“Coarse” stability and bifurcation analysis using time-steppers: A reaction-diffusion example

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099–1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (“coarse”) bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice–Boltzmann model.

Structure and stability of a second molten globule intermediate in the apomyoglobin folding pathway.

Apomyoglobin folding proceeds through a molten globule intermediate (low-salt form; I1) that has been characterized by equilibrium (pH 4) and kinetic (pH 6) folding experiments. Of the eight alpha-helices in myoglobin, three (A, G, and H) are structured in I1, while the rest appear to be unfolded. Here we report on the structure and stability of a second intermediate, the trichloroacetate form of the molten globule intermediate (I2), which is induced either from the acid-unfolded protein or from I1 by > or = 5 mM sodium trichloroacetate. Circular dichroism measurements monitoring urea- and acid-induced unfolding indicate that I2 is more highly structured and more stable than I1. Although I2 exhibits properties closer to those of the native protein, one-dimensional NMR spectra show that it maintains the lack of fixed side-chain structure that is the hallmark of a molten globule. Amide proton exchange and 1H-15N two-dimensional NMR experiments are used to identify the source of the extra helicity observed in I2. The results reveal that the existing A, G, and H helices present in I1 have become more stable in I2 and that a fourth helix--the B helix--has been incorporated into the molten globule. Available evidence is consistent with I2 being an on-pathway intermediate. The data support the view that apomyoglobin folds in a sequential fashion through a single pathway populated by intermediates of increasing structure and stability.