where h is the height and Nflat and Ntilt are the total numbers of tilted and flat MT segments (as given in Table 1), respectively.Kerssemakers et al. 10.1073/pnas.0510400103. |
Fig. 5. Calibration of the FLIC curve. (A) Determination of the FLIC periodicity L via the tilt angle dependence of the x-y-projected intensity periodicity L for 13 MTs. (B) The FLIC parameters changed because of blurring as a function of tilt angle (symbols, data for 320 MTs). Modeled trends using R = 15% and g = 800 nm (solid lines) yielded the best fits to the experimental data. (C) FLIC curve (thin line, height above the Si/SiO2 surface given on the upper x axis) as derived from Eq. 1 using the parameters determined in A and B. Converting the calculated FLIC curve into the x-y-plane and convolving it with the point-spread function (PSF) (dotted line) yielded a good fit (thick line) to an experimentally observed intensity profile of a 10° tilted MT (open squares).
Movie 1. Measuring the height of MTs gliding over a kinesin-coated surface by FLIC microscopy. An immobile MT was fixed in a dilute mesh of agarose under a shallow angle close to a reflecting Si/SiO2 surface. It crossed the vertical FLIC pattern and showed the characteristic interference pattern when imaged on the camera. Unhindered by the agarose, MTs glided over the kinesin-coated surface at constant height. After the appropriate corrections, a comparison of the interference pattern with the intensity of the gliding MTs allowed for the nanometer height measurements.
Movie 2. FLIC imaging of three crossing MTs that glide on a kinesin-coated Si/SiO2 surface. Because the brightness of the MTs scaled with their distance from the surface, it could be easily seen which MTs were passing over the others. For clarity, the video was stopped in the middle.
Movie 3. FLIC imaging of gliding and surface-immobilized MTs at three different heights. The immobile, tilted MT on the left was fixed in a dilute mesh of agarose under a shallow angle close to a reflecting Si/SiO2 surface. It crossed the vertical FLIC pattern and showed the characteristic interference pattern that served as a calibration ruler. Unhindered by the agarose, kinesin-transported MTs glide over the surface. They were obviously brighter, and thus higher, than a surface-immobilized MT (lower right) that had been cross-linked to the surface before. The height difference that led to such an intensity change was »20 nm.
We use tilted MTs to quantify the various parameters in Eq. 1.
To determine the FLIC spacing L, we performed 100-nm-spaced z-sectioning on tilted MTs. From such side views (Fig. 1C) we determined the tilt angles a. From the corresponding intensity profile in the x-y-plane the modulation period L was taken from the average spacing of the first three peaks, obtained via Gaussian fitting. The FLIC periodicity L was then determined by L = L·tan (a). Fig. 5A shows data points of 13 MTs with tilt angles between 9° and 34°. The FLIC periodicity was determined to be 238 ± 10 nm (mean ± SD) for our microscopy system. The difference of this value to the expected value of ~230 nm (mentioned below Eq. 1) indicates the role of the marginal rays when using an objective with high numerical aperture (1).
To determine R and g we used the measured maximum and minimum values i0, i1, and i2 in the x-y-projected intensity modulation (as indicated in Fig. 1B). From Eq. 1 values of R and g were derived for any given tilt angle a, by Ra = i1/i0 and ga = L/ln (i0/i2). However, the recorded images of the tilted MTs are blurred along the length of the MT by the finite resolution of the optical imaging system according to i(x) = I(x)*PSF with PSF being the in-plane point-spread function. Here, it is sufficient to consider the in-plane PSF instead of the 3D PSF because the structures of interest, namely i0, i1, and i2 are vertically separated by not more than ~240 nm. This length is significantly shorter than the vertical elongation of the 3D PSF (i.e., the axial Rayleigh-criterion of 2 l n/NA2) of our imaging system being in the order of 1 mm. The blurring effect can easily be seen in Fig. 1B for increasing tilt angles. As a consequence, Ra and ga strongly varied with tilt angle (symbols in Fig. 5B). These trends can also be modeled: the intensity profile i(x) for any tilt angle a is obtained by (i) inserting h = x·tan(a) in Eq. 1 by using the spacing L determined before, (ii) choosing a set of R and g, and (iii) convolving the result with the PSF (derived by averaging many cross-sections of flat-lying MTs). From the maxima and minima of modeled i(x) the values for Ra and ga could be calculated. By comparing the modeled trend to the experimental data, we found that R = 15% and g = 800 nm (characterized by the solid lines in Fig. 5B) yielded the best fit and these values were consequently chosen for the characterization of our FLIC curve. Fig. 5C shows an example, where the derived parameters are used to reproduce the x-y-projected intensity profile of a MT with tilt angle a = 10°.
The normalized shape of the FLIC curve I(h)/I0 is an invariant for a given imaging system. Therefore, once all of the other parameters were known, only the proportionality factor I0 needed to be determined for each experiment. This was done by again using tilted MTs that were imaged simultaneously with the fluorescent signals from motile MTs. The deconvolved maximum intensity value I0 was determined from the first peak in the intensity modulation of the tilted MTs. We note, that to compare the intensities of gliding and tilted MTs, it was crucial that both types of MTs were illuminated equally and that bleaching was minimized. Thus, we only compared MTs within one image and limited the total illumination time before a measurement to < 1 s. The slightly uneven illumination over the field of view was quantified and corrected for using images of fluorophore-covered surfaces.
Estimation of Statistical Errors
Tilted MT Method
. We measured the height of gliding MTs above a surface via the ratio h of fluorescent intensities of flat and tilted MTs. After correcting for inhomogeneous illumination, the error for a single measurement was mainly caused by variations in the fluorescence intensity for both types of MTs. For directly immobilized or moving MTs, we always divided the MTs into segments of equal length (400 nm). By imaging MTs on glass, we found that the intensities of such segments varied by d= ± 22% (95% confidence range, n = 800). For tilted MTs, we measured the local maxima (and minima) in the zebra stripes. Because the MTs crossed the FLIC fringes under an angle, only a local segment of a few hundred nanometers effectively contributed to each maximum. Thus, it is reasonable to assume the same d for the intensity of one maximum value for one tilted MT. That this is a realistic estimate is shown by the measured values for the ratio Ra = i1 /i0 (solid markers in Fig. 5C). For a single MT we expect an error of Ö2 *22% = 30%. At an average angle (e.g., 15°), we indeed found the points scattering according to R15° = 0.45 ± 0.15, or ~30%.Intensities of both flat MTs and tilted MTs were averaged per image. The resulting intensity ratios were then averaged for all images. The error can thus be easily deduced by: where h is the height and Nflat and Ntilt are the total numbers of tilted and flat MT segments (as given in Table 1), respectively.
Because of the large number of MT segments used for the analysis of the fluorescence intensity, the resulting statistical errors of the measured heights are extremely small and very subtle changes in gliding height can be identified within one assay (as seen in the experiments with crossing MTs and when exchanging the nucleotide composition within one motility assay). However, when repeating similar experiments at different times in different flow chambers the variation of the results increases, probably because of slight variations in the experimental parameters such as SiO2 thickness, temperature, etc. We therefore repeated the standard measurement of the MT elevation on kinesin-1 by using the casein assay four times. The SEM obtained from those experiments was 1.9 nm and is indicated as the principal error for the presented height measurements.
ABCD Method.
The local SiO2 heights of the patches are known with a precision of ± 5 nm (as derived from imaging ellipsometry measurements over the original wafer). For the ABCD method this is the largest error contribution and is hence used as the error margin.1. Lambacher A, Fromherz P (2002) J Opt Soc Am B 19:1435-1453.