#### FRAP principles and axonal mobility of PV*

We used two-photon FRAP to analyse the mobility of Alexa-488-labelled parvalbumin (PV*) in PNs and to search for indications of PV binding partners. PNs were loaded with PV* via a somatic whole-cell patch pipette for at least 30 min before making FRAP recordings. The pipette contained 100 µm PV*, a concentration similar to or smaller than the assumed somatic or axonal concentration of native PV respectively (Kosaka *et al*. 1993). After the 30-min equilibration time, the morphology of the cells (spines, dendrites, soma, nucleus, axon) could be clearly resolved under two-photon excitation, indicating that PV* had free access to all cellular compartments. For FRAP experiments the laser beam was directed to a single point of interest and a brief high-intensity laser pulse was applied to irreversibly bleach the fluorophores within the focal volume (FV). Subsequently, the recovery of the fluorescence, which reflects diffusion of unbleached PV* molecules from neighbouring regions into the FV, was monitored at low laser intensity (Fig. 1).

The size of the FV was determined from the PSF of the microscope, measured with fluorescent microspheres with diameters (100 nm) well below the optical resolution. From the Gaussian profiles of the microsphere signal the radial (*ω*_{r}) and axial (*ω*_{z}) radii of the two-photon spot at the e^{−2} fluorescence intensity were determined to be 0.53 and 1.95 µm respectively. Thus, the axonal radius (∼ 0.5 µm; Fig. 1) is smaller than *ω*_{z} and *ω*_{r}, and FRAP in the axon can be regarded as 1D diffusion in a pipe. In analogy to the 3D diffusion equation in Brown *et al*. (1999), we derived a 1D diffusion equation for the axon (equation 1 in Experimental Procedures) that quantifies diffusional mobility in terms of the apparent diffusion coefficient (*D*). This equation well described individual as well as averaged recordings (Figs 1b and c respectively).

In FRAP experiments, binding of PV* to a large or immobile target would result in a steady-state offset from the prebleach level (Luby-Phelps *et al*. 1995; Star *et al*. 2002; Schmidt *et al*. 2005; for a detailed discussion of FRAP time courses see Reits and Neefjes 2001; Sprague *et al*. 2004). In order to reveal such targets, our diffusion equation included an offset as an additional variable. Normalizing this offset to the maximum decrease in the fluorescence induced during bleaching (bleach depth) quantifies the immobile fraction (IF) of PV* (Luby-Phelps *et al*. 1995; Schmidt *et al*. 2003a, 2005). In the example illustrated in Fig. 1(b), the fluorescence fully returned to the prebleaching level (deviation < 1%), indicating that no IF of PV* was present in the axon. Averaging FRAP recordings obtained at different axonal sites significantly increased the signal-to-noise ratio (Fig. 1c) but yielded indistinguishable offsets. On average (54 FRAP recordings, five cells, three mice), the mean ± SEM IF was 1.7 ± 1.2%. This value was not significantly different from zero (*t*-test against a hypothetical distribution around zero that had a width and sample size identical to the distribution of the measured IFs). Thus, no significant binding of PV* to a large or immobile partner occurred in the axon.

The presence of small or rapid-kinetic binding partners would be characterized by a reduced diffusional mobility with a smaller *D* value but a full recovery of the fluorescence after bleaching (‘retarded’ or ‘effective diffusion’; Crank 1995; Sprague *et al*. 2004). Such interactions are not readily identified in FRAP recordings but require measurements in different cellular compartments as well as with substances that certainly lack cellular binding partners. Thus, we quantified *D* in the individual axonal recordings and derived a median value of 12 (IQR 6–20) µm^{2}/s. By averaging recordings from the same axon, performed with identical protocols and showing similar bleach depths (Fig. 1c), we could narrow the error range to a mean ± SEM of 12 ± 2 µm^{2}/s.

In spiny dendrites of PNs, the *D* of PV* has been reported to be 43 µm^{2}/s (Schmidt *et al*. 2003a), i.e. a value three to four times larger than the axonal *D* found here. In view of this discrepancy, we controlled the purity and quality of PV* by sodium dodecyl sulphate (SDS)–polyacrylamide gel electrophoresis (Fig. 2a). A single protein band with an apparent molecular weight (M_{r}) slightly less than 15 kDa (expected M_{r}∼ 12 kDa) was seen in samples with the unlabelled PV and with PV*. As expected the band in the PV* sample migrated slightly more slowly than PV, in line with the small increase in M_{r} due to the Alexa label. Thus, the reduced axonal mobility of PV does not result from an improper protein.

We further tested whether our experimental approach and the original 3D equation (Brown *et al*. 1999; equation 2 in Experimental Procedures), from which our 1D equation was derived, yields results consistent with published *D* values. To this end, FRAP measurements were performed in an aqueous solution of 40-kDa FD (Fig. 2b). Four independent samples of 50 averaged recordings each were fitted to equation 2, yielding a mean ± SEM *D* of 43 ± 5 µm^{2}/s at 22°C. This estimate is almost identical to the value of 44 ± 5 µm^{2}/s reported by Arrio-Dupont *et al*. (1996).

Taken together, no technical aspects appeared to account for the observed difference between the axonal and dendritic PV* mobility. Thus, the discrepancy could be indicative of retarded diffusion owing to binding of PV* to a mobile axonal target. On the other hand, differences in cytoplasmic properties (i.e. viscosity and tortuosity) of dendrites and axons could also account for the discrepancy (Crank 1995; Sprague *et al*. 2004). In order to distinguish between these two possibilities we performed axonal FRAP experiments with 10-kDa and 40-kDa FD, for which no cellular interaction partner would be expected. As for PV*, individual and averaged recovery curves were well described by our 1D diffusion equation for both dextrans (Fig. 3). For 10-kDa FD, fluorescence recovery occurred with a median *D* of 10 (IQR 6–15) µm^{2}/s (*n* = 20, four cells, three mice), a value not statistically different from *D*_{PV}*. The 40-kDa FD, however, recovered significantly more slowly from bleaching than PV* (*p <* 0.001; Mann–Whitney rank sum test). The median *D* was 5.5 (IQR 3–10) µm^{2}/s (*n* = 17, four cells, three mice). In spiny dendrites *D* values of 32 and 20 µm^{2}/s have been reported for 10-kDa and 40-kDa FD respectively (Schmidt *et al*. 2003a). These values are again three- to four-fold larger than the present axonal values. This indicates that the reduced axonal mobility of PV* was not due to a specific PV interaction, but instead more likely the result of differences in the cytoplasmic properties of dendrites and axons.

During the experiments, we distinguished between the axon initial segment (∼ 20 µm from the soma; Clark *et al*. 2005) and the remainder of the axon. However, no differences were observed in *D* and IF values between these two axonal segments. Consequently, the data were pooled to represent the axonal mobility. Taken together, the data presented show that no large or immobile PV* binding ligand is present in the axon of PNs and further argue against any specific axonal PV* interactions with small and/or rapid-kinetic partners.

#### Somatic and nuclear mobility of PV*

We observed that PV* not only labelled the somata of PNs but also reached the nuclei, indicating that PV can readily pass the pores of the nuclear envelope. Because this property is a prerequisite for a transcription factor, such as the recently identified EF-hand CaBP downstream regulatory element antagonist modulator (DREAM) (Carrion *et al*. 1999), in the next set of experiments we analysed FRAP of PV* in the soma and nucleus (Fig. 4). FRAP in these two compartments is governed by 3D diffusion into the FV. Consequently, we used the previously published 3D diffusion equation (Brown *et al*. 1999) to quantify the somatic and nuclear PV* mobility.

We started by exploring possible immobilization of PV* in the soma or nucleus in terms of the IF that was introduced into the fitting function (see equation 2). The mean ± SEM IF in the soma was found to be −1 ± 1% and that in the nucleus was −1.1 ± 3% (Fig. 5a). Thus, within the scatter range, the fluorescence completely recovered to the prebleaching level in both compartments, indicating that there was no significant interaction of PV with a large or immobile partner in either the soma or nucleus.

As in the axon, we next quantified the somatic and nuclear mobility of PV* in terms of the median *D*. In both compartments, it was found to be 11 µm^{2}/s [somatic IQR 7–16 µm^{2}/s (*n* = 28); nuclear IQR 4–16 µm^{2}/s (*n* = 33); five cells each] (Fig. 5b). This value is similar to *D* in the axon but smaller than that in spiny dendrites (Schmidt *et al*. 2003a). Therefore, we again explored the possibility of retarded diffusion (see above) by performing FRAP experiments with 10- and 40-kDa FD in the soma and the nucleus. Fitting these data with the 3D diffusion equation yielded median *D* values of 9 (IQR 8–12) and 8 (IQR 7–10) µm^{2}/s (*n* = 10, three cells) for somatic and nuclear diffusion of 10-kDa FD respectively (Fig. 5b). For 40-kDa FD, a significantly smaller median *D* of 6 µm^{2}/s was found in both compartments [somatic IQR 5–8 µm^{2}/s (*n* = 17); nuclear IQR 4–8 µm^{2}/s (*n* = 15); three cells each; *p* ≤ 0.001, Mann–Whitney rank sum test). Thus diffusion was three- to four-fold slower than that in dendrites (see above), providing evidence against specific PV* interactions in the soma and nucleus of PNs.

#### Relationship between diffusional mobility and molecular weight

The Stokes–Einstein relationship predicts that in aqueous environments the *D* value of molecules much larger than water molecules is proportional to their hydrodynamic radius. Thus, for relatively large molecules, such as dextrans, *D* should be approximately proportional to the inverse cubic root of the M_{r} (Pusch and Neher 1988; Koch 1999). We previously found that this relationship holds for dextrans in dendrites, but is much steeper for CaBPs (Schmidt *et al*. 2003a, 2005).

We tested for the Stokes–Einstein relationship in axons, somata and nuclei by plotting the logarithms of the obtained *D* values of FDs and PV* against the logarithms of the corresponding M_{r} values (Fig. 5c). For this analysis, the data of the three compartments were pooled. We found a good overlap for the dextran values, with a regression line with the slope set to − 1/3. Similar to the situation in dendrites, a clear (but here non-significant) deviation of PV* from this line was observed. Thus, as in dendrites, the Stokes–Einstein relationships appear to hold approximately when comparing FDs of different M_{r} in axons, somata and nuclei but presumably not in comparing FDs and proteins. This discrepancy is most likely explained by the tertiary structure of FDs and proteins (Kretsinger and Nockolds 1973; Arrio-Dupont *et al*. 1996).

Given that the Stokes–Einstein relationship holds for FDs and an even steeper dependency might be expected for CaBPs (Schmidt *et al*. 2005), we tried to estimate the smallest size of a PV binding ligand that would be revealed in our FRAP experiments with statistical confidence. To this end, we systematically varied the *D* values obtained for 10-kDa FD until we derived a *D* that was significantly smaller than *D*_{PV}*. This ‘statistically distinct *D*’ takes into account the scatter of the data and the sample size. Based on the Stokes–Einstein relationship it was converted to a ‘statistically distinct M_{r}’ for a PV binding partner. This conversion yielded an upper limit for an undetected ligand of 9 kDa.