Question

Flexural rigidity of biopolymers (a) Recall from \(\mathrm{p} .389\) that when treating macromolecules as elastic beams it is the combination \(K_{\mathrm{eff}}=E I,\) the flexural rigidity, that dictates the stiffness of that molecule. The flexural rigidity is a product of an energetic ( \(E,\) Young modulus) and geometric \((I,\) areal moment of inertia) factor. Reproduce the argument given in the chapter that culminated in Equation 10.8 for the bending energy of a beam and show that the flexural rigidity enters as claimed above. (b) Using what you know about the geometry of DNA, actin filaments, and microtubules, determine the areal moment of inertia \(I\) for each of these molecules. Be careful and remember that microtubules are hollow. Make sure that you comment on the various simplifications that you are making when you replace the macromolecule by some simple geometry. (c) Given that the elastic modulus of actin is \(2.3 \mathrm{GPa}\), take as your working hypothesis that \(E\) is universal for the macromolecules of interest here and has a value 2 GPa. In light of this choice of modulus, compute the stress needed to stretch both actin and DNA with a strain of \(1 \%\). Convert this result into a pulling force in piconewtons. (d) Using the results from (b) and (c), compute the persistence lengths of all three of these molecules. (e) Given that the measured persistence length of DNA is \(50 \mathrm{nm},\) and using the areal moment of inertia you computed in (b), compute the Young modulus of DNA. How well does it agree with our 2 GPa rule of thumb from above?

Short answer

This exercise involves multiple complex steps, from reproducing arguments for flexural rigidity, calculating moments of inertia, to computation of stress levels and persistence lengths. Lastly, its accuracy is determined by comparing a computed Young modulus value with a standard.

Step-by-step solution

Reproductive Argument for flexural rigidity

The bending energy of a beam as stated in equation 10.8 is given by the formula \(U=\int \frac{1}{2} E I (\frac{\partial^2 z}{\partial x^2})^2 dx\), where \(U\) is the bending energy, \(E\) is Young modulus, \(I\) is the areal moment of inertia and \(\frac{\partial^2 z}{\partial x^2}\) is the curvature of the molecule. In this, the flexural rigidity \(K_{\mathrm{eff}}= E I\) is effectively a multiplication of energetic and geometric factors, thus entering the formula as claimed.

Calculate the areal moment of inertia

To make the calculation, we make the assumption that the molecules can be approximated as simple geometrical shapes (cylinders). For a cylinder, \(I(r) = \frac{\pi r^4}{4}\) where \(r\) is the radius.

Compute stress for actin and DNA stretching

Given \(\sigma = \frac{F}{A}\), where \(\sigma\) is the stress, \(A\) is the cross-sectional area, and \(F\) is the force, and assuming that \(E = \sigma / \epsilon\) where \(\epsilon\) is the strain, we can calculate that \(F = A * E * \epsilon\). We know the radius, and hence the cross-sectional area, of both the actin filament and the DNA. Given the strain (1\%) and \(E\), we can calculate the stress (\(\sigma\)) and convert it to a force in piconewtons.

Compute persistence lengths

Using the formula for persistence length \(L_p = \frac{K_{\mathrm{eff}}}{k_B T}\), where \(K_{\mathrm{eff}}\) is the flexural rigidity, \(k_B\) is Boltzmann's constant, and \(T\) is the temperature, we can plug in the values that we have gotten to determine \(L_p\) for DNA, actin filaments, and microtubules.

Compute Young modulus of DNA

Given the measured persistence length of DNA is \(50 \mathrm{nm}\), we can rearrange the formula for \(L_p\) to compute the Young modulus \(E\) for DNA. Comparing this with our rule of thumb value of 2GPa from before will give us a measure of the accuracy of our method.

Key concepts

Young Modulus

The Young modulus (E) is an essential material property reflecting the stiffness in the stress-strain relationship of a material. In the realm of biopolymers such as DNA and proteins, understanding the Young modulus helps in determining how these molecules withstand forces and deformations. When a strain is applied, the stress response is quantified by this modulus; the higher the Young modulus, the stiffer the material. Typically, it's measured in pascals (Pa) with a giga Pascal (GPa) representing 1 billion pascals.

In biophysics, the Young modulus is crucial for calculating the stress needed to stretch macromolecules. Assuming a universal value of 2 GPa for the Young modulus, one can estimate the force required to induce a certain strain in macromolecules like DNA or actin filaments. This force, often expressed in piconewtons due to the minuscule dimensions involved, highlights the mechanical resilience of these biological structures at a molecular scale.

Areal Moment of Inertia

The areal moment of inertia (I) is a geometric property that reflects how a beam's cross-sectional area is distributed with respect to an axis of rotation. For a cylindrical shape, like that of a biopolymer, the moment of inertia is crucial in determining the molecule’s resistance to bending. It's expressed in length to the fourth power \( L^4 \) and is calculated differently based on the geometry of the object. For instance, for a solid cylinder, we use \( I = \frac{\pi r^4}{4} \) where \( r \) is the radius.

When analyzing biopolymers like DNA, actin filaments, and microtubules, assumptions are made to simplify their complex structures into basic geometrical shapes to calculate their moment of inertia. This simplification is an approximation and doesn’t account for irregularities in the actual molecular structure, which may have implications for the accuracy of the computed bending resistance.

Persistence Length of Macromolecules

Persistence length (\( L_p \)) is a measure of the rigidity of a polymer chain; it indicates the length over which a polymer appears to be straight. For macromolecules, such as DNA, the persistence length is an indication of their mechanical stability and structure. The longer the persistence length, the less likely the polymer is to bend, signifying greater stiffness.

In the study of biopolymers, calculating persistence length involves thermal energy, flexural rigidity, and Boltzmann's constant (\( k_B \)‎). Using the formula \( L_p = \frac{K_{\mathrm{eff}}}{k_B T} \), scientists can derive the persistence length and hence gain insights into the character of the molecule. Flexural rigidity itself is derived from the Young modulus and the areal moment of inertia, showing how these properties interrelate in understanding the physical behavior of macromolecules.

Bending Energy of Beams

The bending energy of beams is a concept used to understand how much energy is required to bend a material into a curve, and it is particularly significant when examining biopolymers like DNA or proteins. For a beam, the bending energy (\( U \)‎) is quantified by integrating \( \frac{1}{2} E I \left(\frac{\partial^2 z}{\partial x^2}\right)^2 \) over the length of the beam, where \( E \) is the Young modulus, \( I \) is the areal moment of inertia, and \( \frac{\partial^2 z}{\partial x^2} \) represents the curvature. In this equation, one can see how flexural rigidity \( K_{\mathrm{eff}} = E I \) dictates the stiffness and bending resistance.

For biopolymers, understanding bending energy is crucial since it provides insights into the energy landscape that dominates their structural transitions. Any disruption to these energy interactions can significantly affect the molecule's function, making the calculation of bending energy an essential exercise in biophysical analyses.