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3.6 Appendix
3.6.1 The Genetic Material dsDNA
In this chapter we worked with bacteriophages with dsDNA as the genetic material. It would be useful to briefly review the genetic material in this section. Without doubt, the mechanical properties of dsDNA depend primarily on its chemical composition and structure. In this section we will briefly touch upon this aspect and try to argue a bit if the entire details of the structure are relevant for the mechanical behavior we are out to describe.
3.6.1.1 DNA Composition and Structure
DNA stands for deoxyribonucleic acid. It is constructed from individual monomeric units to form an unbranched chain. The elementary monomeric unit of DNA(also RNA), nucleotides, are a troika of phosphate, sugar, and organic base, with the phosphate and sugar units alternating along the backbone. The two different sugars found in the nucleotides, ribose in RNA and deoxyriboses in DNA, are five membered rings differing from each other by only one atom. In either molecule, the OH−1 groups are potential reaction sites for addition of a base. Five organic bases are found in nucleotides, and they fall into two chemically similar groups: purines and pyrimidines. Only four of the five bases are present in a given DNA or RNA molecule, and the onemissing base is different for each:
1) RNA: adenine, guanine, cytosine, uracil.
2) DNA: adenine, guanine, cytosine, thymine.
The reaction of a sugar with a base releases water (an -OH from the sugar plus an H from the base)
Figure 3.16: DNA structure and composition
and produces a sugar-base combination called a nucleoside. Addition of a phosphate to the nucleoside releases water and produces a nucleotide. The nucleotides themselves can polymerize to form DNA and RNA, through linkage between a sugar from one nucleotide and a phosphate from another. In the double-stranded helix of DNA, the bases lie in the interior of the helix and hold the helix together through the hydrogen bonding between base-pairs. As illustrated in Fig. 3.16, each matching base pair on the opposing strands consists of one purine and one pyrimidine: adanine/thymine and guanine/cytosine.
The dsDNA is roughly a cylindrical molecule of diameter 2nm. It consists of a stack of roughly flat plates (the basepairs), each about 0.34nm thick. But the genome total length of λ−phage is 16.5nm, still far bigger than the diameter. Hence, we may hope that the behavior of DNA on such long length scales may not depend very much on the entire details of its structure.
3.6.1.2 Persistence Length and the Elastic Energy of DNA
In the previous section we had a brief argument as to why the entire structural details of the chain may be irrelevant in describing the mechanical properties of DNA. In this section we will model DNA as an elastic rod and introduce the concept of persistence length. If we bend an elastic rod of lengthLand bending-modulusκinto an arc of a circle of radiusR, the elastic energy of the rod is
given by
Earc= 1 2
κL
R2, (3.37)
κ = Y I, where Y is the Young’s modulus, a material property, and I is the area moment of inertia (larger for larger bending cross-section and proportional to R4 for circular cross-section), a geometrical property. The energy of an object in thermal equilibrium fluctuates with time with the thermal energy scale set bykBT (kB = 1.38×10−23J/K is the Boltzmann constant, and T is the absolute temperature). At room temperature the value ofkBT in units of pN-nm is approximately 4.1. One way of quantifying the amplitude of the shape fluctuations at finite temperature is finding the typical distance along the rod over which it loses its tangent-tangent correlation. This length scale must be directly proportional to the bending rigidity κand inversely proportional toT. The combinationκ/kBT has units of length and is called the persistence length.
ξp= κ
kBT (3.38)
So long as its persistence length is large compared to its contour length i.e. ξp L, a filament appears relatively straight.
If the polymer is bent into a curve parametrized by arc length parameters, as in Fig. 3.17 then the following relationships hold.
∂t/∂s = Cn (3.39)
Cn = ∂2r/∂s2. (3.40)
whereCis the curvature at that point and is given by 1/RcwhereRcis the local radius of curvature.
Equation(3.37) gave us an expression for energy per unit length of an elastic rod bent into a circular arc. So the energy of any arbitary curve will be,
Ebend = (κ/2) Z L
0
1/R2cds= (κ/2) Z L
0
(∂t/∂s)2ds= (κ/2) Z L
0
(∂2r/∂s2)2 (3.41)
If we assume that the chain is laterally isotropic, i.e., any rotation perpendicular to the lateral plane does not change its energy, then it can be shown that [92]
ht(0).t(s)i= exp(−s/ξp). (3.42)
r(s) - position
t(s)=unit tangent vector
s=arclength
n
1n
2t
1t
2R
cFigure 3.17: The schematic of an idealized curve describing DNA bending. (a)A point on the curve at arc length s is described by position vectorr(s) and a unit tangent vectorts (b) Two locations are seperated by an arc length ∆s subtending an angle ∆θas a vertex formed by extensions of the unit normalsn1andn2and intersect at a distanceRc from the curve.
The lengthξp in the exponential of the above equation is called persistence length. The persistence length can be related to bending moduliκas [92]
κ=ξpkBT. (3.43)
Similarly, we can also prove that the end-end distance of such a chain is given by
hr2eei= 2ξpL−2ξp2(1−exp(−L/ξp)). (3.44)
Hence, we can say that
hr2eei=
2ξpL ( ifLξp) L2( ifLξp)
(3.45)
The persistence length of DNA is observed to be 50nm [74] under physiological conditions. The typical phage capsid size is around tens to hundreds of nanometers. Thus, the DNA has to be tightly bent inside the phage capsid, and thus it should cost a large amount of bending energy to pack the DNA inside the phage capsid.