Exploring the factors determining the dynamics of different protein folds

Normal mode analyses of homologous proteins at the family and superfamily level show that slow dynamics are similar and are preserved through evolution. This study investigates how the slow dynamics of proteins is affected by variation in the protein architecture and fold. For this purpose, we have used computer-generated protein models based on idealized protein structures with varying folds. These are shown to be protein-like in their behavior, and they are used to investigate the influence of architecture and fold on the slow dynamics. We compared the dynamics of models having different folds but similar architecture and found the architecture to be the dominant factor for the slow dynamics.     

Van der Waals interactions involving proteins.

Van der Waals (dispersion) forces contribute to interactions of proteins with other molecules or with surfaces, but because of the structural complexity of protein molecules, the magnitude of these effects is usually estimated based on idealized models of the molecular geometry, e.g., spheres or spheroids. The calculations reported here seek to account for both the geometric irregularity of protein molecules and the material properties of the interacting media. Whereas the latter are found to fall in the generally accepted range, the molecular shape is shown to cause the magnitudes of the interactions to differ significantly from those calculated using idealized models, with important consequences. First, the roughness of the molecular surface leads to much lower average interaction energies for both protein-protein and protein-surface cases relative to calculations in which the protein molecule is approximated as a sphere. These results indicate that a form of steric stabilization may be an important effect in protein solutions. Underlying this behavior is appreciable orientational dependence, one reflection of which is that molecules of complementary shape are found to exhibit very strong attractive dispersion interactions. Although this has been widely discussed previously in the context of molecular recognition processes, the broader implications of these phenomena may also be important at larger molecular separations, e.g., in the dynamics of aggregation, precipitation, and crystal growth.     

The evolutionary landscape of functional model proteins.

To study the distinct influences of structure and function on evolution, we propose a minimalist model for proteins with binding pockets, called functional model proteins, based on a shifted-HP model on a two-dimensional square lattice. These model proteins are not maximally compact and contain an empty lattice site surrounded by at least three nearest neighbors, thus providing a binding pocket. Functional model proteins possess a unique native state, cooperative folding and tolerance to mutation. Due to the explicit functionality in these models (by design), we have been able to explore their fitness or evolutionary landscapes, as characterized by the size and distribution of homologous families and by the complexity of the inter-relatedness of the functional model proteins. Mindful that these minimalist models are highly idealized and two-dimensional, functional model proteins should nevertheless provide a useful means for exploring the constraints of maintaining structure and function on the evolution of proteins.     

Modelling neutral and selective evolution of protein folding.

We examine a model evolutionary space consisting of genotypes mapped to their corresponding phenotypes. This mapping is derived from a lattice model for proteins which, despite its highly idealized nature, has been shown to share general properties with real proteins. Large evolutionary networks are observed, with genotypes corresponding to non-lethal phenotypes linked by unit mutational steps. Neutral mutations are necessary for traversing the evolutionary networks, and even one neutral mutation in a genotype can change the phenotypes attainable by a unit mutational step.     

A review of morphology-elasticity relationships in human trabecular bone: theories and experiments

In the perspective of predicting mechanical from morphological properties of human trabecular bone, the theoretical and experimental relationships between volume fraction, fabric and elastic properties were reviewed.Five data sets of human trabecular bone and two data sets of idealized cells were obtained from various investigators and analyzed statistically with one isotropic and four anisotropic models. For each model, multiple linear regressions were performed to fit the components of both the compliance and the stiffness tensors using volume fraction and in some cases fabric. The adjusted coefficients of determination of the regressions and the average relative errors of the reported versus the predicted tensor norms were calculated. The three anisotropic models that implied a log transformation of the data showed the best results. Excluding the idealized cell data, the adjusted coefficients of determination of these models ranged from 0.80 to 0.95 for the compliance and from 0.80 to 0.94 for the stiffness tensors, while the average relative errors varied between 16% and 55% for the compliance and between 25% and 62% for the stiffness data. The use of volume fraction alone in the isotropic model decreased the adjusted coefficients of determination by 0.03-0.25 and increased the average relative errors by 5-27%.This review confirms the potential of morphology-elasticity relationships for estimation of elastic properties of human trabecular bone using peripheral quantitative computed tomography or magnetic resonance imaging, but emphasizes the need for standardized measurements of mechanical properties at both continuum and tissue level.     

Stochastic models for sterilization

Probabilistic models for the killing of microorganisms are formulated and described. The utility of the models is illustrated by applying them to the optimization of an idealized fermentation process.     

Statistical analysis of sizes and shapes of virus capsids and their resulting elastic properties

From the analysis of sizes of approximately 130 small icosahedral viruses we find that there is a typical structural capsid protein, having a mean diameter of 5 nm and a mean thickness of 3 nm, with more than two thirds of the analyzed capsid proteins having thicknesses between 2 nm and 4 nm. To investigate whether, in addition to the fairly conserved geometry, capsid proteins show similarities in the way they interact with one another, we examined the shapes of the capsids in detail. We classified them numerically according to their similarity to sphere and icosahedron and an interpolating set of shapes in between, all of them obtained from the theory of elasticity of shells. In order to make a unique and straightforward connection between an idealized, numerically calculated shape of an elastic shell and a capsid, we devised a special shape fitting procedure, the outcome of which is the idealized elastic shape fitting the capsid best. Using such a procedure we performed statistical analysis of a series of virus shapes and we found similarities between the capsid elastic properties of even very different viruses. As we explain in the paper, there are both structural and functional reasons for the convergence of protein sizes and capsid elastic properties. Our work presents a specific quantitative scheme to estimate relatedness between different proteins based on the details of the (quaternary) shape they form (capsid). As such, it may provide an information complementary to the one obtained from the studies of other types of protein similarity, such as the overall composition of structural elements, topology of the folded protein backbone, and sequence similarity.     

Prediction of protein structure from ideal forms

For many years it has been accepted that the sequence of a protein can specify its three-dimensional structure. However, there has been limited progress in explaining how the sequence dictates its fold and no attempt to do this computationally without the use of specific structural data has ever succeeded for any protein larger than 100 residues. We describe a method that can predict complex folds up to almost 200 residues using only basic principles that do not include any elements of sequence homology. The method does not simulate the folding chain but generates many thousands of models based on an idealized representation of structure. Each rough model is scored and the best are refined. On a set of five proteins, the correct fold score well and when tested on a set of larger proteins, the correct fold was ranked highest for some proteins more than 150 residues, with others being close topological variants. All other methods that approach this level of success rely on the use of templates or fragments of known structures. Our method is unique in using a database of ideal models based on general packing rules that, in spirit, is closer to an ab initio approach.     

“Fuzzy oil drop” model applied to individual small proteins built of 70 amino acids

The proteins composed of short polypeptides (about 70 amino acid residues) representing the following functional groups (according to PDB notation): growth hormones, serine protease inhibitors, antifreeze proteins, chaperones and proteins of unknown function, were selected for structural and functional analysis. Classification based on the distribution of hydrophobicity in terms of deficiency/excess as the measure of structural and functional specificity is presented. The experimentally observed distribution of hydrophobicity in the protein body is compared to the idealized one expressed by a three-dimensional Gauss function. The differences between these two distributions reveal the specificity of structural/functional characteristics of the protein. The residues of hydrophobicity deficiency versus the idealized distribution are assumed to indicate cavities with the potential to bind ligands, while the residues of hydrophobicity excess are interpreted as potentially participating in protein-protein complexation. The distribution of hydrophobicity irregularity seems to be specific for particular structures and functions of proteins. A comparative analysis of such profiles is carried out to identify the potential biological activity of proteins of unknown function.     

Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.     

Collagen fiber orientation at the tendon to bone insertion and its influence on stress concentrations

The tendon to bone insertion serves the mechanical role of transferring loads from a relatively compliant tendon to a relatively rigid bone. The details of the mechanism of load transfer are of great importance, since current surgical procedures for tendon reattachment have high failure rates. We hypothesized that the microscopic structure of the insertion is optimized to minimize stress concentrations associated with this load transfer. To explore this, collagen fiber orientation distributions were measured in the supraspinatus tendons of rats. The angular deviation of fibers was fairly uniform across the insertion, and the mean angles of the local distributions deviated mildly from the tendon axis. To explore how these observed property distributions could influence load transfer, these distributions were used to derive material properties for an idealized two-dimensional mechanical model of an insertion. Comparison between stress concentrations in this idealized model and those in three comparison models suggests that the microstructure serves to (1) simultaneously reduce stress concentrations and material mass, and (2) shield the insertion's outward splay from the highest stresses.     

Model testing, prediction and experimental protocols in neuroscience: A case study

In their theoretical and experimental reflections on the capacities and behaviours of living systems, neuroscientists often formulate generalizations about the behaviour of neural circuits. These generalizations are highly idealized, as they omit reference to the myriads of conditions that could perturb the behaviour of the modelled system in real-world settings. This article analyses an experimental investigation of the behaviour of place cells in the rat hippocampus, in which highly idealized generalizations were tested by comparing predictions flowing from them with real-world experimental results. The aim of the article is to identify (1) under what conditions even single prediction failures regarding the behaviour of single cells sufficed to reject highly idealized generalizations, and (2) under what conditions prima facie counter-examples were deemed to be irrelevant to the testing of highly idealized generalizations. The results of this analysis may contribute to understanding how idealized models are tested experimentally in neuroscience and used to make reliable predictions concerning living systems in real-world settings.     

Examination of continuum and micro-structural properties of human vertebral cancellous bone using combined cellular solid models

Two- and three-dimensional structural models of the vertebral body have been used to estimate the mechanical importance of parameters that are difficult to quantify experimentally such as lattice disorder, trabecular thickness, trabecular spacing, connectivity, and fabric. Many of the models that investigate structure–function relationships of the vertebral body focus only on the trabecular architecture and neglect solid–fluid interactions. We developed a cellular solid model composed of two idealized unit cell geometries to investigate the continuum and micro-structural properties of human vertebral cancellous bone in a mathematically tractable model. Using existing histomorphological data we developed structure–function relationships for the mechanical properties of the solid phase, estimated the micro-structural strains, and predicted the fluid flow characteristics. We found that the micro-structural strains may be 1.7 to 2.2 times higher than the continuum level strains between the ages of 40 and 80. In addition, the predicted permeability agrees well with the experimental data.     

Nucleation and decay initiation are the stiffness-sensitive phases of focal adhesion maturation

A cell plated on a two-dimensional substrate forms adhesions with that surface. These adhesions, which consist of aggregates of various proteins, are thought to be important in mechanosensation, the process by which the cell senses and responds to the mechanical properties of the substrate (e.g., stiffness). On the basis of experimental measurements, we model these proteins as idealized molecules that can bind to the substrate in a strain-dependent manner and can undergo a force-dependent state transition. The model forms molecular aggregates that are similar to adhesions. Substrate stiffness affects whether a simulated adhesion is initially formed and how long it grows, but not how that adhesion grows or shrinks. Our own experimental tests support these predictions, suggesting that the mechanosensitivity of adhesions is an emergent property of a simple molecular-mechanical system.     

Prolegomena to a joint model of the contractile system of striated muscle

On the basis of some known properties of the contractile apparatus of muscle and, employing speculative premises, a mathematical model was formulated which makes it possible to express the relations between some macroscopic phenomena of muscle contraction and activities of idealized molecular generators of force quantitatively. With the exception of the sliding filament theory, which is generally accepted, no other published models have, as yet, been taken into consideration.     

Theoretical determination of the CD of proteins containing closely packed antiparallel beta-sheets

Proteins containing two closely packed β-sheets comprise an important class of biopolymers. Rotational and dipole strengths have been determined by the excition coupling model for interacting pairs of two idealized flat β-sheets and for the double β-sheets of seven globular proteins: plastocyanin, human prealbumin, immunoglobulin V_REI, concanavalin A, Cu,Zn superoxide dismutase, staphylococcal nuclease, and elastase. The effects of various geometrical factors on the CD spectra were investigated. Results for the idealized sheets indicate that two sheets display through-space interactions that are large at distances of 5–7 Å and remain significant even at distances typical of the intersheet separations in globular proteins (12–15 Å). The CD spectra are sensitive to the angle (Ω) between the strand directions of the two sheets, with maximum intersheet contributions at Ω = ±45°. Both the intrastrand and interstrand twisting were determined in the seven proteins, and their effects on the calculated CD are discussed. This work represents the first theoretical CD study on the interactions of two regular protein secondary structures, including rotational strength calculations on large sections (up to 135 residues) of globular proteins.     

Design of stable alpha-helical arrays from an idealized TPR motif

The tetratricopeptide repeat (TPR) is a 34-amino acid alpha-helical motif that occurs in over 300 different proteins. In the different proteins, three to sixteen or more TPR motifs occur in tandem arrays and function to mediate protein-protein interactions. The binding specificity of each TPR protein is different, although the underlying structural motif is the same. Here we describe a statistical approach to the design of an idealized TPR motif. We present the high-resolution X-ray crystal structures (to 1.55 and 1.6 A) of designed TPR proteins and describe their solution properties and stability. A detailed analysis of these structures provides an understanding of the TPR motif, how it is repeated to give helical arrays with different superhelical twists, and how a very stable framework may be constructed for future functional designs.     

Dynamic Monte Carlo simulations of a new lattice model of globular protein folding, structure and dynamics

A long-standing problem of molecular biology is the prediction of globular protein tertiary structure from the primary sequence. In the context of a new, 24-nearest-neighbor lattice model of proteins that includes both alpha and beta-carbon atoms, the requirements for folding to a unique four-member beta-barrel, four-helix bundles and a model alpha/beta-bundle have been explored. A number of distinct situations are examined, but the common requirements for the formation of a unique native conformation are tertiary interactions plus the presence of relatively small (but not irrelevant) intrinsic turn preferences that select out the native conformer from a manifold of compact states. When side-chains are explicitly included, there are many conformations having the same or a slightly greater number of side-chain contacts as in the native conformation, and it is the local intrinsic turn preferences that produce the conformational selectivity on collapse. The local preference for helix or beta-sheet secondary structure may be at odds with the secondary structure ultimately found in the native conformation. The requisite intrinsic turn populations are about 0.3% for beta-proteins, 2% for mixed alpha/beta-proteins and 6% for helix bundles. In addition, an idealized model of an allosteric conformational transition has been examined. Folding occurs predominantly by a sequential on-site assembly mechanism with folding initiating either at a turn or from an isolated helix or beta-strand (where appropriate). For helical and beta-protein models, similar folding pathways were obtained in diamond lattice simulations, using an entirely different set of local Monte Carlo moves. This argues strongly that the results are universal; that is, they are independent of lattice, protein model or the particular realization of Monte Carlo dynamics. Overall, these simulations demonstrate that the folding of all known protein motifs can be achieved in the context of a single class of lattice models that includes realistic backbone structures and idealized side-chains.     

Sequence-structure-function relation characterized in silico

Methods for biological function recognition in silico appeared to be useful also for identifying characteristics of structure-to-function relations. The introduction of a three-dimensional Gauss function was assumed to represent the hydrophobic core in a protein molecule. The discrepancy between idealized "fuzzy oil-drop" and the observed one in real proteins appeared to be localized in the ligation site or in the area of biological function related part of protein molecule. The examples of proteins presented in this paper reveal that the structure-function relation can be evaluated and characterized also using the profile of the difference in value between idealized and real hydrophobicity distribution along the polypeptide chain. The specificity of particular polypeptide chain fragments in respect to their biological function and their specific participation in active site creation is discussed in this paper. The scale allowing comparison of different proteins in respect to their ligand-binding sites characteristics is introduced.