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Simulazione di Biomolecole metodi e
applicazioni giorgio colombo colombo_at_ico.mi.cnr.i
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Computational BioChemistry a discipline by
which biochemical problems are solved via
computational methods
Steps 1) a model of the real world is
constructed
2) measurable (and unmeasurable) properties are
computed
3) comparison with experimentally determined
properties
4) validation
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Real World Model
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Computational BioChemistry
Since chemistry concerns the study of properties
of molecular systems in terms of atoms, the
basic challenge is to describe and predict
1) the structure and stability of a molecular
system
2) the (free) energy difference of different
states of the system
3) processes within systems
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Computational BioChemistry
Chemical systems are generally too inhomogeneous
and complex (1023particles) to be treated
analitically
Many particle system
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Computational BioChemistry
Chemical systems are generally too inhomogeneous
and complex to be treated analitically
We need
Numerical simulations of the behaviour of the
system to produce a statistical ensemble of
configurations representing the state of the
system statistical mechanics
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Computational BioChemistry
Outline 1) basic problems of computer
simulation of biological systems 2) Methodology
and applications
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Computer simulations of Molecular systems
Two basic problems 1) the size of the
configurational space accessible to the system -
1023 particles 2) the accuracy of the model or
the interaction potential or the force field used
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Computer simulations of Molecular systems size
of the configurational space
The simulation of molecular systems at non-zero
Temp requires the generation of a statistically
representative set of configurations the
ENSEMBLE
The properties of the system are calculated as
ensemble averages or integrals over the
configuration space generated
For a many particle system the averaging or
integration involves many degrees of freedom as
a result only a part of the configurational space
must be considered
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When choosing a model one should include only
those degrees of freedom on which the property
depends
Increase simplicity speed search power timescale
Decrease complexity accuracy
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Computer simulations of Molecular systems size
of the configurational space
The level of approximation should be chosen such
that the degrees of freedom essential to a proper
evaluation of the property under study can be
sampled
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Computer simulations of Molecular
systems accuracy of molecular model and force
field
If the system has been simulated for long enough
time, the accuracy of the prediction of
properties depends only on the quality of the
interaction potential.
For Biological systems only the atomic degrees of
freedom are considered (no electrons,
Born-Oppenheimer approx). The atomic interaction
function is an effective interaction. The
evolution of the system is described by
classical mechanics
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Computer simulations of Molecular
systems accuracy of molecular model and force
field
Four points to consider
1) Classical mechanics of point masses the
position of one particle depends on the positions
of the others through the effective interaction
function 2) System size and number of degrees of
freedom 3) Sampling and time-scale of the
process 4) Force Field choice
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Computer simulations of Molecular
systems accuracy of molecular model and force
field
Molecular Motions
Time-scale number of atoms
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Computer simulations of Molecular
systems accuracy of molecular model and force
field
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Computer simulations of Molecular
systems accuracy of molecular model and force
field
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Computer simulations of Molecular
systems accuracy of molecular model and force
field
Take home lesson Running and analyzing a
simulation 1) choose an appropriate set of
parameters 2) choose an appropriate interaction
function 3) simulate accordingly to the time
scale of the process or 4) generate a suitable
statistical ensemble.
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Methodology
A typical force field or effective potential for
a system of N atoms with masses mi
(i1,2..N) and cartesian position vectors ri
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Methodology Terms of the potential function
Bond term
Angle term
Improper term
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Methodology Terms of the potential function
Dihedral term
Non-Bonded term
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Methodology treatment of electrostatics
The sums in this term run over all atom pairs in
molecular systems, and it is proportional to N2.
All the other parts of the calculation are
proportional to N.
Several approximations-solutions 1) cutoff
methods 2) continuum methods 3) Periodic methods
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Methodology treatment of electrostatics-Cutoff
methods
R1
All atom pairs(i,j) every step
R2
Force updated every Nc steps
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Methodology treatment of electrostatics-Continuum
methods
If one part of the system is homogeneous, like
the solvent around the solute, the homogeneous
part can be considered a continuum. The system
is divided in two parts 1) an inner region
where charges qi are explicititly treated 2) an
outer region treated as a continuum with
dielectric constant e Poisson-Boltzmann
Equation
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Methodology treatment of electrostatics-Periodic
methods
The system is replicated infinitely. The charge
distribution in the system is represented as
delta functions
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Each point charge is surrounded by a gaussian
charge of opposite sign
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-
The charge interactions become short-ranged. An
error function is used to recover the original
distribution
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Searching the configuration space and generating
the ensemble
Systematic search methods degrees of freedom are
varied systematically (for example torsions), and
the energy V of the new configuration is
calculated. Decane, variation of torsions over
3 values, 7 torsions 37 values of V to calculate
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Searching the configuration space and generating
the ensemble
Random methods a collection of configurations is
generated randomly. From a starting
configuration, a new one is generated by
displacement of some variable Rs1 RS Dr The
energy of the new structure is calculated through
V If E2 lt E1 the conf is accepted else the value
p exp(-(E2-E1)/kT)) is calculated and if it is gt
R it is accepted. R is a random number (0,1)
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Searching the configuration space and generating
the ensemble
Molecular Dynamics
Generates the ensemble of configurations via
application of Natures laws of motion to the
atoms of the molecular system
Advantage dynamical information about the system
is obtained
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Molecular Dynamics
A trajectory ( Ensemble of configurations as a
function of time) is generated by simultaneous
integration of Newtons equations
d2ri(t) / dt2 Fi / mi
Fi - dV(r1, r2, ..rN) / dri
V is the potential function r is the position of
the particle F is the force acting on the
particle
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Molecular Dynamics
d2ri(t) / dt2 Fi / mi
Fi - dV(r1, r2, ..rN) / dri
The integration is performed in small time-steps
1-10 fs Equilibrium quantities can be obtained
by averaging over the sufficiently-long
trajectory Dynamic information is extracted
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Molecular Dynamics
MD can cross potential energy barriers of the
order of kBT kB Boltzmann constant, T Temperature
Energy
Time-scale of the process Number of atoms
Time
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Molecular Dynamics
Natural systems are at Constant-Temperature
Constant-Temperature Molecular Dynamics
Vi velocity of particle i
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Molecular Dynamics
Constant-Temperature Molecular Dynamics weak
coupling to an external bath
The kinetic energy is changed in the time step Dt
by scaling atomic velocities v with a factor l
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Molecular Dynamics
Constant-Temperature Molecular Dynamics
If the heat capacity per degree of freedom is cv,
the change in energy leads to achange in Temp
DT should be equal to the dt of equation (1), and
we obtain
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Molecular Dynamics
Integrating the Equations of motion
Second order differential equations
d2ri(t) / dt2 Fi / mi
Fi - dV(r1, r2, ..rN) / dri
They can be re-written as two first-order
differential equations
dvi(t) dt Fi (ri(t)) / mi
dri(t) / dt vi(t)
Velocity-Verlet Algorithm
ri(tn Dt) 2ri(tn) - ri(tn - Dt) Fi (ri(t))
/ mi (Dt)2
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Molecular Dynamics
Integrating the Equations of motion
Problems
Computational Efficiency
Memory requirements
Velocity
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Molecular dynamics applications
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Molecular dynamics applications
Mechanosensitive Ion Channel response to Pressure
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Molecular dynamics applications
Increasing stretch
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Molecular dynamics applications
Anti-Tumor Peptides structure-activity
correlation