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

  • * Corresponding author: Fabien Crauste

    * Corresponding author: Fabien Crauste
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  • 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.

    Mathematics Subject Classification: 34D20, 34K18, 34K60, 37N25, 92C37.


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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

    $N(t)$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 $\mu_2$
     | Show Table
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    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 $\Omega_M$ $\{(\mu_1, \mu_2) ; 0\leq\mu_1\leq\alpha, \mu_2\geq\beta\}$
     | Show Table
    DownLoad: CSV

    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
    $K2$ $1.37$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
     | Show Table
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