Mechanistic formalism for membrane transport generated by osmotic and mechanical pressure

Since the physical interpretation of practical Kedem-Katchalsky (KK) equations is not clear, we consider an alternative, mechanistic approach to membrane transport generated by osmotic and hydraulic pressure. We study a porous membrane with randomly distributed pore sizes (radii). We postulate that reflection coefficient (sigma p) of a single pore may equal 1 or 0. From this postulate we derive new (mechanistic) transport equations. Their advantage is in clear physical interpretation and since we show they are equivalent to the KK equations, the interpretation of the latter became clearer as well. Henceforth the equations allow clearer and more detailed interpretation of results concerning membrane mass transport. This is especially important from the point of view of biophysical studies on permeation processes across biological membranes, cell membranes including.     

Mechanistic approach to membrane mass transport processes (mini review)

Since the physical interpretation of practical Kedem-Katchalsky equations is not clear, we consider an alternative, mechanistic approach to membrane transport generated by osmotic and hydraulic pressure. We study a porous membrane with randomly distributed pore sizes (radii). We postulate that the reflection coefficient (sigma(p)) of a single pore may equal 1 or 0 only. From this postulate we derive new (mechanistic) transport equations. Their advantage is in clear physical interpretation.     

On the derivation of the Kargol's mechanistic transport equations from the Kedem-Katchalsky phenomenological equations

In the present article, it was demonstrated that--by starting from the so-called adjusted Kedem-Katchalsky (KK) phenomenological equations (Suchanek et al. 2004), i.e. the equations: Jv=LpDeltaP-LpDDeltaPi. JD=-LDpDeltaP+LDDeltaPi it is possible to derive practical transport equations (for the volume flow and the solute flow) in the form of the Kargol s mechanistic transport equations (Kargol and Kargol 2000, 2001, 2003a,b,c, 2004; Kargol 2002). On this basis, it has been found that the KK thermodynamic formalism for membrane transport (practical equations) is in general identical with the mechanistic equations for membrane transport.     

Membrane Transport Generated by the Osmotic and Hydrostatic Pressure. Correlation Relation for Parameters L p, σ, and ω

Standard approach to membrane transport generated by osmotic andhydrostatic pressures, developed by Kedem and Katchalsky, is based onprinciples of thermodynamics of irreversible processes. In this paper wepropose an alternative technique. We derive transport equations from fewfairly natural assumptions and a mechanistic interpretation of the flows.In particular we postulate that a sieve-type membrane permeability isdetermined by the pore sizes and these are random within certain range.Assuming that an individual pore is either permeable or impermeable tosolute molecules, the membrane reflection coefficient depends on the ratioof permeable and impermeable pores. Considering flows through permeableand impermeable pores separately, we derive equations for the total volumeflux, solute flux and the solvent flux across the membrane. Comparing themechanistic equations to the Kedem-Katchalsky equations we find the formereasier to interpret physically. Based on the mechanistic equations we alsoderive a correlation relation for the membrane transport parameters L _p,, and . This relation eliminates the need for experimentaldetermination of all three phenomenological parameters, which in somecases met with considerable difficulties.     

Constructing stochastic models from deterministic process equations by propensity adjustment

Background

Models of biochemical systems are typically complex, which may complicate the discovery of cardinal biochemical principles. It is therefore important to single out the parts of a model that are essential for the function of the system, so that the remaining non-essential parts can be eliminated. However, each component of a mechanistic model has a clear biochemical interpretation, and it is desirable to conserve as much of this interpretability as possible in the reduction process. Furthermore, it is of great advantage if we can translate predictions from the reduced model to the original model.

Results

In this paper we present a novel method for model reduction that generates reduced models with a clear biochemical interpretation. Unlike conventional methods for model reduction our method enables the mapping of predictions by the reduced model to the corresponding detailed predictions by the original model. The method is based on proper lumping of state variables interacting on short time scales and on the computation of fraction parameters, which serve as the link between the reduced model and the original model. We illustrate the advantages of the proposed method by applying it to two biochemical models. The first model is of modest size and is commonly occurring as a part of larger models. The second model describes glucose transport across the cell membrane in baker's yeast. Both models can be significantly reduced with the proposed method, at the same time as the interpretability is conserved.

Conclusions

We introduce a novel method for reduction of biochemical models that is compatible with the concept of zooming. Zooming allows the modeler to work on different levels of model granularity, and enables a direct interpretation of how modifications to the model on one level affect the model on other levels in the hierarchy. The method extends the applicability of the method that was previously developed for zooming of linear biochemical models to nonlinear models.     

The Balance of Fluid and Osmotic Pressures across Active Biological Membranes with Application to the Corneal Endothelium

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.     

Generalization of the Spiegler-Kedem-Katchalsky frictional model equations of the transmembrane transport for multicomponent non-electrolyte solutions.

The Spiegler-Kedem-Katchalsky frictional model equations of the transmembrane transport for systems containing n-component, non-ionic solutions is presented. The frictional interpretation of the phenomenological coefficients of membrane and the expressions connecting the practical coefficients (Lp, sigma i, omega ij) with frictional coefficients (fij) are presented.     

Single Water Channels of Aquaporin-1 Do not Obey the Kedem-Katchalsky Equations

The Kedem-Katchalsky (KK) equations are often used to obtain information about the osmotic properties and conductance of channels to water. Using human red cell membranes, in which the osmotic flow is dominated by Aquaporin-1, we show here that compared to NaCl the reflexion coefficient of the channel for methylurea, when corrected for solute volume exchange and for the water permeability of the lipid membrane, is 0.54. The channels are impermeable to these two solutes which would seem to rule out flow interaction and require a reflexion coefficient close to 1.0 for both. Thus, two solutes can give very different osmotic flow rates through a semi-permeable pore, a result at variance with both classical theory and the KK formulation. The use of KK equations to analyze osmotic volume changes, which results in a single hybrid reflexion coefficient for each solute, may explain the discrepancy in the literature between such results and those where the equations have not been employed. Osmotic reflexion coefficients substantially different from 1.0 cannot be ascribed to the participation of other 'hidden' parallel aqueous channels consistently with known properties of the membrane. Furthermore, we show that this difference cannot be due to second-order effects, such as a solute-specific interaction with water in only part of the channel, because the osmosis is linear with driving force down to zero solute concentration, a finding which also rules out the involvement of unstirred-layer effects. Reflexion coefficients smaller than 1.0 do not necessitate water-solute flow interaction in permeable aqueous channels; rather, the osmotic behaviour of impermeable molecular-sized pores can be explained by differences in the fundamental nature of water flow in regions either accessible or inaccessible to solute, created by a varying cross-section of the channel.     

Modification of the Kedem-Katchalsky equations

The presented modification of the transport equations of Kedem-Katchalsky resulted in the introduction of (omega s/omega) and omega/(omega-Lp sigma[(1-sigma)C1-(1-sigma s)C2]) factors into the Kedem-Katchalsky equations. The above factors determine the influence of boundary layers on transport across the membrane. The modified Kedem-Katchalsky equations were verified for synthetic membranes and it was shown that the value of the (omega s/omega) factor depended on the type of membrane and the membrane configuration system. This modification facilitated a wider range of application of the Kedem-Katchalsky equations to systems in which the solutions were stirred or unstirred.     

Membrane translocation of folded proteins

An ever-increasing number of proteins have been shown to translocate across various membranes of bacterial as well as eukaryotic cells in their folded states as a part of physiological and/or pathophysiological processes. Herein, we provide an overview of the systems/processes that are established or likely to involve the membrane translocation of folded proteins, such as protein export by the twin-arginine translocation system in bacteria and chloroplasts, unconventional protein secretion and protein import into the peroxisome in eukaryotes, and the cytosolic entry of proteins (e.g., bacterial toxins) and viruses into eukaryotes. We also discuss the various mechanistic models that have previously been proposed for the membrane translocation of folded proteins including pore/channel formation, local membrane disruption, membrane thinning, and transport by membrane vesicles. Finally, we introduce a newly discovered vesicular transport mechanism, vesicle budding and collapse, and present evidence that vesicle budding and collapse may represent a unifying mechanism that drives some (and potentially all) of folded protein translocation processes.     

Derivation of unstirred-layer transport number equations from the Nernst-Planck flux equations.

Since the late 1960s it has been known that the passage of current across a membrane can give rise to local changes in salt concentration in unstirred layers or regions adjacent to that membrane, which in turn give rise to the development of slow transient diffusion potentials and osmotic flows across those membranes. These effects have been successfully explained in terms of transport number discontinuities at the membrane-solution interface, the transport number of an ion reflecting the proportion of current carried by that ion. Using the standard definitions for transport numbers and the regular diffusion equations, these polarization or transport number effects have been analyzed and modeled in a number of papers. Recently, the validity of these equations has been questioned. This paper has demonstrated that, by going back to the Nernst-Planck flux equations, exactly the same resultant equations can be derived and therefore that the equations derived directly from the transport number definitions and standard diffusion equations are indeed valid.     

Thermodynamic calculations for biochemical transport and reaction processes in metabolic networks

Thermodynamic analysis of metabolic networks has recently generated increasing interest for its ability to add constraints on metabolic network operation, and to combine metabolic fluxes and metabolite measurements in a mechanistic manner. Concepts for the calculation of the change in Gibbs energy of biochemical reactions have long been established. However, a concept for incorporation of cross-membrane transport in these calculations is still missing, although the theory for calculating thermodynamic properties of transport processes is long known. Here, we have developed two equivalent equations to calculate the change in Gibbs energy of combined transport and reaction processes based on two different ways of treating biochemical thermodynamics. We illustrate the need for these equations by showing that in some cases there is a significant difference between the proposed correct calculation and using an approximative method. With the developed equations, thermodynamic analysis of metabolic networks spanning over multiple physical compartments can now be correctly described.     

Lumenal interactions in nuclear pore complex assembly and stability

Nuclear pore complexes (NPCs) provide a gateway for the selective transport of macromolecules across the nuclear envelope (NE). Although we have a solid understanding of NPC composition and structure, we do not have a clear grasp of the mechanism of NPC assembly. Here, we demonstrate specific defects in nucleoporin distribution in strains lacking Heh1p and Heh2p-two conserved members of the LEM (Lap2, emerin, MAN1) family of integral inner nuclear membrane proteins. These effects on nucleoporin localization are likely of functional importance as we have defined specific genetic interaction networks between HEH1 and HEH2, and genes encoding nucleoporins in the membrane, inner, and outer ring complexes of the NPC. Interestingly, expression of a domain of Heh1p that resides in the NE lumen is sufficient to suppress both the nucleoporin mislocalization and growth defects in heh1Δpom34Δ strains. We further demonstrate a specific physical interaction between the Heh1p lumenal domain and the massive cadherin-like lumenal domain of the membrane nucleoporin Pom152p. These findings support a role for Heh1p in the assembly or stability of the NPC, potentially through the formation of a lumenal bridge with Pom152p.     

Membrane Permeability Modeling: Kedem–Katchalsky vs a Two-Parameter Formalism

The analysis of experiments for the purpose of determining cell membrane permeability parameters is often done using the Kedem–Katchalsky (KK) formalism (1958). In this formalism, three parameters, the hydraulic conductivity (L_p), the solute permeability (P_s), and a reflection coefficient (ς), are used to characterize the membrane. Sigma was introduced to characterize flux interactions when water and solute (cryoprotectant) cross the membrane through a common channel. However, the recent discovery and characterization of water channels (aquaporins) in biological membranes reveals that aquaporins are highly selective for water and do not typically cotransport cryoprotectants. In this circumstance, sigma is a superfluous parameter, as pointed out by Kedem and Katchalsky. When sigma is unneeded, a two-parameter model (2P) utilizing onlyL_pandP_sis sufficient, simpler to implement, and less prone to spurious results. In this paper we demonstrate that the 2P and KK formalism yield essentially the same result (L_pandP_s) when cotransporting channels are absent. This demonstration is accomplished using simulation techniques to compare the transport response of a model cell using a KK or 2P formalism. Sigma is often misunderstood, even when its use is appropriate. It is discussed extensively here and several simulations are used to illustrate and clarify its meaning. We also discuss the phenomenological nature of transport parameters in many experiments, especially when both bilayer and channel transport are present.     

Equations for membrane transport. Experimental and theoretical tests of the frictional model.

Frictional models for membrane transport are tested experimentally and theoretically for the simple case of a solution consisting of a mixture of two perfect gases and a membrane consisting of a porous graphite septum. Serious disagreement is found, which is traced to a missing viscous term. Kinetic theory is then used as a guide in formulating a corrected set of transport equations, and in giving a physical interpretation to the frictional coefficients. Sieving effects are found to be attributable to entrance effects rather than to true frictional effects within the body of the membrane. The results are shown to be compatible with nonequilibrium thermodynamics. Some correlations and predictions are made of the behavior of various transport coefficients for general solutions.     

pH-Dependent Interactions Guide the Folding and Gate the Transmembrane Pore of the β-Barrel Membrane Protein OmpG

The physical interactions that switch the functional state of membrane proteins are poorly understood. Previously, the pH-gating conformations of the β-barrel forming outer membrane protein G (OmpG) from Escherichia coli have been solved. When the pH changes from neutral to acidic the flexible extracellular loop L6 folds into and closes the OmpG pore. Here, we used single-molecule force spectroscopy to structurally localize and quantify the interactions that are associated with the pH-dependent closure. At acidic pH, we detected a pH-dependent interaction at loop L6. This interaction changed the (un)folding of loop L6 and of β-strands 11 and 12, which connect loop L6. All other interactions detected within OmpG were unaffected by changes in pH. These results provide a quantitative and mechanistic explanation of how pH-dependent interactions change the folding of a peptide loop to gate the transmembrane pore. They further demonstrate how the stability of OmpG is optimized so that pH changes modify only those interactions necessary to gate the transmembrane pore.     

Osmotic flow equations for leaky porous membranes

A basic set of equations describing the flows of volume (Jv) and solute (Js) across a leaky porous membrane, coupled to the differences of osmotic and hydrostatic pressures d pi and dP has been derived by using general frictional theory. Denoting the mean pore concentration of solute by c*s and the hydraulic and diffusive conductances by Lp and Ps/RT the equations take the form Jv = LpdP + sigma sLp d pi Js = c*s(1 - sigma f)Jv + Ps d pi/RT sigma s = theta (1 - DsVs/DwVw - Ds/Dos) sigma f = 1 - theta DsVs/DwVw - Ds/Dos in which Dw and Ds are the diffusion coefficients for water and solute in the pore and Dos that for free solution. The relation between the reflection coefficients sigma s and sigma f for osmosis and ultrafiltration is then given by sigma s = sigma f - (1- theta)(1 - Ds/Dos), where theta is the diffusive-driven:pressure-driven flow ratio. These equations follow from the fact that in leaky pores osmosis occurs by diffusion alone and that there cannot be any Onsager symmetry leading to sigma s = sigma f. Symmetry holds in the limits where either the pore is small, when sigma s = sigma f = 1, or where the pore is large when sigma s = sigma f = 0.     

Effect of heteroporosity on flux equations for membranes

An investigation is made of the possible errors in simple integrated equations for solute flux across both non-pieving and sieving porous membranes that can result from variations in the membrane structure. Detailed structural models are used, beginning with a membrane consisting of a parallel array of pores and progressing to series-parallel combinations of pore segments of various lengths and cross-sectional areas, with internal cross connections among pore segments allowed. It is shown that there are both upper and lower mathematical bounds on the possible variations that can be produced in a curve of solute flux versus volume flow by arbitrary variation in the membrane structure, subject only to certain general conditions. In particular, the flux equation for a homoporous membrane is a lower bound- The maximum deviations from this lower bound for a membrane of arbitrary structure are only moderately large, and require rather extreme pore size distributions; most distributions introduce only small errors. Implications of these results in studies of real membrane structure and in the design of experiments are discussed.     

Polyol permeability of the human red cell. Interpretation of glucose transport in terms of a pore

The kinetic equations describing transport through a pore that has a binding site and that undergoes a conformational change are identical to those of a carrier model. Therefore, in order to distinguish between the two models it is necessary to test specific predictions based on detailed mechanistic models. A pore model is described in which the substrate (glucose) is able to reach the single binding site only from the outside when the pore is in conformation I and only from the inside when it is conformation II. On the basis of this model it is predicted that solutes which do not have any specific affinity for the binding site should still have a finite permeability via the glucose transport system if they are the same size or smaller than glucose. This permeability should be proportional to the volume of distribution of the solute in the pore and should therefore decrease with increasing molecular size. A geometric pore volume can be estimated from this size dependence. In order to test these predictions, the glucose-dependent permeability of a series of 4-carbon (erythritol), 5-carbon (d-arabitol, l-arabitol and xylitol) and 6-carbon (d-mannitol, d-sorbitol and $\text{myo-}\text{inositol}$) polyols was measured. The permeability of all the polyols is decreased by the presence of glucose and the $\text{K}{}_{\mathrm{<mtext>I</mtext>}}$ of this “inhibitable” component is similar to that of d-sorbose, suggesting that this component is associated with the glucose transport system. Since these observations could be explained entirely in terms of a specific affinity for a carrier binding site, they do not exclude a carrier mechanism. However, as predicted for the pore model, this “inhibitable” permeability decreased with increasing molecular size and the calculated geometric pore volume was of a size that would be expected for a cell membrane pore.