Previously we described flexible polymers, single chains of monomers:

single-stranded DNA - strings of dA,dT,dG,dC nucleotides
single-stranded RNA - strings of A,U,G,C nucleotides
proteins - strings of amino acids

We talked about the unfolded conformations of these chains in terms of random walks, characterized by a step length, or segment length b » 0.5 nm. The average end-to-end distance squared of our random walk model was just

The elasticity associated with random coil polymers was characterized by the entropic force scale

Three examples of biofilaments which are heavily studied and which we will focus on:

B-DNA - the double-stranded double helix, formed by two complementary-sequence ssDNAs stuck together

actin filaments - 6 nm-thick filaments formed by assembly of proteins (actin `monomers') into filaments; actin filaments form much of the cytoskeleton, and are the tracks along which myosin `motors' move inside muscle fibers to generate forces

microtubules - linear polymers of two types of proteins (a and b tubulin) which form stiff `sticks' about 24 nm thick; microtubules play a structural role in cells, they form long `tracks' which serve as `highways' which `motor proteins' move along

Here are some figures from Molecular Biology of the Cell by Baltimore, Lodish and Darnell to give you an idea of the structures of these things.  There is of course no substitute for reading about them.

These objects are stiff at all at nanometer scales, but dsDNA and actin are flexible enough to be gradually bent by thermal fluctuations over hundreds to thousands of nanometers (tenths to a few microns).   Microtubules are by comparison very stiff, and we will see exactly how stiff and why they are so stiff.

Actin filaments are usually one to a few microns in length in vivo; microtubules are typically ten microns in length in animal cells.

We can use ideas from the theory of elasticity to describe the basic flexibility of these objects and their coupling to external stresses - and then use statistical mechanics to talk about deformations driven by thermal fluctuation. This area has been of the most fruitful areas for biological physics, with many biologically relevant problems that experimentalists and theorists have to work together to understand.

An essential concept that we will develop is persistence length, the distance that a biofilament bends through about a radian (57 degrees, remember 360 degrees = 2 p rads) by random thermal fluctuation. dsDNA, actin filaments, and microtubules have persistence lengths of about 50 nm, 20 microns, and 1 mm, respectively.

There are lots of other biofilaments which are important biologically, many of which have not been terribly heavily studied biophysically (e.g. intermediate filaments, chromatin fiber...). You can start reading about these in any molecular cell biology book like Alberts et al...

The same type of approach can be used to talk about flexibility of cell membranes, as we will discuss in a later lecture.

Suppose we have a rod of equilbrium length L and circular cross section, where the cross-section radius is r. We imagine that the material that the rod is made out of is bonded together well enough that when the rod is deformed and then allowed to relax, it comes back to its original shape and length. We'll call such a material elastic.

Many materials are not elastic, e.g. water and silly putty. However, dsDNA, actin and microtubules are all elastic objects.

We stretch (or compress) the rod by force f. We say that the stress or force per area applied to the interior of the rod is s = f / (pr²).

The starting point for all of elasticity theory is doing this experiment and finding that the fractional change in length of the rod (the elongational strain) is proportional to the applied stress, for small deformations:

The Young modulus describes the stretchability of an elastic material, in an intensive (volume- and shape-independent) way.

A physical interpretation of Y is as the stress (force per area) at which our rod stretched to double its equilibrium length.

Since most materials have nonlinear elasticity which destroys the linear relation between stress and strain, a slightly better thing to think is that 0.01 Y is the stress at which a rod stretches by 1%.

steel                     2 ×10¹¹

glass                     1 ×10¹¹

bone                      2 ×10¹¹

wood                      1 ×10¹⁰

folded protein              10⁹

plexiglass                4×10⁸

dsDNA                     3×10⁸   (Smith et al, Cluzel et al Science 1995)

polymer rubber            10⁶

polymer gel               10⁴

chromosome folded         300     (Poirier et al, Mol. Biol. Cell 2000)
up for cell division

So, folded biopolymers are like little pieces of hard plastic. This is no surprise since both are held together by non-covalent bonds of about the same strength and density.

This leads to a way to think about Y in terms of the cohesive energy per volume holding a material together. Biopolymers are held together by interactions each of which are a few k_B T in strength, with a density of an interaction for every 10 or so cubic angstroms (the volume of a nucleotide or amino acid).

So we would guess
Y » 4×10^-21 J/(10 ×10^-30 m) = 400 MPa

which is intermediate between the actual Y for dsDNA and for folded proteins.

This kind of thinking works for other materials too: crystalline metals and glasses are held together by covalent bonds of eV energies, at a density of a few bonds per cubic angstrom. So we can guess the typical modulus of a covalently bonded crystal:
Y » 10^-19 J/10^-30 m³ = 10¹¹ Pa
which is again not too far from the truth.

Taking the linear stretching law,

f₀ = pr² Y » 3 ·(1×10^-9 nm)² ·(3×10⁸ Pa) = 10^-9 N
= 1 nanonewton (nN) = 1000 piconewtons (pN)

Therefore, a dsDNA will be lengthened by 5% at a force of around 0.05 f₀ = 50 pN.

You can more or see this is true from the following data, which are taken from Smith et al, Science 1995, one of the first experiments directly studying the mechanical properties of single dsDNA molecules.

Now that we know the force vs extension, we can integrate the force to compute the work done on the rod by stretching it. This is elastic energy stored in the rod:

E = (f₀/2) DL² / L = k_B T/2 (equipartition)

so the mean-squared fluctuation is

< (DL)² > = k_B T L / f₀

and the typical fluctuation of length is

We have f₀ = 1000 pN for dsDNA, and we have an average rise of 0.34 nm per base pair. Taking L = 0.34 nm, the fluctuation in rise is

DL = [ 0.34 nm^.4.1 pN·nm/1000 pN]^1/2 = 0.04 nm = 0.4 Å

More careful calculations indicate that the rise fluctuates even more than this - so the 3.4 Å rise per base inferred from DNA crystals and often stated with great seriousness as the rise per base pair, actually is at best a rough average for DNA in solution or in a cell.

Here is a picture of a short dsDNA from a computer simulation which includes thermal agitation and the surrounding water molecules (which are not shown), showing the sort of deformations that people would guess occur at small scales [From simulation data of Michael Feig of the Scripps Institute and Monte Pettitt of the University of Houston]:

In general the concept that biomolecules are soft enough to have their structures fluctuate at » nm length scales is not well appreciated in the molecular-biological literature. This is good since it leaves lots of things for biological physics people to do!

We just stretched a rod. What happens if we bend a rod? Well, the outer edge of the rod gets stretched, while the inner edge gets compressed. Therefore the energy of bending of a rod is again just dependent on its stretching elasticity, i.e. on Y.

We again have a rod of circular cross section, cross section radius r, total length L, now bent in a circular arc of radius R:

We suppose that R >> r so that the deformation of the rod is actually small at the scale r. The outer edge of the rod is lengthened by an amount DL = (r/R)L, the mid-line of the rod stays its unperturbed length L, and the inner edge of the rod is compresed to have DL = -(r/R)L.

As we go across the rod (from z=-r to z=+r), the strain therefore varies (to linear order, appropriate for slight bends) as DL / L = z/ R. Note that this gives a net strain - and therefore a net tensile stress - of zero in each rod cross-section. Therefore the rod as a whole is not being stretched - just bent.

If we integrate the elongational energy per volume, (Y/2) (DL/L)², across the cross-section of the rod, we can find the elastic bending energy per length of rod:

Some arithmetic, and conversion of the integral to a dimensionless variable x = z/r gives

The bending constant has dimensions of energy times length (MKS: J m), or force times length² (MKS: N m²).

The form of the bending energy is in retrospect, obvious. The bending energy should be proportional to 1/R² so that it is minimized when R ® ¥ (for the equilibrium straight rod), and should not depend on the sign of the radius of curvature.

People sometimes talk about curvature k = 1/R which is just the reciprocal of the radius of curvature. Curvature is often useful since it is zero for a straight rod, and the energy is thus simply analytic in the curvature:

B = (p/4) Y r⁴ » (3/4) ×3×10⁸ ×(1×10^-9)⁴ J m » 2×10^-28 J m

This is a very rough estimate - dsDNA is not exactly a uniform elastic rod of 1 nm cross-section radius! But this turns out to be a not bad estimate of the bending constant of dsDNA, as we will see shortly.

Also, note that the fourth power means that B goes up quickly with rod thickness.

In the absence of other information, we guess that the appropriate Young modulus is » 10⁹ Pa, since both actin filaments and microtubules are made of a bunch of folded proteins, polymerized end to end. For actin filaments,

B = (p/4) Y r⁴ = 0.75 ×1 ×10⁹ ×3⁴ ×10^-36 J m = 6×10^-26 J m

For microtubules, we must use the same Y, and we can use the ratios of the radii, 12/3 = 4, and just multiply the actin result by 4⁴ = 256 to get B = 1.5 ×10^-23 J m.

These will turn out to be rather surprisingly close to the experimentally determined values for B for all three of these biofilaments.

Why do people use I-beams and hollow girders in construction?

These numbers for B in J m are a bit hard to understand. We'll see how to visualize the real meaning of B for biofilaments in the next section.

L. Landau and E.M. Lifshitz, Theory of Elasticity, Pergamon Press 1985 - the essential modern elasticity textbook for physicists

Smith SB, Cui YJ, Bustamante C, Science 271 795-799 (1996) and Cluzel P, Lebrun A, Heller C, Lavery R, Viovy JL, Chatenay D and Caron F, Science 271, 792-794 (1996). These two groups simultaneously published experimental data showing the elastic response of single dsDNA molecules. A wealth of data are found in these papers, including the Young modulus of dsDNA and the discovery that B-DNA `switches' to a new long form when under about 65 pN of stretching force.

Gittes F, Mickey B, Nettleton J, Howard J, J Cell Biol 1993 Feb;120(4):923-34, also Ott A, Magnasco M, Simon A, Libchaber A, Phys Rev E 48, R1642-R1645 (1993). After much confusion, careful experiments by these two groups established that the persistence length of actin filaments was about 17 microns. The paper by Gittes et al also discusses microtubule stiffness.

J. Netting, Science News 159, 198 (2001)  - A news report of a new discovery that bacterial have actin-like proteins and some kind of cytoskeleton.  This is a big surprise and comes only a few years after tubulin-like proteins and microtubules were discovered in bacteria.  Every year new discoveries about prokaryote cells show that they are more closely related to eukaryotes than previously thought.   You may enjoy the brand new research article on actin in bacteria, L.J.F. Jones et al, Cell 104, 913-922 (2001).

[Fluorescence image of bacterial cytoskeleton using labeled actin-like proteins]