Ch 24 pages 636-643 Lecture 11 – Equilibrium centrifugation

Ch 24pages 636-643

Lecture 11 – Equilibrium centrifugation

Sedimentation can be used very effectively to separate, purify and analyze all kind of cellular components

It can be understood using the mechanical analogy with flow under gravitation. The steady state velocity at which a particle move under centrifugation is determined by the balance between the angular velocity at which centrifugation occurs and the opposing buoyancy and frictional forces.

Summary of Lecture 10

0 fbg FFF

rsr

f

Vm

dt

drv 2221

We have introduced the sedimentation coefficient:


f

Vms

21

Its dimensions are sec, but a more convenient unit is the Svedberg: S=10-13 s

Values for s are usually referred to pure water at 293K=20oC. Under these conditions, the sedimentation coefficient is indicated as follows:

R

Vms

w

ww

,20

,202,20 6

1

A sedimentation coefficient measured under other conditions, i.e. in a buffered aqueous solution b and/or at another temperature T can be related to standard conditions by the equation


Boundary sedimentation is an equilibrium technique that can be used to separate and analyze macromolecules by sedimentation.

R

Vms

w

ww

,20

,202,20 6

1

w

wbTw

V

Vss

,202

,202,,20

1

1

A macromolecular solution subjected to a centrifugal field will quickly attain a steady state condition in which transport of solute mass occurs at constant velocity and a concentration gradient will be generated

If we spin very fast, eventually the entire macromolecular population will deposit at the bottom of the tube, at a rate that depends on the centrifugal speed and the density and viscosity of the solvent

If we do not spin too fast, centrifugation and diffusion will balance each other out so that the system will attain equilibrium, at which point net transport will cease and transport velocity is zero

Equilibrium Centrifugation

This is because transport by centrifugation and by diffusion will oppose and balance each other: centrifugation will generate a density gradient, and diffusion will try and eliminate such gradient

The rate at which equilibrium is reached depends on kinetic properties (diffusion coefficient, angular velocity etc)

The equilibrium state does not

The concentration at equilibrium is only determined by thermodynamic properties of the system and not by sedimentation coefficients, diffusion etc


The equilibrium concentration can be derived by considering Boltzmann’s distribution. Let us consider again the analogy with gravitation. Molecules in a gravitational field will have different energy depending on whether they are higher or lower; the distribution of molecules in different energy levels is given by Boltzmann’s expression:


kTEEP

Pji

j

i /exp

The probability ratio is equivalent to the ratio of concentrations; if we express the energy per mole instead of per particle by multiplying by Avogadro’s number, we find:

RTEEC

Cji

j

i /exp

Let us return to centrifugation; the centrifugal acceleration for a centrifuge spinning with angular speed is r, the force acting on the particle is


Therefore, the energy is:

rVm 22 )1(

2

)(22

02

0

rmmrdrmmdrFFrU lcentrifugabuoyancy

222 )1(

2rV

mE

Substituting this expression we immediately find the expression for the concentration as a function of the axial distance r:


222 )1(

2rV

mE RTEE

C

Cji

j

i /exp

20

2222

0 2

1)(ln rr

RT

VM

C

rC

Another derivation re-introduces the chemical potential. The condition of equilibrium requires that the free energy, i.e. the system’s chemical potential, which is the sum of the chemical and centrifugal potential, is minimum. If you think about how centrifugation works, what we would like to know is really the concentration profile with respect to the radius r not as a function of time (as in non-equilibrium centrifugation), but rather at equilibrium. We can then impose that the derivative of the chemical potential with respect to r is 0 at equilibrium. In lecture 6, we have introduced the chemical potential in analogy to the classical concept of force as of a potential gradient, to express differences in free energy that induce diffusion


If the concentration of solute C2 is a function of x, the chemical

potential has the general form


)(ln)( 2022 xCRTGxG

A difference in chemical potential exercises a force on the solute molecules, and the force that induces solute flow is related to the chemical potential by the diffusion equation. In the case of centrifugation, we have to add a term that describes the centrifugal field, so that the total chemical potential of the system is:

)()(ln0 rUrCRTGsolutelcentrifugachemtot

Here C is the solute concentration at position r in the centrifuge tube and U is the centrifugal potential at position r in the centrifuge tube. At equilibrium:



0 lcentrifugachemicaltotal

dr

d

dr

d

0)()(ln0

dr

rdU

dr

rCdRT

dr

dGsolute

We can substitute for dU/dr the expression for the centrifugal force provided in the previous lecture (remember, a force is a gradient of a potential), then multiply by Avogadro’s number to obtain the centrifugal potential per mole of solute (which is what the chemical potential will be expressed in) and equate the derivative of the concentration with the derivative in centrifugal potential to obtain:



20

ln ln ( )( )B buoyancy centrifugal

A

d C r d C rRT dU rk T F F m m r

N dr dr dr

Integrating that equation:



20

ln ln ( )( )B buoyancy centrifugal

A

d C r d C rRT dU rk T F F m m r

N dr dr dr

22

2 20 2 2 2 20 0

0

1ln

2 2B

M Vm mCr r r r

C k T RT

2

2 2 2 20

0

1ln

2

M VCr r

C RT

Example : Calculate the weight of carboxy-hemoglobin (cHb) using the following data obtained from an equilibrium centrifugation experiment


At r=4.61 cm C(cHb)=1.220 weight %At r=4.56 cm C(cHb)=1.061 weight %T=293.3K spinning at 8703 revolutions/minuteThe specific volume of cHb is 0.749 cm3/g.The density of water is 1g/cm3

22

02

2

02

1

/ln2

rrV

CCRTM


At r=4.61 cm C(cHb)=1.220 weight %At r=4.56 cm C(cHb)=1.061 weight %T=293.3K spinning at 8703 revolutions/minuteThe specific volume of cHb is 0.749 cm3/g.The density of water is 1g/cm3

7

23 3 1 2 2

2 8.31 10 / 293.3 ln 1.22 /1.06170,900 /

1 0.749 / 1 / 2 8703min min/ 60 (4.61 ) (4.56 )

x x ergs molexKx Kxg mole

cm gx g cm x s cm cm

22

02

2

02

1

/ln2

rrV

CCRTM

This equation means that when a solution reaches equilibrium in a centrifugal field, generated by spinning the sample at an angular frequency , a concentration gradient will be generated of the shape given above

This experiment can be used to measure macromolecular masses or separate components of a mixture

A plot of lnC vs. r2 is a straight line with slope proportional to M. Notice that this method provides absolute molecular weight, while electrophoresis, for example, only provides relative molecular weights


2

2 2 2 20

0

1ln

2

M VCr r

C RT

Example : Consider the centrifugal separation of two gases with molecular weights of 349 g/mole (1) and 352 g/mole (2). How fast do you have to spin the sample to enrich molecule 1 to a level of 1% at r=3cm if its level is 0.7% at r=10cm and T=273K.


20

2222

0, 2

1ln rr

RT

VM

C

C

i

i

Set the buoyancy correction to 1 for gases and subtract the two expressions for C2 and C1 to obtain:

20

2212

10,2

0,12

2ln rr

RT

MM

CC

CC

Then solve for 2


10,2

0,12

20

212

2 ln2

CC

CC

rrMM

RT

7

2 7 2

2 2 2

2 8.31 10 / 273 0.01ln 5.92 10

0.007352 / 349 / 10 3

x x ergs molexKx Kx s

g mole g mole cm

37.69 10 /x radians s

3 11.22 10x s

A type of equilibrium centrifugation involves spinning of a concentrated salt solution at very high speed to generate a density gradient (the density of the solution increases with the salt concentration) and has proven very useful in the study of nucleic acids

If a macromolecule is also present, it will form a boundary at a point in the salt gradient where the macromolecules are buoyant

Equilibrium Sedimentation in a Density Gradient

Suppose a solution of a macromolecule (e.g. DNA) also contains a salt such as CsCl

Initially the salt and the DNA have uniform concentrations.

Once the centrifugation has commenced, the salt quickly reaches equilibrium; the concentration of CsCl will reach equilibrium as described by the equilibrium centrifugation equation:


20

22

0, 2

1ln rr

RT

VM

C

C CsClCsCl

CsCl

CsCl

Because of the equilibrium condition, the density of the solution will vary as a function of r, the distance from the spinning axis. Suppose at r’ the solution has a density


20

22

0, 2

1ln rr

RT

VM

C

C CsClCsCl

CsCl

CsCl

1

2V

where is the specific volume of the macromolecule


20

22

0, 2

1ln rr

RT

VM

C

C CsClCsCl

CsCl

CsCl


At r<r’ the density of the solution is less than 1

2V

the DNA “sinks” to the bottom of the tube, being pulled “downward” by the centrifugal force

At r>r’ the density of the solution is greater than 1

2V

At r=r’ the DNA density increases

the DNA ”floats” “upwards” toward the top of the centrifuge


20

22

0, 2

1ln rr

RT

VM

C

C CsClCsCl

CsCl

CsCl

suppose the solution density gradient is roughly linear, with

dr

drr

Vr

'1

)(2

where d/dr is the density gradient of the solution near r’, and is assumed to be a constant in this region, if we do not go very far from r’


20

22

0, 2

1ln rr

RT

VM

C

C CsClCsCl

CsCl

CsCl

Then the condition for equilibrium of the DNA is:

If the solution density in the buoyancy correction is the function of r defined above, then:

dr

drr

Vr

'1

)(2

r

RT

rVM

dr

Cd DNADNADNA 2

2

)(1ln

r

RT

dr

drr

VVM

dr

Cd DNA

DNADNA

DNA 2

2

'1

1ln


20

22

0, 2

1ln rr

RT

VM

C

C CsClCsCl

CsCl

CsCl

Integrate the equation to find, and after some simple calculus:

Although this equation seems similar to the equilibrium centrifugation equation, it has a term (r-r’)2 instead of r2-r’2, so it differs

dr

drr

Vr

'1

)(2

22

''2)'(

)(ln rrr

dr

d

RT

VM

rC

rC DNADNA

DNA

DNA


The logarithm can be removed to obtain:

dr

drr

Vr

'1

)(2

22

''2)'(

)(ln rrr

dr

d

RT

VM

rC

rC DNADNA

DNA

DNA

2

2

''2

exp)'()( rrrdr

d

RT

VMrCrC DNADNA

DNADNA

2

2

'( ) exp

2DNA

r rC r


This is a gaussian with width:

2

2

2

'exp)'(

rr

rCDNA

drdVMr

RT

DNADNA /'2

2

Notice that the standard deviation increases as the salt gradient d/dr decrease; it also depends on the inverse mass, so that the higher the mass, the sharper the gradient


2

2

2

'exp)'(

rr

rCDNA

drdVMr

RT

DNADNA /'2

2

Many macromolecules have buoyant densities sufficiently different that they can be separated by this technique. Hybrid DNA-RNA was discovered using equilibrium centrifugation in a salt gradient, for example, although they differ in mass by very little.

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Ch 24 pages 636-643 Lecture 11 – Equilibrium centrifugation