Biomolecular Simulation

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Biomolecular Simulation

• Principles and applications

From microscopic interactions to macroscopic
quantities
http//www.biochem.oulu.fi/juffer/
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Theoretical Enzymology
Quantum chemistry/Molecular mechanics (fs).
Protein dynamics
Molecular dynamics (ps to ns).
Interfacial processes
Coarse-grained dynamics (µs to ms).
Protein translation
Stochastic modeling (ms to s).
Tumor growth
Multiple scale modeling (months to years).
Larger length and longer time scales. Less
molecular detail.
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Microscopic world versus macroscopic world
difficult
Statistical Averaging
Classical mechanics
Statistical mechanics
Macroscopic quantities
Quantum mechanics
Biomolecular computer simulation
Observables Dynamic and Thermodynamic
quantities
Microscopic Interactions Phase space Wave function
easy
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Statistical thermodynamics
System in thermal equilibrium with
surroundings (classical version)
Average over states (ensemble) (example here is
U, internal energy)
Canonical partition function
Probability
Momenta and conjugated coordinates
Free energy Entropy
k Boltzmanns constant T Temperature
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Statistical averaging

• Ensemble averages ltAgte
• Time avarages ltAgtt
• If t??, ltAgte ltAgtt
• Ergodic theory Ensemble average is equal to the
  time average.
• Objective of a simulation is the generation of a
  sequence or trajectory of representative states
  (time sequence or an ensemble) from which
  properties can be computed.

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Simulation techniques

• Energy minimization (EM)
• Molecular dynamics (MD)
• Langevin dynamics (LD)
• Brownian dynamics (BD)
• Monte Carlo (MC)

Time averages
Ensemble averages
Stochastic Relies on random sampling
Deterministic Relies on classical mechanics
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Force fields for biomolecular simulation (1)
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Force fields for biomolecular simulation (2)
Total potential energy as a function of all
particle positions.
Describes empirically all interaction between
atoms
Model potential
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Force fields for biomolecular simulation (3)
Bonded interactions
Non-bonded interactions
Harmonic potentials
Coulomb interaction
Lennard-Jones potential
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Force fields for biomolecular simulation (4)
Energy cost in the distortion of bond lengths,
bond angles and dihedral angles.
Long range electrostatic interactions
Lennard-Jones interaction
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Interactions in proteins
Bonded interactions are required to maintain
the integrity of the structure.
Ser120
Non-bonded interaction
Cutinase, 1cus.pdb
Arg208
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Protein energy landscape
Unfolded
Energy
3N dimensional space
Folded
Coordinates r
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Other systems

• Force fields for molecules other than proteins
  (e.g. lipids in biological membranes) rely on
  similar descriptions
• Water is described with Lennard-Jones and Coulomb
  interaction.
• Explicit polarization is usually ignored for
  computational efficiency.

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Monte Carlo

• The aim is to generate a trajectory in phase
  space which samples from a chosen statistical
  ensemble.
• Requires force field to evaluate energies.
• Dynamic quantities cannot be calculated from the
  simulation.

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Monte Carlo
Protein Structure
Protein Thermodynamics Protein Function
Random displacement
Evaluate Energies
Analysis
Update ensemble
Accept or reject
Ensemble averages
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Monte Carlo
Allen and Tildeslev, Computer simulation of
liquids, Clarendon Press, Oxford, 1987
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Monte Carlo

• Importance sampling
• chooses random numbers from a underlying
  probability distribution function r, which allows
  the function evaluation to be concentrated in
  important regions of the phase space.
• sets up a Markov chain of states, which has only
  a limited distribution
• Finite number of trials and outcomes.
• Outcome of trial depends on the outcome of
  previous trial.
• Requires knowledge of underlying transition
  probability
• Importance sampling assumes a certain stochastic
  matrix for it.

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Monte Carlo
Trial move
Ei
Ej
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Monte Carlo
Important part of Coordinate space
Higher probability
Coordinates r
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pKa calculations (1)

• 00001110101000110101001110
• ?
• 00011110111100110101001110
• ?
• 10011110111110110101001110

Sampling of (individual) protonation states
Average occupancy is ?0.66
Juffer et al., J. Phys. Chem. B, 101, 7664-7673
(1997).
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pKa calculations (2)
Juffer, Biochem. Cell. Biol., 76,198-209 (1998).
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Protein-membrane association (1)

• Ions next to flat surface
• carrying a negative surface
• charge density.
• Accumulation of Na.
• Depletion of Cl-.
• Electric moment pointing
• towards flat surface.
• Symmetry along x- and y-axis
• but not along z-axis.

z-axis
Juffer et al., J. Comput. Chem., 17, 1783-1803
(1996).
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Protein membrane association (2)
Statistical distribution of ions is determined by
the physics of the system.
Juffer et al, J. Chem. Phys., 114, 1892-1905
(2001)
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Molecular dynamics

• Employs Newtons Laws of physics Dynamical
  simulation ? Deterministic.
• Generates a sequence of configurations or states
  (an ensemble) as a function of time.
• Requires force field to compute forces on atoms.
• Both thermodynamical and dynamical quantities can
  be computed from the simulation.

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Molecular Dynamics
Protein Thermodynamics Protein Dynamics Protein
Function
Protein Structure
Compute Forces
Update Trajectory
Analysis
Update positions and velocities
Time averages
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Molecular dynamics

• Result of simulation is a trajectory set of
  coordinates and velocities r,v (state) of all
  particles of the system recorded at different
  time.
• Trajectory is employed for analysis, e.g.
• The same quantities as in MC, but also
• Computation of diffusion constant.
• Protein dynamics.

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Molecular dynamics
t0
MD simulation of a solvated protein typically
takes a few ns. Simulation time NDt Dt1-2fs.
Dt
N snapshots of the system
Movie MD simulation Of crambin
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Molecular dynamics software

• Gromacs
• Charmm
• Amber
• Discover, Insight
• Sigma
• NAMD
• ..

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Protein binding to cell membranes
Computer simulation of the diffusional
approach of a peptide towards a lipid
bilayer. Simulation time total 6 ns (shown is
first 2 ns). Total number of atoms in the system
Shepherd et al., Biophys. J. 80, 579-596 (2001)
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Force acting on Sandostatin, MD
While the simulation is deterministic, the force
is sampled from a statistical ensemble
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EM versus MD/MC
Energy barrier
Local minimum
Global minimum
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Multifunctional Enzyme (1)

• ?-oxidation of fatty acids and amino acyl
  coenzyme A
• Import of cytosol proteins into peroxisome matrix

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Multifunctional Enzyme (2)
MFE
SCP-2L
hydratase
dehydrogenase
PTS1
PEX5
lipid-like molecule triggers insertion event
TPRs
PEX13
PEX14
Peroxisomal membrane
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Sterol Carrier Protein Like 2
Peroxisomal targeting signal (PTS)
TPRs of PEX5
inside
ligand
Sterol Carrier Protein type 2 (SCP-2L) Entry
1ikt in PDB
Published structure contains ligand Triton
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Simulation of ligand-free SCP-2L

• MD simulation Total length 50 ns.
• Movie Burial of PTS1 into SCP-2L
• Solvation process of PTS1

Lensink et al, J. Mol. Biol., 323, 99-113, 2002
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Solvent accessible surface of PTS1
10 ns time scale
SCP-2L
rSCP
Total relaxation process takes about 25 ns.
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Protein dynamics

• Dynamics and function are related
• lipases must open lid which covers active site to
  be able to carry out function.
• SCP-2L must make its PTS signal available for
  binding to PEX5.
• High and low frequency motions low frequency
  motions are important for function.
• Domain motions Important for function of
  transport proteins, protein regulation, enzyme
  catalysis.
• Domain motion can be seen as a semi-rigid body
  motion ?A protein is essentially a collection of
  connected rigid bodies.
• Detection of such domains is not trivial.
• Collective or concerted motions in protein
  structures.

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Soft versus hard degrees of freedom

• Hard degrees of freedom usually correspond to
  bond distances and bond angles?very fast motions
• Frequencies of these motions are not very much
  affected for e.g. conformational changes.
• Soft degrees are the remaining set of degrees of
  freedom?can be separated from fast motions.

Typical force field
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Frequency scale of protein motions (1)
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Frequency scale of protein motions (2)

• Computer simulation versus experiment.
• Molecular dynamics versus neutron scattering.

Hinsen and Kneller, J. Chem. Phys., 111,
10766-10769, 1999.
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Methods to detect concerted motions in proteins

• Objective detection of correlation among the
  different motions in proteins.
• Normal mode analysis of protein in vacuum
• Can be applied only when protein resides in a
  potential energy minimum with small fluctuations.
• Principle component analysis of the protein in
  water (for instance essential dynamics)
• More generally applicable.
• Jumping-among-minima (JAM) model separates the
  motions into intra-substate and inter-substate
  motions (catchment regions)

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Normal mode analysis
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Essential dynamics
Amadei et al., Proteins, 17412-425 (1993)
Compute correlation between positional fluctuation
s
MD (gt 1 ns)
Covariance matrix
Smallest eigenvalues (lowest frequecies)
represent largest positional fluctuation
Determine eigenvectors and eigenvalues
Movie First eigenvector of PYP
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Free energy profiles associated with dynamical
modes

• Multiply-hierarchical modes, energy barriers
• singly-hierarchical modes.
• Harmonic modes.

Kitao et al., Proteins, 33496-517 (1998)
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Principal mode category (1)
First 300 modes are termed anharmonic modes.
Kitao et al., Proteins, 33496-517 (1998)
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Principal mode category (2)
Courtesy of D. van Aalten, Dundee University,
Scotland.
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Main conclusions (1)

• Solvent molecules around protein move together
  with protein.
• Local energy surfaces of conformational substates
  are nearly harmonic and mutually similar.
• Subspace spanned by multiply-hierarchical modes
  (essential subspace) is time dependent, whereas
  the subspace spanned by all anharmonic modes is
  time independent.
• Anharmonic modes only a few procent of all modes
  (for lysozyme, 4.5).

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Main conclusions (2)

• Inter-substate motions occur in a
  small-dimensional conformational subspace.
• This subspace is spanned by the multiply- and
  singly-hierarchical modes.
• Two levels of inter-substate motions fast (1-5
  ps) and slow (gt 200 ps.)
• Functional importance of hierarchical modes.

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Langevin dynamics (1)

• Useful when system under investigation is to be
  described in a coarse-grained manner
• That is, when certain degrees of freedom are
  omitted.
• Required are effective force fields.
• Long-time computer simulations
• Time scale accessible is ms to min instead of ps
  to ns Protein folding.

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From MD to LD
Brownian Particle (protein)
i
i
Newtons equation of motion of particle i
Langevins equation of motion for particle i
Random force
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Langevin dynamics (2)
Stochastic differential equation of motion
Effective force due to all Brownian particles
Frictional force
Random force
Represent the effect of omitted degrees of
freedom on dynamics
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Langevin dynamics (3)

• Classical example LD is applicable to cases
  where the mass of the Brownian particles is much
  larger than that of the solvent particles.
• Ri(t) is a random force representing the constant
  bombardment exerted by the solvent.
• -mgv(t) represensents a frictional systematic
  force solvent opposes the motion of the
  particle.
• Solution of the Langevin equation essentially
  results in the probability that the Brownian
  particle will have a certain velocity at a
  certain time.
• If Fi0, the average displacement of a Brownian
  particle is zero.

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Langevin dynamics (4)

• Random force must satisfy certain conditions

Stochastic process defined by Langevin equation
is Markovian Probability of certain state
depends on the state directly preceding it.
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Langevin dynamics (5)
Not Markovian
Generalized Langevin Equation

• M(t-s) is a Memory function.
• Required when
• The mass of the Brownian particle is comparable
  to the solvent.
• If dynamics at shorter time scales are important.