Date Published: December 31, 2015
Publisher: Public Library of Science
Author(s): Xi Cheng, Peter M. Pinsky, Paul J Atzberger.
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. Understanding the pressure balances across membranes is crucial for studying fluid and electrolyte homeostasis in living systems, and is an area of active research. In this study, a set of enhanced Kedem-Katchalsky (KK) equations is proposed to describe fluxes of water and solutes across biological membranes, and is applied to analyze the relationship between fluid and osmotic pressures, accounting for active transport mechanisms that propel substances against their concentration gradients and for fixed charges that alter ionic distributions in separated environments. The equilibrium analysis demonstrates that the proposed theory recovers the Donnan osmotic pressure and can predict the correct fluid pressure difference across membranes, a result which cannot be achieved by existing KK theories due to the neglect of fixed charges. The steady-state analysis on active membranes suggests a new pressure mechanism which balances the fluid pressure together with the osmotic pressure. The source of this pressure arises from active ionic fluxes and from interactions between solvent and solutes in membrane transport. We apply the proposed theory to study the transendothelial fluid pressure in the in vivo cornea, which is a crucial factor maintaining the hydration and transparency of the tissue. The results show the importance of the proposed pressure mechanism in mediating stromal fluid pressure and provide a new interpretation of the pressure modulation mechanism in the in vivo cornea.
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. The movement of these substances is controlled by both passive and active transport processes. Passive transport mechanisms drive water or solutes to move down their concentration gradients without need of energy input, whereas active transport mechanisms propels solutes to move against their concentration gradients at the cost of energy input from metabolic reactions. The interplay between the two mechanisms determines the fluid hydrostatic pressure and osmotic pressure differences across biological membranes, which are important characteristics for biological systems. For example, at the organ level, fluid pressure mediates fluid transport between capillaries and tissues, and facilitates the diffusion of nutrients between the two compartments . At the cellular level, fluid pressure interacts with osmotic pressure to regulate cell volumes at normal state [2, 3] and to drive its shape changes during processes such as protrusion and blebbing [4, 5]. A disturbance of the pressure regulation mechanisms can lead to swelling or shrinking of cells and tissues. A quantitative understanding of fluid and osmotic pressures in living organisms is crucial for studying biological mechanisms such as cell volume regulation and interstitial fluid homeostasis, and is under investigation for various biological systems, e.g. [2, 3, 6, 7]. The goal of the present work is to develop a mathematical description of water and solute transport across membranes and apply it to study the pressure balance conditions in biological systems, characterizing passive and active transport mechanisms and other biological features.
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. Consider a biological membrane that separates two polyelectrolyte solutions with fluid pressure P and P0, solute concentrations Ci and Ci0(i=1,…,N), where N denotes the number of species, and electrostatic potential φ and φ0. We denote one side of the membrane as “inside” and the other as “outside” (see Fig 1) and assume the inside electrolyte solution contains large molecules that carry fixed charges with concentration Cf and valence value zf. The fixed charges are assumed to be “trapped” in the inside solution and the biological membrane is assumed to be impermeable to large molecules . Both solvent and solutes are considered to have finite permeabilities through the membrane (i.e. the membrane is leaky). The classical KK equations  describe the volume flux JV and solute flux Ji(i = 1, …, N) between the two solutions as follows:
where Lp is the hydraulic conductivity, σi and ωi are the reflection coefficient and permeability for species i, respectively, and ΔP and ΔCi are the fluid pressure difference and ionic concentration difference across the membrane, respectively. C¯i denotes the mean ionic concentration, and can be simplified as the arithmetic mean between Ci and Ci0 (i.e. C¯i=(Ci+Ci0)/2). Consider the equilibrium condition in which no fluid flow and no ionic fluxes exist across the membrane, i.e. JV = Ji = 0, Eqs (1, 2) immediately give
which suggests that at equilibrium, ionic concentrations will be balanced and there will be no fluid pressure difference across the membrane. This conclusion is apparently contradicted by the well-known Donnan equilibrium where fixed charges induce imbalance of ionic concentrations and develop an osmotic pressure gradient between the inside and outside environments . This limitation of Eqs (1, 2) is attributed to the fact that they were developed for transport of non-ionic species . Li  derived an extended set of KK equations which incorporate the electrostatic potential difference between separated electrolyte solutions,
Consider a leaky membrane that separates two electrolyte solutions as described above. The movement of solvent and solutes across the membrane can be characterized by a set of fluxes and conjugate forces according to nonequilibrium thermodynamics . The identifications of these quantities are based on the statement of dissipation function Φ, which describes the rate that the free energy is dissipated during transport. Its mathematical formulation is given as [9, 10]
where Jw and Ji (i = 1, …, N) denote fluid flow and solute fluxes, respectively, and Xw and Xi denote the corresponding conjugate forces for fluid and solutes. In their simplest form, Xw and Xi can be written as the electrochemical potential difference across the membrane [10, 12], i.e.
where μw and μi are the electrochemical potential for fluid and solute species i, respectively. In order to derive the mathematical forms for Δμw and Δμi, we first recall the electrochemical potential for water and ions as:
where νw and νi in m3/mol are the partial volume of solvent and solutes, respectively, zi is the valence number for species i, R, T and F are gas constant, temperature and Faraday constant, respectively. The linearized forms for Δμw and Δμi are then given as
where C¯i=(Ci+Ci0)/2 is the average ionic concentration through the membrane.
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 . At thermodynamic equilibrium, no macroscopic flow of fluid or solutes occurs between the two solutions, i.e. JV = Ji = 0, and the dissipation function is zero (see Eq (9)). Applying the KK theory to study this classical condition provides the baseline for the study of the effects of active fluxes. We show that the new theory recovers the Donnan osmotic pressure, and predicts the correct fluid pressure that is required to balance the osmotic pressure. The predicted pressure quantities do not depend on membrane transport properties, indicating that the Donnan equilibrium will be satisfied regardless of the presence of a biological membrane .
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. It is divided into two types, according to the source of energy used, called primary active transport and secondary active transport. In the former category, energy is directly provided by the breakdown of adenosine triphosphate (ATP). In the latter category, energy is derived indirectly from energy stored in the form of ionic concentration differences between the two sides of a membrane. Directly modeling active mechanisms requires identifications of the reaction kinetics, which is only known for a few processes [22, 23]. Alternatively, active mechanisms can be incorporated into the nonequilibrium thermodynamic description of fluid and ionic transport by introducing the affinity of the driving metabolic reaction and its conjugate flux, the rate of reaction per unit membrane area [10, 14] (see details in S1 Appendix A). In the simplest form, the active ionic flux is treated as an independent term that is additive to the passive solute flux equation Eq (32). The net solute flux equation can then be written as
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].
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 , Hodson and Earlam  and Hoshiko and Lindley  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 , 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 .