Systematic derivation of partition functions for ligand binding to two-dimensional lattices

The Ising problem consists in finding the analytical solution of the partition function of a lattice once the interaction geometry among its elements is specified. No general analytical solution is available for this problem, except for the one-dimensional case. Using site-specific thermodynamics, it is shown that the partition function for ligand binding to a two-dimensional lattice can be obtained from those of one-dimensional lattices with known solution. The complexity of the lattice is reduced recursively by application of a contact transformation that involves a relatively small number of steps. The transformation implemented in a computer code solves the partition function of the lattice by operating on the connectivity matrix of the graph associated with it. This provides a powerful new approach to the Ising problem, and enables a systematic analysis of two-dimensional lattices that model many biologically relevant phenomena. Application of this approach to finite two-dimensional lattices with positive cooperativity indicates that the binding capacity per site diverges as Na (N = number of sites in the lattice) and experiences a phase-transition-like discontinuity in the thermodynamic limit N → ∞. The zeroes of the partition function tend to distribute on a slightly distorted unit circle in complex plane and approach the positive real axis already for a 5×5 square lattice. When the lattice has negative cooperativity, its properties mimic those of a system composed of two classes of independent sites with the apparent population of low-affinity binding sites increasing with the size of the lattice, thereby accounting for a phenomenon encountered in many ligand-receptor interactions.

Comparison of x-ray crystal structures of an acyl-enzyme intermediate of subtilisin Carlsberg formed in anhydrous acetonitrile and in water

The x-ray crystal structures of trans-cinnamoyl–subtilisin, an acyl-enzyme covalent intermediate of the serine protease subtilisin Carlsberg, have been determined to 2.2-Å resolution in anhydrous acetonitrile and in water. The cinnamoyl–subtilisin structures are virtually identical in the two solvents. In addition, their enzyme portions are nearly indistinguishable from previously determined structures of the free enzyme in acetonitrile and in water; thus, acylation in either aqueous or nonaqueous solvent causes no appreciable conformational changes. However, the locations of bound solvent molecules in the active site of the acyl- and free enzyme forms in acetonitrile and in water are distinct. Such differences in the active site solvation may contribute to the observed variations in enzymatic activities. On prolonged exposure to organic solvent or removal of interstitial solvent from the crystal lattice, the channels within enzyme crystals are shown to collapse, leading to a drop in the number of active sites accessible to the substrate. The mechanistic and preparative implications of our findings for enzymatic catalysis in organic solvents are discussed.

Third-generation synchrotron x-ray diffraction of 6-μm crystal of raite, ≈Na3Mn3Ti0.25Si8O20(OH)2⋅10H2O, opens up new chemistry and physics of low-temperature minerals

The crystal structure of raite was solved and refined from data collected at Beamline Insertion Device 13 at the European Synchrotron Radiation Facility, using a 3 × 3 × 65 μm single crystal. The refined lattice constants of the monoclinic unit cell are a = 15.1(1) Å; b = 17.6(1) Å; c = 5.290(4) Å; β = 100.5(2)°; space group C2/m. The structure, including all reflections, refined to a final R = 0.07. Raite occurs in hyperalkaline rocks from the Kola peninsula, Russia. The structure consists of alternating layers of a hexagonal chicken-wire pattern of 6-membered SiO4 rings. Tetrahedral apices of a chain of Si six-rings, parallel to the c-axis, alternate in pointing up and down. Two six-ring Si layers are connected by edge-sharing octahedral bands of Na+ and Mn3+ also parallel to c. The band consists of the alternation of finite Mn–Mn and Na–Mn–Na chains. As a consequence of the misfit between octahedral and tetrahedral elements, regions of the Si–O layers are arched and form one-dimensional channels bounded by 12 Si tetrahedra and 2 Na octahedra. The channels along the short c-axis in raite are filled by isolated Na(OH,H2O)6 octahedra. The distorted octahedrally coordinated Ti4+ also resides in the channel and provides the weak linkage of these isolated Na octahedra and the mixed octahedral tetrahedral framework. Raite is structurally related to intersilite, palygorskite, sepiolite, and amphibole.

Theoretical prediction of a crystal structure.

The diffusion equation method of global minimization is applied to compute the crystal structure of S6, with no a priori knowledge about the system. The experimental lattice parameters and positions and orientations of the molecules in the unit cell are predicted correctly.

Projection structure of frog rhodopsin in two crystal forms.

Rhodopsin is the G protein-coupled receptor that upon light activation triggers the visual transduction cascade. Rod cell outer segment disc membranes were isolated from dark-adapted frog retinas and were extracted with Tween detergents to obtain two-dimensional rhodopsin crystals for electron crystallography. When Tween 80 was used, tubular structures with a p2 lattice (a = 32 A, b = 83 A, gamma = 91 degrees) were formed. The use of a Tween 80/Tween 20 mixture favored the formation of larger p22(1)2(1) lattices (a = 40 A, b = 146 A, gamma = 90 degrees). Micrographs from frozen hydrated frog rhodopsin crystals were processed, and projection structures to 7-A resolution for the p22(1)2(1) form and to 6-A resolution for the p2 form were calculated. The maps of frog rhodopsin in both crystal forms are very similar to the 9-A map obtained previously for bovine rhodopsin and show that the arrangement of the helices is the same. In a tentative topographic model, helices 4, 6, and 7 are nearly perpendicular to the plane of the membrane. In the higher-resolution projection maps of frog rhodopsin, helix 5 looks more tilted than it appeared previously. The quality of the two frog rhodopsin crystals suggests that they would be suitable to obtain a three-dimensional structure in which all helices would be resolved.

Crystal structure of a DNA decamer showing a novel pseudo four-way helix-helix junction.

The crystal structure of the decanucleotide d(CGCAATTGCG)2 has been solved by a combination of molecular replacement and heavy-atom procedures and has been refined to an R factor of 20.2% at 2.7 A. It is not a fully base-paired duplex but has a central core of eight Watson-Crick base pairs flanked by unpaired terminal guanosines and cytosines. These participate in hydrogen-bonding arrangements with adjacent decamer duplexes in the crystal lattice. The unpaired guanosines are bound in the G+C regions of duplex minor grooves. The cytosines have relatively high mobility, even though they are constrained to be in one region where they are involved in base-paired triplets with G.C base pairs. The 5'-AATT sequence in the duplex region has a narrow minor groove, providing further confirmation of the sequence-dependent nature of groove width.