Modeling Type 1 Diabetes. in NOD Mice. Joseph M. Mahaffy. Nonlinear 10% of diabetes cases are Type 1, while 90% are Diabetes Oct 2006 p. 9/33. T …
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 1/33
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 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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Cells - Insulin Release
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Type 1 or Juvenile Diabetes - Overview
Diabetes mellitus results from the loss of 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 cells in pancreas Other autoimmune diseases are similar
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Model Scheme for Diabetes
PANCREAS Apoptotic cell
111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 injury 111111 000000 111111 000000
LYMPH NODE
p-MHC activation
naive T cell
peptide p
Cell apoptosis
Dendritic cell
f1
A E
Activated T cell
111111 000000 1111 0000 111 000 111 000 111111 000000 1111 0000 111 000 111 000 111111 000000 1111 0000 111 000 111 000 111111 000000 1111 0000 111 000 111 000 111111 000000 1111 0000 111 000 111 000 111111 000000 1111 0000 111 000 111 000
f2
M
Effector CTL T cells
Memory cells
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Animal
Model for Diabetes
Non-Obese Diabetic or NOD mice undergo apoptosis or programmed cell death of 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 f A 1-f
2 2
M
E
B
p
A Activated T cells M Memory cells E Effector or killer T cells p peptide B Fraction of remaining cells
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Feedback Functions
memory activation
f 2 p
f1 p
k2
k1
p
pn f1 p n k1 p n
Activation function
Inhibition function
m ak2 f2 p m k2 p m
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Complete Model
dA dt dM dt dE dt dp dt dB dt M f1 p - A A - A2 2m1 f2 pA - f1 pM - M M 2m2 1 - f2 pA - E E 
REB - p p -EB
with nonlinear feedback functions
f1 p f2 p pn n k1 p n ak2 m m k2 p m
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2006 p 16/33
Activated T cells
dA dt dM dt dE dt dp dt dB dt M f1 p- A A - A2 2m1 f2 pA - f1 pM - M M 2m2 1 - f2 pA - E E REB - p p -EB
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 M f1 p- A A-A2 2m1 f2 pA - f1 pM - M M 2m2 1 - f2 pA - E E REB - p p -EB
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 Cells
dA dt dM dt dE dt dp dt dB dt M f1 p - A A - A2 2m1 f2 pA - f1 pM - M M 2m2 1 - f2 pA - E E REB - p p -EB
The effector T cells, E, destroy 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 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
M f1 p - A A - A2 2m1 f2 pA - f1 pM - M M 2m2 1 - f2 pA - E E
together with the QSS peptide expression
p RB/p 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
08
06 E 04
02 25 0 1 08 06 04 02 0 M 0 05 A 1 2 15
3
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3D Phase Portrait
01 008 006 004 002 0 02 015 01 005 0 0 02
06 04
08
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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 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 E, p
so p increasing is similar to B decreasing
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Bifurcation Diagram
Y1 2 18 16 14 12 1 08 06 04 02 0 0 05 1 15 2 a15 25 3 35 4
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Bifurcation Diagram
Y1 35
3
25
2
15
1
05
0 2 4 6 8 a15
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10
12
14
16
18
Homoclinic Bifurcation
S S
D
D
L
H
H
S
S
D
D
H
H
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Complete Model - Normal Mouse
Simulated complete model for a normal mouse - Assumed an initial response of Effector T cells - Normal parameter values - Some cells die, but levels at high equilibrium
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Simulation - Normal Mouse
18
16
14
12
1
08
06 B 04 E 02 A 0 M
-02
0
20
40
60
80
100
t
120
140
160
180
200
Link to Homoclinic Diagram
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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 cells until diabetic
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Simulation - Diabetic Mouse
3
25 A 2
15
1 B 05 E 0 M
-05
0
20
40
60
80
100 t
120
140
160
180
200
Link to Experimental dataLink to Homoclinic Diagram
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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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