Simulazione di Biomolecole:metodi e applicazioni

giorgio colombocolombo@ico.mi.cnr.it

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

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