1. Introduction

Sensitive detection of pathogenic nanoparticles, such as viruses, is key to prevention of infectious diseases. Compared to sensing their molecular fingerprints, direct detection of whole particle viruses offers the advantage of easy sample preparation and fast assay. Fluorescence correlation spectroscopy (FCS) is one of the most sensitive methods for detecting suspended nanoparticles [1, 2]. However, the detection limit of FCS, on the order of sub nano-molar [3], does not meet the needs for clinical analysis of viral samples. The use of force fields, such as di-electrophoresis [4, 5] and optical trapping, to concentrate nanoparticles [6, 7], are an appealing approach to lower the FCS detection limit. Lacking an analytical expression for the FCS autocorrelation function, we need to examine experimentally the applicability of existing models for using FCS to measure the concentration of nanoparticles by enriching them in a potential well produced by a force field.

Multiple attempts have been made to interpret FCS data obtained from colloidal nanoparticles in a gradient force field. The experiment of Osborne et al. [8] showed that optical trapping can affect diffusion of molecules in the vicinity of the optical trap. Hosokawa et al. [9] used optically biased diffusion to study the cluster formation of nanoparticles in a highly focused laser beam. They demonstrated that it might be possible to quantify the relationships between the cluster size, trapping energy and the decay time of the FCS autocorrelation function (ACF). More recently, Wang et al. [10] studied the diffusion dynamics of gold nanoparticles in optical confinement and reported a laser-induced rise in temperature changed the diffusion behavior and the average number of gold nanoparticles in the focal area of the laser beam. Ito et al. [11] used a Brownian dynamics simulation to investigate the FCS ACF of colloidal nanoparticles in an optical gradient force field. They found the standard FCS ACF for free diffusing particles is applicable for particles in an optical trap with potential energies lower than 1k_BT. Meng et al. [12] compared the ACF obtained by numerical calculation and Monte Carlo simulation for non-interacting particles in an isotropic Gaussian potential with trapping energies up to 1.8k_BT and showed the ACF amplitude is equal to the inverse of the mean number of particles in the optical trap.

In this paper we report an experimental study that combines optical trapping and FCS to examine the influence of trapping potential energy on the ACF amplitude. We compare the experimental ACF amplitude with the experimentally determined 1/〈N〉 using colloidal suspensions of polystyrene nanoparticles and HIV-1 virus-like particles (VLPs) to verify the predictions by Meng et al. [12]. Our experimental results are also compared with the Brownian dynamics simulation results by Ito et al. [11]. We further explain that the equality of the ACF amplitude to 1/〈N〉 under optical trapping requires that the particle number density fluctuation follows the Poisson statistics.

2. Experimental

2.1. FCS data analysis

A typical FCS experiment measures ACF of the fluorescence intensity from a detection volume defined by the focus of a laser beam. The FCS ACF G(τ) is defined as:

2.2. Optical instrument

A schematic of the optical apparatus is illustrated in Fig. 1. An IR laser (1064nm, TREMC1, Spectra-Physics) is used for trapping fluorescent nanoparticles. A blue laser (488nm) of a confocal laser scanning microscopy (FV-1000, Olympus, Japan) is used for fluorescence excitation. A half-wave plate and a polarization beam splitter are used for adjusting the power of the trapping laser. The power of the 488 nm excitation laser at the focal point is maintained at a low power of 5 μW to minimize photo bleaching of the fluorescent particles in the observation volume. The trapping and excitation beams are combined by a dichromatic mirror and confocally aligned at the detection point. A microscope objective lens (UPlanFluor 100X, NA1.3, Olympus, Japan) is used to focus the laser beams and to collect fluorescence emission. The fluorescence emission passing through a band-pass filter and a beam splitter is detected by two photon-counting avalanche photo diodes (APD: SPCM-AQRH-13-FC, Perkin Elmer). Since the fluorescence emission from the two channels are correlated, while the dark counts produced by thermal noise in the two APD are not, we can minimize the adverse effects due to the APD dark counts by taking the cross-correlation of the two APD outputs to obtain G(τ). The cross-correlation calculation is made by the use of a digital correlator (Flex02-01D, correlator.com, USA). The geometrical parameters of the FCS detection volume w₀, z₀ are determined by using a 10nM solution of Alexa 488 dye. Using the known diffusivity of Alexa 488 molecules in water (430 μm²/s) in fitting Eq. (2) to ACF, w₀ and z₀ are determined to be 0.22 μm and 2 μm, respectively.

 

Fig. 1 Illustration of the experiment apparatus.

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2.3. Sample preparation

2.3.1. Colloidal polystyrene (PS) nanoparticles

An aqueous suspension of 100nm diameter PS particles with a volume fraction of 1% is purchased from Thermo Scientific (Fremont, USA). The PS particles are fluorescently labelled with Firefli red (excitation maximum 543nm, emission maximum 612nm). The ionic (salt) concentration of the stock suspension is estimated to be 2 mM. The stock suspension is diluted 100 times in deionized water for all the FCS measurements. The ionic strength of the diluted sample solution was estimated to be 20 μM. A 20 L sample is held between a microscope cover-glass and a slide and sealed with wax. The chamber thickness of the gap between the two glass substrates is 60 ± 10 μm. FCS measurements are positioned at ∼ 10 μm above the cover glass.

2.3.2. Preparation of virus-like particles (VLP)

HEK 293T cells (ATCC No. CRL-11268) are cultured in DMEM supplemented with 10% fetal calf serum at 37°C and 5% CO₂. The cells at 60% confluency are transfected with the GAG-EGFP plasmid (AIDS reagent program Cat. No. 11468) and incubated for 48 hours. The super-natant containing virus-like particles is then collected, filtered using 0.2 μm filter and centrifuged for 3 hours at 20,000 rpm to concentrate the virus-like particle at the bottom of the test tube by sucrose density centrifugation. The pellet containing the virus-like particles is then resuspended in the cell culture medium. These particles have a similar capsid structure to the wild-type HIV virion, but lack some of the viral genome, which minimizes their pathogenicity. Moreover, these particles are genetically engineered to carry green fluorescent protein (excitation maximum at 488nm and emission maximum at 509nm) tagged to Gag, the primary structural protein of HIV.

SEM and confocal microscope images are acquired to examine the size and fluorescence of the VLPs [13]. To prepare SEM samples, glutaraldehyde is added to the VLPs suspension to a final concentration of 2.5% followed by incubation at room temperature for 1 hour. A 20 μL suspension is then pipetted onto an autoclaved cover glass, dehydrated in ethanol, dried at room temperature and sputter-coated with gold palladium for 60 seconds before imaging.

3. Results and discussion

3.1. Fluorescence correlation spectroscopy in an optical trap

The ACF G(τ) from 110nm PS nanoparticle suspensions in the presence of optical trapping with laser powers between 0 and 18mW are shown in Fig. 2. Each ACF curve represents an average of 10 independent measurements. Fitting Eq. (2) to the trap-free (0mW) curve yields a particle number density 〈N₀〉= 0.165 in the observation volume and diffusivity D = 4.5 μm²/s. The measured diffusivity agrees with that obtained by the Stokes-Einstein equation D = k_BT /(6πηa) = 4.35 μm²/s, where η is the viscosity of water at the ambient temperature and a is the particle radius.

 

Fig. 2 FCS autocorrelation functions of 0.01% (v/v) 110nm PS particle suspensions at selected trapping laser powers. Each curve represents an average of 10 measurements.

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To examine the range of trapping powers at which Eq. (2) can be used to describe the experimental ACFs, we plot in Fig. 3 the same data shown in Fig. 2 and fit curves using Eq. (2) for various trapping energies, 0mW, 6mW, 12mW and 18mW, respectively. The excellent match between Eq. (2) and the experimental ACF data suggests that the conventional ACF can be used to describe the motion of the nanoparticles in a potential well with a depth up to 1.8k_BT. This range of trapping powers exceeds that examined by Ito et al. through Brownian dynamic simulation [11], where Eq. (2) is shown to be applicable for trapping energies only up to 1k_BT. The trapping energy will be discussed later.

 

Fig. 3 The experimental ACFs (black) are fitted using Eq. (2) (red) for trapping laser powers of (a) 0mW ; (b) 6mW ; (c) 12mW ; (d) 18mW.

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In the presence of the optical trap, the average number of particles in the trap can be determined independently by total photon counts: 〈N_trap〉= 〈F_trap〉/ε. Here, ε is the average fluorescence luminosity per particle at fixed excitation light intensity, which can be determined by the ratio $〈F_0〉/G{\left(0\right)}_0^{-1}$, where 〈F₀〉 is the average fluorescence photon counts at zero trapping laser power, and G(0)⁻¹ at zero trapping laser power is equal to the mean number of particle in the FCS observation volume.

Figure 4(a) shows a plot of $G{\left(0\right)}_{\mathit{trap}}^{-1}$ vs. 〈N_trap〉. The linearity in Fig. 4(a) with a slope of 1 suggests $G{\left(0\right)}_{\mathit{trap}}^{-1}$ equals the average number of particles in the presence of optical trapping for all trapping powers tested between 0mW to 18mW.

A semi-log plot of $G{\left(0\right)}_{\mathit{trap}}^{-1}/G{\left(0\right)}_0^{-1}$ vs. the power of trapping laser is shown in Fig. 4(b). Since $G{\left(0\right)}_{\mathit{trap}}^{-1}/G{\left(0\right)}_0^{-1}=〈N_{\mathit{trap}}〉/〈N_0〉$, Fig. 4(b) implies the number of particles in the optical trap follows a Boltzmann distribution with the trapping potential energy:

 

Fig. 4 (a) $G{\left(0\right)}_{\mathit{trap}}^{-1}$ from 3(a) is found to increase linearly with 〈N_trap〉 measured by 〈F_trap〉/ε; the dashed line is a fit with slope = 1 and intercept = 0. (b) Semi-natural log plot of $G{\left(0\right)}_{\mathit{trap}}^{-1}/G{\left(0\right)}_0^{-1}$ vs. the trapping power suggests the average number of particles in the optical trap follows a Boltzmann distribution. (c) $G{\left(0\right)}_{\mathit{trap}}^{-1}/〈N_{\mathit{trap}}〉=1$ for trapping energies up to 1.8k_BT. (d) The decay time $\tau =w_0^2/4D$ decreases with increasing trapping laser power.

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Figure 4(c) shows a constant ratio of $G{\left(0\right)}_{\mathit{trap}}^{-1}/〈N_{\mathit{trap}}〉=1$ for trapping potential energies up to 1.8k_BT, suggesting our experimental result agrees with the theoretical prediction by Meng et al. [12].

The decay time $\tau =w_0^2/4D$ of our experimental ACF decreases with increasing trapping laser power as shown in Fig. 4(d). This trend agrees with the result reported by Hosokawa et al. [9]. However, a theoretical model will be needed to describe the influence of the particle motion in an optical trap to the FCS ACF.

3.2. Optical trapping of VLPs

Since the size of the fluorescent objects could affect the accuracy of the FCS measurements [13], we determine the sizes of the VLPs by SEM imaging and compare them with measurements by FCS. Figure 5(a) shows a SEM image of the VLPs on a glass slide. The average diameter of the particles is 110 ± 11nm (standard deviation from 10 VLPs), which agrees with previous studies [16, 17]. A fluorescence image (Fig. 5(b)) taken by a 100X oil immersion objective lens (UPlanFluor, Olympus, Japan) shows intrinsic green fluorescence.

 

Fig. 5 (a) A SEM image of one VLP. Diameter of the particles is 110 ± 11nm. (b) A fluorescent image of VLPs under 100X objective. (c) Autocorrelation curve of VLPs in culture medium, from which a hydrodynamic size of 115 ± 14nm is obtained. All standard deviation represents 10 independent measurements.

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A FCS autocorrelation curve of the VLPs suspended in culture medium is shown in Fig. 5(c). By fitting the curve by Eq. (2), the ACF yields a diffusion coefficient 3.79 μm²/s. Using the Stokes-Einstein equation D = k_BT /(6πηa) and medium viscosity η = 0.98cP, we determine the diameter of the VLPs to be 115 ± 14nm (standard deviation from 10 independent measurements), which is consistent with results obtained by SEM.

We use the procedures described in Section 3.1 to determine the trapping energy of VLP. Selected ACF curves for different laser powers are shown in Fig. 5(a). A semi-natural log plot of G(0)⁻¹ vs. trapping laser power is shown in Fig. 5(b). For PBS buffer solutions containing 135mM NaCl, the Debye screening length is on the order of 1nm. Since the estimated average distance between two VLPs in our experiments is more than 1 μm, particle interactions can be neglected, and the number of particles in the optical trap follows Eq. (4). A semi-natural log plot of the average number of particles vs. the trapping power is shown in Fig. 5(b). The trapping energy for the VLPs is calculated to be 0.02 ± 0.003k_BT/mW.

Because VLPs are hollow particles surrounded by capsid proteins and a thin shell of phospholipid bilayer their trapping energy is significantly smaller than that of comparably sized solid PS particles. We use a discrete dipole approximation method [18] to numerically calculate the optical trapping energies of these particles. The trapping beam used in the calculation is a fifth-order-corrected monochromatic Gaussian beam with beam waist 0.5 μm. For the 110nm diameter PS spheres (refractive index 1.59) in water (refractive index 1.33), the calculated trapping energy is 0.13k_BT /mW. Modeling VLPs as 100nm inner diameter vesicles with a 10nm lipid bilayer wall (refractive index 1.46) [19], we calculate their trapping energy to be 0.017k_BT /mW, in reasonable agreement with the experimentally determined value for VLP.

Since the relationship G(0) = 〈N〉⁻¹ is derived from the Poisson statistics, the fact that the relation holds for particles in an optical trap implies that the conditions required for Poisson distribution are satisfied. These conditions are i) the probability a particular particle in the trap is much less than one, and ii) the particle movements near the trap are uncorrelated. Thus, for extremely low particle concentrations, one can estimate the trapping energies under which G(0) = 〈N〉⁻¹ is valid. As an example, for a sample of concentration C = 10⁹particle/cm³ in a volume of V = 10⁶μm³, the probability for a particular particle to be in the laser illuminated volume of v = 1 μm³ is p = 10⁻⁶, and the average number of particles in the illuminated volume is 〈N〉= v × C = 10⁻³. Limited by the intrinsic dark-count of a typically APD detector, 〈N〉 ≥ 0.1 is needed to ensure a statistical meaningful FCS measurement, which requires an enhancement of p by a factor of 100. As a result, p is enhanced from 10⁻⁶ to 10⁻⁴, at which the required condition (i) for Poisson distribution is still satisfied. Since the particle interactions are negligible at such low concentration the concentration enhancement by trapping follows a Boltzmann distribution of the trapping potential energy. The trapping energy needed for the concentration enhancement of 100 can be estimated by ln100k_BT = 4.6k_BT.

4. Conclusions

This paper describes an experimental investigation of the use of FCS for analyzing the behavior of colloidal nanoparticles in an optical trap. Our experiments show the standard FCS autocorrelation function for free diffusing particles describes the behavior of particles in an optical trap with trapping energies up to 1.8k_BT. We also found G(0) = 〈N〉⁻¹ to be valid under such trapping energies, which is in good agreement with the results obtained by Meng et al. by Monte Carlo simulation [12].

Our experiments demonstrate that the average number of nanoparticles in the trap increases exponentially with the power of the trapping laser for non-interacting particles in suspension. In this case, the concentration of the particles in the trap follows a Boltzmann distribution. We determine the trapping energy of HIV-1 virus-like particles (VLPs) to be 0.02k_BT per mW of laser power, a value in good agreement with a discrete dipole approximation by modeling VLP as a hollow sphere with a thin shell of phosphor-lipid bilayer. Using probability theory and the requirements for Poisson statistics, we give an example to illustrate how to estimate the range of force-induced concentration enhancement within which the relationship $G{\left(0\right)}_{\mathit{trap}}^{-1}=〈N_{\mathit{trap}}〉$ can be used to determine particle concentration.