in response to graded stimuli 81
4.2 The dynamical model reproduces a continuum of T-bet and GATA-3 expression levels in response to graded stimuli
Classically, mutual inhibition and auto-activation motifs as present in our model are thought to give rise to a bistable system in steady state, which would lead to a digital switch between high T-bet expression and high GATA-3 expression under mixed Th1-Th2 stimuli156, or a quadristable system allowing for an intermediate co-expressing steady state as well as a naive low expressing state (MSc dissertation, E. Pellet). However, our experiments have shown that there is a continuum of expression patterns of T-bet and GATA-3 in response to titrated cytokine amounts during differentiation (cf. Figure 2.9). Therefore, we asked which kind of steady states our parametrised model has.
To analyse the stability of our system upon differentiation in a visually intuitive manner, we considered the phase plane of T-bet and GATA-3 expression under quasi-steady state assumption for STAT1, STAT4 and STAT6, i.e., the assumption that STAT amounts did not vary anymore (S1˙ = ˙S4= ˙S1= 0, Figure 4.7) with fixed values for the pSTAT intensities. We thus reduced the system to two dimensions, T-bet and GATA-3. This approach allows to study the system at steady state and to determine the possible stable expression patterns of T-bet and GATA-3 at the end of differentiation. Using Wolfram Mathematica 10, we computed the T-bet-GATA-3 pairs of values at which T-bet-GATA-3 remains stable by solvingG˙ = 0and T˙ = 0for GATA-3 (GATA-3 nullclines, red lines), and those at which T-bet remains stable by solving for T-bet (T-bet nullclines, blue lines). Thus, the intersection of a T-bet and GATA-3 nullcline represents a steady state for T-bet and GATA-3. We also computed the vector field (blue arrows), which shows if T-bet and GATA-3 expression levels are increasing or decreasing at each point of the T-bet-GATA-3 phase plane. We titrated the input Th1 stimuli IL-12 and IFN-γversus the Th2 stimulus IL-4 and observed the changes in the T-bet-GATA-3 phase plane.
Interestingly, our model has a unique stable steady state that moves along the axes depending on the cytokine stimuli. Thus the model shows a continuum of T-bet-GATA-3 expression patterns in response to mixed stimuli rather than division of the population between GATA-3 and T-bet-expressing cells depending on the polarising stimuli, which would become manifest by two stable steady states on the GATA-3 and T-bet axis, respectively (Figure 4.7). This was indeed what we had previously observed experimentally on day 5 of the IL-12-IL-4 cross-titration in IFN-γ-producing cells: a unimodal population expressing more T-bet in the presence of higher Th1 cytokine concentrations, and more GATA-3 in the presence of higher IL-4 concentrations (see Figure 2.9).
GATA-3 T-bet
IL-12 and IFN-γ stimuli
IL-4 stimulus GATA-3
nullcline T-bet nullcline Stable steady state Vector field
Figure 4.7: Steady-state behaviour of the dynamical model.Phase plane analysis of T-bet and GATA-3 under the assumption of quasi-steady state for all other variables. The system has only one stable steady state; its value depends on cytokine inputs and moves gradually in the T-bet-GATA-3 space.
4.2. The dynamical model reproduces a continuum of T-bet and GATA-3 expression levels
in response to graded stimuli 83
IL-12 stimulus
Figure 4.8: The model quantitatively reproduces independent data from an IL-12-IL-4 crosstitration.
FACS-sorted naive IFN-γ- and IL-4-competent CD4 T cells were activated with APCs and GP61−80peptide under the indicated conditions. IL-12 and IL-4 were titrated against each other. T-bet and GATA-3 expression on day 5 as well the the values predicted by the model are shown as heat maps and dot plots.
Having determined that the dynamical model possessed only one steady state for T-bet and GATA-3 expression, we then considered the quantitative response of the model to graded mixed Th1-Th2 stimuli during differentiation in the form of the expression intensity of the master transcription factors. To do so, we simulated the results of an IL-12-IL-4 cross-titration experiment after five days of differentiation. Under these conditions, IFN-γis produced by the Th cells (and not added in a controlled manner), we extrapolated the IFN-γconcentration based on the T-bet expression on day 5 using the model for IFN-γproduction described in Chapter 3 (see Figure 3.12):
%IFN-γ+= 0.9· T3.2 0.06 +T3.2
Figure 4.8 shows the comparison of the experimental values of T-bet and GATA-3 expression to the values predicted by the model with the best-fit parameters estimated previously. Heatmaps of T-bet and GATA-3 expression show a similar response to IL-12 and IL-4 (Figure 4.8a) between the experimental data and the model; T-bet showed, in both cases, an increase in the presence of IL-12 and IFN-γ and a decrease in the presence of IL-4, while the opposite was true of GATA-3. To verify that the model was able to quantitatively reproduce the data, we computed the correlation between the data and predicted values; the model was able to predict the expression of T-bet and GATA-3 accurately with correlation coefficients R2of 0.92 for T-bet and 0.86 for GATA-3 (Figure 4.8b).
The dynamical model developed in this study has a single, stable fixed point considering the quasi-steady state for T-bet and GATA-3. The value of this fixed point moves continuously along the T-bet and GATA-3 axes depending on the input cytokine stimuli. Furthermore, the model was able to reproduce accurately the continuum of T-bet and GATA-3 expression levels derived from an independent IL-12-IL4 cross-titration experiment.