Orientational Relaxation in Liquid Crystals and Correlation Between Dynamics and Activity of Small Proteins Biman Bagchi Indian Institute of Science Bangalore.

Orientational Relaxation in Liquid Crystals and Correlation Between

Dynamics and Activity of Small Proteins

Biman Bagchi

Indian Institute of Science

Bangalore – 560012, INDIA

Mr. P. Jose, Dr. S.Balasubramanian, Dr. S. Bandyopadhyay, Arnab MukherjeeMr. Subrata Pal, Mr. Sudip Chakrabarty

Power laws in the orientational Power laws in the orientational relaxation near Isotropic-Nematic relaxation near Isotropic-Nematic

phase-transition (INPT)phase-transition (INPT)

New Experimental results (New Experimental results (Fayer et Fayer et al.2002)al.2002)

Temperature dependent 5-OCB data sets displayed on a log plot.

optical Kerr effect data displaying the time dependence of orientational dynamics of the liquid crystal, 5-OCB at 347 K on a log plot.

Time Scales involvedTime Scales involved

Initial exponential decay occurs with a Initial exponential decay occurs with a time constant in 1-5 ps range.time constant in 1-5 ps range.

The long time Landau-de Gennes The long time Landau-de Gennes exponential decay sets in after 100 ns or exponential decay sets in after 100 ns or so, with a time constant few hundred ns.so, with a time constant few hundred ns.

There is a big window between 10 ps to There is a big window between 10 ps to few hundred ns when decay is very slowfew hundred ns when decay is very slow..

Temperature dependent 3-CHBT data sets displayed on a log plot.

The short time portion of the 5-OCB data at 347 K with the exponential contributions removed on a log plot.

Exponent ≈2/3

Mode coupling theory of orientational Mode coupling theory of orientational relaxation near INPTrelaxation near INPT

20 ( 0)

6

LdGR

S k

D

Origin of the slow down in relaxation can be understood from a mean-field theory which gives the following expression for LdG

where S20(k) is the wave number dependent orientational structure factor in the intermolecular frame. DR is the rotational diffusion coefficient. Kerr experiments measure the k= 0 limit of the time derivative of the collective orientational correlation function C2m(k,t).

,(

2

1sl

l

R

C

zz z

Zwanzig - Mori Continued fraction

2 1 Bl

l l k T

I

0

0ztR z dte N N t

Single Particle Rotation

Mode coupling theory Mode coupling theory calculation of rotational frictioncalculation of rotational friction

The baisc idea is that the torque tcf The baisc idea is that the torque tcf on a tagged ellipsoid slows down on a tagged ellipsoid slows down due to the marked slow down in due to the marked slow down in orientational density relaxation.orientational density relaxation.

Unlike in supercooled liquid, this Unlike in supercooled liquid, this happens at small k.happens at small k.

Expression for the torque can be Expression for the torque can be obtained from the DFT free energy obtained from the DFT free energy functional.functional.

( , , ) ' ' ( ', , ') ( ', ', )N r t dr d C r r r t

R

Bz

z

1slC z

z z

2

20 ( )a tC e erfc a t

2

2 2 2 2 3

1 1.3 1.3.5( ) .....

2 (2 ) (2 )(1 )

xeerfc x

x x xx

(JCP (2002))

Collective orientationCollective orientation

20

1,

1

,

cl

B

R

C k zl l f k k T

zI z k z

f20(k) = 1/S20(k)

The time derivative of the theoretical correlation function, C20(t). Also shown is a t -0.63 power law (5-OCB). At short time, thederivative of the theoretical correlation function decays essentially as a power law.

Thus, the leading term in the expansion varies as t-1/2. The above analysis is valid only after the initial short time decay, very close to the INPT.

Fayer et al JCP (2002,2003)

Simulation detailsSimulation details

12 6ˆ4 ( , , )ˆ ˆ( ( , , ) ) ( ( , , ) )

) )( ( ][ s si j

i j s i j s

U r u ur r u u r r u u

2 2ˆ ˆ ˆ ˆ( . . ) ( . . )ˆ ˆ ˆ( , , )

2 1 ( . ) 1 ( . )[1 ( ) ( )]i j i j

i j si j i j

u r u r u r u rr u u

u u u u

2 21 '2 2

0

ˆ ˆ ˆ ˆ( . . ) ( . . )ˆ( , , ) [1 ( . )]

2 1 ( . ) 1 ( . )[ ]1 ( ) ( )i j i j

i j i ji j i j

u r u r u r u rr u u u u

u u u u

The form of the modified inter-molecular Gay-Berne potential is

Where is the unit vector that passes through the center of mass of a pair of molecules, and are unit vectors that passes through the major axis of a pair of ellipsoidal molecules. and give the strength and range of interaction.

r̂

iu

ju

s is double of the minor axis b, gives molecular elongation (aspect ratio), which is the ratio of end-to-end to side-to-side diameters, = e/s. we have worked with aspect ratio 3. In the expression for , can be given in terms of as =

2

2

1

1

where 0 is the energy parameter and ’ =0.382.The scaling used for moment of inertia is I*= I/m0. The density is scaled as *= 0

2 and the temperature is scaled as T* = kbT/0. The equation of motion is integrated with reduced time (t* = (m0

2/ 0)1/2 ) steps with t = 0.002 t*.

The molecular dynamics simulation is run on a system of 576 Gay-Berne ellipsoids in a Micro-Canonical ensemble. The simulations were run at temperatures T*= 1.0, 1.1, 1.2.

The variation of order parameter at different temperatures along the density axis is shown here.

Phase diagram of Gay-Berne ellipsoids with aspect ratio 3.

Isotropic Isotropic-Nematic coexistence

Nematic* *0.34, 1.0, 576T N

* *0.315, 1.0, 576T N

* *0.2, 1.0, 576T N

Static correlation functionsStatic correlation functions

' ' '1( , , )

N N

j i i ji j

g r r r rN

The radial distribution function for non-spherical molecules in a laboratory fixed frame can be expressed in terms of average over delta functions as

where is the translational coordinate in the arbitrary reference system and is the polar angle for the i’th linear molecule in the laboratory fixed reference frame.

r

i

This pair correlation function can be conveniently represented in an inter-molecular reference frame where z axis passes through the center of mass of two ellipsoids.

1 2 1 2

1 2

', , , ,

, ,

, , ( ) ( ) ( )l l m l m l ml l m

g r g r Y Y

', ,g r

This orientational correlation function can be expanded in spherical harmonics

The coefficients of the spherical harmonic expansion of pair correlation function tend to diverge when isotropic nematic phase transitions approached along the density axis.

1, 2 1 2

2 ', , ,16 , , ( ) ( )l l m l m l mg r g r Y Y

These coefficients of expansion of angular pair correlation functions can be calculated from simulation using the expression

..

Single particle orientational time correlation Single particle orientational time correlation functionsfunctions

ˆ ˆ0 .

ˆ ˆ0 . 0

l i iis

l

l i ii

P e e tC t

P e e

The single particle orientational time correlation function of the rank l is defined as

where the is the unit vector or director associated with i th ellipsoid along the major axis of the ellipsoid and is l th rank Legendre polynomial.

lPˆie

The single particle orientational correlation function shows slows down of orientational relaxation function at higher densities.

The linear portion of the single particle orientational correlation function is fitted to a straight line at density = 0.305.

The temperature dependence of the power law relaxation of single particle orientational correlation function is shown here. All are plotted at density =0.31.

Log-Log Plot

Collective orientational correlation Collective orientational correlation functionfunction

ˆ ˆ0 .

ˆ ˆ0 . 0

l i jic

l

l i ji

P e e t

C t

P e e

Slow down in the relaxation of collective orientational time correlation function. The regions where power law relaxation is dominant are fitted to the function

at density=3.1

0.58 1.1-0.014ty

The Log-log plot of derivative of the collective orientational correlation clearly shows the power law relaxation.

The temperature dependence of the collective orientational time correlation function is shown here

Frequency dependent friction against Laplace frequency

~ z-0.6

Compare with the mode coupling theory prediction of growth as z^{-0.5}

Molecular van Hove correlation functionMolecular van Hove correlation function

' ' '1( , , ) 0 0

N N

j i i ji j

G r r r r t tN

1 2 1 2

1 2

', , , ,

, ,

, , , ( , ) ( ) ( )d l l m l m l ml l m

G r t G r t Y Y

1 2

1, 2

', ,2

,

, , ( ) ( ), 16

d l m l m

l l m

G r Y YG r t

Van Hove correlation function for a molecular liquid can be defined as

In a system of linear molecules, the van Hove correlation function can be represented in inter-molecular reference frame as

This correlation function can be expanded in terms of spherical harmonics to get dynamical orientational pair correlation function (DOPCF). The DOPCF gives a direct measure of relaxation of orientational order at different shells around a linear molecule. The DOPCF can be calculated from the simulation using the expression

The dynamic orientational pair correlation function (DOPCF) is plotted at three different densities. In these figure continuous lines starting from top are plotted at time step of 1 in the time interval 1-10. Then dot-dash lines are plotted at a time step of 10 in the time interval 20-100. Finally the dashed lines are plotted at a time step of 100 in time interval 200-1000. All the lines arranged from top to bottom in the increasing order of time. The top sub-figure is at density =0.295, middle sub-figure is at density =0.305 and bottom sub-figure is at density =0.315.

1, 2 1, 2

1, 2 1, 2

, ,

, ,

, ,( )

, 0 ,l l m l l m

p r Rl l m l l m

G r t G r tC t

G r t G r t

In order to study the orientational relaxation of different cages, we can define orientational correlation function in terms of DOPCF.

where R is a fixed distance at which the relaxation is measured.

The correlation function log is plotted against logarithm of time at equal to the first peak of dynamic orientational pair correlation function. The emergence of power is evident at density =0.305.

( )pC t

The log of the correlation function is plotted against log of time at three different temperatures and at constant density = 0.305.

( )pC t

Viscosity increase near Isotropic-Nematic transition

The frequency dependent viscosity at different densities

Zwanzig (1982); Tang and Evans

Dielectric and Rotational Dielectric and Rotational Friction in LiquidsFriction in Liquids

Nee-Zwanzig (1970)Nee-Zwanzig (1970)

Hu-Zwanzig (1974)Hu-Zwanzig (1974)

0 0

0

( )2( )2 ( )DF

kTi

Derivation of Nee-Zwanzig-Derivation of Nee-Zwanzig-Hubbard-Wolynes from MCTHubbard-Wolynes from MCT

Hydration LayerHydration Layer

protein

Hydration layer

It is the dynamics of this layer that is of interest.

Q: What role does this water play in biological activity ?

: What is the magnitude of dielectric friction on rotation of proteins?

How slow is this?

Bulk water

Dynamic Exchange Model

1) Bound water

3) Free Water

Change in potential experienced by the different types of water molecules

Systems StudiedSystems Studied

An anionic micelle (CsPFO)An anionic micelle (CsPFO) A cationic micelle (CTAB)A cationic micelle (CTAB) A small polypeptide, 1ETN A small polypeptide, 1ETN

(Enterotoxin)(Enterotoxin) HP-36HP-36

Snapshot of the micelle evolved after 3 ns from the initial configuration at 350K.

Red spheres are denote the oxygens in the carboxylate

headgroup, while white ones carbon atoms along the

perfluoroctonate chain. Yellow sphere are cesium ions.

The shape of the micelle is spheroidal, more like an oblate, with aspect ratio close to 0.67.

Bulk

Hydrogen bond life time Dynamics of PHG-Water H-bond (CsPFO-water Micelle)

Pal, Bala and Bagchi, PRL (2002)

BulkBulk

Moment-moment and Dipolar TCF (CsPFO –Water System)

Pal, Bala & Bagchi, JCP (2004)

Vibrational SpectrumPower spectrum for oxygen atoms of water molecules lying within 4.5Å .Blue shift in the frequency corresponding to the O…O…O bending mode.

Power spectrum for hydrogen atom of different interfacial water molecules.

Librational mode.Pal,Bala, Bagchi (PRE, 2003)

(SPC/E water)

(Ohmine, Acc. Chem. Res. (1999)

Toxic Domain Of Heat-Stable Enterotoxin Produced By A Pathogenic Strain Of Escherichia Coli (1ETN)

13 Residues Sequence : XCELCCNPAC AGC

1ETN Protein : Characteristics1ETN Protein : Characteristics

13 amino acid residues : Sequence ---13 amino acid residues : Sequence ---

Cys-Cys-Glu-Leu-Cys-Cys-Asn-Pro-Ala-Cys-Ala-Cys-Cys-Glu-Leu-Cys-Cys-Asn-Pro-Ala-Cys-Ala-Gly-CysGly-Cys

■ 1ETN (152 atoms) + 2962 water molecules

The six Cys residues form The six Cys residues form 3 disulfide3 disulfide bridges bridges It has 3 beta turnsIt has 3 beta turns There are 5 intra-molecular hydrogen bonds There are 5 intra-molecular hydrogen bonds ** 13 waters in the crystal structure** 13 waters in the crystal structure**

Hydrogen Bond Life time Dynamics ofProtein – Water HB (1ETN-Water ( TIP3P) System)

Rotational TCF of Water molecules Faster Water Dynamics near 2 ( Active Site)

Water in hydration layer slows Water in hydration layer slows down, may be even by an order down, may be even by an order of magnitude but does in no way of magnitude but does in no way justify assumption of a rigid justify assumption of a rigid hydration layer often assumed in hydration layer often assumed in hydrodynamic calculations.hydrodynamic calculations.

Calculation of the Total Calculation of the Total frictionfriction

We consider two contributions We consider two contributions explicitly:explicitly:

(a) Hydrodynamic friction on an (a) Hydrodynamic friction on an arbitrary ellipsoid (Triaxial method) – arbitrary ellipsoid (Triaxial method) – Extended Hu-Zwanzig.Extended Hu-Zwanzig.

(b) Dielectric friction for arbitrary (b) Dielectric friction for arbitrary charge distribution (Extended Nee-charge distribution (Extended Nee-Zwanzig)Zwanzig)

Dielectric and Hydrodynamic Dielectric and Hydrodynamic Rotational Friction of ProteinsRotational Friction of Proteins

Protein DF HYD Total ExptProtein DF HYD Total Expt BPTI 16.4 75.4BPTI 16.4 75.4 91.8 96.8 91.8 96.8 1ig5 39.7 78.91ig5 39.7 78.9 118.6 125.0 118.6 125.0 UBQT 19.4 85.0UBQT 19.4 85.0 104.4 118.9 104.4 118.9 FCYT 45.1 82.3FCYT 45.1 82.3 127.4 130.1 127.4 130.1 PTCN 69.3 98.9PTCN 69.3 98.9 168.2 149.5 168.2 149.5 RYBO 82.6 155.3 237.9 186.1RYBO 82.6 155.3 237.9 186.1 HLYS 82.3 140.1 222.4 203.6HLYS 82.3 140.1 222.4 203.6 4ake 123.4 387.9 511.3 478.24ake 123.4 387.9 511.3 478.2 3rn3 88.2 150.5 238.7 235.03rn3 88.2 150.5 238.7 235.0 1mbn 164.5 185.0 349.5 246.31mbn 164.5 185.0 349.5 246.3

DF TR Exptotal

Biological ActivityBiological Activity

Correlation between water dynamics Correlation between water dynamics and biological activityand biological activity

1ETN and HP-36 – atomistic MD 1ETN and HP-36 – atomistic MD simulationssimulations

Radial Distribution Function (C - Water) Water molecules are less ordered near 2 (Active Site)

Moment-moment TCF for the Water Molecules in the Protein (1ETN) Solution for different Trajectory length

HP-36HP-36

Chicken villin head piece – a 36 amino acid long actin binding Thermostable protein

RMSD of the different domains RMSD of the different domains of HP-36of HP-36

Side chain motions of ten Side chain motions of ten ResiduesResiduesof helix-3of helix-3

Note theHoppingMotionOf the amino acids

Radial distribution of water Radial distribution of water around the three helicesaround the three helices

Blue waterBlue water = Nearest neighbor = Nearest neighborGreen water = Next-nearest neighborGreen water = Next-nearest neighbor

Mean Square Displacement of Mean Square Displacement of water molecules around three water molecules around three

heliceshelices

Relaxation of dipolar Relaxation of dipolar CorrelationCorrelation

AcknowledgementAcknowledgement

Dr. S. Balasubramanian (JNCASR)

Mr. Subrata Pal (SSCU, IISc)

Dr. S. BandyopadhyayIIT KGP

Mr. Aanab Mukherjee (SSCU, IISc)

Mr. Prasanth. P. Jose (SSCU, IISc)

Mr. Sudip Chakraborty (IIT KGP)

Rotational correlation time of Rotational correlation time of myoglobin in aqueous solution is found myoglobin in aqueous solution is found to be 45 ns in contrast to the Debye-to be 45 ns in contrast to the Debye-Stokes-Einstein estimate of 14 ns from Stokes-Einstein estimate of 14 ns from its radius.its radius.

This has often been taken as evidence This has often been taken as evidence of a rigid hydration layer, although the of a rigid hydration layer, although the hydration layer here needs to be hydration layer here needs to be rather thick to fully account for rather thick to fully account for diffusion!diffusion!

NMR studies found long residence time NMR studies found long residence time of water molecules close to proteins.of water molecules close to proteins.

((Wuthrich, Halle)Wuthrich, Halle)

Time dependent RMSDTime dependent RMSD

Study by Solvation Dynamics : Local Probe with temporal resolution

Water orientational motions along the solvation coordinate together with instantaneous polarization P

Hynes (1985)

Summary of recent Summary of recent experimental resultsexperimental results

Solvation dynamics (SD) of coumarin Solvation dynamics (SD) of coumarin (C480) in(C480) in cyclodextrin cavity – two slow cyclodextrin cavity – two slow components of 109 and 1200 ps. components of 109 and 1200 ps. (Castner, (Castner, Fleming (1995) ).Fleming (1995) ).

Bhattacharyya and coworkers studied SD Bhattacharyya and coworkers studied SD in micelles, lipids, reverse micelles and in micelles, lipids, reverse micelles and microemulsions – very slow decay – microemulsions – very slow decay – may may involve probe displacement. (1996- 2002)involve probe displacement. (1996- 2002)

Photon echo – Photon echo – Fleming (1999)Fleming (1999)

Dynamic Stokes Shift Dynamic Stokes Shift (Pal, Peon, and Zewail (2002)) (Pal, Peon, and Zewail (2002))

Fluorescence Stokes shift of exposed tryptophan in aqueous protein solutions (Pal, Peon & Zewail, 2002)

SC

Dansyl bonded SC

Monellin

Predictions of the Dynamic Exchange Model ( Nandi And Bagchi (1996))

1. The orientational relaxation at protein/micelle surface is non-exponential, with the short time component equal to that for free water while the long time component is equal to the inverse of the rate of bound free transition.

2. Solvation time correlation function is also non- exponential, with the short time component

identical to bulk water but the long time component is again determined by the rate of bound free transition.

Predictions of the dynamic Predictions of the dynamic exchange modelexchange model

1

1

(2 )

( )Rfast

slow bf

D

k

Theoretical Formulation of DEM

• Dynamics of free water molecules

2 12

( , )( , ) ( ) ( ) ( , )

( , )b

b bb b b f f f b f f fBR b b

tt k d k t d

tD t

21

2

( , )( , ) ( , ) ( )

( ) ( , )f

f WR f f f bb

b f b b b

tD t t k d

tk t d

• Dynamics of bound water molecules

,f b

,f b

,W BR R

D D

Orientations of the free and bound water

Fluctuating densities of the free and bound water

Rotational diffusion const. for free and bound water

• Orientation of molecule does not change during the bound free reaction

1 2( ) ( ); ( ) ( )

f b f b b f b fk k

Water dynamics at the surface of an anionic micelle --- atomistic MD simulations of aqueous CsPFO.

Hydrogen bond lifetime dynamics Rotational and Dielectric relaxation Vibrational spectrum

CsPFOSurfactant

Why CsPFO/Water System ?

• Fluorinated amphiphiles are stable, both thermally and chemically, than their hydrocarbon counter parts.

• It is a simple two component system, provides a homogeneous surface.

• Hydrophilic surface can mimic the properties of proteins .

Simulation DetailsSimulation Details

62 surfactants, each with 25 atoms 62 surfactants, each with 25 atoms 10562 water molecules10562 water molecules SPC/E waterSPC/E water Density of bulk water = 1.02 gms/ccDensity of bulk water = 1.02 gms/cc Static dielectric constant of bulk water is 67Static dielectric constant of bulk water is 67 Fully atomistic MD simulations up to 4 ns in Fully atomistic MD simulations up to 4 ns in

NVT ensemble, with time step of 4 fs.NVT ensemble, with time step of 4 fs.

Hydrogen Bond Definition

PHGC-WO dist < 4.35Å WO-WO dist < 3.50ÅPHGO-WO dist < 3.50Å WO-WH dist < 2.45ÅPHG-Water Eng < -6.25Kcal/mole O-O-H Angle < 30°

h(t) = 1, if a pair of atoms are bonded at time t = 0, otherwiseH(t) = 1, if a pair of atoms are continuously bonded between time 0 and t = 0, otherwise

(0) ( )( )

H H tS t

HTCF’s , and

(0) ( )( )

h h tC t

h

Identity, energetics, and Identity, energetics, and environment of surface water environment of surface water

moleculesmolecules

Three types of water : IFW, IBW1, IBW2Three types of water : IFW, IBW1, IBW2 ( based on the hydrogen bonds they form ( based on the hydrogen bonds they form

with the polar head groups of the micelle)with the polar head groups of the micelle)

Ratio : IFW:IBW1:IBW2 = 1:8:1Ratio : IFW:IBW1:IBW2 = 1:8:1

Pal, Balasubramanian and Bagchi JPC(2002,2003)

-- Entropy Effects

Schematic Representation of Interfacial Waters

(a) IBW1

(b) IBW2

PHGO denotes the Oxygen of the polarHead group of the Surfactant, and PHGCis the Carbon atom in the head group.

WO and WH are the Oxygenand Hydrogen atoms of the Water

SP, SB and BB (2002)

Energy Distribution : Interfacial Water Energy Distribution : Interfacial Water ( Single Particle Energy )( Single Particle Energy )

Energy Profile Energy Profile ( Free and Internal Energy )( Free and Internal Energy )

Bond length distribution of the Bond length distribution of the doublydoubly hydrogen bonded hydrogen bonded

speciesspecies

Two hydrogenbonds are notequal

SP,SB and BB (2002)

Water-PHG Hydrogen bond Life TCF (300K)

S(t)S(t)

amp amp τ[ps]τ[ps]

0.059 0.300.059 0.30

0.315 3.60.315 3.6

0.626 9.10.626 9.1

Avg. τ = 6.8psAvg. τ = 6.8ps

C(t)C(t)

amp amp τ[ps] τ[ps]

0.16 3.40.16 3.4

0.63 29.00.63 29.0

0.21 118.50.21 118.5

Avg. τ = 43.7psAvg. τ = 43.7ps

SP,SB and BB, PRL(2002)

Bulk Water-Water Hydrogen bond Life Time Correlation Function

SLWW(t) τavg= 0.52 ps and CL

WW(t) avg ≈ 6.7 ps

Top to bottom, within4.5Å, 6Å, 10Å and far 28Å respectively

Dipolar time correlation function of water molecules at the micellar surface.

t)( )C t

Within 10Å

Temperature Dependence of Dipolar TCF

300K300K

Avg. Avg. τ = 108 psτ = 108 ps

350K350K

Avg. Avg. τ = 57psτ = 57ps

Dielectric Relaxation Real part Imaginary part

Cole-Cole plotWater only

T=300 K

Solvation TCF at the micellar surface. Note the slow decay at the long time.

t)( )sC t

Partial Solvation TCF

Temperature Dependence of Solvation TCF

300K300K

amp amp ττ (ps) (ps)

0.15 0.30.15 0.3

0.15 3.90.15 3.9

0.70 56.40.70 56.4

Avg. Avg. ττ = 40.0ps = 40.0ps

350K350K

amp amp ττ(ps)(ps)

0.15 0.200.15 0.20

0.20 2.600.20 2.60

0.65 30.00.65 30.0

Avg. Avg. ττ = 20.0ps = 20.0ps

Slow translational Diffusion of of interfacial water

Effects of Escherichia Coli (Cholera Toxin)

• E. Coli is a leading cause of foodborne illness. Infection often leads to diarrhea, and occasionally to kidney failure.

The toxic domain of E. Coli is a 13 amino acid domain, where the pathogenic (enzyme binding) part is believed to be the beta turn between 7 (asparagine) and 10 (cysteine) amino acid residues.

Enhanced Friction on Protein Enhanced Friction on Protein Motion due to slow waterMotion due to slow water

The mode coupling theory approach gives The mode coupling theory approach gives an expression for enhanced rotational an expression for enhanced rotational friction in terms of dynamic structure factor friction in terms of dynamic structure factor of the water molecules hydration layer.of the water molecules hydration layer.

Remember that translational modes of the Remember that translational modes of the solvent plays important role in translational solvent plays important role in translational and rotational friction on ions and dipoles and rotational friction on ions and dipoles (Wolynes (1983), Chandra & Bagchi (1989)).(Wolynes (1983), Chandra & Bagchi (1989)).

Dipolar TCF for different Interfacial water molecules

Trajectory of Four Water Molecules

Dwph-G = Shortest distance of each of the water molecules to the micellar surface

Slow Dynamics at theProtein Surface Due toDynamic Exchange BetweenFree and Bound Water.

Orientational correlation function of a local Orientational correlation function of a local directordirector

In a system of ellipsoids it is possible to define a local director in terms of the sum of the unit vectors.

1

1

ˆˆ

ˆ

N

iiN

ii

eU

e

Hence the correlation function is defined as

ˆ ˆ0 .

ˆ ˆ0 . 0

ldl

l

P U U tC t

P U U

Since there is no asymmetry along the axis in the interacting potential sum of unit vectors that initially has a direction will vanish due to the rotation of the directors.This rotational diffusion of the ellipsoid can be modeled in terms of a symmetric double well potential. The arbitrary initial direction of the ellipsoid is in the first well of this symmetric double-well potential and a rotation of from the initial direction, which is indistinguishable from this initial direction is the other well of the symmetric double-well potential. For the rotation the ellipsoid has to over come the potential barrier that is created by it's neighbors. Correlation time can be defined in terms of the relaxation time of the initial direction of the resultant vector. Relaxation of this correlation function can be considered as lose of polarizability of a small region of the system of molecules due to random rotation.

The Log-log plot of the orientational correlation function clearly shows the emergence power law of relaxation.

The linear fit of the second rank OTCFof the local director at density=0.315, and temperature =1 the exponent obtained is 0.34.

The log-log plot of second rank of the OTCF of the local director, is plotted for different temperatures and at constant density=0.315