molecular dynamics summary

8/10/2019 Molecular Dynamics Summary

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of molecules What is going on at the atomic level?

January 14, 2015

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The life of molecules

Calculating whats going on at the atomic level

Florian HseandJose Ramon Villanueva Valencia

Clubes de CienciaGuanajuato - Enero 2015

Contents

1 Integration Algorithms 2

1.1 Principles of Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Numerically calculating forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Forward Difference method . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Central Difference method . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Euler Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Verlet Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Leap Frog Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Force Fields 9

2.1 Important Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.4 Dispersion interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.5 Pauli repulsions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.6 Chemical bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Simple Force Field expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Treatment of non-specific interactions . . . . . . . . . . . . . . . . . . . . . . 15

3 Environmental Controls 18

3.1 Thermodynamic ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Heat bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Andersen thermostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 Berendsen thermostat . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.3 Extended Nose-Hoover thermostat . . . . . . . . . . . . . . . . . . . . 213.4 Pressure control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.1 Berendsen barostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.2 Andersen barostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Principles of diffusion 26

5 Bash commands 27

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1 Integration Algorithms

The key element of every Molecular Dynamics Engine is a stable integration algorithm thataccounts for correct time propagation of the system. As typical MD simulations propagate asystem for about2 fsper step but simulation times of the order of1 sshall be achieved, the

error per step must be very small to assure simulating a physically correct behavior.

1.1 Principles of Classical Mechanics

The first thing to think about when trying to compute the behavior of many particle systemsis how to propagate the system in time. For the following considerations we assume we wouldhave some magical black box method that provides us with the potential every single particlefeels. Additionally we want to assume that the particles behave classically, thus neither spe-cial relativity nor quantum mechanics need to be taken into account when propagating thesystem in time.

Having knowledge about the potential at the position of a particle as well as the close sur-rounding allows for calculating the effective force acting on the particle. Assuming the manyparticles system consists ofN particles, each having coordinates and velocities of

ri =

xiyi

zi

and vi =

vxivyi

vzi

wherei is an index indicating the ith particle the effective force on theith particle Fi(r)canbe calculated via the negative gradient of the potential

Fi(ri) = U(r

i)

ri = U(ri) FxiFyi

Fzi

= xUyUzU

Having computed the forces on the ith particle Newtons second law can be used to calculatethe acceleration of the particle caused by the potential. In order to do so the inertial mass ofthe particle is needed as a parameter of the model.

Fi(ri) =miai(ri)

FxiFyi

Fzi

=mi

axiayi

azi

Acceleration is defined as the rate with which velocity changes over time. Additionally, velocity

is the rate with which the position changes over time. Hence, acceleration is the second totalderivative of position with respect to time. With this relation, the change in position of aparticle can be related to the potential the particles feels at a given time t which allows forestimating the particles position at time t + t.

d2

dt2ri =

1

miFi(ri) =

1

mi

U(ri)

ri

Then only thing we still need to think about is how to integrate this equation of motionefficiently. This is what integration algorithms can be used for.

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1.2 Numerically calculating forces

Before we start thinking about integration algorithms to propagate the system in time we willtry to numerically calculate the spacial derivative of the potential to obtain the forces. Forsimplicity we well consider this problem first in only one dimension and later generalize it to

three dimensions.

1.2.1 Forward Difference method

A very simple method to numerically calculate derivatives can be developed by consideringthe formal definition of the first derivative f(x) of a given function f : x f(x). The firstderivative offat position x = a is defined as

f(a) = limh0

f(a + h) f(a)

h

This expression is also called Differential Quotient. Instead of calculating the DifferentialQuotient, the forward difference method approximates the first derivative by only computingthe Difference Quotient

Definition 1 Given an arbitrary functionf :x f(x), the forward difference first deriva-tivef

FD at a given positionx= a is given by

fFD(a) =f(a + h) f(a)

h

for some constant h > m, where m is the machine precision of the machine used for thecomputation.

Of course this approximation of the first derivative yields a certain error. The error can beestimated by Taylor expanding the expression for the forward difference derivative

fFD(x) =f(x + h) f(x)

h =

f(x) + hf(x) + (h2/2) f(x) + . . . f(x)

h=f(x) + (h/2) f(x) + . . .

For small h the error in this approximation is mostly determined by h itself and the secondderivative of the considered function. Assuming that the function is well-behaved, meaningthat it is sufficiently smooth, every derivative to any order is of about the same order of

magnitude as the function itself. Thus, we may conclude that the error is mostly determinedby the choice ofh.The trade-off in this method as in most of the methods for calculating first derivatives is thaton one hand h cannot be chosen arbitrarily small as we need to be significantly above themachine precision but on the other hand h should be small enough to sufficiently decrease theerror in the computation.It can be shown that for double precision the minimum error being the sum of the round-offerror and the approximation error of 3 108 is achieved for h 4 108

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1.2.2 Central Difference method

The previously discussed forward difference method for approximating the first derivative of afunction can be improved by not considering a given function fat the position the derivativeshould be calculated at and the value of the function in some arbitrary distance h from this

point but rather taking into account the surrounding of this function f around position a.This idea is depicted in figure1. As opposed to the forward difference derivative, the centraldifference derivative is defined as

Definition 2 Given an arbitrary functionf :x f(x), the central difference first deriva-tivef

CD at a given positionx= a is given by

fCD(a) =f(a + h/2) f(a h/2)

h

for some constant h > m, where m is the machine precision of the machine used for thecomputation.

It can be shown that the error made by this method is significantly smaller than the errorof the forward difference method. In order to do so the defining expression is again Taylorexpanded

fCD(x) =f(x + h/2) f(x h/2)

h

=f(x) + h2f

(x) + h2

8 f(x) + h

3

48f(3)(x) + . . . f(x) + h2f

(x) h2

8 f(x) + h

3

48f(3)(x) . . .

h

=f(x) + (h2/24) f(3)(x) + . . .

Again, following the argument that any arbitrary derivative of the function should be of aboutthe same order of magnitude the error if this method is in the order ofh2, which is expectedto be much smaller than h for small h. It turns out that the minimum error of this method,being the sum of the round-off error and the approximation error, is about 3 1011 andcan be achieved for h = 3 105.Having derived two different methods for calculating the first derivative of a given functionat a given position, the central difference method provides already quite good results. Thereare many other methods with even higher accuracy like the extrapolated difference methodbut these methods typically require more potential evaluations. Since potential evaluationsare the bottleneck of MD simulations we want to maintain as few potential evaluations aspossible. Hence, we will stick with the central difference method in our application.

1.3 Euler Algorithm

Knowing how to calculate forces and thus accelerations on a particular particle given thepotential in a small surrounding of the particle we are, in principle, able to solve Newtonsequation of motion. However, doing this numerically yields some challenges that need to beovercome.In order to fully describe one particle in phase space both its position and its momentumhas to be known. Thus, any algorithm should be capable of computing both position andmomenta for future times. The positionr(t + t)and velocity v(t + t)of a particle at some

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(A):f(x)

xa a + h

fFD(a)

f(a)

f(x)

(B):f(x)

xa a + h/2

fCD(a)

f(a)

f(x)

a h/2

Figure 1: Derivative Methods: Depicted are the basic ideas behind the forward differencemethod (A) and the central difference method (B). Both can be used to approximatethe first derivative of a given function but based on the definition of the methods thecentral difference method performs significantly better than the forward difference

method even though the number of evaluations is the same.

timet +t can be calculated from the previous phase space point (r,v)(t) by simply Taylorexpanding the expressions for later times.

r(t + t) =r(t) + tdr(t)

dt +

1

2(t)2

d2r

d2t + . . .

v(t + t) =v(t) + tdv(t)

dt + . . .

Note that truncating the sums after any arbitrary order oft is exact for infinitesimalt. Theoccurring derivatives can be computed from the underlying physics

dr(t)

dt = v(t)

d2r(t)

dt2 =

dv(t)

dt =a(t) =

F(t)

m

where F(t) are the forces acting on the particle at time t. A numerical way to calculate thenext position and velocity is now to consider a finite t t but still truncate the sumsof the Taylor expansion. This way a certain error is made, that is smaller the smallert ischosen. However, in order to achieve interesting time scales within a reasonable simulationtimet should not be chosen too small.Given the position rN and velocity vN at step Nof the simulation, position and velocitiescan be updated via the following scheme

Definition 3 Euler Algorithm

rN+1= rN+ tvN+(t)2

2

FN

m

vN+1= vN+ tFN

m

Note that effectively there is only one force evaluation required for this integration algorithm.Nevertheless, the error made by this algorithm in the position of the particle if of the orderof(t)3 as this is the first order that was neglected in the Taylor expansion.

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The here presented Euler method is known to be a quite inaccurate integration algorithmand should only be used as an academic example. Even though the Euler method can becomparably fast, this algorithm is rather unstable.

1.4 Verlet Algorithm

A more sophisticated method than the Euler Algorithm is the Verlet Algorithm. Similar tocomputing the first derivative with the central difference method the Verlet Algorithm doesnot only calculate the next position by making use of the current position but rather of thecurrent and the previous location of the particle. By including more information into thecalculation the algorithm becomes more stable and the error decreases. A formal derivationof the Verlet Algorithm might look as follows. We consider the position of the particler attimet + t and t t and Taylor expand both expressions.

r(t + t) =r + tdr(t)

dt +

1

2(t)2

d2r(t)

dt2 +

1

6(t)3

d3r(t)

dt3 + . . .

r(t t) =r t dr(t)dt

+12

(t)2 d2r(t)dt2

16

(t)3 d3r(t)dt3

+ . . .

Both expressions can now be added and slightly rearranged, which leads to

r(t + t) = 2r(t) r(t t) + (t)2d2r(t)

dt2 + . . .

Again, note that truncating the sum at any arbitrary order oft still yields a true equationfor infinitesimal t. However, for numerical simulations a finite t t is required. Byconsidering the current and the previous position of the particular an equation could bederived, which is exact up to terms of(t)4, which is already an order of magnitude better

than the Euler Algorithm.Substituting the second time derivative of the position with the underlying physics the VerletAlgorithm can be effectively used to determine the position rN+1 of a particle at step N+ 1of a simulation.

Definition 4 Verlet Algorithm

rN+1= 2rN rN1+ (t)2FN

m

Note that the Verlet algorithm does not explicitly include velocities. However, velocities canbe calculated using the basic idea of the central difference method

r(t + t) r(t) + tv(t) r(t t) r(t) tv(t)

Subtracting the two equations and rearranging the expression leads to

v(t) = r(t + t) r(t t)

2t vN =

rN+1 rN12t

which allows for subsequent computation of the velocities up to an error of(t)2 being veryimportant for thermodynamically controlling the system as we will see later on.

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1.5 Leap Frog Algorithm

In contrast to the Euler Algorithm which propagates positions and velocities one the sametime discretization and opposed to the Verlet Algorithm which does not explicitly includevelocities at all the Leap Frog Algorithm combines advantages of both previously discussed

methods, namely explicitly calculating velocities as well as an error of(t)3

in the positions.The basic idea behind the Leap Frog algorithm is to propagate positions and velocities notat the same points in time but rather along time points which are shifted byt. The math-ematical reasoning behind this idea are widely used midpoint rules which were successfullyapplied to many problems such as numerical integration. Hence, in correspondence to theequations of the Euler Algorithm the Leap Frog Algorithm follows the basic scheme of

r(t + t) =r(t) + v(t + t/2)t

v(t + 3t/2) =v(t + t/2) +F(t + t)

m t

In order to calculate the velocities at times t + nt withn being a natural number, one can

perform a half step integration

v(t + t/2) =v(t) +1

2

F(t)

m

v(t + 3t/2) =v(t + t) +1

2

F(t + t)

m

Using these two equations the Leap Frog algorithm can be formulated in the following way

Definition 5 Leap Frog Algorithm

rN+1= rN+ vNt +(t)2

2

FN

m

vN+1= vN+ t

2m[FN+ FN+1]

Note that even though the force appears to be evaluated twice during one iteration the num-ber of force evaluation reduces to one by storing the previously calculated results. Hence, theLeap Frog Algorithm provides also only one force evaluation.In addition it can be shown that the Leap Frog Algorithm performs a symplecticintegration,meaning that phase space volumes are conserved by this algorithm. This feature arises fromthe fact, that the algorithm propagates positions and velocities on independent time grids.As the Hamiltonian of a single particle consists of one momentum dependent kinetic term andone position dependent potential term, the total energy does not change when propagating asystem with this algorithm. Having this inbuilt feature of energy conservation we are alreadycapable of simulating many particle systems in the microcanonical ensemble.

1.6 Comparison

Aside from the already discussed numerical integration techniques there are many more in-tegration algorithms that for example include higher orders in the Taylor expansion or show

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2 Force Fields

Considering a many particle system with interacting particles, the interactions between theparticles have to be modelled sufficiently accurate to assure correct simulation of the behav-ior of the system. However, to keep the algorithms simple and the parameter space small,

only important interactions shall be included into the model. In order to identify relevantinteractions in biological systems the order of magnitude of certain interactions in the regimebiomolecules exist in will be estimated first to then neglect interactions that are sufficientlyweak compared to others. Based on these considerations, an analytical expression for thepotential energy of a many particle system will be derived.

2.1 Important Interactions

In the following we want to investigate the order of magnitude of particular interactions amongparticles in a biological many particle system as well as interactions between these particles

and their surrounding environment.Before actually calculating typical forces caused by prevalent interactions in biological systemswe should think about how to define strong and weak forces. In order to do so a measurementto which the forces can be compared to is required. Since biological systems exist at ratherconstant temperatures of the order of300 K the canonical ensemble is well suited to describethe thermodynamics of biological systems. Being in the canonical ensemble the system iscoupled to a heat bath while energy between the heat bath and the system can constantly beexchanged. The relevant thermal energy is

Etherm= 1kBT 4.144 pN nm

The size of one atom is typically about 1 = 0.1 nm. Biological relevant proteins and other

biomolecules usually consist of more than 1000 atoms which allows for taking 1 nm as thecharacteristic length of biomolecules. As the system is coupled to a heat bath with a tem-perature of300 K relevant forces must be at least in the order of a few Piconewton. Forcesless than1 pNare usually to weak to cause any actions while forces stronger than1 pNmightinfluence the system.This consideration allows for estimating whether or not certain forces caused by particularinteractions should be taken into account when modelling the interactions relevant for simu-lating biomolecules.

2.1.1 Electrostatics

Biomolecules such as proteins and DNA can, in principle, carry a charge. Aside from that,also within a biomolecule charge distributions are present. For example, there are positivelyand negatively charged amino acids and the phosphate in the backbone of DNA are negativelycharged as well.Maxwells equations tell us that charges cause an electric field which again affects surroundingcharges. The electric field comes along with a potential so that charges in the potential havesome potential energy which might influence the movement of the charges.

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The force of a charge qi in the electric fieldEj caused by another charge qj follows the simplerelation

F = qiEj

Modelling the chargeqj as a point charge1 the electric field and thus the force can be expressed

as follows

Ej = qj4r2ij

F = qiqj4r2ij

Considering the negatively charged phosphates in the backbone of DNA as an example, thedistance between the charges rij is about 2 nm while each phosphate carries one elementarycharge. Given these numbers, the force between the two phosphates is of the order of

Felect 7 pN

Here we see, that this force is well within the order of magnitude of forces that should be taken

into account. Hence, electrostatic interactions are indeed important to model the behavior ofbiomolecules.

2.1.2 Magnetostatics

Nuclei in general carry a magnetic moment in addition to their electric moment. The nu-clei with the greatest magnetic moment are protons, which are present in large number inbiomolecules due to the large amount of hydrogen of which biomolecules comprise. Opposed tothe electrostatic scenario, a magnetic moment B in the magnetic field B of another magneticmoment feel a torque acting on them. The torque depends on both the magnetic moment

and the magnetic field and can be expressed as the product of a force Fand a correspondinglever arm r .

M=B B = F r

Considering the magnetic moment of a proton B = 1.4 1026 Nm/T and a lever arm of

10 nm, the resulting force in a super-strong 1 T magnetic field is

Fmagnet 1.4 107 pN

Magnetic fields are usually weaker than the considered one and the force even with such astrong magnetic field is much smaller than forces which are relevant for biological systems.

Hence, magnetic interactions do not need to be modelled in a Molecular Dynamics simula-tion.

1This assumption is not too bad because the diameter of the nuclei negligibly small compared to Bohrs

radius.

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2.1.3 Gravity

Considering the forces on particles caused by gravity we need the typical mass of one of theparticles. Considering an entire protein, typically consisting of about 300 amino acids, wearrive at a typical mass of about

mprotein,typical 60 1021 g

This mass already appears to be quite small. Furthermore, making the (very accurate) as-sumption that the gravitational potential for proteins on earth does not change along the sizeof the protein, the force acting on the entire protein is of the order of

Fgravitational,typical= mprotein,typical g 6 1010 pN

where g = 9.81 ms2 is the gravitational acceleration. Since gravitational forces are muchsmaller than the relevant force scale gravity can be neglected in any Molecular Dynamicssimulation.

2.1.4 Dispersion interactions

Dispersion forces are intramolecular forces arising from quantum-induces spontaneous po-larization multipoles in molecules. A simple (but not accurate) picture of these types ofinteractions is to think about two atoms with their electron clouds around their nuclei (com-pare figure3). Assuming the existance of fluctuations, one electron cloud of one atom mightdeviate such that the charge centers of positive and negative charges no longer coincide. Thus,a dipole was created2. The other atom is now located in the electric field of a dipole. As theelectron cloud and the nucleus react differently to the just arisen dipole the charge centers ofthe other atom also separate creating another dipole.Eventually there are two dipoles interacting through their electric fields. This classical pic-ture allows for correct modelling of the spacial dependence of this interaction even thougha quantum mechanical description might be more appropriate in this case as it respects thefinite number of electrons and the wave nature of the electrons.Nevertheless, both descriptions predict a potential energy in the form of

Edisp= 3

2

IAIBIA+ IB

AB

R6

Here,Iindicates the first ionization potentials of the atoms and their dipole polarizabilities.Note that this interaction is attractive due to the minus sign and that the spacial dependence

goes as r

6.Dispersion relations are the reason why non-polar molecules stick together even though sep-arated states would be favored entropically. Hence, the assumption that dispersion relationsplay an important role for biomolecules seems valid.

2Considering the finite number of electrons around a nucleus even higher order multipoles can easily be

created but higher order multipoles are typically weaker than dipoles so that they are neglected in this

case.

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A: B: C:

Figure 3: Dispersion Interaction: Depicted is a classical picture of the dispersion interac-tion between two atoms. Nuclei are shown in red while electron clouds are shownin blue. Usually, charge centers coincide (A), but due to fluctuations charge centersmight deviate from one another creating a dipole (B). While these fluctuations areusually short lived for isolated atoms the just formed dipole can induce the creationof another dipole in another atom as positive and negative charges react differentlyto an electric field (C). A system of two induced dipoles has a much longer life timethan a dipole induced by fluctuations alone.

2.1.5 Pauli repulsions

Taking into account the quantum mechanical nature of electrons and nuclei, another in-teraction arises from a very fundamental quantum mechanics principle, the Pauli exclusionprinciple. This principle basically states that two identical fermions3 cannot under any cir-cumstances occupy the same quantum state.Electrons are particles with spin 1/2 and thus fermions. The quantum state of an electronin an atom is defined by the orbital the electron occupies and its spin. Due to the Pauliprinciple, electrons can only occupy the same orbital if they have different spins. Since theonly spin states that are allowed for the electron are +1/2 and 1/2 there can only be twoelectrons per orbital.One important consequence of this principle is that atoms cannot come arbitrarily close to

one another. Since orbitals occupy a certain space around the nuclei the orbitals of two atomsstart to overlap if they get closer and closer. However, electrons with the same spin cannotoccupy the same orbital and thus be in the same space region. This results in an effectiverepulsion.There is no closed expression for this type of repulsion. However, some characteristics of thisrepulsion are known. The repulsion appears on a small length scale but is very strong4. Onepopular mathematical expression used to model this phenomenon is an energy penalty termof the following form

Eexcl = K

r12

whereK is some arbitrary constant. Other methods model the Pauli repulsion with exponen-tial functions or just a hard wall.

2.1.6 Chemical bonds

In biomolecules, many chemical bonds are present. As biomolecules mostly consist of carbon,oxygen, nitrogen, hydrogen, phosphate and sulphur, the chemical bonds are mostly either

3Fermions are particles which carry a half-integer spin4In fact, the repulsion is infinitely strong as the Pauli principle is an exclusion principle.

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of300 K do not change this angle drastically. Taylor expansion of the (unknown) energylandscape yields

E(0+ ) =E(0) + dE(0

d

=0+

1

2()2

d2E(0)

d + . . .

Truncating the sum again allows for modelling the angular potential by a harmonic spring.In addition due to the three dimensional nature of orbitals there is also another term thatneeds to be added for four particle interactions which accounts for the dihedral angle betweenthe four particles. As this term arises from the repulsion of orbitals there is not only but localenergy minimum but many of them. The dihedral angle energy is most commonly modelledvia

E() =n

k(1 + cos(n 0))

Note that this term is rather empirical.

2.2 Simple Force Field expressions

The prior estimates and approximations of prevalent interactions relevant for biological sys-tems now allow for the derivation of an analytical expression which aims to accurately modelthe physics underlying the dynamics of molecules. In general, interactions can be divided intospecific interactions and non-specific interactions. Specific interactions occur between atomsof the molecule that are sequentially close to one another and have chemical bonds estab-lished. Non-specific interactions however potentially occur between all atoms in the systemeven though they are typically larger if the distance between the considered atoms is small.

Coulombic interactions, dispersion interactions and interactions caused by the Pauli exclusionprinciple are among non-specific interactions. To correctly model higher multipole interac-tions, atoms are typically assigned a partial charge corresponding the electron distributionaround the atoms. Dispersion interactions and repulsions are often consolidated to the socalled Lennard-Jones Potential

VLJ(r) = 4

r

12

r

6wherer is the distance between the particles and and are parameters defining the positionand the depth of the minimum.Combining all relevant interactions into one potential energy term V leads to

V =

bonds

kbond(r r0)2 +

angles

kangle( 0)2 +

dihedrals

kdihedral(1 + cos(n 0))

+

charges

qiqj4r

+

particles

4

r

12

r

6

This expression captures the most basic features of Molecular Dynamics even though thereare already many parameters included. The parameters can either be obtained directly fromexperiments or by trying to fit thermodynamic predictions of Molecular Dynamics Simulationto experimental results and correcting for the parameters in the MD model.

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Additionally, there are endeavors to improve force fields by taking into account more com-

r

LJ, Coulomb

Figure 5: Molecular Dynamics Force Field: A simple version of a Molecular DynamicsForce Field consists of specific interactions (red) and non-specific interactions (or-ange). Specific interactions are those that try to mimic the physics of chemicalbonds while non-specific interactions arise from general electrostatic and quantummechanical principles.

plicated interactions. For examples, some force fields allow for changes in the polarization ofmolecules.

2.3 Treatment of non-specific interactions

Even though non-specific interactions can be well modelled by the previously derived expres-sions there is one computational problem arising from the nature of non-specific interactions.Considering typical biomolecule of about10000atoms there are many interactions to calculateduring one force evaluation.In the case of the Lennard-Jones potentials however we deal with a quickly decaying potential(VLJ r

6). This seduces to only consider spatially close atoms for calculating the forcecaused by the Lennard-Jones potentials of other atoms as on one hand atoms far away donot significantly contribute to the total potential and moreover the change in the potential

a large distances is very small and can be neglected. Considering the overall shape of theLennard-Jones potential of a carbon atom (compare figure 6) it can be deduced that a savecut-off distance is larger than about 10. Studies have shown that smaller cut-offs produceartefacts in the simulation.

Instead of simply truncating the potential a slightly more sophisticated method is to shiftthe potential at a defined shift distance to a potential that decays to zero in within a finitedistance. Note that the shifting should be smooth and differentiable to avoid artefacts.The case of the Coulomb potential however is slightly more complicated as the decayingpower is much smaller (VCoulomb r

1). Nevertheless, cut-offs are considered a fast methodto model Coulomb interactions. Typically cut-off distances are chosen slightly larger (12)

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Figure 7: Neighbor Lists: Whenever the force on one particular particle (red) needs to becalculated, certain cut-offs can be applied to reduce the cost of the force evalua-tion. The contribution of particles within a specified cut-off (green) is taken intoaccount while all others are ignored for the force evaluation. However, calculatingdistances between particles is costly and scales badly ( N2) so that a neighborlist is defined containing all particles that are within the force evaluation cut-off(green) and particles which might enter this zone as long as the neighbor list is notupdated (yellow). Applying this method distances between all atoms only need tobe calculated for every update of the neighbor list.

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Note that vi is a one dimensional velocity. Using this expression and knowing the degrees offreedom temperature can be measured numerically during a simulation.Based on this measurement methods can be applied which correct for the temperature anddrive it towards a desired reference temperature.

3.3.1 Andersen thermostat

The Andersen thermostat follows the idea that the velocities of an idealized gas follow aMaxwell-Boltzmann distribution. This distribution is given by

f(v) =

m

2kBT

34v2e

mv

2

2kBT

and can be derived from Statistical Mechanics. When simulating a many particle systemcontrolled by the Andersen thermostat the particles are propagated in time unperturbedaccording to Newtons laws. However, collisions occur at uncorrelated Poissonian distributed

timest, where

P(t) =exp(t)

and is the collision rate. Whenever a collision occurs, the velocity of the colliding particle israndomly drawn from the Maxwell-Boltzmann distribution at the desired reference tempera-ture.Since collisions occur randomly and uncorrelated and velocities are drawn from the correctstatistical mechanics based distribution the Andersen thermostat produces a correct canoni-cal ensemble. However, as velocities are randomly drawn after every collision the Andersenthermostat is not suited for investigating dynamic processes such as diffusion.

3.3.2 Berendsen thermostat

The Berendsen thermostat attempts to control temperature by rescaling velocities at everytime step according to

vnew= vold

1 +

ttime step

T0T 1

1/2It is apparent that this method drives the system towards the desired reference temperaturewith a damping constant . Nevertheless this method is less thermodynamically profound.

It can be shown that the Berendsen thermostat does not necessarily resemble a canonicaldistribution.Despite this great disadvantage the Berendsen thermostat is widely applied for equilibra-tion purposes as it can relatively fast adjust the temperature. Furthermore, the Berendsenthermostat perturbs the system only weakly.

3.3.3 Extended Nose-Hoover thermostat

A more sophisticated thermostat based on extended Lagrangians is the Nose-Hoover thermo-stat. For a simulation of a system with this thermostat the Lagrangian of the system contains

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additional, artificial coordinates which are used for the temperature control. Considering theusual Lagrangian L(r,v of classical mechanics

L(r,v) =T(v) V(r)

with T(v) being the kinetic energy and V(r) being the potential energy. The momentaassociated with r can be obtained via

p=L(r,p)

v

To construct isothermal Molecular Dynamics one can introduce an additional coordinate s inthe Lagrangian of the many particle system

LNose(r,v) =i

mi2

s2v2i V(r) +Q

2s2

g

ln s

The parameterQ acts as an effective mass for the associated coordinate s. The corresponding

momenta are

pi=LNose

v =mis

2vi

ps=LNose

s =Qs

The Hamiltonian of the extended system is then

HNose(r,p) =i

p2i2mis2

+ U(r) + p2s2Q

+ gln s

Considering the extended system a micro-canonical ensemble with 6N+ 2degrees of freedomis generated. The partition function of this ensemble can be written as

ZNose = 1

N!

dpsdsdrdp(HNose E)

= 1

N!

dpsdsdrdp

s3N

i

pi2

2mi+ V(r+

p2s2Q

+ g

ln s E

were p =p/s was introduced. Defining a new Hamiltonian H(r,p) =i

pi

2

2mi+V(r) allows

to rewrite the partition function as8

ZNose = C

N!

drdp exp

3N+ 1

g H(p,r)

8In addition with have to use the following result from distribution theory

[h(s)] =(ss0)/|h(s)|

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where

C=exp[E(3N+ 1)/g]

g

dpsexp

3N+ 1

g

p2s2Q

By setting g = 3N + 1 we see that this partition function corresponds to the canonical

partition function, hence we sample in the canonical ensemble. From the Nose-Hamiltonianthe equations of motion can be derived

dridt

= HNose

pi=

pimis2

dpidt

= HNose

ri=

U(r)

rids

dt =

HNoseps

=psQ

dps

dt =

HNose

s = ip2i

2mi g

s

Using these equations of motion the system can be simulated at a desired reference temper-ature. Even though this thermostat seems to work fine in theory it can be shown that themethod remains non-ergodic for insufficient chaotic systems such as the harmonic oscillator.

3.4 Pressure control

In order to control the pressure in an Molecular Dynamics simulation we first need a measurefor the pressure of a simulated many particle system. To derive a suitable measure we willemploy the virial theorem

qkH

qk

=kBT

Each coordinate derivative in this equation is the negative component of a force fi on theithparticle, so the sum can be rewritten as

1

3

i

rirU

=

1

3

i

ri fi

= N kBT

Here, the forcefi represents both the internal forces and the external forces which are relatedto the external pressure. The effect of the container walls on the system is given by

1

3

i

ri fexti

= pV

In addition, an internal virial can be defined

1

3

i

riiU= 1

3

ri f

interi =W

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Then pressure can be related to

pV =N kBT+ W

Note that in the case of the ideal gas the internal virial vanishes.

3.4.1 Berendsen barostat

The Berendsen barostat is, just like the Berendsen thermostat, a rather simple pressure bath.An extra term is added to the equations of motion to produce a pressure change. The systemis then designed to obey the equation

p

t =

p p0tp

wherep0 is the desired pressure and tp some time constant indicating how strong the pressureis regulated. At each time step the volume of the box is then adjusted by scaling all orthogonal

box vectors by a factor x1/3, where

r =x1/3r

x= 1 Tt

tp(p0 p)

Note that T denotes the isothermal compressibility of the considered medium.

3.4.2 Andersen barostat

The basic idea of the Andersen barostat is to couple the system to an external variable V,

which denotes the volume of the simulation box. The coupling seeks to mimic the action of apiston on a real system, where the piston is designed to have mass Q and has the associatedkinetic energy

TV = 1

2QV2

The potential energy associated with this additional variable arises from the volume energycontribution

UV =pV

The potential and kinetic energies are then expressed through r and v in terms of scaledvariables

r = V1/3s

v= V1/3s

The equations of motion can then be readily obtained from the Lagrangian of the system

LV =T+ TV U UV

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Note that these methods are designed to simulate a system in the isobaric-isoenthalpic en-semble. Only the additional coupling to a temperature bath provides sampling in the desiredisobaric-isothermal ensemble.In addition to the presented environment controlling schemes there are also other, more so-phisticated methods that carry out the presented basic ideas more carefully and thus provide

better results. Nevertheless the historical starting point for correct thermodynamics in many-particle simulations were the coupling schemes presented here.

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molecular dynamics summary

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