Microrheology of Biopolymers at Non-thermal Regimes - Tissue Engineering: Computer Modeling, Biofabrication and Cell Behavior - page 97

could be described using its persistence length (e.g., deoxyribonucleic acid or DNA
molecules) whereas semi-flexible polymers (e.g., actin or microtubules) could be
described by monitoring bending fluctuations along its length [ 10 , 26 - 28 ]. To find
a more robust description that may start without a priori knowledge of the biopoly-
mer's stiffness, the inferred properties could be based on the biopolymer's structure.
In order to experimentally test this approach, the fluctuation of attached probes could
be monitored and analyzed based only on the resulting length and winding turns of
the biopolymer under known tensile strength.
Considering solutions to the Fokker-Plank equation, the probability density is
derived using Feynman path integrals as possible conformations of a biopolymer.
Specific to helical conformations, we derive the winding probability W(n) about its
longitudinal axis [ 18 , 19 ] as follows:

R 4 π
Ll

exp −

2 DR 0

f(s)ds) 2

R 2 l

Ll ( 2 πn +

W(n,L) = (1)

θ 3

4 DR 0

fds

l

where D is a constant drift coefficient, and the following parameters descriptive of
the biopolymer: R = helical radius, L = total length, l = subunit length (such that

L = Nl , N = number of sub-units or monomers in the biopolymer, n =± 1, ± 2,

± )). Here,

f(s) is to be understood as the drift coefficient of each sub-unit. Thus, if there
were N total sub-units, we have f 1 ,f 2 ,... quantify the drift along the length of the
biopolymer as the sum of the drift about each sub-unit:

3, ... , the total number of turns, clockwise ( − ) or counter-clockwise ( +

N

L

f(s)ds =

F f = f m (s) (2)

0

m = 1

the theta function is:

∞

q m 2 cos ( 2 mu) θ 3 (u) = 1 + 2

m = 1

(3)

l
4 DR

u = f(s)ds

exp − Ll/ 4 R 2

q =

Note however, where we have a long biopolymer, Nl 1, the theta function ap-

proaches unity, θ 3 ≈ 1. In this model the drift coefficient f(s) is predicted to mod-

ulate the final conformation of the biopolymer in question. Thus, by choosing an
approximate formulation of f(s) or F f , the conformation is predicted in terms only
of winding-times ( n ) and the total length after stretching L . Note however that elas-
tic moduli are not explicitly incorporated in this approach, which at first instance,
could be considered implicit in f(s) . Since the winding probability W(n,L) is both
afunctionof n and L , the experimental approach involves stretching (variation in L )
and counting winding (variations in n ). Thus, the winding probability W(n,L) cap-
tures a biopolymer's conformational adaptation based on the number of windings n ,
and the measured length L .