Biofilament Dynamics: Line-to-Rod-Level Descriptions - Multiscale Modeling in Biomechanics and Mechanobiology - page 65

We consider a homogeneous filament so that the correlation function depends only
on the distance s between the two points, not s 0 . Continuing the analogy between
the filament contour and the trajectory of a moving particle, l p corresponds to the
correlation time of a particle undergoing Brownian motion. At a given temperature
T , a stiffer filament will remain straighter, hence l p will be longer. For a filament of
a given stiffness, it will bend more at a higher temperature due to stronger thermal
kicks. These two conditions combine to [ 5 , 14 ]

K f
k B T .

l p =

(4.10)

Here, k B is the Boltzmann constant. We ca use a scaling argument to understand
theoriginofEq. 4.10 : Since thermal force bends the filament, the average elastic
energy,

K f
l p

( K f

2 l p l p , ignoring 2), should be equal to the thermal energy k B T , yielding
Eq. 4.10 . If we follow a more rigorous derivation [ 5 , 14 ], Eq. 4.10 is for filaments in
three dimensions where the filament can bend in two orthogonal directions. In two
dimensions, bending can occur only on a plane, so only half of the thermal energy
can bend the filament, making it straighter: l p =

2 K f

k B T (2-dim) [ 26 ].
The definition of l p (Eq. 4.9 ) involves an ensemble average, where we need to
analyze many replicas of the filament. Let us ask a question on de-averaging: For
agiven l p , how does a typical filament look? For this we use computer to generate
example contours. For simplicity, we work in two dimensions. Take a segment of
length s and place it along the x -axis, with its left end at the origin (point A in

Fig. 4. 2 a). When it is bent to an angle

ʔ

ʸ 1 , the length ʔ l 1 of the line joining its two

2 R 1 sin ʸ 2 ends ( AB ) is at an angle ˆ 1 = ʸ 1 / 2 relative to the x -axis. Also, ʔ l 1 = =

2 ʔ s
ʸ 1

sin ʸ 2 , so the coordinate of point B is (ʔ l 1 sin ˆ 1 ,ʔ l 1 cos ˆ 1 ) . Next add the

(a) (c)

(b)

Fig. 4.2 Generating a contour with a given persistence length l p . a Two successive segments of
length ʔ s each. The local bending angle of segment i is ʸ i ( i = 1 , 2 ,... ). b Representing the

i between neighboring segments is ʸ 2 .
c Sample contours starting horizontal at the origin. Dashed l p

contour using straight segments of length ʔ l i . The angle ˆ

= 5, and thick solid l p = 1 . 0. The

length of each contour is 5. ʔ s = 0 . 01 was used