Sigmoid kinetics of protein crystal nucleation

• A novel expression for the nucleation rate, which unifies all steps of nucleation, is devised.
• A logistic function dependence of crystal nuclei number density on nucleation time is found.
• Experimental data on stationary and non-stationary protein crystal nucleation are fitted well.
• Amyloid fibril formation, electrochemical nucleation and glass crystallization obey the same law.
• Perhaps, logistic function dependence of crystal nucleation kinetics is a universal law.

A non-linear differential equation expressing the new phase nucleation rate in the different steps of the process (non-stationary and stationary nucleation and in the plateau region) is derived from basic principles of the nucleation theory. It is shown that one and the same sigmoid (logistic) function describes both nucleation scenarios: the one according to the classical theory, and the other according to the modern two-stage mechanism of protein crystal formation. Comparison to experimental data on both insulin crystal nucleation kinetics and on bovine β-lactoglobulin crystallization indicates a good agreement with the sigmoidal prediction. Experimental data for electrochemical nucleation and glass crystallization obey the same sigmoid time dependence, and suggest universality of this nucleation kinetics law.