**Purpose.**:
Previous experiments showed that mouse lenses have an intracellular hydrostatic pressure that varied from 335 mm Hg in central fibers to 0 mm Hg in surface cells. Model calculations predicted that in larger lenses, all else equal, pressure should increase as the lens radius squared. To test this prediction, lenses of different radii from different species were studied.

**Methods.**:
All studies were done in intact lenses. Intracellular hydrostatic pressures were measured with a microelectrode-manometer–based system. Membrane conductances were measured by frequency domain impedance analysis. Intracellular Na^{+} concentrations were measured by injecting the Na^{+}-sensitive dye sodium-binding benzofuran isophthalate.

**Results.**:
Intracellular hydrostatic pressures were measured in lenses from mice, rats, rabbits, and dogs with radii (cm) 0.11, 0.22, 0.49, and 0.57, respectively. In each species, pressure varied from 335 ± 6 mm Hg in central fiber cells to 0 mm Hg in surface cells. Further characterization of transport in lenses from mice and rats showed that the density of fiber cell gap junction channels was approximately the same, intracellular Na^{+} concentrations varied from 17 mM in central fiber cells to 7 mM in surface cells, and intracellular voltages varied from −45 mV in central fiber cells to −60 mV in surface cells. Fiber cell membrane conductance was a factor of 2.7 times larger in mouse than in rat lenses.

**Conclusions.**:
Intracellular hydrostatic pressure is an important physiological parameter that is regulated in lenses from these different species. The most likely mechanism of regulation is to reduce the density of open Na^{+}-leak channels in fiber cells of larger lenses.

*x*

^{2}= 2

*Dt*, where

*x*is the average distance a particle diffuses in time

*t*, and

*D*is the diffusion coefficient. For biological solutes where

*D*≈ 10

^{−5}cm

^{2}/s, typical numbers are

*t*≈ 0.05 seconds when

*x*= 10 μm, whereas

*t*≈ 2 months when

*x*= 10 cm. Thus diffusion is very fast when distance is short, on the order of the size of a cell, but becomes very slow when distance is long, on the order of the size of an organ. As a consequence, most organs are infiltrated by a dense network of capillaries. In the heart, for example, there is approximately one capillary for each myocyte.

^{ 1 }In contrast, the vertebrate lens lacks a vasculature because the need for transparency precludes the presence of blood vessels, which would scatter light.

^{ 2 }All vertebrate lenses studied have a circulating ionic current that enters at the anterior and posterior poles and exits at the equator.

^{ 3–5 }Work reviewed in Mathias et al.

^{ 6 }suggests the current is carried by Na

^{+}. The fiber cell transmembrane electrochemical gradient for Na

^{+}causes Na

^{+}to move from the extracellular spaces between fiber cells into the intracellular compartment. This creates an extracellular electrochemical gradient for Na

^{+}, causing it to move from outside of the lens into extracellular spaces within the lens and flow toward the lens center. This also creates an opposite intracellular electrochemical gradient for Na

^{+}, causing it to flow back to the lens surface. The intracellular current is directed to the equator by gap junction coupling conductance, which is highest in the peripheral equatorial fiber cells.

^{ 7,8 }At the surface, Na

^{+}is transported out of the lens by the Na/K ATPase, which is concentrated in the equatorial epithelial cells.

^{ 5,9,10 }

^{ 6,11 }suggest a standing circulating Na

^{+}flux would be accompanied by a circulation of fluid. Subsequent experiments

^{ 12,13 }have been consistent with lens fluid transport; however, solute and fluid fluxes were not measured in lenses from the same species or in the same conditions. The properties of solute transport in the mouse lens have been extensively studied.

^{ 2 }Gao et al.

^{ 14 }therefore used mouse lenses to test the hypotheses: “fluid circulates through the lens; the intracellular leg of fluid circulation is driven by hydrostatic pressure and the flow is through gap junction channels; the fluid flow is generated by membrane transport of sodium.” Intracellular hydrostatic pressure was measured as an indicator of fluid flow in wild type lenses or when gap junction coupling or membrane transport of sodium was altered. As described below, their data provided solid support for these hypotheses.

^{ 14 }Hydrostatic pressure was predicted to scale with the group of parameters

*a*

^{2}

*j*

_{Na}/

*N*

_{j}, where

*j*

_{Na}(moles/cm

^{2}s) is the average density of fiber cell transmembrane influx of sodium,

*N*j (cm

^{−2}) is the number of open gap junction channels per area of fiber cell to cell contact, and

*a*(cm) is the lens radius. Using genetically engineered connexin (Cx) knockout/knockin mouse lenses, which were selected because they had normal sodium transport and similar radii, Gao et al.

^{ 14 }show that the intracellular hydrostatic pressure varies inversely with

*N*j. In wild type lenses exposed to acute conditions that had been shown to alter

*j*

_{Na}, Gao et al.

^{ 14 }show that the intracellular pressure varies in proportion to

*j*

_{Na}.

*a*

^{2}arose because of the force/flow relationship in spherical geometry. The purpose of the present study was to experimentally determine the relationship between lens radius and intracellular hydrostatic pressure. To obtain a significant variation in lens size, we used lenses from different species. However, the theoretical dependence on

*a*

^{2}assumes that all lenses have the same density of gap junction channels and the same average sodium influx per unit area of fiber cell membrane, properties which are not guaranteed to be constant among different species. Thus, in the smaller lenses from mice and rats, where electrophysiological studies are possible, we have tried to integrate our hydrostatic pressure data with data on other transport properties.

a | (cm) | Lens radius. |

C_{S} | (F/cm^{2}) | Surface cell membrane capacitance. |

c_{m} | (F/cm^{2}) | Fiber cell membrane capacitance. |

D_{Na} | (cm^{2}/s) | Effective intracellular diffusion coefficient for Na^{+}. |

G_{DF} | (S/cm^{2}) | Coupling conductance per area of cell-to-cell contact for differentiating fibers. |

G_{MF} | (S/cm^{2}) | Coupling conductance per area of cell-to-cell contact for mature fibers. |

g_{m} | (S/cm^{2}) | Fiber cell membrane conductance. |

G_{s} | (S/cm^{2}) | Surface cell membrane conductance. |

J_{Na} | (moles/cm^{2}s) | Radial intracellular flux of Na^{+}. |

j_{Na} | (moles/cm^{2}s) | Average fiber cell transmembrane influx of Na^{+}. |

K | (mm Hg s/ cm^{2} mole) | A group of parameters that are thought to not vary with lens species or size. |

N_{j} | (cm^{−2}) | Density of open fiber cell gap junction channels. |

p_{i} | (mm Hg) | Intracellular hydrostatic pressure. |

r | (cm) | Radial distance from the lens center. |

R_{DF} | (Ω cm) | Effective intracellular resistivity of differentiating fibers. |

R_{MF} | (Ω cm) | Effective intracellular resistivity of mature fibers. |

R_{e} | (Ω cm) | Effective extracellular resistivity. |

S _{m}/V _{T} | (cm^{−1}) | Surface area of membrane per unit volume of tissue. |

u_{i} | (cm/s) | Radial intracellular water flow velocity. |

Δ[Na^{+}]_{i} | (M) | The change in intracellular Na concentration between the lens center and surface. |

Δψ _{i} | (Volts) | The change in intracellular voltage between the lens center and surface. |

Λ_{i} | (cm^{2}/s mm Hg) | The effective intracellular hydraulic conductivity. |

_{2}2, MgCl

_{2}1, HEPES 5, glucose 10, pH 7.4 (normal Tyrode). The lens, suspended by its zonules to a uveal sclera ring of tissue, was dissected from the eye.

^{ 14 }Each lens was transferred and pinned by its sclera to the bottom of a chamber with a Sylgard base. The chamber was mounted on the stage of a microscope and perfused with normal Tyrode solution.

^{ 14 }so only a brief description is given here. The microelectrode resistance (1.5–2.0 MΩ when filled with 3 M KCl) was recorded in solution outside of the lens by injecting serial current pulses and recording the amplitudes of the responding voltage steps, which indicated the resistance of the tip of the electrode. When the electrode was inserted into the lens, positive intracellular pressure pushed cytoplasm into the tip, causing the resistance to increase. A side port on the electrode holder was connected by plastic tubing to a mercury manometer, which allowed adjustment of the pressure within the electrode. The pressure within the electrode was increased to push cytoplasm back into the fiber cell. When pressure within the electrode was equal to intracellular pressure, cytoplasm was just pushed out of the electrode, and the amplitude of the electrode resistance returned to its original value measured in the bathing solution. This was the recorded value of intracellular pressure.

^{+}]

_{i}in the Intact Lens

^{ 15 }SBFI (sodium-binding benzofuran isophthalate, 0.2 mM) was dissolved in the pipette solution. This solution was injected into fiber cells at different depths in the lens by advancing the microelectrode along its track towards the lens center. The ratios of emission at 360/380 nm excitation were compared to Na calibration curves that were determined at seven depths into the lens, as was described for intracellular calcium.

^{ 16 }The ratios versus intracellular sodium concentration, [Na

^{+}]

_{i}, were determined with Equation 1. where

*R*is the experimentally measured ratio at a given depth in the lens, whereas

*K*,

_{d}*R*

_{min}, and

*R*

_{max}were determined by the calibration curves at the same depth. The value of

*K*was essentially independent of depth, but the maximum and minimum ratios,

_{d}*R*

_{max}and

*R*

_{min}, respectively, varied significantly because the lens absorbs light in a wavelength- and depth-dependent manner.

^{ 11 }A centrally placed intracellular microelectrode was used to inject a stochastic current composed of sinusoidal frequencies between 0 Hz and 5 KHz. A peripherally placed intracellular microelectrode was used to record the induced voltage. Both signals were sent to a fast Fourier analyzer (Hewlett Packard, Palo Alto, CA) where the intracellular impedance was calculated in real time. Figure 1 is a structurally based equivalent circuit model of the lens.

^{ 17,18 }This model was used to curve fit the data using the Levenberg-Marquardt nonlinear curve-fitting algorithm.

^{ 19,20 }The overall impedance of the lens (see Fig. 1) is The overall resistance at any frequency is given by the magnitude of the complex impedance

*|Z(jω)|*, and the phase is given by

*θ*(

*ω*) = tan

^{−1}(Im(

*Z*(

*jω*)/

*R*e(Z(

*jω*)).

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

*R*

_{s}in Fig. 1 or Equation 2). The “series resistance” represents the intracellular resistance due to gap junctions between the point of recording the voltage and the surface of the lens.

^{ 21 }By advancing the microelectrode along its track toward the lens center, the series resistance at several depths in each lens can be recorded, and similar data would then be pooled from several lenses to determine gap junction coupling conductance. Thus the high frequency impedance gives a fairly direct measure of gap junction coupling conductance without resorting to nonlinear curve fitting. The relationship between resistance (

*R*

_{S}(

*r*) Ω) and underlying effective intracellular resistivities (

*R*

_{DF}and

*R*

_{MF}Ω-cm) has two forms, depending on whether one is recording in the outer differentiating fibers (DF) or inner mature fibers (MF): Equation 3 was derived for current flow in a conductive sphere with a point source of current at its center; hence the 1/

*r*dependence, where

*r*is the distance from the lens center and

*a*is the lens radius. In mouse and rat lenses, the location

*r*=

*b*(

*b*≈ 0.85

*a*) is the transition between differentiating fibers (DF,

*r*≥

*b*), which have organelles, and mature fibers (MF,

*r*≤

*b*), which lack organelles.

^{ 21 }It is also the site of extensive posttranslational modification of fiber cell membrane proteins, including the Cxs.

^{ 21 }As a consequence,

*R*

_{DF}is less than

*R*

_{MF}, and there is a change in the slope of the resistance versus location curve at

*r*=

*b*.

*Z*

_{L}(

*jω*), depends on the two parallel pathways for current to leave the lens: (a) by crossing surface cell membranes

*Y*

_{s}(

*jω*) S/cm

^{2}and (b) by crossing fiber cell membranes to enter the extracellular spaces and exit the lens

*Y*

_{e}(

*jω*) S/cm

^{2}. Viz:

*G*

_{s}and

*g*

_{m}(S/cm

^{2}) are, respectively, the surface and fiber cell membrane conductances;

*C*

_{s}and

*c*

_{m}(F/cm

^{2}) are, respectively, the surface and fiber cell membrane capacitances;

*R*

_{e}(Ω cm) is the effective resistivity of the extracellular spaces within the lens; and

*S*

_{m}/

*V*

_{T}(cm

^{−1}) is the average surface area of fiber cell membrane per volume of tissue. The parameter

*j*= $\u22121$, and

*ω*represents an arbitrary sinusoidal frequency. The input resistance of the lens,

*R*

_{in}=

*Z*

_{L}(0), is useful because it represents the effects of membrane conductances and lens size in a model-independent manner. In the Results section,

*R*

_{in}is indicated on the impedance data in Figure 6 and is listed in Table 2.

*t*-test.

^{+}or water into fiber cells filling the volume of the lens equals the total efflux leaving surface cells.

*r*dimension) and also angular fluxes around the lens (

*θ*dimension). The resulting equations can be simplified by averaging over the

*θ*dimension and focusing on the radial dependence. The data presented in this paper describe radial gradients that were recorded halfway between the posterior pole and equator, essentially at the location of the angular average of the gradients, so comparison of this “angular average” model with data should be appropriate.

^{ 2 }; hence, the model calculations here focused on Na

^{+}and neglected the roles of other ions. Na

^{+}moves into fiber cells and down its transmembrane electrochemical gradient, which depends on transmembrane voltage and the Nernst potential for Na

^{+}. Thus Na influx is logarithmically related to Na concentration. The equations are, therefore, nonlinear, and their general solutions would require extensive calculations that are beyond the scope of this paper. One particular special case, however, can be easily solved. Assume Na influx (

*j*

_{Na}) is uniform at all radial locations with a value that is the radial average of the actual influx. The resulting equations can then be simply integrated to obtain a quadratic dependence on

*r*for the radial gradients in intracellular hydrostatic pressure, voltage, and Na concentration. These quadratic models fit the data surprisingly well.

*p*

_{i}(0) mm Hg) is the same in lenses of different sizes from several different species, so it is useful to derive the radial variation in pressure in terms of

*p*

_{i}(0). Following the analysis presented in Gao et al.,

^{14}the intracellular hydrostatic pressure is approximately: where

*K*is a constant that depends on lens fiber cell structure and single gap junction channel hydraulic conductivity, factors that are not expected to vary with size or species. The experimental observation is that

*p*

_{i}(0) is the same in lenses with very different radii; hence the prediction is that

*N*

_{j}/

*j*

_{Na}∝

*a*

^{2}. However, in the Results, we show that

*N*

_{j}is the same in mouse and rat lenses, so it is probably the same in the even larger lenses from rabbit and dog. The implication is that

*j*

_{Na}∝ 1/

*a*

^{2}. What is the implication for the effect of size on lens water flow?

*u*

_{i}(

*r*) cm/s) depends on the pressure gradient (i.e., the derivative of

*p*

_{i}with respect to

*r*). Viz: The experimental finding that

*N*

_{j}is independent of size or species implies that

*Λ*

_{j}is also independent of size or species, so water flow will depend on the size dependence of the gradient in intracellular pressure. Differentiating Equation 8 and evaluating the gradient at the lens surface where water flow velocity is maximal gives: Thus, when

*p*

_{i}(0) is independent of lens radius, the water flow velocity at the lens surface is decreased in the larger lenses in proportion to 1/

*a*.

*πa*

^{2}

*u*

_{i}(

*a*), which increases in the larger lenses in proportion to

*a*. Thus, as size increases, the volume of water flow increases but the velocity of flow decreases.

^{+}]

_{i}= [Na

^{+}]

_{i}(0) − [Na

^{+}]

_{i}(

*a*) and Δ

*ψ*

_{i}=

*ψ*(0) −

_{i}*ψ*

_{i}(

*a*). Following the analysis in Wang et al. (2009): Data presented in Results suggest that Δ[Na

^{+}]

_{i}, [Na

^{+}]

_{i}(

*a*), Δ

*ψ*

_{i}and

*ψ*(

_{i}*a*) are each independent of lens size or species. What is the implication for the effect of size on lens Na flux?

*D*

_{Na}and

*Λ*

_{i}depend on

*N*

_{j}and other parameters that are expected to be independent of lens radius or species. Since our experimental data suggest that

*N*

_{j}is independent of lens size or species, the size dependence of

*J*

_{Na}arises through the size dependence of the gradients. Differentiating Equations 12 and 13 and evaluating the gradients at the lens surface where Na flux velocity is maximal gives: Thus, when [Na

^{+}]

_{i}(

*a*), Δ[Na

^{+}]

_{i}, Δ

*ψ*

_{i}, and

*p*

_{i}(0) are each independent of lens radius, the Na flux velocity at the lens surface (

*J*

_{Na}(

*a*)) is predicted to decrease in the larger lenses in proportion to 1/

*a*.

^{+}leaving the lens is given by 4

*πa*

^{2}

*J*

_{Na}(

*a*), which increases in the larger lenses in proportion to

*a*. Thus, just as for water flow, as size increases, the amount of Na

^{+}leaving the lens increases but the velocity of flow decreases.

^{+}leaving the surface cells of the lens equals the total Na

^{+}entering fiber cells throughout the lens volume. Viz: Since the surface-to-volume ratio of a fiber cell is not expected to depend on size or species of lens, combining Equations 15 and 16 leads, once again, to the conclusion In the case of water flow, this conclusion was based on the assumption that transmembrane water flow is proportional to

*j*

_{Na}. Here, we arrive at the same conclusion based entirely on experimental data and the requirement for mass balance.

*a*

^{2}

*j*

_{Na}/

*N*

_{j}.

^{ 14 }Other parameters, such as cell size and gap junction single channel hydraulic conductivity, are predicted to affect intracellular pressure, but we think it is unlikely these factors vary between lenses from different species. Gao et al.

^{ 14 }demonstrated intracellular hydrostatic pressure varied with

*j*

_{Na}/

*N*

_{j}in mouse lenses of similar radii. Here, we are measuring hydrostatic pressure in lenses of very different radii, but from different species. Nevertheless, hydrostatic pressure should still depend on

*j*

_{Na}/

*N*

_{j}, even if either

*j*

_{Na}or

*N*

_{j}is not the same in lenses from different species. Our initial expectation was that they would be similar in lenses from different species, hence the pressures would be significantly higher in lenses with larger radii. Figure 2 illustrates the remarkable observation that hydrostatic pressure measured in the center of lenses of different sizes from different species was always the same.

*p*

_{i}(0)), which did not vary appreciably between these lenses of very different sizes. On average,

*p*

_{i}(0) was 335 ± 6 mm Hg. Indeed, when pressure was graphed as a function of normalized distance from the lens center (

*r*/

*a*), the combined data from 36 lenses from four species all fell on the same curve (Fig. 2A).

*p*

_{i}(0) is the same in lenses of differing radii, water flow velocity, which depends on the gradient in intracellular hydrostatic pressure, decreases in proportion to 1/

*a*, consistent with the data in Figure 2B. Moreover, total water flow leaving the lens surface depends on the surface area times the flow velocity. Total water flow therefore increases in the larger lenses in proportion to

*a*. Thus as size increases, the volume of water flow increases but the velocity of flow decreases.

*N*

_{j}and directly with

*j*

_{Na}.

^{ 14 }Thus the implications of the data shown in Figure 2 are that, in larger lenses from different species, either

*N*

_{j}increases as

*a*

^{2}, or

*j*

_{Na}decreases as

*a*

^{2}, or some combination of both such that

*j*

_{Na}/

*N*

_{j}decreases as

*a*

^{2}. Impedance studies are limited to smaller lenses, so we could not test these possibilities in lenses from all the species represented in Figure 2; however, the impedance of rat and mouse lenses can be measured and compared. The value of

*a*

^{2}for a rat lens is approximately four times greater than that for a mouse lens, so there should be a significant difference in some measureable parameters.

*N*

_{j}in Mouse and Rat Lenses

^{ 18 }This resistance depends on the effective intracellular resistivity, whose value is dominated by gap junctions, which limit cell to cell ion flow. There is a change in the effective intracellular resistivity at the differentiating fiber (DF) to mature fiber (MF) transition (

*r*=

*b*) due to loss of open Cx50 channels in the MF.

^{ 21,22 }

*R*

_{s}in mouse and rat lenses and the curve fits of Equation 3 to each data set are shown in Figure 3. In each species, we assume that

*b*= 0.85

*a*as this has been shown to be a good approximation for the location at which gap junction coupling conductance changes in mouse and rat lenses.

^{ 21 }As can be seen in Figure 3, the resistance curves, when graphed as a function of normalized radial location, are not very different, though some difference is predicted by Equation 3 based on the different values of radius (

*a*cm) in lenses from these two species.

**Figure 3.**

**Figure 3.**

*R*

_{DF}and

*R*

_{MF}) depend inversely on the density of open gap junction channels, but are not directly affected by lens size, though there could be differences in lens fiber cell gap junction expression between the mouse and rat. Table 1 summarizes the best fit values of effective intracellular resistivity, and the coupling conductances per area of cell-to-cell contact (

*G*

_{DF}and

*G*

_{MF}S/cm

^{2}).

*G*= 1/

*wR*, where

*w*= 3 μm is the approximate value of fiber cell width, which is the approximate radial distance between gap junction plaques. The value of coupling conductance should be directly proportional to

*N*

_{j}. In order to account for the pressure data, the coupling conductance in rat would have to be 4-fold greater than in mouse, and that is obviously not the case. Assuming the density of fiber cell gap junction channels is independent of lens size or species, then the pressure data imply that the average fiber cell membrane influx of Na

^{+}should vary as

*j*

_{Na}∝ 1/

*a*

^{2}.

**Table 1.**

**Table 1.**

a (cm) | R _{DF} (KΩ-cm) | R _{MF} (KΩ-cm) | G _{DF} (S/cm^{2}) | G _{MF} (S/cm^{2}) | |

Rat | 0.22 | 4.0 | 6.4 | 0.83 | 0.52 |

Mouse | 0.11 | 3.6 | 4.1 | 0.93 | 0.81 |

**Table 2.**

**Table 2.**

a (cm) | R _{in} (KΩ) | G _{S} (mS/cm^{2}) | g _{m} (μS/cm^{2}) | R _{e} (KΩ-cm) | |

Rat (n = 10) | 0.22 ± 0.002 | 0.9 ± 0.34 | 1.3 ± 0.78 | 2.2 ± 1.23 | 28.1 ± 10.87 |

Mouse (n = 30) | 0.11 ± 0.003 | 3.9 ± 0.63 | 0.3 ± 0.25 | 6.0 ± 2.48 | 21.7 ± 8.18 |

**Table 3.**

**Table 3.**

a* (cm) | g _{m}* (μS/cm^{2}) | g _{Na} (μS/cm^{2}) | g _{Cl} (μS/cm^{2}) | j _{Na} (pmole/(cm^{2} s)) | |

Rat | 0.22 | 2.2 | 1.26 | 0.96 | 0.76 |

Mouse | 0.11 | 6.0 | 5.04 | 0.96 | 3.04 |

^{+}Transport in Mouse and Rat Lenses

^{+}transport across species, then the radial intracellular Na

^{+}flux should decrease as 1/

*a*(see Equation 17) as did the radial intracellular water flow. Three factors drive

*J*

_{Na}: the intracellular Na

^{+}-concentration gradient drives diffusion; the intracellular voltage gradient drives conduction; and the intracellular pressure gradient drives advection (see Equation 16). We have already determined that the center-to-surface change in hydrostatic pressure is the same in mouse and rat lenses, but the rat lens radius is twice that of the mouse lens; hence the pressure gradient and water flow velocity in rat lenses is less than that in mouse lenses by a factor of 2 (

*a*(rat)/

*a*(mouse) = 2). Figure 4 shows that the center-to-surface change in intracellular Na

^{+}concentration is the same in mouse and rat lenses, but the average radius is 2.2 times larger in rat (for [Na

^{+}]

_{i}measurements (

*a*(rat)/

*a*(mouse) = 2.2); hence radial diffusion of Na

^{+}is slower in rat by a factor of 2.2.

**Figure 4.**

**Figure 4.**

*r*/

*a*), they all fall around the same smooth curve. Thus the center-to-surface change in intracellular voltage is also the same in mouse and rat lenses, and hence the voltage gradient and conduction of Na

^{+}are also less in rat by a factor of 2.

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

*J*

_{Na}is reduced by approximately a factor of 2 in rat, it follows that

*J*

_{Na}is slower in rat than mouse lenses by a factor of 2. For mass balance, the total Na flux leaving surface cells of the lens must equal the total Na flux entering fiber cells throughout the volume of the lens (see Equation 16). Invoking mass balance, our data on Na

^{+}transport, like the pressure data, imply that

*j*

_{Na}∝ 1/

*a*

^{2}, or the average influx of Na

^{+}for fiber cells must be approximately a factor of 4 smaller in the rat than the mouse.

^{+}is driven by the transmembrane electrochemical gradient for Na

^{+}and scales with the membrane Na

^{+}conductance. The radial dependence of [Na

^{+}]

_{i}and

*ψ*

_{i}have been measured (Figs. 4 and 5), and their average values are the same in lenses from mice and rats. The transmembrane electrochemical gradient depends also on the average extracellular voltage and Na concentration. Though we have no way to measure extracellular voltages and Na

^{+}concentrations, model calculations

^{ 11 }in connection with the intracellular values presented here suggest that the average transmembrane electrochemical gradient for Na

^{+}should be approximately the same in rat and mouse lenses. A 4-fold reduction in

*j*

_{Na}in rat relative to mouse would therefore most likely be due to a significant reduction in

*g*

_{Na}.

*g*

_{Na}is not known, so it is not possible to directly test this idea by Western blotting or immunostaining. However, whole lens impedance studies provide a comparison of fiber cell membrane conductance

*g*

_{m}(S/cm

^{2}) in rat and mouse lenses. When epithelial cells differentiate into fiber cells, they rapidly lose all of their K

^{+}channels, so fiber cell membrane conductance is selective for Na

^{+}and Cl

^{−}.

^{ 23 }If there is a 4-fold difference in

*g*

_{Na}between rat and mouse lenses, there should be a measurable difference in

*g*

_{m}.

*ω*(radians/s) is applied to a linear circuit, the responding voltage is a sinusoid of the same frequency. Assuming that the applied current is given by

*i*(

*t*) =

*I*sin

*ωt*, then the responding voltage is described by

*v*(

*t*) = sin(

*ωt*+

*θ*). The magnitude of the frequency-dependent resistance of the circuit is

*R*(

*ω*) =

*V*(

*ω*)/

*I*. There is also a phase shift (

*θ*(

*ω*) radians) between the applied current sinusoid and the responding voltage sinusoid. The Methods section describes how the complex frequency-dependent impedance of the lens is measured using Fourier theory. The resulting data presented in Figure 6, however, are exactly the same as intuitively described above, if one replaces the magnitude of the complex impedance with

*R*(

*ω*). These data were curve fit with a structurally based equivalent circuit representation of the lens to obtain the parameter values presented in Table 2.

*R*

_{in}is the experimentally measured input resistance (see Fig. 6). In terms of our equivalent circuit model of the lens given in Equation 4,

*R*

_{in}=

*Z*

_{L}(0).

*G*

_{S}is the surface cell membrane conductance, which represents the total conductance of the surface layer of cells normalized to the surface area of the lens.

*g*

_{m}is the fiber cell membrane conductance per unit area of membrane.

*R*

_{e}is the effective extracellular resistivity, which depends on the volume fraction and tortuosity of the extracellular spaces.

^{ 2 }The parameters in Table 2 are determined by nonlinear curve fitting of impedance data, which are similar to those shown in Figure 6. Of particular interest is the value of

*g*

_{m}, which is a factor of 2.7 times smaller in rat lenses than mouse lenses. As described in the Discussion section, this difference could be due to a 4-fold smaller value of

*g*

_{Na}in fiber cells from rat lenses.

*a*, so the larger the lens, the slower the flow. This conclusion was also surprising, as one would expect that the greater the distance for advection of nutrients, the greater the need for velocity. This brings to mind the question: Why is the central hydrostatic pressure independent of size when this causes reduced advection of nutrients in larger lenses? Some speculations on this question are also discussed below.

^{ 14 }show that hydrostatic pressure in the lens varies in proportion to transmembrane Na

^{+}transport, supporting this hypothesis for fluid flow in the lens. It is therefore not surprising that the results presented here suggest regulation of Na

^{+}influx (

*j*

_{Na}) in the lens is ultimately responsible for maintaining a uniform central hydrostatic pressure of 335 mm Hg in lenses of different sizes from different species.

^{+}influx are presented in Equation 18. Our data suggest that the average transmembrane electrochemical gradient for Na

^{+}is approximately the same in rat and mouse lenses, suggesting that the membrane Na

^{+}conductance (

*g*

_{Na}S/cm

^{2}) may vary as 1/

*a*

^{2}. Since the rat lens radius is approximately 2 times larger than that of the mouse lens,

*g*

_{Na}in mouse lenses should be approximately 4 times larger than in rat lenses. Though we cannot directly measure

*g*

_{Na}, nonlinear curve fitting of impedance data provided an estimate of fiber cell membrane conductance

*g*

_{m}=

*g*

_{Na}+

*g*

_{Cl}.

^{ 23 }There is a significant difference in the best fit values of

*g*

_{m}from rat and mouse lenses, with the average conductance in the mouse lenses being a factor of 2.7 times larger than that in the rat.

*g*

_{m}between rat and mouse could indeed account for a 4-fold higher

*g*

_{Na}in mouse than rat. The parameters with an asterisk in Table 3 have values based on experimental measurements, whereas the values of

*g*

_{Na},

*g*

_{Cl}, and

*j*

_{Na}are estimates. With regard to

*g*

_{Cl}and

*g*

_{Na}, the values were based on the assumption that Cl-channel expression is the same in fiber cell membranes of the mouse and rat lens, but Na-channel expression is 4-fold greater in the mouse than in the rat, with the constraint that the experimental value of

*g*

_{m}=

*g*

_{Na}+

*g*

_{Cl}. There are no data to support our assumption on

*g*

_{Cl}; it is just as likely to differ in rat and mouse lenses, but Table 3 shows one way in which all experimental numbers could add up. If we make an educated guess on average values for the components of the Na

^{+}transmembrane electrochemical gradient (

*ψ*

_{i}= −54 mV,

*ψ*

_{e}= −42 mV, and

*E*

_{Na}= +60 mV), then the values of

*g*

_{Na}in Table 3 give the values shown for

*j*

_{Na}. Our working hypothesis is: fiber cell

*g*

_{Na}is regulated across species to maintain a species-independent central hydrostatic pressure of 335 mm Hg.

^{ 14 }This observation is inconsistent with feedback regulation. It is more likely that lenses from different species are genetically programmed to express different amounts of Na

^{+}-leak channels.

*a*. This brings into question our standing hypothesis. However, in the presence of water flow, advection of nutrients in the aqueous and vitreous humors into extracellular spaces of the lens is inescapable, so rather than discarding this hypothesis, we looked to expand upon it. Moreover, the following caveat applies when comparing the microcirculatory system in different species.

^{ 24 }The correlation is not perfect and there are exceptions; but, if we adopt this view at face value, then the larger the lens, the lower the basal metabolic rate and the less the need for speed in delivering nutrients to central fiber cells. This would explain why decreasing water flow velocity in larger lenses is not deleterious, but it does not explain why the central hydrostatic pressure is so precisely regulated at 335 mm Hg.

^{ 25 }In mice, rats, and rabbits, the refractive index varies from approximately 1.5 in central fiber cells to 1.35 in outer cortical fiber cells,

^{ 26–28 }whereas in the larger lenses of humans and cows, it varies from approximately 1.4 in central fiber cells to 1.35 in outer cortical fiber cells.

^{ 29–31 }Unfortunately, we could find no information on the refractive index of dog lenses. However, in the lenses we have studied here, the refractive index varied parabolically from center to surface, like the pressure gradients shown in Figure 2. And in mouse, rat, and rabbit lenses, the center-to-surface refractive index change was independent of species, like the hydrostatic pressure. While these similarities alone mean little, they are not inconsistent with a relationship. Moreover there are other lines of evidence that suggest a relationship. The first line of evidence is from mathematical modeling of the relationship between hydrostatic and osmotic pressures in fluid transporting epithelia.

^{ 32 }The second line of evidence is based on some recent experiments.

^{ 33 }

^{32}in fluid transporting epithelia there is approximate balance between osmotic and hydrostatic pressures. To provide an intuitive understanding of why this is so, consider the factors generating transmembrane water flow: where

*u*

_{m}(cm/s) is the transmembrane water flow velocity,

*L*

_{m}(cm/s/mm Hg) is the membrane hydraulic conductivity, Δ

*c*

_{m}(moles/cm

^{3}) is the transmembrane osmotic difference, and Δ

*p*

_{m}(mm H

_{g}) is the transmembrane hydrostatic pressure difference. In the absence of a hydrostatic pressure, if

*L*

_{m}becomes very large, water flow velocity increases but not proportionally, because increases in

*u*

_{m}wash away solute, reducing Δ

*c*

_{m}. The theoretical maximum limit for

*u*

_{m}is termed isotonic transport,

^{34}and it occurs when the transported solution has the same osmolarity as that of the bathing medium (

*c*

_{0}moles/cm

^{3}). The osmolarity of the transported solution is given by

*j*

_{m}/

*u*

_{m}(moles/cm

^{3}), where

*j*

_{m}(moles/cm

^{3}) is transmembrane salt flux. Hence the transmembrane water flow velocity is limited by: Combining the above two equations gives: The small parameter

*ε*= (

*j*

_{m}/

*c*

_{0})/(RT

*c*

_{0}

*L*

_{m}) can be thought of as the ratio of membrane salt to water permeability. Where experimental data are available (reviewed in Mathias and Wang

^{34}),

*ε*≈ 10

^{−3}. Assume

*c*

_{0}= 300 mM and Δ

*p*

_{m}= 0, then a transmembrane osmotic gradient of a little less than Δ

*c*

_{m}= 0.3 mM would generate near isotonic transport. However, if a pressure of 335 mm Hg is present in the cell, then Δ

*p*

_{m}/RT = 16.75 mM and for the above inequality to hold, Δ

*c*

_{m}would have to be close to 17 mM. Thus, the experimental observation that membrane water permeability is much larger than salt permeability implies hydrostatic pressure will be approximately balanced by osmotic pressure and transmembrane water flow velocity will be approximately independent of pressure, insofar as transmembrane salt transport is independent of pressure. Transmembrane salt transport depends on Nernst potentials, intracellular voltage and membrane conductances, and should not appreciably change when pressure develops.

*j*

_{m}/

*u*

_{m}would become hypertonic and the cells would begin to accumulate salt. Second, fluid outflow would exceed fluid inflow so the fiber cells would lose water and salt through the gap junctions. But the crystallins are too large to pass through gap junctions so they would be concentrated by the loss of water. Eventually a steady state would be reached where the combined osmotic effect of intracellular salt and crystallins would balance the transmembrane effect of the hydrostatic pressure to within approximately 0.3 mM and fluid circulation would be restored. We are left with the conclusion that the positive intracellular pressure causes the cells to shrink. Though counterintuitive, this idea is consistent with experimental data.

^{ 33 }use MRI to measure the protein to water ratio in bovine lenses. When the lenses were placed in low Na

^{+}, high K

^{+}external solution to block the lens circulation, the gradient in protein-to-water ratio collapsed to approximately 60% of normal in a period of 4 hours. This external solution is first used by Parmelee,

^{ 4 }who uses the vibrating probe to show that it blocks circulating currents in the frog lens. Gao et al.

^{ 14 }use this solution to block the circulation of Na

^{+}in mouse lenses and observe the effect on the central hydrostatic pressure. The central hydrostatic pressure goes to near zero in a period of approximately 2 hours. As the new external solution diffuses into the extracellular spaces within the lens, the transmembrane electrochemical gradient for Na

^{+}is reduced and then eventually eliminated and Na

^{+}influx ceases, leading to elimination of circulating currents, fluid flow, and intracellular hydrostatic pressure. If intracellular hydrostatic pressure causes fiber cells to lose water, then reduction in this pressure should cause water to enter the cells, as measured by Vaghefi et al.

^{ 33 }

^{+}, high K

^{+}solution to have diffused into extracellular spaces at the center of the bovine lenses, so the central fiber cell intracellular hydrostatic pressure was unlikely to have gone to zero. Thus the 40% reduction in the central protein-to-water ratio seen in bovine lenses is probably not a steady state number, so we do not know just how much of the lens protein gradient is associated with the lens circulation. Nevertheless, their data clearly show that the protein concentration gradient is actively maintained, possibly through the link to hydrostatic pressure described above.

^{+}-leak channels. These results lead to the question: What is important about a central pressure of 335 mm Hg? We have discussed the possibility that the pressure gradient creates the refractive index gradient, which is essential for visual acuity. While this is a particularly intriguing answer to our question, there are many gaps in knowledge that will need to be filled before this can be considered a viable hypothesis.

^{ 32 }This is an extracellular compartment, so the pressure–osmolarity balance in the presence of gap junctions and intracellular crystallins has not been quantitatively modeled. More appropriate model calculations are needed.

^{ 14 }show that a low Na

^{+}, high K

^{+}external solution causes the pressure at the center of a mouse lens to go to zero in approximately 2 hours, whereas Vaghefi et al.

^{ 33 }show that the same external solution causes the protein-to-water gradient in a cow lens to drop to 60% normal in approximately 4 hours. A more direct experimental correlation between intracellular hydrostatic pressure and intracellular protein concentration is needed. Moreover, the refractive index gradient in the cow lens goes from approximately 1.4 at the center to 1.35 at the surface, whereas the gradient in the mouse lens goes from approximately 1.5 at the center to 1.35 at the surface. If refractive index and pressure are related, the central pressure in a cow lens might be significantly lower than in a mouse lens, although differences in crystalline subtypes with different refractive power might confound this prediction. Nevertheless, it would be interesting to measure hydrostatic pressure in cow lens.

*. 1989; 64: 742–752. [CrossRef] [PubMed]*

*Circ Res**. 2007; 216: 1–16. [CrossRef] [PubMed]*

*J Membr Biol**. 1982; 2: 843–847. [CrossRef] [PubMed]*

*Curr Eye Res**. 1986; 42: 433–441. [CrossRef] [PubMed]*

*Exp Eye Res**. 2002; 282: C252–C262. [CrossRef] [PubMed]*

*Am J Physiol Cell Physiol**. 1997; 77: 21–50. [PubMed]*

*Physiol Rev**. 1992; 68: 518–529. [CrossRef]*

*Biophys J**. 2009; 88: 919–927. [CrossRef] [PubMed]*

*Exp Eye Res**. 2000; 178: 89–101. [CrossRef]*

*J Mem Biol**. 2003; 44: 4395. [CrossRef] [PubMed]*

*Invest Ophthalmol Vis Sci**. 1985; 48: 435–448. [CrossRef] [PubMed]*

*Biophys J**. 1999; 276: C548–C557.*

*Am J Physiol Cell Physiol**. 2006; 1: 48–57.*

*Physiol Mini-Rev**. 2011; 137: 507–520. [CrossRef] [PubMed]*

*J Gen Physiol**. 2009; 227: 25–37. [CrossRef] [PubMed]*

*J Membr Biol**. 2004; 124: 289–300. [CrossRef] [PubMed]*

*J Gen Physiol**. 1979; 25: 151–80. [CrossRef] [PubMed]*

*Biophys J**. 1981; 34: 61–83. [CrossRef] [PubMed]*

*Biophys J**. 1944; 2: 164–168.*

*Q Appl Math**. 1963; 11: 431–441. [CrossRef]*

*SIAM J Appl Math**. 2010; 90: 179–206. [CrossRef] [PubMed]*

*Physiol Rev**. 2006; 47: 4474–4481. [CrossRef] [PubMed]*

*Invest Ophthalmol Vis Sci**. 2008; 294: C1133–C1145. [CrossRef] [PubMed]*

*Am J Physiol Cell Physiol**. 1985; 62: 299–308. [CrossRef] [PubMed]*

*Am J Optom Physiol Opt**. 1968; 58: 1125–1130. [CrossRef] [PubMed]*

*J Opt Soc Am**. 1984; 24: 409–415. [CrossRef] [PubMed]*

*Vis Res**. 2011; 6: e25140. [CrossRef] [PubMed]*

*PLoS One**. 1989; 66: 822–829. [CrossRef] [PubMed]*

*Optom Vis Sci**. 2005; 45: 2352–2366. [CrossRef] [PubMed]*

*Vis Res**. 1995; 9: 776–782. [CrossRef] [PubMed]*

*Eye (Lond)**. 1985; 47: 823–836. [CrossRef] [PubMed]*

*Biophys J**. 2011; 301: R335–R342. [CrossRef] [PubMed]*

*Am J Physiol Regul Integr Comp Physiol**. 2005; 208: 39–53. [CrossRef] [PubMed]*

*J Membr Biol*