Effect of Mechanical Force on Biopolymers

Analysis of effects of Mechanical

Force on Bio-polymersRupture- Shearing of double stranded DNA

By- Tushar Modi

5th year Engineering Physics

M.Tech IMD

IIT(BHU)

Biopolymers = Bio + Polymers

These are the biologically occurring polymers in nature

Polymers are macromolecules made of certain basic units or

segments called monomers

A monomer itself can be anything from a simple molecule

consisting of a few atoms to a large complex molecule

These polymers are broadly of two kinds:

◦ Homopolymers

◦ Hetropolymers

Biopolymers are classified in three groups based on their

monomers:

◦ Polysaccharides: made of sugars Eg. Amylose

◦ Proteins: made of amino acids, Eg. Myoglobin

◦ Nucleic acids: made of nucleotides, Eg. DNA, RNA

Importance of studying biopolymers:

They are involved in a number of crucial biological reactions.

They have a complex shape and a precise structure taking

part in different reactions. E.g. transcription (during cell

division), metabolism, signaling network in cells, etc.

Hottest topic for R&D is: Drug delivery and genetics.

◦ Drugs are analogous to enzymes. They attach themselves to specific

locations or mix in system, and have some effect. Their delivery system

effects the time and intensity of action

Genetic: Study of Genes. They are the molecular unit of

heredity. They correspond to regions within DNA.

◦ Their studies have proved to be very fruitful in designing of drugs and

genetic engineering. E.g. Marker: this is the gene or sequence of genes

on chromosome which is used to identify some specific trait.

Understanding of above reactions and processes, involve

understanding of the related biopolymers- structure, shape,

composition and kind of neighboring interactions

Single Molecular Force Spectroscopy (SMFS)

Single Molecule Experiments:

It is an experiment that investigate the properties of individual

molecules in a compound structure. They are different from

bulk experiments as in them the individual behavior of

molecules cannot be distinguished, and only average

characteristics could be measured.

They involve targeting some specific molecules and attach

probes to them. Then they are monitored and acted upon by

force and their motion is recorded to study dynamics of the

structure. These can be done my attaching magnetic beads to

molecules or by using atomic force microscope. Such

experiments have resulted in measurements of inter- and intra-

molecular forces in biological systems and have been able to

separate out the fluctuations of individual molecules from the

ensemble average behavior.

SMFS studies have provided unexpected insights into the

strength of the forces driving biological processes and

biological interactions responsible for the mechanical stability

of biological structures.

Surprisingly, with increasing number of experiments and

insights gathered so far, most of the interactions have a

physical and chemical nature and not just mechanical.

Several SMFS experiments these days are focused on free

energy landscape (FEL) of biomolecules; the protein

mechanical stability, experimental aspects of refolding under

quenched force , advances in optical tweezers etc.

Here the concentration is on the mechanical stability of ds-

DNA (double stranded) on application of a pulling force. It

can be in a direction parallel to the DNA axis (Shearing) and

perpendicular to the DNA axis (unzipping), rupturing and

twisting.

Some approaches for Biopolymer force analysis:

Analytic and statistical models:

◦ In analytic models, we incorporate all the types of interactions involved in the system in Hamiltonian.

◦ The Hamiltonian is then used to find the equilibrium states of the system and its properties.

◦ However there are limitations to this approach as in here we have to rely on a lot of assumptions in order to simplify the calculations

◦ A huge progress has been achieved using this approach in the field of thermal denaturation of proteins, DNA, etc.

Coarse grain Simulations:

◦ In this approach, an actual structure of the polymer is constructed using the interactions discussed in Hamiltonian and then we apply the external stimuli in the form of temperature change or mechanical force or some chemical interaction.

◦ In such a simulations, we don’t have to worry about the complexity of the solution as it is handled by computation and thus a lot of assumption can be discarded giving us very accurate results.

All Atomic Simulations :

◦ These are the most accurate type of simulations possible.

◦ In these kind of simulations, individual atomic interactions are

actually emulated by considering their potential (both classically

and quantum mechanically as the need arises). These atomic

models are then substituted to their actual positions in the

structure as it is calculated from experimental results.

◦ Once the structure is made, any kind of external stimuli can be

applied over it in order to analyze the outcome. These

simulations are most accurate and almost don’t require any

assumption regarding the structure. Effects due to solvent

molecules, molecular crowding or even twisting or distortions

can be incorporated.

◦ However, these simulations are pretty costly to perform and

consumes a lot of time. These are usually done using a software

package for molecular dynamics called AMBER (Assisted Model

Building with Energy Refinement)

DNA Rupture

A DNA can be ruptured in three ways:

In my study, I have concentrated upon, the analytic models of

DNA shearing.

a) DNA unzipping

b) DNA rupture

c) DNA Shear

Source: Wikipedia

SHEARING OF ds-DNA

A double stranded DNA consists of two nucleotide polymer chains joined together by hydrogen bonding bases. These nucleotide bases are joined together by covalent bonds.

The interactions among the bases pairs are taken to be of harmonic nature, and thus covalent bonds can be simply replaced with harmonic springs. This assumption can be justified as under a stretching force, the covalent bonds are stretched and thus the entropic contributions are comparatively lower.

The hydrogen bonds potential can be simulated in a number of ways like- harmonic (similar to covalent bonds), Lenard-Jones potential or Morse potential. All these potentials give similar results under the forces discusses here hence harmonic potential is used because of its simplicity.

As the chain is stretched out, there is no need to take the anharmonicity of the covalent bonds into consideration. It is also cross checked with other methods, that even if we take them into consideration, it will give similar results.

Model proposed

The Hamiltonian of this system can be written as:

Another major assumption is taken in the formulation of

expression for the Hamiltonian. The diameter of DNA, i.e. the

space between the two strands are considered to be very small as

compared to the length of the polymer. Hence, the component of

force transverse to the direction of application of force can be

neglected.

Thus keeping the mentioned assumptions and formulations in mind,

following model can be proposed:In the given figure, un represents the displacements in the upper strand having

a spring constant of Q, and U is the spring constant of lower strand where

displacements are represented by vn

Results:The Hamiltonian is differentiated with respect to displacement to find the condition for minimum energy. The study of Hamiltonian gives the following results regarding δn = un – vn (the elongation produced in the nth Hydrogen bond.

The hydrogen bonds between the strands, elongate under the force. Their elongations are given as:

where, A and χ and are the constants depending upon the system configuration given as:

and,

The other constant δ0 is the elongation of the central hydrogen bond and its value for a particular force can be calculated using boundary conditions for conservation of force.

Plot between the Displacements produced in hydrogen bonds in base pairs

with their indices

The force acting being exerted on each base pair is given as = (displacement produced) x (spring constant) (Hooke’s law). Thus the net shear force couple acting on the polymer can be given as the summation of the force exerted by all the base pairs:

The shearing starts from the point of maximum elongation, which can be seen from the elongation equation and applying the required conditions the Force required to shear off the whole DNA chain is given by :

where the constant f1 is the force required to rupture a single hydrogen bond; and l is the total number of base pairs in the DNA.

Plot of the critical shear force with total number of base pairs

Analysis:

The plot between elongations produced in hydrogen bonding

and their indices in the DNA chain was an asymmetric one.

Thus for an applied force, one end will be more stretched

than the other one and hence clearly the shearing will start

from a particular end regardless of the direction in which the

force is applied. It was a ground breaking finding which

explained why the genetic code is preserved in offspring even

when DNA shearing and transcription happen in a very

random fashion during a cell division.

The force required to shear off a DNA, as seen in the plot

between critical force and the total length of the polymer

chain, at first increases linearly with the length and then

reaches a saturation limit after which increase in length has

no effect. Similar results were obtained in experimental

studies of the same. Thus howsoever long DNA sequence is

present in cells it only requires a finite amount of force to

open it up.

Discussion:

The model presented was remarkable because of it’s simplicity with a good prediction of properties, however a lot needs to be done. The model has a major limitation that it does not include the effect of temperature. There is no scope to include thermal noise fluctuations and effect of environmental effects (like solvent and molecular crowding) in the formulations.

We have gone for a publication for this study and have received a good review from Journal of Chemical Physics and are on a fast track to get a final publication.

Apart from the study of shearing, a lot of analytical work as well as statistical formulations are also done in unzipping transitions of DNA where we have used transfer matrix method to solve integrals numerically making C programs for the same. Their results have also shown remarkable similarity with other all atomic as well as experimental studies giving us a very a promising prospective in research for the same.

Currently I am involved in making analytical formulations for shearing taking heterogeneity, temperature effects and structural complexities like helical structure, randomness, etc. into consideration

Thank You