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Article review by Ash Same, April 2013.

R. Antia, V. V. Ganusov, and R. Ahmed. The role of models in understanding CD8^{+} T-cell memory. Nature Reviews. Immunology, 5(2):101-11, Mar, 2005.

Executive Summary

This paper examines past mathematical models and data for the analysis of an organism's ability, using CD8^{+} T-cells, to ‘remember’ pathogens that it has encountered and to respond better the next time they are encountered. It examines the process by which naive T-cells undergo cell division and are converted into effector and memory cells that are connected with a specific pathogen, on exposure to that pathogen. Different models for the conversion and expansion/contraction of these different cell types are compared. It goes on to look at further models, and the relationship between these cell dynamics and the presence of the pathogen and antigens. The processes by which memory cells remain in the system for long time periods is also examined. Models for the dynamics of the different types of memory cells, and those linked to different antigens, are introduced. At the end it discusses the different possible methods by which memory cells provide protection, and how these processes affect pathology.


Immunological memory refers to the ability of an organism to ‘remember’ pathogens that it has encountered and to respond better the next time they are encountered. This paper focuses on the role of past mathematical models in the understanding of immunological memory, particularly CD8^{+} T-cell responses to intra-cellular pathogens after acute infections. One particularly important application of this is vaccinations.

CD8^{+} T-cells can be ‘naive’, in that they are general CD8^{+} T-cells that have not yet been ‘related’ with a specific antigen. These naive cells can divide into many clones. Clones can differentiate into effector T-cells that are related to and fight a specific antigen. They can also form memory cells that may be inactive but are still related to the antigen, and will survive in the host for a long period of time.

The figure below shows the CD8^{+} T-cells related to a specific pathogen. Following initial exposure to the pathogen naive cells undergo exponential clonal expansion (by ~4–5 log), creating effector cells, until the pathogen is eradicated and they undergo exponential clonal contraction (cell-death) (by ~1–2 log) along with conversion to memory cells. Re-exposure occurs after a long period without exposure. The increased cell levels (due to memory cells) at re-exposure provide protection.

Figure 1

Experimental data with lymphocytic choriomeningitis virus (LCMV) has been used to develop mathematical models for the CD8^{+} T-cell levels before and after exposure. Cell-division and cell-death rates for the pathogen and host used were found by matching data from bromodeoxyuridine and D-glucose labels to simple ODE models. Later on CFSE was used, giving extra information about the number of divisions each cell underwent.

Modeling Memory Generation

There are different possibilities for the way in which general CD8^{+} T-cells differentiate into memory T-cells that are related to a specific pathogen. The paper examines two different possibilities, and the models that can arise. Both models set the death-rate of memory cells, \delta _{M}, to zero since these populations are almost constant over time.

PE Model

In this model naive cells are converted to effector cells P_{E} at time T_{{on}} after infection. These effector cells undergo clonal expansion at rate \rho while the pathogen is still present, until time T_{{off}}. After this time expansion ends and clonal contraction occurs at rate \alpha , along with differentiation to memory cells M at rate r.

Figure 2

The following graph shows the cell response to the NP118 epitope of LCMV in BALB/c mice, along with the best fit that can be produced by adjusting parameters in the PE model.

Figure 3

PM Model

In this model naive cells are converted to memory cells P_{M} at time T_{{on}} after infection. These memory cells undergo clonal expansion at rate \rho and differentiate into effector cells E at rate r while the pathogen is still present, until time T_{{off}}. After this time expansion and differentiation end and clonal contraction of effector cells occurs at rate \alpha .

Figure 4

The following graph shows the cell response and the best fit for the PM model, as well as the best fit given a maximum birth-rate constant of \rho = 5 (corresponding to cell division every 3.3 hours).

Figure 5

The best fit for the PE model uses parameters that are biologically reasonable, while the best fit for the PM model requires an unreasonably high birth-rate constant \rho (cell division every 6 minutes) and is inaccurate for constrained \rho . This suggests that the first model is more accurate. Similar results hold for data from different epitopes or different mice species. There have also been experiments that have shown certain effector cells differentiating into memory cells, further supporting the first model.

By determining parameters for the cell division and death rates for different antigens we can find quantitative values representing the immunodominance, or the result of specific antigens within a pathogen being preferentially reacted against by their corresponding T-cells. We can also see that the immunodominance develops in the cell division stage, with large differences in the number of T-cells developed for different antigens.

These models could be improved by including extra terms for the stages that cells go through before they begin division, and during division. But then the data might not be sufficient to estimate all of the extra parameters.

Dependence of Cell Division on the Antigen

Early models (like those above) assumed that, after infection, the T-cells for a specific antigen will only continue to divide so long as the antigen remains present in the system. But more recent experiments have suggested that division will follow the same ‘program’, independent of the length of time that the antigen is present. The idea that the effector cells don’t need information from the antigen is also supported when considering that the cell responses don’t compete with humoral responses for access to the antigen.

There have been numerous accurate models that use this type of system. One model from Antia et al. was designed with certain characteristics so that it would be consistent with existing data. It considers the antigen levels over time. At time t, naive cells N(t) are converted to effector cells at a rate that depends on the antigen level at that time, P(t). Individual effector cells divide, die, and become memory cells independently of both the antigen level and the total number of effector cells, but dependent on the time \tau of conversion of that cell from a naive cell.

y(t,\tau ) is the change in the number of effector cells at time t (from naive cell recruitment, as well as cell division and death) for those cells recruited \tau days earlier. \rho (\tau ) is the division rate of cells recruited \tau days earlier. d(\tau ) is their death rate. All effector cells become memory cells after a set time of being effectors. Y(t) is the total number of T-cells related to the antigen (effector + memory cells).

Figure 6

This model indicates that, along with the antigen-independent components, there are still components of the cell recruitment/expansion that are dependent on antigen levels and presence. As antigen levels and presence increase we get greater naive cell recruitment and therefore more effector cells and more cell division (see graph below, from Kaech and Ahmed). But the only interaction with antigens is during recruitment.

Figure 7

The cell recruitment is also independent of the type of antigen. This shows that immunodominance cannot come from the recruitment phase. The differences may come from the start time of the cell division, following recruitment, which could be affected by the specific antigen.

The paper explains how the lack of communication between effector cells and antigen can be beneficial. Continual communication will allow the required number of cells to be produced, while excess cells must be produced if there is no communication, in order make sure there is enough. But without communication, pathogens will be unable to ‘trick’ the sensors and subvert the immune response. The T-cell ‘program’ will already be in process before the pathogen has a chance do this.

Other models have been developed to study the relationships between cell death-rates and the immunological response, as well as the response following vaccination. Further study could look at how the antigen-dependent and independent ‘instructions’ can be measured, how they are delivered to the cells, and how they each affect the outcome.

Longevity of Memory

For a given pathogen, the number of related memory T-cells can remain high for a very long time, sometimes decades. There have been several different hypotheses to explain immunological memory, as well as corresponding models. Experiments have shown that division of memory cells actually occurs faster than for naive cells. So theories that suggested memory cells don’t divide and simply live for a long time were rejected. Instead, they must have balanced division and death rates.

This division and death of memory cells was compared for infected mice and mice that simply had pathogen-specific memory cells given to them. These processes were concluded to be independent of the presence of antigen (or at least to severe differences in antigen level), despite earlier experiments suggesting otherwise.

The renewal of memory T-cells through cell division and death is believed to occur through some or all of bystander stimulation (where T-cells divide equally, independent of their type), crossreactive stimulation (where only memory cells, not naive cells, divide) and homeostatic regulation (where all T-cells die off in equal proportions, to balance any cell division and keep the number of T-cells constant). Infections result in the clonal expansion of all cells (bystander stimulation), but memory cells may have lower thresholds than naive cells and undergo extra division (crossreactive stimulation). Crossreactive stimulation may also only result in stimulation of memory cells that are related with the current pathogen. So it will cause the creation of extra memory cells that can control the current pathogen, and reduce the numbers of naive cells. Such behavior has been observed experimentally for CD8^{+} memory T-cells.

Figure 8

Different lineages can represent a group of naive cells or a group of memory cells related to a specific antigen. One particular model of different cell populations is discussed. It examines populations of naive cell lineages, along with populations of different memory cell lineages. Memory cells are assumed identical except for the antigen they correspond to. When exposed to a new pathogen that hasn’t been seen before, bystander and crossreactive stimulation occurs on the lineages. Naive cells can also undergo clonal expansion into empty lineages, or be recruited into memory cells for the pathogen, and help to fight it.

After the pathogen is gone homeostatic regulation occurs. If {\hat  {Y}} is the size of the memory that must be retained, and Y is the amount of memory T-cells related to the previous pathogens seen, then \bigtriangleup Y/{\hat  {Y}}\approx -M/{\hat  {Y}}, where M is the number of memory cells related to the new pathogen/antigens.

Over time, as an individual ages, the value {\hat  {Y}} will actually be increasing as well. As {\hat  {Y}} increases, lineage sizes increase proportionally. Putting it all together gives y_{i}(t)=y_{i}(0)exp[-\int _{0}^{t}{\frac  {m(\tau )}{{\hat  {Y}}(\tau )}}d\tau ], where y_{i}(t) is the average number of memory cells in lineage i at time t, {\hat  {Y}}(\tau ) is {\hat  {Y}} at time \tau , and m(\tau ) is M at time \tau . Longevity of immunological memory can be measured by considering the size of the memory cell population over time.

Experiments on CD8^{+} T-cell levels in mice had new pathogens causing an increase in lineages that were part of crossreactive stimulation, but decreases in others, as expected. But other aspects of this model still need to be tested against experimental data. Especially the assumption that memory cells are identical except for the antigen they correspond to. These models could also be extended to humans, as the above is based off the immunological system of mice. The high rate of change of memory cells early on (before many pathogens have been experienced) could also be modeled.

Memory Cells Providing Protection

Vaccinations demonstrate the ability of memory cells to provide protection under re-exposure to pathogens. The paper discusses the different ways in which this can occur – by prevention of infection, or by reducing the magnitude of infection.

Infection could be prevented if memory cell levels cause negative growth of the pathogen, or if they prevent exposures from resulting in infection by decreasing the density of the pathogen and therefore the chance of infection. If infection can’t be prevented, the magnitude could be decreased because of the extra initial T-cells related to the pathogen, because of faster T-cell division, or because of more efficient effector function. All of these factors have shown to be true of memory cells in experimentation.

Assumptions that memory cells cannot kill infected cells, and that effector cells cannot undergo clonal expansion may need to be altered. Experiments have shown that these assumptions are not always true. Models based on these assumptions don’t give the above results when memory cells are increased.

Memory Cells Affecting Level of Pathology

Pathology could be caused by the virus infecting cells, or by the T-cells that kill infected cells. In either case, models show that (for persistent infections) pathology will be maximized when the immune system provides an intermediate response level. There will be higher levels of infected cells than if a high level of response is given, and there will also be a greater level of cell destruction than if a low level of response is given. A more recent study for acute infections has shown that increased memory cells generally decrease pathology. This information can all be taken into account when considering different methods for immunization.

Further Study

The paper discusses ways in which all of these models and this research can be extended to CD4^{+} T-cell responses, and humoral responses to pathogens. CD4^{+} T-cell responses have many similarities, with a few slight differences that could be addressed in the models by altering a few of the parameters and some of the processes as well. Humoral responses are more complex, involving the coordination of different types of cells. But there are still a lot of similarities. Models would need to incorporate interacting models of each separate cell type.

Other areas of development include extension of data from mice to humans, and analysis of the differences in the different species. The dynamics of persistent infections could be studied further.

External Links

R. Antia, V. V. Ganusov, and R. Ahmed. The role of models in understanding CD8^{+} T-cell memory. Nature Reviews. Immunology, 5(2):101-11, Mar, 2005.