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15
Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation
 ANNU. REV. BIOPHYS. BIOMOL. STRUCT
, 1997
"... Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stab ..."
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Cited by 23 (9 self)
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Innovative algorithms have been developed during the past decade for simulating Newtonian physics for macromolecules. A major goal is alleviation of the severe requirement that the integration timestep be small enough to resolve the fastest components of the motion and thus guarantee numerical stability. This timestep problem is challenging if strictly faster methods with the same allatom resolution at small timesteps are sought. Mathematical techniques that have worked well in other multipletimescale contexts—where the fast motions are rapidly decaying or largely decoupled from others—have not been as successful for biomolecules, where vibrational coupling is strong. This review examines general issues that limit the timestep and describes available methods (constrained, reducedvariable, implicit, symplectic, multipletimestep, and normalmodebased schemes). A section compares results of selected integrators for a model dipeptide, assessing physical and numerical performance. Included is our dual timestep method LN, which relies on an approximate linearization of the equations of motion every �t interval (5 fs or less), the solution of which is obtained by explicit integration at the inner timestep �τ (e.g., 0.5 fs). LN is computationally competitive, providing 4–5 speedup factors, and results are in good agreement, in comparison to 0.5 fs trajectories. These collective algorithmic efforts help fill the gap between the time range that can be simulated and the timespans of major biological interest (milliseconds and longer). Still, only a hierarchy of models and methods, along with
Smoothed Langevin dynamics of highly oscillatory systems
, 1996
"... In this paper we generalize a result by Rubin and Ungar on Hamiltonian systems containing a strong constraining potential to Langevin dynamics. Such highly oscillatory systems arise, for example, in the context of molecular dynamics. We derive constrained equations of motion for the slowly varying s ..."
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Cited by 11 (1 self)
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In this paper we generalize a result by Rubin and Ungar on Hamiltonian systems containing a strong constraining potential to Langevin dynamics. Such highly oscillatory systems arise, for example, in the context of molecular dynamics. We derive constrained equations of motion for the slowly varying solution components. This includes in particular the derivation of a correcting forceterm that stands for the coupling of the slow and fast degrees of motion. We will identify two limiting cases: (i) the correcting force becomes, over a finite interval of time, almost identical to the force term suggested by Rubin and Ungar (weak thermal coupling) and (ii) the correcting force can be approximated by the gradient of the Fixman potential as used in statistical mechanics (strong thermal coupling). The discussion will shed some light on the question which of the two correcting potentials is more appropriate under which circumstances for molecular dynamics. In Sec. 7, we also discuss smoothing in the context of constant temperature molecular dynamics.
Integration Methods for Molecular Dynamics
 IN MATHEMATICAL APPROACHES TO BIOMOLECULAR STRUCTURE AND DYNAMICS, IMA VOLUMES IN MATHEMATICS AND ITS APPLICATIONS
, 1996
"... Classical molecular dynamics simulation of a macromolecule requires the use of an efficient timestepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on m ..."
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Cited by 6 (2 self)
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Classical molecular dynamics simulation of a macromolecule requires the use of an efficient timestepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on meaningful time intervals is neither obtainable nor desired, but some restrictions, such as symplecticness, can be imposed on the discretization which tend to imply good long term behavior. The presence of a variety of types and strengths of interatom potentials in standard molecular models places severe restrictions on the timestep for numerical integration used in explicit integration schemes, so much recent research has concentrated on the search for alternatives that possess (1) proper dynamical properties, and (2) a relative insensitivity to the fastest components of the dynamics. We survey several recent approaches.
Torsion Dynamics of Molecular Systems
 Phys. Rev. E
, 1996
"... Based on the concept of free energy, we derive a Hamiltonian formulation for molecular dynamics in torsion space. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced by B.R. Brooks, J. Zhou, and S. Reich in the context of molec ..."
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Cited by 6 (4 self)
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Based on the concept of free energy, we derive a Hamiltonian formulation for molecular dynamics in torsion space. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced by B.R. Brooks, J. Zhou, and S. Reich in the context of molecular dynamics. We consider a few simplifications that allow one to calculate the free energy analytically and to write the corresponding equations of motion as a constraint Hamiltonian system that can conveniently be discretized by the wellknown SHAKE algorithm. The additional computational costs, compared to using the orginal force field and constraining bondlengths and bondangles to their equilibrium value (hard constraints), amount, in general, to less than a complete force evaluation. We show for a single butane molecule that our Hamiltonian formulation yields the correct Boltzmann distribution in the torsion angle while the original Hamiltonian together with hard constraints on the b...
Enhancing Energy Conserving Methods
 BIT
, 1995
"... Recent observations [5] indicate that energymomentum methods might be better suited for the numerical integration of highly oscillatory Hamiltonian systems than implicit symplectic methods. However, the popular energymomentum method, suggested in [3], achieves conservation of energy by a global sc ..."
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Cited by 5 (2 self)
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Recent observations [5] indicate that energymomentum methods might be better suited for the numerical integration of highly oscillatory Hamiltonian systems than implicit symplectic methods. However, the popular energymomentum method, suggested in [3], achieves conservation of energy by a global scaling of the force field. This leads to an undesirable coupling of all degrees of freedom that is not present in the original problem formulation. We suggest enhancing this energymomentum method by splitting the force field and using separate adjustment factors for each force. In case that the potential energy function can be split into a strong and a weak part, we also show how to combine an energy conserving discretization of the strong forces with a symplectic discretization of the weak contributions. We demonstrate the numerical properties of our method by integrating particles that interact through LennardJones potentials. Keywords: Hamiltonian differential equations, energymomentum...
Homogenization Approach To Smoothed Molecular Dynamics
"... This article presents a mathematically rigorous discussion of the limit situation in which the stiffness of the stiff part of the potential is increased to infinity, i.e., of the limit ffl ! 0. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system ..."
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Cited by 4 (0 self)
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This article presents a mathematically rigorous discussion of the limit situation in which the stiffness of the stiff part of the potential is increased to infinity, i.e., of the limit ffl ! 0. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system
Computation of Essential Molecular Dynamics by Subdivision Techniques I: Basic Concept
 IN COMPUTATIONAL MOLECULAR DYNAMICS: CHALLENGES, METHODS, IDEAS
, 1996
"... The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the essential dynamics of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajecto ..."
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Cited by 4 (3 self)
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The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the essential dynamics of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times of conformational changes need to be actually computed. In the language of dynamical systems, such information is contained in the natural invariant measure (infinite relaxation time) or in almost invariant sets ("large" finite relaxation times). The paper suggests the direct computation of these objects via eigenmodes of the associated FrobeniusPerron operator by means of a multilevel subdivision algorithm. The advocated approach is different to both MonteCarlo techniques on the one hand and long term trajectory simulation on the other hand: in our setup long term trajectories are ...
A Free Energy Approach to the Torsion Dynamics of Macromolecules
, 1995
"... Based on the concept of free energy, we give a Hamiltonian formulation for the torsion dynamics of macromolecules. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced in [3] and [14]. We consider a few simplifications that allo ..."
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Cited by 3 (1 self)
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Based on the concept of free energy, we give a Hamiltonian formulation for the torsion dynamics of macromolecules. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced in [3] and [14]. We consider a few simplifications that allow one to calculate the free energy analytically and to write the corresponding equations of motion as a constrained Hamiltonian system. We also discuss a possible stochastic embedding of the reduced dynamics by means of a generalized Langevin approach. 1