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Determinants of the translational mobility of a small solute in cell cytoplasm.

The purposes of this study were: (a) to measure the translational mobility of a small solute in cell cytoplasm; (b) to define quantitatively the factors that determine solute translation; and (c) to compare and contrast solute rotation and translation. A small fluorescent probe, 2,7-bis-(2-carboxyethyl)-5-(and 6-)-carboxyfluorescein (BCECF), was introduced into the cytoplasm of Swiss 3T3 fibroblasts. BCECF translation was measured by fluorescence recovery after photo-bleaching; rotation was measured by Fourier transform polarization microscopy. Diffusion coefficients relative to those in water (D/D0) were determined by comparing mobility in cytoplasm with mobility in standard solutions of known viscosity. At isosmotic cell volume, the relative diffusion coefficients for BCECF translation and rotation in cytoplasm were 0.27 +/- 0.01 (SEM, n = 24, 23 degrees C) and 0.78 +/- 0.03 (n = 4), respectively. As cell volume increased from 0.33 to 2 times isosmotic volume, the relative translational diffusion coefficient increased from 0.047 to 0.32, while the relative rotational diffusion coefficient remained constant. The factors determining BCECF translation were evaluated by comparing rotation and translation in cytoplasm, and in artificial solutions containing dextrans (mobile barriers) and agarose gels (immobile barriers). It was concluded that the hindrance of BCECF translation in cytoplasm could be quantitatively attributed to three independent factors: (a) fluid-phase cytoplasmic viscosity is 28% greater than the viscosity of water (factor 1 = 0.78); (b) 19% of BCECF is transiently bound to intracellular components of low mobility (factor 2 = 0.81); and most importantly, (c) translation of unbound BCECF is hindered 2.5-fold by collisions with cell solids comprising 13% of isosmotic cell volume (factor 3 = 0.40). The product of the 3 factors is 0.25 +/- 0.03, in good agreement with the measured D/D0 of 0.27 +/- 0.01. These results provide the first measurement of the translational mobility of a small solute in cell cytoplasm and define quantitatively the factors that slow solute translation.

the relative rotational diffusion coefficient remained constant. The factors determining BCECF translation were evaluated by comparing rotation and translation in cytoplasm, and in artificial solutions containing dextrans (mobile barriers) and agarose gels (immobile bartiers). It was concluded that the hindrance of BCECF translation in cytoplasm could be quantitatively attributed to three independent factors: (a) fluid-phase cytoplasmic viscosity is 28 % greater than the viscosity of water (factor 1 = 0.78); (b) 19% of BCECF is transiently bound to intracellular components of low mobility (factor 2 = 0.81); and most importantly, (c) translation of unbound BCECF is hindered 2.5-fold by collisions with cell solids comprising 13% of isosmotic cell volume (factor 3 = 0.40). The product of the 3 factors is 0.25 + 0.03, in good agreement with the measured D/D0 of 0.27 + 0.01. These results provide the first measurement of the translational mobility of a small solute in cell cytoplasm and define quantitatively the factors that slow solute translation. C ELL cytoplasm is a complex non-Newtonian fluid comprising an aqueous fluid-phase filling the space within an entangled mesh of filamentous skeletal proteins (cytomatrix) and other macromolecular structures (Bridgman and Reese, 1984;Clegg, 1984;Fulton, 1982;Gershon et al., 1985;Keith, 1973;Porter, 1984). The factors that determine the rotation and translation of solute molecules within this crowded milieu have been the topic of considerable recent interest due to their probable impact on the rates of metabolic reactions. At least three cytoplasmic factors will contribute to solute mobility: (a) fluid-phase cytoplasmic viscosity, i.e., the viscosity in the aqueous space between macromolecules; (b) solute binding to macromolecular structures; and (c) collisional (direct plus hydrodynamic) interactions between the solute and macromolecular obstacles. The relative contributions of these three factors will depend on solute size and the type and extent of solute motion. In this study we focus primarily on the long (relative to the characteristic macromolecular spacing) range translation of small solutes. Such motion plays a role in many cellular processes, including the translocation of cyclic nucleotides in signal transduction, the vectorial transport of solutes across epithelial cells, and the movement of nucleic acids from the cytoplasm to the nucleus for replication and transcription. The translational diffusion of a small solute in cell cytoplasm has not been measured previously by FRAP.
Most studies of dynamics in the cytoplasm have focused on measurement of a single parameter that reflects how rapidly molecules rotate or translate. These measurements thus yield "apparent" cytoplasmic viscosities, which reflect all impediments to motion: the true fluid-phase viscosity, the effects of binding, and the effects of collisions. When the rotational mobility of small molecules in the cytoplasm is characterized in this fashion using electron spin resonance (Keith et al., 1977;Lepoeket al., 1983; or steady-state fluorescence anisotropy techniques Lindmo and Steen, 1977), apparent rotational viscosities in the range 2 to 20 cP (water is 1 cP) are obtained. Similarly, when translational mobilities are characterized via direct observation of the diffusion of microinjected dyes, electron spin resonance, or FRAP, appar-ent translational viscosities are found to lie in the same range (Jacobson and Wojcieszyn, 1984;Kreis et al., 1982;Luby-Phelps et al., 1986Wojcieszyn et al., 1981). The interpretation of these values is subject to several caveats. First, the electron spin resonance results are based on an uncertain estimate of intracellular probe concentration, while the FRAP results are based on an extrapolation of data obtained from probes of different molecular size to zero size. Second, and more importantly, none of these results provide quantitative information about the individual factors contributing to the measured viscosity.
We recently introduced a method of determining fluidphase viscosity (microfluorimetric measurement of picosecond anisotropy decay; Fushimi and Verkman, 1991) that overcomes many of the limitations in previous measurements of solute mobility in cells. The data yield the rotational mobilities of bound and unbound probe, as well as the fraction of probe in each state. These measurements revealed that a significant (15-30%) fraction of fluorophores bind to relatively immobile cellular components. More importantly, it was demonstrated that the fluid-phase cytoplasmic viscosity sensed by the unbound probe was in the range 1.2-1.4 cP for several small fluorophores in Swiss 3T3 fibroblasts  and several types of kidney epithelial cells (Periasamy et al., 1992). This is a surprisingly low viscosity, suggesting that the physical properties of aqueous domain cell cytoplasm are similar to those of water.
The purposes of the present study are: (a) to measure the translational mobility of a small solute in cell cytoplasm; (b) to define quantitatively the factors that determine solute translational mobility; and (c) to compare the rotation and translation of probes in the cytoplasm. A FRAP apparatus was constructed and data analysis methods were developed for quantitative determination of the translational mobilities of small molecules that diffuse rapidly in cell cytoplasm. It was found that the translational diffusion of a small fluorescent probe was approximately four times slower in cell cytoplasm than in water. It is shown that this inhibition of translational diffusion arises because fluid-phase cytoplasmic viscosity is slightly higher than the viscosity of water, a small amount of probe binds to intracellular structures, and, most significantly, because the probe collides with intracellular structures.
were transferred to a 200-#1 perfusion chamber (to facilitate buffer changes) in which the cell-free surface of the coverglass was accessible to a nonimmersion objective or a short-working-distance immersion objective (Chao et al., 1989). All experiments were carried out at 23~

FRAP Apparatus
A FRAP apparatus was constructed to resolve recovery half-times down to 5 ms (Fig. 1). Cells were illuminated with a Gaussian-profile beam (488 nm) from a CR-4 Ar laser (Coherent, Inc., Palo Alto, CA) focused to a 3-#m radius spot through a 16• quartz objective (E. Leitz, Rockleigh, NJ, numerical aperture 0.25) on an inverted epifluorescence microscope. To photobleach the sample, an unattenuated laser pulse (4-5 x 103 times more intense than the probe beam) of 2-3 ms duration was generated using a rapid shutter (Uniblitz, model T132; Vincent Associates, Rochester, NY). To monitor prebleach and postbleach fluorescence, an attenuated probe beam was formed by reflections off of two glass wedge prisms and two frontsurfaced mirrors; a neutral density filter (1.80D U) was positioned between the mirrors to set the attenuation ratio. The mirrors and wedge prisms were mounted on two-axis micropositioners to facilitate alignment of the unattenuated and attenuated beams. Emitted fluorescence passed through a 510nm dichroic mirror, GG515 barrier filter, and rapid shutter (to protect the photomultiplier during the photobleaching pulse; model D122; Uniblitz), and was detected by an R928 photomultiplier (Hamamatsu, Middlesex, NJ) operating at 500-900 V. Photomultiplier signals were amplified (model ll0F; Pacific Instruments, Concord, CA; response time <0.1 ms) and digitized by a 12-bit analog-to-digital converter (Lab Master; Scientific Solutions, Solon, OH) interfaced to a 386 cpu. During each experiment, the first 3,000 data points were collected at a rate of 100/zs -1 ms per point, while the remaining 1,700-1,930 data points were collected at a rate of 10 ms per point.

FRAP Experimental Procedures and Data Analysis
Before each set of experiments, the FRAP instrument was stabilized for 90 min, and the attenuated and unattenuated beams were aligned and steered along the optical axis of the microscope. The recovery curves for a series of 5-8 calibration solutions were measured in quadruplicate before and after every set of experiments to provide an internal standard and to ensure that beam drift did not occur. The calibration solutions, which consisted of 240 /~M BCECF in PBS (pH 7.4) containing sucrose (0-540 g/l), yielded macroscopic viscosities of 1.02 to 23.2 cP as measured by a Cannon-Fenske viscometer (models 50 and 100; Cannon Instru. Co., State College, PA); measured viscosities were in close agreement with data in the CRC handbook 0Neast, 1986). FRAP measurements were performed on calibration solutions of 2-50-#m thickness prepared by depositing a defined microliter volume of solution between an 18-mm-diam round coverglass (above) and a larger rectangular coverglass (below). The solution spread evenly within the area of the round coverglass. Film thickness was confirmed from the z-axis translation required to focus on the upper and lower solution/glass interfaces.
Photobleaching parameters were computed as follows. In each measurement, the prebleach fluorescence, F(-), was determined from the average fluorescence over a 1-s interval prior to photobleaching. The time course of fluorescence recovery after photobleaching, F(t), was described by a 4-parameter biexponential function, F(t) = A1 exp(--t/Tl) + A2 exp(-t/r2) Figure 1. Schematic of F R A P apparatus designed to m e a s u r e recovery times down to 5 ms. See Materials and Methods section for details. + F(oo), where n < 72. F(~o) is the (constant) fluorescence at long times after photobleaching and was determined from the average of F(t) over a 5-s interval 15-20 s after the photobleaching pulse. The F(t) curve was fitted by the successive integration procedure (On-Line Instrument Systems Inc., Jefferson, GA) for t between 7 ms after the end of the photobleaching pulse (well after the emission path shutter was opened) and I00-1000 ms. The fluorescence immediately after bleaching, F(0), was determined by extrapolating F(t) back to zero time. The percent photobleaching was thus equal to 1-F(0)/F(-); the percent recovery was equal to [F(oo)-F(0)]/ [F(-)-F(0)]; and the half-time for recovery, r~2, was determined by numerically solving the equation F0"~) --[F(oo)-F(0)]/2. In *10% of the FRAP studies in cells, there was a slow increase in F(t) after recovery possibly due to continued intracellalar deesterification of BCECF-AM; in these experiments, F(t) was corrected using the measured rate of fluorescence increase determined from the slope of F(t) vs. t for t between 15 and 20 s.
Apparent viscosities for a given sample were determined by comparison of r values for fluorescence recovery after photobleaching in the sample with r values for the calibration solutions. The rationale and justification for this approach are discussed below. The diffusion coefficient in the sample relative to that in water, D/Do, was then computed as the inverse of the apparent viscosity.

Picosecond Anisotropy Measurements
Tmse=resolved fluorescence polarization measurements were carried out by Fourier transform frequency-domain fluorimetry ~'erkman et aL, 1991). Light from a CR-4 Ar laser was impulse modulated by a Pockels cell and components of an SLM 48000 multiharmonic fluorimeter (SLM Instruments; Urbana IL) to give a 6 MHz series of 1--2 nanosecond pulses of plane-polarized light. ,~4 % of the light was split to a reference photomultiplier, while the main be~xa was steered into an inverted epifluorescence microscope and reflected by a 5 lO-nm dichroic mirror for sample excitation through a 40x quartz objective (glycerol immersion, numerical aperture 0.65; Leitz). The beam diameter at the focal plane could be set between 1 ~,m and ,,o25 tan. Emitted fluorescence passed through the dichroic mirror, a GG530 barrier filter (Schott Glass, Duryea, PA), and a Glan-Thompson calcite polarizer that could be rotated through 90 ~ Polarizer alignment, G-factor corrections, and calibration with standards were carried out as described previously Foshimi et al., 1990). Differential phase angles and modulation ratios were measured at 35--40 discrete harmonics of the 6 MHz repetition frequency by cross-correlation multi-harmonic detection. The fluorescence from unlabeled cells was <2% of that from BCECF-loaded ceUs. Phase and modulation data were fitted by a non-linear least-squares procedure (Calafut et al., 1989) to a twocomponent anisotropy-decay model, fit) = g~ exp(-t/r + (l-g0 exp(-th'2c), where rk and r are the shorter and longer rotational correlation times, respectively, and gl is the fractional amplitude corresponding to r In practice, gl represents the fraction of unbound (more rapidly rotating) fluorophore, and (l-g~) represents the fraction of hound (more slowly rotating) fluorophore. The BCECF rotational diffusion coefficient in a given sample relative to that in water, (D/Do)rot, was determined from the ratio of n measured in buffer to that measured in the sample (Foshimi and .

Binding and Stopped-flow Polarization Measurements
Both the extent and the kinetics of probe binding to intracellular struclm'es will enter into the theoretical model developed in the next section. The mC, hodology used to characterize these quantities was as follows.
The extent of BCECF b'mding to cytoplasmic components was determined by picosecond anisotropy measurements, as described above, and by confocal microsco~. For the latter measurements, a monolayer of 31'3 fibroblasts was permeabilized with 40 ~g/nfl digitonin for 30 rain at 5~ and equilibrated with 100 ~ BCECF acid in PBS for 15 min at 23~ BCECF fluorescence in cell cytoplasm and the adjacent solution was measured by microscopy (T~hnical Instmnmnts, San Francisco, CA; 100x objective, 0.6 pan measured z-axis resolution). Binding was determined from the ratio ofintracdlular to extracellular fluorescence, corrected for the nonaqueous cell volume of ~20%.
The kinetics of BCECF binding in a 10% suspension of fibroblast cytoplasm and in a 5% albumip solution was determined by stopped-flow fluorescence polarization using a Hi-Tech SFS1 apparatus, The method exploits the fact that stendy-state fl~ce polarization (or anisoUoiff) increases as probe binds. Equal volunms of a 100/~M BCECF solution and a cytoplasm or albumin solution were mixed in <1 ms in a stopped-flow apparatus. The time course of fluorescence (480-nm excitation, >515 nm emission) was measured at a rate of 0.1 ms/poim. To determine the time course of steady-state fluorescence anisotm~, the incident light was vertically polarized and the emitted fluorescence was measured through vertically and horizontally oriented analyzing polarizers.

Theory for Slowed Diffusion in Cytoplasm
The purpose of this mathematical section is to provide a framework for the analysis of translational diffusion data in cell cytoplasm. It is assumed that cell cytoplasm consists of an aqueous fluid-phase compartment bathing a matrix of mobile and immobile particles that are much larger than the water molecules and small solutes. It is further assumed that three independent factors act to decrease the diffusion coefficient of a small solute in cytoplasm (D) relative to that in water (Do), where F~, F2, and F3 are defined below.
The function Ft (7) describes the slowing of net solute translation due to an increase in true fluid-phase cytoplasmic viscosity. Such an increase would reflect some solute-induced perturbation in solvent slxucmm, which need not be specified. The functional form for F~ follows from the reciprocal relationship between the diffusion coefficient and the viscosity, and can be written where 7/0 is the viscosity of water, and ~ is the true fluidphase microviscosity of cytoplasm. The fluid-phase viscosity ~ can be obtained from measurements of solute rotational dynamics, or potentially of short-range solute translational dynamics. The fluid-pha~ viscosity cannot be obtained from measurements of macroscopic cytoplasmic viscosity'; indeed, the macroscopic viscosity can be many times greater than the fluid-phase viscosity (Furukawa et al., 1991;Scalettar and Abney, 1991). Eq. 2 predicts that if the fluid-phase viscosity increased by a factor of 2, D/D0 would decrease by a factor of 2.
The function Fz(D~,~Db,i,fo,~}) describes the net slowing of solute translation due to transient binding of solute molecules to cytoplasmic structures. For the simplest case in which there is solute binding only to one macromolccular species, the measured FRAP diffusion coefficient has been shown to be a linear combination of the diffusion coefficients for the bound and unbound solute, weighted by the appropriate mole fractions (Elson and Qian, 1989;Elson and Reidler, 1979;Jihnig, 1981;Koppel, 1981). This expression can be generalized to describe multiple bound species; the form for F2 is taken as the ratio of the weighted diffusion coefficient of the bound and unbound solute to the diffusion coefficient of the unbound solute. If D, is the diffusion coefficient of unbound solute, Db.~ the diffusion coefficient of the ith bound solute, and f~.~ the fraction of total solute bound to component i, then It is assumed in Eq. 3 that the kinetics of solute binding and unbinding is fast compared to the characteristic time for translational diffusion in a FRAP experiment. In the FRAP studies described below, Eq. 3 applies for submillisecond binding kinetics. For a simple situation in which 50% of the solute is unboun~l and 50 % is bound at any instant to immobile intracellular structures (f, = 0.5, IDb.,l = 0), Eq. 3 predicts that D/Do would decrease by a factor of 2.
The function F3 ([n~N~l) describes the slowing of solute translation due to coUisional interactions with cytoplasmic structures, where in general there are ni structures of type i, each having volume V~. Two theories have been used to describe the effects of"volume occlusion" by mobile obstacles on diffusion: a stretched-exponential (SE) model (Furukawa et al., 1991;Phillies, 1989) and a free-volume (FV) model (Fujita, 1961;Furukawa et al., 1991;Landry et al., 1988). The SE model has been used to describe the diffusion of relatively large, Brownian particles, whereas the FV model has been used to describe solvent diffusion. Because BCECF is a small (nearly solvent-sized) solute, its motion can be satisfactorily described by both models. We apply both models to demonstrate model independence of the results.
The stretched-exponential model was initially developed to give a good empirical fit to diffusion data in crowded media. Recently, it has been shown that an SE equation can be derived theoretically. The SE model has been found to fit diffusion data well over a wide range of solute and occluding molecule sizes (Phillies, 1989), and has been elegantly used to describe the diffusion of large solutes in cell cytoplasm (Hou et al., 1990). For a single class of obstacles, the SE model predicts where n,V~ is the volume fraction occupied by the occluding molecule, and the prefactor a and the exponent v are scaling parameters. The values of c~ and p can be determined empirically by fitting Eq. 4 to data, and they can be calculated independently from theory. For small solutes, v is usually found to be near unity (Furukawa et al., 1991), while a depends on the concentration units.
In contrast, the free-volume model was derived from a molecular description of diffusion. The FV model is based on the assumption that solute diffusion in crowded media is ratelimited by the availability of "free-volume" i.e., volume sufficiently devoid of obstacles that it can be occupied by solute. The diffusion coefficient is assumed to follow an Arrhenius equation in which the (model-dependent) activation energy reflects the probability of finding appropriate free-volumes. A successful and simple formulation of the FV model was derived by Fujita (1961). For a single class of obstacles, the FV model predicts where r is the free-volume fraction of solvent, ~Oo is ~, in the absence of obstacles, f(0,T) is the temperature-dependent free-volume in the absence of solvent, /3(T) describes the flee-volume difference between solvent and obstacles (in the appropriate reference states), and Bd describes the minimum free volume required for solute displacement, fr0,T) and/3(T) generally have small, positive values, and Bd and ~o0 are taken to be unity. Although they have physical significance, FV parameters are usually determined by parameter regression, rather than by independent experiment.  , demonstrating that both follow the same biexponential time course. The cell data were overlaid onto the calibration data by rescaling the associated ordinate and abscissa, while leaving the exponential amplitudes and time constants (which determine curve shape) unchanged.

Validation of Diffusion Coej~cient Measurement by FRAP
The quantitative determination of BCECF diffusion coefficients in the aqueous phase of ceil cytoplasm was based on the comparison of photobleaching recovery curves in cells with those in calibration solutions. Consistency in cell data therefore hinges critically on consistency in calibration data. fluorescence at long times after photobleaching, F(oo), was nearly equal to that before photobleaching, F(-), indicating that all of the BCECF was mobile. In a total of 188 FRAP measurements made on calibration solutions, the percent recovery was 98 + 1% (SEM). As solution viscosity increased, the rate of recovery decreased. The half-time (r,/2) for recovery provided a quantitative measure of the recovery rate. Because a rigorous theoretical description of the recovery-curve shape would be very complicated and has not yet appeared in the literature (see Discussion), a functional form for the recovery curve shape was chosen empirically. Although a monoexponential function did not fit the data well, a biexponential function fit the data very well as shown by the fitted curves in Fig. 2 A. Plots of residuals showed no systematic deviation of experimental data from the biexponential fit. In addition, the shape of the biexponential curve was nearly the same in all experiments as indicated by the similar ratios of exponential time constants. The invariance of recovery curve provides direct justification for the use of T~:2 as a single parameter measure of recovery rate. Fig. 2 B shows a linear relationship between r~/2 and viscosity for a set of calibration solutions. Although the slope of this calibration plot varied from day to day because of slight variations in beam profile and alignment (slope 21 • 4.2 ms/cP, SD, n = 6), the slope generally changed by <20% from the beginning to the end ofa 6 h set of measurements.
Since Gaussian beams diverge with distance from the focal plane, diffusion coefficients can, in principle, depend on cell thickness. However, for the 16• objective used here, the Rayleigh range (i.e., half the distance over which the beam radius remains within •40% of its minimum value) is quite large, '~50 #m. Not surprisingly, measured rl/2's did not increase as the objective focal point was positioned up to 10 ktm above or below the center of a thin (5/~m) calibration sample, or as sample thickness was increased from 2 to 50 #m. Since the maximum thickness of the 3T3 fibroblasts is <2 tzm, incident beam divergence is not important in our experiments. The possible dependence of diffusion coefficient on beam spot size in considered in the Discussion. Taken together, the findings above establish a rigorous basis for the interpretation of photobleaching curves obtained in cell cytoplasm.

FRAP Experiments in Cells
Photobleaching experiments were conducted using a 3-#m radius beam spot, which covers <1% of the area of a Swiss 3T3 fibroblast (Luby-Phelps et al., 1987). The beam was positioned to photobleach cytoplasm in an area not directly adjacent to the nucleus or cell edge. No more than one photobleaching experiment was performed on a single fibroblast. Cell experiments were calibrated using 5-/~m thick standard solutions; the validity of this approach is demonstrated in Fig. 2 C, in which it is shown that recovery curves follow identical biexponential time courses in calibration solutions and cells. Figs. 3 and 4 show original recovery curves obtained from fibroblasts under a variety of conditions. For a fixed bleaching intensity, the percent bleaching increased with decreasing cell volume and decreasing glycerol concentration. However, the associated recovery times are independent of percent bleaching for the relatively shallow bleaching depths employed, as determined in separate experiments on calibration solutions and cells. Effective cytoplasmic viscosities derived from a series of measurements are summa- In the following sections, the results of these experiments and others will be used to characterize systematically the factors that slow BCECF diffusion in cytoplasm based on the theoretical considerations described in the previous section.

Slowed BCECF Diffusion in Cytoplasm: I. Fluid-phase Viscosity
The measurement of time-resolved anisotropy provides independent information about the rotation of bound and unbound fluorophore. Moreover, because the rotation of small solutes is not hindered by coilisional interactions with intracellular structures when the spacing between structures is much larger than the solute diameter (Drake and Klafter, 1990), the fluid-phase viscosity can be estimated from the rotational diffusion coefficient of the unbound BCECE Time-resolved anisotropy was measured in BCECF-labeled fibroblasts under the conditions used for FRAP measurements as described in Materials and Methods. In four sets of control cells bathed in isosmotic buffer, BCECF rotational correlation times, r and T2~, were 290 + 12 ps and 12 + 3 ns, respectively; the fraction of (bound) BCECF with the longer correlation time was 0,19 + 0.02. In PBS, a single rotational correlation time of 226 ps was measured. These results are in agreement with previous measurements made under similar conditions . At isosmotic cell volume, the relative diffusion coefficients for (A) D0/D for BCECF-loaded 3T3 fibroblasts measured as a function of relative cell volume. Data are mean + SEM for three to seven measurements. The cell data were fired to the SE and FV models by a least-squares procedure that fixed the parameters derived in B and floated as a fittin~ parameter the factor that converts inverse cell volume to dextran concentration. The value of this conversion factor corresponding to an ordinate value of 1 in (A) gives the "effective percentage dextran" associated with isosmotic conditions. The derived effective dextran concentration is 13%; substituting this value into the SE and FV models gives F~ --0. BCECF rotation and translation were 0.78 + 0.03 and 0.27 + 0.01. As cell volume increased from 0.33 to 2 times isosmotic volume, the relative rotational diffusion coefficient remained nearly constant (0.76-0.79), while the relative translational diffusion coefficient increased from 0.047 to 0.32 (Figs. 3 and 6 A). Thus rotation in the cytoplasm is less hindered and less sensitive to changes in cell volume than is translation.
To confirm the sensitivity of BCECF rotation to fluidphase cytoplasmic viscosity and to test the assumption that F, is independent of F2 and F3 (see Theory section), BCECF translation and rotation were measured in cells equilibrated with 28 and 48% glycerol. These glycerol concentrations increase the macroscopic viscosity of aqueous buffers by 2.3-and 5.4-fold (Weast, 1986). As expected, the BCECF rotational diffusion coefficient (obtained from 7zo measurements) decreased by approximately two-and fivefold, confirming the sensitivity of the anisotropy decay measurement. Figs. 4 and 5 show the effect of glycerol addition on the BCECF translational diffusion coefficient. The slope (0.98) of the best-fit line through the data in Fig. 5 demonstrates that the BCECF translational diffusion coefficient is nearly halved for each doubling of fluid-phase viscosity. The parallel increase in fluid-phase viscosity and BCECF translational recovery time provides direct support for the assumption that Ft is independent of F2 and F3.
In smnmary, our data indicate that fluid-phase cytoplasmic viscosity differs to a small but significant extent from the viscosity of water. Inserting the viscosity data into Eq. 2 gives Ft ,~0.78 + 0.03.

Slowed BCECF Diffusion in Cytoplasm: 11. Binding
The slowing of BCECF diffusion due to binding depends on the fraction of bound BCECF in cytoplasm, the mobility of bound BCECF, and the kinetics of binding. The fraction of bound BCECF was measured by two methods. First, the fractional anisotropy loss due to rapid BCECF rotation, 0.81 + 0.01, provides a direct measure of the fraction of BCECF that rotates freely. These data suggest that 19% BCECF is bound, assuming that there is no rapidly rotating "weakly bound" BCECF (see Discussion). AS cell volume changes from 0.33 to 2 times isosmotic volume, there is little change in the percentage of bound BCECF (range 18-21%). Second, the fraction of bound BCECF was estimated by Nipkow wheel confocal rn/croscopy in dlgitonin-permeabilized cells that were equilibrated with I00 ~M BCECF (acid, not acetoxymethylester form). BCECF intensity was measured in cells and in 2-pan-thick films of the BCECF solution used for equilibration. In four experiments, the ratio of intracellular to extracellular BCECF was 1.02, indicating that ,~20% (corrected for cell aqueous volume) of intraceilular BCECF is bound.
The translational mobility of bound BCECF was estimated by assuming that dye binds to cellular protein and thus diffuses like cellular protein. In previous FRAP studies (Jacobson and Wojcieszyn, 1984;Kreis et al., 1982), the cytoplasmic diffusion coefficients of native and exogenous proteins ranging in size from 12 to 440 kD were found to be 50-1,000fold smaller than that measured here for BCECE This result, plus the fact that some dye probably binds to truly immobile structures, suggests that the ratio Db/D~ is very close to zero. An attempt was made to estimate the mobility of bound fluorophore by reacting intracellular proteins non-selectively with carboxyfluorescein succinimidyl ester. Cells were hypotonically loaded with 500 #M carboxyfluorescein succinimidyl ester for 15 rain and then incubated at 37~ in PBS for 60 min. FRAP data obtained from the labeled cells showed small percent bleaching relative to data obtained for aqueous or intracellular BCECF; the fluorescence recovery had a complex multi-exponential time course. The results were difficult to interpret because: (a) free dye apparently was considerably more bleachable than bound dye, and thus the recovery curve was dominated by the diffusion of unbound dye molecules, and (b) the intracellular distribution of the covalentiy bound dye, as determined by fluorescence microscopy, was quite different from that of the non-covalenfiy bound BCECE The kinetics of BCECF binding to intracellular components was estimated by stopped-flow fluorescence polarization as described in Materials and Methods. BCECF binding to a 10% cytoplasmic homogenate and to a 5% albumin solution gave a measurable increase in polarization. 85-90% of the increased polarization occurred within the instrument dead time of 1.5 ms; ,~80% of the remaining increase in polarization had a half-time of 3-5 ms. These data confirm that the binding and unbinding of BCECF occur on a time scale much faster than the half-times for photobleaching recovery.
In summary, the submillisecond BCECF binding kinetics justifies the use of Eq. 3 to describe binding effects. Taken together with the very low mobility of bound dye, Eq. 3 gives F2 '~0.81 + 0.01.

Slowed BCECF Diffusion in Cytoplasm: IlL CoUisional Interactions
The dependence of D/Do on ceil volume provides information that can be used to evaluate the effects of coilisional interactiom on BCECF translational diffusion. Because BCECF rotational diffusion and binding are influenced little by changes in cell volume as shown above, the large effects of cell volume on translational diffusion are due mainly to changes in the fraction of intraceilular volume occupied by the cytomatrix.
The effects of "volume occlusion" by mobile obstacles were investigated empirically by measurement of BCECF diffusion in PBS solutions containing dextrans of different size and composition. Separate studies of anisotropy decay and binding showed little binding of BCECF to dextran and little change in aqueous-phase viscosity with increasing dextran concentration. Fig. 6 B shows that Do/D increases nonlinearly as the solution volume occupied by dextran increases. The dependence of D0/D on percentage dextran was independent of dextran size in the range 10-2,000 kD, in general agreement with results in the literature on the diffusion of small solutes in solutions containing much larger occluding solutes (Blum et al., 1986;Fm'ukawa et al., 1991). The data fitted well to both the stretched-exponential and free-volume models as described in Theory.
The effects of "volume occlusion" by immobile obstacles were also investigated empirically. FRAP experiments were performed in 50-pan-thick films of I, 3, and 5 % low-gelling temperature agarose gels containing 500 #M BCECF in PBS. The Do/D values given in Fig. 6 B (filled circles) were in agreement with the results for equivalent dextran percentages. Unfortunately, gels with agarose concentrations greater than 5 % are difficult to prepare and were not studied. However, the fact that BCECF diffusion is not sensitive to dextran size (and hence dextran mobility) suggests that obstacle mobility is much less important than obstacle concentration in determining D/D0, at least for obstacles with sizes >10 kD. Hence, the dextran data probably provide an adequate model for the effects of volume occlusion on BCECF translation. Fig. 6 A shows the relationship between D0/D for BCECF translation and the reciprocal cell volume, which is assumed to be proportional to the concentration of the cytomatrix. The data were fitted to the stretched-exponentiai and free-volume models using parameters from Fig. 6 B and taking the percentage occluded volume (under isosmotic conditions) as a floating parameter. The data were fitted well when 13 % of isosmotic cell volume is effectively occupied by occluding molecules. Because the majority of occluding molecules in cell cytoplasm are predicted to be in the 10-2,000-kD size range, the dextran solutions provide a reasonable model for cell cytoplasm. From the data in Fig. 6 B, 13% dextran gives an F3 value of 0.40 + 0.04.

Composite Effects of F~, F2, and F3
The product of F1, F2, and F3 determined independently above is 0.25 + 0.03. This value is in good agreement with the D/D0 of 0.27 5= 0.01 measured for BCECF translational diffusion in cell cytoplasm.

Additional Cell Experiments with a Smaller Photobleaching Spot Size
Some cell experiments were performed using a 100• objective (oil immersion, numerical aperture 1.3; Nikon Inc., Garden City, NY) with a 0.4-#m beam radius. FRAP experiments employing the 100 x objective required special technical considerations. First, there was significant dye rearrangement during the 2-3 ms bleaching time, making the effective bleached area much larger than the beam radius. However, the use of standard solutions obviated the need for precisely characterizing the bleached area. Second, the bleach and probe beams generated by the 100x objective diverge substantially. However, we found that r~ did not increase as the objective focal point was positioned up to 2/zm above or below the center of a thin (5 #m) calibration sample, or as sample thickness was increased from 0 to 5 #m. Beam divergence therefore did not affect our experiments on the <2-/zm-thick 3T3 fibroblasts. For these experiments, the standard curves were again linear, but the recovery times were significantly smaller (slope of calibration plot as in Fig.  2 B = 3.5 + 0.7 ms/cP, SD, n = 6). Experiments on control cells yielded D/D0 = 0.24 + 0.01 (SEM, n = 55, 23~ at isosmotic volume. The glycerol experiments (as in Fig. 5) gave D(-glycerol)/D(+glycerol) of 1.0, 2.1, and 5.3 for cells incubated in 0, 28, and 48 % glycerol. The cell volume experiments (as in Fig. 6 A) gavo D0/D of 3.0, 4.2, 13, and 23 for cell volumes of 2, I, 0.5, and 0.33 times isosmotic volume. The latter experiments imply that F3 = 0.44, corresponding to an equivalent dextran concentration of 13%. The product ofF~, F~, and F3 is 0.28 5: 0.03, in good agreement with the value of 0.24 + 0.01 (obtained with the 100x objective).

Discussion
The goals of this study were to measure the translational mobility of a small solute in cell cytoplasm and to construct a consistent physical model of the factors that hinder solute translation. A small solute, rather than a labeled dextran or ficoll, was chosen as the probe molecule so that the results would apply to small intracellular metabolites without the need to extrapolate data obtained for large probes to zero probe molecular size. The translational motion of a small solute in cell cytoplasm has not been quantified previously by photobleaching recovery techniques; because of the very fast recovery times, only a lower limit has been established for the translational diffusion coefficient of fluorescein-type probes (Jacobson and Wojcieszyn, 1984). We found that the fluorescent probe BCECF was mobile in cell cytoplasm and had short photobleaching-recovery times that yielded a translational diffusion coefficient relative to that in water (D/D0) of 0.27 + 0.01. This value is significantly less than unity and less than the BCECF rotational diffusion coefficient relative to that in water (D/D0)~ of 0.78 5: 0.03. Theoretical analysis and experiments on model solutions and cells were carried out to explain these results and to reconcile the differences. It was concluded that the low D/Do for BCECF translation could be quantitatively attributed to the combination of three independent factors: increased fluid-phase cytoplasmic viscosity, intaceUular BCECF binding, and BCECF collisional interactions with cytoplasmic structures. Moreover, the disparity between the relative translational and rotational diffusion coefficients was found to arise because rotation is relatively insensitive to the factor that most significantly perturbs translation: collisions with occluding macromolecules. The value for D/Do and its physical basis have important implications for cellular processes as discussed below.
The quantitative analysis of the photobleaching recovery of a cytoplasmic probe required special considerations. A rigorous theory exists for the calculation of recovery-curve shape for the two-dimensional diffusion of a fluorophore in a spot photobleaching experiment (Axelrod et al., 1976). The theory assumes a Gaussian beam profile, perfect alignment of bleach and probe beams, no movement of unbleached dye into the bleached area during the bleaching pulse, first-order bleaching kinetics, and accurate knowledge of beam diameter. A similar theory describing the threedimensional diffusion of fluorophore in a region illuminated by a diverging Gaussian beam does not exist. Furthermore, studies of two-dimensional systems have shown that results extracted from theory can be greatly affected by difficulties inherent in obtaining reproducible beam profiles and alignment (Barisas, 1980), bleach pulses that are short relative to the recovery kinetics (Bertch and Koppel, 1988), and purely first-order bleaching kinetics (Bjarneson and Peterson, 1991). For these reasons, our approach was to compare recovery curves for cells with curves obtained for a series of thin films of aqueous buffers having known viscosity. Recovery curves were obtained in the calibration solutions before and after every set of experiments to ensure stability of beam profile and alignment, and to provide a calibration relation for determination of apparent viscosity from recovery half-times. The use of calibration standards makes unnecessary the characterization of beam profile and alignment because the standard and samples are bleached and probed under identical conditions. Another important design criterion was the use of a very brief bleach pulse to minimize diffusion of unbleached dye into the bleached area during the bleaching pulse. In these experiments, a 2-3 ms bleach pulse with 5,000:1 attenuation ratio for bleach-to-probe beam intensity was generally used.
Photobleaching experiments monitor diffusion over micronscale distances. The distance probed is approximately proportional to (~'L~) 1~, and thus the distance scale probed here with the 16x objective (3 #m) is probably about twice that probed with the 100x objective. (The characteristic distance scale monitored by the 100x objective is significantly larger than its spot size because dye moves extensively during the bleaching pulse for this objective.) The consistency between the diffusion coefficients obtained with the 16x (D/D0 = 0.27 5-0.03) and 100x (D/D0 = 0.24 5-0.03) objectives suggests that diffusion within the cytoplasm is not sensitive to distance on the micron distance scale. This observation is in accord with theoretical predictions that the diffusion coefficient should be sensitive to distance only in a regime that is comparable to the obstacle spacing (Pusey and Tough, 1985;Scalettar and Abney, 1991). The distance scale studied here is typical of that studied in spot photobleaching experiments of cytoplasmic diffusion (1.5-6.0 #m; Jacobson and Wojcieszyn, 1984;Kreis et al., 1982;Luby-Phelps et ai., 1986Wojcieszyn et al., 1981) and smaller than that studied in pattern photobleaching experiments (11.8/~m to one-half cell size; Blatter and Wier, 1990;Wang et al., 1982).
The rotational motion of small fluorophores was described by an anisotropy decay model containing two rotational correlation times that differed by a factor of >20. From studies of aqueous buffers containing glycerol to increase viscosity and proteins to bind fluorophore , the shorter correlation time was assigned to rotation of unbound dye and the longer correlation time to rotation of bound dye. The agreement between estimates of intracellular BCECF binding by anisotropy decay and intracellular partitioning (measured by confocal microscopy) supports the assignment of the shorter correlation time to the rotation of unbound dye. However, there are several caveats in the interpretation of anisotropy data that deserve mention. Although the two-component anisotropy decay model provided a good statistical fit to phase and modulation data as judged by X ~ analysis, it is likely that heterogeneity exists in dye rotation associated with the short and long rotational correlation times; if the data were sufficiently well-resolved, a model containing a bimodal distribution of rotational correlation times might be superior. Furthermore, it must be recognized that the measured fluid-phase cytoplasmic viscosity is valid only in aqueous compartments to which the solute has access; no direct information is available about the possible existence and properties of compartments that exclude solutes. If compartments that exclude solute do exist, then their physical properties are probably not important for cellular enzymatic and metabolic events. However, such compartments may act as additional obstacles that will further slow solute diffusion.
The translational motion of a small solute is governed by 3 factors: the instantaneous velocity of the solute when it is in motion, the fraction of time spent in motion, and the route between initial and final positions. The first factor is deter-mined by an impediment to motion that acts over short time (or distance) scales, i.e., viscosity. The second factor is determined by the fraction of solute molecules that are bound (not in motion) at any instant in time. The third factor is determined by the solute trajectory in space, which depends upon the nature and distribution of obstacles. It is conceptually reasonable that (to a first approximation) binding is independent of viscosity and collisions. The results of the experiments in which glycerol was added to increase fluidphase viscosity demonstrate experimentally that viscosity is independent of binding and collisions. Finally, solute-solvent interactions, which determine the viscosity, are much stronger than solute-solute interactions, which determine the collisional effects (Pusey and Tough, 1985). Factors 1 and 3 thus influence diffusion over very different time (or distance) scales and are largely independent of one another (Pusey and Tough, 1985). The good agreement between the product of the three factors that govern BCECF diffusion in cytoplasm, as determined independently, and the observed rate of BCECF diffusion, provides further support for the separation of the overall barrier to diffusion into three independent components.
The translational motion of a small solute between an initial and a final position can be compared to the motion of an auto on a road, providing useful heuristic insight into the factors that slow solute motion. The time required for an auto to reach a target destination depends on its speed, the fraction of time it is in motion, and its route. The auto's speed (assumed to be constant when the auto is in motion) is analogous to fluid-phase solute diffusion (F0 driven by thermal energy. The fraction of time the auto is in motion (as opposed to stopped at red lights, etc.) is analogous to intracellular binding (F~). Finally, the auto's route is analogous to the effects of obstacles (F3), and can include traffic congestion (fluctuations in obstacle concentration) and cul-de-sacs (dead ends among immobile obstacles).
In addition, shortcomings in the solute/auto analogy provide useful insight into the differences between driven motion and random, diffusive motion. For the auto, the driver consciously chooses both the speed and the route, and can navigate the shortest distance between two points. However, the cost of this precision is that energy must be expended. For the Brownian solute, the speed is dictated by temperature and solvent viscosity, and the route traveled is random. The random route means that the time taken to travel between two points varies, and that the solute may not reach the second point at all. However, the advantage of this imprecision is that no energy must be expended. Brownian motion thus provides a cost-effective transport mechanism useful whenever precision and timing need not be exact.
The results and analyses reported in this study have direct relevance to other studies of the dynamics of solutes in crowded biological systems. A large body of related work has focused on the determinants of the diffusion of large molecules in the cytoplasm (reviewed by  as well as lipids (reviewed by Blackwell and Whitmarsh, 1990) and proteins (reviewed by Jacobson et al. [1987] and Scalettar and Abney [1991]) in biological membranes. Membrane proteins provide a particularly striking example of hindered diffusion: their translation can be more than 100-fold slower in biological membranes than in dilute artificial membranes. Less than one-tenth of this decrease can be attributed to the factor herein called F3 (Scalettar and Abney, 1991). The present study would suggest that this is not surprising and that F1, F2, and F3, only when taken together, can account for the total decrease in protein mobility in biological membranes.
In summary, there are two principle conclusions to this study. First, the translational diffusion coefficient of a small solute (BCECF) in cytoplasm relative to that in water, D/Do, is 0.27 + 0.01. Second, this value can be modeled as a product (D/D0 = FIF2F3 = 0.25 + 0.03)of three independent factors reflecting increased fluid-phase cytoplasmic viscosity (F~ = 0.78), BCECF binding to relatively immobile cellular components (F2 = 0.81), and, most importantly, collisional interactions of BCECF with cytoplasmic components (F3 = 0.40). Therefore, for a small metabolitesized solute that does not bind to cellular components, the diffusion coefficient for long-range translation would be '~2 • 10 -6 cme/s. The characteristic times associated with solute diffusion over 10 nm, 1/~m, and 10/xm distances would thus be 80 ns, 800 ~s, and 80 ms, respectively.