Dynamics of Immunological Models

O R I G I N A L A R T I C L E

Dynamics of Immunological Models

A. A. Pinto^•N. J. Burroughs^•M. Ferreira^• B. M. P. M. Oliveira

Received: 17 June 2010 / Accepted: 5 July 2010 / Published online: 4 August 2010 Springer Science+Business Media B.V. 2010

Abstract We analyse the effect of the regulatory T cells (Tregs) in the local control of the immune responses by T cells. We obtain an explicit formula for the level of antigenic stimulation of T cells as a function of the concentration of T cells and the parameters of the model. The relation between the concentration of the T cells and the antigenic stimulation of T cells is an hysteresis, that is unfold for some

A. A. Pinto (&)

Department of Mathematics, Faculty of Sciences, University of Porto;

LIAAD-INESC Porto LA, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal e-mail: aapinto1@gmail.com

A. A. Pinto

Center of Mathematics and Department of Mathematics and Applications, School of Sciences, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal

N. J. Burroughs

Mathematics Institute and Warwick Systems Biology Centre, University of Warwick, Coventry CV4 7AL, UK

e-mail: njb@maths.warwick.ac.uk M. Ferreira

ESEIG, Instituto Polite´cnico do Porto, Rua D. Sancho I, number 981, 4480-876 Vila do Conde, Portugal

e-mail: migferreira2@gmail.com M. Ferreira

LIAAD-INESC Porto LA, University of Porto, Porto, Portugal M. Ferreira

University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal B. M. P. M. Oliveira

Faculdade de Cieˆncias da Nutric¸a˜o e Alimentac¸a˜o, Universidade do Porto;

LIAAD-INESC Porto LA, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal e-mail: bmpmo@fcna.up.pt

DOI 10.1007/s10441-010-9117-6

parameter values. We study the appearance of autoimmunity from cross-reactivity between a pathogen and a self antigen or from bystander proliferation. We also study an asymmetry in the death rates. With this asymmetry we show that the antigenic stimulation of the Tregs is able to control locally the population size of Tregs. Other effects of this asymmetry are a faster immune response and an improvement in the simulations of the bystander proliferation. The rate of variation of the levels of antigenic stimulation determines if the outcome is an immune response or if Tregs are able to maintain control due to the presence of a transcritical bifurcation for some tuning between the antigenic stimuli of T cells and Tregs. This behavior is explained by the presence of a transcritical bifurcation.

Keywords Immunology TregsSecretion inhibitionODE model

1 Introduction

This chapter surveys, essentially, the works presented in the papers [2,3,4,5,6] and [7].

The primary function of the immune system is the protection of the host from pathogen invasion. During such an invasion, T cells specific to the antigen proliferate and under most circumstances successfully remove the pathogen. However, the immune system can also target self antigens (autoimmunity) and cause tissue damage and death. Regulatory T cells, or Tregs, have emerged as a fundamental component of the T cell repertoire, being generated in the thymus under positive selection by self peptides [10]. The Treg repertoire is as diverse as conventional T cells [10] and perform vital immune suppressive functions. Removal of Tregs, eg by (cell sorted) adoptive transfer experiments cause a variety of autoimmune disorders in rodents, whilst many autoimmune diseases can be associated with a misregulation of Tregs, eg IPEX [17]. Under exposure to their specific antigen, conventional T cells are activated leading to secretion of growth cytokines (predominantly interleukine 2, denoted IL2), and expression of the interleukine 2 receptor which triggers cytokine driven proliferation. However, in the presence of active Tregs the growth of conventional T cells is inhibited. Part of this growth inhibition is the inhibition of IL2 secretion by T cells [18,23]. Significantly, addition of IL2 abrogates inhibition, whilst IL2 appears to be a key intermediary of the dynamics between Tregs and conventional T cells [9,16, 21]. The process of Treg signalling to conventional T cells is still a matter of debate, evidence exists for both cell:cell mediated inhibition and soluble mediators such as TNFband IL10 [17]. It is likely that multiple methods of regulation are involved.

Further, most studies indicate that regulation is not T cell specific, i.e. Tregs inhibit all conventional T cells independent of their antigen specificity [22], although a different report suggests the contrary [20]. Tregs clearly function to limit such autoimmune responses with a delicate balance between appropriate immune activation and immune response suppression being achieved. How such a balance is established and controlled is the central focus of some papers [2]. The motivation is the observation that T cell proliferation through cytokines already has such a control structure;

cytokine driven growth exhibits a quorum population size threshold [1]. We propose in

[2] that Tregs locally adjust these thresholds by inhibiting IL2 secretion. The immune response-suppression axis is then a balance between the local numbers of activated T cells (e.g. from a pathogen encounter) and activated Tregs. This balance can be altered by cross reactivity [2] or through bystander proliferation [6]. We present our immune response model as a set of six ordinary differential equations. The analysis of the model is discussed in Sect.2, where we show that this model has a bistability region bounded by two thresholds of antigenic stimulation of T cells and we discuss the immune responses when there is cross reactivity between a pathogen and a self antigen. A modified immune response model is presented in Sect.3. In this model we consider that T cells in different states have distinct death rates. We observe that this model has a better behaviour than the model with symmetric death rates in the simulations of a bystander immune response. This model has a transcritic bifurcation for some tuning between the antigenic stimulation of the T cells and the antigenic stimulation of Tregs. As a consequence, the rate of increase of antigenic stimulation of T cells may determine, for parameter values near the transcritic bifurcation, if an immune response arises or not.

2 Immune Response Model

There are a number of different (CD4) T cell regulatory phenotypes reported; we propose a model of Tregs, which are currently identified as CD25^?T cells, although this is not a definitive molecular marker. At a genetic level, these Tregs express Foxp3, a master regulator of the Treg phenotype inducing CD25, CTLA-4 and GITR expression, all correlating with a suppressive phenotype [17].

We model a population of Tregs (denoted R,R*) and conventional T cells (T,T*) with processes shown schematically in Fig.1. Both populations require antigenic stimulation for activation, Tregs being activated by self antigens. Levels of antigenic stimulation are denoted a and b for Tregs and conventional T cells respectively. On activation conventional T cells both secrete IL2 and acquire

Fig. 1 Model schematic showing growth, death and phenotype transitions of the Treg populationsR,R*, and autoimmune T cellT,T* populations. Cytokine dynamics are not shown: IL2 is secreted by activated T cellsT* and adsorbed by all the T cell populations equally

proliferative capacity in the presence of IL2 while Tregs proliferate in the presence of IL2, although less efficiently than normal T cells [23], and they do not secrete IL2. Activated Tregs suppress IL2 secretion [23] thereby inhibiting T cell growth. However, if IL2 is present (CD4) T cells can still proliferate [17,18]. In the model we assume that T cells activated by exposure to their specific antigen have a cytokine secreting state (a normal activated state) and a non secreting state to which they revert at a constant ratek; thus in absence of antigen growth halts.

Tregs also induce a transition to the (inhibited) nonsecreting state, this transition rate is assumed proportional to the Treg population density. This transition can either be through direct cell:cell contact or be induced by soluble inhibitors [17], both of which give identical mass action kinetics over suitable density ranges. T cells regain secretion status on CD28 coreceptor stimulation [21], which we assume correlates with antigen exposure through an increased conjugate formation rate. Thus in the presence of costimulation and Tregs, the T cell population would be a mixture of partially inhibited, and normal T cells. Although we assume an antigen dependent rate of secretion inhibition reversion, similar results would be obtained with a constant reversion rate for example, i.e. if costimulation exposure is independent of antigen density. Note that exogenous IL2 does not reverse the suppressed phenotype, i.e. secretion status is not reacquired on cell proliferation [21].

Regulatory T cells are assumed to be in homeostasis, thus Treg density is controlled through some type of (nonlinear) competition. Homeostatic mecha- nisms of Tregs are currently poorly understood; in [2] we use a generic mechanism that utilises a cytokine (denoted J), analogous to interleukine 7 which is known to homeostatically regulate memory T cells [19]. We assume the cytokine is secreted by the local tissues, thereby sustaining a local population of Tregs activated by a probably tissue specific profile of self antigens. Tregs compete for this cytokine by adsorption and thus population homeostasis is achieved.

In Sect. 3 we discuss an asymmetry in the model where we assume that the secreting T cellsT* have a lower death rate than the non secreting T cells T and that the active Tregs R* also have a lower death rate than inactive Tregs R [4].

We also consider an inflow R_input of tregs instead of the J cytokine (both mechanisms yield similar results). This asymmetry in the death rates and the inflow R_input of Tregs allow the antigenic stimulation a of Tregs to regulate the size of the local population of Tregs. We also include a growth limitation mechanism; we use a (quadratic) Fas-FasL death mechanism [15], that is assumed to act on all T cells equally. Results will be similar with any saturation mechanism. Finally, we include an influx of (auto) immune T cells into the tissue (T_inputin cells per ml per day), which can represent T cell circulation or naive T cell input from the thymus.

A set of ordinary differential equations is employed to study the dynamics, with a compartment for each T cell population (inactive Tregs R, active Tregs R*, non secreting T cellsT, secreting activated T cellsT*), interleukine 2 densityIand the homeostatic Treg {\frac{{dJ}}{{dt}}} cytokineJ,

dR

dt ¼ ðqðIþJÞbðRþRþTþTÞ dÞR^ þkðR^ aRÞ;

dR

dt ¼ ðqðIþJÞ bðRþRþTþTÞ dÞR^ kðR^ aRÞ;

dJ

dt ¼^rðS ð^aðRþRÞ þ^dÞJÞ;

dT

dt ¼ ðqIbðRþRþTþTÞ dÞTþkðTbTþcRTÞ þTinput; dT

dt ¼ ðqIbðRþRþTþTÞ dÞTkðTbTþcRTÞ; dI

dt¼rðT ðaðRþRþTþTÞ þdÞIÞ:

ð1Þ

Parameters are defined in [2]. Our model has components that have been used in previous models, for instance cytokine dependent growth [1,14], cytokine kinetics [24], Fas-FasL mediated death [8], and positive feedback of T cells on Tregs [11, 12], in our model this is explicitly though IL2.

The only important aspects of this model are a mechanism to sustain a population of Tregs, secretion inhibition of T cells with a rate that correlates with Treg population size, and growth and competition for IL2 with a higher growth rate of T cells relative to Tregs. Other aspects of the model can be altered with only quantitative differences in behaviour.

2.1 The Fine Balance Between Tregs and T Cells

We study the equilibria of the immune system in a neighbourhood of the default values for the parameters and variables. The concentration of T cells varies between a minimum value corresponding to thehomeostasisconcentration of T cellsT_hom, ie when there is no antigenic stimulation of T cells (b=0), and a maximum value, that we call the capacity of T cells Tcap, which is obtained for high levels of antigenic stimulation of T cells (b= ??). Using Eq. (4) and the parameters from Table 1 in [2], the values T_hom and T_capare implicitly determined as zeros of a polynomial. In particular, for the default values of the parameters, these values are given byT_hom=9.6910²cells per ml andT_cap=9.7910⁶cells per ml. When

Fig. 2 The equilibria manifold for Thymic inputsTinput

[[1,10,000] with the other parameters at their default values. Low values ofbare bluishand higher values are reddish

the system is at equilibrium, we present, in Eq. (2), the relation between the concentration of T cells and the concentration of the Tregs for values of the concentration of T cells betweenT_homandT_cap.

LetY₁,Y₂,Y₃be the following polynomials Y1ðxÞ ¼ ^aCðxÞ b^dBðxÞ Y2ðxÞ ¼2^abBðxÞ

Y3ðxÞ ¼Y₁²ðxÞ 2ðdCðxÞ qSxÞY2ðxÞ;

whereB(x)=(1-e)xandC(x)=eT_input?B(x)(bx?d).

LetX₁,X₂,X₃be the following polynomials X₁ðyÞ ¼BCðyÞDðyÞ qS X2ðyÞ ¼ 2bBCðyÞ

X3ðyÞ ¼X₁²ðyÞ 2TinputCðyÞX2ðyÞ;

whereB=1-eandCðyÞ ¼^ayþ^d;andD(y)=by?d.

Letx=T?T* be the total concentration of T cells andy=R?R* be the total concentration of Tregs.

When the system is at equilibrium, the concentration of Tregs y=R?R* is given by theTreg curve

yðxÞ ¼Y1ðxÞ þ ffiffiffiffiffiffiffiffiffiffiffi Y3ðxÞ p

Y₂ðxÞ ; ð2Þ

wherex=T?T* is the total concentration of T cells. Conversely, the concen- tration of T cellsx(y) is determined by

xðyÞ ¼X1ðyÞ ffiffiffiffiffiffiffiffiffiffiffi X3ðyÞ p

X₂ðyÞ ; ð3Þ

wherey=R ?R* is the total concentration of Tregs (see [5]).

The maximum concentrationRmaxof Tregs is a zeroX2(y)=0 of a fourth order polynomialX2(y) and, so,Rmaxhas an explicit solution. In particular, for the default values of the parameters, the maximum concentration R_max of Tregs is given by R_max=2.1910⁴cells per ml, and the corresponding concentration of T cells is given by x(R_max)=1.9910⁴ cells per ml. The minimum concentration R_min of Tregs is given byR_min=156 cells per ml, and the corresponding concentration of T cells is given byT_cap=9.7910⁶cells per ml.

2.2 The Equilibria Manifold

Let theantigen function b(x,y) be given by

bðx;yÞ ¼ uðx;yÞðkxð1þcAyÞ þTinputÞ

kð1Þqx³ð^ayþ^dÞ kxuðx;yÞ; ð4Þ where A=a/(1?a) and uðx;yÞ ¼ ðqSxT_inputð^ayþ^dÞÞðaðxþyÞ þdÞ. Let x(y) andy(x) be as in Eqs. 2and3. The level of the antigenic stimulation of T cells

is given byb(x,y(x)), or, equivalently, by b(x(y),y), when the system is at equi- librium (stable or unstable). Conversely, given an antigenic stimulation levelbof T cells, the concentrationxof T cells and the concentrationyof Tregs are zeros of the twelfth order polynomials that can be explicit constructed (see [3,5]). In Eq.5, we describe the level of the antigenic stimulation of T cells from the concentrationxof the T cells, for the simplified model without Tregs (i.e. y=0). Let bðxÞ~ be the antigen function in the absence of Tregs given by

bðxÞ ¼~ ðaxþdÞðkxþT_inputÞðbx²þdxT_inputÞ

kxðqx²þ ðTinputbx²dxÞðaxþdÞÞ: ð5Þ The level of the antigenic stimulation of T cells is given bybðxÞ;~ when the system is at equilibrium (stable or unstable). Conversely, given an antigenic stimulation level b, the concentrationxof T cells is a zero of the fourth order polynomial that can be explicit constructed (see [5]).

2.3 Sensibility of the Antigenic Threesholds

The antigenic stimulation is likely to change in time, either by fluctuations or caused by pathogens. If the antigenic stimulation rises above the thresholdb_H, an immune response is triggered. Reversion to the controlled state is only acquired when the antigenic stimulation decreases to a value bellow the thresholdb_L\b_H. Figure 3 in [2] illustrates the bifurcation plot with respect to antigenic stimulation b. When the system is at equilibrium, the threshold values of antigen stimulation b_L and b_H of T cells are determined using zeros of a polynomial. The antigen threshold function V(x,y,z) is equal to V₆(y,z) x⁶?...? V₀(y,z), where the functionsV₀(y,z), ...,V₆(y,z) are the polynomials presented in [5]:

V0ðy;zÞ ¼kC²F²T_input³ V1ðy;zÞ ¼2kCFT_input² H

V2ðy;zÞ ¼kTinputðH²þCFTinputð3qCEcACFkz2aGÞÞ V₃ðy;zÞ ¼kT_inputð2FGðaGqCEþcBCFkzÞ þ2qCEFkð1þcAyÞ

aCTinputð2cAFkzþqEz2qEþ2aGÞÞ

V₄ðy;zÞ ¼kðCðGðqEðaTinputð1zÞ þkFð1þcAyÞÞ 4kzacAFTinputÞ kCTinputðaqEðz1Þð1þcAyÞ þzcAðqEFþa²TinputÞÞÞ þGðkzcBF²Gþa²GTinputqz^aEFTinputÞÞ

V5ðy;zÞ ¼qGkðcBFkð^dqE2aGÞ þqEðkaCð1þcAyÞ k^aFa^aTinputÞÞz V₆ðy;zÞ ¼aGk²ðacAGþEqðc^dA^aÞÞz;

where

A¼a=ð1þaÞ;B¼bdad;CðyÞ ¼^ayþ^d;DðyÞ ¼byþd;E¼1epsilon; FðyÞ ¼ayþ d;G¼qS andH(y)=aC(y)T_input-F(y)G.

A threshold of the antigenic stimulationb_Mof T cells exists, if, and only if, there is a zerox_M [[T_hom,T_cap] of the antigen threshold functionV(x,y(x),y⁰(x)) such

thatb_M=b(x_M,y(x_M)), whereM [{L,H}. The equality V(x,y(x),y⁰(x))=0 is equivalent toVðxÞ ¼~ 0;whereVðxÞ~ is a polynomial that can be explicit constructed (see [5]). If the antigenic stimulation b of T cells rises above the threshold b_H control is lost and autoimmunity arises. After an autoimmune response, the control state is recovered when the stimulation falls below a lower threshold b_L. This phenomena is due to the equilibria manifold being an hysteresis. For the parameters that unfold the hysteresis the antigenic threesholdsb_Landb_Hform a cusp. Thecusp bifurcationat the antigenic stimulationb_Cof T cells is an origin of the unfold of the hysteresis, with respect to a parameter, and, so, biologically relevant. The concentrationxCof the T cells corresponding to levels of the antigenic stimulation b_C=b(x_C,y(x_C)) satisfies the following equalities

Vðx;yðxÞ;y⁰ðxÞÞ ¼0 and

Wðx;yðxÞ;y⁰ðxÞ;y⁰⁰ðxÞÞ ¼0; ð6Þ where

Wðx;y;z;vÞ ¼Xⁿ

i¼1

Viðy;zÞxⁱ¹ i þXⁿ

i¼0

oViðy;zÞ

oy xⁱzþoViðy;zÞ oz xⁱv:

2.4 Dynamics of Cross-Reactive Proliferation

Humans are continually exposed to pathogens which invoke T cell activation and immune responses. This has consequences on Treg control of autoimmune states since immune responses produce IL2 and thus can lead to bystander growth and loss of Treg homeostasis, or in the case of cross reactivity direct stimulation of auto reactive T cells.

We simulate a pathogen exposure in a tissue with initial Treg control of an autoimmune population where there is cross reactivity between the pathogen specific T cells and self (details in [2]). Depending on the strength of the autoimmune antigen we observe either a loss of control, i.e. Tregs fail to reacquire control, or autoimmune suppression is reinstated post infection. The former case

Fig. 3 The hysteresis of the equilibria manifold for Thymic inputsTinput [1,10,000], with the other parameters at their default values. The hysteresis unfolds for high values of the parameterTinput.aLow values ofy=R?R* are bluish and higher values are reddish. bLow values ofx=T?T* are bluish and higher values are reddish

corresponds to autoimmune antigenic stimulation levels bL\b\bH, although escape is not guaranteed as the immune response must be of sufficient duration to allow autoimmune T cells numbers to rise and sufficiently outnumber Tregs (about 5.5 days in our simulations). Autoimmune T cells withb\b_Lare always controlled post infection, Fig.4a. The length of the period post infection for autoimmune control to be reestablished is determined by the slow death rate of T cells; we did not include a specific downsizing mechanism or memory differentiation in the model which would reduce this period. Escape dynamics can also show a delay in the onset of autoimmunity when infection duration is short, Fig.4c. Cross reactivity enhances escape; however pure bystander growth can also lead to escape although longer infection periods and higher b are required for escape. We also note that infections transiently enhance Treg populations, either eventually returning to homeostasis, or in the case of autoimmune responses being themselves suppressed;

in our model this is because of their susceptibility to Fas-FasL mediated apoptosis.

Other population saturation models, specifically assumptions of the impact on Tregs of T cell saturation, can give different levels of Tregs.

3 Asymmetry Between the Active and Inactive T Cells

In this section, we consider an asymmetry reflecting that the difference between the growth and death rates can be higher for the active T cells and the active Tregs than for the inactive T cells and inactive Tregs as in [4]. This asymmetry can be explained by the effect of memory T cells. The memory T cells last longer than the other T cells and react more promptly to their specific antigen. This leads to a positive correlation between the antigen stimulation and the difference between growth rate and the death rate of T cells, as we assume in this model. This asymmetry brings up the relevance of the antigenic stimulation of Tregs in the control of the local Treg population size. Hence, under homeostasis, larger antigenic

(A) (B) (C)

Fig. 4 T cell cross reactivity between a pathogen and self. aCross reactive autoimmune T cells on pathogen infection can return to a controlled state if antigenic stimulation from self (b=0.1) is low.

bFor highb=0.5 stimulation from self, an initially controlled autoimmune T cell clone escapes when infection occurs with a cross reacting pathogen. cIdentical to B. except infection interval is 7 days. A pathogen infection is modelled as a step increase inbto 5000 for day 0–10 (a, b), 0–7 (c) to model both the increase in antigen (cross reactive) and costimulation.Red solid: relative concentration of Tregs as a function of timey(t)=(R?R*)/Tcap.Black dashed: relative concentration of T cells as a function of timex(t)=(T?T*)/Tcap

stimulation of Tregs results in larger Treg population size. Furthermore, there is a positive correlation between the antigenic stimulation of Tregs and the thresholdsb_L andb_Hof antigenic stimulation of T cells. Hence, organs with different antigenic stimulation of Tregs have different levels of protection against the development of an immune response by T cells, under the presence of their specific antigen. In [4], it is described the effect of the positive correlation between the antigenic stimulation of T cells and the antigenic stimulation of Tregs, due to the antigen presenting cells (APC), such as dendritic cells, to present both self antigens and foreign antigens. In [4], with a structurally different model, it is attained the same conclusions, as in [13], due to the asymmetry in the difference between the growth and death rates.

This behavior is mathematically explained by the presence of a transcritical bifurcation. Slow rates of increase of the antigenic stimuli do not lead to an immune response, but fast rates of increase of the antigenic stimuli can trigger an immune response. The model proposed consists of a set of ordinary differential equations and is employed to study the dynamics, with a compartment for each T cell population (inactive Tregs R, active Tregs R*, non secreting T cellsT, secreting activated T cells T*) and interleukine 2 density I, similar to Eq.1 but it is considered an influx of Tregs into the tissue (R_inputin cells per ml per day), similarly to the influxT_inputof T cells for the Tregs:

dR

dt ¼ ðqIbðRþRþTþTÞ d_RÞRþkðR^ aRÞ þR_input; dR

dt ¼ ðqIbðRþRþTþTÞ dRÞRkðR^ aRÞ;

dT

dt ¼ ðqIbðRþRþTþTÞ dTÞTþkðTbTþcRTÞ þ Tinput; dT

dt ¼ ðqIbðRþRþTþTÞ dTÞTkðTbTþcRTÞ;

dI

dt¼rðT ðaðRþRþTþTÞ þdÞIÞ;

The parameters range are defined in [2] and the new parameters considered here areR_input=T_input,d_R=d_T=0.1,d_R^*=d_T^*=0.01. The new aspect of this model, comparing with the model presented in [2], is the asymmetry in the difference between the growth and death rates.

3.1 Transcritical Bifurcation

In [4], it is studied the positive correlation between the antigenic stimulationa of Tregs and the the antigenic stimulationbof T cells. Since both are presented by the antigen presenting cells (APC). For simplicity, it is presented a linear relation between these stimuli in the form: a=a₀?m b. If the the levels of antigenic stimulationaof the Tregs and the levels of antigen stimulationbof the T cells are independent, i.e. the slopemis equal to zero, an hysteresis is present regardless of the value of the antigenic stimulationaof the Tregs. The two antigenic thresholds bLandbHbound the bistability region of the hysteresis. Similarly to the case when the antigenic stimulationaof Tregs is independent of the antigenic stimulationbof

T cells, an hysteresis is present for small values 0Bm\m_Sof the slope m. We observe, when m=m_S, the appearance of a saddle-node bifurcation point, disconnected from the hysteresis. For values of the slopem_S\m\m_X, a loop is present, disconnected from the hysteresis. Whenm=mX, is appears a transcritical bifurcation when the loop touches the hysteresis at the thresholdbHof the hysteresis (see Fig.5).

For values ofm[m_Xthe loop merges with the hysteresis, corresponding to a system with a wider hysteresis.

Choosingm[m_X, the slopemcan be related to autoimmunity, since a reduction of the slope will decrease the antigenic thresholdb_H of T cells, and in particular when the slopem=m_Xthere is a discontinuity inb_Hwith slopes belowm_Xcreating thresholdsb_H400 times lower than the thresholdsb_Hfor slopes abovem_X. 3.2 Fast and Slow Variation of Antigenic Stimulation

The antigenic stimulation of T cells and Tregs is likely to change with time either rapidly, e.g. due to an infection with the consequent an immune response as an outcome; or slowly, caused by the natural modifications of the organism with age and with the T cells being kept in control by the Tregs.

In [4], it is modeled these situations by considering that the antigenic stimulation b of T cells increases from b₀\b_L to b_?[b_H in an interval of time. It is considered that the antigen stimulation a of Tregs varies with the antigen stimulationbof T cells in the linear relation:a(b)=a₀?m b, witha₀=1/2.4 and slopem=2[m_X. If the slopemis above the value of the transcritical bifurcation m[m_X=1.64..., it is observed that a fast variation of the antigenic stimulation triggers an immune response, i.e. (see Fig.6a) the concentrationT?T* of T cells is high, near the capacity valueT_cap, with a transient increase in the concentration R?R* of Tregs. On the other hand, in a slow variation of the antigenic stimulation, (see Fig.6b) the concentrationT?T* of T cells has a temporary small increase and the increase in the concentrationR ?R* of Tregs is able to sustain a controlled

Fig. 5 Equilibria for different tuningsa=a0?mbof the antigenic stimulations, witha0=1/2.4 and m=1.64….Black lines: concentration of T cellsx(b) as a function of the self antigenic stimulationb;

Red lines: concentration of Tregsy(b) as a function of the self antigenic stimulationb.Solid linesindicate stable equilibria anddashed linesindicate unstable equilibria. A transcritical bifurcation is present for b=1.88910^-1andx=1.36910^-3andy=8.60910^-4

state, thus resulting in a subclinical behaviour that enhances the protection of the tissue against autoimmunity.

3.3 Dynamics of Bystander Proliferation

The asymmetry maintains the basic features of the model, namely, the existence of a controlled stable steady state (with low concentration of T cells) and the existence of an immune response stable steady state (with high concentration of T cells), depending of the antigenic stimulationb of T cells and the initial conditions. The thresholdsb_L and b_H of antigenic stimulation of T cells, are similar to the ones presented in Sect.2.3. We observe that the modified immune response model has slightly faster growth rate of the T cells, in particular for high antigenic stimulations bof T cells due to the lower average death rate of T cells.

In the case of the cross reactive direct stimulation, the results are analogous to the immune response model (see Sect. 2.4): the final state of the model is either a controlled state or an immune response state, the last one being achieved if the stimulation of the autoimmune antigenb is betweenb_Landb_Hand the duration of the immune response is of sufficient duration.

The simulations of the bystander proliferation have differences between the immune response model and the model presented in this Section. In the simulations of a bystander proliferation, we considered two lines of T cellsT_b;T_bandT_c;T_cthat have, respectively, pathogenic stimulationband autoimmune antigenic stimulation c(equations presented in [6]).

We consider a tissue with initial controlled state of both T cells lines and we simulate a pathogen infection as a step increase inbfrom 0 to 1000 between days 0 and 7 (other choices of suitable pathogen dynamics give analogous results). In the

(A) (B)

Fig. 6 Effect of different rates of increase ofb. The level of antigenic stimulationbof T cells varies between 0.01 and 10 in a log-sigmoid function of time with steepness factors. The antigenic stimulation aof Tregs isa(b)=1/2.4?2b.Black dashes: total concentration of the T cells as a function of time x(t)=(T?T*)/Tcap.Red line: total concentration of the Tregs as a function of timey(t)=(R?R*)/

Tcap.Blue dash-dots: antigenic stimulationbof T cells as a function of time (presented asb(t)/100 to fit in the window scale).Vertical grey dots: median time of the variation (100 days). aFast variations of the antigenic stimulation (s=10 days) give an immune response. bFor slow variations of the antigenic stimulation (s=100 days) the T cells remain controlled

simulations in Fig.7we choose the autoimmune antigenic stimulationc=0.1 to be a constant value between the thresholdsc_Landc_H of antigenic stimulation. If the autoimmune stimulation of T cells was low (c\c_L) the concentration of autoimmune secreting T cells would not reach the necessary value to sustain autoimmunity after pathogen clearance and Tregs would be able to regain control.

On the other hand, if the autoimmune stimulation of T cells was high (c[c_H) it would be impossible to have an initial controlled autoimmune state. In the simulations presented in Fig.7we observe that the concentration of T cells from the autoimmune lineTc;T_cincreases with the concentration of the T cells responding to the pathogenTb;T_b due to the increase in the secretion of I cytokines byT_b.

In the end of the pathogen exposure (after 7 days) the secreting autoimmune T cells T_c generate enough I cytokine to sustain the population of autoimmune T cells T_c;T_c in high concentrations, thus developing an autoimmune response caused by their bystander proliferation. There is also a transient increase in the population of Tregs due to theIcytokine followed by suppression of Tregs due to the Fas-FasL mediated apoptosis. The onset of autoimmunity depends both of the duration of the pathogen exposure and of the model considered.

In the symmetric case withdT ¼dR¼dR¼dT¼0:1 per day, we observe that the autoimmune response stabilizes around 10 weeks after the pathogen exposure (see Fig.7a). The autoimmunity arises approximately after 8 weeks after the pathogen exposure in the asymmetric case (see Fig.7) where we considered a lower death rate of secreting T cells and active Tregs (d_T=d_R=0.1 per day anddR¼ dT¼0:01 per day).

Acknowledgements We thank LIAAD-INESC Porto LA, Calouste Gulbenkian Foundation, PRODYN- ESF, POCTI and POSI by FCT and Ministe´rio da Cieˆncia e da Tecnologia, and the FCT Pluriannual Funding Program of the LIAAD-INESC Porto LA and of the Research Center of Mathematics of

(A) (B)

Fig. 7 Bystander proliferation for different models. aThe immune response model: the T cells not responding to the pathogen have equal concentration to the T cells responding to the pathogen. bThe modified immune response model: the T cells not responding to the pathogen have lower concentration than the T cells responding to the pathogen.Black dots: relative concentration of T cells that respond to the pathogen as a function of timexðtÞ ¼ ðTbþT_bÞ=Tcap.Green dashes: relative concentration of T cells that do not respond to the pathogen as a function of timezðtÞ ¼ ðTcþT_cÞ=Tcap.Red solid: relative concentration of Tregs as a function of timey(t)=(R?R^*)/Tcap

University of Minho, for their financial support. M. Ferreira also acknowledges the financial support from the FCT grant with reference SFRH/BD/27706/2006. Part of this research was developed during a visit by the authors to the CUNY, IHES, IMPA, MSRI, SUNY, Isaac Newton Institute, University of Berkeley and University of Warwick. We thank them for their hospitality.

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