A Method to Quantify Molecular Diffusion within Thin Solvated Polymer Films: A Case Study on Films of Natively Unfolded Nucleoporins

We present a method to probe molecular and nanoparticle diffusion within thin, solvated polymer coatings. The device exploits the confinement with well-defined geometry that forms at the interface between a planar and a hemispherical surface (of which at least one is coated with polymers) in close contact and uses this confinement to analyze diffusion processes without interference of exchange with and diffusion in the bulk solution. With this method, which we call plane–sphere confinement microscopy (PSCM), information regarding the partitioning of molecules between the polymer coating and the bulk liquid is also obtained. Thanks to the shape of the confined geometry, diffusion and partitioning can be mapped as a function of compression and concentration of the coating in a single experiment. The method is versatile and can be integrated with conventional optical microscopes; thus it should find widespread use in the many application areas exploiting functional polymer coatings. We demonstrate the use of PSCM using brushes of natively unfolded nucleoporin domains rich in phenylalanine–glycine repeats (FG domains). A meshwork of FG domains is known to be responsible for the selective transport of nuclear transport receptors (NTRs) and their macromolecular cargos across the nuclear envelope that separates the cytosol and the nucleus of living cells. We find that the selectivity of NTR uptake by FG domain films depends sensitively on FG domain concentration and that the interaction of NTRs with FG domains obstructs NTR movement only moderately. These observations contribute important information to better understand the mechanisms of selective NTR transport.

The position of the glass cover slip was kept fixed, the position of the glass rod was adjusted in x, y and z by the micromanipulator, and the objective's z focus was adjusted as required using the microscope's focus wheel. Prior to coarse approach the position where the glass cover slip interfaced the bulk solution was located: this interface was readily identified by imaging in reflection mode (pinhole size of 5 airy units) as a laterally uniform maximum in intensity at this z position. As the glass rod and, simultaneously, the focus position were lowered the radius of the ring decreased. (B3) Newtonian rings emerged when the hemi-spherical cap was sufficiently close to the glass cover slip (i.e. within a distance of a few microns) for interference of light reflected from the two interfaces to occur; interference fringes were best visible with the objective focusing a few microns below the interface, and the micrograph shows the two surfaces in contact. All scale bars are 100 μm; for comparison, the glass rod has a diameter of approximately 6 mm. 2) before ('pre-bleach') and after ('post-bleach') bleaching of a circular region of 23 μm diameter (indicated by the dashed white circle in the post-bleach image). As in Fig. 2B, the fluorescence intensity at the centre of the plane-sphere contact is already reduced before bleaching. However, bleaching entails a further reduction in fluorescence intensity. This is best seen in the graph which shows the normalised mean fluorescence intensity within the bleached region as a function of time. No recovery is observed in the bleached area, indicating the Nsp1 FG domains do not diffuse laterally.  Figure S1 for the film formation process). Arrows on top of the graph indicate the start and duration of incubation with 2 µM GFP solutions in working buffer; during remaining times, the surface was exposed to plain working buffer. (E) Fluorescence micrographs of GFP NTR on a FG Nsp1 functionalized surface (corresponding to the scenario in (C-D)). A central circular region (45 µm in diameter) was photo-bleached and the presented images were recorded before bleaching (i), just after bleaching (t = 0 s; ii) and 20 s post bleaching (iii); (iv) shows the fluorescence recovery in the bleached area as a function of time. Note that fluorescence signal in this assay stems from GFP NTR located in the FG Nsp1 film as well as in the nearby bulk solution, and that recovery in the bulk solution is faster than the time resolution of this disk FRAP assay. Moreover, the rate of recovery of GFP NTR in the film is also enhanced owing to rapid exchange with the bulk solution. This explains the relatively low apparent degree of photo-bleaching at t = 0 min despite the extended bleach phase (marked in grey in (iv)). From the QCM-D data in (A-D), it is evident that GFP Std did not adsorb to the bare Ni 2+ -EDTA surface or the FG Nsp1 film. In contrast, GFP NTR and GFP Inert did adsorb to the bare Ni 2+ -EDTA surface to different extents, and this binding was largely resistant to rinsing in working buffer. However, such undesired binding to the substrate was largely reduced in the presence of the FG Nsp1 films: whilst GFP Inert showed only minor binding, GFP NTR binding was pronounced yet largely reversible as expected for a specific interaction with FG Nsp1 . The FRAP data in (E) confirm that virtually all GFP NTR is mobile and thus reversibly bound in FG Nsp1 films. Figure S5. Diffusion of GFP Std in Nsp1 FG domain films. Line FRAP data for GFP Std in FG Nsp1 films are displayed analogous to Fig. 6A-C. Note that the fluorescence intensity in the region of overlapping FG domain films is reduced for GFP Std as compared to GFP NTR , owing to its much lower partition coefficient. To enhance the signal, the present measurement was therefore performed with a 10-fold increased bulk concentration of GFP Std (20 µM) and data were averaged over a wider radial range (7.4 µm; horizontal error bars in C) for line FRAP analysis. The best-fit line in B corresponds to D = 5.3 ± 1.9 μm 2 /s, k = 0.91 ± 0.02 and K0 = 0.52 ± 0.06 and the mean of the data in C is 6.5 ± 1.9 μm 2 /s. This value should be considered an estimate because the recovery time (τr = 11 ± 4 ms) was comparable to the bleaching time (27 ms) for this data set.
S7 Figure S6. Glass surfaces are smooth on the nm scale. Atomic force micrographs (0.5  0.5 µm 2 ) of the surface of (A) a glass rod (hemi-spherical part), and (B) a glass cover slip functionalized with EDTA. Insets show height profiles taken along the white dashed lines. The root-mean-square (rms) roughness values of these surfaces are 0.39 nm and 0.26 nm, respectively, demonstrating the glass surfaces are smooth on the nm scale, without EDTA and after EDTA coating. Analysis was performed with a Nanoscope Multimode 8 (Bruker, CA, USA) AFM system. Micrographs of the surface topography were acquired in air using Peak Force Tapping mode using sharpened triangular Si3N4 cantilevers (nominal spring constant 0.06 N/m; NP-S, Bruker). Images were secondorder plane fitted (without noise filtering or sharpening) and surface roughness was analysed using Nanoscope Analysis Software. Intensity profile obtained by normalizing the data in (A) and subsequent averaging along the horizontal axis (black dots). The red line represents the best fit with a sum of five Gaussian 'holes' and is seen to reproduce the data well. (C) Peak width, expressed as 2 (where 2 is the variance), obtained from equivalent fits using a range of bleaching iterations (the red arrow highlights the number of iterations used for line FRAP). Error bars represent standard deviations across multiple images (n = 10 for 10 bleach iterations, n = 2 otherwise). Up to 20 bleach iterations, the peak width does not depend significantly on the number of bleach iterations; this indicates that the bleached line width does not depend on the bleaching level, implying that the bleached line width equals the imaging resolution, over this range. The convolution of the bleached lines with the imaging resolution gives (2 ) 2 = 0e 2 + 0c 2 , and with 0e = 0c , we obtain 0e = 0c = √2 . Averaging the data in (C) up to 20 bleach iterations gives 2 = 0.71 ± 0.06 , and thus 0e = 0c = 0.50 ± 0.04 μm.

RICM analysis of a hemi-sphere pressing on a planar surface
Geometry of the interface. The geometry of the interface is schematically shown in Fig. S8A. In the absence of force applied by the hemi-sphere, the cover slip is ideally flat (ℎ 1 ( ) = 0) and the end of the rod is ideally hemi-spherical (ℎ 2 ( ) = − √ 2 − 2 ). Because the radius of the hemi-sphere is large ( = 3 mm) compared to the size of the area of interest in the image ( max ≈ 100 μm), the shape of the hemi-sphere can be approximated by a parabola for RICM intensity calculation (ℎ 2 ( ) ≈ 2 2 ) and 'nonlocal' curvature effects can be neglected. The phase in the RICM profile can thus be written as where is the refractive index of the medium between the surfaces and λ is the wavelength of light.
Here any defocus is neglected as only very large changes in focus would modify the RICM pattern (vide infra).

S9
Applying a force on the hemi-sphere, two effects will modify the geometry of the setup:  The cover slip will flex at large scale. The change in height due to flexural deformation is (Eq. 6.6.25 in ref. 4) is the flexural rigidity of the cover slip along with its Young's modulus 1 , Poisson ratio 1 and thickness . This equation assumes a point-like contact which is not valid around the contact area but gives a good order of magnitude of the large scale deformation. In particular, the maximum deformation is located at the centre of the cover slip and scales as ℎ flex (0) = − w 2 16 . [S3] With 1 = 72.9 GPa, 1 = 0.208 and = 175 μm (provided by the supplier), we obtain = 0.034 Pa•m 3 . As shown in Fig. S8B (blue line), this deformation is of small amplitude (ℎ flex < 120 nm) for ≤ 10 mN. Moreover, this small vertical deflection is applied over a large distance ( w = 5 mm ≫ 100 μm, over which the RICM pattern is observed). Hence the contribution of the flexural deformation to the change in the RICM pattern can be neglected.  At smaller scale around the contact point, both surfaces are deformed and this deformation can be described by Hertz's theory. Fig. S8B shows that this deformation is much more significant, and hence it is the only one considered in the following analyses.

Hertz contact deformation of the surfaces and resulting RICM pattern.
A detailed treatment of Hertz contact mechanics can be found, for example, in ref. 5. When applying a force , both the hemi-sphere and the coverslip will deform to give rise to a contact area of radius (from Eqs The gap distance is and hence the RICM phase becomes Fig. S8B (orange line) we can estimate that c is in the micrometre range for applied forces in the mN range. A good approximation of the RICM pattern (which is typically fitted over the range 0-100 μm) is then obtained by looking at the limit ≫ c . In this case one obtains Fitting this phase with a quadratic formula including a constant offset Φ 0 = 4 λ ℎ off (as usually done to account for a distance ℎ off between the two reflecting surfaces) one thus gets Using = 3 mm, 1 = 72.9 GPa, 1 = 0.208, 2 = 64 GPa and 2 = 0.2 (as provided by the suppliers), one gets * = 35.56 GPa and The above analysis is confirmed by a more rigorous fitting of the RICM profile derived from the exact expression of the phase (Eq. S6B). The results are shown in Fig. S8C (grey dots). For small forces ( < 1 mN), Eq. S9 fits the simulation well (Fig. S8C, blue line). For larger forces, there is a deviation and the simulation is better fitted by the formula (Fig. S8C, orange line) with an error of less than 0.02 nm across 0 < < 10 mN. The simulations also showed that the fit is robust to small errors (a few percent) in as well as to small differences (a few μm) between the upper surface of the cover slip and the imaging plane (data not shown).
The above analysis was performed for two glass surfaces in direct contact. Equivalent simulations with a 6 nm thick interlayer of il = 1.47 (i.e., the equivalent of two FG Nsp1 films considering also the EDTA surface functionalisation, vide infra) were well fitted by ℎ off ≈ 6 nm × with an error of less than 0.25 nm across 0 < < 10 mN.

Determination of the compressive force and interface geometry from RICM.
Using Eqs. S10 one can readily estimate the compressive force from experimentally measured RICM heights. Eqs. S4 and S5 provide a description of the geometry of the sphere-plane interface for any given applied force.
We note here that the additional FG domain interlayer affects the shape of the ℎ off ( ) curve only marginally. From a comparison of Eq. S10B with Eq. S10A, it is clear that there is a positive constant offset of 6 nm × 1.47 1.334 ≈ 6.6 nm to ℎ off . Any additional change in ℎ off , however, is marginal: over the relevant force range of 10 mN, it remains below 1 nm. For simplicity, we have neglected the additional change, as it is below the resolution limit of ℎ off in our setup, but we did take into account the constant offset when calculating the applied force from RICM data.

S11
Generalisation of RICM analysis. The example above demonstrated how the applied force can be extracted from RICM data for a compressed interlayer of known optical thickness ℎ il opt = ℎ il il (where ℎ il and il are the geometrical thickness and the refractive index, respectively, of the optically homogeneous interlayer; for interlayers with a refractive index gradient along the surface normal, the value can be calculated as ℎ il opt = ∫ il (ℎ)dℎ ℎ il 0 ). Analogously, it is also possible to determine the optical thickness of the (compressed) interlayer from the RICM data for a given applied force. Whilst a detailed procedure is not presented here, we highlight that it is generally convenient to make the ansatz [S11] where and are positive functions that describe the effects of the interlayer and of the applied force, respectively. From Eq. S10B, we can identify = ℎ il opt ⁄ as the 'effective' interlayer thickness. We recall that ≈ 2 ⁄ (Eq. S8), i.e. to a first approximation is defined by the radius of contact, as determined by the applied force and the geometry and mechanical properties of the apposed surfaces. ℎ off can be positive or negative, and the sign indicates whether the (positive) effect of the interlayer or the (negative) effect of the force dominate.

Estimate of compressive forces between hemi-sphere and plane
As an alternative to the above, we also considered the mechanics of our system to obtain a rough estimate of the applied forces. Whilst the setup as a whole is mechanically complex, we recognised that the lever arm that connects the rod to the micromanipulator is likely to be the most compliant element. We determined the spring constant of the lever arm to be karm  250 N/m using linear regression analysis of its deformation under the load of a set of defined weights (Fig. S9). With this lever spring constant, a lever deflection of 16 μm (i.e. from 'soft' to 'hard' contact) corresponds to a difference in force of 4 mN. The order of magnitude compares favourably with the RICM analysis which gave 7 mN over the same range. These values are in reasonable agreement if one considers a resolution limit of ±2 nm in the ℎ off determination.

FG domain film thickness under strong compression
The FG domain of Nsp1 is an intrinsically disordered polypeptide chain. Upon compression, solvent is squeezed out of the FG Nsp1 brush, and in the limit of very strong compression it can be expected that the FG Nsp1 brush resembles a virtually solvent-free and incompressible polymer melt. 6 With a grafting density Γ = 5 ± 1 pmol/cm 2 , a molecular weight Mw = 64.1 kDa of Nsp1 FG domains, and an effective density ρ = 1.4 g/cm 3 for the compacted polypeptide, we find that the fully compressed film has a thickness of dmin = Γ Mw / ρ = 2.3 ± 0.5 nm.
To estimate the force required for full compression, we consider the osmotic pressure Π osm of a polymer solution according to Flory-Huggins where B is the thermal energy, the Kuhn segment length, χ the Flory interaction parameter, and φ the polymer volume fraction. The entropy of polymer mixing was here neglected since the polymer chains are confined through grafting. In the limit of strong compression φ is approximately constant across the brush, and relates to the brush thickness as φ = min . [S13] In our experiments a force is applied and brushes are confined between a hemi-spherical and a planar surface. This can be translated into an equivalent pressure in the geometry of two parallel planar surfaces using Derjaguin's approximation [S14] where is the radius of the hemi-sphere. Balancing external and osmotic pressures (Π osm = Π) results in [S15] Integration of Eq. S4 gives ]. [S16] In the limit of → min = 2 B 3 min (1 − χ). [S17] With = 0.76 nm for polypeptide chains, = 3 mm and χ > 0, we find that the brush becomes fully compressed at forces > 0.4 mN. Since PSCM operates at forces in the mN range, we can thus conclude that the brush becomes fully compressed at the centre of the sphere-plane contact area.