Solid-liquid Equilibrium and Free Energy Calculation of Hard-sphere, Model protein, and

Solid-liquid Equilibrium and Free Energy Calculation of Hard-sphere, Model protein, and

Lysozyme Crystals

Jaeeon Chang

Center for Molecular and Engineering Thermodynamics Department of Chemical Engineering

University of Delaware

Overview

Determination of liquid-solid transition using histogram reweighting method and expanded ensemble MC simulations

Fluid-solid phase equilibria of patch-antipatch protein model

The combined simulation approach of atomistic and continuum models for the thermodynamics of lysozyme crystals

Objective

To develop a generic method to predict liquid-solid phase equilibria from Monte Carlo simulations.

Motivation

Methods involving particle insertion scheme used in Gibbs ensemble MC are not applicable to dense liquids and solids.

The previous method based on equation of state requires fitting of simulation data to an assumed form of EOS.

Kofke’s Gibbs-Duhem integration method requires one known point on the coexistence curve.

Hard sphere fluid and fcc crystal (Test model)

The simplest nontrivial potential modelno vapor-liquid transition, athermal solid-liquid transition

Reference system for perturbation theory Model for colloid system

63*

3*

V

N

kT

pp

Canahan-Starling equation of state for liquid phase

3

32

*

*

1

1

p

NkT

pV

Outline of the Methodology

Basic principle

Tliq = Tsol pliq = psol liq = sol

A series of NPT MC simulations are performed to separately construct equations of state for liquids and solids using histogrm reweighting method

To obtain the chemical potentials of liquid branch an accurate estimate at a particular density should be provided either from direct simulations (Widom method, Free energy perturbation method, Bennett acceptance method) or from the integration of the equation of state from from zero density to the liquid density.

For the chemical potentials of solid branch, the free energy at a particular density is obtained using Einstein crystal and the expanded ensemble method.

Histograms from NPT Monte Carlo simulations

K

EVh

TpN

pVEEVNEVP

,

,,

exp,,,

Probability for a single histogram reweighting

h

V, E

Hard sphere N=256

V / N

0.90 0.95 1.00 1.05 1.10

Pro

ba

bili

ty

1112

13

11

12p* = 13

Hard spheres

Histogram Reweighting Method in NPT ensemble

Composite probability for multiple histograms (not normalized)

jjjjj

R

j

i

R

i

CVpEK

pVEEVh

TpEVP

exp

exp,

,;, V E

iii TpEVPC ,;,exp

Chemical potential

iiiii CCTpNG 0,,ln

A known value of free energy is required to specify C0

Average properties

EVPEVPEEV EV E

,, EVPEVPVVV EV E

,,

Ci’s are determined in a self-consistent manner

MC

CS EOS

Histogram

Hard Sphere Fluid-Solid Transition ( n=256, 5M cycle)p = 11.79 liq = 0.9437 sol = 1.0447 = 16.30

0.8 0.9 1.0 1.1 1.2

p 3

/ k

T

8

10

12

14

16

18

20

MCLiquid Solid CS EOS

Construction of Equation of State for Hard spheres

Free Energy of Solid

The classical Einstein crystal as a reference

Variation of potential from the reference to the system of interest

Einstein crystal (reference)

Repulsive core turned on

Einstein field turned off

The expanded ensemble method: : hopping over subensembles

iji

jij ww

P

PAA ln

i

ii TVNZwTVNZ ,,exp,,

20)( i

iiU rrr

ln2

3

N

A

Ref) Lyubartsev et al., J. Chem. Phys. 96, 1776 (1992). Chang and Sandler, J. Chem. Phys. 118, 8390 (2003).

density = 1.04086

0 200 400 600 800 1000

Aex

/ N

6

7

8

9

10

0 5 10 15 205.9

6.0

6.1

6.2

Free Energy of Hard-sphere Fcc Solid

04086.1*

MC

CS EOS

Histogram

Hard Sphere Fluid-Solid Transition ( n=256, 5M cycle)p = 11.79 liq = 0.9437 sol = 1.0447 = 16.30

0.8 0.9 1.0 1.1 1.2

p 3

/ k

T

8

10

12

14

16

18

20

MCLiquid Solid CS EOSPhase Eq

CS EOS

Histogram

0.8 0.9 1.0 1.1 1.2e

x /

kT +

ln

3

12

14

16

18

20

22

24

CS EOS LiquidSolidPhase Eq

Determination of Solid-liquid Transition

Equilibrium properties

p* = 11.79, liq* = 0.944, sol* =1.045, * = 16.30

c.f) Lee and Hoover (1968)

p* = 11.70, liq* = 0.943, sol* =1.041

Conclusions

• Using the combination of the histogram reweighting and expanded

ensemble simulation methods a new generic algorithm for predicting liquid-

solid equilibria is proposed.

• The liquid-solid equilibria for hard-sphere systems of varying size up to 1372

particles are studied, and the limit for the infinitely large system is accurately

determined.

Overview

Determination of liquid-solid transition using histogram reweighting method and expanded ensemble simulations



Introduction

Background

3-D protein structure by X-ray crystallography

Crystallization windows correlated with slightly negative values of B2

L-L equilibria described by isotropic short-range interaction

Anisotropic model necessary for F-S equilibria

Objectives of this work

Understanding of the role of anisotropic interactions on fluid-solid equilibria of protein solutions from ‘exact’ computer simulations

Comparison of the free energies of different crystal structures and phase diagram involving multiple solid phases

Concentration

Tem

pera

ture

SF

L-L

r /

0.6 0.8 1.0 1.2 1.4

u(r)

/

-6

-4

-2

0

2

4

isotropicanisotropic p* = 5

Patch-antipatch potential model of globular proteins

Ref; Hloucha et al., J. Crystal Growth, 232, 195 (2001)

12

4

5

3

12

4

5

3

35140

1165.2rr

ruru ppatiso

Three patch-antipatch pairs in perpendicular directions

A narrow range of orientation :

ap = 12º

ruruSruru isopatnm

nmisopair ,

,,

22

21

21,

p

n

p

mnm aa

S

pp *

4~5 for chymotrypsinogen

Crystal structures

simple cubic SC

orientationally disordered face centered cubicFCC(d)

orientationally ordered face centered cubicFCC(o)

High T Low T Low T

Free energy (Fluid)

Thermodynamic integration method over the equation of state

*

0 2*

**

*

*1

d

p

TNkT

AA id

*

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

p*

0

2

4

6

8

10

12

14

Fluid

Free energy of the Einstein crystal

0ln

2exp1

2ln I

N

AEino

cos1cos120 rrr t

Einu

ln3ln

2

3

t

Eint

N

A

Constraining potential

Free energy

Free energy (Solid)

The classical Einstein crystal as a reference


Ref) J. Chang and S. I. Sandler, J. Chem. Phys. 118, 8390 (2003)

Einstein crystal (reference)

Repulsive core turned on

Attractions turned on

Einstein field turned off

The expanded ensemble method: a direct measure of free energy difference

iji

jij ww

P

PAA ln

i

ii TVNZwTVNZ ,,exp,,

Coexistence at a fixed temperature

*

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

p*

0

2

4

6

8

10

12

14

FluidFCC(d)HistogramCoexistence

T*=1, eps(patch) =0

p

7 8 9 10 11 12 13

10

11

12

13

14

15

16

FluidFCC(d)

T* = 1.0, p* = 1 (isotropic model)

T*

0.3 0.5 0.7 0.9 1.1p*

0.001

0.01

0.1

1

10

FCC

Fluid

*

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

T*

0.3

0.5

0.7

0.9

1.1

Fluid

FCC

Coexistence for isotropic model

Density Osmotic pressure

Coexistence for anisotropic model with p* = 5

*

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

T*

0.3

0.5

0.7

0.9

1.1

0.00 0.05 0.100.32

0.36

0.40

SCFluid

FCC(d)

FCC(o)

T*

0.32 0.34 0.36 0.38 0.40p*

0.0001

0.001

0.01

0.1

1

10

FCC(o)

SC

FCC(d)

Fluid

SC-FCC(d)

Density Osmotic pressure

The ordered phases [ SC and FCC(o) ] are stable at low temperatures

T*

0.32 0.34 0.36 0.38 0.40

p*

0.0001

0.001

0.01

0.1

1

10

FCC(o)

SC

FCC(d)

Fluid

SC-FCC(d)

Phase diagram for anisotropic model with p* = 5

Chemical potential of solid beyond saturation pressure

sat

p

psat ppVVdpTGPTG

sat

,

T*

0.32 0.34 0.36 0.38 0.40

p*

0.0001

0.001

0.01

0.1

1

10

FCC(d)

Fluid

SC

FCC(o)

Saturated solids Phase diagram

FCC(o) cannot be prepared directly from solutions

Conclusions

• More than one crystal structures are compatible even with the

simple anisotropic protein model.• Phase diagram involving multiple solid phases are determined from

the histogram reweighting method.

J. Phys. Chem. B, 109, 19507 (2005)

Overview

Determination of liquid-solid transition using histogram reweighting method and expanded ensemble simulations



Introduction

Protein crystal

• Protein crystals are used to determine three-dimensional structure of proteins.

• Trial-and-error screening methods to find crystallization conditions

• Weak attractive interactions for the crystallization to occur

Lysozyme

• Natural antibiotic enzyme to break the cell wall of bacteria

• Several crystal forms depending upon the solution conditions ( T, pH, Ionic strength and species)

• Solubility, heat of crystallization are known experimentally.

• Phase transition between tetragonal (low T) and orthorhombic (high T) forms occurs near the room temperature

Hen egg white lysozyme (193L)

Objective and approach

• To compare the thermodynamic properties of tetragonal and orthorhombic crystals of hen egg white lysozyme

Crystallographic structure

NVT Monte Carlo simulation

Expanded ensemble MC simulation

Thermodynamic properties

Boundary element method

Elec. U, S

vdw U

vdw A

Gibbs-Helmholtz relation

Elec. A

Elec. A

NVT Monte Carlo simulations

tetragonal form (PDB 193L) orthorhombic form (PDB 1F0W)

Potential model (implicit water)• Semi-empirical model of Asthagiri et al., Biophys. Chem. (1999)• 0.5 OPLS force field, r < 6Å• Hamaker interactions ( H = 3.1 kT), r > 6Å

System• 16 protein molecules (rigid body) • NVT MC at experimental density and at 298 K• Translation ~ 0.1Å, Rotation ~ 1º

Free energy of crystal from expanded ensemble MC


Ref) Lyubartsev et al., J. Chem. Phys. 96, 1776 (1992). Chang and Sandler, J. Chem. Phys. 118, 8390 (2003).

Einstein crystal(known free energy)

Einstein field turned offFull interaction

The expanded ensemble MC method: hopping over subensembles

12

1

2

12ln

ww

P

PAA

TVNZwTVNZ ,,exp,,

i

Eini

jiij uuU 1

= 0 0 < < 1 = 1

Boundary element method for electric potential

i

r2

k k

k

i

ii q

dSGn

dSn

G

rrrrr

r

rr

r

rrrr

4

1

rrrr 41iG

22

kTzce eii 2022

0

rrr

r

rr

r

rrrr dSG

ndS

n

G ep

ep

l m n

lmnee

p GG rrrrr rrrrrr 4expeG

ze

zetanh22 rr

222 cosh ze 0

20

2

2

4exp

zc

zccZcZze

pppp

• Protein domain: dielectric const. 4

• Aqueous domain ( a single protein in solution ): dielectric const. 80

• Aqueous domain of the Crystal is in the Donnan equilibrium with the solution. Optimally linearized Poisson-Boltzmann equation

• Electrostatic free energy

kk

kqF r2

1

Boundary elements and charge distribution in the protein

• 1,000-2,000 Triangular elements were obtained using Connolly’s program with subsequent simplification procedure.

• The charges are placed at 30 ionizable residues (Asp, Glu, Arg, His, Lys, Tyr) and the C and N termini.

• The Henderson-Hasselbalch equation using experimental pKa data.

pH

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Cha

rge

of ly

sozy

me

-15

-10

-5

0

5

10

15

20

193L 1JIS 1LZA 1F0W 1HSX 1JJ1 1JPO 1BGI

VD

W E

nerg

y (k

J/m

ol)

-160

-120

-80

-40

0

Tetragonal Orthorhombic

4 A

4.5A

5 A

6 A

4.2A

VDW

Hamaker

Hamaker

OPLS

VDW energy from MC simulations

• The VDW interactions in the tetragonal form are less attractive than in the orthorhombic form.

• Noticeable variations in the energy among the PDB structures are observed due to the variations in the side chain conformations.

• The Lennard-Jones interactions within 6 Å dominate over the water-mediated Hamaker interactions at longer distances.


A -

Ao (

kJ/m

ol)

-100

-80

-60

-40

-20

0

VDWVDW + ElecExperiment

Free energy of lysozyme crystal (pH 4.5 and I = 0.36 M)


• Standard state at 1mol/L : A°/NkT = ln(Λ3/1660) – 1

• The experimental values are close to each other since the transition occurs near 298 K.

• Electrostatic contribution to the free energy is repulsive.

• The predicted Helmholtz energies are less than the experimental values.

Crystal structure


Ene

rgy

of c

ryst

alliz

atio

n (k

J/m

ol)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

VDWVDW + ElecExperiment

Energy of crystallization (pH 4.5 and I = 0.36 M)


• The VDW contributions play a dominant role in both crystals.

• For the tetragonal crystal, the predicted energy is acceptable considering the wide range of the reported experimental values from –40 to –140 kJ/mol.

• The less attractive energy from experiment suggests energetically unfavorable release of water molecules from crystal contacts.

• The larger disagreement for the orthorhombic crystal form indicates a large difference in the solvation structure.

water release


S -

So (

J/m

ol/K

)

-400

-300

-200

-100

0

ExperimentTheoryVDWVDW+Elec

Entropy of lysozyme crystal (pH 4.5 and I = 0.36 M)


• The VDW entropy for the tetragonal form is in good agreement with the mean field theory.

• The electrostatic contributions to the entropy are negative, arising from the reorganization of water molecules and ions.

• For the tetragonal form, there should be a release of about 4 water molecules upon crystallization ( the entropy change on the melting of ice ~ 22 J/mol/K).

water release

Tetragonal Orthorhombic 193L 1JIS 1F0W 1HSX

No. of hydrated water molecules per protein

142

142

137

153

No. of bridging water molecules between proteins

r < 3.6 Å r < 4.0 Å

28 46

23 41

12 28

15 29

>

Experimental evidence of distinct hydration structures

• A water molecule is counted as a bridging molecule if it is also close to another protein.

• Whereas the total number of hydrated water molecules is almost the same, there is a decrease of about ten bridging water molecules for each protein in the orthorhombic crystal forms.

• Additional water molecules are expelled from between the contacting surfaces when a lysozyme molecule becomes part of an orthorhombic crystal, which is an energetically less favorable but entropically more favorable process.

~

Conclusions

• We have carried out Monte Carlo simulations of the hen egg white lysozyme crystals at the atomistic level and the boundary element calculations to solve the Poisson-Boltzmann equation for the electrostatic interactions.

• The crystallization energy of the tetragonal structure agrees reasonably well with experimental data, while there is a considerable disagreement for the orthorhombic form.

• A large difference in the experimental energy of crystallization between the two crystals indicates energetically unfavorable solvation in the orthorhombic form.

• The much higher value of the entropy of the orthorhombic crystal is explained in terms of the entropy gain of the water molecules released during the crystallization.

• National Science Foundation• Department of Energy

• Prof. Stanley Sandler • Prof. Abraham Lenhoff• Dr. Jeffrey Klauda (NIH)• Dr. Stephen Garrison (NIST)• Mr. Gaurav Arora

Acknowledgements

Solid-liquid Equilibrium and Free Energy Calculation of Hard-sphere, Model protein, and

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