Prostaglandin E2 enhances the sodium conductance of exocrine glands in isolated frog skin (Rana esculenta)

Summary Prostaglandins are known to stimulate the active transepithelial Na⁺ uptake and the active secretion of Cl^– from the glands of isolated frog skin. In the present work the effect of prostaglandin E₂ (PGE₂) on the glandular Na⁺ conductance was examined. In order to avoid interference from the Na⁺ uptake and the glandular Cl^– secretion the experiments were carried out on skins where the Cl^– secretion was inhibited (the skins were bathed in Cl^– Ringer's solution in the presence of furosemide, or in NO ₃ ^– Ringer's solution), and the active Na⁺ uptake was blocked by the addition of amiloride. Transepithelial current, water flow and ion fluxes were measured. A negative current was passed across the skins (the skins were clamped at –100 mV, basolateral solution was taken as reference). When PGE₂, was added to the skins under these experimental conditions, the current became more negative; this was mainly due to an increase in the Na⁺ efflux. Together with the increase in Na⁺ efflux a significant increase of the water secretion was observed. The water secretion was coupled to the efflux of Na⁺, and when one Na⁺ was pulled from the basolateral to the apical solution via this pathway 230 molecules of water follwed. From the data presented it is suggested that this pathway for Na⁺ is confined to the exocrine glands.     

The Control of the Membrane Potential of Muscle Fibers by the Sodium Pump

Frog sartorius muscles were made Na-rich by immersion in K-free sulfate Ringer's solution in the cold. The muscles were then loaded with Na²⁴ and the extracellular space cleared of radioactivity. When such Na-rich muscles were transferred to lithium sulfate Ringer's solution at 20°C, Na efflux was observed to increase with time, to reach a maximum about 15 minutes after the transfer of the muscles to Li₂SO₄, and then to decline. The decline in efflux from these muscles was proportional to ([Na]_i)⁸ over a considerable range of [Na]_i. The membrane potential of Na-rich muscles was about -48 mv in K-free sulfate Ringer's at 4°C but changed to -76 mv in the same solution at 20°C and to -98 mv in Li₂SO₄ Ringer's at 20°C. By contrast, muscles with a normal [Na]_i showed a fall in membrane potential when transferred from K-free sulfate Ringer's to Li₂SO₄ Ringer's solution. The general conclusions from this study are (a) that Na extrusion is capable of generating an electrical potential, and (b) that increases in [Na]_i lead to reversible increases in P_Na of muscle fibers.     

The Influence of Sodium-Free Solutions on the Membrane Potential of Frog Muscle Fibers

The membrane potential of frog sartorius muscle fibers in a Cl- and Na-free Ringer's solution when sucrose replaces NaCl is about the same as that in normal Ringer's solution. The K⁺ efflux is also about the same in the two solutions but muscles lose K and PO₄ in sucrose Ringer's solutions. The membrane potential in sucrose Ringer's solution is equal to that given by the Nernst equation for a K⁺ electrode, when corrections are made for the activity coefficients for K⁺ inside and outside the fiber. For a muscle in normal Ringer's solution, the measured membrane potential is within a few millivolts of E_K. This finding is incompatible with a 1:1 coupled Na-K pump. It is consistent with either no coupling of Na efflux to K influx, or a coupling ratio of 3 or greater.     

Effect of ATP on the Calcium Efflux in Dialyzed Squid Giant Axons

Dialysis perfusion technique makes it possible to control the internal composition of squid giant axons. Calcium efflux has been studied in the presence and in the virtual absence (<5 µM) of ATP. The mean calcium efflux from axons dialyzed with 0.3 µM ionized calcium, [ATP]_i > 1,000 µM, and bathed in artificial seawater (ASW) was 0.24 ± 0.02 pmol·cm^-2·s^-1 (P/CS) (n = 8) at 22°C. With [ATP]_i < 5 µM the mean efflux was 0.11 ± 0.01 P/CS (n = 15). The curve relating calcium efflux to [ATP]_i shows a constant residual calcium efflux in the range of 1–100 µM [ATP]_i. An increase of the calcium efflux is observed when [ATP]_i is >100 µM and saturates at [ATP]_i > 1,000 µM. The magnitude of the ATP-dependent fraction of the calcium efflux varies with external concentrations of Na⁺, Ca⁺⁺, and Mg⁺⁺. These results suggest that internal ATP changes the affinity of the calcium transport system for external cations.     

Calcium Efflux from Barnacle Muscle Fibers : Dependence on External Cations

Calcium-45 was injected into single giant barnacle muscle fibers, and the rate of efflux was measured under a variety of conditions. The rate constant (k) for ⁴⁵Ca efflux into standard seawater averaged 17 x 10^–4 min^–1 which corresponds to an efflux of about 1–2 pmol/cm²·s. Removal of external Ca (Ca_o) reduced the efflux by 50%. In most fibers about 40% of the ⁴⁵Ca efflux into Ca-free seawater was dependent on external Na (Na_o); treatment with 3.5 mM caffeine increased the magnitude of the Na_o-dependent efflux. In a few fibers removal of Na_o, in the absence of Ca_o, either had no effect or increased k; caffeine (2–3.5 mM) unmasked an Na_o-dependent efflux in these fibers. The Na_o-dependent Ca efflux had a Q₁₀ of about 3.7. The data are consistent with the idea that a large fraction of the Ca efflux may be carrier-mediated, and may involve both Ca-Ca and Na-Ca counterflow. The relation between the Na_o-dependent Ca efflux and the external Na concentration is sigmoid, and suggests that two, or more likely three, external Na⁺ ions may activate the efflux of one Ca⁺². With a three-for-one Na-Ca exchange, the Na electrochemical gradient may be able to supply sufficient energy to maintain the Ca gradient in these fibers. Other, more complex models are not excluded, however, and may be required to explain some puzzling features of the Ca efflux such as the variable Na_o-dependence.     

Volume Reabsorption, Transepithelial Potential Differences, and Ionic Permeability Properties in Mammalian Superficial Proximal Straight Tubules

This paper describes experiments designed to evaluate Na⁺ and Cl^- transport in isolated proximal straight tubules from rabbit kidneys. When the perfusing solution was Krebs-Ringer buffer with 25 mM HCO₃^- (KRB) and the bath contained KRB plus 6% albumin, net volume reabsorption (J_v, nl min^-1 mm^-1 was -0.46 ± 0.03 (SEM); V_e, the spontaneous transepithelial potential difference, was -1.13 ± 0.05 mV, lumen negative. Both J_v, and V_e, were reduced to zero at 21°C or with 10^-4 M ouabain, but J_v, was not HCO₃^- dependent. Net Na⁺ reabsorption, measured as the difference between ²²Na⁺ fluxes, lumen to bath and bath to lumen, accounted quantitatively for volume reabsorption, assuming the latter to be an isotonic process, and was in agreement with the difference between lumen to bath ²²Na⁺ fluxes during volume reabsorption and at zero volume flow. The observed flux ratio for Na⁺ was 1.46, and that predicted for a passive process was 0.99; thus, Na⁺ reabsorption was rationalized in terms of an active transport process. The Cl^- concentration of tubular fluid rose from 113.6 to 132.3 mM during volume reabsorption. Since V_e, rose to +0.82 mV when tubules were perfused with 138.6 mM Cl^- solutions, V_e may become positive when tubular fluid Cl^- concentrations rise during volume reabsorption. The permeability coefficients P_Na and P_Cl computed from tracer fluxes were, respectively, 0.23 x 10^-4 and 0.73 x 10^-4 cm s^-1. A P_Na/P_Cl ratio of 0.3 described NaCl dilution potentials at zero volume flow. The magnitudes of the potentials were the same for a given NaCl gradient in either direction and P_Na/P_Cl was constant in the range 32–139 mM NaCl. We infer that the route of passive ion permeation was through symmetrical extracellular interfaces, presumably tight junctions, characterized by neutral polar sites in which electroneutrality is maintained by mobile counterions.     

Isolated epithelial cells of the toad bladder. Their preparation, oxygen consumption, and electrolyte content

Epithelial cells of the toad bladder were disaggregated with EDTA, trypsin, hyaluronidase, or collagenase and were then scraped free of the underlying connective tissue. In most experiments EDTA was complexed with a divalent cation before the tissue was scraped. Q _OO2, sucrose and inulin spaces, and electrolytes of the isolated cells were measured. Cells disaggregated by collagenase or hyaluronidase consumed O₂ at a rate of 4 µl hr^-1 dry wt^-1. Q _OO2 was increased 50% by ADH (100 U/liter) or by cyclic 3'',5''-AMP (10 mM/liter). Na⁺-free Ringer''s depressed the Q _OO2 by 40%. The Q _OO2 of cells prepared by trypsin treatment or by two EDTA methods was depressed by Na⁺-free Ringer''s but was stimulated relatively little by ADH. Two other EDTA protocols produced cells that did not respond to Na⁺ lack or ADH. The intracellular Na⁺ and K⁺ concentrations of collagenase-disaggregated cells were 32 and 117 mEq/kg cell H₂O, respectively. Cation concentrations of hyaluronidase cells were similar, but cells that did not respond to ADH had higher intracellular Na⁺ concentrations. Cells unresponsive to ADH and Na⁺ lack had high sucrose spaces and low transcellular membrane gradients of Na⁺, K⁺, and Cl^-. The results suggest that trypsin and EDTA disaggregation damage the active Na⁺ transport system of the isolated cell. Certain EDTA techniques may also produce a general increase in permeability. Collagenase and hyaluronidase cells appear to function normally.     

Abscisic Acid Stimulated Osmotic Adjustment and Its Involvement in Adaptation of Tobacco Cells to NaCl

Osmotic adjustment of cultured tobacco (Nicotiana tabacum L. var Wisconsin 38) cells was stimulated by 10 micromolar (±) abscisic acid (ABA) during adaptation to water deficit imposed by various solutes including NaCl, KCl, K₂SO₄, Na₂SO₄, sucrose, mannitol, or glucose. The maximum difference in cell osmotic potential (Ψπ) caused by ABA treatment during adaptation to 171 millimolar NaCl was about 6 to 7 bar. The cell Ψπ differences elicited by ABA were not due to growth inhibition since ABA stimulated growth of cells in the presence of 171 millimolar NaCl. ABA caused a cell Ψπ difference of about 1 to 2 bar in medium without added NaCl. Intracellular concentrations of Na⁺, K⁺, Cl⁻, free amino acids, or organic acids could not account for the Ψπ differences induced by ABA in NaCl treated cells. However, since growth of NaCl treated cells is more rapid in the presence of ABA than in its absence, greater accumulation of Na⁺, K⁺, and Cl⁻ was necessary for ion pool maintenance. Higher intracellular sucrose and reducing sugar concentrations could account for the majority of the greater osmotic adjustment of ABA treated cells. More rapid accumulation of proline associated with ABA treatment was highly correlated with the effects of ABA on cell Ψπ. These and other data indicate that the role of ABA in accelerating salt adaptation is not mediated by simply stimulating osmotic adjustment.     

Characterization of amino Acid efflux from isolated soybean cells

Cells from reproductive soybean (Glycine max [L.] Merr.) plants were isolated using a mechanical-enzymic technique that produced a high yield of uniform, physiologically active cells. Cells were incubated in a pH 6.0 buffered solution and subjected to various treatments in order to determine the nature of net amino acid efflux. Total net amino acid (ninhydrinreactive substances) efflux was not affected by the following conditions: (a) darkness, (b) aeration, (c) K⁺ concentrations of 0.1, 1.0, 10, or 100 millimolar and (d) pH 4, 5, 6, 7, or 8. The Q₁₀ for net amino acid efflux between 10°C and 30°C was 1.6. Thus, it seems that net amino acid efflux requires neither current photosynthetic energy nor a pH/ion concentration gradient. Amino acid analyses of the intra-and extracellular fractions over time showed that each amino acid was exported linearly for at least 210 minutes, but that export rate was not necessarily related to internal amino acid pools. Amino acids that were exported fastest were alanine, lysine, leucine, and glycine. Addition of the inhibitor p-chloromercuriphenyl sulfonic acid, 3(3,4-dichlorophenyl)-1,1-dimethylurea, or carbonylcyanide p-trifluoromethoxyphenylhydrazone increased the rate of total amino acid efflux but had specific effects on the efflux of certain amino acids. For example, p-chloromercuriphenyl sulfonic acid greatly enhanced efflux of γ-aminobutyric acid, which is not normally exported rapidly even though a high concentration normally exists within cells. The data suggest that net amino acid efflux is a selective diffusional process. Because net efflux is the result of simultaneous efflux and influx, we propose that efflux is a facilitated diffusion process whereas influx involves energy-dependent carrier proteins.     

Effect of temperature on electrolyte metabolism of isolated frog skin

A study is presented on the effect of temperature on unidirectional active ion transport, resting electrolyte equilibrium (electrolyte composition), and oxygen consumption in isolated frog skin. The aims were twofold: first, to find out whether the rate of active transport can be changed without affecting the Na⁺ and K⁺ balance of skin itself; second, to arrive at minimal ΔNa/ΔO₂ values by correlating quantitatively inhibition of active ion transport with inhibition of O₂ consumption. NaCl transport was maximal at 20°C. At 28° and at temperatures below 20°, rate of NaCl transport was diminished. In many instances NaCl transport was diminished in skins which maintained their normal Na⁺ and K⁺ content. In several cases, however, neither rate of transport nor resting electrolyte equilibrium was affected; in other cases, both were. O₂ consumption decreased when lowering the temperature over the range from 28 to 10°C. From a plot of log Q_OO2 against 1/T an activation energy of µ 13,700 cal. was calculated, valid for the range from 10 to 20°C. It appeared that µ was smaller for temperatures above 20°C. Working between 10 and 20°, it was found that, on the average, 4 to 5 equivalents of Na⁺ were transported for one mole of O₂ consumed in skins with undisturbed resting electrolyte equilibrium.     

Tolerance of Hordeum marinum accessions to O2 deficiency, salinity and these stresses combined

Background and Aims

When root-zone O₂ deficiency occurs together with salinity, regulation of shoot ion concentrations is compromised even more than under salinity alone. Tolerance was evaluated amongst 34 accessions of Hordeum marinum, a wild species in the Triticeae, to combined salinity and root-zone O₂ deficiency. Interest in H. marinum arises from the potential to use it as a donor for abiotic stress tolerance into wheat.

Methods

Two batches of 17 H. marinum accessions, from (1) the Nordic Gene Bank and (2) the wheat belt of Western Australia, were exposed to 0·2 or 200 mol m⁻³ NaCl in aerated or stagnant nutrient solution for 28–29 d. Wheat (Triticum aestivum) was included as a sensitive check species. Growth, root porosity, root radial O₂ loss (ROL) and leaf ion (Na⁺, K⁺, Cl⁻) concentrations were determined.

Key Results

Owing to space constraints, this report is focused mainly on the accessions from the Nordic Gene Bank. The 17 accessions varied in tolerance; relative growth rate was reduced by 2–38 % in stagnant solution, by 8–42 % in saline solution (aerated) and by 39–71 % in stagnant plus saline treatment. When in stagnant solution, porosity of adventitious roots was 24–33 %; salinity decreased the root porosity in some accessions, but had no effect in others. Roots grown in stagnant solution formed a barrier to ROL, but variation existed amongst accessions in apparent barrier ‘strength’. Leaf Na⁺ concentration was 142–692 µmol g⁻¹ d. wt for plants in saline solution (aerated), and only increased to 247–748 µmol g⁻¹ d. wt in the stagnant plus saline treatment. Leaf Cl⁻ also showed only small effects of stagnant plus saline treatment, compared with saline alone. In comparison with H. marinum, wheat was more adversely affected by each stress alone, and particularly when combined; growth reductions were greater, adventitious root porosity was 21 %, it lacked a barrier to ROL, leaf K⁺ declined to lower levels, and leaf Na⁺ and Cl⁻ concentrations were 3·1–9-fold and 2·8–6-fold higher, respectively, in wheat.

Conclusions

Stagnant treatment plus salinity reduced growth more than salinity alone, or stagnant alone, but some accessions of H. marinum were still relatively tolerant of these combined stresses, maintaining Na⁺ and Cl⁻ ‘exclusion’ even in an O₂-deficient, saline rooting medium.Key words: Aerenchyma, combined salinity and waterlogging, leaf Cl⁻, leaf K⁺, leaf Na⁺, radial O₂ loss, salt tolerance, salinity–waterlogging interaction, sea barleygrass, waterlogging tolerance, wheat, wild Triticeae     

Propagation Speed in Myelinated Nerve: I. Experimental Dependence on External Na+ and on Temperature

Conduction speed (θ) in single myelinated Rana pipiens sciatic nerve fibers has been precisely measured using intracellular recording and on-line digital computer techniques. The dependence of relative speed on external Na concentration at 15°C has been found to be ln(θ₁/θ₂) = 0.524 (±0.018) ln ([Na⁺]₁/[Na⁺]₂) + 0.003. Thus θ has very close to a square root dependence on [Na⁺]₀ for these fibers. This experimental finding is not in complete agreement with a theoretical prediction based on a solution of the Hodgkin-Huxley (H.H.) equations. The effect of small temperature variations around 15°C on θ has also been measured for Rana fibers in Ringer's solution. θ has close to an exponential dependence on T and a Q₁₀ of 2.95 has been estimated.     

Effect of Temperature on the Potential and Current Thresholds of Axon Membrane

The effect of temperature on the potential and current thresholds of the squid giant axon membrane was measured with gross external electrodes. A central segment of the axon, 0.8 mm long and in sea water, was isolated by flowing low conductance, isoosmotic sucrose solution on each side; both ends were depolarized in isoosmotic KCl. Measured biphasic square wave currents at five cycles per second were applied between one end of the nerve and the membrane of the central segment. The membrane potential was recorded between the central sea water and the other depolarized end. The recorded potentials are developed only across the membrane impedance. Threshold current values ranged from 3.2 µa at 267deg;C to 1 µa at 7.5°C. Threshold potential values ranged from 50 mv at 26°C to 6 mv at 7.5°C. The mean Q₁₀ of threshold current was 2.3 (SD = 0.2), while the Q₁₀ for threshold potentials was 2.0 (SD = 0.1).     

Studies on chloride permeability of the skin ofLeptodactylus ocellatus: II. Na+ and Cl− effect of inward movements of Cl−

Summary At low concentration (1mm) of Cl^– in the outer solution, the influx of chloride through the isolated skin (J ₁₃ ^Cl ) of the South American frogLeptodactylus ocellatus (L.) seems to be carried by two mechanisms: (i) a passive one that exhibits the characteristics of an exchange diffusion process, and (ii) an active penetration. Studies of the influx and efflux of chloride (J ₁₃ ^Cl andJ ₃₁ ^Cl ) indicate, that the presence of a high (107mm) concentration of Cl^– in the outer solution activates the translocation of this ion through the cells. Studies of the unidirectional flux of Cl^– across the outer barrier (J ₁₂ ^Cl ) indicate that Na⁺ out stimulates the penetration of Cl^– at this level. Cl^– out, in turn, stimulates, theJ ₁₂ ^Na , but this effect is only detected at low concentrations of Na⁺ out.     

Different Rates of Synthesis and Degradation of Two Chloroplastic Ammonium-Inducible NADP-Specific Glutamate Dehydrogenase Isoenzymes during Induction and Deinduction in Chlorella sorokiniana Cells

The kinetics of accumulation (per milliliter of culture) of the α- and β- subunits, associated with chloroplast-localized ammonium inducible nicotinamide adenine dinucleotide phosphate-specific glutamate dehydrogenase (NADP-GDH) isoenzymes, were measured during a 3 hour induction of synchronized daughter cells of Chlorella sorokiniana in 29 millimolar ammonium medium under photoautotrophic conditions. The β-subunit holoenzyme(s) accumulated in a linear manner for 3 hours without an apparent induction lag. A 40 minute induction lag preceded the accumulation of the α-subunit holoenzyme(s). After 120 minutes, the α-subunit ceased accumulating and thereafter remained at a constant level (i.e. steady state between synthesis and degradation). From pulsechase experiments, using ³⁵SO₄ and immunochemical procedures, the rate of synthesis of the α-subunit was shown to be greater than the β-subunit during the first 80 minutes of induction. The α- and β-subunits had different rates of degradation during the induction period (t_½ = 50 versus 150 minutes, respectively) and during the deinduction period (t_½ = 5 versus 13.5 minutes) after removal of ammonium from the culture. During deinduction, total NADP-GDH activity decreased with a half-time of 9 minutes. Cycloheximide completely inhibited the synthesis and degradation of both subunits. A model for regulation of expression of the NADP-GDH gene was proposed.     

Sodium flux in the smooth muscle of frog stomach

Sodium efflux from rings of frog stomach muscle was measured at 5° and 15°C in three different steady states. After incubation in normal, K-free, or ouabain (10^-4 M) solutions, intracellular cations stabilized at markedly differing levels. At 5°C, inhibition of Na extrusion was shown in the rate coefficients for ²²Na efflux, which were slightly smaller in K-free than in normal solutions, and much smaller in ouabain. Due to the intracellular Na concentration differences, total Na efflux was similar in K-free and ouabain solutions, and only ⅕ as large in normal solution. At 15°C, normal total Na flux was only 1/7;–1/10 inhibitors, and may be underestimated. The total flux differences may involve dependence of the Na pump and Na permeation on internal Na concentration. The Q ₁₀ of the steady-state fluxes was 3.7 in ouabain, 2.8 in K-free solution, and 1.9 in normal solution. The high temperature dependence of influx as well as efflux suggests transport mechanisms other than simple diffusion. Sodium turnover in the cell water was 46–66 mM/hr in inhibitors at 15°C, and a high rate of Na extrusion in normal muscle is suggested. However, cell volume:surface ratio is only 1.6 µ and all estimates of Na flux were under 3 pmoles/cm² per sec, indicating low Na permeability.     

Influence of Oxygen and Temperature on the Dark Inactivation of Pyruvate, Orthophosphate Dikinase and NADP-Malate Dehydrogenase in Maize

The influence of oxygen and temperature on the inactivation of pyruvate, Pi dikinase and NADP-malate dehydrogenase was studied in Zea mays. O₂ was required for inactivation of both pyruvate, Pi dikinase and NADP-malate dehydrogenase in the dark in vivo. The rate of inactivation under 2% O₂ was only slightly lower than that at 21% O₂. The in vitro inactivation of pyruvate, Pi dikinase, while dependent on adenine nucleotides (ADP + ATP), did not require O₂.

The postillumination inactivation of pyruvate, Pi dikinase in leaves was strongly dependent on temperature. As temperature was decreased in the dark, there was a lag period of increasing length (e.g. at 17°C there was a lag of about 25 minutes) before inactivation proceeded. Following the lag period, the rate of inactivation decreased with decreasing temperature. The half-time for dark inactivation was about 7 minutes at 32°C and 45 minutes at 17°C. The inactivation of pyruvate, Pi dikinase in vitro following extraction from illuminated leaves was also strongly dependent on temperature, but occurred without a lag period. In contrast, NADP-malate dehydrogenase was rapidly inactivated in leaves (half-time of approximately 3 minutes) during the postillumination period without a lag, and there was little effect of temperature between 10 and 32°C. The results are discussed in relation to known differences in the mechanism of activation/inactivation of the two enzymes.

     

Contraction and Action Potentials of Frog Heart Muscles Soaked in Sucrose Solution

Isolated auricles or ventricles from the frog continue to contract, either spontaneously or when stimulated, for from 2 to 4 hours after they are placed in isotonic sucrose solution. After the muscles stop contracting in sucrose solution, contractility is partially restored when the muscles are placed in chloride Ringer's. However, contractility is usually not restored if the muscles are placed in sulfate Ringer's. Ventricles soaked in sucrose solution at 4–7°C continue to contract for 12 to 24 hours and during the first few hours in sucrose solution the contractions often are enhanced. Several types of experiment indicate that the sucrose solution does replace the Ringer's in the extracellular space. Auricles and ventricles also continue to conduct action potentials, with an overshoot, for from 30 to 360 minutes after being placed in sucrose solution. Muscles soaked in sucrose until they are inexcitable rapidly recover in chloride Ringer's but often fail to recover in sulfate Ringer's. The results are discussed in relation to theories about the generation of the action potential in cardiac muscle, and the role of the extracellular fluid in contraction.     

Temperature Dependence of Accommodation and Excitation in Space-Clamped Axons

Accommodation and excitation in space-clamped squid axons were studied with the double sucrose gap technique, using linear current ramps, short (50 µsec) square wave pulses, and rheobasic square wave pulses as stimuli. The temperature was varied from 5° to 35°C. Experimental results showed a Q₁₀ for accommodation which was 44% higher than that for excitation. Yet calculations on the basis of the Hodgkin-Huxley equations predict equal Q₁₀'s for excitation and accommodation. Although the Hodgkin-Huxley equations are spectacularly successful for so many nerve phenomena, the differences between calculations of accommodation and these experiments, which were designed to test the equations, show that the equations need modification in this area.