1. Introduction to mathematical modeling … in immunology
PhD. Karina García Martínez
Center of Molecular Immunology
Universidad de Oriente
June, 2016

2. Contents:
A brief introduction to Immunology
Mathematical modeling in Immunology:
Cross-regulation model of T cell dynamics

3. Contents:
A brief introduction to Immunology
Mathematical modeling in Immunology:
Cross-regulation model of T cell dynamics

4. Simple scheme of the immune responseB
Pathogen B B B B
Antigen B Th
(+) Th Th Th Tc Th Tc Tc Tc Tc Tc Tc
(+)
The problem of immunology is to understand the dynamics of the immune response

5. Regulatory T cells in the immune responseB
Pathogen B B B B
Antigen B Th
(+) Th Th Th R Tc Tc Tc Th Tc Tc Tc Tc
(+)
Regulatory T cell blocks the immune response by inhibiting the activation and proliferation of Helper T cells

6. Dominant tolerance: existence of Regulatory T cells
CD25+CD4+T cells
CD25- CD4+ T cells
Autoimmune
Healthy or R E T
(Effector cells)
(Regulatory cells)
Experimental reports indicate that populations of both regulatory and effector cells coexist in tolerant animals as can be reveald by selective induction of autoimmunity or tolerance in recipient animals receiving different subpopulations of cells. This observations in athymic animals strongly favors the mechanisms in which the growth of the regulatory cells population is dependent on continuous interactions with effector cells.
In normal animals there exist autoreactive T cells capable of causing autoimmunity
Regulatory T cells control the autoreactive cells

7. Possible regulatory mechanisms
Model 1 Th R
Model 2 R R Th
(-) Th
Model 3 R R Th
(+)
(-) Th
A mathematical model can help us distinguish between these hypotheses?

8. Contents:
A brief introduction to Immunology
Mathematical modeling in Immunology:
Cross-regulation model of T cell dynamics

9. This was the first model describing the formation of multicellular conjugates of T cells and APCs. No mathematical model has been developed to study the quantitative implications of such interactions on cell population dynamics.

10. Antigen Presenting Cells
Basic postulates of the model
Variables:
APC Th R
Effector cells
Regulatory cells
Antigen Presenting Cells
Assumptions:
Regulatory and Effector cells interact only during simultaneous conjugation with an APC
Antigen Presenting Cells (APCs) are a homogeneous population with fixed size
Each APC has a finite and fixed number of conjugation sites, which can be occupied by a single cell, irrespective of its phenotype

11. T cells – APCs conjugates
: number of E and R cells per APC
, R C A
i,j
: number of APC cells conjugated with “i” E cells and “j” R cells A 1 , 2 E C R E R F F
We assume quasi-steady state equilibrium: j = s å
A = i A
i,j
(E,R) s
: total number of conjugation sites per APC

12. T cells – APCs conjugates
: number of E and R cells per APC
, R C A
i,j
: number of APC cells conjugated with “i” E cells and “j” R cells E R F F
Obtaining the total number of EC and RC per APC:
F Ke
1+F Ke
Ec = E
F Kr
1+F Kr
Rc = R
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

13. T cells – APCs conjugates
: number of E and R cells per APC
, R C A
i,j
: number of APC cells conjugated with “i” E cells and “j” R cells E R F F
Obtaining the total number of EC and RC per APC:
F Ke
1+F Ke
Ec = E
F Kr
1+F Kr
Rc = R
F= s.A – Ec - Rc
We use the classical combinatorial problem of sampling without replacement, expressed in terms of the hyper-geometric distribution.
1ra hipergeometrica: probability of having an APC with i+j T cells of any kind
2da: probability that within a random sample of i+j T cells, drawn from a population of Ec and Rc cells, “i” are E cells and “j” are R cells
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]

14. Qualitatively different mechanisms of suppressionå s å s = s + a e i , j ( ) × A ( E , R ) - k d .E f d t e i , j i = 1 j = d R å s å s a r i , j ( ) = s + × A ( E , R ) - k d .R d t r i , j f j = 1 i =
Can be achieved by setting the pair of parameters in such a way that they represent the net effect of different processes (hypothetical mechanisms by which regulatory cells may suppress effector cells while both participate in multicellular conjugates with the APC)

15. Qualitatively different mechanisms of suppressionå s å s = s + a e i , j ( ) × A ( E , R ) - k d .E f d t e i , j i = 1 j = d R å s å s a r i , j ( ) = s + × A ( E , R ) - k d .R d t r i , j f j = 1 i =
Model 1 a e
Model 2 R a ( i , j ) = p × i ( i , ) = p × i e e e ( i , j ) = a ( i , j ) = p × j r r ( i , j ) = p × i r r
Model 3 a ( i , ) = p × i e e a ( i , j ) = e a ( i , j ) = m × i × j r

16. Model 1 d E = s + p E - k .E d t d R = s + p R - k .R d tf e c d t e d R = s + p R - k d .R r c d t r f
Stable states interpreted as autoimmune or tolerance!
Phase plane: nullclines for both differential equations.
Region I: the growth capacity of E cells is higher than the one of R cells such that the latter is excluded (region II: the opposite)
Competition: R cells are simply cells that are themselves unable to trigger effector function but may interfere with E cells by competing with them for the use of some limited growth factor.
The model has always a globally stable equilibrium
Only one of the subpopulations of T cell will persist, out competing the other one

17. Model 2 d E = s + p s Ec - k .E d t d R p = s + R - k .R d t
Hyp(0, Rc , s A, s) = s + p
s Ec - k d .E f d t e e
s A - Rc d R p = s + R - k d .R d t r r c f
The model has a parameter region where bistability exists
Two equilibria composed exclusively of either R or E cells

18. Model 3 d E = s + p s Ec - k .E d t d R m (s-1) = s + Ec Rc - k .R d t
Hyp(0, Rc , s A, s)
s Ec - k d .E f d t e e
s A - Rc d R m
(s-1) = s +
Ec Rc - k d .R d t r s f
The model has a parameter region where bistability exists, and both R or E cells can coexist in equilibrium

19. Model 1
Model 2 R NO
Bistability
Coexistence of E and R
Bistability NO
Coexistence of E and R
YES
Model 3
Accounting for the quantitative details of adoptive transfers of tolerance requires models with bistable regimes in which either regulatory cells or effector cells dominate the steady state
Bistability
YES
Coexistence of E and R
Only the mechanism proposed in Model 3 is compatible with experimental observations

20. Concluding remarks
We provide a simple mathematical model to study cell interactions dependent on their co-localization on multicellular conjugates.
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

21. Concluding remarks IL-2 is a good candidate
We provide a simple mathematical model to study cell interactions dependent on their co-localization on multicellular conjugates.
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
We developed another model to study the effect of this molecule in the regulatory mechanism
IL-2 is a good candidate
Conference of Dr. Kalet León next Thursday!