A simple mathematical formula for stoichiometry quantification of viral and nanobiological assemblage using slopes of log/log plot curves

Abstract

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+2.4193 T−0.001746 T 2 [ 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 5×10 8 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.