## Algorithmic challenges in computational molecular biophysics (1999)

Venue: | Journal of Computational Physics |

Citations: | 33 - 3 self |

### BibTeX

@ARTICLE{Schlick99algorithmicchallenges,

author = {Tamar Schlick and Robert D. Skeel and Axel T. Brunger and Laxmikant V. Kalé and John A. Board and Jan Hermans and Klaus Schulten},

title = {Algorithmic challenges in computational molecular biophysics},

journal = {Journal of Computational Physics},

year = {1999},

volume = {151},

pages = {9--48}

}

### OpenURL

### Abstract

A perspective of biomolecular simulations today is given, with illustrative applications and an emphasis on algorithmic challenges, as reflected by the work of a multidisciplinary team of investigators from five institutions. Included are overviews and recent descriptions of algorithmic work in long-time integration for molecular dynamics; fast electrostatic evaluation; crystallographic refinement approaches; and implementation of large, computation-intensive programs on modern architectures. Expected future developments of the field are also discussed. c ○ 1999 Academic Press Key Words: biomolecular simulations; molecular dynamics; long-time integration; fast electrostatics; crystallographic refinement; high-performance platforms.

### Citations

3888 | Optimization by simulated annealing
- Kirkpatrick, Gelatt, et al.
- 1983
(Show Context)
Citation Context ...54]. That is, such gradient descent schemes find a minimum near the starting point and therefore cannot sample molecular conformation space well to find the most optimal solution. Simulated annealing =-=[73]-=- as an optimization technique is thus a viable alternative. Unlike gradient descent methods, simulated annealing can cross barriers between minima and thus explore a greater volume of parameter space ... |

2491 |
Equations of state calculations by fast computing machines
- Metroplis, Rosenbluth, et al.
- 1953
(Show Context)
Citation Context ... move, from one set of parameters to another so as to be consistent with the Boltzmann distribution at the given temperature. The two most widely used generation mechanisms are Metropolis Monte Carlo =-=[94]-=- and molecular dynamics simulations. For X-ray crystallographic refinement, molecular dynamics has proved to be extremely successful [30]. 6.4. Torsion-Angle Molecular Dynamics for Refinement A suitab... |

1515 |
Practical Optimization
- Gill, Murray, et al.
- 1981
(Show Context)
Citation Context ...ing the global minimum cannot be assured by any current method. The existence of many local minima confounds local gradient optimization techniques such as conjugate gradient or least-squares methods =-=[54]-=-. That is, such gradient descent schemes find a minimum near the starting point and therefore cannot sample molecular conformation space well to find the most optimal solution. Simulated annealing [73... |

783 | A fast algorithm for particle simulations
- Greengard, Rokhlin
- 1987
(Show Context)
Citation Context ...strophysical literature for solving the gravitational N-body problem [5, 8]; the FMA of Greengard and Rokhlin appeared in 1987, placing this class of algorithms on a more rigorous theoretical footing =-=[50]-=-. The FMA was soon adapted to the needs of electrostatic particle simulation for MD [23, 118]. The multipole-accelerated algorithms overcome the O(N 2 ) complexity of the N-body problem by using a hie... |

596 |
VMD-visual molecular dynamics
- Humphrey, Dalke, et al.
- 1996
(Show Context)
Citation Context ... the folded domain; surrounding water molecules are not shown. The four figures at extensions 10, 17, 150, and 285 ˚A correspond, respectively, to regions I to IV defined in (a). Figure created using =-=[61]-=-. In the application of SMD to the force-induced unfolding of the immunoglobin domains of titin [88], the starting structure for the SMD simulations was constructed on the basis of a nuclear magnetic ... |

389 | Numerical Integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes
- Ryckaert, Ciccotti, et al.
- 1977
(Show Context)
Citation Context ... the equations of motion via Lagrange multipliers has made possible an increase from 0.5 or 1 fs to around 2 fs, sometimes slightly higher, with modest added cost. The algorithm known widely as SHAKE =-=[109]-=- and its variant RATTLE [3] are implemented in many molecular packages. The Verlet-based constrained MD algorithm becomes the iterative sequence V n+1/2 = V n −1 �t − M 2 ∇ E(X n ) + G(X n ) T λ n X n... |

377 |
Understanding Molecular Simulation from Algorithms to
- Frenkel, Smit
- 2002
(Show Context)
Citation Context ...atic protocols in a protein dynamics application. Future perspectives are described in the final section. Readers are referred to general introductions to molecular and biomolecular modeling found in =-=[2, 24, 26, 43, 52, 82, 95]-=-. 1.1. Interpretation of Experiments Regarding Unfolding of Immunoglobin Domains of a Muscle Protein Molecular recognition and specific ligand/receptor interactions are central to many biochemical pro... |

257 |
Molecular dynamics with coupling to an external
- Berendsen, Postma, et al.
- 1984
(Show Context)
Citation Context ...ange described below has begun but not completed (after 1.5 ns). 4.5. Lysozyme Dynamics Results The 6-ns simulation performed isothermally, with coupling to a heat bath following the Berendsen method =-=[16]-=-, reveals that the protein has locally stable domains connected flexibly [Geoff Mann, University of North Carolina]. Lysozyme consists of three principal domains: an N-terminal lobe, a C-terminal lobe... |

237 |
A smooth particle mesh Ewald method
- Essmann, Perera, et al.
- 1995
(Show Context)
Citation Context ...a more complete derivation of the basic method is given in [124]. 3.4.1. Particle–Mesh Ewald One of the most efficient algorithms known for evaluating the Ewald sum is the PME method of Darden et al. =-=[33, 34]-=-. The use of Ewald’s “trick” of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fou... |

216 |
A hierarchical o(n log n) force-calculation algorithm
- Barnes, Hut
- 1986
(Show Context)
Citation Context ...ir implementation on parallel machines. 3.2. Multipole-Accelerated Methods Multipole-accelerated algorithms first appeared in the astrophysical literature for solving the gravitational N-body problem =-=[5, 8]-=-; the FMA of Greengard and Rokhlin appeared in 1987, placing this class of algorithms on a more rigorous theoretical footing [50]. The FMA was soon adapted to the needs of electrostatic particle simul... |

212 | NAMD2: greater scalability for parallel molecular dynamics
- Kale, Skeel, et al.
- 1999
(Show Context)
Citation Context ...longer times. Such advances may also permit study of the refolding process of the Ig domains. Presently, a simulation of 10 ns on a cluster of eight PC workstations using the simulation program NAMD2 =-=[72]-=- requires 3 weeks.sALGORITHMIC CHALLENGES 13 An alternative solution to overcome the problem described is suggested in another article in this issue [55]. 1.2. Protein/DNA Interactions In addition to ... |

181 |
Molecular Modelling: Principles and Applications
- Leach
- 2001
(Show Context)
Citation Context ...atic protocols in a protein dynamics application. Future perspectives are described in the final section. Readers are referred to general introductions to molecular and biomolecular modeling found in =-=[2, 24, 26, 43, 52, 82, 95]-=-. 1.1. Interpretation of Experiments Regarding Unfolding of Immunoglobin Domains of a Muscle Protein Molecular recognition and specific ligand/receptor interactions are central to many biochemical pro... |

172 |
Computer experiments on classical fluids. I. Thermodynamical properties of Lennard–Jones molecules
- Verlet
- 1967
(Show Context)
Citation Context ...stem. The trajectories generated by Newtonian and Langevin models are different. 2.3. The Leap-frog/Verlet Method and Its Limitations The typical integrator used for MD is the leap-frog/Verlet method =-=[127]-=-. Its generalization to the Langevin equation above is given by the iteration sweep V n+1/2 = V n −1 �t + M 2 [−∇ E(X n ) − γ MV n + R n ] X n+1 = X n + �tV n+1/2 (2.5) V n+1 = V n+1/2 −1 �t + M 2 [−∇... |

160 |
Pathways to a protein folding intermediate observed in a 1-ms simulation in aqueous solution
- Duan, Kollman
- 1998
(Show Context)
Citation Context ... increasing system sizes. Typical simulation times and system sizes are shown in the table, but there are extreme recent examples of unusually long or large simulations (e.g., 1 µs of a small protein =-=[38]-=-, or nearly a million atoms). Thus, the development of more efficient simulation algorithms and computational strategies is continuously motivated by the remaining temporal gap needed to describe prot... |

154 | A parallel hashed oct-tree N-body algorithm", Supercomputing
- Warren, Salmon
- 1993
(Show Context)
Citation Context ... in [80]. The hierarchy of approximations in the multipole-accelerated algorithms leads naturally to the use of hierarchical data structures such as the oct-tree for representing the simulation space =-=[129]-=-. The variants of multipole-accelerated algorithms [20] differ primarily in details of how this tree is traversed and which rules are used to decide when two sets of particles are sufficiently distant... |

151 |
Free R value: a novel statistical quantity for assessing the accuracy of crystal structures. Nature 355:472–474
- Brünger
- 1992
(Show Context)
Citation Context ...IC CHALLENGES 39 reduced by “cross validating” σA values, i.e., by computing them from a randomly selected test set which is excluded from the summation on the right hand-side of Eqs. (6.2) and (6.3) =-=[28, 74]-=-. The expected values of 〈|Fo|〉 and the corresponding variance (σ 2 ML ) are derived from σA, the observed |Fo|, and calculated |Fc| [98]. These quantities can be used to derive a Gaussian approximati... |

148 | An efficient program for many-body simulation
- Appel
- 1985
(Show Context)
Citation Context ...ir implementation on parallel machines. 3.2. Multipole-Accelerated Methods Multipole-accelerated algorithms first appeared in the astrophysical literature for solving the gravitational N-body problem =-=[5, 8]-=-; the FMA of Greengard and Rokhlin appeared in 1987, placing this class of algorithms on a more rigorous theoretical footing [50]. The FMA was soon adapted to the needs of electrostatic particle simul... |

121 |
Interaction models for water in relation to protein hydration
- Berendsen, Postma
- 1981
(Show Context)
Citation Context ...nd-length forces were not calculated, as the bonds were kept at fixed lengths using the SHAKE method. (This is necessary because the empirical SPC water force field had been developed for rigid bonds =-=[17, 56]-=-.) The used MTS scheme results in a roughly equal distribution of the load over four components repeated at successively longer intervals: 2, 6, 12, and 24 fs. The MTS protocol results in a saving of ... |

112 |
Particle mesh ewald: an N log(N) methods for Ewald sums in large systems
- Darden, York, et al.
- 1993
(Show Context)
Citation Context ...a more complete derivation of the basic method is given in [124]. 3.4.1. Particle–Mesh Ewald One of the most efficient algorithms known for evaluating the Ewald sum is the PME method of Darden et al. =-=[33, 34]-=-. The use of Ewald’s “trick” of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fou... |

104 | Rattle: a ‘velocity’ version of the shake algorithm for molecular dynamics calculations
- Andersen
- 1983
(Show Context)
Citation Context ...Lagrange multipliers has made possible an increase from 0.5 or 1 fs to around 2 fs, sometimes slightly higher, with modest added cost. The algorithm known widely as SHAKE [109] and its variant RATTLE =-=[3]-=- are implemented in many molecular packages. The Verlet-based constrained MD algorithm becomes the iterative sequence V n+1/2 = V n −1 �t − M 2 ∇ E(X n ) + G(X n ) T λ n X n+1 = X n + �tV n+1/2 g(X n+... |

103 | Reversible unfolding of individual titin immunoglobulin domains by AFM. Science 276:1109–1112 - Rief, Gautel, et al. - 1997 |

96 |
Reversible multiple time scale molecular dynamics
- Tuckerman, Berne, et al.
- 1990
(Show Context)
Citation Context ...ined dynamics [85] and the impulse–MTS method [18]. 2.7. MTS Schemes and Resonance Problems Symplectic and time-reversible MTS methods for biomolecular dynamics have been described in the early 1990s =-=[47, 126]-=-. A natural splitting for MD simulations is the following three-class hierarchy: bonded forces are considered “fast” and resolved at an inner timestep �τ; forces within a certain cutoff (e.g., 6 ˚A) a... |

85 |
Charm++: Parallel Programming with MessageDriven Objects,” in Parallel Programming using C
- Kale, Krishnan
- 1996
(Show Context)
Citation Context ...ata structures. As a result of all these factors, NAMD performs faster than some of the Fortran-based programs for equivalent tasks. Most of the parallel modules in NAMD are implemented using Charm++ =-=[70]-=-, a C++ based parallel programming library. The parallelization strategy employed in NAMD, as described above, leads to having multiple patches and force objects on each processor. Each of these objec... |

69 |
Die Berechnung optischer und elektrostatischer Giuerpotentiale. A nn
- Ewald
- 1921
(Show Context)
Citation Context ...ional cost of just a few percent over the cost of the multipole methods applied to an original simulation cell of size of order 10 ˚A [81]. 3.4. Ewald Methods The Ewald summation was invented in 1921 =-=[42]-=- to permit the efficient computation of lattice sums arising in solid state physics. The desirability of using periodic boundary conditions in MD simulations leads to use of Ewald methods where the en... |

67 |
Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions. Molecular Simulation
- Grubmüller, Heller, et al.
- 1991
(Show Context)
Citation Context ...ined dynamics [85] and the impulse–MTS method [18]. 2.7. MTS Schemes and Resonance Problems Symplectic and time-reversible MTS methods for biomolecular dynamics have been described in the early 1990s =-=[47, 126]-=-. A natural splitting for MD simulations is the following three-class hierarchy: bonded forces are considered “fast” and resolved at an inner timestep �τ; forces within a certain cutoff (e.g., 6 ˚A) a... |

59 | an interoperable framework for parallel programming
- KALE
- 1996
(Show Context)
Citation Context ...ing threads, while much of the initialization code was implemented using a simple message passing style. Integrating such disparate modules in a single program was a challenge. The Converse framework =-=[71]-=- developed at the University of Illinois is used to support multiple languages and libraries in a single application. Languages and libraries such as Charm++ and PVM are implemented on top of Converse... |

56 |
Computer Simulation Using Particles, McGraw-Hill
- Hockney
- 1989
(Show Context)
Citation Context ...ional manner, evaluating the erfc terms within a cutoff radius. To speed up the Fourier space calculation, PME borrows from the particle–particle, particle–mesh (P 3 M) method of Hockney and Eastwood =-=[59]-=- to interpolate all the randomly spaced particles onto a regular mesh. The Fourier sum is then computed via FFT techniques on the regular mesh and the results are interpolated back to the actual parti... |

55 | Ligand binding and molecular mechanics calculation of the streptavidin-biotin rupture force - Grubmüller, Heymann, et al. - 1996 |

50 | Symplectic numerical integrators in constrained Hamiltonian systems
- Leimkuhler, Skeel
- 1994
(Show Context)
Citation Context ...ysics and mathematics communities spread quickly to the computational chemistry community. It was shown that the popular Verlet method is symplectic, as are the RATTLE scheme for constrained dynamics =-=[85]-=- and the impulse–MTS method [18]. 2.7. MTS Schemes and Resonance Problems Symplectic and time-reversible MTS methods for biomolecular dynamics have been described in the early 1990s [47, 126]. A natur... |

48 |
NAMD—a parallel object-oriented molecular dynamics program
- Nelson, Humphrey, et al.
- 1996
(Show Context)
Citation Context ...: one side of the DNA was identical to the target sequence and the other side was altered, containing two mutations in the sequence (see Fig. 2). The simulations were performed using the program NAMD =-=[97]-=- on the solvated protein/DNA complex involving approximately 36,600 atoms (Fig. 2). The 100-ps simulation required 22 days in 1996 on eight HP 735 workstations. The full electrostatic interactions are... |

47 | Long-time-step methods for oscillatory differential equations
- Garcı́a-Archilla, Sanz-Serna, et al.
- 1999
(Show Context)
Citation Context ... half the fastest period (see Fig. 1 of [89]). The resonant timesteps are method dependent [115]. The instability problem for �t nearly equal to the half-period was later reported for the impulse–MTS =-=[12, 46]-=-. Detailed analyses of resonance artifacts in extrapolative versus impulse–MTS schemes are presented in [12, 110], where hybrid schemes are proposed to balance stability with accuracy. Resonance insta... |

44 |
Molecular dynamics study of unbinding of the avidin-biotin complex
- Izrailev, Stepaniants, et al.
- 1997
(Show Context)
Citation Context ...nitial position in the protein and allowing the ligand to explore new contacts along its unbinding path. SMD has already been successfully applied to studies of the dissociation of biotin from avidin =-=[48, 66]-=-, the unbinding of retinal from bacteriorhodopsin [64] and of retinoic acid from its nuclear hormone receptor [77], and the extraction of lipids from membranes [91, 122]. A review of the method is pro... |

43 | Unfolding of titin immunoglobulin domains by steered molecular dynamics simulation
- Lu, Isralewitz, et al.
- 1998
(Show Context)
Citation Context ..., 150, and 285 ˚A correspond, respectively, to regions I to IV defined in (a). Figure created using [61]. In the application of SMD to the force-induced unfolding of the immunoglobin domains of titin =-=[88]-=-, the starting structure for the SMD simulations was constructed on the basis of a nuclear magnetic resonance (NMR) structure of the Ig domain I27 of the cardiac titin I-band [63]. The 300-ps simulati... |

36 |
Longer time steps for molecular dynamics
- Izaguirre, Reich, et al.
(Show Context)
Citation Context ...for the impulse method, for LongLinearAverage, and for another mollified impulse method Equilibrium that defines A(X) to be a projection onto the equilibrium value of the bond lengths and bond angles =-=[65]-=-. (The pseudo-energy is the actual energy with Eslow(X) replaced with Eslow(A(X)).) The energy rises rapidly for the impulse method but is nearly flat for the Equilibrium version of the mollified impu... |

33 |
Overcoming stability limitations in biomolecular dynamics. I. Combining force splitting via extrapolation with Langevin dynamics in
- Barth, Schlick
- 1998
(Show Context)
Citation Context ...d every �t = k2tm = k1�τ, the outermost timestep. The ratio r = k1k2 = �t/�τ determines the overall speedup, since the majority of the work in the total force evaluation stems from the slow component =-=[11]-=-. Typical timestep regimes are �τ = 0.5 fs, �tm around 1 or 2 fs, and �t = 4 fs. The requirement of symplecticness unfortunately necessitates the merging of the slow forces with remaining terms via pe... |

31 |
A Recursive Formulation for Constrained Mechanical System Dynamics
- Bae, Kuhl, et al.
- 1988
(Show Context)
Citation Context ...hods for stiff differential equations, like the implicit-Euler, for general biomolecular dynamics [116, 135]. Other approaches for integrating the equations of motion exactly in the constrained space =-=[6, 7, 68, 107]-=- are described in the context of crystallography and NMR refinement (Section 6). 2.5. Other Ways to Achieve Speedup in MD Additional ways to reap speedups from MD simulations are simplified treatments... |

31 |
Extrapolation versus impulse in multiple-timestepping schemes. II. linear analysis and applications to Newtonian and Langevin dynamics
- Barth, Schlick
- 1998
(Show Context)
Citation Context ...more closely (see Fig. 6). This suggests that the parameter γ should be kept as small as possible, just sufficient to guarantee numerical stability of the method. An estimate for such a γ is given in =-=[12]-=- and refined ins24 SCHLICK ET AL. FIG. 6. BPTI spectral densities derived from the cosine Fourier transforms of the velocity autocorrelation function for all atoms for LN versus the Verlet generalizat... |

31 |
The crystal structure of the light-harvesting complex II (B800-850) from Rhodospirillum molischianum
- Koepke, Hu, et al.
- 1996
(Show Context)
Citation Context ...imilar results were obtained for a test case at lower (2.8 ˚A) resolution and for two real cases involving new crystal structures starting with an initial model built into a poor electron density map =-=[1, 75]-=-.s42 SCHLICK ET AL. 7. FUTURE PERSPECTIVES It is impressive to look back 20 years and appreciate how rapidly the field of biomolecular modeling has progressed from the first dynamic simulation of a sm... |

29 |
Crystallographic R factor refinement by molecular dynamics
- Brünger, Kuriyan, et al.
- 1987
(Show Context)
Citation Context ...ng can cross barriers between minima and thus explore a greater volume of parameter space to find better models (deeper and/or wider minima). Following its introduction to crystallographic refinement =-=[30]-=-, there have been major improvements of the original method in four principal areas: the measure of model quality, the search of the parameter space, the target function, and the modeling of conformat... |

29 |
A portable distributed implementation of the parallel multipole tree algorithm
- Rankin, Board
- 1995
(Show Context)
Citation Context ...ole code D-PMTA, the distributed parallel multipole tree algorithm, is a message passing code which runs both on workstation clusters and on tightly coupled parallel machines such as the Cray T3D/T3E =-=[20, 21, 104]-=-. Reporting performance numbers for any of these algorithms is always complicated by the fact that there are many variable parameters (simulation and computer dependent) which govern both the accuracy... |

27 |
Crystallographic refinement by simulated annealing: Methods and applications
- Brünger
- 1997
(Show Context)
Citation Context ...asonable initial models (obtained by standard crystallographic techniques) produces significantly better final solutions compared to those produced by least-squares or conjugate-gradient minimization =-=[27, 30, 45, 78, 107]-=-. In recent tests [1, 107], arbitrarily “scrambled” models were generated from an initial model of α-amylase inhibitor built using experimental phase information from multiple isomorphous replacement.... |

27 |
Multiple time step methods in molecular dynamics
- Streett, Tildesley, et al.
- 1978
(Show Context)
Citation Context ...ortant in force-splitting approaches, which update forces in different classes at varying, appropriately chosen frequencies. Such multiple-timestep (MTS) schemes were suggested two decades ago for MD =-=[123]-=- and have recently become popular in biomolecular dynamics simulations with the introduction of symplectic variants. 2.6. Symplectic Integrators Important mathematical contributions regarding symplect... |

25 | Algorithms for constrained molecular dynamics
- Barth, Kuczera, et al.
- 1995
(Show Context)
Citation Context ...amework of iterative nonlinear-system solvers such as SOR, an analogy which helps explain convergence properties, as well as improve convergence via introduction of an additional relaxation parameter =-=[9]-=-. Unfortunately, the SHAKE approach cannot be extended to freeze the next fastest vibrational mode, heavy-atom bond angles, because the overall motion is altered significantly [10, 53]. This strong vi... |

25 |
Dangers of multiple-time-step methods
- Biesiadecki, Skeel
- 1993
(Show Context)
Citation Context ...s spread quickly to the computational chemistry community. It was shown that the popular Verlet method is symplectic, as are the RATTLE scheme for constrained dynamics [85] and the impulse–MTS method =-=[18]-=-. 2.7. MTS Schemes and Resonance Problems Symplectic and time-reversible MTS methods for biomolecular dynamics have been described in the early 1990s [47, 126]. A natural splitting for MD simulations ... |

23 |
Difficulties with multiple timestepping and the fast multipole algorithm in molecular dynamics
- Bishop, Skeel, et al.
- 1997
(Show Context)
Citation Context ...ot explained by resonance in the papers which first applied these MTS variants to biomolecules [60, 130], large energy growth has been reported beyond this threshold; see also recent illustrations in =-=[19]-=-. Barth and Schlick found that the introduction of stochasticity into MTS schemes delays resonance artifacts of impulse–MTS to an outer timestep near the period (10 fs) rather than half-period; see vi... |

23 |
Accelerated molecular dynamics simulation with the parallel fast multipole algorithm
- Jr, Causey, et al.
- 1992
(Show Context)
Citation Context ...Greengard and Rokhlin appeared in 1987, placing this class of algorithms on a more rigorous theoretical footing [50]. The FMA was soon adapted to the needs of electrostatic particle simulation for MD =-=[23, 118]-=-. The multipole-accelerated algorithms overcome the O(N 2 ) complexity of the N-body problem by using a hierarchy of approximations to represent the effect of increasingly distant groups of charged pa... |

23 |
The effect of long-range electrostatic interactions in simulations of macromolecular crystals—a comparison of the Ewald and truncated list methods
- York, Darden, et al.
- 1993
(Show Context)
Citation Context ...tials, where all interactions past a certain distance were simply ignored. Though acceptable for some simulation experiments, this cutoff approximation leads to unphysical behavior in many situations =-=[131]-=-. An alternative “brute force” quadratic method tailored to specialized parallel hardware [25, 62] increased the size of systems that could be simulated “exactly,” but the quadratic complexity eventua... |

22 |
A fast recursive algorithm for molecular dynamics simulation
- Jain, Vaidehi, et al.
- 1993
(Show Context)
Citation Context ...hods for stiff differential equations, like the implicit-Euler, for general biomolecular dynamics [116, 135]. Other approaches for integrating the equations of motion exactly in the constrained space =-=[6, 7, 68, 107]-=- are described in the context of crystallography and NMR refinement (Section 6). 2.5. Other Ways to Achieve Speedup in MD Additional ways to reap speedups from MD simulations are simplified treatments... |

22 |
New spherical-cutoff methods for long-range forces in macromolecular simulation
- Steinbach, Brooks
- 1994
(Show Context)
Citation Context ...tion 3). The rapidly decaying van der Waals forces can still be treated by cutoff techniques, though more sophisticated schemes than straightforward truncation—smooth switching and shifting functions =-=[121]-=-— are needed to tame the artifacts that might otherwise result from abrupt changes in energy. Another practical procedure for cutting the work per MD timestep is to use a nonbonded pairlist and update... |

21 |
Torsion angle dynamics: reduced variable conformational sampling enhances crystallographic structure refinement. Proteins
- Rice, Brünger
- 1994
(Show Context)
Citation Context ...hods for stiff differential equations, like the implicit-Euler, for general biomolecular dynamics [116, 135]. Other approaches for integrating the equations of motion exactly in the constrained space =-=[6, 7, 68, 107]-=- are described in the context of crystallography and NMR refinement (Section 6). 2.5. Other Ways to Achieve Speedup in MD Additional ways to reap speedups from MD simulations are simplified treatments... |