Accurate hydrodynamic

transport properties for biomolecules

Sergio Aragon

San Francisco State University

Dept. of Chemistry and Biochemistry

CalTech PASI Jan 4-16, 2004

Acknowledgements

Tilman Rosales

Martin Perez

David Hahn, Post-Doctoral Fellow

Funding:

NIH MBRS SCORE Grant SO6 GM52588 (Aragon).

NIH: MBRS-RISE (Rosales & Perez))

I. INTRODUCTION

How do we find the size and shape of molecules in solution?Simple expressions everyone knows.

II. HOW DO WE KNOW HYDRODYNAMICS WORKS? Stick boundary conditions. Slip boundary conditions.

III. HYDRODYNAMICS WITHOUT BEADS Focusing on the SURFACE

V. SUMMARY & OUTLOOK

OUTLINE

IV. APPLICATION TO BIOMOLECULESProteinsNucleic Acids

MOLECULAR SIZE AND SHAPE IN MOLECULAR SIZE AND SHAPE IN SOLUTION?SOLUTION?

IN SOLUTION: Diffusion

Tether + atomic force microscope

Optical Tweezers

SOLIDS: x-Ray Diffraction, Microscopy

DIFFUSION SENSITIVE METHODS:

Dynamic Laser Light ScatteringTransient Electric or Magnetic BirefringenceFlow BirefringenceFluorescence Polarization AnisotropyFluorescence photobleaching recoveryMagnetic Resonance

Molecular Sizes: Nano-scale

• Small molecules are less than 2 nm in scale.• Globular proteins range from 2 to 10 nm in

scale.• DNA ranges from 3 to 1000 nm in scale.• Transport properties in the nano-scale can

obtained to high accuracy using classical hydrodynamics (with appropriate boundary conditions).

F i c k ’ s F i r s t L a w

F = - D c x t

x( , )

F i c k ’ s S e c o n d L a w o r , t h e “ D i ff u s i o n E q u a t i o n ”

Dt

txcx

txc

),(2

),(2

E i n s t e i n R e l a t i o n

D = k T / f D = k T f - 1

S t o k e s - E i n s t e i n R e l a t i o n

D = k T / ( 6 R )

f ~ g / s ; D ~ c m 2 / s

Simple Expressions Everyone Knows

IgG3

Semi-flexible polymerRod polymer

SOME COARSE GRAINED BEAD MODELSSOME COARSE GRAINED BEAD MODELS

I. SLIP boundary conditions are not accessible at present.

Small to intermediate sized molecules in non-hydrogen bonding solvents cannot be accurately treated.

II. Hydrodynamic interaction tensors are approximate

Greatest errors occur for touching and overlapping beads at the Rotne-Prager level. Use of high order infinite series instead, runs into matrix inversion problems.

III. No hydrodynamic interaction expressions are available for unequal sized overlapping beads. This makes atomistic level modeling difficult and leads to coarse grained models or ad hoc bead resizing to avoid problems.

LIMITATIONS OF BEAD MODELSLIMITATIONS OF BEAD MODELS

HYDRODYNAMICS WITHOUT BEADS

• Hydrodynamics occurs at the SURFACE

• An Exact solution can be written down for the Stokes Equations in terms of the Oseen tensor.

• Both SLIP and STICK boundary conditions can be exactly formulated.

• Solvent size can be taken into account

Youngren & Acrivos, J. Chem. Phys. 63, 3486 (1975).

Theory: 3.88 ps/cpExperiment = 3.53 + 0.07 ps/cp

Alms et al. J. Chem. Phys. 59,5570 (1973).

10% discrepancy!

BENZENE TUMBLING IN NON-POLAR SOLVENTSBENZENE TUMBLING IN NON-POLAR SOLVENTS

Slip boundary conditions

HYDRODYNAMICS

Navier-Stokes Equations -- (complicated!)

Stokes “Creeping Flow Equations”

HYDRODYNAMIC BOUNDARY CONDITIONS

2u= P

u0

STICK => Fluid layer sticks to surface

SLIP => Fluid layer slips by surface

BASIC HYDRODYNAMICS

Valid for small Reynolds number.

Reynolds Number

• Definition of Reynolds Number

Flow through tube Diffusive flow

• Re >> 2000 turbulent flow

• Re < 2000 laminar flow

• Re << 1 Stokes flow

η

ρ vdRe

a η 6π

kTD

a η π6

ρ Tk Re

2

η

ρ DRe

Reynold’s Number as a function of Radius

/a(A)10 x 1.1Re 3

0.5 1 1.5 2

0.05

0.1

0.15

0.2

0.25

Radius a in Angstroms

Rey

nold

s nu

mbe

r

YOUNGREN-ACRIVOS METHOD

)(

))((

8

1),(

)().,()()( 0

xf

yx

yxyx

yxyx

dSxxfyxyuyvSp

2IT

T

is the unknown Surface Stress Force.

For Stick Boundary conditions: u(y) = vp + xrp

J. Fluid Mech.. 69,377(1975)

Discretize the surface:

inversion. matrix by solve

...

...

.........

...

)(

...

)(

matrix 3x3 ),( )(f.)(

element surface on constant )(f

1

13

1

331

111

13

1

j1

j1

vf

f

f

yv

yv

dSxyxTyyv

yS

NxNNNxNNN

N

NxN

kjkj

N

jjkk

jj

N

jjp

j

G

GG

GG

GG

HOW DO WE CALCULATE THE TRANSPORT TENSORS?HOW DO WE CALCULATE THE TRANSPORT TENSORS?

Ellipsoids

Triangulations for an oblate ellipsoid of axial ratio 4 with 528 triangles.

Triangulations for a prolate ellipsoid of axial ratio 1/4 with 504 triangles.

Calculate the Total Force and Torque:

zyxpp

zyxpp

ppj

N

jjp

ppj

N

jj

v

vvvv

vfxrT

vfF

,0,0,0,,0,0,0,,0

,0,0,0,,0,0,0,,0

..

..

1

1

rrrt

trtt

KK

KK

Do 6 BE calculations: Note that G matrix is the same for all!

This makes the K6x6 matrix. Invert it to obtain the D6x6 matrix.

DONE!

GIVEN THE SURFACE STRESSES, WE HAVE EVERYTHING!GIVEN THE SURFACE STRESSES, WE HAVE EVERYTHING!

Ellipsoid Extrapolations

0.0006 0.0008 0.0012x

0.4905

0.491

0.4915

0.492

Data & Response

0.0006 0.0008 0.0012x

-1.5 ´ 10 -6

-1 ´ 10 -6

-5 ´ 10-7

5´ 10-7

1´ 10-6

Residuals

:ParameterTable®

Estimate SE TStat PValue1 0.488534 0.0000106824 45732.5 0x 2.70891 0.0240266 112.747 0.0000786579x2 - 132.873 12.3924 - 10.7221 0.00858652

,

RSquared® 0.999997, AdjustedRSquared® 0.999994,

EstimatedVariance® 3.93299 ´ 10- 12>

Ellipsoids: Comparison with Theory

OBLATE

p Dtt % Dtt|| % Drr % Drr|| %

2 0.84104 0.004 0.73655 0.02 0.22105 0.06 0.17728 0.01

4 0.48853 0.004 0.38441 0.02 0.033933 0.07 0.027774 0.03

8 0.26664 0.01 0.19514 0.02 4.5054 10-3 0.07 3.9623 10-3 0.04

30 0.076401 0.02 0.052341 0.02 8.7111 10-5 0.02 8.3765 10-5 0.07

PROLATE

p Dtt % Dtt|| % Drr % Drr|| %

1 1.33356 0.02 1.33356 0.02 1.0003 0.03 1.0003 0.03

1/2 0.96702 0.007 1.10782 0.03 0.3323 0.02 0.61999 0.03

1/4 0.64839 0.006 0.83443 0.003 0.073631 0.01 0.34671 0.003

1/8 0.40954 0.0007 0.57596 0.04 0.013296 0.008 0.18224 0.04

1/30 0.15318 0.01 0.23973 0.06 3.9941 10-5 0.02 0.049895 0.13

Space-filled model of lysozyme (6LYZ), smooth quartic surface and tesselation of lysozyme.

Table I. Specific volumes of selected proteins.

ProteinSpecific Volume

(mL/g)

  VDW4 Exp5,6 This work

Lysozyme (6LYZ) 0.526 0.696 0.691

ChymotrypsinogenA (2CGA) 0.527 0.703 0.697

Myoglobin (1MBO) 0.569 0.721 0.720

Ribonuclease (7RSA) 0.539 0.745 0.747

Histogram ofTriangle areas for Ovalbum in

0

50

100

150

200

250

1.104 1.546 1.988 2.43 2.872 3.313 3.755 4.197 4.639 5.081 5.522 5.964 6.406 6.848 7.289 7.731 8.173 8.615 9.057 9.498 9.94

Patch Area in A^2

Nu

mb

er

of

pa

tch

es

Coalesce Histogram

Extrapolation for Lysozyme

Estimate SE TStat

1 1.10262 10-6 2.68559 10-10 4105.68

x 2.38376 10-5 4.60052 10-5 51.815

RSquared -> 1.

EstimatedVariance -> 1.22447 10-20

Diffusion Coefficients depend on hydration thickness

0.8 0.9 1.1 1.2 1.3 1.4 1.5

1.01 ´ 10 -6

1.02 ´ 10-6

1.03 ´ 10 -6

0.8 0.9 1.1 1.2 1.3 1.4 1.5

1.55 ´ 10 7

1.575 ´ 10 7

1.625 ´ 10 7

1.65 ´ 10 7

1.675 ´ 10 7

1.7 ´ 10 7

1.725 ´ 10 7

Drr

Dtt

Table II. Comparison of transport properties and hydration content = 1.1 +- 0.1

  Exp. Data2 Calc. DataComputed Hydration

(NMR freeze3)

Computed Hydration

VDW4

Protein Dt Dr Dt Dr|| Dr trace (gH2O/gprotein) (gH2O/gprotein)

  107cm2/s 105 s-1 107cm2/s 105 s-1 105 s-1    

Lysozyme (6LYZ)11.2(.2) 2.0(.1) 11.0 1.9 2.16 0.325

(0.34)0.386

Chymotrypsinogen (2CGA)

9.2(.2) 1.28(.01) 9.24 1.22 1.26 0.303(0.34)

0.401

Myoglobin (1MBO)

10.4(.8) 1.67(.05) 10.2 1.62 1.74 0.314(0.42)

0.399

RibonucleaseA (7RSA)

10.68(.1) 2.2(.1) 10.2* 1.87 2.11 0.388*(0.34)

0.381

Protein Volume/Hydsub-

units Mass

(kDa)

V(cm3/g)

h(g/g)

Calc. Exp. Ref. % Err. Calc. Exp. Ref. % Err.

BPTI (5PTI) 1 6.5 0.699 0.718 1 -3 0.414

Cytochrome c (1HRC) 1 12.4 0.706 0.715 1 -1 0.336 0.35 5 -4

Ribonuclease (7RSA) 1 13.7 0.687 0.703 1 -2 0.360

Lysozyme (6LYZ) 1 14.3 0.699 0.703 1 -1 0.325 0.34 5 -4

-Lactalbumin (1HFX) 1 14.4 0.692 0.704 2 -1 0.329 0.362 9 -9

Myoglobin (1MBO) 1 17.2 0.726 0.745 1 -3 0.348 0.42 5 -17

Trypsin (1TPO) 1 23.2 0.728 0.727 1 0 0.286

Trypsinogen (1TGN) 1 24.0 0.702 0.73 3 -4 0.290

Chymotrypsinogen A (2CGA) 1 25.7 0.728 0.721 1 1 0.304 0.34 5 -11

Elastase (1EST) 1 25.9 0.732 0.73 1 0 0.294

Subtilysin (1SUP) 1 27.5 0.722 0.731 1 -1 0.260

Carbonic Anhydrase B (2CAB) 1 28.7 0.703 0.731 1 -4 0.283

Taka - Amylase A (6TAA) 1 54.0 0.733 0.700 4 2 0.223

Apo Ovotransferrin (1AIV) 1 75.4 0.722 0.732 11 -1 0.328 0.28 12 18

Transferrin (1H76) 1 76.0 0.711 0.725 5 -2 0.289

-Lactoglobulin (1BEB) 2 36.7 0.705 0.751 5 -6 0.294 0.29 5 0

Oxyhemoglobin (1HHO) 4 64.6 0.727 0.749 6 -3 0.295 0.42 5 -29

Alkaline Phosphatase (1ALK) 2 94.7 0.740 0.725 7 2 0.219

Citrate Synthase (1CTS) 2 97.9 0.711 0.733 8 -3 0.245 0.339 10 -28

Lactate Dehydrogenase (6LDH) 4 146.2 0.772 0.741 2 4 0.231 0.362 9 -36

Aldolase (1ADO) 4 156.0 0.754 0.743 5 1 0.258

Catalase (4BLC) 4 232.0 0.746 0.73 5 2 0.205 0.290 9 -29

Protein Transport

sub-

unit

s

Mass (kDa)

Dt(10-7cm2/s) Dr(107s-1)

Calc. Exp. Ref.%

Err.Calc.(1) Calc.(2) Calc. Exp.

Ref.

% Err.

BPTI (5PTI) 1 6.5 13.66 14.4 1 -5 4.96 3.48 3.98 4.25 33 -6

Cytochrome c (1HRC) 1 12.4 11.63 11.1 - 11.6 2 - 4 3 2.59 2.36 2.46

Ribonuclease (7RSA) 1 13.7 10.84 10.68 5 2 2.34 1.73 1.93 2.01 48 -4

Lysozyme (2CDS) 1 14.3 10.99 10.9 6, 39 - 41 1 2.62 1.82 2.09 2.04 42 2

-Lactalbumin (1HFX) 1 14.4 10.84 10.57 7 2 2.48 1.73 1.98

Profilin (1PNE) 1 14.8 10.74 10.6 8 1 2.15 1.84 1.95 1.57 8 24Myoglobin (1MBO) 1 17.2 10.24 10.4 9 -2 1.88 1.56 1.67 1.46 56 13

Leghemoglobin (1LH1) 1 17.3 10.26 10.0 10 3 1.99 1.53 1.68

-Lactoglobulin (3BLG) 1 18.4 10.07 1.74 1.56 1.62 1.61 51 1

Cellulase (2ENG) 1 22.0 9.63 9.8 62 -2 1.50 1.37 1.41

Somatotropin (1HGU) 1 22.1 8.84 8.88 11 0 1.31 0.95 1.07

Trypsin (1TPO) 1 23.3 9.50 9.3 12 2 1.51 1.28 1.35 1.13 53 19

Trypsinogen (1TGN) 1 24.0 9.49 9.68 13 -2 1.49 1.28 1.35

Chymotrypsinogen A (2CGA) 1 25.7 9.04 9.23 14 -2 1.25 1.14 1.17 1.1 47 6

Elastase (1EST) 1 25.9 9.06 9.5 15 -5 1.28 1.13 1.18

Savinase (1SVN) 1 26.7 9.35 1.36 1.27 1.30 1.35 46 -4

Subtilysin (1SUP) 1 27.3 9.10 9.04 16 1 1.25 1.17 1.20

Carbonic Anhydrase B (2CAB) 1 28.7 8.84 8.89 17 -1 1.20 1.04 1.09 1.08 50 1

Taka - Amylase (6TAA) 1 54.0 7.22 7.37 18 -2 0.765 0.506 0.592

Human Serum Albumin (1AO6) 1 69.0 6.17 6.15 58 0 0.412 0.330 0.357 0.349 57 2apo Ovotransferrin (1AIV) 1 75.4 5.86 6.14 52 -5 0.0408 0.0259 0.0309 0.0217 52 42Transferrin (1H76) 1 76.0 5.96 5.73 - 6.0 19 - 21 1 0.422 0.259 0.313 0.3 61 0

Multi-Subunit protein Transport

-Lactoglobulin (1BEB) 2 36.7 7.74 7.3 22 - 24 5 1.004 0.579 0.721 0.75 51 -4

Oxyhemoglobin (1HHO) 4 68.0 6.95 6.5 - 6.9 24 - 25 4 0.594 0.509 0.537 0.52 21 4

KDPG Aldolase (1EUN) 3 69.2 6.22 5.6 26 11 0.415 0.369 0.383

Alkaline Phosphatase (1ALK) 2 94.7 5.92 5.7 27 4 0.439 0.269 0.32

5

Concanavalin (2CTV) 4 96.2 5.75 5.6 24, 28 4 0.303 0.290 0.295

Citrate Synthase (1CTS) 2 97.9 5.82 5.8 29 0 0.380 0.276 0.310

Glucose Oxidase (1GPE) 2 133.7 5.45 5.02 - 5.13 30, 59 7 0.297 0.238 0.25

8

Canavalin (2CAV) 3 141.0 5.32 5.10 24 4 0.243 0.215 0.225

Lactate Dehydrogenase (6LDH) 4 145.2 5.08 4.99 31 2 0.216 0.201 0.20

6 0.20 55 5

Aldolase (1ADO) 4 156.0 4.66 4.29 - 4.8 32 - 35 3 0.166 0.147 0.15

3

Nitrogenase MoFe (2MIN) 4 220.0 4.41 4.0 36 10 0.166 0.120 0.135

Catalase (4BLC) 4 230.3 4.49 4.1 24, 37 10 0.156 0.138 0.144

Xanthine Oxidase (1FIQ) 6 270.0 3.94 3.9 38 0 0.133 0.0747 0.0940

Protein

Mass kDa

[] (cm3/g) Dt(10-7cm2/s)

Calc. Exp. Ref.%Err

Calc. Exp. Ref.%Er

.

Cytochrome c (1HRC) 1 12.4 3.04 2.74 1 12 11.63 11.1 - 11.6 29 - 31 3

Ribonuclease (7RSA)a 1 13.7 3.52 3.30 - 3.50 2, 38 3 10.84 10.68 32 2

Lysozyme (2CDS) 1 14.3 3.22 2.98 - 3.00 3, 4 8 11.04 10.9 33 – 36 1

-Lactalbumin (1HFX) 1 14.4 3.38 3.4 5 0 10.84 10.57 45 2

Myoglobin (1MBO) 1 17.2 3.37 3.15 - 3.25 6, 7 5 10.24 10.4 46 -2

Trypsinogen (1TGN) 1 24.0 3.00 2.96 8 1 9.49 9.68 47 -2

Chymotrypsinogen A (2CGA) 1 25.7 3.23 3.12 10 3 9.04 9.23 48 -2

Carbonic Anhydrase B (2CAB) 1 28.7 3.08 2.76 - 3.2 11, 44 3 8.84 8.89 49 -1

Taka - Amylase A (6TAA) 1 51.2 3.23 3.3 12 -2 7.22 7.37 50 -2

Human Serum Albumin (1AO6) 1 66.2 3.92 3.9 41, 42 0 6.17 6.15 43 0

Ovotransferrin (1OVT) 1 75.5 3.86 3.8 13 2 6.03 5.72 51 5

Transferrin (1H76) 1 76.0 3.85 4.0 14 -4 5.96 5.73 – 6.0 52, 53 1

Insulin (9INS)2

6.4 3.15 2.9 66 9 14.45

-Lactoglobulin (1BEB) 2 36.7 3.65 4.0 16 – 18 -9 7.74 7.3 54 5

α-Chymotrypsin (5CHA) 2 49.7 3.27 4.1 39 -20 7.24 7.40 40 -2

Ricin (2AAI) 2 61.7 3.35 2.96 67 14 6.61 6.0 68 10

Oxyhemoglobin (1HHO) 4 68.0 2.87 3.6 - 4.0 19, 21 -18 6.95 6.5 – 6.9 55, 56 7

Alkaline Phosphatase (1ALK) 2 94.7 3.17 3.4 23 -7 5.92 5.7 57 4

Citrate Synthase (1CTS) 2 97.9 3.20 3.95 24 -20 5.82 5.8 58 0

Glucose Oxidase (1GPE) 2 133.7 2.83 4.0 25 -29 5.45 5.13 59 6

Lactate Dehydrogenase (6LDH) 4 145.2 3.21 3.8 26 -16 5.08 4.99 60 2

Aldolase (1ADO) 4 156.0 3.87 4.0 27, 36 -3 4.66 4.29 - 4.8 61 - 64 5

Catalase (4BLC) 4 230.3 3.15 3.9 28, 37 -19 4.49 4.1 65 10

The Intrinsic Viscosity of Proteins.

The Ubiquitin Problem

Translational and rotational diffusion coefficients calculated for ubiquitin, ubiquitin modified and ubiquitin clipped with a hydration layer of 1.5 Å. Dt and Dr experimental are 14.9 x107 cm2/s and

4.02x107 s-1

 Protein Dt 107 cm2/s Dr 107 s-1

UBQ 12.58 3.10

UBQ modified 12.96 3.46

UBQ clipped 13.32 3.76

DNA oligomer tessellation

Uniform hydration Non-uniform hydration

Table IV: Uniform Hydration of a DNA oligomer

Uniform inflation

-10.00

-8.00

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Infl ation to all atoms(A)

% error rotation

% error translation

Table V: Non-uniform Hydration of a DNA oligomer

Nitrogen Inflation

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0 1 2 3 4

Inflation to Nitrogens(A)

% E

rro

r

% error rotation

% error translation

Nitrogen inflation of 2.6 A yields a 1% error in both Dt and Dr.

Conclusions I

• Extrapolation is needed for high accuracy.• A single calculation yields a transport property

within 2% of the extrapolated value.• Numerical accuracy is better than 0.1%• Precision is 0.1 % for Dt and 0.3% for Dr with

extrapolation.• Hydration content and specific volumes are

obtainable, in good agreement with experiment• Using 1.1 A hydration, we can match transport

properties for a broad range of proteins within their experimental error.

Conclusions II

• Uniform hydration layer describes hydrodynamic transport of proteins well.

• We appear to be detecting a difference between the crystal and solution structures for multi-subunit proteins.

• Nucleic acids are better described by non-uniform hydration, with more water in the grooves.

• This work demonstrates the effectiveness of the boundary element method for the calculation of the transport properties of biomolecules, and their intrinsic advantage over traditional bead methods.

References

1.Antiosewicz,J. and Porschke,D. J. Phys.Chem. 93,5301-5305 (1989)2.Youngren,G.K. and Acrivos,A,J. Fluid. Mech. 69,377-402 (1975)3.Connolly,M.L., J. Mol. Graphics 11, 139-141 (1993)4.Zhou,H-X.,Biophys.J. 69, 2286-2297 (1995)5.Squire,P.G. and Himmel,M.E. Arc. Bioc. Bioph.,196,165-177 (1979)6.Kuntz,I.D. and Kauzman, W. Ad. Protein Chem.,28,239-345 (1974)7.Bull,H.B. and Breese,Arch.Biochem. Biophys.,128,497-502 (1968)8. Eimer, W. and Pecora, R., J.Chem.Phys., 94, 2324 (1991)

Martin Perez, M.S. 2003, Ph.D. candidate, UC San Diego

Tilman Rosales. M.S. 2002, Ph.D. candidate, U. Maryland/NIH

Chris Zimmer, M.S. 2003, Ph.D. candidate, UC Davis.

Ryan Moffet, B.S. 2002, Ph.D. candidate, UC San Diego

Chris Potter, B.S. Chem

Heather Harding, B.S. Chem

David Hahn, Ph.D. Postdoctoral Fellow

ARAGON GROUP

Visualization & Interactive Computation Projects

1. Triangulation Visualization & smart triangulator?

Rotate a triangulation real time to visualize shape. Generate a smart triangulator to make Coalesce obsolete.

2. Manually assisted hydration program

Need to avoid a very very expensive computation first.

3. Web based interactive computation.

Integration of fortran and visualization via a web accesible user interface. Enable non-expert to perform high precision hydrodynamic computations.

Fig. 2 Lysozyme: Explicit Hydration

256 waters are included within a cutoff distance of 3.25 A from the molecular surface. [Solvate Program]

Table III. Comparison of Uniform Hydration and Explicit Hydration for Lysozyme

A^2 A*3 10^6 cm^2/s 1/s Hydrationlysozyme molec surface volume dt dr1 dr2 dr3 patches h # Waters

1.3 5837.369 24663.206 11.31 1.967E+07 2.018E+07 2.843E+07 29501.4 5868.519 25354.493 11.23 1.927E+07 1.973E+07 2.772E+07 28141.5 5909.729 25990.992 11.17 1.903E+07 1.950E+07 2.719E+07 2812 0.445 354

Dt Expt: 11.2Explicit hydrationcutoff distancelysozyme 3.24 7195.532 22867.055 11.06 1.867E+07 1.937E+07 2.622E+07 3921 0.301 245lysozyme 3.25 7200.115 23188.109 11.36 1.894E+07 1.968E+07 2.622E+07 4001 0.315 256lysozyme 3.26 7265.007 23444.381 11.04 1.844E+07 1.911E+07 2.706E+07 3942 0.327 266

lysozyme 3.25 7200.115 23188.109 11.14 1.896E+07 2.020E+07 2.842E+07 4653 0.315 256