The stiffness of biomolecules - from DNA to actin

When engineers talk about the stiffness of materials they usually describe the Young's modulus, E, the ratio of stress to strain. This is constant for any geometry of that particular material. For example the Young's modulus of a particular type of steel might be 200 GPa and that of a bone might be 30 GPa.  So how would one describe the stiffness of tiny soft and floppy biomolecules? These are so puny that they are constantly being buffeted by their environment and thermal fluctuations. They can also adopt a myriad of different conformations.

We use a parameter called the PERSISTENCE LENGTH, to describe the stiffness of molecules like DNA and actin.

So what exactly is the persistence length which is often given the symbol ξ ? Essentially it is the length over which a polymer remains straight. When considering biomolecules we talk about the competition between thermal fluctuations and deterministic energy costs, Our thermal energy is KT (fluctuations) and this must be compared to the elastic energy of the polymer, that is:

here K is the famous Boltzmann's constant, T is the temperature, E is the Young's modulus, I is the geometric moment, and R is the radius  of curvature. L is the length of the polymer. Defining the persistence length to be that length for which the radius of curvature (of the bend) is equal to the length of the polymer means we can sett  L and R equal to ξ . Thus we obtain

There is a more elegant way to derive the same result using tangent correlation functions but this approach is nice and simple. So in terms of persistence length how stiff is Actin? Is DNA stiff or floppy. What about spaghetti? The values in the table below are approximate!

What do these values mean? Well spaghetti would need to be very very long for thermal fluctuations at room temperature to bend it. It turns out that DNA is very stiff for a molecule of its size. Reassuringly microtubules and actin are stiff for length scales exceeding that of the cells that they reinforce.