A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems

We propose a general procedure for solving incomplete data estimation problems. The procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model, the nonlinear structural measurement error model, and the random effects model. The procedure is based on the general principle of stochastic approximation and the Markov chain Monte-Carlo method. Applying the theory on adaptive algorithms, we derive conditions under which the proposed procedure converges. Simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model.

Dynamic weighting in Monte Carlo and optimization

Dynamic importance weighting is proposed as a Monte Carlo method that has the capability to sample relevant parts of the configuration space even in the presence of many steep energy minima. The method relies on an additional dynamic variable (the importance weight) to help the system overcome steep barriers. A non-Metropolis theory is developed for the construction of such weighted samplers. Algorithms based on this method are designed for simulation and global optimization tasks arising from multimodal sampling, neural network training, and the traveling salesman problem. Numerical tests on these problems confirm the effectiveness of the method.

Enhanced conformational sampling in Monte Carlo simulations of proteins: application to a constrained peptide.

A Monte Carlo simulation method for globular proteins, called extended-scaled-collective-variable (ESCV) Monte Carlo, is proposed. This method combines two Monte Carlo algorithms known as entropy-sampling and scaled-collective-variable algorithms. Entropy-sampling Monte Carlo is able to sample a large configurational space even in a disordered system that has a large number of potential barriers. In contrast, scaled-collective-variable Monte Carlo provides an efficient sampling for a system whose dynamics is highly cooperative. Because a globular protein is a disordered system whose dynamics is characterized by collective motions, a combination of these two algorithms could provide an optimal Monte Carlo simulation for a globular protein. As a test case, we have carried out an ESCV Monte Carlo simulation for a cell adhesive Arg-Gly-Asp-containing peptide, Lys-Arg-Cys-Arg-Gly-Asp-Cys-Met-Asp, and determined the conformational distribution at 300 K. The peptide contains a disulfide bridge between the two cysteine residues. This bond mimics the strong geometrical constraints that result from a protein's globular nature and give rise to highly cooperative dynamics. Computation results show that the ESCV Monte Carlo was not trapped at any local minimum and that the canonical distribution was correctly determined.

The effect of protein concentration on ion binding.

The concentration of protein in a solution has been found to have a significant effect on ion binding affinity. It is well known that an increase in ionic strength of the solvent medium by addition of salt modulates the ion-binding affinity of a charged protein due to electrostatic screening. In recent Monte Carlo simulations, a similar screening has been detected to arise from an increase in the concentration of the protein itself. Experimental results are presented here that verify the theoretical predictions; high concentrations of the negatively charged proteins calbindin D9k and calmodulin are found to reduce their affinity for divalent cations. The Ca(2+)-binding constant of the C-terminal site in the Asn-56 --> Ala mutant of calbindin D9k has been measured at seven different protein concentrations ranging from 27 microM to 7.35 mM by using 1H NMR. A 94% reduction in affinity is observed when going from the lowest to the highest protein concentration. For calmodulin, we have measured the average Mg(2+)-binding constant of sites I and II at 0.325, 1.08, and 3.25 mM protein and find a 13-fold difference between the two extremes. Monte Carlo calculations have been performed for the two cases described above to provide a direct comparison of the experimental and simulated effects of protein concentration on metal ion affinities. The overall agreement between theory and experiment is good. The results have important implications for all biological systems involving interactions between charged species.

Ionophores and receptors using cation-π interactions: Collarenes

Cation-π interactions are important forces in molecular recognition by biological receptors, enzyme catalysis, and crystal engineering. We have harnessed these interactions in designing molecular systems with circular arrangement of benzene units that are capable of acting as ionophores and models for biological receptors. [n]Collarenes are promising candidates with high selectivity for a specific cation, depending on n, because of their structural rigidity and well-defined cavity size. The interaction energies of [n]collarenes with cations have been evaluated by using ab initio calculations. The selectivity of these [n]collarenes in aqueous solution was revealed by using statistical perturbation theory in conjunction with Monte Carlo and molecular dynamics simulations. It has been observed that in [n]collarenes the ratio of the interaction energies of a cation with it and the cation with the basic building unit (benzene) can be correlated to its ion selectivity. We find that collarenes are excellent and efficient ionophores that bind cations through cation-π interactions. [6]Collarene is found to be a selective host for Li+ and Mg2+, [8]collarene for K+ and Sr2+, and [10]collarene for Cs+ and Ba2+. This finding indicates that [10]collarene and [8]collarene could be used for effective separation of highly radioactive isotopes, 137Cs and 90Sr, which are major constituents of nuclear wastes. More interestingly, collarenes of larger cavity size can be useful in capturing organic cations. [12]Collarene exhibits a pronounced affinity for tetramethylammonium cation and acetylcholine, which implies that it could serve as a model for acetylcholinestrase. Thus, collarenes can prove to be novel and effective ionophores/model-receptors capable of heralding a new direction in molecular recognition and host-guest chemistry.

Physical reasons for the unusual α-helix stabilization afforded by charged or neutral polar residues in alanine-rich peptides

We have carried out conformational energy calculations on alanine-based copolymers with the sequence Ac-AAAAAXAAAA-NH2 in water, where X stands for lysine or glutamine, to identify the underlying source of stability of alanine-based polypeptides containing charged or highly soluble polar residues in the absence of charge–charge interactions. The results indicate that ionizable or neutral polar residues introduced into the sequence to make them soluble sequester the water away from the CO and NH groups of the backbone, thereby enabling them to form internal hydrogen bonds. This solvation effect dictates the conformational preference and, hence, modifies the conformational propensity of alanine residues. Even though we carried out simulations for specific amino acid sequences, our results provide an understanding of some of the basic principles that govern the process of folding of these short sequences independently of the kind of residues introduced to make them soluble. In addition, we have investigated through simulations the effect of the bulk dielectric constant on the conformational preferences of these peptides. Extensive conformational Monte Carlo searches on terminally blocked 10-mer and 16-mer homopolymers of alanine in the absence of salt were carried out assuming values for the dielectric constant of the solvent ɛ of 80, 40, and 2. Our simulations show a clear tendency of these oligopeptides to augment the α-helix content as the bulk dielectric constant of the solvent is lowered. This behavior is due mainly to a loss of exposure of the CO and NH groups to the aqueous solvent. Experimental evidence indicates that the helical propensity of the amino acids in water shows a dramatic increase on addition of certain alcohols, such us trifluoroethanol. Our results provide a possible explanation of the mechanism by which alcohol/water mixtures affect the free energy of helical alanine oligopeptides relative to nonhelical ones.