A mathematical journey through optical biosensors
Date
2016
Authors
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Publisher
University of Delaware
Abstract
Many chemical reactions in nature involve a stream of reactants ( ligand molecules) flowing through a fluid-filled volume, over a surface to which other reactants (receptors) are confined. Such surface-volume reactions occur during processes like blood platelet adhesion, drug absorption, and DNA damage repair. To measure rate constants associated with such reactions, scientist use optical biosensors. Correct interpretation of biosensor data requires having an accurate mathematical model, and models have been successfully proposed and progressively refined throughout the years. Although such models are typically limited to situations involving only a single type of reaction on the surface (e.g. bimolecular), many chemical reactions involve more than one step or molecule. One example is the polymerase switch which occurs during DNA translesion synthesis, which is a reaction thought to be critical to DNA damage repair. We formulate and study a mathematical model for this multiple-component process. We derive physically relevant numerical and analytic results, including a nonlinear set of Ordinary Differential Equations (i.e. Effective Rate Constant Equations) which may be used to estimate reaction rate constants from raw data. We also demonstrate that our mathematical model can aid in eliminating ambiguity from sensogram data. We extend the theory for bimolecular reactions as well. Modeling these reactions results in a nonlinear differential equation, and we can find an approximate solution to this equation using traditional perturbation methods in the reaction-limited parameter regime. However, outside of this parameter regime these methods fail. By coupling the strained coordinates technique with the Homotopy Perturbation Method, a method motivated by concepts from topology, we are able to find an expression for the approximate solution of our equation despite the presence of a strong nonlinearity