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We present a variational approach for the simulation of large conformational changes of proteins (including multiple protein chains/ligands) which takes advantage of their cross-linked one-dimensional nature, a structure which often occurs in civil engineering. Conformational changes are computed by incremental energy minimisation. We use an efficient finite element method for finding equilibria of complexes composed of inter-linked chains; this method is based on recent advances in the description of one-dimensional elasticity. Protein backbone elasticity, van der Waals repulsions, hydrogen bonds and salt bridges are taken into account, together with user-defined geometric distance constraints that may be imposed for purposes of simulating various binding processes based on chemical knowledge. These computational methods have been integrated into a system, Proteinmorphosis, which includes interactive visualisation. The conformational change of calmodulin upon peptide binding is examined as a first experiment. Allostery in hemoglobin, which consists of a cooperative oxygen binding mechanism, is a second, more sophisticated, numerical experiment. Different modelling strategies are designed to understand the allostery. The results for both molecules are consistent with existing hypotheses, and reproduce the known atomic positions after binding to within the experimental error. The modelling system is part of an on-going program to model structural biology, from protein structure to cell and tissue properties.

We use the rotation group and its algebra to provide a novel description of deformations of special Cosserat rods or thin rods that have negligible shear. Our treatment was motivated by the problem of the simulation of catheter navigation in a network of blood vessels, where this description is directly useful. In this context, we derive the Euler differential equations that characterize equilibrium configurations of stretch-free thin rods. We apply perturbation methods, used in time-dependent quantum theory, to the thin rod equations to describe incremental deformations of partially constrained rods. Further, our formalism leads naturally to a new and efficient finite element method valid for arbitrary deformations of thin rods with negligible stretch. Associated computational algorithms are developed and applied to the simulation of catheter motion inside an artery network.

Proteinmorphosis is a physically-based interactive modeling system for simulating large or small conformational changes of proteins and protein complexes. It takes advantage of the cross-linked one-dimensional nature of protein chains. The user can, based on her chemical knowledge, pull pairs of points (lying either on a single protein or on different molecules) together by specifying geometric distance constraints. The resulting conformation(s) of the molecule(s) of interest is computed by an efficient finite element formalism taking into account elasticity of the protein backbone, van der Waals repulsions, hydrogen bonds, salt bridges and the imposed distance constraints. The conformational change is computed incrementally and the result can be visualized as an animation; complete interactivity is provided to position and view the proteins as desired by the user. Physical properties of regions on the protein can also be chosen interactively. The conformational change of calmodulin upon peptide binding is examined as a first experiment. It is found that the result is satisfactory in reproducing the conformational change that follows on peptide binding. We use Proteinmorphosis to study the cooperative hemoglobin oxygen binding mechanism in a second, more sophisticated, experiment. Different modeling strategies are designed to understand the allosteric (cooperative) binding process in this system and the results are found to be consistent with existing hypotheses.