Synchronizing beta cells in the pancreas.

The secretion of insulin from the pancreas relies on both gap junctions and subpopulations of beta cells with specific intrinsic properties.

Do oscillations in pancreatic islets require pacemaker cells?

The pancreatic islets of Langerhans are biomedically important because they are home to the beta cells that secrete insulin and are hence important for understanding diabetes. They are also an important case study for the mechanisms of bursting oscillations and how these oscillations emerge from the electrical coupling of highly heterogeneous cells. Early work has pointed to a voting/democratic paradigm, where the islet properties are a nonlinear average of the cell properties, with no 'conductor leading the orchestra'. Recent experimental work has uncovered new facets of this heterogeneity, and has identified small world networks dominated by a small subset of cells with a high degree of functional connectivity, assessed via correlations of calcium oscillations. It has also been suggested that these connectivity hubs act as pacemakers necessary for islet oscillations. We reviewed modeling studies that have confirmed the existence of small worldness, and we did not find evidence for obligatory pacemakers. We conclude that democracy rather than oligarchy remains the most likely organizing principle of the islets.

Flipping the switch on the hub cell: Islet desynchronization through cell silencing.

Pancreatic ß cells, responsible for secreting insulin into the bloodstream and maintaining glucose homeostasis, are organized in the islets of Langerhans as clusters of electrically coupled cells. Gap junctions, connecting neighboring cells, coordinate the behavior of the islet, leading to the synchronized oscillations in the intracellular calcium and insulin secretion in healthy islets. Recent experimental work has shown that silencing special hub cells can lead to a disruption in the coordinated behavior, calling into question the democratic paradigm of islet insulin secretion with more or less equal input from each ß cell. Islets were shown to have scale-free functional connectivity and a hub cell whose silencing would lead to a loss of functional connectivity and activity in the islet. A mechanistic model representing the electrical and calcium dynamics of ß cells during insulin secretion was applied to a network of cells connected by gap junctions to test the hypothesis of hub cells. Functional connectivity networks were built from the simulated calcium traces, with some networks classified as scale-free, confirming experimental results. Potential hub cells were identified using previously defined centrality measures, but silencing them was unable to desynchronize the islet. Instead, switch cells, which were able to turn off the activity of the islet but were not highly functionally connected, were found via systematically silencing each cell in the network.

Challenges and opportunities for the simulation of calcium waves on modern multi-core and many-core parallel computing platforms.

State-of-the-art distributed-memory computer clusters contain multicore CPUs with 16 and more cores. The second generation of the Intel Xeon Phi many-core processor has more than 60 cores with 16 GB of high-performance on-chip memory. We contrast the performance of the second-generation Intel Xeon Phi, code-named Knights Landing (KNL), with 68 computational cores to the latest multicore CPU Intel Skylake with 18 cores. A special-purpose code solving a system of nonlinear reaction-diffusion partial differential equations with several thousands of point sources modeled mathematically by Dirac delta distributions serves as realistic test bed. The system is discretized in space by the finite volume method and advanced by fully implicit time-stepping, with a matrix-free implementation that allows the complex model to have an extremely small memory footprint. The sample application is a seven variable model of calcium-induced calcium release (CICR) that models the interplay between electrical excitation, calcium signaling, and mechanical contraction in a heart cell. The results demonstrate that excellent parallel scalability is possible on both hardware platforms, but that modern multicore CPUs outperform the specialized many-core Intel Xeon Phi KNL architecture for a large class of problems such as systems of parabolic partial differential equations.

Development and Analysis of a Quantitative Mathematical Model of Bistability in the Cross Repression System Between APT and SLBO Within the JAK/STAT Signaling Pathway.

Cell migration is a key component in development, homeostasis, immune function, and pathology. It is important to understand the molecular activity that allows some cells to migrate. Drosophila melanogaster is a useful model system because its genes are largely conserved with humans and it is straightforward to study biologically. The well-conserved transcriptional regulator Signal Transducer and Activator of Transcription (STAT) promotes cell migration, but its signaling is modulated by downstream targets Apontic (APT) and Slow Border Cells (SLBO). Inhibition of STAT activity by APT and cross-repression of APT and SLBO determines whether an epithelial cell in the Drosophila egg chamber becomes motile or remains stationary. Through mathematical modeling and analysis, we examine how the interaction of STAT, APT, and SLBO creates bistability in the Janus Kinase (JAK)/STAT signaling pathway. In this paper, we update and analyze earlier models to represent mechanistically the processes of the JAK/STAT pathway. We utilize parameter, bifurcation, and phase portrait analyses, and make reductions to the system to produce a minimal three-variable quantitative model. We analyze the manifold between migratory and stationary steady states in this minimal model and show that when the initial conditions of our model are near this manifold, cell migration can be delayed.

Clustered cell migration: Modeling the model system of Drosophila border cells.

In diverse developmental contexts, certain cells must migrate to fulfill their roles. Many questions remain unanswered about the genetic and physical properties that govern cell migration. While the simplest case of a single cell moving alone has been well-studied, additional complexities arise in considering how cohorts of cells move together. Significant differences exist between models of collectively migrating cells. We explore the experimental model of migratory border cell clusters in Drosophila melanogaster egg chambers, which are amenable to direct observation and precise genetic manipulations. This system involves two special characteristics that are worthy of attention: border cell clusters contain a limited number of both migratory and non-migratory cells that require coordination, and they navigate through a heterogeneous three-dimensional microenvironment. First, we review how clusters of motile border cells are specified and guided in their migration by chemical signals and the physical impact of adjacent tissue interactions. In the second part, we examine questions around the 3D structure of the motile cluster and surrounding microenvironment in understanding the limits to cluster size and speed of movement through the egg chamber. Mathematical models have identified sufficient gene regulatory networks for specification, the key forces that capture emergent behaviors observed in vivo, the minimal regulatory topologies for signaling, and the distribution of key signaling cues that direct cell behaviors. This interdisciplinary approach to studying border cells is likely to reveal governing principles that apply to different types of cell migration events.

Mathematical Modeling of Nuclear Trafficking of FOXO Transcription Factors.

Nuclear cytoplasmic flux of Foxo transcription factors is paramount in cellular gene regulation. For example, excluding Foxo from skeletal muscle nuclei is necessary to avoid muscle wasting through elevated protein breakdown. Constructing a mathematical model of the signaling process leading to alteration of Foxo nuclear cytoplasm ratio is useful in predicting and interpreting such ratio changes. In this chapter we derive a general mathematical model for nuclear cytoplasmic flux. We apply this model to Foxo flux and take advantage of rapid phosphorylation approximation and conservation conditions to reduce the Foxo flux model. We constrain our model with data from mouse skeletal muscle with applied IGF. This procedure provides an example of what might be called the central approach of mathematical modeling: The cycling of a biological question through mathematical formulation and back to biological interpretation.

Modeling of glucose-induced cAMP oscillations in pancreatic ß 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 ß 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 ß 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.

Tissue landscape alters adjacent cell fates during Drosophila egg development.

Extracellular signalling 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) signalling in the epithelium. High STAT signalling 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.

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.

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.

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.

Vitamin D and DBP: the free hormone hypothesis revisited.

The last five years have witnessed a remarkable renaissance in vitamin D research and a complete re-evaluation of its benefits to human health. Two key factors have catalyzed these changes. First, it now seems likely that localized, tissue-specific, conversion of 25-hydroxyvitamin D (25OHD) to 1,25-dihydroxyvitamin D (1,25(OH)2D) drives many of the newly recognized effects of vitamin D on human health. The second key factor concerns the ongoing discussion as to what constitutes adequate or optimal serum vitamin D (25OHD) status, with the possibility that vitamin D-deficiency is common to communities across the globe. These two concepts appear to be directly linked when low serum concentrations of 25OHD compromise intracrine generation of 1,25(OH)2D within target tissues. But, is this an over-simplification? Pro-hormone 25OHD is a lipophilic molecule that is transported in the circulation bound primarily to vitamin D binding protein (DBP). While the association between 25OHD and DBP is pivotal for renal handling of 25OHD and endocrine synthesis of 1,25(OH)2D, what is the role of DBP for extra-renal synthesis of 1,25(OH)2D? We hypothesize that binding to DBP impairs delivery of 25OHD to the vitamin D-activating enzyme 1α-hydroxylase in some target cells. Specifically, it is unbound, 'free' 25OHD that drives many of the non-classical actions of vitamin D. Levels of 'free' 25OHD are dependent on the concentration of DBP and alternative serum binding proteins such as albumin, but will also be influenced by variations in DBP binding affinity for specific vitamin D metabolites. The aim of this review will be to discuss the merits of 'free 25OHD' as an alternative marker of vitamin D status, particularly in the context of non-classical responses to vitamin D. This article is part of a Special Issue entitled '16th Vitamin D Workshop'.

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.

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)(2)D) and its precursor 25-hydroxyvitamin D (25OHD), but accurate analysis of DBP-bound and free 25OHD and 1,25(OH)(2)D is difficult. To address this, two new mathematical models were developed to estimate: 1) serum levels of free 25OHD/1,25(OH)(2)D based on DBP concentration and genotype; 2) the impact of DBP on the biological activity of 25OHD/1,25(OH)(2)D in vivo. The initial extracellular steady state (eSS) model predicted that 50 nM 25OHD and 100 pM 1,25(OH)(2)D), <0.1% 25OHD and <1.5% 1,25(OH)(2)D 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)(2)D via the enzyme CYP27B1, as well as binding of 1,25(OH)(2)D to the vitamin D receptor (VDR). The iSS model was optimized to 25OHD/1,25(OH)(2)D-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)(2)D. 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.

How pancreatic beta-cells discriminate long and short timescale cAMP signals.

The translocation of catalytic protein kinase A (cPKA) in response to cyclic-adenosine mono-phosphate (cAMP) depends on the pattern of stimulus applied to the cell. Experiments with IBMX have shown that 1), sustained cAMP elevation is more effective than oscillations of cAMP at getting cPKA into the nucleus; and 2), cPKA enters the nucleus by diffusion. We constructed mathematical models of cAMP activation of cPKA and their diffusion in order to study nuclear translocation of cPKA, and conclude that hindered diffusion of cPKA through the nuclear membrane by a rapid-binding process, but not globally reduced diffusion, can explain the experimental data. Perturbation analysis suggests that normal physiological oscillations of glucose would not result in nuclear translocation, but chronically high glucose that produces extended calcium plateaus and/or chronic glucagonlike peptide-1 stimulation could result in elevated levels of nuclear cPKA.

Initiation and propagation of a neuronal intracellular calcium wave.

The ability to image calcium movement within individual neurons inspires questions of functionality including whether calcium entry into the nucleus is related to genetic regulation for phenomena such as long term potentiation. Calcium waves have been initiated in hippocampal pyramidal cells with glutmatergic signals both in the presence and absence of back propagating action potentials (BPAPs). The dendritic sites of initiation of these calcium waves within about 100 microm of the soma are thought to be localized near oblique junctions. Stimulation of synapses on oblique dendrites leads to production of inositol 1,4,5-trisphosphate (IP(3)) which diffuses to the apical dendrite igniting awaiting IP(3) receptors (IP(3)Rs) and initiating and propagating catalytic calcium release from the endoplasmic reticulum. We construct a reduced mathematical system which accounts for calcium wave initiation and propagation due to elevated IP(3). Inhomogeneity in IP(3) distribution is responsible for calcium wave initiation versus subthreshold or spatially uniform suprathreshold activation. However, the likelihood that a calcium wave is initiated does not necessarily increase with more calcium entering from BPAPs. For low transient synaptic stimuli, timing between IP(3) generation and BPAPs is critical for calcium wave initiation. We also show that inhomogeneity in IP(3)R density can account for calcium wave directionality. Simulating somatic muscarinic receptor production of IP(3), we can account for the critical difference between calcium wave entry into the soma and failure to do so.

A kinetic model of oxygen regulation of cytochrome production in Escherichia coli.

Recent experimental work has identified the principal components arrayed by Escherichia coli in its sensing of, and response to, varying levels of oxygen. This apparatus may be leveraged/modified by the metabolic engineer to identify nonuniform oxygen and glucose regimens that deliver better yields than their uniform counterparts. Toward this end we build and analyse a mathematical model that captures the role played by oxygen in the regulation of cytochrome production in E. coli.