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NOVEL MULTISCALE ALGORITHMS FOR MOLECULAR DYNAMICS

A Dissertation

Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

by

Qun Ma, B.S., M.S.

Jesús A. Izaguirre, Director

Department of Computer Science and Engineering Notre Dame, Indiana July 2003

c Copyright by

QUN MA 2003 All rights reserved

NOVEL MULTISCALE ALGORITHMS FOR MOLECULAR DYNAMICS

Abstract by Qun Ma In post-genomic computational biology and bioinformatics, long simulations of the dynamics of molecular systems, particularly biological molecules such as proteins and DNA, require advances in time stepping computational methods. The most severe problem of these algorithms is instability. The objective of this dissertation is to present original work in constructing multiscale multiple time stepping (MTS) algorithms for molecular dynamics (MD) that allow large time steps. First, through nonlinear stability analysis and numerical experiments, we reveal that MTS integrators such as Impulse suffer nonlinear overheating when ∆t = T /3 or possibly ∆t = T /4 when constant–energy MD simulations are attempted, where ∆t is the longest step size and T is the shortest period of the modes in the system. Second, we present Targeted MOLLY (TM), a new multiscale integrator for MD simulations. TM combines an efficient implementation of B-spline MOLLY exploiting analytical Hessians of energies and a self–consistent dissipative leapfrog integrator. Results show that TM allows very large time steps for slow forces (and thus multiscale) for the numerically challenging flexible TIP3P water systems (Jorgensen, et al. J. Chem. Phys., vol 79, pp 926–935, 1983) while still computing the dynamical and structural properties accurately. Finally, we show yet another new MOLLY integrator, the Backward Euler (BE) MOLLY in which hydrogen bond forces can easily be included in the averaging and thus stability might be further improved.

To my parents, my wife and my kids

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CONTENTS

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 1: OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . 1.1 Classical Molecular Dynamics . . . . . . . . . . . . . . . . 1.2 Multiple Time Stepping . . . . . . . . . . . . . . . . . . . . 1.3 Multiscale Approaches for Molecular Dynamics . . . . . . . 1.4 Numerical Instabilities and Nonlinear Overheating . . . . . 1.5 Targeted Mollified Impulse . . . . . . . . . . . . . . . . . . 1.5.1 B-spline Averaging . . . . . . . . . . . . . . . . . . 1.5.2 Stochasticity and Long Time Dynamics . . . . . . . 1.5.3 Possible Applications of Targeted Mollified Impulse 1.6 Backward Euler Mollified Impulse . . . . . . . . . . . . . . 1.7 Software Engineering Issues . . . . . . . . . . . . . . . . . 1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 3 5 7 8 10 11 12 13 14 15 17

CHAPTER 2: NONLINEAR OVERHEATING OF IMPULSE 2.1 Stability Limitations of Impulse . . . . . . . . . . . 2.2 Nonlinear Stability Analysis . . . . . . . . . . . . . 2.2.1 Assumptions and Procedure . . . . . . . . . 2.2.2 Application to Multiple Time Stepping . . . 2.2.3 Proof of Main Result . . . . . . . . . . . . . 2.2.4 Implication of 4:1 Resonance . . . . . . . . 2.2.5 Justification of Analysis . . . . . . . . . . . 2.3 Numerical Experiments . . . . . . . . . . . . . . . . 2.3.1 Model Systems . . . . . . . . . . . . . . . . 2.3.2 Power Spectrum Analysis . . . . . . . . . . 2.3.3 Simulation Protocol . . . . . . . . . . . . .

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2.4

2.3.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 44

CHAPTER 3: TARGETED MOLLIFIED IMPULSE . 3.1 The Maximum Allowable Step Size . . . . . 3.2 Langevin Related Methods . . . . . . . . . . 3.3 Targeted Mollified Impulse . . . . . . . . . . 3.3.1 B-spline Averaging . . . . . . . . . . 3.3.2 Self–Consistent Dissipative Leapfrog 3.3.3 Pairwise Targeted Langevin Coupling 3.3.4 Sampling Properties . . . . . . . . . 3.3.5 Implementation Issues . . . . . . . . 3.4 Numerical Experiments . . . . . . . . . . . . 3.4.1 Simulation Protocol . . . . . . . . . 3.4.2 Results and Discussion . . . . . . . .

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48 49 51 53 54 57 59 59 63 65 65 66

CHAPTER 4: BACKWARD EULER MOLLY . . 4.1 Backward Euler Averaging . . . . . . . 4.2 Introducing a Heuristic Approximation . 4.3 Results and Discussion . . . . . . . . .

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75 76 77 78

CHAPTER 5: SUMMARY AND FUTURE WORK . . . . . . . . . . . . . . . . . 5.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 83

APPENDIX A: ADDITIONAL DETAILS . . . . . . . A.1 Potentials for Flexible Water . . . . . . . . . A.2 Metric of Instabilities in MD . . . . . . . . . A.3 Analytical Hessians for CHARMM Potentials A.4 Ewald Summation . . . . . . . . . . . . . . . A.5 Names and Acronyms . . . . . . . . . . . . . A.6 ProtoMol Units . . . . . . . . . . . . . . . .

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85 85 86 87 92 95 96

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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FIGURES

1.1

A schematic of Impulse scheme.

1.2

Collaboration between an integrator object and force objects.

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2.1

(a) Wave-numbers and (b) periods of flexible water systems. . . . . . . . . . .

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2.2

(a) Wave-numbers and (b) periods of flexible Melittin protein systems. . . . . .

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2.3

(a) Wave-numbers and (b) periods of the rigid Melittin/water system.

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2.4

Energy drift for flexible water simulations at 0.015 K using Impulse. Each data point is obtained from a 500 ps simulation. Notice that the peaks at step sizes of 2.57 fs and 3.33 fs show evidence of milder 4:1 resonance and unstable 3:1 resonance (overheating). . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Block-averaged drift of total energy for 500 ps of molecular dynamics simulation of the small flexible water system using Impulse. It illustrates the 3:1 nonlinear resonance at one third of the fastest period near zero K. . . . . . . . . . . . .

41

2.5

2.6

Same as Fig. 2.5 except that it shows evidence of a 4:1 nonlinear resonance.

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42

2.7

Energy drift for flexible water simulations at 300 K using Impulse. Each data point is obtained from a 500 ps simulation. Notice that the peaks at step sizes of 2.40 fs and 3.33 fs show evidence of 4:1 resonance and 3:1 resonance (overheating). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.8

Same as in Fig. 2.5 except that the temperature here is 300 K.

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2.9

Same as in Fig. 2.8 except that it shows evidence of a possible 4:1 resonance. . .

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2.10 Energy drift for flexible Melittin protein simulations at 300 K using Impulse. Each data point is obtained from a 10 ns simulation. Notice that the peaks at step sizes of ∆t = 3.00, 3.27, 3.78 fs, show evidences of 3:1 overheating. . . . . . .

46

2.11 Energy drift of simulations of explicitly solvated Melittin system at 300 K using

3.1

SHAKE-Impulse. Each point is obtained from a 500 ps MD simulation. Most likely these drifts correspond to the combined 4:1 and 3:1 resonances associated with the remaining modes. . . . . . . . . . . . . . . . . . . . . . . . . . .

47

The maximum allowable step size for systems with different cutoff distances. . .

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3.2

(a) Phase space trajectory (the dots), and (b) velocity distribution (the squares) of a harmonic oscillator with Hamiltonian given by Eq. (3.24) (x(0) = 0, x(0) ˙ = 1) using self–consistent dissipative leapfrog with γ = 0.1. . . . . . . . . . . . .

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(a) Phase space trajectory (the dots), and (b) velocity distribution (the squares) of a Fermi–Pasta–Ulam problem whose Hamiltonian is given by Eq. (3.25) (x(0) = 0, x(0) ˙ = 1) using Targeted MOLLY with γ = 0.01. . . . . . . . . . . . . .

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(a) Phase diagram, and (b) velocity distribution of the Fermi–Pasta–Ulam problem whose Hamiltonian is given by Eq. (3.25) (x(0) = 0, x(0) ˙ = 1) using Targeted MOLLY with γ = 1.0 × 10−5 . . . . . . . . . . . . . . . . . . . .

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3.5

Energy drift for flexible water simulations of 200 ps at 300 K in vacuum. . . . .

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3.6

Energy drift for flexible water simulations of 200 ps at 300 K with normal or periodic boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . .

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Radial distribution functions for the O-H interactions for leapfrog (δt = 1 fs) and Targeted MOLLY (TM, ∆t = 16 fs and δt = 2 fs). . . . . . . . . . . . .

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Radial distribution functions for the H-H interactions for leapfrog (δt = 1 fs) and Targeted MOLLY (TM, ∆t = 16 fs and δt = 2 fs). . . . . . . . . . . . .

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Two hydrogen bonded water molecules with librational motion and stretching motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Energy drift of 100 ps molecular dynamics simulation of the small flexible water system using BE MOLLY. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The structure of the estrogen receptor (ER) and the raloxifen complex (ER/RAL). The drawing methods for the protein and raloxifen are Cartoon (cylinders for helices/coils, tubes for turns and directional sheets for β sheets) and VDW, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3

3.4

3.7 3.8

4.1 4.2

5.1

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TABLES

2.1

THE SHORTEST PERIODS FOR FLEXIBLE WATER SYSTEM. . . . . . . .

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3.1

SELF-DIFFUSION COEFFICIENT (D) FROM FLEXIBLE TIP3P WATER SIMULATIONS WITH PERIODIC BOUNDARY CONDITIONS WITH EWALD ACCURACY OF = 10−12 . THE ERROR BAR IS GIVEN BY TWICE OF THE STANDARD DEVIATION. . . . . . . . . . . . . . . . . . . . . . . .

72

3.2

CPU TIME (T ) AND SPEED UP (η) of 400 PS OF MD SIMULATIONS OF THE SMALL FLEXIBLE WATER MODEL SYSTEM WITH PERIODIC BOUNDARY CONDITIONS WITH EWALD ACCURACY OF = 10−12 . . . . . . . 72

3.3

CPU TIME (T ) AND SPEED UP (η) of 10 PS OF MD SIMULATIONS OF THE MIDIUM AND THE LARGE FLEXIBLE WATER MODEL SYSTEMS WITH PERIODIC BOUNDARY CONDITIONS WITH EWALD ACCURACY OF = 10−7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

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OVERHEAD OF MOLLIFICATION FROM THE SMALL FLEXIBLE FLEXIBLE WATER SYSTEM SIMULATIONS (4 PS EACH) USING B-SPLINE MOLLY. IN(x,y,z), x DENOTES THE OUTER TIME STEP (FS), y THE INNER TIME STEP (FS), AND z THE TIME STEP (FS) FOR MOLLY AVERAGING. . . . 74

A.1 UNITS USED IN PROTOMOL . . . . . . . . . . . . . . . . . . . . . . . .

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SYMBOLS

A(q) Aq (q) H ∇ ∆t ρ(A) δ γ δrel δt ij rel θ0 kB ξ(x, p) π(x, p) σij C I KA KB M T U (q) Drel lO−H lk p(t) q(t) qH qi rc t tchar xij

Averaged positions Averaging Jacobian (force filter) Hamiltonian Gradient operator Long time step Spectral radius of matrix A Dirac delta function Langevin dynamics collision parameter Percent relative root mean square deviation Short time step Dielectric constant between atoms i and j Estimated relative error Reference angle Boltzmann constant Transformation of positions Transformation of momenta Location of Lennard-Jones minimum between atoms i and j Electrostatic constant Identity matrix Angle force constant Bond force constant Mass matrix Langevin heat bath temperature Potential energy Percent relative drift of total energy Bond length (O–H) Reference bond length between atoms i and j for constraint k Momentum vector (R3N ) Position vector (R3N ) Charge for H Charge for atom i Cutoff distance Time, independent variable Characteristic time Distance between atoms i and j

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ACKNOWLEDGEMENTS

First of all, I would like to thank my advisor, Dr. Izaguirre, for his brilliant insights and ideas, lots of invaluable advice, many helpful discussions, lots of encouragement and support. Thank you so much for introducing me to this vibrant research area. I thank my Ph.D. Committee members, Dr. Karavelas, Dr. Gezelter and Dr. Madey, for their time and valuable comments. I also thank Dr. Hu for reading my Ph.D. research proposal. I am grateful to all of you for providing assistance in my career placement. I thank Dr. Thierry Matthey for perfecting the framework of ProtoMol, without which my research would never be as smooth. I thank Thierry again for his contributions in running simulations during the development of Targeted MOLLY. I will never forget other members of the Laboratory for Computational Life Sciences (LCLS), current and past. I have enjoyed very much working with all of you. Thank you for making LCLS such a collegial environment. I would express my deepest love and thanks to my wife, Yao, for her support and sacrifices, and my children, Jesse and Hannah, for bringing us lots of joy. I would like to thank Ms. Ginny Johns. Thank you so much for all your help! This research was supported in part by NSF Grant No. BIOCOMPLEXITY-PHY0083653, NSF CAREER award No. ACI-0135195 and a grant from the University of Notre Dame. The numerical experiments were performed in part using Norwegian supercomputing facilities through a Norwegian Research Council grant, and the BOB and Hydra computer clusters of the University of Notre Dame.

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CHAPTER 1 OVERVIEW

The advance in genome sequencing projects in recent years has resulted in a rapid increase of the protein sequence data which far exceeds the known protein structures. Thus, predicting the 3D native structures of proteins from the known amino acid sequence, i.e., protein folding, has become pressing in structural genomics and computational biology. It is plausible to use molecular dynamics (MD) simulations to study the folding of proteins because the full atomic representation of both protein and solvent (water) employed by MD simulation provides the highest level of resolution and accuracy [22,59–62]. However, the multiscale nature of biological molecular systems and the severe restrictions on the time steps of MD simulations make simulation of protein folding using MD with current time stepping techniques an almost impossible enterprise. As a notable historical example in 1998, it took over 100 days on a CRAY T3D/T3E supercomputer using all 256 CPUs to perform the 1 microsecond MD simulation of a small 36-residue protein called the villin headpiece subdomain by UCSF researchers only to observe the early stages of folding. In nature the folding from totally denatured structure to the native structure takes tens of microseconds or even longer even for small proteins whereas it may take several seconds for large proteins. In a MD system, long range Coulombic forces are slow in nature whereas bonded forces are fast. Meanwhile, the slow force evaluation is computationally more expensive than the bonded one. Multiple time stepping (MTS) integrators have been introduced to

1

take advantage of the above-mentioned fact such that slow forces are incorporated into the solutions less frequently than the fast forces. Though in general MTS integrators are a great success, their potential is not fully exploited since the time steps allowed by MTS integrators are still limited by instabilities. Multiscale integrators can help effectively and efficiently simulate the multiscale dynamics of biomolecular systems. With a multiscale integrator, we can resolve the dynamics of the systems of interest with appropriate accuracies, e.g., slow processes of interest are resolved more accurately than fast processes. The overall objective of this dissertation is to present multiscale MTS algorithms (integrators) for MD. These integrators, due to the use of larger time steps, would allow long time MD simulations of large systems to be done reasonably quickly using moderate computing resources. We present the nonlinear analysis on Impulse, a state-of-the-art MTS integrator, in Chapter 2. The analysis together with the numerical experiment reveal the underlying mechanism of additional numerical instability – nonlinear overheating – for MTS integrators. Then in Chapter 3, we present the Targeted Mollified Impulse (Targeted MOLLY, or TM) method, a new multiscale integrator for molecular dynamics simulations, which combines an efficient implementation of B-spline MOLLY exploiting analytical Hessians of energies and a self-consistent dissipative leapfrog integrator. Results show that TM allows very large time steps for slow forces (and thus multiscale) while still computing the dynamical and structural properties properly. Finally, in Chapter 4 we show yet another new MOLLY integrator, the Backward Euler (BE) MOLLY in which hydrogen bond forces can easily be included in the averaging and thus stability might be further improved. The main contribution of this dissertation is the Targeted MOLLY method. While it brings to light a deeper understanding of the numerical difficulties of the popular methodologies in MD, nonlinear overheating of Impulse serves mainly as the motivation that leads to the development of TM in the context of this writing.

2

1.1 Classical Molecular Dynamics In classical molecular dynamics (MD), starting with the atomic coordinates, the molecular connectivity and force field parameters, we compute trajectories 1 by solving Newton’s equations of motion, Eq. (1.1), which are a system of second order ordinary differential equations (ODE). M

d2 q(t) = −∇U(q(t)). dt2

(1.1)

where q is the position vector, M is the diagonal mass matrix, U(q) is the potential energy, and −∇U(q) is the force. In practice, we solve the following system of first order ODEs q˙ = M−1 p,

p˙ = −∇U(q),

(1.2)

where p is the momentum vector. The potential energy, U, is typically given by U = U bonded + U nonbonded , U bonded = U bond + U angle + U dihedral + U improper , U nonbonded = U Lennard-Jones + U electrostatic .

(1.3) (1.4) (1.5)

Details of the above terms are provided in Appendix A.1. Note that the U bond and U angle terms are harmonic (linear and angular spring) potentials that model covalent bond interactions, and the U electrostatic term is the Coulomb potential, U Lennard-Jones term models a van der Waals attraction and hard core repulsion. The coefficients in these potentials are determined experimentally and often aided by theoretical approaches such as ab initio quantum mechanical calculations. It is inappropriate to expect that accurate trajectories be computed for long time intervals in MD simulations because MD trajectories are chaotic. Consequently, one can only 1

A MD trajectory is a collection of the time evolution of the Cartesian coordinates for each particles (atoms) in 3-d space.

3

expect that the trajectories have the correct statistical properties, which can be verified by using the initial velocities that are randomly generated from a Maxwell distribution. For a Hamiltonian system 2 , H = H(p, q, t), when a symplectic integrator is used to solve the equations of motion, any given energy surface in phase space is only changed slightly and statistical properties of long-time dynamics are retained. A transformation in phase space from positions x and momenta p x¯ = ξ(x, p),

p¯ = π(x, p)

(1.7)

is said to be symplectic if its Jacobian matrix (of partial derivatives) satisfies 

T 









 ξx ξp   0 I   ξx ξp   0 I      = . −I 0 πx πp πx πp −I 0

(1.8)

Symplectic transformations preserve volume in phase space. The biggest benefit of symplecticness is stability because energy is nearly conserved. An integrator should also preserve time-reversibility of Eq. (1.2). In [19], it is stated that any lack of perfection of such reversal should be due to rounding-off errors only, not the program. Many symplectic integrators are time reversible. The use of symplectic integrators is also favored in hybrid Monte Carlo methods due to their volume preserving property and time-reversibility which help achieve detailed balance and thus valid sampling as observed in [79]. Eq. (1.2) is usually solved using the Verlet or leapfrog algorithm. One step of discretized of leapfrog is given below: If H = H(p, q, t) is a sufficiently smooth real function defined in Ω × I, where Ω is a domain in the oriented Euclidean space R2n of the points (p, q) = (p1 , . . . , pn , q1 , . . . , qn ) and I is an open interval of the real line R of the variable t (time), then the Hamiltonian system of differential equations with Hamiltonian H is, by definition, given by 2

∂H dpi =− , dt ∂qi

∂H dqi = , dt ∂pi

i = 1, . . . , n.

(1.6)

The integer n is called the number of degrees of freedom and Ω is the phase space. The product of Ω × I is the extended phase space.

4

half a kick pn−1+ = pn−1 − a drift

∆t ∇U(qn−1 ). 2

(1.9)

qn = qn−1 + ∆tM−1 pn−1+ .

(1.10)

∆t ∇U(qn ). 2

(1.11)

half a kick pn = pn−1+ −

Algorithm 1: One step of leapfrog discretization The symbol pn−1+ represents the momenta just after the (n − 1)st kick. qn−1 and qn are the positions. ∆t is the integration time step, which is typically 1 fs in MD. Alternatively, leapfrog can be represented by M

∞ X d2 q = − ∆t δ(t − n0 ∆t)∇U(q), 2 dt 0

(1.12)

n =−∞

where δ is the Dirac delta function. 1.2 Multiple Time Stepping MD is a very challenging problem: the size of the system can be very large (up to tens of thousands or even millions of atoms) and the force evaluation is very expensive. The time complexity of each type of bonded force is linear in terms of N, the total number of atoms in the system, i.e., O(N), whereas the time complexity of each type of nonbonded forces is quadratic, i.e., O(N 2 ) using all-pair calculations. Though fast algorithms for electrostatics exist, which have lower time complexity, e.g., Ewald (O(N 3/2 ), Particle Mesh Ewald (PME) (O(N log N)) [21, 53] and MultiGrid (O(N)) [103], unfortunately, they all have large coefficients and thus they show their superiority to the direct method only when the system size is large enough. Thus the force evaluation at each time step is very expensive, especially that for the slow forces. Meanwhile, these fast electrostatics algorithms do not parallelize as well as more brute-force approaches.

5

Biological systems such as proteins and DNAs are important target application areas of MD. These systems are multiscale in nature. For example, the dynamics of proteins contain motions over vastly different time scales, from atomic vibrations in the order of femtoseconds to collective motions that may occur in the order of milliseconds or even seconds, a span of fifteen orders of magnitude [17, pp. 19,20]. Consider a MD simulation using leapfrog with time step of 1 fs, a simulation of one second would need to advance the equations of motion by quadrillion (1015 ) steps! Even for a moderate system size, the computational challenge for such a long simulation is very demanding. In order to enable long simulations of large systems, multiple time stepping (MTS) integrators have been a subject of research since the late 1980s [7,14,28,29,33,34,43–47,58, 68–70,93,106,110,111]. In MTS MD, the forces are split into fast and slow components, F fast and F slow , and evaluates the former more frequently than the latter to speed up simulations 3 . A state-of-the-art MTS algorithm is the Verlet-I [33,34]/r-RESPA [106]/Impulse integrator (hereafter referred to as Impulse). A schematic of Impulse is shown in Fig. 1.1. The discretization of this problem using Impulse with step size ∆t for the slow part is given by Algorithm (2).

Figure 1.1. A schematic of Impulse scheme. 3

For biological molecules, the bonded forces are considered “fast” ones which have periods on the order of 10 fs, the long range nonbonded forces are considered “slow” ones which have characteristic time greater than 300 fs, and the short range nonbonded interactions are considered “medium” ones.

6

1 2

kick: p+ n−1 = pn−1 +

∆t slow F (qn−1 ), 2

(1.13)

oscillate: Propagate qn−1 and p+ n−1 by integrating q˙ = M−1 p,

p˙ = Ffast (q)

(1.14)

for an interval ∆t using leapfrog to get qn and p− n. 1 2

kick: pn = p− n +

∆t slow F (qn ). 2

(1.15)

Algorithm 2: Discretization using Verlet-I/r-RESPA/Impulse. Although the exact propagator of a system of ordinary differential equations (ODEs) yields exact solutions, in practice only the numerical solution to the system of ODEs can be obtained using a numerical integrator. Such a numerical solution can be regarded as the exact solution of a slightly different system of ODEs. If the given system is a Hamiltonian system, e.g., the constant-energy NVE ensemble in MD, then the slightly different system is Hamiltonian if and only if the integrator is symplectic [94]. Impulse is one such symplectic integrator. 1.3 Multiscale Approaches for Molecular Dynamics Because of the chaotic nature of MD trajectories, it is expected only that the trajectories have the correct statistical properties, cf. [97] and [57, p. 354], as the computed trajectories are overwhelmed by the effect of finite time step and finite precision. Accurate and statistically correct trajectories can be obtained using small time steps and symplectic integrators (for leapfrog, ∆t < 2.2 fs, for Impulse, ∆t < 3.3 fs for flexible water systems or explicitly solvated proteins/DNAs) [69,70,100]. We should also be able to obtain less accurate but statistically correct trajectories with ∆t one or more orders of magnitude larger than that of the Impulse. In many important applications, this con-

7

trolled accuracy reduction is still acceptable since these integrators still ensure sampling of a correct ensemble. We have developed the Targeted Mollified Impulse (TM) method as a multiscale integrator for MD simulations, which is the subject of Chapter 3. Other examples of multiscale integrators are: First, the Asynchronous Variational Integrator (AVI) approach of Marsden and co-workers, which produces symplectic integrators that avoid resonances by advancing different degrees of freedom asynchronously in time. Although it is a very promising method, it has not yet been implemented and validated for MD, cf. [51, 63]. Second, the nonsymplectic time reversible integrator of Haire and Lubich [36, 37]. Finally, the nonsymplectic reversible averaging integrator of Reich and Leimkhuler for the special case of separable slow and fast variables [58]. Besides multiscale integrators, the other multiscale approach for MD is coarsening. Coarsening relies on model reduction, which needs careful validation and is harder to control in a precise manner. Two notable examples of coarsened model approaches are: First, Bai and Brandt’s coarsened Monte Carlo, which coarsens simple polymer chains and performs Monte Carlo simulations at each spatial level to verify that the coarsened model reproduces the probability distribution function of interest [4]. It is not obvious how to produce an automatic coarsening procedure for biological macromolecules though. Second, Balaeff and Schulten’s hybrid continuum/MD models, such as an elastic rod model of a DNA loop in the lac operon [5]. These coarsening approaches are less general though very powerful. 1.4 Numerical Instabilities and Nonlinear Overheating Due to the chaotic nature of the MD trajectories, the numerical solution to Eq. (1.2) is said to be stable if the total energy is bounded. This is what the metric of measuring instabilities – percent drift of total energy – is based on, cf. Appendix A.2. Also, as a rule

8

of thumb, we say a classical MD simulation is acceptably stable if the percent relative drift of total energy is less than, say, 3%. Otherwise we say it suffers from numerical instabilities or overheating. Note that the above metric is absolutely indicative only if a symplectic integrator is used for the solution of the trajectories from the constant energy (NVE) ensemble. For other ensembles, e.g., constant temperature (NVT) ensemble, usually additional degrees of freedom are introduced and the equations for an extended system are solved. In such cases, the use of the energy drift as metric to measure instability is inappropriate and can be misleading. More empirical evidence is needed to ensure the numerical solution of these ensembles to be meaningful. Impulse has severe instability when the longest step size ∆t is a multiple of the period of the fastest motion, which happens because natural resonances are excited. Impulse also has a numerical instability at one half of the shortest period (the linear stability barrier, this is 5 fs for flexible water systems and explicitly solvated proteins/DNAs). These instabilities have been very well studied and reported [6, 14, 15, 93, 101]. To make matters even worse, there exists a numerical instability when the longest step size is one third, and possibly even one fourth of the shortest period. To avoid confusions, we shall call this type of nonlinear instabilities overheating or nonlinear overheating. The overheating is characterized by a mild but systematic drift in the energy when time steps are within certain ranges. We use a 1-d nonlinear model and express the stability conditions in terms of the parameters of the model. The nonlinear analysis shows that Impulse is practically unstable when the longest step size equals to one third of the shortest period of the system, an overheating termed as 3:1 instability. A milder overheating may or may not happen when the largest step size equals to one fourth of the shortest period depending on the state of the system. Numerical experiments on flexible water systems, flexible protein systems, and SHAKE-constrained protein/water system are performed, representing a wide range practical protocols adopted by molecular modelers. These numerical

9

results perfectly match those theoretical predictions. The overheating of Impulse due to nonlinear resonances is the subject of Chapter 2. Nonlinear overheating makes the stability limit of MTS algorithms such as Impulse much tighter than the previously known linear instabilities, and for problems without a wide gap on time scales may render MTS schemes such as Impulse not a much better choice than the single time stepping (STS) integrators such as leapfrog 4 . Serious considerations must be made in choosing an appropriate step size in MTS MD simulations. This is increasingly important as the computer power still increases according to Moore’s law, i.e., computer power increases by ten-fold every five years. As longer and longer simulations are enabled solely because of the sheer increase of computer power, numerical results may become meaningless if proper attention is not paid to nonlinear overheating. 1.5 Targeted Mollified Impulse We can estimate the largest achievable time steps for MTS integrators assuming accuracy does not degenerate due to resonances or other mechanisms. We estimate these time steps can be on the order of 100fs, cf. Section 3.1. This estimate rationalizes the effort of constructing multiscale integrators. We have developed a novel multiscale stochastic integrator called Targeted Mollified Impulse (TM) for molecular dynamics simulations. We have implemented TM in ProtoMol, a component-based framework for MD simulations [46, 74–77]. ProtoMol has a modular design that allows for easy prototyping of complex methods, and it is freely available on the web 5 . Details are given in Chapter 3. TM is an MTS integrator that uses in its outermost level the Mollified Impulse method (MOLLY, currently using B-spline MOLLY) [28, 29, 44–47], and in the innermost level the self-consistent dissipa4

Nonlinear overheating restricts step size of Impulse: ∆tmax = 3.3 fs, and ∆tmax = 2.2 fs for flexible water and explicitly solvated proteins, cf. [69, 70]. 5 ProtoMol website: http://www.nd.edu/˜lcls/protomol.

10

tive leapfrog (SCD-leapfrog), which is commonly used in Dissipative Particle Dynamics [13, 24, 32, 56, 84, 85]. TM uses targeted Langevin coupling in the SCD-leapfrog, i.e., introducing a random force and a dissipative force only for interacting pairs of atoms associated with the fastest normal modes in the system. The linear momentum of the system is conserved in TM. Using Langevin coupling to stabilize an MD integrator is called Langevin Stabilization [45]. A substantial speedup is obtained using TM with larger time steps. As an additional benefit, because the time step is larger, this new integrator is more scalable than less stable integrators for parallel MD simulations. This is due to the difficulty in parallelizing several of the popular fast electrostatic methods such as particle mesh Ewald, cf. [21, 53]. This benefit may be smaller with modern multi-grid methods for fast electrostatics [55, 92, 103]. Larger time steps come at a price, since significant testing of the dynamics as well as the integrator’s parameter space must be performed, cf. [12, p. 185]. This can be alleviated by automating this exploratory process, cf [55], and is justified when performing multiple MD simulations with different initial conditions, or when running a single long simulation, which are the target applications of this work. 1.5.1 B-spline Averaging To extend the stability limit of Impulse, the Mollified Impulse (MOLLY) method has been introduced in [29]. MOLLY is actually a family of methods in which the accuracy reduction present in Impulse is avoided or reduced. In MOLLY, the slow part of the potential energy is defined at time-averaged positions. The time average is obtained by doing dynamics over vibrations using forces that produce those vibrations. Thus, U slow (q) becomes U slow (A(q)),

(1.16)

and the force is computed using the chain rule, −∇U slow (q) is replaced by − Aq (q)T ∇U slow (A(q)), 11

(1.17)

where Aq (q) is a sparse Jacobian matrix. Modifying potential rather than force ensures MOLLY be symplectic [94]. The pre-factor Aq (q)T improves the stability of Verlet-I/rRESPA by working as a filter that eliminates components of the slow force impulse in the directions of the fast forces. Note that it has been shown that statistical properties are not sensitive to perturbations in the Hamiltonian, H, [20] which justifies the MOLLY approach. Any bias introduced by MOLLY when used as part of sampling method can be removed using an umbrella sampling criterion, for example, in combination with hybrid Monte Carlo [40, 52]. MOLLY can successfully overcome the linear stability barrier of Impulse, allowing larger time steps (6 fs) for pure flexible water MD simulations, a 50% asymptotic speedup over Impulse [44, 46, 47]. The B-spline averaging functions have been suggested in [28] due to their compact local support. The prototype of MOLLY integrators using B-spline weight functions was constructed using a numerical automatic differentiation tool called ADOL-C [31], due to the lack of the analytical Hessians at that time [44, 47]. To improve the efficiency of B-spline MOLLY, we developed the analytical Hessians of energies for the CHARMM force field [71,72], cf. Appendix A.3. We implemented B-spline MOLLY using the analytical Hessians in ProtoMol. Section 3.3.1 details the B-spline MOLLY. 1.5.2 Stochasticity and Long Time Dynamics For a limited time interval the numerical solution of MD simulations is very near the exact solution of a modified Hamiltonian when using symplectic integrators [100]. However, from examination of the numerical trajectories of one-dimensional systems, it is not clear whether this result extends to very long time intervals [98]. Given that multiple time stepping symplectic integrators suffer from severe instabilities that limit the largest time step possible, one frequently needs to introduce some form of damping when using large time steps in order to stabilize the integrator. Intro-

12

ducing stochasticity into MD simulations is the least harmful way to restore energy lost through damping. Stochasticity may be introduced in the initial conditions, the boundary conditions, or the integrator itself [109, p. 299]. In the current work, we introduce the stochasticity in the integrator itself in a targeted manner in order to minimize damage to the dynamics introduced by stochasticity. In order to reason about long time dynamics using stochastic MD, one may rely on the concept of weak convergence of stochastic processes, which states that only underlying probability distributions need to be close, cf. [54, pp. 23,326]. 1.5.3 Possible Applications of Targeted Mollified Impulse TM naturally samples the canonical ensemble, which is also supported by our analysis in Section 3.3.4. Three possible applications of TM are: (i) long-time kinetics, for which trajectories can be accurate only statistically, for long time limit of either a microcanonical or canonical distribution; (ii) short-time kinetics, for which trajectories can be accurate in a strong sense, and (iii) thermodynamics, such as structural and free energy calculations. First, TM is suitable for long-time simulations of chaotic or mixing Hamiltonian systems due to its computational efficiency through large time steps. Thus one can generate multiple trajectories using TM to calculate kinetics quantities such as the transition rates for structural changes. One can also use TM to facilitate the study of ensemble dynamics in protein folding, e.g., folding@home [96]: Initial conditions are chosen from some prescribed distribution and a large number of trajectories can be calculated with TM. If the governing equations of motion for the MD system are stochastic due to stochastic boundary conditions or implicit solvent, the long time limit is likely a canonical distribution; on the other hand, if the governing equations of motion are deterministic, the long time limit is likely a microcanonical distribution. In the latter case, the use of artificial Langevin coupling is speculative, since the stochastic artifacts may obscure delicate features of the

13

dynamics, such as the decay rate of correlation functions. Such features might require a smaller time step in any case. However, in the case of large number of atoms, N, the canonical ensemble is approximately the same as the microcanonical. More precisely, the long time distribution for Langevin coupling differs from constant energy by only O(1/N), see [1, Section 2.3]. Second, TM can be used for the calculation of short-time dynamical quantities from a canonical ensemble. Ideally, one should use Newtonian dynamics with initial conditions drawn from an NPT ensemble. In practice, dynamical data is gathered with temperature control still on. Targeted Langevin reduces the dynamical artifacts by only targeting the fastest interactions and conserving linear momentum. We use TM to compute the selfdiffusion coefficient from simulations of model systems of TIP3P water with flexible bonds and angles [48]. We are able to use a large time step of 16 fs for the outer time step and 2 fs for the inner time step. The baseline for comparison is the leapfrog method with time step of 1 fs. Third, TM can be used to compute structural quantities such as the radial distribution functions from a canonical ensemble. The targeted Langevin coupling allows these quantities to be computed more accurately as compared to other Langevin approaches because it imposes less randomness in the slower modes and thus helps to cross energy barriers. Results show that the kinetic temperature is properly bounded using TM with large time steps. We present results of the correct computation of radial distribution functions for the same system of flexible TIP3P water using large time steps. 1.6 Backward Euler Mollified Impulse We show a new MOLLY method, the Backward Euler MOLLY, in which the MOLLY averaging is based on backward Euler formulation. We numerically solve the following

14

nonlinear equation to get the averaged positions, A(q) ˜ A(q) = q + τ 2 M−1 F(A(q)),

(1.18)

˜ where q is the vector of original positions of atoms, τ is the time step, F(A(q)) is the force evaluated at A(q). With this formulation, the convergence to the solution is very ˜ More importantly, a subset fast at least when only bonded forces are included in F. of nonbonded forces, which mimic the librational modes and stretching modes in the hydrogen bonds, can be easily incorporated into the equations. The purpose of adding nonbonded forces into the averaging is to yield more stability. Initial results show that BE MOLLY is more stable than B-spline MOLLY when same fastest forces are included in the averaging (bonds and angles), cf. Chapter 4. As an additional benefit, we expect that greater step sizes can be expected as compared to what TM currently allows by combining Backward Euler MOLLY with targeted Langevin coupling. Implicit methods are generally not preferred in the context of heavy numerical computation. Here we use the implicit BE MOLLY mainly to investigate the theoretical step size limit of MOLLY integrators when a subset of nonbonded forces are included in the averaging, which mimic the librational and stretching motions in H-bonded molecules. 1.7 Software Engineering Issues The implementation of the integrators mentioned in this thesis and many other popular algorithms is made easy due to a good design of ProtoMol. Good software engineering practices include an object-oriented design and the use of inheritance and templates, which greatly increase code-reuse, and readability, maintainability, and reliability of the software. In particular, for the design of integrators, we use inheritance and MTS and STS integrators are two different main branches. Inheritance is well-suited to capture the unique and shared behaviors of integrators with great flexibility and negligible effect on run-time efficiency [46, 74, 75]. In ProtoMol, the integrators represent the middle layer. 15

The collaboration of an integrator object with its associated force objects is based on composition and encapsulation, see Fig. 1.2. The diagram illustrates the integrator hierarchy with two concrete integrator classes representing MTS and STS integrators. The force hierarchy is pruned at the top level. The forces to be used by an integrator object (representing one level in an MTS integrator chain) are encapsulated in ForceGroup. n Integrator #myPositions : CoordinateBlock * #myVelocities : CoordinateBlock *

ForceGroup 1

m +evaluateSystemForces() +evaluateExtendedForces()

#myForces : CoordinateBlock * #myEnergies : EnergyStructure * +run(numTimesteps) Force

ExtendedForce

StandardIntegrator

+evaluate() +parallelEvaluate()

+run(numTimesteps) #doHalfKick() #doDriftOrNextIntegrator() #calculateForces()

SystemForce

+evaluate() +parallelEvaluate()

MTSIntegrator

STSIntegrator

#myNextIntegrator : StandardIntegrator * #myCycleLength : int

−myTimestep : Real #doDriftOrNextIntegrator()

#doDriftOrNextIntegrator()

Leapfrog HybridMC +doDrift()

−myInitialTemperature : Real

+doHalfKick()

−myOldPositions : CoordinateBlock *

+doKickDoDrift()

−myOldTotalEnergy : Real

+run()

−acceptNewPositions() −calcRandVelocity() −returnToOldPositions()

doHalfKick(); doDriftOrNextIntegrator(); calculateForces(); doHalfKick();

+run() −updatePositions()

myForceGroup−>evaluateSystemForces() myForceGroup−>evaluateExtendedForces()

Figure 1.2. Collaboration between an integrator object and force objects.

16

1.8 Conclusions The original contributions of this dissertation work are summarized here, including nonlinear analysis of Impulse, efficient B-spline MOLLY, Backward Euler MOLLY, and Targeted MOLLY. Nonlinear Overheating of Impulse We have been able to pinpoint the mechanism of instability in MTS molecular dynamics – the nonlinear overheating – when the largest time step equals to one third, and possibly one fourth of the shortest period of the system. We call them 3:1 and 4:1 nonlinear overheating, respectively. Theoretical analysis and numerical experiments match exactly, cf. [69, 70]. B-spline MOLLY We derived analytical Hessians for various potential energies defined in the CHARMM force field, including bonds, angles, Lennard Jones and electrostatics,. We implemented the B-spline MOLLY using these analytical Hessians in ProtoMol. We show that B-spline MOLLY successfully overcome the linear stability barrier of Impulse (∆t = 5 fs for flexible water systems and explicitly solvated proteins/DNAs), cf. [46]. Backward Euler MOLLY We implemented a new MOLLY integrator based on the numerical solution of the backward Euler formulation of the averaged forces. Results show that BE MOLLY is more stable than B-spline MOLLY when same fastest forces (bonds and angles) are included in the MOLLY averaging. Such a formulation also allows a subset of nonbonded forces to be easily included in the MOLLY averaging, thus boosting the stability of the integrator itself.

17

Targeted MOLLY We designed this novel multiscale MTS integrator for MD simulations, which allows ∆t = 16 fs for flexible water simulations while still computing the dynamical and structural properties correctly as compared to leapfrog at ∆t = 1 fs, cf. [66–68, 75]. The large time step results in significant speedup of the simulations. This is an important result because Targeted MOLLY may be suitable for many different applications where long MD simulations are needed. Longer time steps enable long simulations to finish quicker using the same computing resources.

18

CHAPTER 2 NONLINEAR OVERHEATING OF IMPULSE

This chapter discusses additional stability limitations of multiple time stepping (MTS) integrators for molecular dynamics (MD) that attempt to bridge time scale 1 . In particular, it is shown that when constant–energy (NVE) simulations of Newton’s equations of motion are attempted using the MTS integrator Verlet-I [34]/r-RESPA [106]/Impulse (hereafter referred to as Impulse), there is nonlinear overheating, whose effect is a mild but systematic drift in the energy, when the longest step size is one third or possibly one fourth of the period(s) of the fastest motion(s) in the system. This is demonstrated both through the analysis of a nonlinear model problem and through a thorough set of computer experiments. The predicted and observed instabilities match exactly. A linear instability for Impulse at around half of the period of the fastest motion has been identified and explained by previous work [28, 101]. Nonlinear resonances in single time stepping MD integrators was discovered by Mandziuk and Schlick [73], empirical nonlinear overheating analysis for the single time stepping integrator leapfrog is in [95], but unstable nonlinear resonances for MTS integrators have not been reported by researchers other than us [69, 70]. 1

The rest of this chapter comes mainly from Q. Ma et al. Nonlinear instabilities in multiple time stepping molecular dynamics, cf. [69], and Q. Ma et al. Verlet-I/r-RESPA/Impulse is limited by nonlinear instability, cf. [70].

19

2.1 Stability Limitations of Impulse Molecular dynamics solves the equations of motion given in Eq. (1.2). In an attempt to bridge the time scale gap between simulations and the phenomena of interest, multiple time stepping integrators have been an area of active research for more than a decade [7, 14, 28, 29, 33, 34, 43–47, 58, 68–70, 93, 106, 110, 111]. The prototypical algorithm is the Impulse integrator, which splits the forces into fast and slow components, and evaluates the former more frequently than the latter: ∞ X d2 fast M 2q = F − ∆t δ(t − n0 ∆t)∇U slow (q). dt 0

(2.1)

n =−∞

Assuming the fast force is harmonic, the discretization of Eq. (1.2) using Impulse with step size ∆t for the slow part and analytical solution of the fast part with a spring constant of Ω2 is given by Algorithm 3. 1 2

kick:

∆t ∇U(q), 2 oscillate: Let s0 = sin hΩ and c0 = cos hΩ, we have s0 q1 q c0 Ω = p+ p+ −Ωs0 c0 0 1/2 p+ 0 = p−

1 2

(2.2)

(2.3)

kick: p1 = p+ 1/2 −

∆t ∇U(q). 2

(2.4)

Algorithm 3: The Impulse discretization with step size ∆t for the slow part and analytical solution of the fast part.

Impulse exhibits severe instability when the longest step size ∆t is a multiple of the shortest period and a numerical linear instability at half of the shortest period. These results have been confirmed through numerical experiments [15] and using simple linearforce model problems [6,14,93,101]. For explicitly solvated biological molecules without constraining of bond lengths, the shortest period of vibrational modes is around 10 fs and 20

the linear instability occurs at about 5 fs. Thus the five–femtosecond time step, equal to half of the shortest period of most biological molecules, has been termed as the “linear stability barrier” for allowable time steps [101]. A similar linear stability analysis of a 1-d linear problem reconfirms the linear stability barrier of Impulse, cf. [46]. Impulse also suffers nonlinear overheating which manifests itself by a mild but systematic drift in the energy when time steps are within certain ranges. Systematic drift can be observed in simulations reported in the literature even when using longest time steps around 3 or 4 fs, cf. Fig. 2 in [15], Fig. 3 in [112], and [42]. We show that there is a 3:1, and possibly a 4:1, nonlinear overheating in Impulse for both unconstrained and constrained dynamics, that significantly limits the stability region of the method. More precisely, there is a 3:1 unconditionally unstable resonance and a 4:1 conditionally stable resonance in Impulse. The nonlinear analysis is presented in Section 2.2. We also present empirical evidence of the nonlinear overheating through precise computer experiments in Section 2.3. Two sets of the flexible water simulations are performed: one under controlled conditions resembling the equilibrium points of the MTS integrator, which are close to, but not exactly, the state at zero temperature, where the KAM stability theory holds [97, p. 132–133] 2 . the other at room temperature. Both sets of experiments clearly reveal the 3:1 instability and the 4:1 resonance. One set of flexible protein (not solvated and not constrained) simulations reveal the 3:1 instability that correlates to several fastest modes that are very close to each other. The modes associated with stretching of bonds of polar hydrogens in the bio-molecules and the bond stretching and angle bending in the solvent (water) molecules can be eliminated altogether with proper constraining using SHAKE [11, 107] and RATTLE [3, 82]. 2

Kolmogorov-Arnold-Moser (KAM) Theorem was first outlined in 1954 by Kolmogorov and was subsequently proved in the 1960s by Arnold and Moser [105, p.109]. It gives conditions under which chaos is restricted in extent. A useful paraphrase of the KAM theorem is, ”For sufficiently small perturbation, almost all tori (excluding those with rational frequency vectors) are preserved.” Conditions for applicability of the KAM theorem are: small perturbations, smooth perturbations, and sufficiently irrational map winding number.

21

We show that constrained dynamics simulations of explicitly solvated bio-molecules using Impulse with SHAKE or RATTLE as the inner-most integrator still exhibit instabilities (4:1 and 3:1 nonlinear overheating) when outer time steps are greater than 4 fs for long simulations. Note that we may get instability even for longer step sizes than those suffering nonlinear overheating. This is so not only because we will be approaching the linear instability at half of the period, which manifests itself in the neighborhood of that step size, but also because that at non-zero temperatures different normal modes are mixed through energy transfer. Increasing computer power requires stricter stability properties. An early example of this phenomenon was Milne’s algorithm which was an efficient method until the use of electronic computers revealed the unfortunate long term effects of the weak (linear) instability [80, 81]. Thus, the nonlinear overheating of Impulse which is the focus of this chapter is likely to be very significant in long MD simulations due to the tenfold increase in computer power every five years (the Moore’s law) [78] and the desire to simulate longer processes that are of biological relevance and can be experimentally verified, such as the folding of proteins. 2.2 Nonlinear Stability Analysis Given here is a procedure for analyzing the stability of a reversible symplectic map, which extends the analysis of [91]. Stability conditions for Impulse are obtained by applying this procedure to multiple time stepping. 2.2.1 Assumptions and Procedure Let yn+1 = M(yn ) be the map of interest. In the present context M depends on a step size parameter h, so we may at times write Mh (y) instead of M(y). Reversible means

22

that RM(RM(y)) = y where R = diag(1, −1). Most practical reversible symplectic integrators, including simple implicit ones [104], can be expressed M(y) = RN −1 (RN(y))

(2.5)

where N(y) = Nh/2 (y) is itself an area-preserving map 3 . It is easily verified that M(y) is indeed reversible. Given here are the stability conditions for this important special case of reversible maps in terms of the simpler map N(y). The analysis is valid only in some neighborhood of a fixed point y ∗ = M(y ∗ ) of the map. We assume that the Jacobian matrix M 0 (y ∗) is power-bounded, which is necessary for stability. Also assume that y ∗ = (q ∗ , 0), which will be the case except possibly for values of h so large so as not to be of practical interest (see Section 2.2.5, Proposition 1). A procedure is described below: Step 1 Express 



 a11 a12  ∗ ∗ 2 N(y) = N(y ∗ ) +   (y − y ) + O(ky − y k ). −a21 a22

(2.6)

For stability it is necessary that either 0 < a11 a22 < 1 or a11 = a22 = 0 or a12 = a21 = 0 (see Section 2.2.5, Proposition 2). The symplectic property implies that the determinant a11 a22 + a12 a21 = 1. Step 2 Choose α 6= 0, β 6= 0 so that the map NY (Y ) = diag(α, 1/α) (N(y ∗ + diag(β, 1/β)Y ) − N(y ∗ )) satisfies



(2.7)



 γ σ  2 NY (Y ) =   Y + O(kY k ) −σ γ

(2.8)

For conventional methods the momentum reversal RNh/2 (Ry) is identical to the time reversal −1 −1 (Ry) is the same as the adjoint, N−h/2 (y) [39]. N−h/2 (y) and hence RNh/2 3

23

where σ 2 + γ 2 = 1. This can be done as follows (see Section 2.2.5, Proposition 3): α=

a21 a22 a11 a12

1/4 ,

β=

α= α=

a21 a12 a22 a11

a12 a22 a11 a21

1/4 if 0 < a11 a22 < 1,

(2.9)

1/2 1/2

β,

if a11 = a22 = 0,

(2.10)

1 , β

if a12 = a21 = 0.

(2.11)

Step 3 Express the map NY (Q, P ) in complex form as Nz (z, z¯) = µz + iµr(z, z¯)

(2.12)

where z = Q + iP , µ = γ − iσ, and r(z, z¯) = c1 z 2 + 2¯ c1 z¯ z + c2 z¯2 + c3 z 2 z¯ + c4 z¯3 + U.T.s.

(2.13)

The U.T.s (unimportant terms) are defined to be the z 3 term, the z¯ z 2 term, and those of degree 4 or more. This can always be done (see Section 2.2.5, Proposition 4). Express cj = aj + ibj where aj and bj are real, and define a = 2a1 ,

(2.14)

c = 2a2 ,

(2.15)

f = 2a3 − 12a1 b1 + 4a2 b2 ,

(2.16)

g = 2a4 + 4a2 b1 + 4a1 b2 .

(2.17)

Conclusion (See Section 2.2.5 for proof) Let λ = µ2 . 1. Third order resonance. Suppose λ3 = 1 but λ 6= 1. The map is stable at equilibrium if c = 0 and −σf − 3γa2 6= 0, and it is unstable if c 6= 0. Hence, third order resonance is normally unstable.

24

2. Suppose λ3 6= 1. Let F = −σ(4γ 2 − 1)f − 3γ(4γ 2 − 1)a2 − γ(4γ 2 − 3)c2 ,

(2.18)

G = −σ(4γ 2 − 1)g + 2γac.

(2.19)

(a) Fourth order resonance. Suppose λ4 = 1. The map is stable at equilibrium if |G| < |F |, and it is unstable if |G| > |F |. (b) Suppose λ4 6= 1. The map is stable at equilibrium if F 6= 0. 2.2.2 Application to Multiple Time Stepping We apply the analysis procedure just outlined to a nonlinear model problem to obtain the nonlinear stability conditions on multiple time stepping algorithms. Assume a 1-d nonlinear model problem with potential energy given by 1 1 U(q) = Ω2 q 2 + Aq 2 + |2 {z } | 2 oscillate

1 3 1 4 Bq + Cq +O(q 5), 3{z 4 }

(2.20)

kick

where the splitting between the oscillate and kick step for Impulse is done as indicated. The discretization of this problem using the first half of Impulse is given by: 1 2

kick: h 2 3 4 p+ 0 = p − (Aq + Bq + Cq ) + O(q ), 2

1 2

oscillate: Let s0 = sin hΩ and c0 2   q1/2  p1/2

= cos hΩ , we have 2     s0 0 q    c Ω  =  . −Ωs0 c0 p+ 0

Let λ = µ2 , where µ = γ − iσ in which     1 − h s0 0 A 1/2 c0 , 2 Ωc γ=   0, 25

c0 6= 0, c0 = 0,

(2.21)

(2.22)

    1+

and σ=

1/2 h c0 A 2 Ωs0

s0 ,

  0,

We assume that either −(s0 )2
2hB 2 s0 c0 /Ω. It is unstable if 0 < C < 2hB 2 s0 c0 /Ω. Thus, Impulse may or may not be stable at the fourth order resonance. This fourth order resonance is observed in our numerical experiments, although our experiments are not conclusive regarding whether this is an unstable nonlinear resonance. Using the stability conditions for fourth order resonance in MTS integrator (C > 2hB 2 s0 c0 /Ω), we obtain the result for the single time stepping leapfrog integrator for the case Ω → 0 and A = ω 2 : C>

2 2 B . ω2

(2.23)

Note that this is so because the largest allowable time step, ∆t, for leapfrog is given by ω∆t < rather than ω∆t < 2, cf. [97]. 26

√

2

(2.24)

2.2.3 Proof of Main Result Step 1 Eq. (2.22) can be rewritten as    h s0 0  q1/2   c − 2 ΩA   =  p1/2 −Ωs0 − h2 c0 A

s0 Ω

c0

 









  q  h  0  2 3   (Bq + Cq )    −  2 1 p

+ O(q 4).

(2.25)

The elements of the matrix determine the linear stability condition stated in the result. Step 2 In the case of −(s0 )2
0 and that there is a stable equilibrium at r = r∗ > 0. Writing r = r∗ + q, we have 1 U(r∗ + q) = U(r∗ ) + Ω2 q 2 + kr∗−3 q 2 − kr∗−4 q 3 + kr∗−5 q 4 + O(q 5), 2

(2.39)

which yields A = 2kr∗−3 ,

B = −3kr∗−4 ,

C = 4kr∗−5.

(2.40)

The condition given for stability becomes k < 0 or

h2 k 2 r∗−3

sin hΩ 4 < k, hΩ 9

(2.41)

and the above condition is satisfied if 4 h2 kr∗−3 < . 9

(2.42)

This relation can be interpreted in terms of the error due to the finite step size h used to sample the slow force. From [64, Eq. (10)] it follows that discretization introduces an error

1 2 h (−kr∗−2 )2 24

in the potential energy, and comparing this to the potential energy

kr∗−1 itself we get the quantity of estimated relative error rel =

1 2 h kr∗−3 . 24

With this

definition the condition for stability can be expressed rel
0 as long as Mδ0 (yδ∗ ) 6= I assuming consistency of Mδ , U 0 (q0∗ ) = 0, and U 00 (q0∗ ) > 0 where U refers to the potential energy as given by Eq. (2.20).

a11 a12 Proposition 2 Let N (y ) = be the Jacobian matrix of the map N at −a21 a22 the fixed point y ∗ . For stability of Mh , it is necessary that either 0 < a11 a22 < 1 or a11 = a22 = 0 or a12 = a21 = 0. 0

∗

30

Proof. Multiplying both sides of Eq. (2.44) by R and then applying the map N, we have N(RMh (y)) = RN(y).

(2.49)

Forming the Jacobian matrix on both sides at the fixed point y ∗ , we have N 0 (Ry ∗ )RM 0 (y ∗) = RN 0 (y ∗ ),

(2.50)

which leads to −1

M 0 (y ∗) = RN 0 (y ∗ ) RN 0 (y ∗ ).

(2.51)

The symplecticness property of N implies a11 a22 + a12 a21 = 1. The inverse of the Jacobian is given by

 N 0 (y ∗ )

−1

 a22 = a21

 −a12  . a11

(2.52)

Thus, we have 

  a11 a22 − a12 a21 Mh0 (y ∗ ) =  −2a11 a21

2a12 a22 a11 a22 − a12 a21

 .

(2.53)

¯ of Because M 0 (y ∗) is power bounded and detM 0 (y ∗ ) = 1, it has two eigenvalues λ, λ unit modulus 4 and hence |trace(M 0 (y ∗))| ≤ 2. If the trace is less than 2 in magnitude, then 0 < a12 a21 < 1. If its magnitude is 2, then power-boundedness implies that the off-diagonal elements of M 0 (y ∗) are both zero, which further implies that a11 = a22 = 0 or a12 = a21 = 0. 2 Proposition 3 Let

NY (Y ) = D1 (N(y ∗ + D2 Y ) − N(y ∗ ))

(2.54)

where D1 = diag(α, 1/α) and D2 = diag(β, 1/β). We can choose D1 and D2 so that NY0 (0) is a rotation matrix and so that MY (Y ) = RNY−1 (RNY (Y )) is stable at Y = 0 if and only if M(y) is stable at y = yh∗ . 4

√ The eigenvalues of M 0 (y ∗ ) are λ1,2 = a11 a22 − a12 a21 ± 2i a11 a12 a21 a22 .

31

Proof. First, y ∗ = RN −1 (RN(y ∗ )),

(2.55)

N(Ry ∗ ) = RN(y ∗ ).

(2.56)

which implies

Hence, since y ∗ = (q ∗ , 0), we have RN(y ∗ ) = N(Ry ∗ ) = N(y ∗ ).

(2.57)

Multiplying both sides of Eq. (2.54) by D1−1 , we have N(y ∗ + D2 Y ) = D1 −1 NY (Y ) + N(y ∗ ).

(2.58)

From the above we have Y = D2 −1 (N −1 (D1 −1 NY (Y ) + N(y ∗ )) − y ∗).

(2.59)

which implies that the inverse of the map NY (Y ) is given by NY −1 (Y ) = D2 −1 (N −1 (D1 −1 Y + N(y ∗ )) − y ∗).

(2.60)

Then MY (Y ) = RNY−1 (RNY (Y )) = RD2−1 (N −1 (D1−1 (RD1 (N(y ∗ + D2 Y ) −N(y ∗ ))) + N(y ∗ )) − y ∗) | | {z } {z } cancel

(2.61)

cancel

Because D1−1 RD1 = R and Eq. (2.57), this becomes MY (Y ) = RD2−1 (N −1 (RN(y ∗ + D2 Y )) − y ∗)

(2.62)

MY (Y ) = D2−1 M(y ∗ + D2 Y ) − y ∗ ,

(2.63)

or

32

which is a symplectic transformation, y = y ∗ + D2 Y of the map M. A symplectic transformation preserves the stability property. Finally NY0 (0) = D1 N 0 (y ∗ )D2 ,

(2.64)

which can be verified to be a rotation matrix. 2 Proposition 4 Let

NY (Y ) =

γ −σ

σ γ

Y + O(kY k2 )

(2.65)

be symplectic. Let z = Q + iP, and let µ = γ − iσ. Then the map becomes def

z 7→ Nz (z, z¯) = µz + iµr(z, z¯)

(2.66)

r(z, z¯) = c1 z 2 + 2¯ c1 z¯ z + c2 z¯2 + c3 z 2 z¯ + c4 z¯3 + U.T.s.

(2.67)

where

Proof. See [91], Eq. (2.15).

2

Proof of Conclusion. Perform a symplectic change of variables X = NY (Y ) and the map becomes X 7→ MX (X) = NY (RNY−1 (RY )), def

(2.68)

which can be expressed as X1 = MX (X0 ) where X1 = NY (X1/2 ),

X0 = RNY (RX1/2 ).

(2.69)

The new map MX has the same stability properties at the origin as the map MY , and it is also reversible. Expressed in complex form z1 = Nz (z, z¯),

¯z (z, z¯), z0 = Nz (¯ z , z) = N

(2.70)

¯z is Nz with its coefficients complex conjugated. The map MX satisfies the where N hypotheses of Lemma 4.2 of [91] with λ = eiφ = µ2 , so its complex equivalent has the form z1 = µ2 z0 + iµL(µz0 , µ ¯z¯0 ), 33

(2.71)

where L(z, z¯) = az 2 + 2az¯ z + c¯ z 2 + (f + i(a2 − c2 ))z 2 z¯ + g¯ z 3 + U.T.s

(2.72)

and a, c, f , g are real. Substituting Eq. (2.70) into Eq. (2.71) gives ¯z (z, z¯), µ ¯z (z, z¯) + iµL(µN Nz (z, z¯) = µ2 N ¯Nz (¯ z , z)).

(2.73)

From Eq. (2.66), we have ¯z (z, z¯) = µ N ¯z − i¯ µr¯(z, z¯),

(2.74)

and substituting both equations into Eq. (2.73) gives r(z, z¯) = −¯ r (z, z¯) + L(z − i¯ r(z, z¯), z¯ + ir(¯ z , z)).

(2.75)

Expanding this and using Eq. (2.72) gives L(z, z¯) = r(z, z¯) + r¯(z, z¯) + iψ(z, z¯) + U.T.s,

(2.76)

ψ(z, z¯) = 2 (a(z + z¯)¯ r(z, z¯) − (az + c¯ z )r(¯ z , z)) .

(2.77)

where

Using Eqs. (2.67) and (2.72), we equate coefficients to get a = 2a1 ,

(2.78)

c = 2a2 ,

(2.79)

f = 2a3 − 12a1 b1 + 4a2 b2 ,

(2.80)

g = 2a4 + 4a2 b1 + 4a1 b2 .

(2.81)

The conclusion follows from Theorem 4.3 of [91] using eiφ/2 = µ = γ − iσ.2

34

2.3 Numerical Experiments Numerical results on the step-size-related, nonlinear overheating are obtained for three model systems: the small flexible water system, the flexible Melittin protein system, the SHAKE–constrained solvated Melittin protein system. In the interest of reproducibility of our results, we provide additional details of the flexible water systems in Appendix A.1. The numerical experiments in this section show that there is overheating at around a third or possibly a fourth of the period(s) of the fastest motion(s) when integrating Newton’s equations of motion using Impulse. All simulations use the CHARMM force field [71, 72]. The numerical experiments also show that for realistic biological systems such as explicitly solvated proteins, the step size is also limited by nonlinear overheating even when the bonds of polar hydrogens in the proteins and the bonds and angles in waters are made rigid using the SHAKE or RATTLE algorithm. The justification of freezing the almost decoupled high frequency stretching can be found in [88]. Unstable resonances usually manifest themselves in the neighborhood of a certain step size: There is a definite range of step sizes that cause unbounded energy drift even if the neighboring step sizes do not seem to cause any problems. Examples of this resonance phenomenon are presented in [15]. 2.3.1 Model Systems The real world problems are interesting and important, but they are usually too large for algorithm prototyping and extensive evaluations. We choose some of the simpler and smaller problems as our model problems for algorithm testing because testing is very tedious and time–consuming. The numerical challenges remain unchanged or even more pronounced (such as stability restrictions) but the turn-around time for testing is significantly reduced using small model problems. Numerical experiments have been performed using flexible water, based upon the

35

TIP3P model [48], with flexibility incorporated by adding bond stretching and angle bending harmonic terms, cf. [57, p. 184], flexible proteins, and proteins in SHAKEconstrained rigid waters. Flexible water also is termed as “neat water” [113]. Potentials for flexible water are detailed in Appendix A.1 Experiments such at those in [15] suggest that flexible water models are particularly sensitive to de-stabilizing artifacts in numerical integrators. System of flexible water has fastest motions with periods of around 10 fs. ˚ radius sphere with 423 atoms. We have used Model-S, the “small” water system, 10 A The same system has also been used for testing the MOLLY and Targeted MOLLY meth˚ for this model system although in ods, cf. Chapter 3. The cutoff distance is set to 6.5 A ˚ Setting a smaller cutoff distance will allow instabilities to practice one can set it to 8.0 A. appear earlier, if any, because the slow force contributions are bigger. For stability testing, we also used a flexible protein system with PDB name 2mlt [30] (Melittin) as our model system. This system has two proteins, each containing 434 atoms. The shortest periods are about 9 fs, 10 fs and 11 fs which correspond to O-H, N-H and C-H stretching, respectively. For stability testing, we also used as a model system an explicitly solvated, rigid wa˚ 38 A× ˚ ter/protein system: Melittin (model system B) immersed in a box (about 58 A× ˚ of rigid TIP3P water molecules. This system contains 5143 atoms. In the simula25 A) tions of this system, the bonds of polar hydrogens in the protein, and the O-H bonds and H-O-H angles in waters are made rigid by using the SHAKE method. The periods of the remaining fastest modes are in the range of 18 to 24 fs, which correspond to the H-X-H angle bending (where X represents a non-hydrogen atom) and C=C stretching. 2.3.2 Power Spectrum Analysis All possible vibrations of a molecular system can be described as a superposition of fundamental oscillations which are called normal modes for the molecules. Power

36

spectrum analysis of the systems of interest forms the basis of correlating the time step related nonlinear overheating with one or many of the normal modes. We perform power spectrum analysis of the time history of the energy of the simulations to reveal the characteristic frequencies of the normal modes of the system. Then we are able to correlate the normal modes with the step size related energy drift. Wave-number unit is typically reported in the literatures, which is the number of waves per centimeter 5 . The wave-numbers and periods of the fastest normal modes of the systems of interest are presented in Fig. 2.1 for the small water system, Fig. 2.2 for the flexible protein system and Fig. 2.3 for the rigid water/protein system. In these simulations, Impulse uses an inner time step of 1 fs and outer time step of 2 fs and each simulation is 200 ps. 0.4

FD, Impulse (∆ t, δ t) = (2.0,0.1)

0.4

0.35

Power spectrum for KE


0.35 0.3 0.25 0.2 0.15

0.3 0.25 0.2 0.15

0.1

0.1

0.05

0.05

0 1000

1500

2000 2500 3000 Wave number [cm−1]

3500

0 5

4000

(a)

10

15 Period [fs]

20

25

(b)

Figure 2.1. (a) Wave-numbers and (b) periods of flexible water systems. Not surprisingly, the step size used in the MD simulations affects the accuracy of the frequencies (and thus periods and wave-numbers) of the system. For the flexible water system, the frequencies are obtained through different simulations with different step sizes and are summarized in Table 2.1. The relationship between the wave-number, Nwave−number ([cm−1 ]) and the period, T ([fs]) is given as follows: Nwave−number = 105 /(3T ). 5

37

1 0.9

0.8

0.8 Power spectrum for TE

Power spectrum for TE

1 0.9

0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0 2000

2500

3000 −1 Wave number (cm )

3500

0 5

4000

6

7

8 9 10 11 12 Periods of leading modes

(a)

13

14

15

(b)

0.025

0.025

0.02

0.02 Power spectrum for KE


Figure 2.2. (a) Wave-numbers and (b) periods of flexible Melittin protein systems.

0.015

0.01

0.005

0 1000

0.015

0.01

0.005

1200

1400 1600 −1 Wave number [cm ]

1800

0 15

2000

(a)

20

25 Period [fs]

30

35

(b)

Figure 2.3. (a) Wave-numbers and (b) periods of the rigid Melittin/water system. 2.3.3 Simulation Protocol Each simulation of flexible waters has a length of 500 ps. The system was minimized using 10000 steps of conjugate-gradient minimization. Then the system was equilibrated for 100 ps. One system was equilibrated at 0.015 K and the other was equilibrated at 300 K. The temperature of the former is close to zero K under which the KAM theorem is applicable. Each simulation of the flexible Melittin proteins has a length of 10 ns. The system was minimized using 80000 steps of conjugate-gradient minimization. Then the system was equilibrated for 200 ps at 300 K. Each simulation of the explicitly solvated 38

Table 2.1. THE SHORTEST PERIODS FOR FLEXIBLE WATER SYSTEM. Integrator (∆t, δt) Impulse (2.0, 0.1) Impulse (2.0, 1.0) Leapfrog (−, 2.0)

Periods for Bond Stretching 9.87, 10.07 9.71, 9.91 9.12, 9.33

2mlt proteins system has a length of 500 ps. The bonds of polar hydrogens in the protein, the O-H bonds and H-O-H angles in water are made rigid using the SHAKE method. The system was minimized using 30000 steps of conjugate-gradient minimization. Then the system was equilibrated for 200 ps at 300 K. We used the program NAMD2.3 [49] to minimize and equilibrate the flexible water system. Then we ran simulations of this system using ProtoMol [46, 74–77]. For the protein and the solvated protein systems, we used the program NAMD2.5 to minimize and equilibrate them, and used the same program for the actual simulation runs. To measure instabilities, we use the metric of percent relative drift of total energy for every simulation, which is detailed in Appendix A.2. 2.3.4 Summary of Results Here we summarize of the numerical results obtained for flexible water simulations at close to zero temperature and room temperature, and room temperature simulations of flexible protein and rigid protein/water systems. Flexible Water at Near Zero Temperature For the simulations of flexible water at 0.015 K, the overheating associated with outer step sizes is plotted in Fig. 2.4. Each data point is obtained from a 500 ps simulation with a step size ∆t given by the x-axis, and an innermost step size δt equal or very close to 0.1 fs so that that the overheating of the simulations are not due to the errors in the inner integrator, cf. [10]. The peaks at step sizes of 2.57 fs and 3.33 fs show evidence of milder 39

4:1 resonance and unstable 3:1 resonance (overheating).

Percent Relative Drift of Total Energy

1

0.8

0.6

0.4

0.2

0

2

2.2

2.4

2.6 2.8 3 Time step (fs)

3.2

3.4

3.6

Figure 2.4. Energy drift for flexible water simulations at 0.015 K using Impulse. Each data point is obtained from a 500 ps simulation. Notice that the peaks at step sizes of 2.57 fs and 3.33 fs show evidence of milder 4:1 resonance and unstable 3:1 resonance (overheating).

Figs. 2.5 and 2.6 show the details of the energy drift of the simulations by plotting the block-averaged energy output to visualize the nonlinear overheating for simulations at near zero temperature. In Fig. 2.5, the percent relative drift of total energy of the three simulations is 0.01%, 0.82% and 0.01% for ∆t = 3.20 fs, ∆t = 3.33 fs and ∆t = 3.50 fs, respectively. In Fig. 2.6, the percent relative drift of total energy of the three simulations is 0.004%, 0.119% and −0.038% for ∆t = 2.43 fs, ∆t = 2.57 fs and ∆t = 2.70 fs, respectively. All curves in both figures have been shifted for clarity. 40

−1642.9672 (∆ t, δ t) = (3.20000, 0.10000) (∆ t, δ t) = (3.33330, 0.11111) (∆ t, δ t) = (3.50000, 0.10000) −1642.9673 ∆ t = 3.3333

Total Energy (kcal/mol)

−1642.9673 ∆ t = 3.5000 −1642.9674

−1642.9674

−1642.9675

−1642.9675

∆ t = 3.2000

0

0.5

1

1.5

2

2.5 Time (fs)

3

3.5

4

4.5

5 5

x 10

Figure 2.5. Block-averaged drift of total energy for 500 ps of molecular dynamics simulation of the small flexible water system using Impulse. It illustrates the 3:1 nonlinear resonance at one third of the fastest period near zero K. Flexible Water at Room Temperature We also explored whether the overheating effect is present at room temperature using the same system, except that now it has been equilibrated at 300 K. We are able to pinpoint the same overheating as in the simulations near the equilibrium point. The results are shown in Fig. 2.7 in which the peaks at step sizes of 2.40 fs and 3.33 fs show evidence of 4:1 resonance and 3:1 overheating. Fig. 2.7 also shows that one may get overheating even for longer step sizes in the neighborhood of nonlinear resonances (the last few data points with 3.33 < ∆t ≤ 3.80 (fs)). Similarly, the details of the energy drift for room temperature simulations are shown 41

−1642.97599 (∆ t, δ t) = (2.4336, 0.1014) (∆ t, δ t) = (2.5700, 0.1028) (∆ t, δ t) = (2.7000, 0.1000) −1642.97600


∆ t = 2.5700

−1642.97601

∆ t = 2.7000

−1642.97602

∆ t = 2.4336

−1642.97603

−1642.97604

0

0.5

1

1.5

2

2.5 Time (fs)

3

3.5

4

4.5

5 5

x 10

Figure 2.6. Same as Fig. 2.5 except that it shows evidence of a 4:1 nonlinear resonance. in Figs. 2.8 and 2.9. In Fig. 2.8, the percent relative drift of total energy of the three simulations is 0.70%, 5.42% and 1.07% for ∆t = 3.10 fs, ∆t = 3.33 fs and ∆t = 3.50 fs, respectively. In Fig. 2.9, the percent relative drift of total energy of the three simulations is 0.11%, 1.20% and 0.24% for ∆t = 2.33 fs, ∆t = 2.40 fs and ∆t = 2.47 fs, respectively. All curves in both figures have been shifted for clarity. Flexible Protein at Room Temperature In addition to simulations of flexible waters, we perform simulations of the flexible Melittin protein. The overheating associated with outer step sizes is plotted in Fig. 2.4 in which each data point is obtained from a 10 ns simulation with a step size ∆t given by the x-axis, and an inner step size δt equal to 1/3 of outer step size. The peaks at step 42


6 5 4 3 2 1 0 −1

2

2.2

2.4

2.6

2.8 3 3.2 Time step (fs)

3.4

3.6

3.8

Figure 2.7. Energy drift for flexible water simulations at 300 K using Impulse. Each data point is obtained from a 500 ps simulation. Notice that the peaks at step sizes of 2.40 fs and 3.33 fs show evidence of 4:1 resonance and 3:1 resonance (overheating). sizes of ∆t = 3.00, 3.27, 3.78 fs, show evidences of 3:1 overheating, which correspond to one third of the periods of O-H, N-H and C-H stretching, respectively. PME is used for Coulomb force evaluation [8, 9, 89, 90, 113]. Rigid Water/Protein at Room Temperature Finally, we perform simulations the solvated 2mlt system SHAKE-constraining all the bonds of polar hydrogens in the protein and the waters. The results are shown in Fig. 2.11 in which each point is obtained from a 500 ps MD simulation with a step size ∆t given by

43

(∆ t, δ t) = (3.1000, 0.1000) (∆ t, δ t) = (3.3363, 0.1011) (∆ t, δ t) = (3.5000, 0.1000) −755

∆ t = 3.3363


−760

−765 ∆ t = 3.5000 −770

−775 ∆ t = 3.1000 −780

−785

0

0.5

1

1.5

2

2.5 Time (fs)

3

3.5

4

4.5

5 5

x 10

Figure 2.8. Same as in Fig. 2.5 except that the temperature here is 300 K. the x-axis, and an inner step size δt in the range of 0.82 to 1 fs. PME is used for Coulomb force evaluation. Simulations with outer time step greater than 4 fs are unstable. It is hard to make any specific identifications of nonlinear resonance just from these figures because the remaining modes are continuous. Most likely these drifts correspond to the combined effects of 4:1 and 3:1 resonances associated with the remaining modes including angle bending, C=C stretching, and some of the non-bonded interactions. 2.4 Discussion In protein simulations, there are possibly several other factors that may also contribute to instability and overheating. One such factor could be that the cutoff radii for the short44

−776 (∆ t, δ t) = (2.3322, 0.1014) (∆ t, δ t) = (2.4000, 0.1000) (∆ t, δ t) = (2.4672, 0.1028)

−777


−778

−779 ∆ t = 2.4672 −780

∆ t = 2.4000

−781

−782

∆ t = 2.3322

−783

−784

0

0.5

1

1.5

2

2.5 Time (fs)

3

3.5

4

4.5

5 5

x 10

Figure 2.9. Same as in Fig. 2.8 except that it shows evidence of a possible 4:1 resonance. /intermediate-/long-range forces for Coulomb interactions in Ewald splitting [8, 9, 89, 90, 113] are not perfectly matched. Larger ∆t are allowed with Impulse through the use a true distance based splitting for Coulombic forces evaluated using PME method, cf. [113, Section IV] and [90]. Nonetheless, the step size related nonlinear instabilities should not be neglected. In particular, although 4:1 nonlinear instability could be eliminated by designing a switching function that satisfies the inequality equation, Eq. (2.42), 3:1 nonlinear instability is a general phenomenon. In some applications, accuracy limits the time step, but in the important cases shown here, the time step is limited by stability. The nonlinear overheating and linear instabilities severely limit the time steps of Im-

45


7 6 5 4 3 2 1 0 2.8

3

3.2 3.4 Time step (fs)

3.6

3.8

4

Figure 2.10. Energy drift for flexible Melittin protein simulations at 300 K using Impulse. Each data point is obtained from a 10 ns simulation. Notice that the peaks at step sizes of ∆t = 3.00, 3.27, 3.78 fs, show evidences of 3:1 overheating. pulse integrator for MD simulations of biological molecular systems. Though the mollified Impulse (MOLLY) methods are proven to overcome the linear instabilities of Impulse effectively [46, 47], methodologies for stabilizing nonlinear overheating have not been well studied. We propose an effective way of alleviating the nonlinear overheating – a two-part solution – through the combination of MOLLY and Targeted Langevin Coupling, which is the subject of next chapter.

46

12 10 8 6 4 2

4.5

5

5.5

2

5.5

1.8

5



0 4

1.6 1.4 1.2 1 0.8 0.6

4.5

4.6

4.7

4.8 4.9 Time step (fs)

5

5.1

6 6.5 Time step (fs)

7.5

8

10

4.5 4 3.5 3 2.5 2 1.5

7 Percent Relative Drift of Total Energy


14

9.5 9 8.5 8 7.5 7 6.5 6 5.5

5.5

5.6

5.7

5.8

5.9 6 6.1 Time step (fs)

6.2

6.3

6.4

6.5

5

6.7

6.8

6.9

7 7.1 7.2 Time step (fs)

7.3

7.4

7.5

Figure 2.11. Energy drift of simulations of explicitly solvated Melittin system at 300 K using SHAKE-Impulse. Each point is obtained from a 500 ps MD simulation. Most likely these drifts correspond to the combined 4:1 and 3:1 resonances associated with the remaining modes.

47

CHAPTER 3 TARGETED MOLLIFIED IMPULSE

We have shown that Impulse is not only limited by linear instability but also nonlinear overheating. This discovery motivated us to introduce Langevin coupling technique into the integrators to stabilize the integrator while retaining some accuracy. Previous success in constructing multiscale integrators for molecular dynamics (MD) includes MOLLY [28, 29, 46, 47] and Langevin stabilization of MOLLY (LM) [44, 45]. MOLLY methods themselves successfully break the linear stability barrier [46, 47], but may be still subjected to nonlinear overheating due to the existence of fast vibrational modes. In LM, a mild damping and random noise is added to the system to remove instabilities while still retaining some accuracy in the dynamics. Though not known by the time of LM was introduced, the nonlinear overheating may have been suppressed by the use of Langevin forces in the LM method. Using the idea of Langevin stabilization as well as the idea of pairwise Langevin interactions in Dissipative Particle Dynamics (DPD) [13, 32, 84, 85], we have devised a new integrator, Targeted MOLLY (TM) 1 . TM is a combination of two integrators: MOLLY as the outer integrator and a self–consistent dissipative leapfrog as the inner integrator in which targeted Langevin coupling is used to remove some instabilities (nonlinear overheating) while preserving linear momenta and thus retaining more accuracy in dynamics. In the current evaluation, we use B-spline 1

The rest of this chapter comes from Q. Ma and J. Izaguirre, Long time step molecular dynamics using targeted Langevin stabilization, cf. [66], and Q. Ma and J. Izaguirre Targeted Mollified Impulse – a multiscale stochastic integrator for molecular dynamics, cf. [68].

48

MOLLY that exploits analytical Hessians of energies. MOLLY’s enhanced stability plus the targeted Langevin coupling allow accurate computation of dynamics with ∆t larger than previous methods. Results from the evaluation of TM on model systems of TIP3P water with flexible bonds and angles [48] show that TM allows up to 16 fs for the outer time step while still computing the dynamical and structural properties correctly. Section 3.1 lists a simple procedure to estimate the maximum allowable stepsizes for MTS schemes. Section 3.2 covers some background of Langevin–related methods including Langevin dynamics, Langevin stabilization and the Dissipative Particle Dynamics (DPD). Section 3.3 details the design of TM and the study of its sampling properties. Section 3.4 presents the numerical results of TM. 3.1 The Maximum Allowable Step Size In theory, how large can the outer time step be for MTS schemes? This question becomes pressing as one tries to push up the step sizes in MTS schemes in multiscale MD simulations. Here we outline a simple procedure to address this question. The largest allowable time steps based on accuracy concerns can be estimated according to the characteristic time of the system [64]. tchar=2π(ρ(M−1/2 Uslow −1/2 ))−1/2 xx M

(3.1)

where ρ(A) is the spectral radius of the matrix A. The characteristic time corresponds to the fastest normal mode of the slow interactions, which happens between two atoms separated by cutoff distance, rc . To retain enough accuracy in incorporating the slow force contributions to the solution to the governing equations of motion, it is necessary to incorporate the slow force contributions every one tenth of the characteristic time of the slow forces. The value of one tenth of tchar for the time step ∆T gives an error of 1.5% in the fastest normal mode of a quadratic potential.

49

We estimate the characteristic time of a solvated molecular system through the use of a 1-d model. The model consists of two atoms who interact with each other through the Coulombic potential, Eq. (A.43). And these two atoms each belong to a different water molecule and are separated by a cutoff distance, rc . In Eq. (A.43), we have 14 = 1 and C/0 = 8.98755 × 109 Nm2 C−2 . Other known variables and constants are: mH = 1.008 amu, mO = 15.9994 amu, qH = 0.417 e and qO = 0.814 e (1amu = 1.6605402 × 10−27 kg, 1e = 1.60217733 × 10−19 C.) For our simple model, the energy function and Hessian reduce to very simple forms. Assume the equation of the Coulomb energy is given by U(x1 , x2 ) = C1 / |x1 − x2 | ,

(3.2)

where C1 = C14 qi qj /0 . For different interactions, this constant is C1H−H = 4.0118 × 10−29 , C1H−O = 8.0235 × 10−29 and C1O−O = 1.6047 × 10−28. Without loss of generality, assume x1 > x2 . Then the Hessian, Uxx , becomes the following:   Uxx

 = 2C1 

1 (x1 −x2 )3

−1 (x1 −x2 )3

−1 (x1 −x2 )3

1 (x1 −x2 )3

 .

(3.3)

Assume the mass matrix is M = diag(m1 , m2 ). Thus   B = 2C1 



1 (x1 −x2 )3 m1

−1√ (x1 −x2 )3 m1 m2

−1√ (x1 −x2 )3 m1 m2

1 (x1 −x2 )3 m2

whose spectral radius is ρ(B) = max{λ1 , λ2 } =

2C1 (m1 +m2 ) . m1 m2 (x1 −x2 )3

 ,

(3.4)

By making x1 − x2 = rc ,

and plugging in the known variables, we can compute the spectral radius and thus the characteristic time for different rc values very easily. O-H interactions are the fastest among O-H, H-H and O-O interactions. Thus, we have Eq. (3.5) to compute the maximum allowable step sizes for different cutoff values, see also Figure 3.1. ∆tmax = 32.61(rc/6.5)1.5 .

50

(3.5)

˚ and Examples of maximum allowable time steps are ∆tmax ≈ 32 fs for rc = 6.5 A, ˚ Note that this procedure can be used to estimate charac∆tmax ≈ 96 fs for rc = 13.5 A. teristic of other systems as well.

110 100 90

rc

∆ t max (fs)

80 70

+ +

60

--

--

50

←∆t max = 32.61 (rc / 6.5)1.5

40 30 20 10 3

4

5

6

7

8

9

10

11

12

13

14

Cutoff distance, rc (angst) Figure 3.1. The maximum allowable step size for systems with different cutoff distances.

3.2 Langevin Related Methods Langevin stabilization of Newtonian dynamics has been proven to be successful in lengthening the time steps allowed by the integrators [47]. Here we briefly review the techniques related to Langevin dynamics including LN, LM and DPD. LN is a 3-level MTS integrator, cf. [93], for Langevin dynamics governed by the 51

following stochastic differential equation: MdV (t) = −∇U(X(t))dt − γMV (t)dt +

p 2γkB T M 1/2 dW (t),

(3.6)

where X(t) is the position, V (t) is the velocity, γ is the collision parameter, W (t) is a vector of independent standard Wiener processes 2 . kB is the Boltzmann constant, and T is the Langevin bath temperature. These equations correspond to a constant temperature ensemble. LN uses constant extrapolation (CE) on the outer time step, midpoint extrapolation (ME) on the medium time step, and position Verlet with BBK discretization [18] on the innermost time step. LN’s outer integrator has a linear instability that is removed by damping and the Langevin coupling [93]. The damping coefficient, γ, needs to be 5-20 ps−1 to stabilize LN; this damping destroys many important dynamical properties of the system. While it is not possible to use LN for short-time kinetics, this method is one of the best, however, if Langevin dynamics appropriately models the problem at hand, such as the case of long-time kinetics using implicit solvent or thermodynamic computations. Langevin coupling can also be used to stabilize integrators for Newtonian dynamics, termed as Langevin Stabilization, in which the damping needs to be smaller than the natural decay rate of the system (e.g., γ = 0.2 ps−1 ) so that dynamical information obtained using such methods is very close to MD. Langevin stabilization of Equilibrium MOLLY (LM) uses Langevin-Impulse [45, 102] and MOLLY, and Langevin forces are introduced in the inner integrator to stabilize MOLLY. Much lower values of γ, can be used to stabilize LM. And more importantly, because the damping coefficient is small, the numerical artifacts are reduced such that dynamical properties such as the self-diffusion coefficient could still be computed correctly as compared to those obtained by using leapfrog method with time step δt = 1 fs [45]. MOLLY itself allows outer time step ∆t to be 6 fs with A standard Wiener process, W = {W (t), t ≥ 0} , is a process with independent increments for which W (0) = 0 with probability 1, E(W (t)) = 0, Var(W√(t) − W (s)) = t − s for all 0 ≤ s ≤ t. An important computational property is that W (tn ) = W (tn−1 )+ tn − tn−1 Z n where Z n is a sequence of independent standard Gaussian random variables with mean 0 and variance 1 [54, p. 28]. 2

52

accuracy comparable to Impulse at ∆t = 3 fs [47], and LM allows outer time step ∆t to be 12 fs [45] (a 300% asymptotic speedup). Dissipative particle dynamics (DPD), a method similar to MD, is another successful example in lengthening the time steps [13, 32, 84, 85]. It is a “coarse-grained” simulation technique. Particle positions and velocities are continuous and their values are determined by knowledge of the forces acting on the particles. Besides coarse-graining, DPD introduces the Langevin force in a pairwise manner which results in momentum conservation in simulations. It has been shown that in order to recover the proper thermodynamic equilibrium for a DPD fluid at a prescribed temperature T, it is necessary to impose a fluctuation-dissipation relation between the dissipative and random forces. DPD can reproduce Boltzmann statistics and Navier-Stokes behavior, which make it an attractive numerical technique to simulate complex fluids [85]. The original DPD model has serious problems: essentially all traditional integration schemes lead to distinct deviations from the true equilibrium behavior, including an unphysical systematic drift of the temperature from the value predicted by the fluctuation-dissipation theorem, and artificial structures in the radial distribution function. Pagonabarraga et al. improved the DPD model by introducing self-consistency iterations to the update process of the velocities and velocity-dependent dissipative forces [85]. This new scheme reduces numerical artifacts. DPD has been very successful in simulations of polymeric systems and in reproducing equilibrium properties. 3.3 Targeted Mollified Impulse Targeted Mollified Impulse (Targeted MOLLY, or TM) is a new multiscale MTS integrator for molecular dynamics. In TM, the inner-most level leapfrog integrator is substituted with a self–consistent dissipative leapfrog, and the outer level Impulse integrator is substituted with the B-spline MOLLY. Note that if one can keep the outer level Impulse

53

integrator but uses the self–consistent dissipative leapfrog in the inner level, one can get Targeted Impulse (TI). By using the Langevin forces to stabilize the outer integrator, the step sizes can be lengthened and some accuracy is maintained. In particular, TM reduces the artifacts by only targeting the fastest interactions and conserving linear momentum. Not preserving momentum implies that the macroscopic behavior will be diffusive [24]. Conserving linear momentum is an improvement over traditional Langevin integrators such as BBK [18] and Langevin-Impulse [102]. In the same sense, TM is an improvement over LM [45]. The TM discretization scheme is shown in Algorithm 4. 1 2

mollified kick: p+ n−1 = pn−1 +

∆t slow F (¯ qn−1 ), 2

(3.7)

oscillate: Propagate qn−1 and p+ n−1 by integrating q˙ = M−1 p,

p˙ = Ffast (q) + FR (q) + FD (q)

(3.8)

for an interval ∆t to get qn and p− n using the SCD-leapfrog scheme, Algorithm 5. R D F and F are the random and dissipative forces, respectively, and are defined in Eqs. (3.23) and (3.22). ¯ n and a a time averaging: Calculate a temporary vector of time-averaged positions q ¯ n . The time averaging function uses only the fastest Jacobian matrix, Jn = ∇q q forces Freduced (q). 1 2

mollified kick: pn = p− n +

∆t slow F (¯ qn ). 2

(3.9)

Algorithm 4: Targeted MOLLY discretization.

3.3.1 B-spline Averaging MOLLY defines the slow part of the potential energy at time-averaged positions, and the force is made a gradient of the potential energy [28, 29, 44–47]. The time average is obtained by doing dynamics over vibrations using forces that produce those vibrations.

54

Thus, U slow (q) becomes U slow (A(q)),

(3.10)

with the force defined as a gradient of this averaged potential, −∇U slow (q) is replaced by − Aq (q)T ∇U slow (A(q)),

(3.11)

where Aq (q) is a sparse Jacobian matrix. The discretization of Eq. (1.2) using MOLLY with step size ∆t for the slow part is same as Algorithm 4 except that MOLLY solves the Newtonian dynamics using leapfrog in the “oscillate” step. Intuitively, Fslow is evaluated at the averaged positions rather than the “impulse” positions, it is more accurate in describing the rapidly varying Fslow (q(t)). MOLLY is still symplectic since we perturb the potential rather than the force [94]. Acting as a filter, the pre-factor, Aq (q)T , improves the stability by effectively “mollifying” components of Fslow (Aq ) that might induce instability. MOLLY integrators have different stability and accuracy properties when different averaging functions are used. Two different averaging methods have been developed. One is B-spline MOLLY which is based on explicit time averaging and B-spline weight functions. Results for a prototype of B-spline MOLLY are reported in [101] in which a numerical differentiation tool is used to compute the Hessians needed in the implementation. Another one is Equilibrium MOLLY which in the case of linear forces is a nearly perfect filter [47]. Equilibrium MOLLY has been implemented in NAMD 2 [50] and in ProtoMol [46, 74–77]. We have implemented B-spline MOLLY efficiently using the analytical Hessians in PROTO M OL. The MOLLY averaging can be done by numerically integrating an auxiliary, reduced problem: 1 A(q) = ∆t

Z

∞

φ 0

55

t ∆t

˜ (t)dt q

(3.12)

where φ

t ∆t

˜ (t) solves an auxiliary problem is a weight function, and q M

d2 ˜ = F reduced (˜ q q), 2 dt

˜ (0) = q, q

d ˜ (0) = 0. q dt

(3.13)

B-spline weight functions are non-zero over a short interval. They have compact support in time and thus make the method computationally feasible. Thus the use of B-spline functions in MOLLY averaging is suggested in paper [28]. One such B-spline weight function that has been tested is called L ONG AVERAGE: φ(s) = 1, 0 ≤ s < 1;

φ(s) = 1/2, s = 1;

φ(s) = 0, s < 0 or s > 1.

(3.14)

A shorter averaging that is a scaling of L ONG AVERAGE is called S HORTAVERAGE: φ(s) = 2, 0 ≤ s < 1/2;

φ(s) = 1, s = 1/2;

φ(s) = 0, s < 0 or s > 1/2. (3.15)

We explicitly compute the A(q). Meanwhile, we compute Aq (q) using the chain rule. Assuming that the leapfrog method with time step δt used, the procedure of computing A(q) and Aq (q) can be shown as follows: Initialization is given by X := q,

P := 0,

B := 0,

t := 0,

(3.16)

Xq := I, Pq := 0, Bq := 0, and step by step integration by P := P + 12 δtF reduced (X), Pq := Pq + 12 δtFXreduced (X)Xq , B := B + 12 δtXφ(t/∆t),

Bq := Bq + 12 δtXq φ(t/∆t),

X := X + δtM−1 P,

Xq := Xq + δtM−1 Pq ,

(3.17)

t := t + δt, B := B + 12 δtXφ(t/∆t),

Bq := Bq + 12 δtXq φ(t/∆t),

P := P + 12 δtF reduced (X), Pq := Pq + 12 δtFXreduced (X)Xq . The value (1/∆t)B is used for A(q) and (1/∆t)Bq for Aq (q). We continue the above integration until we reach a value of t such that φ(t/∆t) is zero at this value and remains 56

zero for larger values of t. Note that δt should be chosen such that the leapfrog scheme reduced is stable and that H ≡ −FX = UXX (X) is the Hessian of energy. Apparently the use

of analytical Hessians is the key to efficiently compute Aq (q) in the above procedure, cf. Appendix A.3. Note that the Hessian-vector product can be performed efficiently using methods like those in [65]. 3.3.2 Self–Consistent Dissipative Leapfrog The self–consistent dissipative leapfrog (SCD-leapfrog) integrator incorporates pairwise Langevin forces and random forces in its force evaluation processes and updates velocities and velocity-dependent dissipative forces in a self–consistent manner to moderate the numerical instabilities associated with the fast motions. The SCD-leapfrog has only been used in DPD simulations, and is well studied and presented in [13, 85]. It is included here for completeness. The discretization for one step is given in Algorithm (5), which is same as that in [13, 85] except that the dissipative and random forces are computed for the pairs of atoms that have fast interactions instead of for all DPD particles. In Algorithm (5), FC is the conservative force vector acted on each atoms. For flexible waters, it includes bonds, angles, and possibly short range electrostatic and Lennard-Jones forces. FD is the dissipative force introduced, which depends on the relative positions and velocities of the atoms being considered and FR is the random force. These forces are calculated using the following formulae: FD (rij , vij ) = −γω D (rij )(vij · ˆrij )ˆrij , FR (rij ) = σω R (rij )ξij δt−1/2 ˆrij ,

(3.22) (3.23)

where we have used the notation rij ≡ rj − ri and vij ≡ vj − vi , and ˆrij denotes a unit vector in the direction of rij ; ri and vi are the position and velocity of atom i, respectively. The amplitude of the dissipative force is characterized by γ which is related to the effectiveness of damping. The random force, FR , is characterized by its amplitude, 57

1 2

kick: p+ n−1 = pn−1 +

oscillate:

δt C (F + FD + FR ), 2

qn = qn−1 + M −1 p+ n−1 δt,

(3.18)

(3.19)

R evaluate: Calculate FC (qn ), FD (qn , p+ n−1 ), and F (qn ). 1 2

kick, part (a): + ˆ− p n = pn−1 +

1 2

δt C (F + FR ), 2

(3.20)

kick, part (b): ˆ− pn = p n +

δt D F , 2

(3.21)

evaluate: Calculate FD i (rj , vj ) and go back to Eq. (3.21) until convergence (or stop after a certain number of self-consistency iterations). Algorithm 5: self–consistent dissipative leapfrog for MD simulations. σ, the direction, rîj , the Gaussian distributed random variable, ξ, with zero mean and unit variance, and the time step, δt. The reason for the appearance of the term δt−1/2 in √ 2 Eq. (3.23) was shown in [32, Section II]. As a result, hfi2 i ≡ h(FR / δt) i = σ 2 /δt, i.e., i the spread of the random force becomes wider as the time step becomes smaller, which holds true if we interpret the random force as a Wiener process, see [54, p. 28]. The weight functions, ω D and ω R in this protocol are both set to 1. Thus ω R =

√

ωD

is satisfied, which ensures that the probability to observe a particular configuration of the targeted molecular dynamics particles is given by the Boltzmann distribution in equilibrium. We require that γ and σ satisfy kB T = σ 2 /(2γ) where kB is the Boltzmann constant and T the equilibrium temperature. In our implementation, the unit of γ is amu/fs. Note that if only one self-consistency iteration is used, this method reduces to the dissipative leapfrog method. In practice, we allow 2 to 8 iterations. The regular dissipative leapfrog scheme displays pronounced unphysical artifacts in the radial distribution function, g(r), and thus does not produce the correct equilibrium properties [13]. 58

3.3.3 Pairwise Targeted Langevin Coupling TM uses pairwise targeted Langevin couplings, i.e., Langevin forces are targeted at particular pairs of atoms that have interactions which we know would cause instability. Momentum preserving implies that the macroscopic behavior will be hydrodynamic [24]. Atoms in those pairs, including covalently bonded pairs (bonds and angles) and possibly hydrogen-bonded pairs, have fast interactions. These are an O(N) subset of O(N 2 ) pairs of atoms. As an example, for water, there should be at least three pairs of Langevin forces: one for each of the two hydrogen-oxygen pairs to damp the bond stretching and one for the hydrogen-hydrogen pair to damp the “imaginary” bond stretching and thus the angle bending. A small number (≤ 4) of pairs might arise from hydrogen bonds formed with neighboring molecules. 3.3.4 Sampling Properties Since SCD leapfrog satisfies the fluctuation-dissipation relation, it reproduces the correct equilibrium behavior of the DPD model [85]. DPD model can reproduce Boltzmann statistics if the fluctuation-dissipation relation is satisfied [24]. TM is just an MTS version of self–consistent dissipative leapfrog with a small perturbation of the potential: MOLLY, consistent with Impulse in the limit of infinitesimal time step, uses a potential that is just a small perturbation of that used by Impulse. The above suggests that TM samples from the canonical ensemble at least at small time steps. For systems with small number of atoms, N, if the Langevin coupling is weak, convergence to the canonical ensemble may take a long time. For large N, the canonical ensemble is approximately the same as the microcanonical, and the strength of the Langevin coupling is not relevant. We investigate the sampling properties of the self–consistent dissipative method and TM using two simple model problems: a harmonic oscillator and a Fermi–Pasta–Ulam Problem. Results show that in both cases parameters

59

for TM can be chosen so that it samples the canonical ensemble for the model problems. We first consider a harmonic oscillator consisting of a unit mass point connected to a linear spring. The Hamiltonian is given as follows: 1 1 H(x, x) ˙ = x˙ 2 + Ω2 x2 , 2 2

(3.24)

where x is the displacement of the unit point mass, x˙ is the velocity, Ω2 is the spring constant. The harmonic oscillator is often chosen to test algorithms because the equations of motion can be solved analytically. Let x(0) = 0, x(0) ˙ = 1 be the initial conditions, Ω2 = 100, and T = 1/kB be the prescribed equilibrium temperature where kB is the Boltzmann constant. The motion of the point mass is then propagated using SCD-leapfrog with δt = 0.05τ where τ is the period of the fast spring oscillation given by τ =

2π . Ω

The unit for the damping

coefficient, γ, is [mass]/[time] which is somewhat arbitrary depending on the units for mass and time. All runs are 2.0 × 107 time steps. It is found that for the given initial conditions and same number of time steps, convergence to the canonical ensemble can only be reached if γ > 1.0 × 10−4 . A typical phase diagram is presented in Fig. 3.2, showing that convergence to the canonical ensemble is reached with γ = 0.1, in which the dots in (a) correspond to consecutive points separated by 10000 time steps, and the outer ellipse is the micro-canonical ensemble solution obtained with γ = 0 whereas the fitted solid line in (b) is the exact result which is Gaussian. Now let us consider a model problem consisting of a unit mass point connected by a linear spring and a nonlinear spring. The Hamiltonian is the following: 1 1 1 H(x, x) ˙ = x˙ 2 + Ω2 x2 + (x − 1)4 , 2 2 4

(3.25)

where x is the displacement of the point mass, x˙ is the velocity, and Ω2 is the spring constant of the hard spring. This is one of the simplest Fermi–Pasta–Ulam problems, proposed in [38, p. 17]. Note that by expanding the the derivative of the the last term in 60

1

1.5

0.5

P(V)

V

1 0

0.5 −0.5

−1

−0.1

−0.05

0 X

0.05

0 −1

0.1

−0.5

0 V

0.5

1

(b)

(a)

Figure 3.2. (a) Phase space trajectory (the dots), and (b) velocity distribution (the squares) of a harmonic oscillator with Hamiltonian given by Eq. (3.24) (x(0) = 0, x(0) ˙ = 1) using self– consistent dissipative leapfrog with γ = 0.1.

Eq. (3.25), we get the following: (x − 1)3 = −1 + 3x − 3x2 + x3 .

(3.26)

Thus the soft nonlinear spring consists of a constant part, a linear part, and two nonlinear parts. We choose to study such a system because this system has wide separation of time scales, where an MTS scheme is best suited; and because the potential of the system is only dependent on a single variable, x(t), making the B-spline averaging computationally simple: The averaged position based only on the fast forces with x(0) = x, and x(0) ˙ =0 is given by A(x) = x

sin(Ω∆t) , Ω∆t

(3.27)

where ∆t is the outermost time step of the MTS integrator. The mollification factor is Ax (x) =

sin(Ω∆t) Ω∆t

(3.28)

Let x(0) = 0, x(0) ˙ = 1 be the initial conditions, Ω2 = 100 be the spring constant, and T = 1/kB be the prescribed equilibrium temperature. The motion of the point mass is then propagated using TM. Note that for this particular problem, in order to best show 61

the dynamical nature of the system, we need to effectively incorporate the nonlinear contributions to the solution using the TM scheme. We choose ∆t = 0.1τ and δt = 0.05τ where τ is the period of the fast spring oscillation given by τ =

2π . Ω

Though the long time

step is only twice of the short time step, the above choice serves the purpose of showing the sampling properties of TM very well. If, for example, we had chosen ∆t = k2 τ where k = 1, 2, . . . , m, then the system reduces to a harmonic oscillator subjected to a periodic forcing with constant magnitude and same direction, which is totally different from the original dynamical system. All runs are 2.0 × 107 time steps. It is found that for the given initial conditions and same number of time steps, convergence to the canonical ensemble can only be reached if γ > 1.0 × 10−4 . A typical phase diagram is presented in Fig. 3.3, showing that convergence to the canonical ensemble is reached with γ = 0.01, in which the dots in (a) correspond to consecutive points separated by 10000 time steps, the outer ellipse is the micro-canonical ensemble solution obtained with γ = 0, whereas the fitted solid line in (b) is the exact result which is Gaussian. It is seen that with a proper value of γ, TM generates the canonical ensemble. 1

1.5

0.8 0.6 0.4

P(V)

V

0.2 0

1

−0.2 −0.4

0.5

−0.6 −0.8 −1 −0.1

−0.05

0

0.05

0 −1

0.1

X

(a)

−0.5

0 V

0.5

1

(b)

Figure 3.3. (a) Phase space trajectory (the dots), and (b) velocity distribution (the squares) of a Fermi–Pasta–Ulam problem whose Hamiltonian is given by Eq. (3.25) (x(0) = 0, x(0) ˙ = 1) using Targeted MOLLY with γ = 0.01. Numerical experiments show that with weak Langevin coupling, e.g. γ = 1.0 × 10−5 , 62

convergence to the canonical ensemble still occurs, nonetheless the convergence takes a longer time. Results (phase diagram and velocity distribution) for the simulation of the FPU problem with γ = 1.0 × 10−5 are presented in Fig. 3.4 in which the dots in (a) correspond to consecutive points separated by 25000 time steps, and the total number of steps is 1.0 × 108 . It is seen that with a weak Langevin coupling, the convergence to the canonical ensemble has been reached after 1.0 × 108 time steps. Note that convergence to the canonical ensemble has not been reached after 2.0 × 107 time steps. 1.5

1

0.5

0

V

V

1

0.5 −0.5

−1 −0.1

−0.05

0

0.05

0 −1

0.1

X

(a)

−0.5

0 X

0.5

1

(b)

Figure 3.4. (a) Phase diagram, and (b) velocity distribution of the Fermi–Pasta–Ulam problem whose Hamiltonian is given by Eq. (3.25) (x(0) = 0, x(0) ˙ = 1) using Targeted MOLLY with γ = 1.0 × 10−5 .

3.3.5 Implementation Issues In order to compute the averaged positions and the prefactor matrix, we will need data structures for heavy-atom groups (HAGs). A HAG is a group of atoms in which one is a heavy atom and others are hydrogen atoms, and there are covalent bonds from hydrogen atoms (daughters) to the heavy atom (mother). When only bonds and angles are considered as the fast forces in the MOLLY averaging, HAGs are decoupled from each other and thus averaging and mollification can be done HAG by HAG. A list of HAGs can be generated in the beginning of the execution of the simulation program once for all, 63

thus the overhead associated with generating HAGs is negligible. Assume an HAG consists of 3 hydrogens (atoms 1, 2 and 3) and one carbon (atom 0). We consider 3 polar C-H bonds, i.e., B01 for 0-1, B02 for 0-2 and B03 for 0-3, and 3 angles, i.e., A102 for 1-0-2, A103 for 1-0-3 and A203 for 2-0-3), then the total Hessian (upper half), H, is given in Eq. (3.29) (symmetric, only the upper half is shown): H[0][0] = B01[0][0] + B02[0][0] + B03[0][0]+ A102[1][1] + A103[1][1] + A203[1][1], H[0][1] = B01[0][1] + A102[1][0] + A103[1][0], H[0][2] = B02[0][1] + A102[1][2] + A203[1][0], H[0][3] = B03[0][1] + A102[1][2] + A203[1][2], H[1][1] = B01[1][1] + A102[0][0] + A103[0][0],

(3.29)

H[1][2] = A102[0][2], H[1][3] = A103[0][2], H[2][2] = B02[1][1] + A102[2][2] + A203[0][0], H[2][3] = A203[0][2], H[3][3] = B03[1][1] + A103[2][2] + A203[2][2]. For the flexible water systems, the above reduces to simpler forms. Consider a water molecule with the oxygen numbered as atom j, and the hydrogens as i and k. Let the bd bond Hessians for atoms i and j, and j and k be Hijbd and Hjk , and the angle Hessian be a Hijk , then the total Hessian, H, for this molecule is given in Eq. (3.30): a H[0][0] = Hijk [0][0] + Hijbd [0][0], a H[0][1] = Hijk [0][1] + Hijbd [0][1], a H[0][2] = Hijk [0][2],

H[1][1] =

a [1][1] Hijk

(3.30)

+

Hijbd [1][1]

+

a bd H[1][2] = Hijk [1][2] + Hjk [0][1], a bd H[2][2] = Hijk [2][2] + Hjk [1][1].

64

bd Hjk [0][0],

3.4 Numerical Experiments We perform numerical experiments using B-spline MOLLY and Targeted MOLLY (TM) integrators. We show that B-spline MOLLY successfully break the linear stability barrier, and TM allows outer time step of up to 16 fs while still computing the dynamical properties such as self-diffusion coefficient and structural properties such as radial distribution function correctly. The numerical experiments reported here are all performed on water systems since the current implementation of TM is based on the prototype of B-spline MOLLY in ProtoMol, which takes advantage of the natural decoupling of the molecules in pure flexible TIP3P water systems [48]. Further evaluation of TM should include simulations of large biological molecules using the generalized B-spline MOLLY. 3.4.1 Simulation Protocol We perform simulations of three flexible TIP3P water systems: Model-S as described in Section 2.3.1, Model-M and Model-L. The latter two systems are larger and mainly used for timing purposes such that the speed up for relatively larger systems can be mea˚ radius sphere with 3243 atoms. sured. Model-M is A “medium” water system, a 20 A ˚ radius sphere with 30006 atoms. Model-L is a “large” water system, 42 A All the systems are equilibrated using NAMD 2.3 [50] during 100 ps of simulation time by minimization followed by temperature re-scaling to 300 K. The smallest system is equilibrated by another 250 ps simulation using ProtoMol [75,76] with periodic boundary conditions before the production runs. By equilibrating we avoid highly improbable values of different contributions to energies. We then ran simulations using P ROTO M OL. Results discussed in this section can be reproduced using ProtoMol. All simulations use periodic boundary conditions and Ewald sum for computing full electrostatic interactions [1, 23, 25, 27, 35, 89, 108].

65

We use a 2-level Targeted MOLLY integrator for the simulations with outer time step of 16 fs and inner time step of 2 fs. The B-spline MOLLY integrator in the outer level uses bonds and angles in the averaging, and it uses L ONG AVERAGE averaging function. The SCD-leapfrog in the inner level uses pairwise Langevin interactions of bonds and angles. The MOLLY averaging step size is 2 fs. The Langevin coupling coefficient, γ, is 4 amu/ps. The number of iterations for self-consistency for all simulations is 2. The Ewald accuracy, ≡ exp(−p), is 10−12 for the small water system, and 10−7 for the two larger water systems. We use the k-space part of Ewald sum as the long force. Short forces include bonded forces plus the real part of the Ewald sum with corrections. Lennard Jones forces use a C 2 switching function. For the smallest system, the parameters ˚ and cutoff of 6.5 A; ˚ for the two larger systems, the parameters are are switchon = 4 A ˚ and cutoff of 8.0 A. ˚ switchon = 6.5 A In ProtoMol it is straightforward to specify what integrators and forces to use at each level using an Integrator Definition Language (IDL). A detailed discussion on IDL and MTS implementation can be found in [46, 75]. Program 1 illustrates the two-level Targeted MOLLY integrator as described above. The actual frequency of a force at a given level is recursively defined by the product of the current cycle length and the frequency of the next inner integrator. 3.4.2 Results and Discussion Among dynamical quantities of the system, we chose to compute the self-diffusion coefficient, D, which characterizes temporal correlations between the displacements or velocities of the molecules. The Einstein relation, valid at long times, is 1 2tD = h|ri (t) − ri (0)|2 i 3

(3.31)

where ri (t) is the position of the mass center of the molecule [2]. These averages are computed for each of the N particles in the simulation, the results added together, and 66

Integrator { level 1 BSplineMOLLY { # Long-range electrostatics cyclelength 8 BSplineType long force Coulomb -algorithm FullEwald -reciprocal -accuracy 1.e-12 } level 0 SCDLeapfrog { # Fast varying forces timestep 2.0 # [fs] gamma 4.0 temperature 300 numIter 2 force Bond, Angle,Improper, Dihedral force Coulomb -algorithm FullEwald -real -correction -accuracy 1.e-12 force LennardJones -algorithm NonbondedCutoff -switchingFunction C2 -cutoff 6.5 -switchon 4.0 } }

Program 1: Two-level Targeted MOLLY integrator in ProtoMol. The step sizes for fast and slow forces are 2 fs and 16 (=2·8) fs, respectively. divided by N to improve statistical accuracy. Another technique for improving statistical accuracy is block-averaging in which one averages over many intervals of trajectories. Approximation to the position of the center of mass of small molecules is obtained using the positions of the heaviest atoms within these molecules. We show that TM recovers the correct radial distribution functions and the selfdiffusion coefficient even when very large time steps are used for simulating flexible waters while maintaining a properly bounded temperature. We also show that the computational overhead associated with mollification is low. Although TM recovers correct radial distribution functions and self-diffusion coefficient for flexible waters at very large time steps, we should note that TM is not a substitute for Hamiltonian dynamics unless the system size, N, is large enough, given that the random and dissipative forces alters the nature of the dynamics. Overcoming Linear Stability Barrier with B-spline MOLLY We first show results to confirm that B-spline MOLLY is able to break the linear stability barrier of Impulse. We show the energy drift during a 200 ps MD simulation of

67

−700

Impulse (∆ t, δ t) = (5, 1) BSplineMOLLY using Short (∆ t, δ t) = (5, 1) BSplineMOLLY using Long (∆ t, δ t) = (5, 1)

−710 −720


−730 −740 −750 −760 −770 −780 −790 −800 −810

0

0.2

0.4

0.6

0.8

1 Time (fs)

1.2

1.4

1.6

1.8

2 5

x 10

Figure 3.5. Energy drift for flexible water simulations of 200 ps at 300 K in vacuum. the Model-S system with normal (vacuum) boundary conditions (NBC/VBC) using direct method for Coulomb force evaluations, see Fig. 3.5. With outer time step of 5 fs and inner time step of 1 fs, simulations using B-spline MOLLY with L ONG AVERAGE is stable. Bspline MOLLY with L ONG AVERAGE with ∆t = 6 fs are not stable with typical drift of total energy greater than 6% over 200 ps of MD simulation. We also perform simulations of the same system using B-spline MOLLY with period boundary conditions (PBC) with Ewald sum accuracy of 10−7. The relative drift of total energy for a 200 ps simulation with L ONG AVERAGE is 0.22 ± 0.11% using ∆t = 5 fs, and and 3.53 ± 0.38% using ∆t = 6 fs, which indicate that B-spline MOLLY with LongAverage is more stable with periodic boundary conditions, see Fig. 3.6.

68

−740 Impulse (∆ t, δ t) = (6, 1), NBC B−spline MOLLY (∆ t, δ t) = (6, 1), NBC B−spline MOLLY (∆ t, δ t) = (6, 1), PBC


−750 −760 −770 −780 −790 −800 −810 0

50

100 Time (ps)

150

200

Figure 3.6. Energy drift for flexible water simulations of 200 ps at 300 K with normal or periodic boundary conditions.

Properly Bounded Temperature From the simulations with TM (with outer time step of 16 fs and inner time step of 2 fs), we show that the kinetic temperature of the system is properly bounded around the prescribed equilibrium temperature. The length of each simulation is 400 ps. We measure the relative drift of molecular temperature, denoted by ∆Tr , in percent, with respect to mean temperature, T¯ , in Kelvin. In the simulation of the small water system, ∆Tr = −1.58 ± 3.78% with mean temperature of 297.30 K (the prescribed equilibrium temperature is 300 K). We also used only the non-hydrogen atoms (i.e., oxygens for waters) to approximate the molecular temperature.

69

Correct Radial Distributional Functions Because TM imposes less randomness in slower modes, which helps to cross barriers, it is expected that TM recovers the structural quantities such as the radial distribution functions more accurately than other Langevin methods such as LM. This property is illustrated by showing the agreement between TM (with outer time step of 16 fs and inner time step of 2 fs) and leapfrog (with step size of 1 fs) on the radial distribution function, g(r), computed from the simulations, see Figs. 3.7 and 3.8. The length of each simulation is 400 ps. It is seen that the structural property of the system is very well recovered using TM even with large time steps.

Leapfrog ∆ t = 1 fs TM ∆ t = 16 fs, δ t = 2 fs

2

RDF G(OH)

1.5

1

0.5

0

1.5

2

2.5

3

3.5

4

4.5

5

Distance R(OH) (Angstroms) Figure 3.7. Radial distribution functions for the O-H interactions for leapfrog (δt = 1 fs) and Targeted MOLLY (TM, ∆t = 16 fs and δt = 2 fs).

70

1.8

Leapfrog ∆ t = 1 fs TM ∆ t = 16 fs, δ t = 2 fs

1.6

RDF G(HH)

1.4 1.2 1 0.8 0.6 0.4 0.2

2

2.5

3

3.5

4

4.5

5

Distance R(HH) (Angstroms) Figure 3.8. Radial distribution functions for the H-H interactions for leapfrog (δt = 1 fs) and Targeted MOLLY (TM, ∆t = 16 fs and δt = 2 fs).

Correct Self-Diffusion Coefficient The self-diffusion coefficients computed from different simulations are summarized in Table 3.1. The diffusion coefficient is computed using Einstein’s relation for nonoverlapping 4.8 ps blocks of 400 ps trajectories, averaging over all time origins of oxygen atoms only. From Table 3.1 we can see that the self-diffusion coefficient is computed correctly using TM as compared with the result using leapfrog. The CPU times of different simulations (simulation length for the smallest system is 400 ps and that for the two larger systems is 10 ps) are summarized in Tables 3.2 and 3.3, from which we can see a substantial speedup using TM. All timing was obtained on a single node of the Hydra

71

Table 3.1. SELF-DIFFUSION COEFFICIENT (D) FROM FLEXIBLE TIP3P WATER SIMULATIONS WITH PERIODIC BOUNDARY CONDITIONS WITH EWALD ACCURACY OF = 10−12 . THE ERROR BAR IS GIVEN BY TWICE OF THE STANDARD DEVIATION. Integrator ∆t Type (fs) leapfrog TM 16

δt (fs) 1 2

γ D (amu/ps) (10−5 cm2 s) 3.69 ± 0.01 4 3.68 ± 0.01

Table 3.2. CPU TIME (T ) AND SPEED UP (η) of 400 PS OF MD SIMULATIONS OF THE SMALL FLEXIBLE WATER MODEL SYSTEM WITH PERIODIC BOUNDARY CONDITIONS WITH EWALD ACCURACY OF = 10−12 . Integrator ∆t Type (fs) leapfrog Impulse 4 TM 16

δt (fs) 1 1 2

γ T η (amu/ps) (hr) 68.28 1.00 23.50 2.91 4 11.40 5.99

cluster (SUN OS 2.7, 32 nodes each of which is 440 MHz dual-processor SunSparc 10 with a 256 MB shared memory). Compared to leapfrog with time step of 1 fs, TM with ∆t = 16 fs and δt = 2 fs achieves a six-fold computational speedup, whereas Impulse with outer time step of 4 fs and inner time step of 1 fs only achieves a three-fold speedup. In contrast, LM only computes self-diffusion coefficient correctly with ∆t ≤ 12 fs [45] .

Note that one needs to tune the parameters for TM in order to recover correct dynamTable 3.3. CPU TIME (T ) AND SPEED UP (η) of 10 PS OF MD SIMULATIONS OF THE MIDIUM AND THE LARGE FLEXIBLE WATER MODEL SYSTEMS WITH PERIODIC BOUNDARY CONDITIONS WITH EWALD ACCURACY OF = 10−7 .

Integrator ∆t Type (fs) leapfrog Impulse 4 TM 16

δt (fs) 1 1 2

γ 3243 atoms 30006 atoms (amups) T (hr) η T (hr) η 20.02 1.00 1021.31 1.00 13.01 1.54 538.92 1.89 4 6.28 3.20 237.91 4.29

72

ics. As an example, after choosing an inner time step of δt = 2 fs, the proper values for γ are 0.7 and 0.9 amu/ps for ∆t = 8 fs and ∆t = 12 fs, respectively. These results suggest that the appropriate value of γ is a nonlinear function of ∆t. Low Computational Overhead of Mollification The overhead of mollification is estimated under 3.5% of the total execution time with high Ewald accuracy (10−12 ), see Table 3.4. All simulations are performed on a single node of a Linux cluster composed of processors of Intel (R) Xeon (TM) 1.7 GHz, 1GB memory, and 2.4.17 kernel. The overhead depends on the cost of non-bonded force evaluations. The larger the cost of the non-bonded force evaluation, the smaller percentage of the overhead incurred by the mollification because this overhead is linear in terms of number of atoms with a very small constant. For TM with ∆t = 16 fs and δt = 2 fs, the overhead associated with mollification is estimated as about 1 − 2% of the total execution time in which both bonds and angles are included in the MOLLY forces. Larger Time Steps with Better Force Splitting The fast and slow forces splitting of Ewald sum used in these simulations is easy to implement [88], it is suboptimal for accuracy. Therefore more appropriate splittings in Ewald sum have been suggested [8, 9, 90, 113]. We expect that TM should allow even larger time steps using a more appropriate splitting.

73

Table 3.4. OVERHEAD OF MOLLIFICATION FROM THE SMALL FLEXIBLE FLEXIBLE WATER SYSTEM SIMULATIONS (4 PS EACH) USING B-SPLINE MOLLY. IN(x,y,z), x DENOTES THE OUTER TIME STEP (FS), y THE INNER TIME STEP (FS), AND z THE TIME STEP (FS) FOR MOLLY AVERAGING.

Force method MOLLY forces Ewald −6 (10 ) Bond

Bond+Angle

Ewald (10−12 )

Bond

Bond+Angle

Integrator Run time (s) Impulse(4,1,-) 311.12 Impulseulse(4,2,-) 168.31 MOLLY(4,1,1) 321.30 MOLLY(4,1,2) 316.91 MOLLY(4,2,2) 172.69 MOLLY(4,1,1) 329.72 MOLLY(4,1,2) 320.47 MOLLY(4,2,2) 176.66 Impulse(4,1,-) 584.40 Impulse(4,2,-) 320.61 MOLLY(4,1,1) 597.80 MOLLY(4,1,2) 594.71 MOLLY(4,2,2) 326.70 MOLLY(4,1,1) 606.30 MOLLY(4,1,2) 603.80 MOLLY(4,2,2) 328.72

74

Overhead(%) 3.17 1.83 2.54 5.67 2.92 4.73 2.24 1.73 1.87 3.61 3.21 2.47

CHAPTER 4 BACKWARD EULER MOLLY

As seen in the preceding chapter, TM is successful in the attempt of constructing multiscale integrators for molecular dynamics (MD). We also believe that we may push the step size limit even further by combining a stabler MOLLY integrator with targeted Langevin coupling. In this chapter, we investigate whether or not a MOLLY method would be more stable if we include a subset of nonbonded forces in the averaging in addition to the bonds and angles. Observations of instabilities of SHAKE–Impulse such as those reported in Chapter 2 clue us how to improve the stabilities of MOLLY: inclusion of nonbonded forces in MOLLY averaging. Even in fully constrained dynamics simulations (constraining all bonds and angles), there still exists fast vibrational modes such as the librational (i.e., hindered rotational or caged) motions hydrogen bonded molecules in a rigid water system. These librational modes are mainly due to electrostatic forces with periods centered around 50 fs (by experiments) or 45 (by simulations) [16]. Modes of H-bond stretching due to electrostatic force and Lennard-Jones force have longer periods. It is logical ˚ < r < 3.5A, ˚ to think that by adding a subset (range) of nonbonded forces, e.g., 1.5A into the averaging, we may get more stable integrators [43]. However, neither B-spline MOLLY nor Equilibrium MOLLY allow this extension to be done easily. Here we propose a new multiple time stepping (MTS) method, Backward Euler MOLLY that allows this extension to be easily included in the averaging.

75

4.1 Backward Euler Averaging The MOLLY averaged position, A(q), can be expressed as below using the backward Euler formulation: ˜ A(q) = q + τ 2 M−1 F(A(q)),

(4.1)

˜ where q is the vector of original positions of atoms, τ is the time step, F(A(q)) is the force evaluated at A(q). Differentiate both sides of Eq. (4.1) w.r.t. q, we have ˜ A(q) (A(q))Aq (q). Aq (q) = I + τ 2 M−1 F

(4.2)

Rearranging the above equation yields the mollification factor as: −1 ˜ A(q) (A(q)) Aq (q) = I − τ 2 M−1 F .

(4.3)

Eq. (4.1) is a nonlinear set of equations which can be solved by Newton–Raphson method which converges very fast if a good initial guess is provided. Eq. (4.1) is rewritten as follows: ˜ F (q) ≡ A − τ 2 M−1 F(A) − q = 0,

(4.4)

where A ≡ A(q) is the unknown. From the above equation a nonlinear set of equations for the corrections, δA, are obtained as: J · δA = −F ,

(4.5)

where J is the Jacobian matrix. The corrections are added to the solution vector Anew = Aold + δA.

(4.6)

Unlike previous attempts of solving the whole systems of MD equations involving all the nonbonded forces using the implicit methods such as the work of Zhang and Schlick [111], the computational cost of solving the systems of nonlinear equations, Eq. (4.1), is greatly simplified since only a small subset of non-bonded forces are considered. 76

In case of quadratic potentials such as those of harmonic oscillators, it only takes one iteration to reach convergence. To illustrate this, we consider the same Fermi–Pasta– Ulam (FPU) model problem as used in Section 3.3.4. The averaged position using the backward Euler formulation and the MOLLY factor can be easily obtained: A(x) =

x , 1 + τ2

Ax (x) =

1 . 1 + τ2

(4.7)

Thus the mollified nonlinear force is Fmollified = −

1 x ( − 1)3 . 2 2 1+τ 1+τ

(4.8)

4.2 Introducing a Heuristic Approximation In order to solve the implicit equations faster, we introduce an approximation: We assume that heavy atoms do not move in the averaging interval. This heuristic ensures a block diagonal Hessian matrix which would be used in the solving process. Besides the numerical advantages, the above approximation is justified in the case of librational motions where the heavy atoms are essentially not moving. Without this approximation, the Hessian of the bonded and hydrogen-bonded energies for the whole system would be a sparse matrix with irregular filling. With this approximation, however, J in Eq. (4.5) becomes block diagonal and can be readily inverted. As an example, assume two hydrogenbonded atoms as shown in Figure 4.1: atom 3 is an oxygen and atom 2 is a hydrogen. The exact Hessian for this interaction potential would look like the following:    H00 H01  HH−bond =   H10 H11

(4.9)

whereas the in the approximated Hessian the only non-zero block is H11 . With this approximation, we will only need to incorporate H11 into the appropriate blocks in the total Hessian for each of the heavy atom group, and in this case, a water molecule.

77

1 4

3 2 0 Stretching Libration

5

Figure 4.1. Two hydrogen bonded water molecules with librational motion and stretching motion.

This approximation enables us to eliminate the use of sparse matrix representation of the total Hessian, and only one entry of the block representing the water molecule or heavy-atom group in the total Hessian of the system has to be modified. This speeds up the computation while maintaining a right approximation of the averaging and mollification. 4.3 Results and Discussion Convergence Testing To test the convergence properties, we use a single water model in which the atoms are placed such that the water is off its equilibrium configuration. We then use the proposed Newton–Raphson method to solve for the averaged position using a large value of τ, e.g.,20 to 200 fs), and only using bonds and angles for the averaging forces. The results show that the positions of the atoms quickly converge to the near equilibrium configuration in a mere 2 to 4 iterations. It takes about 15 to 20 iterations to reach the preset convergence criteria (the two-norm of the difference between two consecutive iterations less than 10−6 in variables and functions). This suggests that the solution process is numerically correct, and that for the water system, the equations for the MOLLY averaging 78

−793 −794

Etotal (Kcal/mol)

−795

∆ t = 5 fs, block averaged ∆ t = 5 fs, fitted ∆ t = 6 fs, block averaged ∆ t = 6 fs, fitted

−796 −797 −798 −799 −800 −801 −802 0

20

40

60

80

100

Time (ps) Figure 4.2. Energy drift of 100 ps molecular dynamics simulation of the small flexible water system using BE MOLLY.

is well behaved. Stability Testing We run simulations of the small flexible water model system. We allow 15 iterations to ensure a converged solution for the averaged positions and only use bonds and angles for the averaging forces. Results show that the simulation is absolutely stable when ∆t = 5 fs. When ∆t = 6 fs, the simulation has more drift of total energy, but is still stable within relatively long period of time (100 ps), which is an improvement over the B-spline MOLLY method. Energy drift for two particular simulations are shown in Fig. 4.2 in which for ∆t = 5 fs, the drift of total energy is only 0.29 ± 0.14% over 100 ps period, 79

whereas for ∆t = 6 fs, the drift is 1.60 ± 0.31% over the same period of time. These simulations were performed with vacuum boundary conditions using direct method for Coulomb force evaluations. Coulombic energies were split into fast and slow by multi˚ Lennard-Jones energies plying the C1 switching function of Eq. (A.4) where rc = 6.5 A. were split into fast and slow by multiplying the C2 switching function of Eq. (A.6) where ˚ r0 = 4.0 A. ˚ rc = 6.5 A, Discussion We implemented BE MOLLY in ProtoMol such that a subset of Coulombic forces, or a subset of both Coulombic and Lennard–Jones forces can be added in the averaging. Such an implementation is straightforward – in the MOLLY averaging step, a range of nonbonded forces such as Coulombic forces are included in every force evaluation. 1 In the meantime, the Hessians of the interacting nonbonded pairs are evaluated. Since only the heavy atoms, e.g., oxygens, are considered not moving in the librational motions, only one block of the Hessian is needed. These single blocks of Hessians are included in the total Hessian of each of the heavy atom groups (HAGs) in the appropriate blocks. However, we have not found improvement on stability by introducing the nonbonded forces in the averaging. The effect on stability by introducing nonbonded forces into the averaging is still under investigation.

1

ProtoMol supports calculation of range forces by using range switching functions.

80

CHAPTER 5 SUMMARY AND FUTURE WORK

Molecular dynamics (MD) is an important tool for computational biology and bioinformatics. It is becoming a routine tool for computational biologists to conduct their research. The size of the systems being simulated is becoming larger and larger, e.g., over 100, 000 atoms, and the simulations are getting longer, e.g., 5 to 100 ns or even 1 to 10 µs. However, there has not been much progress in the time stepping algorithms – the step sizes for such simulations are still very restricted (for Impulse, ∆tmax = 3.3 fs, for nonsymplectic LM, ∆tmax = 12 fs for flexible water and explicitly solvated proteins/DNAs). Similarly, force evaluation is still very expensive. Though some methods such as MultiGrid allow O(N) time complexity, they all have large constants which make them still quite expensive to evaluate. This thesis presents original work that leads to deeper understanding of numerical instabilities in multiple time stepping (MTS) MD and to the construction of multiscale integrators for MD that allow larger time steps than previous methods. 5.1 Summary of Contributions First, we uncovered the nonlinear overheating of Impulse MTS scheme. We demonstrated through a rigorous nonlinear analysis of a nonlinear model problem solved using MTS scheme and a thorough set of numerical experiments. From the analysis and experiments, we found that MTS integrators such as Impulse suffer nonlinear overheating when

81

the largest time step equals to one third, and possibly one fourth of the shortest period of the system, which we have called 3:1 and 4:1 nonlinear overheating, respectively. For many important biological systems, the shortest period is around 10 fs which limits the the largest step size used by Impulse to less than 3.3 fs, cf. [69, 70]. This sets up a new rule of thumb for choosing appropriate time steps for MD, i.e., we suggest a step size for such systems no larger than 3 fs to avoid overheating caused by unstable nonlinear resonances, whereas the popular rule of thumb is that ∆t ≤ 4 fs. It is important to adhere to this new criterion especially for long simulations. Otherwise, even though the rapid increase of computer power following Moore’s law will enable longer and longer simulations, these simulations may give invalid results since too aggressive choice of step sizes may ruin the whole simulation. Second, we derived analytical Hessians for various CHARMM potentials, including bonds, angles, Lennard Jones and electrostatics. We re-implemented the B-spline MOLLY using these analytical Hessians in ProtoMol. Not only does this new implementation allow B-spline MOLLY to successfully overcome the linear stability barrier of Impulse, but also makes B-spline MOLLY very efficient, cf. [46]. Under this new implementation, it is a simple matter to run a simulation of several hundred picoseconds, whereas under the old implementation using the automatic differentiation tool, ADOL-C, it is even a luxury to run a simulation of a few thousand femtoseconds. Third, we designed a novel multiscale MTS integrator, Targeted MOLLY, for MD simulations, which allows ∆t = 16 fs for flexible water simulations while still computing the dynamical and structural properties correctly. The large time step results in significant speedup of the simulations. Targeted MOLLY is a combination of the new implementation of B-spline MOLLY exploiting the analytical Hessians and the targeted Langevin coupling scheme. Langevin coupling that only targets at pairs of fast interactions to stabilize MD simulations is proven to be very successful, cf. [66–68, 75].

82

Finally, we introduced the Backward Euler MOLLY, a new MOLLY integrator based on the numerical solution of the backward Euler formulation of the averaged forces. Results show that this integrator allows smaller drift than B-spline MOLLY when only bonds and angles are included as the fast forces in MOLLY averaging. Stability might be improved upon including a subset of nonbonded forces in the MOLLY averaging, which can be easily done due to the particular formulation it adopted. Moreover, we may combine this MOLLY integrator with the targeted Langevin coupling technique to yield an integrator that allows even larger time steps. 5.2 Future Work Extensive testing of Targeted MOLLY for general molecular systems will be necessary to provide valuable information for assessment of the validity of the method. This entails the validation of the generalization of TM to handle systems other than those containing only water, and parameter sweeping for systems of interest, such as the estrogen receptor (ER) and raloxifen (RAL) complex (ER/RAL) [87], see Fig. 5.1 which is generated using VMD [41]. Effects on stability by including a subset of nonbonded forces into averaging need more investigation. The BE MOLLY provides a good starting point. Such an investigation is important to understand the theoretical step size limit of MOLLY integrators. For stability monitoring, it is going to be more efficient to use the shadow Hamiltonian approach [100]. In order to fully reap the benefits of the shadow Hamiltonian approach, we will need to generalize this approach to handle multiple time stepping MD simulations. The generalized MTS shadow Hamiltonian will also serve as an invaluable technique for enhancing the sampling of hybrid Monte Carlo (HMC) methods. Another promising direction is the use of cheaper, mixed implicit/explicit methods for MD simulations as proposed in [110]. The idea is to use methods that are implicit only

83

Figure 5.1. The structure of the estrogen receptor (ER) and the raloxifen complex (ER/RAL). The drawing methods for the protein and raloxifen are Cartoon (cylinders for helices/coils, tubes for turns and directional sheets for β sheets) and VDW, respectively.

in the fastest motions, and explicit in the rest. This idea has been successfully applied in the solution of reaction–diffusion partial differential equations. In [110] the methods were tested in a simple nonlinear model problem, and the idea to create MTS mixed implicit/explicit methods for MD simulations was proposed but not implemented. In addition, we propose to use proper constraining to preserve the topology. It may also be applicable to use the targeted Langevin coupling approach to enhance the stability of the mixed implicit/explicit methods.

84

APPENDIX A ADDITIONAL DETAILS

Here we report additional details closely related to the numerical experiments reported in this thesis for the purpose of completeness. A.1 Potentials for Flexible Water Here we detail the potentials used for the flexible water systems, including bonds, angles, Coulombic and Lennard–Jones potentials. The energy for a bond interaction is 1 Ukbond = KB (xij − lk )2 , 2

(A.1)

where KB is a bond force constant and lk is a reference bond length between atoms i and j for constraint k. Finally, the energy for an angle interaction is 1 Ukangle = KA (θk − θ0 )2 , 2

(A.2)

where KA is an angle force constant, and θk and θ0 are the current value of the angle and the reference angle for angle constraint k. ˚ 2 , qH = For flexible water, KA = 55 kcal mol−1 degrees2 , KB = 450 kcal mol−1 A ˚ and θ0 = 104.52 degrees. The Lennard0.417 e, qO = −0.834 e, lO−H = 0.957 A, ˚ σO−O = 3.1506 A, ˚ σO−H = 1.75253 A, ˚ H−H = Jones parameters are σH−H = 0.4 A, 0.046 kcal mol−1 , O−O = 0.1521 kcal mol−1 , O−H = 0.08365 kcal mol−1.

85

The potential energy function for an electrostatic interaction with a C1 switching function [99] is given by Uijelectrostatic = C

qi qj 1 C (rij ), rij

(A.3)

where rij = krj − ri k is the distance between atoms i and j, qi is the charge for atom i, and C = 332.0636 kcal mol−1 K−1 . Note that in Ewald summations with periodic boundary conditions, the switching function is not used. The switching function splits Coulomb energy into fast and slow parts, as given by    1 − ( 3 |~rij |rc 2 − 1 |~rij |3 )rc −3 2 2 1 C (~rij ) =   0

if|~rij | ≤ rc ,

(A.4)

if|~rij | > rc ,

where rc ≡ cutoff is the cutoff distance where the function value becomes zero. The energy for a Lennard–Jones interaction with a C2 switching function [42] is given by

UijLennard−−Jones = 4ij

σij rij

12 −

σij rij

6 ! C 2 (rij ),

(A.5)

where ij and σij are the Lennard–Jones energy minimum and cross over point (where the LJ function is zero). And     1     2 2 2 2 rij |2 − 3r0 2 ) c + 2|~ C 2 (~rij ) = (|~rij | − rc ) (r 3 2 2  (rc − r0 )      0

if|~rij | ≤ r0 , ifr0 ≤ |~rij | < rc ,

(A.6)

if|~rij | > rc ,

where r0 ≡ switchon that where it becomes active. A.2 Metric of Instabilities in MD For simulations using symplectic methods such as Impulse or MOLLY, we use the “Percent Relative Drift of Total Energy,” Drel , as a metric to measure the instabilities [47], which is given as follows: Drel = 100bL/K, 86

(A.7)

where b is the slope of the linear curve fit of the block–averaged total energy, L is the simulation length, and K is the average kinetic energy throughout the simulation. For a fixed simulation length, the bigger the value of Drel , the more overheating the simulation undergoes. In order to measure the goodness of the linear curve fit, we define the error bars as two times the “Percent Relative Root Mean Square Deviation,” δrel , which is given as follows: δrel

v u N X 100 u t (y − y˜ )2 /N, = i i K i=1

(A.8)

where yi is the block–averaged total energy at time ti , N is the number of data points of yi , and y˜i is the value of the fitted straight line at ti . A.3 Analytical Hessians for CHARMM Potentials Here we derive the analytical Hessians of different potentials (energies) for the CHARMM force field [71,72] used in the implementation of B-spline MOLLY integrators in ProtoMol. These analytical Hessians can also be used to construct other MD integrators. Bonds describe a linear bond between two atoms [83]. These bonds are described by a simple harmonic springs. The energy of a bond between atoms i and j is given by: Ebond = k (|~rij | − r0 )2

(A.9)

where k is the spring constant, ~rij = ~rj − ~ri , and r0 is the rest distance of the bond. The gradient and Hessian of bond energy are given as following. Erbond = 2k(|~rij | − r0 )[−rîj , rîj ]T (A.10)     T −rîj rîj T  |~rij | − r0  I −I  2 k r0  rîj vîj bond Err = 2k  . (A.11)  +  |~rij | |~rij | T T −rîj vîj rîj rîj −I I Angle interactions describe angular bonds between three atoms. These bonds are modeled as harmonic angular springs. The energy of such a bond between atoms i, j, and

87

k is given by: Eangle = Eθ + Eub ,

(A.12)

Eθ = kθ (θ − θ0 )2 ,

(A.13)

(A.14) Eub = kub (|~rik | − rub )2 , ~rij · ~rkj where kθ is the force constant, θ = cos−1 , θ0 is the rest angle of this bond, |~rij ||~rkj | kub is Urey–Bradley (UB) constant, ~rik = ~rk − ~ri , |~rik | is the calculated distance between atoms i and k and rub is rest distance for the UB term. The Hessian of the UB energy is as follows:    T  I 0 −I   rîk rîk − I   2 kub rub  ub  + Err = 2kub  0 0 0 0    |~ r | ik    −I 0

Let C(α, β, γ) =

−rîk rîk T

I

α+β−γ √ √ 2 α β



0 −rîk rîk + I    . (A.15) 0 0   + I 0 rîk rîk T − I T

where α, β and γ are scalars, and α = |~rij |2 , β = |~rkj |2 , and

γ = |~rki|2 . The Eθ part of the angle energy can be expressed as Eθ (C(α, β, γ)) = kθ (cos−1 (C(α, β, γ)) − θ0 )2 .

(A.16)

The second derivative of the above equation is then obtained as follows: rr z }| { 2 kθ [sin θ − (θ − θ0 ) cos θ] = EC Crr + Cr CrT sin3 θ | {z }

Ea

θ Err

(A.17)

b Err

(θ−θ0 ) √ , g = −α+β+γ √ where EC = − 2kθsin , Cr = f αr +g βr +h γr in which f = 4α−β+γ ,h = θ α3/2 β 4 α β 3/2 √ √ √ − 2 √α1 √β , αr = (−2, 2, 0)T α rîj , βr = (0, 2, −2)T β rˆkj , and γr = (2, 0, −2)T γ rˆki .

Thus Cr now can be expressed as follows:   −2f~rij + 2h~rki  Cr =   2f~rij + 2g~rkj  −2g~rkj − 2h~rki 88

   .  

(A.18)

and the upper half of the symmetric matrix Cr CrT is given by Cr CrT [0][0] = 4f 2~rij ~rij T − 4f h(~rij ~rki T + ~rki~rij T ) + 4h2~rki~rki T

(A.19)

Cr CrT [0][1] = 4gh~rki~rkj T − 4f g~rij ~rkj T + 4f h~rki~rij T − 4f 2~rij ~rij T

(A.20)

Cr CrT [0][2] = 4f g~rij ~rkj T + 4f h~rij ~rki T − 4gh~rki~rkj T − 4h2~rki~rki T

(A.21)

Cr CrT [1][1] = 4f 2~rij ~rij T + 4f g(~rij ~rkj T + ~rkj ~rij T ) + 4g 2~rkj ~rkj T

(A.22)

Cr CrT [1][2] = −4f g~rij ~rkj T − 4f h~rij ~rki T − 4gh~rkj~rkiT − 4g 2~rkj ~rkj T (A.23) Cr CrT [2][2] = 4g 2~rkj ~rkj T + 4gh(~rkj ~rki T + ~rki~rkj T ) + 4h2~rki~rkiT .

(A.24)

The Crr part is computed as follows: Ca

rr z }| { Crr = (f αrr + g βrr + h γrr ) + (αr frT + βr grT + γr hTr ), | {z }

(A.25)

b Crr

where  αrr





 2I −2I 0    , = −2I 2I 0     0 0 0



0   0 0   , βrr =  0 2I −2I     0 −2I 2I





 2I 0 −2I     γrr =  0 0 0     −2I 0 2I (A.26)

and fr = fα αr + fβ βr + fγ γr ,

gr = gα αr + gβ βr + gγ γr ,

hr = hα αr + hβ βr + hγ γr

(A.27)

where −α + 3β − 3γ −α − β − γ 1 √ , fγ = 3/2 √ , fβ = 3/2 3/2 5/2 8α β 8α β 4α β 3α − β − 3γ 1 √ 5/2 , gγ = √ 3/2 , = fβ , gβ = 8 αβ 4 αβ

fα =

(A.28)

gα

(A.29)

hα = fγ ,

hβ = gγ ,

hγ = 0.

89

(A.30)

a In Equation (A.25), the first part, Crr , becomes  −2f I −2hI  2(f + h)I  a Crr = 2(f + g)I −2gI  −2f I  −2hI −2gI 2(g + h)I

   ,  

(A.31)

b whereas the second part, Crr , becomes b = fα αr αrT + fβ αr βrT + fγ αr γrT + gα βr αrT + gβ βr βrT + Crr

gγ βr γrT + hα γr αrT + hβ γr βrT + hγ γr γrT

(A.32)

which is equivalent to b Crr = fα αr αrT + gβ βr βrT + fβ (αr βrT + βr αrT ) + fγ (αr γrT + γr αrT ) +

gγ (βr γrT + γr βrT ),

(A.33)

b thus upper half of the symmetric matrix, Crr , now can be expressed as follows: b [0][0] = 4fα~rij ~rij T − 4fγ (~rij ~rkiT + ~rki~rij T ) Crr

(A.34)

b Crr [0][1] = −4fα~rij ~rij T − 4fβ ~rij ~rkj T + 4fγ ~rki~rij T + 4gγ ~rki~rkj T

(A.35)

b Crr [0][2] = 4fβ ~rij ~rkj T + 4fγ ~rij ~rki T − 4gγ ~rki~rkj T

(A.36)

b Crr [1][1] = 4fα~rij ~rij T + 4fβ (~rij ~rkj T + ~rkj ~rij T ) + 4gβ ~rjk~rjk T

(A.37)

b Crr [1][2] = −4gβ ~rjk~rjk T − 4fβ ~rij ~rkj T − 4fγ ~rij ~rki T − 4gγ ~rkj ~rki T

(A.38)

b [2][2] = 4gβ ~rjk~rjk T + 4gγ (~rkj ~rkiT + ~rki~rkj T ). Crr

(A.39)

The Lennard–Jones (van der Waals) interactions describe the forces resulting from local interactions of atoms. The van der Walls energy between two atoms i and j is described by Evdw =

A B 12 − |~rij | |~rij |6

(A.40)

where A and B are constants specified for a pair of atom types explicitly in the parameter file, ~rij = ~rj − ~ri , the vector from atom i to atom j, |~rij | is the length of vector ~rij . 90

The first derivative of the van der Waals energy is as follows. ! 6B 12 A Er = [−r~ij , r~ij ]T . 8 − 14 |~rij | |~rij | The Hessian of van der Waals energy is as follows.     T T −rîj rîj   I −I   rîj rîj Err = C1   + C2  . T T −I I −rîj rîj rîj rîj

(A.41)

(A.42)

6B 12 A −48 B 168 A 8 − 14 , and C2 = 8 + 14 . |~rij | |~rij | |~rij | |~rij | Electrostatics describes the force resulting from the interaction between two charged

where C1 =

particles. The electrostatic energy between two atoms i and j is described by the Coulomb’s Law as: Eelect =

14 C qi qj 0 |~rij |

(A.43)

where 14 is scaling factor for 1 − 4 interactions, C = 2.31 × 10−19 J nm, qi , qj are charges for atom i and j, 0 is dielectric constant, ~rij and |~rij | are same as defined before. The first derivative of the electrostatic energy is as follows. Er = −

14 C qi qj [−r~ij , r~ij ]. 0 |~rij |3

The Hessian of electrostatic energy is as follows.   T −I + 3rîj rîj T  14 C qi qj  I − 3rîj rîj Err = −  . T T 0 |~rij |3 −I + 3rîj rîj I − 3rîj rîj

(A.44)

(A.45)

A switching function is often applied to a nonbonded energy computation to suppress the destabilizing factor introduced by the cutoff approximations. When the effective energy (Ee ) is taken as the raw energy (E) multiplied by a switching function (Y ), i.e., Ee = E Y, the Hessian of the effective energy is given as the following using chain rule: T ∂2E ∂Y ∂E ∂2Y ∂ 2 Ee = Y + 2 + E ∂r 2 ∂r 2 ∂r ∂r ∂r 2 91

(A.46)

There are two popular switching functions, C 1 and C 2 . C 1 is defined by Equation (A.4). The first derivative of C 1 is given by Cr1 = −

3 3|~rij |2 (−rîj , rîj )T + (−rîj , rîj )T , 2rc 2rc3 

(A.47)



3 3|~rij |  I −I  ) +  2rc |~rij | 2rc3 −I I   T T −rîj rîj  3|~rij |  rîj rîj 3 ) + +(   2rc |~rij | 2rc3 −rîj rîj T rîj rîj T

1 Crr = (−

(A.48)

The other switching function, C 2 , is defined by Equation (A.6), which is more expensive to evaluate. The first derivative of C 2 is given by Cr2

12(|~rij |2 − rc 2 )(|~rij |2 − r0 2 ) = (−~rij , ~rij )T 2 2 3 (rc − r0 )

(A.49)

The Hessian is given by two parts: a

2 2 2 Crr = Crr + Crr

where

b

 2 Crr

a

2 Crr

b

(A.50)



12(|~rij |2 − rc 2 )(|~rij |2 − r0 2 )  I −I   , (rc 2 − r0 2 )3 −I I   T T 2 2 2 2 −rîj rîj  24|~rij | (2|~rij | − r0 − rc )  rîj rîj =  . 2 2 3 (rc − r0 ) T T −rîj rîj rîj rîj =

(A.51)

(A.52)

A.4 Ewald Summation In the simulations reported in this thesis, we used Ewald summation method [1,23,25, 27,35,89,108] to evaluate the Coulomb forces with periodic boundary conditions. The details of implementation and choices of parameters can be found on the ProtoMol [75, 76] 92

website. The fast electrostatics algorithms including Ewald, PME [21, 53] and MultiGrid [103] have been implemented in ProtoMol by Dr. Thierry Matthey [74]. It is included here for completeness. The Coulomb energy, U, is given by N N 1 X† X X erfc(α|rij + n|) U = qi qj 4π0 n i=1 j=i+1 |r ij + n| | {z } Real–space term  2 2  N N   1 X 1 − k22 X X 4α + e q cos(k · r ) + q sin(k · r ) i i i i   0 V k2 i=1 i=1 k>0 | {z } Reciprocal–space term Nm X Nm 1 X†−1 X erf(α|rκλ |) − 3 − qnκ qnλ 4π0 n=1 κ=1 λ=κ+1 |r κλ | 4π 2 0 i=1 | {z } | {z } Point self–energy Intra–molecular self energy 2  2  N N 1 1 X X   − q + q r , i i i 80 V α2 i=1 60 V i=1 | {z } | {z }

α

N X

M

qi2

Charged system term

(A.53)

Surface dipole term

where “daggered” († ) summation indicates a given exclusion scheme to omit, modify site −1

pairs i, j if n = 0. The “inverse daggered” († ) summation indicates to consider only site pairs i, j, which are normally omitted, modified by the exclusion scheme. The rest of the symbols are: n, lattice vector of periodic cell images; k, reciprocal lattice vector of periodic cell images; k, modulus of k; i, j, absolute indices of all charged sites; κ, λ, indices of sites within a single molecule; N, total number of charged sites, M, total number of molecules; Nm , number of sites on molecule m; qi , qj , charge on absolute site i, j; qmκ , charge on site κ of molecule m; r i , Cartesian coordinate of site i; r ij , r j − r i ; α, real/reciprocal space partition parameter; and V , volume of one periodic cell. The force on charge i is given by Eq. (A.54). Both the real– and reciprocal–space series (the sums over n and k) converge fairly rapidly so that only a few terms need be evaluated. We define the cutoff distances rc and kc so that only terms with |rij + n| < rc 93

and |k| < kc are included. f i = −∇r i U N erfc(α|rij + n|) rij + n qi X† X 2α −α2 |r ij +n|2 = qj +√ e 4π0 n j=1 |rij + n| |r ij + n|2 π j6=i | {z } 2 X k − k22 + qi 2 e 4α 0 V k k>0 |

(

Real–space term

sin(k · r i )

N X j=1

qj cos(k · r j ) − cos(k · r i )

N X

) qj sin(k · r j )

j=1

{z

}

Reciprocal–space term M Nm X Nm 2α −α2 |rκλ |2 erf(α|rκλ |) r κλ qi X†−1 X + qj √ e − 4π0 n=1 κ=1 λ=κ+1 π |rκλ | |rκλ |2 | {z } Intra-molecular term " !# N X qi + qj r j . 60 V j=1 | {z }

(A.54)

Surface dipole term

The parameter α determines how rapidly the terms decrease and the values of rc and kc needed to achieve a given accuracy. For a fixed α and accuracy the number of terms in the real–space sum is proportional to the total number of sites, N but the cost of the 3

reciprocal–space sum increases as N 2 . An overall scaling of N 2 may be achieved if α varies with N. The optimal value of α is [26, 86] 1 √ tR N 6 α= π tF V 2

(A.55)

where tR and tF are the execution times needed to evaluate a single term in the real– and reciprocal–space sums respectively. If we require that the sums converge to an accuracy of = exp(−p) the cutoffs are then given by √ p √ rc = , kc = 2α p. α

(A.56)

A representative value of tR /tF specific to ProtoMol has been established as 5.5. Though this will vary on different processors and for different potentials its value is not critical since it enters the equations as a sixth root. 94

A.5 Names and Acronyms Names and acronyms used in this thesis are summarized here for clarity. • DPD. Dissipative Particle Dynamics, a method for studying complex fluids. • HMC. Hybrid Monte Carlo, a sampling method that alternates between molecular dynamics (MD) simulation and Monte Carlo (MC) selection. • KAM. Kolmogorov–Arnold–Moser, a name related to the theory of stability of chaos. • MOLLY. The mollified Verlet-I/r-RESPA/Impulse method, a stabler version of Impulse due to mollification of the slow forces. • RATTLE. An iterative method involving constraining the velocities in addition to SHAKE–constraining positions. • SHAKE. An iterative method for solving constrained differential algebraic equations (DAE) in MD such that the positions of atoms satisfy distance constraints. • TIP3P. A popular rigid water solvent model consisting of 3 sites, proposed by Jorgensen et al. in 1983. In this thesis, we used a modified TIP3P model in which we do not impose the distance constraints, a model of water with flexible bonds and angles, which adds stiffness to the systems of equations and thus imposes greater challenges to numerical methods. • TM. Targeted MOLLY, a multiscale stochastic integrator with targeted Langevin stabilization. • Verlet-I/r-RESPA. Reversible Reference System Propagator Algorithm, also called Impulse. It is a generalization of the single time stepping (STS) Verlet method, multiple time stepping (MTS) and symplectic. 95

Table A.1. UNITS USED IN PROTOMOL Quantity Time Length Energy Force Mass Temperature Charge

A.6

ProtoMol Unit fs ˚ A kcal/mol ˚ kcal/mol/A amu K e

SI 10−15 s 10−10 m 4184/(6.022045 × 1023 ) J 4184/(6.022045 × 1013 ) N 1.6605 × 10−27 Kg K 1.602 × 10−19 C

ProtoMol Units

Due to the legacy of molecular dynamics simulations, ProtoMol uses a particular combination of units such that it is consistent with other MD packages such as NAMD. In √ particular, the velocity unit is 103 4.184 m/s due to the use of PDB factor, 20.45482706. Velocities in PDB files are scaled to get a more accurate representation since the PDB format has a limited representation of floating numbers.

96

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