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.

Clusters of charged residues in protein three-dimensional structures.

Statistically significant charge clusters (basic, acidic, or of mixed charge) in tertiary protein structures are identified by new methods from a large representative collection of protein structures. About 10% of protein structures show at least one charge cluster, mostly of mixed type involving about equally anionic and cationic residues. Positive charge clusters are very rare. Negative (or histidine-acidic) charge clusters often coordinate calcium, or magnesium or zinc ions [e.g., thermolysin (PDB code: 3tln), mannose-binding protein (2msb), aminopeptidase (1amp)]. Mixed-charge clusters are prominent at interchain contacts where they stabilize quaternary protein formation [e.g., glutathione S-transferase (2gst), catalase (8act), and fructose-1,6-bisphosphate aldolase (1fba)]. They are also involved in protein-protein interaction and in substrate binding. For example, the mixed-charge cluster of aspartate carbamoyl-transferase (8atc) envelops the aspartate carbonyl substrate in a flexible manner (alternating tense and relaxed states) where charge associations can vary from weak to strong. Other proteins with charge clusters include the P450 cytochrome family (BM-3, Terp, Cam), several flavocytochromes, neuraminidase, hemagglutinin, the photosynthetic reaction center, and annexin. In each case in Table 2 we discuss the possible role of the charge clusters with respect to protein structure and function.

Chemotactic signal integration in bacteria.

Chemotactic signaling in Escherichia coli involves transmission of both negative and positive signals. In order to examine mechanisms of signal processing, behavioral responses to dual inputs have been measured by using photoactivable "caged" compounds, computer video analysis, and chemoreceptor deletion mutants. Signaling from Tar and Tsr, two receptors that sense amino acids and pH, was studied. In a Tar deletion mutant the photoactivated release of protons, a Tsr repellent, and of serine, a Tsr attractant, in separate experiments at pH 7.0 resulted in tumbling (negative) or smooth-swimming (positive) responses in ca. 50 and 140 ms, respectively. Simultaneous photorelease of protons and serine resulted in a single tumbling or smooth-swimming response, depending on the relative amounts of the two effectors. In contrast, in wild-type E. coli, proton release at pH 7.0 resulted in a biphasic response that was attributed to Tsr-mediated tumbling followed by Tar-mediated smooth-swimming. In wild-type E. coli at more alkaline pH values the Tar-mediated signal was stronger than the Tsr signal, resulting in a strong smooth-swimming response preceded by a diminished tumbling response. These observations imply that (i) a single receptor time-averages the binding of different chemotactic ligands generating a single response; (ii) ligand binding to different receptors can result in a nonintegrated response with the tumbling response preceding the smooth-swimming response; (iii) however, chemotactic signals of different intensities derived from different receptors can also result in an apparently integrated response; and (iv) the different chemotactic responses to protons at neutral and alkaline pH may contribute to E. coli migration toward neutrality.