From Dynamics to Thermodynamics using Molecular Simulationkkrk.chem.elte.hu/molim/lectures/DvdSpoel-dynamic-to-thermodyna… · • 1000 molecules/box 10 ns production sims ... where

From Dynamics to Thermodynamics using Molecular

Simulation

David van der Spoel

tisdag 18 december 12

Computational Chemistry

Physical models to describe molecules

Software to evaluate models and do predictions - GROMACS

Model validation and development

Applications in Chemistry - Water & other liquids

(Applications in Biology - PitGP, Virus, Malaria)



Ab-initio methods

Density functional theory

Semi-empirical

Atomistic Force field

Coarse grained FF / QSAR / Continuum descriptions


Cost

Acc

urac

y

Ab Initio

DFT

Semi-Empirical

Force Field

Coarsegraining



Log(Cost)

Acc

urac

y

Ab Initio

DFT

Semi-Empirical

Force Field

Coarsegraining

Computational Chemistry Accuracy per $


Log(Cost)

Acc

urac

y

Ab Initio

DFT

Semi-Empirical

Force Field

Coarsegraining


0

Depends on the problem!tisdag 18 december 12

What is Molecular Dynamics• Coordinates of atoms/particles as a function of time

• Time correlation functions

• Experimental: Nuclear Magnetic Resonance spectroscopy

• Computer simulations at different levels of spatial and temporal accuracy

• Sampling the conformational space - partition function


M.Levitt, Nat. Struct. Biol. 8 (2001), p. 392-393tisdag 18 december 12

How Many Lennard-Jones Parameters for a Protein?

Two parameters per interaction

One atom type per element (C,H,N,O,S) : 10 with combination rules, 40 without

With four atom types per element: 40 with combination rules, 760 without


The Physical Model

What determines the quality?

How can we test it?


Our Gr8t Software


http://www.gromacs.org




Liquid FF Benchmark

• Test available force field for how good they reproduce properties for ~ 150 organic liquids

• 1000 molecules/box 10 ns production sims

• OPLS/AA, GAFF + CGenFF from literature



Simple props

500 1000 1500 2000 2500lExper. (g/l)

500

1000

1500

2000

2500

l Cal

c. (

g/l)

GAFFOPLS/AACGenFF

20 40 60 80 1006Hvap,Exper. (kJ/mol)

20

40

60

80

100

6H

vap,

Cal

c. (k

J/m

ol)

GAFFOPLS/AACGenFF

Density Enthalpy of Vaporization


Not so simple props

0 10 20 30 40 50 60aExper. (10-3N/cm)

0

10

20

30

40

50

60

a Cal

c. (1

0-3N

/cm

)

GAFFOPLS/AA

1 10 100¡0,Exper.

1

10

100

¡ 0,Calc.

GAFFOPLS/AA

Surface Tension Dielectric Constant

Due to cut-off of Van der Waals forces? Polarization?


Complex props

0 100 200 300cV,Exper. (J/mol K)

0

100

200

300

c V,C

alc. (J

/mol

K)

GAFFOPLS/AA

0 0.5 1 1.5 2gT,Exper. (10-3/K)

0

0.5

1

1.5

2

g T,C

alc. (

10-3

/K)

GAFFOPLS/AA

Heat capacity Compressibility

2 Phase Thermodynamics method!


Page 70

(table continued on next page)

Página 1 de 1Document

12/10/2003http://emedia.netlibrary.com/nlreader/nlreader.dll?bookid=17984&filename=Page_70.html

Yihzak Marcus: The Properties of Solvents


Handbook of Chemistry & Physics


Science in the 21st century...


360 380 400 420 440T (K)

40

45

50

55

60

65

70!(0

)

FitData used in fit

ethanamide - 60-35-5

0 100 200 300 400 500 600T (K)

0

20

40

60

80

!(0

)

FitData used in fit

methanol - 67-56-1

100 150 200 250 300T (K)

10

12

14

16

18

20

!(0

)

FitData used in fitData not used in fit

bromomethane - 74-83-9

280 290 300 310 320 330T (K)

9

9.5

10

10.5

11

!(0

)

FitData used in fitData not used in fit

1,1-dichloroethane - 75-34-3


Statistics

• Quantitative evaluation per property.

Table 2: Statistics of a linear fit of calculated to experimental values according to y = ax+ b.Uncertainties in the simulation results are used as weights in the fit. The number of (experimental)data points N is given for each property. Root mean square deviation (RMSD) from experimentalvalues, average relative deviation in % and the correlation coefficient R2 are given. OPLS/AAresults from ref.,70 and CGenFF results from ref.36 (using the so called CHARMM GeneralizedForce Field) are also listed for comparison.

Force field N a b RMSD % Dev. R2ρ (g/l)

GAFF 235 0.96 58.6 82.9 4 97%OPLS/AA 235 0.98 21.3 40.3 2 99%CGenFF36 111 1.03 -36.0 26.0 2 99%OPLS/AA70 9 1.01 -24.0 45.3 4 96%

ΔHvap (kJ/mol)GAFF 231 1.07 0.8 10.6 17 83%OPLS/AA 231 0.96 3.4 6.5 11 89%CGenFF36 95 0.94 2.4 4.7 7 84%

γ (10!3N/m)GAFF 155 0.75 0.9 8.6 23 70%OPLS/AA 155 0.97 -5.5 7.3 22 89%

ε(0)GAFF 170 0.21 0.6 23.4 52 48%OPLS/AA 183 0.16 0.7 22.7 57 56%

αP (10!3/K)GAFF 221 0.90 0.3 0.3 24 67%OPLS/AA 221 0.91 0.3 0.3 21 75%OPLS/AA70 9 0.53 0.8 0.7 42 39%

κT (1/GPa)GAFF 60 0.56 0.1 0.3 31 64%OPLS/AA 60 0.61 0.2 0.3 21 76%OPLS/AA70 8 0.93 0.0 1.1 59 84%

cP (J/mol K)cV (J/mol K)cclassP (J/mol K)

33


Benchmark is Foundation for Future FF

• > 100,000 experimental data points gathered in a SQL database

• Allows rational FF derivation ... at last

No moreof this

http://virtualchemistry.org/


http://virtualchemistry.org

http://virtualchemistry.org

Can Quantum Chemistry Help?

• Compute properties for which there is no experimental data

• Electric properties (multipole moments) polarizabilities are reproduced well

• Energies ?


Enthalpy of Formation

-400 -200 0 200 400 600Exper. (kJ/mol)

-40

-20

0

20

40

Cal

c. -

Expe

r.

G2G3G4CBS-QB3


Fear not

• Nature is linear...

05

1015202530

Aldehydes Esters-Methyl-AlkanoatesBenzene derivatives

05

1015202530

Cyclic mono ethers Cycloalkanes Difluoroalkanes

05

1015202530

Pola

rizab

ility

(A3 ) Esters-Alkyl-Methanoates Alkanes Ethers-Methoxy-Alkanes

05

1015202530

Ketones Perfluoroalkanes Primary Alcohols

05

1015202530

Primary Thiols Primary Carboxylic AcidsPrimary Nitroalkanes

0 5 1005

1015202530

Primary Amines

0 5 10# Carbon

Secondary Amines

0 5 10

Tertiary Amines

H2O

O2

NH3


Summary Force Field

• Intermolecular energies within ~ 5-7 kJ/mol in force fields - for supported atom types

• A lot of work going on in force field tuning and development

• Many properties can be computed based on long classical simulations


References

• Carl Caleman, Paul van Maaren, Minyan Hong, Jochen Hub, Luciano Costa, David van der Spoel, J. Chem. Theor. Comput. 8 (2012) 61-74

• David van der Spoel, Paul van Maaren, Carl Caleman, Bioinformatics (2012) doi: 10.1093/bioinformatics/bts020

• Oliver Lange, David van der Spoel, Bert L. de Groot - Biophysical Journal 99 (2010) 647-655


The Two Phase Thermodynamics Method

David van der Spoel, Uppsala University


What’s the problem?

• For ideal gases theories exist from which one can derive thermodynamics properties

• For solids another kind of theory exists

• For liquids there is no such thing


1st Ansatz

• Treat Liquid just like a Solid - system of harmonic oscillators


Downloaded 18 Jul 2009 to 130.238.41.194. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

Berens et al. JCP 79 (1983) pp2375tisdag 18 december 12



The trick

• S(v) can be derived from the velocity auto-correlation function

• Assume the frequencies in the classical simulation are correct (and the amplitudes)

• Reweigh the S(v) to get a corresponding Quantum S(v)


Example - C2H6SO

0 20 40 60 80 100 120ν (1/ps)

0

2000

4000

6000

8000

S(ν)

Density of states

BondsAngles &Dihedrals

Collective modestisdag 18 december 12

where M is the number of equivalent particles and m is their mass.The 2PT method approximates

DoSliquid(!) = DoSgas(!) +DoSsolid(!) (S4)

and if we would have an analytical expression for DoSgas(!) – note thatPascal et al. [3] use the subscript “di!” rather than “gas” as we do forconsistency here – we could subtract this from the DoS(!) obtained fromthe simulation (Eq. S2). Indeed, the gaseous (di!usive) component of thedensity of states DoS(!)gas, described as a gas of hard spheres (Equation24, reference [1]) can be written as:

DoSgas(!) = DoS(0)

!

1 +"

DoS(0)"!

6fN

#2$

!1

(S5)

where f is the fluidicity, the fraction of the 3N degrees of freedom thatcorresponds to the di!usional parts of the system, or in other words

%

"

0DoSgas(!)d! = 3fN, (S6)

independent of DoS(0). A derivation of the equation needed to solve forf can be found in reference [1], the final result being (Equation 12, refer-ence [2]):

2(f5/"3)3/2 ! 6(f5/"3)! (f7/"3)1/2 + 6(f5/"3)1/2 + 2f = 2 (S7)

where " is the dimensionless di!usion constant defined by (Equation 13,reference [2]):

" =2DoS(0)

9N

&

"kBTN

M

'1/2 &N

V

'1/3 & 6

"

'2/3

(S8)

with V is the volume of the system. Eq. S7 is monotonous between 0 and 1(cf. Figure 2 in reference [1]) and hence a bisection algorithm can be usedto solve for f given ".

Now we will try to comprehend the solid component of the density ofstates, DoSsolid(!). Here, each mode is considered to be harmonic and weuse the partition function for a quantum harmonic oscillator as given byMcQuarrie [8]:

qqHO(!) =e!!h"/2

1! e!!h"(S9)

where h is Planck’s constant, # = 1/kBT . Since the canonical partionfunction Q of a system is given by the product of the components (Equations3.18-3.20, reference [6]):

Q =N(

i=1

qi (S10)

and therefore

lnQ =N)

i=1

ln qi (S11)

8

Caleman et al. JCTC 8 (2012) 61-74tisdag 18 december 12

Final Result 1



2nd Ansatz

• Treat a liquid as a combination of solid and ideal gas - 2 phase thermodynamics

• Gas can be treated as a hard-sphere gas for which analytical thermodynamical results exist

Lin et al. JCP 2003, JPCB 2012, Pascal PCCP 2011tisdag 18 december 12

Final Result 2

• Determine “fluidicity” - how fluid is the simulation between 0 and 1

• Modified weighting function - same as Berens for the solid part + another for the gas part

Lin et al. JCP 2003, JPCB 2012, Pascal PCCP 2011tisdag 18 december 12

How does it work?

• Do MD simulation with flexible bonds, 20-100 ps (time step 0.2 fs)

• Save velocities every 4-5 fs

• Compute mass-weighted velocity ACF and compute properties S, E, A, Cv by weighting the S(v) and integrating!


Heat Capacity Cv

0 100 200 300c

V,Exper. (J/mol K)

0

100

200

300c V

,Calc

. (J/

mol K

)GAFFOPLS/AA

Caleman et al. JCTC 8 (2012) 61-74tisdag 18 december 12

Hydrophobic Segregation, Phase Transitions and the Anomalous Thermodynamics of Water/Methanol Mixtures

Tod A. Pascal and William A Goddard, IIIJ. Phys. Chem. B http://dx.doi.org/10.1021/jp309693d

Hydrophobic Segregation, Phase Transitions and the AnomalousThermodynamics of Water/Methanol MixturesTod A. Pascal*,† and William A Goddard, III*,†,‡

†Materials and Process Simulation Center, California Institute of Technology, Pasadena, California 91125, United States‡World Class University Professor, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea

ABSTRACT: When water and methanol are mixed, the entropy of mixingdecreases, whereas mixing simple liquids normally leads to an increase inentropy. One speculation on the origin of the anomaly involves formation ofwater icebergs next to the hydrophobic methanol group, while more recenttheories point to nanoscale clustering of methanol molecules. To elucidate theorigin of this effect, we carried out extensive molecular dynamics calculations onwater/methanol mixtures ranging from 0 to 100% and applied the 2PT methodto extract the entropy and free energy changes of each component as a functionof concentration. We find that water molecules lose at most 1/35 of their liquidentropy in mixtures. Methanol molecules, on the other hand, lose 3 times as much entropy as the water molecules, and theirrotational entropy contains the signature of the entropic loss. We find that methanol has a discontinuous specific heat profile inthese mixtures with a maximum at 40% methanol. These results do not support the iceberg model of immobilized waters andinstead suggests a molecular mechanism of hydrophobic segregation at low methanol concentration where ordering of themethanol molecules bury the hydrophobic group away from the water phase. For higher methanol concentrations, there isinsufficient water to accomplish this effect, and the system freely mixes and transitions to one better described as water dissolvedinto methanol.

1. INTRODUCTIONThe solubility and mixing of organics in water plays a criticalrole in biological life on this planet, and solvent extractiontechniques underlie a plethora of industrial processes includingmany in the beer and wine industries. Mixtures of alcohols suchas methanol and ethanol with water would appear to be thesimplest examples of solvated systems with “hydrophobic”groups. However, these mixtures show unusual entropy andfree energy with increasing alcohol concentration1 (xMeOH).When two ideal liquids i and j are mixed, the entropy of

mixing Smix necessarily increases,2 i.e.:

= ! + > ! +S kN x x x x kN x x( ln ln ) ( )i i j j i jmix (1)

since the mole fraction (number density) xi of each componentis less than unity (N = !iNi is the total number of moleculesand k is Boltzmann’s constant). However, for mixtures ofmethanol and water the entropy decreases, a fact that has beenknown experimentally as early as 1937 from the partial pressuremeasurements of Butler et al.,3 who showed that excess freeenergy has a harmonic dependence on xMeOH. Further earlyexperimental evidence came from the studies of Lama and Lu,4

who showed that the entropy of mixing dominates thethermodynamics while remaining always negative over theentire concentration range.This anomalous thermodynamic behavior has stimulated

numerous theoretical investigations, the earliest of which arethe studies of Frank and Evans5 who introduced the so-called“Iceberg” model of solvation in 1945. This model posits thatthe negative entropy of water/methanol mixtures arise from the

immobilization of the waters molecules solvating the hydro-phobic "CH3 groups of methanol. Some recent experimentalresults6 have supported this model; however, it has beenchallenged both experimentally7"10 and theoretically (forexample see refs11 and 12). A more recent explanationsuggests that incomplete mixing and molecular segregation isresponsible for the entropy profile.12

Previous theoretical works have used molecular dynamics(MD) and Monte Carlo simulations to elucidate the hydrogenbonding, diffusion, and partial volumes effects of water/methanol mixtures (see ref13 and the references therein).There have also been numerous studies of the enthalpy ofmixing (see for example refs 14"17). Far less attention hasbeen paid to the entropy and free energy of mixing,12,18,19

however. In this paper we report the excess entropy, enthalpy,and heat capacity of water/methanol mixtures at 17 pointsalong the concentration curve, and we report separately thewater and methanol contributions to these thermodynamicsproperties. Our studies use MD simulations and the two-phasethermodynamics20,21 (2PT) method.

2. THEORY AND COMPUTATIONAL METHODS2.1. The 2PT Method Applied to Polymolecular

Systems. Methods of efficiently accessing entropies of waterfrom molecular dynamics trajectories22,23 are becoming more

Received: September 29, 2012Revised: November 5, 2012

Article

pubs.acs.org/JPCB

© XXXX American Chemical Society A dx.doi.org/10.1021/jp309693d | J. Phys. Chem. B XXXX, XXX, XXX"XXX


Summary 2PT

• Promising new method

• Lowest frequencies contribute most - may also work with constraints

• Very recently also for mixtures

• More development and testing needed


References• Carl Caleman, Paul J. van Maaren, Minyan Hong, Jochen S. Hub, Luciano T.

Costa, David van der Spoel: Force Field Benchmark of Organic Liquids: Density, Enthalpy of Vaporization, Heat Capacities, Surface Tension, Compressibility, Expansion Coefficient and Dielectric Constant J. Chem. Theor. Comput. 8 pp. 61-74 (2012)

• Lin, S. T.; Blanco, M.; Goddard, III, W. A. J. Chem. Phys. 2003, 119, 11792–11805.

• Lin, S.-T.; Maiti, P. K.; Goddard, III, W. A. J. Phys. Chem. B 2010, 114, 8191–8198.

• Pascal, T. A.; Lin, S.-T.; Goddard, III, W. A. Phys. Chem. Chem. Phys. 2011, 13, 169–181.

• Huang, S.-N.; Pascal, T. A.; Goddard, III, W. A.; Maiti, P. K.; Lin, S.-T. J. Chem. Theory Comput. 2011, 7, 1893–1901.

• Berens, P. H.; Mackay, D. H. J.; White, G. M.; Wilson, K. R. J. Chem. Phys. 1983, 79, 2375–2389


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From Dynamics to Thermodynamics using Molecular Simulationkkrk.chem.elte.hu/molim/lectures/DvdSpoel-dynamic-to-thermodyna… · • 1000 molecules/box 10 ns production sims ... where