A Mathematical Formulation and Solution of the CoPhMoRe Inverse Problem for Helically Wrapping Polymer Corona Phases on Cylindrical Substrates

Corona phase molecular recognition (CoPhMoRe) is a new technique that generates a nanoparticle-coupled polymer phase, capable of recognizing a specific molecule with high affinity and selectivity. CoPhMoRe has been successfully demonstrated using polymer wrapped single walled carbon nanotubes, resulting in molecular recognition complexes, to date, for dopamine, estradiol, riboflavin, and L-thyroxine, utilizing combinatorial library screening. A rational alternative design to this empirical library screening is to solve the mathematical formulation that we introduce as the CoPhMoRe inverse problem. This inverse problem seeks a linear function representing the position of monomers or functional groups along a polymer backbone that results in a 3-dimensional structure capable of recognizing a specific molecule when mapped to a nanoparticle surface. The potential solution space for such an inverse problem is infinite in general, but for the specific constraint of a helically wrapping polymer, mapped to a cylindrical nanoparticle, we show in this work that two types of inverse problems are exactly solvable. In one case, the polymer pitch and composition can be designed to allow for the specific binding of a small molecule analyte in the occluded space on the nanotube surface. In the other, a larger macromolecule can interact with a deformed helix, which partially conforms to it. A simplified, coarse-grained molecular model of a helically wrapping polymer demonstrates the inhomogeneous binding potential formed by a wrapping with a given pitch. Calculating the potential maps for various pitch values illustrates that there is an optimal pitch that enables the selective and specific binding of the target analyte. An additional coarse-grained model of a helical wrapping by a polymer consisting of alternating hydrophobic hydrophilic segments demonstrates the resulting deformed helix corona around the nanotube, which forms accessible binding pockets between the hydrophilic loops. While these are the idealized forms of actual CoPhMoRe phases, the formation and solution of such inverse problems may serve to reduce the dimensionality of library screening for CoPhMoRe discoveries, as well as provide a theoretical basis for understanding certain types of CoPhMoRe recognition.