Using MathematicaTM to draw energy landscape funnels


Protein folding funnels are an intuitive way to think about the energy landscape of protein conformational space.  The surface of the funnel represents the energy of individual members of a conformational ensemble whose conformation is represented by the x,y coordinate of a particular location on the surface.  Conformations that are most like the folded conformation have x,y coordinates near the center (minimum) of the funnel while highly disordered unfolded conformations are found at the periphery.  Conformational space of real proteins consists of subensembles represented by the small wells around the funnel that are “local minima” in the energy.  The folding mechanism of a particular member of the unfolded ensemble, found high up on the funnel amounts to a path down the funnel, the way a marble might role down the surface.  It might fall into local minima on the way to the folded state and needs enough kinetic energy to make its way uphill and out of the well.

Mathematica is a powerful graphical rendering tool that can be use to generate figure like the one above with only a few lines of code.  The notebook you can download by right-clicking here and saving to your computer will let you define the shape of the funnel, the number and shape of the local minima and the appearance of the surface.  The positions and depth of the minima are chosen with a random number generator, so no two funnels you generate will look the same.  Have fun and let us know what you think.  If you use a funnel figure generated with this notebook in a talk or paper, please let us know and give credit to the Oas Lab.