Rotamer optimization for protein design through MAP estimation and problem‐size reduction

The search for the global minimum energy conformation (GMEC) of protein side chains is an important computational challenge in protein structure prediction and design. Using rotamer models, the problem is formulated as a NP‐hard optimization problem. Dead‐end elimination (DEE) methods combined with systematic A* search (DEE/A*) has proven useful, but may not be strong enough as we attempt to solve protein design problems where a large number of similar rotamers is eligible and the network of interactions between residues is dense. In this work, we present an exact solution method, named BroMAP (branch‐and‐bound rotamer optimization using MAP estimation), for such protein design problems. The design goal of BroMAP is to be able to expand smaller search trees than conventional branch‐and‐bound methods while performing only a moderate amount of computation in each node, thereby reducing the total running time. To achieve that, BroMAP attempts reduction of the problem size within each node through DEE and elimination by lower bounds from approximate maximum‐a‐posteriori (MAP) estimation. The lower bounds are also exploited in branching and subproblem selection for fast discovery of strong upper bounds. Our computational results show that BroMAP tends to be faster than DEE/A* for large protein design cases. BroMAP also solved cases that were not solved by DEE/A* within the maximum allowed time, and did not incur significant disadvantage for cases where DEE/A* performed well. Therefore, BroMAP is particularly applicable to large protein design problems where DEE/A* struggles and can also substitute for DEE/A* in general GMEC search. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Maximizing Algebraic Connectivity Over Unicyclic Graphs

We consider the class of unicyclic graphs on n vertices with girth g, and over that class, we attempt to determine which graph maximizes the algebraic connectivity. When g is fixed, we show that there is an N such that for each n>N, the maximizing graph consists of a g cycle with n?g pendant vertices adjacent to a common vertex on the cycle. We also provide a bound on N. On the other hand, when g is large relative to n, we show that this graph does not maximize the algebraic connectivity, and we give a partial discussion of the nature of the maximizing graph in that situation.     

Carbon chains and the (5,5) single-walled nanotube: structure and energetics versus length

Reliable thermochemistry is computed for infinite stretches of pure-carbon materials including acetylenic and cumulenic carbon chains, graphene sheet, and single-walled carbon nanotubes (SWCNTs) by connection to the properties of finite size molecules that grow into the infinitely long systems. Using ab initio G3 theory, the infinite cumulenic chain (:C[double bond]C[double bond]C[double bond]C:) is found to be 1.9+/-0.4 kcal/mol per carbon less stable in free energy at room temperature than the acetylenic chain (.C[triple bond]C-C[triple bond]C.) which is 24.0 kcal/mol less stable than graphite. The difference between carbon-carbon triple, double, and single bond lengths (1.257, 1.279, and 1.333 A, respectively) in infinite chains is evident but much less than with small hydrocarbon molecules. These results are used to evaluate the efficacy of similar calculations with the less rigorous PM3 semiempirical method on the (5,5) SWCNT, which is too large to be studied with high-level ab initio methods. The equilibrium electronic energy change for C(g)-->C[infinite (5,5) SWCNT] is -166.7 kcal/mol, while the corresponding free energy change at room temperature is -153.3 kcal/mol (6.7 kcal/mol less stable than graphite). A threefold alternation (6.866, 6.866, and 6.823 A) in the ring diameter of the equilibrium structure of infinitely long (5,5) SWCNT is apparent, although the stability of this structure over the constant diameter structure is small compared to the zero point energy of the nanotube. In general, different (n,m) SWCNTs have different infinite tube energetics, as well as very different energetic trends that vary significantly with length, diameter, and capping.     

Arithmetic Properties of Overpartitions into Odd Parts

In this article, we consider various arithmetic properties of the function which denotes the number of overpartitions of n using only odd parts. This function has arisen in a number of recent papers, but in contexts which are very different from overpartitions. We prove a number of arithmetic results including several Ramanujan-like congruences satisfied by and some easily-stated characterizations of modulo small powers of two. For example, it is proven that, for n ≥ 1, (mod 4) if and only if n is neither a square nor twice a square. Received March 17, 2005     

Absolute rate coefficient of the OH + CH(3)C(O)OH reaction at T = 287-802 K. The two faces of pre-reactive H-bonding

The rate constants for the reaction OH + CH3C(O)OH --> products (1) were determined over the temperature range 287-802 K at 50 and 100 Torr of Ar or N2 bath gas using pulsed laser photolysis generation of OH by CH3C(O)OH photolysis at 193 nm coupled with OH detection by pulsed laser-induced fluorescence. The rate coefficient displays a complex temperature dependence with a sharp minimum at 530 K, indicating the competition between a reaction proceeding through a pre-reactive H-bonded complex to form CH3C(O)O + H2O, expected to prevail at low temperatures, and a direct methyl-H abstraction channel leading to CH2C(O)OH + H2O, which should dominate at high temperatures. The temperature dependence of the rate constant can be described adequately by k1(287-802 K) = 2.9 x 10(-9) exp{-6030 K/T} + 1.50 x 10(-13) exp{515 K/T} cm3 molecule(-1)(s-1), with a value of (8.5 +/- 0.9) x 10-13 cm3 molecule(-1)(s-1) at 298 K. The steep increase in rate constant in the range 550-800 K, which is reported for the first time, implies that direct abstraction of a methyl-H becomes the dominant pathway at temperatures greater than 550 K. However, the data indicates that up to about 800 K direct methyl-H abstraction remains adversely affected by the long-range H-bonding attraction between the approaching OH radical and the carboxyl -C(O)OH functionality.     

Infrared and Raman spectra, ab initio calculations, conformational stability and vibrational assignment of 1-bromo-1-silacyclopentane

Infrared and Raman spectra (3500-60 cm(-1)) of gas and/or liquid and solid 1-bromo-1-silacyclopentane (c-C4H8SiBrH) have been recorded and the vibrational data indicate the presence of a single conformer with no symmetry which is consistent with the twisted form. Ab initio calculations with a variety of basis sets up to MP2(full)/6-311+G(2df,2pd) predict the envelope-axial and envelope-equatorial conformers to be saddle points with nearly the same energies but approximately 900 cm(-1) (5.98 kJ/mol) lower in energy than the planar conformer. Density functional theory calculations by the B3LYP method predict slightly lower energies for the two envelope forms and considerably lower energy for the planar form compared to the MP2 predictions. By utilizing the MP2(full)/6-31G(d) calculations the force constants, frequencies, infrared intensities, band contours, Raman activities, and depolarization values have been obtained to support the vibrational assignment. Estimated r0 structural parameters have been obtained from adjusted MP2(full)/6-311+G(d,p) calculations. These experimental and theoretical results are compared to the corresponding quantities of some other five-membered rings.     

Homo- and heteroleptic complexes of four-membered group 13 metal(I) N-heterocyclic carbene analogues with group 10 metal(0) fragments

A series of complexes between recently developed four-membered group 13 metal(I) heterocycles and group 10 metal(0) fragments have been prepared and structurally characterized. One prepared complex, [Pt{Ga[N(Ar)]2CNCy2}3] (Ar = C6H3Pri2-2,6; Cy = cyclohexyl), possesses the shortest Pt-Ga bonds yet reported, the covalent components of which are suggested by theoretical studies to have significant pi character.     

Highly regioselective Buchwald–Hartwig amination at C-2 of 2,4-dichloropyridine enabling a novel approach to 2,4-bisanilinopyridine (BAPyd) libraries

The highly regioselective Buchwald–Hartwig amination at C-2 of the cheap and readily accessible reagent, 2,4-dichloropyridine with a range of anilines and heterocyclic amines is described. This new methodology is robust and provides a facile access to 4-chloro-N-phenylpyridin-2-amines on 0.25 mol scale. These intermediates undergo a further Buchwald–Hartwig amination at higher temperature to enable rapid exploration of the chemical space at C-4 and to provide a library of 2,4-bisaminopyridines.     

Sixteen Eisenstein series

S. Ramanujan gave fourteen families of series in his Second Notebook in Chap. 17, Entries 13–17. In each case he gave only the first few examples, giving us the motivation to find and prove a general formula for each family of series. The aim of this paper is to develop a powerful tool (four versatile functions f ₀,f ₁,f ₂, and f ₃) to collect all of Ramanujan’s examples together.