Figure 1.
The Dynamics of the Changes in the Numbers of Cells in Different Naive and Memory Cell Lineages upon Exposure to the jth Pathogen
Boxes represent the populations of naive (xi) and memory (yi) cells in the ith lineage. Shaded boxes represent lineages that are occupied prior to exposure to the pathogen (most naive and a few memory lineages are occupied), and shading indicates the relative number of cells in a given naive and memory lineage. Red boxes indicate lineages that the pathogen is able to stimulate (i.e., lineages for which fij or gij equals one). In this example, the pathogen causes the expansion of naive cells from the x12 lineage to form memory cells in the previously unoccupied y12 lineage, as well as the cross-reactive expansion of cells in the memory lineages y0 and bystander activation (dotted lines) of cells in occupied memory lineages (y2, y6, and y9).
Table 1.
Definition of Symbols Used in This Paper
Table 2.
Estimates of Population Sizes and Repertoires Used in the Simulations (Corresponding to the Spleen of a Mouse)
Figure 2.
Simulations of the Change in Memory following Exposure to Pathogens
In the left panels we follow the change in size of representative memory clones shown in different colors (the thicker blue line represents many clones together). We mark the average decline in memory per exposure, d, defined as a decline in the total number of cells in memory lineages that were occupied prior to exposure to the pathogen, normalized to the total number of memory cells. In the right panels we show the frequency distribution of the size of these lineages at the beginning (open bars) and end (filled bars) of the simulation. We consider 75 exposures to new pathogens.
(A) We set cross-reactivity to zero.
(B) Memory lineages have the same average cross-reactivity, but we assume there is no competition between the expansions of cells in different lineages.
(C) Memory lineages have the same average cross-reactivity, and we add competition for expansion as described in the text.
(D) Memory lineages have different levels of cross-reactivity (but keep the average cross-reactivity unchanged), and there is no competition for expansion.
Parameters are as in Table 2 with on average 50 naive and 10 memory lineages specific for each pathogen (i.e., f = 5 · 10−5 and g = 2 · 10−3).
(A) We set g = 0 and let naive cells expand 200-fold, resulting in the expansion factor m = 2·107/106 × 200 = 4·103. The total expansion of naïve cells is M = 50m = 2 · 105.
(B) We let naive cells expand 200-fold and memory 2-fold (i.e., c = 1).
(C) The total expansion is kept the same as in (A), T = 2 · 105, but there is competition between the expansion of naive and memory cells as described in the text.
(D) Cross-reactivity is log-normally distributed, resulting in ḡ ≈ 2 · 10−3 and variance , for 5 · 103 memory clones.
Figure 3.
Analytic Approximations for the Average Decline in Memory
We plot the average change in memory lineages (defined by Equation 7) following exposure to a pathogen. The change in memory is proportional to Mj, the number of memory cells of new specificities generated by the pathogen (A); and inversely proportional to Ŷ, the total size of the memory compartment (B). The simulations for all the cases considered in the Results section were indistinguishable from the lines shown and are thus not explicitly plotted. Parameters: Same as previously, with Ŷ = 2 · 107 in (A) and Mj = 2 · 105 in (B).
Figure 4.
Cross-Reactivity Can Lower the Loss of Memory if There Is Competition
We plot the effect of changing the average cross-reactivity on the decline in memory per exposure to a new pathogen in the presence and absence (□) of competition between the expansion of naive and memory cells described by Equation 6. Symbols represent the results obtained by computer simulations (for the introduction of 100 pathogens) and lines represent the analytical approximation as described in the text. Parameters are as in Figure 2B and 2C (absence and presence of competition).
Figure 5.
Variance of the Natural Logarithm of Size of Memory Lineages as the Function of the Number of Exposures in the Absence (squares,
0) and Presence (diamonds,
) of Variation in Cross-Reactivity between Different Memory Lineages
Other parameters are the same as in Figure 2B and 2D, and the mean cross-reactivity is kept the same at g ≈ 2 · 10−3. Lines show the predictions according to Equation 9, and
).