Click on any of these titles to go directly to the corresponding section: To see publications that appeared during this period: Publications 2000


This report describes work carried out during the 2000 fiscal year in the Mathematical Research Branch, NIDDK. Our work consists of theoretical modeling of systems of biomedical interest, carried out in collaboration with experimental scientists at NIH and elsewhere.

Comings and Goings: Greg Smith left to join the Mathematics Department at Arizona State University. Victor Matveev came from the laboratory of Xiao-Jing Wang at Brandeis to take up a position as a Fogarty Visiting Associate. Chip Zimliki (JHU) joined as a Special Volunteer to complete the theoretical component of his Ph. D. thesis with A. Sherman. Julie Stern was here as a summer student, and former Chief, John Rinzel, spent the summer in residence.

Meetings: R. Mejia organized, with M. Marcano of the University of Puerto Rico, a mini-symposium on Mathematical Models in Physiology, at the SIAM Annual Meeting, July 10-14, 2000 in Rio Mar, PR. A. Sherman was an invited speaker at workshops on "The Nonlinear Dynamics of Calcium in Living Organisms" (LANL, Santa Fe NM), March, 2000, and "Large-Scale Computations in the Simulation of Materials (CMU, Pittsburgh, PA), March, 2000. R. Mejia was elected, with T. Elston of North Carolina State University, to chair the Gordon Conference on Theoretical Biology and Biomathematics in 2004.

Honors: Bo Yan received an NIH FARE award, which funded his trip to the Protein Society Symposium, San Diego, CA, August 5-9, 2000.

Synaptic Physiology

Computer modeling and analysis of buffered diffusion of calcium and its binding to neurosecretory triggers: We have developed a software package for solving numerically the 3D reaction-diffusion equations (RDEs) describing calcium entry through a set of channels into a presynaptic (or neurosecretory) terminal, its diffusion and reaction with endogenous and exogenous buffers. RDEs are solved using the alternating-direction implicit method (ADI). The modeling program also provides a simultaneous solution to the set of ordinary differential equations (ODEs) describing the binding of calcium ions to the putative exocytic calcium sensors. The program is used to test numerically ifferent hypotheses explaining the experimentally observed short-term activity-dependent enhancement of synaptic response, in particular the free-calcium and bound-calcium models of synaptic facilitation. Our results reveal the importance of the interplay between the kinetics of calcium buffering and the binding kinetics of neurosecretory calcium sensors. We find that endogenous fixed and mobile buffers exert opposing influences on the magnitude of the short-term enhancement of neurosecretory response. (V. Matveev and A. Sherman)

Secretory Physiology

Enhancement of emergent bursting due to heterogeneity: Beta cells in the pancreatic islets of Langerhans exhibit bursting oscillations of membrane potential with concommitant oscillations of calcium that have a period of 10 - 60 seconds. These oscillations are central to control of insulin secretion. Isolated beta cells, however, exhibit either bursting with much faster (or slower, see next paragraph) periods, or merely continuous spiking. A long term project is to explain this emergent behavior. Here we focus on the spiking cells and extend the observation of Sherman and Rinzel (PNAS, 1992) that electrical coupling of continuous spiking cells can allow them to become bursters. The phenomenon is delicate - it persists for only weak coupling. We have shown now that if the cell are heterogeneous, the emergent bursting is much more robust in that it persists for much stronger coupling. A paper is in preparation. (A. Sherman and G. de Vries)

Phantom bursting in pancreatic beta cells. This model seeks to explain why electrically coupled beta cells in the islets of Langerhans exhibit burst periods that are intermediate between the fast (< 10 second) and slow (> 1 minute) oscillations seen in isolated cells. The hypothesis is that the intermediate rhythm results from the interaction of (relatively) fast and slow negative feedback mechanisms. This may also explain why no negative feedback process operating on the intermediate time scale has been identified. Click here for an animation of the model. The model predicts that fast isolated cells can be induced to burst at a slower islet-like rate by injecting an inward conductance (using Dynamic Clamp) with a time constant comparable to that of the putative fast negative feedback process. This prediction was confirmed by L. Satin and colleagues. A paper has appeared in Biophysical Journal. (A. Sherman (MRB), R. Bertram and J. Previte (Penn State-Erie), L. Satin, T. Kinard, and P. Goforth (MCV-VCU))

Control of action potential firing patterns in GT1 neurons: GT1 neurons are a cell line derived from hypothalamic, GnRH-secreting neurons. They represent the interface between neuronal and secretory cells and share properties inherent to both groups of cells. These cells are of interest because they represent a key locus in the neuronal control of major endocrine functions, in this case the reproductive axis.  We have constructed a detailed model of GT1 neuron electrophysiology and calcium signaling.  The model has been used to study two specific features of signaling in GT1 neurons.  Firstly, in close combination with experimental approaches, the model was used to investigate the underlying causes of the change in morphology of action potentials, from high amplitude, short duration spikes to long duration, low amplitude spikes, after application of various agonists of GT1 neurons.  This work showed that a small, agonist-induced depolarization of the interpulse membrane potential caused an increase in the degree of sodium channel inactivation.  This led to a transition from sodium/calcium action potentials (which are high amplitude but brief) to essentially calcium-only action potentials.  Although the latter are of lower amplitude, they are prolonged, and cause an increased amount of calcium entry per spike.  This work was presented in a paper in the Biophysical Journal.

The second major piece of work from the GT1 neuron model was a detailed study of the role of pacemaker currents in the generation of action potential firing patterns in response to agonists such as GnRH (which activates the phosphoinositide-derived signaling system) and agonists that activate the cyclic AMP (cAMP) signaling system.  We found that in order to simulate the effects of these agonists with good fidelity, it was necessary to include three pacemaker channels:  1) a calcium-activated potassium current (SK-type), which acts as a brake to regulate high levels of stimulation; 2) a store-operated calcium current, which controls the firing frequency when the stores have been depleted by GnRH, and 3) a cAMP-activated, calcium inhibited cation channel, which regulates basal firing rate and underlies the response to cAMP-generating agonists.  This channel is switched off during the sustained elevation of calcium after GnRH application.  This work is in press in the Journal of Neuroscience and we have constructed an animation of the effects of the pacemaker currents.

Analysis of bursting mechanism in pituitary somatotroph cells:  Somatotroph cells, the growth-hormone secreting cells of the anterior pituitary gland, fire bursts of action potentials during basal activity.  Positive and negative agonists (growth-hormone releasing hormone, GHRH, and somatostatin, SRIF, respectively) convert the bursting to continuous activity, or quiescence, respectively.  Additionally, somatotrophs express large conductance calcium-activated potassium channels (BK-type) and blocking these channels converts the bursting to single spiking.  We are currently developing a model to investigate the generation of the bursting activity, and how the agonists and BK blockers modulate the electrical activity.  (AP LeBeau, A Sherman, and F Van Goor, S Stojilkovic: ERRB/NICHD, and Yue-Xian Li, Math Dept., University of British Columbia, Vancouver, Canada)

Biological Free Energy Transduction

Theoretical studies on the mechanism of activation  of muscle contraction by Ca2+:  Kinetics of binding of myosin subfragment 1 to regulated actin is used to discriminate among different models for muscle activation.  The study involves extensive mathematical analyses and Monte Carlo simulation  programming. The study shows that the allosteric model of Hill can not be ruled  out based on equilibrium and kinetic data of binding myosin to regulated actin alone; other structural information of the actomyosin complex during the activation process is needed. (Y. Chen, B. Yan)

Theoretical formulation on the motility of kinesin motors:  This project deals with the derivation of formalisms useful for modeling the mechanisms of kinesin motors. In most motility assays, it is the movement of a large plastic bead (not the motor; too small to be in the assay) that is measured. The formalism we derive directly connects the biochemical kinetics of the motor and the movement of the plastic bead measured in an in vitro motility assay and is therefore useful in elucidating the molecular mechanisms of the motor based on measured motility data. Using the formalism, we show that the recently measured motility data of Visscher et al (1999) can be simulated qualitatively using a simple two-state hand-over-hand model for the kinesin motor.  (Y.Chen, B. Yan)

Molecular mechanics calculations on the structure and function of kinesin motors: The main purpose of the third project is to study how a kinesin motor moves or walks on a microtubule theoretically using molecular mechanics simulations. That is, instead of modeling the kinesin's movement mechanism from the motility data as we do in our second project, we are trying to directly study the mechanism using the molecular modeling method. As a first step toward this goal, we have studied the conformations of the kinesin motor in solution, especially in the linker area that links the motor domain and the neck domain of the motor, as a function of the bound nucleotide. It is found that the conformation of the kinesin agrees with those found experimentally using NMR or fluorescence methods.

Biased movement of a brownian particle moving in a spatially periodic potential : We study biased movement of a Brownian particle moving in a spatially periodic potential whose shape in each period alternates between a flat (off) and an asymmetric linear saw-tooth (on) form. A flat potential corresponds to zero force; and a linear potential to a constant force. The alternation in form is periodic in time. The analysis is intended to complement recent Monte Carlo simulations of Chen, Yan, and Miura (Phys. Rev. E, in press). An arbitrary initial state in an off period evolves by ordinary diffusion until the saw-tooth potential turns on. The then current concentration will evolve according to the forced diffusion PDE until the saw-tooth potential is turned off. If this off-on cycle is repeated many times, the dissipative nature of diffusion ensures that all memory of the initial state will disappear; and after many repetitions of the off-on cycle, the time-dependent concentration profiles within each cycle will repeat.The associated eigenvalue-eigenfunction expansions required in the analysis have been determined and different numerical procedures are being considered. (R. Rubin and Y. Chen)

Microtubules and Statistical Mechanics

Dynamic instability of microtubules:  In an early publication [R. J. Rubin, PNAS 85, 446 (1988)],  and more recently [D. J. Bicout and R. J. Rubin Phys. Rev. E 59, 913 (1999)], we have investigated the dynamic properties of microtubule polymerization using Hill's discrete one-dimensional model [T. L. Hill, PNAS 81, 6728 (1984)] and the continuum approximation to Hill's model formulated by Dogterom and Leibler [M. Dogterom and S. Leibler, Phys. Rev. Letters, 70, 1347 (1993)].  Mean life-times of microtubules were investigated in each of these models using a simple absorbing boundary condition.  More recently, Bicout and Rubin (manuscript in preparation) have calculated the average length as a function of the time of a microtubule starting to grow on a bare nucleating site. This calculation was carried out in the framework of the continuum model where the analysis is considerably more simple than for Hill's discrete model. However, in contrast to the discrete Hill model [Rubin unpublished], special care is  required in formulating the appropriate boundary condition at a nucleating site in the continuum version. For this reason, Rubin has carried out a detailed comparison of the moments of the Green's function solutions of the discrete and continuum models by considering several special cases of the model parameters (manuscript in preparation). One of the special cases treated is of independent interest. The special case was the subject of a paper entitled, "Hill's Microtubule Dynamic Instability Model, Giddings-Eyring Chromatography Model, and Two-state Random Walks", which was presented at the March 2000 meeting of the American Physical Society [Bull. of the Amer. Phys. Soc. 45 (1), 191 (2000)].  (R. Rubin and D. Bicout)

 Diffusive transfer in cell-cell fusion systems: Effect of pore size on membrane to membrane and cytoplasm to cytoplasm transfer:   We have devised a simple 3-compartment model to analyse and compare intercompartment diffusive transfer through a small pore in 2D and 3D. The 3D (cytoplasm) version of the model consists of two hemispherical compartments, A and B, with a common center and base plane and a small connecting spherical "pore" compartment. The analog in the 2D (membrane) case is obvious. Assuming that diffusant is initially present only in A (and uniformly distributed in A), that the pore radius is negligible compared to the radii of A and B, and that the diffusion coefficient in the pore compartment is infinite (a rapid mixing assumption), the appropriate coupled diffusion equations are solved. A simple one-exponential approximation for the amount of diffusant in A in excess of its final value is obtained. Our 2D result agrees with a more realistic two sphere model calculation [R. J. Rubin and Y. Chen, Biophys. J. 58, 1157-1167 (1990)]. A manuscript is in preparation.   (R. J. Rubin and Y. Chen)

Structural Biology

Integral membrane proteins comprise receptors and transporters in epithelial cells that form the mammalian kidney. Neural net models that are trained using well-characterized proteins (e.g., G-protein coupled receptors) have been developed to identify the location of protein segments relative to the membrane or cytosole.  Several proteins of importance in epithelial transport have been studied this year, including human 11-cis retinol dehydrogenase, PKD1, PKD2, and SLC4A2.  (R Mejia)

Renal Physiology

Concentration mechanism in the renal inner medulla. The mechanism by which the mammalian kidney concentrates solutes in the inner medulla has yet to be established, in particular, because active solute transport is considered unlikely due to limited blood flow. We have developed models to test several hypotheses. These have shown that an elastic epithelial cell with two bathing solutions and distinct apical and basolateral transport properties will not concentrate fluid passively under peristaltic stress.  A model of the inner medulla that includes both cellular and tubular compartments shows modest gradients due to peristalsis and larger gradients obtained with inclusion of the cellular compartment.  This is due in part to osmotic driving forces that develop in the cell, including osmolytes produced to protect the cells against osmotic shock.  (R Mejia, M Knepper: NHLBI)

Immunomorphometry.  The anatomy and histology of the kidney have been studied for decades, most recently in an effort to understand the "passive concentrating mechanism".  We use immunofluorescent immunolabeling of renal tissue sections to identify both structure and function.  Transporters being used to identify and quantify structure include AQP1 (located in descending Henle's limb (DL)), AQP2 (located in collecting duct (CD)), UT (located in both DL and CD); a Cl- channel protein (located in ascending Henle's limb); and von Willebrand factor located in the vasa recta. (R Mejia, J Wade: UMD,

Parameters for experimentalists and modelers. A database of renal parameters with author, title, abstract, address and source continues to be compiled.  Physiologic and geometric parameters as well as conditions of measurement are included. This database, which is in extended MEDLINE format, was formerly made available to investigators via a gopher server.  This server was discontinued early in 1998, and alternatives for availability of the database on the web are being considered; currently parameter references are available upon request to  (R Mejia and M Knepper: NHLBI)


A database has been designed and implemented to serve as a learning tool to obtain curated information for the design of microarray probes to scan collecting duct tissues from human, rat, and mouse.  A web page ( contains proteins grouped according to type and function.  The goal is to provide the most accurate information available for the gene, DNA sequence, est sequences, etc. in order to develop arrays that contain genes of interest against which mRNA expression levels can be quantitatively assessed.  The database contains links to the literature and links to curated sequences that access data dynamically. (J. Legato, M. Knepper: NHLBI, R. Star: NIDDK, R. Mejia)

Cell Energetics

We have previously described the effect of changing transport work on the concentration profiles of high energy hosphate compounds within single cells using a reaction-diffusion model. A model with cylindrical geometry (pdes in two space dimensions and time) is used to investigate the biological hypothesis that ATP concentration is not limiting and that there can be significant ADP concentration gradients within the cell.  We presently test the hypothesis that there are local changes near the membrane that modulate channel function without significant changes in global cystolic concentration, and we do this by incorporation of vesicular compartments into the model.  (R Mejia and R Lynch: U of Arizona, Lynch Lab)

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