Allan deviation tells the binding properties in single-molecule sensing with whispering-gallery-mode optical microcavities

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Allan deviation tells the binding properties in single-molecule sensing with whispering-gallery-mode optical microcavities

Deshui Yu and Frank Vollmer

Phys. Rev. Research 3, 023087 – Published 3 May 2021

Abstract

We propose a scheme of exploring the kinetic properties of the receptor-ligand binding process by means of the frequency stability measurement of a whispering-gallery-mode optical microcavity. In this scheme, the probe beam is directly locked to the microcavity and the time resolution of the sensing detection is determined by the feedback loop response time $10 μ s$ , which is much shorter than the current limit of about 20 ms. The receptor-ligand interactions are entirely mapped onto the probe light frequency and can be recovered through the analysis of its Allan deviation. As a result, one can investigate a chemical and biological reaction at the single-molecule level without the need for resolving the occurrence and duration of individual binding events. We numerically study this sensing scheme based on a practical optoplasmonic platform, where several gold nanorods are adsorbed onto the surface of a spherical microcavity that is located in the aqueous environment. The microsphere has a typical radius of $40 μ m$ and quality factor of $10^{6}$ (finesse of $2.6 \times 10^{3}$ ) at the wavelength of 785 nm. This powerful scheme may extend single-molecule sensing to the kinetic regime of the biochemical reactions.

Received 16 December 2020
Revised 24 March 2021
Accepted 29 March 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.023087

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Whispering gallery mode resonators

Atomic, Molecular & Optical

Authors & Affiliations

Deshui Yu and Frank Vollmer

Living Systems Institute, Physics and Astronomy, University of Exeter, Exeter EX4 4QD, United Kingdom

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Vol. 3, Iss. 2 — May - July 2021

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Figure 1
(a) Single-molecule detection based on the optoplasmonic hybridization. A silica microsphere with a radius of $R_{cav} = 40 μ m$ is situated in the aqueous environment. Several gold nanorods are adsorbed onto the microsphere's surface. The TE-polarized cw and ccw WGMs at $λ_{cav} = 785$ nm are excited by a probe laser through an optical prism. The transmitted light is fed back into the probe laser, locking its frequency to the microcavity. The Allan deviation of the probe laser is measured by the beat note with a reference laser. (b) Longitudinal LSPR spectrum of a gold nanorod and the electric-field intensity distribution of LSPR at $λ_{cav}$ . The circular cross section of the nanorod has a diameter of 10 nm and the nanorod's length is 35 nm. The regions around two ends of the nanorod are termed the hotspots, where the near-field intensity is strongly enhanced by a factor $Λ = 800$ . (c) Evanescent interaction between the microsphere and a gold nanorod. The receptor molecules, which interact with the ligand molecules, are permanently attached within the nanorod's hotspots. (d) Schematic diagrams of the time traces of the wavelength shift of a WGM. The spike events correspond to the transient receptor-ligand interactions while the step events denote the permanent binding between single receptor and ligand molecules.
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Figure 2
(a) Frequency relations. The cw and ccw WGMs are degenerate for the bare microsphere in the water. The resonance frequency of two WGMs is $ω_{cav}$ . When the gold nanorods are adsorbed to the surface of the microsphere, the resonance frequency $ω_{ccw}$ of the ccw WGM is shifted from $ω_{cav}$ by an amount of $δ_{s, 0}$ . When the ligand molecules further bind to the gold nanorods, the total frequency shift of the ccw WGM is $δ_{s}$ , i.e., $ω_{ccw} = ω_{cav} + δ_{s}$ . (b) Typical power spectral density $S_{y} (f)$ of the frequency noise in $ω_{cav} (t)$ measured in experiment. The bare microsphere, i.e., without the gold nanorods and in the absence of the receptor-ligand binding interactions, is immersed in the water. (c) Allan deviation $σ_{y} (τ)$ . The square symbol denotes the stability of the experimentally-measured frequency $ω_{cav}$ of the microsphere in the aqueous environment. The corresponding numerical simulation result is given by the dashed line. The circle symbol is the numerical result of the spike-event-perturbed frequency $ω_{ccw}$ with $R = 250 s^{- 1}$ , $T = 244$ s, and $τ_{ave} = 10^{- 2}$ s. The diamond symbol corresponds to the numerical simulation result of the spike-event-induced frequency shift $δ_{s}$ . The solid curves denote the asymptotes.
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Figure 3
(a) Numerically simulated traces of the spike-event-induced mode shift $δ_{s} (t)$ in units of fm. The constant bias $δ_{s, 0}$ caused by the gold nanorods has been removed. A ligand binding to a receptor causes an about 10-fm shift. (b) Power spectral density $S_{(δ_{s} / ω_{0})} (f)$ of the numerically simulated $δ_{s} (t)$ . The curve fitting illustrates $S_{(δ_{s} / ω_{0})} (f) = h_{0}^{'} f^{0} + h_{- 2}^{'} f^{- 2}$ with $h_{0}^{'} = 2.14 \times 10^{- 17} {Hz}^{- 1}$ and $h_{- 2}^{'} = 5.20 \times 10^{- 15}$ Hz. (c) Distribution of spike heights extracted from the numerical simulation. The bins around 10 fm correspond to the single-molecule events. (d) Probability of the single-molecule events extracted from the numerical simulation vs the average duration $τ_{ave}$ for different event rates $R$ .
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Figure 4
(a) Dependence of the average spike height $Δ_{ave}$ in a time trace of $δ_{s} (t)$ on the average duration $τ_{ave}$ with different event rates $R$ . The solid lines correspond to the results of the empirical formula (18). (b) Upper panel: White frequency $h_{0}^{'}$ and random walk $h_{- 2}^{'}$ parameters vs $R$ . The solid lines correspond to the results of the empirical formula (19) while the dashed lines denote the results of the empirical equation (22). Lower panel: Cross times $τ_{c 1}$ and $τ_{c 2}$ as a function of $τ_{ave}$ with $R = 250 s^{- 1}$ . (c) White frequency $h_{0}^{'}$ and random walk $h_{- 2}^{'}$ parameters vs $τ_{ave}$ . In all plots, the filled square and circle symbols denote the numerical simulation results.
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