Multi-type Galton–Watson Processes with Affinity-Dependent Selection Applied to Antibody Affinity Maturation
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We analyze the interactions between division, mutation and selection in a simplified evolutionary model, assuming that the population observed can be classified into fitness levels. The construction of our mathematical framework is motivated by the modeling of antibody affinity maturation of B-cells in germinal centers during an immune response. This is a key process in adaptive immunity leading to the production of high-affinity antibodies against a presented antigen. Our aim is to understand how the different biological parameters affect the system’s functionality. We identify the existence of an optimal value of the selection rate, able to maximize the number of selected B-cells for a given generation.
KeywordsMulti-type Galton–Watson process Germinal center reaction Affinity-dependent selection Evolutionary landscapes
Mathematics Subject Classification60J85 60J80 92B99 92D15
This work was supported by the Labex inflamex, ANR Project 10-LABX-0017.
- Anderson SM, Khalil A, Uduman M, Hershberg U, Louzoun Y, Haberman AM, Kleinstein SH, Shlomchik MJ (2009) Taking advantage: high-affinity B cells in the germinal center have lower death rates, but similar rates of division, compared to low-affinity cells. J Immunol 183(11):7314–7325CrossRefGoogle Scholar
- Balelli I, Milisic V, Wainrib G (2016) Branching random walks on binary strings for evolutionary processes in adaptive immunity. arXiv preprint arXiv:1607.00927
- Keşmir C, De Boer RJ (1999) A mathematical model on germinal center kinetics and termination. J Immunol 163(5):2463–2469Google Scholar
- Minc H (1988) Nonnegative matrices. John Wiley & Sons, New YorkGoogle Scholar
- Murphy KM, Travers P, Walport M et al (2012) Janeway’s immunobiology, vol 7. Garland Science, New YorkGoogle Scholar
- Pang W, Wang K, Wang Y, Ou G, Li H, Huang L (2015) Clonal selection algorithm for solving permutation optimisation problems: a case study of travelling salesman problem. In: International Conference on Logistics Engineering, Management and Computer Science (LEMCS 2015). Atlantis PressGoogle Scholar
- Shannon M, Mehr R (1999) Reconciling repertoire shift with affinity maturation: the role of deleterious mutations. J Immunol 162(7):3950–3956Google Scholar
- Shen WJ, Wong HS, Xiao QW, Guo X, Smale S (2012) Towards a mathematical foundation of immunology and amino acid chains. arXiv preprint arXiv:1205.6031
- Shlomchik MJ, Watts P, Weigert MG, Litwin S (1998) Clone: a Monte-Carlo computer simulation of B cell clonal expansion, somatic mutation, and antigen-driven selection. In: Kelsoe G, Flajnik MF (eds) Somatic diversification of immune responses. Springer, Berlin, Heidelberg, pp 173–197CrossRefGoogle Scholar
- Xu H, Schmidt AG, O’Donnell T, Therkelsen MD, Kepler TB, Moody MA, Haynes BF, Liao HX, Harrison SC, Shaw DE (2015) Key mutations stabilize antigen-binding conformation during affinity maturation of a broadly neutralizing influenza antibody lineage. Proteins Struct Funct Bioinf 83(4):771–780CrossRefGoogle Scholar