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Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. The method is applied mostly in chemical physics, materials science, and biophysics. Because molecular systems typically consist of a vast number of particles, it is impossible to determine the properties of such complex systems analytically; MD simulation circumvents this problem by using numerical methods. However, long MD simulations are mathematically ill-conditioned, generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and parameters, but not eliminated. For systems that obey the ergodic hypothesis, the evolution of one molecular dynamics simulation may be used to determine the macroscopic thermodynamic properties of the system: the time averages of an ergodic system correspond to microcanonical ensemble averages. MD has also been termed "statistical mechanics by numbers" and "Laplace's vision of Newtonian mechanics" of predicting the future by animating nature's forces and allowing insight into molecular motion on an atomic scale. MD was originally developed in the early 1950s, following earlier successes with Monte Carlo simulations—which themselves date back to the eighteenth century, in the Buffon's needle problem for example—but was popularized for statistical mechanics at Los Alamos National Laboratory by Marshall Rosenbluth and Nicholas Metropolis in what is known today as the Metropolis–Hastings algorithm. Interest in the time evolution of N-body systems dates much earlier to the seventeenth century, beginning with Isaac Newton, and continued into the following century largely with a focus on celestial mechanics and issues such as the stability of the solar system. Many of the numerical methods used today were developed during this time period, which predates the use of computers; for example, the most common integration algorithm used today, the Verlet integration algorithm, was used as early as 1791 by Jean Baptiste Joseph Delambre. Numerical calculations with these algorithms can be considered to be MD done "by hand". As early as 1941, integration of the many-body equations of motion was carried out with analog computers. Some undertook the labor-intensive work of modeling atomic motion by constructing physical models, e.g., using macroscopic spheres. The aim was to arrange them in such a way as to replicate the structure of a liquid and use this to examine its behavior.

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1 A Brief Introduction To Molecular Dynamics And NAMD Presented By -- Rumela Adhikary
2 What is Molecular Dynamics?? Molecular Dynamics is a computer simulation method, where the physical movements of the atoms and molecules are studied over a certain and fied period of time. In MD, for a certain time, the atoms and molecules of interest, are allowed to interact and a view of dynamic evolution is obtained by predictive trajectory. The basic idea for molecular dynamics is, to solve equation of motion, along with proper force feld, model potential and boundary conditions for the system of interest, to obtain bulk properties of the system, at equlibrium.
3 Things need to know before proceeding in Molecular Dynamics -- 1) Boundary Conditions 2) Model systems and interaction potential 3) Algorithms for solving diferential equations
4 Boundary conditions: There are two major types of boundary conditions -- • Isolated boundary condition (IBC) : It is ideally suited for studying clusters and molecules. • Periodic boundary condition (PBC): It is suited for studying bulk liquids and solids.
5 Periodic Boundary Conditions(PBC) a) Need of periodic boundary conditions: During computer simulations of liquids, if the properties of liquid is measuerd assuming the liquid as a unit cell of liquid molecules, then there is a serious problem occurs – the huge surface:bulkphase ratio (~ 0.953:1 for 1000 molecules) . To avoid this problem, PBC is used. b) How this PBC works: Since we can assume liqids as a huge collection of molecules, so a very interesting assumption is taken. The lquid is assumed as a pseudo infnite lattice, where the central unit cell is the actual boi containing real molecules. The other boies surrounding it, contains the images of the molecules present in the central unit cell. So, if any molecule moves from one side of central boi to it’s neighbouring cell, then instanteniously that molecule’s image enter to the central unit cell from opposite side with eiact same velocity. Thus number density of molecules in the central unit cell is always constant.
6 A 3-D periodic system, where molecules can enter and leave the cells by any of cube faces A torus , a simpler way to assume the pseudo infnte lattice Around each particle of the central boi,there can be considerd a sphere of radius rc which is <= l/2 where, l is the length of the central boi.
7 Model systems and interaction potential For any system, to understand it’s properties, one has to solve the corresponding equations of motion(EOM) , which can be either in the form of Hamiltonian, Lagrangian or Newtonian. In the most fundamental form, written in Hamiltonian is --- H(q,p) = K(p) + V(q) Where K(p) is the kinetic energy form and V(q) is the potential energy form. We are actually interesting in the potential energy part as, this part contain all type of interaction terms and which can be solved using model systems.
8 The potential energy terms can be written in this way --- Where, The frst term ie. v1 represents the efect of eiternal feld on the system The second term ie. v2 represents the pair potentials, and this is the most contributing and most impotrtant term in the equation. The third term, ie. v3 involvs triplets of the molecules, which is actually signifcant in liquid densities.
9 There are many types of potentials involved in eiploring properties of liquids. But, there are some idealized potentials which are though very unrealistic but, a lot of simpler and convenient to use . These are -- ● Hard sphere potential ● Square well potential ● Soft sphere potential
10 Hard Sphere potential The hard sphere potential has the following form -- Where,V is the intermolecular potential between the two atoms or molecules, σ is the distance at which the intermolecular potential between the two particles is zero, r is the distance of separation between centres of both particles.
11 Square well potential The form of square well potential is following--- Where,v(r) is the intermolecular potential between the two atoms or molecules, is the ε well depth and a measure of how strongly the two particles attract each other, σ1 is the distance at which the intermolecular potential between the two particles becomes zero then goes to negative and σ2 is the distance from where the the potential again becomes zero, r is the distance of separation between centres of both particles.
12 Soft sphere potential Soft sphere potential has following form-- Where , is the distance at which σ the intermolecular potential between the two particles is zero, is the ε well depth and a measure of how strongly the two particles attract each other is the ν hardness parameter =12 ν
13 Along with thee idealized potentials there is another well known potential --- Lennard-Jones Potential The form of L-J potential is as below -- where V is the intermolecular potential between the two atoms or molecules, is the ε well depth and a measure of how strongly the two particles attract each other, is the σ distance at which the intermolecular potential between the two particles is zero, r is the distance of separation between centres of both particles.
14 We have discussed some of known potentials used to frame the potential energy in the Hamiltoniun..So, for a type of molecule one has to build a model potential and then adjust it. But, for simulation of a complicated system, where many atoms(diferent) are involved, then this job of building potential and adjusting it becomes difcult, so this concept of potentials are modifed with the concept of Force Field.
15 Algorithms for Solving Diferential equations In molecular dynamics, one deals with classical equations of motion generated for N-particles clssical system interacting via a potential v. To solve these equations, we need suitable algorithms which follows some perticular criteria.. a) Hamiltoniun is a constant of motion provided H doesn’t depent explicitly on time, so the force, acting on the system, must not be explicitly time dependent or velocity dependent. b) The equations are reversible in time, i.e. vector quantities like velocities and momentum, when change sign, they retrace to their trajectories.The actual point to do this is,to keep the paticle trajectories in the appropriate constant-energy hypersurface, so that the ensemble average are correct. c) The potential v, acting on the system, may be a continous funtion of particle position, ie. they can be expand into Taylor series, or they can be disontinous at some velocities like hard sphere potential or square well potential. So, this two diferent types of velocity change must be distinguised.
16 There are some algorithms keeping in mind these criteria -- 1) Verlet algorithm: This algorithm is based on a second order system solving by using it’s current position ri(t) and forces fi(t) and previous positions ri (t – Δt). The derivation for this algoritm is straight forward: ri (t – Δt) = ri (t) – Δt vi (t) + (Δt2/2mi)*fi (t) – Δt3/3! (d3ri (t)/ dt3 ) + O (Δt4 ) From further calculation, we get , vi (t) = [ri (t +Δt) - ri (t – Δt) ] / 2Δt ] + O (Δt3 ) Where velocities used to calculate kinetic energy. But this algorithm gives numerical imprecision and also the knowledge of ri (t +Δt) required.
17 2) Leap-frog algorithm This algoritm is actually slightly modifed verson of Verlet algorithm. In this case, half advancing in timestep is counted. This also a good way for maintaining reversibility. vi (t + Δt/2) = vi (t - Δt/2) + (Δt/mi ) fi (t) ri (t +Δt) = ri (t ) + Δt vi (t + Δt/2) In this algorithm, the velocities are updated at mid time steps and “leap” ahead their positions. The velocity obtained from this algorithm is as follows --- vi (t) = [vi (t -Δt/2) + vi (Δt + Δt/2) ] /2 This algorithm is better than classical Verlet algorithm, but still velocities are not handled in satisfactory manner.
18 3)Velocity Verlet Algorithm This algorithm is algebrically equivalent to classical Verlet algorithm. ri (t + Δt) = ri (t) + Δt vi (t) + (Δt2 /mi ) fi (t) + O (Δt3) vi (t + Δt) = vi (t) + (Δt /2mi ) [fi (t) + fi (t + Δt)] + O (Δt 3) The actual Verlet algorithm can derived from this. Though this algorithm is efectively simple, but this is actually the most widely used algorithm in Molecular dynamics. This scheme preserves volume in Phase-space.
19 Upto this all the types of algorithms discussed are for microcannonical ensemble(NVE). But, to study a real system, we have to study cannonical ensemble (NVT) or NPT ensemble. I) Connonical Ensemble (NVT) a) Andersen Thermostat At constant temperature, the system is assumed in contact with a heat bath of temperature T. So there is energy fuctuations present. To resolve this stochastic forces are used which act on atoms of the sampleto change their kinetic energy by collision. So, in practical case, to generate the required equation of motion of N- particles for NVT ensemble, in Andersen Thermostat, the hamiltoniun is supplimented with the stocastic collision term, which is an instantenious event which afects the momentum of the particle. This collision occurs in Poisson distribution.
20 b) Langevine Themostat This kind of thermostat also mimics the coupling of the system to a heat bath. In this case the Hamiltoniun is modifed with a ‘random force’ (much like stochastic one in Andersen thermostat) and a deterministic “frictional force” proportional with particle velocities. This terms are actually connected with Langevine’s fuctuation -dissipation equation. Along with these two, there is another thermostat known as Lowe- Andersen thermostat, which is a modifed version of Andersen thermostat.
21 II)Isothermal – isobarical ensemble (NPT) Andersen Barostat At constant temperature and pressure, the energy, pressure and enthalpy of an N-particle system fuctuate. So, to simulate such system, these fuctuations should be simulate. So, in this case, the Hamiltoniun is modifed by adding instanteneous stochastic collistion. And in betwwen two stochastic collision, the state of the sytem evolves according to it’s corresponfding EOM like following-- * Image is collected from Molecular dynamics simulations at constant pressure and/or temperature by H.C. Andersen Where, ρij is a dimentionless number equals to rij /V1/3 Q is a variable which can be interpret as volume, more precisely it can be eiplained as the coordinate of the piston. This whole equation is for scaled system, whose momentum conjugate of Q is Π And ᴫi is the momentum conjugate pi
22 A short trip to History of MD simulation : ● In 1956, Alder and Wainwright frst reported molecular dynamics (MD) simulation of hard spheres. (1958: First X-ray structure of a protein) ● In 1960, Vineyard group did the simulation of damaged Cu crystal. Now, there comes a breakthrough in liquid simulation using MD...... ● In 1964, Aneesur Rahman did MD simulation of liquid Ar using L-J potential ● Later, in 1971, Rahman and Stillinge worked on MD simulation of water. In pot 80’s, the etended hamiltonian methods were introduced in the feld of molecular dynamics simulatuion and some of the pioneering works are --- ● In 1980, by H. C. Andersen on MD method for NPH, NVT, NPT ensembles. ● In 1986, R. Car and M. Parrinello’s work on Ab initio MD (includes electronic degrees of freedom)
23 NAMD and molecular dynamics simulations: In NAMD simulations, some emperical force felds are used, that approiimate actual atomic forces in biopolymer or biological systems. For large systems, where lots of atoms are involved, so simulations require long time . So, to reduce the computational cost , NAMD is used , which uses parallization of computer. Nanoscale Molecular Dynamics (NAMD) is a perticularly well suited software, very much useful to workstation clusters as well as paraller computers and very efcient in running the MD simulation. It is written using the Charm++ parallel programming model, using CHARMM as force feld. But, it also can use other force felds like AMBER, GROMACS.
24 Now before, proceeding to NAMD, let’s see what’s a ‘Force feld’ ?? A force feld, is basically refers to -- The functional forms, with respect to a computer simulation, which is used to describe the intra- and inter-molecular potential energy of a collection of atoms, and the corresponding parameters that will determine the energy of a given confguration. These functions and parameters, in force feld, have been derived from eiperimental work on molecules and from accurate quantum mechanical calculations. They are often refned by the use of computer simulations to compare calculated condensed phase properties with eiperiment.
25 The general equation of force feld is given below-- Where the frst term corresponds to the sum over all bonds, with an equilibrium bond- length r0 and there is one term for every pair ij of directly connected atoms. The second term is a sum over all bond angles and there is one term for each set of three connected atoms ijk and it usually has a quadratic form. The third term is the sum over all torsions involving four connected atoms ijkl. In principle, this is an eipansion in trigonometric functions with diferent values of n, the multiplicity (i.e. the number of minima in a rotation of 2π around the j–k bond),many force felds fi n = 3. This term can also include improper torsions, where the four atoms defning the angle are not all connected by covalent bonds. The fourth term is a sum over the non-bonded interactions (between molecules and within molecules). In particular, it describes the electrostatic and repulsion–dispersion interactions.
26 In previous slide, it has been informed that, NAMD is a well suited software for molecular dynamics simulation . Then, it must have some important features, which make it suitable. Lets know those..... • Force Field Compatibility • Efcient Full Electrostatics Algorithms • Multiple Time Stepping • Input and Output Compatibility • Dynamics Simulation Options • Easy to Modify and Extend • Interactive MD simulations • Load Balancing
27 1. Force Field Compatibility: In NAMD , the force feld used, is same with that of, what is used in the programs CHARMM and X-PLOR. The speciality in this case is that – this force feld includes the local interaction terms consisting of bonded interaction between the 2nd , 3rd , 4th atom and also the pair wise interactions including electrodstatic and van der Waals forces. 2. Efcient Full Electrostatics Algorithms: NAMD incorporates the Particle Mesh Ewald (PME) algorithm, to account the full electrostatic interactions. This algorithm reduces the computational compleiity of electrostatic force evaluation from O(N2 ) to O(N log N). 3. Multiple Time Stepping In molecular dynamics, the velocity Verlet algorithm is used to advance the positions and velocities of the atoms in time for it’s simplicity and stability . To further reduce the cost of the evaluation of long-range electrostatic forces, a multiple time step scheme is employed. The local interactions like, bonded, van der Waals and electrostatic interactions (within a specifed distance, as it’s a long range force) are calculated at each time step.
28 4. Input and Output Compatibility The input and output fle formats used by NAMD are easy to understand. Input formats include coordinate fles in PDB format , structure fles in X-PLOR PSF format, and energy parameter fles are correspond to CHARMM forcefeld. 5. Dynamics Simulation Options MD simulations may be carried out using several options, including – Constant energy dynamics – Constant temperature dynamics via ∗ Velocity rescaling ∗ Velocity reassignment ∗ Langevin dynamics – Periodic boundary conditions – Constant pressure dynamics via ∗ Berendsen pressure coupling ∗ Nos´e-Hoover Langevin piston – Energy minimization, – Fiied atoms – Rigid waters – Rigid bonds to hydrogen – Harmonic restraints – Spherical or cylindrical boundary restraints.
29 5. Easy to Modify and Eitend Eitensibility and maintainability is a primary objective for NAMD. In order to achieve this, NAMD is designed in an object-oriented style with C++ language. NAMD’s modular design allows one to integrate and test new algorithms easily. 6. Interactive MD simulations A system undergoing simulation in NAMD may be viewed and altered with VMD. So, the user must have requisite knowledge of both VMD and NAMD. 7. Load Balancing An important factor in parallel applications is the equal distribution of computational load among the processors. NAMD uses a simple uniform spatial decomposition where the entire model is split into uniform cubes of space called patches. An initial load balancer assigns patches and the calculation of interactions among the atoms within them to processors such that the computational load is balanced as much as possible. During the simulation, an incremental load balancer monitors the load and performs necessary adjustments.
30 To run a simulation what types of fle we need?? 1) pdb (Protein Data Bank) fle, where atomic coordinates and/or velocities of system are recorded. 2) psf (Protein Structure File), where the structural informations of the protein ate stored. 3) Force feld parameter fle, like CHARMM, AMBER, GROMACS 4) Force feld topology fle, containing information on atom types, charges etc. 5) NAMD confguration fle
31 Generate or download A pdb fle( remember that pdb doesn’t contain H) Using proper tcl script and Running it with psfgen generate Another set of new pdb and psf fle with coordinates of H-atom Incorporate water, using “ SOLVATE” package in VMD tk-console, for the system or ,by writing proper script in Tcl, with respet to the system size. So that the molecule be completely immersed in liquid . Create a confguration File with required parameters Run with NAMD. Get the diiferent types of output fles and start ANALYSIS Flow Chart for running NAMD
32 Step -1 Step -1 Generating pdb fle PDB fles are indispensable parts for running MD simulations of any system. They can be created or downloaded from http://www.pdb.org . For small molecules, one can create PDB by generating structute in any type of Molecular visualization Software like Avogadro, GaussView etc. . This is an eiample of PDB fle for the protein Ubiquitin
33 Step -2 Step -2 Generating PSF fle PSF fles are also very important for properly viewing a molecule and doing it’s MD simulation. PSF’s can be generated either by using VMD’s inbuild psf-generator “ Automatic psf Builder”, or by writing proper tcl script which can be run using “psfgen” obtained with NAMD software package. This is the psf fle generated by psfgen software using tcl script. It is important here to mention that, in psf fle the residue-ID’s are according to the force feld topology fle, which is used to generate the psf fle.
34 Now, we have a pdb fle and a psf fle.. Ubiquin with all H-atom which was actually missing in pdb fle. Highlighting diferent secondary structures of Ubiquitin.
35 Step-3 Step-3 Incorporating water molecules around the protein molecule Water addition is important to study dynamics of a biomolecule. This can be done by two ways -- I. By using proper tcl script II. By using “solvate” package in VMD tk-console But, before putting the molecule in a boi of water,the size of the boi must be checked, so that the molecule doesn’t interact with it’s image in the neit cell(From Periodic boundary Condition requirement). To build up a proper system, the total system must be neutral. So, if the protein molecule carries eiess charge, then some ions must have to be added to the system to make a proper, neutral and stable system.
36 Command: This command is written on VMD tk-console. After solvation complete--
37 After addition of water boi, the system looks like this ---
38 There is another thing called “Implicit Solvent” Sometimes for large molecules, using requisite number of solvent make the system- size so large, that speed and computationbnal cost becomes an issue. To solve this problem “Implicit Solvent Model” is used. In this method, the efects of solvent is included in the inter-atomic force calculation, and thereby nullifng the need of eiplicit solvent molecules. This method is designed to behave like original solvent. Like, polar solvents, which act as dielectric and screens the electrostatic interactions, so their cooresponding implicit solvent model behave same ways. For more appropriate result using implicit solvent model, Generalized Born Implicit Solvent (GIBS) models are used.
39 Just one thing left to run an MD simulation-- NAMD Confguration File The NAMD confguration fle (also called a confg fle, .conf fle, or .namd fle) is the most important thing to run a correct and proper MD simulation. This is actually given to NAMD on the command line. This fle specifes virtually everything about the simulation to be done.
40 This is a general confguration fle ---
41
42
43 Now a system is ready for simulation. To run the simulation the following command should be written in command line --- namd2 ubq_wb_eq.conf > ubq_wb_eq.log &
44 Output fles In NAMD, there are 11 output fles generated--- 1) *.log 7) *.restart.vel 2) *.coor 8) *.restart.xsc 3) *.vel 9) *.restart.coor.old 4) *.xsc 10) *.restart.vel.old 5) *.dcd 11) *.restart.xsc.old 6) *.restart.coor
45 Informations about the output fles --- ● Among this 11 fles, there are seven fles contain binary data. These are : *.coor , *.vel, *.dcd, *.restart.coor, *.restart.vel, *.restart.coor.old and *.restart.vel.old ● The other 4 fles are eitended system confguration fle (*.isc, *.restart.isc, *.restart.isc.old and *.log fle) This 4 fles store the periodic cell dimensions of the system and the time steps along with the informations asked in confguration fle.
46 *.log File
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moleculardynamics_idea_workflow_namd-ppt.pdf

  • 1. 1 A Brief Introduction To Molecular Dynamics And NAMD Presented By -- Rumela Adhikary
  • 2. 2 What is Molecular Dynamics?? Molecular Dynamics is a computer simulation method, where the physical movements of the atoms and molecules are studied over a certain and fied period of time. In MD, for a certain time, the atoms and molecules of interest, are allowed to interact and a view of dynamic evolution is obtained by predictive trajectory. The basic idea for molecular dynamics is, to solve equation of motion, along with proper force feld, model potential and boundary conditions for the system of interest, to obtain bulk properties of the system, at equlibrium.
  • 3. 3 Things need to know before proceeding in Molecular Dynamics -- 1) Boundary Conditions 2) Model systems and interaction potential 3) Algorithms for solving diferential equations
  • 4. 4 Boundary conditions: There are two major types of boundary conditions -- • Isolated boundary condition (IBC) : It is ideally suited for studying clusters and molecules. • Periodic boundary condition (PBC): It is suited for studying bulk liquids and solids.
  • 5. 5 Periodic Boundary Conditions(PBC) a) Need of periodic boundary conditions: During computer simulations of liquids, if the properties of liquid is measuerd assuming the liquid as a unit cell of liquid molecules, then there is a serious problem occurs – the huge surface:bulkphase ratio (~ 0.953:1 for 1000 molecules) . To avoid this problem, PBC is used. b) How this PBC works: Since we can assume liqids as a huge collection of molecules, so a very interesting assumption is taken. The lquid is assumed as a pseudo infnite lattice, where the central unit cell is the actual boi containing real molecules. The other boies surrounding it, contains the images of the molecules present in the central unit cell. So, if any molecule moves from one side of central boi to it’s neighbouring cell, then instanteniously that molecule’s image enter to the central unit cell from opposite side with eiact same velocity. Thus number density of molecules in the central unit cell is always constant.
  • 6. 6 A 3-D periodic system, where molecules can enter and leave the cells by any of cube faces A torus , a simpler way to assume the pseudo infnte lattice Around each particle of the central boi,there can be considerd a sphere of radius rc which is <= l/2 where, l is the length of the central boi.
  • 7. 7 Model systems and interaction potential For any system, to understand it’s properties, one has to solve the corresponding equations of motion(EOM) , which can be either in the form of Hamiltonian, Lagrangian or Newtonian. In the most fundamental form, written in Hamiltonian is --- H(q,p) = K(p) + V(q) Where K(p) is the kinetic energy form and V(q) is the potential energy form. We are actually interesting in the potential energy part as, this part contain all type of interaction terms and which can be solved using model systems.
  • 8. 8 The potential energy terms can be written in this way --- Where, The frst term ie. v1 represents the efect of eiternal feld on the system The second term ie. v2 represents the pair potentials, and this is the most contributing and most impotrtant term in the equation. The third term, ie. v3 involvs triplets of the molecules, which is actually signifcant in liquid densities.
  • 9. 9 There are many types of potentials involved in eiploring properties of liquids. But, there are some idealized potentials which are though very unrealistic but, a lot of simpler and convenient to use . These are -- ● Hard sphere potential ● Square well potential ● Soft sphere potential
  • 10. 10 Hard Sphere potential The hard sphere potential has the following form -- Where,V is the intermolecular potential between the two atoms or molecules, σ is the distance at which the intermolecular potential between the two particles is zero, r is the distance of separation between centres of both particles.
  • 11. 11 Square well potential The form of square well potential is following--- Where,v(r) is the intermolecular potential between the two atoms or molecules, is the ε well depth and a measure of how strongly the two particles attract each other, σ1 is the distance at which the intermolecular potential between the two particles becomes zero then goes to negative and σ2 is the distance from where the the potential again becomes zero, r is the distance of separation between centres of both particles.
  • 12. 12 Soft sphere potential Soft sphere potential has following form-- Where , is the distance at which σ the intermolecular potential between the two particles is zero, is the ε well depth and a measure of how strongly the two particles attract each other is the ν hardness parameter =12 ν
  • 13. 13 Along with thee idealized potentials there is another well known potential --- Lennard-Jones Potential The form of L-J potential is as below -- where V is the intermolecular potential between the two atoms or molecules, is the ε well depth and a measure of how strongly the two particles attract each other, is the σ distance at which the intermolecular potential between the two particles is zero, r is the distance of separation between centres of both particles.
  • 14. 14 We have discussed some of known potentials used to frame the potential energy in the Hamiltoniun..So, for a type of molecule one has to build a model potential and then adjust it. But, for simulation of a complicated system, where many atoms(diferent) are involved, then this job of building potential and adjusting it becomes difcult, so this concept of potentials are modifed with the concept of Force Field.
  • 15. 15 Algorithms for Solving Diferential equations In molecular dynamics, one deals with classical equations of motion generated for N-particles clssical system interacting via a potential v. To solve these equations, we need suitable algorithms which follows some perticular criteria.. a) Hamiltoniun is a constant of motion provided H doesn’t depent explicitly on time, so the force, acting on the system, must not be explicitly time dependent or velocity dependent. b) The equations are reversible in time, i.e. vector quantities like velocities and momentum, when change sign, they retrace to their trajectories.The actual point to do this is,to keep the paticle trajectories in the appropriate constant-energy hypersurface, so that the ensemble average are correct. c) The potential v, acting on the system, may be a continous funtion of particle position, ie. they can be expand into Taylor series, or they can be disontinous at some velocities like hard sphere potential or square well potential. So, this two diferent types of velocity change must be distinguised.
  • 16. 16 There are some algorithms keeping in mind these criteria -- 1) Verlet algorithm: This algorithm is based on a second order system solving by using it’s current position ri(t) and forces fi(t) and previous positions ri (t – Δt). The derivation for this algoritm is straight forward: ri (t – Δt) = ri (t) – Δt vi (t) + (Δt2/2mi)*fi (t) – Δt3/3! (d3ri (t)/ dt3 ) + O (Δt4 ) From further calculation, we get , vi (t) = [ri (t +Δt) - ri (t – Δt) ] / 2Δt ] + O (Δt3 ) Where velocities used to calculate kinetic energy. But this algorithm gives numerical imprecision and also the knowledge of ri (t +Δt) required.
  • 17. 17 2) Leap-frog algorithm This algoritm is actually slightly modifed verson of Verlet algorithm. In this case, half advancing in timestep is counted. This also a good way for maintaining reversibility. vi (t + Δt/2) = vi (t - Δt/2) + (Δt/mi ) fi (t) ri (t +Δt) = ri (t ) + Δt vi (t + Δt/2) In this algorithm, the velocities are updated at mid time steps and “leap” ahead their positions. The velocity obtained from this algorithm is as follows --- vi (t) = [vi (t -Δt/2) + vi (Δt + Δt/2) ] /2 This algorithm is better than classical Verlet algorithm, but still velocities are not handled in satisfactory manner.
  • 18. 18 3)Velocity Verlet Algorithm This algorithm is algebrically equivalent to classical Verlet algorithm. ri (t + Δt) = ri (t) + Δt vi (t) + (Δt2 /mi ) fi (t) + O (Δt3) vi (t + Δt) = vi (t) + (Δt /2mi ) [fi (t) + fi (t + Δt)] + O (Δt 3) The actual Verlet algorithm can derived from this. Though this algorithm is efectively simple, but this is actually the most widely used algorithm in Molecular dynamics. This scheme preserves volume in Phase-space.
  • 19. 19 Upto this all the types of algorithms discussed are for microcannonical ensemble(NVE). But, to study a real system, we have to study cannonical ensemble (NVT) or NPT ensemble. I) Connonical Ensemble (NVT) a) Andersen Thermostat At constant temperature, the system is assumed in contact with a heat bath of temperature T. So there is energy fuctuations present. To resolve this stochastic forces are used which act on atoms of the sampleto change their kinetic energy by collision. So, in practical case, to generate the required equation of motion of N- particles for NVT ensemble, in Andersen Thermostat, the hamiltoniun is supplimented with the stocastic collision term, which is an instantenious event which afects the momentum of the particle. This collision occurs in Poisson distribution.
  • 20. 20 b) Langevine Themostat This kind of thermostat also mimics the coupling of the system to a heat bath. In this case the Hamiltoniun is modifed with a ‘random force’ (much like stochastic one in Andersen thermostat) and a deterministic “frictional force” proportional with particle velocities. This terms are actually connected with Langevine’s fuctuation -dissipation equation. Along with these two, there is another thermostat known as Lowe- Andersen thermostat, which is a modifed version of Andersen thermostat.
  • 21. 21 II)Isothermal – isobarical ensemble (NPT) Andersen Barostat At constant temperature and pressure, the energy, pressure and enthalpy of an N-particle system fuctuate. So, to simulate such system, these fuctuations should be simulate. So, in this case, the Hamiltoniun is modifed by adding instanteneous stochastic collistion. And in betwwen two stochastic collision, the state of the sytem evolves according to it’s corresponfding EOM like following-- * Image is collected from Molecular dynamics simulations at constant pressure and/or temperature by H.C. Andersen Where, ρij is a dimentionless number equals to rij /V1/3 Q is a variable which can be interpret as volume, more precisely it can be eiplained as the coordinate of the piston. This whole equation is for scaled system, whose momentum conjugate of Q is Π And ᴫi is the momentum conjugate pi
  • 22. 22 A short trip to History of MD simulation : ● In 1956, Alder and Wainwright frst reported molecular dynamics (MD) simulation of hard spheres. (1958: First X-ray structure of a protein) ● In 1960, Vineyard group did the simulation of damaged Cu crystal. Now, there comes a breakthrough in liquid simulation using MD...... ● In 1964, Aneesur Rahman did MD simulation of liquid Ar using L-J potential ● Later, in 1971, Rahman and Stillinge worked on MD simulation of water. In pot 80’s, the etended hamiltonian methods were introduced in the feld of molecular dynamics simulatuion and some of the pioneering works are --- ● In 1980, by H. C. Andersen on MD method for NPH, NVT, NPT ensembles. ● In 1986, R. Car and M. Parrinello’s work on Ab initio MD (includes electronic degrees of freedom)
  • 23. 23 NAMD and molecular dynamics simulations: In NAMD simulations, some emperical force felds are used, that approiimate actual atomic forces in biopolymer or biological systems. For large systems, where lots of atoms are involved, so simulations require long time . So, to reduce the computational cost , NAMD is used , which uses parallization of computer. Nanoscale Molecular Dynamics (NAMD) is a perticularly well suited software, very much useful to workstation clusters as well as paraller computers and very efcient in running the MD simulation. It is written using the Charm++ parallel programming model, using CHARMM as force feld. But, it also can use other force felds like AMBER, GROMACS.
  • 24. 24 Now before, proceeding to NAMD, let’s see what’s a ‘Force feld’ ?? A force feld, is basically refers to -- The functional forms, with respect to a computer simulation, which is used to describe the intra- and inter-molecular potential energy of a collection of atoms, and the corresponding parameters that will determine the energy of a given confguration. These functions and parameters, in force feld, have been derived from eiperimental work on molecules and from accurate quantum mechanical calculations. They are often refned by the use of computer simulations to compare calculated condensed phase properties with eiperiment.
  • 25. 25 The general equation of force feld is given below-- Where the frst term corresponds to the sum over all bonds, with an equilibrium bond- length r0 and there is one term for every pair ij of directly connected atoms. The second term is a sum over all bond angles and there is one term for each set of three connected atoms ijk and it usually has a quadratic form. The third term is the sum over all torsions involving four connected atoms ijkl. In principle, this is an eipansion in trigonometric functions with diferent values of n, the multiplicity (i.e. the number of minima in a rotation of 2π around the j–k bond),many force felds fi n = 3. This term can also include improper torsions, where the four atoms defning the angle are not all connected by covalent bonds. The fourth term is a sum over the non-bonded interactions (between molecules and within molecules). In particular, it describes the electrostatic and repulsion–dispersion interactions.
  • 26. 26 In previous slide, it has been informed that, NAMD is a well suited software for molecular dynamics simulation . Then, it must have some important features, which make it suitable. Lets know those..... • Force Field Compatibility • Efcient Full Electrostatics Algorithms • Multiple Time Stepping • Input and Output Compatibility • Dynamics Simulation Options • Easy to Modify and Extend • Interactive MD simulations • Load Balancing
  • 27. 27 1. Force Field Compatibility: In NAMD , the force feld used, is same with that of, what is used in the programs CHARMM and X-PLOR. The speciality in this case is that – this force feld includes the local interaction terms consisting of bonded interaction between the 2nd , 3rd , 4th atom and also the pair wise interactions including electrodstatic and van der Waals forces. 2. Efcient Full Electrostatics Algorithms: NAMD incorporates the Particle Mesh Ewald (PME) algorithm, to account the full electrostatic interactions. This algorithm reduces the computational compleiity of electrostatic force evaluation from O(N2 ) to O(N log N). 3. Multiple Time Stepping In molecular dynamics, the velocity Verlet algorithm is used to advance the positions and velocities of the atoms in time for it’s simplicity and stability . To further reduce the cost of the evaluation of long-range electrostatic forces, a multiple time step scheme is employed. The local interactions like, bonded, van der Waals and electrostatic interactions (within a specifed distance, as it’s a long range force) are calculated at each time step.
  • 28. 28 4. Input and Output Compatibility The input and output fle formats used by NAMD are easy to understand. Input formats include coordinate fles in PDB format , structure fles in X-PLOR PSF format, and energy parameter fles are correspond to CHARMM forcefeld. 5. Dynamics Simulation Options MD simulations may be carried out using several options, including – Constant energy dynamics – Constant temperature dynamics via ∗ Velocity rescaling ∗ Velocity reassignment ∗ Langevin dynamics – Periodic boundary conditions – Constant pressure dynamics via ∗ Berendsen pressure coupling ∗ Nos´e-Hoover Langevin piston – Energy minimization, – Fiied atoms – Rigid waters – Rigid bonds to hydrogen – Harmonic restraints – Spherical or cylindrical boundary restraints.
  • 29. 29 5. Easy to Modify and Eitend Eitensibility and maintainability is a primary objective for NAMD. In order to achieve this, NAMD is designed in an object-oriented style with C++ language. NAMD’s modular design allows one to integrate and test new algorithms easily. 6. Interactive MD simulations A system undergoing simulation in NAMD may be viewed and altered with VMD. So, the user must have requisite knowledge of both VMD and NAMD. 7. Load Balancing An important factor in parallel applications is the equal distribution of computational load among the processors. NAMD uses a simple uniform spatial decomposition where the entire model is split into uniform cubes of space called patches. An initial load balancer assigns patches and the calculation of interactions among the atoms within them to processors such that the computational load is balanced as much as possible. During the simulation, an incremental load balancer monitors the load and performs necessary adjustments.
  • 30. 30 To run a simulation what types of fle we need?? 1) pdb (Protein Data Bank) fle, where atomic coordinates and/or velocities of system are recorded. 2) psf (Protein Structure File), where the structural informations of the protein ate stored. 3) Force feld parameter fle, like CHARMM, AMBER, GROMACS 4) Force feld topology fle, containing information on atom types, charges etc. 5) NAMD confguration fle
  • 31. 31 Generate or download A pdb fle( remember that pdb doesn’t contain H) Using proper tcl script and Running it with psfgen generate Another set of new pdb and psf fle with coordinates of H-atom Incorporate water, using “ SOLVATE” package in VMD tk-console, for the system or ,by writing proper script in Tcl, with respet to the system size. So that the molecule be completely immersed in liquid . Create a confguration File with required parameters Run with NAMD. Get the diiferent types of output fles and start ANALYSIS Flow Chart for running NAMD
  • 32. 32 Step -1 Step -1 Generating pdb fle PDB fles are indispensable parts for running MD simulations of any system. They can be created or downloaded from http://www.pdb.org . For small molecules, one can create PDB by generating structute in any type of Molecular visualization Software like Avogadro, GaussView etc. . This is an eiample of PDB fle for the protein Ubiquitin
  • 33. 33 Step -2 Step -2 Generating PSF fle PSF fles are also very important for properly viewing a molecule and doing it’s MD simulation. PSF’s can be generated either by using VMD’s inbuild psf-generator “ Automatic psf Builder”, or by writing proper tcl script which can be run using “psfgen” obtained with NAMD software package. This is the psf fle generated by psfgen software using tcl script. It is important here to mention that, in psf fle the residue-ID’s are according to the force feld topology fle, which is used to generate the psf fle.
  • 34. 34 Now, we have a pdb fle and a psf fle.. Ubiquin with all H-atom which was actually missing in pdb fle. Highlighting diferent secondary structures of Ubiquitin.
  • 35. 35 Step-3 Step-3 Incorporating water molecules around the protein molecule Water addition is important to study dynamics of a biomolecule. This can be done by two ways -- I. By using proper tcl script II. By using “solvate” package in VMD tk-console But, before putting the molecule in a boi of water,the size of the boi must be checked, so that the molecule doesn’t interact with it’s image in the neit cell(From Periodic boundary Condition requirement). To build up a proper system, the total system must be neutral. So, if the protein molecule carries eiess charge, then some ions must have to be added to the system to make a proper, neutral and stable system.
  • 36. 36 Command: This command is written on VMD tk-console. After solvation complete--
  • 37. 37 After addition of water boi, the system looks like this ---
  • 38. 38 There is another thing called “Implicit Solvent” Sometimes for large molecules, using requisite number of solvent make the system- size so large, that speed and computationbnal cost becomes an issue. To solve this problem “Implicit Solvent Model” is used. In this method, the efects of solvent is included in the inter-atomic force calculation, and thereby nullifng the need of eiplicit solvent molecules. This method is designed to behave like original solvent. Like, polar solvents, which act as dielectric and screens the electrostatic interactions, so their cooresponding implicit solvent model behave same ways. For more appropriate result using implicit solvent model, Generalized Born Implicit Solvent (GIBS) models are used.
  • 39. 39 Just one thing left to run an MD simulation-- NAMD Confguration File The NAMD confguration fle (also called a confg fle, .conf fle, or .namd fle) is the most important thing to run a correct and proper MD simulation. This is actually given to NAMD on the command line. This fle specifes virtually everything about the simulation to be done.
  • 40. 40 This is a general confguration fle ---
  • 41. 41
  • 42. 42
  • 43. 43 Now a system is ready for simulation. To run the simulation the following command should be written in command line --- namd2 ubq_wb_eq.conf > ubq_wb_eq.log &
  • 44. 44 Output fles In NAMD, there are 11 output fles generated--- 1) *.log 7) *.restart.vel 2) *.coor 8) *.restart.xsc 3) *.vel 9) *.restart.coor.old 4) *.xsc 10) *.restart.vel.old 5) *.dcd 11) *.restart.xsc.old 6) *.restart.coor
  • 45. 45 Informations about the output fles --- ● Among this 11 fles, there are seven fles contain binary data. These are : *.coor , *.vel, *.dcd, *.restart.coor, *.restart.vel, *.restart.coor.old and *.restart.vel.old ● The other 4 fles are eitended system confguration fle (*.isc, *.restart.isc, *.restart.isc.old and *.log fle) This 4 fles store the periodic cell dimensions of the system and the time steps along with the informations asked in confguration fle.
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