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Departments of * Chemical and Biological Engineering and Biological Sciences, University at Buffalo, State University of New York, Buffalo, New York
Correspondence: Address reprint requests to Professor Johannes M. Nitsche, Dept. of Chemical and Biological Engineering, Furnas Hall, University at Buffalo, State University of New York, Buffalo, NY 14260-4200. Tel.: 716-645-2911 ext. 2213; Fax: 716-645-3822; E-mail: nitsche{at}eng.buffalo.edu.
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ABSTRACT |
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 TOP ABSTRACT GLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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GLOSSARY |
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TOPABSTRACTGLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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V)j
V)total
pore
Greek symbols
pore
, 
Subscripts and other affixes
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INTRODUCTION |
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TOPABSTRACTGLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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; Kumar and Gilula, 1996; Simon and Goodenough, 1998; Yeager et al., 1998; Harris, 2001). They are formed when two hemichannels (half-pores, connexons), each comprising six connexin subunits (of which >20 types are currently known; Harris, 2001; Eiberger et al., 2001), dock in the intercellular medium. Such pores represent key features of multicellular organisms because they provide the only documented means for the direct exchange of small metabolites between cells. As such, they have been implicated in a multitude of normal physiological and disease processes including electrical synchronization of heart beat, homeostasis, tumor suppression, and direction of early developmental processes (Edelson, 1990; Lo, 1996). Most cells express multiple connexin types, which may associate to form hemichannels that are either homomeric (comprising only one type) or heteromeric (comprising multiple types). Complete gap junctions may also be either homotypic (if the two constituent hemichannels are identical) or heterotypic (if they are not).
Although gap junctions were once often regarded simply as indiscriminate aqueous conduits between cells, a growing body of evidence in the literature indicates that they exhibit significant selectivity based on a complex interplay of physicochemical factors (Flagg-Newton et al., 1979; Brink and Dewey, 1980; Brink and Ramanan, 1985; Traub et al., 1994; Elfgang et al., 1995; Veenstra et al., 1995; Veenstra, 1996; Cao et al., 1998; Nicholson et al., 2000; Gong and Nicholson, 2001; Harris, 2001). Permselectivity features of connexins are not restricted to simple size or charge discrimination, and are likely to significantly influence their function in biological systems. This fact is graphically illustrated in recent studies of Goldberg et al. (1999, 2002), where the rates of transmission of specific endogenous metabolites through gap junctions composed of different connexins expressed in C6 glioma cell monolayers were compared. The surprising conclusion from this comparison was that two connexins (i.e., Cx43 and Cx32), which form channels with similar dye permeability, showed as much as 300-fold differences in permeability to ATP, and lower levels of relative selectivity for other metabolites including ADP, AMP, glutamate, and glutathione. Bevans and Harris (1999) also observed a dramatic shift in selectivity between cAMP and cGMP in reconstituted hemichannels when the connexin composition (ratio of Cx32/Cx26 subunits) was changed. Clearly, quantitative descriptions of gap junctional selectivity for a variety of compounds varying in different physical parameters will be needed if one is to ultimately elucidate the underlying molecular mechanisms and develop generalizable rules for the permeability features of a given connexin.
An important avenue toward this end is provided by experiments in which dye is introduced into one member of a pair (Veenstra et al., 1995; Cao et al., 1998; Valiunas et al., 2002), a chain (Simpson et al., 1977; Schwarzmann et al., 1981; Brink and Ramanan, 1985; Zimmerman and Rose, 1985), or a monolayer (Flagg-Newton et al., 1979; Schwarzmann et al., 1981; Safranyos and Caveney, 1985; Steinberg et al., 1994; Traub et al., 1994; Elfgang et al., 1995; Goldberg et al., 1995; Cao et al., 1998) of cells, and observed to spread linearly or radially into the neighboring cell(s) as a function of time. The ultimate goal of such experiments is to deduce absolute, or at least relative, values of unitary (single-pore) junctional permeabilities Ppore of various channel types to probes of varying size, shape, charge, and other physicochemical properties. This microscopic parameter quantifies the diffusive flow 
(moles/time) of dye through a single channel according to the relation
 | (1) |
, pp. 296–298, 337–341; Nitsche, 1999, p. 480). We refer to the product AporePpore, representing the constant of proportionality between concentration difference and resulting molecular flow, simply as (AP)pore, because the two factors usually appear together.
There generally exist two complications in the translation of observed dye transfer rates into unitary junctional area-times-permeability factors (AP)pore. The first is the fact that the intercellular membrane permeability Pjunc is not directly indicative of (AP)pore, because it represents the collective outcome of many unitary channel transport processes proceeding in parallel, as described by the equation
 | (2) |
; Traub et al., 1994; Veenstra et al., 1995; Cao et al., 1998; Valiunas et al., 2002) in which dye transfer measurements are accompanied by intercellular electrical conductance measurements, so that the total number of channels can be counted if their unitary conductance is known.
A second, more serious complication arises from the fact that observed dye transfer rates represent the net outcome of the membrane resistance actually sought, and a mass transfer resistance associated with diffusion through cellular cytoplasm to and from the membrane. If the cells employed are sufficiently small, or if the membrane has sufficiently low permeability, then the cytoplasm is effectively well mixed and the dye transfer rate is directly indicative of the (rate-limiting) value of Pjunc. This parameter can then be deduced from a data analysis scheme in which individual cells are treated as coupled, well-mixed compartments (Zimmerman and Rose, 1985; Cao et al., 1998; Valiunas et al., 2002). Generally, however, a diffusion model is needed to analyze data and deconvolute membrane from cytoplasmic transport effects. Ample precedent for such diffusion models exists, as has been reviewed recently (Nitsche, 1999). The study of fluorescent dye transfer between septate giant axons of the earthworm by Brink and Ramanan (1985) exemplifies a rigorous analysis of this type. These authors determined values of Pjunc and the cytoplasmic diffusivity Dcyt for three dyes, and found that transfer of dichlorofluorescein (but not carboxyfluorescein) was accompanied by a significant cytoplasmic diffusion resistance. A decrease of the apparent transport coefficients for Lucifer Yellow with time was indicative of significant dye binding to components of the cytoplasm. Similar and more complex models of diffusion within single cells (e.g., Horowitz et al., 1970; Kargacin and Fay, 1991) and cell aggregates (e.g., Ramanan and Brink, 1990; Christ et al., 1994) exist.
Apparently the only determination to date of absolute values of junctional permeability on a per-channel basis has been reported recently by Valiunas et al. (2002). Their combined measurements of fluorescence intensity and conductance yielded unitary transfer rates for Lucifer Yellow (LY) in HeLa cells coupled by junctions comprising rat Cx43 and Cx40 connexins. The smallness of their system tends to minimize the physical factors discussed above.
This article addresses a new series of experiments (Weber et al., 2004) based on a novel system in which passage of fluorescent dye from a "donor" Xenopus oocyte to a coupled "acceptor" oocyte is quantified by digital video images (Nicholson et al., 2000), with concurrent measurement of the electrical conductance between the same cell pair. The specific purpose is to develop the modeling infrastructure needed to deduce unitary gap junctional permeabilities from raw data in the form of the ratio of acceptor-cell/donor-cell fluorescence intensities as a function of time. Equations describing the transient, three-dimensional cytoplasmic and transmembrane diffusion process are formulated and then solved using a finite difference technique. A separate single-cell version of the model, fitted to data for uncoupled oocytes, is used to deduce values of the cytoplasmic diffusivity Dcyt, as well as two parameters characterizing binding to the cytoplasm, which figure in the full double-cell model. The theory ultimately yields curves for the acceptor/donor concentration ratio that fit the raw data well, and lead to self-consistent values of (AP)pore. The efficacy of the approach is demonstrated with reference to passage of three Alexa-series dyes through gap junction channels composed of a number of connexin types. The outcome is a set of results for (AP)pore at a level of quantitation surpassing previous more-qualitative analyses in the literature. These results demonstrate both dye and connexin dependencies of channel permeability. Derived unitary permeability data are found to be consistent with a microscopic model embodying an interplay between hindered diffusion and a permeant-pore affinity factor, the latter making the pore energetically favorable for the dye, thereby increasing in-pore concentration and flux levels. Valiunas et al. (2002) noted that the flux values they measured for LY were below those one would expect for efficient propagation of labile signals in multicellular networks. The values measured here for the Alexa-series dyes, however, are more consistent with what one might expect for propagation of such signals.
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SUMMARY OF EXPERIMENTS ANALYZED |
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TOPABSTRACTGLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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). Two Xenopus oocytes are immersed in medium within a well created by pushing the end of a plastic microfuge tube into a layer of agar at the bottom of a petri dish. A relatively small (41.4 nl) bolus of 10 mM fluorescent dye solution (Alexa 350, Alexa 488, or Alexa 594 in the experiments analyzed here, Molecular Probes, eugene, OR) is introduced by micropipette at a prescribed injection spot xspot within one of these cells. Fluorescence intensity is averaged over two square imaging boxes, respectively positioned to reflect dye concentrations within "donor" and "acceptor" regions, at a number of discrete time points.
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) are of two types. For any particular dye, single-cell experiments aim to quantify the cytoplasmic diffusivity Dcyt, and the binding (forward) rate coefficient kcyt and equilibrium constant for any reversible binding to elements of the cytoplasm, without the complication of intercellular transfer. Although oocyte pairs are still employed to maintain a geometry identical to that of subsequent double-cell experiments, no connexins are expressed, making the intercellular membrane between the coupled cells effectively impermeable. The point of injection xspot = xspot,single lies at one side of one of the oocytes. The donor and acceptor imaging windows (0.43 x 0.43 mm) are positioned at opposite sides of this injected oocyte to quantify the equilibration process as dye spreads across it by diffusion and binds to the cytoplasm, in principle eventually settling down to a spatially uniform distribution. Double-cell experiments, in which given connexins are expressed, address the junctional permeability of the intercellular membrane. The point of injection xspot = xspot,double lies near the center of one oocyte, and dye diffuses to the other oocyte through the intercellular membrane, whose permeability is the only remaining unknown parameter to be determined by data fitting. Donor and acceptor imaging windows (0.86 x 0.86 mm) are centered on the injected cell and its neighbor, respectively, to track this process.
Typical raw data recorded from the imaging system are shown in Fig. 2 (Weber et al., 2004), which presents snapshots of the dye distribution at selected times after injection in cases of (A) single-cell and (B) double-cell experiments. Fluorescence intensity is encoded in terms of hue; it increases in the order of: black (zero) 
violet blue green yellow red white (highest). These images show clearly the spreading of dye across the cellular cytoplasm (A and B) and through the intercellular membrane (B). They also give a feel for the inhomogeneity of the cellular cytoplasm, and the possible variability from experiment to experiment.
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FORMULATION OF THE MODEL AND METHODS OF CALCULATION |
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TOPABSTRACTGLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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Geometry
As shown in Fig. 3, the shapes of the two oocytes are idealized in terms of identical truncated (intersecting) ellipsoidal surfaces with prescribed semiaxes and positions chosen to match dimensions measured from a number of images of the system. The two ellipsoids intersect along an ellipse representing the perimeter of the planar intercellular membrane, comprising the apposed, junctionally coupled portions of the two cellular membranes at z = 0. According to the coordinate system used here, the direction up in the laboratory is equivalent to the -y direction. A view from the bottom (Fig. 2)—looking up at the oocytes through the petri dish—corresponds to the view employed experimentally with the inverted microscope. This represents a view from the positive y axis, and reveals the half-length (1.06 mm) and width (1.33 mm) of the cell pair, as well as the width of the intercellular membrane (0.78 mm). A side view (Fig. 1) corresponds to a view along the x axis and reveals the thickness of the oocytes (1.14 mm). The cell receiving the initial injection of dye (at a prescribed point xspot) is taken to be the "-" cell below the plane z = 0, so that diffusion occurs primarily in the +z direction. In a single-cell experiment (with xspot = xspot,single) the dye stays inside this "-" cell, whereas in a double-cell experiment (with xspot = xspot,double) dye enters the "+" cell through the intercellular membrane. According to the assumed truncated ellipsoidal shape, the volume Vcell of each oocyte is 
0.90 mm3 and the area Amem of the intercellular membrane is 0.41 mm2. Also represented in Fig. 3, B–E, are the imaging boxes and optical paths.
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" affix. Thus, for instance, 
, 
, etc. Required for subsequent analysis is a representation of the "+" and "-" oocyte surfaces in spherical coordinates based at the origin (which coincides with the center of the elliptical intercellular membrane). (Polar and azimuthal angles and are measured from the positive z and x axes, respectively, as in the usual definition of spherical coordinates; see Bird et al., 2002, p. 826.) On these surfaces, the (dimensionless) radial distance 
varies with and as given by a function 
defined in Appendix A. The regions of space occupied by these respective oocytes are denoted by
 | (3) |
 | (4) |
is denoted by 
and corresponds to the coordinate value = 
/2. The uncoupled cellular membranes are the surfaces given by 
for /2 
and 
for 0 /2, respectively denoted by 
and 
.
Governing transport equations
Theoretical analysis focuses on the concentrations c- and c+ of freely diffusing dye within the "-" and "+" oocytes, which are functions of position x = (x, y, z) and time t, and are defined over the respective spatial domains - and + for all t 
0. Also considered are populations of dye molecules bound to the cytoplasm, for which the corresponding concentration fields are denoted by the symbols b- and b+.
A number of physicochemical parameters enter the model and determine the predicted outcome of a dye transfer experiment. Individual cells are characterized by a diffusivity Dcyt (mm2/s) of dye within the cytoplasm, as well as a forward rate constant kcyt (s-1) and equilibrium constant Kcyteq (dimensionless) for reversible binding of dye to the cytoplasm. The latter two parameters appear in rate expressions of the form
 | (5) |
 | (6) |
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. All dye concentrations are made dimensionless using a characteristic value c0 (defined in the next subsection), and are regarded as functions of dimensionless position and time. Thus, we deal with 
, 
, etc.
The time-dependent intracellular diffusion and binding process is governed by the dimensionless equations (compare to Bird et al., 2002, Chap. 19; Cussler, 1997, pp. 319–320; Deen, 1998, pp. 54–56)
 | (7) |
 | (8) |
 | (9) |
 | (10) |
is a dimensionless binding rate coefficient. Dye transfer through the intercellular membrane is described by the equation
 | (11) |
denotes a dimensionless membrane permeability.
The images recorded from the experimental system show essentially no leakage for the dyes Alexa 488 and Alexa 594. Although some such leakage into the external solution is evident for Alexa 350, the concentrations involved are small compared with the observed intracellular concentrations. Therefore, although our model allows for an arbitrary permeability of the uncoupled portions of the two cellular membranes, actual calculations are performed with this permeability set to zero. The additional boundary conditions effectively imposed are thus
 | (12) |
 | (13) |
Characterization of dye injections and initial conditions
For all dye diffusion studies, injections by micropipette introduced 41.4 nl (0.0414 mm3) of a 10 mM dye solution into one oocyte, amounting to 4.14 x 10-10 mol of dye (Weber et al., 2004). The characteristic concentration c0 is specifically defined in terms of this mole number as c0 = (4.14 x 10-10 mol of injected dye)/L3 = 4.14 x 10-10 mol/mm3 = 0.414 mM.
The injected volume is small but finite (roughly one-twentieth of the cell volume), and the insertion and removal of the pipette undoubtedly causes some mixing of the cellular contents. Therefore, the injection process produces an initial dye distribution within the "-" cell that is highly concentrated around the point of injection, but is not a perfectly sharp Dirac delta distribution. We model it using a multivariate Gaussian (normal) distribution,
 | (14) |
( 0.21) is set by the reasonable order-of-magnitude criterion that the volume of injected dye equal the volume of a sphere with radius equal to the standard deviation, 
. The position vector 
represents the epicenter of the injection, which differs between single- and double-cell experiments, distinguished by the subscript expt (either single or double). The coordinates assumed in the model are 
and 
, based on a separate series of injections (mimicking those in the actual dye transfer experiments) specifically aimed at locating the epicenters (Weber, 2003). These points are marked by tiny spheres in Fig. 3. The factor spot, expt is a normalization factor computed such that the Gaussian distribution is normalized (has unit volume integral) over the injected ("-") cell. Its numerical values turn out to be spot, single 1.68 and spot, double 1.11 for single- and double-cell cases, respectively. (The Gaussian distribution without the factor spot, expt is normalized over all space.)
The remaining initial conditions reflect the facts that, at the instant of injection, no dye has diffused into the "+" cell, and binding has not had a chance to occur:
 | (15) |
 | (16) |
 | (17) |
Finite difference solution
Equations 7–17 collectively constitute a coupled set of initial boundary value problems to be solved for the position and time dependencies of the intracellular concentrations 
, 
, 
, and 
. Given the location of the injection spot on the yz plane, the solution of these equations must be symmetric around this plane, so that attention can be restricted to the interval -/2 /2. Our approach to their solution involves a new radial coordinate representing the fractional distance from the origin to the cell surface in each direction defined by the polar and azimuthal angles and , namely
 | (18) |
= 1. Within each, the triple of coordinates (, , ) defines a coordinate system which is nonorthogonal, but has the attractive feature that the oocyte domain is given by one of the simple expressions
 | (19) |
 | (20) |
in spherical coordinates (Bird et al., 2002, p. 836) adopts a rather lengthy form in terms of first and second derivatives with respect to , , and , given in Appendix A, which also provides explicit expressions for the normal derivatives 
appearing in the boundary conditions (Eqs. 11–13), as well as other requisite properties of our coordinate system.
The intervals 0 1, 0 , and -/2 /2 are respectively divided into Nr, 2N, and 2N subdivisions. Fig. 4 shows the resulting spatial discretization (at the cross section y = 0) for (Fig. 4 A) a coarse mesh with Nr = N = N = 6 and (Fig. 4 B) a more refined mesh with Nr = N = N = 12. Discrete values of the dye concentrations 
, 
, 
, and 
are defined on either of these meshes at each time (i, j, and k, respectively, indexing the values of , , and ).
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spot,expt for the Gaussian initial distribution 
is computed by approximating the required volume integral using Simpson's rule. For interior nodes (nodes for which 
/2 and < 1), the right-hand side of Eq. 7 or 9 is computed using second-order (three-point) central difference approximations for all derivatives appearing in (Appendix A, Eq. 36), and Euler's method with a prescribed time step 
is used to advance the nodal concentration values in time. Symmetry conditions are incorporated into the process of imposing the differential equation at nodes for which = -/2 or /2. For nodes at the physical boundaries (intercellular membrane, = /2, and uncoupled cellular membranes, = 1), the boundary conditions (Eqs. 11–13) are applied to compute concentration values consistent with the updated interior values, using asymmetric (one-sided) formulas for normal derivatives. All nodal values of the bound dye concentrations are updated according to Eqs. 8 and 10 by Euler's method using the complete set of current values and . Further details of the procedure are given by Chang (2003).
The preceding calculation (including optical computations discussed in the next subsection) was coded in Fortran and run on several PCs. For reference, timing data for runs on a PC are included in the captions of Figs. 5 and 6 below. Execution time increases very rapidly with increasing degree of mesh refinement, because of the increasing number of nodes, and—further—a concomitantly decreasing time step needed to maintain numerical stability (determined empirically, and surprisingly small). Fitting of cytoplasmic properties to single-cell data (Fig. 5 below) was carried out using the refined mesh (Nr = N = N = 12, see Fig. 4 B; 
), because here it is especially important to resolve intracellular concentration gradients accurately as dye diffuses from one side of the oocyte to the other. Fitting of the intercellular membrane permeability to double-cell data (Fig. 6 below) was performed using the coarse mesh (Nr = N = N = 6, see Fig. 4 A; 
), because here the intracellular diffusion process is less critical, especially when the intercellular membrane controls the rate of dye transfer.
Optical analysis of the model
Consider a given donor or acceptor imaging box marked, say, in the xz plane (Fig. 3, B and C). Of special relevance is the subset of the cytoplasm comprising all points whose projections in the +y or -y directions (which are respectively downwards or upwards in the laboratory) onto the x,z plane fall inside the box (see optical paths marked in Fig. 3, D and E). We assume that the average fluorescence intensity measured from such a box is proportional to the total amount of dye (mobile and bound) contained in this subset, i.e., visible through the window defined by the imaging box. At any time, this quantity (divided by the box area, and made dimensionless with c0L) is given by an integral of the form
 | (21) |
and 
denote functions, defined in Appendix A, giving the 
coordinates of the upper and lower oocyte surfaces, respectively, in terms of 
and 
. Numerical approximation of the nested integrals in Eq. 21 is effected using Simpson's rule. Required values of the intracellular concentrations are interpolated from the nodal values 
and 
defined on the finite difference mesh using a three-dimensional extension of two-dimensional bilinear interpolation.
At any time, the calculated quantity directly comparable with the measured acceptor-box/donor-box fluorescence intensity ratio is
 | (22) |
Counting of channels
Equation 2 makes it possible to determine the unitary channel area-times-permeability factor, (AP)pore, from the membrane permeability Pjunc, provided one has the ability to count open channels between oocytes. Toward this end, dye transfer measurements were accompanied by measurements of the macroscopically observable intercellular electrical conductance Gobs (Weber et al., 2004), defined as
 | (23) |
V)total denotes the total cell-to-cell voltage drop. The conductance Gobs yields the number of open channels Npore with knowledge of the unitary channel conductance pore. However, Gobs is not simply equal to Nporepore owing to sources of electrical resistance other than the membrane channels (e.g., cytoplasmic resistance), which cause (V)total to exceed the voltage drop (V)j actually occurring across the intercellular membrane.
Sophisticated models exist for dual voltage-clamp measurement of electrical conductance between cells, addressing junctional access resistance and other factors (e.g., Wilders and Jongsma, 1992; Van Rijen et al., 1998). For present purposes, an estimate was obtained experimentally for the relation between the fraction (V)j/(V)total of the total voltage drop occurring across the intercellular membrane, and Gobs, as described by Weber et al. (2004) and presented in their Fig. 5 E. A simple cell-pair conductance model was then developed to support an empirical correlation of these data (Appendix B). It leads to the equations
 | (24) |
 | (25) |
12,800 S-1 in units of reciprocal siemens or ohms (S-1 = ), this type of relation provides a reasonable description of the data (Weber et al., 2004, their Fig. 5 E). With this fitted value of ß, Eq. 25 allows the number of open channels Npore to be estimated from the measured intercellular conductance Gobs. Further discussion of the parameter ß is provided in Appendix B. |
RESULTS |
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TOPABSTRACTGLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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Analysis of single-cell experiments—characterization of cytoplasmic diffusion and binding
Our first step is to ascertain values of the model parameters characterizing the cellular cytoplasm (Table 1). This determination is made by fitting of the measured curves showing acceptor-box/donor-box fluorescence intensity ratio as a function of time in the single-cell experiments (Weber et al., 2004), in which dye diffuses across the cytoplasm of only one (the "-") oocyte. The apparent starting point of each data set is matched by adjusting BG. The initial rapid rise of a calculated curve is controlled by the cytoplasmic diffusivity Dcyt. With a given value of Dcyt, the shape of the "shoulder" (i.e., the region of high curvature between the initial rapid rise and the later leveling off) depends mainly upon kcyt. Given Dcyt and kcyt, the choice of Kcyteq then determines the calculated acceptor/donor level at longer times. Judicious use of these facts expedites the fitting process. As a general philosophy we use the minimum Kcyteq consistent with the data. Fig. 5 compares the computed and measured curves. In cases where two experimental curves were measured and fitted (for Alexa 350 and Alexa 594, represented in parts A and C), the values of Dcyt, kcyt, and Kcyteq listed in Table 1 represent averages of the respective values belonging to the individual curves. The dimensionless binding rate coefficient 
is computed by making the average kcyt dimensionless using the average Dcyt. Alexa 488 differs from the other two dyes in the respect that all three data sets exhibit a steeper initial rise and a more abrupt leveling off, indicative of a higher Dcyt for this dye. They are fitted collectively by the single theoretical curve shown in part B. The model parameter showing the greatest variability among fits to different data sets for a given dye (and therefore the greatest uncertainty) is Kcyteq. For Alexa 594 the two fitted values of Kcyteq differ by a factor of 
5, owing to significant differences between the measured acceptor/donor levels at longer times (see Fig. 5 C).
Included for reference in Table 1 are values of the bulk aqueous diffusivity Daq of each dye at 25°C, estimated from the molecular structure using the Wilke-Chang correlation, together with Schroeder's rule as a predictor of the molar volume (Poling et al., 2001, pp. 4.33–4.35, 11.21–11.23). The Stokes-Einstein equivalent radius a follows from Daq according to the formula a = kT/(6µDaq) (Deen, 1987; Poling et al., 2001, p. 11.21).
Analysis of double-cell experiments—determination of intercellular membrane and unitary channel permeability
For a double-cell experiment the parameter to be determined is the apparent macroscopic permeability of the intercellular membrane (
in dimensionless form), which depends upon the unitary channel permeability and the degree of intercellular coupling (i.e., number of functional channels); see Eq. 2. For all combinations of dye and type of connexin expressed in the oocytes, this parameter was fitted by trial and error to each of a number of measured curves giving the acceptor-box/donor-box fluorescence intensity ratio as a function of time (Weber et al., 2004), producing values in the range 
. Data sets were screened to ensure that they conformed to pre-established criteria. Initial junctional conductance had to be between 5 and 50 µS to obtain sufficient signal while avoiding artifacts from cytoplasmic bridges, and was not allowed to increase more than twofold over the 6-h duration of the experiment. Fig. 6 shows examples of fits of the model to the data. Some data sets contained more scatter or other nonidealities, as detailed by Chang (2003), but the model could offer a reasonable representation of all the data. In all, 189 double-cell data sets were analyzed.
Aside from , one (and sometimes two) parameters varied between double-cell experiments. The estimated starting point of each curve was matched approximately by adjusting the background parameter BG appearing in Eq. 22. For a minority (approximately one-quarter) of data sets, Kcyteq was reduced (usually by a factor of 2 or 4) because the observed high rate of cell-to-cell dye transfer (
) was consistent with a lower degree of cytoplasmic binding. The implied variability in agreed with that already observed in fitting the single-cell data (which showed variations in by a factor of 5).
In double-cell control experiments, the preparatory step of injection with connexin RNA and antisense oligonucleotide to endogenous Xenopus Cx38 was replaced by an injection of antisense oligonucleotide alone (so that no channels would be expressed in the cellular membrane). A small fluorescence intensity sometimes observed in the acceptor imaging box represents a background likely to arise from refraction or optical imperfections in the system, or some residual endogenous channel formation. As an order-of-magnitude check on the possible effects of such imperfections, they may be characterized in terms of an "equivalent" degree of intercellular membrane permeability that would give rise to the same rate of change of the acceptor/donor. The conclusion of this analysis (Chang, 2003) is that values of 
(which arise for data sets showing low dye transfer rates) might not be significant as they fall under the possible level of optical noise in the system. This threshold value of applies to Alexa350; the average noise level seems to be lower for the other two dyes.
For each experimental curve to which a value of was fitted, Npore was estimated from the measured intercellular conductance using Eq. 25 together with the known unitary channel conductance. Table 2 lists unitary conductances for the channel types considered. It agrees well with a recent approximate tabulation of unitary conductances "in 120–150 mM salt" (Harris, 2001, p. 383), and is roughly applicable to currents carried by the natural cytoplasmic medium. The final result of our analysis is the quantity AmemPjunc/Npore = (AP)pore (compare to Eq. 2). It represents the effective constant of proportionality between a macroscopic dye concentration difference across the intercellular membrane, and the resulting molecular flow (moles/time), reckoned on a per-channel basis. Fig. 7 shows the variation of (AP)pore with unitary channel conductance for each dye. Because of variations over more than an order of magnitude, we report mean values (with error bars indicating mean ± SE) of the logarithm of (AP)pore. For each combination of dye and channel type, the logarithm of the ratio of membrane permeability to channel number was averaged over all data sets analyzed.
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(8 and 0.1, respectively) discussed above indicate that the oocyte system is capable of determining intercellular membrane permeabilities over a range spanning roughly two orders of magnitude. They are translated approximately into corresponding limits on (AP)pore in Appendix C. The conclusion is that values of (AP)pore exceeding the order of 1.4 x 10-9 mm3/s (log10[(AP)pore/(mm3/s)] - 8.9) are probably not reliably indicated. In this regime the dye transfer rate is limited by intracellular diffusion and does not reflect the (high) membrane permeability. Noise in the system may be characterized in terms of an equivalent permeability, and renders possibly insignificant values of (AP)pore below the order of 1.8 x 10-11 mm3/s (log10[(AP)pore/(mm3/s)] - 10.7). |
DISCUSSION |
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TOPABSTRACTGLOSSARYINTRODUCTIONSUMMARY OF EXPERIMENTS ANALYZEDFORMULATION OF THE MODEL...RESULTSDISCUSSIONAPPENDIX A: MATHEMATICAL DETAILS...APPENDIX B: SIMPLE CELL-PAIR...APPENDIX C: APPROXIMATE RELATION...APPENDIX D: CALCULATION OF...ACKNOWLEDGEMENTSREFERENCES |
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in terms of functional comparisons between different gap junctions. Here we focus on what our analysis and the results say about the physics of the intercellular transfer process.
Factors affecting the macroscopically observable rate of dye transfer
Xenopus oocytes furnish a good system for the quantification of dye transfer rates (Nicholson et al., 2000; Weber et al., 2004). However, they present the challenge that the desired junctional permeability is convoluted with a number of other obscuring physical factors. It is worthwhile to recap the effects these factors have on our derived values of (AP)pore.
The most important nonjunctional phenomenon influencing intercellular transfer seems to be binding of dye to components of the cytoplasm. We assume reversible binding because all attempts at describing the data with an irreversible binding model failed. The fitted values of , ranging from 6 to 10 (Table 1), indicate that the bound state is strongly preferred for all three dyes (because they significantly exceed unity). The characteristic binding times 
are on the order of 10–20 min. Although the precise microscopic origin of the binding process remains to be clearly defined, this type of gradual phenomenon has been indicated in other dye transfer studies (Brink and Ramanan, 1985). It is worth noting that the levels of the acceptor/donor curves in Fig. 5 (significantly below unity) after one-half hour represent a transient phenomenon. Because of the reversibility of binding, these curves would ultimately reach values around unity after a much longer elapsed time, i.e., the final equilibrium state is a spatially uniform distribution of dye.
An analysis not explicitly accounting for binding would erroneously ascribe the consequent slowness of dye transfer to lower apparent values of intercellular membrane (and unitary channel) permeability. The double-cell experiments alone provide no means of deconvoluting the effects of intercellular membrane (and ultimately junctional) diffusional resistance, and binding. The single-cell data (Fig. 5) provide the independent information needed to characterize the latter, and hence achieve the deconvolution. Two illustrative calculations performed to test the effects of binding indicate that, for cases of moderate and high membrane permeability, ignoring binding would decrease the derived values of (AP)pore by factors of 10 and 60, respectively. We could fit the double-cell data in this way, but to do so would be to ignore the very strong and consistent evidence for a significant degree of binding embodied in Fig. 5.
Because of the O(1 mm) path length across an oocyte, the role played by cytoplasmic diffusional resistance is also significant. Its quantitative importance is made clear by values of the dimensionless parameter , representing the ratio of intercellular membrane to cytoplasmic permeabilities, found to be around unity or greater in many cases (see, e.g., labels on curves in Fig. 6). Mobilities of dye molecules are noticeably lower in cytoplasm than in bulk water. Values of Dcyt range from 30 to 90% of the corresponding values of Daq (Table 1), reflecting hindered mobility in the
