Molina-Par?s C., Lythe G. (editors) Mathematical Models and Immune Cell Biology

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14 Dynamics of Peripheral Regulatory and Effector T Cells 303

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Chapter 15

Mathematical Models of the Role of IL-2

in the Interactions Between Helper

and Regulatory CD4

T Cells

Kalet Le

on and Karina Garc´ıa-Mart´ınez

Abstract Mathematical models for the role of IL-2 in the dynamic interplay

between CD4

helper and regulatory T cells are studied. These models are ex-

tensions of the crossregulation model of CD4

T cell dynamics including IL-2

molecules. The goal is to understand how the immune system is dynamically or-

ganized, structured by this interaction with self antigens, and how such organization

might determine its overall function. We consider two model variants. In the ﬁrst,

regulatory T cells suppress helper T cells by competing with them for IL-2 in the

lymph node. The second variant adds a direct inhibition of helper T cell activation

which requires their co-localized activation on the APCs. We use the models to study

the impact of treatments that either sequester or inject IL-2 in the immune system

response. We show that treatment sequestering IL-2 could be used in particular con-

ditions, both to render tolerant a preexistent immune/autoimmune system or to break

a preexistent tolerant state, inducing an immune response. However, IL-2 injections

will always reinforce the preexistent state, further expanding either the regulatory

or helper T cells for a preexistent tolerant or immune state.

Interleukin-2 (IL-2) was the ﬁrst T cell growth factor to be identiﬁed and molecu-

larly cloned. Since its discovery, it was shown to promote T cell proliferation and

survival in vitro [1], and to enhance T cell immunity in vivo in particular instance

of viral infection [2] and vaccination [3–5]. However, this classical role of IL-2

in promoting T cell immunity has been recently challenged by experiments using

the IL-2 or IL-2R KO mice [6], which have shown that in vivo IL-2 is an essen-

tial homeostatic growth factor for the CD4

CD25

FoxP3

(regulatory) T cells,

supporting an additional role for this cytokine in the maintenance of natural and

induced tolerance.

CD4

CD25



FoxP3



(helper) T cells have been identiﬁed as the principal source

of IL-2 in vivo, while CD4

CD25

Foxp3

regulatory T cells are unable to pro-

duce this cytokine [7], suggesting that they have to sequester the IL-2 produced by

K. Le´on (



)

Center of Molecular Immunology, Havana, Cuba

e-mail: kalet@cim.sld.cu

C. Molina-Par´ıs and G. Lythe (eds.), Mathematical Models and Immune Cell Biology,

DOI 10.1007/978-1-4419-7725-0

15,

 Springer Science+Business Media, LLC 2011

305

306 K. Le´on and K. Garc´ıa-Mart´ınez

the helper cells in order to proliferate and survive. Moreover, in vitro and in vivo

experiments have shown that regulatory T cells inhibit the production of IL-2 in

the responder’s helper cells [8–10], limiting also in this way their own source of

this cytokine. Altogether, the latter facts reveal a complex dynamic relationship be-

tween IL-2 and CD4

T cells. On one hand, IL-2 promotes the proliferation of the

helper T cells, which may drive effective immunity and foster IL-2 production. But,

on the other hand, it promotes the expansion of regulatory T cells, which may turn

off the immune reaction, as well as the IL-2 production on its own. The dynamic

balance between these opposing forces might result in a complex response to IL-2

modulation treatments, such as those ones being investigated in clinical [11,12]and

preclinical [2,13] studies.

In this chapter, mathematical models describing the regulatory role of IL-2 in

CD4

T cell dynamics are studied. We review modelling of the dynamical aspects of

interleukin-2 interaction with T cells. Then, recently developed mathematical mod-

els, which were designed to study the interplay of IL-2 with helper and regulatory

CD4

T cells, are described. These models are used to compare different hypothe-

ses regarding the speciﬁc role of IL-2 in the mechanism by which regulatory T cells

suppress helper T cells, and to explore the effect of IL-2 modulating therapies on

the dynamics. Overall, this chapter will illustrate, through these particular examples,

how mathematical models can be used to analyze a speciﬁc biological problem, ex-

plaining known experimental facts and deriving predictions that might guide new

experimentation.

Brief Review of Mathematical Models of IL-2 Dynamics

Several mathematical models have addressed molecular and cellular aspects of

IL-2 interaction with its speciﬁc receptor at the T cell surface. IL-2 was one of

the ﬁrst biological molecules discovered to signal through interactions with a re-

ceptor composed of multiple protein chains, the ˛,theˇ and the  chains [14].

Debate continues [15] on the detailed dynamics of IL-2 molecular interaction with

these receptor chains. Some evidence suggests, for instance, the existence of a pre-

assembled form of the ˛ and ˇ subunit of IL-2 receptor; while other evidence

suggests that the latter dimer is dynamically formed upon IL-2 ligations. Mathe-

matical models like the one developed by Goldstein et al. [16], call attention to the

potential pitfalls of using classical Scatchard plot technique to study the binding

properties of dynamically assembled receptors. Later models [17] directly analyzed

experimental data of the IL-2/IL-2 receptor interaction.

Mathematical models from the group of Lauffenberger [18] have signiﬁcantly

contributed to elucidating the dynamics of IL-2 and IL-2 receptor internalization

and degradation by T cells upon signalling. These models help to understand and

quantify the dynamics and have practical implications, predicting the effect of par-

ticular IL-2 mutants, which, by altering the latter dynamics, exhibits super-agonist

effects [19,20].

15 IL-2, Helper and Regulatory CD4

T Cells 307

Svirshchevskaya et al. [21] developed a mathematical model of IL-2 dependent

proliferation for the CTLL2 T cell line. This model can be used to quantify IL-2 or

IL-2-inhibitor activity in an in vitro proliferation assay. Borisova et al. [22]integrate

different aspect of IL-2 and IL-2 receptor dynamics, in a model for the cell-cycle

of T cells. This model was used to analyze experimental data [23], predicting the

effect of different concentrations of IL-2 in the cultures, both for the expression of

IL-2/IL-2 receptor complexes at T cell surface and the overall dynamics of cell cycle

progression. Morel et al. [24] also developed a mathematical model of lymphocyte

proliferation, including the effect of both IL-2 and IL-4 cytokines. They used the

model to study synergistic and antagonistic effects of these cytokines, which might

derive from the competition of these molecules at the cell surface for the common

 chain that integrate their receptors. A more recent model, developed by De Boer

et al. [25], analyzes also experimental data of T cell proliferation in vitro in the

presence of different levels of IL-2. However they use data from a more informa-

tive technique to score T cell proliferation. This technique uses a ﬂuorescent dye,

known as CFSE, to allow a dissection of the generational structure of proliferating

T cell populations. The analysis of available data, with the model, suggests that IL-2

levels signiﬁcantly inﬂuence T cell death rate in the culture, but as a function of the

division history of individual T cells (i.e. the number of division cycles each T cell

has undergone).

Published modelling results, either on the regulatory role of IL-2 in the dynamics

of helper and regulatory CD4

T cells or on the biological impact of therapies based

on modulating IL-2 activity, are rare. Burroughs et al. [26], proposed a model to

study the dynamic interaction of helper and regulatory T cells, including IL-2 as

a homeostatic factor shared by these T cell populations. This model predicts an

interesting quorum threshold effect, which regulates helper T cell activation and

is dynamically controlled by the presence of regulatory T cells in different situa-

tions. However, the latter model relies on some biologically unrealistic assumptions,

which might limit its applications (it assumes the existence of an unknown home-

ostatic cytokine which is exclusively used by regulatory T cells in the absence of

IL-2, contradicting existing experimental observations). Our group has recently de-

veloped a mathematical model for helper and regulatory CD4

T cell dynamics

[29], accounting for many aspects of IL-2 dynamics. This model is an extension of

the crossregulation model of immunity [27,28], used to investigate the dynamics of

the interactions between regulatory and helper CD4

T cells. The following sec-

tion in this chapter will review the mathematical formulation and properties of this

model.

IL-2 in Helper and Regulatory CD4 T Cell Dynamics

The models introduced in this section are a natural extension of the crossregula-

tion model [29]. They model the interaction between helper and regulatory T cell

populations inside a lymph node, which contain a constant amount of their cognate

308 K. Le´on and K. Garc´ıa-Mart´ınez

antigen presenting cells (APCs) and homeostatic cytokines. The processes of IL-2

production, degradation and function, which are relevant to CD4

T cell dynam-

ics, are explicitly included. Two model variants, with different mechanisms for the

role of IL-2 in T cell mediated suppression, are studied. In the ﬁrst model variant,

regulatory T cells suppress helper T cell proliferation by effectively reducing IL-2

concentration, producing competition for this cytokine [8–10, 30, 31]. The regula-

tory T cells exploit, therefore, their natural over-expression of the high-afﬁnity IL-2

receptor to dominate the system. The second model variant includes, in addition to

competition for IL-2, that the regulatory T cells inhibit the activation of helper cells

on the APCs, perhaps by temporally conditioning the APCs [32–34], as postulated

in the original crossregulation model.

Postulates of the Models

The models rely on eight postulates which capture current knowledge regarding the

relationship between IL-2 and CD4

T cells. Bellow we enunciate each postulate;

provide a brief biological justiﬁcation with relevant references, and explain how

each postulate is implemented in the models.

1. Helper and regulatory cells go through at least three different functional states

during their life cycle.

The cell cycle for T cells has been extensively studied [35]. It is well estab-

lished that T cells require signalling through their T cell receptor (TCR) in order

to leave the G0 phase and enter the G1 phase of cell cycle. However TCR sig-

nalling is not enough to proceed into the cell cycle. These cells are arrested at the

G1 checkpoint, if insufﬁcient levels of relevant cytokines are encountered. IL-2,

IL-4, IL-7 and other  chain family of cytokines are potent growth factors for

T cells.

The models here assume that both helper (E) and regulatory (R) T cells pass

through the same three functional states during their life cycles. These functional

states are:

(a) Resting T cells which have not received signal through TCR recently (since

its last division or its de novo generation)

(b) Activated T cells which are in the functional state derived from the interac-

tion of resting T cells and cognate APCs

activated T cells that receive enough cytokine-related signal

2. Helper cells produce IL-2, upon activation with APCs, being the major source

for this cytokine in vivo.

In vitro studies have shown that CD4

CD25



T cells, cultured with APC and

anti-CD3, produce large amounts of IL-2 [36]. Moreover, in vivo experiments

[37]haveshownthatCD4

CD25



FoxP3



(helper) T cells capable of pro-

ducing IL-2 are mandatory to produce the IL-2, which sustain a functional

15 IL-2, Helper and Regulatory CD4

T Cells 309

CD4

CD25

FoxP3

(regulatory) T cell population. In the models, only helper

T cells produce IL-2. This IL-2 production is explicitly modelled here as a burst

associated with the transition of helper T cells, from the resting state to the ac-

tivated state (although qualitatively similar results are obtained in the model if

IL-2 is assumed to be produced continuously by helper T cells while in the acti-

vated state). No other source of IL-2 is considered, since we are interested on the

interplay between helper and regulatory T cells.

3. Helper T cells proliferate and survive, upon TCR ligation on APCs and sig-

nalling received from IL-2 or other homeostatic cytokines.

In vitro experiments have shown that IL-2 is a growth/survival factor for activated

CD4

T cells [1]. However, normal T cells response can be obtained in vivo in

IL-2 deﬁcient mice [38,39], suggesting that helper T cells can use other cytokines

to proliferate/survive in vivo additionally to IL-2 (perhaps IL-7 [40–42], IL-15

[42, 43]orIL-21[44]). In the model, resting helper T cells are activated after

conjugation with APCs, which provide MHC-peptide and other costimulatory

signals. Once activated, these cells can proceed into the cell cycle if enough

cytokine signal is received. This required signal comes, for this cell type, either

from available IL-2 or from an alternative cytokine, referred as IL-˛,whichis

present at a constant homeostatic level in the lymph node. The amount of IL-˛

is a control parameter of the model, whose value is assumed to be set by cellular

subsets not included in the model.

4. Regulatory T cells do not produce IL-2, but their proliferation and survival is

strictly dependent on both TCR ligation and IL-2-derived signalling.

Regulatory T cells do not express IL-2 mRNA in vitro [36]orinvivo[7],

even after activation. In vitro, they proliferate when stimulated simultaneously

with APCs and external IL-2. The required IL-2 mediated signal can be recov-

ered by co-culturing regulatory cells with helper cells capable of producing

IL-2 [36]. Moreover, in vivo experiments have shown that the absence of

CD4

CD25



FoxP3



T cells capable of producing IL-2 leads to the absence

of regulatory T cell population and furthermore to the development of autoim-

munity [36]. Thus, in the model regulatory T cells do not produce IL-2. Like

helper T cells, they are activated after conjugation with cognate APCs. How-

ever, to proceed into the cell cycle, they need a cytokine related signal provided

exclusively by available IL-2.

5. Regulatory T cells inhibit IL-2 production by helper T cells upon their co-

localized activation on the APCs. This interaction might also inhibit other

processes on helper T cell proliferation.

Regulatory T cells have been shown to inhibit the proliferation of helper T cells

bothinvitro[8,10,45,46] and in vivo [6,47,48]. In vitro, this interaction requires

cell to cell contact between Regulatory T cells, Helper T cells and APCs [8,

10, 46, 49]. This interaction reduces IL-2 production and mRNA expression by

CD4

CD25



T cells [8, 10]. However, regulatory T cells have been shown to

control autoimmunity [6, 47] and to inhibit in vivo expansion of CD4

CD25



T cells from IL-2 or IL-2R KO mice [48], suggesting that inhibition is more than

just suppression of IL-2 production.

310 K. Le´on and K. Garc´ıa-Mart´ınez

Accounting for the suppressive interaction exerted by regulatory T cells over

the helper T cell is the core of the crossregulation model [28, 50]. In that model

suppression was considered as dependent on multicellular conjugate formation.

That is, helper T cells activation is assumed effective only when these cells

are conjugated to APCs that are not simultaneously conjugated to regulatory

T cells. However, as discussed in [27], such a modelling strategy accounts for

many detailed mechanisms of suppression, which assume that suppression oc-

curs upon the colocalized activation of helper and regulatory T cells. The two

model variants in this chapter use the latter formalism to treat suppression. The

ﬁrst model variant assumes that suppression exerted by regulatory cells does not

affect the rate of helper T cell activation (transitions from resting to activated

state), but does affect the burst of IL-2 production associated with it. The second

model variant assumes, however, that suppression does reduce the activation rate

of helper T cells, also affecting IL-2 production.

6. IL-2 signal upon ligation to speciﬁc receptors which are differentially

expressed on helper and regulatory T cells.

In the model, a single class of IL-2 receptor is included, resembling the high

afﬁnity form of the receptor, which is the most relevant for CD4

T cell biology.

The numbers of IL-2 receptors on helper and regulatory T cells in different

activation states (resting, activated and cycling) are controlled by independent

parameters, which may be chosen to fulﬁl desired biological constraints. Sig-

nalling through IL-2 receptors is taken to be equivalent to its binding to IL-2. The

model equation computes the mean number of receptors bound per cell, for each

cell type at a given time. The fraction of particular cells taking action, depending

on the IL-2 derived signal, is computed with a sigmoid function (a Hill function)

of the mean level of bound IL-2 receptors per cell. The sigmoid type of function

was selected following in vitro studies [51], which have shown this type of dose

response curve for activated T cell proliferation in response to external IL-2.

The signalling machineries of helper and regulatory T cells are assumed to be

similar. Thus we use the same sigmoid type of function for these cells: i.e with

a similar value of Hill coefﬁcient h. Differences between IL-2 signalling on

these cell types are assumed to rely mainly on their differential expression of

IL-2 receptor and in the value of the control parameters S

and S

which set the

sensitivity threshold of each speciﬁc cell type to the signal. In the case of helper

T cells, IL-˛ is assumed to provide an additional signal, which is equivalent to

the IL-2 mediated signal. Therefore, we modelled the presence of a constant

level of this cytokine in the system as equivalent to the signalling through a ﬁxed

number of IL-2 receptors.

7. IL-2 is internalized and degraded upon ligation to its receptor.

In vitro kinetic experiments have shown internalization of IL-2 by T cells [18].

A fraction of the internalized IL-2 is recycled back, bound to an ˛ chain of the

receptor, while another fraction is degraded along with the ˇ and  chain of

the receptor [19,52]. Mathematical models [18] and experiments have estimated

rates of internalization and degradation of 0:04 min

1

[19]and0:035 min

1

[53]. Consequently, the model includes, in the equations for IL-2 dynamics,

15 IL-2, Helper and Regulatory CD4

T Cells 311

a degradation term proportional to the total number of IL-2 molecules bounded

to the receptors. This term sums the net effect of internalizing and recycling IL-2

at the cell surface. Different cell types may have different IL-2 degradation rates.

8. Processes, such as diffusion inside the Lymph Node, IL-2 association and

dissociation to receptors, or T cell conjugation and dissociation from APCs,

are assumed to be quasi steady-state equilibrium.

Characteristic free movement of T cells inside the lymph node has a typical

observed velocity of 10 m/min [54]. Thus if the diameter of a Lymph Node is

around 1 mm, a lymphocyte can go through it in less than 2 h. IL-2 association

and dissociation to its receptor is a process with a time scale of seconds, while

T cell conjugation to APCs can last for around 1 h [54]. These processes are,

in any case, faster than those cellular processes in the model, which alter T cell

population sizes, for instance cell proliferation (8–12 h) [55–58] and cell life

span (at least 7 days). Quasi steady-state assumptions are common in mathe-

matical biology [59]. They rely on identifying fast and slow processes in the

same dynamics. The approximation allow then to uncouple the fast and slow

dynamics, essentially by following the slow dynamic with ordinary differential

equations, where the variables of the fast dynamics are computed at equilibrium

as function of the slow dynamical variables for each given time. Quasi steady

state assumptions were already included in the crossregulation model [28],

taking T cells movement and conjugation to APCs as the fast dynamics process.

In the models here, this assumption is extended by assuming processes of IL-2

diffusion and its binding to IL-2 receptor also as fast processes.

Equations for the Dynamics of T Cell Population

in the Lymph Node

The model equations for the dynamics of CD4

helper (E) and regulatory (R) T cells

are:

D 

 K

1 

.s1/

C 2K

C ˛K



IL2



C .I l˛/

 K

(15.1)

D K

1 

.s1/

 K

(15.2)

D K



IL2



C .I l˛/



IL2



C .I l˛/

 K

(15.3)

312 K. Le´on and K. Garc´ıa-Mart´ınez

D 

 K

C 2K

C ˛K



IL2



 K

(15.4)

D K

 K

(15.5)

D K



IL2





IL2



 K

: (15.6)

Variables and parameters are deﬁned in Tables 15.1 and 15.2. Note that, two model

variants are encoded in these equations, being its realization dependent of the value

of switching parameter .

Equations (15.1)–(15.6) represent the following processes and interactions which

implement models postulates (see Fig. 15.1):

(a) Resting E and R cells are produced by an external source term, the thymus [ﬁrst

term of (15.1)and(15.4)]; They die at a constant rate [last term of (15.1)and

Table 15.1 Description of model variables and intermediate quantities

Item Description

Independent variables

Total number of resting, activated and cycling E Cells

Total number of resting, activated and cycling R cells

IL-2 Total number of IL-2 molecules in the Lymph Node

Intermediate quantities

Total number of resting, activated and cycling E Cells conjugated to

APCs

D E

Number of resting, activated and cycling E cells not conjugated to the

APCs (note that, free cells Dtotal cells-conjugated cells)

Total number of resting, activated and cycling R Cells conjugated to

APCs

D R

Number of resting, activated and cycling R cells not conjugated to the

APCs (note that, free cells Dtotal cells-conjugated cells)

IL2

;IL2

Mean number of IL-2 molecules bound per cell, respectively for the

resting, activated or cycling E cells

IL2

;IL2

Mean number of IL-2 molecules bound per cell, respectively for the

resting, activated or cycling R cells

IL2

Total IL-2 bound to receptors (D IL2

CIL2

IL2

CIL2

)

IL2

Number of IL-2 molecules not conjugated to the receptors (D IL2 

IL2

)