Material Characterization of Porcine Lenticular ... - ACS Publications

May 10, 2008 - Department of Veterans Affairs, 915 North Grand Boulevard, St. ... This unique protein system directly enables proper visual function o...
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Biomacromolecules 2008, 9, 1519–1526

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Material Characterization of Porcine Lenticular Soluble Proteins Matthew A. Reilly,†,‡ Brian Rapp,†,§ Paul D. Hamilton,† Amy Q. Shen,| and Nathan Ravi*,‡,⊥,# Department of Veterans Affairs, 915 North Grand Boulevard, St. Louis, Missouri 63106, and Department of Energy, Environmental, and Chemical Engineering, Department of Biomedical Engineering, Department of Mechanical and Aerospace Engineering, and Department of Ophthalmology and Visual Sciences, Washington University in St. Louis, 1 Brookings Drive, St. Louis, Missouri 63130 Received November 8, 2007; Revised Manuscript Received February 29, 2008

The soluble proteins present in the ocular lens impart important optical and dynamic mechanical properties on the lens. The short-range order of crystallin proteins grants transparency to a very concentrated protein solution. This unique protein system directly enables proper visual function of the eye. These proteins were investigated in steady and oscillatory shear. Steady shear data were fitted with a modified Herschel-Bulkley yield stress model that allows for a Newtonian plateau at low shear rates. The Cox-Merz rule was used in conjunction with large amplitude oscillatory shear to give insight into the degradation of the fluid structure with increasing strain. The shear thinning viscoelastic behavior of these proteins gives rise to beneficial mechanical properties and results from the same short-range order granting optical transparency.

1. Introduction The ocular lens is the pivotal tissue in the process of visual accommodation: the dynamic changing of focus from far to near. This is accomplished by action of the ciliary muscle, which exerts a force on the lens capsule via the zonular fibers, thereby changing the lens curvature and its optical power.1–3 The lens capsule’s elastic modulus is on the order of 105-106 Pa, depending on the age and species of the eye,4–7 while the modulus of the lens fiber matrix is on the order of 101-104 Pa.5,8–11 The decapsulated lens matrix initially maintains its shape, but creeps slowly under its own weight, behaving like a viscoelastic solid. Thus, the lens capsule is able to transduce force applied to it by the ciliary muscle to shape the lens fibers, thereby changing the lens’ optical power for accommodation.3 Presbyopia, the gradual loss of accommodation amplitude with age, is clinically diagnosed when the near point of accommodation becomes more remote than the normal near distance of the patient and occurs in all humans in the fifth decade of life. The dynamics of accommodation have been measured in vivo and modeled.12–14 These investigations have revealed that the time scale for accommodation is on the order of 10-1 s. Small fluctuations in steady state accommodation have also been observed and linked to heartbeat and neurological origins.15 To ascertain the origins of the dynamics of accommodation, various groups have undertaken the study of the tissues involved in accommodation. Van Alphen and Graebel demonstrated that the elastic modulus of the lens is 102-103 times lower than all other tissues involved, indicating that much of the elastic behavior of the system is achieved by other tissues.5 This indicates that the overall viscoelastic behavior of the tissues involved in accommodation may originate primarily in the lens. * To whom correspondence should be addressed. Tel.: (314) 289-6470. E-mail: [email protected]. † Department of Veterans Affairs. ‡ Department of Energy, Environmental, and Chemical Engineering. § Department of Biomedical Engineering. | Department of Mechanical and Aerospace Engineering. ⊥ Chief of Staff, Department of Veterans Affairs. # Department of Ophthalmology and Visual Sciences.

Studies of the isolated ex vivo lens have confirmed viscoelastic behavior,16–18 indicating that the dynamics of this process depend not only on the neurophysiology of the accommodative function,19 but also on the material properties of the lens. More recently, extensive in vitro viscoelastic studies on human lenses of varying ages indicate a relaxation spectrum consisting of approximately three time constants in the physiological range.9,20 The intrinsic relaxation time constants of the lens matrix are on the order of 30, 300, and 3000-30000 ms.9,12,17,18,20 This viscoelastic behavior is essential to the proper function of the dynamic accommodation feedback mechanism.21 However, the molecular origins of these viscoelastic processes in the lens remain unknown. Understanding the origins of these processes requires a basic understanding of the complex architecture of the lens. The human ocular lens is an aspherical biconvex ellipsoid with an equatorial radius of 8-10 mm and an axial thickness of 3-5 mm, depending on age. An elastic collagen capsular bag contains a matrix of tightly packed, well-organized lens fibers. These individual cells arch through the equator, spanning the entire axial length of the lens.22 The lens fiber cells are formed like long, hollow tubes of cytoskeletal elements and are filled with soluble proteins, primarily globular crystallin proteins. Each fiber is several millimeters long and has a very small cross-section, measured in microns, with a hexagonal cross-sectional shape. The cell wall is primarily composed of cytoskeleton and membrane-bound proteins, such as actin, CP-49, and other intermediate filaments. The cell cytoplasm is filled with soluble proteins, primarily lens crystallins.23 The lens crystallin proteins have been studied extensively with respect to their role in the pathogenesis of cataracts, with a recent comprehensive review of their structure, stability, and function. Briefly, the crystallins are a nonhomogeneous mixture with three major components, R -, β -, and γ-crystallins, each with several subfractions. Each of these fractions are distinguished by molecular weight and electrostatic charge, with R- and β-crystallins existing as oligomers, while the γ-crystallins existing primarily as monomers.24

10.1021/bm701229t CCC: $40.75  2008 American Chemical Society Published on Web 05/10/2008

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For proper function, the lens must have a high refractive index (RI) and be transparent to visible light.24 The high RI of the lens is achieved with concentrated solutions of crystallin proteins, requiring that the crystallins be highly soluble. Protein-protein interactions generally lead to aggregation and light scattering (i.e., a loss of transparency) at such high concentrations. However, aggregation of the lens proteins is prevented by the chaperone activity of the R-crystallins, which also impart a short-range order on the crystallin structure, maintaining lens transparency.25,26 Small-angle X-ray scattering (SAXS) experiments show that above 200 mg/mL, light scattering actually decreases, indicating that transparency is due to short-range order between highly concentrated crystallins.25 This short-range order is a direct result of the electric charge distribution found in crystallin solutions27–29 and implies that three body interactions give structure to the fluid.30–32 Proper function of the dynamic mechanism of accommodation also requires viscous damping behavior. In general, high viscosity solutions of globular proteins are achieved by increasing the concentration to the onset of aggregation.33–35 However, as the lens requires transparency for proper function, short-range order controlled by the electrostatic charges of the various crystallin fractions gives a high viscosity without aggregation.36–38 Specific integer ratios of the individual crystallin fractions suggest important concentration-related structural effects.39 Bieri and Kiefhaber reported that protein folding exhibits viscositydependent behavior.40 Ikeda and Nishinari propose a detailed model for the rheological behavior of bovine serum albumin (BSA), a ubiquitous protein with many characteristics in common with lens crystallins.41 Windhab also discussed the relationship between fluidized volume fraction and the viscosity of complex fluids.42 The viscoelastic behavior of the unfractionated lens crystallin solution has not been studied. However, based on the study of similar concentrated protein solutions (such as BSA), we hypothesize that the lens crystallin solution is a shear-thinning viscoelastic fluid at physiological concentration. Shear-thinning behavior would decrease the force required to achieve a rapid step change in accommodation. Viscoelastic fluid-like behavior would allow the eye to maintain sharp focus when subjected to perturbations. We examined this hypothesis using capillary and cone and plate rheometry to evaluate the rheological behavior of physiologically concentrated solutions of lens crystallins.

2. Experimental Section 2.1. Sample Preparation and Preparative Chromatography. Approximately 30 lenses were needed to generate a single sample. Porcine lenses and their components have been shown to be very similar to human lenses in terms of molecular, immunological, optical, and biometric properties43–49 and, thus, were adopted as an animal model. All analyses were performed within 36 h of animal death. Lenses were decapsulated and sodium azide was added in powdered form at 0.1% (weight to volume) final concentration (Aldrich Chemical, Milwaukee, WI) as a preservative. Material was homogenized using a dounce homogenizer to destroy the lens architecture. The lens homogenate was then spun in a Beckmann model LM-60 ultracentrifuge (Beckman Coulter Instruments, Fullerton, CA) using a SW41 rotor at 30000 rpm for 60 min (100000 × g). The less dense insoluble cell fraction formed a band at the surface. The soluble fraction was acquired by pipetting the sample from underneath the insoluble fraction, yielding the unfractionated crystallin solution at or near the physiological concentration. Samples were then stored overnight at 4 °C. Fractionation of individual crystallin classes was achieved by diluting the unfractionated lens crystallins in column buffer, (50 mM Tris, 50

Reilly et al. mM NaCl, 1 mM EDTA, 1 mM DTT, and 0.1% Na azide) to approximately 3% concentration, 1.2 g of the soluble fraction was loaded on a 5 × 90 cm Pharmacia Sephacryl S-300 column (Pfizer Inc., New York, NY), and fractions were collected every 20 min with a flow rate of 1.3 mL/min. Samples were maintained in column buffer until used for rheological study or analyzed by HPLC. Absorbance was measured at 280 nm using an ISCO UA-5 monitor (Teledyne Isco, Inc., Lincoln, NE). Fractions were pooled into their respective peaks of R-, βhigh-, βlow-, and γ-crystallins and initially concentrated using an Amicon DC2 concentrator (Amicon Corp., Danvers, MS) with a 10 kDa molecular weight cutoff (MWCO) hollow fiber. Final concentration was performed using a Centriprep YM-10 concentrator (Millipore, Billerica, MA) with a 10 kDa MWCO filter. The concentration of the final crystallin solutions and the total lens crystallin was determined by lyophilizing 150 µL aliquots of the solution in duplicate using a positive displacement pipettor. The weight of the aliquots and the dry weight after lyophylization were determined. Dry weight of buffer aliquots were also determined and subtracted from the weights of the crystallin aliquots. 2.2. Analytical Chromatography. High performance liquid chromatography (HPLC) and gel permation chromatography (GPC) analyses were performed on three types of samples: unfractionated total lens crystallins, fractionated lens crystallins, and individually purified crytallin fractions. The fractionated crystallin samples were recombined at the ratios calculated from their respective refractive index areas in the unfractionated crystallin results. The HPLC/GPC system used was a Spectraseries P200 (Thermo Separation Products, Waltham, MA) with a MetaChem degasser (Torrance, CA) and was equipped with an RI detector (Viscotek Corp., Houston, Texas). Multiple columns were connected in series (A4000PWXL, G4000PWXL, and the G6000PWXL Tosoh Biosep, Montgomery Ville, PA). Viscotek Trisec software (Viscotek Corp.) was used to calculate RI area. The diluted total lens crystallins were injected onto the column at a concentration of 15 mg/ mL at 25 and 100 µL volumes. Two HPLC column buffers were used: 20 mM Tris, 0.1% NaN3, at pH 7.4, and 50 mM Tris, 100 mM NaCl, 1 mM EDTA, 1 mM DTT, 0.1% NaN3, also at pH 7.4. The relative areas of distinct RI peaks were used to estimate the physiological ratios of specific crystallin concentrations. To confirm the reliability of this method in determining the ratio of the crystallin fractions, the purified crystallin fractions were mixed in these ratios. This reconstituted mixture was then injected onto the HPLC/GPC columns and the UV trace was compared with that of the unfractionated lens crystallins obtained directly from homogenization and ultracentrifugation. Individual crystallin fractions were injected onto the HPLC/ GPC columns and were analyzed for molecular weight, intrinsic viscosity, and radius of gyration. Two unfractionated crystallin samples were analyzed using the HPLC/GPC method, and the areas of the RI peaks were determined using Viscotek software. Average values are reported and were used for reconstitution. Number-average molecular weight (MN), intrinsic viscosity ([η]), and hydrodynamic radius (Rh) were determined for each crystallin fraction by Viscotek Corp., using low-angle light scattering (LALS), RI, and ultraviolet detectors in tandem. Using distilled water for a density calibration standard, crystallin solutions were pipetted in duplicate, using high precision positive displacement pipettes. Aliquots were weighed before and after lyophilization to determine the water weight and dry weight, respectively. Weight-to-weight and weight-to-volume calculations were made based on these findings. 2.3. Electrophoresis. Isoelectrofocusing (IEF) was carried out on a PhastSystem (Amersham Biosciences Corp., Piscataway, NJ). All materials were purchased and used as part of the system. PhastGel IEF 3-9 media are 5% polyacrylamide gels containing ampholytes that form a linear pH gradient in the gels. Proteins migrate under the electric field, theoretically unhindered by gel tortuosity, to a point in the pH gradient that corresponds to their isoelectric point (pI). Standard IEF markers spanning pI 3.5-9.3 were applied and run concurrently with

Porcine Lenticular Soluble Proteins samples. Experiments were performed as suggested by the instrument manufacturer by applying a 2000 V electric field for 75 V · h prior to introduction of the sample. Samples were applied in the center of the gel, using an eight-well comb, at a concentration of 1.5 µg/µL and a volume of 1µL/lane. IEF was achieved by maintaining a 2000 V electric field for 410 V · h after the introduction of the samples. The IEF gels were then stained with commassie blue and photographed. Samples were added in a nondenaturing environment and were presumed to be in their native state. 2.4. Rheometry. A stress-controlled rheometer with an aluminum cone and plate geometry (diameter 60 mm, cone angle 1°, truncation 29 µm; TA Instruments, Inc., New Castle, DE) and a capillary rheometer (diameter 1.024 × 10-2 m, length 63.1 mm; Vilastic-3, Vilastic Scientific, Inc., Austin, TX) were used to characterize the rheological behavior of the lens soluble protein dispersion under a variety of conditions. Simple viscoelastic and yield stress models were fitted to the data to characterize the phenomenological behavior of the fluid. The cone and plate rheometer torque range was 0.1-200 µN · m, and the minimum measurable stress was 0.0008 Pa. Temperature was controlled at 37 °C by means of a Peltier plate. After the experiment, the Peltier plate was flushed with water to remove the sample, and then dried and cleaned using isopropyl alcohol before adding new sample. In addition, the cone and plate rheometer was first calibrated using 0.960 Pa · s calibration oil (Brookfield Engineering, Middleboro, MA) to determine the range of shear rates which may be accurately examined in the neighborhood of the protein dispersion viscosity. The sample was maintained by surrounding the instrument geometry with an Environmental Control Chamber (TA Instruments, Inc., New Castle, DE) supplied with deionized water. Samples were allowed 10 min to equilibrate to 37 °C. Oscillatory shear was applied at 1 Hz for 10 min to ensure sample homogeneity. A 15 min relaxation period was allotted between each treatment to allow the proteins to relax completely to their equilibrium state. Steady state viscosity was then determined by incrementally increasing the shear rate from 10-4 to 102 s-1. After waiting 15 min, the steady-state viscosity of the same sample was again tested to check for hysteresis. A shear strain scan was performed at constant angular frequency of 1 rad per min to determine an appropriate strain amplitude for small angle oscillating shear (SAOS) such that the material would behave in a linearly viscoelastic fashion. Analysis was then performed at constant shear strain of 1.0% and varying frequency from 10-2 to 102 Hz. Further analyses were carried out at higher strain levels to examine the effects of structural degradation on the fluid’s storage (G′) and loss (G′′) moduli. However, because these larger strain amplitudes were outside of the linear viscoelastic range, the values for G′ and G′′ are not quantitative. Rather, because the stress waveforms deviated somewhat from the assumed form of linear viscoelasticity, they may only be used to draw qualitative conclusions regarding the behavior of the fluid. Creep experiments were also performed at a variety of stress levels far below, near, and far above the yield stress. All analyses were performed in duplicate to determine intersample deviations. Results given display data for all iterations. Oscillatory shear results with a raw phase greater than 150° were discarded due to the importance of inertial effects. Finally, one sample was repeatedly tested in steady shear and oscillatory shear with approximately one minute between iterations to examine intrasample hysteresis. The capillary rheometer was first calibrated using distilled water as directed by the manufacturer. Each sample was heated in a water bath at 37 °C for 30 min and centrifuged at 3000 rpm for 10 min to remove air bubbles. Sample chamber temperature was maintained at 37 °C using a Thermo Electron HAAKE DC30/K10 control unit and circulator (Thermo Electron Corp., Karlsruhe, Germany) throughout testing. Analyses were performed at shear rates from 0.05 to 850 s-1 at various fixed frequencies, as well as frequencies from 0.02-15 Hz at constant strain amplitudes. Further measurements were performed on individual

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Table 1. Determination of Total Crystallin Concentration water cal wt (mg)

total crystallin wt (mg)

dry wt (mg)

wt/wt (%)

wt/vol (%)

148.8 148.8 149a

165.1 164.8 164 ( 0.21

51.7 50.9 51.3 ( 0.6

31.3 30.9 31.1 ( 0.3

34.7 34.2 34.4 ( 0.4

a

Data derived from density measurements at 20 °C.

crystallin fractions at near physiological concentrations by varying the shear rate from 0.05 to 850 s-1 and holding the frequency constant at 2 Hz. 2.5. Rheological Modeling. Equilibrium viscosity measurements were modeled using three different constitutive models. The viscosity of the three-parameter rate process-based Powell-Eyring model is described by

η - η∞ sin h-1(λγ˙ ) , ) η0 - η ∞ λγ˙

(1)

where η is the viscosity at shear rate γ˙ , η0, and η∞ are the asymptotic viscosities as γ˙ approaches zero and infinity, respectively, and λ is a characteristic time scale of the dispersion’s viscous processes.44 λ is also the inverse of the critical shear rate (that is, the shear rate at which shear thinning begins). The Cross model used by Tiffany and Koretz50 has the form

η - η∞ 1 ) η0 - η∞ 1 + (λγ˙ )n

(2)

where n is a power law parameter and all other parameters have the same meaning as in the Powell-Eyring model.51 The yield stress of the fluid was characterized by considering a modified form of a Herschel-Bulkley power law model for yield stress fluids. We augmented the Herschel-Bulkley relation to allow for a Newtonian plateau at low shear rates. This modified constitutive relation has the form

{

()

τy

τ)

γ˙ γ˙ c

τy + k

If γ˙ e γ˙ c

( ) γ˙ - γ˙ c γ˙ c

(3)

n

If γ˙ g γ˙ c

where τy ) η0γ˙ c is the yield stress, k is the power-law consistency, n is a power law exponent, and γ˙ c is the critical shear rate at which shear thinning begins. The Cox-Merz rule relates the linear dynamic viscosity η*(ω) to the steady shear flow viscosity η(γ˙ ).52 It was used to assess the role of fluid structure to its rheological behavior since this empirical correlation breaks down when the fluid structure changes due to flow conditions.41 It states that η and the magnitude of the complex viscosity |η/(ω)| are related as

η(γ˙ ) ) |η/(ω)|ω)γ˘

(4)

for fluids with intact structures. Individual crystallin fractions were modeled as purely Newtonian fluids (i.e., their viscosity was independent of shear rate) at physiological concentrations.

3. Results and Discussion 3.1. Chromatography. Chromatographic analysis of the porcine crystallins has been included to give insight into aspects of the makeup of the crystallin mixture, showing the ratios of the major fractions and their differences in size and charge. Data for the determination of the total crystallin concentration are given in Table 1.

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Figure 1. Comparison between the ultraviolet absorbance at 280 nm from HLPC/GPC columns of the unfractionated crystallins (s) and the reconstituted crystallin sample comprised of purified fractionated crystallins (---). The slight shift may be caused by the absence of noncrystallin components (such as salts) in the reconstituted solution. These species may act as a buffer, preventing aggregation. Table 2. Determination of Individual Crystallin Fraction Concentrations crystallin fraction RI area (%) MN (105 Da) [η] (m3/kg) Rh (nm) RH RL βH βL γ

31.06 31.06 32.38 17.00 19.54

15.97 7.92 1.38 0.45 0.22

0.065 0.047 0.041 0.034 0.027

12.14 8.45 4.77 2.92 2.11

Partial analysis of the porcine crystallins has been included to give insight into aspects of the makeup of the crystallin mixture, showing the ratios of the major fractions and their differences in size and charge. The reconstituted crystallin solution HPLC/GPC data indicate that the concentrations of the individual crystallin fractions selected were very similar to those of the natural lens (Figure 1). The slight shift toward larger retention volumes is indicative of aggregation, which is likely a result of the removal of some natural noncrystallin components of the lens during reconstitution that help maintain the shortrange order. This aggregation is probably due to the concentration procedure, occurring when the crystallins were returned to physiological concentrations for rheological testing, despite maintaining the crystallins in the column buffer throughout the process. In particular, γ-crystallins are known to aggregate, depending on buffer conditions, protein concentration, solution viscosity, and the absence of R-crystallins.24 The ratios found are similar to those reported by Robinson et al. for young human lenses.39 The measurements of individual crystallin fraction concentrations (which were used for reconstitution) are given in Table 2. 3.2. Electrophoresis. IEF (Figure 2) indicated that the R-crystallin fraction was electrically neutral at pI 6.0. βH and βL fractions were neutral at pI 7.0. Six subfractions of γ-crystallins were observed, each achieving neutrality at pI greater than 7. The high molecular weight (1500-2000 kDa) R-crystallins did not migrate, as indicated by the stained band at the injection point of the R-crystallin lane. This is likely indicative that their radius is larger than the pore dimension of the IEF gel. However, as the high and low molecular weight R-crystallins are comprised of identical monomer units, the true pI should be similar for both varieties. Vidal and Cabezas-Cerrato studied porcine lens crystallins using two-dimensional electrophoresis.27 They drew compari-

Figure 2. IEF of lens crystallin fractions. The rightmost column gives the pI values and locations of several standard proteins for comparison. The R-crystallins had the lowest pI, while β-crystallins were virtually neutral, and the γ-crystallins had higher values. This suggests that the crystallins may behave as colloids when combined in solution and that the charge distributions may contribute significantly to the development of fluid structure.

sons between the porcine and the bovine crystallins, which were quite similar and indicated IEF points ranging from pH 5.7 to 7.4. One-dimensional electrophoresis indicated very similar subfraction molecular weights to those reported by Vidal and Cabezas-Cerrato. In a thorough study on isoelectric focusing of calf lens crytallins by Bours et al., 13 bands were noted, with values ranging from 7.05-9.05.28 These results show a similar range in values to that which we observed for the porcine crystallins Additionally, a further comparison study showed the isoelectric points of crystallins of rat, dogfish, calf, and human to be within the range of pH 6.80-8.87.29 At physiological pH, it appears that electrostatic attraction would exist between the negatively charged crystallins and positively charged crystallins. These electrostatic charges likely play an essential role in the development of short-range crystallin order31 and lens transparency.24 These charge distributions may give rise to the threebody interactions implicated in the generation of short-range order in colloidal dispersions.30–32 3.3. Rheology. No intrasample hysteresis was observed in consecutive analyses of steady shear viscosity or oscillatory shear testing after the 15 min relaxation period. Additional testing with an approximately 1 min delay also indicated no hysteresis, indicating that the fluid rapidly recovered its shortrange order. This was somewhat unexpected due to the large values of relaxation time (λ) found for all three of the viscosity models, though the relaxation times for oscillatory tests were much shorter (shown below). In any case, this result indicated that the fluid structure reformed very rapidly upon the cessation

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Figure 3. Steady state shear viscosity data (O) fitted with various models (s): (a) the Powell-Eyring model, (b) the Cross model, and (c) the yield stress model of eq 3. Note the Newtonian plateau at low shear rates, followed by strong shear thinning with increasing shear rate, and the development of a final plateau at high shear rates. (d) Cox-Merz rule comparison of steady shear viscosity with complex viscosity at various strain amplitudes. The agreement between the steady and the complex viscosities diminished rapidly as the strain level increased, indicating that the fluid structure was increasingly broken down as the strain amplitude increased. Table 3. Best-Fit Parameters for Steady Shear Rheological Models model

η0 (Pa · s)

Powell-Eyring Cross yield stress mean values

× × × ×

3.13 4.80 2.83 3.58

1

10 101 101 101

η∞ (Pa · s) -2

2.30 × 10 1.79 × 10-2 N/A 2.04 × 10-2

k (Pa · sn+1) N/A N/A 3.41 × 10-3 3.41 × 10-3

γ˙ (s-1)

n N/A 7.74 × 10-1 5.98 × 10-1 6.86 × 10-1

of flow. Steady shear viscosity measurements (Figure 3) indicated strong shear-thinning behavior and were fitted well with the Powell-Eyring model (eq 1), Cross model (eq 2), and the modified Herschel-Bulkley yield stress model (eq 3). Bestfit parameters and other implied quantities are given in Table 3. The Cross model significantly overpredicted the low shear rate asymptotic viscosity η0 relative to the other two models, though gave a quantitative fit (as measured by the R2 value) approximately equal to the other two. All models gave similar results over the range of intermediate shear rates. Both Powell-Eyring and Cross models overpredict the viscosity at high shear rates. The yield stress model does not predict a high shear rate plateau, which is physically unrealistic. The yield stress model gives the best fit over the broadest range of shear rates, failing only at very high, physiologically irrelevant values and is, therefore, the best choice for most applications. The magnitude of η0-η∞ may be used as a measure of the importance of structure in determining the fluid’s flow properties.31 This quantity was on the order of 104 Pa · s for the lens soluble proteins, indicating that the structure of the fluid was

6.16 8.08 1.70 1.04

× × × ×

λ (s) -4

10 10-4 10-3 10-3

1.62 1.24 5.87 1.15

× × × ×

G0 (Pa)

τy(Pa) 3

10 103 102 103

1.93 3.88 4.82 3.54

× × × ×

-2

10 10-2 10-2 10-2

1.93 3.88 4.82 3.54

× × × ×

10-2 10-2 10-2 10-2

very important in determining its rheological character. A first approximation of a characteristic strain rate that might be expected in vivo could be taken as the velocity V of the lens equator divided by the undeformed radius r0 or

γ˙ ≈

V r0

(5)

taking V ≈ 5 mm/s and r0 ≈ 5 mm,12,20 this gives γ˙ ≈ O (1 s-1), which is well above the onset of shear thinning. This naive method for determining a characteristic strain rate indicates that the role of shear thinning may be significant in decreasing the forces required for accommodation. The sigmoidal shape of the equilibrium viscosity to shear rate data indicate two primary flow regimes. At low shear rates, these interparticle forces dominate, resulting in a very high viscosity. At high shear rate, viscous forces dominate, destroying noncovalent or weak interparticle forces, and the viscosity approaches that of the solvent. The excellent fit of the Powell-Eyring model indicates that the dispersion behaves as a viscoelastic biphasic suspension. Tiffany and Koretz used a

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Figure 4. First normal stress coefficient as a function of shear rate. The presence of a normal stress difference indicated that the solution is not purely viscous and exhibits anisotropy. The data were well fitted with a simple power law (R2 ) 0.991). The normal stress difference is indicative of fluid anisotropy. Thus, these data indicate a breakdown in fluid structure as inertial forces increase.

concentric cylinder rheometer to determine the steady shear viscosity of various concentrations of R-crystallin solutions at a variety of temperatures.51 Their data was fitted with a Cross model, which is a variant of the Ree-Eyring constitutive model commonly employed for fitting the viscous behavior of colloidal suspensions under steady shear.31 Unfortunately, they displayed their results in figures using linear scales, making comparison difficult. They also reported the inability to measure the asymptotic viscosities and neglected to give the fitting parameters for the Cross model. In the present study, the Cross model failed to fit the low shear rate plateau, while the other two models both fitted it well. This may indicate that the Cross model is insufficient for describing the steady shear flow of this solution and why Tiffany and Koretz had difficulty measuring the value of the low shear rate asymptote. Individual physiologically concentrated crystallin fractions exhibited Newtonian behavior when evaluated using capillary rheometry. The Newtonian viscosities of the R-, βH-, βL-, and γ-crystallins were 1.40 × 10-3 Pa · s, 1.31 × 10-3 Pa · s, 8.76 × 10-4 Pa · s, and 8.80 × 10-4 Pa · s, respectively. While one might expect a significantly higher viscosity for R-crystallin relative to the other crystallin fractions at the same concentration, its relatively low volume fraction diminishes the effect of its large molecular dimension. It is also important to note that, while the unfractionated crystallins exhibited dramatic shear thinning, no shear thinning was observed in the individual crystallin fractions. This again indicates that it is the interaction between the differently charged particles that gives rise to the structure of the fluid. The first normal stress coefficient ψ1 was also measured during steady shear using the cone and plate rheometer (Figure 4). This parameter is indicative of the level of anisotropy present at the test shear rate.53 The dramatic decrease in ψ1 with increasing γ˙ was again indicative of the change of fluid structure with increasing inertial forces. These results were well fitted with a simple power power law (ψ1 ) 329γ˙ -1.98; R2 ) 0.991). No low-shear asymptote was observed for ψ1, which appears to be common for high molecular weight systems.51 The value of the exponent in the power law fit is very close to -2, indicating that the stress difference is nearly constant across all shear rates. Significant decrease in G′ was observed with increasing strain and shear rate amplitudes, while G′′ was relatively independent

Reilly et al.

Figure 5. Storage and loss moduli for the soluble lens proteins measured at various strain amplitudes. Note that the moduli values for strains above 1% are approximate because the strain waveform deviated somewhat from the pure sinusoidal wave expected for linear viscoelasticity. The moduli consistently decreased with increasing strain to a strain of approximately 50%, above which the moduli values remained unchanged. This result indicated that the structure of the fluid was broken down completely at these high strain levels.

of shear rate and depended significantly on strain only at large amplitudes. (Figure 6). Viscoelastic parameters G′ and G′′ exhibited a crossover frequency of approximately 2.5 Hz in the linear viscoelastic regime (Figure 5), corresponding to a relaxation time of about 0.4 s. The crossover frequency at 2.5% strain decreased to about 0.15 Hz, corresponding to a relaxation time of about 6.7 s. At 5% strain and above, no crossover frequency could be measured within the tested frequency range and G′′ was always greater than G′. At 50% and higher strains, G′ was always greater than G′′. However, at strains greater than 5%, the stress waveforms were dissimilar to the linear viscoelastic waveforms, such that only the approximate amplitude of the complex modulus G/ (where G/ ) √G′2 + G′′2 ) was reliable and the quantitative values of G′ and G′′ were unreliable. Still, it was clear that significant strain softening occurred as the material structure was degraded. The trend of increasing relaxation time with increasing strain agreed with the notion that the fluid structure is disrupted when subjected to large or very rapidly applied strains. The dynamic moduli at 1% strain were not well fitted by any reasonable number of linear viscoelastic relationships. They were well fitted by a three-mode Giesekus model (not shown), but the resulting parameters did not give agreement with the steady shear data. Confirmation of the validity of the oscillatory measurements at low strains was offered by the method suggested by Cross in which η0 and λ may be used to compute the value of the shear modulus at the low shear asymptote, G0, as G0 ) η0/λ.55 This result gives a value for G0 of approximately 0.05 Pa, which coincides with the asymptote at low shears seen in a shear rate scan at 1 Hz (Figure 6). Creep experiments gave further insight into the behavior of the fluid as stress increased (Figure 7). At stresses far below the yield stress, the sample behaved as a viscoelastic solid with a compliance plateau at equilibrium. Near the yield point, the compliance curve qualitatively changes to a more liquid-like shape, though the compliance values are similar to the lower stress measurements. Finally, far above the yield stress, the sample behaves in an almost purely viscous fashion with much higher compliance values, indicating that the structure was fully

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Figure 6. Dependence of storage modulus (G′, O) and loss modulus (G′′, 0) on (a) strain amplitude measured at 1 Hz with the cone and plate rheometer and (b) shear rate amplitude measured at 1 Hz with the capillary rheometer. In both cases, the behaviors of the dynamic moduli are qualitatively similar. The loss modulus decreases slightly with increasing strain, with a more dramatic decrease occurring at a strain of approximately 3, while it is constant across the entire range of shear rates tested. The storage modulus exhibits similar behavior, but the onset of rapid strain softening occurs at a lower strain amplitude of approximately 0.4. The storage modulus exhibits a rapid decline at shear rates above 8 s-1 .

Figure 7. Creep shear compliance measured at a variety of shear stress levels well below the yield stress (O), near the yield stress (0), and well above the yield stress (x). Note the qualitative differences in the creeping behavior of the fluid as it approaches the yield stress. Far below the yield point, the compliance exhibits a solid-like plateau at long time scales. Near the yield point, the solution behaves as a fluid, losing its elastic-like plateau, despite the fact that the compliance values are similar to those far below the yield point. Well above the yield point, the sample is much more compliant and behaves as a purely viscous fluid.

degraded. This gives further insight into the nature of the transition of this solution around the yield point as it changes from a viscoelastic solid to a nearly purely viscous liquid.

4. Conclusions Based on a series of systematic investigations involving chromatography, electrophoresis, and rheometry, our experimental data clearly support the hypothesis that the lens crystallins are a liquid-like shear-thinning viscoelastic material. These characteristics are desirable for achieving stable accommodation in the presence of a variety of potential disturbances and are granted by the fluid’s structure. Thus, the fluid’s structure is vital for both transparency of the lens and neural control of dynamic accommodation. Delaye and Tardieu demonstrated short-range order of crystallins at high protein concentration.37,56 This result, combined with the findings of the present study, suggest that the

electrostatic charge distribution of lens crystallins prevents the formation of large, light scattering aggregates. Thus, the same structure giving rise to the transparency of the lens also gives rise to several desirable mechanical characteristics, such as diminished force requirements and damping. The soluble proteins’ high RI (1.3996) indicates that they contribute to the optical properties of the lens.57 Because the total intact lens behaves as a viscoelastic solid, the elastic mechanical properties may originate in the structural proteins which form the lens fiber membranes. Accommodation and disaccommodation are the result of forces rapidly applied to the lens substance by the elastic lens capsule.3 This implies a high shear rate at the capsule-lens interface. As the shear rate increases, the short-range order of the crystallins is broken down, decreasing the force requirements of dynamic accommodation. The slight yield stress is helpful in maintaining focus when perturbed by small amplitude oscillations. If the lens crystallins behaved in a purely elastic manner, the forces required for accommodation would be too great for the feeble ciliary muscle and achieving high quality, stable vision would be impossible.15 Additionally, the neural feedback mechanism controlling accommodation may require the damping behavior of the lens for proper function.14 Acknowledgment. Supported by a merit review grant from the Department of Veterans Affairs to N.R., by awards to the Washington University Department of Ophthalmology and Visual Sciences by Research to Prevent Blindness, Inc., and the NIH (P30 EY 02687) Core Grant.

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