Co-operative association, in which a protein subunit is held simultaneously by two bonds, is enormously more favorable than association forming either bond alone. A theoretical framework for calculating the effect of co-operativity is developed here, which should have a broad application to protein-protein and protein-DNA associations. The theory is applied in detail to actin. Fragmentation of an actin filament is extremely unfavorable: the association constant for annealing-fragmentation is estimated here to be at least 10(13) M-1. In contrast to these very strong bonds within the filament, subunits are loosely attached at the end, with an association constant of 2 x 10(5) M-1. The eight orders of magnitude difference between annealing-fragmentation and end association can be attributed to the co-operative formation of one additional protein-protein bond in the annealing reaction. This observation, and a quantitative analysis of the co-operativity, lead to an important conclusion: the longitudinal bond, which connects subunits in the long-pitch helix, must be substantially stronger than the diagonal bond, which connect subunits between these helices. This conclusion contradicts some recent models based on Fourier construction, in which the longitudinal bond is weak or absent. Prominent longitudinal bonds also require a rigidity of the actin filament that must be reconciled with previous reports of torsional flexibility. A hinge within the actin subunit is suggested, separating it into two flexibly attached domains. In one possible model the two domains are oriented radially: the inner domains are connected by longitudinal and diagonal bonds to form a relatively rigid helical backbone, and the outer domains are attached to this backbone by flexible hinges, permitting them to move through angles of 10 degrees to 20 degrees or more. Flexibility of the outer, myosin-binding domain should be functionally important, permitting attachment of myosin cross-bridges over a range of angles.