Introduction to linear and nonlinear programming

The term molecular mechanics (MM) refers to the use of simple potential-energy functions (e.g., harmonic oscillator or Coulombic potentials) to model molecular systems. Molecular mechanics approaches are widely applied in molecular structure refinement, molecular dynamics (MD) simulations, Monte Carlo (MC) simulations, and ligand-docking simulations.

Typically, molecular mechanics models consist of spherical atoms connected by springs which represent bonds. Internal forces experienced in the model structure are described using simple mathematical functions. For example, Hooke’s law is commonly used to describe bonded interactions, and the nonbonded atoms might be treated as inelastic hard spheres or may interact according to a Lennard-Jones potential. Using these simple models, a molecular dynamics simulation numerically solves Newton’s equations of motion, thus allowing structural fluctuations to be observed with respect to time.

Dynamic simulation methods are widely used to obtain information on the time evolution of conformations of proteins and other biological macromolecules1–4 and also kinetic and thermodynamic information. Simulations can provide fine detail concerning the motions of individual particles as a function of time. They can be utilized to quantify the properties of a system at a precision and on a time scale that is otherwise inaccessible, and simulation is, therefore, a valuable tool in extending our understanding of model systems. Theoretical consideration of a system additionally allows one to investigate the specific contributions to a property through “computational alchemy”,5 that is, modifying the simulation in a way that is nonphysical but nonetheless allows a model’s characteristics to be probed. One particular example is the artificial conversion of the energy function from that representing one system to that of another during a simulation. This is an important technique in free-energy calculations.6 Thus, molecular dynamics simulations, along with a range of complementary computational approaches, have become valuable tools for investigating the basis of protein structure and function.

This review offers an outline of the origin of molecular dynamics simulation for protein systems and how it has developed into a robust and trusted tool. This review then covers more recent advances in theory and an illustrative selection of practical studies in which it played a central role. The range of studies in which MD has played a considerable or pivotal role is immense, and this review cannot do justice to them; MD simulations of biomedical importance were recently reviewed.4 Particular emphasis will be placed on the study of dynamic aspects of protein recognition, an area where molecular dynamics has scope to provide broad and far-ranging insights. This review concludes with a brief discussion of the future potential offered to advancement of the biological and biochemical sciences and the remaining issues that must be overcome to allow the full extent of this potential to be realized.

1.1. Historical Background
MD methods were originally conceived within the theoretical physics community during the 1950s. In 1957, Alder and Wainwright7 performed the earliest MD simulation using the so-called hard-sphere model, in which the atoms interacted only through perfect collisions. Rahman8 subsequently applied a smooth, continuous potential to mimic real atomic interactions. During the 1970s, as computers became more widespread, MD simulations were developed for more complex systems, culminating in 1976 with the first simulation of a protein9,10 using an empirical energy function constructed using physics-based first-principles assumptions. MD simulations are now widely and routinely applied and especially popular in the fields of materials science11,12 and biophysics.

As will be discussed later in this review, a variety of experimental conditions may be simulated with modern theories and algorithms. The initial simulations only considered single molecules in vacuo. Over time, more realistic or at least biologically relevant simulations could be performed. This trend is continuing today.

The initial protein MD simulation, of the small bovine pancreatic trypsin inhibitor (BPTI), covered only 9.2 ps of simulation time. Modern simulations routinely have so-called equilibration periods much longer than that, and production simulations of tens of nanoseconds are routine, with the first microsecond MD simulation being reported in 1998.13 In addition, the original BPTI simulation included only about 500 atoms rather than the 104-106 atoms that are common today. While much of this advancement results from an immense increase in availability of computing power, major theoretical and methodological developments also contribute significantly.

The number of publications regarding MD theory and application of MD to biological systems is growing at an extraordinary pace. A single review cannot do justice to the recent applications of MD. Using data from ISI Web of Science, the authors estimate that during 2005 at least 800 articles will be published that discuss molecular dynamics and proteins. The historical counts are shown in Figure 1.






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Figure 1


Articles matching ISI Web of Science query “TS=(protein) AND TS=(molecular dynamics)”.

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Molecular dynamics: survey of methods for simulating the activity of proteins.

Surface infrared spectroscopy

We have investigated the sources of error in bond lengths and dissociation energies computed from ab initio effective potentials derived from Phillips–Kleinman type pseudo‐orbitals. We propose an alternate pseudo‐orbital, effective potential treatment with the primary objective of agreement with all‐electron molecular calculations. This new treatment forces the pseudo‐orbitals to match precisely the Hartree–Fock orbitals in the valence region and thereby eliminates the major cause of error in the earlier calculations. Effective core potentials derived from these revised pseudo‐orbitals were used to compute potential energy curves for the ground states of F2, Cl2, and LiCl and the results are compared with previous all‐electron and effective potential calculations. Our effective potentials yield dissociation energies and bond lengths which are in excellent agreement with the all‐electron values. Furthermore, in contrast to other procedures, our revised effective potentials result in an excellent descriptio...

/pdf/improved-ab-initio-effective-core-potentials-for-molecular-205idlr66v.pdf

IMPROVED Ab Initio EFFECTIVE CORE POTENTIALS FOR MOLECULAR CALCULATIONS

The atomic structure of the Si(001) surface has been examined with use of scanning tunneling microscopy (STM). The STM images reveal a dimer-type reconstruction and are inconsistent with chain and vacancy models. Both buckled and nonbuckled dimers are observed, giving rise to regions of (2×1), c(4×2), and p (2×2) symmetry. The surface has a high density of vacancy-type defects, which appear to induce or stabilize buckling of the dimers at room temperature. The STM images also reveal the atomic structure at steps and defects. At high annealing temperature the step density increases dramatically, eventually leading to faceting.

Scanning tunneling microscopy of Si(001).

Theoretical cluster calculations for the Si(100) surface show that the use of doubly occupied orbital wavefunctions, such as the closed‐shell Hartree–Fock (HF), lead to an asymmetric dimer description of the surface. The inclusion of electron correlation produces a symmetric dimer description with a ground state ∠1.0 eV below the minimum of the HF buckled dimer. There are two low‐lying states of the symmetric dimer (a singlet and a triplet) with very different geometries. Energy minimization calculations indicate that a (2×1) structure is favored over a c(2×2) structure. We also report ionization potentials for surface and Si(2p) core electrons that are consistent with current experimental data.

/pdf/electronic-correlation-and-the-si-100-surface-buckling-36ve24qcvu.pdf

Electronic correlation and the Si(100) surface: Buckling versus nonbuckling

We have carried out theoretical studies (generalized valence bond) for chemisorbed O atom and O2 molecule on Si(111) surfaces using clusters of atoms to model the surface. For the perfect surface we find that O2 molecule binds to a single surface Si with an Si–O bond length of 1.68 A (bulk SiO2RSi–O = 1.61 A), an Si–O–O angle of 116 °, and an O–O bond distance of 1.32 A. We have also calculated the shifts in the Si(2p) and O(1s) core level energies for chemisorbed O and O2 on the surface. The Si(2p) shift is +1.1 eV (higher binding energy) for the O atom and +1.5 eV for O2, in agreement with the shifts observed for low oxygen exposure by Spicer and co‐workers.

/pdf/oxidation-of-silicon-surfaces-5evnwpqxwe.pdf

Oxidation of silicon surfaces

Using ab initio quantum chemical methods (generalized valence bond), we examine (i) the electronic states of Si (111) and GaAs (110) surface, (ii) the relaxation of the Si (111) surface, (iii) the reconstruction of the GaAs surface, (iv) the initial steps in the chemisorption of O_2 on Si (111), and (v) the bonding of O atom to Ga and As centers.

/pdf/theoretical-studies-of-si-and-gaas-surfaces-and-initial-3xp3ak5jmv.pdf

Antonio Redondo

Papers

Electronic correlation and the Si(100) surface: Buckling versus nonbuckling

Oxidation of silicon surfaces

Theoretical studies of Si and GaAs surfaces and initial steps in the oxidation

The peroxy radical model for the chemisorption of O2 onto silicon surfaces

Ab initio effective potentials for silicon