Beyond the code: the mechanical properties of DNA as they relate to mitosis

Appendix

Explanatory box: life at the nanoscale

Life inside a cell is dominated by viscosity and thermal motion. Weight and inertia, properties familiar in our world as the force of gravity is acting upon us at all times, are hardly relevant. The most important expression for the role of viscous forces at the nanoscale comes from Osborne Reynolds (1883). Reynolds number (F _i/F _v) is the expression of inertial force (F _i) to viscous force (F _v). Inertial force, the classical Newtonian physics law of F = ma, is very small at the cellular level as mass is infinitesimal. At low Reynolds number, viscosity trumps inertia. When a bacterial flagellum ceases to turn, the bacterium coasts less than an angstrom (the width of an H-bond; Purcell 1977). Elastic, viscous, and thermal forces dominate at this scale. The force equation for understanding how cellular materials respond to being pulled or pushed is the elastic modulus of the material (E = Young’s modulus) times the area, force = EA. To solve this equation, we must know the physical properties and the size of the materials in question.

The Young’s modulus is the relation of stress to strain. Stress is the distribution of force per unit area (F/A; inset A, below) and is a measure of how a material reacts to external load. Strain is the geometric expression of deformation (ΔL/L) of the material. Microtubules and DNA have a Young’s modulus of approximately 1–2 GPa, much like hard plastics that have a Young’s modulus of 1–2 GPa and in contrast to rubber that is much lower, 1–10 MPa. However, we know by observing microtubules and DNA that these molecules behave very differently in cells. In general, microtubules appear straight or curved, while DNA is highly coiled and compacted. The parameter that characterizes the fluctuation in shape of a flexible filament is its persistence length. Persistence length describes a filament’s resistance to thermal force and is the distance over which the correlation of the direction of the two ends of a polymer is lost (see inset B below). Fragments shorter than the persistence length of a material are most likely to be linear, while fragments longer than the persistence length are disordered (inset B, below). Microtubules have a persistence length on the order of 5–10 mm (Howard 2001), while DNA has a persistence length of 50 nm. Consider the consequences of microtubule polymers with persistence length in millimeters, while cellular dimensions are on the order of micrometers. One quickly realizes that microtubule-based motors likely influence the mechanical deformation state of the polymers to which they bind. Likewise, histone proteins significantly influence the trajectory of DNA (with a persistence length of 50 nm = ~150 bp at 0.33 nm/bp) as DNA of one persistence length wraps one and a half times around a histone octamer. Histones (and other chromosomal proteins) dramatically alter the material properties of the chromosome, as noted in the large decrease in Young’s modulus relative to DNA (see Table 3).

Table 3 Elastic modulus and persistence length of the major structural elements in the mitotic spindle

Persistence length (L _p) is related to the Young’s modulus (E) in the following way, L _p = EI / k _B T, where I = second moment of area, k _B = Boltzmann constant, and T = room temperature (Kelvin). I is a measure of a material’s resistance to bending, k _B T is the relevant energy scale for all molecular interactions inside a cell (at room temperature, k _B T = 4.1 pN nm). As a material’s flexural rigidity increases (EI), the longer the distance before thermally induced bending (L _p) becomes large.

For a molecule like DNA, where the persistence length is very short relative to the length of an average chromosome (in yeast, the average chromosome is $1 \times {10^{7} } \mathord{\left/ {\vphantom {{10^{7} } {16}}} \right. \kern-\nulldelimiterspace} {16} = \sim 0.2\,{\text{mm}}$; hence, L>>L _p), there are important mechanical consequences. Namely, DNA behaves as an entropic spring. The short rigid domains, being linked via flexible joints, will adopt a state of greatest disorder (entropy), as illustrated in inset B (below). From Hooke’s law, we know that F = ĸx, ĸ = spring constant (Newton/meter), x = change in distance. For small forces, F = 3k _B Tx / n(2L _p)². The spring constant of this freely jointed chain is equal to 3k _B T / n(2L _p)² (n = number of segments). For DNA length 10 kb, the spring constant = 0.036 fN/nm, small indeed. As force is applied to DNA, there is a corresponding decrease in the number of states of disorder, hence a decrease in entropy. In the absence of force, the chain will return to a state of highest disorder. This freely jointed chain has a number of mechanical properties in common with rubber-like materials, namely, that as temperature increases, the spring constant increases and the chains tend to shorten. A demonstration of an entropic spring can be found at the Department of Materials Science and Metallurgy (http://www.doitpoms.ac.uk/tlplib/stiffness-of-rubber/index.php).

Explanatory inset 1: polymer physics