INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS THEORY RESULTS DISCUSSION REFERENCES

Fluorescence correlation spectroscopy (FCS) can be used to study kinetic properties of single molecules (Schwille et al., 1997; Widengren and Rigler, 1998; Schwille, 2001). If such measurements are done in cells or cell processes, the volume from which fluorescence is gathered is important because it can be on the same order of magnitude or smaller than the volume of the excitation laser focus. In such systems, the confinement of diffusion by the plasma membrane has to be taken into account (Gennerich and Schild, 2000).

We investigate diffusion of single molecules in dendrites after loading cultured neurons with tetramethylrhodamine (TMR)-dextran through a patch pipette. Surprisingly, autocorrelation analysis of dendritic fluorescence clearly suggested the presence of more than one molecular species, although only one species was loaded into the cytosol.

We solve this apparent contradiction by extending the FCS model of confined diffusion to the case of anisotropic diffusion. Fitting dendritic FCS data to this diffusion model indicated the presence of just one molecular species. Anisotropic diffusion in dendrites could be expected because the known intradendritic microtubular network might hamper diffusion orthogonally to the microtubules while leaving it largely unaffected in the longitudinal direction.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS THEORY RESULTS DISCUSSION REFERENCES

Cell culture

Cultured neurons of the olfactory bulb (OB) of Xenopus laevis were prepared as described previously by Bischofberger and Schild (1995). Briefly, larvae of Xenopus laevis (stage 48-54, Nieuwkoop and Faber, 1956) were anesthetized with Tricain (100 mg/l), and the OBs were extirpated. The tissue was incubated at 22°C for 90 min in a dissociation solution containing EDTA (1 mM), papain (30 U/ml), and cysteine (1.5 mM). The resulting pieces were triturated with an Eppendorf pipette. The cells were plated onto dishes coated with poly-L-lysine (50 µg/ml) and laminin (20 µg/ml) in a drop of medium (50 µl, L15 (Leibovitz), Life Technologies, Karlsruhe, Germany) containing 70% L15, 10% horse serum, and 50 µg/ml gentamycin. After 20 h, 0.1 ml growth medium, which contained 75% L15, 5% horse serum, and 50 µg/ml gentamycin, was added, allowing the cells to condition their environment. Measurements were carried out within 2 weeks after plating.

The cultured cells were characterized by the use of antibodies against glial cells and GABAergic neurons. Mitral cells were identified by injection of fluorescent beads into the lateral olfactory tract and subsequent retrograde labeling (Bischofberger et al., 1995). The results were very similar to those reported for rat-cultured OB cells (Trombley and Westbrook, 1990); mitral cells appeared as the largest neurons in the culture and were multipolar, whereas glutamic acid decarboxylase-positive cells were smaller and of ellipsoidal shape. Both cell types had large nuclei filling most of the soma. All FCS measurements taken in the soma were thus performed in the nucleus.

Electrophysiology

Standard patch clamp equipment was used as described (Czesnik et al., 2001). To load the neurons with the fluorescent dye TMR-dextran (10 kDa, Molecular Probes, Leiden, the Netherlands), the neurones were patch clamped in the whole cell configuration using borosilicate glass pipettes having pipette resistances of ~6 M. To check for the stationarity of the experimental conditions we measured the holding current and cell resistance over the entire duration of an FCS recording. The compostition of the bath solution was (in mM): NaCl 102, KCl 2, MgCl₂ 2, CaCl₂ 2, Glucose 20, HEPES 10, 240 mOsm, pH 7.8, whereas that of the pipette solution was NaCl 2, KCl 103, MgCl₂ 3, EGTA 1, HEPES 10, K₂-ATP 1, Na₂-GTP 0.01, 220 mOsm, pH 7.8.

Transmission electron microscopy

For transmission electron microscopy, the tadpoles were perfused with 1.5% glutaraldehyde (Sigma-Aldrich Chemie, Deisenhofen, Germany) in 0.05 M cacodylic buffer (pH 7.4) through the tail vein for 15 min. Brains were removed and stored in the same fixative for several days at 10°C. Forebrains were dissected, postfixed in 2% osmium tetroxide in the same buffer for 2 h at room temperature, dehydrated through alcohol and propylene oxide, and embedded into Poly/Bed 812 Embedding Media (Polysciences, Warrington, PA). Ultrathin sections were cut with diamond knives on a SORVALL MT2-B Ultra-Microtome (DuPont, Wilmington, DE), collected on copper slot grids provided with formvar support films, counterstained with uranilacetate and lead citrate and examined in JEM100-CX transmission electron microscope (JEOL, Tokyo, Japan). Negatives were scanned using a flatbed scanner at 600 DPI (SNAPSCAN 1236, AGFA, Köln, Germany), and imported into Adobe Photoshop 6.0 for layout.

FCS set-up

The FCS apparatus was attached to a modified inverted Zeiss Axiovert 35 microscope (Carl Zeiss, Göttingen, Germany) with a C-Apochromat 40/1.2 W (Carl Zeiss). A HeNe cw-laser (2.2 mW) at 543.5 nm was used as excitation source (LK 54015, Laser Graphics, Dieburg, Germany). The back aperture was not overilluminated. The laser intensity in the focal plane was set to 3.14 kW/cm². This power affected neither the appearance of the neuron under investigation nor its electrophysiological properties. Tandem galvanometer mirrors (GD120DT, GSI Lumonics, Unterschleissheim, Germany) were used for x-y positioning, and a piezo-driven objective holder (P-721.10, Physik Instrumente, Waldbronn, Germany) was used for z positioning. The voltages to the xy scanner and the z piezo were controlled either by potentiometers or by a custom program written in C and running on a Siemens microcontroller (MCB-167, Keil Elektronik, Grasbrunn, Germany), to which a 12-bit dual-channel DAC (DAC 2813AP, Burr & Brown, Tucson, AZ) was latched. The detection pinhole had a diameter of 50 µm. The beamwaist radius and the structure factor were determined to be r_xy = 0.24 µm and S = r_z/r_xy = 7 by measuring the translational three-dimensional (3D)-diffusion of TMR (T-5646, Sigma-Aldrich Chemie) in water, assuming a diffusion constant of D = 2.8 × 10⁶ cm²/s (Rigler et al., 1993). The dark count rate of the avalanche photodiode used (SPCM-AQ-141, EG&G, Optoelectronics, Dumberry, Canada) was 100 s¹, its photon-detection efficiency was 70-80%. Back- reflexion (_exc = 543.5 nm) was blocked by an interference filter (HQ 582/50, OD6, AF Analysetechnik, Pfrondorf, Germany) put in front of the photodiode. The output pulses of the photon-counting module were fed to a correlator board (ALV-5000/E, ALV, Langen, Germany).

Data analysis

Autocorrelation functions (ACFs) were calculated by the correlator board and saved as ASCII files. The ACFs were analyzed either with Origin (Microcal Software, Northhampton, MA) in case of analytical fitting functions or with Mathematica (Mathematica 4.0, Wolfram Research, Champaign, IL) in case of nonanalytical functions (e.g., Eq. 6).

Size of dendrites

The geometry of dendrites was determined in two different ways. First, the dendritic diameter was entered as a free parameter into the model of confined diffusion and thus resulted from the fit (see below, Theory). Second, a line-scan profile across a dendrite was taken as an independent measure of dendritic size.

After an FCS measurement, the cultured neurons were incubated for ~3 min in 20 µM di-8-ANNEPS (Molecular Probes) dissolved in the bath solution. After flushing the bath, we scanned along a line orthogonal to the dendrite. An example of the resulting intensity profile is shown in Fig 1 A. This curve was best described by a y-line scan profile for a rectangular dendritic cross section s_rect(y) as given by Gennerich and Schild (2000),

(1)

where

(2)

with C_s being the average number of dye molecules per surface element. g accounts for the overall optical losses of the emission pathway, including the efficiency of the photoavalanche diode, and Q is the quantum efficiency of the fluorescent dye. I₀ is the maximum laser intensity in the focus. y = y₀ and z = 0 are the center of the dendritic cross section with width d_y and height d_z. Assuming a circular dendritic cross section s_circ(y) (Gennerich and Schild, 2000) gave a much worse fit to the data (Fig. 1 A), presumably because the adhesion between dendrite and Petri dish leads to an approximately rectangular shape at the base of the dendrite.

View larger version (17K): [in this window] [in a new window]   FIGURE 1   Line-scan profiles taken from dendrites of cultured mitral cells. (A) Line-scan through a dendrite stained with di-8-ANNEPS (noisy trace); scan velocity: 32 nm/s. The solid curve shows the fitted theoretical line-scan profile srect(y) (Eq. 1) for a rectangular dendritic cross section. Result of the fit:  = 15.31 kcps, dy = 0.895 µm, dz = 0.761 µm, y0 = 1.026 µm, and IB = 8.464 kcps, respectively (fixed parameters: rxy = 0.236 µm and rz = 1.652 µm). Fitting the theoretical line-scan profile scirc(y) for a circular cross section (dashed) gave  = 23.97 kcps and d = 1.22 µm, respectively (fixed parameters: rxy = 0.236 µm, rz = 1.652 µm, y0 = 1.026 µm, and IB = 8.464 kcps). (B) Line-scan through a thick dendrite that was homogeneously filled with 10 kDa TMR-dextran (noisy trace); scan velocity: 38 nm/s. The theoretical line-scan profile s'circ (y) (Eq. 6) for a circular dendritic cross section was fitted (solid curve) to the experimental line-scan profile. Result of the fit:  = 384.71 kcps/µm2, R = 1.636 µm, and y0 = 2.053 µm, respectively. Fitting the theoretical line-scan profile s'rect (y) (Eq. 8) for a rectangular cross section (dashed curve) gave  = 105.5 kcps, dy = 3.035 µm, and y0 = 2.058 µm (fixed parameters: rxy = 0.236 µm, S = 7, and IB = 2.2 kcps).

Occasionally, there were dendrites that were best modeled by a circular dendritic cross section (Fig. 1 B). For a dendrite homogeneously filled with a fluorescent dye, the theoretical profile follows from the convolution product,

(3)

of the excitation volume, given by the detectable emission intensity distribution I_E(x, y, z) (Rigler et al., 1993),

(4)

and a circular boundary function in the y-z plane orthogonal to the dendrite,

(5)

whereby we have used cylinder coordinates, x', y' = r cos  and z' = r sin . For the circular cross section, we thus have

(6)

where

(7)

and y = y₀ and z = 0 give the center of the dendritic cross section. Fitting this function to the profile shown in Fig. 1 B shows that this profile is well described by a circular cross section. In this case, the alternative fit for a rectangular dendritic cross section, (Gennerich and Schild, 2000),

(8)

with

(9)

was unsatisfactory. This was presumably due to the low adhesion of the dendritic plasma membrane to the Petri dish.

	THEORY

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS THEORY RESULTS DISCUSSION REFERENCES

Standard FCS model

The ACF for Brownian diffusion of m noninteracting fluorescent species in the case of an open Gaussian-shaped detection volume V_d, V_d = ^3/2rr_z, is given by (Aragón and Pecora, 1976; Rigler et al., 1993)

(10)

where T and _T account for the fractional part of molecules being in the triplet state and the triplet-state decay time constant, respectively (Widengren et al., 1995). _j, is the fractional weighting factor (Elson and Magde, 1974) and _diffj := r/4D_j the characteristic diffusion time constant for the jth species with diffusion constant D_j, respectively.

FCS model for confined diffusion

In the case of a diffusion space with a width or height on the same order of magnitude or smaller than the detection volume V_d, the confinement of diffusion has to be taken into account. In case of a sufficiently small dendritic height d_z, i.e., d_z/r_z  0.833 (Gennerich and Schild, 2000), the diffusion in z direction (optical axis) can be neglected. In case of FCS measurements made in dendrites with diameters of 1 µm or less, diffusion in the axial direction was neglected, because, with r_xy  0.24 µm and S = 7, i.e., r_z = 1.7 µm, d_z/r_z was smaller than 0.59.

Under these conditions, the ACF model for diffusion in small dendrites for m noninteracting fluorescent species becomes

(11)

with m standard ACF terms for diffusion along the dendrite, i.e., the x axis,

(12)

and m ACF terms () for confined diffusion along the y axis,

(13)

with

(14)

The confinement parameter Y is given by the ratio of the width d_y of the dendrite to the radius r_xy of the open detection volume in the focal plane, i.e., Y := d_y/r_xy (for more details, and for an expression of (), Eq. 13, that does not require the calculation of error functions, see Gennerich and Schild, 2000).

FCS models for anisotropic diffusion

In anisotropic media, the diffusion along and perpendicular to the x axis is characterized by different diffusion constants D_x = D and D_y = D_z = D, and the ACF model for unconfined diffusion of one species becomes

(15)

with _diff := r/4D and _diff := r/4D. For diffusion detectable only along the x axis, this model reduces to

(16)

The model for confined diffusion of one fluorescent species (Eq. 11, m = 1)

(17)

can easily be extended to a model for anisotropic diffusion by replacing Eqs. 12 and 13 by

(18)

and

(19)

FCS model for single transition events

The linear transition of a single molecule, e.g., a large autofluorescent species, through the detection volume is a deterministic process. It can cause a detectable intensity spike and, therefore, lead to a transient ACF contribution. The intensity change i(t) upon a linear translation of a fluorescent particle through the focus volume is given by the convolution product of the excitation volume I_E(x, y, z) (Eq. 4) and the boundary function of the fluorescent particle. If a small particle (diameter d r_xy) travels along the x axis, the intensity change is given by

(20)

if the particle crosses the center of the detection volume at t = 0 and v is the particle velocity.

In this case, the ACF contains a transient component proportional to

(21)

where the characteristic "transition" time _t is defined by _t := r_xy/v. Due to the averaging over time, this component fades with time. The same expression was obtained by Magde et al. (1978) for the stationary stochastic process of uniform translations of fluorescent particles.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS THEORY RESULTS DISCUSSION REFERENCES

Time constants of dendritic diffusion

We measured the ACF of fluorescence fluctuations of 10 kDa TMR-dextran in dendrites of cultured neurons of the OB. The ACFs of subsequent measurements in the same dendritic compartment are shown in Fig. 2 (the corresponding line-scan profile is shown in Fig. 1 A). They had variations that were, at least for large , larger than the noise of the ACF. At first glance, dendritic ACFs could be fitted by a one-component model (Eq. 11 for m = 1, Fig. 3 A), but the fit was never satisfactory (Fig. 3 C). Using a two-component model (Eq. 11, m = 2) (Fig. 3 B) led to considerably smaller residues (Fig. 3 D), apparently indicating the existence of two molecular species.

View larger version (14K): [in this window] [in a new window]   FIGURE 2   Autocorrelation plots of fluorescence fluctuations of 10 kDa TMR-dextran emitted from a dendrite of a cultured neuron of the OB (the line-scan profile of this dendrite is shown in 1 A). The duration of each FCS measurement was 10 s. The time delay between two measurements was ~10 s. The measurements were carried out in the same dendritic compartment after diffusion had reached a steady state.

View larger version (19K): [in this window] [in a new window]   FIGURE 3   Autocorrelation function calculated from fluorescence fluctuations of 10 kDa TMR-dextran emitted from a dentrite of a cultured neuron of the OB (identical to the lowest trace shown in Fig. 2). (A) Fitting a one-component model for confined diffusion (Eq. 11, m = 1, dashed curve) gave N = 2.164, T = 0.223, T = 6.05 µs, and diff1 = 0.515 ms (fixed parameter: Y = dy/rxy = 3.79, taken from the line-scan measurement of this dendrite, Fig. 1 A). (B) Fitting a two-component model (Eq. 11, m = 2, dashed curve) gave Ntot = 2.028, T = 0.208, T = 3.01 µs, 1 = 0.598, diff1 = 0.185 ms, and diff2 = 1.422 ms (fixed parameter: Y = 3.79). (C) Residuals of the least square fit shown in A. (D) Residuals of the fit shown in B. The dendritic dye concentration was calculated to be 25 nM (C = Ntot/V*dNA, NA = 6.02 × 1023/Mol, and V*d = 0.133 fl obtained from V*d = 3/2rxyrz[erf(Y/ )]2/ erf(Y) × [erf(Z/ )]2/erf(Z), and Z = dz/rz = 0.461, see Gennerich and Schild, 2000).

In many cases, however, even a two-component model did not converge well and a third component had to be assumed. Figure 4 gives an example of such an ACF showing a characteristic shoulder for large . A stationary ACF shoulder of this shape is characteristic of stationary uniform translation processes (Magde et al., 1978), whereas a sudden onset and subsequent decay is caused by a single transient event (STE, see above). In fact, a satisfactory approximation was obtained using an ACF model with two terms for confined diffusion according to Eq. 11 (summation index j = 1, 2), and another one for an STE according to Eq. 21 (summation index j = 3) (Fig. 4).

View larger version (10K): [in this window] [in a new window]   FIGURE 4   ACF with STE component, calculated from fluorescence fluctuations of 10 kDa TMR-dextran emitted from a dentrite of a cultured neuron of the OB (identical to the upper trace of Fig. 2). Fitting a three-component model with two components of normal, confined diffusion (Eq. 11, summation index j = 1, 2) and one component for an STE (Eq. 21, summation index j = 3) gave Ntot = 2.012, T = 0.134, T = 4.64 µs, 1 = 0.499, diff1 = 0.209 ms, 2 = 0.375, diff2 = 1.573 ms, 3 = 0.114, and t3 = 239.5 ms (fixed parameter: Y = 3.79).

Evaluating dendritic FCS data consistently led to two or three components with relative contributions ₁, ₂, and ₃ and time constants _diff1, _diff2 and _t3 or _diff3, respectively. Plotting the time constants _diff1 (first component), _diff2 (second component), and _t3 or _diff3 (third component) for the six subsequent recordings of the experiment shown in Fig. 2 revealed that two of the time constants show a surprisingly small variance and average values of 189 µs and 1.626 ms, respectively (Fig. 5, A and B), whereas the third one fluctuated markedly. In this case, the third component could be described either by a third diffusional component (₃, _diff3) or by an STE (₃, _t3).

View larger version (11K): [in this window] [in a new window]   FIGURE 5   Parameter representation of the subsequent ACFs shown in Fig. 2. (A) Fitted diffusion and transition time constants of six individual recordings (recording index i). The two horizontal lines are the mean values diff1 = 0.189 ms and diff2 = 1.626 ms of the first and second diffusion time constants, respectively. (B) Relative contributions j of the fluorescent species to the ACFs as a function of fitted time constants.

Plotting the relative contributions ₁, ₂, and ₃ as a function of the time constant  showed that the narrowly distributed contributions ₁ and ₂ are on the order of magnitude of 55 and 40%, whereas the third component (_diff3 in case of diffusion, _t3 in case of an STE), which varied between 10 ms and more than 200 ms, contributed about 10% or less to the ACF (Fig. 5 B). Note that the third component was absent in some recordings (e.g., Fig. 5 A, recording index i = 2).

Taken together, analyzing dendritic diffusion using the model for confined diffusion leads to the confusing result that two or three molecular species have to be assumed though just one had been added to the cytosol. This apparent contradiction raises some questions, which we will answer in the following sections:

1.	Is there anything particular about dendritic diffusion that cannot be observed in somata? This question led us to perform FCS measurements in somata and to compare them to results obtained in dendrites.
2.	Are there autofluorescent species in addition to the exogenous fluorophore that contribute to the ACF? To test this we did experiments in cells with no TMR-dextran added.
3.	Does the dendritic cytoarchitecture, in particular the microtubulus network, introduce an anisotropy? If so, intradendritic diffusion must be described by a model that distinguishes diffusion along the dendrite from diffusion across the dendrite. We set up such a model (Eqs. 15-19) and applied it to data taken from dendrites.
4.	Does TMR-dextran bind to cytosolic constituents or the plasma membrane? We investigated this possibility by measuring diffusion in thin dendrites, which can be modeled as standard one-dimensional (1D) diffusion.
5.	Is our model for confined diffusion possibly flawed in a way that leads to apparent fluorescent species? We checked this by measuring ACFs in thick dendrites where the standard model for two-dimensional (2D) or 3D diffusion can be used. We then determined the number of components under these conditions.

Diffusion in somata differs from diffusion in dendrites

Somatic ACFs, like dendritic ACFs, showed marked fluctuations (Figs. 6 A and 7 A), which partly appeared to be based upon STEs of large molecules (Fig. 6 B). Because the somatic width and height (d_y, d_z  6 µm) are sufficiently larger than the width and hight of the detection volume (with r_xy  0.24 µm and S  7, we find Y = d_y/r_xy  25 and Z = d_z/r_z  3.53, see Gennerich and Schild, 2000), the confinement of diffusion due to the somatic plasma membrane can be neglected. We therefore fitted the somatic ACFs using the standard diffusion model (Eq. 10). In contrast to our findings in dendrites, only one fluorescent component of the soma was constant over subsequent recordings (Figs. 6 C and 7 C). This component contributed 75-95% (Fig. 6 D) and 90-100% (Fig. 7 D) to the total ACF, respectively, whereas the fluctuating second and third diffusion time constants contributed little to the total ACF (Figs. 6 D and 7 D). Similar results were observed in 67 somata. In many cases, a one-component FCS model (Eq. 10, m = 1) with ₁ = 1 (Fig. 7 D) was therefore sufficient to obtain a satisfactory fit (Fig. 7, B and C, recording index i = 1, 3).

View larger version (16K): [in this window] [in a new window]   FIGURE 6   ACFs calculated from fluorescence fluctuations of 10 kDa TMR-dextran emitted from a soma of a cultured neuron of the OB. Each of the subsequent ACFs was measured over 10 s in the same somatic compartment (the time delay between two measurements was ~10 s). The measurements were carried out after diffusion had reached a steady state. (A) ACFs of six subsequent measurements. (B) ACF with STE component. Fitting an ACF model with two components according to Eq. 10 (summation index j = 1, 2) and one STE component (Eq. 21, summation index j = 3) gave Ntot = 5.14, T = 0.18, T = 1.73 µs, 1 = 0.76, diff1 = 0.246 ms, 2 = 0.153, diff2 = 5.57 ms, 3 = 0.07, and t3 = 360.62 ms (fixed parameter: S = 7). (C) Fitted diffusion and transition time constants of the ACFs shown in A (recording index i). The horizontal line denotes the mean diff1 = 0.239 ms of the first diffusion time constant diff1. (D) Relative contributions j of the fluorescent species as a function of the fitted time constants plotted in C. The dye concentration within the soma was ~15 nM.

View larger version (13K): [in this window] [in a new window]   FIGURE 7   ACFs calculated from fluorescence fluctuations of 10 kDa TMR-dextran emitted from a soma of a cultured neuron of the OB. The ACFs were measured over 2.6-7.7 s in the same somatic compartment (the time delay between two measurements was ~10 s). The measurements were carried out after diffusion had reached a steady state. (A) ACFs of four subsequent measurements. (B) ACF identical to one of the lower curves shown in A (measurement time: tm = 2.6 s). Fitting an ACF model with one component according to Eq. 10 gave: N = 31.21, T = 0.109, and diff1 = 0.216 ms (fixed parameter: T = 4 µs and S = 7). (C) Fitted diffusion time constants of the ACFs shown in A (recording index i). The horizontal line denotes the mean value diff1 = 0.206 ms of the first diffusion time constant diff1. (D) Relative contributions j of the fluorescent species as a function of the fitted time constants plotted in C. The dye concentration within the soma was calculated to be ~100 nM (C = N/VdNA, Vd = 0.51 fl).

The predominant first time constant _diff1 was found to be 1.2-2.6 times larger with respect to aqueous solution (_diff = 0.16 ms for our set-up), indicating a relatively free and rapid diffusion of a macromolecule-sized solute in somata. The fluctuating components observed in both somata and dendrites may be brought about by large autofluorescent particles (e.g., autofluorescent mitochondria, arrow in Fig. 9 A; see also, Brock et al., 1998) diffusing or being transported through the focal volume. Note, however, that the second stationary component _diff2 as measured in dendrites was never observed in somata. It therefore appears to be specific for dendritic diffusion.

Transient autofluorescent particles

Many recordings showed infrequent fluorescence peaks that led to additional transient ACF contributions that were best fitted by Eq. 21. The fits resulted in large transition times _t (e.g., _t3 = 360.62 ms for the ACF shown in Fig. 6 B, and below _t = 292 ms and _t = 222 ms for the two upper curves shown in Fig. 11 A). We tentatively interpreted these events as large autofluorescent intracellular particles passing through the focus volume at random times. This hypothesis was confirmed by recordings from dendrites of cultured neurons that had not been filled with TMR-dextran. Figure 8 shows ACFs recorded under such conditions. The lower ACF, which corresponds to the fluorescence count rate in the left inset (0-7 s) contains just a noise component brought about by uncorrelated back-scattering light of the exciting laser beam. However, the fluorescence emission peaked at random times (Fig. 8, right inset, 18-26 s), and whenever this happened the ACF made a "jump" to assume transiently a shape as shown in the upper trace of Fig. 8. Such ACFs could be approximated by the FCS model for STEs (Eq. 21) with time constants in the range of 5-1200 ms, i.e., identical to the STEs observed in dendrites filled with TMR-dextran.

View larger version (21K): [in this window] [in a new window]   FIGURE 8   Autocorrelation curves calculated from autofluorescence fluctuations emitted from a dentrite of a cultured OB neuron that was not filled with TMR-dextran. The lower curve shows the ACF calculated from the fluctuations shown in the left inset (0-7 s, background intensity IB  0.8 kcps), and the upper curve shows the ACF obtained from the fluorescence fluctuations shown in the right inset (18-26 s). Fitting the upper curve with an STE component (Eq. 21) gave t = 9.57 ms (dashed curve).

Anisotropic dendritic diffusion

In mitral-cell dendrites, there is a regular network of microtubuli (Fig. 9) along the dendrite, suggesting approximately free diffusion in the direction along the dendrite but hampered diffusion across the dendrite. We therefore assumed that dendritic diffusion was anisotropic and approximated dendritic FCS data using a model for anisotropic diffusion (Eq. 17). Fitting this model to dendritic ACFs without STE contribution (e.g., Fig. 2, lowest curve; see also Fig. 5 A, recording index i = 2) indeed suggest one component with ₁ = 1 (Fig. 10). The data shown in Fig. 2 lead to characteristic time constants _diff and _diff for diffusion parallel and orthogonal to the dendrite, _diff = (0.179 ± 0.011) ms and _diff = (14.62 ± 2.75) ms. In this example, the mobility of 10 kDa TMR-dextran, as compared to its mobility in water (_diff = 0.16ms), was thus reduced by a factor of 1.1 along the dendrite and by a factor of 91 across the dendrite. Diffusion is thus markedly slower in directions orthogonal to the dendritic axis. Similar results were measured in 61 dendrites. Table 1 gives a summary of the measured dendritic and somatic diffusion constants. D_aq of 10 kDa TMR-dextran (100 nM) measured in a drop of water gave the value D_aq = (8.5 ± 0.3) × 10⁷/cm²/s.

View larger version (140K): [in this window] [in a new window]   FIGURE 9   Transmission electron micrographs of a mitral cell dendrite. (A) Low magnification of the soma (lower left) and a dendrite with mitochondrium (arrow). The nucleus (nu) occupies most of the volume of the soma. Scale bar: 2 µm. (B) Higher magnification of the region indicated in A. Note the parallel arrangement of microtubuli along the dendrite. Scale bar: 100 nm. The inset shows a higher magnification with digitally increased contrast of the region indicated in B. L denotes the distance between adjacent microtubuli and dMT the microtubular diameter.

View larger version (9K): [in this window] [in a new window]   FIGURE 10   Autocorrelation function calculated from fluorescence fluctuations of 10 kDa TMR-dextran emitted from a dentrite of a cultured neuron of the OB (identical to the lowest trace shown in Fig. 2). Fitting the model of anisotropic confined diffusion (Eqs. 17, 18, and 19, dashed curve) gave N = 2.074, T = 0.207, T = 4.23 µs, diff = 0.172 ms, and diff = 15.85 ms (fixed parameter: Y = dy/rxy = 3.79, taken from the line-scanning measurement of this dendrite, Fig. 1 A).

We tried to support our interpretation of anisotropic diffusion by the following experiment: We incubated the cultured cells adding nocodazole or colchicine (10 µg/ml each, Sigma-Aldrich Chemie) to the culture medium to disrupt the microtubules. Both drugs inhibit polymerization of tubulin. After two days of incubation, the microtubules started to disaggregate as evidenced by antibody staining against -tubulin (not shown). At the same time, the dendritic cross sections became irregular and FCS measurements led to inconsitent results. Occasionally, the geometry of a dendritic compartment could still be approximated by a circular or rectangular shape. In those cases, there was just one characteristic diffusion time constant, which was in the same range as the predominant time constant found in somata (not shown).

Binding of TMR-dextran to cellular elements

Though the above interpretation of anisotropic dendritic diffusion may appear plausible, two potential artifacts seem possible. First TMR-dextran may bind to or interact with cytosolic elements or the plasma membrane, and second, our model for confined diffusion itself might introduce an apparent time constant. We therefore carried out experiments in dendrites with diameters of 200 nm or less. In such dendrites, diffusion in the lateral direction (y-z plane) can be neglected because, with r_xy = 0.24 µm and r_z = 1.7 µm, we have Y = d_y/r_xy  0.83 and Z = d_z/r_z < 0.83, i.e., the dendritic width and height are sufficiently smaller than the width and height of the detection volume (Gennerich and Schild, 2000). In these cases, the 1D model for diffusion along the dendritic axis must be applied (Eq. 16). Figure 11 A shows an example of subsequent FCS measurements taken from a dendrite with a diameter of ~190 nm (line-scan profile, Fig. 11 D). Here we could easily distinguish between stationary ACFs and ACFs reflecting STEs (two upper curves in Fig. 11 A). The stationary ACFs (lower curves in Fig. 11 A) could be well fitted with a one-component model (1D-ACF G_x(), Eq. 16) (Fig. 11 B), suggesting only one fluorescent species. For the two upper curves, we obtained a satisfactory approximation by using an ACF model with one term for 1D diffusion and a second one for an STE (Eq. 21; _t = 292 ms for the upper curve and _t = 220 ms for the second curve from the top, see Fig. 11 C). For the stationary curves, the characteristic diffusion time constant gave a value of _diff = (0.233 ± 0.013) ms, which is in the range of the first stationary diffusion time constant _diff1 found in thicker dendrites and analyzed with a 2D diffusion model.

View larger version (24K): [in this window] [in a new window]   FIGURE 11   FCS and line-scan measurements carried out on a thin dendrite of a cultured neuron of the OB that was filled with 10 kDa TMR-dextran. (A) ACFs, normalized by the average number of molecules N, of the subsequent measurements. The duration of each FCS measurement was between 4 and 8 s. The two upper curves correspond to the fluorescence fluctuations shown in the inset (upper ACF: solid intensity trace; second ACF from the top: dashed intensity trace; background intensity IB = 0.6 kcps). The time delay between two measurements was ~10 s. (B) One of the lower stationary ACFs shown in A (non-normalized). Fitting the one-component 1D model Gx () (Eq. 16) gave N = 1.893, T = 0.107, T = 4.27 µs, and diff = 0.217 ms. (C) ACF containing an STE component (identical to the second curve from top shown in A). Fitting a two-component model with one component for normal 1D diffusion (Eq. 16) and one component for an STE (Eq. 21) gave Ntot = 1.548, T = 0.156, 1 = 0.707, diff = 0.318 ms, and t = 221.74 ms (fixed parameter: T = 4 µs). (D) Line-scan through a thin dendrite of a cultured neuron that was homogeneously filled with TMR-dextran (10 kDa) (noisy trace) (the corresponding FCS measurements are shown in A); scan velocity: 38 nm/s. The theoretical line-scan profile s'rect(y) (Eq. 8) for a rectangular dendritic cross section was fitted (solid curve) to the experimental line-scan profile. Result of the fit:  = 10.45 kcps, dy = 0.19 µm, y0 = 0.589 µm, and IB = 0.599 kcps, respectively (fixed parameter: rxy = 0.236 µm).

A second stationary diffusion time constant _diff2, as it would arise from TMR-dextran bound to some other molecule, was never observed in these experiments. Binding to cytosolic elements or the plasma membrane can thus be ruled out. The second diffusion time constant _diff2 could therefore be attributed to dendritic 2D diffusion as discussed above.

Diffusion in thick dendrites

As a last source of apparent fluorescent species, we considered a potential flaw of our model for confined diffusion (Eq. 11). The data shown so far were all taken from dendrites the line-scan profiles of which were best decribed by rectangular cross-sections (see Fig. 1 A). "Best described" does, of course, not mean perfectly described, and thus the boundary conditions of the diffusion model are not perfectly correct. This might be responsible for the appearance of an additional time constant, which should thus be absent in thick dendrites, where the standard diffusion model can be applied (for critical values, see Gennerich and Schild, 2000).

We carried out FCS measurements in thick dendrites. Figure 12 A shows ACFs of a dendrite the radius of which was 1.64 µm (Fig. 1 B gives the line-scan profile of this dendrite). Characteristic diffusion times were assessed using the standard 3D diffusion model (Eq. 10) or its analog for (unconfined) anisotropic diffusion (Eq. 15). Figure 12 B shows the characteristic time constants obtained from part A of the figure and it is obvious that two of them are essentially unchanged over time. The third time constant (upper traces in Fig. 12 A) presumably reflect STEs. The STE-free traces in Fig. 12 A could all be fitted using the standard model for two fluorescent species (Fig. 12 C). Fitting the one-component model was not satisfactory (Fig. 12 C). Therefore, the appearance of two time constants is not brought about by the model of confined diffusion. The above interpretation of anisotropic diffusion is thus the most plausible explanation of our data. In fact, the STE-free traces obtained in thick dendrites (lower traces in Fig. 12 A) could be fitted using the model of unconfined anisotropic diffusion (Eq. 15) of one fluorescent species (Fig. 12 D).

View larger version (18K): [in this window] [in a new window]   FIGURE 12   Autocorrelation plots and parameter representation from FCS measurements carried out on a thick dendrite of a cultured neuron of the OB. (A) ACFs, normalized by the average number of molecules Ntot, calculated from fluorescence fluctuations of 10 kDa TMR-dextran emitted from the center of the dendrite (Fig. 1 B shows the line-scan profile of this dendrite). The duration of each FCS measurement was ~5 s to reduce the detection probability of STEs. The time delay between two measurements was ~10 s. (B) Fitted diffusion and transition time constants plotted over the recording index i. The two horizontal lines are the mean values diffI = 0.217 ms and diff2 = 1.71 ms of the first and second diffusion time constants with the contributions 1 and 2 of 60 and 40%, respectively. Due to the confinement of diffusion along the z axis (Z = dz/rz = 2R/rz = 1.981, with rz = 1.652 µm) together with the large structure factor of S = 7, virtually identical time constants resulted from the standard two-component 2D-ACF model (not shown). Because the dendritic width was sufficiently larger than the width of the detection volume (with dy = 2R = 3.27 µm and rxy = 0.236 µm, we find Y = dy/rxy = 13.86), the confinement of diffusion along the y axis was neglected. (C) ACF (noisy trace), identical to one of the lower stationary traces shown in A. Fitting the standard two-component model (Eq. 10, m = 2, dashed curve) gave Ntot = 47.38, T = 0.175, 1 = 0.621, diff1 = 0.211 ms, and diff2 = 1.99 ms (fixed parameters: T = 4 µs and S = 7). Fitting the standard one-component model (Eq. 10, m = 1, solid curve) gave N = 49.64, T = 0.233, and diff1 = 0.561 ms (fixed parameters: T = 4 µs and S = 7). (D) Fitting the model for anisotropic nonconfined diffusion (Eq. 15, dashed curve) to the ACF shown in C gave N = 47.98, T = 0.191, diff = 0.187 ms, and diff = 4.169 ms (fixed parameters: T = 4 µs and S = 7). The dendritic dye concentration was calculated to be ~170 nM (C = Ntot/V*dNA, with V*d = 3/2rxyrz[erf(Z/)]2/erf(Z) = 0.467 fl, see Gennerich and Schild, 2000).

	DISCUSSION

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