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Time after force is applied
short period (10-9 sec) to reach constant velocity
Resisting force fvfrom viscous drag Driving force Fx
€
− f ⋅v+Fx = 0, or Fx = fv
Directed movement in solution induced by an external force (gravitation, centrifugation, electric field)
⇒ if we measure the velocity of motion produced by a known force we can determine the friction coefficient and diffusion coefficient
Determination of sedimentation- and diffusioncoefficients by analytical ultracentrifugation (AUC)
absorbancemeasurements
Determination of molecular weight and hydrodynamic shape (sedimentation-, diffusions- and friction coefficient) of biological macromolecules in solution
Ff: frictional force
Fc: centrifugal force
Fb: buoyant force
Fc
Fb
Ff
Idealized sedimentation velocity profile in the absence of diffusion
t0
t1
t2
t3
t0
t1t2
t3
radial position r
Sedimentation velocity analysis
t0
t1
t2
t3
t0
t1t2
t3
radial position r
- velocity of band movement -> sedimentation coefficient- spreading of band boundary -> diffusion coefficient- sedimentation and diffusion coefficient -> molecular weight
• mass M of the molecules• density ρ of buffer, 0.9982 g ml-1 for water at 20 ˚C• viscosity η of buffer, 1.002 mPa second for water at 20 ˚C
Parameters that describe the hydrodynamicproperties of macromolecules in solution
D= kTf
• sedimentation coefficient s
• frictional coefficient f
• diffusion coefficient D
v =∂v∂m
• partial specific volume v bar ( ) protein: 0.73 ml g-1, DNA: 0.55 ml g-1
€
v €
s= dr dtϖ 2r
=M ⋅(1-vρ)NA ft
Relations that involve the frictional coefficient
A sphere has radius r and volume V.
€
ft =6 ⋅π ⋅η⋅r
€
Vsphere =43⋅π ⋅r3
For the translation the resulting frictional coefficient ff in a medium with viscosity η can be calculated according to Stokes law.
The volume of a spherical particel can be calculated from its molecular mass, the partial specific volume v bar and Avogadros number NA.
€
Vsphere =M ⋅vNA
Determination of sedimentation- and diffusioncoefficients by analytical ultracentrifugation (AUC)
absorbancemeasurements
Ff: frictional force
Fc: centrifugal force
Fb: buoyant force
Fc
Fb
Ff
€
Fc =ω2 ⋅r ⋅m
€
m0 =m ⋅ν ⋅ρwith
€
Fb = – ω2 ⋅r ⋅mo
€
Ff =– f ⋅v
proportional to the mass m,distance r from the center and angular velocity ω (2π/60 x rpm)
proportional to the mass of the displaced solvent m0,which can be calculated from the density of the solvent ρ and the partial specific volume v bar of the molecule
Determination of the sedimentation coefficient
D= kTf
With the molecular weight M the diffusion coefficient D can be calculated from s or f.
ω2 ⋅r ⋅m – ω2 ⋅r ⋅mo – f ⋅v= 0
At constant velocity the sum of all forces is zero.
M⋅(1-ν ⋅ρ)NA⋅ f
= vω2 ⋅r
= sRearranging and using the molecular weight M of avogadro number NA (=1 mol) particles.
This defines the sedimentation coefficient s in Svedberg (10-13 sec =1 S) as the ratio of velocity to field strength.
€
Fc + Fb + Ff =0
Experimental determination of thesedimentation coefficient
€
v = drbdt= rb ⋅ω
2 ⋅s
€
ln rb t( )rb t0( )
=ω2 ⋅s⋅ t– t0( )
€
M ⋅(1-ν ⋅ρ)NA ⋅ f
= vω 2 ⋅r
= s Definition of the sedimentation coefficient s in Svedberg (10-13 sec =1 S) as derived before
The speed v is determined from the movement of the boundary
after integration
How to included information on spreading of boundary(= diffusion) during sedimentation velocity experiment?
Determination of the diffusion coefficient from a distribution of apparent sedimentation coefficients g(s*)
€
s* = 1ω 2 ⋅t
⋅ln rrm
converting the distance traveled by the particle after time t into a sedimentation coefficient, yields a distribution of apparent sedimenation coefficient g(s*) or c(s*)
t0
tConc
entra
tion
radial position r
The mobility of a particle at a certain time corresponds to a sedimentation coefficient
higher apparent s dueto forward diffusion
lower apparent s dueto forward diffusion
“true” s valuewithout diffusion
0
0.1
0.2
0.3
1 2 3 4 5
g(s*
) (A 2
60Sv
edbe
rg-1
)
s* (Svedberg)
The distribution of apparent sedimentation coefficientsfor two short DNA duplexes of 32 and 59 base pairs
DNA 59 bp
DNA 32 bp
Lamm equation to describe the temporal changes of the concentration distribution of a molecule during sedimentation
s: sedimentation coefficient
D: diffusion coefficient
x: distance from the center of the rotorω: angular velocity
or
€
∂c∂t
=1x∂∂x
x D ∂c∂x− sω2xc
€
∂c∂t
= D ∂ 2c∂x2
+1x∂c∂x
− sω2 x ∂c
∂x+ 2c
no analytical solution but can be solved for specific cases or by using numerical methods to derive s and D from the change of the concentration gradient over time.
Determination of the molecular weight from s and D
D can be determined directly from the shape of the sedimentation band.
M⋅(1-ν ⋅ρ)NA⋅ f
= vω2 ⋅r
= s The sedimentation coefficient s in Svedberg (10-13 sec =1 S)
€
sD=M(1−v⋅ρ) NA f
RT NA f=M(1−v⋅ρ)
RTThis is the Svedberg equation according to which the molecular weight M can be calculated from s and D.
€
D= kTf= RTNA f
Frictional coefficients of spheres
A sphere has radius r and volume V.
€
f =6 ⋅π ⋅η⋅r
€
Vsphere =43⋅π ⋅r3
For the translation the resulting frictional coefficient f in a medium with viscosity η can be calculated according to Stokes law.
The sedimentation coefficient s
The volume of a spherical particel can be calculated from its molecular mass, the partial specific volume v bar and Avogadros number NA.
€
Vsphere =M ⋅vNA
s=M⋅(1-ν ⋅ρ)NA⋅ f
Frictional coefficients of spheres
€
s= M(1−v⋅ρ)NA ⋅6π ηr€
Vsphere =43⋅π ⋅r3 =M ⋅v
NA⇔r3M ⋅v
NA⋅ 34π
⇒ r= 3Mv4 π NA
13
substitute r from equation given above
€
s= M(1−v⋅ρ)
NA ⋅6π η3Mv4 π NA
13
= M23(1−v⋅ρ)
6π η NA
23
34 π
13v13
for s spherical molecule s is proportional to M2/3€
s=0.012 M23(1−v⋅ρ)
v13
after substituting all the constants and water viscosity
Calculated sedimentation coefficients for spherical proteins
1
10
100
1000
1 10 100 1000 104sedi
men
tatio
n co
effic
ient
s20。C
, w (S
)
molecular weight (kDa)
forbidden region
real proteins(nonglobular, hydrated)
hydrated globular proteins
theoretical for unhydrated spheres
6.3S3.9S 9.9S
50 200
15.8S
400
globular, hydrated
Svergun, D.I. et al 1998, Proc. Natl. Acad. Sci. USA. 95:2267-72.
First 3Å hydration layer around lysozyme ~10% denser than bulk water
About 0.3 to 0.4 g H2O per g of protein
Protein hydration
Kuntz, I.D., Jr., and W. Kauzmann. 1974. Hydration of proteins and polypeptides. Adv Protein Chem. 28:239-345.
The amount of protein hydration can be calculatedfrom the amino acid composition
Amino acid Hydration
ionic
Asp- 6
ionicGlu- 7
ionicArg+ 3
ionic
Lys+ 4
polarAsn, Gln, Ser, Thr,Trp 2
polarPro, Tyr 3
NonpolarAla, Gly, Val,IIe, Leu,
Met2
NonpolarPhe 0
DNA “Spine” of hydrationin the minor groove of DNA
because of its high negative charge density DNA is strongly hydrated
about 0.8-1.0 g H2O per g of DNA
per base pair about 22-24 molecules water in direct contact with the DNA
another 16-18 H2O are thought to be also in the primary hydration shell
0.9 g H2O per g DNA corresponds to about 30 molecules water/base pair
Molecular weight determination bysedimentation equilibrium centrifugation
-0.01
0
0.01
ES-1 DNA(32 bp)
ES-2 DNA(59 bp)
NtrC protein NtrC + ES-1
NtrC + ES-2
NtrC-P + ES-2
Analysis of sedimentation equilibrium centrifugation -determination of M independent of D or f
€
Pi∝gi ⋅exp-E ikT
€
PjPi=exp -Ej
kT
exp -E ikT
€
cjci= exp −
E j−E iRT
Meff =M -M0 =M -M ⋅v ⋅ρ=M ⋅(1-v ⋅ρ)
Fz =Meff ⋅ω2 ⋅r=M -(1-v⋅ρ)⋅ω2 ⋅r
Wz =-Meff ⋅ω2 ⋅r ⋅(rj -ri )
€
Wz = -M eff ⋅ω2 ⋅r⋅dr
ri
rj∫
=-M eff ⋅ω2 ⋅
12r 2
ri
rj
=-M eff ⋅ω2 ⋅1
2⋅ rj
2−ri2
€
Ej -Ei =M ⋅(1-ν ⋅ρ)⋅ω2 ⋅12⋅ rj
2−ri2
€
cjci= exp
M ⋅(1-ν ⋅ρ)⋅ω 2 ⋅ rj2−ri
2( )2⋅RT
€
Ar =A0 ⋅expM ⋅(1-ν ⋅ρ)⋅ω 2 ⋅ r 2−r0
2( )2⋅RT
+E
(1)
(2) (3)
Frictional coefficients forellipsoids of revolution
Ellipsoid Volume V a, b half length of two axes
€
V = 43⋅π ⋅a ⋅b2
axial ratio
€
p= ab
Re: radius of sphere with the same Volume
€
Ft =ft
6 π η Re
L
2b
€
p= L2b
cylinder random coil of N segments with length b, Rg radius of gyration
f proportional to ≈ N1/2 or L1/2
b
Friction coefficients for different shapes
1/2
f is proportional to ≈ L1/2
contour lengthL = b·N
€
s=M ⋅(1-vρ)NA ft
∝ MM1/2
∝M1/2
Frictional coefficients for oligomers and polymers
Assume a polymer with N segment and frictional coefficient f1 per segment is fixed and the fluid is moving
Without hydrodynamic interactions the frictional coeffcient would be fN = N · f1
In the presence of hydrodynamic interactions fN < N · f1 This is because, on the average, each segment decreases the fluid velocity near it, and thus each experiences a smaller frictional force
Kirkwood approximation to calculate thefriction coefficients for complex shapes
fn = n⋅ f1⋅ 1+f1
6⋅π ⋅η⋅n 1
Rijj≠i
n∑
i=1
n∑
−1For a complex of n spheres of identical size and frictional coeffizient f1 the total frictional coefficient can be estimated according to the Kirkwood approximation.
DNA-protein complex of NtrC with its enhancer binding sites
Rippe, K., N. Mücke, and A. Schulz. 1998. Association states of the transcription activator protein NtrC from E. coli determined by analytical ultracentrifugation. J. Mol. Biol. 278:915-933.
Kirkwood approximation to calculate thefriction coefficients for complex shapes
fn = n⋅ f1⋅ 1+f1
6⋅π ⋅η⋅n 1
Rijj≠i
n∑
i=1
n∑
−1
For a complex of n segments with frictional coeffizient f1 the total frictional coefficient can be estimated according to the Kirkwood approximation. In this equation Rij is the distance between segments and every distance is counted twice according to the summation. For an object that consists of identical spheres with radius r according to Stokes law we obtain
€
fnf1
=n 1+rn
1Rijj≠i
n∑
i=1
n∑
−1
Kirkwood approximation for a dimer
For direct contact R12 = R21 = 2 r
€
fnf1
= 2 ⋅ 1+r2
12r
+12r
−1
= 2 ⋅ 1+12
−1
= 2 ⋅0.66 =1.33
€
fn =2 ⋅ f1 ⋅0.66 i. e. 66 % of two spheres
€
fnf1
=n 1+rn
1Rijj≠i
n∑
i=1
n∑
−1
Kirkwood approximation for adimer with a long linker
for a long friction free linker R12 and R21 is very large so that
€
fN =2 ⋅ f1 i. e. that of two separate spheres€
r2
1R12
+1R21
≈0
€
fnf1
= 2 ⋅ 1+r2
1R12
+1R21
−1
Kirkwood approximation to calculate thesedimentation coefficient s for bead models
see van Holde p. 205
€
sns1
=1+ rn
1Rijj≠i
n∑
i=1
n∑
Dependence of s and D on molecular mass
€
s= dr dtϖ 2r
=M ⋅(1-vρ)NA ft
∝ MM1/2 or M1/3
sedimentation coefficient s increases with mass
protein (sphere)DNA
protein: DNA: double mass M => 0.8 fold lower D double mass M => 0.7 fold lower D
!
D"M#1
3
!
D"M#1
2
What do we know and want to know
Sedimentation velocity (3 samples, 5-6 h)• species present in the mixture• sedimentation coefficient s• diffusion/frictional coefficient D or f
Sedimentation equilibrium (9 samples, 24 h)• mass M of the complex• Equilibrium dissociation constant (if in ~µM
range)
Calculate from the sequence• mass M of monomer (units)• extinction coefficient• partial specific volume v bar• density ρ of buffer at exp. Temp• viscosity η of buffer at exp. Temp
AUC - Sample Cells & Rotor
• Samples are loaded into cells with clear windows
(quartz or sapphire) (sample vs. reference)
• Cells are placed in a rotor with vertical holes
• N.B. - Balance is critical.Low throughput
Cell assembly for sedimentation equilibrium (left) and velocity (right) runs with 12 mm centerpieces
Screw Ring
(301922)
Screw-Ring
Washer
(362328)
Window Holder
(305037)
Window Holder
(305037)
Window Liner
(362329)
Window Liner
(362329)
Window
quartz
(301730)
sapphire
(307177)
Window
quartz
(301730)
sapphire
(307177)
Centerpiece
(see Table 1)
filling holes (2)
Cell Housing
(334602, includes
plugs and plug
gaskets)
Keyway
Housing Plug (2)
(362327)
Plug Gasket (2)
(327022)
Window Gasket
(327071)
Window Gasket
(327071)
Gasket (aluminum
centerpiece only)
(330446)
Gasket (aluminum
centerpiece only)
(330446)
Screw Ring
(301922)
Screw-Ring
Washer
(362328)
Window Holder
(305037)
Window Holder
(305037)
Window Liner
(362329)
Window Liner
(362329)
Window
quartz
(301730)
sapphire
(307177)
Window
quartz
(301730)
sapphire
(307177)
Centerpiece
(366755)
filling holes (6)
Cell Housing
(368115, includes
plugs and plug
gaskets)
Keyway
Housing Plugs (6)
(362327)
Plug Gaskets (6)
(327022)
Window Gasket
(327071)
Window Gasket
(327071)
Equilibrium
External-Fill Double-Sector
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