Monte Carlo (MC) Simulation Methods. Elisa Fadda

Transcription

1 Monte Carlo (MC) Simulation Methods Elisa Fadda 1011-CH328, Molecular Modelling & Drug Design 2011

2 Experimental Observables A system observable is a property of the system state. The system state i is defined by its energy E i OESY spectrum All accessible states at a certain temp contribute to the experimental observable values. A method that allows us to generate states at low energy will enable us to evaluate molecular properties and experimental observables accurately.

3 Experimental Values The value that we measure experimentally is an average of the property A over the time of the measurement and it is therefore known as time average.

4 Exp. vs Comp. Timescales Experiments s s 10-9 s 10-6 s 10-3 s 1 s Time (s) 4

5 Back to the Conformational Space? 3 degrees of freedom space to explore starting from a partially unfolded structure: How many conformations are accessible to the system at room temperature? 5

6 How to Explore the Phase Space A single system evolving in time is replaced by a large number of replications of the system that are considered simultaneously. Time averages can be replaced by ensemble averages. In the ergodic hypothesis time averages and ensemble averages are identical. 6

7 Monte Carlo (MC) Methods MC simulations generate configurations by making random changes to the positions of the species present, together with their orientations. 7

8 Monte Carlo (MC) Simulations Unlike in MD simulations, in MC we sample from a 3-dimensional phase space (no momentum contributions). Potential energies and thermodynamic properties are calculated from the positions of the atoms in states at low energy. V ( r ) = dr V ( r ) ρ( r ) Probability of obtaining the configuration r ρ( r ) = dr exp [ V r kt ] ( ) exp V ( r ) kt == exp [ V r kt ] ( ) Z Configurational integral 8

9 Monte Carlo Integration Method Is a numerical integration using random numbers. We can determine the area of the circle, therefore π, by generating random numbers within the square. r 9

10 Monte Carlo Integration Method Square s area = 4r 2 Then the area of the circle can be estimated by the ratio of the points inside the circle to the total number of points, yielding an approximation for π. r Circle Total Circle Total π = 3.2 4r 2 = 40 = 50 = A Circle = π r 2 10

11 In Order to Calculate Z(T) 1. Randomly generate a configuration (3 coordinates) 2. Calculate the potential energy for such configuration V(r ) 3. Calculate the Boltzmann factor exp [ V ( r ) kt ] 4. Add the Boltzmann factor to an accumulated sum and the V(r ) contribution to its accumulated sum, and go back to 1 5. After a number of trials the mean value of the potential energy is given by: V ( r ) trials V ( r i i= 1 = trials i= 1 exp )exp [ ] V ( r ) kt [ ] V ( r ) kt i i 11

12 Monte Carlo (MC) Integration In MC simulations the location, orientation and perhaps the geometry of a molecule or a collection of molecules are chosen according to a statistical distribution. Many possible conformations of a molecule could be examined by choosing the conformation angles randomly. If enough iterations are done and the results are weighted by a Boltzmann distribution this gives a statistically valid result. 12

13 Issues with straight MC Integration Straight MC integration leads to the evaluation of high energy as well as low energy conformations, which contribute negligibly to the thermodynamic properties. Only low energy conformations coincide with the physically observed phases. These conformation are characterized by a high probability of occurring and are the only states that make a significant contribution to the integral. The Metropolis MC method biases the generation of configurations towards those that make the most significant contribution to the integral: Metropolis MC generates only the most probable configurations and counts them equally MC integration generates random configurations and then weights them based on their probability.

14 Boltzmann Distribution For a single system at a well-defined temperature it gives the probability that the system has a specific energy (it is in a specified state). In a more general case the Boltzmann distribution indicates the fraction of particles occupying a set of states: States degeneracy Energy Total number of particles Partition function umber of particles 14

15 Metropolis Monte Carlo In many thermodynamic properties of a molecular system, those states with a high probability ρ(r ) are also the ones which make a significant contribution to the integral. In the Metropolis method the generation of configuration is biased towards the ones which make the highest contribution, i.e. states with probability exp[ V ( r ] ) kt and then counts each of them equally (trials). 15

16 Implementation At each iteration of the simulation a new configuration is generated: 1. A random number 0 ζ 1 is chosen and the coordinates are changed as: x y z new new new = = = x y z old old old ( 2ζ 1) rmax ( 2ζ 1) r ( 2ζ 1) rmax max 2. The energy of the new configuration is calculated V(r ). 3. Check Point: a. If the energy of the new configuration is lower than the old then the new configuration is used for the next iteration. b. If the energy of the new configuration is higher than the old then the Boltzmann factor is compared to a random number between 0 and 1. If the Boltzmann factor is greater than the random number then the new configuration is accepted, if not it is rejected. 16

17 Main Differences Between MD and MC The Metropolis algorithm generates a Markov Chain of states: 1. The outcome of each trial depends OLY upon the preceding trial and not upon any previous trials (stochastic method). 2. Each trial belongs to a finite set of possible outcomes. Contrary to MC, in MD all states are connected in time: MD is a deterministic method. o temporal relationship between successive MC configurations (no temporal properties, e.g. no transport coefficients) In MD there s a kinetic energy contribution to the energy, whereas in MC the total energy is determined OLY from the potential energy function. 17

18 Main Differences Between MD and MC MC simulations converge quite fast and accurate values of properties can be obtained more rapidly than in MD. However. The very essence of the M-MC makes the treatment of large flexible molecules rather difficult, unless some of the internal degrees of freedom are frozen. In MC simulation it s easier than in MD simulation to hold P and T constant. MC is more suitable than MD for the simulation of lattice models. In appropriate system the ability of MC to make non-physical moves facilitates the identification of large conformational changes, however it lacks the ability of MD to explore extensively local sites in the phase space. 18