Modeling Type 1 Diabetes

Transcription

1 p. 1/3 Modeling Type 1 Diabetes in NOD Mice Joseph M. Mahaffy Nonlinear Dynamical Systems Group Computational Sciences Research Center Department of Mathematical Sciences San Diego State University October 2006

2 University of British Columbia Sabbatical leave with host Leah Edelstein-Keshet Diabetes Oct 2006 p. 2/3

3 Outline Biology - Diabetes and Immune Response Mathematical Model Bifurcation Analysis Simulations Discussion and Conclusions Diabetes Oct 2006 p. 3/3

4 Glucose Metabolism Ingest food - Breaks down to simple sugars Blood absorbs sugar - Raises blood glucose concentration β cells in pancreas respond - Insulin released Cells increase glucose uptake - Insulin facilitates glucose transport across cell membranes, especially in skeletal muscles Glucose converted to glycogen - Preferred energy storage of cells Blood sugar level decreases - Body tightly regulates glucose levels Diabetes Oct 2006 p. 4/3

5 β Cells - Insulin Release Diabetes Oct 2006 p. 5/3

6 Type 1 or Juvenile Diabetes - Overview Diabetes mellitus results from the loss of β cells - An auto-immune disease Insulin production is severely reduced Hereditary disease - about 4-20 per 100,000 people Peak diagnosis occurs around age 14 10% of diabetes cases are Type 1, while 90% are Type 2 (where cells become insulin resistant, mostly in obese individuals) Treatment is regular injections of insulin - transplants are usually attacked by immune system Diabetes Oct 2006 p. 6/3

7 Type 1 or Juvenile Diabetes - Symptoms Classic Symptoms - Polyphagia (hungry) - Polydipsia (thirsty) - Polyuria (frequent urination) Other Symptoms - Blurred vision - Fatigue - Weight loss - Poor wound healing Diabetes Oct 2006 p. 7/3

8 Type 1 or Juvenile Diabetes - Disease Increased heart disease - Atherosclerosis from low insulin Blindness (retinopathy) - Increased pressure in eye Nerve damage (neuropathy) Kidney damage (nephropathy) Diabetes Oct 2006 p. 8/3

9 T Cell Activation T cells mature in the thymus - Cross-react with self-protein to prevent autoimmunity T cells migrate to Lymph nodes - Interact with antigen presenting cells (APCs) - APCs present antigen protein fragment (about 9 AAs) inside MHC (major histocompatibility complex) - The peptide-mhc complex interacts with T cells surface receptors - T cells with appropriate specificity become activated Most antigens are foreign proteins from viruses and bacteria Diabetes Oct 2006 p. 9/3

10 T Cell Immune Response Activated T cells proliferate about 6 cell divisions Most become Effector cells (cytotoxic T-lymphocytes or CTLs) - CTLs are efficient specific killers, destroying target cells - Relatively short-lived Some become Memory cells - No immediate effect - Long-lived cells - New exposure to same antigen, rapidly activated - Strategy for vaccines Type 1 diabetes when CTLs attack β cells in pancreas Other autoimmune diseases are similar Diabetes Oct 2006 p. 10/3

11 Model Scheme for Diabetes PANCREAS LYMPH NODE p MHC naive T cell β Cell injury apoptosis Apoptotic β cell peptide p activation Dendritic cell f 1 Activated T cell A E Effector (CTL) T cells f 2 M Memory cells Diabetes Oct 2006 p. 11/3

12 Animal Model for Diabetes Non-Obese Diabetic or NOD mice undergo apoptosis or programmed cell death of β cells in the pancreas shortly after birth Clearance of apoptotic cells by macrophages is reduced - Possibly forms self-antigen - Experiments suggest a fragment from IGRP (glucose-6-phosphate catalytic subunit-related protein) produces a dominant antigen Experiments designed to find autoreactive CD8+ T cells in pancreas of NOD mice Observed three waves of CD8+ T cells before mice became diabetic around 16 weeks Diabetes Oct 2006 p. 12/3

13 NOD Mice Data Link to Model Simulation Diabetes Oct 2006 p. 13/3

14 Simple Model Schematic f 1 f 1 A f 2 M 1 f 2 E B p A = Activated T cells M = Memory cells E = Effector or killer T cells p = peptide B = Fraction of remaining β cells Diabetes Oct 2006 p. 14/3

15 Feedback Functions memory f 2 (p) activation f 1(p) k 2 k 1 p Activation function Inhibition function f 1 (p) = p n k n 1 + pn f 2 (p) = akm 2 k m 2 + pm Diabetes Oct 2006 p. 15/3

16 Complete Model da dm de dp db = (σ + αm)f 1 (p) (β + δ A )A ǫa 2 = β2 m 1 f 2 (p)a f 1 (p)αm δ M M = β2 m 2 (1 f 2 (p))a δ E E = REB δ p p = κeb with nonlinear feedback functions f 1 (p) = f 2 (p) = p n k n 1 + pn ak 2 m k m 2 + pm Diabetes Oct 2006 p. 16/3

17 Activated T cells da dm de dp db = (σ + αm)f 1 (p) (β + δ A )A ǫa 2 = β2 m 1 f 2 (p)a f 1 (p)αm δ M M = β2 m 2 (1 f 2 (p))a δ E E = REB δ p p = κeb The production of activated T cells, A, from naive T cells and memory cells. Diabetes Oct 2006 p. 17/3

18 Activated T cells da dm de dp db = (σ + αm)f 1 (p) (β + δ A )A ǫa 2 = β2 m 1 f 2 (p)a f 1 (p)αm δ M M = β2 m 2 (1 f 2 (p))a δ E E = REB δ p p = κeb The production of activated T cells, A, from naive T cells and memory cells. The loss of activated T cells, A, becoming effector and memory T cells, decaying, and competing with others. Diabetes Oct 2006 p. 17/3

19 Effector T Cells and β Cells da dm de dp db = (σ + αm)f 1 (p) (β + δ A )A ǫa 2 = β2 m 1 f 2 (p)a f 1 (p)αm δ M M = β2 m 2 (1 f 2 (p))a δ E E = REB δ p p = κeb The effector T cells, E, destroy β cells producing the protein that activates T cells. Diabetes Oct 2006 p. 18/3

20 Complete Model - Discussion 5-Dimensional Model - Highly nonlinear - Difficult to analyze 17 Physiological parameters - Many are known or have good estimates - Constrains possible solutions Time Scale - The peptide, p, has fast reaction kinetics - This allows Quasi-Steady State Approximations - The β cells, B, have slow dynamics - This allows consideration of slow changing parameter Diabetes Oct 2006 p. 19/3

21 Quasi-Steady State Model The model for analysis consists of three equations: da dm de = (σ + αm)f 1 (p) (β + δ A )A ǫa 2 = β2 m 1 f 2 (p)a f 1 (p)αm δ M M = β2 m 2 (1 f 2 (p))a δ E E together with the QSS peptide expression p (RB/δ p )E 3-D system of differential equations permits a more complete analysis. Diabetes Oct 2006 p. 20/3

22 QSS Model - Discussion Equilibria - There are 1-5 equilibria - With physiological parameters, 3 equilibria exist Linear Analysis of 3 Equilibria - Origin is a Stable Node - Diseased State is a Node that is Stable or Unstable depending on parameters - Third equilibrium is a Saddle Node Diabetes Oct 2006 p. 21/3

23 3D Phase Portrait Diabetes Model E M A 2 Diabetes Oct 2006 p. 22/3

24 3D Phase Portrait * Diabetes Oct 2006 p. 23/3

25 QSS Model - Parameter Study Parameters - Experimental data compiled by Marée, Santamaria, and Edelstein-Keshet - Physiological range of parameters limited by their study for most parameters in the model - Several parameters remain unknown, so varied to obtain desired behaviour - Sensitivity of the parameters was studied Diabetes Oct 2006 p. 24/3

26 QSS Model - Bifurcation Study Bifurcation Analysis - Many parameters investigated using AUTO with XPP - Chose peptide clearance rate δ p as it is believed that poor clearance could induce diabetes - In the normal range of clearance, the most solutions approached the Origin - When halved, the many solutions oscillated about the Diseased State - The QSS approximation is p RB δ p E, so δ p increasing is similar to B decreasing Diabetes Oct 2006 p. 25/3

27 Bifurcation Diagram Y Diabetes Oct 2006 p. 26/ a15

28 Bifurcation Diagram Y a15 Diabetes Oct 2006 p. 27/3

29 Homoclinic Bifurcation S S D D L H H S S D D H H Diabetes Oct 2006 p. 28/3

30 Complete Model - Normal Mouse Simulated complete model for a normal mouse - Assumed an initial response of Effector T cells - Normal parameter values - Some β cells die, but levels at high equilibrium Diabetes Oct 2006 p. 29/3

31 Simulation - Normal Mouse E A M B t Link to Homoclinic Diagram Diabetes Oct 2006 p. 30/3

32 Complete Model - Diabetic Mouse Simulated complete model for a diabetic mouse - Assumed an initial response of Effector T cells - Lower peptide clearance - Increasing spikes of Activated T cells - Waves of short-lived Effector T cells - High Memory cell populations allow new response - Slow decline of β cells until diabetic Diabetes Oct 2006 p. 31/3

33 Simulation - Diabetic Mouse A B M 0.5 E t Link to Experimental datalink to Homoclinic Diagram Diabetes Oct 2006 p. 32/3

34 Discussion and Conclusions Designed a reasonable model for NOD mice Parameters fit physiological data Simulations indicate parameters and initial conditions may be too sensitive Excellent qualitative behaviour of the model Good example of a homoclinic bifurcation Model supports biological theory of defective clearance after apoptosis Diabetes Oct 2006 p. 33/3