Skip to main content

Rare Event Simulation for T-cell Activation

Abstract

The problem of statistical recognition is considered, as it arises in immunobiology, namely, the discrimination of foreign antigens against a background of the body’s own molecules. The precise mechanism of this foreign-self-distinction, though one of the major tasks of the immune system, continues to be a fundamental puzzle. Recent progress has been made by van den Berg, Rand, and Burroughs (J. Theor. Biol. 209:465–486, 2001), who modelled the probabilistic nature of the interaction between the relevant cell types, namely, T-cells and antigen-presenting cells (APCs). Here, the stochasticity is due to the random sample of antigens present on the surface of every APC, and to the random receptor type that characterises individual T-cells. It has been shown previously (van den Berg et al. in J. Theor. Biol. 209:465–486, 2001; Zint et al. in J. Math. Biol. 57:841–861, 2008) that this model, though highly idealised, is capable of reproducing important aspects of the recognition phenomenon, and of explaining them on the basis of stochastic rare events. These results were obtained with the help of a refined large deviation theorem and were thus asymptotic in nature. Simulations have, so far, been restricted to the straightforward simple sampling approach, which does not allow for sample sizes large enough to address more detailed questions. Building on the available large deviation results, we develop an importance sampling technique that allows for a convenient exploration of the relevant tail events by means of simulation. With its help, we investigate the mechanism of statistical recognition in some depth. In particular, we illustrate how a foreign antigen can stand out against the self background if it is present in sufficiently many copies, although no a priori difference between self and nonself is built into the model.

This is a preview of subscription content, access via your institution.

References

  1. Arstila, T., Casrouge, A., Baron, V., Even, J., Kannelopoulos, J., Kourilsky, P.: A direct estimate of the human α β T cell receptor diversity. Science 286, 958–961 (1999)

    Article  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  3. Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)

    MATH  Google Scholar 

  4. Borovsky, Z., Mishan-Eisenberg, G., Yaniv, E., Rachmilewitz, J.: Serial triggering of T cell receptors results in incremental accumulation of signaling intermediates. J. Biol. Chem. 277, 21529–21536 (2002)

    Article  Google Scholar 

  5. Bucklew, J.A.: Introduction to Rare Event Simulation. Springer, New York (2004)

    MATH  Google Scholar 

  6. Davis, S.J., Ikemizu, S., Evans, E.J., Fugger, L., Bakker, T.R., van der Merwe, P.A.: The nature of molecular recognition by T cells. Nat. Immunol. 4, 217–224 (2003)

    Article  Google Scholar 

  7. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer, New York (1998)

    MATH  Google Scholar 

  8. den Hollander, F.: Large Deviations. Am. Math. Soc., Providence (2000)

    MATH  Google Scholar 

  9. Dieker, A., Mandjes, M.: On asymptotically efficient simulation of large deviation probabilities. Adv. Appl. Probab. 37, 539–552 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  10. Dushek, O., Coombs, D.: Analysis of serial engagement and peptide-MHC transport in T cell receptor microclusters. Biophys. J. 94, 3447–3460 (2008)

    Article  Google Scholar 

  11. Gonzalez, P.A., Carreno, L.J., Coombs, D., Mora, J.E., Palmieri, E., Goldstein, B., Nathenson, S.G., Kalergis, A.M.: T-cell receptor binding kinetics required for T cell activation depend on the density of cognate ligand on the antigen-presenting cell. Proc. Natl. Acad. Sci. USA 102, 4824–4829 (2005)

    Article  ADS  Google Scholar 

  12. Hlavacek, W.S., Redondo, A., Wofsy, C., Goldstein, B.: Kinetic proofreading in receptor-mediated transduction of cellular signals: receptor aggregation, partially activated receptors, and cytosolic messengers. Bull. Math. Biol. 64, 887–911 (2002)

    Article  Google Scholar 

  13. Hunt, D.F., Henderson, R.A., Shabanowitz, J., Sakaguchi, K., Michel, H., Sevilir, N., Cox, A.L., Appella, E., Engelhard, V.H.: Characterization of peptides bound to the class I MHC molecule HLA-A2.1 by mass spectrometry. Science 255, 1261–1263 (1992)

    Article  ADS  Google Scholar 

  14. Kalergis, A.M., Boucheron, N., Doucey, M.A., Palmieri, E., Goyarts, E.C., Vegh, Z., Luescher, I.F., Nathenson, S.G.: Efficient T cell activation requires an optimal dwell-time of interaction between the TCR and the pMHC complex. Nat. Immunol. 2, 229–234 (2001)

    Article  Google Scholar 

  15. Kronmal, R.A., Peterson, A.J.: On the alias method for generating random variables from a discrete distribution. Am. Stat. 33, 214–218 (1979)

    MATH  Article  MathSciNet  Google Scholar 

  16. Lancet, D., Sadovsky, E., Seidelmann, E.: Probability model for molecular recognition in biological receptor repertoires: Significance to the olfactory system. Proc. Natl. Acad. Sci. USA 90, 3715–3719 (1993)

    Article  ADS  Google Scholar 

  17. Lord, G.M., Lechler, R.I., George, A.J.: A kinetic differentiation model for the action of altered TCR ligands. Immunol. Today 20, 33–39 (1999)

    Article  Google Scholar 

  18. Madras, N.: Lectures on Monte-Carlo Methods. Am. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  19. Mason, D.: A very high level of crossreactivity is an essential feature of the T-cell receptor. Immunol. Today 19, 395–404 (1998)

    Article  Google Scholar 

  20. McKeithan, T.W.: Kinetic proofreading in T-cell receptor signal transduction. Proc. Natl. Acad. Sci. USA 92, 5042–5046 (1995)

    Article  ADS  Google Scholar 

  21. Rabinowitz, J.D., Beeson, C., Wulfing, C., Tate, K., Allen, P.M., Davis, M.M., McConnell, H.M.: Altered T-cell receptor ligands trigger a subset of early T cell signals. Immunity 5, 125–135 (1996)

    Article  Google Scholar 

  22. Rosenwald, S., Kafri, R., Lancet, D.: Test of a statistical model for molecular recognition in biological repertoires. J. Theor. Biol. 216, 327–336 (2002)

    Article  Google Scholar 

  23. Ross, S.M.: Simulation. Academic Press, San Diego (2002)

    Google Scholar 

  24. Rothenberg, E.V.: How T-cells count. Science 273, 78–80 (1996)

    Article  ADS  Google Scholar 

  25. Sadowsky, J.S., Bucklew, J.A.: On large deviations theory and asymptotically efficient Monte Carlo estimation. IEEE Trans. Inf. Theory 36, 579–588 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  26. Sousa, J., Carneiro, J.: A mathematical analysis of TCR serial triggering and down-regulation. Eur. J. Immunol. 30, 3219–3227 (2000)

    Article  Google Scholar 

  27. Stevanovíc, S., Schild, H.: Quantitative aspects of T cell activation—peptide generation and editing by MHC class I molecule. Semin. Immunol. 11, 375–384 (1999)

    Article  Google Scholar 

  28. Utzny, C., Coombs, D., Muller, S., Valitutti, S.: Analysis of peptide/MHC-induced TCR downregulation: deciphering the triggering kinetics. Cell Biochem. Biophys. 46, 101–111 (2006)

    Article  Google Scholar 

  29. Valitutti, S., Lanzavecchia, A.: Serial triggering of TCRs: a basis for the sensitivity and specificity of antigen recognition. Immunol. Today 18, 299–304 (1997)

    Article  Google Scholar 

  30. Valitutti, S., Muller, S., Cella, M., Padovan, E., Lanzavecchia, A.: Serial triggering of many T-cell receptors by a few peptide-MHC complexes. Nature 375, 148–151 (1995)

    Article  ADS  Google Scholar 

  31. van den Berg, H.A., Molina-París, C.: Thymic presentation of autoantigens and the efficiency of negative selection. J. Theor. Med. 5, 1–22 (2003)

    MATH  Google Scholar 

  32. van den Berg, H.A., Rand, D.A.: Antigen presentation on MHC molecules as a diversity filter that enhances immune efficacy. J. Theor. Biol. 224, 249–267 (2003)

    Article  Google Scholar 

  33. van den Berg, H.A., Rand, D.A.: Quantitative theory of T-cell responsiveness. Immunol. Rev. 216, 81–92 (2007)

    Google Scholar 

  34. van den Berg, H.A., Rand, D.A., Burroughs, N.J.: A reliable and safe T-cell repertoire based on low-affinity T-cell receptors. J. Theor. Biol. 209, 465–486 (2001)

    Article  Google Scholar 

  35. Viola, A., Lanzavecchia, A.: T-cell activation determined by T-cell receptor number and tunable thresholds. Science 273, 104–106 (1996)

    Article  ADS  Google Scholar 

  36. Zint, N., Baake, E., den Hollander, F.: How T-cells use large deviations to recognize foreign antigens. J. Math. Biol. 57, 841–861 (2008)

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ellen Baake.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lipsmeier, F., Baake, E. Rare Event Simulation for T-cell Activation. J Stat Phys 134, 537–566 (2009). https://doi.org/10.1007/s10955-008-9672-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9672-2

Keywords

  • Immunobiology
  • Statistical recognition
  • Large deviations
  • Rare event simulation