Slip complexity in dynamic models of earthquake faults.

We summarize recent evidence that models of earthquake faults with dynamically unstable friction laws but no externally imposed heterogeneities can exhibit slip complexity. Two models are described here. The first is a one-dimensional model with velocity-weakening stick-slip friction; the second is a two-dimensional elastodynamic model with slip-weakening friction. Both exhibit small-event complexity and chaotic sequences of large characteristic events. The large events in both models are composed of Heaton pulses. We argue that the key ingredients of these models are reasonably accurate representations of the properties of real faults.

Thermodynamic parameters for the helix-coil transition of oligopeptides: molecular dynamics simulation with the peptide growth method.

The helix-coil transition equilibrium of polypeptides in aqueous solution was studied by molecular dynamics simulation. The peptide growth simulation method was introduced to generate dynamic models of polypeptide chains in a statistical (random) coil or an alpha-helical conformation. The key element of this method is to build up a polypeptide chain during the course of a molecular transformation simulation, successively adding whole amino acid residues to the chain in a predefined conformation state (e.g., alpha-helical or statistical coil). Thus, oligopeptides of the same length and composition, but having different conformations, can be incrementally grown from a common precursor, and their relative conformational free energies can be calculated as the difference between the free energies for growing the individual peptides. This affords a straightforward calculation of the Zimm-Bragg sigma and s parameters for helix initiation and helix growth. The calculated sigma and s parameters for the polyalanine alpha-helix are in reasonable agreement with the experimental measurements. The peptide growth simulation method is an effective way to study quantitatively the thermodynamics of local protein folding.

Modeling and optimization of populations subject to time-dependent mutation.

It has become clear that many organisms possess the ability to regulate their mutation rate in response to environmental conditions. So the question of finding an optimal mutation rate must be replaced by that of finding an optimal mutation schedule. We show that this task cannot be accomplished with standard population-dynamic models. We then develop a "hybrid" model for populations experiencing time-dependent mutation that treats population growth as deterministic but the time of first appearance of new variants as stochastic. We show that the hybrid model agrees well with a Monte Carlo simulation. From this model, we derive a deterministic approximation, a "threshold" model, that is similar to standard population dynamic models but differs in the initial rate of generation of new mutants. We use these techniques to model antibody affinity maturation by somatic hypermutation. We had previously shown that the optimal mutation schedule for the deterministic threshold model is phasic, with periods of mutation between intervals of mutation-free growth. To establish the validity of this schedule, we now show that the phasic schedule that optimizes the deterministic threshold model significantly improves upon the best constant-rate schedule for the hybrid and Monte Carlo models.