Population Dynamics of Regulatory T-Lymphocytes
B RYNJA K OHLER AND J AMES P. K EENER
University of Utah, Salt Lake City, Utah
Introduction
Relation to Autoimmunity
Steady State Analysis
Our modeling goals are to better understand mechanisms behind the onset and progression
of autoimmune disorders such as multiple sclerosis.
A healthy immune response to a pathogen (typically an infectious microbe) includes an
explosion in effector T cells (TE ).
Regulatory T cells (TR ) play an important role in preventing autoimmune responses. Clinical intervention to boost regulatory cell response may be an effective therapy for autoimmune disease.
With I
= 0,
AA = 0, TE = 0, AR
k2 kR > k1 kE
= 1,
T
Stability of the trivial state
=
=
)
+k2 , TR
=
kR (+kR ) is a stable steady state provided that
1
1
R
A
! k A + kA(1A, A ) + = k1 , KpT+ET + TE + TE
E A R
A
E
E
k
(1 , AA )
R
TR = (k A + k (1 , A ) + )
E A R
A
=) We can find steady states in the AA , TE plane.
2
Naive CD4+ T cells must receive appropriate signalling from APCs to differentiate into the effector type.
The activation state of the antigen
presenting cell is critical in properly
priming a TE response.
“AA nullcline”
2
E A
Biological Background
2
TE
b) regulation by TR cells
c) regulation fails and clinical disease results
d) possible clinical immune intervention
Model Extensions
For a range of parameters two stable steady states are possible. One corresponding to low TE
and AA , which is considered a regulated response, and one corresponding to high TE and AA
which is a proper, active immune response.
TE
a) a local autoimmune process initiates
“TE nullcline”
The AA , TE plane
Feedback through cytokines or direct
activity changes the balance of the
APC activation state.
TE
parameter contraint.
Other steady states satisfy
A
! TE = k (k A k+kkR A
,I
E A R (1 , AA ) + ) k
T = k A + k (1 , A ) + We present a mathematical model for the TR population and its influence on effector T cell
(TE ) dynamics through interactions with antigen presenting cells (APCs).
The development of autoimmune disease may result from a disturbed equilibrium between
and TR cells [2].
TE
Suppression mechanisms
TR cells are thought to suppress TE through a reducing the proliferation rate. This
reduction is blocked by sufficient IL-6. To model this replace the proliferation term
with
+ AA )2
TE
p (K(K+sA
)2 + T 2 K + T
where and Ks are parameters.
s
A
R
E
How can viruses trigger autoimmunity? [3]
– Inflammation is a function of viral load, AA /TE cytokines, and bystander signals.
– Previously we assumed antigen presentation sites on APCs were proportional to the
cell population fractions AA and AR . We can, however, include dynamics on the
amount and quality of antigen presented:
I=0 (no inflammation)
AA
AA
I=2 (bistability)
I=3 (high inflammation)
– Let SA and SR represent antigen presentation sites on AA and AR , respectively.
AA
– Let V represent the viral population.
dV
dt
dSA
dt
dSR
dt
Bifurcation Diagram
Schematic diagram illustrating how APCs control TR development and activation based on
recent experimental findings [1].
The saddle-node bifurcations indicated in the above phase-planes are illustrated here.
Aa
1.50
Question: Does this system exhibit switch behavior?
=
=
rV , dT TE V , dA AA V
(
V + b)AA , SA
V + b)AR , SR
= (
The basic model provides a framework for further study of the above mentioned extensions.
5
1.25
4
7
6
1.00
Model Equations
Future Research
0.75
0.50
A A + AR = 1
dAA = IA + k T A , k T A
R 1 E R 2 R A
dt
dT = ,k A T , k A T + , T
E A
R R
dt
dTE = k A T + pTE2 , T 2 , T
E
E A
E
dt
K + TE
dTR = k A T , T
R R
R
dt
AA , AR – APC, active/resting state
T – Naive CD4+ T cells
TE , TR – Effector/regulatory T cells
I – inflammation
k1 , k2 , kE , kR – transition rates
– source (from thymus)
– natural death of T cells
p,K – proliferation parameters
– non-linear death parameter for TE
Model Assumptions
Cells “well-mixed” as in a lymph node.
Total antigen presentation remains constant during the time modeled.
Non-linear proliferation rate of TE due to cytokine production (IL-2).
Non-linear death of TE due to activation induced cell death (by Fas/Fas-ligand interactions), but is small.
Inflammation/infection (I ) is a given time-dependent function, and treated like a parameter.
3
2
0.25
1
0.00
-1.
0.
1.
2.
3.
4.
5.
6.
inflammation
Improve parameter estimates for quantitative results to compare with experimental allergic
encephalomyelitis (EAE) and animal models of viruses which mimic demyelinating disease.
Add spatial compartments for cell interactions which occur in the lymph nodes, circulation,
and target tissue.
Consider a range of cross-reacting T cells of different specificities.
Indeed Powrie and Maloy’s description can exhibit “switch” behavior.
References
Implications
[1] K.J. Maloy, R. Powrie. 2003. Regulating the Regulators Science. 299: 1030-1031.
Consistent with experiment: a critical dose of Freud’s adjuvant required for activating and
immune responses.
[2] M.G. von Herrath, L.C. Harrison. 2003. Antigen-induced Regulatory T Cells in Autoimmunity. Nature Reviews: Immunology. 3: 223-232.
System can exhibit hysteresis. The implications here require a more precise notion of inflammation.
[3] R.S. Fujinami. 2001. Viruses and autoimmune disease – two sides of the same coin? TRENDS
in Microbiology. 9(8): 377-381.
Parameter constraints derived in the stability condition.
Parameter sensitivity, particularly to p and K imply there may necessarily be additional
suppression mechanisms at work to make the switch robust.
Acknowledgements
Thanks to Robert Fujinami, Fred Adler, and the Math Physiology Group at the University of Utah