Mathematical limits of multilocus models: the genetic transmission of bipolar disorder.

We describe a simple, graphical method for determining plausible modes of inheritance for complex traits and apply this to bipolar disorder. The constraints that allele frequencies and penetrances lie in the interval 0-1 impose limits on recurrence risks, KR, in relatives of an affected proband for a given population prevalence, KP. We have investigated these limits for KR in three classes of relatives (MZ co-twin, sibling, and parent/offspring) for the general single-locus model and for two types of multilocus models: heterogeneity and multiplicative. In our models we have assumed Hardy-Weinberg equilibrium, an all-or-none trait, absence of nongenetic resemblance between relatives, and negligible mutation at the disease loci. Although the true values of KP and the KR's are only approximately known, observed population and family data for bipolar disorder are inconsistent with a single-locus model or with any heterogeneity model. In contrast, multiplicative models involving three or more loci are consistent with observed data and, thus, represent plausible models for the inheritance of bipolar disorders. Studies to determine the genetic basis of most bipolar disorder should use methods capable of detecting interacting oligogenes.