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Simulazione di Biomolecole: metodi e applicazioni giorgio colombo colombo@ico.mi.cnr.it

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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 WorldModel

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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 (10 23 particles) to be treated analitically CrystallineLiquid stateGas phase solid statemacromolecules Quantumpossible still impossible possible Classical easy computer simulations trivial 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 - 10 23 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 m i (i=1,2..…N) and cartesian position vectors r i :

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Methodology: Terms of the potential function Bond term Angle term Improper term b

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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 N 2. 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 q i are explicititly treated 2) an outer region treated as a continuum with dielectric constant 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 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

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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 3 7 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 R s+1 = R S + r 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)

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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 d 2 r i (t) / dt 2 = F i / m i F i = - V(r 1, r 2, …..r N ) / r i 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 d 2 r i (t) / dt 2 = F i / m i F i = - V(r 1, r 2, …..r N ) / r i 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 k B T k B Boltzmann constant, T Temperature Energy Time Time-scale of the process Number of atoms

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Molecular Dynamics Natural systems are at Constant-Temperature Constant-Temperature Molecular Dynamics V i 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 t by scaling atomic velocities v with a factor

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Molecular Dynamics Constant-Temperature Molecular Dynamics T should be equal to the dt of equation (1), and we obtain If the heat capacity per degree of freedom is c v, the change in energy leads to achange in Temp

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Molecular Dynamics Integrating the Equations of motion d 2 r i (t) / dt 2 = F i / m i F i = - V(r 1, r 2, …..r N ) / r i Second order differential equations They can be re-written as two first-order differential equations dv i (t)] dt = F i (r i (t)) / m i dr i (t) / dt = v i (t) Velocity-Verlet Algorithm r i (t n + t) = 2r i (t n ) - r i (t n - t) + F i (r i (t)) / m i ( t) 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

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