Accurate hydrodynamic transport properties for biomolecules Sergio Aragon San Francisco State University Dept. of Chemistry and Biochemistry CalTech PASI Jan 4- 16, 2004
Jan 20, 2016
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