Brownian dynamics of a compressed polymer brush model. Off-equilibrium response as a function of surface coverage and compression rate.

We study the compressive behaviour of a polymer-covered surface (i.e., a "polymer brush") using Brownian dynamics simulations. The model consists of grafted chains with variable flexibility, variable intra- and inter-chain interactions, as well as different surface coverage. We discuss the polymer brush response to confinement by considering variable rates of compression under a hard plane. Our results show a small degree of inter-chain entanglement, regardless of whether the interaction is attractive or merely excluded volume. We observe that the molecular shape depends strongly on the surface coverage. Dense brushes exhibit a limited degree of lateral deformation under compression; instead, chains undergo a transition that produces a local patch with near-solid packing. This effect due to surface density can be undone partially by increasing the attractive nature of the chain interaction, by modulating the rate of compression, or by allowing "soft anchoring", i.e., the possible Brownian drift of the grafting bead on the surface. We have also studied the polymer brush relaxation while maintaining the compressing plane, as well as after its sudden removal. We find evidence that also the relaxation depends on surface density; dense brushes appear to be configurationally frustrated at high compression and are unable to undergo swelling, regardless of the pressure applied.

Off-equilibrium response of grafted polymer chains subject to a variable rate of compression.

We present Brownian dynamics simulations of single grafted semiflexible chains (i.e., "polymer mushrooms") with varying persistence lengths, intra-chain interactions, and subject to confinement. The results from different rates of compression are presented in the cases of an approaching infinite plane and a paraboloid tip. We discuss the different behaviour observed for grafted chains with strong and weak self-attraction (i.e., "hard" and "soft" polymer mushrooms). In both cases the effect on the size and shape is more pronounced for a slow compression rate, especially for "hard mushrooms". We have also studied the relaxation of the chain while the compressing plane is maintained, and when it is removed suddenly. We find that the response depends strongly on the time allowed for relaxation in the compressed state. When using instead a paraboloid tip, the overall effects are similar yet less pronounced because the chain can dodge the confining object via an "escape transition."

Structural transitions in neutral and charged proteins in vacuo.

In vacuo proteins provide a simple laboratory to explore the roles of sequence, temperature, charge state, and initial configuration in protein folding. Moreover, by the very absence of solvent, the study of anhydrous proteins in vacuo will also help us to understand specific environmental effects. From the experimental viewpoint, these systems are now beginning to be characterized at low resolution. Molecular dynamics (MD) simulations, in combination with tools for protein shape analysis, can complement experiments and provide further insights on the folding-unfolding transitions of these proteins. We review some aspects of this issue by using the results from a detailed MD study of hen egg-white lysozyme. For lysozyme ions, unfolding can be triggered by Coulombic repulsion. In neutral lysozyme, unfolding can be induced by centrifugal forces and also by weakening the monomer-monomer interaction. In both cases, the resulting unfolded transients can be used as initial configurations for relaxation dynamics. All trajectories are analyzed in terms of global molecular shape features of the backbone, including its anisometry and chain entanglement complexity. This strategy allows us to quantify separately the degree of polymer collapse and the evolution of large-scale folding features. Using these last two notions, we discuss some basic questions regarding the nature of the accessible paths associated with unfolding from, and refolding into, compact conformers.

Proteins in vacuo: denaturing and folding mechanisms studied with computer-simulated molecular dynamics.

Mounting evidence from experiments suggests that the native fold in solution is metastable in dehydrated proteins. Results from a number of experiments that use mass spectrometry indicate also that folding-unfolding transitions take place in protein ions even in the absence of water. These observations on anhydrous proteins call for a re-evaluation of our understanding of the folding transition. In this context, computer-assisted simulations are an important complementary tool. Here, we provide an overview of recent progress on the simulation of proteins in vacuo. In particular, we discuss the response of proteins and protein ions to perturbations that trigger unfolding and re-folding transitions. By comparing the general patterns emerging from theory and experiment, we propose a series of new measurements that could help to validate, and improve, current simulation models.

Characterization of fold diversity among proteins with the same number of amino acid residues.

Chain entanglements provide a simple and global measure of folding in a macromolecule. The complexity of these entanglements can be expressed by the pattern of projected bond-bond crossings, or "overcrossings", associated with the molecular backbone. In this work, we use this approach to characterize quantitatively the range of tertiary folds observed in proteins with a given chain length. To discriminate among folding features, we use two shape descriptors derived from the probability distribution of overcrossings: the mean overcrossing number, N, and the most probable overcrossing number, N*. The values of N and N* relate to the content of secondary structure in a protein as well as its global three-dimensional organization. We propose a measure of folding diversity based on the properties of these descriptors. In addition, we discuss the application of our method to study how tertiary folds evolve during protein dynamics.

Electron-density-dependent fused-sphere surfaces derived from pseudopotential calculations.

Fused-sphere surfaces can be used to mimic a molecular "boundary" associated with a constant value of the electron density. The simplest of such fused-sphere models are constructed by using the atomic radii for the spherical isodensity surfaces of individual atoms. In this work, we discuss the extension of this model to molecules containing atoms beyond the second row. In these many-electron systems, the computation of electron densities is usually simplified by adopting a pseudopotential (or effective-core potential) approach. Here, we discuss the performance of large- and small-core pseudo-potential calculations as a tool to derive atomic radii. Our results provide an optimum set of variable radii that can be used to build fused-sphere surfaces. This continuum of surfaces provides a simple approximation to the low-electron-density regions around molecules with heavy atoms.

Addendum to "Quantitative measure of folding in two-dimensional polymers".

Recently, we introduced a measure of folding complexity for two-dimensional polymers, N macro, the mean radial intersection number [Phys. Rev. E 59, 4209 (1999)]. In this addendum, we expand on three aspects of the previous work. First, we provide an analytical expression for N macro that is valid for two-dimensional networks. Second, we show that the power-law scaling N macro approximately equal to n(beta), with n the number of monomers, has a different critical exponent for random and self-avoiding walks. Finally, we find that the folding features in optimized projections of experimental three-dimensional (native) protein backbones fall between the latter limit models.

Unfolded in vacuo lysozyme folds into native, quasinative, and compact structures.

We show that the relaxation dynamics of unfolded in vacuo lysozyme is not random. Analyses of molecular dynamics trajectories in a convenient space of molecular shape descriptors reveal a "favored" pattern of transitions leading to stable conformations. The relaxation paths exhibit a balanced change in shape features: globular spheroids are formed slowly enough to allow the proper entanglement of secondary-structural elements. The present study shows that a protein in vacuo can actually (re)fold into native and quasinative structures. The driving force for these transformations is intrinsic to the polypeptide chain.

Heuristic lipophilicity potential for computer-aided rational drug design.

In this contribution we suggest a heuristic molecular lipophilicity potential (HMLP), which is a structure-based technique requiring no empirical indices of atomic lipophilicity. The input data used in this approach are molecular geometries and molecular surfaces. The HMLP is a modified electrostatic potential, combined with the averaged influences from the molecular environment. Quantum mechanics is used to calculate the electron density function rho(r) and the electrostatic potential V(r), and from this information a lipophilicity potential L(r) is generated. The HMLP is a unified lipophilicity and hydrophilicity potential. The interactions of dipole and multipole moments, hydrogen bonds, and charged atoms in a molecule are included in the hydrophilic interactions in this model. The HMLP is used to study hydrogen bonds and water-octanol partition coefficients in several examples. The calculated results show that the HMLP gives qualitatively and quantitatively correct, as well as chemically reasonable, results in cases where comparisons are available. These comparisons indicate that the HMLP has advantages over the empirical lipophilicity potential in many aspects. The HMLP is a three-dimensional and easily visualizable representation of molecular lipophilicity, suggested as a potential tool in computer-aided three-dimensional drug design.

Three-dimensional lipophilicity characterization of molecular pores and channel-like cavities.

Molecular lipophilicity is a useful property for assessing molecular similarity or complementarity within the context of computer-aided drug design. As well, local contributions to solvent affinity help us to understand both dynamics and conformational stability in biomolecules. In this work, we discuss an approach to characterize the local contributions to hydrophobicity by using one- and two-dimensional representations of molecular channel-like cavities. The method monitors how a phenomenological lipophilicity potential (based on fragmental atom contributions) changes over a continuum of "molecular tubes" used for modeling channels and pores. Our results convey a relatively detailed picture of the spatial distribution of water affinity. The procedure can then be used as a complement to the hydrophobicity scales based on averaging contributions from single amino acids. In addition, we can study how the water affinity changes for inner and outer regions of the pores. As an application, we compute the 3D distribution of lipophilicity in the "pore conformation" of gramicidin A. The qualitative trends indicated by our results are broadly consistent with computer simulations of the gramicidin channel in the presence of hydrated ions. The behavior revealed by the simulations can then be incorporated to produce an improved, simple 2D model for water-channel interactions.

Modeling lipophilicity from the distribution of electrostatic potential on a molecular surface.

Molecular lipophilicity L is represented as a function of four surface electrostatic potential descriptors: L = f(B+F, B-F, B+R, B-R). Each B descriptor is computed from the products of elements of molecular surface area, delta(si), and the molecular electrostatic potential (MEP), V(ri), at the center of an area element: B = sigma(i) delta(si) V(r(i)). Octanol-water partition coefficients (P(ow)) are correlated with these four surface-MEP descriptors: log P(ow) = c0 + c1B+F + c2B-F + c3B+R + c4B-R. Good correlations are obtained for homologous series of aliphatic alcohols, amines and acids, as well as for a set of aromatic compounds with various functional groups. Within this approach, we find that the molecular fragment contributions of surface-MEP descriptions to log P are approximately additive. We have computed the values for the following fragments: -CH2-, -CH3, _COOH, -OH and -NH2. These contributions can be used to estimate the molecular lipophilicity and partition coefficients of new compounds, without additional quantum-mechanical calculations. The proposed approach provides a reasonably accurate tool that can be useful in quantitative structure-activity relations for computer-aided rational drug design. More importantly, the correlation model is conceptually simpler than previous work in the literature and can be improved systematically.

Interrelation between electrostatic and lipophilicity potentials on molecular surfaces.

Molecular electrostatics and lipophilicity are two important properties included in quantitative structure-activity relationships (QSARs) employed for rational drug design. The molecular electrostatic potential (MEP) provides information on the position, distribution, and extent of electrophilic and nucleophilic regions around a molecule. Similarly, the solvent affinity can be represented by a local phenomenological potential of semiempirical nature: the molecular lipophilicity potential (MLP). Although a simultaneous, three-dimensional display of MEP and MLP is possible, it may not provide a practical tool for comparing molecules. In this work, we deal with the simpler two-dimensional maps of the entire molecular surface projected onto an MEP-MLP plane. We analyze how these maps change with the following factors: (1) composition and molecular geometry, (2) the quality of the computation of MEP, and (3) the parameter set used for evaluating the lipophilicity potential. The approach is used to compare series of pyrazole derivatives. The methodology is useful in assessing molecular similarity, as well as in establishing the nature of differences between compounds.

Overcrossing spectra of protein backbones: characterization of three-dimensional molecular shape and global structural homologies.

A procedure is developed and applied to characterize the global shape and folding features of the backbone of a chain molecule. The methodology is based on the following concept: the probability of observing a rigid placement of a backbone in 3-space as a projected curve with N overcrossings. The numerical computation of these probabilities allows one to construct the overcrossing spectrum of a macromolecule at a given configuration. Although the spectrum is built from the knowledge of the nuclear geometry of the main-chain atoms, the shape descriptor overlooks local geometrical features (such as distances and contacts) and provides a characterization of essential (topological) features of the overall fold, such as its compactness and degree of entanglement. In contrast with other shape descriptors, the present approach gives an absolute characterization of the configuration considered, and not one that is relative to an arbitrarily chosen reference structure. Moreover, it is possible to discriminate between folding features that otherwise may not be distinguished when using other geometrical or topological global descriptors. The overcrossing spectrum is proposed as a tool that complements current structural analyses of macromolecules, especially when monitoring structural homologies within a group of related or unrelated polymers. In this work, we apply the methodology to the analysis of proteins having the globin fold. The results are compared with those of other proteins exhibiting similar size and number of residues. Some basic properties of the spectra as a function of the chain length are also discussed.

Global characterization of protein secondary structures. Analysis of computer-modeled protein unfolding.

Analyses of structural and molecular shape changes undergone by a protein during an unfolding process are presented. The procedure, based on a spherical shape map method, provides a topological description of a three-dimensional macromolecular structure. Local properties of the backbone are used to derive a global characterization of its fold. A spherical shape map of backbone crossings is associated with a given macromolecular conformation. The map is built by classifying each point on the sphere according to the crossing pattern obtained when the backbone is observed along a direction defined by the center of the sphere and the chosen point. The surface of the sphere can be divided in equivalence classes. All the points within a given class correspond to directions from which the backbone has the same overcrossing pattern. Automatic computation and display of these equivalence classes is discussed, as is the implementation of the technique on a computer graphics workstation. The graphical manipulation simplifies the analysis of these maps when following a change in the conformation of the backbone. The procedure is illustrated with the results of a molecular dynamics computer simulation of the unfolding of the bacteriophage T4 glutaredoxin protein (in the form of its polyglycine model). The method gives a novel description of the differential structural stability for the characteristic secondary structural elements (alpha-helices and beta-sheets) present in the protein. Recognition of the persistence of structural elements over the simulation time is performed in an unbiased manner.

Implementing knot-theoretical characterization methods to analyze the backbone structure of proteins: application to CTF L7/L12 and carboxypeptidase A inhibitor proteins.

In this work we apply a recently developed method for characterizing the shape of the tertiary structure of proteins. The approach is based on a combination of graph- and knot-theoretical characterizations of Cartesian projections of the space curve describing the protein backbone. The proposed technique reduces the essential shape features to a topologically based code formed by a sequence of knot symbols and polynomials. These polynomials are topological invariants that describe the overcrossing and knotting patterns of curves derived from the molecular space curve. These descriptors are algorithmically computed. The procedure is applied to describe the structure of the carboxy terminal fragment of the L7/L12 chloroplast ribosomal protein (CTF L7/L12) and the potato carboxypeptidase A inhibitor protein (PCI), which has a set of three disulfide bridges. In the former case, we describe the protein's shape features in terms of its alpha-helices, and a backbone simplified by considering helices without internal structure. An extension of the methodology to describe disulfide bridges is discussed and applied to PCI. Changes in the knot-theoretical characterization due to possible uncertainties in the resolution of the X-ray structure, as well as the inclusion of low-frequency motions of the backbone, are also discussed.

A method for the characterization of foldings in protein ribbon models.

The ribbon model of chain macromolecules is a useful tool for analyzing some of the large-scale shape features of these complex systems. Up to now, the ribbon model has been used mostly to produce graphical displays, which are usually analyzed by visual inspection. In this work we suggest a computational method for characterizing automatically, in a concise and algebraic fashion, some of the important shape features of these ribbon models. The procedure is based on a graph-theoretical and knot-theoretical characterization of three well-defined projections of a space curve associated with the ribbon. The labeled graphs can be characterized by the handedness of the crossovers in the ribbon that are the vertices of the graph. The method can be used to provide a fully algebraic representation of the changes occurring when a molecule, such as a protein, undergoes conformational rearrangements (folding), as well as to provide a shape comparison for a pair of related molecular ribbons. This algebraic representation is well suited for easy storage, retrieval, and computer manipulation of the information on the ribbon's shape. Illustrative examples of the method are provided.