Global stochastic optimization in hierarchical modeling of ligand protein binding profiles

We present a detailed explanation of a mathematical method and numerical technique applied to solve an irregular non-linear fitting problem that results from attempts to model the calorimetric profiles generated by the binding of phenolic ligands to the insulin hexamer. The method employed uses a non-traditional approach to modeling data. Rather than start with a simplified model, we use a hierarchical tree of physical models with different degrees of sophistication. Starting with the model of highest dimension, we work our way to an optimum model which is of a lower dimension and is less complex. The algorithm uses two complementary techniques. First, a sensitivity analysis in the vicinity of the optimal point for each model is used to estimate errors in the parameters; that, in turn, provides the user with insight for model simplification. Second, we utilize the optimized model in the prediction of new experimental curves. The core of the method combines a strategy based on the proper split of the initial global numerical task into three locally independent subtasks, and induces a specific split in the search space. The application of three different optimization techniques (two parametric and one variational) with an alternating objective function defined in corresponding subspaces, in combination with the search along the hierarchical tree of mathematical models, enables us to overcome difficult computational problems, including over-parametrization. We have obtained very accurate fits to a number of calorimetric curves, resulting in a quantitative description of intrinsic functional (free ligand concentration) and vector (equilibrium coefficients and enthalpies of binding) parameters. These quantitative results can now be used to improve the stability of insulin formulations. We believe that, with small modifications to the model, the method and algorithms presented in this article can be applied to other protein-ligand systems.