14.1 Physicochemical structure of DNA [DNA top]

DNAexists in a number of double-strandedforms, including A-DNA, B-DNA, and Z-DNA, as well as in single-stranded form. In this chapter and throughout this text, we focus on the (most common) B-DNA double-stranded form, with only brief mention of single-stranded DNA. Throughout this chapter, then, “DNA” and “dsDNA” imply double-stranded B-DNA unless otherwise specified.

Double stranded B-DNA’s physical properties are well studied and are prototypical of linear (unbranched) macromolecules. It thus is a model system for exploring introductory polymer physics. Figure 14.1 highlights features of DNA at several length scales. DNA in aqueous solution appears as a spaghetti-like linear polymer with a characteristic radius. If the local orientation of the polymer chain is viewed in more detail, say on 10–100 nm length scales, the relative rigidity of the polymer chain is measured using apersistence length ℓ_p. Below 10 nm, the polymer has a diameter of approximately 2 nm, with a molecular structure characterized by a double-helix with a negatively-charged sugar structure on the exterior and hydrogen bonding between complementary base pairs on the interior. Mathematical definitions of ℓ_p and  are presented in Section 14.1.2.

14.1.1 Chemical structure of DNA

A DNAmolecule has a hydrophilic sugar (deoxyribose) backbone with negatively-charged phosphate groups and a sequence of nitrogenous bases. This sequence consists of the bases adenine (A), guanine (G), cytosine (C), and thymine (T). Biologically, this sequence of bases codes for protein sequences in organisms. Hydrogen bonding between A and T bases and between C and G bases causes two strands of DNA with complementary sequences to spontaneously bind and form a double helix structure, in which the nitrogenous bases are hydrogen bonded to each other in the interior of the structure, while the phosphate groups are pendant on the outside structure. The stability of this structure in aqueous solutions confers relatively high chemical stability todouble-stranded DNA in biological systems.

The hydrogen bonds that cause two strands of DNA to come together can be overcome by thermal energy, a process called melting or denaturing the DNA. When this happens, the two strands of DNA fall apart and become single-stranded DNA(ssDNA), a form that is significantly less chemically stable. The temperature at which melting occurs is referred to as the meltingtemperature of the DNA molecule, which varies depending on sequence and solution conditions, but is typically 45-60^∘C.

Melting is reversible–when the hydrogen bonding overcomes thermal energy, two complementary single-stranded molecules bind with each other to form a double-stranded molecule, a process calledannealing. When the annealing process is observed between single-stranded molecules from separate sources, we refer to it as hybridization. Measuring the degree to which an unknown ssDNA or ribonucleic acid(RNA) sample hybridizes with a known ssDNA molecule is a useful way to identify the genetic makeup of an unknown sample. These hybridization assays, includingSouthern blotting, northernblotting, and DNAmicroarrays, are powerful analytical tools.

14.1.2 Physical properties of dsDNA

Double-stranded DNA is a prototypical linear chain polymer, and it has a number of physical properties and characteristic lengths, described below, illustrated in Figure 14.2, and listed in Table 14.2. We consider DNA as a dilute component in a goodsolvent, and thus we consider DNA’s interactions with the solvent only. Water and aqueous solutions are “good” solvents for DNA, and thus this chapter implicitly treats the dynamics of linear polymers in good solvents. The behavior of DNA in abad solvent, e.g., an nonpolar organic solvent, would be quite different. Of note, the variation between bases (A, T, C, and G) has little effect on the physical properties of DNA. For most DNA molecules, the polymer properties can be approximated as independent of the base sequence. The physical properties of DNA are, for the most part, a function of the length of the DNA strand only.

In this section, we start by defining physical and mathematical terminology for flexible linear (unbranched) polyelectrolytes, for which DNA is an excellent prototype. These tools are useful when we develop models to relate intrinsic polymer properties (sequence, contour length) to observed transport properties (diffusivity, electrophoretic mobility).

Definition of linear (unbranched) polymer properties and length scales

We start by modeling a linear polymer such as DNA mathematically as an idealized contour through space, and we describe the properties and length scales of this contour. Intuitively, we can think of the DNA molecule as behaving somewhat like a microscopic piece of spaghetti suspended in water. In this approximation, we ignore all details of the chemical structure other than the contour of the chemical backbone, and subsume all chemical properties into contour parameters such as the persistence length and radius of gyration, to be discussed below. The geometry of the DNA molecule (as defined by the backbone contour) fluctuates with time owing to the thermal fluctuations in the system. Because of this, we define the molecular state using both instantaneous properties of the contour (which fluctuate) as well as time- or ensemble averages (which do not).

Key length scales and definitions for DNA include the diameter, contour length, persistence length, end-to-end length, and radius of gyration. dsDNA has a spatially-uniform diameter of approximately 2 nm. Because DNA has a significant negative charge, the electrical double layer surrounding the DNA leads to electrostatic repulsion between components of the polymer, and the effective diameter can be much larger.

Thecontour length ℓ_c of a DNA molecule is the arclength of the backbone contour, i.e., the distance we would travel if we moved along the curved backbone from one end of the molecule to the other. In dsDNA, the base pair spacingis approximately 0.34 nm, and thus the contour length of a DNA molecule with N_bp base pairs is ℓ_c ≃ 0.34 nm×N_bp. DNA molecules can range in length from just a few base pairs (these molecules are typically referred to as oligomers) to hundreds of thousands or millions of base pairs (see Table 14.1). The contour length and the base sequence are the only parameters intrinsic to the DNA molecule, and since the sequence has little effect on the physical properties, the contour length is the only molecular parameter that significantly affects DNA physical properties in aqueous solutions.

We define the coordinate system and notation as follows: we define the scalar s as the arclength (i.e., the distance along the polymer backbone, where s = 0 corresponds to one end and s = ℓ_c corresponds to the other end. ℓ_c is the contour length describedabove. We also use s₁ and s₂ to denote two specific points along the polymer, and use Δs defined as to denote the arclength between these two points. Since the polymer contour is in general curved, the arclength is not equal to the linear distance between the points. We also define (s) as the position vector of a point on the backbone with respect to the coordinate system origin. The unit vector tangent to the polymer backbone is , and the vector quantifying the magnitude and direction of the local curvature of the backbone is proportional to .

Thepersistence length ℓ_p is a measure of the rigidity of a linear polymer, and is evaluated by determining the distance two points of the DNA polymer need to be from each other for their orientation to become statistically uncorrelated. We use a statistical measure because the position and orientation of a DNA molecule in an aqueous solution is always fluctuating with time owing to thermal perturbations. The persistence length is a measure of the rigidity of the polymer backbone. If the backbone is stiff (imagine uncooked spaghetti), then the components of the backbone tend to point in the same direction. If the backbone is flexible (now imagine cooked spaghetti), then the parts of the polymer backbone point in random directions.

Persistence length has a precise mathematical definition, namely

(14.1)

where brackets denote the time-average of a fluctuating property. The persistence length ℓ_p is meaningful only if it is independent of Δs. The persistence length can equivalently be defined using

(14.2)

This relation compares two points along the polymer backbone that are separated by a distance Δs along the backbone. Over time, it takes the unit tangent vector to the polymer backbone at each of these two points, compares them by taking the dot product of the two vectors (effectively evaluating the cosine of the angle between the two tangents), and averaging the result. This relation asserts that the time-averaged cosine of the angle decays exponentially as the arclength between the two points increases. Two proximal points are perfectly correlated, the angle between their tangents is zero, and the dot product between their unit tangent vectors is one. Two points separated by a large arclength are uncorrelated, and the cosine of the angle between them varies randomly between -1 and 1, eventually averaging to zero. In between, the correlation of tangent angles decays exponentially as the arclength separating the points increases. The persistence length is the characteristic length of this exponential decay. dsDNA, for example, has apersistence length approximately equal to 50 nm in concentrated electrolyte solutions at room temperature. Structural biological molecules are stiffer and have a higher persistence length (for example, F-actin’s persistence length has been measured to be 17 μm [139]).

The persistence length is important when developing equations to relate DNA transport properties to intrinsic polymer properties and solution conditions. The persistence length provides a means for understanding DNA’s configuration, i.e., the shape of the DNA molecule as a function of contour length and external environment. While linear polymers such as DNA typically have a well-defined ℓ_p, many models that describe these polymers do not.

Theend-to-end length ℓ_e of a DNA molecule is a scalar measure of the linear distance (not the arclength) between the two endpoints of the molecule at any instant. This can thus be written as

(14.3)

where the vertical bars denote the magnitude of the vector. Often, we are actually more concerned with , the time-averaged value of this property, given by

(14.4)

Theradius of gyration  of a DNA molecule is a statistical measure of the linear distances between different points on the DNA backbone, and ³ is thus an approximate measure of the volume that encloses the DNA molecule. Because the light scattered off of a DNA molecule in solution in certain limits is proportional to , the radius of gyration is usually the most easily measured DNA property in solution. The radius of gyration  is defined as the time average of the root mean square of the linear distances between the elements of the contour and the polymer centroid.

(14.5)

The size of a microfluidic domain, when compared to , tells us whether the configuration of a DNA molecule in aqueous solution is affected by the fluid boundaries.

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Symbol	Property	When/How Observed	When/How Calculated
d	diameter	observed using x-ray crystallography	from molecular models of DNA structure
ℓ c	contour length	not directly observed	known from number of base pairs
ℓK	Kuhn length (see modeling section)	not a physical length	chosen to make idealized models match physical observables
ℓe	end-to-end length	microscopy of fluorophores attached to ends	fluctuates; only  predicted by models
< ℓe >	ensemble-averaged end-to-end length	microscopy of fluorophores attached to ends	predicted by models
ℓ p	persistence length	not directly observed	predicted by some models
	radius of gyration	experimentally observed with light scattering or fluorescence microscopy [140]	predicted by models
D	diffusivity	observed directly using fluorescence microscopy of diffusing molecules[141, 142]	related to contour length using  and Zimm dynamics
μEP	electrophoretic mobility	observed directly with EP velocity if EOF is well-characterized and subtracted	related to surface charge density by double layer theory and Rouse dynamics
	extended length	fluorescence microscopy of molecules in nanochannels see [143, 144]	see [143, 144]

Table 14.2: Modes of observation and calculation of physical properties and length scales of DNA.

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