Electrostatics in computational protein design

Catalytic activity and protein-protein recognition have proven to be significant challenges for computational protein design. Electrostatic interactions are crucial for these and other protein functions, and therefore accurate modeling of electrostatics is necessary for successfully advancing protein design into the realm of protein function. This review focuses on recent progress in modeling electrostatic interactions in computational protein design, with particular emphasis on continuum models.     

A continuum method for determining membrane protein insertion energies and the problem of charged residues

Continuum electrostatic approaches have been extremely successful at describing the charged nature of soluble proteins and how they interact with binding partners. However, it is unclear whether continuum methods can be used to quantitatively understand the energetics of membrane protein insertion and stability. Recent translation experiments suggest that the energy required to insert charged peptides into membranes is much smaller than predicted by present continuum theories. Atomistic simulations have pointed to bilayer inhomogeneity and membrane deformation around buried charged groups as two critical features that are neglected in simpler models. Here, we develop a fully continuum method that circumvents both of these shortcomings by using elasticity theory to determine the shape of the deformed membrane and then subsequently uses this shape to carry out continuum electrostatics calculations. Our method does an excellent job of quantitatively matching results from detailed molecular dynamics simulations at a tiny fraction of the computational cost. We expect that this method will be ideal for studying large membrane protein complexes.     

Sampling rate effects on measurements of correlated and biased random walks

When observing the two-dimensional movement of animals or microorganisms, it is usually necessary to impose a fixed sampling rate, so that observations are made at certain fixed intervals of time and the trajectory is split into a set of discrete steps. A sampling rate that is too small will result in information about the original path and correlation being lost. If random walk models are to be used to predict movement patterns or to estimate parameters to be used in continuum models, then it is essential to be able to quantify and understand the effect of the sampling rate imposed by the observer on real trajectories. We use a velocity jump process with a realistic reorientation model to simulate correlated and biased random walks and investigate the effect of sampling rate on the observed angular deviation, apparent speed and mean turning angle. We discuss a method of estimating the values of the reorientation parameters used in the original random walk from the rediscretized data that assumes a linear relation between sampling time step and the parameter values.     

Microscopic and semimacroscopic redox calculations: what can and cannot be learned from continuum models

 The major role of electrostatic effects in the control of redox potentials in proteins is now widely appreciated. However, the evaluation and conceptualization of the actual electrostatic contributions is far from trivial, and some models still overlook the nature of electrostatic effects in proteins. This commentary considers different contributions to redox potentials of proteins and discusses the ability of different models to capture these contributions. It is pointed out that macroscopic models which consider the protein as a medium of uniform low dielectric constant cannot reproduce the proper physics of redox proteins. In particular, it is pointed out that the crucial effects of the protein permanent dipoles must be taken into account explicitly and that these permanent dipoles involve effective dielectric constants that are different from those for ionized residues. It is also argued that the reorganization of the protein upon change of oxidation states or ionization of protein residues should be taken into account in redox calculations. The role of water penetration and the inadequacy of describing electrostatic effects by solvent accessibility is briefly mentioned. The nature and the meaning of the "dielectric constant" that should be used in redox calculations are also discussed. It is pointed out that the "dielectric constant" ε_p used in current discretized continuum (DC) models is simply a representation of the contributions which are treated implicitly and not the proper dielectric constant of the protein. It is then explained that the need to use a large "dielectric constant" in DC models reflects, among other factors, the implicit representation of the reorganization of permanent dipoles, and that even an explicit treatment of the fluctuations of ionized surface residues will lead to incorrect results when one uses ε_p=εˉ in continuum treatments. Finally, it is suggested that although the discussion and classification of different contributions to redox potentials is very useful, only the evaluation of the totality of the protein contributions (rather than an arbitrary subset) can lead to a quantitative understanding of redox proteins. Received, accepted: 26 November 1996     

A Continuum Three-Zone Model for Swarms

We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popular biological models of aggregations. In continuum models, individuals are replaced with density and velocity functions. Individual behavior is modeled with convolutions acting within three interaction zones corresponding to repulsion, orientation, and attraction, respectively. We begin with a variable-speed first-order model in which the velocity depends directly on the interactions. Next, we present a variable-speed second-order model. Finally, we present a constant-speed second-order model that is coordinated with popular individual-based models. For all three models, linear stability analysis shows that the growth or decay of perturbations in an infinite, uniform swarm depends on the strength of attraction relative to repulsion and orientation. We verify that the continuum models predict the behavior of a swarm of individuals by comparing the linear stability results with an individual-based model that uses the same social interaction kernels. In some unstable regimes, we observe that the uniform state will evolve toward a radially symmetric attractor with a variable density. In other unstable regimes, we observe an incoherent swarming state.     

Experimental and theoretical studies of mechanical unfolding of different proteins

Mechanical properties of proteins are important for a wide range of biological processes including cell adhesion, muscle contraction, and protein translocation across biological membranes. It is necessary to reveal how proteins achieve their required mechanical stability under natural conditions in order to understand the biological processes and also to use the knowledge for constructing new biomaterials for medical and industrial purposes. In this connection, it is important to know how a protein will behave in response to various impacts. Theoretical and experimental works on mechanical unfolding of globular proteins will be considered in detail in this review.     

Transport rates of proteins in porous materials with known microgeometry.

Many biological and biotechnological systems involve the diffusion of macromolecules through complicated macroporous (pore size on the order of 10-100 microns) environments. In this report, we present and evaluate an experimental system for measuring the rate of protein transport in an inert, macroporous membrane. For this particular membrane system, the microgeometry was characterized in terms of distribution of pore size, position, and orientation. Although the rate of protein desorption was much less than expected based on continuum diffusion models, we demonstrate that the measured transport rates are consistent with diffusion of protein in a complex, interconnected network of water-filled pores. The porous systems exhibit transitional behavior in quantitative agreement with the behavior of percolation lattices (mean square error 7%, n = 29). Predictive mathematical models of the diffusion process were developed: these models used percolation concepts to describe pore topology, continuum models of diffusion/dissolution to describe protein movement at each single pore, and measured pore size distributions. Effective diffusion coefficients for protein transport in aqueous, constricted macropores were predicted by this technique. Predicted diffusion coefficients, based on measured and derived microstructural parameters, agree with experimentally measured diffusion coefficients within a factor of 2. This approach may be useful in the design of porous polymer systems for biological applications and for evaluating other biological systems where conduction of mass, heat, momentum, or charge occurs in a heterogeneous environment.     

Protocol for MM/PBSA molecular dynamics simulations of proteins

Continuum solvent models have been employed in past years for understanding processes such as protein folding or biomolecular association. In the last decade, several attempts have been made to merge atomic detail molecular dynamics simulations with solvent continuum models. Among continuum models, the Poisson-Boltzmann solvent accessible surface area model is one of the oldest and most fundamental. Notwithstanding its wide usage for simulation of biomolecular electrostatic potential, the Poisson-Boltzmann equation has been very seldom used to obtain solvation forces for molecular dynamics simulation. We propose here a fast and reliable methodology to implement continuum forces in standard molecular mechanics and dynamics algorithms. Results for a totally unrestrained 1 ns molecular dynamics simulation of a small protein are quantitatively similar to results obtained by explicit solvent molecular dynamics simulations.     

子宫内膜异位症动物模型研究进展

为研究子宫内膜异位症的病因、发病机制、治疗和预防对策,需要构建实验动物模型。本文通过分析各种模型的优点和缺点,为选择性应用动物模型提供指导,从而更利于对该病的研究。     

Aptamers as therapeutic and diagnostic agents

Aptamers are oligonucleotides derived from an in vitro evolution process called SELEX. Aptamers have been evolved to bind proteins which are associated with a number of disease states. Using this method, many powerful antagonists of such proteins have been found. In order for these antagonists to work in animal models of disease and in humans, it is necessary to modify the aptamers. First of all, sugar modifications of nucleoside triphosphates are necessary to render the resulting aptamers resistant to nucleases found in serum. Changing the 2'OH groups of ribose to 2'F or 2'NH2 groups yields aptamers which are long lived in blood. The relatively low molecular weight of aptamers (8000-12000) leads to rapid clearance from the blood. Aptamers can be kept in the circulation from hours to days by conjugating them to higher molecular weight vehicles. When modified, conjugated aptamers are injected into animals, they inhibit physiological functions known to be associated with their target proteins. A new approach to diagnostics is also described. Aptamer arrays on solid surfaces will become available rapidly because the SELEX protocol has been successfully automated. The use of photo-cross-linkable aptamers will allow the covalent attachment of aptamers to their cognate proteins, with very low backgrounds from other proteins in body fluids. Finally, protein staining with any reagent which distinguishes functional groups of amino acids from those of nucleic acids (and the solid support) will give a direct readout of proteins on the solid support.     

Continuum secondary structure captures protein flexibility

The DSSP program assigns protein secondary structure to one of eight states. This discrete assignment cannot describe the continuum of thermal fluctuations. Hence, a continuous assignment is proposed. Technically, the continuum results from averaging over ten discrete DSSP assignments with different hydrogen bond thresholds. The final continuous assignment for a single NMR model successfully reflected the structural variations observed between all NMR models in the ensemble. The structural variations between NMR models were verified to correlate with thermal motion; these variations were captured by the continuous assignments. Because the continuous assignment reproduces the structural variation between many NMR models from one single model, functionally important variation can be extracted from a single X-ray structure. Thus, continuous assignments of secondary structure may affect future protein structure analysis, comparison, and prediction.     

An experimental and computational investigation of the post-yield behaviour of trabecular bone during vertebral device subsidence

Interbody fusion device subsidence has been reported clinically. An enhanced understanding of the mechanical behaviour of the surrounding bone would allow for accurate predictions of vertebral subsidence. The multiaxial inelastic behaviour of trabecular bone is investigated at a microscale and macroscale level. The post-yield behaviour of trabecular bone under hydrostatic and confined compression is investigated using microcomputed tomography-derived microstructural models, elucidating a mechanism of pressure-dependent yielding at the macroscopic level. Specifically, microstructural trabecular simulations predict a distinctive yield point in the apparent stress–strain curve under uniaxial, confined and hydrostatic compression. Such distinctive apparent stress–strain behaviour results from localised stress concentrations and material yielding in the trabecular microstructure. This phenomenon is shown to be independent of the plasticity formulation employed at a trabecular level. The distinctive response can be accurately captured by a continuum model using a crushable foam plasticity formulation in which pressure-dependent yielding occurs. Vertebral device subsidence experiments are also performed, providing measurements of the trabecular plastic zone. It is demonstrated that a pressure-dependent plasticity formulation must be used for continuum level macroscale models of trabecular bone in order to replicate the experimental observations, further supporting the microscale investigations. Using a crushable foam plasticity formulation in the simulation of vertebral subsidence, it is shown that the predicted subsidence force and plastic zone size correspond closely with the experimental measurements. In contrast, the use of von Mises, Drucker–Prager and Hill plasticity formulations for continuum trabecular bone models lead to over prediction of the subsidence force and plastic zone.     

Computational sidechain placement and protein mutagenesis with implicit solvent models

Structure prediction and computational protein design should benefit from accurate solvent models. We have applied implicit solvent models to two problems that are central to this area. First, we performed sidechain placement for 29 proteins, using a solvent model that combines a screened Coulomb term with an Accessible Surface Area term (CASA model). With optimized parameters, the prediction quality is comparable with earlier work that omitted electrostatics and solvation altogether. Second, we computed the stability changes associated with point mutations involving ionized sidechains. For over 1000 mutations, including many fully or partly buried positions, we compared CASA and two generalized Born models (GB) with a more accurate model, which solves the Poisson equation of continuum electrostatics numerically. CASA predicts the correct sign and order of magnitude of the stability change for 81% of the mutations, compared to 97% with the best GB. We also considered 140 mutations for which experimental data are available. Comparing to experiment requires additional assumptions about the unfolded protein structure, protein relaxation in response to the mutations, and contributions from the hydrophobic effect. With a simple, commonly-used unfolded state model, the mean unsigned error is 2.1 kcal/mol with both CASA and the best GB. Overall, the electrostatic model is not important for sidechain placement; CASA and GB are equivalent for surface mutations, while GB is far superior for fully or partly buried positions. Thus, for problems like protein design that involve all these aspects, the most recent GB models represent an important step forward. Along with the recent discovery of efficient, pairwise implementations of GB, this will open new possibilities for the computational engineering of proteins.     

From continuum Fokker-Planck models to discrete kinetic models

Two theoretical formalisms are widely used in modeling mechanochemical systems such as protein motors: continuum Fokker-Planck models and discrete kinetic models. Both have advantages and disadvantages. Here we present a "finite volume" procedure to solve Fokker-Planck equations. The procedure relates the continuum equations to a discrete mechanochemical kinetic model while retaining many of the features of the continuum formulation. The resulting numerical algorithm is a generalization of the algorithm developed previously by Fricks, Wang, and Elston through relaxing the local linearization approximation of the potential functions, and a more accurate treatment of chemical transitions. The new algorithm dramatically reduces the number of numerical cells required for a prescribed accuracy. The kinetic models constructed in this fashion retain some features of the continuum potentials, so that the algorithm provides a systematic and consistent treatment of mechanical-chemical responses such as load-velocity relations, which are difficult to capture with a priori kinetic models. Several numerical examples are given to illustrate the performance of the method.     

The discrete nature of trabecular bone microarchitecture affects implant stability

Small endosseous implants, such as screws, are important components of modern orthopedics and dentistry. Hence they have to reliably fulfill a variety of requirements, which makes the development of such implants challenging. Finite element analysis is a widely used computational tool used to analyze and optimize implant stability in bone. For these purposes, bone is generally modeled as a continuum material. However, bone failure and bone adaptation processes are occurring at the discrete level of individual trabeculae; hence the assessment of stresses and strains at this level is relevant. Therefore, the aim of the present study was to investigate how peri-implant strain distribution and load transfer between implant and bone are affected by the continuum assumption. We performed a computational study in which cancellous screws were inserted in continuum and discrete models of trabecular bone; axial loading was simulated. We found strong differences in bone-implant stiffness between the discrete and continuum bone model. They depended on bone density and applied boundary conditions. Furthermore, load transfer from the screw to the surrounding bone differed strongly between the continuum and discrete models, especially for low-density bone. Based on our findings we conclude that continuum bone models are of limited use for finite element analysis of peri-implant mechanical loading in trabecular bone when a precise quantification of peri-implant stresses and strains is required. Therefore, for the assessment and improvement of trabecular bone implants, finite element models which accurately represent trabecular microarchitecture should be used.     

Adaptation de l’irradiation à l’activité tumorale en radiothérapie conformationnelle avec modulation d’intensité pour les cancers tête et cou. Étude préliminaire sur fantômes

The aim of this study was to index delivered doses of irradiation to tumor activity and not only to tumor geometry. The elaboration of a new highly precise treatment protocol, based on tumor activity required a specific configuration in order to use this type of irradiation. Using a Treatment Planning System and two head and neck phantoms specially created, we have performed system characterization according to different treatment plans. Two models were created and used: a simplistic and an anatomical model. Our results showed that high-precision radiotherapy in limited zones is possible with intensity modulated radiation therapy when several conditions are met such as location, number of organs at risk, distance between Planning Target Volume and organs at risk, presence, volume and location of the tumor activity, number of fields. In order to use this irradiation method adapted to the tumor activity, a precise geometry will be necessary. However such high total and fractionated doses should be carefully evaluated before being prescribed clinically.     

The hierarchical continuum concept

Abstract. Two general models have been proposed to explain the structure of the plant community: the community-unit model of Clements and the continuum model of Whittaker and Curtis, the latter based on Gleason's individualistic distribution of species. It is generally assumed that most ecologists now accept the continuum model. Empirical evidence suggests, however, that the continuum in its current form does not fully describe the observed patterns of vegetation along environmental gradients. In this paper, we introduce the hierarchical continuum as a general concept to represent dynamic community structure along regional spatial gradients. The hierarchical continuum is derived from a combination of the individualistic distribution of species, hierarchical assemblage structure, and the core-satellite species hypothesis. The hierarchical continuum concept predicts that the distribution of species across sites in a region will be polymodal, which reflects hierarchical structure, and that the distribution and abundance of species within and between sites will be spatially and temporally dynamic. Regional distribution of plant species in North American tallgrass prairie, southeastern flood-plain hardwood forests, northern upland hardwood forests, and boreal forests were either bimodal or polymodal as predicted by the hierarchical continuum concept. Species in tallgrass prairie were spatially and temporally dynamic with an average turnover of 8–9 species per 50 m² yr¹. In addition, the hierarchical continuum concept predicts the potential for fractal (self-similar) patterns of community structure, and provides a framework for testable hypotheses concerning species distributions along environmental gradients.     

Using molecular dynamics simulations on crambin to evaluate the suitability of different continuum dielectric and hydrogen atom models for protein simulations

Molecular dynamics simulations of enzymes with enough explicit waters of solvation to realistically account for solute-solvent interactions can burden the computational resources required to perform the simulation by more than two orders of magnitude. Since enzyme simulations even with an implicit solvation model can be imposing for a supercomputer, it is important to assess the suitability of different continuum dielectric models for protein simulations. A series of 100-picosecond molecular dynamics simulations were performed on the X-ray crystal structure of the protein crambin to examine how well computed structures, obtained using seven continuum dielectric and two hydrogen atom models, agreed with the X-ray structure. The best level of agreement between computed and experimental structures was obtained using a constant dielectric of 2 and the all-hydrogen model. Continuum dielectric models of 1, 1r, and 2r also led to computed structures in reasonably good agreement with the X-ray structure. In all cases, the all-hydrogen model gave better agreement than the united atom model, although, in one case, the difference was not significant. Dielectric models of 4, 80, and 4r with either hydrogen model yielded significantly poorer fits. It is especially noteworthy that the observed trends did not semiquantitatively converge until about 50 picoseconds into the simulations, suggesting that validation studies for protein calculations based on energy minimizations or short simulations should be viewed with caution.     

Predictive ecology: systems approaches

The world is experiencing significant, largely anthropogenically induced, environmental change. This will impact on the biological world and we need to be able to forecast its effects. In order to produce such forecasts, ecology needs to become more predictive--to develop the ability to understand how ecological systems will behave in future, changed, conditions. Further development of process-based models is required to allow such predictions to be made. Critical to the development of such models will be achieving a balance between the brute-force approach that naively attempts to include everything, and over simplification that throws out important heterogeneities at various levels. Central to this will be the recognition that individuals are the elementary particles of all ecological systems. As such it will be necessary to understand the effect of evolution on ecological systems, particularly when exposed to environmental change. However, insights from evolutionary biology will help the development of models even when data may be sparse. Process-based models are more common, and are used for forecasting, in other disciplines, e.g. climatology and molecular systems biology. Tools and techniques developed in these endeavours can be appropriated into ecological modelling, but it will also be necessary to develop the science of ecoinformatics along with approaches specific to ecological problems. The impetus for this effort should come from the demand coming from society to understand the effects of environmental change on the world and what might be performed to mitigate or adapt to them.