ON RATES AND OFF RATES A linear polymer of protein molecules, such as an actin flament or a microtubule, assembles (polymerizes) and disassembles (depolymerizes) by the addition and removal of subunits at the ends of the polymer. The rate of addition of these subunits (called monomers) is given by the rate constant kon, which has units of M–1 sec–1. The rate of loss is given by koff (units of sec–1). subunit + polymer (with n subunits) polymer (with n +1 subunits) kon koff NUCLEATION A helical polymer is stabilized by multiple contacts between adjacent subunits. In the case of actin, two actin molecules bind relatively weakly to each other, but addition of a third actin monomer to form a trimer makes the entire group more stable. Further monomer addition can take place onto this trimer, which therefore acts as a nucleus for polymerization. For tubulin, the nucleus is larger and has a more complicated structure (possibly a ring of 13 or more tubulin molecules)—but the principle is the same.    The assembly of a nucleus is relatively slow, which explains the lag phase seen during polymerization. The lag phase can be reduced or abolished entirely by adding premade nuclei, such as fragments of already polymerized microtubules or actin flaments. THE CRITICAL CONCENTRATION The number of monomers that add to the polymer (actin flament or microtubule) per second will be proportional to the concentration of the free subunit (konC), but the subunits will leave the polymer end at a constant rate (koff) that does not depend on C. As the polymer grows, subunits are used up, and C is observed to drop until it reaches a constant value, called the critical concentration (Cc). At this concentration, the rate of subunit addition equals the rate of subunit loss. At this equilibrium, kon C = koff so that Cc = (where Kd is the dissociation constant; see Figure 3–44). koff      kon = TIME COURSE OF POLYMERIZATION The assembly of a protein into a long helical polymer such as a cytoskeletal flament or a bacterial fagellum typically shows the following time course: The lag phase corresponds to time taken for nucleation. The growth phase occurs as monomers add to the exposed ends of the growing flament, causing flament elongation. The equilibrium phase, or steady state, is reached when the growth of the polymer due to monomer addition precisely balances the shrinkage of the polymer due to disassembly back to monomers. time amount of polymerLAG PHASE GROWTH PHASE EQUILIBRIUM PHASE PLUS AND MINUS ENDS The two ends of an actin flament or microtubule polymerize at different rates. The fast-growing end is called the plus end, whereas the slow-growing end is called the minus end. The difference in the rates of growth at the two ends is made possible by changes in the conformation of each subunit as it enters the polymer. free subunit subunit in polymer SLOW FAST minus end plus end This conformational change affects the rates at which subunits add to the two ends. Even though kon and koff will have different values for the plus and minus ends of the polymer, their ratio koff/kon—and hence Cc—must be the same at both ends for a simple polymerization reaction (no ATP or GTP hydrolysis). This is because exactly the same subunit interactions are broken when a subunit is lost at either end, and the fnal state of the subunit after dissociation is identical. Therefore, the ˜G for subunit loss, which determines the equilibrium constant for its association with the end, is identical at both ends: if the plus end grows four times faster than the minus end, it must also shrink four times faster. Thus, for C > Cc, both ends grow; for C < Cc, both ends shrink.    The nucleoside triphosphate hydrolysis that accompanies actin and tubulin polymerization removes this constraint. Kd PANEL 16–2: The Polymerization of Actin and Tubulin 902