Sampling and convergence in free energy calculations of protein-ligand interactions: The binding of triphenoxypyridine derivatives to factor Xa and trypsin

The binding of a set of 10 triphenoxypyridine derivatives to two serine proteases, factor Xa and trypsin, has been used to analyze factors related to sampling and convergence in free energy calculations based on molecular dynamics simulation techniques. The inhibitors investigated were initially proposed as part of the Critical Assessment of Techniques for Free Energy Evaluation (CATFEE) project for which no experimental results nor any assessment of the predictions submitted by various groups have ever been published. The inhibitors studied represent a severe challenge for explicit free energy calculations. The mutations from one compound to another involve up to 19 atoms, the creation and annihilation of net charge and several alternate binding modes. Nevertheless, we demonstrate that it is possible to obtain highly converged results (+/- 5-10 kJ/mol) even for such complex multi-atom mutations by simulating on a nanosecond time scale. This is achieved by using soft-core potentials to facilitate the creation and deletion of atoms and by a careful choice of mutation pathway. The results show that given modest computational resources, explicit free energy calculations can be successfully applied to realistic problems in drug design.     

Periodic solitons in nonlocal nonlinear media

We identify periodic solitons in nonlocal nonlinear media: multi-hump soliton solutions propagating in a fully periodic fashion. We also demonstrate recurrences and breathers whose evolution is statistically periodic and discuss why some systems support periodic solitons while others do not.     

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let be a finite dimensional division algebra having center , and let be a normal subgroup of finite index. Suppose is not solvable. Then we may assume that is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property . This property includes the requirement that the diameter of the commuting graph should be , but is, in fact, stronger. Another ingredient is to show that if the commuting graph of has Property , then is open with respect to a nontrivial height one valuation of (assuming without loss of generality, as we may, that is finitely generated). After establishing the openness of (when is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of over its prime subfield to eliminate as a possible quotient of , thereby obtaining a contradiction and proving our main result.

     

Pattern formation in a cavity longer than the coherence length of the light in it

We study, theoretically and experimentally, the evolution of patterns in a passive nonlinear cavity that is longer than the coherence length of the light circulating in it. The patterns exhibit spatial line narrowing as the feedback is increased, resembling the line narrowing in lasers.     

Collisions between optical spatial solitons propagating in opposite directions

We formulate the theory describing the evolution and interactions between optical spatial solitons that propagate in opposite directions. We show that coherent collisions between counterpropagating solitons give rise to a new focusing mechanism resulting from the interference between the beams, and that interactions between such solitons are insensitive to the relative phase between the beams.