Measurement of hindered diffusion in complex geometries for high-speed single-molecule experiments

In a high-speed single-molecule experiment, a protein is tethered between two substrates that are manipulated to exert force on the system. To avoid nonspecific interactions between the protein and nearby substrates, the protein is usually attached to the substrates through long, flexible linkers. This approach precludes measurements of mechanical properties with high spatial and temporal resolution, for rapidly exerted forces are dissipated into the linkers. Because mammalian hearing operates at frequencies reaching tens to hundreds of kilohertz, the mechanical processes that occur during transduction are of very short duration. Single-molecule experiments on the relevant proteins therefore cannot involve long tethers. We previously characterized the mechanical properties of protocadherin 15 (PCDH15), a protein essential for human hearing, by tethering an individual monomer through very short linkers between a probe bead held in an optical trap and a pedestal bead immobilized on a glass coverslip. Because the two confining surfaces were separated by only the length of the tethered protein, hydrodynamic coupling between those surfaces complicated the interpretation of the data. To facilitate our experiments, we characterize here the anisotropic and position-dependent diffusion coefficient of a probe in the presence of an effectively infinite wall, the coverslip, and of the immobile pedestal.


Introduction
A protein under tension exhibits both entropic and enthalpic elasticity, a behavior that can be measured by observing the elongation of a single molecule while applying mechanical force. In such an experiment, the molecule is placed between two substrates, at least one of which is part of an elastic transducer through which forces can be delivered, for example an optically trapped, micrometer-sized bead. To avoid non-specific interactions between the protein and the substrates to which it is attached, the protein is usually We developed a novel single-molecule assay that did not require long, flexible spacers. The protein was instead stretched directly between a diffusing probe-a 1 µmdiameter plastic bead to which force could be applied by optical tweezers-and an immobile glass pedestal-a 2 µm-diameter bead fixed to the coverslip. The protein's ends were attached to the two beads through distinct, short, and relatively inelastic linkers. 4 This approach allowed us to characterize the equilibrium mechanics of protocadherin 15 (PCDH15), a protein whose properties implicate it as part of a molecular spring important for hearing 2 . Determining the entropic and enthalpic stiffness of the protein is crucial for our understanding of the molecular basis of mechanotransduction by the inner ear. Human ears can detect sounds at frequencies up to 20 kHz, and some bats and dolphins have a hearing range exceeding 200 kHz. The protein machinery that underlies hearing must therefore be capable of responding to very fast stimuli that likely produce mechanical responses far from thermal equilibrium. For two reasons, this highfrequency behavior has not been explored through single-molecule experiments. First, in the presence of flexible linker molecules, high-frequency force stimuli are largely filtered before they can elongate a protein of interest. Second, even in the absence of flexible linkers, the mechanical response of a protein is filtered owing to the drag on the bead and the stiffness of the optical potential that confines it. If these filtering effects are not too large compared to the time constant of the protein's response, and if the drag on the bead is known, it is nevertheless possible to compensate for the filtering. In this study we characterize the anisotropic and position-dependent diffusion coefficient of a bead in our single-molecule assay in the presence of an effectively infinite wall, the coverslip, and of an immobile spherical obstacle, the pedestal. The results should facilitate analysis of high-speed single-molecule experiments relevant to auditory transduction.

Correction of the position signal for light scattered by the pedestal
Determining the drag near a coverslip and pedestal requires high-precision measurement of the three-dimensional diffusion of a probe confined in a weak, position-sensing optical trap. The probe's position can be estimated with sub-nanometer precision and microsecond temporal resolution by interfering the light scattered forward by the probe with the unscattered portion of the trapping beam on a quadrant photodiode 3 (Fig. 1A).
The diode's difference signals are then linearly related to the probe's position along the two axes perpendicular to the optical axis, and the signal summed over all four quadrants is proportional to the probe's axial position.
When the probe and pedestal are in close proximity-as is the case in singlemolecule experiments without long linkers-the position-sensing beam is scattered not only by the probe, but also by the pedestal (Fig. 1B). Although this effect complicates estimation of the probe's position, the diode's total signal !"!#$ can be approximated to first order as the sum of two independent signals 4,5 : the signal %&'&(!#$ owing to the pedestal in the absence of the probe and the signal %)"*& owing to the probe in the absence of the pedestal: Here the vectors represent the position coordinates of the probe and pedestal, which are the displacements of the respective objects from their positions when the photodiode's output is zero. The offset %&'&(!#$ is sensitive to the precise value of the distance as well as to the shape of the pedestal itself, and must therefore be determined at the beginning of each experiment.
In a typical experiment, the pedestal is fixed at least 1.5 µm from the focal spot of the position-sensing beam, a distance determined by the radii of the pedestal and probe.
The probe's diffusion is confined by the beam's trapping potential and is centered on the focal spot. The pedestal's signal thus constitutes a constant offset added to the probe's signal. If the magnitude of this offset is known, it can be subtracted from the total signal to yield the signal of the probe alone 2 .
To visualize the contributions of the two independent signals, we independently recorded the signals for displacements of the probe and the pedestal, then displayed them offset by 1.5 µm relative to one another (Fig. 1C). This procedure reflected the case in which the probe was at the center of the position-sensing optical trap, defined as x = 0, In order to demonstrate that we could successfully correct for the influence of the pedestal, we next used the stimulus trap to hold the probe at the center of the positionsensing trap (x = 0). We recorded the photodiode's total signal and recovered the position of the probe by subtracting the offset caused by the pedestal. The position signal after compensation was nearly zero (Fig. 1E). If the offset correction was not performed and the total signal on the detector was calibrated without subtraction of the pedestal's influence, a significant systematic position error arose that depended sensitively on the distance between the pedestal and the center of the position-sensing optical trap. All the data presented in the remainder of this work were corrected by this means.

Localization of the pedestal's surface by thermal-noise imaging
Before assessing the hydrodynamic drag near a pedestal, it was necessary to localize the pedestal's surface. We accomplished this by the super-resolution technique of thermal-noise imaging 6 . The spatial probability density of a probe diffusing in a weak optical trap was a three-dimensional Gaussian distribution with an ovoid iso-probability surface ( Fig. 2A). When a pedestal intersected the optical trap, a portion of its volume 7 became inaccessible to the probe's diffusion: the forbidden volume in the probe's spatial probability density then provided a negative image of the pedestal (Fig. 2B). We computed a line profile along the x-axis through the probe's spatial probability density and converted the result by Boltzmann statistics to an energy landscape (Fig. 2E). We defined the wall of infinite energy as the impenetrable boundary of the pedestal and set x = 0 at this location.
Diffusion was further restricted when the probe was attached to the pedestal by a short peptide that represented the concatenation of the two linkers used in an experiment to attach a PCDH15 monomer to the probe and pedestal (Fig. 2C). In an actual experiment, the monomer was attached at each end by one of the linker peptides ( Fig. 2D). In both instances, the energy functions became steeper as the probe was confined both by the optical trap and by the tether (Fig. 2E). The slopes of the three energy functions defined the position-dependent forces exerted on the probe (Fig. 2F).

Determination of local diffusion constants
When a bead diffuses close to a boundary, its mean squared displacement becomes  (Fig. 3A) 11,12 . We then made use of the fact that, for each voxel, the slope relating the mean squared displacement along each axis to the time lag is twice the probe's local diffusion constant along that axis. Although the resulting three-dimensional spatial map of diffusion constants was limited in spatial extent by the width of the trapping volume, larger volumes could be explored by displacing the optical trap in steps smaller than the width of the trapping volume and recording partially overlapping diffusion maps that were subsequently fused.
In our single-molecule assay of PCDH15 molecules, the protein was stretched along the x-axis. Because we were therefore mainly interested in how the associated diffusion constant Dx changed with extension from the pedestal, we moved the optical trap along that axis in 100 nm steps and determined the diffusion constant at each position (Fig. 3B). Although the focal spot of the optical trap remained fixed during each measurement, the trap was weak enough that the probe could diffuse along all three axes with respect to that point. We also computed the diffusion constants for motion along the y-axis, tangential to the pedestal but at a fixed height above the coverslip (Dy, Fig. 3C) as well as those along the z-axis, tangential to the pedestal but perpendicular to the coverslip (Dz, Fig. 3D). These results were determined for a probe maintained at a distance of 500 nm from the coverslip, so that the average z-position of the probe corresponded to the equator of the pedestal (Fig 3E).
Assuming that the coverslip acted as an infinite wall to which the probe's diffusion Voxel size contributed to the resolution of our measurements, for smaller voxels permitted a more granular mapping of the local diffusion and drag. However, this benefit had to be balanced with the need to obtain sufficient data points from each voxel, the probability of which decreases as voxel size declines 12 . Another consideration for measurements of local diffusion constants was the choice of time lags at which to measure the mean squared displacement. This value plateaus beyond a characteristic autocorrelation time = ( 5 ) ⁄ as a result of the probe's confinement in the optical trap, resulting in a measured value smaller than that for a free particle. To capture the motion of the probe while it approximated free diffusion, the time lag accordingly had to be much smaller than this autocorrelation time. (2 ) : in which k is the spring constant of the optical trap, which in our experiments was 4 µN·m -1 along the x-and y-axes and 1.2 µN·m -1 along the z-axis. a is the displacement from the trap's center, t the time lag, the viscous drag coefficient of the probe given by Stokes' law, and D the local diffusion constant. kB and T are respectively the Boltzmann constant and thermodynamic temperature. For the ratio to remain small such that essentially free diffusion occurs, the time lag had to be set much smaller than the characteristic drift time . Combining the two effects of optical trapping, the minimum of t and tD determines the timescale at which the probe's motion deviated from free diffusion.
If t < tD, as was the case in our system for excursions of less than 150 nm from the trap's center along any axis, then the influence of the gradient force on the mean squared displacement was negligible. If instead t > tD, then the measured value would have exceeded that of a freely diffusing particle for intermediate time lags.

Methods
The photonic-force microscope used in these experiments was capable of measuring the position of a micrometer-sized probe bead with an integration time of 1 µs, sampled at