Exploring the origins of topological frustration: Design of a minimally frustrated model of fragment B of protein A

Topological frustration in an energetically unfrustrated off-lattice model of the helical protein fragment B of protein A from Staphylococcus aureus was investigated. This Gō-type model exhibited thermodynamic and kinetic signatures of a well-designed two-state folder with concurrent collapse and folding transitions and single exponential kinetics at the transition temperature. Topological frustration is determined in the absence of energetic frustration by the distribution of Fersht φ values. Topologically unfrustrated systems present a unimodal distribution sharply peaked at intermediate φ, whereas highly frustrated systems display a bimodal distribution peaked at low and high φ values. The distribution of φ values in protein A was determined both thermodynamically and kinetically. Both methods yielded a unimodal distribution centered at φ = 0.3 with tails extending to low and high φ values, indicating the presence of a small amount of topological frustration. The contacts with high φ values were located in the turn regions between helices I and II and II and III, intimating that these hairpins are in large part required in the transition state. Our results are in good agreement with all-atom simulations of protein A, as well as lattice simulations of a three- letter code 27-mer (which can be compared with a 60-residue helical protein). The relatively broad unimodal distribution of φ values obtained from the all-atom simulations and that from the minimalist model for the same native fold suggest that the structure of the transition state ensemble is determined mostly by the protein topology and not energetic frustration.

Extension of the thermodynamics of small systems to open metastable states: An example

By using a simplified model of small open liquid-like clusters with surface effects, in the gas phase, it is shown how the statistical thermodynamics of small systems can be extended to include metastable supersaturated gaseous states not too far from the gas–liquid equilibrium transition point. To accomplish this, one has to distinguish between mathematical divergence and physical convergence of the open-system partition function.

A model of excitation and adaptation in bacterial chemotaxis

Bacterial chemotaxis is widely studied because of its accessibility and because it incorporates processes that are important in a number of sensory systems: signal transduction, excitation, adaptation, and a change in behavior, all in response to stimuli. Quantitative data on the change in behavior are available for this system, and the major biochemical steps in the signal transduction/processing pathway have been identified. We have incorporated recent biochemical data into a mathematical model that can reproduce many of the major features of the intracellular response, including the change in the level of chemotactic proteins to step and ramp stimuli such as those used in experimental protocols. The interaction of the chemotactic proteins with the motor is not modeled, but we can estimate the degree of cooperativity needed to produce the observed gain under the assumption that the chemotactic proteins interact directly with the motor proteins.

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).