Analytical ultracentrifugation

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

Measuring the velocity with which the boundary moves

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

Logarithmic plot of boundary versus time

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

Hydration of nucleosome core particle

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

Preferred hydration sites of DNA bases

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)

Making hydrodynamic models

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

Experimental strategies

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

Absorbance optical system of analytical

ultracentrifuge

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