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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., (Copyright © 2014 Elsevier Inc. All rights reserved.)

In nanotechnology, biomolecular assemblies serve not only as model systems for the construction of nanodevices, but they can also be used directly as templates for the formation of nanostructures. Biological nano-building blocks can either be isolated as complete functional units from living cells or viruses (biological “Top down” approach) or formed by biomolecular assembly from recombinant or synthetic components (“Bottom up” approach). In both cases, rational design of nanostructures requires knowledge of the stoichiometry of the biological structures, which frequently occur as multimers, i.e., the morphological complex is composed of multiple copies of one or more macromolecules. In this paper, a method is described for the stoichiometric quantification of molecules in bio-nanostructures. The method is based on using dilution factors and relative concentrations rather than absolute quantities, which are often difficult to determine, especially in short-lived assembly intermediates. The approach exploits the fact that the larger the stoichiometry of the component is, the more dramatic is the influence of the dilution factor (decrease in concentration) on the reaction. We established and used the method to determine the stoichiometry of components of bacterial virus phi29. The log of dilution factors was plotted against the log of reaction yield. The stoichiometry Z was determined with the equation Z=−1.58+2.4193T−0.001746T2 [T ∈ (0,1000), or ∠α ∈ (0°, 89.9°)], where T is the slope of the curve (tangent of ∠α, which is the angle between the x-axis and the concentration dependent curve). Z can also be determined from a standard table given in this report. With the bacteriophage phi29 in vitro assembly system, up to 5×108 infectious virions per ml can be assembled from 11 purified components, giving our method a sensitivity of nine orders of magnitude. We confirmed the stoichiometries of phi29 components that were determined previously with microscopic approaches. The described method also responded to programmed stoichiometry changes, which were generated by assembling the phi29 DNA packaging motor from modified pRNA (DNA-packaging RNA) molecules forming a trimer of dimers or a dimer of trimers, instead of the wild-type hexamer. [Copyright &y& Elsevier]