Thermodynamics of information transfer between subunits in oligomeric enzymes and kinetic cooperativity. 1. Thermodynamics of subunit interactions, partition functions and enzyme reaction rate

The strength of quaternary constraints between two subunits of a polymeric enzyme depends upon the number of neighboring subunits and upon whether these subunits are liganded or not. These quaternary constraints between two subunits of a complex polymeric enzyme may be expressed, however, in terms of quaternary constraints that exist within ideal dimers. The influence of quaternary constraints on the reaction rate of a complex polymeric enzyme may thus be expressed in terms of the intersubunit strain that exists within dimers. This conclusion, that was far from evident, appears to be the consequence of the postulates of structural kinetics, and derive as well from usual thermodynamic principles. The structural steady-state equations may be expressed in terms of partition and sub-partition functions. As applied to structural kinetic models, a partition function expresses how, during the steady state, the energy of a population of enzyme molecules is distributed over n states. Similarly a sub-partition function describes how, during the steady state, the energy of these enzyme molecules is partitioned among only n-k of these states. Although the concept of partition function was initially formulated for equilibrium processes, it may be extended without any loss of generality to non-equilibrium processes. Moreover it is reminiscent of the concept of binding polynomial presented some years ago by Wyman for the equilibrium binding of a ligand to a protein. With this formalism derived from statistical mechanics, a structural rate equation may be derived from the ratio of a sub-partition function of degree n-1 and of a partition function of degree n. Again these properties are the consequence of the postulates of structural kinetics associated with simple ideas derived from statistical thermodynamics.     

A Partition Function Approximation Using Elementary Symmetric Functions

In statistical mechanics, the canonical partition function can be used to compute equilibrium properties of a physical system. Calculating however, is in general computationally intractable, since the computation scales exponentially with the number of particles in the system. A commonly used method for approximating equilibrium properties, is the Monte Carlo (MC) method. For some problems the MC method converges slowly, requiring a very large number of MC steps. For such problems the computational cost of the Monte Carlo method can be prohibitive. Presented here is a deterministic algorithm – the direct interaction algorithm (DIA) – for approximating the canonical partition function in operations. The DIA approximates the partition function as a combinatorial sum of products known as elementary symmetric functions (ESFs), which can be computed in operations. The DIA was used to compute equilibrium properties for the isotropic 2D Ising model, and the accuracy of the DIA was compared to that of the basic Metropolis Monte Carlo method. Our results show that the DIA may be a practical alternative for some problems where the Monte Carlo method converge slowly, and computational speed is a critical constraint, such as for very large systems or web-based applications.     

Hill’s small systems nanothermodynamics: a simple macromolecular partition problem with a statistical perspective

Using a simple example of biological macromolecules which are partitioned between bulk solution and membrane, we investigate T.L. Hill’s phenomenological nanothermodynamics for small systems. By introducing a system size-dependent equilibrium constant for the bulk-membrane partition, we obtain Hill’s results on differential and integral chemical potentials μ and [^(m)]\hat{\mu} from computations based on standard Gibbsian equilibrium statistical mechanics. It is shown that their difference can be understood from an equilibrium re-partitioning between bulk and membrane fractions upon a change in the system’s size; it is closely related to the system’s fluctuations and inhomogeneity. These results provide a better understanding of nanothermodynamics and clarify its logical relation with the theory of statistical mechanics.     

Noninteracting local-structure model of folding and unfolding transition in globular proteins. I. Formulation

A statistical-mechanical model (a noninteracting local structure model) of folding and unfolding transition in globular proteins is described and a formulation is given to calculate the partition function. The process of transition is discussed in this model within the framework of equilibrium statistical mechanics. In order to clarify the range of applicability of such an approach, the characteristics of the folding and unfolding transition in globular proteins are analyzed from the statistical-physical point of view. A theoretical advantage is pointed out in studying folding and unfolding processes taking place as conformational fluctuations in individual protein molecules under macroscopic equilibrium at the melting temperature. In this case, paths of folding and unfolding are shown to be identical in the statistical sense. A key to the noninteracting local structure model lies in the concept of local structures and the assumption of the absence of interactions between local structures. A local structure is defined as a continuous section of the chain which takes the same or similar local conformation as in the native conformation. The assumption of the absence of inter-actions between local structures endows the model with the remarkable character that its partition function can be calculated exactly; thereby the equilibrium population of various conformations along the folding and unfolding paths can be discussed only by a knowledge of the folded native conformation.     

Prediction of multimolecular assemblies by multiple docking

The majority of proteins function when associated in multimolecular assemblies. Yet, prediction of the structures of multimolecular complexes has largely not been addressed, probably due to the magnitude of the combinatorial complexity of the problem. Docking applications have traditionally been used to predict pairwise interactions between molecules. We have developed an algorithm that extends the application of docking to multimolecular assemblies. We apply it to predict quaternary structures of both oligomers and multi-protein complexes. The algorithm predicted well a near-native arrangement of the input subunits for all cases in our data set, where the number of the subunits of the different target complexes varied from three to ten. In order to simulate a more realistic scenario, unbound cases were tested. In these cases the input conformations of the subunits are either unbound conformations of the subunits or a model obtained by a homology modeling technique. The successful predictions of the unbound cases, where the input conformations of the subunits are different from their conformations within the target complex, suggest that the algorithm is robust. We expect that this type of algorithm should be particularly useful to predict the structures of large macromolecular assemblies, which are difficult to solve by experimental structure determination.     

A grammatical theory for the conformational changes of simple helix bundles.

Polymers, including biomolecules such as proteins, have two particularly important types of single-molecule transitions: a helix-coil transition, driven by interactions that are local in the chain, and a collapse transition, driven by nonlocal interactions. A long-standing challenge of polymer statistical mechanics has been to deal with both types of transition in a single theoretical framework. The simplest paradigmatic problem would be a theory of helix-bundle folding. Here, we show how the machinery of formal grammars, originally developed in the context of linguistic analysis and programming-language compilation, provides a simple and general way to combine the Zimm-Bragg model of alpha-helices with the model of Chen and Dill for nonlocal interactions in antiparallel polymeric systems. We use a well-known construction in the theory of formal grammars to give the statistical mechanical partition function for two-helix bundles. Predictions are shown to be quite good in comparison to exact enumerations within a lattice model.     

Isolation of tropomyosin particles from cultured cell cytosol and their protein composition assay

The presence of an actin-binding protein, tropomyosin, in particles or protein complexes not bound with actin structures were found during an assay of structural rearrangements of actin cytoskeleton. To study the composition and properties of these protein complexes, a novel method of their isolation without destroying cytoskeleton structures has been elaborated. The protein composition of isolated tropomyosin particles was assessed by gel filtration, electrophoresis, and Western blotting. It was demonstrated that they are about 700-kDa multimolecular complexes. In addition to tropomyosin and actin, these complexes contained Hsp70, Hsp90, and myosin-9 identified by mass spectrometry. It was found that the deacetylase inhibitor, trichostatin A, which induced actin cytoskeleton rearrangements, changed the number of tropomyosin particles and caused redistribution of tropomyosin between cytosol and cytoskeleton. These results demonstrate that these multimolecular complexes may participate in the process of reorganization of actin microfilaments.     

Isolation of tropomyosin particles from the cytosol of cultured cells and their protein composition analysis

The presence of actin-binding protein, tropomyosin, shaped as particles or protein complexes that have no bonds with actin structures were found while the analisys of structural rearrangements of actin cytoskeleton. However, their functioning is still unknown. To study the composition and properties of these protein complexes a novel method of their separation from the cells without destroying the structures of the cytoskeleton have been developed. The protein composition of isolated tropomyosin particles has been analised by gel filtration, electrophoresis and Western blotting. They appeared to be a multimolecular complexes of about 700 kDa. Beside the tropomyosin and actin these complexes also contain the Hsp70, Hsp90 and myosin-9 identified by mass spectrometry analisys. Also, under inhibition of deacetylases by trichostatin A, changes in the number of particles and redistribution of tropomyosin between cytosol and cytoskeleton take place along with actin cytoskeleton rearrangements. The results obtained give a reason to assume that these multimolecular complexes may participate in the process of reorganization of the actin microfilaments.     

Supramolecular organization of the subsarcolemmal cytoskeleton of adult skeletal muscle fibers. A review

This review is focused on the composition and organization of the junctional subsarcolemmal cytoskeleton of adult muscle fibers. The cytoskeleton of muscle fibers is organized in functionally distinct compartments and the subsarcolemmal cytoskeleton itself can be broadly divided into junctional (myotendinous junction, neuromuscular junction and costameres) and non-junctional domains. In junctional zones three different multimolecular cytoskeletal complexes coexist: the focal adhesion-type, the spectrin-based and the dystrophin vs utrophin-based membrane skeleton systems. These complexes extend over several levels, from intracytoplasmic to subsarcolemmal and transmembranous; their common feature is the anchorage of actin filaments emanating from the intracytoplasmic level. The different cytoskeletal proteins, their putative roles and their interactions with various signaling pathways are presented here in detail. The subsarcolemmal cytoskeleton complexes are thought to play distinct physiological roles (membrane stabilization, force transmission to extracellular matrix, ionic channel anchorage, etc) but their colocalization on the three sarcolemmal junctional domains strongly suggests interrelated or common functions.     

Statistical mechanical analysis of interfacial tension of black lipid membrane

The thermodynamic state of the black (bimolecular) lipid membrane (BLM) is discussed by the theoretical extension of the equilibrium adsorption of the surfactant monolayer on the air-water interface. The formula of the interfacial tension of the BLM and its variations by the surfactant concentration and the temperature are given by the use of the statistical mechanics. The BLM is shown to exist as a metastable (supersaturated) liquid state for appropriate solvent conditions. For the different situations, the BLM may exist in a true equilibrium state with dispersed monomers and micelles. As a result, it is shown that experimental studies of the interfacial tension of the BLM by Tien (1967, 1968) can be understood on this scheme consistently.     

SMC complexes in bacterial chromosome condensation and segregation

Bacterial chromosomes segregate via a partition apparatus that employs a score of specialized proteins. The SMC complexes play a crucial role in the chromosome partitioning process by organizing bacterial chromosomes through their ATP-dependent chromatin-compacting activity. Recent progress in the composition of these complexes and elucidation of their structural and enzymatic properties has advanced our comprehension of chromosome condensation and segregation mechanics in bacteria.     

Discrimination of the native from misfolded protein models with an energy function including implicit solvation.

An essential requirement for theoretical protein structure prediction is an energy function that can discriminate the native from non-native protein conformations. To date most of the energy functions used for this purpose have been extracted from a statistical analysis of the protein structure database, without explicit reference to the physical interactions responsible for protein stability. The use of the statistical functions has been supported by the widespread belief that they are superior for such discrimination to physics-based energy functions. An effective energy function which combined the CHARMM vacuum potential with a Gaussian model for the solvation free energy is tested for its ability to discriminate the native structure of a protein from misfolded conformations; the results are compared with those obtained with the vacuum CHARMM potential. The test is performed on several sets of misfolded structures prepared by others, including sets of about 650 good decoys for six proteins, as well as on misfolded structures of chymotrypsin inhibitor 2. The vacuum CHARMM potential is successful in most cases when energy minimized conformations are considered, but fails when applied to structures relaxed by molecular dynamics. With the effective energy function the native state is always more stable than grossly misfolded conformations both in energy minimized and molecular dynamics-relaxed structures. The present results suggest that molecular mechanics (physics-based) energy functions, complemented by a simple model for the solvation free energy, should be tested for use in the inverse folding problem, and supports their use in studies of the effective energy surface of proteins in solution. Moreover, the study suggests that the belief in the superiority of statistical functions for these purposes may be ill founded.     

The cytoplasmic plaque of tight junctions: A scaffolding and signalling center

The region of cytoplasm underlying the tight junction (TJ) contains several multimolecular protein complexes, which are involved in scaffolding of membrane proteins, regulation of cytoskeletal organization, establishment of polarity, and signalling to and from the nucleus. In this review, we summarize some of the most recent advances in understanding the identity of these proteins, their domain organization, their protein interactions, and their functions in vertebrate organisms. Analysis of knockdown and knockout model systems shows that several TJ proteins are essential for the formation of epithelial tissues and early embryonic development, whereas others appear to have redundant functions.     

The cytoplasmic plaque of tight junctions: a scaffolding and signalling center

The region of cytoplasm underlying the tight junction (TJ) contains several multimolecular protein complexes, which are involved in scaffolding of membrane proteins, regulation of cytoskeletal organization, establishment of polarity, and signalling to and from the nucleus. In this review, we summarize some of the most recent advances in understanding the identity of these proteins, their domain organization, their protein interactions, and their functions in vertebrate organisms. Analysis of knockdown and knockout model systems shows that several TJ proteins are essential for the formation of epithelial tissues and early embryonic development, whereas others appear to have redundant functions.     

The chemical potential in solutions and the free energy of reaction

The expression for the chemical potential of a species in a multi-component solution is usually derived by a thermodynamic argument based on ideal gas theory. A direct statistical mechanical derivation is given below, using the methods of the preceding paper. The analysis is formulated in such a way that there is no need to introduce the notion of standard states. It is made clear why certain species (water, H⁺ in buffered systems) may be ignored in calculating the affinity (negative free energy change) of a chemical reaction. The equilibrium constant is expressed in terms of generalized phase integrals or partition functions.     

Ligand binding systems at equilibrium: specificity, heterogeneity, cross-reactivity, and site-site interactions

The simplest ligand binding model that can account for systems which consist of heterogeneous receptors that show site-site interactions and can react with several types of ligands was presented. This model was based on the grand canonical partition function whose properties and potentialities for obtaining binding functions are briefly illustrated. A computer simulation of this model was carried out in order to examine the contributions of site-site interactions on the shape of binding isotherms and on specificity curves. The shape of the binding isotherms was shown to be independent regardless of whether the path of binding was considered. However, the shape of the specificity curves was strongly dependent on site-site interactions and on the particular sequence of ligand-receptor configurations formed during the ligand binding process, leading to the conclusion that site-site interactions may influence the specificity of a system more than even very strong cross-reactions. A method based on the Ising model of statistical mechanics was also described and was used for obtaining binding functions applicable to infinite lattices which show site-site interactions and competitive binding. The results presented here point out possible errors that can arise if standard statistical methods are used to fit binding data in order to prove a given binding model.     

The complement system and its biological activities

The complement system is a non specific humoral defence mechanism which can be triggered by various effectors : immune-complexes, extrinsic proteases, bacterial cell-wall polysaccharides, viral membranes. Different peptides or multimolecular complexes are formed by a cascade of proteolytic cleavages; they take an active part in the inflammatory response and may contribute to different pathological manifestations.