Anita T. Layton

Modeling vesicle traffic reveals unexpected consequences for Cdc42p-mediated polarity establishment

BACKGROUND Polarization in yeast has been proposed to involve a positive feedback loop whereby the polarity regulator Cdc42p orients actin cables, which deliver vesicles carrying Cdc42p to the polarization site. Previous mathematical models treating Cdc42p traffic as a membrane-free flux suggested that directed traffic would polarize Cdc42p, but it remained unclear whether Cdc42p would become polarized without the membrane-free simplifying assumption. RESULTS We present mathematical models that explicitly consider stochastic vesicle traffic via exocytosis and endocytosis, providing several new insights. Our findings suggest that endocytic cargo influences the timing of vesicle internalization in yeast. Moreover, our models provide quantitative support for the view that integral membrane cargo proteins would become polarized by directed vesicle traffic given the experimentally determined rates of vesicle traffic and diffusion. However, such traffic cannot effectively polarize the more rapidly diffusing Cdc42p in the model without making additional assumptions that seem implausible and lack experimental support. CONCLUSIONS Our findings suggest that actin-directed vesicle traffic would perturb, rather than reinforce, polarization in yeast.

High-order multi-implicit spectral deferred correction methods for problems of reactive flow

Models for reacting flow are typically based on advection-diffusion-reaction (A-D-R) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three sub-processes can be widely different, leading to disparate time-step requirements for robust and accurate time-integration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this time-scale disparity. The proposed methods are high-order multi-implicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of A-D-R equations. Spectral deferred correction methods compute a high-order approximation to the solution of a differential equation by using a simple, low-order numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several sub-processes implicitly but independently, while avoiding the splitting errors present in traditional operator-splitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semi-implicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the sub-process time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific time-scales ordering, the generalization to any ordering combination is straightforward.

A numerically efficient and stable algorithm for animating water waves

Water motion can be realistically captured by physically based fluid models. We begin by presenting a survey on fluid simulation models that are based on fluid dynamics equations, from the most comprehensive Navier–Stokes equations to the simple wave equation. We then present a model that is based on the two-dimensional shallow water equations. The equations are integrated by a novel numerical method – the implicit semi-Lagrangian integration scheme – which allows large timesteps while maintaining stability, and which is described in detail in this paper. Gentle wave motions, the superposition of waves, drifting objects, and obstacles and boundaries of various shapes can be efficiently simulated with this model.

Role of three-dimensional architecture in the urine concentrating mechanism of the rat renal inner medulla

Recent studies of three-dimensional architecture of rat renal inner medulla (IM) and expression of membrane proteins associated with fluid and solute transport in nephrons and vasculature have revealed structural and transport properties that likely impact the IM urine concentrating mechanism. These studies have shown that 1) IM descending thin limbs (DTLs) have at least two or three functionally distinct subsegments; 2) most ascending thin limbs (ATLs) and about half the ascending vasa recta (AVR) are arranged among clusters of collecting ducts (CDs), which form the organizing motif through the first 3-3.5 mm of the IM, whereas other ATLs and AVR, along with aquaporin-1-positive DTLs and urea transporter B-positive descending vasa recta (DVR), are external to the CD clusters; 3) ATLs, AVR, CDs, and interstitial cells delimit interstitial microdomains within the CD clusters; and 4) many of the longest loops of Henle form bends that include subsegments that run transversely along CDs that lie in the terminal 500 microm of the papilla tip. Based on a more comprehensive understanding of three-dimensional IM architecture, we distinguish two distinct countercurrent systems in the first 3-3.5 mm of the IM (an intra-CD cluster system and an inter-CD cluster system) and a third countercurrent system in the final 1.5-2 mm. Spatial arrangements of loop of Henle subsegments and multiple countercurrent systems throughout four distinct axial IM zones, as well as our initial mathematical model, are consistent with a solute-separation, solute-mixing mechanism for concentrating urine in the IM.

Mechanistic mathematical model of polarity in yeast

We use a novel mathematical modeling framework to dissect the contributions of diffusive transport and directed vesicular transport in polarity establishment. The findings call into question the role of vesicle-mediated Cdc42 traffic in polarity maintenance.

A mathematical model of O2 transport in the rat outer medulla. I. Model formulation and baseline results

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O(2) supply to the renal medulla barely exceeds its O(2) requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O(2) distribution. We extended the region-based mathematical model of the rat OM developed by Layton and Layton (Am J Physiol Renal Physiol 289: F1346-F1366, 2005) to incorporate the transport of RBCs, Hb, and O(2). We considered basal cellular O(2) consumption and O(2) consumption for active transport of NaCl across medullary thick ascending limb epithelia. Our model predicts that the structural organization of the OM results in significant Po(2) gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O(2), at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po(2) between regions, limits O(2) reabsorption from long descending vasa recta, and helps preserve O(2) delivery to the inner medulla. Under baseline conditions, significantly more O(2) is transferred radially between regions by capillary flow, i.e., advection, than by diffusion. In agreement with experimental observations, our results suggest that 79% of the O(2) supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

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.

Functional implications of the three-dimensional architecture of the rat renal inner medulla

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal inner medulla (IM) was used to investigate the significance of transport and structural properties revealed in recent studies that employed immunohistochemical methods combined with three-dimensional computerized reconstruction. The model simulates preferential interactions among tubules and vessels by representing two concentric regions. The inner region, which represents a collecting duct (CD) cluster, contains CDs, some ascending thin limbs (ATLs), and some ascending vasa recta; the outer region, which represents the intercluster region, contains descending thin limbs, descending vasa recta, remaining ATLs, and additional ascending vasa recta. In the upper portion of the IM, the model predicts that interstitial Na(+) and urea concentrations (and osmolality) in the CD clusters differ significantly from those in the intercluster regions: model calculations predict that those CD clusters have higher urea concentrations than the intercluster regions, a finding that is consistent with a concentrating mechanism that depends principally on the mixing of NaCl from ATLs and urea from CDs. In the lower IM, the model predicts that limited or nearly zero water permeability in descending thin limb segments will increase concentrating effectiveness by increasing the rate of solute-free water absorption. The model predicts that high urea permeabilities in the upper portions of ATLs and increased contact areas of longest loop bends with CDs both modestly increase concentrating capability. A surprising finding is that the concentrating capability of this region-based model falls short of the capability of a model IM that has radially homogeneous interstitial fluid at each level but is otherwise analogous to the region-based model.

The Mammalian Urine Concentrating Mechanism: Hypotheses and Uncertainties

The urine concentrating mechanism of the mammalian kidney, which can produce a urine that is substantially more concentrated than blood plasma during periods of water deprivation, is one of the enduring mysteries in traditional physiology. Owing to the complex lateral and axial relationships of tubules and vessels, in both the outer and inner medulla, the urine concentrating mechanism may only be fully understood in terms of the kidneys three-dimensional functional architecture and its implications for preferential interactions among tubules and vessels.

Modeling oxygen consumption in the proximal tubule: effects of NHE and SGLT2 inhibition.

The objective of this study was to investigate how physiological, pharmacological, and pathological conditions that alter sodium reabsorption (TNa) in the proximal tubule affect oxygen consumption (QO2 ) and Na(+) transport efficiency (TNa/QO2 ). To do so, we expanded a mathematical model of solute transport in the proximal tubule of the rat kidney. The model represents compliant S1, S2, and S3 segments and accounts for their specific apical and basolateral transporters. Sodium is reabsorbed transcellularly, via apical Na(+)/H(+) exchangers (NHE) and Na(+)-glucose (SGLT) cotransporters, and paracellularly. Our results suggest that TNa/QO2 is 80% higher in S3 than in S1-S2 segments, due to the greater contribution of the passive paracellular pathway to TNa in the former segment. Inhibition of NHE or Na-K-ATPase reduced TNa and QO2 , as well as Na(+) transport efficiency. SGLT2 inhibition also reduced proximal tubular TNa but increased QO2 ; these effects were relatively more pronounced in the S3 vs. the S1-S2 segments. Diabetes increased TNa and QO2 and reduced TNa/QO2 , owing mostly to hyperfiltration. Since SGLT2 inhibition lowers diabetic hyperfiltration, the net effect on TNa, QO2 , and Na(+) transport efficiency in the proximal tubule will largely depend on the individual extent to which glomerular filtration rate is lowered.