1. Introduction

The correlation between the mechanical properties of cells and their physiological conditions has been the focus of extensive research for decades. When a living cell is executing a specific series of functions or undergoing a series of biochemical processes, the functions or the processes often correlate fairly strongly with its morphology and/or mechanical architecture. Cytoskeleton, an intra-cellular polymer network, plays an important role in cellular mechanics, especially in supporting the mechanical strength and the morphology of the cell as well as in the deformability (or rigidity) of the cell; it is involved in many cellular functions, and is characteristically altered in many diseases, including cancer [1]. The cellular deformability is the combined result of several mechanical and geometrical properties, such as visco-elasticity of the cellular membrane and cytosol [2] as well as the cell morphology.

Quantitative measurement of the mechanical properties of cells, and tracking their dynamic changes over time in response to various external stimulants is an important issue that is difficult to resolve by conventional biochemical methods. For example, at the molecular level, it is clear that ligand-receptor interactions on cellular membrane could respond to changes in mechanical forces by altering their enzymatic functions [3]. Forces associated with these biological events are typically very small, ranging from a few pico-Newton (pN) to tens of pN. Forces associated with cellular activities have been measured by several methods such as atom force microscopy, micropipette suction, and optical tweezers [4–7]. These methods allow us to probe cellular forces on the order of sub-pN to 100pN [8], and to characterize the tension of lateral diffusion of receptors in the plasma membrane. The value of such tension is generally in the range of 1 - 10pN/μm, which resembles the characteristics of an extremely soft spring [9, 10].

Since the first successful demonstration of optical tweezers as a force transducer for the study of biological forces by Ashkin [11], optical tweezers have been used extensively for the study of the mechanics of cells. Specifically, optical forced oscillation of a trapped particle has been implemented in different experimental configurations and applied to the measurement of interaction between different kinds of samples, including biological objects [12-15].

In this paper, we report the experimental demonstration of an optical force oscillator via a set of twin optical tweezers to study the protein-protein interaction by tracking the forced-oscillatory motion of a Concanavalin A (ConA - Succinylated Concanavalin A; Vector Laboratories Inc.) lectin-coated polystyrene bead interacting with the glyprotein protein on the cellular membrane of a Chinese hamster ovary (CHO) cell and monitoring the dynamics of the binding of the bead onto the cell membrane. By measuring the amplitude of the oscillatory motion as a function of frequency and time over a frequency range (~10Hz to 600Hz), we analyzed the dynamics of ligand-receptor interactions on cell membranes and measured the force constant (or the elastic constant) of such interaction.

2. Experimental configuration and procedures

A schematic diagram of the optical setup is shown in Fig. 1. A laser beam (λ=532nm from a solid-state diode-pumped, frequency-doubled Nd:YVO₄ laser; Verdi, Coherent Inc.) was expanded and collimated via a spatial filter/beam expander (SF/BE) and divided into two, namely Beam I (denoted B1) and Beam II (denoted B2) with approximately equal optical power (~5.6mW) by a cube beamsplitter (CBS1). Beam I in one of the arms was chopped by an optical chopper. The two beams were recombined by a polarizing beam-splitting (PBS) cube and directed into an oil-immersion objective lens (100×, NA=1.25 Olympus Inc.) for optical trapping of micro-particles (2.83μm diameter polystyrene bead; Polysciences Inc, Warrington, PA) suspended in either de-ionized water or in a proper buffer solution inside a closed chamber (50mm × 18mm × 0.2mm between a pair of cover glasses) mounted on a sample stage. At the focal plane of the microscope objective Beam I was shifted about 1.6μm away from Beam II by adjusting the orientation of mirrors M1, M2, and M3. Besides, the optical powers of the two beams exiting the microscope objective lens were equalized by fine tuning the two half-wave (λ/2) plates, one in each arm. A third beam (Beam III, λ=633nm, from a He-Ne laser; denoted B3) was also directed into the same objective through a pair of telescope lenses (L1 and L2), a dichroic mirror (D1), a mirror (M3), and a cube beam-splitter (CBS2) to track the position of the trapped particle via a quadrant photodiode (QPD; Model: OT301, Precision Position Sensing Amplifier, ON-TRAK, CA) in conjunction with a lock-in amplifier (Model 7225, DSP, Signal Recovery). The telescopic lenses L1 and L2 were used to fine-tune the focal position of the tracking B3 inside the sample chamber at the output of the microscope objective lens. Beam III was kept at very low power (~0.1mW) to minimize its effect on the motion of the trapped particle. Forward-scattering of the probe beam (i.e., Beam III) by the trapped particle was collected by a condenser and projected onto the QPD to track the displacement of the trapped bead in the transverse x-y plane. The output of the QPD was fed to the lock-in amplifier whose reference signal was provided by the optical chopper. A narrow band-pass filter (N3) was placed in front of the QPD to transmit the probe beam (B3), block-off theextensively?for the study trapping beams (B1 and B2), and minimize the back ground. The sample was illuminated by a white light source for incoherent imaging through the microscope objective lens, a cube beam-splitter (CBS2), and a mirror (M4), via a CCD camera. Two notch filters (N1 and N2) were placed in front of the CCD camera to block-off both the trapping and the probing laser beams. When Beam I was chopped at a selected chopping frequency (typically ~ 10Hz to 600Hz), the trapped particle was forced by the modulating optical potential to oscillate at the same frequency in the steady state; the amplitude of the oscillating particle at each driving frequency was measured by the QPD in conjunction with the lock-in amplifier with an appropriate integration time.

3. Experimental results and discussion

We trapped a polystyrene micro-sphere (d = 2.83μm) suspended in de-ionized water (viscosity η = 7.0×10^-4 Nsm^-2) with the dual-tweezers setup described above, chopped Beam I at selected frequencies (~ 10Hz to 600Hz), and measured the relative amplitude of the particle oscillation as a function of frequency. The experimental data and the theoretical fits of the normalized oscillation amplitude vs. frequency for optical forced oscillation under different optical power of a polystyrene micro-sphere (diameter = 2.83μm) suspended in de-ionized water is depicted in Fig. 2, where the solid lines are the theoretical fits based on Eq. (7) (given in the appendix) with the transverse optical force constant k as the single fitting parameter. In Eq. (7), m is the mass of the particle, the angular frequency ω = 2πf, where f is the oscillation frequency in Hz, the parameter β = 6πηr, where η is the viscosity of the surrounding medium, and r is the radius of the particle. Experimental parameters associated with our experiments include the diameter of polystyrene bead d = 2r = 2.83μm, viscosity of de-ionized water η = 7.0x10^-4 Nsm^-2, viscosity of Dulbecco’s Modified Eagle medium (DMEM) η = 8.4×10^-4 Nsm^-2, mass of the polystyrene micro-sphere m = 4πρr³/3 = 1.32×10^-14kg. The transverse optical force constant k deduced from the best fits were 1.74 pN/μm, 4.06 pN/μm, and 5.73 pN/μm for trapping power of 5mW, 10mW, and 15mW, respectively. Increasing trapping power leads to a proportional increase in the values of force constant as is depicted in Fig. 3.

 

Fig. 1. A schematic diagram of the experiment setup.

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Fig. 2. Normalized oscillation amplitude vs. frequency for optical forced oscillation of a polystyrene micro-sphere (diameters = 2.83μm) in de-ionized water when the optical power was 15mW (open square), 10mW (open triangle), and 5mW (open circle). The solid lines are the theoretical fits based on Eq. (7) given in the appendix.

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Fig. 3. The transverse force constant as a function of laser power for optical trapping of a polystyrene particle (d = 2.83μm) suspended in de-ionized water.

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Using the same technique, we also measured the transverse optical force constants for two other cases, namely, an uncoated polystyrene particle (diameter = 2.83μm) in de-ionized water (viscosity η = 7.0×10^-4 Nsm^-2) and a ConA (lectin)-coated polystyrene particle (diameter = 2.83μm) in DMEM (viscosity η = 8.4×10^-4 Nsm^-2). Our experimental results are summarized in Table I for three different cases, namely an uncoated polystyrene particle in de-ionized water, an uncoated polystyrene particle in DMEM solution, and a ConA-coated polystyrene particle in DMEM solution. In all cases the diameter of the particle was 2.83μm, and the optical trapping power was 5mW in each trapping beam.

To study the dynamics of the interaction of ConA (lectin) with glycoprotein on the plasma membrane of a Chinese hamster ovary (CHO) cell we repeated the experiment by bringing a CHO cell in the sample chamber to the vicinity of a ConA-coated polystyrene particle as the particle was trapped and oscillated by the twin tweezers. For comparison, identical experiments were also repeated with a Bovine Serum Albumin (BSA)-coated polystyrene particle. A comparison of the time dependence of the oscillation amplitude of a ConA-coated bead (denoted by solid squares) with that of a BSA-coated bead (denoted by *), both oscillating at 50Hz in the vicinity of a CHO cell, are depicted in Fig. 4(a). In the case of the ConA-coated particle, the oscillation amplitude decayed from an initial steady state value to a lower final steady state value as would be expected from the interaction between ConA and glycoprotein on the cell membrane. In contrast, no such decay in oscillation amplitude was observed in the case of the BSA-coated bead. When the final steady state was reached, we measured the relative oscillation amplitude as a function of frequency over a frequency range from 10Hz to 70Hz for the case of the ConA-coated particle; the experimental data are depicted in Fig 4(b) along with the theoretical fit, based on Eq. (10) given in the appendix, with a single fitting parameter k′ = (3/2)k + k_int, where k is the transverse optical force constant of each trap in the absence of the cell, and k_int is the force constant representing the particle-cell interaction. [Goodness of the fit: R²: 0.9149, RMSE (root mean squared error) = 0.06191]. From the best fit, we obtained k′ = 5.76 pN/μm and k_int = k′ - (3/2)k = (5.76 - 3.23) pN/μm = 2.53pN/μm. When a particle is simultaneously acted upon by an optical force and a cellular force, we assume that the interaction of the particle with the cell can be represented by another linear spring in parallel to the linear spring associated with the optical trap as is illustrated in Fig. 6 given in the appendix. The relatively poor fit of the experimental data to the theoretical curve (compared with the data in the absence of the cell) is attributed to the volatility (or semi-fluidity) of the cell membrane which is not taken into account in our simplified (linear spring) model. Furthermore, the results are only semi-quantitative in a relative sense, since they varied fairly widely from cell to cell and also for particle-cell interaction at one specific position of the cell compared with that at another position of the same cell.

Table 1. A comparison of the transverse force constant for three different cases. In all cases the diameter of the polystyrene particle was 2.83μm, and the optical trapping power was 5mW

View Table

 

Fig. 4. (a) The time-dependence of the relative amplitude of a polystyrene particle (diameter = 2.83μm) executing optical forced oscillation at 50Hz in the vicinity of a CHO cell; the data for ConA-coated particle are denoted by solid squares and the data for a BSA-coated particle are denoted by “*”.

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Fig. 4. (b).Normalized amplitude versus frequency for the oscillation of a ConA-coated particle in the vicinity of a CHO cell after the final steady state was reached; the solid line is the theoretical fit based on Eq. (10) given in the Appendix.

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To further investigate how the cytoskeleton would affect the interaction force constant determined by the approach described above, we repeated the experiment and treated the CHO cell with Latrunculin A, a drug used extensively as an agent to sequester monomeric actin in living cells and to destruct the cytoskeleton. The dynamic response of the CHO cell membrane to Latrunculin A was probed by a ConA-coated bead laterally attached to the CHO cell membrane and oscillating at 50Hz (driven by the twin tweezers). Fig. 5(a) shows the oscillation amplitude as a function of time prior to the drug-treatment (denoted by open squares) and the corresponding data (denoted by “*”) taken one minute after the drug treatment with the concentration of Latrunculin A = 50μM (Latrunculin A/DMEM). We repeated the same experiments with a much higher concentration of Latrunculin A [125μM (Latrunculin A/DMEM)] and a much longer treating time (5 minutes); the oscillation amplitude as a function of time before the drug-treatment (denoted by open squares) and after the drug treatment (denoted by “*”) are shown in Fig. 5(b). The significantly larger oscillation amplitude after the drug treatment reflects the softening of the cellular membrane due to the destruction of the cytoskeleton by the drug.

4. Summary and conclusion

We have developed an optical forced oscillation system via a set of twin tweezers to study the interaction of a ConA (lectin)-coated polytyrene particle with glycoprotein on the cellular membrane of a Chinese hamster ovary (CHO) cell. From the relative oscillation amplitude as a function of frequency in the range of 10Hz to 600Hz, the force constant of the interaction was determined to be ~ 2.5pN/μm. At the oscillation frequency of 50Hz, we also measured the oscillation amplitude as a function of time to monitor the dynamic of the ConA (lectin)-glycoprotein interaction before and after the CHO cell was treated with 50μM of Latrunculin A. About one minute after the drug treatment, the relative oscillation amplitude increased to nearly 3 times the original value in about 50 seconds, revealing the softening of the cellular plasma membrane by Latrunculin A. In summary, we have demonstrated that the measurement of the amplitude of optical force oscillation (as a function of frequency and/or as a function of time) of a protein-coated particle interacting with a cellular membrane can be used to study semi-quantitatively the dynamics of protein-protein interaction at the cellular membrane in real-time.

 

Fig. 5. (a). Relative oscillation amplitude as a function of time prior to the drug-treatment (denoted by open squares) and 1 minute after the CHO cell was treated with 50μM of Latrunculin A (Latrunculin A/DMEM), denoted by “*”; the solid lines represent the average general trend of the experimental data.

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Fig. 5. (b). Relative oscillation amplitude as a function of time prior to the drug-treatment (denoted by open squares) and 5 minute after the CHO cell was treated with 125μM of Latrunculin A (Latrunculin A/DMEM), denoted by “*”.

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Appendix

A simple theoretical model of optical forced oscillation via a set of twin optical tweezers.

In the linear spring model of optical trap, the one-dimensional equation of motion of a particle trapped in a set of twin tweezers [See Fig. 6 below] can be written as

(1)$$m\ddot{x}=-\beta \dot{x}-k_1\left(x-x_1\right)-k_2\left(x-x_2\right)$$

where m is the mass of the particle, the parameter β = 6πηr, where η is the viscosity of the surrounding medium, and r is the radius of the particle, k_i and x_i are the force constant and the center position of i^th optical tweezers, respectively; for a twin set of symmetric optical tweezers, k₁ = k₂ = k. When one of the trapping beams (say beam #1) is chopped periodically at a fundamental frequency ω while the second beam remains stationary,

(2)$$k_1=\frac{k(e^{i\omega t}+1)}{2}$$

(3)$$k_2=k$$

and Eq. (1) becomes

(4)$$m\ddot{x}=-\beta \dot{x}-\frac{k(e^{i\omega t}+1)}{2}(x-x_1)-k(x-x_2)$$

Ignoring the transient solution and the higher harmonic frequency components, we assume a steady state solution x(t) of Eq. (4) in the form of

(5)$$x(t)=D\left[e^{i(\omega t-\phi )}\right]+x_0$$

where D, ϕ, and x₀ are the oscillation amplitude, the relative phase (with respect to the phase of the chopping signal), and the center position of the oscillating particle.

Solving the differential equation [Eq. (4)] with the steady state solution [Eq. (5)], we obtain

(6)$$D(\omega )=\frac{\frac{1}{3}kd}{\sqrt{{(\frac{3}{2}k-m{\omega }^2)}^2+{(\beta \omega )}^2}}$$

(7)$$\frac{D(\omega )}{D(0)}=\frac{\frac{3}{2}k}{\sqrt{{(\frac{3}{2}k-m{\omega }^2)}^2+{(\beta \omega )}^2}}$$

When a cell is brought into contact with the particle trapped in the equilibrium position of the stationary optical tweezers in the right (see Fig. 6), we assume that the particle-cell interaction can be represented by another linear spring (with spring constant k_int) in parallel to the optical trap. Eq. (1) above can now be re-written as

(8)$$m\ddot{x}=-\beta \dot{x}-k_1(x-x_1)-k_2(x-x_2)+k_{\mathrm{int}}(x-x_2)$$

Eq. (4) becomes

(9)$$m\ddot{x}=-\beta \dot{x}-\frac{k(e^{i\omega t}+1)}{2}(x-x_1)-(k+k_{\mathrm{int}})(x-x_2)$$

and Eq. (7) is replaced by

(10)$$\frac{D(\omega )}{D(0)}=\frac{\frac{3}{2}k\text{'}}{\sqrt{{(k\text{'}-m{\omega }^2)}^2+{(\beta \omega )}^2}}$$

where k′ = (3/2)k + k_int.

 

Fig. 6. A simplified linear spring model of a particle simultaneously acted upon by optical forces from a set of twin optical tweezers and a cellular interactive force when a cell is in contact with the particle in the equilibrium trapping position of the tweezers on the right.

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Acknowledgments

This work was supported by the National Science Council of the Republic of China Grants NSC 95-2752-E010-001-PAE, NSC 94-2120-M-010-002, NSC 94-2627-B-010-004, NSC 94-2120-M-007-006, NSC 94-2120-M-010-002, and NSC 93-2314-B-010-003, and the Grant 95A-C-D01-PPG-01 from the Aim for the Top University Plan supported by the Ministry of Education of the Republic of China.

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