Concentrating engines and the kidney. III. Canonical mass balance equation for multinephron models of the renal medulla.

The canonical mass balance relation derived for the central core model of the renal medulla is extended to medullary models in which an arbitrary assemblage of renal tubules and vascular capillaries exchange with each other both directly and via the medullary interstitium and in which not all of the vascular loops or loops of Henle extend to the papilla. It is shown that if descending limbs of Henle and descending vasa recta enter the medulla at approximately plasma osmolality, the concentration ratio is given by: r = 1/[1 - ft(1 - fu)(1 - fw)], where ft is fractional solute transport out of ascending Henle's limb, fu is fractional urine flow, and fw is fractional dissipation; fw is a measure of the solute returned to the systemic circulation without its isotonic complement of water. A modified equation that applies to the diluting as well as the concentrating kidney is also derived. By allowing concentrations in interstitium and vascular capillaries to become identical at a given medullary level, conservation relations are derived for a multinephron central core model of the renal medulla.     

Application of the local-bulk partitioning and competitive binding models to interpret preferential interactions of glycine betaine and urea with protein surface

Two thermodynamic models have been developed to interpret the preferential accumulation or exclusion of solutes in the vicinity of biopolymer surface and the effects of these solutes on protein processes. The local-bulk partitioning model treats solute (and water) as partitioning between the region at/or near the protein surface (the local domain) and the bulk solution. The solvent exchange model analyzes a 1:1 competition between water and solute molecules for independent surface sites. Here we apply each of these models to interpret thermodynamic data for the interactions of urea and the osmoprotectant glycine betaine (N,N,N-trimethylglycine; GB) with the surface exposed in unfolding the marginally stable lacI HTH DNA binding domain. The partition coefficient K(P) quantifying accumulation of urea at this protein surface (K(P) approximately equal 1.1) is only weakly dependent on urea concentration up to 6 M urea. However, K(P) quantifying exclusion of GB from the vicinity of this protein surface increases from 0.83 (extrapolated to 0 M GB) to 1.0 (indicating that local and bulk GB concentrations are equal) at 4 M GB (activity > 40 M). We interpret the significant concentration dependence of K(P) for GB, predicted to be general for excluded, nonideal solutes such as GB, as a modest (8%) attenuation of the GB concentration dependence of solute nonideality in the local domain relative to that in the bulk solution. Above 4 M, K(P) for the interaction of GB with the surface exposed in protein unfolding is predicted to exceed unity, which explains the maximum in thermal stability observed for RNase and lysozyme at 4 M GB (Santoro, M. M., Liu, Y. F., Khan, S. M. A., Hou, L. X., and Bolen, D. W. (1992) Biochemistry 31, 5278-5283). Both thermodynamic models provide good two-parameter fits to GB and urea data for lacI HTH unfolding over a wide concentration range. The solute partitioning model allows for a full spectrum of attenuation effects in the local domain, encompasses the cases treated by the competitive binding model, and provides a somewhat better two-parameter fit of effects of high GB concentration on lacI HTH stability. Parameters of this fit should be applicable to isothermal and thermal unfolding data for all proteins with similar compositions of surface exposed in unfolding.     

Scaling phloem transport: information transmission

Sieve tubes are primarily responsible for the movement of solutes over long distances, but they also conduct information about the osmotic state of the system. Using a previously developed dimensionless model of phloem transport, the mechanism behind the sieve tube's capacity to rapidly transmit pressure/concentration waves in response to local changes in either membrane solute exchange or the magnitude and axial gradient of apoplastic water potential is demonstrated. These wave fronts can move several orders of magnitude faster than the solution itself when the sieve tube's axial pressure drop is relatively small. Unlike the axial concentration drop, the axial pressure drop at steady state is independent of the apoplastic water potential gradient. As such, the regulation of whole‐sieve tube turgor could play a vital role in controlling membrane solute exchange throughout the translocation pathway, making turgor a reliable source of information for communicating change in system state.     

Temperature dependence of the solubility of non-polar gases in water

An explanation is provided for the experimentally observed temperature dependence of the solubility and the solubility minimum of non-polar gases in water. The influence of solute size and solute-water attractive interactions on the solubility minimum temperature is investigated. The transfer of a non-polar solute from the ideal gas into water is divided into two steps: formation of a cavity in water with the size and shape of the solute and insertion of the solute in this cavity which is equivalent to 'turning on' solute-water attractive interactions. This two step process divides the excess chemical potential of the non-polar solute in water into repulsive and attractive contributions, respectively. The reversible work for cavity formation is modeled using an information theory model of hydrophobic hydration. Attractive contributions are calculated by modeling the water structure in the vicinity of non-polar solutes. These models make a direct connection between microscopic quantities and macroscopic observables. Moreover, they provide an understanding of the peculiar temperature dependences of the hydration thermodynamics from properties of pure water; specifically, bulk water density and the water oxygen-oxygen radial distribution function.     

The variance of the distribution of traversal times in a capillary bed

A random walk model of capillary tracer transit times is developed that treats simulataneously: plug flow in the capillary, radial and axial diffusion in the capillary cylinder and tissue annulus, and endothelial barriers to solute transport. The mean transit time is simply the volume of distribution divided by blood flow. Variance of transit times has additive terms for radial, axial, and barrier influences that are reduceable to variances of simpler models of capillary exchange. The dependence of variance on the solute diffusion coefficient is not monotonic, but has a minimum near 0·5 × 10?6 cm²/s for reasonable parameters and no barrier, Small molecules like inert gases are expected to have larger variances with higher diffusion coefficients, while larger molecules and barrier limited solutes will have the reverse dependence. Available literature data indicates that capillary heterogeneity will have a major influence on whole-body variance of transit times.     

Effects of metal ion and solute conformation change on hydration of small amino acid

The effects of metal ion and solute conformation change on the structures, energetic and dynamics of water molecules in the first hydration shell of amino acid were studied, using three forms of alanine (Ala) and Li(+)/Ala as model molecules. The theoretical investigations were started with construction of the test-particle model (T-model) potentials for all molecules involved and followed by molecular dynamics (MD) simulations of [Ala](aq) and [Li(+)/Ala](aq) at 298 K. The MD results showed that the hydrogen bond (H-bond) networks of water at the functional groups of Ala are strengthened by the metal ion binding, whereas the rotation of the N-C(alpha) bond from the angle phi=0 degrees to 180 degrees brings about smaller effects which cannot be generalized. It was also shown that the dynamics of water molecule in the first hydration shell of amino acid could be estimated from the total-average potential energy landscapes and the water exchange diagrams. The MD results suggested inclusion of an additional dynamic step in the water exchange process, in which water molecule moves inside a channel within the first hydration shell of solute, before leaving the channel at some point. The theoretical results reported in the present work iterated the necessity to include explicit water molecules in the model calculations.     

Analysis of time hierarchy in plant cell volume changes

A simple model of plant cell volume changes is presented. It is based on Kedem-Katchalsky equations for water and solute transport and on linear approximation of the dependence of intracellular hydrostatic pressure on the cell volume. Active transport of solute is also included. The time hierarchy within the system is analyzed by appropriate normalization of variables and by the assessment of the numerical values of model coefficients. The dynamics of the system comprises a slow process of solute exchange and a fast process of water transport. This explains the wellknown biphasic response of the cell volume to a sudden change in external conditions. An approximation of equations describing the system behaviour on the basis of the Tikhonov's theorem is proposed. The approximative solution is compared with the exact numerical solution of the original equations. The approximation is very good under physiological conditions, but it ceases to hold when the solute permeability of the cell membrane increases causing the breakdown of the entire time hierarchy within the system.     

Homogenization Theory for the Prediction of Obstructed Solute Diffusivity in Macromolecular Solutions

The study of diffusion in macromolecular solutions is important in many biomedical applications such as separations, drug delivery, and cell encapsulation, and key for many biological processes such as protein assembly and interstitial transport. Not surprisingly, multiple models for the a-priori prediction of diffusion in macromolecular environments have been proposed. However, most models include parameters that are not readily measurable, are specific to the polymer-solute-solvent system, or are fitted and do not have a physical meaning. Here, for the first time, we develop a homogenization theory framework for the prediction of effective solute diffusivity in macromolecular environments based on physical parameters that are easily measurable and not specific to the macromolecule-solute-solvent system. Homogenization theory is useful for situations where knowledge of fine-scale parameters is used to predict bulk system behavior. As a first approximation, we focus on a model where the solute is subjected to obstructed diffusion via stationary spherical obstacles. We find that the homogenization theory results agree well with computationally more expensive Monte Carlo simulations. Moreover, the homogenization theory agrees with effective diffusivities of a solute in dilute and semi-dilute polymer solutions measured using fluorescence correlation spectroscopy. Lastly, we provide a mathematical formula for the effective diffusivity in terms of a non-dimensional and easily measurable geometric system parameter.     

Water T2 relaxation in sugar solutions

1H spin-spin relaxation times of water were measured with the CPMG sequence in dilute aqueous solutions of glucitol, mannitol, glycerol, glycol, the methyl D-pyranosides of alpha-glucose, beta-glucose, alpha-galactose, beta-galactose, alpha-xylose, beta-xylose, beta-arabinose and sucrose, alpha,alpha-trehalose, beta-maltose, maltotriose and maltoheptaose. The relaxation-time dispersion was measured by varying the CPMG pulse spacing, tau. These data were interpreted by means of the Carver-Richards model in which exchange between water protons and labile solute hydroxyl protons provides a significant contribution to the relaxation. From the dependences on temperature and tau, parameters characteristic of the pool of hydroxyls belonging to a given solute were extracted by nonlinear regression, including: the fraction of exchangeable protons, P, the chemical-shift difference between water protons and hydroxyl protons, deltaomega, the intrinsic spin-spin relaxation time, T2, and the chemical exchange rate, k. These solute-specific parameters are related, respectively, to the concentration, identity, mobility and exchange life-time of the hydroxyl site. At 298 K, values of deltaomega, T2 and k were found to be of the order of 1 ppm, 100 ms and 1000 s(-1), respectively. Effects of molecular size, conformation and solute concentration were investigated. The exchange mechanism was characterised by Eyring activation enthalpies and entropies with values in the ranges 50-70 kJ mol(-1) and -10 to 60 J K(-1)mol(-1), respectively.     

Introducing Titratable Water to All-Atom Molecular Dynamics at Constant pH

Recent development of titratable coions has paved the way for realizing all-atom molecular dynamics at constant pH. To further improve physical realism, here we describe a technique in which proton titration of the solute is directly coupled to the interconversion between water and hydroxide or hydronium. We test the new method in replica-exchange continuous constant pH molecular dynamics simulations of three proteins, HP36, BBL, and HEWL. The calculated pK_a values based on 10-ns sampling per replica have the average absolute and root-mean-square errors of 0.7 and 0.9 pH units, respectively. Introducing titratable water in molecular dynamics offers a means to model proton exchange between solute and solvent, thus opening a door to gaining new insights into the intricate details of biological phenomena involving proton translocation.     

Measurements of gross and net ecosystem productivity and water vapour exchange of a Pinus ponderosa ecosystem,and an evaluation of two generalized models

Net ecosystem productivity (NEP), net primary productivity (NPP), and water vapour exchange of a mature Pinus ponderosa forest (44°30′ N, 121°37′ W) growing in a region subject to summer drought were investigated along with canopy assimilation and respiratory fluxes. This paper describes seasonal and annual variation in these factors, and the evaluation of two generalized models of carbon and water balance (PnET‐II and 3‐PG) with a combination of traditional measurements of NPP, respiration and water stress, and eddy covariance measurements of above‐and below‐canopy CO₂ and water vapour exchange. The objective was to evaluate the models using two years of traditional and eddy covariance measurements, and to use the models to help interpret the relative importance of processes controlling carbon and water vapour exchange in a water‐limited pine ecosystem throughout the year. PnET‐II is a monthly time‐step model that is driven by nitrogen availability through foliar N concentration, and 3‐PG is a monthly time‐step quantum‐efficiency model constrained by extreme temperatures, drought, and vapour pressure deficits. Both models require few parameters and have the potential to be applied at the watershed to regional scale. There was 2/3 less rainfall in 1997 than in 1996, providing a challenge to modelling the water balance, and consequently the carbon balance, when driving the models with the two years of climate data, sequentially. Soil fertility was not a key factor in modelling processes at this site because other environmental factors limited photosynthesis and restricted projected leaf area index to ～1.6. Seasonally, GEP and LE were overestimated in early summer and underestimated through the rest of the year. The model predictions of annual GEP, NEP and water vapour exchange were within 1–39% of flux measurements, with greater disparity in 1997 because soil water never fully recharged. The results suggest that generalized models can provide insights to constraints on productivity on an annual basis, using a minimum of site data.     

A Biomechanical Triphasic Approach to the Transport of Nondilute Solutions in Articular Cartilage

Biomechanical models for biological tissues such as articular cartilage generally contain an ideal, dilute solution assumption. In this article, a biomechanical triphasic model of cartilage is described that includes nondilute treatment of concentrated solutions such as those applied in vitrification of biological tissues. The chemical potential equations of the triphasic model are modified and the transport equations are adjusted for the volume fraction and frictional coefficients of the solutes that are not negligible in such solutions. Four transport parameters, i.e., water permeability, solute permeability, diffusion coefficient of solute in solvent within the cartilage, and the cartilage stiffness modulus, are defined as four degrees of freedom for the model. Water and solute transport in cartilage were simulated using the model and predictions of average concentration increase and cartilage weight were fit to experimental data to obtain the values of the four transport parameters. As far as we know, this is the first study to formulate the solvent and solute transport equations of nondilute solutions in the cartilage matrix. It is shown that the values obtained for the transport parameters are within the ranges reported in the available literature, which confirms the proposed model approach.     

Application of a fiber-matrix model to transport in renal tubules

The effects of tight junction structure on water and solute fluxes across proximal tubular epithelium were examined with fiber-matrix equations previously derived by Curry and Michel (1980. Microvascular Research. 20:96-99). Using plausible estimates of tight junction fiber length and width the model predicts solute (Ps) and water permeability (Lp) coefficients that agree with the measured values. When fiber-matrix and pore models were compared for physiologically relevant ranges of matrix void fraction (80-98%) and pore radii (0-20 A), the fiber-matrix model predicted a 10-fold higher Lp/Ps ratio. Lp/Ps was most sensitive to small changes in tight junction structure when void fractions exceeded 90%. Void fractions of 96.5% and 97.1% predicted previously measured values for Lp and solute permeabilities in rat and rabbit proximal tubules. These values are consistent with void fractions and permeabilities of artificial membranes. The fiber-matrix tight junction model was incorporated into a model of reabsorption from the rat proximal tubule developed by Weinstein (1984). American Journal of Physiology. 247:F848-F862.) A void fraction of 98% predicted the experimental results for isosmotic reabsorption driven by active transport. Changing void fraction over the range of 97-99% produced a 50-75% change in predicted volume reabsorption with active transport. According to the fiber-matrix model: (a) solute permeabilities alone cannot be used to predict Lp, (b) previously measured solute permeabilities in the proximal tubule are compatible with significant water reabsorption through a water-permeable tight junction, and (c) hydraulic and solute permeabilities may be sensitive to small changes in tight junction fiber length and diameter or ionic strength within the tight junction.     

Permeation of protons, potassium ions, and small polar molecules through phospholipid bilayers as a function of membrane thickness.

Two mechanisms have been proposed to account for solute permeation of lipid bilayers. Partitioning into the hydrophobic phase of the bilayer, followed by diffusion, is accepted by many for the permeation of water and other small neutral solutes, but transient pores have also been proposed to account for both water and ionic solute permeation. These two mechanisms make distinctively different predictions about the permeability coefficient as a function of bilayer thickness. Whereas the solubility-diffusion mechanism predicts only a modest variation related to bilayer thickness, the pore model predicts an exponential relationship. To test these models, we measured the permeability of phospholipid bilayers to protons, potassium ions, water, urea, and glycerol. Bilayers were prepared as liposomes, and thickness was varied systematically by using unsaturated lipids with chain lengths ranging from 14 to 24 carbon atoms. The permeability coefficient of water and neutral polar solutes displayed a modest dependence on bilayer thickness, with an approximately linear fivefold decrease as the carbon number varied from 14 to 24 atoms. In contrast, the permeability to protons and potassium ions decreased sharply by two orders of magnitude between 14 and 18 carbon atoms, and leveled off, when the chain length was further extended to 24 carbon atoms. The results for water and the neutral permeating solutes are best explained by the solubility-diffusion mechanism. The results for protons and potassium ions in shorter-chain lipids are consistent with the transient pore model, but better fit the theoretical line predicted by the solubility-diffusion model at longer chain lengths.     

Physiology of the interaction of angiosperm parasites and their higher plant hosts

Abstract. Interactions between parasitic angiosperms and their hosts occur at the level of seed germination, haustorial development and resource transfer. Chemicals released from the host function as cues for host recognition, and trigger germination as well as haustorial initiation. Transpiration is a key process regulating solute transfer from host to parasite, and some parasitie plants have unusual stomatal characteristics. Although solute transfer is apoplastic, the haustorium appears to play a role in regulating solute composition. Host responses to infection are reviewed, and it is concluded that competition for water and solutes are unlikely to play a major role in determining reductions in host productivity: metabolic incompatability is suggested to be the major cause of this.     

Integral equation theories for predicting water structure around molecules

Water plays a crucial role in the structure and function of proteins and other biological macromolecules; thus, theories of aqueous solvation for these molecules are of great importance. However, water is a complex solvent whose properties are still not completely understood. Statistical mechanical integral equation theories predict the density distribution of water molecules around a solute so that all particles are fully represented and thus potentially both molecular and macroscopic properties are included. Here we discuss how several theoretical tools we have developed have been integrated into an integral equation theory designed for globular macromolecular solutes such as proteins. Our approach predicts the three-dimensional spatial and orientational distribution of water molecules around a solute. Beginning with a three-dimensional Ornstein-Zernike equation, a separation is made between a reference part dependent only on the spatial distribution of solvent and a perturbation part dependent also on the orientational distribution of solvent. The spatial part is treated at a molecular level by a modified hypernetted chain closure whereas the orientational part is treated as a Boltzmann prefactor using a quasi-continuum theory we developed for solvation of simple ions. A potential energy function for water molecules is also needed and the sticky dipole models of water, such as our recently developed soft-sticky dipole (SSD) model, are ideal for the proposed separation. Moreover, SSD water is as good as or better than three point models typically used for simulations of biological macromolecules in structural, dielectric and dynamics properties and yet is seven times faster in Monte Carlo and four times faster in molecular dynamics simulations. Since our integral equation theory accurately predicts results from Monte Carlo simulations for solvation of a variety of test cases from a single water or ion to ice-like clusters and ion pairs, the application of this theory to biological macromolecules is promising.     

A mathematical model of solute coupled water transport in toad intestine incorporating recirculation of the actively transported solute

A mathematical model of an absorbing leaky epithelium is developed for analysis of solute coupled water transport. The non-charged driving solute diffuses into cells and is pumped from cells into the lateral intercellular space (lis). All membranes contain water channels with the solute passing those of tight junction and interspace basement membrane by convection-diffusion. With solute permeability of paracellular pathway large relative to paracellular water flow, the paracellular flux ratio of the solute (influx/outflux) is small (2-4) in agreement with experiments. The virtual solute concentration of fluid emerging from lis is then significantly larger than the concentration in lis. Thus, in absence of external driving forces the model generates isotonic transport provided a component of the solute flux emerging downstream lis is taken up by cells through the serosal membrane and pumped back into lis, i.e., the solute would have to be recirculated. With input variables from toad intestine (Nedergaard, S., E.H. Larsen, and H.H. Ussing, J. Membr. Biol. 168:241-251), computations predict that 60-80% of the pumped flux stems from serosal bath in agreement with the experimental estimate of the recirculation flux. Robust solutions are obtained with realistic concentrations and pressures of lis, and with the following features. Rate of fluid absorption is governed by the solute permeability of mucosal membrane. Maximum fluid flow is governed by density of pumps on lis-membranes. Energetic efficiency increases with hydraulic conductance of the pathway carrying water from mucosal solution into lis. Uphill water transport is accomplished, but with high hydraulic conductance of cell membranes strength of transport is obscured by water flow through cells. Anomalous solvent drag occurs when back flux of water through cells exceeds inward water flux between cells. Molecules moving along the paracellular pathway are driven by a translateral flow of water, i.e., the model generates pseudo-solvent drag. The associated flux-ratio equation is derived.     

A model of epithelial water transport. The corneal endothelium.

To try to understand how an epithelial tissue can transport water between bathing solutions of equal tonicity and how intracellular solute and protein concentration are related to the structural specialization of the cell membrane at its apical, basal, and lateral margins, we have formulated and solved, using approximate analytical techniques, a new model which combines the detailed transport of local osmotic flow in extracellular channel with the multicompartment approach of thermodynamic models requiring the overall conservation of water and solute for the entire cell layer. Thus, unlike most previous models, which dealt exclusively with either the average properties of the cell layer or the local transport in the extracellular channel, we are able to solve simultaneously for the interaction of the cell with its environments across its apical, basal, and lateral cell membranes as well as the detailed transport in the extracellular channel. The model is then applied to corneal endothelium to obtain new insight into the water flow movement in this tissue under in vitro and in vivo conditions. Then in vitro solution shows that the cell at 297 mosmol/liter is slightly hypotonic to the 300-mosmol/liter external bathing solutions which drive water equally out both the aqueous (apical) and stromal (basal) cell faces. This water is replaced from the extracellular channel. There is a net flow of water because more water enters the channel through its open stromal end than through the higher resistance tight junction. In vivo, the solution predicts that the stromal swelling pressure forces water through the tight junctions towards the stroma so that there is no net flow. The interesting new features of our solution are the water recirculation pattern and the role of the osmotically active proteins in making the cell hypertonic relative to the channel.     

Hydration and conformational equilibria of simple hydrophobic and amphiphilic solutes.

We consider whether the continuum model of hydration optimized to reproduce vacuum-to-water transfer free energies simultaneously describes the hydration free energy contributions to conformational equilibria of the same solutes in water. To this end, transfer and conformational free energies of idealized hydrophobic and amphiphilic solutes in water are calculated from explicit water simulations and compared to continuum model predictions. As benchmark hydrophobic solutes, we examine the hydration of linear alkanes from methane through hexane. Amphiphilic solutes were created by adding a charge of +/-1e to a terminal methyl group of butane. We find that phenomenological continuum parameters fit to transfer free energies are significantly different from those fit to conformational free energies of our model solutes. This difference is attributed to continuum model parameters that depend on solute conformation in water, and leads to effective values for the free energy/surface area coefficient and Born radii that best describe conformational equilibrium. In light of these results, we believe that continuum models of hydration optimized to fit transfer free energies do not accurately capture the balance between hydrophobic and electrostatic contributions that determines the solute conformational state in aqueous solution.