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

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