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November  2018, 23(9): 3969-4002. doi: 10.3934/dcdsb.2018120

A multiscale model of the CD8 T cell immune response structured by intracellular content

1. 

Univ Lyon, Ecole centrale de Lyon, CNRS UMR 5208, Institut Camille Jordan, 36 avenue Guy de Collonge, 69134 Ecully Cedex, France

2. 

Inria, Univ Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

3. 

Univ Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

* Corresponding author: Fabien Crauste

Received  May 2017 Revised  November 2017 Published  April 2018

During the primary CD8 T cell immune response, CD8 T cells undergo proliferation and continuous differentiation, acquiring cytotoxic abilities to address the infection and generate an immune memory. At the end of the response, the remaining CD8 T cells are antigen-specific memory cells that will respond stronger and faster in case they are presented this very same antigen again. We propose a nonlinear multiscale mathematical model of the CD8 T cell immune response describing dynamics of two inter-connected physical scales. At the intracellular scale, the level of expression of key proteins involved in proliferation, death, and differentiation of CD8 T cells is modeled by a delay differential system whose dynamics define maturation velocities of CD8 T cells. At the population scale, the amount of CD8 T cells is represented by a discrete density and cell fate depends on their intracellular content. We introduce the model, then show essential mathematical properties (existence, uniqueness, positivity) of solutions and analyse their asymptotic behavior based on the behavior of the intracellular regulatory network. We numerically illustrate the model's ability to qualitatively reproduce both primary and secondary responses, providing a preliminary tool for investigating the generation of long-lived CD8 memory T cells and vaccine design.

Citation: Loïc Barbarroux, Philippe Michel, Mostafa Adimy, Fabien Crauste. A multiscale model of the CD8 T cell immune response structured by intracellular content. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3969-4002. doi: 10.3934/dcdsb.2018120
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show all references

References:
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A. Abbas, A. Lichtman and S. Pillai, Basic Immunology: Functions and Disorders of the Immune System, Fourth edition edition, Elsevier/Saunders, Philadelphia, PA, 2014. Google Scholar

[2]

D. AlemaniF. PappalardoM. PennisiS. Motta and V. Brusic, Combining cellular automata and Lattice Boltzmann method to model multiscale avascular tumor growth coupled with nutrient diffusion and immune competition, J. Immunol. Methods, 376 (2012), 55-68.  doi: 10.1016/j.jim.2011.11.009.  Google Scholar

[3]

G. Altan-Bonnet and R. Germain, Modeling T cell antigen discrimination based on feedback control of digital ERK responses, PLoS Biology, 3 (2005), e356. doi: 10.1371/journal.pbio.0030356.  Google Scholar

[4]

R. AntiaC. BergstromS. PilyuginS. Kaech and R. Ahmed, Models of CD8+ Responses: 1. What is the Antigen-independent Proliferation Program, J. Theor. Biol., 221 (2003), 585-598.  doi: 10.1006/jtbi.2003.3208.  Google Scholar

[5]

R. AntiaV. Ganusov and R. Ahmed, The role of models in understanding CD$ 8^{+} $ T-cell memory, Nat. Reviews, 5 (2005), 101-111.  doi: 10.1038/nri1550.  Google Scholar

[6]

V. Appay and S. Rowland-Jones, Lessons from the study of T-cell differentiation in persistent human virus infection, Semin. Immunol., 16 (2004), 205-212.  doi: 10.1016/j.smim.2004.02.007.  Google Scholar

[7]

J. ArsenioB. KakaradovP. MetzS. KimG. Yeo and J. Chang, Early specification of CD8+ T lymphocyte fates during adaptive immunity revealed by single cell gene-expression analyses, Nat. Immunol., 15 (2014), 365-372.  doi: 10.1038/ni.2842.  Google Scholar

[8]

M. Bernaschi and F. Castiglione, Design and implementation of an immune system simulator, Comput. Biol. Med., 31 (2001), 303-331.  doi: 10.1016/S0010-4825(01)00011-7.  Google Scholar

[9]

M. BernaschiS. Succi and F. Castiglione, Large-scale cellular automata simulations of the immune system response, Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics, 61 (2000), 1851-1854.  doi: 10.1103/PhysRevE.61.1851.  Google Scholar

[10]

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[11]

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[12]

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[13]

A. CappuccioP. Tieri and F. Castiglione, Multiscale modelling in immunology: A review, Brief. Bioinformatics, 17 (2016), 408-418.  doi: 10.1093/bib/bbv012.  Google Scholar

[14]

J. ChangE. Wherry and A. Goldrath, Molecular regulation of effector and memory T cell differentiation, Nat. Immunol., 15 (2014), 1104-1115.  doi: 10.1038/ni.3031.  Google Scholar

[15]

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[16]

F. CrausteJ. MafilleL. BoucinhaS. DjebaliO. GandrillonJ. Marvel and C. Arpin, Identification of nascent memory cd8 t cells and modeling of their ontogeny, Cell Systems, 4 (2017), 306-317.  doi: 10.1016/j.cels.2017.01.014.  Google Scholar

[17]

F. CrausteE. TerryI. Le MercierJ. MafilleS. DjebaliT. AndrieuB. MercierG. KanekoC. ArpinJ. Marvel and O. Gandrillon, Predicting pathogen-specific CD8 T cell immune responses from a modeling approach, J. Theor. Biol., 374 (2015), 66-82.  doi: 10.1016/j.jtbi.2015.03.033.  Google Scholar

[18]

W. CuiN. JoshiA. Jiang and S. Kaech, Effects of signal 3 during CD8 T cell priming: Bystander production of IL-12 enhances effector T cell expansion but promotes terminal differentiation, Vaccine, 27 (2009), 2177-2187.  doi: 10.1016/j.vaccine.2009.01.088.  Google Scholar

[19]

L. Davis and F. Adler, Mathematical models of memory CD8+ T-cell repertoire dynamics in response to viral infections, Bull. Math. Biol., 75 (2013), 491-522.  doi: 10.1007/s11538-013-9817-6.  Google Scholar

[20]

R. De BoerM. OpreaR. AntiaK. Murali-KrishnaR. Ahmed and A. Perelson, Recruitment times, proliferation, and apoptosis rates during the CD8(+) T-cell response to lymphocytic choriomeningitis virus, J. Virol., 75 (2001), 10663-10669.   Google Scholar

[21]

K. E. EwingsC. M. Wiggins and S. J. Cook, Bim and the pro-survival Bcl-2 proteins: Opposites attract, ERK repels, Cell Cycle, 6 (2007), 2236-2240.   Google Scholar

[22]

O. Feinerman, G. Jentsch, K. Tkach, J. Coward, M. Hathorn, M. Sneddon, T. Emonet, K. Smith and G. Altan-Bonnet, Single-cell quantification of IL-2 response by effector and regulatory T cells reveals critical plasticity in immune response, Mol. Syst. Biol., 6 (2010), p437. doi: 10.1038/msb.2010.90.  Google Scholar

[23]

A. FriedmanC. Kao and C. Shih, Asymptotic phases in a cell differentiation model, Journal of Differential Equations, 247 (2009), 736-769.  doi: 10.1016/j.jde.2009.03.033.  Google Scholar

[24]

A. FriedmanC. Kao and C. Shih, Asymptotic limit in a cell differentiation model with consideration of transcription, Journal of Differential Equations, 252 (2012), 5679-5711.  doi: 10.1016/j.jde.2012.02.006.  Google Scholar

[25]

V. Ganusov, Discriminating between different pathways of memory CD8+ T cell differentiation, J. Immunol., 179 (2007), 5006-5013.  doi: 10.4049/jimmunol.179.8.5006.  Google Scholar

[26]

X. Gao, C. Arpin, J. Marvel, S. Prokopiou, O. Gandrillon and F. Crauste, IL-2 sensitivity and exogenous IL-2 concentration gradient tune the productive contact duration of CD8(+) T cell-APC: a multiscale modeling study, BMC Syst. Biol., 10 (2016), p77. doi: 10.1186/s12918-016-0323-y.  Google Scholar

[27]

A. D. GuerreroR. L. WelschhansM. Chen and J. Wang, Cleavage of anti-apoptotic Bcl-2 family members after TCR stimulation contributes to the decision between T cell activation and apoptosis, The Journal of Immunology, 190 (2013), 168-173.  doi: 10.4049/jimmunol.1201610.  Google Scholar

[28]

J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993.  Google Scholar

[29]

N. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, Journal of the London Mathematical Society, 25 (1950), 226-232.  doi: 10.1112/jlms/s1-25.3.226.  Google Scholar

[30]

D. HildemanY. ZhuT. MitchellJ. Kappler and P. Marrack, Molecular mechanisms of activated T cell death in vivo, Curr. Opin. Immunol., 14 (2002), 354-359.  doi: 10.1016/S0952-7915(02)00335-7.  Google Scholar

[31]

T. Höfer, O. Krichevsky and G. Altan-Bonnet, Competition for IL-2 between regulatory and effector T cells to chisel immune responses, Front. Immunol., 3 (2012). doi: 10.3389/fimmu.2012.00268.  Google Scholar

[32]

A. IntlekoferE. Wherry and S. Reiner, Not-so-great expectations: Re-assessing the essence of T-cell memory, Immunol. Rev., 211 (2006), 203-213.  doi: 10.1111/j.0105-2896.2006.00396.x.  Google Scholar

[33]

S. Kaech and R. Ahmed, Memory CD8+ T cell differentiation: Initial antigen encounter triggers a developmental program in naive cells, Nat. Immunol., 2 (2001), 415-422.  doi: 10.1038/87720.  Google Scholar

[34]

S. Kaech and W. Cui, Transcriptional control of effector and memory CD8+ T cell differentiation, Nat. Rev. Immunol., 12 (2012), 749-761.  doi: 10.1038/nri3307.  Google Scholar

[35]

P. KimP. Lee and D. Levy, Modeling regulation mechanisms in the immune system, J. Theor. Biol., 246 (2007), 33-69.  doi: 10.1016/j.jtbi.2006.12.012.  Google Scholar

[36]

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Figure 1.  Schematic representation of the differentiation scheme used in this work, including cell differentiation states and their interactions. The different subpopulations mainly distinguish themselves by their abilities to survive, to proliferate and to address the infection. Dotted lines for death represent smaller death rates than regular lines
Figure 2.  Simplified interaction network between Ki67, Bcl2 and the pathogen. Upon stimulation of the CD8 T cell by an APC, Ki67 and Bcl2 levels of expression rise. Ki67 sustains its own expression with a fast positive feedback loop while a slow negative feedback loop kicks in late and is associated with exhaustion of proliferating cells. Bcl2 sustains its expression independently of other stimuli and its expression is inhibited by Ki67
Figure 3.  Illustration of the location process of newly activated cells at $t = t_k$ with $n = 4$. The pink disk represents the support of the source of activated cells. The density of newly activated cells at time $t_k$ is approximated by $n^2$ piles of cells. The black disc in each square represents the location of the corresponding pile of cells and its surface represents its weight. That weight is proportional to the intersection between the square and the pink disc
Figure 4.  Expected behavior of the multiscale model in the maturity space. Before the primary response, the main domain is empty and all the CD8 T cells are naive. Then (Panel A), APC stimulate the naive cells which start to differentiate into activated cells, migrate in $\Omega_A$ and move in the ($\mu_1, \mu_2$) maturity plane following $(v_1, v_2)$, as described by (3). For the secondary response (Panel B), there is no naive cell left, but there already are cells in the domain: memory cells. The appearance of APC activates the quiescent memory cells which move once again following $(v_1, v_2)$
Figure 5.  Eigenvalues of the characteristic equation (12) for $\tau\in[0;30]$. Parameter values were chosen such that $f'(\overline{x})\overline{x} = 0.3$, $g'(\overline{x})\overline{x} = 0.4$. For $\tau$ small enough, all branches are in the half plane $Re(\lambda)<0$. Then, the first branch crosses the imaginary axis at $(0, \omega)$ where $\omega = \overline{x}\sqrt{g'(\overline{x})^2-f'(\overline{x})^2}$. All branches cross at the same point as shown in Proposition 2. For $\tau$ large enough, there even exist strictly positive eigenvalues
Figure 6.  Simulation of $\rho_\varepsilon$ in the case where (1) has a strictly positive stable steady state. Red dashed lines represent the limits of domains $\Omega_A$, $\Omega_{EE}$, $\Omega_{LE}$ and $\Omega_M$ ($\alpha$ and $\beta$ values). The pictures show $\rho_\varepsilon$ at different time steps: following a primary stimulation, CD8 T cells are activated and differentiate into early effector cells $(a)$, then to late effector cells $(b)$, and finally to memory cells $(c)$; upon re-stimulation by the same pathogen, memory CD8 T cells are activated $(d)$, and start a new cycle, differentiating into early and late effector cells $(e)$ and again in memory cells $(f)$
Figure 7.  Subpopulation counts for the primary response displayed in Figure 6.(a) to 6.(c). A log-scale is used for the y-axis, a standard representation of CD8 T cell counts. Naive (a), activated (b), early effector (c), late effector (d) and memory (e) cell counts are represented over 20 days post-infection (the infection occurs at day 0), as well as the total CD8 T cell counts (f). Experimental data come from Crauste et al.[16], CD8 T cell counts have been measured in mice infected with vaccinia virus
Figure 8.  Simulation of $\rho_\varepsilon$ in the case where (1) has an unstable positive steady state. Legend is similar to Figure 6, except that in the memory phase (Figures $(c)$ and $(f)$) one clearly observes that memory cells present more heterogeneity thanks to the limit cycle appearing in the $(\mu_1, \mu_2)$-space
Table 1.  Main variables of the model
VariablesDescription
$N(t)$Naive CD8 T cell count at time $t$
$P(t)$Pathogen count at time $t$
$E(t)$Early and Late Effector CD8 T cell count at time $t$
$\mu_1(t)$Level of expression of Ki67 at time $t$
$\mu_2(t)$Level of expression of Bcl2 at time $t$
$\rho(t, \mu_1, \mu_2)$Density of CD8 T cells with maturities $(\mu_1, \mu_2)$ at time $t$
$v_1$Maturation velocity of $\mu_1$
$v_2$Maturation velocity of $\mu_2$
VariablesDescription
$N(t)$Naive CD8 T cell count at time $t$
$P(t)$Pathogen count at time $t$
$E(t)$Early and Late Effector CD8 T cell count at time $t$
$\mu_1(t)$Level of expression of Ki67 at time $t$
$\mu_2(t)$Level of expression of Bcl2 at time $t$
$\rho(t, \mu_1, \mu_2)$Density of CD8 T cells with maturities $(\mu_1, \mu_2)$ at time $t$
$v_1$Maturation velocity of $\mu_1$
$v_2$Maturation velocity of $\mu_2$
Table 2.  Definition of the CD8 T cell subsets as functions of maturity levels
Cell SubsetDomainDefinition
Activated cells $\Omega_A$ $\{(\mu_1, \mu_2) ; \mu_1\geq\alpha, \mu_2\geq\beta\}$
Early Effector cells $\Omega_{EE}$ $\{(\mu_1, \mu_2) ; \mu_1\geq\alpha, 0\leq\mu_2\leq\beta\}$
Late Effector cells $\Omega_{LE}$ $\{(\mu_1, \mu_2) ; 0\leq\mu_1\leq\alpha, 0\leq\mu_2\leq\beta\}$
Memory cells $\Omega_M$ $\{(\mu_1, \mu_2) ; 0\leq\mu_1\leq\alpha, \mu_2\geq\beta\}$
Cell SubsetDomainDefinition
Activated cells $\Omega_A$ $\{(\mu_1, \mu_2) ; \mu_1\geq\alpha, \mu_2\geq\beta\}$
Early Effector cells $\Omega_{EE}$ $\{(\mu_1, \mu_2) ; \mu_1\geq\alpha, 0\leq\mu_2\leq\beta\}$
Late Effector cells $\Omega_{LE}$ $\{(\mu_1, \mu_2) ; 0\leq\mu_1\leq\alpha, 0\leq\mu_2\leq\beta\}$
Memory cells $\Omega_M$ $\{(\mu_1, \mu_2) ; 0\leq\mu_1\leq\alpha, \mu_2\geq\beta\}$
Table 3.  Parameters values used for numerical results displayed in Figures 6 to 8. Between Figures 6 and 8, only values of $\tau$, $\Delta t$, and the time step $\delta t$ are modified, but the ratio $\tau/\delta t$ is preserved. Between Figures 6 and 7, only the total simulation time $T$ is modified, as Figure 7 only describes CD8 T cell counts for the primary response. N.U. means 'no unit'
Parameter type Parameter Value Unit
Figure 6Figure 7Figure 8
Naive$\gamma_{N}$ $0.7$d$^{-1}$
$\theta_{N}$ $10^3$cells
$N_{0}$ $2.10^3$cells
Pathogen$\gamma_{E}$ $1$d$^{-1}$
$\theta_{E}$ $5.10^{4}$cells
$k_{P}$ $0.5$d$^{-1}$
$P_{0}$ $10^{5}$cells
Intracellular$\gamma_{r1}$ $2.51$d$^{-1}$
$\theta_{r1}$ $0.063$pM
$\gamma_{l1}$ $3$d$^{-1}$
$\theta_{l1}$ $0.2383$pM
$k_{1}$ $0.58$d$^{-1}$
$\gamma_{P1}$ $1.6$d$^{-1}$
$\theta_{P1}$ $3.10^{4}$cells
$\tau$$0.45$$0.89$days
$K2$ $1.37$pM
$r2$ $0.91$d$^{-1}$.cell$^{-1}$
$k2$ $0.078$d$^{-1}$
$\gamma_{P2}$ $1.1$d$^{-1}$
$\theta_{P2}$ $10^{5}$cells
$k_{12}$ $2.27$d$^{-1}$.pM$^{-1}$
Domains$\alpha$ $0.5$pM
$\beta$ $0.5$pM
Source$x_{m}$ $0.8$pM
$y_{m}$ $0.75$pM
$\sigma$ $0.05$pM
Population$\gamma_{d}$ $0.01$pM$^{-2}$.d$^{-1}$
$\gamma_{AICD}$ $0.1$pM.d$^{-1}$
$\theta_{AP}$ $5.10^{4}$cells
$\theta_{A2}$ $0.5$pM
$\gamma_{div}$ $2.5$d$^{-1}$
$\theta_{div}$ $0.01$pM
$\gamma_{Fas}$ $1$d$^{-1}$
$\theta_{Fas}$ $5.10^{4}$cells
Simulation$n$ $5$N.U.
$\delta t$$0.22$$0.45$days
$\Delta t$ $0.22$$0.45$days
$T$$50$$20$$50$days
Parameter type Parameter Value Unit
Figure 6Figure 7Figure 8
Naive$\gamma_{N}$ $0.7$d$^{-1}$
$\theta_{N}$ $10^3$cells
$N_{0}$ $2.10^3$cells
Pathogen$\gamma_{E}$ $1$d$^{-1}$
$\theta_{E}$ $5.10^{4}$cells
$k_{P}$ $0.5$d$^{-1}$
$P_{0}$ $10^{5}$cells
Intracellular$\gamma_{r1}$ $2.51$d$^{-1}$
$\theta_{r1}$ $0.063$pM
$\gamma_{l1}$ $3$d$^{-1}$
$\theta_{l1}$ $0.2383$pM
$k_{1}$ $0.58$d$^{-1}$
$\gamma_{P1}$ $1.6$d$^{-1}$
$\theta_{P1}$ $3.10^{4}$cells
$\tau$$0.45$$0.89$days
$K2$ $1.37$pM
$r2$ $0.91$d$^{-1}$.cell$^{-1}$
$k2$ $0.078$d$^{-1}$
$\gamma_{P2}$ $1.1$d$^{-1}$
$\theta_{P2}$ $10^{5}$cells
$k_{12}$ $2.27$d$^{-1}$.pM$^{-1}$
Domains$\alpha$ $0.5$pM
$\beta$ $0.5$pM
Source$x_{m}$ $0.8$pM
$y_{m}$ $0.75$pM
$\sigma$ $0.05$pM
Population$\gamma_{d}$ $0.01$pM$^{-2}$.d$^{-1}$
$\gamma_{AICD}$ $0.1$pM.d$^{-1}$
$\theta_{AP}$ $5.10^{4}$cells
$\theta_{A2}$ $0.5$pM
$\gamma_{div}$ $2.5$d$^{-1}$
$\theta_{div}$ $0.01$pM
$\gamma_{Fas}$ $1$d$^{-1}$
$\theta_{Fas}$ $5.10^{4}$cells
Simulation$n$ $5$N.U.
$\delta t$$0.22$$0.45$days
$\Delta t$ $0.22$$0.45$days
$T$$50$$20$$50$days
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