Models of immune memory: on the role of cross-reactive stimulation, competition, and homeostasis in maintaining immune memory

Abstract

There has been much debate on the contribution of processes such as the persistence of antigens, cross-reactive stimulation, homeostasis, competition between different lineages of lymphocytes, and the rate of cell turnover on the duration of immune memory and the maintenance of the immune repertoire. We use simple mathematical models to investigate the contributions of these various processes to the longevity of immune memory (defined as the rate of decline of the population of antigen-specific memory cells). The models we develop incorporate a large repertoire of immune cells, each lineage having distinct antigenic specificities, and describe the dynamics of the individual lineages and total population of cells. Our results suggest that, if homeostatic control regulates the total population of memory cells, then, for a wide range of parameters, immune memory will be long-lived in the absence of persistent antigen (T1/2 > 1 year). We also show that the longevity of memory in this situation will be insensitive to the relative rates of cross-reactive stimulation, the rate of turnover of immune cells, and the functional form of the term for the maintenance of homeostasis.

Figure 1

A schematic view of the dynamics of an antigen-specific immune response after stimulation.

Figure 2

The duration of immune memory in the single compartment model. We show how the duration of immune memory, plotted as the half-life of the population of pathogen-specific cells after stimulation, T_1/2, will depend on the average frequency of infection with different pathogens, average clonal expansion per pathogen, and input from the thymus. Parameters: Total number of cells X = 10⁸, total input from thymus 10⁵ cells per day, and rates of stimulation with pathogens and extent of expansion of pathogen specific memory cells as indicated.

Figure 3

The dynamics of the immune repertoire for a single compartment model. We plot the repertoire of immune cells (scaled so that the maximum repertoire equals unity) as a function of time. We find that the maintenance of the repertoire over a long duration depends on the input from the thymus. Parameters: In the simulation, we examined the dynamics of 10⁵ lineages of cells with a total of ≈10⁶ cells (this amounts to ≈1/100th of the immune system of a mouse). Probability of stimulation per lineage per day, q = 10⁻⁶; input from thymus stochastic with probabilities, a = 0.0, 0.001, and 0.01 per lineage per day; background death rate, d = 0.003 per day; cross-reactivity, c = 0.001 per day; homeostasis being incorporated by a logistic function, S(X) = s(1 − X/k), with s = 1.0 per day and k = 10⁶ lymphocytes; magnitude of the specific immune response (burst size), m = 10⁴ cells per lineage.

Figure 4

The duration of immune memory for a two-compartment model with independent homeostatic regulation in naive and memory compartments. In this model, the naive and memory cells differ in their properties, but the homeostasis acts independently in both compartments. We plot how the duration of immune memory (represented by the half-life of the population of pathogen-specific cells after stimulation, T_1/2) will depend on the frequency of infection with different pathogens and the extent of clonal expansion to memory cells per pathogen. Parameters as in Fig. 2, with the carrying capacity of 10⁸ cells equally divided in naive and memory compartments and input from thymus set to zero.

Figure 5

Dynamics of naive and memory repertoires in the two-compartment model. We plot the repertoire in the naive and memory compartments and the total number of lineages stimulated as a function of time. We find that independent homeostasis in naive and memory compartments allows the repertoire to be maintained even in the absence of input from the thymus, which is set to zero in the simulations. In a, the naive and memory compartments are assumed to have a fixed size, and in b, the size of the naive compartment decreases and that of the memory compartment increases linearly with time. Parameters as in Fig. 3 except input from thymus, a = 0; levels of cross-reactivity, c_x = 0.001 (naive) and c_y = 0.05 (memory); background death rates, d_x = 0.003 and d_y = 0.05; the total population, k_x + k_y = 10⁶ cells, is assumed to be divided equally between naive and memory compartments (k_x = k_y = 5 × 10⁵ cells) in a whereas in b there is a slow shift from naive to memory populations, with 80% of the cells having a naive phenotype initially and then declining to 20% by the end of the simulations.