Cells dynamically regulate adhesions and ultimately traction force in response to mechanical stimuli [6
]. This process can be seen as a feedback system consisting of a cycle of force transduction, biochemical signalling and generation of a mechanical response. During neuronal development, cells can adhere to each other through cell adhesion receptors (e.g.
apCAM) by mutually adapting their mechanical behaviors. To mimic a cell-cell interaction, we developed a closed-loop force measurement system which adjusts the optical forces applied to each bead to compensate for the cell's mechanical response. Closed-loop force measurements offer additional advantages such as increasing the effective trap stiffness at low frequency and lengthening the experiment [25
]. Here we describe the implementation of multiplexed closed-loop force measurements, quantify the efficacy of our system and apply the technique to a live cell.
Our proportional-gain position clamp is implemented in MATLAB. The position of multiple optically trapped beads is determined using our centroid tracking algorithms. The optical traps are then steered according to
is the trap position, Δt
is the feedback loop time, bead
is the bead position, Gp
is the proportional gain and set point
is the setpoint position. Each bead is treated independently and simultaneously. The feedback loop time ranges from 0.1 to 0.3 s depending on the size of the field of view and the number of traps.
4.2. Quantification of feedback loop efficiency
To test our position clamp, a piezoelectric stage applies a sinusoidal drag force to ten beads trapped at about 10 μm from the bottom coverslip ((Media 2)). We tune the drag force in order to mimic the force and time scales of the cell. In the experiment shown in , beads, strongly coupled to the cell, were moving at about 3 μm/min within the trap resulting in a loading rate of about 0.55 pN/s. To achieve a similar velocity and loading rate, the beads are suspended in a poly(ethylene glycol) solution with a viscosity η = 0.08 Pa.s, about 80 times higher than water. The trap stiffnesses are about 15 pN/μm. With these parameters, the average velocity is about 2 μm/min and the maximum loading rate is 0.5 pN/s. A setpoint for each trap is defined and the proportional gain is set to 0.3, the highest possible value which does not cause the bead to oscillate. The feedback loop time is measured to be 0.26 s, the rate limiting steps being image processing and hologram calculation. Nine of the beads are position-clamped and one bead is held in a stationary trap as a reference to account for external forces and any possible instrumental drift ().
Fig. 3 Position clamp efficiency (a) (Media 2) A sinusoidal drag force is applied to 10 trapped beads with holographic optical tweezers. Nine beads are position-clamped (Gp = 0.3) and one is in a stationary trap (Gp = 0) (bottom center). The arrows represent (more ...)
Under the sinusoidal drag force, the root mean-squared displacement of the bead held in a stationary trap is 236 nm and the average root mean-squared displacement of the position-clamped beads is 48 ± 8 nm (average ± standard deviation) (). To quantify the frequency dependence of the feedback system, the power spectral density of each clamped-bead is compared to the power spectral density of the reference bead as shown in . For frequencies lower than 0.1 Hz, the amplitudes of the displacements of the position-clamped beads are attenuated. At the lowest frequency, the amplitude of the bead motion is attenuated thirty-fold. At higher frequencies, the position clamp is ineffective.
4.3. Modeling the feedback loop efficiency
To understand the measured position clamp efficiency, we derive the expected power spectral density of a position-clamped bead with respect to the power spectral density of a reference bead in the same conditions. Newton's second law for a trapped bead undergoing an external force (ignoring inertia) is
is the hydrodynamic drag and Fext
is the external force (drag force and thermal noise). In a proportional-gain position clamp, the trap is steered according to Eq. (1)
. The time delay, Δt
, between acquisition of the image and the update of the hologram is readily accounted for in the frequency domain. Fourier transforming Eqs. (1)
, we get
where τ0 = γ/k is the relaxation time of the trap.
Combining Eqs. (3)
, we find the power spectral density of a position-clamped bead with respect to the power spectral density of a reference bead is
To compare our experimental data to our model, we use Δt = 0.26 s, τ0 = 0.10 s (calculated from the measured trap stiffness and the Stokes drag from measured viscosity) and Gp = 0.3 (, solid line). Without any free parameters, we found a good agreement between our model and the experimental data.
Our video-based feedback is slower than feedback systems that use a quadrant photodetector for motion detection and can operate in the KHz range [8
]. However, quadrant detector systems cannot readily detect more than two beads. In conditions mimicking the time and force scales of the cell, our feedback system operating at 4 Hz can control the motion of nine beads up to 0.1 Hz. Based on our model, control of bead motion up to 1 Hz may be achieved by speeding up the feedback loop to 30 Hz, which is within the range of speeds for conventional CCD cameras, as indicated by the dashed line in .
4.4. Closed-loop force measurements on live cells
We apply our feedback system to measure coupling dynamics on a live cell. Two apCAM-coated beads are placed near the leading edge of a growth cone with holographic optical tweezers. One of the beads is controlled actively while the other one is kept in a stationary trap as a reference ((Media 3)). To recover beads that move too far, the position clamp switches reversibly to a force clamp whenever the forces generated by the cell reach the maximum optical force (). In a force clamp, the trap is kept at a fixed distance from the bead so that the maximum optical force is constantly applied to the bead. The direction of the optical forces points from the bead to the setpoint. In this experiment, the measured stiffness of each trap is 75 pN/μm and the proportional gain is set at 0.3.
Fig. 4 Closed-loop force measurement on a live cell (a) (Media 3) Left bead is in a fixed trap (Gp = 0), right bead is in an active trap (Gp = 0.3). Both beads are pulled toward the axon shaft (arrow represents the optical forces). Movie: sped up 30 times, 42 (more ...)
When the bead is held on the membrane of the cell, the optical forces are adapted in response to the forces applied by the cell on the bead. At first, the position-clamped bead remains within 50 ± 38nm from the setpoint while the optical forces are dynamically changing (). In some cases, fast relaxation of the bead into the center of the trap is observed causing the beads to overshoot pass the setpoint. In other cases, if the optical forces reach the maximum value, 75 pN, the feedback loop switches into force clamp. Held under constant force, the bead moves at a constant rate toward the axon shaft revealing a strong mechanical coupling between the bead and the cell. However, transient slippage events are also observed suggesting that the coupling to the underlying cytoskeleton weakens intermittently.