LSPR sensor response calculation

LSPR biosensors detect a target molecule by monitoring plasmon resonance frequency changes due to the proximity of the target to the metallic nanoparticle. We present a proof of concept calculation that shows how our approach can be used to model the response of LSPR biosensors.

Problem setup

We place a $r = 10 \,nm$ silver sphere in water, surrounded by six $r = 4 \,nm$ spherical proteins at a distance $d$ along the principal axes (x,y,z).

Fig 1. Setup for response calculation. There are two proteins that are not represented in this figure: they are placed at a distance $d$ from the silver sphere on the axis going into the paper (1)

Mesh convergence analysis

Before running the response calculations we perform a mesh convergence analysis to ensure that the results are not going to be affected by changing the size of the mesh.

We run the case presented in Fig. 1 for a wavelength of $380 \, nm$ and a distance $d\,=\,1 \,nm$ for meshes of 512-128, 2048-512, 8192-2048 and 32768-8192 elements where the first number corresponds to the metallic sphere and the second one to the protein.

The values for the dielectric constants for water and silver were obtained by interpolation of experimental data (2,3):

  • Water : 1.7972 + 8.5048e-09j
  • Silver: -3.3877 + 0.1922j

For the protein (modeled as a sphere) we use Bovine Serum Albumina (BSA) data extracted from the functional relationship provided by Pahn, et al. (4)

  • Protein: 2.7514 + 0.2860j

The error calculation uses the Richardson extrapolation value of the extinction cross-section as a reference, $Cext\,=\, 2663.476 \, nm^2$


Fig 2. Convergence of extinction cross-section of a $r=10\, nm$ silver sphere with six spherical proteins located at $1\,nm$ away along the principal axis, versus the average of number of elemnts per quare nanometer.

We obtain a 1/N convergence (Fig. 2), proving that the numerical solutions computed with PyGBe are correctly resolved by the meshes.

avg_dens % error
0.52 28.09
2.08 6.56
8.32 1.64
33.26 0.41

Runtime and Hardware specs

Runs performed in a GPU card NVIDIA Tesla C2075:

avg_dens time [s]
0.52 57.82
2.08 302.37
8.32 1294.63
33.26 7170.62

For reference we also run the previous cases in a newer GPU card, NVIDIA Tesla K40:

avg_dens time [s]
0.52 24.72
2.08 137.70
8.32 649.50
33.26 3695.76

Sensor response POC

To analyze how the LSPR response and the sensor-analyte distance are related, we use the mesh of 8192 elements for the silver sphere and 2048 elements for the protein spheres.

We vary the distance between the silver sphere (sensor) and the analytes (small spheres i.e. proteins) and we calculate the extinction cross-section as a function of wavelength for the different distances:

  • $d = \infty$ i.e. we only have the sensor (silver sphere r=10)
  • $d=4 \,nm$
  • $d=2 \,nm$
  • $d=1 \,nm$


Fig 3. Extinction cross-section versus wavelength for a $r=10\,nm$ silver sphere with six spherical proteins located at diferent distances away form the surface.

Note

The runs were performed for wavelngths in the range [380-400] nm, each 0.5 nm, what gives a total of 41 runs.

Discussion

Figure 3 shows the variation of the extinction cross-section with respect to the wavelength, for each of the distances shown above. The red shift in the resonance frequency and the decrement of the peak in the presence of the analyte agrees with the behavior observed in experiments perform by Tang et al. (5). This indicates that our boundary element method approach, using electrostatic approximation, is capable of reproducing characteristic resonance frequency shift in LSPR biosensors.

Experiments suggests that the distance between the nanoparticle and the analyte affects the sensitivity of the sensor (6), and these proof-of-concept calculations show that our apprach can be use to study sensitivity versus distance, using electrostatics.

Work in progress

Our next step is to study the interaction between a realistic model of the BSA protein (non-spherical) and the nanoparticle; more specifically how the relative position between them affects sensitivity.


Fig 4. Setup for response calculation using crystal structure of real protein.

PyGBe can solve problems with arbitrary geometries which will allow us to also study the sensitivity of previously unexplored nanoparticle geometries.

References

(1) Cooper Villagran, Christopher David. Biomolecular electrostatics with continuum models: a boundary integral implementation and applications to biosensors. Diss. 2015.

(2) Hale, G. M. and Querry, M. R. (1972). Optical constants of water in the 200-nm to 200-μm wavelength region. Appl. Opt., 12(3):555–563.

(3) Johnson, P. B. and Christy, R. W. (1972). Optical constants of nobble metals. Phys. Rev. B, 12(6):4370–4379.

(4) Phan, Anh D., et al. "Surface plasmon resonances of protein-conjugated gold nanoparticles on graphitic substrates." Applied Physics Letters 103.16 (2013): 163702.

(5) Tang, Bin, et al. "Two-dimensional correlation localized surface plasmon resonance spectroscopy for analysis of the interaction between metal nanoparticles and bovine serum albumin." The Journal of Physical Chemistry C 114.49 (2010): 20990-20996.

(6) Haes, A. J., Zou, S., Schantz, G. C., and Van Duyne, R. P. (2004). A nanoscale optical biosensor: The long range distance dependence of the localized surface plasmon resonance of noble metal nanoparticles. J. Phys. Chem B, 108:109–116.

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