As active transport mechanisms are presented and consume energy, the zero flux conditions correspond to a non-equilibrium steady state [ 26 ]. Eqs 35 , 36 then become 43 An essentially linear relationship is observed between the active ionic flux and the three pressure estimations. In this section an example application of Eqs 31 , 40 is presented to study the endothelial transport process of the in vivo human cornea see Fig 3. The exchange of fluid and ions across the endothelium controls the level of corneal hydration, which is a crucial factor for maintaining the transparency of the tissue [ 21 , 24 ].
Fixed charges are associated with sulphated proteoglycans in the stroma the bulk layer of the tissue , and generate osmotic pressure by Donnan effect [ 8 , 27 ]. The active ionic transport processes located in the endothelium reduce the osmotic pressure by pumping ions out from the tissue. Furthermore, metabolic reactions take place in the in vivo cornea, rendering nonzero transendothelial fluxes for metabolic species glucose, bicarbonate and lactate ions [ 25 , 28 , 29 ].
Fixed charges are associated with sulphated proteoglycans that are attached to the collagen fibrils, which form a lamella by assembling as parallel arrays following the direction of the lamella. The fixed charges give rise to a swelling tendency of the cornea [ 8 ] by the Donnan effect [ 17 ]. In addition, glucose transports from aqueous humor to stroma for the metabolic activities needed by corneal cells, and lactate ion transports in the opposite direction [ 25 ]. The underlying molecular mechanisms of corneal endothelial pumping are still under investigation [ 31 ].
At steady state, only the volume flux is zero, with the steady solute fluxes being nonzero in general. This is a more general situation than that discussed in the preceding section.
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The concentrations of these species in the outside solution aqueous humor are listed in table 1. The reported reflection coefficients and solute permeabilities of corneal endothelium are given in table 2. Additional model input includes the net solute fluxes for metabolic species and the active bicarbonate flux [ 30 ], which were estimated by Leung et al. The effects of fixed charges on fluid pressure and solute concentrations, given by Eqs 51 , 52 , are illustrated in Fig 4a and 4b in the range of , where mM is the measured fixed charge concentration in corneal stroma [ 8 ].
On the other hand, the predicted anion concentrations chloride, bicarbonate and lactate ions show an opposite trend by decreasing with C f. The contrast between the curves of cation and anions is attributed to the Donnan effect, which requires accumulation of cations to compensate the concentration of negative fixed charges.
As expected, the calculated glucose concentration is invariant with C f since it is a neutral species. The effect of active bicarbonate flux on solute concentrations is presented in Fig 5a in the range of , where denotes the reported value of bicarbonate pumping flux see table 3. The chloride and lactate ion concentrations, on the other hand, increase mildly with.
The osmotic pressure, which is a measure of the total osmolarity including all the solutes, decreases as bicarbonate ions are pumped out from stroma to aqueous humor see Fig 5b. This result confirms the necessity of the active transport mechanism to maintain the hydration of the tissue.
Ionic Transport Processes: in Electrochemistry and Membrane Science
The proposed KK Eqs 31 , 40 present a general framework to describe coupled transport through biological membranes. This work may be viewed as an extension from Li [ 16 ], Hodson and Earlam [ 18 ] and Hoshiko and Lindley [ 14 ] by including the effect of fixed charges, multi-component solutes both ionic and non-ionic species and active transport mechanisms. While the derivations follow the standard practice of the KK theory [ 10 ], the key step is to take account of the fixed charge concentration. The theory resolves the difficulties of existing KK equations in predicting transmembrane fluid pressure for charged electrolyte solution.
The recovery of the fluid pressure and osmotic pressure in Donnan equilibrium provides verification of the proposed theory, and explains the swelling tendency of tissues like cornea with fixed charge inside with or without the presence of a bounding membrane, a phenomena that has been observed experimentally and explained qualitatively [ 18 ].
As water transport is primarily passive in biological systems [ 32 ], conventional analysis treats water as being effectively in thermodynamic equilibrium [ 3 ], with the water potential difference across membrane assumed to be zero. The analyses presented in the preceding two sections have shown that this may not be the case when active ionic transport processes are present. This implies that the values of the water potential across the separating membrane will not be equal. Specifically for the cornea, the pressure Eq 57 suggests a new interpretation of the stromal fluid pressure, which is an important quantity in corneal biomechanics because it interacts with the stromal collagen and modulates the fluid content of the tissue [ 21 ].
It assumes negative stromal fluid pressure, which contributes a fluid leak from the aqueous humor to the stroma, and negative osmotic pressure, which results in a fluid leak in the other direction. This view of a positive stromal fluid pressure is corroborated by a recent structural analysis of the cornea [ 21 ], which needs further experimental verification. An alternative way of interpretating the membrane transport is to view the conjugate forces as a result of mechanical friction among solute water and solutes [ 33 ].
It will be unity only if the membrane is strictly impermeable to solute species i. Extensions on the current work include two aspects. First, the new KK theory can be incorporated into continuum models for cells or tissues to describe the interactions between membrane transport and the bulk electrolyte inside. A multiphasic model for the living cornea that accounts for hydration, fixed charges and endothelial transport is under development. Second, it should be noted that the assumption of an independently specified active ionic flux in the above theory, for example in the preceding section, is employed for simplicity [ 24 ].
The consideration of active ionic flux, and its coupling with the solvent and solutes can be elaborated by taking into account the underlying metabolic reactions [ 9 ]. This development would require identification of the molecular mechanisms, but could potentially enhance the understanding of active mechanism involvement in membrane transport. Performed the experiments: XC.
Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract The movement of fluid and solutes across biological membranes facilitates the transport of nutrients for living organisms and maintains the fluid and osmotic pressures in biological systems. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Data Availability: All relevant data are within the paper and its Supporting Information files.
Introduction The exchange of fluid and solutes across biological membranes facilitates the transport of substances needed for living organisms to maintain their metabolic activities, and regulates pressure balances across bounding membranes to maintain the structural integrity of biological systems. Limitations of existing Kedem-Katchalsky theories In this section we describe a limitation of existing KK theories [ 12 , 16 ] which are unable to recover the Donnan equilibrium when fixed charges exist on one side of the membrane.
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Modified KK equations for passive transport Consider a leaky membrane that separates two electrolyte solutions as described above. The corresponding conjugate forces and are given as 17 18 Note that the sum of all volume fractions is unity, i. Application to equilibrium conditions: recovery of Donnan equilibrium In this section the modified KK Eqs 31 and 32 are used to study the Donnan equilibrium in which unequal distributions of ionic concentration and fluid pressure are developed between two ionic solutions separated by a membrane [ 17 ].
Extension to include the effect of active ionic transport Active transport across cell membranes enables solute movement against their concentration gradient and is one of the major factors for keeping homeostasis within the body. The net solute flux equation can then be written as 40 It is noted that Eq 40 recovers the solute flux equation as used in [ 21 , 24 , 25 ] to describe the effect of active ionic flux without considering the underlying molecular mechanisms. Fig 2. Application to endothelial transport in the in vivo cornea In this section an example application of Eqs 31 , 40 is presented to study the endothelial transport process of the in vivo human cornea see Fig 3.
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Fields values in the aqueous humor [ 25 ]. Table 2. Endothelial membrane properties [ 25 ]. Table 3. Reported solute fluxes across corneal endothelium [ 25 ]. Table 4. Comparisons of the solute concentrations and stationary fluid pressure at stromal-endothelial interface of the cornea by the new KK equations and that by [ 25 ]. Fig 4.
The effect of fixed charge concentration on a the stromal uid pressure and b the solute concentration differences across the corneal endothelium. Fig 5. Seller Inventory FU Brand new book, sourced directly from publisher.
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