Externally driven countercurrent multiplication in a mathematical model of the urinary concentrating mechanism of the renal inner medulla

Substitution of measured permeabilities into mathematical models of the concentrating mechanism of the renal inner medulla yields less than the known urine osmolalities. To gain a better understanding of the mechanism we analyse a model in which a force of unspecified origin [expressed as fraction, ɛ, of entering descending thin limb (DTL) concentration] drives fluid from DTL to interstitial vascular space (CORE), thus concentrating the solution in DTL. When flow in the DTL reverses at the hairpin bend of the loop of Henle, the high solute permeability of ascending thin limb (ATL) permits solute to diffuse into the CORE thus permitting ɛ to be multiplied many-fold. Behavior of the model is described by two non-linear differential equations. In the limit for infinite salt permeability of ATL the two equations reduce to a single equation that is formally identical with that for the Hargitay and Kuhn multiplier, which assumes fluid transport directly from DTL to ATL (Z. Electrochem. Angew. Phys. Chem. 55, 539, 1951). Solutions of the equations describing the model with parameters taken from perfused thin limbs show that urine osmolalities of the order of 5000 mosm L⁻¹ can be generated by forces of the order of 20 mosm L⁻¹. It seems probable that mammals including desert rodents use some variant of this basic mechanism for inner medullary concentration.     

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.     

A mathematical model of the urine concentrating mechanism in the rat renal medulla. II. Functional implications of three-dimensional architecture

In a companion study [Layton AT. A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol. (First published November 10, 2010). 10.1152/ajprenal.00203.2010] a region-based mathematical model was formulated for the urine concentrating mechanism in the renal medulla of the rat kidney. In the present study, we investigated model sensitivity to some of the fundamental structural assumptions. An unexpected finding is that the concentrating capability of this region-based model falls short of the capability of models that have radially homogeneous interstitial fluid at each level of only the inner medulla (IM) or of both the outer medulla and IM, but are otherwise analogous to the region-based model. Nonetheless, model results reveal the functional significance of several aspects of tubular segmentation and heterogeneity: 1) the exclusion of ascending thin limbs that reach into the deep IM from the collecting duct clusters in the upper IM promotes urea cycling within the IM; 2) the high urea permeability of the lower IM thin limb segments allows their tubular fluid urea content to equilibrate with the surrounding interstitium; 3) the aquaporin-1-null terminal descending limb segments prevent water entry and maintain the transepithelial NaCl concentration gradient; 4) a higher thick ascending limb Na(+) active transport rate in the inner stripe augments concentrating capability without a corresponding increase in energy expenditure for transport; 5) active Na(+) reabsorption from the collecting duct elevates its tubular fluid urea concentration. Model calculations predict that these aspects of tubular segmentation and heterogeneity promote effective urine concentrating functions.     

A region-based model framework for the rat urine concentrating mechanism

The highly structured organization of tubules and blood vessels in the outer medulla of the mammalian kidney is believed to result in preferential interactions among tubules and vessels; such interactions may promote solute cycling and enhance urine concentrating capability. In this study, we formulate a new model framework for the urine concentrating mechanism in the outer medulla of the rat kidney. The model simulates preferential interactions among tubules and vessels by representing two concentric regions and by specifying the fractions of tubules and vessels assigned to each of the regions. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model equations, which are derived in dynamic form, are solved to obtain steady-state solutions by means of a stable and efficient numerical method, based on the semi-Lagrangian semi-implicit method and on Newton’s method. In this application, the computational cost scales as O(N ²), where N is the number of spatial subintervals along the medulla. We present representative solutions and show that the method generates approximations that are second-order accurate in space and that exhibit mass conservation.     

Role of structural organization in the urine concentrating mechanism of an avian kidney

The organization of tubules and blood vessels in the quail medullary cone is highly structured. This structural organization may result in preferential interactions among tubules and vessels, interactions that may enhance urine concentrating capability. In this study, we formulate a model framework for the urine concentrating mechanism of the quail kidney. The model simulates preferential interactions among renal tubules by representing two concentric cores and by specifying the fractions of tubules assigned to each of the concentric cores. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model results suggest that the preferential interactions among tubules enhance the urine concentration capacity of short medullary cones by reducing the diluting effect of the descending limbs on the region of the interstitium where the collecting ducts are located; however, the effects on longer cones are unclear.     

Calculation of effective diffusivities for biofilms and tissues.

In this study we describe a scheme for numerically calculating the effective diffusivity of cellular systems such as biofilms and tissues. This work extends previous studies in which we developed the macroscale representations of the transport equations for cellular systems based on the subcellular-scale transport and reaction processes. A finite-difference model is used to predict the effective diffusivity of a cellular system on the basis of the subcellular-scale geometry and transport parameters. The effective diffusivity is predicted for a complex three-dimensional structure that is based on laboratory observations of a biofilm, and these numerical predictions are compared with predictions from a simple analytical solution and with experimental data. Our results indicate that, under many practical circumstances, the simple analytical solution can be used to provide reasonable estimates of the effective diffusivity.     

A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal medulla was used to investigate the significance of transport and structural properties revealed in anatomic studies. The model simulates preferential interactions among tubules and vessels by representing concentric regions that are centered on a vascular bundle in the outer medulla (OM) and on a collecting duct cluster in the inner medulla (IM). Particularly noteworthy features of this model include highly urea-permeable and water-impermeable segments of the long descending limbs and highly urea-permeable ascending thin limbs. Indeed, this is the first detailed mathematical model of the rat urine concentrating mechanism that represents high long-loop urea permeabilities and that produces a substantial axial osmolality gradient in the IM. That axial osmolality gradient is attributable to the increasing urea concentration gradient. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict that the interstitial NaCl and urea concentrations in adjoining regions differ substantially in the OM but not in the IM. In the OM, active NaCl transport from thick ascending limbs, at rates inferred from the physiological literature, resulted in a concentrating effect such that the intratubular fluid osmolality of the collecting duct increases ~2.5 times along the OM. As a result of the separation of urea from NaCl and the subsequent mixing of that urea and NaCl in the interstitium and vasculature of the IM, collecting duct fluid osmolality further increases by a factor of ~1.55 along the IM.     

A sparse matrix method for renal models

A sparse matrix method for the numerical solution of nonlinear differential equations arising in modeling of the renal concentrating mechanism is given. The method involves a renumbering of the variables and equations such that the resulting Jacobian matrix has a block tridiagonal structure and the blocks above and below the main diagonal have a known set of complementary nonzero columns. The computer storage for the method is O(n). Results of some numerical experiments showing the stability of the method are given.     

Stability properties of elementary dynamic models of membrane transport

Living cells are characterized by their capacity to maintain a stable steady state. For instance, cells are able to conserve their volume, internal ionic composition and electrical potential difference across the plasma membrane within values compatible with the overall cell functions. The dynamics of these cellular variables is described by complex integrated models of membrane transport. Some clues for the understanding of the processes involved in global cellular homeostasis may be obtained by the study of the local stability properties of some partial cellular processes. As an example of this approach, I perform, in this study, the neighborhood stability analysis of some elementary integrated models of membrane transport. In essence, the models describe the rate of change of the intracellular concentration of a ligand subject to active and passive transport across the plasma membrane of an ideal cell. The ligand can be ionic or nonionic, and it can affect the cell volume or the plasma membrane potential. The fundamental finding of this study is that, within the physiological range, the steady states are asymptotically stable. This basic property is a necessary consequence of the general forms of the expressions employed to describe the active and passive fluxes of the transported ligand.     

Breakthrough performance of linear-DNA on ion-exchange membrane columns

Breakthrough performance of linear-DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves were not significantly affected by flow-rate or inlet solution concentration. In the theoretical analysis a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine coupled to the solution of the partial differential model equations was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant and the forward interaction rate-constant, which are the parameters of the membrane transport model. Through this approach a good prediction of the adsorption phenomena is obtained for inlet concentrations and flow rates greater than 0.2 mg/ml and 0.16 ml/min.     

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.     

A numerical method for renal models that represent tubules with abrupt changes in membrane properties

 The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method. Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002 This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases, grant DK-42091.) Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45 Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit     

Role of UTB Urea Transporters in the Urine Concentrating Mechanism of the Rat Kidney

A mathematical model of the renal medulla of the rat kidney was used to investigate urine concentrating mechanism function in animals lacking the UTB urea transporter. The UTB transporter is believed to mediate countercurrent urea exchange between descending vasa recta (DVR) and ascending vasa recta (AVR) by facilitating urea transport across DVR endothelia. The model represents the outer medulla (OM) and inner medulla (IM), with the actions of the cortex incorporated via boundary conditions. Blood flow in the model vasculature is divided into plasma and red blood cell compartments. In the base-case model configuration tubular dimensions and transport parameters are based on, or estimated from, experimental measurements or immunohistochemical evidence in wild-type rats. The base-case model configuration generated an osmolality gradient along the cortico-medullary axis that is consistent with measurements from rats in a moderately antidiuretic state. When expression of UTB was eliminated in the model, model results indicated that, relative to wild-type, the OM cortico-medullary osmolality gradient and the net urea flow through the OM were little affected by absence of UTB transporter. However, because urea transfer from AVR to DVR was much reduced, urea trapping by countercurrent exchange was significantly compromised. Consequently, urine urea concentration and osmolality were decreased by 12% and 8.9% from base case, respectively, with most of the reduction attributable to the impaired IM concentrating mechanism. These results indicate that the in vivo urine concentrating defect in knockout mouse, reported by Yang et al. (J Biol Chem 277(12), 10633–10637, 2002), is not attributable to an OM concentrating mechanism defect, but that reduced urea trapping by long vasa recta plays a significant role in compromising the concentrating mechanism of the IM. Moreover, model results are in general agreement with the explanation of knockout renal function proposed by Yang et al.     

The drift approximation solves the Poisson, Nernst-Planck, and continuum equations in the limit of large external voltages

Nearly linear current-voltage curves are frequently found in biological ion channels. Using the drift limit of the substantially non-linear Poisson-Nernst-Planck equations, we explain such behavior of diffusion-controlled charge transport systems. Starting from Gauss' law, drift, and continuity equations we derive a simple analytical current-voltage relation, which accounts for this deviation from linearity. As shown previously, the drift limit of the Nernst-Planck equation applies if the total electric current is dominated by the electric field, and integral contributions from concentration gradients are small. The simple analytical form of the drift current-voltage relations makes it an ideal tool to analyze experiment current-voltage curves. We also solved the complete Poisson-Nernst-Planck equations numerically, and determined current-voltage curves over a wide range of voltages, concentrations, and Debye lengths. The simulation fully supports the analytical estimate that the current-voltage curves of simple charge transport systems are dominated by the drift mechanism. Even those relations containing the most extensive approximations remained qualitatively within the correct order of magnitude. Received: 24 September 1998 / Revised version: 22 January 1999 / Accepted: 22 January 1999     

Accounting for thermodynamic non-ideality in the Guinier region of small-angle scattering data of proteins

Hydrodynamic studies of the solution properties of proteins and other biological macromolecules are often hard to interpret when the sample is present at a reasonably concentrated solution. The reason for this is that solutions exhibit deviations from ideal behaviour which is manifested as thermodynamic non-ideality. The range of concentrations at which this behaviour typically is exhibited is as low as 1–2 mg/ml, well within the range of concentrations used for their analysis by techniques such as small-angle scattering. Here we discuss thermodynamic non-ideality used previously used in the context of light scattering and sedimentation equilibrium analytical ultracentrifugation and apply it to the Guinier region of small-angle scattering data. The results show that there is a complementarity between the radially averaged structure factor derived from small-angle X-ray scattering/small-angle neutron scattering studies and the second virial coefficient derived from sedimentation equilibrium analytical ultracentrifugation experiments.     

Random environments and stochastic calculus

This paper investigates the use of heuristically derived stochastic differential equations (SDEs) as models in population biology. It is stressed that these equations are best viewed as approximations for more realistic, but often analytically intractable, models. A criterion is presented for determining which interpretation (e.g., Ito or Stratonovich) is likely to serve as the most useful approximation. Several limit theorems are presented which illustrate the use and implications of this criterion. In particular, it is shown that the solutions to sequences of models which approach a given SDE may converge to a diffusion process which corresponds to no solution of the SDE. However, arguing that in population biology the SDEs are generally serving as approximations to stochastic difference equations with autocorrelated noise, it is shown for a variety of models that the Ito calculus may provide a more useful approximation than the Stratonovich. Important limitations of this result and on the use of SDEs are indicated. These findings and observations are compared with those of several papers in the recent literature.     

Structure-based classification of 45 FK506-binding proteins

The FK506-binding proteins (FKBPs) are a unique group of chaperones found in a wide variety of organisms. They perform a number of cellular functions including protein folding, regulation of cytokines, transport of steroid receptor complexes, nucleic acid binding, histone assembly, and modulation of apoptosis. These functions are mediated by specific domains that adopt distinct tertiary conformations. Using the Threading/ASSEmbly/Refinement (TASSER) approach, tertiary structures were predicted for a total of 45 FKBPs in 23 species. These models were compared with previously characterized FKBP solution structures and the predicted structures were employed to identify groups of homologous proteins. The resulting classification may be utilized to infer functional roles of newly discovered FKBPs. The three-dimensional conformations revealed that this family may have undergone several modifications throughout evolution, including loss of N- and C-terminal regions, duplication of FKBP domains as well as insertions of entire functional motifs. Docking simulations suggest that additional sequence segments outside FKBP domains may modulate the binding affinity of FKBPs to immunosuppressive drugs. The docking models also indicate the presence of a helix-loop-helix (HLH) region within a subset of FKBPs, which may be responsible for the interaction between this group of proteins and nucleic acids.     

Membranes with the same ion channel populations but different excitabilities

Electrical signaling allows communication within and between different tissues and is necessary for the survival of multicellular organisms. The ionic transport that underlies transmembrane currents in cells is mediated by transporters and channels. Fast ionic transport through channels is typically modeled with a conductance-based formulation that describes current in terms of electrical drift without diffusion. In contrast, currents written in terms of drift and diffusion are not as widely used in the literature in spite of being more realistic and capable of displaying experimentally observable phenomena that conductance-based models cannot reproduce (e.g. rectification). The two formulations are mathematically related: conductance-based currents are linear approximations of drift-diffusion currents. However, conductance-based models of membrane potential are not first-order approximations of drift-diffusion models. Bifurcation analysis and numerical simulations show that the two approaches predict qualitatively and quantitatively different behaviors in the dynamics of membrane potential. For instance, two neuronal membrane models with identical populations of ion channels, one written with conductance-based currents, the other with drift-diffusion currents, undergo transitions into and out of repetitive oscillations through different mechanisms and for different levels of stimulation. These differences in excitability are observed in response to excitatory synaptic input, and across different levels of ion channel expression. In general, the electrophysiological profiles of membranes modeled with drift-diffusion and conductance-based models having identical ion channel populations are different, potentially causing the input-output and computational properties of networks constructed with these models to be different as well. The drift-diffusion formulation is thus proposed as a theoretical improvement over conductance-based models that may lead to more accurate predictions and interpretations of experimental data at the single cell and network levels.     

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.