R. Antia and V. V. Ganusov conceived the basic ideas. R. Ahmed helped formulate the problem and consider the implications for experimental work. R. Antia and V. V. Ganusov derived analytical approximations. V. V. Ganusov performed the simulations. V. V. Ganusov and R. Antia wrote the paper. S. S. Pilyugin contributed to the derivations and helped to modify the text.
¤ Current address: Theoretical Biology, Utrecht University, Utrecht, Netherlands
The authors have declared that no competing interests exist.
Immunological memory—the ability to “remember” previously encountered pathogens and respond faster upon re-exposure is a central feature of the immune response in vertebrates. The cross-reactive stimulation hypothesis for the maintenance of memory proposes that memory cells specific for a given pathogen are maintained by cross-reactive stimulation following infections with other (unrelated) pathogens. We use mathematical models to examine the cross-reactive stimulation hypothesis. We find that: (i) the direct boosting of cross-reactive lineages only provides a very small increase in the average longevity of immunological memory; (ii) the expansion of cross-reactive lineages can indirectly increase the longevity of memory by reducing the magnitude of expansion of new naive lineages which occupy space in the memory compartment and are responsible for the decline in memory; (iii) cross-reactive stimulation results in variation in the rates of decline of different lineages of memory cells and enrichment of memory cell population for cells that are cross-reactive for the pathogens to which the individual has been exposed.
Immunological memory—the ability to “remember” previously encountered pathogens and respond faster on re-exposure—is a central feature of the immune response of vertebrates. Exposure to a pathogen results in the clonal expansion of a few relatively rare clones of immune cells which are specific for the pathogen to form a population large enough to control the pathogen. Immunological memory arises from the maintenance of an elevated numbers of these pathogen-specific immune cells. There has been much debate on the contribution of different processes such as the persistence of antigen, cross-reactive stimulation, and homeostasis to the maintenance of the elevated number of “memory” cells. Models have been useful in understanding the contributions of these various processes to the maintenance of memory. The models have shown that the decline rate of memory specific for previously encountered pathogens arises due to exposure to new pathogens—this causes the replacement of a fraction of “old” memory cells with memory cells specific for new pathogens. In this paper Ganusov, Antia, and colleagues use mathematical models to explore how the ability of cross-reactive memory cells to respond to the antigens on more than one pathogen can help in the maintenance of immunological memory.
Immunological memory—the ability to “remember” previously encountered pathogens and respond faster upon re-exposure—is a central feature of the immune response of vertebrates. This more rapid response arises, in large part, from an increase in the number of B and T cells specific to the pathogen, and usually (but not always, see [
There are a number of ways in which elevated numbers of “memory” CD8+ T cells, and thus immune memory, could be maintained [
One possibility is that memory resulted in the generation of a population of non-dividing “memory” cells with a long lifespan. This hypothesis was rejected by the elegant experiments that demonstrated that memory T cells incorporate BrdU, indicating that this population is undergoing division [
Given that memory cells are undergoing proliferative renewal (turnover), the central question is whether this turnover is antigen-dependent or antigen-independent. There has been an extensive debate on the role of antigen for the maintenance of the memory cell population. The current view is that the maintenance in the population of memory cells does not require the persistence of antigen [
What can maintain the proliferative renewal of immune cells in the absence of specific antigenic stimulation? Three possibilities have been proposed: bystander stimulation, cross-reactive stimulation, and homeostatic regulation of turnover. The bystander-stimulation hypothesis was based on the observation that infections result not only in the expansion of cells specific for the antigens expressed by the pathogen but also bystander cells with other specificities [
These three hypotheses are not mutually exclusive: there can be contributions to proliferation by bystander and cross-reactive stimulation while the total population of cells is homeostatically regulated. Mathematical models are a useful tool for dissecting the relative contributions of these processes to the maintenance of memory [
The earlier model suggests that the longevity of memory is independent of the extent of bystander stimulation. The biological explanation is that the effects of bystander stimulation are compensated for by homeostatic regulation of the population of memory cells. The earlier model, however, did not consider the effects of different levels of cross-reactivity between the responses to different pathogens. This limitation is highlighted by recent experimental studies described by Welsh and colleagues [
In this paper we explicitly include different levels of cross-reactivity into the models describing the dynamics of naive and memory cells following exposure to pathogens. This allows us to determine how the longevity of memory depends on the extent of cross-reactivity, and to explore the hypothesis that cross-reactivity plays an important role in the maintenance of memory.
We formulate a model that describes the long-term dynamics of naive and memory CD8+ T cells following exposure to pathogens. The model builds on an earlier model [
Exposure to a pathogen stimulates the cells in specific naive lineages and cross-reactive memory lineages. Stimulated naive cells expand in numbers, some die, and others differentiate into memory cells. This encompasses the expansion and contraction phases of the immune response following acute infections [
Homeostatic regulation brings the naive and memory compartments back to their nominal sizes by proportional changes in the population of cells in each lineage (we essentially assume that all cells are identical except for their antigenic specificities). The cell numbers in all lineages remain constant until the next challenge.
As seen in
Boxes represent the populations of naive (
Definition of Symbols Used in This Paper
We consider the following processes which operate on these cell populations. In the absence of pathogenic challenges, the naive and memory cell populations are homeostatically regulated. For simplicity, we ignore the immigration of naive cells from the thymus. Upon exposure to a pathogen, the naive and memory compartments undergo short-term changes resulting from the net expansion of cells in naive and memory lineages to generate memory cells. We let
Infection with the
The expansion phase is followed by the homeostatic regulation where the naive and memory lineages are scaled proportionately to restore the homeostatic equilibrium in both compartments,
The model incorporates cross-reactive stimulation in a very general way that includes the possibilities that different pathogens share the same epitope and that memory lineages have lower thresholds for stimulation than naive cells. We define the level of cross-reactivity of memory lineages for the
We explore the behavior of our model using a combination of simulations and analytical approximations. Because we are focusing on the role of cross-reactivity for the maintenance of memory, we keep the total population size of memory population constant over time. The consequences of changing the size of the total memory population have been considered earlier [
The simulations we run are illustrative—their role is to allow us to get an idea of how cross-reactivity can affect the longevity of immunological memory. We begin with simple scenarios and add additional features stepwise.
The initial conditions for the simulations are shown in
Estimates of Population Sizes and Repertoires Used in the Simulations (Corresponding to the Spleen of a Mouse)
We begin with the simplest scenario with the following two assumptions. First, the expansion of all lineages is independent of each other. Specifically, there is no competition between the expansion of cells in different lineages including naive and memory cells. Second, all memory lineages have the same average cross-reactivities, that is,
On stimulation with
In
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,
(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
(A) We set
(B) We let naive cells expand 200-fold and memory 2-fold (i.e.,
(C) The total expansion is kept the same as in (A),
(D) Cross-reactivity is log-normally distributed, resulting in
In
We plot the average change in memory lineages (defined by
In reality we might not expect the two simplifying assumptions given above to be met. First, we expect competition between the expansion of cells of different lineages in response to a given pathogen. Second, we expect that different memory lineages may have different levels of cross-reactivity dependent on their T cell receptor. We examine the consequences of introducing these below.
In
We plot the effect of changing the average cross-reactivity on the decline in memory per exposure to a new pathogen in the presence
In
Variance of the Natural Logarithm of Size of Memory Lineages
Other parameters are the same as in
In this section we consider the decline of memory following exposure to a new pathogen in more detail. As seen in the previous section, this decline may be different for different lineages and will be dependent on a number of factors such as cross-reactivity and competition. In the first section we consider the average decline in memory, in the second section we consider the effect of competition on the loss of memory, and in the third section we consider the effect of variations in the extent of cross-reactivity.
As a measure for the “average” loss of memory, we consider the decline in the total number of cells in memory lineages that were occupied prior to exposure to the pathogen. As is shown in
This result suggests that the average decline of memory following exposure to a pathogen is: (i) directly proportional to
We use
How might cross-reactive stimulation affect the average loss of memory on exposure to new pathogens? As the loss of memory is determined by
In a more realistic scenario competition would be much more complex and involve factors such as different activation thresholds for different lineages and for naive and memory cells. However complex the term for competition may be, the effect of competition on memory will depend on how much it reduces the generation of memory cells in previously unoccupied memory lineages.
What effect does the introduction of cross-reactivity have on the loss of memory on exposure to new pathogens? Our simulations (Results section) suggested that in the absence of cross-reactivity (i.e., at
Here we explore how the first three factors, namely, cross-reactive expansion (i.e., parameters
As we show in
In contrast, when there is a distribution of cross-reactivity level between different memory lineages (or distribution in the level of cross-reactive expansion
We have used mathematical models to investigate the relative contribution of multiple processes, including cross-reactive stimulation and homeostasis, on the longevity of immunological memory. Mathematical models are required because it is hard to rigorously follow complex interactions between these processes using only verbal descriptions, and mathematics simply provides us with the appropriate tools to do so. Indeed in this paper the models show how the constraint imposed by homeostasis limits the ability of cross-reactive stimulation to boost memory cell populations in a way which increases the average longevity of memory.
In this section we begin by giving an intuitive summary of our results, and integrate them with previous studies on the longevity of immunological memory. We then describe the relevance of our model to current experimental work on the role of cross-reactive stimulation for the maintenance of immunological memory.
Our model extends the previous studies and models [
First, we show that the direct boosting of cross-reactive memory cells does not, on average, increase the longevity of immunological memory. This result arises because of the constraint imposed by homeostasis. The direct boosting of some lineages increases their frequencies, but homeostasis requires that this boost in frequencies is compensated for by a decline in the frequency of other memory lineages. A careful accounting shows that the average longevity of immunological memory is independent of the extent of cross-reactivity.
Second, we suggest an alternative mechanism by which cross-reactive stimulation can increase the longevity of memory. This occurs because of competition between the expansion of cells in hitherto unstimulated naive lineages and cells in cross-reactive memory lineages. The larger the population of cells in cross-reactive memory lineages, the smaller the expansion of cells from naive lineages. We show that this competition results in the average longevity of memory increasing with increasing cross-reactivity (see
Third, we find that the introduction of cross-reactivity results in different lineages of memory cells having different rates of decline. The rate of decline of a particular lineage will depend on a number of factors including: (i) the intrinsic cross-reactivity of the lineage; and (ii) the specific pathogens encountered (see
We now discuss the relevance of our model to current experimental work on the role of cross-reactive stimulation for the maintenance of immunological memory.
Our results are consistent with the pioneering experiments of Welsh and colleagues which describe cross-reactive stimulation during the generation of immune responses and immunological memory [
The model can be subject to additional tests which include testing the assumptions of the model and its predictions. A key assumption which we need to verify is whether all the memory CD8+ T cells have similar properties except for their antigenic specificities. A key prediction is whether the decline in existing memory is determined by the number of memory cells of new specificities generated following infection by a pathogen. While some studies have observed results consistent with this prediction [
Consider the decline of memory following exposure to a new pathogen. As seen in the main text, this decline may be different for different lineages and will be dependent on a number of factors such as cross-reactivity and competition. As a measure for the “average” loss of memory, we consider the decline in the total number of cells in memory lineages that were occupied prior to exposure to the pathogen.
Following stimulation with the
Lineages with
lineages with
lineages with
After the expansion phase and restoration of homeostasis, the corresponding memory lineages will have the following values
The total number of cells in previously occupied memory lineages is given by
We note that the first step in the model (the generation of memory cells) comprises both the expansion and contraction phases of the immune responses to acute infections, and the second step (the restoration of homeostasis) could take a longer time.
Consider the case of infection of inbred C57Bl/6 or BALB/c mice with the Armstrong strain of the (LCMV) [
From the simulations, we noticed that all memory lineages most of the time decline in size exponentially with the rate
We assume that each memory lineage has a fixed cross-reactivity
Then the probability that a given memory lineage will have size ln
After summing and integrating, we obtain the following expressions for the mean and variance of size of memory lineages
In the case when all lineages have the same size at
For our analysis as in simulations shown in
If
The expected decline of memory following exposure to a pathogen then is given by
The probability
We thank the three referees for very helpful comments.
Pichinde virus