On the critical concentration for net assembly of dynamically unstable polymers

Cytoskeletal and cytomotive filaments are protein polymers that move molecular cargo and organize cellular contents in all domains of life. A key parameter describing the self-assembly of many of these polymers -including actin filaments and microtubules- is the minimum concentration required for polymer formation. This 'critical concentration for net assembly' ($$) is easy to calculate for eukaryotic actins but more difficult for dynamically unstable filaments such as microtubules and some bacterial polymers. To better understand how cells (especially bacteria) regulate assembly of dynamically unstable polymers I investigate the microscopic parameters that influence their critical concentrations. Assuming simple models for spontaneous nucleation and catastrophe I derive expressions for the monomer-polymer balance. In the absence of concentration-dependent rescue, fixed catastrophe rates do not produce clear critical concentrations. In contrast, simple ATP-/GTP-cap models with concentration-dependent catastrophe rates, generate phenomenological critical concentrations that increase linearly with the rate of nucleotide hydrolysis and decrease logarithmically with the rate of spontaneous nucleation.