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

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

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

Simple scheme of the immune response B 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

Regulatory T cells in the immune response B 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

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

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?

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

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.

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

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

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

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]

Qualitatively different mechanisms of suppression å 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)

Qualitatively different mechanisms of suppression å 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

Model 1 d E = s + p E - k .E d t d R = s + p R - k .R d t f 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

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

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

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

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

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!