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Introduction to mathematical modeling
… in immunology
PhD. Karina García Martínez
Center of Molecular Immunology
Universidad de Oriente
June, 2016
Contents:
1. A brief introduction to Immunology
2. Mathematical modeling in Immunology:
 Cross-regulation model of T cell dynamics
Contents:
1. A brief introduction to Immunology
2. Mathematical modeling in Immunology:
 Cross-regulation model of T cell dynamics
Simple scheme of the immune response
Pathogen
B
B
B
Antigen
B
B
B
(+)
Th
Th
Th
Th
Th
Th
Th
Th
Tc
Tc
Tc
Tc
Tc
Tc
Tc
(+)
The problem of immunology is to understand the dynamics of the immune response
Regulatory T cells in the immune response
Pathogen
B
B
B
Antigen
B
B
B
(+)
Th
Th
Th
Th
Th
Th
Th
R
Th
Tc
Tc
Tc
Tc
Tc
Tc
(+)
Regulatory T cell blocks the immune response by inhibiting the
activation and proliferation of Helper T cells
Tc
Dominant tolerance: existence of Regulatory T cells
Autoimmune
CD25- CD4+ T cells
E
E
T E ET
T
T
T
(Effector cells)
Healthy
or
Autoimmune
CD25+CD4+T cells
R R
R
R
Healthy
R
R
R
R
(Regulatory cells)
 In normal animals there exist autoreactive T cells capable of causing autoimmunity
 Regulatory T cells control the autoreactive cells
Possible regulatory mechanisms
Model 1
Model 2
R
R
R
R
Th
Th
(-)
Th
Th
Model 3
R
R
Th
(+)
(-)
Th
A mathematical model can help us distinguish between these hypotheses?
Contents:
1. A brief introduction to Immunology
2. Mathematical modeling in Immunology:
 Cross-regulation model of T cell dynamics
Basic postulates of the model
Variables:
Th
Effector
cells
R
Regulatory
cells
APC
Antigen
Presenting Cells
Assumptions:
1. Regulatory and Effector cells interact only during simultaneous
conjugation with an APC
2. Antigen Presenting Cells (APCs) are a homogeneous population with
fixed size
3. Each APC has a finite and fixed number of conjugation sites, which can
be occupied by a single cell, irrespective of its phenotype
T cells – APCs conjugates
E ,R
C
C
A
i,j
: number of E and R cells per APC
: number of APC cells conjugated with “i” E cells and “j” R cells
E
E
F
C
R
F
A
1,2
We assume quasi-steady state equilibrium:
s s
A =   A (E,R)
j = 0 i =0 i,j
s : total number of conjugation sites per APC
R
C
T cells – APCs conjugates
E ,R
C
C
A
i,j
: number of E and R cells per APC
: number of APC cells conjugated with “i” E cells and “j” R cells
E
F
R
F
Obtaining the total number of EC and RC per APC:
F Ke
Ec =
E
1+F Ke
F Kr
Rc =
R
1+F Kr
F= s.A – Ec - Rc
0 =(Ke Kr ) F3+ (Ke + Kr+ Ke Kr (s.A-E-R)) F2 + (1 – Ke (s.A-E) - Kr (s.A-R)) F – s.A
T cells – APCs conjugates
E ,R
C
C
A
i,j
: number of E and R cells per APC
: number of APC cells conjugated with “i” E cells and “j” R cells
E
F
R
F
Obtaining the total number of EC and RC per APC:
F Ke
Ec =
E
1+F Ke
F Kr
Rc =
R
1+F Kr
F= s.A – Ec - Rc
0 =(Ke Kr ) F3+ (Ke + Kr+ Ke Kr (s.A-E-R)) F2 + (1 – Ke (s.A-E) - Kr (s.A-R)) F – s.A
Distributing EC and RC among the individual APC sites:
Ai,j = Hyp[i+j, Ec+Rc, sA, s] . Hyp[i, Ec, Ec+Rc, i+j]
Qualitatively different mechanisms of suppression
s
s
dE
= se +  ae (i, j ) A i, j ( E ,R) - kd .E f
dt
i =1 j =0
s
s
dR
= sr +  ar (i, j ) A i, j ( E ,R) - kd .R f
dt
j =1 i =0
Qualitatively different mechanisms of suppression
s
s
dE
= se +  ae (i, j ) A i, j ( E ,R) - kd .E f
dt
i =1 j =0
s
s
dR
= sr +  ar (i, j ) A i, j ( E ,R) - kd .R f
dt
j =1 i =0
Model 1
Model 2
ae(i, j) = pe i
ae (i,0) = pe i
ae (i, j) = 0
a r(i, j) = pr  j
ar (i, j) = pr i
R
Model 3
ae (i,0) = pe i
ae (i, j) = 0
ar (i, j) = m  i  j
dE
= se + pe Ec - kd .E f
dt
dR
= sr + pr Rc - kd .R f
dt
 The model has always a globally stable equilibrium
 Only one of the subpopulations of T cell will persist, out competing the other one
dE
= se + pe Hyp(0, Rc , s A, s) s Ec - kd .E f
s A - Rc
dt
dR
= sr + pr Rc - kd .R f
dt
 The model has a parameter region where bistability exists
 Two equilibria composed exclusively of either R or E cells
dE
= se + pe Hyp(0, Rc , s A, s) s Ec - kd .E f
s A - Rc
dt
dR
= sr + m (s-1) Ec Rc - kd .R f
s
dt
 The model has a parameter region where bistability exists,
and both R or E cells can coexist in equilibrium
Model 1
R
Model 2
NO Bistability
NO Coexistence of E and R
YES Bistability
NO Coexistence of E and R
Model 3
YES Bistability
YES Coexistence of E and R
Only the mechanism proposed in Model 3 is compatible with
experimental observations
Concluding remarks
1.
We provide a simple mathematical model to study cell interactions
dependent on their co-localization on multicellular conjugates.
2.
Our results strongly support a mechanism for suppression where:
- Regulatory cells actively inhibit Effector cell proliferation upon
their co-localization on multicellular conjugates with APCS.
- Effector cells act as a “growth factor” for the Regulatory cells
Concluding remarks
1.
We provide a simple mathematical model to study cell interactions
dependent on their co-localization on multicellular conjugates.
2.
Our results strongly support a mechanism for suppression where:
- Regulatory cells actively inhibit Effector cell proliferation upon
their co-localization on multicellular conjugates with APCS.
- Effector cells act as a “growth factor” for the Regulatory cells
IL-2 is a good candidate
Conference of Dr. Kalet León next Thursday!