1. Soheil Fatehiborougeni , Jesan Morales, Sergio Lopez, Sachin Goyal , School of Engineering, University of California, Merced
Bending and twisting of bio-filaments such as DNA (Deoxyribonucleic acid)
are crucial to their biological functions. For example, as shown in figure 1,
gene expression is governed by the sequence-dependent looping behavior of
its “non-coding” DNA segment (portion of DNA that does not contain genes).
Background figure (http://mashable.com/2013/06/13/supreme-court-dna-patents/)
Figure 1. Role of looping of the non-coding DNA in regulating genes, namely,
LacZ, LacY and LacA. The looping is mediated by binding with a V-shaped
protein, the Lactose-Repressor. The depicted loop shape is a prediction from
continuum-rod model simulation. Harish et al. [1]
Thus the deformability or the constitutive law that a bio-filament follows, which
depends on its atomistic structure, must be an important trait for biological
evolution of its chemical sequence. Unfortunately, there is no simple formula
to derive the constitutive law from a given atomistic structure. Instead, the
information for estimating constitutive law has to be extracted from the actual
deformations of the bio-filaments. Recently, Hinkle et al. [2] proposed an
approach that uses a continuum-rod model with static deformation data
generated from discrete-structure simulation which has shown encouraging
success in estimating the constitutive law. The method is adaptable to use
high-fidelity Molecular Dynamics (MD) simulations to generate input
deformation data. We extended the scope of the method in order to identify
the constitutive law from dynamic state deformations of filament in a wide
variety of loading scenarios including distributed loading and three
dimensional deformations.
The dynamics of deformation for a slender elastic rod are characterized by
two-axis bending and torsion. In order to track the orientation of each cross-
section with respect to an inertial reference frame {𝑒}𝑖, a body-fixed frame
{𝑎}𝑖 is fixed to the mass center of each cross-section.
Figure 2. Rod cross-section with body-fixed frame and the position vector 𝑅
evaluated from inertial frame, arc length parameter shown with s, external force
F and external moment Q, restoring internal force 𝑓 and internal moment 𝑞.
Cross-sectional rigid body dynamics are described by the translational velocity 𝑣, and angular
velocity w vectors, and the latter is equal to rotation of cross-section fixed frame, relative to inertial
frame per unit time. Deformations are captured by curvature and twist vector 𝜅, defined as the
rotation of body-fixed frame relative to inertial frame per unit arc length. Gradient of 𝑅 along s are
denoted by 𝑟. The model is developed by employing Kirchhoff-Clebsch elastic rod theory described
in [3]. Application of the equilibrium of linear and angular momentum yield equations 1 and 2.
Equations 3 and 4 are compatibility equations regarding the fact that both centerline position
vector R, and transformation from body-fixed frame to inertial frame are continuous in space and
time.
𝜕 𝑓
𝜕𝑠
+ 𝑘 × 𝑓 = 𝑚
𝜕𝑣
𝜕𝑥
+ 𝑚(𝑤 × 𝑣) − 𝐹 (1)
𝜕𝑞
𝜕𝑠
+ 𝑘 × 𝑞 = 𝐼 ∙
𝜕𝜔
𝜕𝑡
+ 𝜔 × 𝐼 ∙ 𝜔 + 𝑓 × 𝑟 − 𝑄 (2)
𝜕𝑣
𝜕𝑥
+ 𝑘 × 𝑣 =
𝜕 𝑟
𝜕𝑥
+ 𝜔 × 𝑟 (3)
𝜕𝜔
𝜕𝑠
+ 𝑘 × 𝜔 =
𝜕𝑘
𝜕𝑡
(4)
sConstitutive law in its general form is an algebraic differential equation can be expressed as,
𝜓 𝜅, 𝑞 = 0 (5)
Rod model equations are a set of four
differential equations with five vector
unknowns namely 𝜔, 𝑣, 𝑓, 𝜅, and 𝑞 .
Discrete-structure or MD simulations can
be used to simulate the deformation of
the filament. For example MD simulations
use atom by atom description of the
filament and take into account atomic-
level interactions for calculating the
position of each atom. This position data
can be differentiated to generate the
curvature and twist vector 𝜅. When the
curvature and twist vector 𝜅, is obtained
the four differential equations of rod
model are solved numerically for four
other unknowns 𝜔, 𝑣, 𝑓, and 𝑞 with the
order shown in figure 3.
Figure 3. Algorithm for calculation of internal force 𝑓 and
internal moment 𝑞.
After solving the rod model equations
numerically and finding the values of
𝜔, 𝑣, 𝑓, and 𝑞 a function approximation
tool is employed to find the functional
relationship between restoring internal
moment q and curvature and twist vector
𝜅. This method is capable of simulate
filament for a variety of loading scenarios
including distributed external force and
three-dimensional deformations. In the
following some representative case
studies are shown. In all cases the
filament is modeled as a cantilever fixed
at one end and free at the other end as
shown in figure 4.
Figure 4. Schematic view of filament modeled as cantilever
beam with all possible external interactions shown in 2D.
In section (a) of the figure 5 a
distributed shear force is
applied on every cross-section
of the rod and calculated
moment versus input curvature
data is plotted. A least-square
function approximation method
is employed to fit a polynomial
through the scattered data. The
input constitutive law for
generating curvature data is
polynomial.
Where 𝐶1 = 2500.
In section (b) of figure 5 the
results of estimated constitutive
law is shown when the filament
is under a net bending
moment. The input constitutive
law in this case is :
Where 𝐶1 = 500.
The fitted polynomial through
the scattered is benchmarked
with the input constitutive law.
In order to calculate the input curvature data for these simulations a
computational rod model has been developed that is capable to solve the rod
model equation along with the relationship between curvature and restoring
moment as the constitutive law. This input constitutive law can be of any
functional form.
Figure 5. (a) Estimated polynomial constitutive
law for the filament under distributed shear
force. (b) Estimated trigonometric constitutive
law for the filament under bending moment
load.
𝑞2 𝑠, 𝑡 = 𝐶1(𝜅2 𝑠, 𝑡 −
𝜅2 𝑠, 𝑡 3 + 𝜅2 𝑠, 𝑡 5 (6)
𝑞2 𝑠, 𝑡
= 𝐶1 sin
𝜅2 𝑠, 𝑡 2 + 𝜅1 𝑠, 𝑡
2
(7)
[1] Palanthandalam-Madapusi, H. J., and Goyal, S., 2011. “Robust estimation of nonlinear
constitutive law from static equilibrium data for modeling the mechanics of dna”. Automatica,
47(6), June, pp. 1175–1182.
[2] Hinkle, A. R., Goyal, S., and Palanthandalam-Madapusi, H. J., 2012. “Constitutive-Law
Modeling of Microfilaments From Their Discrete-Structure Simulations - A Method Based on
an Inverse Approach Applied to a Static Rod Model”. ASME Journal of Applied Mechanics, 79,
Sep, p. 051005.
[3] Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity. 4 ed. 1944, New York:
Dover Publications.
The continuum-rod theory has shown encouraging results in modeling
dynamics of bio-filaments deformations such as DNA. However efficiency of
the rod model is critically restricted with the lack of knowledge about the
constitutive law. Discrete-structure simulations such as molecular dynamics
simulations can provide high-fidelity data for identifying the constitutive law.
So far we extended the scope of a recently proposed inverse approach [2] to
estimate the constitutive law from dynamic deformations. Several
representative case studies with a variety of loading scenarios are defined to
show the efficiency of the estimation. One of the future steps of this research
is to adapt the method to use input curvature data that is generated by
Molecular Dynamics simulations in order to yield more realistic results.
(a)
(b)