UMBC REU Site: Computational Simulations of Pancreatic Beta Cells

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1 UMBC REU Site: Computational Simulations of Pancreatic Beta Cells Sidafa Conde 1, Teresa Lebair 2, Christopher Raastad 3, Virginia Smith 4 Kyle Stern, 5 David Trott 5 Dr. Matthias Gobbert 5, Dr. Bradford Peercy 5, Dr. Arthur Sherman 6 1 U. Mass. Dartmouth, 2 St. Vincent College, 3 U. Washington, 4 U. Virginia, 5 UMBC, 6 NIH Acknowledgments: REU, HPCF, NSF, UMBC Mathematics and Statistics, UMBC 1 / 14

2 UMBC REU Site: Computational Simulations of Pancreatic Beta Cells Background and Model Numerical Methods and Results Application Simulations Conclusions Mathematics and Statistics, UMBC 2 / 14

3 Background: Diabetes and Beta Cells What is Diabetes? What are β-cells? Islets of Langerhans Insulin Secretion and Electrical Activity Why do we care? Mathematics and Statistics, UMBC 3 / 14

4 Background: Beta Cell Metabolic Activity 1 glucose enters β-cell 2 glucose metabolizes into ATP in Mitochondria 3 ATP causes potassium channels to close 4 depolarization of cell membrane 5 causes the calcium channel to open 6 calcium influx causes release of insulin 0 Nunemaker and Satin, 2005 Mathematics and Statistics, UMBC 4 / 14

5 Mathematical Model: Equations dv = (I K + I Ca + I K(Ca) + I K(ATP) ) dt C m (1) dn dt = (n n) τ n (2) d[ca] dt = f cyt (J mem + J er ) (3) d[ca er ] = σ V f er J er dt { ( )} (4) d[adp] [ATP] [ADP] exp (r + γ) 1 [Ca] r 1 = dt τ a (5) d[g6p] dt = κ R GK R PFK ) (6) d[fbp] dt = κ (R PFK 0.5 R GPDH ) (7) Mathematics and Statistics, UMBC 5 / 14

6 Mathematical Model: Discrete Model Dual Oscillator Model Islet model is a 3D N N N cubic grid of β-cells Cell state has 7 variables: V n Ca ADP Caer G6P FBP Each cell has a slow or fast parameter assignment Nearest Neighbor Coupling between β-cells Time evolution governed by system of Ordinary Differential Equations (ODE) Mathematics and Statistics, UMBC 6 / 14

7 Numerical Methods: Implementation Vectorized standard initial value problem: dy dt = f(t, y) + Cy, 0 < t t f, y(0) = y 0. y = (V, n, [Ca], [Ca er], [ADP], [G6P], [FBP]) V = (V 1,..., V N 3),, [FBP] = ([FBP] 1,..., [FBP] N 3) C is the coupling matrix term Very Large problem size, N = 10 implies DOF = 7 N 3 = 7000 stiff ODE: With final time t f = 500,000 ms and resolution of order of 1 ms, the final iteration count is 40-50,000 Matlab ode15s: sophisticated ODE solver designed for stiff problems with variable method order. Performance issues with Matlab s default ode15s designed for small system size speed loss: inefficient memory re-allocations memory burden: gradient approximations saved for all iterations Mathematics and Statistics, UMBC 7 / 14

8 Numerical Methods: Our Contribution and Results Our Contributions: Resolved issues with ode15s specific to this problem Enabled random coupling between cells Extended slow and fast assignment schemes Simplified, Vectorized, Generalized Code Numerical Results: reduced wall clock time by 30 50% more easily devise simulations testing slow fast distributions and and coupling schemes faster turn around time for simulation study easier to do science, less time hacking UMBC Tara Cluster: 2.66 GHz, 24GB Memory 82 compute nodes, 656 cores N DOF No Mod Mem Mod :00:51 00:00: :01:37 00:01: :05:41 00:03: :21:32 00:13: :02:44 00:34: :40:18 01:25: :22:46 03:46: :09:50 08:37: :36:36 18:32: Mathematics and Statistics, UMBC 8 / 14

9 Application Simulations: Slow and Fast Bursting Cell Independent β-cells (No Coupling) Slow cell period 4-5 minutes Fast cell period seconds Mathematics and Statistics, UMBC 9 / 14

10 Application Simulations: Coupled vs. Independent Cells Mathematics and Statistics, UMBC 10 / 14

11 Application Simulations: Bursting Period vs Fast/Slow Distribution 1000 picosieman coupling Slow and fast cells have the nearly identical traces Increasing number of slow cells increases burst period Cleaner trace, (less jaggedness) Mathematics and Statistics, UMBC 11 / 14

12 Application Simulations: Burst Period vs. Number of Fast Cells Average Burst Period of Cells with Varying Coupling Constants 7 Average Burst Period (min) Percentage of Fast Cells in Islet 200 ps 600 ps 1000 ps Adding fast cells decreases burst period Hump around 20% fast cells is related to coupling strength and islet size Right shift in graph occurs with increased islet size Mathematics and Statistics, UMBC 12 / 14

13 Conclusions: Conclusions and Future Work Conclusions: Adding just a few slow cells greatly increases burst period Adding fast cells does not have as great an effect on burst period Future work: Explore stochastic coupling in more detail. Explore more slow and fast cell assignment schemes (shells, layers, stochastic) Explore coupling in more parameters Mathematics and Statistics, UMBC 13 / 14

14 Conclusions: Thank You! For Further Exploration: Visit the HPC REU Website Technical Report Visit our Poster Mathematics and Statistics, UMBC 14 / 14