Binomial distribution for quantification of protein subunits in biological nanoassemblies and functional nanomachines

Living systems produce ordered structures and nanomachines that inspire the development of biomimetic nanodevices such as chips, MEMS, actuators, sensors, sorters, and apparatuses for single-pore DNA sequencing, disease diagnosis, drug or therapeutic RNA delivery. Determination of the copy numbers of subunits that build these machines is challenging due to small size. Here we report a simple mathematical method to determine the stoichiometry, using phi29 DNA-packaging nanomotor as a model to elucidate the application of a formula ∑M=0(Z)((Z)M)p(Z-M)q(M), where p and q are the percentage of wild-type and inactive mutant in the empirical assay; M is the copy numbers of mutant and Z is the stoichiometry in question. Variable ratios of mutants and wild-type were mixed to inhibit motor function. Empirical data were plotted over the theoretical curves to determine the stoichiometry and the value of K, which is the number of mutant needed in each machine to block the function, all based on the condition that wild-type and mutant are equal in binding affinity. Both Z and K from 1-12 were investigated. The data precisely confirmed that phi29 motor contains six copies (Z) of the motor ATPase gp16, and K=1. From the clinical editor: To determine copy numbers of subunits that form nanomachines in living organisms is a daunting task due to the complexities and the inherently small sizes associated with such systems. In this paper, a simple mathematical method is described how to determine the stoichiometry of copies in biomimetic nanodevices, using phi29 DNA-packaging nanomotor as a model.