Molecular simulations in chemistry

Adam Liwo

Room B325

adam@sun1.chem.univ.gda.pl

• 30 lecture hours

• 2 hrs/week; Tuesdays, 8:15 – 10:00 am

• Completion requirements• Project• Exam

Scope

1. Purpose, time-, and size-scales of molecular simulations.

2. Energy surfaces of molecules.

3. All-atom force fields: purpose, derivation, and parameterization

4. Treatment of solvent in force fields. Models of water.

5. Metropolis Monte Carlo.

6. Molecular dynamics.

7. Calculating ensemble-averages and error estimation in simulations.

8. Umbrella-sampling simulations and the weighted-histogram analysis method.

9. Generalized-ensemble simulations.

10.Enlarging the time- and size-scale of simulations: coarse-grained models. The CABS and UNRES force fields.

11.Thermodynamics and kinetics of protein folding from simulations.

12.QM/MM simulations.

Literature

Daan Frenkel, Berend Smit, „Understanding Molecular Simulation: From Algorithms to Applications” Academic Press, San Diego, 1996

D.C.A. Rapaport, „The Art of Molecular Dynamics Simulations”, Cambridge University Press, 1998.

A.R. Leach: „Molecular Modeling: Principles and Applications”, Pearson Education EMA, 2001.

Learning Nature – how does Science work?

Experiment

Model(equations)

Exact solution

Simulations

No model(pysicochemical

tables)

Equations (approximate) – exact solutions

„I am really longing for those good old times when a theorist didn’t need anything but a piece of paper, a pencil, and own brains”.

Quotation from a late Professor of Physical Chemistry.

Not possible anymore…unless we want to consider spherical horses in vacuo to model horse race.

Feynman’s dream that we will be able to ‘see’ the solutions of equations someday does not seem to ever come true.

Successful examples of the „exact solution” approach

• Chemical Thermodynamics (phenomenological).

• Chemical Kinetics.

• Modeling electrochemical processess.

• Quantum Chemistry.

• Kinetic theory of gases.

• Application of Statistical Mechanics in Chemistry.

What are ‘Simulations’?

Modeling (computing) the behavior of complex systems by applying a given description (e.g., Newton’s equations of motion).

‘Das ganze Tschechische Volk ist eine Simulantenbande’ – Dr. Gruenstein of K.u.K military draft office

Where do the ‘molecular simulations’ enter into play?

- Condensed systems composed of many particles (e.g.,

a protein + solvent).

- Strong interactions between system’s components.

The partition function cannot be separated.

- The time evolution has Lyapunov instability depending

on the initial conditions.

- Therefore, we actually need to compute system’s

behavior for given initial/boundary condition rather than

analyze the solutions in terms of those.

Are simulations another versionof experiment?

No, we do not deal with a real system but with a ‘virtual’ one.

However, the results depend on starting point and are subject to statistical error as the experiemental results.

(Pre) History• Lord Kelvin (early 1900’s): hand computations of hard-

sphere collisions.

• Manhattan Project (Ulam; 1940’s – 1950’s): hand and computer simulations of nuclear fission (ENIAC computer).

• J.D. Bernal (1950’s): mechanical models of liquid particles from rubber/styrofoam balls connected with metal rods.

• G. Vineyard (1950’s): computer simulation of radiation damage in crystalline Cu.

• Rosenbluth, Rosenbluth, Metropolis, Teller (1950’s): Formulation of the Metropolis Monte Carlo algorithm.

• Alder and Wainwright (1957): MD simulations of hard-sphere liquids.

Types of simulations

• Monte Carlo (MC): need only energy).

• Molecular dynamics (MD): time evolution; need forces).

• Combination thereof.

What systems do we treat and what are the limits?

Individual components

System level(Networks)

Avera

gin

g o

ver individ

ual co

mp

one

nts

PDEs to describe reaction/diffusion

Network graphs

Fully-detailed

Atomistically-detailed

Coarse-grained

QM

QM/MM

All-atom

United-atom

Residue level

Molecule/domain

level

Avera

gin

g o

ver „less im

porta

nt” deg

rees of

freed

omDescription

level

10-15

femto10-12

pico10-9

nano10-6

micro10-3

milli100

secondsbond

vibrationloop

closure

helixformation

folding of-hairpins

proteinfolding

all atom MD step

sidechainrotation

MD Package

Explicit Solvent

Implicit Solvent

AMBERa

1 fs 2 fs

CHARMMb

3 fs 4-5 fs

TINKERc

1 fs 2 fs

Time step t for some standard MD packages

a http://amber.scripps.edu/

b http://www.charmm.org/

c http:// dasher.wustl.edu/tinker/

Energy surfaces of molecular systems and their properties

From Schrödinger equation to analytical all-atom potentials

),...,,;,...,,(ˆ

ˆ

2121 nN

EH

HE

rrrRRR

elN

ba ji ijai ai

a

ab

ba

a iia

a

HH

rr

Z

r

ZZ

mH

ˆˆ

11ˆ

The Born-Oppenheimer approximation

elelelel

NelN

el

elji ijai ai

ael

N

ba ab

ba

NNN

nNelNN

nN

EH

EEE

E

rr

Z

E

r

ZZE

ˆ

),...,,(

1

)()...()(),...,,(

),...,,;,...,,(),...,),(

),...,,;,...,,(

2

22

2122

212

RRR

RRRRRR

rrrRRRRRR

rrrRRR

1

11

11

1

HNCHCN

Conversion of iso-hydrogen cyanide into hydrogen cyainde

AM1 energy hypersurface of the conversion of iso-hydrogen cyanide into hydrogen cyanide

Energy [kcal/m

ol]

HCNHNC

Transition structure

Contour plot of the PES

HCN

HNC

struktura przejściowa

HNCHCN

H

N -C

E

E╪

reaction coordinate

en

erg

y [kcal/m

ol]

Jean-Louis David

Napoleon

Propane PES as a function of the two dihedral angles

Conformational-energy map of terminally-blocked alanine

(degrees)

(deg

rees

)

**,

******2

1

****,,

2

2

2222

2

2

yyxxO

yyy

Exxyy

xy

Eyyxx

yx

Exx

x

E

yyy

Exx

x

EyxEyxE

Energy expansion about the stationary point

The derivatives are zero at a stationary point but it need not be a stable point (Coulomb’s egg problem).

2

2

22

22

2

2

***

***

2

1*,

yyy

Exxyy

xy

E

yyxxyx

Exx

x

E

EyxE

2

22

2

2

2

y

E

yx

Eyx

E

x

E

H Matrix H is termed energy Hessian

22

21

2

1

2

1*,,

*

*

*

***,

2

1*,

EEyxEyy

xx

yy

xxyyxxEyxE

T

T

V

VVH

H

V – eigenvector matrix

Neighborhood of the minimum corresponding to the HCN molecule

Neighborhood of the transition point

Energy [kcal/mol]

Case study: the układu HCN – HNC system

2

2

2

2

1

2

2

2

22

2

12

21

2

21

2

21

2

nnn

n

n

x

E

xx

E

xx

E

xx

E

x

E

xx

Exx

E

xx

E

x

E

H

**2

1

*

*

*

*,,*,*2

1*,,,

1 1

22

11

2221121

jj

n

i

n

jiiij

nn

nn

xxxxh

xx

xx

xx

xxxxxxExxxE

H

Generalization on n coordinates

*

*

*

2

1

*,,,*,,,

22

11

2

1

2

1

2222

211

2121

nn

T

n

T

n

nn

nn

xx

xx

xx

EEExxxE

Vξ

VVH

Minimum: all Hessian eigenvalues > 0

Corresponds to a stable state of a system.

First-order saddle point: 1<0, 2, …,n >0

Corresponds to the transition state in a reaction. Higher-order transition points are not interesting.

A maximum: all Hessian eigenvalues < 0.

Summary pf critical points