A stochastic model of cytotoxic T cell responses

https://doi.org/10.1016/j.jtbi.2003.12.011Get rights and content

Abstract

We have constructed a stochastic stage-structured model of the cytotoxic T lymphocyte (CTL) response to antigen and the maintenance of immunological memory. The model follows the dynamics of a viral infection and the stimulation, proliferation, and differentiation of naı̈ve CD8+ T cells into effector CTL, which can eliminate virally infected cells. The model is capable of following the dynamics of multiple T cell clones, each with a T cell receptor represented by a digit string. MHC–viral peptide complexes are also represented by strings and a string match rule is used to compute the affinity of a T cell receptor for a viral epitope. The avidities of interactions are also computed by taking into consideration the density of MHC–viral peptides on the surface of an infected cell. Lastly, the model allows the probability of T cell stimulation to depend on avidity but also incorporates the notion of an antigen-independent programmed proliferative response. We compare the model to experimental data on the cytotoxic T cell response to lymphocytic choriomeningitis virus infections.

Introduction

Cytotoxic T lymphocytes (CTL) play a crucial role in the immune system's defense against viral infections. After the first exposure to a virus, T cells rapidly replicate and attack infected cells in a primary response. The T cell population then decreases, leaving behind a population of long-lived memory cells. These memory cells allow the immune system to respond to subsequent encounters with similar infections more efficiently in a secondary response. A lifetime of exposure to pathogens shapes an organism's repertoire of memory cells, making the states of individual immune systems unique.

Based on a variety of experimental results, we constructed a stochastic model of the CTL response to antigens. Our ultimate goal is to use the model to provide insight into the pathology and possible treatments of diseases such as AIDS, influenza, cancer, and autoimmune disorders. In the process of building the model we were able to situate experimental data from multiple experiments in a coherent framework that forms a relatively complete and consistent interpretation of T cell behavior. We take a computationally efficient stage-structured modeling approach which allows us to incorporate biologically realistic features of T cell proliferation and differentiation relatively easily, resulting in a model that makes quantitative predictions. In the sections that follow, we summarize relevant CTL biology, describe our model, then present preliminary results.

Section snippets

T cell biology

CTL reside in tissue or circulate through the body via the blood and lymph to detect cells that have been compromised by foreign organisms, such as viruses. We present a summary of the CTL biology relevant to our model. Many essential components of the immune response, such as the innate immune system, dendritic cells, and CD4+ T cells, are intentionally omitted; their roles in facilitating the CTL response are implicit in the model.

The model

Our model consists of two interacting parts: a stage-structured model of the T cell activation, proliferation and differentiation and a model of viral infection. The models are coupled in that infected cells stimulate naı̈ve T cells and are killed by effector T cells (depicted in Fig. 1). In addition, our model includes a representation of TCR binding and a realistic-sized T cell repertoire.

Results

Our model reproduces population-level phenomena seen experimentally in laboratory mice, and we describe some of these results below. We begin with experiments that illustrate the basic differences between primary and secondary responses and the dynamics of CTL of different affinities, then proceed to describe simulation runs that replicate results found in mouse experiments.

Conclusion

We have presented a stochastic stage-structured model of the CTL response to viral infections that features realistic behavior on the level of the individual cells yet is more efficient than standard agent-based approaches to modeling. Our model incorporates antigen- and affinity- dependent stimulation of naı̈ve cells as well as an antigen- independent programmed proliferative response and differentiation into effector cells as suggested by recent experiments. A benefit of our approach is that

Acknowledgements

DLC and SF gratefully acknowledge the support of the National Science Foundation (ANIR-9986555), the Office of Naval Research (N00014-99-1-0417), the Defense Advanced Research Projects Agency (AGR F30602-00-2-0584), the Intel Corporation. MPD is supported by the James S. McDonnell Foundation (21st Century Research Awards/Studying Complex Systems). ASP acknowledges the support of the National Institute of Health (grants R37 AI28433 and R01 RR06555). Portions of this work were done under the

References (86)

  • B.F. Dibrov et al.

    Mathematical model of immune processes

    J. Theor. Biol.

    (1977)
  • S. Ehl et al.

    The impact of variation in the number of CD8+ T-cell precursors on the outcome of virus infection

    Cell Immunol.

    (1998)
  • J.D. Farmer et al.

    The immune system, adaption and machine learning

    Physica D

    (1986)
  • S.M. Kaech et al.

    Molecular and functional profiling of memory CD8 T cell differentiation

    Cell

    (2002)
  • J.W. Kappler et al.

    T cell tolerance by clonal elimination in the thymus

    Cell

    (1987)
  • J.L. Maryanski et al.

    Single-cell PCR analysis of TCR repertoires selected by antigen in vivoa high magnitude CD8 response is comprised of very few clones

    Immunity

    (1996)
  • K. Murali-Krishna et al.

    Counting antigen-specific CD8+ T cellsa reevaluation of bystander activation during viral infection

    Immunity

    (1998)
  • A.S. Perelson et al.

    Theoretical studies of clonal selectionminimal antibody repertoire size and reliability of self-non-self discrimination

    J. Theor. Biol.

    (1979)
  • A.S. Perelson et al.

    Modeling immune reactivity in secondary lymphoid organs

    Bull. Math. Biol.

    (1992)
  • P.E. Seiden et al.

    A model for simulating cognate recognition and response in the immune system

    J. Theor. Biol.

    (1992)
  • D.J. Smith et al.

    Deriving shape space parameters from immunological data

    J. Theor. Biol.

    (1997)
  • D.J. Smith et al.

    Using lazy evaluation to simulate realistic-size repertoires in models of the immune system

    Bull. Math. Biol.

    (1998)
  • Armstrong, C., Lillie, R., 1934. Experimental lymphocytic choriomeningitis of monkeys and mice produced by a virus...
  • T.P. Arstila et al.

    A direct estimate of the human αβ T cell receptor diversity

    Science

    (1999)
  • M.F. Bachmann et al.

    Distinct kinetics of cytokine production and cytolysis in effector and memory T cells after viral infection

    Eur. J. Immunol.

    (1999)
  • V.P. Badovinac et al.

    Programmed contraction of CD8+ T cells after infection

    Nat. Immunol.

    (2002)
  • D.L. Barber et al.

    Cutting edgerapid in vivo killing by memory CD8 T cells

    J. Immunol.

    (2003)
  • W. Barchet et al.

    Direct quantitation of rapid elimination of viral antigen-positive lymphocytes by antiviral CD8+ T cells in vivo

    Eur. J. Immunol.

    (2000)
  • M. Blackman et al.

    The role of the T cell receptor in positive and negative selection of developing T cells

    Science

    (1990)
  • J.N. Blattman et al.

    Evolution of the T cell repertoire during primary, memory, and recall responses to viral infection

    J. Immunol.

    (2000)
  • J.N. Blattman et al.

    Estimating the precursor frequency of naive antigen-specific CD8 T cells

    J. Exp. Med.

    (2002)
  • P. Bousso et al.

    The composition of a primary T cell response is largely determined by the timing of recruitment of individual T cell clones

    J. Exp. Med.

    (1999)
  • P. Bousso et al.

    Facing two T cell epitopesa degree of randomness in the primary response is lost upon secondary immunization

    J. Immunol.

    (2000)
  • D.H. Busch et al.

    Evolution of a complex T cell receptor repertoire during primary and recall bacterial infection

    J. Exp. Med.

    (1998)
  • A.M. Byers et al.

    Cutting edgerapid in vivo CTL activity by polyoma virus-specific effector and memory CD8+ T cells

    J. Immunol.

    (2003)
  • A. Casrouge et al.

    Size estimate of the αβ TCR repertoire of naive mouse splenocytes

    J. Immunol.

    (2000)
  • Chao, D.L., Davenport, M.P., Forrest, S., Perelson, A.S., 2003. Stochastic stage-structured modeling of the adaptive...
  • M. Cohn et al.

    A computerized model for the self-non-self discrimination at the level of the T(h) (Th genesis). I. The origin of ‘primer’ effector T(h) cells

    Int. Immunol.

    (2002)
  • M.P. Davenport et al.

    Clonal selection, clonal senescence, and clonal successionthe evolution of the T cell response to infection with a persistent virus

    J. Immunol.

    (2002)
  • R.J. De Boer et al.

    Recruitment times, proliferation, and apoptosis rates during the CD8+ T-cell response to lymphocytic choriomeningitis virus

    J. Virol.

    (2001)
  • M. Derby et al.

    High-avidity CTL exploit two complementary mechanisms to provide better protection against viral infection than low-avidity CTL

    J. Immunol.

    (2001)
  • V. Detours et al.

    The paradox of alloreactivity and self restrictionquantitative analysis and statistics

    Proc. Natl Acad. Sci. USA

    (2000)
  • R.W. Dutton et al.

    T cell memory

    Annu. Rev. Immunol.

    (1998)

    • Cited by (87)

    • Modelling cross-reactivity and memory in the cellular adaptive immune response to influenza infection in the host

      2017, Journal of Theoretical Biology
      Citation Excerpt :

      We can either model the expansion of CD8+ T cells specific to epitopes which are experimentally determined to be dominant, or generate a large range of epitopes with different CD8+ T cell avidities and abundances. In the latter case, the model would have to be expanded to take into account that there are many different CD8+ T cell clones which can respond to a particular epitope with different avidities, as modelled by Chao et al. (2004). Our work could also be extended to model the changes in epitope-specific CD8+ T cell numbers over an individual's lifetime (Quinn et al., 2016), either due to natural infection or vaccination.

    • The Evolution of the Algorithms for Collective Behavior

      2016, Cell Systems
      Citation Excerpt :

      Desert plants accelerate soil respiration in response to brief episodes of rainfall (Potts et al., 2014). Cytokines create chemical interactions that T cells follow, with the outcome that numbers of T cells increase at a location where pathogens are patchy and concentrated in time and space (Fricke et al., 2015; Chao et al., 2004). An interesting question is whether the reverse is also true.

    • Bifurcations of a mathematical model for HIV dynamics

      2016, Journal of Mathematical Analysis and Applications

    • Applying the skin sensitisation adverse outcome pathway (AOP) to quantitative risk assessment

      2014, Toxicology in Vitro

    • Stability in distribution for age-structured HIV model with delay and driven by Ornstein–Uhlenbeck process

      2021, Studies in Applied Mathematics

    • A Validated Mathematical Model of the Cytokine Release Syndrome in Severe COVID-19

      2021, Frontiers in Molecular Biosciences
    View all citing articles on Scopus
    View full text
    Copyright © 2004 Elsevier Ltd. All rights reserved.