Structure and interactions of biological helices

Helices are essential building blocks of living organisms, be they molecular fragments of proteins ($\alpha$-helices), macromolecules (DNA and collagen), or multimolecular assemblies (microtubules and viruses). Their interactions are involved in packing of meters of genetic material within cells and phage heads, recognition of homologous genes in recombination and DNA repair, stability of tissues, and many other processes. Helical molecules form a variety of mesophases in vivo and in vitro. Recent structural studies, direct measurements of intermolecular forces, single-molecule manipulations, and other experiments have accumulated a wealth of information and revealed many puzzling physical phenomena. It is becoming increasingly clear that in many cases the physics of biological helices cannot be described by theories that treat them as simple, unstructured polyelectrolytes. The present article focuses on the most important and interesting aspects of the physics of structured macromolecules, highlighting various manifestations of the helical motif in their structure, elasticity, interactions with counterions, aggregation, and poly- and mesomorphic transitions.

Figure 1

Atomic structure of (a) a sample polypeptide $\alpha$-helix, (b) a double-stranded helix of $B$-DNA, and (c) triple-helix collagen. The gray ribbons are guides to the eye showing the protein or DNA helical backbones. The two grooves separating the sugar-phosphate backbone strands (ribbons) in DNA are often referred to as minor and major grooves based on their size in the $B$ form of DNA.

Figure 2

X-ray-diffraction pattern from calf thymus $B$-DNA (107) showing the location of layer lines, Eq. (9), on the scattering wave-vector axis. The lines of constant $k_z$ (thin solid lines) are slightly curved due to the specific geometry of x-ray film mounting. From 107.

Figure 3

Convention used for x-ray analysis of helical molecules. The location of each molecule is given by the lateral $\left({\mathbf{R}}_{\nu }\right)$ and axial coordinates $\left(Z_{\nu }\right)$ in the laboratory frame $(X,Y,Z)$. The local $(r,z,\varphi )$ coordinates of the atomic scattering centers in the molecule $\nu$ have their origin at the location $({\mathbf{R}}_{\nu },Z_{\nu })$. The molecular azimuthal orientation ${\Phi }_{\nu }$ of a helix is determined by a cut through molecular origin $({\mathbf{R}}_{\nu },Z_{\nu })$ parallel to the $XY$ plane.

Figure 4

Sketch of a supercoil (coiled coil) formed by two helical molecules (strands).

Figure 5

Schematic drawing of the pattern of fractional $(\sim 0.5e)$ charges on amide (+) and carbonyl (−) residues on two interlocked $\alpha$-helix backbones. The coil of carbonyls is shifted axially from the coil of amides by $\approx H∕2$ so that the adjacent, hydrogen-bonded amide and carbonyl are axially aligned. There are about 3.7 carbonyl and amide residues per helical turn in each coil $(H∕h\approx 3.7)$. ${\Phi }_1$, ${\Phi }_2$ and $Z_1$, $Z_2$ are the azimuthal orientations and axial coordinates of each helix, correspondingly. They are defined at the point of origin on each helix, which is selected in the middle between the hydrogen-bonded amide and carbonyl. The azimuthal orientation of each helix is the angle between the $x$ axis and the vector pointing from the centerline to the point of origin. The relative azimuthal angle between the two $\alpha$-helices is defined as $\delta \varphi ={\Phi }_1-{\Phi }_2-2\pi (Z_1-Z_2)∕H$. The net charge of these zwitterionic helices is zero.

Figure 6

Wigner crystal of ordered condensed counterions on uniformly negatively charged cylinders. The interaction energy is minimized when correlated counterions on one cylinder oppose correlation holes (negatively charged spaces between adsorbed counterions) on the other. Such a configuration may result in attraction even if there is a net charge on the cylinders.

Figure 7

Schematic diagram for interacting parallel DNA molecules with ideal, negatively charged double helical backbones (phosphate strands) and counterions absorbed in the major and minor grooves. The azimuthal ${\Phi }_{\nu }$ and axial $Z_{\nu }$ coordinates of each molecule are defined with respect to a reference cross section as the azimuthal coordinate of the middle of the minor groove and the height of the cross section, respectively. The reference cross sections for the two molecules are selected consistently so that the backbone of one molecule is completely superimposed onto the backbone of the other molecule upon a lateral shift by $\mathbf{R}$, azimuthal rotation by ${\Phi }_2-{\Phi }_1$, and axial shift by $\delta Z$. The interaction depends only on $\delta \varphi ={\Phi }_1-{\Phi }_2-2\pi \delta Z∕H$. In this diagram, a value of $\delta \varphi =0$ is shown, which minimizes the interaction energy at large $R$ by maximizing energetically favorable interactions between negatively charged strands on one molecule and positively charged grooves on the other molecule. The ideal helical structure of the backbones results in the same alignment repeating with the axial periodicity $H$, creating an electrostatic zipper. Unlike the simpler zipper that locks in opposing negatively and positively charged groups on single-stranded helices at $\delta \varphi =0$ regardless of $R$ (cf. Fig. 5), the optimal alignment between the double helices depends on $R$. The alignment with $\delta \varphi \ne 0$ becomes energetically favorable at small $R$ [Eq. (60)].

Figure 8

Electrostatic contributions to the interaction energies given by Eqs. (51, 58, 59). The interaction energies shown demonstrate the relative importance of including the higher-order helical harmonic contributions in the overall interaction. Here it is assumed that counterions compensate 95% of the DNA phosphate charges, i.e., $\theta =0.95$, and the helical harmonics are set to ${\zeta }_n=1$. (For this calculation and subsequent calculations, unless noted, the radius of DNA is taken to be $a=9.5\mathrm{\AA }$ and the Debye screening length is set to ${\kappa }_D^{-1}=7\mathrm{\AA }$.) Note the faster decay for higher-order helical harmonics.

Figure 9

Interaction energy between two DNA molecules with ideal helical backbones as a function of the interaxial separation $R$ and the relative azimuthal orientation $\delta \varphi$ between molecules calculated at $\theta =0.9$ and ${\zeta }_n$ corresponding to 40%:60% minor:major groove distribution [Eq. (40)]. The surface plot shows a bifurcation point at $R\sim 35\mathrm{\AA }$, $\delta \varphi =0$, resulting from azimuthal frustration in the interaction potential due to the competition between the first and second helical harmonics ($u_1$ and $u_2$) in Eq. (57). At larger $R$, a single $\delta \varphi =0$ optimizes the energy. At smaller $R$, the energy is optimized at two finite $\delta \varphi$ given by Eq. (60). At $R<25\mathrm{\AA }$ (not shown), the interaction becomes repulsive due to the contribution of image forces and possible hard-core clashes between residues on opposing molecules.

Figure 10

(a) Pair interaction energy and (b) corresponding recognition energy for nonhomologous torsionally flexible ($C_t=3\times {10}^{-19}\mathrm{erg}\mathrm{cm}$, solid line) and rigid ($C_t=\infty$, dashed line) molecules (52). The recognition energy [Eq. (66)] is defined as the difference in the pair energy between nonhomologous and homologous (short dashed line) molecular pairs. Results are shown for $\theta =0.8$, ${\zeta }_n$ corresponding to 30%:70% minor:major groove distribution of condensed counterions [Eq. (40)], interaxial separation $R=30\mathrm{\AA }$, and a helical coherence length ${\lambda }_c\sim 300\mathrm{\AA }$. As shown, torsional flexibility reduces the recognition energy but does not totally eliminate it.

Figure 11

Frequency of homologous recombination as a function of the base-pair length for homologous DNA molecules. Recombination becomes frequent only for sequences with 50–200 base pairs. Replotted from the data of 402.

Figure 12

Schematic illustration of two helices skewed at an angle of $\psi$. (a) The length of effective juxtaposition $\left(L_{\mathrm{eff}}\right)$ represents the region of significant overlap between electric fields created by molecules, which determines the electrostatic interaction energy. (b) Relative orientation of strands and grooves within the effective juxtaposition region determines the chiral torque between molecules. The helix with lighter markings lies behind that with darker markings. As shown here, right-handed helices prefer a right-handed twist to prevent crossing of the helical strands (dashed dark lines on the back of the front helix and solid gray lines on the front of the helix in the rear). For cholesteric liquid crystals, such a preferred twist will result in a right-handed cholesteric chirality. In the case of supercoils (or coiled coils), this preferred twist will yield a left-handed supercoil since such a twist will lead to helices wrapping around each other in the chiral sense opposite to their own handedness.

Figure 13

Cross-sectional diagram of a hexagonal columnar assembly of cylinders. The hexagonal Wigner-Seitz cell (dotted line) about each molecule can be approximated as a cylinder with radius $R_s=R\sqrt{\sqrt{3}∕2\pi }$, where $R$ is the lattice spacing between nearest-neighbor cylinders.

Figure 14

X-ray studies of columnar DNA assemblies at different densities. (a) Diffraction patterns of hydrated DNA fibers (cf. Fig. 2) at three different interaxial spacings $d_{\mathrm{int}}$ between adjacent double helices. The interaxial spacings are calculated from the positions of the Bragg peaks (the two dark spots on the equator), $K=k_{\text{Bragg}}$. The locations of the diffraction maxima on the first layer line are marked by white arrows, highlighting their change with $d_{\mathrm{int}}$. These maxima follow the Bragg peaks, suggesting a large intermolecular contribution to the scattering. (b) Variation in the locations ${\mathbf{K}}_n$ of the diffraction maxima on the first $(n=1)$, second $(n=2)$, and third $(n=3)$ layer lines with separation $\left(d_{\mathrm{int}}\right)$ between DNA. The bold lines show expected locations of the intramolecular maxima [Eq. (101)]. Thin solid lines show the locations of the diffraction peaks in hexagonal crystals with the same azimuthal orientation of all molecules; dashed lines, in crystals with alternating orientations of molecules in consecutive rows; dotted lines, in crystals with different azimuthal orientations of all three molecules in the unit cell (see Fig. 20). The corresponding Miller indices are shown near each line. Intermolecular scattering appears to determine ${\mathbf{K}}_1$ at all separations. ${\mathbf{K}}_2$ are dominated by intramolecular scattering at $d_{\mathrm{int}}>35\mathrm{\AA }$ resulting in flattening of the observed dependence. ${\mathbf{K}}_3$ are consistent with intramolecular scattering at all $d_{\mathrm{int}}$. Reproduced from 182.

Figure 15

Simplified side view of an elementary building block (unit cell) that has the characteristic symmetry of a cholesteric liquid crystal. The cholesteric liquid crystal can be viewed as a stack of molecular layers where local packing of molecules is hexagonal and the molecules in each layer are twisted by an angle $\Psi$ with respect to another layer. The unit cell consists of a molecular triad with two parallel molecules (labeled 2 and 3) and a third molecule (labeled 1) twisted in the direction parallel to the plane formed by the first two molecules. The liquid crystal is an assembly of such triads where molecules may have random shifts (not shown) along their axes.

Figure 16

Cholesteric pitch shown as a function of the interaxial distance between 146 base-pair DNA molecules with radii $a=9\mathrm{\AA }$, as found from Eq. (103) at $\theta =0.7$. The different curves were calculated at ${\zeta }_n$ corresponding to different distributions of condensed counterions [Eq. (40)]: (1) $f_1=0.3$, $f_2=0.6$, $f_3=0$; (2) $f_1=0.35$, $f_2=0.6$, $f_3=0$; (3) $f_1=0.35$, $f_2=0.55$, $f_3=0$; (4) $f_1=0.4$, $f_2=0.55$, $f_3=0$ (190). Inset: The pitch measured experimentally for DNA molecules of the same size in solutions with different salt concentrations (reproduced from 363). The estimated and measured pitch vs distance curves share the same qualitative behavior, i.e., a minimum at intermediate separations, a sharp rise at shorter separations (higher densities), and a gradual rise at larger separations (lower densities).

Figure 17

Experimental and theoretical plots of osmotic stress in columnar DNA assemblies. (a) Measured osmotic stress vs interaxial separation in DNA solutions with different salts at different concentrations and temperatures (320, data kindly provided by D. Rau). For monovalent NaCl, repulsion is always seen and can be fitted by the PB theory. However, for $\mathrm{Mn}{\mathrm{Cl}}_2$, at intermediate separations the system is metastable or unstable (collapses) at constant pressure (dotted lines). At higher temperatures, the range of the attraction increases, which may occur due to redistribution of ${\mathrm{Mn}}^{2+}$ between binding sites in minor and major grooves (51). (b) Calculated osmotic pressure vs separation in aggregates of torsionally flexible, nonideal DNA (Sec. 5D, 5E, 6B, 6C). The top curve (thick solid line) shows the pressure calculated at $0.15M$ monovalent salt concentration assuming $\theta =0.1$. The lower plots show the pressure at $50\mathrm{m}M$ concentration of 2:1 electrolyte, $\theta =0.8$, ${\lambda }_c=300\mathrm{\AA }$, and different torsional rigidities $C_t$ and partitioning of condensed counterions between minor and major grooves [Eq. (40)]: $C_t=3\times {10}^{-19}\mathrm{erg}\mathrm{cm}$ and 40%:60% minor:major groove distribution (thin solid curve); $C_t=0.5\times {10}^{-19}\mathrm{erg}\mathrm{cm}$ and the same counterion distribution (dashed curve); and $C_t=0.5\times {10}^{-19}\mathrm{erg}\mathrm{cm}$ and 20%:80% minor:major groove distribution (dash-dotted curve). The dotted tie lines represent first-order transitions, resulting in part from the azimuthal frustration of the pair interaction potential (Secs. 5D, 6I).

Figure 18

Mean repulsive force per unit length between guanosine molecules from experiment (symbols) (254, data kindly provided by P. Mariani) and from theoretical fits (lines) using ground-state calculations, Eqs. (107, 108, 109, 110) (330). Data are shown for solutions with 0.1 (open circles, dashed lines) and $0.5M$ (open triangles, solid lines) NaCl salt concentrations. Theoretical fits were made assuming an effective radius of each guanosine stack of $13.2\mathrm{\AA }$ where $\sim 84\%$ of the guanosine charge was compensated by condensed counterions.

Figure 19

Measured (open circles) and predicted (thin solid line) repulsive forces between collagen triple helices at $p\mathrm{H}$ 6.0 and low ionic strength (reproduced from 184). The prediction was based on the mean-field theory of hydration forces (184). The corresponding expression for the repulsive hydration force can be obtained from Eqs. (105, 108, 110) using the correlation length in water $(\approx 4\mathrm{\AA })$ instead of ${\kappa }^{-1}$ and replacing the factor $4l_B{\zeta }_{N_s}^2∕l_c^2$ with a phenomenological constant (determined by fitting the data). However, electrostatic image repulsion of partial charges on the collagen backbone from dielectric cores of surrounding collagen molecules fits the data as well (assuming $\kappa =0$ and $5⩽\epsilon ⩽80$ so that $7⩽l_B⩽100\mathrm{\AA }$).

Figure 20

Different spin configurations and lattice types for columnar assemblies of DNA in the ground state (138). The FM (ferromagnetic) state corresponds to when the spins (azimuthal orientations of the molecules) are all aligned. The RHO AFI state corresponds to an antiferromagnetic, rhombic lattice where spins on one sublattice have one value and those on the other sublattice have another. The rhombic packing of this lattice reduces less favorable interactions between like spins. The AFP is a hexagonal antiferromagnetic Potts state that has three sublattices with different spin orientations. In both the AFI and AFP lattices, the relative spin angles tend to increase with increasing molecular density.

Figure 21

Estimated torsional ${\Delta }_{\varphi }$ (solid squares) and rigid-body ${\Delta }_{\Phi }$ (open circles) fluctuations in an assembly of $500\text{−}\mathrm{\AA }$-long DNA fragments at $C_t=3.0\times {10}^{-19}\mathrm{erg}\mathrm{cm}$, a DNA charge compensation $\theta =0.7$, and ${\zeta }_n$ corresponding to 30%:70% minor:major groove distribution [Eq. (40)] (221). The break at $\sim 25\mathrm{\AA }$ corresponds to the transition from the ferromagnetic state to the rhombic antiferromagnetic state (at smaller molecular separations).

Figure 22

Comparison of attractive interaction energies per base pair in a hexagonal aggregate calculated within different models at complete charge compensation ($\theta =1$, no image charge repulsion). (1) (Long dashed) Wigner crystal model [Eqs. (52, 53, 54)] with $2+$ counterions (calculated as described in Sec. 5C2, but with the complete set of harmonics). (2) (Short dashed) Wigner-crystal model with $3+$ counterions. (3) (Solid line) Electrostatic zipper model [calculated from Eqs. (57, 59)] at 30:70 minor:major groove ratio of condensed:bound counterions; ${\zeta }_1^2=0.5$ and ${\zeta }_2^2=3.3$, Eq. (40). Inset: The same results on a logarithmic scale, demonstrating the slower decay of the energy within the electrostatic zipper model.

Figure 23

Aggregation energy (per base pair) for a hexagonal aggregate of long flexible DNA as a function of charge compensation fraction $\theta$, calculated for the electrostatic zipper mechanism from Eqs. (58, 59) (parallel nonideal helical pairs from Sec. 5E) and Eqs. (97, 98, 99) (aggregates of parallel molecules from Sec. 6) at ${\lambda }_c=300\mathrm{\AA }$, $C_t=3\times {10}^{-19}\mathrm{erg}\mathrm{cm}$. The values of ${\zeta }_1$ and ${\zeta }_2$ were selected to match the counterion adsorption model [Eq. (40)] with 50:50 (long dashed), 30:70 (solid curve), and 20:80 (dotted curve) minor:major groove ratios of condensed counterions.