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Simulazione di Biomolecole:
metodi e applicazioni
giorgio colombo
[email protected]
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
Real World
Model
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
Computational BioChemistry
Chemical systems are generally too
inhomogeneous and complex (1023particles)
to be treated analitically
Crystalline
solid state
Quantum
Classical
possible
easy
Liquid state
macromolecules
still impossible
computer simulations
Many particle
system
Gas phase
possible
trivial
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
Computational BioChemistry
Outline:
1) basic problems of computer simulation
of biological systems
2) Methodology and applications
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
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
When choosing a model one should include only those
degrees of freedom on which the property depends
Model
Degrees of freedom
Left
Example of Property
Removed
Predicted
Force Field
Quantum Nuclei,
mechanical electrons
nucleons
Reactions
Coulomb
All atoms,
polariz
Atoms
dipoles
electrons
Binding charged
ligands
Ionic
models
All atoms
Solute +
dipoles
solvent atoms
hydration
GROMOS
All solute
atoms
Solute atoms
solvent
Gas phase
conformation
MM2
Groups of
atoms as
balls
Atom groups
Individual
atoms
Folding topology LW
of macromolecules
Increase:
simplicity
speed
search power
timescale
Decrease:
complexity
accuracy
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
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
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
Computer simulations of Molecular systems:
accuracy of molecular model and force field
Molecular Motions
Time-scale
number of atoms
Computer simulations of Molecular systems:
accuracy of molecular model and force field
Computer simulations of Molecular systems:
accuracy of molecular model and force field
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.
Methodology
A typical force field or effective potential for a system
of N atoms with masses mi (i=1,2..…N)
and cartesian position vectors ri:
V (r1 , r2 ,.....rN ) 


dihedrals
1
1
1
2
2
2






K
b

b

K




K



 b 0 angles
2 
 
0
0
bonds 2
improp 2
K 1  cos(n   ) 
 C
dihedrals
12
6
(
i
,
j
)
/
r

C
(
i
,
j
)
/
r
12
ij
6
ij  qi q j /( 4 0 r rij )
pairs ( i , j )

Methodology:
Terms of the potential function
Bond term
1
2


K
b

b

b
0
bonds 2
b
Angle term
1
2


K





0
2
angles
Improper term
1
2


K





0
improp 2
dihedrals

Methodology:
Terms of the potential function
Dihedral term

K 1  cos(n   )

dihedrals
Non-Bonded term
 C
12
6
(
i
,
j
)
/
r

C
(
i
,
j
)
/
r
12
ij
6
ij  qi q j /(4 0 r rij )
pairs ( i , j )

Methodology:
treatment of electrostatics
 C
12
6
(
i
,
j
)
/
r

C
(
i
,
j
)
/
r
12
ij
6
ij  qi q j /(4 0 r rij )

pairs ( i , j )
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
Methodology:
treatment of electrostatics-Cutoff methods
R1
R2
All atom
pairs(i,j) every step
Force updated
every Nc steps
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 
Poisson-Boltzmann Equation:
  (r )  k  (r )
2
2
Methodology:
treatment of electrostatics-Periodic methods
The system is replicated infinitely.
The charge distribution in the system is
represented as delta functions
+
+
+
-
Each point charge is surrounded by a
gaussian charge of opposite sign
The charge interactions become
short-ranged.
An error function is used to recover the
original distribution
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
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
Rs+1= RS + Dr
The energy of the new structure is calculated through V
If E2 < E1 the conf is accepted
else the value p= exp(-(E2-E1)/kT)) is calculated and if it
is > R it is accepted. R is a random number (0,1)
Searching the configuration space
and generating the ensemble
Molecular Dynamics
Generates the ensemble of configurations via application of
Nature’s laws of motion to the atoms of the molecular system
Advantage:
dynamical information about the system is obtained
Molecular Dynamics
A trajectory ( Ensemble of configurations as a function of
time) is generated by simultaneous integration of Newton’s
equations
d2ri(t) / dt2 = Fi / mi
Fi = - V(r1, r2, …..rN) / ri
V is the potential function
r is the position of the particle
F is the force acting on the particle
Molecular Dynamics
d2ri(t) / dt2 = Fi / mi
Fi = - V(r1, r2, …..rN) / ri
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
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
Molecular Dynamics
Natural systems are at Constant-Temperature
Constant-Temperature Molecular Dynamics
N
1 2
1
Ekin (t )   mi vi (t )  N df k BT (t )
2
i 1 2
Vi velocity of particle i
Molecular Dynamics
Constant-Temperature Molecular Dynamics:
weak coupling to an external bath
dT (t ) / dt   T T0  T (t )
1
The kinetic energy is changed in the time step Dt
by scaling atomic velocities v with a factor l
1
Ekin (t )  (l  1) N df k BT (t )
2
2
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


df 1
df v
DT  N c
DEkin
DT should be equal to the dt of equation (1), and we obtain


l  1  c k B / 2 Dt T0 / T (t )  1
df
v
1
1
T
1/ 2
Molecular Dynamics
Integrating the Equations of motion
Second order differential equations
d2ri(t) / dt2 = Fi / mi
Fi = - V(r1, r2, …..rN) / ri
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
Molecular Dynamics
Integrating the Equations of motion
Problems:
Computational Efficiency
Memory requirements
Velocity
Molecular dynamics:
applications
Molecular dynamics:
applications
Mechanosensitive Ion Channel: response to Pressure
Molecular dynamics:
applications
Increasing stretch
Molecular dynamics:
applications
Anti-Tumor Peptides: structure-activity correlation