Analysis of linear trade models and relation to scale economies

We discuss linear Ricardo models with a range of parameters. We show that the exact boundary of the region of equilibria of these models is obtained by solving a simple integer programming problem. We show that there is also an exact correspondence between many of the equilibria resulting from families of linear models and the multiple equilibria of economies of scale models.

Brownian dynamics simulations of protein folding: Access to milliseconds time scale and beyond

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion–collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion–collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Large-scale protein structure modeling of the Saccharomyces cerevisiae genome

The function of a protein generally is determined by its three-dimensional (3D) structure. Thus, it would be useful to know the 3D structure of the thousands of protein sequences that are emerging from the many genome projects. To this end, fold assignment, comparative protein structure modeling, and model evaluation were automated completely. As an illustration, the method was applied to the proteins in the Saccharomyces cerevisiae (baker’s yeast) genome. It resulted in all-atom 3D models for substantial segments of 1,071 (17%) of the yeast proteins, only 40 of which have had their 3D structure determined experimentally. Of the 1,071 modeled yeast proteins, 236 were related clearly to a protein of known structure for the first time; 41 of these previously have not been characterized at all.

Slip complexity in earthquake fault models.

We summarize studies of earthquake fault models that give rise to slip complexities like those in natural earthquakes. For models of smooth faults between elastically deformable continua, it is critical that the friction laws involve a characteristic distance for slip weakening or evolution of surface state. That results in a finite nucleation size, or coherent slip patch size, h*. Models of smooth faults, using numerical cell size properly small compared to h*, show periodic response or complex and apparently chaotic histories of large events but have not been found to show small event complexity like the self-similar (power law) Gutenberg-Richter frequency-size statistics. This conclusion is supported in the present paper by fully inertial elastodynamic modeling of earthquake sequences. In contrast, some models of locally heterogeneous faults with quasi-independent fault segments, represented approximately by simulations with cell size larger than h* so that the model becomes "inherently discrete," do show small event complexity of the Gutenberg-Richter type. Models based on classical friction laws without a weakening length scale or for which the numerical procedure imposes an abrupt strength drop at the onset of slip have h* = 0 and hence always fall into the inherently discrete class. We suggest that the small-event complexity that some such models show will not survive regularization of the constitutive description, by inclusion of an appropriate length scale leading to a finite h*, and a corresponding reduction of numerical grid size.

The parsec-scale jet in M87.

We briefly review the observed structure and evolution of the M87 jet on scales less, similar1 parsec (pc; 1 pc = 3.09 x 10(16) m). Filamentary features, limb-brightening, and side-to-side oscillation are common characteristics of the pc-scale, and kpc-scale jets. The most prominent emission features on both the pc and subpc scales appear stationary (v/c < 0.1). Nonetheless, based on the jet's flux evolution, the presence of kpc-scale superluminal motion, and the absence of a visible counter-jet, we argue for the presence of an underlying relativistic flow, consistent with unified models. The initial jet collimation appears to occur on scales <0.1 pc, thus favoring electromagnetic processes associated with a black hole and accretion disk.