Department of Computer Science and Engineering
Prof. Jesus Izaguirre
Alice Ko

1. What is biomolecular modeling?
2. Historical perspective
3. Theory and experiments
4. Simulation procedures
5. MDSimAid

• Application of computational models to understand the structure, dynamics, and thermodynamics of biological molecules
• The models must be tailored to the question at hand: Schrodinger equation is not the answer to everything! Reductionist view bound to fail!
• This implies that biomolecular modeling must be both multidisciplinary and multiscale

1. 1946 MD calculation
2. 1960 force fields
3. 1969 Levinthal’s paradox on protein folding
4. 1970 MD of biological molecules
5. 1971 protein data bank a. 1998 ion channel protein crystal structure b. 1999 IBM announces blue gene project

1. Born-Oppenheimer approximation (fixed nuclei)
2. Force field parameters for families of chemical compounds
3. System modeled using Newton’s equations of motion
4. Examples: hard spheres simulations (alder and
Wainwright, 1959); Liquid water (Rahman and
Stillinger, 1970); BPTI (McCammon and
Karplus); Villin headpiece (Duan and Kollman,
1998)

• The computational study of atomic fluctuations in BPTI and other proteins has shown that :
– Directional character of active-site fluctuations in enzymes contributes to catalysis
– Small amplitude fluctuations are “lubricant”
– It may be possible to extrapolate from short time fluctuations to larger-scale protein motions

• Collective motions particularly important for biological function, e.g., displacements for transition from inactive to active
– Extended nature of these motions makes them sensitive to environment: great difference between vacuum and solution simulations
– Collective motions transmit external solvent effects to protein interior

• For the transport protein hemoglobin there are several important motions:
– Oxygen binding produces tertiary structural change
– A quaternary structural change from deoxy (low oxygen affinity) to oxy configuration takes place. This transmits information over a long distance
– From the X-ray deoxy and oxy structures, a stochastic reaction path has been found. Detailed ligand binding has been performed using MD . A statistical mechanical model has provided coupling between these two processes

• For the related storage protein, myoglobin:
– Fluctuations in the globin are essential to binding: the protein matrix in X-ray is so tightly packed that there is no low energy path for the ligand to enter or leave the heme pocket
– Only through structural fluctuations can the barriers be lowered sufficiently
– Demonstrated through energy minimization and molecular dynamics

• Three open problems are the following:
1. Ion channel gating: highly correlated fluctuations are likely to be of great importance. Long time dynamics problem
2. Flexible docking: for MMP, enzymes, etc., fluctuations enter into thermodynamics and kinetic of reactions. Sampling problem
3. Protein folding: too complicated for full treatment but for smallest proteins, beyond current methodology.
Coarsening problem

Macroscopic properties are often determined by molecule-level behavior.
Quantitative and/or qualitative information about macroscopic behavior of macromolecules can be obtained from simulation of a system at atomistic level.
Molecular dynamics simulations calculate the motion of the atoms in a molecular assembly using Newtonian dynamics to determine the net force and acceleration experienced by each atom. Each atom i at position r i
, is treated as a point with a mass m i and a fixed charge q i
.
NOTE: This material is courtesy of Klaus Schulten, www.ks.uiuc.edu

1) Build realistic atomistic model of the system
2) Simulate the behavior of your system over time using specific conditions (temperature, pressure, volume, etc)
3) Analyze the results obtained from MD and relate to macroscopic level properties

• What is a forcefield?
• How to prepare your system for MD?
• What specific conditions (temperature, pressure, volume, etc) will be used in MD?

In molecular dynamics a molecule is described as a series of charged points
(atoms) linked by springs
(bonds).
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.

Bond Angle
Dihedral Improper

Ubond = oscillations about the equilibrium bond length
Uangle = oscillations of 3 atoms about an equilibrium angle
Udihedral = torsional rotation of 4 atoms about a central bond
Unonbond = non-bonded energy terms (electrostatics and Lenard-Jones)

Topolgy files contain:
• atom types are assigned to identify different elements and different molecular orbital environments
• charges are assigned to each atom
• connectivities between atoms are established
Parameter files contain:
• force constants necessary to describe the bond energy, angle energy, torsion energy, nonbonded interactions (van der Waals and electrostatics)
• suggested parameters for setting up the energy calculations

MASS HS 1.0080 ! thiol hydrogen
MASS C 12.0110 ! carbonyl C, peptide backbone
MASS CA 12.0110 ! aromatic C
........ (missing data here)
!-----------------------------------------------------------
AUTOGENERATE ANGLES=TRUE DIHEDRALS=TRUE END
!-----------------------------------------------------------
RESIDUE ALA
GROUP
ATOM N TYPE=NH1 CHARGE= -.4700 END !
|
ATOM HN TYPE=H CHARGE= .3100 END !
N--HN
ATOM CA TYPE=CT1 CHARGE= .0700 END !
| HB1
ATOM HA TYPE=HB CHARGE= .0900 END !
| /
GROUP ! HA-CA--CB-HB2
ATOM CB TYPE=CT3 CHARGE= -.2700 END !
ATOM HB1 TYPE=HA CHARGE= .0900 END !
ATOM HB2 TYPE=HA CHARGE= .0900 END !
ATOM HB3 TYPE=HA CHARGE= .0900 END !
GROUP !
ATOM C TYPE=C CHARGE= .5100 END
ATOM O TYPE=O CHARGE= -.5100 END
!END GROUP
BOND CB CA
BOND N HN
| \
| HB3
O=C
|
BOND N CA
BOND O C
BOND C CA
BOND CA HA
BOND CB HB1
BOND CB HB2
BOND CB HB3
DONOR HN N
ACCEPTOR O C
END {ALA }
HA
N
CA
O
C
HN
HB1
CB
HB3
HB2

!BOND PARAMETERS: Force Constant, Equilibrium Radius
BOND C C 600.000 {SD=.022} 1.335 ! ALLOW ARO HEM
BOND CA CA 305.000 {SD=.031} 1.375 ! ALLOW ARO
!ANGLE PARAMETERS: Force Constant, Equilibrium Angle,
Urie-Bradley Force Const., U.-B. equilibrium (if any)
ANGLE CA CA CA 40.00 {SD=.086} 120.0000 UB 35.000 2.416
ANGLE CP1 N C 60.00 {SD=.070} 117.0000 ! ALLOW PRO
!DIHEDRAL PARAMETERS: Energy Constant, Periodicity, Phase Shift, Multiplicity
DIHEDRAL C CT2 NH1 C 1.60 {SD=.430} 1 180.0000 ! ALLOW PEP
DIHEDRAL C N CP1 C .80 {SD=.608} 3 .0000 ! ALLOW PRO PEP
!IMPROPER PARAMETERS: Energy Constant, Periodicity(0), Phase Shift(0)
! Improper angles are introduced for PLANARITY maintaining
IMPROPER HA C C HA 20.00 {SD=.122} 0 .0000 ! ALLOW PEP POL ARO
IMPROPER HA HA C C 20.00 {SD=.122} 0 180.0000 ! ALLOW PEP POL ARO
! -----NONBONDED-LIST-OPTIONS-------------------------------
CUTNB= 13.000 TOLERANCE= .500 WMIN= 1.500 ATOM
INHIBIT= .250
! -----ELECTROSTATIC OPTIONS--------------------------------
EPS= 1.000 E14FAC= 1.000 CDIELECTRIC SHIFT
! -----VAN DER WAALS OPTIONS--------------------------------
VSWITCH
! -----SWITCHING /SHIFTING PARAMETERS-----------------------
CTONNB= 10.000 CTOFNB= 12.000
! -----EXCLUSION LIST OPTIONS-------------------------------
NBXMOD= 5
! ------------
! EPS SIGMA EPS(1:4) SIGMA(1:4)
NONBONDED C .1100 4.0090 .1100 4.0090 ! ALLOW PEP POL ARO
NONBONDED CA .0700 3.5501 .0700 3.5501 ! ALLOW ARO

• What is a forcefield?
• How to prepare your system for MD?
• What specific conditions (temperature, pressure, volume, etc) will be used in MD?

What you have up to now:
• pdb file,
• topology file
• parameter file
What you need next: a program capable of reading and manipulating this information
Programs:
X-PLOR, CHARMm, NAMD2, AMBER, GROMOS, EGO,
PROTOMOL, etc.

Minimization
Energy
The energy of the system can be calculated using the forcefield.
The conformation of the system can be altered to find lower energy conformations through a process called minimization.
Conformational change
Minimization algorithms:
• steepest descent (slowly converging – use for highly restrained systems
• conjugate gradient (efficient, uses intelligent choices of search direction – use for large systems)
• BFGS (quasi-newton variable metric method)
• Newton-Raphson (calculates both slope of energy and rate of change)

Solvation
Biological activity is the result of interactions between molecules and occurs at the interfaces between molecules (protein-protein, protein-
DNA, protein-solvent, DNA-solvent, etc).
Why model solvation?
• many biological processes occur in aqueous solution
• solvation effects play a crucial role in determining molecular conformation, electronic properties, binding energies, etc
How to model solvation?
• explicit treatment: solvent molecules are added to the molecular system
• implicit treatment: solvent is modeled as a continuum dielectric

• What is a forcefield?
• How to prepare your system for MD?
• What specific conditions (temperature, pressure, volume, etc) will be used in MD?

MD = change in conformation over time using a forcefield
Energy
Energy supplied to the minimized system at the start of the simulation
Conformation impossible to access through MD
Conformational change

Energy function:
used to determine the force on each atom:
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

Constant energy, constant number of particles (NE)
Constant energy, constant volume (NVE)
Constant temperature, constant volume (NVT)
Constant temperature, constant pressure (NPT)
Choose the ensemble that best fits your system and start the simulations
Happy computing!

1) Build realistic atomistic model of the system
2) Simulate the behavior of your system over time using specific conditions (temperature, pressure, volume, etc)
3) Analyze the results obtained from MD and relate to macroscopic level properties

Ion channels are membrane spanning proteins that form a pathway for the flux of inorganic ions across cell membranes.
Potassium channels are a particularly interesting class of ion channels, managing to distinguish with impressive fidelity between K + and Na + ions while maintaining a very high throughput of K + ions when gated.

• retrieve the PDB (coordinates) file from the Protein Data Bank
• use topology and parameter files to set up the structure
• add hydrogen atoms using
X-PLOR
• minimize the protein structure using NAMD2

lipids
Simulate the protein in its natural environment: solvated lipid bilayer

gaps
Inserting the protein in the lipid bilayer
Automatic insertion into the lipid bilayer leads to big gaps between the protein and the membrane => long equilibration time required to fill the gaps.
Solution: manually adjust the position of lipids around the protein

solvent
Kcsa channel protein
(in blue) embedded in a (3:1) POPE/POPG lipid bilayer. Water molecules inside the channel are shown in vdW representation.
solvent

Summary of simulations:
• protein/membrane system contains 38,112 atoms, including
5117 water molecules, 100 POPE and 34 POPG lipids, plus K+ counterions
• CHARMM26 forcefield
• periodic boundary conditions, PME electrostatics
• 1 ns equilibration at 310K, NpT
• 2 ns dynamics, NpT
Program: NAMD2
Platform: Cray T3E (Pittsburgh Supercomputer Center)

RMS deviations for the KcsA protein and its selectivity filer indicate that the protein is stable during the simulation with the selectivity filter the most stable part of the system.
Temperature factors for individual residues in the four monomers of the KcsA channel protein indicate that the most flexible parts of the protein are the N and C terminal ends, residues 52-60 and residues 84-90. Residues 74-80 in the selectivity filter have low temperature factors and are very stable during the simulation.

In SMD simulations an external force is applied to an atom or a group of atoms to accelerate processes, for example, passing of ions through a channel protein.
In the SMD simulations of the K channel, a moving, planar harmonic restraint, with a force constant of 21 kJ/mol/A, was applied to one of the ions in the channel. The restraint was applied along the z-axis only, allowing the ion to drift freely in the plane of the membrane. To avoid local heating caused by applied external forces, all heavy atoms were coupled to a
Langevin heat bath with a coupling constant of 10/ps.

Steered ion 78
77
76
75
After 400 ps, the leading ion moves into chamber 77. The intervening chamber is left unfilled by the trailing ion for ~ 100 ps. At 500 ps, the trailing ion moves into chamber 76 behind the leading ion, overcoming its attraction for Thr75 and Val76 but moving into a more favorable interaction position with Gly77. The trailing ion overcomes an energy barrier of ~50 kJ/mol due to interactions with the backbone. Just after this trailing ion moves in, PRO3 isomerizes to bring the carbonyl oxygen of Gly77 into favorable alignment with both potassium ions. The isomerization stabilizes the backbone by ~30 kJ/mol. Around 795-800 ps, the two ions make a concerted transition to the next pair of chambers. Almost simultaneously, Gly77 carbonyl oxygen swings back to its former position.

• Quality of the forcefield
• Size and Time – atomistic simulations can be performed only for systems of a few tenths of angstroms on the length scale and for a few nanoseconds on the time scale
• Conformational freedom of the molecule – the number of possible conformations a molecule can adopt is enormous, growing exponentially with the number or rotatable bonds.
• Only applicable to systems that have been parameterized
• Connectivity of atoms cannot change during dynamics – no chemical reactions

Grubmuller (1989)
Grubmuller, Heller, Windemuth & Schulten (1991)
Tuckerman, Berne & Martyna (1992)
F_fast F_fast F_fast F_fast F_fast F_fast
1.
Fast/slow force splitting (Grubmuller ’89, ’91, and Tuckerman ’92)
2.
Use switching functions on potentials/forces (Tuckerman ’92)
F_fast
Time
F_slow
F_slow F_slow
Time

Ma, Izaguirre & Skeel (2001)
• Even for MTS, theory and experiment indicate that energy growth occur unless longest  t < 1/3 period .
• Note: This resonance is purely numerical artifact, natural resonance only happens at longest  t = period.
.

Table 1. Diffusion coefficient computed using different methods for 400 ps simulation of 141 water molecules, PBC, Ewald
Type  t
(fs)
LF -
 t
(fs)
1

(ps -1 )
D(10 -5 cm 2 /s)
 = 10 -7
D(10 -5 cm 2 /s)
 = 10 -12
3.64 ± 0.05 3.71 ± 0.02
VI 4 1 3.62 ± 0.06 3.77 ± 0.09
DM 16 2 4.05
3.65 ± 0.05 3.70 ± 0.05

Table 2. Speedup comparisons for different methods for 400 ps simulation of 141 water molecules, PBC, Ewald
Type  t
(fs)
 t
(fs)
LF 1
(ps

-
-1 )
Time and Speedup
(hours,  = 10
49.83, 1.00
-7 )
Time and Speedup
(hours,  = 10
68.28, 1.00
-12 )
VI 4 1 -
DM 16 2 4.05
18.68, 2.67
10.09, 4.94
23.50, 2.91
11.40, 5.99

• Time reversible, momentum preserving and temperature preserving MTS integrator for molecular dynamics.
• Computes dynamics correctly using step sizes up to 16 fs.
• Substantial speedup for sequential executions.

Simulation Procedure Overview

• Setup
1. PDB file
– Protein Databank ( http://www.rcsb.org/pdb/ )
2. PSF file
– Generated specifically for the molecule
– Contains the detailed composition and connectivity of the molecule(s) of interest

• Setup
1. Topology file
– information for putting molecules together, such as what atoms are to be used, which of these atoms are bonded to each other, and the sets of atoms that form bond angles
2. Parameter file
– physical parameters (force constants, van der
Waals forces, bonds, angles, etc.)

• Solvation
– Create water box or shell to enclose the molecule
• Minimization
– Minimize the energy of the system in order to reach the most favorable configuration

• Heating
– Initial velocities are assigned at a low temperature. Periodically, new velocities are assigned at a slightly higher temperature and the simulation is allowed to continue. This is repeated until the desired temperature is reached.

• Equilibration
– The point of the equilibration phase is to run the simulation until the structure, pressure, temperature and energy become stable with respect to time.

• Dynamics
– Normal/Periodic boundary condition
– Single/Multiple time stepping
– Integrators
– Electrostatics

• Analysis
– Mean energy
– RMS difference between two structures

• Visualization
– molecular graphics programs
• VMD
• InsightII

• Purpose
– To help setup molecular simulations
– To avoid being overwhelmed by the amount of technical details
– To choose optimal parameters based on desire accuracy

• Python interpreter
• CHARMM
– Bob.nd.edu
– http://yuri.harvard.edu/
• ProtoMol
– Bob.nd.edu
– www.nd.edu/~lcls/ProtoMol

• PSF generation
• Solvation
• Minimization
• Heating
• Equilibration
• Dynamics

• Extend to multiple time stepping methods
• Multiple ensembles (currently NVE)
• Interface to more programs (NAMD,
CHARMm – for dynamics)
• Collaborations, incorporate suggestions
• GUI

• Protein Data Bank
– http://www.rcsb.org/pdb/
• This talk
– http://www.nd.edu/~izaguirr/webpage_files/presentations.htm
• CHARMM
– http://yuri.harvard.edu/
• ProtoMol
– http:// www.nd.edu/~lcls
• VMD
– http://www.ks.uiuc.edu/Research/vmd/

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Langevin dynamics. J. Chem. Phys.
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(1982), 495-500.
[3] 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.
[4] B. Garcya-Archilla, J. M. Sanz-Serna, and R. D. Skeel. The mollified impulse method for oscillatory differential equations. In D. F. Griffiths and G.
A. Watson, editors, Numerical Analysis 1997, pages 111–123, Pitman, 1998
[5] R. D. Groot and P. B. Warren. Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys.
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107(11):4423–4435, Sep 15 1997.
[6] H. Grubmuller. Dynamiksimulation sehr grober Makromoleckule auf einem Parallelrechner, Master’s thesis, Physik-Dept. der Tech. Univ. Munchen,
Munich, 1989
[7] 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.
[8] J. A. Izaguirre, D. P. Catarello, J.M.Wozniak, and R. D. Skeel. Langevin stabilization of molecular dynamics. J. Chem. Phys.
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Feb. 1, 2001.
[9] J. A. Izaguirre, Q. Ma, T. Matthey, J. Willcock, T. Slabach, B. Moore, and G. Viamontes. Overcoming instabilities in verlet-i/r-RESPA with the mollified impulse method. In T. Schlick, editor, Proceedings of International Workshop on Methods for Macromolecular Modeling , volume
24, Berlin-New York, 2002. Springer Verlag. In press, preprint at http://www.nd.edu/˜ .
[10] J. A. Izaguirre, S. Reich, and R. D. Skeel. Longer time steps for molecular dynamics. J. Chem. Phys.
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[11] 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.
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[15] 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
[16] R. D. Skeel & J. A. Izaguirre. An impulse integrator for Langevin dynamics, submitted, 2002.
[17] M. Tuckerman, B. J. Berne, and G. J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys 97 (1992), no. 3, 1990-2001
[18] 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.
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The LCLS would like to thank the following--
National Science Foundation Biocomplexity grant PHY-0083653
Department of Computer Science and Engineering, Univ. of Notre Dame and our Collaborators:
• Dr. Mark Alber, Mathematics and Center for Applied Mathematics, Notre Dame
• Dr. Petter E. Bjorstad, Institutt for Informatikk, U. of Bergen, Norway
• Dr. Gabor Forgacs, Physics and Biology, University of Missouri-Columbia
• Dr. James A. Glazier, Physics, Notre Dame
• Dr. George Hentschel, Physics, Emory University
• Dr. Edward Maginn, Chemical Engineering, Notre Dame
• Dr. J. Andrew McCammon, Chemistry & Biochemistry, University of California, San
Diego
• Dr. Stuart Newman, Cell Biology and Anatomy, New York Medical College
• Dr. Martin Tenniswood, Biological Sciences and Walther Cancer Institute, Notre Dame
• Dr. Robert Skeel, Computer Science and Beckman Institute, University of Illinois at
Urbana-Champaign