The Dynamics of the Changes in the Numbers of Cells in Different Naive and Memory Cell Lineages upon Exposure to the <i>j^th</i> Pathogen

<p>Boxes represent the populations of naive (<i>x_i</i>) and memory (<i>y_i</i>) cells in the <i>i^th</i> 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 <i>f_ij</i> or <i>g_ij</i> equals one). In this example, the pathogen causes the expansion of naive cells from the <i>x</i>₁₂ lineage to form memory cells in the previously unoccupied <i>y</i>₁₂ lineage, as well as the cross-reactive expansion of cells in the memory lineages <i>y</i>₀ and bystander activation (dotted lines) of cells in occupied memory lineages (<i>y</i>₂, <i>y</i>₆, and <i>y</i>₉).</p

Simulations of the Change in Memory following Exposure to Pathogens

<div><p>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, <i>d,</i> 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.</p><p>(A) We set cross-reactivity to zero.</p><p>(B) Memory lineages have the same average cross-reactivity, but we assume there is no competition between the expansions of cells in different lineages.</p><p>(C) Memory lineages have the same average cross-reactivity, and we add competition for expansion as described in the text.</p><p>(D) Memory lineages have different levels of cross-reactivity (but keep the average cross-reactivity unchanged), and there is no competition for expansion.</p><p>Parameters are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-t002" target="_blank">Table 2</a> with on average 50 naive and 10 memory lineages specific for each pathogen (i.e., <i>f</i> = 5 · 10⁻⁵ and <i>g</i> = 2 · 10⁻³).</p><p>(A) We set <i>g</i> = 0 and let naive cells expand 200-fold, resulting in the expansion factor <i>m</i> = 2·10⁷/10⁶ × 200 = 4·10³. The total expansion of naïve cells is <i>M</i> = 50<i>m</i> = 2 · 10⁵.</p><p>(B) We let naive cells expand 200-fold and memory 2-fold (i.e., <i>c</i> = 1).</p><p>(C) The total expansion is kept the same as in (A), <i>T</i> = 2 · 10⁵, but there is competition between the expansion of naive and memory cells as described in the text.</p><p>(D) Cross-reactivity is log-normally distributed, resulting in <i>ḡ</i> ≈ 2 · 10⁻³ and variance , for 5 · 10³ memory clones. </p></div

Cross-Reactivity Can Lower the Loss of Memory if There Is Competition

<p>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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-e006" target="_blank">Equation 6</a>. 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-g002" target="_blank">Figure 2</a>B and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-g002" target="_blank">2</a>C (absence and presence of competition).</p

Figure 5

<div><p>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</p><p>Other parameters are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-g002" target="_blank">Figure 2</a>B and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-g002" target="_blank">2</a>D, and the mean cross-reactivity is kept the same at <i>g</i> ≈ 2 · 10⁻³. Lines show the predictions according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-e009" target="_blank">Equation 9</a>, and ). </p></div

Analytic Approximations for the Average Decline in Memory

<p>We plot the average change in memory lineages (defined by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.0020055#pcbi-0020055-e007" target="_blank">Equation 7</a>) following exposure to a pathogen. The change in memory is proportional to <i>M_j</i>, 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 <i>M_j</i> = 2 · 10⁵ in (B).</p

Two-epitope EMM.

<p>Panel A: A schematic for the two-epitope EMM. The HA antigen has two epitopes: <i>X</i> on the head and <i>S</i> on the stem. Binding of antibodies specific for these epitopes masks them and masked epitope is indicated by <i>O</i>. Panel B-D: We plot for the two-epitope model how pre-existing antibody to the stem of HA, <i>A</i>_<i>S</i>, affects boosting (fold increase) in the antibody to both the head (<i>A</i>_<i>X</i>) and the stem (<i>A</i>_<i>S</i>) of HA following immunization. In the basic model (Panel B) boosting is independent of the level of pre-existing antibody. In the ACM (Panel C) prevaccination antibody to the stem clears the antigen and causes an equal reduction in boosting of antibodies to both the head and stem of HA. In the FIM (in the absence of epitope masking) (Panel D), prevaccination antibody rapidly binds antigen and these antigen-antibody complexes downregulate B cell proliferation to both epitopes. In the EMM (Panel E) pre-existing antibody to the stem masks only the stem epitope, thus reducing only the boosting of antibody to the stem of HA (and boosting of antibody to the head remains unaffected). Corresponding models equations are shown in <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1005692#sec010" target="_blank">Methods</a> section. Parameters are in the <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1005692#ppat.1005692.t001" target="_blank">Table 1</a>. For ACM parameter <i>d</i>_<i>b</i> is equal 3, for FIM parameter <i>α</i> = 0.01 and <i>α</i> = 0 for other models.</p

Model parameters and initial values for variables.

<p>Model parameters and initial values for variables.</p

Illustration of steric interference between antibodies to the epitopes on the head of HA in the multi-epitope model.

<p>We describe antigenic drift by changing only epitope <i>Y</i> on the head of HA between the two virus strains. Antibody to <i>X</i> generated in response to a previously experienced strain sterically hinders efficient stimulation of B cells specific for the new epitope <i>Y</i>.</p

Boosting of antibodies to the head and stem epitopes of HA following vaccination with inactivated H5N1.

<p>Panel A shows IgG titers against HA head (red) and stem (blue) epitopes measured prevaccination and 30 days post-vaccination. Panel B shows the fold-increase in IgG antibody titers against HA head (red) and stem (blue) epitopes calculated from the data in panel A. Panel C shows the relationship between the pre- and post-vaccination antibody titers. In the absence of boosting, we expect the data to fall on the dashed line (slope = 1). If the degree of boosting is independent of the initial titer, boosting would result in the data falling on a line parallel to (and above) the dashed line. The solid line, representing the best fit line, has slope less than one (least squares; slope = 0.28; 95% CI = [0.090;0.476]), indicating that there is less boosting when initial antibody titers are high. Data are from [<a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1005692#ppat.1005692.ref020" target="_blank">20</a>].</p

Dynamics of the immune response during primary and booster immunizations in the one-epitope model.

<p>Panel A shows a schematic and the equations for the basic one-epitope model with addition of enhanced antibody-bound antigen clearance (in green), Fc<i>γ</i>R-mediated inhibition (in blue), or epitope masking (in orange). Panels B-E show the dynamics of antigen and immune responses following primary and secondary immunization in these models. Panel B shows that in the basic model primary and secondary immunizations result in identical boosts (fold increases in antibody). Panels C, D and E show that in the ACM, FIM and EMM, respectively, the antibody generated during the primary response reduces the boosting of antibody following the second immunization. Parameters are shown in <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1005692#ppat.1005692.t001" target="_blank">Table 1</a>, <i>d</i>_<i>b</i> = 3 for the ACM, <i>α</i> = 0 for basic, ACM, EMM and <i>α</i> = 0.01 for FIM.</p