Order parameters for the backbone N-H and C alpha-H bond vectors have been calculated from a 150 ps molecular dynamics (MD) simulation of human type-alpha transforming growth factor in H2O solvent. Two kinds of 'crankshaft motions' of the polypeptide backbone are observed in this MD trajectory. The first involves small-amplitude rocking of the rigid peptide bond due to correlated changes in the backbone dihedral angles psi i-1 and phi i. These high-frequency 'librational crankshaft' motions are correlated with systematically smaller values of motional order parameters for backbone N-H bond vectors compared to C alpha-H bond vectors. In addition, infrequent 'crankshaft flips' of the peptide bond from one local minimum to another are observed for several amino acid residues. These MD simulations demonstrate that comparisons of N-H and C alpha-H order parameters provide a useful approach for identifying crankshaft librational motions in proteins.

The nonlocal competition in prey is incorporated into a diffusive predator-prey model with additional food in predator and time delay. The local stability of the coexisting equilibrium is studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is also investigated by using time delay as bifurcation parameter. Some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions are given by utilizing the normal form method and center manifold theorem. Our results suggest that nonlocal competition together with time delay can induce spatially inhomogeneous bifurcating periodic solutions in the diffusive predator-prey model.

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In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation.

In this paper, we study a delayed diffusive predator-prey model with nonlocal competition in prey and habitat complexity. The local stability of coexisting equilibrium are studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is investigated by using time delay as bifurcation parameter. We give some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution by utilizing the normal form method and center manifold theorem. Our results suggest that only nonlocal competition and diffusion together can induce stably spatial inhomogeneous bifurcating periodic solutions.