Mimicking the Strategy of the Immune System: Insight Gained from Mathematics

Abstract

The humoral immune response to antigenic challenge involves a Darwinian process of somatic mutations and selection for B-cells carrying higher affinity antibodies. Insufficient knowledge about this process may hamper the use genetically engineered antibodies for vaccination. The aim of the present work is to circumvent some of the experimental difficulties in the study of affinity maturation by investigating this system mathematically. Using dynamic programming methods we look for mutation rate as a function of clone size, which maximizes the probability that the required antibody structure is generated before the pathogen kills the host. We show analytically that the globally optimal strategy for a fast production of high affinity antibodies is to utilize a step-function mutation rate, i.e., a minimal mutation rate in early stages of the immune response, followed by the maximal possible rate when the proliferating B-cells population size exceeds a given threshold. Based on the methodology we have followed it may be concluded that the good performance of this simple, thrifty, two-stage strategy cannot be improved by evolutionary development of a more complex control of the response. Laboratory experiments are suggested for testing the validity of our theoretical results.