Simulazione di Biomolecole: metodi e applicazioni giorgio colombo [email protected].
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Transcript of Simulazione di Biomolecole: metodi e applicazioni giorgio colombo [email protected].
Simulazione di Biomolecole:metodi e applicazioni
giorgio [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 ofmolecular 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 Liquid state Gas phasesolid state macromolecules
Quantum possible still impossible possible
Classical easy computer simulations trivial
Many particlesystem
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 Temprequires 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 thosedegrees of freedom on which the property depends
Model Degrees of freedom Example of Property
Left Removed Predicted Force Field
Quantummechanical
Nuclei,electrons
nucleons Reactions Coulomb
All atoms,polariz
Atomsdipoles
electrons Binding chargedligands
Ionicmodels
All atoms Solute +solvent atoms
dipoles hydration GROMOS
All soluteatoms
Solute atoms solvent Gas phaseconformation
MM2
Groups ofatoms asballs
Atom groups Individualatoms
Folding topologyof macromolecules
LW
Increase:simplicity
speedsearch power
timescale
Decrease:complexityaccuracy
Computer simulations of Molecular systems:size of the configurational space
The level of approximation should be chosen such thatthe 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-scalenumber 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 parameters2) choose an appropriate interaction function 3) simulate accordingly to the time scale of the process or4) generate a suitable statistical ensemble.
Methodology
A typical force field or effective potential for a systemof N atoms with masses mi (i=1,2..…N)
and cartesian position vectors ri:
)4/(/),(/),()cos(1
2
1
2
1
2
1),.....,(
06
612
12),(
20
20
2021
ijrjiijijjipairsdihedrals
dihedralsimpropangles
bbonds
N
rqqrjiCrjiCnK
KKbbKrrrV
Methodology:Terms of the potential function
202
1bbKb
bonds
Bond term
Angle term
202
1 Kangles
Improper term
202
1 K
dihedralsimprop
b
Methodology:Terms of the potential function
Dihedral term
Non-Bonded term
)cos(1 nKdihedrals
)4/(/),(/),( 06
612
12),(
ijrjiijijjipairs
rqqrjiCrjiC
Methodology:treatment of electrostatics
)4/(/),(/),( 06
612
12),(
ijrjiijijjipairs
rqqrjiCrjiC
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 methods2) continuum methods3) Periodic methods
Methodology:treatment of electrostatics-Cutoff methods
R1
All atom pairs(i,j) every stepR2
Force updated every Nc steps
Methodology:treatment of electrostatics-Continuum methods
If one part of the system is homogeneous, like the solventaround 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 dielectricconstant
Poisson-Boltzmann Equation: )()( 22 rkr
Methodology:treatment of electrostatics-Periodic methods
++
+ --
-
The system is replicated infinitely.The charge distribution in the system isrepresented 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 theoriginal distribution
Searching the configuration spaceand 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 torsions37 values of V to calculate
Searching the configuration spaceand 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 variableRs+1= RS + r
The energy of the new structure is calculated through V
If E2 < E1 the conf is acceptedelse 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 spaceand 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 functionr 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 kBTkB Boltzmann constant, T Temperature
Energy
Time
Time-scale of the processNumber of atoms
Molecular Dynamics
Natural systems are at Constant-Temperature
Constant-Temperature Molecular Dynamics
)(2
1)(
2
1)( 2
1
tTkNtvmtE Bdfii
N
ikin
Vi velocity of particle i
Molecular Dynamics
Constant-Temperature Molecular Dynamics:weak coupling to an external bath
)(/)( 01 tTTdttdT T
The kinetic energy is changed in the time step t by scaling atomic velocities v with a factor
)(2
1)1()( 2 tTkNtE Bdfkin
Molecular DynamicsConstant-Temperature Molecular Dynamics
kindfvdf EcNT
1
T should be equal to the dt of equation (1), and we obtain
2/1
011 1)(/2/1 tTTtkc TB
dfv
If the heat capacity per degree of freedom is cv, the change in energy leads to achange in Temp
Molecular DynamicsIntegrating the Equations of motion
d2ri(t) / dt2 = Fi / mi
Fi = - V(r1, r2, …..rN) / ri
Second order differential equations
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 + t) = 2ri(tn) - ri(tn - t) + Fi (ri(t)) / mi (t)2
Molecular DynamicsIntegrating 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