15.3 DNA

225

where upper I Subscript sIs is the moment of inertia (equals pi r Superscript 4 Baseline divided by 4= πr⁴/4 for a cylinder of radius rr), k Subscript upper BkB is Boltz-

mann’s constant, and upper TT is the absolute temperature. For DNA, r almost equals 1.2r ≈1.2 nm and

upper E almost equals 10 Superscript 6E ≈10⁶ N/m, giving German p almost equals 60p ≈60 nm. The radius of gyration upper R Subscript gRg of the polymer (length

upper LL) as a Gaussian coil is given by left parenthesis upper L German p divided by 6 right parenthesis Superscript 1 divided by 2(Lp/6)^1/2.

A mixture of different molecules of DNA is usually separated into its components

using gel electrophoresis, in which the DNA is driven by an electric field through a

hydrogel (usually polyacrylamide or agarose). Recently, model environments have

been created from arrays of precisely positioned microfabricated pillars. Long poly-

mers in such confined media move by reputation (rather like a snake moving through

tall stiff grass—it is constrained laterally but can move along its length), in which they

are confined to sliding along an imaginary tube between the pillars. The diffusivity

upper DD is, as usual,

upper D equals k Subscript upper B Baseline upper T divided by delta commaD = kBT/δ ,

(15.3)

where deltaδ is the drag coefficient and is equal to 2 pi eta upper L2πηL, etaη being the viscosity of the

solvent. The time for the polymer to diffuse out of its tube of length upper LL is

tau equals upper L squared divided by left parenthesis 2 upper D right parenthesisτ = L²/(2D)

(15.4)

but, in that interval, the polymer would have moved a distance equal to upper R Subscript gRg if it had

formed a Gaussian coil; the effective diffusion coefficient in the gel is then found

from upper D Subscript normal g normal e normal l Baseline divided by upper D equals left parenthesis upper R Subscript g Baseline divided by upper L right parenthesis squaredDgel/D = (Rg/L)²; hence,

upper D Subscript normal g normal e normal l Baseline equals StartFraction German p k Subscript upper B Baseline upper T Over 12 pi eta upper L squared EndFraction periodDgel =

pkBT

12πηL^{2 .}

(15.5)

Under the action of a relatively weak electric field and provided upper LL is not too great,

the mobility of the DNA in the gel is

mu equals StartFraction sigma German p Over StartRoot 12 EndRoot pi eta upper L EndFraction commaμ =

σp

√

12πηL

,

(15.6)

where sigmaσ is the charge per unit length of the DNA.³

Problem. On the basis of the above, devise a laboratory technique (including quan-

titative estimation of parameters for design and operation) for separating different

molecules of DNA.

3 For polymers confined by their congeners, a given chain can slowly escape from its tube by

Brownian motion: The mobilitymuμ of the whole chainupper NN monomers long ismu 1 divided by upper Nμ1/N, wheremu 1μ1 is the

mobility of one monomer. Hence, from the Einstein relationupper D Subscript normal t normal u normal b normal e Baseline equals mu 1 k Subscript upper B Baseline upper T divided by upper NDtube = μ1kBT/N and the relaxation

time (to which viscosity is proportional) for tube length upper LL (tilde upper N∼N) to be lost and created anew,

tau Subscript normal t normal u normal b normal e Baseline tilde upper L squared divided by upper D equals upper N upper L squared divided by left parenthesis mu 1 k Subscript upper B Baseline upper T right parenthesis tilde upper N cubedτtube ∼L²/D = N L²/(μ1kBT ) ∼N ³, in contrast to small molecules not undergoing reptation,

for whichtau tilde upper Nτ ∼N.