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PHYS 461 & 561, Fall 2011-2012 110/18/2011
Lecture 8: Random Walks and the Structure of
Macromolecules
Lecturer:Brigita Urbanc Office: 12909(Email: [email protected])
Course website: www.physics.drexel.edu/~brigita/COURSES/BIOPHYS_20112012/
PHYS 461 & 561, Fall 2011-2012 210/18/2011
Description of the macromolecular structure as random walks
motivation: calculate entropic cost of DNA packed into a cell; description of DNA stretching by optical tweezers
atomic coordinates deposited on databases such as Protein Data Bank (pdb format of an ascii file with x,y,z coordinates of known protein structures)
(r1, r
2, … r
N) with r
i = (x
i, y
i, z
i)
data derived based on Xray crystallography or NMR
statistical measure of the size of a macromolecule: Radius of Gyration R
G
example of DNA characterization r(s) where s is the distance along the contour of the molecule
PHYS 461 & 561, Fall 2011-2012 310/18/2011
Random walk model of a polymer:series of rigid rods (the Kuhn segments) connected by flexible hinges
lattice model in 1D lattice model in 3D
PHYS 461 & 561, Fall 2011-2012 410/18/2011
DNA molecule as a random walk:Every macromolecular configuration is equally probable
AFM image
PHYS 461 & 561, Fall 2011-2012 510/18/2011
What is a random walk? 1D random walker: p
R = p
L = ½ at every step, independently
for N random walk steps, there are 2N possible macromolecular configurations
the mean distance of the walker from its point of departure:
<R> = < ∑i x
i> = ∑
i <x
i> = 0
the variance of the probability distribution:
<R2> = < ∑i∑
j x
i x
j > = ∑
i<x
i2> + ∑
i≠j<x
ix
j>
<R2> = N a2
thus the size of the area covered by the random walk is:
√<R2> = √N a
PHYS 461 & 561, Fall 2011-2012 610/18/2011
Derivation Based on the Probability Theory: a macroscopic state: many microscopic states
N steps in 1D random walk: N = n
R + n
L, each with
a probability of ½
a probability associated with each microstate of N steps is: (1/2)N
multiplicity W associated with n
R steps is:
W = N!/[nR! (N n
R)!]
PHYS 461 & 561, Fall 2011-2012 710/18/2011
Probability of an overall departure nR from the origin:
p(nR; N) = N!/[n
R! (N n
R)!] (1/2)N
This distribution is normalized:
∑nR
p(nR; N) = 1, n
R∈{0,N}
Probability distribution for the endtoend distance: R = a (n
Rn
L) = a (2n
R N) ; n
R= N/2 + R/(2a)
such thatp(R; N) = (1/2)N N!/{[N/2 + R/(2a)]! [N/2 – R/(2a)]!}which takes the form of a Gaussian distribution for R « Na: p(R; N) = 2/√2N exp[R2/(2Na2)]
PHYS 461 & 561, Fall 2011-2012 810/18/2011
To obtain the probability distribution function for the endtoenddistance of a freely jointed chain P(R; N), p(R; N) needs to be divided by the number of integer R values per unit length (=2a):
P(R; N) = 2/√2Na2 exp[R2/(2Na2)]
Note that <R> = 0 and <R2> = Na2, independent of the spaceDimension. This can be used to derive P(R; N) in 3D space:
P(R; N) = A exp(R2);A = [3/(2Na2)]3/2; = 3/(2Na2)
Use normalization condition in 3D and calculate the variance to determine A and (integration in 3D).
PHYS 461 & 561, Fall 2011-2012 910/18/2011
Endtoend distance probability distribution for 1D random walk:Binomial (dots) versus Gaussian (curve) distribution for N=100
PHYS 461 & 561, Fall 2011-2012 1010/18/2011
Persistence Length versus the Kuhn Length
persistence length: the length scale P over which the polymer
remains approximately straight the Kuhn length: the length of a step a in the random walk model
<R2> = La persistence length
P can be calculated from the unit tangent
vector t(s) where s is a distance along the polymer: <t(s) t(u)> = exp(|su|/
P)
to find the relationship between a and P we use R = ∫
0L ds t(s):
<R2> = <∫0
L ds t(s) ∫0L du t(u)> = ∫
0L∫
0L ds du <t(s) t(u)> =
2 ∫0
L ds ∫sL du exp[(su)/
P] = 2 ∫
0L ds ∫
0ꝏ dx exp(x/
P) = 2L
P
a = 2P
PHYS 461 & 561, Fall 2011-2012 1110/18/2011
<R2> = 2LP
<RG
2> = 1/3 LP
(use definition)
L = 0.34 nm NBP
NBP …
# of basepairs
RG = 1/3 √N
BP
P nm
Size of genomic DNA in solution