Protein Crystallization and

Protein-Protein Interactions

Xueyu Song

Department of Chemistry

Iowa State University

Why Study Protein Crystallization?

• Protein Crystallization as a Model of Colloidal System Phase Transitions

• Protein Crystal as the Starting Point of Structural Determination

• Diseases due to Protein Crystallization

Cataract due to Protein Aggragates

Human Genome Pilot Projecthttp://proteome.bnl.gov/progress.html

Progress Toward Structure Solution by X-ray Crystallography

By January 18,2005:

Cloned Express Soluble Purified Good Crystal Structure

124 115 91 63 19 21

From Clone to Purified Protein 51%

From Clone to Good Crystal 15%

From Purified Protein to Good Crystal 30%

Experimental Status in TargetDB by by worldwide Structure Genomics Centers http://targetdb.pdb.org/statistics/TargetStatistics.html

Protein Structure of LH2

Classical Nucleation Theory

R*The nucleation is an activated process. The nucleation barrier can be estimated by the following way

where is the chemical potential difference between the solid phase and the meta-stable liquid phase and is negative. is the surface tension and R is the radius of the nucleus. The probability to have a nucleus with size R is proportional to exp(-G). A critical nucleus is where G(R) is at its maximum. The size of the critical nucleus is ,and the activation barrier is

which results from a spherical critical nucleus assumption.

3 24( ) 4

3G R R R

2R

3

2

16

3 ( )G

The meta-stable critical region may offer optimal crystallization window.

V. Talanquer and D. Oxtoby, J. Chem. Phys. 109, 203(1998).O. Galkin and P.G. Vekilov, Proc. Natl. Acad. Sci. USA, 97, 6277(2000)

Phase diagram for lysozyme with 3% NaCl at pH=4.5 in 0.1M NaAc buffer.Muschol and Rosenberger, J. Chem. Phys. 107, 1953(1997)

Phase diagram for crystallin Asherie, Lomakin and Benedek, Phys. Rev. Lett. 77, 4832(1996).

The effect of anisotropic interactions on

colloidal phase and nucleation behaviors

• Most of studies employ isotropic potenitals.

• Typical “globular proteins” are “football-shape” molecules.

Molecular surface of -crystallin

Do isotropic models capture all of qualitative features of protein crystallization?

Crystal nucleation in anisotropic colloids

Orientaional-dependent interfacial free energies may favor highly non-spherical critical nucleus. Thus, the anisotropy of the interaction may change the nucleationdynamics qualitatively.

Direct observation of critical nuclei in apoferritin crystallization

S.T. Yau and P.G. Vekilov, Nature 406, 494(2000)

Molecular surafce of apofferitin: quasi-spherical, but with highly inhomogeneous surface charges; 24-mer of subunits in rhombic dodecahedron symmetry.

Experimentally, critical nuclei are found to be planar rather than spherical.

It seems moderate anisotropy can play an important role in protein crystallization

Orientationally ordered crystals are the useful crystal for diffraction

Hard dumbbell fluidsD=hard sphere diameter; L=bond length.

We can define an “anisotropy parameter”, L*=L/d, to characterize the effect of anisotropy of the interaction.

L*=0 will be the hard sphere limit.

Vega, Paras and Monson, J. Chem. Phys. 97, 8543(1992)

Plastic solid (PS) is fcc and ordered solid (OS) is hexagonal.

0<L*<0.38: Fluid PS OS

L*>0.38: Fluid OS. PS unstable.

L*=0.38, a triple point of fluid-PS-OS.

Fluid

PS

OS

Solid L* γ100 γ110 γ111

PC 0.0 0.59(1) 0.58(1) 0.57(1)

0.15 0.57(1) 0.57(1) 0.57(1)

0.30 0.60(1) 0.60(1) 0.60(1)

γs1 γs2 γs3

CP1 0.40 2.19(2) 2.19(2) 2.48(2)

0.60 2.21(2) 2.21(2) 2.43(2)

1.0 2.65(2)

All of the interfacialfree energies are in the unit of kBT/σ2,

where σ is the diameter of a hard sphere with the same volume as the hard dumbbell, just as the reduced coexistence density, ρ*.Yan and Song, Phys. Rev. E 74,031611(2006)

Density functional theories of freezing

Crystalline solids treated as a self-sustained inhomogeneous “fluids”:Ramakrishinan and Yusouff, Phys. Rev. B 19, 2775(1979).

Oxtoby, Haymet, Torazona, Ashcroft, Baus,etc. further developed the theory with the well-known liquid properties as input, esp. weighted density approximation.

Such applications mainly to isotropic potenitals, straight-forward extensions to anisotropic models remain problematic.

Fundamental measure functional theory:Rosenfeld, Phys. Rev. Lett. 63, 980(1989).

Again, applications are mainly to isotropic fluids and extentions to anisotropicfluids remain as an active research area.

Density functional + Mean field theory of weakly anisotropic fluids

H.J. Woo and X. Song, Phys. Rev. E, 63, 051501(2001) J. Chem. Phys. 116, 4587(2002)

Potential separation:

0 1( , ; , ) ( ) ( , ; , )i i j j ij i i j jv r r v r v r r leads to free energy separation: f=f0+f1

f0: free energy of the lattice formation of the isotropic reference system calculated from conventional density functional theory

f1: orientational correaction calculated from a mean field theory

Such a simple and efficient method seems to capture the effect of anisotropic interactions on freezing.

Application of Our theory to hard dumbbell freezing

Isotropic reference system: effective hard sphere model

Density functional: modified weighted density approximation

PS or PC: fcc, OS or OC:hcp

Triple point: theoretical L*=0.36 vs. simulation 0.38

Freezing of soft dumbbell systemsDumbbells with site-site hard sphere Yukawa potential:

( )( )

r

rU r

e rr

specifies the interaction strength characterized the attraction rangeL* indicates the extent of our anisotropic interaction

Especially, =∞ or β=/kBT=0 leads back to hard dumbbell models.

Applications of our theory and Monte Carlo simulations yield the detailed phase diagrams of the above system. Suchphase behaviors leads to new perspectives on the roles ofanisotrpic interactions in protein crystallization. X.Song, Phys. Rev. E, 66, 011909(2002)

k=4.0, L*=0.6 k=4.0, L*=0.3

κ=9.0, L*=0.6 κ=9.0, L*=0.3

Kiefersauer, Than, Dobbek, Gremer, Melero, Strobl, Dias, Soulimaneand Huber, J. Appl. Cryst. 33, 1223(2000).

Using five proteins, they show that the hydrationand dehydration are in most cases reversible processes, associated with changes in shape,unit-cell dimensions and crystalline order. Thus,the variation of humidity can be used to improveThe quality of protein crystals.

Protein-Protein InteractionsComplimentary Hydrophobic Surface

Highly Specific for Recognition

Nonspecific Crystal Contact

Highly Sensitive to Solution Conditions

Crystal Contacts of Lysozymes

We found that crystal contacts can be formed using various surface residues depending the solution conditions.

The PDB files used6LYT,1LZT,2LKR,0LZT and 1LYS

Protein-Protein Interaction Model at Residue Level

Each Residue Carries a Permanent Dipole and a Polarizable Diople. The Charge of a Residue Depends on pH of the Solution. X. Song, Mol. Simulations. 29, 643(2003) .

The Boundary Element Method to Solve Poisson-Boltzmann Equation

Our Calculations show thatthe effective interaction is highly anisotropic and sucha high anisotropy could be used to stabilize the orientationally ordered crystal.X. Song, Mol. Simulations. 29,643(2003) .

Van der Waals Interactions

1 260

3( ) ( ) ( )U R d i i

R

For two atoms London(1937) show that

Where α1,2(ω) are the polarizabilities of the first and second atoms.

Lifshitz (1956) extended it to two macrobodies in a dielectric medium.McLachalan(1963) extended it to two molecules in a dielectric medium.But all with simple geometries such as spheres or slabs.

We recently formulated a theory which accounts for the arbitrary geometries of molecules in a dielectric medium. X. Song and X. Zhao, J. Chem. Phys. 120, 2005(2004)

The van der Waals interaction between two protein molecules in an electrolyte solution

Based upon our residue model the effective action in Fourier space of the two proteinmolecules in a solution can be written as

, , , , ', , ',' ',

1[ ] ( ') ( , ')

2 2 2

n n n

r n r n r n r n r n r n n r nr n r r n r r nr n

S m m m m T r r m m R r r m

αr,n the frequency-dependent polarizability of a residue located at rT(r-r’) dipole-dipole interaction tensorRn(r,r’) the reaction filed tensor at frequency ωn=2πn/βħ, which is determined by the protein and solution dielectric function.

The total polarizabilty of a residue can be written as

2( )

1 / 1 ( / )nu e

n nn rot n I

i

αnu the static nuclear polarizability of a residueωrot the characteristic frequency of the nuclear collective motionαe the electronic polarizability of a residue ωI the ionization frequency of a residue as in traditional Drude model

An Inhomogeneous Dielectric Model of Proteins (X. Song, J. Chem. Phys. 116, 9359,2002)

Van der Waals Interactions between Proteins

The Van der Waals interaction between two BPTI molecules at three different relative orientations and distance fromour theory based on our model at residue level. The anisotropy can be aslarge as tens of kBT.

X. Song and X. Zhao, J. Chem. Phys. 120, 2005(2004)

The Validity of Our Model: The calculation of the binding constant of BPTI-Trypsin complex

We have calculated the binding constant between BPTI with various mutations at the P1 position with trypsin and a good agreement with experimental measurements is achieved.

Model System: BPTI• The Crystal Structure of the Complexes Between BPTI-Trypsin and Ten P1

Mutants of BPTI, Helland et al, J. Mol. Biol., 287, 923-942, 1999 Resolution: 1.75 – 2.05 Å

Interscaffolding Additivity;Binding of P1 Mutants of Bovine Pancreatic Trypsin Inhibitor to Four Serine Proteases, Krowarsch et al, J. Mol. Bio., 289, 175-186, 1999

Protein-Protein Interactions for BPTI - Trypsin System

PDB code Experimental K Desc

3BTK 1.70E+13  P1=Lys(WT)

3BTG 1.50E+04  P1=Gly

3BTW 7.60E+06  P1= Trp

3BTE 2.10E+06  P1=Glu

3BTF 1.20E+08 P1=Phe 

3BTH 6.20E+06  P1=His

3BTM 3.90E+07  P1=Met

3BTQ 2.30E+06  P1-=Gln

3BTT 2.90E+05  P1=Thr

Comparison between Experimental and Calculated Results

PDB code

Calculated Binding Energy

Experimental Binding Energy

3BTK(WT) --17.5 -17.8

3BTG -5.3 -5.6

3BTW -12.3 -9.3

3BTE -8.2 -8.5

3BTF -12.4 -10.9

3BTH -8.7 -9.2

3BTM -14.1 -10.3

3BTQ -13.4 -8.6

3BTT -13.9 -7.4

Phase diagram for lysozyme with 3% NaCl at pH=4.5 in 0.1M NaAc buffer.Muschol and Rosenberger, J. Chem. Phys. 107, 1953(1997)

Phase diagram for crystallin Asherie, Lomakin and Benedek, Phys. Rev. Lett. 77, 4832(1996).

Debye-Hückel approximations

• Potential of mean force (w1s): force acting on particle 1 of type s

• Substitute into radial distribution function, obtain Poisson-Boltzmann (PB) equation.

• Approximate RHS of PB equation (linearize by expanding exponents) to get the linearized PB.

1 1( )s s sw r q r q r

s

rqss

seqcr

14

12

rcqceqcs

ss

ssrq

ss

ss

121

1( )r

Basics of Debye-Hückel Theory

• Linear Poisson-Boltzmann equation (outside the ion)

• Laplace’s equation (inside the ion)

• Solve these equations for (r), using boundary conditions

rr

12

12

012 r

ss cq 22 4

where

2 is related to ionic strength

1 1

1 1

q qr

r a

ar

eqr

ar

11

1

0<r < a

r>a

The Maxwell Equations1

· 0, 0

1 4· 4 , .

c t

c t c

BB E

DD B J

Using Fourier Transformation: ·i t ie k r

· 0, 0c

k B k E B

4· 4 , .

ii i

c c

k D k B D J

Constitutive Relations( , )· ,

4( , ) ( , ) ( , )

i

D k E

k k kò

For an isotropic system,

2 2( , ) ( , )( ) ( , ) .t lk k

k k

kk kkk I

Dispersion RelationsIf there is no external charge and current, the condition for a set of homogenous equation to have non-trivial solution is:

2 22

2 2

22

2 2

· ( · ).

( ) ( , ) 0

kc c

kk c

D E E k k E

kkI k

The dispersion relations:

22

2( , ) 0; ( , ) 0t l

ck k k

Connection to Debye-Hückel TheoryAssuming the root that is closest to the real axis is kD, we have

From the longitudinal part of the homogeneous equation

It leads to the linearized Poisson-Boltzmann equation for the potential

which is the starting point of the Debye-Hückel theory if the imaginary part of kD is identified as the conventional Debye screening length.

2 2( ) ( ) 0.rD lk k k

2( ) · 0l k

k

kkE

2 2( ) ( ) 0.Dk k k

The Debye Screening

Let's consider that a unit external charge potential

is applied to the ionic fluid, the screened potential is given by

0 ( ) 1/r r

( )r 0 ( )( )( )l

kk

k

Fourier transformation of the above equation yields

eff

( ) ~Dik re

rr

ò

eff

( )1

2[ ]

D

lk K

d kk

dk

òwhere

2

2( ) (1 )DH

l eff

kk

k Traditional Debye-Hückel model

Solvation Dynamics in Ionic Fluids: A Dielectric Continuum Model

Theoretical Formulation: Caillol, Levesque and Weis, J. Chem. Phys. 85, 6645(1985)

If P(t) is the total dipole moment at time t and J(t) is the current of the ions then dielectric function measured experimentally, εeff(ω), can be written as

With appropriate models of solvents we can compute the above dielectric function directly from simulations and at the same time the solvation correlation function can also be obtained with a solute and the same solvent model from simulations.

eff0

0 0

4 ( ) 4 ( 0)( ) lim( ( ) )

4 1P(0) P(0) P(0) P( ) J(0) J( ) ( 1)

3i t i t

B

i i

i t e dt t e dtVk T

Comparison of C(t) between a direct simulation and a dielectric continuum model using the dielectric function from

the same solvent model (Tosi-Fumi) C153 (Maroncelli) in NaCl melt at 1300K

Comparison of C(t) between a direct simulation and a dielectric continuum model using the dielectric function from

the same solvent model (Margulis and Berne) C153 (Maroncelli) in BMIM PF6 at 1000K

Computing interfacial free energies of liquid-crystal interface: From the nucleation of simple systems to protein crystallization

How to obtain the interfacial free energies of crystal-meltExperimentally:● Measure nucleation rate to extract the interfacial free energy via classical nucleation theoryD. Turnbull, J. Appl. Phys. 21, 1022(1950)M. E. Glicksman, C. Vold, Acta Metall. 17, 1(1969)

● Measure the shape of nucleus to extract the interfacial free energy via Wulff construction, very difficultR.E. Napolitano, S. Liu, and R. Trivedi, Interface Science 10, 217(2002)

Al-4.0wt%Cu

Solid-liquid interface

Liquiddroplet

Theoretically:● Density functional theory is the main tool, but accurate estimate is still very challenging W. A. Curtin, Phys. Rev. B 39, 6775(1989).D. W. Marr and A. P. Gast, Phys. Rev. E 47, 1212(1993). R. Ohnesorge, H. Lowen and H. Wagner, Phys. Rev. E 50, 4801(1995).V.B. Warshasky and X. Song, Phys. Rev. E 73, 031110(2006)

Interface Simulations Previous DFT Our Work100 0.62(2) 0.35 0.79 110 0.62(2) 0.30 0.89 111 0.61(2) 0.26 0.87

Simulations:Cleaving Potential Method:The interfacial free energy of a crystal-melt interface equals to the reversible work required to create the interface.J. Q. Broughton and G. H. Gilmer, J. Chem. Phys. 84, 5759(1986).R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 85, 4751(2000).R. L. Davidchack and B. B. Laird, J. Phys. Chem. B 109,17802(2005).Y. Mu and X. Song, J. Chem. Phys. 124, 034712(2006)Y. Mu and X. Song, Phys. Rev. E . 74,031611(2006)

Capillary Wave Method:The crystal-melt interfacial free energy is related to the height fluctuations of the interface.J. J. Hoyt, M. Asta and A. Karma, Phys. Rev. Lett. 86, 5530(2001).J.J. Hoyt, M. Asta and A. Karma, Mater. Sci. Eng. R41, 121(2003).E. Chacon and P. Tarazona, Phys. Rev. Lett. 91, 166103(2003).J.R. Morris, Z.Y. Lu, Y.Y. Ye and K.M. Ho, Interf. Sci. 10, 143(2002).J. R. Morris, Phys. Rev. B 66, 144104(2002).J. R. Morris and X. Song, J. Chem. Phys. 119, 3920(2003).Y. Mu, A. Houk and X. Song, J. Phys. Chem. B 109, 6500(2005).X. Feng and B. B. Laird, J. Chem. Phys. 124, 044707(2006).

Capillary Wave Method (CWM)

q ijexp( q.r )x yij

ij

h h iA

From the order parameter we cancalculate the height function of the interface and its Fourier transform.You need ~50000 particles to have Good statistics.

22 1

q

( )| |

B

qh

k T

Interfacial free energy

Our Work

(CWM)

Davidchack and Laird

(Cleaving Potential)

Lennard-Jones

100

110

111

0.361(8)

0.369(8)

0.361(8)

0.355(8)

0.366(3)

0.371(3)

0.360(3)

0.347(3)

Hard Sphere

100

110

111

0.62(2)

0.64(2)

0.62(2)

0.61(2)

0.573(5)

0.592(7)

0.571(6)

0.557(7)

Cleaving Potential Method

This approach was introduced by Broughton and Gilmer in 1986 using thermodynamic integration to create the interface reversibly.

J. Q. Broughton and G. H. Gilmer, J. Chem. Phys. 84, 5759(1986).

Davidchack and Laird improved the cleaving potential method through the use of specially designed “cleaving wall” which consists of one or two ideal crystal layers in the chosen orientation.

R. L. Davidchack and B. B. Laird, Phys. Rev. Lett. 85, 4751(2000).

R. L. Davidchack and B. B. Laird, J. Chem. Phys. 118, 7651(2003).

We improved multi-step thermodynamic perturbation method to avoid the hysteresis of the thermodynamic integration, thus, greatly improve the efficiency.

Y. Mu and X. Song, J. Chem. Phys. 124, 034712(2006)

Y. Mu and X. Song, Phys. Rev. E 74,031611(2006).

Cleaving Potential Procedure

Step 1:Split the crystal bulk phase with cleaving walls while maintaining the periodic boundary conditions. Step 2: Split the liquid bulk phase

in a similar way.

Step 3: Juxtapose the cleaved crystal and liquid systems by rearranging the boundary conditions while maintaining the cleaving potentials.

Step 4: Remove the cleaving wall from the combined system. 1 2 3 4w w w w A

Cleaving Potential Method using non-equilibrium work measurements

The Jarzynski identity (Phys. Rev. Lett. 78, 2977(1997)) is :

here is the reciprocal temperature, is the difference in free energy between two system states A and B, and is the work involved in a process taking the system from A to B. The angular brackets indicate that the quantity is an ensemble average performed on the initial equilibrium system state A.

exp( ) exp( )A B AF W

TkB1AB FFF

BAW

However, straightforward employment of thermodynamic perturbation method tends to give large statistical errors in the results, especially for the cases where the free energy difference between two equilibrium states is relatively large.

In order to minimize the statistical errors of the results, we use the Bennett acceptance ratio method to circumvent such difficulties.

Bennett method:J. Comput. Phys. 22, 245(1976),

1 1

1 exp ( ) 1 exp ( )A B B AA BW F W F

Simulation results for LJ systems:

Dependence of the free energy difference in each stage on the number of thermodynamic perturbation steps used for the (100) crystal orientation. It can be seen clearly that the reliable result can be obtained only with a few perturbation steps, which is in contrast to thermodynamic integration where the increment must be very small in order to ensure the reversibility of the integral path.

By summing the results of the four stages, we obtained the interfacial free energies of different crystal orientations for the Lennard-Jones system ( T = 0.617):

Interface Our results Davidchack&Laird (TI)

100 0.371(3) 0.371(3)

110 0.361(3) 0.360(3)

111 0.354(3) 0.347(3)

Yan Mu and Xueyu Song, J. Chem. Phys. 124, 034712(2006).

Crystal structures

A Snapshot of the combined system consisting of orientationally ordered crystal and its melt with s1 interface.

Solid L* γ100 γ110 γ111

PC 0.0 0.59(1) 0.58(1) 0.57(1)

0.15 0.57(1) 0.57(1) 0.57(1)

0.30 0.60(1) 0.60(1) 0.60(1)

γs1 γs2 γs3

CP1 0.40 2.19(2) 2.19(2) 2.48(2)

0.60 2.21(2) 2.21(2) 2.43(2)

1.0 2.65(2)

All of the interfacialfree energies are in the unit of kBT/σ2,

where σ is the diameter of a hard sphere with the same volume as the hard dumbbell, just as the reduced coexistence density, ρ*.Yan and Song, Phys. Rev. E 74,031611(2006)

Acknowledgement

Hyung-June WooXuefeng ZhaoZhigang Yu $$$: PRF, NSF, DOE Vadim WarshavskyYan MuAndrew HoukBongkeun KimJie LuoQiang Zhang

Collaborators: Jake Petrich, Jamie Morris, Mark Hargrove, Dan Armstrong