Immunology and mathematics: crossing the divide

Abstract

'It's high time molecular biology became quantitative, it cries out to a physicist ... for modeling. Modeling isn't a crutch, it's the opposite; it's a way of suggesting experiments to do, to fill gaps in your understanding.' John Maddox, Editor of Nature 1966-73, and 1980-95.

Figure 1

Non-linear response of IL-2-dependent T-cell line to increasing doses of IL-2. At low doses, there is little or no response. As the IL-2 concentration increases, T-cell proliferation increases exponentially and then plateaus and in some cases decreases at high concentrations of IL-2.

Figure 2

The three body problem. The trajectories of two satellites (A and B) beginning near the right hand planet with similar but not identical starting conditions under the influence of two planets (•). Initially the two trajectories are quite similar and proportional to the difference in starting position but they soon diverge dramatically and in an unpredictable manner. This illustration of sensitive dependency shows how very large effects can result from very small differences in initial conditions. In fact, immeasurably small (approaching infinitely small) differences in the starting conditions can lead to the same divergence in trajectories.

Figure 3

T-cell receptor recognition of peptide MHC complexes on antigen presenting cells. Typical model in immunology text books is of a single TCR binding to a high affinity peptide MHC complex on APC (a). A more realistic model is of multiple TCR binding to a range of different peptide MHC complexes with different affinities, some low and some high (b). Discrimination between high and low affinity binding and the contribution to T-cell activation of low affinity binding is taken into account by mathematical models.

Figure 4

Hysteresis of TCR activation. Hysteresis is an important concept to understand and is defined in the text. In the example shown here, as the TCR signal strength increases from left to right, there comes a point (↑₂) when T-cell activation, which could be measured for example by measured by phosphorylation of T-cell signalling proteins such as Lck, suddenly increases and the cell becomes activated (A). In the reverse direction, the T cell remains activated even when the TCR signal has decreased below the threshold point at (A) until the lower threshold point (B) is reached (↑₁) when the T cell returns to its resting state. This shows how prolonged T-cell activation could occur in response to low affinity ligands that generate an activation signal above (↑₁) but below the level of (↑₂) induced by the initial activation event with a high affinity ligand. See reference .

Figure 5

Cytokines acting between cells help determine Th1 and Th2 differentiation. This is not by any means a complete illustration of the cytokines involved but it makes the point that cytokines produced by differentiating Th1 and Th2 cells work in both positive and negative feed back loops to influence the differentiation process.

Figure 6

Signalling events that determine T-bet and GATA-3 expression in a single T cell. The essential features that determine the outcome of the differentiation process are the positive and negative feed back loops indicated in the diagram. Mathematical modelling of these show hysteresis enabling the cell to behave as a bistable switch as shown in Fig. 7.

Figure 7

A schematic diagram of how a single cell responds to a Th2 signal by expressing GATA-3. The curve represents the steady state GATA-3 expression reached by the cell in response to Th2 promoting signals. The x-axis represents the strength of the external stimuli that induces expression of GATA-3 such as TCR signalling in the presence of IL4. Here there is no Th1 stimulus and so T-bet expression remains low. Recently activated cells start with low levels of T-bet and GATA-3 expression. Increasing stimulation by extrinsic signals up to a threshold level (↑₂) increases GATA-3 levels slowly (region A). At the threshold, the cell rapidly approaches a state of high GATA-3 expression (region C) in which autoactivation of GATA-3 occurs at its maximum rate. In this state, the level of GATA-3 is relatively insensitive to fluctuations in the external stimulus. However if the stimulus is reduced below the lower threshold (↑₁) the cell reverts to low level GATA-3 expression. Continued extrinsic signalling at a level greater than (↑₁) but below the upper threshold (↑₂) is able to sustain high levels of GATA-3 during the first few rounds of division.

Figure 8

Homeostasis of the T memory cell compartment. A stable memory T-cell compartment requires that the input is exactly balanced by the output, but how is this achieved?

Figure 9

T-cell fratricide model of homeostasis. In this simplified model, only the resting and proliferating Tm cell compartments are considered. Input from antigen-activated naïve cells or memory cells has been considered elsewhere and does not alter the general conclusions. In this simplified model, differential equations are used to describe the rate of change in the resting and proliferating Tm compartments as a consequence of resting Tm cells undergoing homeostatic proliferation induced for example by IL-15 or IL-7 (a), reversion back to the resting state (r), T cell proliferation within the cycling compartment (c), death or loss of cells within the resting compartment (d) and fratricide by T cells within the cycling compartment (fy²).

Figure 10

Model of Tm homeostasis by competition for resources. Again, only the resting and proliferating Tm cell compartments are considered. Input from antigen activated naïve cells or memory cells does not alter the general conclusions. As with the fratricide model, differential equations are used to describe the rate of change in the resting and proliferating Tm compartments but a saturating term ax/(k + x) is used to describe the rate at which resting Tm cells undergo homeostatic proliferation induced for example by IL-15 or IL-7 and death by fraticide is replaced by the linear term δy.

Figure 11

The fratricide and competition models can be distinguished by following the number of cycling cells in the circulation after adoptive transfer of resting Tm cells. In the fratricide model, addition of Tm cells () results in a rapid increase in the number of cycling cells and then a slow return of the cycling and resting cells back to the normal equilibrium point (black filled circle). In contrast, in the competition model, addition of Tm cells does not significantly increase the number of cycling cells and the number of resting cells slowly returns back to the equilibrium point (•).