a T-cell and a DC during stage 1 in a lymph node [85, 231]. Thus, for every time-step we generate a Bernoulli random variable with mean 0.25 and if it is 1 the T-cell moves on (if T is the waiting time to generate a 1 then T −1 is geometrically distributed and one can show that E(T) = 0.251 ). We have to point out that in the literature one can find different values for this mean binding/scanning time and this question is currently under investigation [14, 15, 13]. An exponential or Poisson distribution might be more realistic than the Bernoulli distribution. The other mode of a T-cell is the ’movement mode’. The T-cell is not bound to any cell and moves freely from node to node. This is implemented by randomly choosing one of the three coordinates of the T-cell and randomly in- or decrementing this coordinate by 1. Thereby, a kind of random walk is established. As mentioned before we assume periodic boundary conditions. If a T-cell leaves the lattice on one side it enters the lattice on the opposite.
We now have a basic T-cell migration model for the thymic medulla at hand. It is easily possible to change this model by changing the mean scanning time of T-cells, change T-cell movement from totally random to perhaps a directed random walk, change the settings for the environment or allow the simultaneous scanning of several DCs/mTECs by one T-cell. It is also easy to extend the model by for example introducing a special scanning and/or activation model and/or equip the mTECs and DCs with self antigens.
7.2 Results
Although the model is very basic, there are already questions that can be asked and answered by it. We can compare our artificial medullary section to real microscopic images of medullary sections from experiments and we can estimate the number of (dif-ferent) DCs and mTECs a single T-cell meets during a 5 day cycle of migration. We also introduce a first extension of the model by equipping the mTECs with tissue restricted antigens. This allows us to test if the mTECs alone are sufficient to guarantee that a T-cell meets all these TRAs during negative selection. This is crucially important because otherwise autoimmune reactions are quite probable as shown in several experiments, see for example [117, 194, 199].
In our model we did not specify an environment in the medulla but generate it ran-domly under some constrains. In Figure 7.1 one such realisation is shown from two dif-ferent perspectives. We do not show the dendritic cells as these are distributed uniformly over the free nodes, but concentrate on the mTECs. It is evident that the desired struc-ture of different mTEC areas and rare single mTECs outside these areas is generated.
This kind of visualisation gives a good overview over the three-dimensional composition of our artificial medullary section. However, it is not suitable for comparisons with re-sults from experiments. In experiments the rere-sults are visualised by microscopic images of thin slices of the thymic medulla. Therefore, we imitate this procedure by keeping one coordinate constant and visualising the other coordinates in a 2D image as can be seen in Figure 7.2 for different constant x1 coordinates. A comparison of our four example images with images from [162] is difficult. If we take Figures 1g and 1j from [162] we get
120 A model for T-cell migration in the thymic medulla
Figure 7.1: Example of a medullary section generated randomly by our simulation from two different perspectives. Only the mTECs are shown.
a picture of the mTEC/DC composition of a medullary area in two dimensions. This looks roughly qualitatively similar to medullary areas in our images. For future work a more specific comparison with experimentally generated images is of course desireable but for a first model our generated environments seem to be sufficient.
In the second step we introduce the T-cells. These are randomly placed on the lattice and start to move in a random fashion when the simulation is started. Figure 7.3 shows the movement of a T-cell during 7200 time steps. This corresponds to a migration time of 5 days. For reasons of visibility we again did not plot the dendritic cells. The right figure is a zoomed-in version of the left figure. There, we visualised the points where a T-cell was connected to an mTEC/DC for 3 time steps or longer by circles. The figures show the random walk like movement of the T-cells for the timesteps where it is not bound to mTECs or DCs as well as the jumps at the boundaries because of the periodic boundary conditions. These assumptions are therefore met and the simulation behaves in the intended way. Although the dendritic cells are not shown it is evident from the circles that a T-cells does not move freely very much but is bound to either DCs or mTECs.
The T-cells seem to use the time they have efficiently. Long times of movement without DC/mTEC encounters would lead to a very inefficient negative selection process. T-cell migration and the development of the medullary microenvironement should be guided in a way to enable as many encounters as possible such that a T-cell sees as many self antigens as possible. However, we can also see that a T-cell revisits some positions and thereby some mTECs or DCs. At first glance this seems inefficient, but if we assume that a T-cell only scans parts of a DC this looks different. Most probably the T-cell just sees another part of the self antigen repertoire on the DC surface.
We repeated our simulations for 10 randomly generated microenvironments and 100 randomly placed T-cells and estimated the number of mTECs/DCs a T-cell encounters.
The results can be seen in Figure 7.4. We compare the number of mTEC/DC hits for the four different scenarios emerging from taking either the big or small lattice and
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Figure 7.2: Four cuts through the example model in Figure 7.1. The cuts where done at x1 = 10,20,30,40. The black dots are again the mTECs, whereas the black crosses are dendritic cells. A comparison to the Figures in [162] is difficult. If we take Figures 1g and 1j together we get a picture of the distribution of mTECs and DCs in an islet and this seems qualitatively similar to mTEC islet regions in our figures.
assuming a total medulla size of 0.1cm3 or 0.2cm3. The histograms show that in all cases the number of hits seem to approach a Gaussian curve. The only difference is the mean of this curve, which is for the smaller medulla about 1800 and for the bigger medulla about 1700. This is an interesting result, because by doubling the size of the medulla, the number of mTECs and DCs are halved for our section. It follows, that although our lattice is less crowded with mTECs and DCs, it is crowded enough such that a single T-cell meets only 100 fewer mTECs/DCs. Furthermore, we see that the maximal average number of hits, 2400, is never reached in our small test simulations.
This might be important for estimations in models of negative selection. For our model in the previous sections we assumed for example 2000 APC meetings during negative selection. Another lesson to learn from the histograms is that the results for the small and the big lattice are not too different. Therefore, the smaller lattice, which leads to a
122 A model for T-cell migration in the thymic medulla
Figure 7.3: Visualisation of one T-cell movement trajectory through our artificial medullary environ-ment. The right picture is a zoomed version of the left picture.
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Figure 7.4: Histograms of the number of mTEC/DC hits of a T-cell calculated from 10 different microenvironments with 100 T-cells that migrated for 5 days. Upper left: 30×3 lattice, 0.02 total medulla size; Upper right: 30×3 lattice, 0.01 total medulla size; Lower left: 50×3 lattice, 0.02 total medulla size; Lower right: 50×3 lattice, 0.01 total medulla size.
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Figure 7.5: Histograms of the number of mTEC hits of a T-cell calculated from 10 different microen-vironment with 100 T-cells that migrated for 5 days. Upper left: 30×3lattice, 0.02 total medulla size;
Upper right: 30×3 lattice, 0.01 total medulla size; Lower left: 50×3 lattice, 0.02 total medulla size;
Lower right: 50×3 lattice, 0.01 total medulla size.
speed up of the simulation, should be suitable for many simulatory test scenarios.
Having clarified these first general facts of our simulation, we now turn to the inves-tigation of promiscuous gene expression in our model. In a first attempt, we assume that only the mTECs are involved in presenting tissue restricted antigens. In Figure 7.5 we show the number of only the mTEC hits for the same experimental settings as in the last paragraph. In contrast to Figure 7.4 the number of mTEC hits follows quite different distributions for the different parameter settings. For a medulla size of 0.01 (upper and lower right figure) the hit distribution looks quite similar with a mean hit number of about 50. Here, a Gaussian-like shape is not met, in contrast to the case where we have the 30×3 lattice and a 0.02cm3 medulla size. The mean number of mTEC hits is also higher with about 70. The biggest discrepancy occurs for the 50×3 lattice and 0.02cm3 medulla size. The mTEC hit distribution is shaped similar to an exponential distribution with a mean hit number of about 10.
Given the fact, that under our assumptions a T-cell has to meet as many mTECs as possible, the variance in all hit distributions and the differences of the mean number of mTEC hits are significant. Obviously, the number of mTEC hits is very much dependent on the model parameters and the positioning of the individual T-cells on the lattice.
124 A model for T-cell migration in the thymic medulla
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Figure 7.6: Histograms of the number different tissue restricted antigens a T-cell sees. Calculated from 10 different microenvironment with 100 T-cells that migrated for 5 days. Upper left: 30×3 lattice, 0.02 total medulla size; Upper right: 30×3 lattice, 0.01 total medulla size; Lower left: 50×3lattice, 0.02 total medulla size; Lower right: 50×3lattice, 0.01 total medulla size.
This can be even better illustrated if we go one step further. Until now we only observed mTEC and DC encounters. However, we can equip the mTECs with tissue restricted antigens and estimate how many of these a T-cell sees during negative selection.
From the literature we know that every TRA is expressed by about 3% of all mTECs. It follows that one mTEC expresses about 90 randomly chosen TRAs. In our simulation we can follow how many of the 3000 different TRAs one T-cell observes. Furthermore every TRA on an mTEC is replaced by another every 20 hours. This is especially helpful for the purpose of our simulations, because the T-cells cannot leave the lattice but reenter when leaving and will most probably meet several mTECs/DCs more than one time on their journey. The results can be seen in Figure 7.6. The scenario settings are again the same as described before. The most eyecatching message from this Figure is that (nearly) no T-cell sees all tissue restricted antigens, given that only mTECs present them. Even more, there is a wide variance in the number of seen TRAs over all T-cells.
This contradicts the overall goal to see all TRAs almost surely. In any circumstances it is obvious that promiscuous gene expression just by mTECs does not work. This speaks strongly for the mechanism of crosspresentation of TRAs from mTECs to dendritic cells.
By this mechanism TRAs are transferred to DCs and presented by them.
7.3 Discussion 125