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

– p. 1/3

University of British Columbia

Sabbatical leave with host

Leah Edelstein-Keshet

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Outline • Biology - Diabetes and Immune Response • Mathematical Model • Bifurcation Analysis • Simulations • Discussion and Conclusions

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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

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β Cells - Insulin Release

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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

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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

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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)

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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

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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

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Model Scheme for Diabetes

LYMPH NODE

PANCREAS Apoptotic β cell

peptide p activation

β Cell

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 injury 000000 111111 000000 111111

naive T cell

p−MHC

apoptosis Dendritic cell

f1

A E

111 000 0000 1111 000 111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111 000 111 0000 1111 000 111 000 111 000 111

Effector (CTL) T cells

Activated T cell

f2

M

Memory cells

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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

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NOD Mice Data

Link to Model Simulation

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Simple Model Schematic f1 f1

M f

2

A 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

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Feedback Functions memory

activation

f 2 (p)

f1 (p)

k2

Activation function

Inhibition function

k1

p pn f1 (p) = n k1 + p n

ak2m f2 (p) = m k2 + p m

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Complete Model dA dt dM dt dE dt dp dt dB dt

= (σ + αM )f1 (p) − (β + δA )A − ǫA2 = β2m1 f2 (p)A − f1 (p)αM − δM M = β2m2 (1 − f2 (p))A − δE E = REB − δp p = −κEB

with nonlinear feedback functions f1 (p) =

pn k1n + pn

f2 (p) =

ak2 m k2m + pm

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Activated T cells dA dt dM dt dE dt dp dt dB dt

= (σ + αM )f1 (p)−(β + δA )A − ǫA2 = β2m1 f2 (p)A − f1 (p)αM − δM M = β2m2 (1 − f2 (p))A − δE E = REB − δp p = −κEB

The production of activated T cells, A, from naive T cells and memory cells.

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Activated T cells dA dt dM dt dE dt dp dt dB dt

= (σ + αM )f1 (p)−(β + δA )A−ǫA2 = β2m1 f2 (p)A − f1 (p)αM − δM M = β2m2 (1 − f2 (p))A − δE E = REB − δp 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.

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Effector T Cells and β Cells dA dt dM dt dE dt dp dt dB dt

= (σ + αM )f1 (p) − (β + δA )A − ǫA2 = β2m1 f2 (p)A − f1 (p)αM − δM M = β2m2 (1 − f2 (p))A − δE E = REB − δp p = −κEB

The effector T cells, E , destroy β cells producing the protein that activates T cells.

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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

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Quasi-Steady State Model The model for analysis consists of three equations: dA dt dM dt dE dt

= (σ + αM )f1 (p) − (β + δA )A − ǫA2 = β2m1 f2 (p)A − f1 (p)αM − δM M = β2m2 (1 − f2 (p))A − δE E

together with the QSS peptide expression p ≈ (RB/δp )E

3-D system of differential equations permits a more complete analysis.

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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

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3D Phase Portrait Diabetes Model

1

0.8

E

0.6

0.4

3

0.2 2.5 2

0 1

1.5 0.8

1

0.6 0.5

0.4 0.2 0

0

A

M

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3D Phase Portrait

0.1 0.08 0.06 0.04 0.02 0 0.2

*

0.15

0.8 0.6

0.1

0.4 0.05

0.2 0

0

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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

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QSS Model - Bifurcation Study • Bifurcation Analysis - Many parameters investigated using AUTO with XPP - Chose peptide clearance rate δp 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 RB E, p≈ δp

so δp increasing is similar to B decreasing

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Bifurcation Diagram Y1 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2 a15

2.5

3

3.5

4

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Bifurcation Diagram Y1 3.5

3

2.5

2

1.5

1

0.5

0 2

4

6

8

10

12

14

16

18

a15

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Homoclinic Bifurcation S

S

D

H

H

S

S

D

D

H

L

D

H

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Link to Diabetic SimulationLink to Normal Simulation

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

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Simulation - Normal Mouse 1.8

1.6

1.4

1.2

1

0.8

0.6 B 0.4 E

M

0.2 A 0

−0.2

0

20

40

Link to Homoclinic Diagram

60

80

100 t

120

140

160

180

200

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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

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Simulation - Diabetic Mouse 3

2.5 A 2

1.5

1 M

B 0.5 E 0

−0.5

0

20

40

60

80

100 t

120

Link to Experimental dataLink to Homoclinic Diagram

140

160

180

200

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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

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