Renewal theory, Geiger counters, and the maximum number of receptors bound to a randomly haptenated polymer chain.

Polymers containing small chemical groups (haptens) covalently attached at random along the chain are commonly used to initiate an immune response. Properties of the polymer such as its length, the spacing of the haptens, and the total number of haptens along the chain, correlate with its immune reactivity. Here we model the ability of many finite-sized cell surface receptors to bind simultaneously the haptens conjugated to a polymer chain. The binding sites on two different receptors or on separate parts of a multivalent receptor cannot be arbitrarily close to one another; so, in general, not all haptens along a polymer chain can be simultaneously bound by receptors. We develop an analogy between the steric hindrance among receptors detecting randomly placed haptens and the temporary locking of a Geiger counter that has detected a radioactive decay. Using renewal theory, we compute the probability distribution, and its moments, for the maximum number of haptens that can be simultaneously bound by monovalent receptors. We also model flexible bivalent receptors and obtain the mean and variance of the maximum number of receptors bound to randomly haptenated polymers, and the mean and variance of the maximum number of haptens bound. We demonstrate the importance of our results by applying them to immunological data and showing that, for polymers used in immunology, the effective valence of a polymer may be as much as 50% smaller than its nominal valence.