Metadynamics

Metadynamics is a computer simulation method in computational physics, chemistry, and biology. It is used to estimate the free energy and other state functions of a system, where ergodicity is hindered by the form of the system's energy landscape.

1. METADYNAMICS VISHAL KUMAR Reg no-16MSCSCC01 MSc Computational Chemistry Department of Computational Sciences

2. What can we do if we only know the starting point but not the end point of a reaction? METADYNAMICS It was first suggested by Alessandro Laio  and  Michele Parrinello  in 2002 Source-http://people.sissa.it/~ laio / Source-https://www.hpc-ch.org/marcel-benoist-prize-for-michele-parrinello/

3. Time scale problem Activated events This A-B can be : Chemical reaction Phase transition between liquid and solid Conformational change, etc …… Molecular dynamics can access only a limited time scale. Branduardi , D., Gervasio , F. L., & Parrinello , M. (2007). From A to B in free energy space. The Journal of chemical physics, 126 (5), 054103.

4. Potential energy surface is rough Bolhuis , P. G., Dellago , C., & Chandler, D. (2000). Reaction coordinates of biomolecular isomerization . Proceedings of the National Academy of Sciences, 97 (11), 5877-5882. Source: Geology, mountains, peaks, alps – Fürstentum LiechtensteinFürstentum Liechtenstein 1140 × 410 Search by image Mountains Liechtenstein

5. Proposed solutions to the sampling problem Enhanced sampling- Parallel sampling, replica exchange, simulated tempering. Trajectory based schemes- NEB, Transition path sampling, Forward flux finite temperature string method. Bias potential- Umbrella sampling, Local elevation, Conformational flooding, Adaptive bias force, Self-healing umbrella sampling, Multicanonical MD, Wang-Landau, Metadynamics.

6. Objectives of Meta dynamics The philosophy of meta dynamics , it is two fold in which we want one end to solve taking a problem going from one medium to another by simulation method, start from one conformer to another from A to B gives right statistical distribution of this conformer. To bring down the complexity of the system which is made up of ‘N’ degree of freedom bring down to a level which we can understand the system why system go from A to B, that will enhance the understanding of the problem.

7. METADYNAMICS Whenever we go put a “small gaussian” Always move in the direction that minimizes the sum of F(s) and all the Gaussian potential The system try to minimize its energy We will add small gaussian potential (Repulsive gaussian). Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81. B A C

8. Source: https://youtube.com/watch?v=CtIrLkx6aNo

9. Example of meta dynamics in 1-D potential Laio , A., & Gervasio , F. L. (2008). Metadynamics : a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Reports on Progress in Physics, 71 (12), 126601.

10. Applications of Meta dynamics Chemical reactions Protein folding Molecular docking Phase transitions Encapsulation of DNA onto hydrophobic and hydrophilic single-walled carbon nanotubes

11. METHOD It aims to enhance the exploration of the free energy surface , F(S (R) ) of a limited set of collective variables S(R) For a canonical ensemble fundamental thermodynamic function is Helmholtz free energy The probability to find the system on a hyperplane, S’ of atomic positions, R , in phase space is given by here are the atomic velocities, H is the Hamiltonian of the system, δ is the delta function. Continue………… Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81.

12. Molecular dynamics (MD) simulation is used to sample phase space, where we can derive a model Hamiltonian of the molecular system from the Lagrangian , Now model potential or force field Continue………… Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81.

13. Such complicated phenomena are modeled well when the interatomic forces are computed from the instantaneous electronic structure via an interface with an ab initio method. Density functional theory (DFT) is a particularly efficient electronic structure method , where the electronic(Kohn- Sham) energy, which is a functional of the electronic density , is constructed from the one-electron wave function The DFT potential, Terms involved- Electronic kinetic energy, Electron-nuclei interaction, Coulombic potential, Configuration of atomic position R Ground state wave function is obtained from wave function It minimize ,which are readily found from a self-consistent matrix diagonalization.   Continue…………

14. In stead of diagonalization at every step, The wave function can be updated using an extended Langrangian technique Car- Parrinello MD (CPMD) The wave functions are treated as fictitious particles with a mass , follow the nuclei adiabatically for small . The potential V DFT (instead of V MM ) evolves. The last term ensures wave function orthogonality through the Lagrange multipliers .   Continue………… Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81.

15. Now we will use a hybrid quantum mechanics/molecular mechanics(QM/MM) method Chemically active part treated with QM Rest treated with a MM force field via a mixed Langrangian Last term electrostatic interaction between the MM and QM parts of the system Continue………… Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81.

16. After introducing the model Langrangians for classical MD and first principle CPMD, Now we will focus our attention on enhancing the sampling of collective variables S(R). for extended Lagrangian methods, we introduce a fictitious particle, , for each collective variable, S ( R ), with a mass , which interacts with the system via a harmonic spring with force constant attached to (R ),   Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81.

17. With relatively large mass and stiff spring constant, Fictitious particles slowly ‘‘roll” in the bottom of the initial free energy well. At every time interval δ t we add a relatively small Gaussian-shaped repulsive potential At current point S(t) to the biasing potential (s , t), it discourages the system from revisiting this point. The history dependent potential builds up until it counter-balances the underlaying free energy well. It allowing the system to escape via a saddle point to a nearby local minimum. Procedure is repeated at local minima. When all the minima are ‘‘filled” with Gaussian potential ‘‘hills” the system will move barrier free.   Ensing, B., De Vivo, M., Liu, Z., Moore, P., & Klein, M. L. (2006). Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Accounts of chemical research, 39 (2), 73-81.

18. A Simple example: Alanine dipeptide Laio , A., & Gervasio , F. L. (2008). Metadynamics : a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Reports on Progress in Physics, 71 (12), 126601.

19. Examples of Collective variables(CV) Distances Angles- bonding and torsional Coordination number- between individual atoms or between different species Solvation energy Electric field Reaction path

20. The choice of collective variables The reliability of meta dynamics is strongly influenced by the choice of the CVs. Ideally the CVs should satisfy three properties: They should clearly distinguish between the initial state , the final state and the intermediates. They should describe all the slow events that are relevant to the process of interest. Their number should not be too large, otherwise it will take a very long time to fill the free energy surface.

21. what happens if a relevant CV is neglected? Laio , A., & Gervasio , F. L. (2008). Metadynamics : a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Reports on Progress in Physics, 71 (12), 126601.

22. The algorithm A practical example of the Fortran code The routine performs three tasks: It computes the value of the CVs=S(x). Every time steps , it stores the value of s in an array that contains the centers of all Gaussians. It computes the derivative of (S(x) , t) with respect to x using chain rule:   These derivatives are then added to the usual forces on the atoms, biasing the dynamics of the system. Laio , A., & Gervasio , F. L. (2008). Metadynamics : a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Reports on Progress in Physics, 71 (12), 126601.

23. Source: https://youtube.com/watch?v=IzEBpQ0c8TA

24. Enhancing the fluctuation Barducci, A., Bonomi, M., & Parrinello, M. (2011). Metadynamics. Wiley Interdisciplinary Reviews: Computational Molecular Science, 1 (5), 826-843.

25. Conclusion Metadynamics offers a very promising technique to study activated physical/chemical processes! Advantages Efficient exploration of reaction pathways Provides insights in to the mechanism Allows free energy estimation Disadvantages Comes with little additional computational overhead.