an improved hybrid monte carlo method for conformational sampling of proteins
DESCRIPTION
An improved hybrid Monte Carlo method for conformational sampling of proteins. Jesús A. Izaguirre and Scott Hampton Department of Computer Science and Engineering University of Notre Dame March 5, 2003 - PowerPoint PPT PresentationTRANSCRIPT
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An improved hybrid Monte Carlo method for
conformational sampling of proteins
Jesús A. Izaguirre and Scott HamptonDepartment of Computer Science and Engineering
University of Notre Dame
March 5, 2003
This work is partially supported by two NSF grants (CAREER and BIOCOMPLEXITY) and two grants from University of Notre Dame
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Overview1. Motivation: sampling conformational space of proteins
2. Methods for sampling (MD, HMC)
3. Evaluation of new Shadow HMC
4. Future applications
3
Protein: The Machinery of LifeNH2-Val-His-Leu-Thr-Pro-Glu-Glu-Lys-Ser-Ala-Val-Thr-Ala-Leu-Trp-Gly-Lys-Val-Asn-Val-Asp-Glu-Val-Gly-Gly-Glu-…..
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Protein Structure
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Why protein folding? Huge gap: sequence data and 3D structure data
EMBL/GENBANK, DNA (nucleotide) sequences 15 million sequence, 15,000 million base pairs
SWISSPROT, protein sequences120,000 entries
PDB, 3D protein structures20,000 entries
Bridging the gap through prediction Aim of structural genomics:
“Structurally characterize most of the protein sequences by an efficient combination of experiment and prediction,” Baker and Sali (2001)
Thermodynamics hypothesis: Native state is at the global free energy minimum
Anfinsen (1973)
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Questions related to folding I Long time kinetics:
dynamics of folding only statistical
correctness possible ensemble dynamics e.g., folding@home
Short time kinetics strong correctness
possible e.g., transport
properties, diffusion coefficients
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Questions related to folding II Sampling
Compute equilibrium averages by visiting all (most) of “important” conformations
Examples: Equilibrium
distribution of solvent molecules in vacancies
Free energies Characteristic
conformations (misfolded and folded states)
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Overview1. Motivation: sampling conformational space of proteins
2. Methods for sampling (MD, HMC)
3. Evaluation of new Shadow HMC
4. Future applications
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Classical molecular dynamics Newton’s
equations of motion:
Atoms Molecules CHARMM force
field(Chemistry at Harvard Molecular Mechanics)
'' ( ) ( ). - - - (1)U Mq q F q
Bonds, angles and torsions
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What is a Forcefield?
To describe the time evolution of bond lengths, bond angles and torsions, also the nonbond van der Waals and elecrostatic interactions between atoms, one uses a forcefield.The forcefield is a collection of equations and associated constants designed to reproduce molecular geometry and selected properties of tested structures.
In molecular dynamics a molecule is described as a series of charged points (atoms) linked by springs (bonds).
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Energy Terms Described in the CHARMm forcefield
Bond Angle
Dihedral Improper
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Energy Functions
Ubond = oscillations about the equilibrium bond lengthUangle = oscillations of 3 atoms about an equilibrium angleUdihedral = torsional rotation of 4 atoms about a central bondUnonbond = non-bonded energy terms (electrostatics and Lennard-Jones)
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Molecular Dynamics –what does it mean?MD = change in conformation over time using a forcefield
Conformational change
EnergyEnergy supplied to the minimized system at the start of the simulation
Conformation impossible to access through MD
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MD, MC, and HMC in sampling Molecular Dynamics takes long steps in phase
space, but it may get trapped Monte Carlo makes a random walk (short
steps), it may escape minima due to randomness
Can we combine these two methods?
MCMDHMC
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Hybrid Monte Carlo We can sample from a distribution with
density p(x) by simulating a Markov chain with the following transitions: From the current state, x, a candidate state x’
is drawn from a proposal distribution S(x,x’). The proposed state is accepted with prob.min[1,(p(x’) S(x’,x)) / (p(x) S(x,x’))]
If the proposal distribution is symmetric, S(x’,x)) = S(x,x’)), then the acceptance prob. only depends on p(x’) / p(x)
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Hybrid Monte Carlo II Proposal functions must be reversible:
if x’ = s(x), then x = s(x’) Proposal functions must preserve
volume Jacobian must have absolute value one Valid proposal: x’ = -x Invalid proposals:
x’ = 1 / x (Jacobian not 1) x’ = x + 5 (not reversible)
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Hybrid Monte Carlo III Hamiltonian dynamics preserve volume in
phase space Hamiltonian dynamics conserve the Hamiltonian
H(q,p) Reversible symplectic integrators for
Hamiltonian systems preserve volume in phase space
Conservation of the Hamiltonian depends on the accuracy of the integrator
Hybrid Monte Carlo: Use reversible symplectic integrator for MD to generate the next proposal in MC
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HMC Algorithm
Perform the following steps:1. Draw random values for the momenta p from
normal distribution; use given positions q2. Perform cyclelength steps of MD, using a
symplectic reversible integrator with timestep t, generating (q’,p’)
3. Compute change in total energy H = H(q’,p’) - H(q,p)
4. Accept new state based on exp(- H )
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Hybrid Monte Carlo IV
Advantages of HMC: HMC can propose and accept distant points
in phase space, provided the accuracy of the MD integrator is high enough
HMC can move in a biased way, rather than in a random walk (distance k vs sqrt(k))
HMC can quickly change the probability density
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Hybrid Monte Carlo V As the number of atoms
increases, the total error in the H(q,p) increases. The error is related to the time step used in MD
Analysis of N replicas of multivariate Gaussian distributions shows that HMC takes O(N5/4 ) with time step t = O(N-1/4) Kennedy & Pendleton, 91
System size N
Max t
66 0.5
423 0.25
868 0.1
5143 0.05
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Hybrid Monte Carlo VI The key problem in scaling is the accuracy of
the MD integrator More accurate methods could help scaling Creutz and Gocksch 89 proposed higher
order symplectic methods for HMC In MD, however, these methods are more
expensive than the scaling gain. They need more force evaluations per step
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Overview1. Motivation: sampling conformational space of proteins
2. Methods for sampling (MD, HMC)
3. Evaluation of new Shadow HMC
4. Future applications
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Improved HMC Symplectic integrators conserve exactly
(within roundoff error) a modified Hamiltonian that for short MD simulations (such as in HMC) stays close to the true Hamiltonian Sanz-Serna & Calvo 94
Our idea is to use highly accurate approximations to the modified Hamiltonian in order to improve the scaling of HMC
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Shadow Hamiltonian
Work by Skeel and Hardy, 2001, shows how to compute an arbitrarily accurate approximation to the modified Hamiltonian, called the Shadow Hamiltonian
Hamiltonian: H=1/2pTM-1p + U(q) Modified Hamiltonian: HM = H + O(t p) Shadow Hamiltonian: SH2p = HM + O(t 2p)
Arbitrary accuracy Easy to compute Stable energy graph
Example, SH4 = H – f( qn-1, qn-2, pn-1, pn-2 ,βn-1 ,βn-2)
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See comparison of SHADOW and ENERGY
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Shadow HMC Replace total energy H with shadow
energy SH2m = SH2m (q’,p’) – SH2m (q,p)
Nearly linear scalability of sampling rateComputational cost SHMC, N(1+1/2m), where
m is accuracy order of integrator Extra storage (m copies of q and p) Moderate overhead (25% for small
proteins)
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Example Shadow Hamiltonian
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ProtoMol: a framework for MD
Front-end
Middle layer
back-end
libfrontend
libintegrators
libbase, libtopologylibparallel, libforces
Modular design of ProtoMol (Prototyping Molecular dynamics).Available at http://www.cse.nd.edu/~lcls/protomol
Matthey, et al, ACM Tran. Math. Software (TOMS), submitted
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SHMC implementation Shadow Hamiltonian
requires propagation of β
Can work for any integrator
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Systems tested
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Sampling Metric 1 Generate a plot of dihedral angle vs.
energy for each angle Find local maxima Label ‘bins’ between maxima For each dihedral angle, print the label
of the energy bin that it is currently in
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Sampling Metric 2 Round each dihedral angle to the
nearest degree Print label according to degree
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Acceptance Rates
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More Acceptance Rates
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Sampling rate for decalanine (dt = 2 fs)
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Sampling rate for 2mlt
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Sampling rate comparison Cost per conformation is total
simulation time divided by number of new conformations discovered (2mlt, dt = 0.5 fs) HMC 122 s/conformation SHMC 16 s/conformation HMC discovered 270 conformations in
33000 seconds SHMC discovered 2340 conformations in
38000 seconds
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Conclusions SHMC has a much higher acceptance
rate, particularly as system size and timestep increase
SHMC discovers new conformations more quickly
SHMC requires extra storage and moderate overhead.
SHMC works best at relatively large timesteps
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Future work Multiscale problems for rugged energy surface
Multiple time stepping algorithms plus constraining Temperature tempering and multicanonical
ensemble Potential smoothing
System size Parallel Multigrid O(N) electrostatics
Applications Free energy estimation for drug design Folding and metastable conformations Average estimation
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Acknowledgments Dr. Thierry Matthey, lead developer of
ProtoMol, University of Bergen, Norway Graduate students: Qun Ma, Alice Ko,
Yao Wang, Trevor Cickovski Dr. Robert Skeel, Dr. Ruhong Zhou, and
Dr. Christoph Schutte for valuable discussions
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Multiple time stepping Fast/slow force splitting
Bonded: “fast” Small periods
Long range nonbonded: “slow” Large characteristic time
Evaluate slow forces less frequently Fast forces cheap Slow force evaluation expensive
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The Impulse integratorGrubmüller,Heller, Windemuth and Schulten,
1991 Tuckerman, Berne and Martyna, 1992
The impulse “train”
Time, t
Fast impulses, t
Slow impulses, t
How far apart can we stretch the impulse train?
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Stretching slow impulses t ~ 100 fs if accuracy does not degenerate
1/10 of the characteristic time MaIz, SIAM J. Multiscale Modeling and Simulation, 2003
(submitted)
Resonances (let be the shortest period) Natural: t = n , n = 1, 2, 3, … Numerical:
Linear: t = /2 Nonlinear: t = /3
MaIS_a, SIAM J. on Sci. Comp. (SISC), 2002 (in press)MaIS_b, 2003 ACM Symp. App. Comp. (SAC’03), 2002 (in press)
Impulse far from being multiscale!
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3rd order nonlinear resonance of Impulse MaIS_a, SISC, 2002; MaIS_b, ACM
SAC’03, 2002
Fig. 1: Left: flexible water system. Right: Energy drift from 500ps MD simulation of flexible water at room temperature revealing 3:1 and 4:1 nonlinear resonance (3.3363 and 2.4 fs)
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
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Objective statement
Design multiscale integrators that are not limited by nonlinear and linear instabilities Allowing larger time steps Better sequential performance Better scaling
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
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Targeted MOLLY (TM)
TM = MOLLY + targeted Langevin coupling
Mollified Impulse (MOLLY) to overcome linear instabilities
Izaguirre, Reich and Skeel, 1999
Stochasticity to stabilize MOLLYIzaguirre, Catarello, et al, 2001
MaIz, 2003 ACM Symp. App. Comp. (SAC’03), 2002 (in press)MaIz, SIAM J. Multiscale Modeling and Simulation, 2003 (submitted)
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Mollified Impulse (MOLLY)
MOLLY (mollified Impulse) Slow potential at time averaged positions,
A(x) Averaging using only fastest forces Mollified slow force = Ax(x) F(A(x)) Equilibrium and B-spline
B-spline MOLLY Averaging over certain time interval Needs analytical Hessians Step sizes up to 6 fs (50~100% speedup)
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Introducing stochasticity Langevin stabilization of MOLLY (LM)
Izaguirre, Catarello, et al, 2001
12 fs for flexible waters with correct dynamics Dissipative particle dynamics (DPD):
Pagonabarraga, Hagen and Frenkel, 1998
Pair-wise Langevin force on “particles” Time reversible if self-consistent
Vi Vj
FRi, FD
i
FRj = - FR
i
FDj = - FD
i
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Targeted Langevin coupling
Targeted at “trouble-making” pairs Bonds, angles Hydrogen bonds
Stabilizing MOLLY Slow forces evaluated much less
frequently Recovering correct dynamics
Coupling coefficient small
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TM: main results
16 fs for flexible waters Correct dynamics
Self-diffusion coefficient, D. leapfrog w/ 1fs, D = 3.69+/-0.01 TM w/ (16 fs, 2fs), D = 3.68+/-0.01
Correct structure Radial distribution function (r.d.f.)
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TM: correct r.d.f.
Fig. 4. Radial distribution function of O-H (left) and H-H (right) in flexible waters.
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TM: discussion Much larger time steps (~100 fs)
Better force splitting Zhou, Harder, etc. 2001
More stable MOLLY integrators Conformational transition rate
folding@home: ensemble dynamics Celestial systems Graphics rendering in gaming
applications
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
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Lengthening time scales: SRP
1 msec simulation of an enzyme using stochastic reaction path (SRP) method. Courtesy of R. Elber at Cornell Univ.. SRP requires initial and final configuration.
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What do we learn from SRP?
Cons: final configuration not always available
Pros: large time steps/long simulation Motions whose char. time is less than the
step size filtered out Essential (interesting) motions remain
What are other approaches for filtering out fast motions?
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Lengthening time scales: MUSICO
Ideas: Splitting into nearly linear and nonlinear
parts Implicit integration of linear part with
constraining of internal d.o.f. Explicit treatment of highly nonlinear part Optional pairwise stochasticity for stability
MUSICO (in progress) Multi-Scale Implicit-explicit Constrained
molecular dynamics
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MUSICO: the anticipations
Larger step sizes: ps, ns Implicit methods generally more stable
Schlick, et al (1997)
Greater scalability Hessian computations mostly local Low in communication
Real speedup Linear parts cheap 100 fold speedup sequentially?
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MUSICO: the implicit part Implicit equations for nearly linear part
Newton-Raphson method
Require Hessians Krylov subspace method
1/ 2
21/ 2 1/ 2 1/ 2
1 1/ 2
1 1/ 2 1
1. / 2,
2. ( ),
3. ,
4. / 2
n n n
n n n
n n n
n n n
t
U t
t
t
x x p
F x F
p p F
x x p
20 1 1 1/ 2 1/ 2 1/ 2( , ,..., ) ( ) 0,
.N n n nf f f U t
F F x F
J f F
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
62
Applications of approximate MD
Combining w/ other algorithms Steered/interactive MD Monte Carlo (MC)
Escaping from local minima Quicker exploring conformational space
Coarsening Large system and/or long simulations
Protein simulations: Folding/Unfolding/Equilibrium Drug design Other
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Approx. MD in drug design
Designing drugs that regulate proteins Examples: drugs that bind to
Enzymes Receptors Ion channels and transporter systems
Evaluation of binding affinity using approximate MD Very long simulations and better sampling
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Approx. MD in drug design (cont.)
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ProtoMol: a framework for MD
Front-end
Middle layer
back-end
libfrontend
libintegrators
libbase, libtopologylibparallel, libforces
Modular design of ProtoMol (Prototyping Molecular dynamics).Available at http://www.cse.nd.edu/~lcls/protomol
Matthey, et al, ACM Tran. Math. Software (TOMS), submitted
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
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Summary Introduction Nonlinear instabilities (original) New algorithms
Targeted MOLLY (original) Larger time steps and speedup Correct dynamics and structures
MUSICO (original, in progress) Applications of approx. MD (in progress)
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
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Future work Short term
MUSICO / Krylov solver Simulation of small proteins in vacuum
Mid term Optimization of MUSICO / Krylov solver Simulation of large proteins in water Binding affinity in drug design
Long term Distributed protein folding and drug design Infrastructure for heterogeneous distributed
computing – similar to folding@home
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Future work (cont.)
Funding opportunities NSF NIH DOE Pharmaceutical industry
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Key references[1] Izaguirre, Ma, et al. Overcoming instabilities in Verlet-I/r-RESPA with the
mollified impulse method. Vol. 24 of Lecture Notes in Comput. Sci. & Eng., pages 146-174, Springer-Verlag, Berlin, New York, 2002
[2] Ma, Izaguirre and Skeel. Verlet-I/r-RESPA/Impulse is limited by nonlinear instability. SIAM J. Scientific Computing, 2002 (in press).
[3] Ma and Izaguirre. Targeted mollified impulse method for molecular dynamics. SIAM J. Multiscale Modeling and Simulation, submitted
[4] Matthey, Cickovski, Hampton, Ko, Ma, Slabach and Izaguirre. PROTOMOL, an object-oriented framework for prototyping novel applications of molecular dynamics. ACM Tran. Math. Software (TOMS), submitted
[5] Ma, Izaguirre and Skeel. Nonlinear instability in multiple time stepping molecular dynamics. 2003 ACM Symposium on Applied Computing (SAC’03). Melborne, Florida. 2002 (in press)
[6] Ma and Izaguirre. Long time step molecular dynamics using targeted Langevin Stabilization. Accepted by the 2003 ACM SAC’03. Melborne, Florida. 2002 (in press)
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Overview Introduction
Molecular dynamics (MD) in action Classical MD Protein folding
Nonlinear instabilities of Impulse integrator Approximate MD integrators
Targeted MOLLY MUSICO Applications
Summary Future work Acknowledgements
73
Acknowledgements
People Dr. Jesus Izaguirre Dr. Robert Skeel, Univ. of Illinois at Urbana-
Champaign Dr. Thierry Matthey, University of Bergen, Norway
Resources Hydra and BOB clusters at ND Norwegian Supercomputing Center, Bergen, Norway
Funding NSF BIOCOMPLEXITY-IBN-0083653, and NSF CAREER Award ACI-0135195
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THE END. THANKS!
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Key references[1] J. A. Izaguirre, Q. Ma, T. Matthey, et al. Overcoming instabilities in Verlet-I/r-RESPA
with the mollified impulse method. In T. Schlick and H. H. Gan, editors, Proceedings of the 3rd International Workshop on Algorithms for Macromolecular Modeling, Vol. 24 of Lecture Notes in Computational Science and Engineering, pages 146-174, Springer-Verlag, Berlin, New York, 2002
[2] Q. Ma, J. A. Izaguirre, and R. D. Skeel. Verlet-I/r-RESPA/Impulse is limited by nonlinear instability. Accepted by the SIAM Journal on Scientific Computing, 2002. Available at http://www.nd.edu/~qma1/publication_h.html.
[3] Q. Ma and J. A. Izaguirre. Targeted mollified impulse method for molecular dynamics. Submitted to the SIAM Journal on Multiscale Modeling and Simulation, 2003.
[4] T. Matthey, T. Cickovski, S. Hampton, A. Ko, Q. Ma, T. Slabach and J. Izaguirre. PROTOMOL, an object-oriented framework for prototyping novel applications of molecular dynamics. Submitted to the ACM Transactions on Mathematical Software (TOMS), 2003.
[5] Q. Ma, J. A. Izaguirre, and R. D. Skeel. Nonlinear instability in multiple time stepping molecular dynamics. Accepted by the 2003 ACM Symposium on Applied Computing (SAC’03). Melborne, Florida. March 2003
[6] Q. Ma and J. A. Izaguirre. Long time step molecular dynamics using targeted Langevin Stabilization. Accepted by the 2003 ACM Symposium on Applied Computing (SAC’03). Melborne, Florida. March 2003
[7] M. Zhang and R. D. Skeel. Cheap implicit symplectic integrators. Appl. Num. Math., 25:297-302, 1997
76
Other references[8] J. A. Izaguirre, Justin M. Wozniak, Daniel P. Catarello, and Robert D. Skeel. Langevin
Stabilization of Molecular Dynamics, J. Chem. Phys., 114(5):2090-2098, Feb. 1, 2001. [9] T. Matthey and J. A. Izaguirre, ProtoMol: A Molecular Dynamics Framework with Incremental
Parallelization, in Proc. of the Tenth SIAM Conf. on Parallel Processing for Scientific Computing, 2001.
[10] H. Grubmuller, H. Heller, A. Windemuth, and K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long range interactions, Molecular Simulations 6 (1991), 121-142.
[11] M. Tuckerman, B. J. Berne, and G. J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys 97 (1992), no. 3, 1990-2001
[12] J. A. Izaguirre, S. Reich, and R. D. Skeel. Longer time steps for molecular dynamics. J. Chem. Phys., 110(19):9853–9864, May 15, 1999.
[13] L. Kale, R. Skeel, M. Bhandarkar, R. Brunner, A. Gursoy, N. Krawetz, J. Phillips, A. Shinozaki, K. Varadarajan, and K. Schulten. NAMD2: Greater scalability for parallel molecular dynamics. J. Comp. Phys., 151:283–312, 1999.
[14] R. D. Skeel. Integration schemes for molecular dynamics and related applications. In M. Ainsworth, J. Levesley, and M. Marletta, editors, The Graduate Student’s Guide to Numerical Analysis, SSCM, pages 119-176. Springer-Verlag, Berlin, 1999
[15] R. Zhou, , E. Harder, H. Xu, and B. J. Berne. Efficient multiple time step method for use with ewald and partical mesh ewald for large biomolecular systems. J. Chem. Phys., 115(5):2348–2358, August 1 2001.
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Other references (cont.)[16] E. Barth and T. Schlick. Extrapolation versus impulse in multiple-time-stepping schemes:
Linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys., 1997.[17] I. Pagonabarraga, M. H. J. Hagen and D. Frenkel. Self-consistent dissipative particle dynamics
algorithm. Europhys Lett., 42 (4), pp. 377-382 (1998).[18] G. Besold, I. Vattulainen, M. Kartunnen, and J. M. Polson. Towards better integrators for
dissipative particle dynamics simulations. Physical Review E, 62(6):R7611–R7614, Dec. 2000.[19] R. D. Groot and P. B. Warren. Dissipative particle dynamics: Bridging the gap between
atomistic and mesoscopic simulation. J. Chem. Phys., 107(11):4423–4435, Sep 15 1997.[20] I. Pagonabarraga and D. Frenkel. Dissipative particle dynamics for interacting systems. J.
Chem. Phys., 115(11):5015–5026, September 15 2001.[21] Y. Duan and P. A. Kollman. Pathways to a protein folding intermediate observed in a 1-
microsecond simulation in aqueous solution. Science, 282:740-744, 1998. [22] M. Levitt. The molecular dynamics of hydrogen bonds in bovine pancreatic tripsin inhibitor
protein, Nature, 294, 379-380, 1981 [23] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press,
New York, 1987.[24] E. Hairer, C. Lubich and G. Wanner. Geometric numerical integration: structure-preserving
algorithms for ordinary differential equations. Springer, 2002[25] C. B. Anfinsen. Principles that govern the folding of protein chains. Science, 181, 223-230,
1973
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Flood of data – genes and proteins
http://www.ncbi.nlm.nih.gov/Genbank/genbankstats.html
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Flood of data (cont.)
http://us.expasy.org/sprot/relnotes/relstat.html
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Flood of data (cont.)PDB Content Growth
http://www.rcsb.org/pdb/holdings.html
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Flood of data (cont.)
http://us.expasy.org/sprot/relnotes/relstat.html
82
Pseudo-code of an MD simulation
MD Simulation:
(1) Pre-processing: Construct initial configuration of x0, v0 and F0;(2) loop 1 to number of steps
(a) Update velocities (by a half step)(b) Update positions (by a full step)(c) Evaluate forces on each atom(d) Update velocities (by a half step)
(3) Post-processing
Algorithm 1: Pseudo-code of an MD simulation.
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Why protein folding? Huge gap: sequence data and 3D structure data
EMBL/GENBANK, DNA (nucleotide) sequences 15 million sequence, 15,000 million base pairs
SWISSPROT, protein sequences120,000 entries
PDB, 3D protein structures20,000 entries
Bridging the gap through prediction Aim of structural genomics:
“Structurally characterize most of the protein sequences by an efficient combination of experiment and prediction,” Baker and Sali (2001)
Thermodynamics hypothesis: Native state is at the global free energy minimum
Anfinsen (1973)
84
Folding: challenging time scale gap
Longest fully solvated: Duan and Kollman, 1998 1 s single trajectory/100 days Cray T3E
IBM’s Blue Gene project: 1999 -- 2005 Petaflop (1015 flops per second) computer
for folding proteins by 2005
85
Stömer/Verlet/Leapfrog Discretization of Eq. (1) as
Stömer (astronomy, its higher order variants, 1907, aurora borealis), Verlet (molecular dynamics, 1967), Leapfrog (partial differential equations)
An explicit one-step method:
Second order accurate, symplectic, and time reversible
21 12 ( )n n n nq q q h f q
1 1
1/ 2 1 1/ 2 1/ 2 1/ 2 1
: ( , ) ( , ) :
( ), , ( )2 2
Vh n n n n
n n n n n n n n n
q v q v
h hv v f q q q hv v v f q
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Verlet-I/r-RESPA/ImpulseGrubmüller,Heller, Windemuth and Schulten,
1991 Tuckerman, Berne and Martyna, 1992
The state-of-the-art MTS integrator Splitting via switching functions 2nd order accurate, time reversible
slow1/ 2half kick: ( ) / 2
oscillate: update positions and momentum
using leapfrog ( /2, much smaller time steps)
n n nv v t f q
t
Algorithm 1. Half step discretization of Impulse integrator
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Nonlinear resonance of Impulse
Approach Analytical: Stability conditions using only model
parameters for a simple nonlinear problem Numerical: Long simulations differing only in outer
time steps; correlation between step size and instability
Results: energy growth occurs unless longest t < 1/3 shortest period.
Unconditionally unstable 3rd order resonance Flexible waters: outer time step less than 3~3.3fs Proteins w/ SHAKE: time step less than 4~5fs
Ma, Izaguirre and Skeel (2003)
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Nonlinear instability: analytical Approach:
1-D nonlinear model problem, in the neighborhood of stable equilibrium
MTS splitting of potential:
Analyze the reversible symplectic map Express stability condition in terms of problem
parameters Result:
3rd order resonance stable only if “equality” met 4th order resonance stable only if “inequality” met Impulse unstable at 3rd order resonance in practice
2 2 2 3 4 5oscillate kick( ) ( / 2) ( / 2 / 3 / 4) ( ).U q q Aq Bq Cq O q
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Nonlinear: analytical (cont.) Main result. Let
1. (3rd order) Map stable at equilibrium if and unstable if
Impulse is unstable in practice. 2. (4th order) Map stable if
and unstable if
May be stable at the 4th order resonance.
2 , where , where
1 ' /(2 ') if ' 0, 0 if ' 0, and
1 ' /(2 ') if s '/ 0, 0 if s '/ 0, and
' sin( / 2) and c ' cos( / 2).
i
hs A c c c
hc A s
s h h
0, 0,B C 0.B
20 or 2 ' '/ ,C C hB s c 20 2 ' '/ .C hB s c
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Fig. 2: Left: flexible melittin protein (PDB entry 2mlt). Right: Energy drift from 10ns MD simulation of flexible 2mlt at room temperature revealing 3:1 nonlinear resonance (at 3, 3.27 and 3.78 fs).
Nonlinear: numerical (cont.)
91
Nonlinear: numerical (cont.)
Energy drift from 500ps SHAKE-constrained MD simulation of explicitly solvated 2mlt at room temperature revealing combined 4:1 and 3:1 nonlinear resonance.
92
B-spline MOLLY (cont.)Total energy(Kcal/mol) vs. time (fs)
Relative drift of toal energy:
< 1.0% for t = 5.0 fs,
< 3.5% for t = 6.0 fs.
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TM: background (cont.) Dissipative particle dynamics:
Coarsening Pair-wise random/dissipative force on
elements (thus momentum preserving) Time reversible if using self-consistent
scheme (self-consistent dissipative leapfrog, SCD-leapfrog)
Vi Vj
FRi, FD
i
FRj = - FR
i
FDj = - FD
i
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Self-consistent dissipative leapfrog
1. 1/2 kick: ( ) /(2 ),
2. Oscillate:
3. Evaluate: ( ), ( , ) ( ) , ( ) /
4. 1/2 kick (a): ( ) /(2 )
5. 1/2 kick (b):
C D Ri i i i i
i i i
C D Ri j i j j ij ij ij i j ij ij
C Ri i i i
ii
v v t F F F m
r r v t
F r F r v v e e F r e t
v v t F F m
v v t F
/(2 )
6. Evaluate: ( , )
Iterate steps 5 and 6 k times to obtain consistency of and .
Di
Di j j
Di ij
m
F r v
F v
Algorithm 2. One step of the self-consistent dissipative leapfrog discretization
Pagonabarraga, Hagen and Frenkel, 1998
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Binding effectiveness computation Structure based drug design
First, study molecular details of a function or condition
Then, find a protein target for a disease Third, design the drug that binds to the
protein target Determine protein structure (experiments,
prediction) Search drug databases – screening Effectiveness of binding using molecular dynamics
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Small proteins
Beta hairpin protein Villin head piece protein
http://www.stanford.edu/group/pandegroup/Cosm/phase2.html
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Targeted MOLLY discretization
One step of Targeted MOLLY:(1) Update velocities w/ mollified “slow” forces (by a half long-step)(2) Propagate positions and velocities w/ “fast” forces and pairwise
targeted Langevin coupling (by a full long-step)(3) Do a time-averaging of positions using “fastest” forces and
evaluate the mollified “slow” force(4) Update velocities w/ mollified “slow” forces (by a half long-step)
Algorithm 1: Pseudo-code of one step of Targeted MOLLY
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TM: one step of discretization
slow1 1
1 1
1 fast R D
-n
1 MOLLIFIED KICK: / 2,
2OSCILLATE: Propagate and by integrating
, ( ) ( ) ( ) for an interval to get and p
n n
n n
n
p p t F
q p
q M p p F q F q F qt q
_
_reduced
_slow
using SCD-leapfrog.
A TIME AVERAGING: Calculate time-averaged positions and
a Jacobian matrix, using only the fastest forces, ( ).1
MOLLIFIED KICK: (2
n
n q n
n n n
q
J q F q
p p t F q
) / 2.
Algorithm 2. One step of the Targeted MOLLY (TM) discretization
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TM: canonical ensemble sampling
Fig. 3: Phase diagram (left) and velocity distribution (right) of the FPU problem [Hairer, C. Lubich and G. Wanner 2002] using TM with Langevin coupling coefficient of 0.01 showing TM recovers canonical ensemble after 2E07 steps with initial conditions x(0) = 0, v(0) = 1.
2 2 2 4 2( , ) / 2 / 2 ( 1) / 4, with (0) 0, (0) 1, 100.H x v v x x x v
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MD in action: coalescence
Coalescence of water droplets w/ surfactants in gas phase. http://www.cse.nd.edu/~lcls/protomol
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ProtoMol: object oriented framework for MD Matthey and Izaguirre (2001), Matthey, et al (2003)
http://www.cse.nd.edu/~lcls/protomol
Kale, Skeel, et al (1999)
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ProtoMol (cont.)
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Approx. MD: ion channel gating
KcsA potassium channel, courtesy of the Theoretical and Computational Biophysics Group at UIUC, http://www.ks.uiuc.edu/Research/smd_imd/kcsa/
104
MD: Verlet Method
Newton’s equation represents a set of N second order differential equations which are solved numerically at discrete time steps to determine the trajectory of each atom.
Advantage of the Verlet Method: requires only one force evaluation per timestep
Energy function:
used to determine the force on each atom:
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What is a Forcefield?
To describe the time evolution of bond lengths, bond angles and torsions, also the nonbond van der Waals and elecrostatic interactions between atoms, one uses a forcefield.The forcefield is a collection of equations and associated constants designed to reproduce molecular geometry and selected properties of tested structures.
In molecular dynamics a molecule is described as a series of charged points (atoms) linked by springs (bonds).
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Energy Terms Described in the CHARMm forcefield
Bond Angle
Dihedral Improper
107
Energy Functions
Ubond = oscillations about the equilibrium bond lengthUangle = oscillations of 3 atoms about an equilibrium angleUdihedral = torsional rotation of 4 atoms about a central bondUnonbond = non-bonded energy terms (electrostatics and Lenard-Jones)
108
Multigrid I
109
Multigrid II
110
Results (10-4 rPE)
111
Simulation Results for Melittin
PME requires about 3% of the CPU time (17 days 20 hours) when measured against Ewald
MG in pbc requires only about 1% MG is about 66% faster than PME