Because these dynamics are highly multifactorial and nonlinear, mathematical models are required to precisely investigate this issue. Presented here is such a model (Fig. 1) describing the dynamics between an intracellular pathogen and a specific CD8⁺ T cell response. The model, which is based on previously published approaches (2-4), takes into account CD8⁺ T cell clones directed against different epitopes of the pathogen population. The rate of generation and expansion of the CD8⁺ T cell responses is proportional to antigen load. In the absence of antigen, memory CD8⁺ T cells decay at a defined rate, which is thought to be low. The model assumes that CD8⁺ T cells lyse infected cells and secrete soluble mediators, such as IFN-, that enhance immunity and interfere with viral replication (5).

The outcome of the model is determined by the efficacy of the immune system. Because the model is deterministic, reduction of pathogen load to zero is not possible in the context of the model. If the immune response is efficient, however, pathogen load can be reduced to extremely low values that, in practical terms, correspond to elimination of the pathogen (reduction of load below one bacterial cell or one virus particle). If the immune response is less efficient, pathogen load attains higher values, corresponding to persistent replication.

In the present context, three immune system parameters are important for interpreting the experimental data (Fig. 1): (i) c, the rate of antigen-driven CD8⁺ T cell proliferation; (ii) p, the rate of CD8⁺ T cell-mediated lysis of infected cells; and (iii) q, the rate of nonlytic CD8⁺ T cell-mediated inhibition of microbial replication (e.g., by IFN-). Under the model, the level of specific CD8⁺ T cells is mainly determined by the rates of lytic and nonlytic pathogen inhibition. Weaker inhibition results in higher pathogen load and a higher level of CD8⁺ T cells (Fig. 1). Unless CD8⁺ T cell proliferation saturates at low densities, the model predicts that the elevation of CD8⁺ T cells is significantly higher than the elevation of pathogen load (not shown in Fig. 1), as observed in the experiments of Badovinac et al. (1).

On the other hand, theory suggests that the immunodominance hierarchy of the specific CD8⁺ T cell clones is governed by competition for antigenic stimulation. This is determined by the magnitude of antigen-driven CD8⁺ T cell proliferation (6). The more efficient the rate of antigen-driven expansion of a given CD8⁺ T cell clone, the better its competitive ability, because less antigen is needed for stimulation (7, 8). Because IFN- can influence the rate of CD8⁺ T cell proliferation (through regulation of antigen presentation), a reduction of IFN- can result both in higher pathogen load and in shifted immunodominance (Fig. 1).

Because absence of IFN- can also result in a higher rate of microbial replication, the model presented here further predicts a reduced death phase of the CD8⁺ T cell response due to prolonged antigen persistence, even if the pathogen becomes undetectable. With fast microbial replication, the CD8⁺ T cell response initially reduces pathogen load to low levels; a prolonged phase of limited replication then follows before equilibrium is reached. This interpretation is supported by recent data from mice deficient in IFN- that were infected with a strain of lymphocytic choriomeningitis virus (LCMV Armstrong): Compared with wild-type mice, the knockout animals were characterized not only by higher levels of CD8⁺ T cells in the memory phase, but also by increased and persistent viral load that remained low and nonpathogenic, albeit in this case above the limit of detection (9).

In summary, this brief analysis shows that, contrary to the arguments of Badovinac et al. (1), their main observations can indeed be explained by the basic mechanisms by which perforin and IFN- control infection. Of course, mathematical modeling does not preclude the existence of additional and more complicated regulatory effects of these molecules. When interpreting experimental data, however, it is important to keep in mind the most parsimonious mechanism that can lead to the observed results.

Dominik Wodarz
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08540, USA
E-mail: wodarz{at}ias.edu

REFERENCES AND NOTES

1.	V. P. Badovinac, A. R. Tvinnereim, J. T. Harty, Science 290, 1354 (2000) [Abstract/Free Full Text] .
2.	M. A. Nowak and C. R. Bangham, Science 272, 74 (1996) [Abstract] .
3.	R. J. De Boer and A. S. Perelson, J. Theor. Biol. 190, 201 (1998) [CrossRef] [Web of Science] [Medline] .
4.	D. Wodarz and M. A. Nowak, J. Theor. Med. 2, 113 (2000) .
5.	Details of the model are given by Wodarz and Nowak (4). Essentially, the model is given by four differential equations: where x is the quantity of uninfected host cells, y is the quantity of infected host cells, z1 is the quantity of specific CD8+ T cell clone 1, and z2 is the quantity of specific CD8+ T cell clone 2. Uninfected cells are produced at a rate  and die at a rate d; they become infected by the pathogen at a rate  and die at a rate a. CD8+ T cells lyse infected cells at rate pi; they also secrete IFN-, which inhibits microbial replication at a rate qi. Two factors determine the responsiveness of the CD8+ T cells: the expansion rate of the cells, ci, and the aforementioned secretion of IFN-, which enhances the rate of CD8+ T cell proliferation at a rate fqi. The rate of CD8+ T cell proliferation saturates at high CD8+ T cell abundances, as determined by the parameter , and CD8+ T cells die at a rate b. Reduction of perforin and IFN- is represented in the model as a reduction in the parameters pi and qi, respectively.
6.	D. Wodarz and M. A. Nowak, Eur. J. Immunol. 30, 2704 (2000) [CrossRef] [Web of Science] [Medline] .
7.	J. A. Borghans, L. S. Taams, M. H. Wauben, R. J. de Boer, Proc. Natl. Acad. Sci. U.S.A. 96, 10782. (1999).
8.	M. A. Nowak, et al., Nature 375, 606 (1995) [CrossRef] [Medline] .
9.	C. Bartholdy, J. P. Christensen, D. Wodarz, A. R. Thomsen, J. Virol. 74, 10304 (2000) [Abstract/Free Full Text] .
10.	Parameters for the model run were chosen as follows:  = 10; d = 0.1;  = 0.02; a = 0.1; p1 = 1; p2 = 0.1; q1 = 30; q2 = 5; c1 = 0.01; c2 = 0.02; b = 0.05; e = 10; f = 0.001. For PKO, p1 = 0.2; p2 = 0.02. For GKO, q1 = 0.5; q2 = 0.5. The model is robust, however, and does not depend on the particular parameter values chosen.

Response: The utility of mathematical models to explain complex biological processes is related to how well their assumptions fit the in vivo situation and whether they accurately encompass actual experimental findings. The model of Wodarz is based on three assumptions: (i) increased CD8⁺ T cell responses in perforin-deficient (PKO) mice result from delayed clearance of infection; (ii) persistent infection accounts for aberrant CD8⁺ T cell decline in IFN--deficient (GKO) mice; and (iii) persistent infection accounts for altered hierarchies of immunodominance in GKO mice by eliminating competition for antigen.

In contrast to these assumptions, our results show that clearance of attenuated Listeria is identical at early and late times after infection of wild-type and PKO mice. Moreover, immunization of PKO mice with peptide-coated dendritic cells also resulted in increased CD8⁺ T cell response. By 10 days after infection, clearance of the attenuated Listeria infection (less than 100 bacteria, or less than one infected cell per gram of liver) occurred in both GKO and wild-type mice (1). In subsequent studies, we have found that the response to another subdominant epitope from Listeria is not elevated in GKO mice (2), a result that is not consistent with the generalized increase in subdominant responses predicted by the Wodarz model based on persistent infection and antigen competition.

The Wodarz model also relies on the assumption that persistent infection (y > 0) is required for survival of memory T cells, a contention that is clearly not supported by experimental results (3-5). Thus, the Wodarz model is of limited utility because of assumptions that are not consistent with the data and because of its failure to account for the observed results.

*Also Department of Mathematics, University of Iowa.

Vladimir P. Badovinac
Amy R. Tvinnereim
Herbert W. Hethcote
John T. Harty
Department of Microbiology
University of Iowa
Iowa City, IA 52242, USA

REFERENCES