CE mechanisms con-tinue to be a subject of much research (32). Since the EE is larger,it is beneficial to focus on it in designing optimal structures. Ingeneral, high EE is observed for reporters confined betweenmetal domains (33). It has been suggested that alternating dielec-tric and metal shells around a metal sphere could lead to a highEE effect at the core surface (34), and alternating metal and di-electric shells around a dielectric core could lead to a high EEeffect at the shell surfaces (35). The potential of such multishellconstruct for designing and controlling SERS for sensing, how-ever, has been surprisingly underappreciated though it hasreceived attention as a method to tune scattering and plasmon-related absorption (36). Multishell spheres can be considered to

Author contributions: A.K.K. and R.B. designed research; A.K.K. performed research;A.K.K. and X.L. contributed new reagents/analytic tools; A.K.K. and R.B. analyzed data;and A.K.K. and R.B. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1To whom correspondence should be addressed at: Department of Bioengineering andBeckman Institute for Advanced Science and Technology, 405 North Mathews Avenue,Urbana, IL 61801. E-mail: rxb@illinois.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1003926107/-/DCSupplemental.

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be a spherically symmetric class of nanolayered alternatingmetal-dielectric probes (nano-LAMPs). Spherical nano-LAMPsare not only attractive for their directionally invariant responsebut also serve as a convenient model to study critical issues indesigning SERS-based probes. Here, we propose the use of sphe-rical nano-LAMPs as a means to designing SERS probes withcontrolled enhancement and exceptional multiplexing capability.A theoretical model is used to predict the limits of enhancementas a function of probe size and examine the spectral response ofdesigned structures.

ResultsThe schematic use of nano-LAMPs is illustrated in Fig. 1A. TheSERS probe’s “tail” links to specific molecular analytes ofbiochemical interest and transduces its concentration. The SERSprobe’s “head” consists of nanostructured metal, dielectric, andreporter components. The spectral response of the reporter is en-hanced and recorded. The probe head (Fig. 1B) in our proposedconfiguration consists of dielectric layers with embedded reportermolecules alternating with metal layers. Several features of thisconfiguration are important and help narrow the scope of thetheoretical model. First, the dielectric embedment shields repor-ter molecules from direct contact with the metal surfaces and re-duces CE to negligible levels. Second, probes are modeled with aprotective outer silica shell for biocompatibility as well as shield-ing molecules outside the probe from enhancement (37). Sincethe outer layer is fixed, third, odd number-layered LAMPs willnecessarily have a dielectric core, whereas even number-layeredLAMPs will have a metal core. Finally, the probe’s sphericalsymmetry ensures enhancement uniformity with respect to thedirection of illuminating radiation and likely simplifies fabrica-tion. Hence, the analyte concentration can be directly relatedto the signal arising from the probe.

If there were no SERS enhancement within the probe, a simplenumerical enhancement (NE) would still result as a large numberof molecules could be placed within the probe for every analytemolecule. The enhancement here, however, arises from the sumof the surface-enhanced signal arising from every reporter mole-cule in the probe. To quantify signal enhancement, hence, wedefine a net enhancement factor (NEF) as the ratio of a probes’

total Raman signal to that of a single analyte molecule. Withoutloss of generality, we can assume that the analyte and unen-hanced reporter have equivalent spectra and do not considerresonance Raman enhancement (RRE). Assuming that the re-porter is uniformly distributed in the dielectric layers at low con-centration, the NEF is defined by the volumetric integral as

NEF ¼ZZZ

jERamanðωSÞj2jElocðωoÞj2crdV ; [1]

where ERaman is the electric field at the shifted frequency ωs, Elocis the field at the incident laser frequency ωo, cr is the concentra-tion of reporter, and V is the volume of the dielectric reporter.The low concentration of the reporter reduces directly the NEFbut is essential to prevent variability arising from surface adsorp-tion. Following the treatment considered earlier (38), the EE foreach reporter molecule scales as G ¼ jERamanðωSÞj2jElocðωoÞj2≈jElocj4. Hence, the task is reduced to determining the local elec-tric field at all locations within the probe, which is calculatedusing Mie theory for multilayered spheres as detailed in Methods(and Fig. S1). In this report, for the purposes of demonstrativecalculations, we discuss the NEFs achievable with silver-silicanano-LAMPs. For a given nano-LAMP size, thinner metal shellsallow larger dielectric shells (and higher reporter loading). Thinmetal shells, however, result in a smaller mean free path for elec-trons that dampens and broadens the surface plasmon resonances(smaller EE). Thicker metal shells result in higher fields (largerEE) but smaller dielectric volumes (lower reporter loading) andhigher metal absorption losses. The ultimate constraints on shellsizes, further, are determined by common fabrication limits ofnanostructures (39, 40). The problem of determining enhance-ment for any LAMP structure is, thus, bounded. We systemati-cally maximized and minimized the enhancement of probes ofdifferent diameters, at different excitation wavelengths, usinggenetic algorithms (GAs), as detailed in Methods (41).

Each LAMP size can have even or odd layers. The thickness ofany given layer in an optimized structure then depends on theoverall configuration of the LAMP to maximize the counteractingeffects arising from coupled plasmonic interactions and reporterloading. Optimized NEFs, hence, depend on both the total probesize (R) and the number of layers (L) for odd- or even-layeredLAMPs, as shown in . 1 C and E. The contribution of reporterloading, or NE, is simply linear as shown in . 1 D and F and ismuch lower than the overall NEF. Hence, the enhanced probesignal arises primarily from the EE and not simply from morereporters per analyte. From . 1 C and E, it can be seen thatthe enhancement in signal achieves a plateau at ∼100 nm (for785-nm excitation). A distribution of the enhancement withinLAMPs (Fig. 2 and Fig. S3) indicates that the optimal structurereorganizes the surface plasmons to create local hot spots with ahigh NEF density at the core and appreciable contributions fromelsewhere. This interplay of molecular and plasmon density de-termines the NEF. Interestingly, enhancement beyond a criticalsize is not predicted to improve by the complexity of additionallayers or loading. The largest size of an optimal SERS probe,hence, is bounded by enhancement considerations, whereasthe lower size limit is bounded by fabrication constraints. The de-sired probe size within this size range is likely dependent on theapplication and may be determined by considerations of fabrica-tion, toxicity, or uptake. For example, in human cervical cancercells (42), 50-nm sized gold spheres were found to be taken upmore quickly compared to other sizes in 10- to 100-nm range.For a given size, then, SERS enhancement is bounded anddepends on the number of layers. Hence, both complexity of po-tential nano-LAMPS and their sizes are predicted to be bounded,providing practitioners in this area an operating window forparticles’ synthesis and use.

Fig. 1. Structure and electromagnetic characteristics of nano-LAMPs. (A)Typical application of LAMPs. (B) Nano-LAMP configurations for odd andeven number of layers. LAMPs with two layers have reporter embeddedin protective outer layer, whereas all others have a protective, unlabeled out-ermost layer. The maximum NEF for even (C) and odd (E) layered silver-silicaLAMPs of different sizes for 785-nm excitation (other wavelengths are shownin Fig. S2). (d, f) Purely numerical enhancement is significantly smaller; hence,the net amplification in signal arises from a couple of the NE and EE. Thecolor bar code for C, D, E, and F is given below D and F.

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We also note that NEFs of ∼1012 can be achieved for ordinary,nonresonant organic reporters. This availability is substantiallyhigher than that encountered in using self-assembled monolayers(SAMs) for consistent CE or resonant dyes in simple geometries.The elimination of CE here is counterintuitive and would likelyhave been detrimental to enhancement, but structuring the probebalances the loss by enabling higher EE. Hence, consistency ofprobe response is freed of the constraints of using reporters thatform SAMs. Further, as opposed to using resonant dyes in simplegeometries, structuring of the particle can be used to obtain largeenhancements. Hence, the probe is freed of the constraint ofusing a reporter resonant at the excitation frequency. The releaseof these two constraints considerably enhances the multiplexingcapability of SERS probes as almost any organic molecule is nowenabled to be used as a reporter and can be used to providedesired enhancements, subject only to the constraints in Fig. 1.Further increases beyond the upper limits of Fig. 1 are alsopossible following the conventional route of RRE. For example,a NEF of ∼108 is attained at 785-nm excitation for a probe of size50 nm by using four layers. Using a resonant dye at the same ex-citation frequency as a reporter would lead to NEFs of ∼1014,levels that allow for assaying up to single molecule levels. Asopposed to the localization of molecules within nanoscale inter-stitial spaces, hence, the LAMP configuration can provide singlemolecule sensitivity for molecules outside the probe. Finally,most analytes can be tagged using existing linking mechanismssuch as antibody-antigen, cDNA binding, and probe functionali-zation, thus generalizing the sensing capabilities of LAMPs.

While the capability required for single molecule assays andlarge-scale multiplexing will be possible with LAMPs, measuringmultiple epitopes in biological samples often requires a large con-centration range to be measured. Hence, tunable and large NEFsmust be coaddressed with the challenge of devising a large dy-namic range. Just as we enhanced the signal, we optimize the EEto quench (43) the Raman signal. Thus, the response of a parti-cularly abundant analyte can be reduced, whereas a sparse

analyte can be enhanced to measure both on the same platformand during the same experiment. Examining quenching as afunction of size using GAs, we determined that quenching canbe achieved using three-layered LAMPs by designing the struc-ture to enhance E fields in the reporterless protective silica shell.Here, neither the reporter in the inner layers is enhanced, nor isthe medium outside the probe. Three-layered LAMPs werefound to be optimal because it becomes difficult to reorganizeplasmonic interactions from several layers into the outer layer.Alternately LAMPs may be constructed with only a very lowconcentration in one of the inner layers. This possibility is notexamined here for brevity. Quenching is illustrated in Fig. 3(and Fig. S4), where distributions in minimized configurationsare depicted in the plane of incident illumination. Finally, the en-hanced and quenched probe pair for a given size and excitationwavelength provides the allowable dynamic range as shown inFig. 4A (and Fig. S2). The shaded regions represent the viabletuning space of probe signals for any common organic used asa reporter. As noted above, using a lower concentration orweaker Raman scattering reporter, the total signal could be evenlower. Similarly, the upper limit could be higher by using a reso-nant dye. For example, choosing LAMPs of size 100 nm, onecould achieve a maximum NEF of ∼1011 and quenched NEFof ∼10−2 at 785-nm excitation laser for nonresonant molecules.By using resonant molecules and lowering dye concentrations,hence, analytes over 20 orders of magnitude in concentrationmay be measurable in a single assay. Finally, an interesting con-sequence of the interacting plasmons arises in the spectral beha-vior of LAMPs. The structuring of a particle into the LAMPconfiguration provides an opportunity to tailor the response ofcomplex probes to be spectrally flat (Fig. 4B) compared to sim-pler structures. The spectral uniformity indicates that the sameLAMPs can be excitable by multiple laser wavelengths, and en-hancements at shifted frequencies do not experience a drop-offfrom the excitation laser frequency that is often selected to co-incide with the spectral plasmon resonance peak of SERS probes.

Fig. 2. Enhancement distributions in the incident plane of illumination for even-layered silver-silica LAMPs of various configurations that are designed tomaximize enhancements at an excitation wavelength of 785 nm. From top to bottom, the number of layers increases from 2 to 10. From left to right the totalsize increases as shown.

13622 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1003926107 Kodali et al.

Hence, the LAMP configuration promises sensitive, tailored,reporter-independent, and spectrally consistent responses whilebeing amenable to fabrication (44).

Finally, we place in perspective the potential advance withLAMPs compared to several other notable and demonstratedprobe technologies. The concept of a SERS probe-based assaywas first demonstrated (using a SERS-active dye molecule) byimmobilizing an antibody on an Ag surface and labeling the de-tection antibody with a SERS-active dye for a 104-fold enhance-ment (45). Silica-coated SERS-active gold colloids with resonantreporters functionalized to gold surfaces were reported (3) toshow SERS intensities with total enhancement factors on the or-der of 1013–1014. A SERS enhancement of 106 has been reported(46) using nanoshells to enhance para-mercaptoanailine, and aSERS enhancement of 1012 is calculated when reabsoprtion ofRaman emission by surrounding nanoparticles is taken into ac-count. Thus, the concept of interactions among particles is notedand shown to be very effective. High enhancements (1011–1014)(47) have also been reported for RuBPy-embedded SiO2-core-Au

shell nanoparticles while using SAMs on Au nanoshells instead ofspheres increases signal ∼176-fold (48). These enhancementlevels are within the range of the structures proposed hereand, indeed, the methods used to optimize geometries here willlikely benefit these existing approaches.

DiscussionIn summary, we have theoretically demonstrated the possibility ofdesigning SERS probes (via nano-LAMPs) that possess designedenhancement that can be tuned over a permissible range, wave-length-tuning capability, and potential for multiwavelength exci-tation. The layered structure permits both the reduction ofvariance by shielding reporters from direct interaction with metalsurfaces as well as being spherically symmetric. In principle,the configuration offers unlimited multiplexing using simple re-porters, whose enhancements do not depend on their scatteringcross-section but can be independently tuned by the probe. Theuse of any reporter, tailored enhancement, and reproducible res-ponse will open the possibility of measuring multiple molecularspecies in complex samples. A practical translation of this tech-nology will facilitate the development of a large dynamic rangeand suppression of the signal from the surrounding media, whichare both a limiting factor in the use of SERS probes. While theconcept of enhancements in signal has been previously examined,the concepts of containing the enhancement within the particlefor amplified signals and designing probes for quenching in theoptimization makes this work especially practical. Just as we haveenhanced the response of the reporter, the response of the probesmay also be tuned to enhance or suppress out of probe enhance-ment to any desirable level. An enhancement of the region out-side the probe, for example, has been recently used for sensitivemeasurements (49). In this case, the use of LAMPs will be morepowerful and generally applicable than the use of solid substrates.The proposed structures are amenable to modern fabricationmethods (50), and a number of biocompatibility and moleculartargeting strategies are available for nanoparticles (51) that canbe employed for the proposed structures. At the same time, stra-

Fig. 4. Allowable probe signal and multiple excitation capability. (A) Rangeof NEF attainable via silver-silica nano-LAMP configurations for 785-nm ex-citation as a function of probe size. (B) LAMPs can be made capable of ex-citation at multiple frequencies by tailoring the internal structure. Here, asix-layer LAMP provides similar signal at multiple frequiencies for the samesize (50 nm) as a three-layered LAMP, while maintaining high enhancement.

Fig. 3. Enhancement distributions in the incident plane of illumination for even-layered silver-silica LAMPs of various configurations that are designed toquench internal enhancements at an excitation wavelength of 785 nm. From top to bottom, the number of layers increases from 2 to 10, whereas from left toright the total size increases as shown.

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tegies to use spectroscopic signatures for quantitative analyses viaefficient computer algorithms (52) are available to analyze thedata. Hence, the realization and use of nano-LAMP probes, asdesigned in this manuscript, can enable progress in a wide varietyof fields ranging from clinical diagnostics, environmental sensing,to basic biomedical research.

MethodsLayered Mie Theory. Owing to the symmetry of LAMPs (Fig. S1), the responsedue to a laser excitation can be simulated by evaluating the fields within thesphere when a plane wave is incident, without the loss of generality, byextending Mie theory. We consider here an L-layered nanosphere withembedding medium denoted by Lþ 1, the field in jth layer can be repre-sented in terms of vector spherical harmonics as

Ej ¼ ∑N

n¼1

En

�−inðnþ 1Þ

ðκjrÞ2πnðcos θÞ sin θ cosϕUj

nðκjrÞer

þ cosϕðκjrÞ

fπnðcos θÞV jnðκjrÞ − iτnðcos θÞðUj

nðκjrÞÞ0geθ

−sinϕðκjrÞ

fτnðcos θÞV jnðκjrÞ − iπnðcos θÞðUj

nðκjrÞÞ0geϕ�; [2]

Hj ¼ −κjω∑

N

n¼1

En

�inðnþ 1ÞðκjrÞ2

πnðcos θÞ sin θ sinϕV jnðκjrÞer

−sinϕðκjrÞ

fπnðcos θÞUjnðκjrÞ − iτnðcos θÞðVj

nðκjrÞÞ0geθ

−cosϕðκjrÞ

fτnðcos θÞUjnðκjrÞ − iπnðcos θÞðV j

nðκjrÞÞ0geϕ�; [3]

where the functionsUn and Vn are given in terms of Ricatti-Bessel functions ψand ξ as follows:

UnðκjrÞ ¼ djnfψnðκjrÞ − AjnξnðκjrÞg; [4]

VnðκjrÞ ¼ cjnfψnðκjrÞ − BjnξnðκjrÞg: [5]

For the field in the core and the incident field, to be well defined at origin:an1 ¼ bn

1 ¼ 0 and for the field in the embedding medium to be well definedat infinity: cnLþ1 ¼ dn

Lþ1 ¼ 1. The other coefficients of expansion can be ob-tained using the following continuity conditions between j and (j þ 1)thlayer, for j ¼ 1;…L.

ðEjþ1 − EjÞ × er ¼ 0; [6]

ðHjþ1 −HjÞ × er ¼ 0. [7]

Applying these boundary conditions and using orthogonality of the func-tions involved, a recursive formulation can be obtained. In the followinglogarithmic Bessel functions Dn

ð1Þð¼ ψ 0n∕ψnÞ, Dn

ð3Þð¼ ξ0n∕ξnÞ and ratio func-tion Rnð¼ ψn∕ξnÞ are used for stability and accuracy of calculating higherorder coefficients.

The coefficients An and Bn are obtained by upward recursion and thecoefficients as follows:

A1n ¼ 0; B1

n ¼ 0 and for j ¼ 2;…L; [8]

Gjn ¼ 1

mj

�RnðκjrjÞDð1Þ

n ðκjrjÞ − AjnD

ð3Þn ðκjrjÞ

RnðκjrjÞ − Ajn

�; [9]

Hjn ¼ mj

�RnðκjrjÞDð1Þ

n ðκjrjÞ − BjnD

ð3Þn ðκjrjÞ

RnðκjrjÞ − Bjn

�; [10]

Ajþ1n ¼ −Rnðκjþ1rjÞ

mjþ1Gjn −Dð1Þ

n ðκjþ1rjÞmjþ1G

jn −Dð3Þ

n ðκjþ1rjÞ; [11]

Bjþ1n ¼ −Rnðκjþ1rjÞ

Hjn −mjþ1D

ð1Þn ðκjþ1rjÞ

Hjn −mjþ1D

ð3Þn ðκjþ1rjÞ

: [12]

The coefficients cn and dn are obtained by proceeding in a downward recur-sion as follows:

cLþ1n ¼ 1; dLþ1

n ¼ 1 and for j ¼ L;…1; [13]

Ujn ¼ djþ1

n ξnðκjþ1rjÞfRnðκjþ1rjÞ þ Ajþ1n g; [14]

V jn ¼ cjþ1

nmj

mjþ1

ξnðκjþ1rjÞfRnðκjþ1rjÞ þ Bjþ1n g; [15]

djn ¼Uj

nðκjrjÞξnðκjrjÞfRnðκjrjÞ þ Aj

ng; [16]

cjn ¼ V jnðκjrjÞ

ξnðκjrjÞfRnðκjrjÞ þ Bjng

: [17]

Numerical integration of electromagnetic enhancement is performed with aweighted sum of integrand values using weights derived appropriate forspherical shells (53).

Probe Parameters. Since the Raman enhancements observed so far are great-est for silver, it is chosen as the metal and silica is chosen as both the dielectricand protective layer. The core of LAMPs is constrained to be larger than10 nm in diameter and shells to be thicker than 2 nm and 1 nm for the metaland dielectric material, respectively. A 1% loading and a molecular volume of0.5 nm3 for the reporter are considered. Whole number shell sizes subject tosize constraints are encoded using floating point numbers in [0, 1] interval.

Genetic Algorithm and Parameters. The adapted operators for the algorithmare tournament selection without replacement (s ¼ 4) (54), simulated binarycrossover (SBX) (55, 56) with ηc ¼ 10, crossover probability pc ¼ 0.9, a poly-nomial mutation (57) with η ¼ 20, and mutation probability p ¼ 0.1. TheGA was typically run for about 30–50 iterations after which all cases con-verged. In tournament selection without replacement and with tournamentsize s, s chromosomes (probe structures) are chosen at random without re-placement and entered into a tournament against each other. The best(low error) individual in the group of s wins the tournament and is selectedinto a mating pool for evolving new solutions. In SBX, individuals in themating pool are divided into random pairs, and each pair undergoes recom-bination with a probability pc. For each pair participating in the crossover,each gene (or shell size) undergoes a contracting or expanding crossoveroperation with a probability 0.5. Therefore, for each pair of chromosomesundergoing recombination, on average half of the genes are modified usingeither contracting or expanding crossover operations. The operations aredesigned to mimic crossover operator behavior on binary domains. The poly-nomial mutation is similar to SBX, and the only difference is in the computa-tion of the polynomial probability. Instead of using genotypic distancebetween two parents as in SBX, the distance between a gene and itscorresponding upper or lower bound, whichever is closer, is considered incomputing the contracting and expanding probability distributions. In poly-nomial mutation, each gene (or variable) undergoes contracting or expand-ing operation with a probability p.

ACKNOWLEDGMENTS. This work was supported by a grant from the BeckmanInstitute for Advanced Science and Technology and by the National ScienceFoundation support through Teragrid resources and the Faculty FellowsProgram of the National Center for Supercomputing Applications at theUniversity of Illinois.

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