Transport of proteins to and from cilia is crucial for normal cell function and survival, and interruption of transport has been implicated in degenerative and neoplastic diseases. It has been hypothesized that the ciliary axoneme and structures adjacent to and including the basal bodies of cilia impose selective barriers to the movement of proteins into and out of the cilium. To examine this hypothesis, using confocal and multiphoton microscopy we determined the mobility of the highly soluble photoactivatable green fluorescent protein (PAGFP) in the connecting cilium (CC) of live Xenopus retinal rod photoreceptors, and in the contiguous subcellular compartments bridged by the CC, the inner segment (IS) and the outer segment (OS). The estimated axial diffusion coefficients are DCC = 2.8 ± 0.3, DIS = 5.2 ± 0.6, and DOS = 0.079 ± 0.009 µm2 s−1. The results establish that the CC does not pose a major barrier to protein diffusion within the rod cell. However, the results also reveal that axial diffusion in each of the rod’s compartments is substantially retarded relative to aqueous solution: the axial diffusion of PAGFP was retarded ∼18-, 32- and 1,000-fold in the IS, CC, and OS, respectively, with ∼20-fold of the reduction in the OS attributable to tortuosity imposed by the lamellar disc membranes. Previous investigation of PAGFP diffusion in passed, spherical Chinese hamster ovary cells yielded DCHO = 20 µm2 s−1, and estimating cytoplasmic viscosity as Daq/DCHO = 4.5, the residual 3- to 10-fold reduction in PAGFP diffusion is ascribed to sub-optical resolution structures in the IS, CC, and OS compartments.

Cilia are evolutionarily conserved subcellular organelles present on most cells of the body, and they serve many essential physiological functions, including vision, hearing, olfaction, and mechanosensation (Davis et al., 2006). The functions of many cilia remain undetermined, yet their importance is evident from the fact that mutations in genes encoding ciliary proteins give rise to devastating hereditary and often multi-organ disease, including Bardet-Biedl syndrome, Usher syndrome, combined polycystic kidney disease, and retinal degeneration (Pazour and Rosenbaum, 2002; Badano et al., 2006). Thus, a quantitative understanding of cilia construction, maintenance, and function is important for cell biology, physiology, and medicine.

All cilia are organized around a microtubule-based axoneme that is surrounded by a plasma membrane specialized with tissue-specific molecular sensors and ion channels (Fig. 1). The axoneme extends from and is anchored to basal bodies consisting of the centrioles that organize the cytoskeleton of the ciliary axoneme and of the spindle apparatus during cell division. Recent proteomic analysis of primary cilia shows that they comprise several hundred (Blacque et al., 2005; Pazour et al., 2005) to a couple of thousand proteins (Liu et al., 2007), depending on the species. Cilia do not contain protein and membrane synthetic machinery; thus, a central problem in the study of cilia has been to determine the mode of delivery of proteins to and removal from this relatively isolated compartment.

A distinctive form of molecular motor transport, intraflagellar transport (IFT), is a hallmark of primary cilia (Rosenbaum and Witman, 2002). Originally discovered in Chlamydomonas (Kozminski et al., 1995), IFT is highly conserved in eukaryotes, involving kinesin and dynein-based transport of large, multi-molecular particles along the microtubules of the ciliary axoneme (Rosenbaum and Witman, 2002). IFT transport has been proposed to subserve the coordinated delivery of both nonmembrane-bound and membrane-bound molecular components to, and their removal from, the cilium, allowing it to grow and be renewed, despite its remove from the nucleus, ER, and Golgi. A hypothesis implicit in the proposed function of IFT is that both the membrane and the interior of the cilium are effectively separated from the plasma membrane and cytoplasm of the cell body. Indeed, it has been explicitly proposed that the basal bodies and the associated structures at the base of the cilium, including the transitional fibers that extend to the plasma membrane, provide either sieving mechanisms that limit the size of soluble molecules allowed to pass into and out of cilia to ∼10 kD (Jensen et al., 2004; Praetorius and Spring, 2005), or serve as a gating mechanism that controls the passage of proteins between the ciliary compartment and the bulk cytoplasm (Deane et al., 2001; Trojan et al., 2008). Moreover, parallels have been drawn between the basal body complex and nuclear pore complexes in terms of control of protein permeability (Deane et al., 2001). These proposals rest primarily on ultrastructural studies of cilia and flagella and on studies where IFT components have been genetically ablated or reduced, resulting in shortening, disorganization, or complete loss of the cilia or flagella. However, the structure of the primary cilium (Fig. 1), with its apparently patent central core, suggests that the cilium may not be isolated from the bulk cytoplasm and that a mode of transport distinct from IFT, namely diffusion, could also be available for transport of soluble proteins within cilia.

Here, for the first time, we report measurements of the diffusion of a soluble protein within a primary cilium, the connecting cilium (CC) of retinal rod photoreceptor. Photoreceptors provide both a rationale and a useful preparation for the investigation of diffusional movement of proteins in primary cilia. A rationale comes from the massive light-dependent translocation of soluble phototransduction proteins, including arrestin (∼48 kD), transducin (α subunit, ∼39 kD; βγ, ∼46 kD), and recoverin (∼23 kD) between the inner segment (IS) and outer segment (OS) compartments of rods (Broekhuyse et al., 1985; Philp et al., 1987; Peet et al., 2004; Strissel et al., 2005). The translocation of these proteins has been hypothesized to be governed by either IFT (Marszalek et al., 2000) or by diffusion driven by a gradient created by light-dependent changes in local binding (Nair et al., 2005; Calvert et al., 2006). Whether by IFT or by diffusion, the proteins must pass through the CC, i.e., a short segment of the whole photoreceptor sensory cilium that joins the OS and IS compartments, and which preserves the classical ultrastructure of a primary cilium with plasma membrane in close apposition to the axoneme (Fig. 1 D).

The utility of the retinal rod as a preparation for investigating diffusional movement through a primary cilium arises from the relatively large diameter of the rod cell on either side of the much narrower CC, and from two additional felicitous features of the adjoining IS and OS. The large diameter of the rod allows accurate, high resolution confocal measurement of fluorescent protein concentrations on either side of the CC (e.g., Peet et al., 2004). One of the felicitous features of rods is that the cytoplasm of the IS proximate to the CC, the myoid, is relatively unstructured, so that (as data presented here will demonstrate) it very rapidly equilibrates upon sudden, local changes in fluorescent protein concentration. In contrast, on the OS side of the CC the lamellar disc stack, which occupies most of the cross section of the rod, greatly retards the longitudinal diffusion of proteins so that concentration changes in the IS are not rapidly dissipated from the base of the OS after propagating there through the CC. It follows from these features that an experimenter, using appropriate methods of confocal microscopy, can quickly create a change in concentration of a soluble fluorescent protein in the IS, and then, by measuring the gradient of fluorescent protein across the CC and its redistribution between the two segments, estimate its apparent diffusion coefficient within the CC using Fick’s first law or its embodiment in the full solution of appropriate diffusion equations.

### Generation of transgenic Xenopus laevis expressing photoactivatable green fluorescent protein (PAGFP)

A plasmid containing the coding sequence of PAGFP (Patterson and Lippincott-Schwartz, 2002) was provided by G.H. Patterson and J. Lippincott-Schwartz (National Institutes of Health [NIH], Bethesda, MD). Transgenic Xenopus laevis were generated using the REMI method (Kroll and Amaya, 1996). In brief, the coding sequence of PAGFP was placed downstream from the Xenopus opsin promoter, which confined expression specifically to the rod photoreceptors (Mani et al., 2001). The plasmid was linearized with XhoI endonuclease and incubated with isolated Xenopus laevis sperm nuclei that were similarly digested. Eggs were fertilized by injection of the sperm nuclei. Transgenic embryos were identified by epifluorescent fundus imaging and allowed to develop into tadpoles and adult frogs.

### Tissue preparation

Xenopus tadpoles, stage 42–60, expressing PAGFP in rods were dark-adapted for at least 2 h before experiment. All subsequent procedures were performed under infrared illumination to minimize activation of the light receptor, rhodopsin. Tadpoles were anesthetized by bathing in 0.05% tricaine (Ethyl 3-aminobenzoate methanesulfonate; Sigma-Aldrich) and decapitated. Eyes were removed and retinas were dissected into frog Ringer’s solution (in mM: 120 NaCl, 2 KCl, 10 HEPES, 1.6 MgCl2, 10 glucose, 0.03 EDTA, and 1.0 CaCl2). Retinas were oriented ganglion cell side down in a 50-µl bubble of Ringer’s on a polypropylene sheet and sliced into strips ∼50–100-µm wide and 100–200-µm long. Slices were transferred to an imaging chamber that consisted of a 35-mm polystyrene Petri dish, into the center of which a 5-mm diameter hole was drilled and then covered with a No. 1 glass coverslip attached with tackywax, which formed the bottom of the chamber. A moist sponge was placed in the Petri dish and the lid was loosely applied, such that humidity prevented evaporation of the Ringer’s solution while allowing free gas exchange. The chamber was then place onto the stage of the inverted confocal/multiphoton microscope. All procedures and experiments were performed at 21°C within 2 h of retinal dissection. Experiments were conducted in accordance with the NIH Guide for the Care and Use of Laboratory Animals.

### Multiphoton photoconversion of PAGFP and imaging in live Xenopus rods

Imaging and protein flux measurements were performed with a custom-built confocal/multiphoton microscope described previously (Peet et al., 2004; Calvert et al., 2007) that was modified by the addition of a custom LabVIEW-based software interface that allows for rapid (up to 12 Hz) image acquisition and protocols that control the laser intensity and positioning for photoconversion of fluorescent molecules (designed and implemented in conjunction with Michael Coleman, Coleman Technologies Inc.).

Transgenic expression of proteins under the Xenopus opsin promoter results in heterogeneous protein levels across rods in a given animal’s retina (Peet et al., 2004). Thus, 3-D scans of the retinal slices were performed before experiment to identify rods with sufficient PAGFP expression and that were well oriented with their long axis as close to parallel to the x–y image plane as possible. The initial 3-D scans were performed with visible, confocal scanning using the 488-nm line of an argon-ion laser (model 163C; Newport Corp.) focused to the diffraction limit with a 60×, 1.2 NA, water-immersion objective (Plan Apo VC; Nikon). The 3-D coordinates for the photoconversion pulse were manually selected from these initial scans using the LabVIEW interface.

PAGFP was photoconverted at the specified coordinates by multiphoton excitation from the Ti:S laser (Mai Tai HP; Newport Corp.) tuned to 820 nm and focused to the diffraction limit. The use of multiphoton excitation allowed the production of spatially well-defined fields of photoconverted molecules (Zipfel et al., 2003) and minimized the exposure of the tissue to visible light. The laser exposures ranged from 0.1 to 100 ms and 10 to 20 mW (average power). The equilibration of the photoconverted PAGFP was then monitored with serial x–y scans with 488-nm confocal excitation that intersected the photoconversion site. Rapid focus corrections were made before and after the photoconversion pulse to account for the measured focus difference between 820- and 488-nm illumination caused by objective chromatic aberration (Calvert et al., 2007).

Raw images were processed using custom MATLAB (The MathWorks) routines to correct for slight field inhomogeneities inherent to the optical system and for nonlinearities in the photon detectors as follows. Full field scans with each laser in the system of solutions of fluorescein were averaged to generate image field–flattening maps. Except where noted, no other image processing was done.

The 3-D point spread functions (psf) were estimated from spatial fluorescence distribution patterns obtained during 3-D scans of 0.1-µm fluorescent microspheres (Polysciences, Inc.), as described previously (Calvert et al., 2007). The fluorescence profiles in x,y and x,z were approximated by peak normalized Gaussians with σx,y = 0.16 and σz1 = σz2 = 0.68 for the focused Ti:sapphire laser tuned to 820 nm (multiphoton excitation, cf. Eq. 6 in Theory section), and σx,y = 0.16 and σz = 0.61 for the 488-nm line of the argon ion laser.

### Online supplemental material

Although the approaches used to model the diffusional movement of PAGFP in the rod are presented in the Theory section, additional theoretical details of the methods used to obtain numerical solutions to equations, process and fit the data with theoretical curves generated by the model, as well as video clips of PAGFP diffusing in the IS and OS compartments and equilibrating throughout the rod cytoplasm are available at http://www.jgp.org/cgi/content/full/jgp.200910322/DC1.

In this section, we present three distinct approaches to the use of diffusion theory to characterize the movement of soluble proteins in living photoreceptors or in other elongated cells, such as many neurons that possess distinct, interconnected cytoplasmic compartments containing structural elements of variable densities. First, we present a general 3-D model applicable for determination of diffusivities within relatively homogeneous cytoplasm; this model is applied to the IS compartment. We also present two approaches that reduce the problem of diffusion between interconnected cytoplasmic compartments to 1-D; the application of these approaches depends on the hypothesis of “rapid radial equilibration,” which is examined experimentally. One of these approaches is a direct application of Fick’s first law to movement through the CC, whereas the second involves longitudinal or axial diffusion in and between all the interconnecting compartments and incorporates position-specific diffusivities and cross-sectional areas. The overall modeling incorporates two additional important features. The first is the description of the source of photoconverted GFP created by the Ti:sapphire laser, specified in terms of the measured multiphoton point-spread function. The second is a description of the analysis by which the confocal measurement of fluorescence intensity is transformed into local concentration (as required by the diffusion models) in cellular compartments with structurally inhomogeneous cytoplasm.

### 3-D model of PAGFP excitation and diffusional equilibration in the rod cytoplasm

A model describing the dynamic equilibration of proteins along a concentration gradient produced by the photoactivation of PAGFP in the cytoplasm of rods was developed to assess the mobility of molecules within and between rod compartments. We begin with the description of the distribution of PAGFP in the rod before, during, and after photoconversion. The photoconversion of PAGFP is treated as a rapid, irreversible change of state of the molecule from a basal, B, form to a photoconverted, C, form upon multiphoton absorption after illumination with a pulsed Ti:sapphire laser tuned to 820 nm,

$B→hνC.$
(1)

The time-dependent concentration distributions of PAGFP in the two forms are represented as b(x,y,z,t), the basal or unconverted form of PAGFP, and c(x,y,z,t), the converted form. (In the presentation that follows it is helpful to have names to distinguish the two forms; thus, we shall refer to them as “basal-PAGFP” and “converted-PAGFP” or the “b-form” and “c-form.”) As will be explained further, these concentrations are referenced to the cytoplasm. With these considerations, the general system of equations governing the photoconversion and diffusion of PAGFP in three dimensions may be written:

$∂b∂t=D∇2b−Qs$
(2)
$∂c∂t=D∇2c+Qs$
(3)
$Qs=αm*γ*Im(x,y,z,t) b,0≤t≤ΔT=0,t≥ΔT$
(4)
$b(x,y,z,t)=b0,t≤0c(x,y,z,t)=0,t≤0$
(5)

where D is the diffusion coefficient of PAGFP, which is assumed to be identical for the basal and photoconverted states of the molecules, ∇2 is the Laplacian operator, and Qs is a source term that describes the generation of photoconverted molecules. The source term Qs embodies the intensity- and time-dependent photoconversion of PAGFP by a Ti:sapphire laser pulse of duration ΔT, where $αm*$ is the effective multiphoton absorption cross section of the basal form of PAGFP, γ* is the quantum efficiency of photoconversion, and I(x,y,z,t) is the photon flux density at spatial coordinate (x,y,z) and time, t. The superscript m is used to generalize the equation to include flux profiles determined by and describing multiphoton absorption (see below). Eq. 5 describes the initial concentration distributions of the two forms of PAGFP.

### The distribution of the basal form of PAGFP in the rod

In addition to its role as the substrate creating the source from which the c-form of PAGFP diffuses, the b-form is of importance in several ways for the experiments and theory. First, because it is, albeit weakly, fluorescent with 488-nm excitation, it serves to identify rods with high levels of expression of PAGFP, which are the most suitable for experimentation. Second, on the hypothesis (presented in detail below) that before photoconversion the b-form is in equilibrium with the water space of the cell, measuring its initial 3-D distribution in a rod targeted for photoconversion serves as a critical baseline for monitoring the progress toward equilibrium.

### The multiphoton point spread function and PAGFP photoconversion

The Ti:sapphire laser used to photoconvert PAGFP is focused into the sample to the diffraction limit; thus, the spatial profile of the photon flux density, I(x,y,z), may be approximated as the following Gaussian function normalized to the flux density at the laser focus, I0 = I(x = 0, y = 0, z = 0),

$Im(x,y,z)I0m = 1(2π)3/2σ2xyexp(−12x2+y2σxy2)× {az1σz1exp(−12z2σ2z1)+az2σz2exp(−12z2σ2z2)} ,azi> 0,$
(6)

where σ’s are the standard deviations of the Gaussian profiles in the direction of the indicated axis, and azi are scaling factors for the sum of two Gaussians used in the z dimension.

The superscript m used in Eqs. 4 and 6 is used to express the fact that the photoconversion of PAGFP, and indeed the measurement of the I(x,y,z) profile, is achieved by multiphoton absorption by fluorescent molecules. Thus, the effective multiphoton excitation profile is a power law function of the photon flux profile (Zipfel et al., 2003), such that the exponent m denotes the number of (infrared) photons that must be nearly simultaneously absorbed for excitation of a fluorescent molecule (cf. Calvert et al., 2007). Thus, we refer to Im(x,y,z) as the “multiphoton point spread function,” or mppsf.

### Source intensity rate parameter

In Eq. 4, the product $αm*γ*I0m$ has the unit s−1 and thus may be treated as a composite “source intensity rate” parameter. This parameter is related to the “bleach depth” parameter of FRAP studies, but it is more general and has the distinct advantage in the context of Eqs. 24 that the photoconversion pulse does not need to be short relative to the diffusion of the photoconverted species (cf. Calvert et al., 2007).

### Coordinate system

The rod OS is approximately a right-circular cylinder, and other portions of the rod have circular symmetry (Fig. 2 B); thus, it is convenient to describe 3-D positional information within the rod in cylindrical coordinates: two orthogonal coordinate axes, a longitudinal or axial (z) and a radial (r), and an angular coordinate, θ, that describes the rotation about z (Fig. 2 B). The spatial coordinate systems in Cartesian and cylindrical coordinates are:

$r⇀=(x,y,z)(Cartesian coordinates)r⇀=(r cos(θ),r sin(θ),z)(cylindrical coordinates)$

where θ is the azimuthal angle about the z axis and $r=|r⇀|=x2+y2$, with distance specified in µm. The cylindrical coordinate system allows different values of D for molecular diffusion along the two orthogonal axes, allowing for analysis of anisotropic movement. To implement potential difference in these diffusivities, we have included independent radial, Dr, and axial, Dz, variable diffusion coefficients in our analyses (supplemental text).

Note that the z dimension of the psf introduced above refers to the optical light propagation axis and is distinct from the z dimension of the cylindrical coordinate system used in analysis of diffusion in the rod. All discussion of the model output refers to the latter cylindrical coordinate system.

### Boundary conditions (BCs)

PAGFP cannot cross the plasma membrane; the addition of 0 flux BCs at the plasma membrane is thus required to complete the system of equations:

$∂c(r,z=l,t)∂z = ∂c(r,z=L,t)∂z = ∂c(r=0,z,t)∂r=∂c(r=r0,z,t)∂r= 0.$
(7)

Eq. 7 describes the case in which photoactivation is centered at r = 0.

### Conservation of PAGFP

Eqs. 25, when combined with the no-flux BCs, Eq. 7, imply that that the total quantity of PAGFP in the cell, which includes both unconverted and photoconverted forms, should remain constant. Moreover, after a photoconversion exposure, given negligible bleaching by the 488-nm scanning beam, the quantity of photoconverted PAGFP should remain constant as it redistributes throughout the cell. Testing for conservation is an important control for system and cell stability.

### Numerical solution of the system equations

To solve the system defined by Eqs. 27, we used the numerical methods of lines (Schiesser, 1991; Schiesser and Griffiths, 2009). In brief, the terms of the Laplacian (∇2) operator, expressed in cylindrical coordinates, were replaced by appropriate algebraic (finite difference) approximations. The resulting system of ordinary differential equations (ODEs) in time was coded in MATLAB and integrated with a sparse integrator in the MATLAB function library (ode15s). Various tests were used to determine the accuracy of the solutions, including comparison to analytical solutions of special cases of the model (supplemental text). To compare the model output to fluorescence data, the blurring of the distribution of fluorescent molecules in the cell caused by the scanning psf was accounted for by convolving the model output with a kernel based on the measured psf (Materials and methods). This allowed the estimation of diffusion coefficients within relatively homogeneous cell compartments large enough to accommodate the “volume” of the psf.

### Diffusion in the CC: alternative approaches

Some specific problems arise in the application of the general model (Eqs. 27) to the CC per se. First, the dimensions of the CC are below the resolution limit of the confocal microscope. Second, the CC presents a discontinuity in the cell radius r0 between the IS and OS compartments that is difficult to accommodate in the full 3-D model. To circumvent these problems, we used two alternative theoretical approaches. These approaches rest on the assumption that PAGFP molecules equilibrate in the radial dimension of the rod much more quickly than in the longitudinal, thus reducing the problem to a single spatial dimension, the rod axis. This assumption is reasonable for small molecules in the OS of rods (Lamb et al., 1981; Olson and Pugh, 1993) and, as will be shown in Results, can be experimentally justified for PAGFP diffusion in the IS and OS compartments.

### Approach 1: Fick’s law

The diffusivity of PAGFP in the CC can in principle be estimated with a straightforward application of Fick’s first law:

$fluxCC(t)=DCC ACC [cIS(t)−cOS(t)]LCC.$
(9)

In this expression, fluxCC(t) is the net flux of the c-form of PAGFP into the OS through the CC at time t after a photoconversion exposure in the IS, cIS(t) and cOS(t) are the concentrations of the protein in the IS and OS, respectively, at the locations adjacent to the CC, LCC is the length and ACC the cross-sectional area of the cilium patent to diffusion, and DCC is the diffusion coefficient. To apply Eq. 9, the concentrations cIS(t) and cOS(t), as will be shown below, can be extracted with Eq. 20, and the parameters LCC and ACC are estimated from electron micrographs (Table I). The net flux into the OS can be estimated from the total mass of the c-form in the OS; thus, if MOS(t) is the total mass of the c-form in the OS at time t, estimated from the spatially integrated fluorescence, then fluxCC(t) = dMOS(t)/dt . Thus, we have the following relation from which DCC can be estimated:

$dMOS/dt[cIS(t)−cOS(t)]=const= DCC ACCLCC.$
(10)

An important feature of Eq. 10 is that it predicts that the ratio of two measureable time-varying quantities should be a time-independent constant.

### Approach 2: 1-D model of the rod

The assumption of rapid radial diffusion allows the photoreceptor to be modeled in one spatial dimension, with appropriate accommodations for the variable cross-sectional area (Fig. 2, A and B). Thus, Fick’s first law of diffusion in one spatial dimension may be expressed as follows (cf. Crank, 1975):

$fA=−D∂c∂z,$
(11)

where f is the flux of molecules (number of molecules per unit time, t) passing through a cross section of area A and, $∂c/∂z$ is the gradient of the diffusing species along the axial (z) coordinate. Consideration of mass balance then allows one to write the rate of change of concentration in any volume element along z as the spatial derivative of Eq. 11,

$∂c∂t=−∂(fA)∂z.$
(12)

Rearrangement and substitution readily yield Fick’s second law of diffusion in one spatial dimension for the case in which both the cross-sectional area and diffusion coefficient are dependent on the axial position (see supplemental text):

$∂c∂t=1A(z)∂(A(z)⋅D(z)⋅∂c∂z)∂z.$
(13)

Solution of Eq. 13 allows the time course of equilibration of a diffusing molecular species in the rod to be calculated, given experimental determinations of A(z), D(z), and $∂c/∂z$.

### Initial and BCs

The natural physical BCs for the problem are “0 flux” at the rod synaptic terminal and at the tip of the OS (Fig. 2 B). However, because the OS and IS differ radically in their material density and structure, and because the IS equilibrates far more rapidly than the OS, it is convenient and warranted to impose a “pseudo boundary” at a z position (identified as z = 0) of our choosing. c(z = 0,t) may be derived from the measured fluorescence intensity at z = 0 and used as a time-varying Dirichlet BC. These BCs are implemented as follows:

$∂c(z=L,t)∂z=0$
(14a)
$c(z=0,t)=f(t)$
(14b)

Eq. 14a states that there is no flux at either end of the cell, i.e., the synaptic terminal (LST) or the tip of the OS (LOS) of the rod, whereas Eq. 14b is the Dirichlet BC that states that the concentration at the specific axial position z = 0 is a known function of time, f(t).

As initial condition, the model assumes PAGFP to be completely equilibrated before the application of the photoconversion exposure and there to be none of the photoconverted form:

$b(z,t=0)=b0 for all z, andc(z,t=0)=c0=0.$
(15)

Evidence supporting this assumption will be presented.

### Solutions

Eqs. 1315 were solved by means of the numerical method of lines (Schiesser, 1991; Schiesser and Griffiths, 2009). Further explanation of the BCs and details of the method of solution are presented in the supplemental text.

### The relationship between measured fluorescence and PAGFP concentration

The application of alternative approaches 1 or 2 to measure the diffusion of PAGFP through the CC requires determination of the concentration gradient between the IS and OS compartments (Eqs. 10 and 13). The simplest approach to estimating the gradient would be to assume that the concentration of converted-PAGFP is everywhere proportional to the specific fluorescence attributable to that form of protein, with the same proportionality constant. This assumption is reasonable for cells such as Chinese hamster ovary (CHO) cells, whose cytosol is largely homogeneous (Calvert et al., 2007) but clearly invalid for rods (Fig. 2, B and C). Thus, the rod interior is to varying degrees occupied by organelles such as mitochondria in the IS or structural elements such as the discs of the OS, which exclude PAGFP. When the psf exceeds the size of such structures, a spatially varying fluorescence distribution is expected, even if PAGFP is present at the same concentration throughout the cytosol (Fig. 2 C) (Peet et al., 2004).

### The enhanced green fluorescent protein (EGFP)/PAGFP equilibration hypothesis

In a previous paper (Peet et al., 2004), we proposed and found support for a hypothesis about EGFP that provides a basis for resolving the problem of inferring concentrations from fluorescence in the presence of subcellular structures or organelles that cannot be resolved with confocal/multiphoton microscopy. The hypothesis proposes that EGFP equilibrates with the communicating water spaces of cells, such that at equilibrium each volume element of the cell contains a quantity of EGFP proportional to its accessible water space. Because PAGFP diffuses both in aqueous solutions and in live CHO cells with diffusion coefficients identical to those of EGFP (Calvert et al., 2007), it is reasonable to hypothesize that both the b-form and c-form of PAGFP equilibrate with the cytoplasm in the same manner as EGFP; therefore, here we generalize the name of the hypothesis to the “EGFP/PAGFP equilibration hypothesis.”

The specific fluorescence F recorded for 488-nm excitation when the psf is centered at a location (x0,y0,z0) is the convolution of the psf and the concentrations of the two forms of the PAGFP:

$F(x0,y0,z0)=K∭I(x−x0,y−y0,z−z0) [αbγb b (x,y,z) +αcγc c (x,y,z) ] dx dy dz=K I0 ∭psf(x−x0,y−y0,z−z0) [αbγb b (x,y,z) +αcγc c (x,y,z)] dx dy dz$
(16)

In Eq. 16, K is a system constant that embodies the collection efficiency of the objective, and the optical transmission and capture of fluorescence emission by the system detectors, αb and αc are the effective molecular cross sections of the fluorescent species at the excitation wavelength, γb and γc are the quantum efficiencies of fluorescence given absorption, and b(x,y,z) and c(x,y,z) are the concentrations of the b-form and c-form of PAGFP in the elemental volume dx dy dz, understood to be a very small fraction of the psf volume. The second line of Eq. 16 restates the first in terms of the psf, where $psf(x,y,z)≡I(x,y,z)/I0$ and I0 is the photon flux density at the center of the psf.

The efficiency αbγb of exciting fluorescence from basal-PAGFP with 488-nm illumination is ∼1% that of the photoconverted state, αcγc (Patterson and Lippincott-Schwartz, 2002). Nonetheless, the initial distribution b(x,y,z) of the b-form can be measured, provided care is taken to exclude autofluorescence artifacts (Calvert et al., 2007). Before a photoconverting exposure, c(x,y,z) in Eq. 16 is 0, and in post-processing scan data, the fluorescence attributable to the initial distribution b(x,y,z) can be subtracted voxel by voxel to extract c(x,y,z,t).

The EGFP/PAGFP equilibration hypothesis can be expressed in the terms of Eq. 16. Initially, before photoconversion, when only the b-form of PAGFP is present in the cell and equilibrated in the cytoplasm, the hypothesis is expressed as

$F0(x0,y0,z0) ≡ Kαbγb ∭I(x−x0,y−y0,z−z0) b0(x,y,z) dx dy dz=Kαbγbbaq,0 I0∭psf(x−x0,y−y0,z−z0) faq(x,y,z) dx dy dz ,$
(17)

whereas after a photoconversion exposure when the cell has again equilibrated, the hypothesis applied to the combined fluorescence from the b-form and c-form is

$F∞(x0,y0,z0) ≡K ∭I(x−x0,y−y0,z−z0) (αbγb b∞(x,y,z)+αcγc c∞(x,y,z))dx dy dz =K(αbγb baq,∞+αcγc caq,∞) I0 ∭psf(x−x0,y−y0,z−z0) faq(x,y,z) dx dy dz.$
(18)

In Eqs. 17 and 18, the subscripts “0” and “∞”represent the initial equilibrated prephotoconverted state and the final post-photoconverted state after reequilibration, respectively, and faq is the aqueous fraction of the volume element. The “≡” signs in the first lines of Eqs. 17 and 18 indicate that the expression is definitional, whereas the second line embodies the equilibration hypothesis. Thus, in Eq. 17, baq,0 represents the initially equilibrated concentration of the b-form of PAGFP, whereas in Eq. 18, baq,∞ and caq,∞ are the finally equilibrated concentrations of the two forms in a volume element of pure cytosol, and $b∞(x,y,z), c∞(x,y,z)$ are the equilibrated concentrations in a partially obstructed volume element. It follows from Eq. 18 that in an equilibrated cell, the fractional water space at location (x0,y0,z0) can be estimated as

$f¯aq(x0,y0,z0)≡F∞(x0,y0,z0)/Fmax,∞=∭psf(x−x0,y−y0,z−z0) faq(x,y,z) dx dy dz ∭psf(x−x0,y−y0,z−z0) dx dy dz$
(19)

(cf. Peet et al., 2004); in the denominator of Eq. 19, faq is presumed to be unity, that is, maximal fluorescence is obtained from volume elements that are “pure cytosol.” Thus, $f¯aq$ is a volumetrically weighted average estimate of $faq(x,y,z)$, with the psf acting as the 3-D weighting function.

Consider now a time t after a photoconversion exposure, and let ΔF(x,y,z,t) be the incremental fluorescence, i.e., the measured fluorescence with the preexposure fluorescence due to the basal form of PAGFP subtracted. Then, the concentration of the c-form of the fluorescent protein in the cytosolic component of a volume element can be estimated as

$c¯(x,y,z,t)=ΔF(x,y,z,t)I0 K f¯aq(x,y,z)αcγc (1−δ),$
(20)

where $δ=αbγb/αcγc≈0.01$ and the psf volume is normalized. Two caveats for the application need to be mentioned. First, the volume of the psf sets the scale of relevant structural inhomogeneities. Second, there is no a priori guarantee that there exists in the cell a cytosolic volume the size of the psf that is not to some degree obstructed.

### Prediction of uniform scaling of fluorescence after photoconversion

Because the application of the diffusion model rests on the validity of the EGFP/PAGFP equilibration hypothesis via Eq. 20, it is worthwhile to make and test quantitative predictions of the hypothesis that do not depend on the model. The first such prediction follows readily from Eqs. 17 and 18 and relates the distribution of fluorescence $F∞(x,y,z)$ of the equilibrated c-form of PAGFP after a photoconversion exposure, with the initial fluorescence distribution $F0(x,y,z)$ of the b-form:

$F∞(x,y,z)=u F0(x,y,z),$
(21)

with u = [(αcγccaq,∞+ αbγbbaq,∞)/αbγbbaq,0)] a scalar quantity. This relationship follows because all distributions of fluorescence arising from equilibrated PAGFP must be proportional to one another, as they depend only on $faq(x,y,z)$ and constants of the system and protein. A closely related prediction is that the scale factor u in Eq. 21 can be obtained from the total or spatially integrated fluorescence:

$Ftot,∞=u Ftot,0,$
(22)

where $Ftot=∭F(x,y,z) dx dy dz$ is the integrated cell fluorescence. Confirmation of these predictions provides support of the PAGFP equilibration hypothesis, and of the estimation of $f¯aq(x,y,z)$ in the rod before exposure to a photoconversion stimulus, allowing c(x,y,z,t) to be derived from Eq. 20.

### Equilibration of photoconverted PAGFP between rod cytoplasmic compartments

Retinal slices were scanned in 3-D at 488 nm to find rod cells that had suitably high PAGFP expression levels and that were well aligned for imaging, i.e., with their longitudinal axis nearly parallel to the imaging plane (Fig. 3 A). The central-most image plane in the z dimension that contained a suitable cell was then selected as the focal plane (z coordinate) at which to photoconvert and monitor PAGFP over time (Fig. 3 B). The x–y coordinates for PAGFP photoconversion were then manually selected from the image based on spatial cues, such that the photoconversion pulse would be localized to the myoid region of the IS (Fig. 3 B, red dot in the IS region), which was identified as the midpoint between the nucleus, the protuberance just distal to the synapse, and the ellipsoid, the region of lower florescence just proximal to the OS (Peet et al., 2004). Proper placement of the photoconversion pulse was confirmed after experiment from examination of the pattern of diffusing, photoactivated PAGFP. In a few cases, PAGFP was inadvertently activated in the nucleus, which was evident by rapid nuclear filling. These cells were excluded from the analysis.

### Time course of photoactivated PAGFP equilibration

To quantitatively examine the kinetics of photoactivated PAGFP equilibration throughout the entire cytoplasm, the fluorescence changes were evaluated in two subregions of the cell at ∼10-s intervals, one that encompassed the OS and the other that included the IS photoconversion site, the nucleus, and the synapse (Fig. 3 C; images just before and at several times after the photoconversion exposure are shown in Fig. 3 D; see Video 1). In this analysis, the final time course image (25.5 min for the cell shown in Fig. 3) was used to delineate the regions of fluorescence to be integrated because the boundary between the OS and the IS compartments was more clearly evident. The integrated florescence (F) within the regions in each time course image was normalized to the integrated fluorescence in the respective regions in the prephotoconversion image (F0) and plotted as a function of time from the photoconversion pulse onset (Fig. 3 E).

### Conservation of photoactivated PAGFP mass

To test mass conservation, the total fluorescence increase in the x,y images taken after the photoactivation pulse (Ftot) was normalized to the total fluorescence in the pre-conversion image (Ftot,0) and examined over the time course of the experiment (Fig. 3 F). Conservation was used as a criterion for cell integrity over the time course of the experiment, and cells failing this test were rejected from further analysis. The total fluorescence in the cell depicted in Fig. 3 increased 5.3 ± 0.10–fold (mean ± SD) and varied little (SD < 2% of the time-averaged fluorescence increase), indicating that the mass of photoconverted PAGFP was stable throughout the experiment.

### Half-time of PAGFP equilibration between IS and OS

The ratio Ftot/Ftot,0 provides the expected scaling factor for the fluorescence increase in any local region of the rod upon equilibration of PAGFP after a focal photoconversion exposure (Eqs. 21 and 22). For the rod of Fig. 3, Ftot/Ftot,0, = 5.3 (Fig. 3 E, dashed line). This value was used to estimate the half-time of equilibration of the OS after the photoconversion pulse. For the rod of Fig. 3, the time to 50% equilibration was 9.2 min (Fig. 3 E), and it was ∼85% equilibrated at 25.5 min. On average, the time to half equilibration was 7.0 ± 1.0 min (Table II). This result is in contrast to the time course of GFP equilibration in mouse rods after photobleaching in rod OSs at room temperature, which was reported to be complete within ∼3 min (T1/2, ∼30 s) (Nair et al., 2005). This difference can be explained on the basis of the different sizes of these cells (see Discussion).

### PAGFP diffusion between compartments proximal to the OS–IS junction

The diffusion of photoactivated PAGFP between IS compartments was examined by selecting smaller regions of interest (ROIs) within three major IS structures: the myoid, synaptic region, and nucleus (Fig. 4). The synaptic region equilibrated with the myoid relatively rapidly, within ∼5–10 min. More precisely, the fluorescence in the synaptic region rose to a peak level approximately equal to that in the myoid within ∼5 min, and then declined in parallel with the fluorescence in the myoid. This indicates that the cytoplasmic spaces of the myoid and the synaptic region are contiguous and relatively unobstructed. In contrast, the movement of PAGFP into the nucleus and OS compartments occurred much more slowly, indicating barriers to diffusion.

It has been shown that GFP and its variants enter the nuclei of Xenopus photoreceptors (Peterson et al., 2003; Peet et al., 2004). The mechanism for nuclear filling is not understood. Some have posited that GFP and its variants are actively transported to the nucleus. Results shown in Fig. 4 reveal for the first time that the movement of PAGFP across the nuclear membrane is relatively slow, taking ∼20 min to equilibrate with the neighboring myoid and synapse compartments. Ongoing experiments and analyses are aimed at testing the hypothesis that this slow movement is purely passive. PAGFP transport into the nucleus provides the interesting prospect of using it as a tracer for the regulation of nuclear pore complex permeability and active protein transport across the nuclear membrane in living retinal photoreceptors, studies that are currently underway.

### Test of the prediction of uniform scaling of fluorescence after photoconversion

The EGFP/PAGFP equilibration hypothesis (see Theory) predicts that upon equilibration after a photoconversion pulse, the fractional change in fluorescence (u) at all locations within the cell (Eq. 21) will be the same as the fractional increase in total cellular fluorescence (Eq. 22). This prediction is borne out as shown in Fig. 5, which compares the florescence of PAGFP in the rod depicted in Figs. 3 and 4 at points along a line that intersects each of the cell’s major compartments before and 50 min after photoactivation in the cell’s myoid. The fluorescence varies considerably along the line in both images. This variation is interpreted to indicate differences in the aqueous fraction in the different compartments (see Theory; Eq. 19). The bottom panel in Fig. 5 B, which plots the fluorescence after internal normalization to the fluorescence of brightest fifth percentile of voxels in each original image, shows that the distribution of fluorescence is essentially the same before and after photoconversion, after sufficient time for equilibration. Moreover, the ratio of the fluorescence of the brightest voxels was ∼5.3-fold, the same as the mean Ftot/Ftot,0 in Fig. 3 F. Similar results were obtained with six cells examined in this way.

### Radial equilibration of PAGFP is rapid relative to intercompartmental transport in both the IS and OS

The rates of radial equilibration in the OS and IS compartments (Figs. 6 and 7, respectively) were assessed using a “point blast–ROI scan” protocol (Calvert et al., 2007). Thus, the x,y coordinates for a photoconversion pulse were selected within a confocal scan at the central z plane of a rod cell, such that it would occur at the radial center of the cell. A 2-D ROI centered on the pulse site was then defined (Figs. 6 A and 7 A, green boxes). The ROI was scanned repeatedly before and after a multiphoton photoconversion pulse (Figs. 6 C and 7 C, and Videos 2 and 3). F/F0 in smaller regions within the acquired ROI (subROIs) near the plasma membrane boundaries were then compared with F/F0 measured for subROIs that encompassed the photoconversion sites at the radial center of the compartments (Figs. 6 B and 7 B, red boxes). Radial equilibration time was defined as the time after the photoconversion pulse when the radial F/F0 values merged with that of the photoconversion site (this definition was used because there is a much slower component of equilibration when OS intercompartmental transport is taken into consideration). The average radial equilibration time in the myoid region of the IS was 2.9 ± 0.6 s (n = 13 cells), and in the OS it was 1.9 ± 0.2 s (n = 12 cells). These values were not significantly different at the P = 0.05 level. Thus, PAGFP equilibrates radially within a few seconds in both the IS and OS compartments, much more quickly than the time required for equilibration between these compartments, which is on the order of tens of minutes (Fig. 3). This result justifies the assumption of rapid radial equilibration in both IS and OS compartments used in the 1-D models applied to measure PAGFP diffusion within the CC.

### PAGFP diffusion is isotropic in the myoid and anisotropic in the OS

In addition to radial equilibration, axial equilibration time was assessed. SubROIs were defined at distances equal to those between the photoconversion site ROI and the radial ROIs, but at positions along the orthogonal, axial dimension of the rod (Figs. 6 B and 7 B, axial ROIs). This allowed a direct comparison of the equilibration times in the radial and axial dimensions for a given compartment. In the myoid region, the average axial equilibration time was 3.4 ± 0.8 s (n = 13 cells), a value not significantly different from the radial equilibration time at the P = 0.05 level. To our knowledge, this is the first assessment of the directional uniformity of molecular diffusion in the IS compartment of rods. In contrast, and not surprisingly, the mean axial equilibration time of 14.5 ± 1.2 s (n = 12 cells) in the OS compartment was nearly eightfold slower than that of the radial, presumably due to the highly tortuous axial diffusion path caused by membranous discs in the OS (cf. Fig. 2 B). It should be noted that as defined here, equilibration times are figures of merit and not directly convertible to diffusion coefficients, which must be determined by the application of an appropriate model.

### Flux of PAGFP between IS and OS compartments: estimation of the diffusion coefficient in the CC

The flux of PAGFP between the OS and IS compartments was measured according to the direct application of Fick’s first law (Eq. 10 and Fig. 8). Three regions (Fig. 8 A) were selected for analysis in sequential time series images. Region 1, which encompassed the entire OS compartment, defined the area over which OS fluorescence was integrated. Regions 2 and 3, which were chosen within the IS and OS compartments in close proximity to the IS–OS junction, defined regions over which the fluorescence was averaged and to which Eq. 20 was applied to arrive at the water space–corrected relative concentration of protein (Fig. 8 B).

Using the radial symmetry of the OS, the time course of the flux of photoactivated PAGFP into the OS compartment, dMOS(t)/dt, was estimated from the time-dependent change in the mass of activated PAGFP detected in the x,y scan plane, Mxy, as follows:

$dMOS(t)dt=(Mxy(ti+1)−Mxy(ti))⋅VOSVxyti+1−ti,$
(23)

where $Mxy(t)=∬(F(x,y,t)−F0(x,y))⋅Vvoxdxdy$, Vvox is the volume of the voxel defined by the x,y spatial sampling frequency, and the psf dimension in z, VOS/Vxy is the ratio of total OS envelope volume and the x,y scan plane volume. VOS was determined directly from 3-D scans of the cell as described in the supplemental text. dMOS(t)/dt and cIS(t)−cOS(t) for one cell are shown in Fig. 8 C. Eq. 10 predicts that the ratio $(dMOS/dt)/[cIS(t)−cOS(t)]$ will be constant over time, and this indeed appears to be the case through ∼6 min after the photoconversion pulse (Fig. 8 D), after which the magnitude of the difference between them becomes too small to accurately measure. The average value of $(dMOS/dt)/[cIS(t)−cOS(t)]$ over the initial 6 min after the photoconversion pulse for the cell depicted was 0.52 µm3s−1. Considering the average dimensions of the CC (Table I), we obtain an axial diffusion coefficient of PAGFP in the CC of DCC = 3.5 µm2 s−1. Similar values were obtained from six cells (Table II).

### The diffusion coefficient of PAGFP in the myoid region of the IS

The diffusion coefficient DIS in the myoid region of the IS was estimated using the point blast–line scan method (Fig. 9) (cf. Calvert et al., 2007), which provides much higher temporal resolution than the point blast–ROI scan method presented in Figs. 6 and 7. A brief multiphoton photoconversion pulse was applied to the radial center of the myoid, which was followed by rapid (∼1-KHz) line scans that intersected the photoconversion site (Fig. 9 A, left). The result is a spatiotemporal map of the relaxation of the photoconverted PAGFP within the IS cytoplasm (Fig. 9 A, right). We then applied the 3-D diffusion model described in Theory to the relaxation map varying the diffusion coefficient, D, which was assumed to be uniform for the radial and axial directions (Fig. 9 B). The root mean square difference error between the model and the data was calculated (supplemental text) and plotted as a function of D to arrive at the optimal estimate of DIS for the given cell. On average, DIS = 5.16 ± 0.62 µm2 s−1 (n = 13 cells).

### The axial diffusion coefficient of PAGFP in the OS compartment

To estimate the coefficient of diffusion, DOS, of PAGFP along the axial dimension of the OS, the fluorescence levels of voxels within time course images were corrected according to Eq. 20. The corrected fluorescence, F′(x,z,t), was then averaged within a region that was confined to the OS along the radial dimension in each time course image, and F′(z,t = 0) was subtracted to arrive at a spatiotemporal map of PAGFP concentration change, F′(z,t) (Fig. 10).

Predicted spatiotemporal maps of PAGFP equilibration, Fm(z,t), in the OS compartment were then produced from the 1-D diffusion model presented in Theory (Eqs. 1315). Time-varying Dirichlet BCs in Eq. 14b, FDB(t), were derived from the averaging of n proximal F′(z,t) values from each of the time course images,

$F′DB(t)=∑z=1nF′(z(n),t)n,$
(24)

and fitted with a sixth-order polynomial, the result of which was used in model calculations (Fig. 10 B).

The OSs of pre-metamorphic Xenopus rods are not generally perfectly cylindrical, instead tapering proximo-distally along the axial dimension of the OS. To account for the tapering in model calculations, the area of cross section as a function of the rod axial position (A(z) in Eq. 13) was determined with a MATLAB routine (see supplemental text). The axial diffusion coefficient in Eq. 13, D(z), was assumed to be constant over all z.

Predicted profiles were thus calculated (Fm(z,t)) (Fig. 10 D) for a range of axial diffusion coefficients and compared with F′(z,t). DOS for PAGFP was determined as the value that produced the smallest ERMS between model prediction and data. For the rod depicted in Fig. 10, DOS = 0.075 µm2 s−1 produced the model output that best fit the data. DOS values obtained from this and other cells are given in Table II.

### Alternative approach to estimating DCC

The 1-D diffusion model with variable cross-sectional area and variable D (Eqs. 1315) offers an alternative approach to quantifying the diffusion coefficient of PAGFP in the CC. The present analysis is similar to that for the axial diffusion of PAGFP in the OS outlined in the previous section, except that the region defined for analysis was extended to include the portion of the cell just proximal to the interface between the IS and OS compartments, thus including the length of rod containing the CC (Fig. 11 A). The predicted Fm(z,t) was then calculated with diffusion coefficients that varied along the axial dimension of the rod. Three zones of differing diffusion coefficients used in the model, DM(z), were defined as follows:

$DM(z)=DIS,zz(CCL),$
(25)

where DIS is the diffusion coefficient of PAGFP in the IS, the average value determined from point blast-line scan experiments in the IS described in Fig. 9 was used, DCC is the axial diffusion coefficient of PAGFP in the CC, DOS is the diffusion coefficient of PAGFP in the OS determined for the specific cell under investigation as shown in Fig. 10, and z(CCl) and z(CCL) are the axial boundaries of the CC. Fm(z,t) was calculated for a range of DCC, and the best value of DCC was considered to be that which produced the minimum value of ERMS. DCC = 2.5 µm2 s−1 provided the best fit for the rod depicted in Fig. 11. Values for DCC from this and other cells using this method are presented in Table II.

### Summary: reduction of diffusion in the rod’s compartments

A useful summary of the results can be made by comparing the effective diffusion coefficients estimated for PAGFP in the different rod compartments with that measured in aqueous solution and in the relatively unstructured, passed CHO cells. A useful metric capturing the comparison is the “fold-reduction” of the effective diffusion coefficient Deff from that in aqueous solution: (Daq/Deff) (Table III).

Quantitative characterization of the steady-state distributions, concentrations, and movements of soluble proteins in living cells is a goal of broad importance for understanding many cellular functions, including cell signaling kinetics and protein renewal mechanisms. Complex polarized cells such as sensory receptors pose many problems to achieving such characterization, for their subcellular “compartments” are differentiated ultrastructurally and often connected by or terminated in narrow constricted structures, such as cilia or axons, whose diameter is near or below the resolution of light microscopy (Figs. 1 and 2). Using the Xenopus rod photoreceptor as a model system and PAGFP as a reporter, we have shown how many of the problems can be addressed and in particular have, for the first time, measured the diffusion coefficient of a protein in a primary cilium, as well as in the other major compartments of a highly polarized cell, and also the overall time for the protein concentration to equilibrate within and between compartments.

### The CC poses a modest barrier to protein diffusion

The basal body–axoneme complex has been compared with the nuclear pore complex in terms of regulating protein access to the cilium (Deane et al., 2001), and it has been proposed that the permeability of the basal body–axoneme complex to protein transport may be regulated in a signal-dependent manner (Wolfrum et al., 2002). The 36-fold reduction in the effective diffusion coefficient of PAGFP in the CC relative to aqueous solution (Table III) provides the first direct evidence that these structures do indeed impede soluble protein movement between the cell body and the cilium, and leaves open the possibility of signal-dependent control of cilium permeability. However, because DCC is only twofold smaller than DIS (Table III) and PAGFP diffusion is more rapid in the IS than anywhere else in the cell, it is difficult to posit that the CC constitutes a highly distinctive barrier to protein movement.

The most distinctive feature of the CC seen in electron micrographs that would be expected to impact protein movement is the 9 + 0 microtubule axoneme and associated structures, but the core of the axoneme appears to offer an unobstructed path for diffusion (cf. Fig. 2). Although the axonemal components will reduce the effective cross-sectional area available for diffusion, as the maximal diameter of each microtubule doublet is ∼37 ± 1 nm (Rosenkranz, 1977), the entire axonemal structure reduces the effective cross-sectional area by only ∼7%, not nearly enough to account for the 36-fold reduction from Daq. It remains possible that the cross-sectional area of the CC available for diffusion of PAGFP is much smaller than apparent from transmission electron micrographs (EMs). Electron tomographic examination of the centrosomal region of chondrocyte cilia (Jensen et al., 2004) shows it to be crowded with fine filamentous material, transitional fibers, and vesicles. Moreover, the lumen of the basal bodies contains large vesicles and electron-dense “plugs” that may occlude or substantially reduce the cross-sectional area available for diffusion. However, as we now discuss, retarded diffusion elsewhere in the rod suggests that more general principles of differentiated cells are at play.

### Protein diffusion in the IS is substantially retarded

The IS comprises the region of the rod between the ellipsoid and the nucleus. This region contains the major protein- and lipid-synthesizing and -degrading machinery of the rod and is expected to be homologous to similar regions in most cell types. Diffusion in the IS of PAGFP is isotropic (Fig. 9) and faster than in any other region of the rod (DIS = 5.2 µm2 s−1). Nonetheless, PAGFP diffusion is substantially retarded in the IS, being 18-fold lower than in aqueous solution and fourfold below that in CHO cells (Table III).

### Axial diffusion of PAGFP in the OS is retarded 50-fold more than predicted by geometry

The effective coefficient for axial diffusion of PAGFP in the OS, DOS, is reduced 1,125-fold from that in aqueous solution (Table III). The OS comprises densely stacked discs, paired membrane bilayers spaced at ∼30 nm, which greatly reduce the patent cross-sectional area, so that most paths for axial diffusion are highly tortuous (Fig. 2) (cf. Rosenkranz, 1977). The impact of tortuosity on DOS may be predicted from a straightforward relation (Lamb et al., 1981):

$τ=FVFA.$
(26)

Here, FV is of the fraction of the OS volume occupied by the discs, and FA is the fraction of the cross section accessible to molecular diffusion (Lamb et al., 1981; Olson and Pugh, 1993). FV has long been estimated to be ∼0.5 (Rosenkranz, 1977; see also Peet et al., 2004), although a recent cryoelectron microscopy study of mouse rods suggests it could be as low as 0.3 (Nickell et al., 2007). FA is determined by the gap between the discs and the plasma membrane (10–20 nm), and by the incisures, infoldings of the plasma membrane that invaginate the discs (Fig. 12). On average, a frog disc contains 27 incisures with lengths of 2.7 µm (Tsukamoto, 1987) and gaps of ∼10 nm (according to measurements made in salamander; Olson and Pugh, 1993). Accordingly, the area of a 7-µm diameter frog rod OS available for axial diffusion is ∼1.0 µm2, 2.6% of the OS cross-sectional area. Thus, Eq. 23 gives τ = 0.5/0.026 ∼20, and so predicts that tortuosity alone should reduce axial PAGFP diffusion 20-fold from Daq. If the incisures were inaccessible to PAGFP diffusion, Eq. 26 would yield τ ∼40. Removing the contribution of tortuosity, we are left with an unaccounted 28- to 56-fold reduction in DOS from Daq, and 6- to 12-fold reduction from DCHO (Table III).

### Mechanisms underlying the retarded diffusion of PAGFP in the rod

GFP and its variants, particularly EGFP, have been used in benchmark studies of protein diffusion in cellular aqueous compartments (for review see Verkman, 2002; see also Dix and Verkman, 2008). We have recently shown that the GFP variant used here, PAGFP, is indistinguishable from EGFP with respect to its diffusion in aqueous solutions of varied viscosity and in CHO cells, but provided substantially superior performance with respect to signal/noise and light load (Calvert et al., 2007). The diffusion coefficient of GFP variants in aqueous compartments of cultured cells is generally reduced approximately three- to fivefold from its value in water or physiological saline (Swaminathan et al., 1997; Partikian et al., 1998; Dayel et al., 1999; Calvert et al., 2007). It is thus striking that PAGFP diffusion is much more retarded in the aqueous compartments of rods (Table III) than in those of most cells previously investigated (for exceptions, see Elowitz et al., 1999; Konopka et al., 2009), and calls for some explanation.

Retarded diffusion of macromolecules, including GFP variants, in aqueous compartments of cells has been ascribed primarily to four factors: fluid-phase cytoplasmic viscosity, macromolecular crowding, binding interactions, and hydrodynamic boundary effects (Hou et al., 1990; Kao et al., 1993; Minton, 1997; Luby-Phelps, 2000; Verkman, 2002; Dix and Verkman, 2008; Zhou et al., 2008). The viscosity of the aqueous cytoplasm is governed by the concentration of small solutes and organizing effects of macromolecules on water (Clegg, 1984; Parsegian and Rau, 1984). Crowding reduces the volume fraction available for translational diffusion and thus the number of possible paths available to a diffusing molecule. Binding interactions include all inelastic collisions of the diffuser with nonsolvent particles and intracellular surfaces (e.g., ER and SER) and microfilaments. Retarding hydrodynamic friction occurs when a significant fraction of the paths available to the diffuser are put in close contact with solvent that is less mobile. (These factors are not completely distinct; see the citations just given.)

The retardation of PAGFP diffusion by several of these factors could be greater in photoreceptors than in previously investigated cells (e.g., CHO cells). Photoreceptors may use a relatively high density of cytoskeletal elements to maintain their highly structured, polarized morphology. Crowding by macromolecules may be exceptionally high in the cytoplasm of the IS and OS: interferometric measurements of cytoplasm, which probe particulate density, indicate that the refractive index of the OS is 1.41 (Sidman, 1957) and the core of the myoid region of cones has an index of 1.42 (Rowe et al., 1996). These values contrast with those of cultured mammalian cells, with reported refractive indices in the 1.36 to 1.37 range (Lanni et al., 1985). The high refractive index in the OS is due to dense packing of lipids and phototransduction proteins, and in the IS it may reflect denser packing of protein-synthetic machinery needed to support the high turnover of phototransduction components in the OS, which, in the case of frogs, renews ∼2% of its large complement of proteins daily (Young, 1967). Finally, the lamellar membranes of the OS and the putative high density cytoskeletal components offer an increased surface area that may retard the diffusion of PAGFP by weak binding interactions or hydrodynamic friction.

Strong binding interactions between PAGFP and subcellular structures do not appear to be involved because in the cells examined, the diffusion coefficients in the CC and OS did not correlate with the concentration of expressed PAGFP, which varied more than 200-fold in the rods investigated (see supplemental text). Another possible mechanism for slowed diffusion of PAGFP is self-crowding. However, the highest expression level in the cells examined was ∼100 µM, at least 10-fold below the theoretical threshold for crowding effects (see supplemental text).

The value of DOS (0.079 µm2 s−1) reported here for PAGFP differs considerably from the value (3.3 µm2 s−1) recently reported for EGFP expressed in Xenopus rods (Wang et al., 2008). Several factors may explain this discrepancy, including the harsh conditions used to isolate the cells and an inadequate description of the FRAP bleach pattern. The manipulations used by Wang et al. (2008) to isolate, immobilize, and permeabilize the cells could cause cell swelling (increasing the water space) and depolymerization of cytoskeleton components (decreasing crowding). Our approach strictly maintains physiological intactness of the photoreceptors, as we imaged only rods that were in retinal slices and maintained normal morphology. The diffusion model applied by Wang et al. (2008) assumes uniform bleaching along the entire width of the rod OS, which was fixed to the coverslip. However, both bleaching and scanning were performed at a single focus level with a 1.3-NA oil-immersion objective. The radius of the diffraction-limited illumination volume along the axis of focus (the Rayleigh or Abbe limit) is r = λ/NA2 ∼0.3 µm at 488 nm. Given an average Xenopus rod diameter of 6–7 µm, a substantial fraction of the OS width would have been minimally bleached; therefore, the fluorescence recovery signals were likely much sped up by unbleached molecules diffusing radially into the scan plane from out of focus regions (cf. Figs. 6, 7, and 9).

### Could advection due to active transport within the CC contribute to soluble PAGFP movement into the OS?

An alternative mechanism to diffusive transport of PAGFP to the OS compartment that warrants consideration is transport by advective flow of cytoplasm that may be generated by motor-based transport carriers. At a concentration of ∼3 µM in the OS, rhodopsin is the most abundant protein in the rod and, given the well-established rate of renewal of the OS disc membranes (∼14 d in Xenopus), can be estimated to be transported to the OS compartment via post-Golgi vesicles at a maximal rate of approximately one ∼300-nm diameter vesicle per second. It is controversial whether these vesicles transit the CC to reach the OS, or rather fuse with the plasma membrane in the IS near the cilium transition zone and are carried to the OS via IFT. However, if the mechanism of transport of rhodopsin-laden membrane were shown to be motor-based movement of vesicles through the core of the CC, the ∼300-nm diameter vesicles would be large enough to occlude most of the CC lumen, and could possibly produce a peristaltic flow of cytoplasm toward the OS. To examine the possibility of such an “advective” transport of PAGFP, the 1-D model of diffusion in the rod cell described in Theory was extended to include an advective term. This advection/diffusion model predicts that the steady-state PAGFP concentration in the OS compartment would exceed that in the IS compartment by >30% (see supplemental text). Contrary to this prediction, in none of the cells examined in the present paper, nor in the large population expressing EGFP examined in a previous paper (Peet, et al. 2004), did the steady-state concentration of PAGFP or EGFP in the OS compartment exceed that of the IS. It is thus unlikely that advection plays a significant role in the transport of soluble proteins between rod photoreceptor compartments.

### Impact of variation in cell dimensions on the time course of soluble protein equilibration

Although diffusivities govern the local movements of protein in the cells, the overall time to equilibrate is greatly affected by cellular and subcellular dimensions. Consider the cases of frog and mouse rod photoreceptors. The CC of frog and mouse rods are the same diameter, yet the diameter of the cells are very different: frog rods are on average ∼7 µm in diameter, whereas those of mice are ∼1.4 µm. As a consequence, the frog rod cilium involves an ∼330-fold reduction in the cross section available for diffusion, whereas the corresponding reduction in the mouse rod is only ∼19-fold. In a three-compartment simulation, these differences resulted in equilibration times that differ approximately sevenfold (Fig. 13).

We similarly examined the equilibration time predicted for a ciliated epithelial cell where the cilium diameter is constant but attached to a large cytoplasmic reservoir, as, for example, for an olfactory cilium (cf. Fig. 1). Assuming the diffusion coefficients of the cell body and the cilium to be equal and ∼5 µm2 s−1, the equilibration was remarkably fast (Fig. 13 D). Although the equilibration time varied significantly with cilium length, equilibration half-times were <1 min for lengths up to 20 µm (Fig. 13 D), suggesting that diffusion could be a viable means of soluble protein transport into all primary cilia.

The authors wish to thank Drs. Mohammad Haeri and Barry Knox for valuable assistance in generating transgenic Xenopus, Dr. Kathy Battaglia and Cheryl Gretzula for plasmid preparation, and Jason Stewart for valuable technical assistance.

P.D. Calvert is recipient of a Career Development Award from Research to Prevent Blindness (RPB). This work was supported by the Lions Clubs of Central New York and by grants from the Karl Kirchgessner Foundation (to P.D. Calvert) and the National Institutes of Health (grant EY018421 to P.D. Calvert and grants EY016453 and EY02660 to E.N. Pugh). The SUNY UMU Department of Ophthalmology is the recipient of an unrestricted grant from RPB.

Alan S. Verkman served as guest editor. (Journal policy regarding manuscripts submitted by the Editor or Associate Editors is available at http://jgp.rupress.org/misc/policies.shtml#submitted.)

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Abbreviations used in this paper:

• BC

boundary condition

•
• CC

connecting cilium

•
• CHO

Chinese hamster ovary

•
• EFGP

enhanced green fluorescent protein

•
• EM

electron micrograph

•
• IFT

intraflagellar transport

•
• IS

inner segment

•
• ODE

ordinary differential equation

•
• OS

outer segment

•
• PAGFP

photoactivatable green fluorescent protein

•
• ROI

region of interest

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