Simulated effect of stochastic channel noise at basal and stimulatory glucose conditions.

<p><b>A).</b> Fraction of cells showing significant [Ca²⁺] elevations (left) and representative time-courses of [Ca²⁺] (right) for simulations at 5.5mM glucose with g_coup = 120pS, with and without stochastic channel noise. <b>B).</b> As in A for simulations at 11mM glucose and g_coup = 120pS. <b>C).</b> As in A for simulations at 5.5mM glucose and <i>g</i>_<i>coup</i> = 0pS. <b>D).</b> As in A for simulations at 11mM glucose and <i>g</i>_<i>coup</i> = 0pS. <b>E).</b> As in A for simulations at 5.5mM glucose and <i>g</i>_<i>coup</i> = 10pS. Vertical scale bars represent 20% increase in simulated [Ca²⁺]. <b>F).</b> Time courses of K_ATP noise factor (<i>S</i>), K_ATP open probability (<i>p</i>_{<i>0(KATP)</i>}) and membrane potential (<i>V</i>_<i>m</i>) in a representative cell for <i>g</i>_<i>coup</i> = 120pS (left) and <i>g</i>_<i>coup</i> = 0pS (right). Arrows indicate excursions in <i>S</i> and <i>p</i>_{<i>0(KATP)</i>} that correspond to substantial membrane depolarization. All simulations were run with <i>P</i>_<i>mu</i>t = 1. Data in A,C,E is presented as mean for n = 10 simulations with different random number seeds.</p

Decreases in Gap Junction Coupling Recovers Ca²⁺ and Insulin Secretion in Neonatal Diabetes Mellitus, Dependent on Beta Cell Heterogeneity and Noise

<div><p>Diabetes is caused by dysfunction to β-cells in the islets of Langerhans, disrupting insulin secretion and glucose homeostasis. Gap junction-mediated electrical coupling between β-cells in the islet plays a major role in coordinating a pulsatile secretory response at elevated glucose and suppressing insulin secretion at basal glucose. Previously, we demonstrated that a critical number of inexcitable cells can rapidly suppress the overall islet response, as a result of gap junction coupling. This was demonstrated in a murine model of Neonatal Diabetes Mellitus (NDM) involving expression of ATP-insensitive K_ATP channels, and by a multi-cellular computational model of islet electrical activity. Here we examined the mechanisms by which gap junction coupling contributes to islet dysfunction in NDM. We first verified the computational model against [Ca²⁺] and insulin secretion measurements in islets expressing ATP-insensitive K_ATP channels under different levels of gap junction coupling. We then applied this model to predict how different K_ATP channel mutations found in NDM suppress [Ca²⁺], and the role of gap junction coupling in this suppression. We further extended the model to account for stochastic noise and insulin secretion dynamics. We found experimentally and in the islet model that reductions in gap junction coupling allow progressively greater glucose-stimulated [Ca²⁺] and insulin secretion following expression of ATP-insensitive K_ATP channels. The model demonstrated good correspondence between suppression of [Ca²⁺] and clinical presentation of different NDM mutations. Significant recoveries in [Ca²⁺] and insulin secretion were predicted for many mutations upon reductions in gap junction coupling, where stochastic noise played a significant role in the recoveries. These findings provide new understanding how the islet functions as a multicellular system and for the role of gap junction channels in exacerbating the effects of decreased cellular excitability. They further suggest novel therapeutic options for NDM and other monogenic forms of diabetes.</p></div

Simulated decline in insulin secretion due to Kir6.2 and SUR1 mutations.

<p><b>A).</b> Total time-averaged insulin secretion for simulations of Kir6.2^{[ΔN30,K185Q]} expression in a fraction of cells indicated (<i>P</i>_<i>Mut</i>), with <i>g</i>_<i>coup</i> = 120pS (black), and 0pS (red). <b>B).</b> Representative time-courses of insulin secretion averaged across the simulated islet for each <i>P</i>_<i>Mut</i> and <i>g</i>_<i>coup</i> in A. <b>C).</b> Total time-averaged insulin secretion for simulations that include mutant K_ATP channel activity for <i>g</i>_<i>coup</i> = 120pS and <i>g</i>_<i>coup</i> = 0pS, with stochastic channel noise. Simulations include the characterized mutations indicated where reported <i>k</i>^<i>’</i>_<i>1/2</i> changes and reported <i>α</i> (if any) are accounted for. <b>D).</b> Representative time-courses of insulin secretion averaged across the simulated islet for indicated mutations and <i>g</i>_<i>coup</i> in C. <b>E).</b> Total time-averaged insulin secretion for simulations that include mutant K_ATP channel activity, as in C, with and without stochastic channel noise for <i>g</i>_<i>coup</i> = 0pS. <b>F).</b> Representative time-courses of insulin secretion averaged across the simulated islet for indicated mutations and noise in E. <b>G).</b> Total time-averaged insulin secretion for simulations that include mutant K_ATP channel activity for <i>g</i>_<i>coup</i> = 120pS and <i>g</i>_<i>coup</i> = 0pS, with stochastic channel noise. Simulations include the characterized mutations indicated where reported <i>k</i>^<i>’</i>_<i>1/2</i> changes and estimated <i>α</i> are accounted for. <b>H).</b> As in G for <i>g</i>_<i>coup</i> = 120pS and <i>g</i>_<i>coup</i> = 10pS. <b>I).</b> Fraction of cells showing significant [Ca²⁺] elevations for simulations that include selected mutant K_ATP channel activity, as in G, for <i>g</i>_<i>coup</i> = 120pS (left) and <i>g</i>_<i>coup</i> = 0pS (right) and with stochastic channel noise, as <i>p’</i>_<i>0</i> is reduced to model sulfonylurea action. <b>J).</b> Total time-averaged insulin secretion for simulations as in H, that include selected mutant K_ATP channel activity, as in G, for <i>g</i>_<i>coup</i> = 0pS with stochastic channel noise, as <i>p’</i>_<i>0</i> is reduced to model sulfonylurea action. Results are arranged in order of clinical severity, with the clinical classification indicated in G. * indicates mutations where sulfonylurea therapy is reported to be ineffective. Simulations in A,B were run at 20mM glucose, where all others were run at 11mM glucose, <i>P</i>_<i>mu</i>t = 1. Data in A,C,E,G,H,I is presented as mean±SD for n = 3 simulations with different random number seeds.</p

Simulated dependence of islet [Ca²⁺] on electrical coupling upon reduced K_ATP ATP sensitivity.

<p><b>A).</b> Quantification of [Ca²⁺], as expressed by the fraction of cells showing significant [Ca²⁺] elevations, for simulations of Kir6.2^{[ΔN30,K185Q]} expression in a percentage of cells (<i>P</i>_<i>Mut</i>), at 20mM glucose, with <i>g</i>_<i>coup</i> = 120pS (black), 50pS (blue), and 0pS (red). Inset represents islet where specific cells show increased K_ATP activity modelling mosaic Kir6.2^{[ΔN30,K185Q]} expression (green). <b>B).</b> Fraction of cells showing significant [Ca²⁺] elevations for simulations of diazoxide application uniformly in all cells (<i>α</i>) at 11mM glucose, <i>P</i>_<i>mu</i>t = 1, with <i>g</i>_<i>coup</i> as in A. Inset represents islet where cells show variable K_ATP activity modelling intrinsic heterogeneity (green). <b>C).</b> Representative time courses from simulations with increased <i>k</i>^<i>’</i>_<i>1/2</i> values (reduced ATP-sensitivity) as indicated, far left for <i>g</i>_<i>coup</i> = 120pS. <b>D).</b> As in C for <i>g</i>_<i>coup</i> = 50pS. <b>E).</b> As in C for <i>g</i>_<i>coup</i> = 0pS. Scale bars represent 20% increase in simulated [Ca²⁺]. <b>F).</b> Fraction of cells showing significant [Ca²⁺] elevations for increased <i>k</i>^<i>’</i>_<i>1/2</i> values uniformly in all cells at 11mM glucose, <i>P</i>_<i>mu</i>t = 1, for <i>g</i>_<i>coup</i> = 120pS (black), 50pS (blue), 20pS (purple), 10pS (green), and 0pS (red). <b>G).</b> Mean [Ca²⁺] duty cycle for increased <i>k</i>^<i>’</i>_<i>1/2</i> values as in F. <b>H).</b> <i>k</i>^<i>’</i>_<i>1/2</i> value at which 50% of cells show significant [Ca²⁺] elevations for values of <i>g</i>_<i>coup</i> over 5 simulated islets. <b>I).</b> Mean±SD cell parameters for active and inactive cells at <i>g</i>_<i>coup</i> = 0pS for simulations in A (<i>P</i>_<i>mut</i>), B (<i>α</i>), F (<i>k</i>^<i>’</i>_<i>1/2</i>). All data is representative of n = 5 simulations with different random number seeds.</p

Simulated effect of stochastic channel noise upon Kir6.2 and SUR1 mutations.

<p><b>A).</b> Fraction of cells showing significant [Ca²⁺] elevations (left) and mean [Ca²⁺] duty cycle (right), for simulations with and without stochastic channel noise, for increasing <i>k</i>^<i>’</i>_<i>1/2</i> with <i>g</i>_<i>coup</i> = 120pS (<i>α</i> = 0). <b>B).</b> As in A for increasing <i>k</i>^<i>’</i>_<i>1/2</i> with <i>g</i>_<i>coup</i> = 0pS (<i>α</i> = 0). <b>C).</b> As in A for increasing <i>α</i> with <i>g</i>_<i>coup</i> = 0pS (<i>k</i>^<i>’</i>_<i>1/2</i> = 1). <b>D).</b> Fraction of cells showing significant [Ca²⁺] elevations (left) and mean [Ca²⁺] duty cycle (right), for simulations with and without stochastic channel noise, that include mutant K_ATP channel activity for <i>g</i>_<i>coup</i> = 120pS. Simulations include the characterized mutations indicated where reported <i>k</i>^<i>’</i>_<i>1/2</i> changes and reported <i>α</i> (if any) are accounted for. <b>E).</b> As in D for simulations with and without stochastic channel noise, that include mutant K_ATP channel activity for <i>g</i>_<i>coup</i> = 0pS. <b>F).</b> Representative time courses for simulations as in E, with and without stochastic channel noise for <i>g</i>_<i>coup</i> = 0pS. Vertical scale bars represent 20% increase in simulated [Ca²⁺]. All simulations were run at 11mM glucose, <i>P</i>_<i>mu</i>t = 1. Data in D,E is presented as mean±SD for n = 3–5 simulations with different random number seeds.</p

Simulated dependence of [Ca²⁺] on electrical coupling following modulation of other K_ATP parameters.

<p><b>A).</b> False-color scale map displaying the fraction of cells showing significant [Ca²⁺] elevations for <i>g</i>_<i>coup</i> = 120pS upon variation in both <i>k</i>^<i>’</i>_<i>1/2</i> (ATP-sensitivity) and <i>p</i>^<i>’</i>_<i>o</i> (open channel conductance) values, as indicated. <b>B).</b> As in A, for <i>g</i>_<i>coup</i> = 0pS. <b>C).</b> Fraction of cells showing significant [Ca²⁺] elevations upon increased <i>p</i>^<i>’</i>_<i>o</i> values at <i>k</i>^<i>’</i>_<i>1/2</i> = 1, for <i>g</i>_<i>coup</i> = 120pS, 50pS, and 0pS. <b>D).</b> As in C for <i>k</i>^<i>’</i>_<i>1/2</i> = 4. <b>E).</b> Fraction of cells showing significant [Ca²⁺] elevations upon decreased <i>p</i>^<i>’</i>_<i>o</i> values at <i>k</i>^<i>’</i>_<i>1/2</i> = 6 for <i>g</i>_<i>coup</i> = 120pS and 0pS. <b>F).</b> Mean [Ca²⁺] duty cycle upon decreased <i>p</i>^<i>’</i>_<i>o</i> values at <i>k</i>^<i>’</i>_<i>1/2</i> = 6, as in E. <b>G).</b> Fraction of cells showing significant [Ca²⁺] elevations for <i>g</i>_<i>coup</i> = 120pS upon increased <i>k</i>^<i>’</i>_<i>1/2</i> values for different <i>H</i> values, as indicated. <b>H).</b> as in G for <i>g</i>_<i>coup</i> = 0pS. All simulations were run at 11mM glucose, <i>P</i>_<i>mu</i>t = 1. Data is representative of n = 5 simulations with different random number seeds.</p

Calcium imaging and modeling in islets with varying Cx36 and Kir6.2^{[ΔN30,K185Q]} expression.

<p><b>A).</b> Image of autofluorescence in GFP channel in islet lacking Kir6.2^{[ΔN30,K185Q]} expression (left); with representative time courses of Rhod2 fluorescence in islets isolated from Cx36^+/+ and Cx36^-/- mice at 20mM glucose (middle); and time courses from simulations representing Cx36^+/+ (<i>g</i>_<i>coup</i> = 120pS) and Cx36^-/- (<i>g</i>_<i>coup</i> = 0pS) islets at 20mM glucose (right). Scale bars represent 20% increase in Rhod2 fluorescence or 20% increase in simulated [Ca²⁺] respectively. <b>B).</b> As in A, for islets with ~50% Kir6.2^{[ΔN30,K185Q]} expression, as indicated by GFP fluorescence; or simulated to have ~50% Kir6.2^{[ΔN30,K185Q]} expression. <b>C)</b>. Quantification of [Ca²⁺], as expressed by the fraction of cells showing significant [Ca²⁺] elevations, in islets from Cx36^+/+ mice with increasing Kir6.2^{[ΔN30,K185Q]} expression (pr with linear regression (solid grey line, dashed grey lines indicate 95% CI.); together with corresponding simulation results with <i>g</i>_<i>coup</i> = 120pS (solid black line). <b>D).</b> As in C, for islets from Cx36^+/- mice and simulation results with <i>g</i>_<i>coup</i> = 50pS. <b>E).</b> As in C, for islets from Cx36^-/- mice and simulation results with <i>g</i>_<i>coup</i> = 0pS. Each data point in C-E represents an average over n = 2–5 islets per mouse, with a total of N = 24 mice (Cx36^+/+), 19 mice (Cx36^+/-) or 22 mice (Cx36^-/-).</p

Simulated decline in [Ca²⁺] due to Kir6.2 and SUR1 mutations, and recovery following decreased electrical coupling.

<p><b>A).</b> Fraction of cells showing significant [Ca²⁺] elevations for simulations that include mutant K_ATP channel activity, for <i>g</i>_<i>coup</i> = 120pS (black) and <i>g</i>_<i>coup</i> = 0pS (red). Simulations include the characterized mutations indicated where reported <i>k</i>^<i>’</i>_<i>1/2</i> changes and reported <i>α</i> (if any) are accounted for. <b>B).</b> Mean [Ca²⁺] duty cycle for simulations that include mutant K_ATP channel activity as in A. <b>C)</b>. Fraction of cells showing significant [Ca²⁺] elevations for simulations that include mutant K_ATP channel activity, for <i>g</i>_<i>coup</i> = 120pS (black) and <i>g</i>_<i>coup</i> = 0pS (red). Simulations include the characterized mutations indicated where reported <i>k</i>^<i>’</i>_<i>1/2</i> changes and estimated <i>α</i> (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005116#pcbi.1005116.s002" target="_blank">S2 Table</a>) are accounted for. <b>D).</b> Mean [Ca²⁺] duty cycle for simulations that include mutant K_ATP channel activity as in C. All simulations were run at 11mM glucose, <i>P</i>_<i>mu</i>t = 1. Data is presented as mean±SD for n = 3–5 simulations with different random number seeds. Results are arranged in order of clinical severity, with the clinical classification indicated: T2D- (Type2 Diabetes; TNDM- Transient Neonatal Diabetes Mellitus; PNDM- Permanent Neonatal Diabetes Mellitus; DEND- PNDM with Developmental Delay and Neurological features, including iDEND. * indicates mutations where sulfonylurea therapy is reported to be ineffective.</p

Boolean network model describes [Ca^2<b>+</b>]_i suppression as a function of coupling conductance.

<p>A) Percent cells showing [Ca²⁺]_i elevations in islets from Kir6.2^[AAA]-expressing mice as a function of gap junction conductance, together with Boolean network model fit. Filled squares indicate mean(±s.e.m.) experimental data, solid line represents mean of simulations that best fit data for wild-type coupling value <i>p₀</i> = 0.38 and <i>Sp</i> = 0.15 (χ² = 0.416), dashed lines represents 95% confidence intervals of simulation fit. Gap junction conductance for each data point was normalized to the wild-type conductance and scaled by the fitted <i>p₀</i>. B) Mean(±s.e.m.) experimental data with Boolean network simulations for varying threshold of inactive cells <i>Sp</i>.</p

Phase Transitions in the Multi-cellular Regulatory Behavior of Pancreatic Islet Excitability

<div><p>The pancreatic islets of Langerhans are multicellular micro-organs integral to maintaining glucose homeostasis through secretion of the hormone insulin. β-cells within the islet exist as a highly coupled electrical network which coordinates electrical activity and insulin release at high glucose, but leads to global suppression at basal glucose. Despite its importance, how network dynamics generate this emergent binary on/off behavior remains to be elucidated. Previous work has suggested that a small threshold of quiescent cells is able to suppress the entire network. By modeling the islet as a Boolean network, we predicted a phase-transition between globally active and inactive states would emerge near this threshold number of cells, indicative of critical behavior. This was tested using islets with an inducible-expression mutation which renders defined numbers of cells electrically inactive, together with pharmacological modulation of electrical activity. This was combined with real-time imaging of intracellular free-calcium activity [Ca²⁺]_i and measurement of physiological parameters in mice. As the number of inexcitable cells was increased beyond ∼15%, a phase-transition in islet activity occurred, switching from globally active wild-type behavior to global quiescence. This phase-transition was also seen in insulin secretion and blood glucose, indicating physiological impact. This behavior was reproduced in a multicellular dynamical model suggesting critical behavior in the islet may obey general properties of coupled heterogeneous networks. This study represents the first detailed explanation for how the islet facilitates inhibitory activity in spite of a heterogeneous cell population, as well as the role this plays in diabetes and its reversal. We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and coordinate a robust insulin response with high dynamic range. These findings also give new insight into emergent multicellular dynamics in general which are applicable to many coupled physiological systems, specifically where inhibitory dynamics result from coupled networks.</p></div