Coagulation-Fragmentation with a Finite Number of Particles: Models, Stochastic Analysis, and Applications to Telomere Clustering and Viral Capsid Assembly



Coagulation-fragmentation processes with a finite number of particles is a recent class of mathematical questions that serves modeling some cell biology dynamics. The analysis of the models offers new challenging questions in probability and analysis: the model is the clustering of particles after binding, the formation of local subclusters of arbitrary sizes and the dissociation into subclusters. We review here modeling and analytical approaches to compute the size and number of clusters with a finite size. Applications are clustering of chromosome ends (telomeres) in yeast nucleus and the formation of viral capsid assembly from molecular components. The methods to compute the probability distribution functions of clusters and to estimate the statistical properties of clustering are based on combinatorics and hybrid Gillespie-spatial simulations. Finally, we review models of capsid formation, the mean-field approximation, and jump processes used to compute first passage times to a finite size cluster. These models become even more relevant for extracting parameters from live cell imaging data.



This research was supported by a Marie-Curie grant. We thank S. Manley for discussions and sharing with us the GAG super-resolution data. We also thank the hospitality of the Newton Institute in Cambridge during the year 2016.


  1. 1.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Reprint of the 1972 edn. (Dover, New York, 1992)Google Scholar
  2. 2.
    D.J. Aldous, Deterministic and stochastic models for coalescence (aggregation, coagulation): review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    G.E. Andrews, The Theory of Partitions. Encyclopedia of Mathematics and Its Applications, vol. 2 (Addison-Wesley, Reading, MA, 1976)Google Scholar
  4. 4.
    R. Becker, W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen. Ann. Phys. 24 719–752 (1935)CrossRefMATHGoogle Scholar
  5. 5.
    K. Bystricky, P. Heun, L. Gehlen, J. Langowski, S.M. Gasser, Long-range compaction and flexibility of interphase chromatin in budding yeast analyzed by high-resolution imaging techniques. Proc. Natl. Acad. Sci. U. S. A. 101(47), 16495–16500 (2004)CrossRefGoogle Scholar
  6. 6.
    T. Carlsson, T. Ekholm, C. Elvingson, Algorithm for generating a Brownian motion on a sphere. J. Phys. A Math. Theor. 43(50), 505001 (2010)Google Scholar
  7. 7.
    S. Chandrasekar, Stochastic problems in physics and astrophysics. Rev. Mod. Phys. 15, 1–89 (1943)CrossRefGoogle Scholar
  8. 8.
    J.F. Collet, Some modelling issues in the theory of fragmentation-coagulation systems. Commun. Math. Sci. 1, 35–54 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    C.R. Doering, D. Ben-Avraham, Interparticle distribution functions and rate equations for diffusion-limited reactions. Phys. Rev. A 38, 3035 (1988)CrossRefGoogle Scholar
  10. 10.
    R. Durrett, B.L. Granovsky, S. Gueron, The equilibrium behavior of reversible coagulation-fragmentation processes. J. Theor. Probab. 12, 447–474 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    S. Gueron, The steady-state distributions of coagulation-fragmentation processes. J. Math. Biol. 37, 1–27 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    J. Gunzenhäuser, R. Wyss, S. Manley, A quantitative approach to evaluate the impact of fluorescent labeling on membrane-bound HIV-Gag assembly by titration of unlabeled proteins. PLoS One 9(12), e115095 (2014)Google Scholar
  13. 13.
    D. Holcman, N. Hoze, Z. Schuss, Analysis and interpretation of superresolution single-particle trajectories. Biophys. J. 109, 1761–1771 (2015)CrossRefGoogle Scholar
  14. 14.
    N. Hoze, D. Holcman, Coagulation–fragmentation for a finite number of particles and application to telomere clustering in the yeast nucleus. Phys. Lett. A 376, 845–849 (2012)CrossRefMATHGoogle Scholar
  15. 15.
    N. Hoze, D. Holcman, Residence times of receptors in dendritic spines analyzed by stochastic simulations in empirical domains. Biophys. J. 107, 3008–3017 (2014)CrossRefGoogle Scholar
  16. 16.
    N. Hoze, D. Holcman, Modeling capsid kinetics assembly from the steady state distribution of multi-sizes aggregates. Phys. Lett. A 378, 531–534 (2014)CrossRefMATHGoogle Scholar
  17. 17.
    N. Hoze, D. Holcman, Kinetics of aggregation with a finite number of particles and application to viral capsid assembly. J. Math. Biol. 70, 1685–1705 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    N. Hoze, D. Holcman, Stochastic coagulation-fragmentation processes with a finite number of particles. Ann. Appl. Probab. (2016, accepted)Google Scholar
  19. 19.
    N. Hoze, D. Nair, E. Hosy, C. Sieben, S. Manley, A. Herrmann, J.B. Sibarita, D. Choquet, D. Holcman, Heterogeneity of receptor trafficking and molecular interactions revealed by superresolution analysis of live cell imaging. Proc. Natl. Acad. Sci. U. S. A. 109, 17052–17057 (2012)CrossRefGoogle Scholar
  20. 20.
    N. Hoze, M. Ruault, C. Amoruso, A. Taddei, D. Holcman, Spatial telomere organization and clustering in yeast Saccharomyces cerevisiae nucleus is generated by a random dynamics of aggregation–dissociation. Mol. Biol. Cell 24, 1791–1800 (2013)CrossRefGoogle Scholar
  21. 21.
    S. Jacquot, A historical law of large numbers for the Marcus-Lushnikov process. Electron. J. Probab. 15, 605–635 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    F.P. Kelly, Reversibility and Stochastic Networks. Wiley Series in Probability and Mathematical Statistics (Wiley, Chichester, 1979)Google Scholar
  23. 23.
    P. Krapivsky, S. Redner, E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010)CrossRefMATHGoogle Scholar
  24. 24.
    A.A. Lushnikov, Coagulation in finite systems. J. Colloid Interface Sci. 65, 276–285 (1978)CrossRefGoogle Scholar
  25. 25.
    A. Marcus, Stochastic coalescence. Technometrics 10, 133–143 (1968)MathSciNetCrossRefGoogle Scholar
  26. 26.
    H.G. Rotstein, Cluster-size dynamics: a phenomenological model for the interaction between coagulation and fragmentation processes. J. Chem. Phys. 142, 224101 (2015)CrossRefGoogle Scholar
  27. 27.
    H. Schober, et al., Controlled exchange of chromosomal arms reveals principles driving telomere interactions in yeast. Genome Res. 18, 261–271 (2008)CrossRefGoogle Scholar
  28. 28.
    Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach. Applied Mathematical Sciences, vol.170 (Springer, New York, 2010)Google Scholar
  29. 29.
    Z. Schuss, Diffusion and Stochastic Processes: An Analytical Approach (Springer, New York, 2010)MATHGoogle Scholar
  30. 30.
    Z. Schuss, Nonlinear Filtering and Optimal Phase Tracking. Applied Mathematical Sciences, vol. 180 (Springer, New York, 2011)Google Scholar
  31. 31.
    P. Therizols, T. Duong, B. Dujon, C. Zimmer, E. Fabre, Chromosome arm length and nuclear constraints determine the dynamic relationship of yeast subtelomeres. Proc. Natl. Acad. Sci. U. S. A. 107, 2025–2030 (2010)CrossRefGoogle Scholar
  32. 32.
    C.J. Thompson, Classical Equilibrium Statistical Mechanics (Oxford University Press, Oxford, 1988)Google Scholar
  33. 33.
    B.R. Thomson, Exact solution for a steady-state aggregation model in one dimension. J. Phys. A 22, 879–886 (1989)MathSciNetCrossRefGoogle Scholar
  34. 34.
    M. von Smoluchowski, Drei Vorträge über Diffusion Brownsche Molekularbewegung und Koagulation von Kolloidteichen. Phys. Z. 17, 557–571 (1916)Google Scholar
  35. 35.
    J.A. Wattis, An introduction to mathematical models of coagulation–fragmentation processes: a discrete deterministic mean-field approach. Physica D 222, 1–20 (2006)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    R. Yvinec, M.R. D’Orsogna, T. Chou, First passage times in homogeneous nucleation and self-assembly. J. Chem. Phys. 137, 244107 (2012)CrossRefGoogle Scholar
  37. 37.
    A. Zlotnick, Theoretical aspects of virus capsid assembly. J. Mol. Recognit. 18, 479–490 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut für Integrative Biologie, ETHUniversitätstrasse 16ZürichSwitzerland
  2. 2.Institute for Biology École Normale SupérieureApplied Mathematics and Computational BiologyParisFrance
  3. 3.Churchill CollegeUniversity of Cambridge, Storey’s WayCambridgeUK

Personalised recommendations