Bradford E. Peercy

Vitamin D Binding Protein and Monocyte Response to 25-Hydroxyvitamin D and 1,25-Dihydroxyvitamin D: Analysis by Mathematical Modeling

Vitamin D binding protein (DBP) plays a key role in the bioavailability of active 1,25-dihydroxyvitamin D (1,25(OH)2D) and its precursor 25-hydroxyvitamin D (25OHD), but accurate analysis of DBP-bound and free 25OHD and 1,25(OH)2D is difficult. To address this, two new mathematical models were developed to estimate: 1) serum levels of free 25OHD/1,25(OH)2D based on DBP concentration and genotype; 2) the impact of DBP on the biological activity of 25OHD/1,25(OH)2D in vivo. The initial extracellular steady state (eSS) model predicted that 50 nM 25OHD and 100 pM 1,25(OH)2D), <0.1% 25OHD and <1.5% 1,25(OH)2D are ‘free’ in vivo. However, for any given concentration of total 25OHD, levels of free 25OHD are higher for low affinity versus high affinity forms of DBP. The eSS model was then combined with an intracellular (iSS) model that incorporated conversion of 25OHD to 1,25(OH)2D via the enzyme CYP27B1, as well as binding of 1,25(OH)2D to the vitamin D receptor (VDR). The iSS model was optimized to 25OHD/1,25(OH)2D-mediated in vitro dose-responsive induction of the vitamin D target gene cathelicidin (CAMP) in human monocytes. The iSS model was then used to predict vitamin D activity in vivo (100% serum). The predicted induction of CAMP in vivo was minimal at basal settings but increased with enhanced expression of VDR (5-fold) and CYP27B1 (10-fold). Consistent with the eSS model, the iSS model predicted stronger responses to 25OHD for low affinity forms of DBP. Finally, the iSS model was used to compare the efficiency of endogenously synthesized versus exogenously added 1,25(OH)2D. Data strongly support the endogenous model as the most viable mode for CAMP induction by vitamin D in vivo. These novel mathematical models underline the importance of DBP as a determinant of vitamin D ‘status’ in vivo, with future implications for clinical studies of vitamin D status and supplementation.

A mathematical model of collective cell migration in a three-dimensional, heterogeneous environment.

Cell migration is essential in animal development, homeostasis, and disease progression, but many questions remain unanswered about how this process is controlled. While many kinds of individual cell movements have been characterized, less effort has been directed towards understanding how clusters of cells migrate collectively through heterogeneous, cellular environments. To explore this, we have focused on the migration of the border cells during Drosophila egg development. In this case, a cluster of different cell types coalesce and traverse as a group between large cells, called nurse cells, in the center of the egg chamber. We have developed a new model for this collective cell migration based on the forces of adhesion, repulsion, migration and stochastic fluctuation to generate the movement of discrete cells. We implement the model using Identical Math Cells, or IMCs. IMCs can each represent one biological cell of the system, or can be aggregated using increased adhesion forces to model the dynamics of larger biological cells. The domain of interest is filled with IMCs, each assigned specific biophysical properties to mimic a diversity of cell types. Using this system, we have successfully simulated the migration of the border cell cluster through an environment filled with larger cells, which represent nurse cells. Interestingly, our simulations suggest that the forces utilized in this model are sufficient to produce behaviors of the cluster that are observed in vivo, such as rotation. Our framework was developed to capture a heterogeneous cell population, and our implementation strategy allows for diverse, but precise, initial position specification over a three- dimensional domain. Therefore, we believe that this model will be useful for not only examining aspects of Drosophila oogenesis, but also for modeling other two or three-dimensional systems that have multiple cell types and where investigating the forces between cells is of interest.

Mathematical modeling reveals modulation of both nuclear influx and efflux of Foxo1 by the IGF-I/PI3K/Akt pathway in skeletal muscle fibers

Foxo family transcription factors contribute to muscle atrophy by promoting transcription of the ubiquitin ligases muscle-specific RING finger protein and muscle atrophy F-box/atrogin-1. Foxo transcriptional effectiveness is largely determined by its nuclear-cytoplasmic distribution, with unphosphorylated Foxo1 transported into nuclei and phosphorylated Foxo1 transported out of nuclei. We expressed the fluorescent fusion protein Foxo1-green fluorescent protein (GFP) in cultured adult mouse flexor digitorum brevis muscle fibers and tracked the time course of the nuclear-to-cytoplasmic Foxo1-GFP mean pixel fluorescence ratio (N/C) in living fibers by confocal imaging. We previously showed that IGF-I, which activates the Foxo kinase Akt/PKB, caused a rapid marked decline in N/C, whereas inhibition of Akt caused a modest increase in N/C. Here we develop a two-state mathematical model for Foxo1 nuclear-cytoplasmic redistribution, where Foxo phosphorylation/dephosphorylation is assumed to be fast compared with nuclear influx and efflux. Cytoplasmic Foxo1-GFP mean pixel fluorescence is constant due to the much larger cytoplasmic than nuclear volume. Analysis of N/C time courses reveals that IGF-I strongly increased unidirectional nuclear efflux, indicating similarly increased fractional phosphorylation of Foxo1 within nuclei, and decreased unidirectional nuclear influx, indicating increased cytoplasmic fractional phosphorylation of Foxo1. Inhibition of Akt increased Foxo1 unidirectional nuclear influx, consistent with block of Foxo1 cytoplasmic phosphorylation, but did not decrease Foxo1 unidirectional nuclear efflux, indicating that Akt may not be involved in Foxo1 nuclear efflux under control conditions. New media change experiments show that cultured fibers release IGF-I-like factors, which maintain low nuclear Foxo1 in the medium. This study demonstrates the power of quantitative modeling of observed nuclear fluxes.

Coupled Cell Model of Border Zone Arrhythmias

Border zones between normal and ischemic tissue have been implicated as a cause of arrhythmic cardiac activity. A variety of experiments with coupled cells and strips of tissue has been designed to understand the arrhythmogenic effects of border zone currents. In this paper, we use systems of differential equations to model an ischemic (depolarized) cell (or region) coupled to a normal cell under a variety of conditions. For two ionic models (reduced Hodgkin-Huxley and Luo-Rudy I), we find the boundary in parameter space between oscillatory and nonoscillatory solutions. We find that there are regions in parameter space for which the ischemic cell (region) is stable and inexcitable when uncoupled, but when coupled to a normal, excitable cell the cells oscillate. We state a general principle that relates the oscillation of a forced single cell to oscillations of the coupled system. Furthermore, in modeling drug-modified experimental dynamics we are able to reproduce early after depolarization (EAD)-like phenomena, which has implications in locating oscillations in the drug-free experiment. Finally, we describe a mechanism by which oscillations in the transmembrane potential may be encountered during reperfusion.

Tissue landscape alters adjacent cell fates during Drosophila egg development

Extracellular signaling molecules control many biological processes, but the influence of tissue architecture on the local concentrations of these factors is unclear. Here we examine this issue in the Drosophila egg chamber, where two anterior cells secrete Unpaired (Upd) to activate Signal Transducer and Activator of Transcription (STAT) signaling in the epithelium. High STAT signaling promotes cell motility. Genetic analysis shows that all cells near the Upd source can respond. However, using upright imaging, we show surprising asymmetries in STAT activation patterns, suggesting that some cells experience different Upd levels than predicted by their location. We develop a three-dimensional mathematical model to characterize the spatio-temporal distribution of the activator. Simulations show that irregular tissue domains can produce asymmetric distributions of Upd, consistent with results in vivo. Mutant analysis substantiates this idea. We conclude that cellular landscape can heavily influence the effect of diffusible activators and should be more widely considered.

Modeling of glucose-induced cAMP oscillations in pancreatic β cells: cAMP rocks when metabolism rolls.

Recent advances in imaging technology have revealed oscillations of cyclic adenosine monophosphate (cAMP) in insulin-secreting cells. These oscillations may be in phase with cytosolic calcium oscillations or out of phase. cAMP oscillations have previously been modeled as driven by oscillations in calcium, based on the known dependence of the enzymes that generate cAMP (adenylyl cyclase) and degrade it (phosphodiesterase). However, cAMP oscillations have also been reported to occur in the absence of calcium oscillations. Motivated by similarities between the properties of cAMP and metabolic oscillations in pancreatic β cells, we propose here that in addition to direct control by calcium, cAMP is controlled by metabolism. Specifically, we hypothesize that AMP inhibits adenylyl cyclase. We incorporate this hypothesis into the dual oscillator model for β cells, in which metabolic (glycolytic) oscillations cooperate with modulation of ion channels and metabolism by calcium. We show that the combination of oscillations in AMP and calcium in the dual oscillator model can account for the diverse oscillatory patterns that have been observed, as well as for experimental perturbations of those patterns. Predictions to further test the model are provided.

Insight into spontaneous recurrent calcium waves in a 3-D cardiac cell based on analysis of a 1-D deterministic model

Calcium dynamics in a cardiac cell are described by a system of 3-D non-linear stochastic partial differential equations. To obtain solutions that have biophysical properties, it is necessary to explore the model parameter space. To decrease the complexity of the parameter search, we reduce the 3-D stochastic model to a 1-D deterministic model. The reduction of the problem from 3-D to 1-D is done through an asymptotic approximation after non-dimensionalization and based on rational biophysical assumptions of the 3-D domain; the stochastic to deterministic transformation is based on the regular property of the 3-D solution. The result of the model reduction proves very effective in reducing the time required to get qualitative as well as quantitative information about parameter regions in the 3-D stochastic model including calcium dynamics (sparks, wave propagation, and recovery) observed in cardiac cells.

Time-stepping techniques to enable the simulation of bursting behavior in a physiologically realistic computational islet

Physiologically realistic simulations of computational islets of beta cells require the long-time solution of several thousands of coupled ordinary differential equations (ODEs), resulting from the combination of several ODEs in each cell and realistic numbers of several hundreds of cells in an islet. For a reliable and accurate solution of complex nonlinear models up to the desired final times on the scale of several bursting periods, an appropriate ODE solver designed for stiff problems is eventually a necessity, since other solvers may not be able to handle the problem or are exceedingly inefficient. But stiff solvers are potentially significantly harder to use, since their algorithms require at least an approximation of the Jacobian matrix. For sophisticated models, systems of several complex ODEs in each cell, it is practically unworkable to differentiate these intricate nonlinear systems analytically and to manually program the resulting Jacobian matrix in computer code. This paper demonstrates that automatic differentiation can be used to obtain code for the Jacobian directly from code for the ODE system, which allows a full accounting for the sophisticated model equations. This technique is also feasible in source-code languages Fortran and C, and the conclusions apply to a wide range of systems of coupled, nonlinear reaction equations. However, when we combine an appropriately supplied Jacobian with slightly modified memory management in the ODE solver, simulations on the realistic scale of one thousand cells in the islet become possible that are several orders of magnitude faster than the original solver in the software Matlab, a language that is particularly user friendly for programming complicated model equations. We use the efficient simulator to analyze electrical bursting and show non-monotonic average burst period between fast and slow cells for increasing coupling strengths. We also find that interestingly, the arrangement of the connected fast and slow heterogeneous cells impacts the peak bursting period monotonically.

Dynamics of Computational Islet Simulations: Islets with majority mutated open KATP channels retain bursting

Abstract The study of pancreatic beta-cells comprises a crucial part of the study of the group of diseases known as diabetes. These cells exist in groups known as islets of Langerhans and are responsible for storing and producing insulin. They exhibit electrical bursting behavior during insulin production that correlates with the rate at which insulin is secreted into the bloodstream. Coupling is a natural process within islets that enables the cells to communicate with one another and transfer various ions and electrical currents; coupling of both voltage and metabolites can occur. We model multicellular islets using an existing system of seven ordinary differential equations to model beta cell function. We simulate cells with mutated KATp channels that remain open indefinitely, which have been described in experimental studies but not yet modeled. Simulations ran with these mutations reveal the existence of a bursting death threshold, described by the least percentage of cells in the islet that must be mutated for electrical bursts to completely disappear. We determine that this threshold is independent of coupling strengths, cell distribution, and possibly islet dimension; however, we also determined that this threshold is dependent on the glucose influx rate.

Extending the IP3 receptor model to include competition with partial agonists.

The inositol 1,4,5-trisphosphate (IP(3)) receptor is a Ca(2+) channel located in the endoplasmic reticulum and is regulated by IP(3) and Ca(2+). This channel is critical to calcium signaling in cell types as varied as neurons and pancreatic beta cells to mast cells. De Young and Keizer (1992) created an eight-state, nine-variable model of the IP(3) receptor. In their model, they accounted for three binding sites, a site for IP(3), activating Ca(2+), and deactivating Ca(2+). The receptor is only open if IP(3) and activating Ca(2+) is bound. Li and Rinzel followed up this paper in 1994 by introducing a reduction that made it into a two variable system. A recent publication by Rossi et al. (2009) studied the effect of introducing IP(3)-like molecules, referred to as partial agonists (PA), into the cell to determine the structure-function relationship between IP(3) and its receptor. Initial results suggest a competitive model, where IP(3) and PA fight for the same binding site. We extend the original eight-state model to a 12-state model in order to illustrate this competition, and perform a similar reduction to that of Li and Rinzel in the first modeling study we are aware of considering PA effect on an IP(3) receptor. Using this reduction we solve for the equilibrium open probability for calcium release in the model. We replicate graphs provided by the Rossi paper, and find that optimizing the subunit affinities for IP(3) and PA yields a good fit to the data. We plug our extended reduced model into a full cell model, in order to analyze the effects PA have on whole cell properties specifically the propagation of calcium waves in two dimensions. We conclude that PA creates qualitatively different calcium dynamics than would simply reducing IP(3), but that effectively PA can act as an IP(3) knockdown.