Basics of molecular dynamics simulations Reduction of the quantum problem to a classical one Parameterization of force fields Boundaries, energy minimization,
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Basics of molecular dynamics simulations
Reduction of the quantum problem to a
classical one
Parameterization of force fields
Boundaries, energy minimization, integration
algorithms
Analysis of trajectory data
Lecture notes are at :
www.physics.usyd.edu.au/~serdar/ws1
Quantum, classical and stochastic description of molecular systems
Quantum mechanics (Schroedinger equation)Quantum mechanics (Schroedinger equation)
The most fundamental approach but feasible only for few atoms (~10).The most fundamental approach but feasible only for few atoms (~10).
Approximate methods (e.g. density functional theory) allows treatment Approximate methods (e.g. density functional theory) allows treatment
of larger systems (~1000) and dynamic simulations for picoseconds.of larger systems (~1000) and dynamic simulations for picoseconds.
Classical mechanics (Newton’s equation of motion)Classical mechanics (Newton’s equation of motion)
Most atoms are heavy enough to justify a classical treatment (except Most atoms are heavy enough to justify a classical treatment (except
H). The main problem is finding accurate potential functions (force H). The main problem is finding accurate potential functions (force
fields). fields).
MD simulation of 100,000 atoms for many nanoseconds is now feasible.MD simulation of 100,000 atoms for many nanoseconds is now feasible.
Stochastic mechanics (Langevin equation)Stochastic mechanics (Langevin equation)
Most biological processes occur in the range of microsec to millisec. Most biological processes occur in the range of microsec to millisec.
Thus to describe such processes, a simpler (coarse-grained) Thus to describe such processes, a simpler (coarse-grained)
representation of atomic system is essential (e.g. Brownian dynamics).representation of atomic system is essential (e.g. Brownian dynamics).2
,
2
22
2
22
2
2
2
,,
i i
ine
e
ji ji
ji
ii
in
iineen
ezU
em
H
ezz
MH
EUHH
rR
rr
RR
rRrR
Many-body Schroedinger eq. for a molecular system
Here m and Mi are the mass of the electrons and nuclei,
r and Ri denote the electronic and nuclear coordinates,
and and i denote the respective gradients.
nuclear hamiltonian
electronic hamiltonian
elect-nucl. interact.
(1)
3
rRRrR ,, ieini
rRRrRR ,, ieinieinneen EUHH
Separation of the electronic wave function
Nuclei are much heavier and hence move much slower than
electrons. This allows decoupling of their motion from those of
electrons. Introduce the product wave function:
Substituting this ansatz in the Schroedinger eq. gives
For fixed nuclei, the electronic part gives
rRRrR ,, ieieienee EUH (2)
4
rRRrRRR ,, ieinieinien EEH
Substitute the electronic part back in the Schroedinger eq.
Eqs. (2, 3) need to be solved simultaneously, which is a
formidable
problem for most systems. For two nuclei, there is only one
coordinate
for R (the distance), so it is feasible. But for three-nuclei, there
are
4 coordinates (in general for N nuclei, 3N-5 coordinates are
required),
which makes numerical solution very difficult.
Born-Oppenheimer (adiabatic) approximation consists of neglecting
the cross terms arising from
(which are of order m/M), so that the nuclear part becomes
rR ,ienH
ininien EEH RRR (3)
5
ieji ji
jii
iii
i
Eezz
U
NiUdt
dM
RRR
R
RR
2
2
2
,,1
Classical approximation for nuclear motion
Nuclei are heavy so their motion can be described classically, that
is,
instead of solving the Schroedinger Eq. (3), we solve the
corresponding
Newton’s eq. of motion
At zero temperature, the potential can be minimized with respect to
the
nuclear coordinates to find the equilibrium conformation of
molecules.
At finite temperature, Eqs. (2) and (4) form the basis of ab initio
MD
(ignores quantum effects in nuclear motion and electronic exc. at
finite T)
(4)
6
Methods of solution for the electronic equation
Two basic methods of solution:
1. Hartree-Fock (HF) based methods: HF is a mean field theory.
One finds the average, self-consistent potential in which electrons move.
Electron correlations are taken into account using various methods.
2. Density functional theory: Solves for the density of electrons.
Better scaling than HF (which is limited to ~10 atoms); 100’s of atoms.
Car-Parrinello MD (DFT+MD) has become popular in recent years.
(2)
rRR
rRrRrr
,
,2 ,
222
2
ieie
iei i
i
i
E
ezem
Electronic part of the Schroedinger eq. (2) has the form
7
een
)()( rrr
)]([''
)'()(
2)]([
)(
)(
332
32
,
2
rrr
rrr
rrR
rrrR
nErrddnne
nTH
rdnez
ezU
UHE
xce
i i
i
ei i
iene
eneeee
Density functional theory (DFT)
In DFT one uses the density function of electrons, n(r) instead
of their wavefunction
Expectation value of the electronic energy in the ground state becomes
8
Nidt
dM
N
ijiji
ii ,,1,
2
2
FF
r
Classical mechanics
Molecular dynamics (MD) is the most popular method for
simulation studies of biomolecules. It is based on Newton’s
equation of motion.
For N interacting atoms, one needs to solve N coupled Diff. Eq’s:
Force fields are determined from experiments and ab initio
methods.
Analytically this is an intractable problem for N>2,
but we can solve it easily on a computer using numerical
methods.
Current computers can handle N ~ 105-6 particles, which is large
enough for description of most biomolecules.
Integration time, however is still a bottleneck (106 steps @ 1 fs =
1 ns)
9
iiiiii
i mdt
dm RvF
r
2
2
Stochastic mechanics
In order to deal with the time bottle-neck in MD, one has to
simplify the simulation system (coarse graining). This is
achieved by describing parts of the system as continuum with
dielectric constants.
Examples:
• transport of ions in electrolyte solutions (water →
continuum)
• protein folding (water → continuum)
• function of transmembrane proteins (lipid, water →
continuum)
To include the effect of the atoms in the continuum, modify
Newton’s eq. of motion by adding frictional and random forces:
Langevin
equation10
Molecular dynamics (MD) simulations
MD programs consist of over 100,000 lines of coding. To
understand
what is going on in an MD simulation, we need to consider
several
topics:
1. Force fields (potential functions among atoms)
2. Boundaries and computation of long-range interactions
3. Molecular mechanics (T=0, energy minimization)
4. MD simulations (integration algorithms, ensembles)
5. Analysis of trajectory data to characterize structure and
function of proteins11
Empirical force fields (potential functions)
Problem: Reduce the ab initio potential energy surface among the atoms
ji kji
kjijii UUU ),,(),( 32 RRRRRR
To a classical many-body interaction in the form
ieji ji
jii E
ezzU R
RRR
2
Such a program has been tried for water but so far it has failed to
produce a classical potential that works. In strongly interacting systems,
it is difficult to describe the collective effects by summing up the
many-body terms.
Practical solution: Truncate the series at the two-body level and assume
(hope!) that the effects of higher order terms can be absorbed in U2 12
Non-bonded interactions
Interaction of two atoms at a distance R = |Ri - Rj| can be
decomposed
into 4 pieces
1. Coulomb potential
2. Induced polarization
3. Attractive dispersion (van der Waals) (1/R6)
4. Short range repulsion (eR/a)
The last two terms are combined into a 6-12 Lennard-Jones
potential
Combination rules for different atoms:
612
4RR
ULJ
2/)( jiijjiij 13
R
qqU ji
04Coul
12-6 Lennard-Jones potential (U is in kT, r in Å)
3 3.5 4 4.5 5-0.5
0
0.5
1
1.5
2
2.5
r
U(r)
AkT 3,41
14
Because the polarization interaction is many-body and requires
iterations,
it has been neglected in current generations of force fields.
The non-bonded interactions are thus represented by the Coulomb
and
12-6 LJ potentials.Model RO-H (Å) HOH qH (e) (kT) (Å) (D) eps (T=298)
SPC 1.0 109.5 0.410 0.262 3.166 2.27 65±5
TIP3P 0.957 104.5 0.417 0.257 3.151 2.35 97±7
Exp. 1.86 80
SPC: simple point chargeTIP3P: transferable intermolecular potential with 3 points
Popular water models (rigid)
15
Covalent bonds
In molecules, atoms are bonded together with covalent bonds,
which arise from sharing of electrons in partially occupied
orbitals. If the bonds are very strong, the molecule can be
treated as rigid as in water molecule. In most large molecules,
however, the structure is quite flexible and this must be taken
into account for a proper description of the molecules. This is
literally done by replacing the bonds by springs. The nearest
neighbour interactions involving 2, 3 and 4 atoms are described
by harmonic and periodic potentials
02
02
0 cos122
ijklijkl
ijklijk
ijkijk
ijij
rij
bond nVk
rrk
U
bond stretching bending torsion (not very good)16
Interaction of two H atoms in the ground (1s) state can be described
using Linear Combinations of Atomic Orbitals (LCAO)
)1()1( 21 scsc BA
)1()1(2
1ss BA
From symmetry, two solutions with lowest and highest energies are:
+ Symmetric
- Anti-symmetric
17
The R dependence of the potential energy is approximately
given by the Morse potential
2)( 01 RRe eDU
Morse
20 )(
21
RRkUbond
where
De: dissociation energy
R0: equilibrium bond distance
Controls the width of the potential
Classical representation:
eDk 2/
k
18
An in-depth look at the force fields
Three force fields, constructed in the eighties, have come to
dominate
the MD simulations
1. CHARMM (Karplus @ Harvard)
Optimized for proteins, works well also for lipids and nucleic
acids
2. AMBER (Kollman & Case @ UCSF)
Optimized for nucleic acids, otherwise quite similar to CHARMM
3. GROMOS (Berendsen & van Gunsteren @ Groningen)
Optimized for lipids and does not work very well for proteins
(due to smaller partial charges in the carbonyl and amide
groups)
The first two use the TIP3P water model and the last one, SPC
model.
They all ignore the polarization interaction (polarizable versions
are under construction)
19
Charm parameters for alanine
Partial charge (e)
ATOM N -0.47 |
ATOM HN 0.31 HN—N
ATOM CA 0.07 | HB1
ATOM HA 0.09 | /
GROUP HA—CA—CB—HB2
ATOM CB -0.27 | \
ATOM HB1 0.09 | HB3
ATOM HB2 0.09 O==C
ATOM HB3 0.09 |
ATOM C 0.51
ATOM O -0.51
Total charge: 0.0020
Bond lengths :
kr (kcal/mol/Å2) r0 (Å)
N CA 320. 1.430 (1)
CA C 250. 1.490 (2)
C N 370. 1.345 (2) (peptide bond)
O C 620. 1.230 (2) (double bond)
N H 440. 0.997 (2)
HA CA 330. 1.080 (2)
CB CA 222. 1.538 (3)
HB CB 322. 1.111 (3)
1. NMA (N methyl acetamide) vibrational spectra
2. Alanine dipeptide ab initio calculations
3. From alkanes
20 )( rrkU r bond
21
Bond angles :
k (kcal/mol/rad2) 0 (deg)
C N CA 50. 120. (1)
C N H 34. 123. (1)
H N CA 35. 117. (1)
N CA C 50. 107. (2)
N CA CB 70. 113.5 (2)
N CA HA 48. 108. (2)
HA CA CB 35. 111. (2)
HA CA C 50. 109.5 (2)
CB CA C 52. 108. (2)
N C CA 80. 116.5 (1)
O C CA 80. 121. (2)
O C N 80. 122.5 (1)
20 )( kUangle
Total 360 deg.
Total 360 deg.
22
Basic dihedral configurations trans cis
Definition of the dihedral
angle for 4 atoms A-B-C-D
)cos(12 0 nV
U ndihedDihedrals:
23
Dihedral parameters:
Vn (kcal/mol) n 0 (deg) Name
C N CA C 0.2 1 180.
N CA C N 0.6 1 0.
CA C N CA 1.6 1 0.
H N CA CB 0.0 1 0.
H N CA HA 0.0 1 0.
C N CA CB 1.8 1 0.
CA C N H 2.5 2 180.
O C N H 2.5 2 180.
O C N CA 2.5 2 180.
)cos(12 0 nV
U ndihed
24
BoundariesIn macroscopic systems, the effect of boundaries on the
dynamics of biomolecules is minimal. In MD simulations,
however, the system size is much smaller and one has to
worry about the boundary effects.
• Using nothing is not realistic for bulk simulations.
• Minimum requirement: water beyond the simulation box must
be treated using a continuum representation
• Most common solution: periodic boundary conditions.
The simulation box is replicated in all directions just like in a
crystal. The cube and rectangular prism are the obvious
choices for a box shape.
25
Periodic boundary conditions in two dimensions: 8 nearest neighbours
Particles in the box freely move to the next box, which means
when one moves out another appears from the opposite side of
the same box. In 3-D, there are 26 nearest neighbours.26
Treatment of long-range interactions
Problem: the number of non-bonded interactions grows as N2
where N is the number of particles. This is the main
computational bottle neck that limits the system size.
Because the Coulomb potential between two unit charges, has
long range [U=560 kT/r (Å)], use of any cutoff is problematic and
should be avoided.
All the modern MD codes use the Ewald sum method to treat the
long-range interactions without cutoffs. Here one replaces the
point charges with Gaussian distributions, which leads to a much
faster convergence of the sum in the reciprocal (Fourier) space.
The remaining part (point particle – Gaussian) falls exponentially in
real space, hence can be computed easily.27
Molecular mechanics
Molecular mechanics deals with the static features of
biomolecular systems at T=0 K, that is, the energy is given
solely by the potential energy of the system.
Two important applications of molecular mechanics are:
1. Energy minimization (geometry optimization):
Find the coordinates for the configuration that minimizes the
potential energy.
2. Normal mode analysis:
Find the vibrational excitations of the system built on the
absolute minimum using the harmonic approximation.
28
Molecular dynamics
In MD simulations, one follows the trajectories of N particles
according to Newton’s equation of motion:
where U(r1,…, rN) is the potential energy function
consisting of bonded and non-bonded interactions.
We need to consider:
• Integration algorithms, e.g., Verlet, Leap-frog
• Initial conditions and choice of the time step
• MD simulations at constant temperature and pressure
• Constrained dynamics for rigid molecules (SHAKE)
),,(,2
2
Niiiiii Udt
dm rrFFr
29
2
2
)()()()( t
m
tttttt
i
iiii
Fvrr
2
2)(
)()()(
)()()(
tmt
ttttt
tmt
ttt
i
iiii
i
iii
Fvrr
Fvv
Integration algorithms
Given the position and velocities of N particles at time t,
integration of Newton’s equation of motion yields at t+t
In the popular Verlet algorithm, one eliminates velocities using
the positions at t-t,
Adding eq. (*) yields: 2)()()(2)( t
m
tttttt
i
iiii
Frrr
(*)
30
tttttt
tm
ttttttt
iii
i
iiii
)2/()()(
)()2/()()( 2
vrr
Fvrr
tttttt
tm
ttttt
iii
i
iii
)2/()()(
)()2/()2/(
vrr
Fvv
In the Leap-frog algorithm, the positions and velocities are
calculated at different times separated by t/2
To show its equivalence to the Verlet algorithm, consider
Subtracting the two equations yields the Verlet result.
If required, velocity at t is obtained from:
)]2/()2/([21
)( ttttt iii vvv31
To iterate these equations, we need to specify the initial
conditions.
• The initial configuration of biomolecules can be taken from the
Protein Data Bank (PDB) (if available).
• In addition, membrane proteins need to be embedded in a
lipid bilayer.
• All the MD codes have facilities to hydrate a biomolecule, i.e.,
fill the void in the simulation box with water molecules at the
correct density.
• Ions can be added at random positions.
After energy minimization, these coordinates provide the
positions at t=0.
Initial velocities are sampled from a Maxwell-Boltzmann
distribution: kTvm
kT
mvP ixi
iix 2exp
2)( 2
2/1
32
MD simulations at constant temperature and
pressure
MD simulations can be performed in the NVE ensemble, where
all 3 quantities are constant. Due to truncation errors, keeping
the energy constant in long simulations can be problematic. To
avoid this problem, the alternative NVT and NPT ensembles are
employed. The temperature of the system can be obtained
from the average K. E.Thus an obvious way to keep the temperature constant at T0 is
to scale the velocities as:
)(),()( 0 tTTtvtv ii
NkTK23
Because K. E. has considerable fluctuations, this is a rather crude method. 33
A better method which achieves the same result more smoothly
is the Berendsen method, where the atoms are weakly coupled
an external heat bath with the desired temperature T0
If T(t) > T0 , the coupling term is negative, which invokes a
viscous force slowing the velocity, and vice-versa for T(t) < T0
Similarly in the NPT ensemble, the pressure can be kept
constant by simply scaling the volume. Again a better method
(Langevin piston), is to weakly couple the pressure difference
to atoms using a force as in above, which will maintain the
pressure at the desired value (~1 atm).
dt
d
tT
Tm
dt
dm i
iiiiir
Fr
1)(
02
2
34
Analysis of trajectory data
1. Fundamental quantities
Total energy, temperature, pressure, volume (or density)
2. Structural quantities
Root Mean Square Deviation (RMSD)
Distribution functions (e.g., pair and radial)
Conformational analysis (e.g., Ramachandran plots,
rotamers)
3. Dynamical quantities
Time correlation functions & transport coefficients
Free energy calculations using perturbation, umbrella
sampling and steered MD methods 35
Fundamental quantities
• Total energy: strictly conserved but due to numerical errors it
may drift.
• Temperature: would be constant in a macroscopic system
(fluctuations are proportional to 1/N, hence negligible). But
in a small system, they will fluctuate around a base line.
Temperature coupling ensures that the base line does not drift
from the set temperature.
• Pressure: similar to temperature, though fluctuations are
much larger compared to the set value of 1 atm.
• Volume (or density): fixed in the NVT ensemble but varies in
NPT. Therefore, it needs to be monitored during the initial
equilibration to make sure that the system has converged to
the correct volume. Relatively quick for globular proteins but
may take a long time (~1 ns) for systems involving lipids
(membrane proteins).36
Root Mean Square Deviation
For an N-atom system, variance and RMSD at time t are defined
as
Where ri(0) are the reference coordinates. Usually they are taken
from the first frame in an MD simulation, though they can also be
taken from the PDB or a target structure. Because side chains in
proteins are different, it is common to use a restricted set of atoms
such as backbone or C. RMSD is very useful in monitoring the
approach to equilibrium (typically, signalled by the appearance of
a broad shoulder).
It is a good practice to keep monitoring RMSD during production
runs
to ensure that the system stays near equilibrium.
RMSD,)0()(1
)(1
2N
iii tN
t rr
37
Evolution of the RMSD for the backbone atoms of ubiquitin
38
rr
rNrg waterion
24
)()(
Distribution functions
Pair distribution function gij(r) gives the probability of finding a
pair of atoms (i, j) at a distance r. It is obtained by sampling the
distance rij in MD simulations and placing each distance in an
appropriate bin, [r, r+r]. Pair distribution functions are used in
characterizing correlations between pair of atoms, e.g.,
hydrogen bonds, cation-carbonyl oxygen. The peak gives the
average distance and the width, the strength of the interaction.
When the distribution is isotropic (e.g., ion-water in an
electrolyte), one samples all the atoms in the spherical shell, [r,
r+r]. Thus to obtain the radial distribution function (RDF), the
sampled number needs to be normalized by the volume ~4r2r
(as well as density)39
Conformational analysisIn proteins, the bond lengths and bond angles are more or less
fixed.
Thus we are mainly interested in conformations of the torsional
angles. As the shape of a protein is determined by the backbone
atoms, the torsional angles, (N−C) and (C−C), are of
particular interest. These are conveniently analyzed using the
Ramachandran plots. Conformational changes in a protein
during MD simulations can be most easily revealed by plotting
these torsional angles as a function of time.
Also of interest are the torsional angles of the side chains,
etc.
Each side chain has several energy minima (called rotamers),
which are separated by low-energy barriers (~ kT). Thus
transitions between different rotamers is readily achievable, and
they could play functional roles, especially for charged side
chains.
40
Time correlation functions & transport coefficients
At room temperature, all the atoms in the simulation system are
in a constant motion characterized by their average kinetic
energy: (3/2)kT. Free atoms or molecules diffuse according to
the equation Dtt2 6)( r
dttD
0
)()0(31
vv
While this relationship can be used to determine the diffusion
coefficient, more robust results can be obtained using correlation
functions, e.g., D is related to the velocity autocorrelation function
as
Similarly, conductance of charged particles is given by the Kubo formula
ii
iqdttVkT vJJJ
,)()0(3
1
0
(current)
41
2)0(
)()0()(
ttC
Bound atoms or groups of atoms fluctuate around a mean value.
Most of the high-frequency fluctuations (e.g. H atoms), do not
directly
contribute to the protein function. Nevertheless they serve as
“lubricant” that enables large scale motions in proteins (e.g.
domain motions) that do play a significant role in their function.
As mentioned earlier, large scale motions occur in a ~ ms to s
time scale, hence are beyond the present MD simulations.
(Normal mode analysis provides an alternative.) But torsional
fluctuations occur in the ns time domain, and can be studied in
MD simulations using time correlation functions, e.g.
These typically decay exponentially and such fluctuations can be
described using the Langevin equation. 42
Free energy calculations
Free energy is the most important quantity that characterizes a
dynamical process. Calculation of the absolute free energies is
difficult in MD simulations. However free energy differences can
be estimated more easily and several methods have been
developed for this purpose. The starting point for most
approaches is Zwanzig’s perturbation formula for the free energy
difference between two states A and B:
If the two states are not similar enough, there is a large hysteresis
effect and the forward and backward results are not equal.
)()(
/)](exp[ln)(
/)](exp[ln)(
ABGBAG
kTHHkTABG
kTHHkTBAG
BBA
AAB
43
Alchemical transformation and free energy
perturbation
To obtain accurate results with the perturbation formula, the
energy difference between the states should be < 2 kT, which is
not satisfied for most biomolecular processes. To deal with this
problem, one introduces a hybrid Hamiltonian
and performs the transformation from A to B gradually by
changing the parameter from 0 to 1 in small steps. That is, one
divides [0,1] into n subintervals with {i, i = 0, n}, and for each i
value, calculates the free energy difference from the ensemble
average
BA HHH )1()(
ikTHHkTG iiii /))]()((exp[ln)( 11
44
1
01)(
n
iiiGG
The total free energy change is then obtained by summing the
contributions from each subinterval
The number of subintervals is chosen such that the free energy
change at each step is < 2 kT, otherwise the method may lose
its validity.
1
0
)(
dH
G
Thermodynamic integration : Another way to obtain the free energy
difference is to integrate the derivative of the hybrid Hamiltonian:
This integral is evaluated most efficiently using a Gaussian quadrature.45
Absolute free energies from umbrella samplingHere one samples the densities along a reaction coordinate and
determines the potential of mean force (PMF) from the Boltzmann
eq.
)(
)(ln)()()()(
00
/)]()([0
0
z
zkTzWzWezz kTzWzW
Here z0 is a reference point, e.g. a point in bulk where W vanishes.
In general, a particle cannot be adequately sampled at high-
energy
points. To counter that, one introduces harmonic potentials, which
restrain the particle at desired points, and then unbias its effect.
For convenience, one introduces umbrella potential at regular
intervals along the reaction coordinate (e.g. ~0.5 Å). The PMF’s
obtained in each interval are unbiased and optimally combined
using the Weighted Histogram Analysis Method (WHAM).
46
Steered MD (SMD) simulations and Jarzynski’s
equationSteered MD is a more recent method where a harmonic force is
applied to an atom on a peptide and the reference point of this
force is pulled with a constant velocity. It has been used to
study unfolding of proteins and binding of ligands. The discovery
of Jarzynski’s equation in 1997 enabled determination of PMF
from SMD, which has boosted its applications.
)]([,. 0
//
tkW
ee
f
i
kTWkTF
vrrFdsF
Jarzynski’s equation:
Work done by the harmonic force
This method seems to work well in simple systems and when F is large
but beware of its applications in complex systems! 47
A critical look at simulation methods
Molecular dynamics is the “standard model” of biomolecules.
• A higher level (quantum) description is simply not feasible for
any biomolecular system.
• A lower level description sacrifices the atomic detail and
cannot be expected to succeed without guidance from MD.
Nevertheless, MD simulations alone cannot provide a complete
description of biomolecules, and hence we need to appeal to
both
higher and lower level theories to make progress.
• Quantum description (in a mixed QM/MM scheme) is required
for
1. Development and testing of force fields
2. Enzyme reactions that involve making or breaking of
bonds
3. Transport of light particles (e.g., protons and electrons)
48
• Stochastic description and coarse-graining are required for any
process that is beyond the current capabilities of MD simulations,
that is, more than 105-6 atoms and longer than microseconds.
1. Protein-protein interactions, protein aggregation (water in
continuum, protein may be rigid, coarse-grained, semi-flexible
or flexible)
2. Aggregation of membrane proteins (water and lipid in
continuum, protein as above)
3. Ion channels and transporters: modelling of ion permeation
(water and lipid in continuum, protein may be rigid or semi-
flexible)
4. Protein folding (water in continuum, protein coarse-grained)
5. Self assembly of lipid bilayers and micelles (water and lipid
molecules coarse-grained)
6. DNA condensation (water in continuum, DNA coarse-grained)
49
The guiding principle: Occam’s razor
Use the simplest method that yields the answer, provided:
• The method is justified (domain of validity)
• The parameters employed can be derived from a more
fundamental theory.
Most of the people who use molecular dynamics or other
simulation
methods treat them as black boxes and apply them to
biomolecules
without worrying too much whether they are valid or relevant.
You also need to watch out for experts who publish only the
good
results!
A cautionary tale: Continuum description of ion channels using
the
Poisson-Nernst-Planck (drift-diffusion) equations.
50
Here, : potential, r: charge density, J: flux, n: concentration of ions
PNP equations are solved simultaneously, yielding potential and flux
Assumption: Ions are represented by an average charge density, zvenv
which invokes a mean-field approximation.
This is alright in a bulk electrolyte, but in ion channels with radius
smaller than the Debye length (8 Å for 0.15 M), one should worry about
the validity of PNP equations.
Poisson-Nernst-Planck (PNP) Equations
kT
enznDJ
enzext0 Poisson equation
Nernst-Planck equation
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+
+
+
+
+
+
+
= 2 = 80
Induced chargesat the water-protein interface
Image force on an ion
In application of PNP to Ca channels, diffusion coeff. is reduced by 10-4 !
The problem: Ions induce like-charges at a water-protein interface, which
repel them. When ions are represented as a continuous charge density,
this image force (or dielectric self-energy) is severely underestimated.
water protein
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A simple test: PNP vs Brownian dynamics (BD)
Control study:
Set artificially ε = 80 in the protein. No induced charges on the
boundary, hence no discrepancy between the two methods
r = 4 Å
Na +
Cl -
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Realistic case:
In the realistic case (εε = 2 = 2 in the protein), ions do not enter
the channel in BD due to the dielectric self-energy barrier.
Only in large pores (r > 10 Å), validity of PNP is restored.
r = 4 Å
PNP
BD
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