Continuity of states between the cholesteric → line hexatic transition and the condensation transition in DNA solutions

Abstract

A new method of finely temperature-tuning osmotic pressure allows one to identify the cholesteric → line hexatic transition of oriented or unoriented long-fragment DNA bundles in monovalent salt solutions as first order, with a small but finite volume discontinuity. This transition is similar to the osmotic pressure-induced expanded → condensed DNA transition in polyvalent salt solutions at small enough polyvalent salt concentrations. Therefore there exists a continuity of states between the two. This finding, together with the corresponding empirical equation of state, effectively relates the phase diagram of DNA solutions for monovalent salts to that for polyvalent salts and sheds some light on the complicated interactions between DNA molecules at high densities.

Figure 1. First-order peaks in the 1D x-ray diffraction intensity profiles of the DNA samples, when [NaCl] = 0.1 M, after a linear background is subtracted.

The 1D intensity profiles (i.e., I(q) vs. q) are obtained by radial integration of the intensity distributions in the 2D raw x-ray images of the samples. Intensity distributions are fitted (black lines) to either one Lorentzian in the cholesteric (blue) and LH (red) phases or the sum of two Lorentzians in the coexistence region (green). The procedure used for processing x-ray diffraction data and peak fits is described in detail elsewhere. I(q) is the scattering intensity, with q being the scattering wave vector, i.e., q = (4π/λ)sin(θ/2), where θ is the scattering angle and λ is the x-ray wavelength. The interaxial distances between the neighboring DNA chains (d_int) are determined from the peak positions (q_max) as , where d_Bragg = 2π/q_max. (a): First-order diffraction peaks when [PEG] = 20 wt%, 22 wt%, 25 wt%, 30 wt%, and 40 wt% with temperature fixed at T = 40°C (with corresponding pressures Π = 5.3 atm, 6.7 atm, 9.5 atm, 15.6 atm, and 38.3 atm) from left to right, respectively. At low pressures (Π = 5.3 atm and 6.7 atm), the DNA precipitate is in the cholesteric phase, where the full width at half-maximum (FWHM) of the first-order peak is (increases with increasing d_int). Instrumental resolution and experimental error in the determination of FWHM of the first-order diffraction peaks are ≈0.001Å⁻¹ FWHM and vary slightly with q. At high pressures (Π = 15.6 atm and 38.3 atm), DNA bundles are in the LH phase, which is characterized by a narrow first-order peak, i.e., (increases with decreasing d_int). When Π = 9.5 atm, the narrow LH peak is superimposed with the broad cholesteric peak in the diffraction profile. The two distinct types of peaks coexist over a small range of Π, i.e., coexistence region. (b): Phase coexistence observed when [PEG] = 22 wt% and T = 30°C (with corresponding pressure Π = 7.7 atm). (c): Phase coexistence observed when [PEG] = 20 wt% and T = 15°C (with corresponding pressure Π = 7.4 atm).

Figure 2. Examples of the use of temperature variations for fine tuning the osmotic pressure to induce and measure the cholesteric → LH transitions.

(a), (b), and (c): [NaCl] = 0.1 M, 0.2 M, and 0.3 M, respectively. In (b) and (c), the left and right axes show temperature variations and the corresponding osmotic pressures Π, respectively, when [PEG] = 20 wt%. It is clearly observed in the [NaCl] = 0.1 M data shown in (b) and (c) in Fig. 1 and a large number of similar measurements at various [NaCl] (i.e., for 0.05 M ≤ [NaCl] ≤ 0.4 M) that the only impact of T (i.e., for 15°C ≤ T ≤ 45°C) on the DNA-DNA interactions is through its effect on Π. Temperature does not have a detectable effect on the DNA-DNA interactions in the absence of CoHex over the range of osmotic pressures considered in this study. Under certain concentrations of CoHex, the effect of temperature on the DNA-DNA interactions is also negligible and we can use the same methodology for the fine-measuring of the cholesteric → LH transitions (see caption to Fig. 7). DNA samples are equilibrated at each temperature at least one hour before the measurements. Temperature is controlled before and during the measurements using a Peltier device. The biggest interaxial spacings in LH phase () are determined from x-ray diffraction profiles at the lowest pressures in the coexistence region. Cholesteric phase data points given here are from the diffraction profiles characterized by only the broad peak (without the narrow LH peak), i.e., cholesteric phase data are not shown in the coexistence region. In (a), temperature variations (right axes) and the corresponding pressures (left axis) are shown for two different [PEG], i.e., [PEG] = 20 wt% (blue solid circles) and [PEG] = 22 wt% (purple right-facing triangles). The two right axes in (a) showing the temperature variations are for [PEG] = 20 wt% and [PEG] = 22 wt% from left to right, respectively. As explained in the text (see also the caption to Fig. 1), the variations of d_int and FWHM with pressure are independent of temperature for all [NaCl]. Note that (a) is adapted from ref. .

Figure 3. Osmotic pressure data for different [NaCl], shown for .

Cholesteric phase data are shown with filled symbols while unfilled symbols represent LH phase data. At low pressures, DNA bundles are in the cholesteric phase. Cholesteric → LH transitions take place at transition pressures Π_tr ≈ 7.4, 6.3, 6.0, 5.8 atm for [NaCl] = 0.1 (blue circles), 0.2 (red squares), 0.3 (green triangles), 0.4 M (brown inverted triangles), respectively with abrupt changes in d_int (from to ) at the transition. Π_tr, , and do not vary significantly for [NaCl] ≥ 0.4 M. The interaxial separations and are given in the top axes for [NaCl] = 0.1, 0.2, 0.3, 0.4 M from bottom to top, respectively. Horizontal lines show the transitions. The overall error in the determination of the interaxial separations in the LH phase is about 0.1Å. The overall error in the cholesteric phase is bigger (as big as ≈0.2Å) due to the positional disorder and broadening of diffraction peaks. Upon increasing osmotic pressure in the LH phase, d_int decreases monotonically, and osmotic pressure curves for all [NaCl] converge. Here data are shown up to the pressure (Π ≈ 72 atm) where the differences between the measured d_int for the given ionic conditions are ≈0.1Å, i.e., close to the uncertainty in the determination of d_int. Therefore, in the fits of LH phase data to Π₀, data from to d_int ≈ 26Å are used. The dashed lines represent the fits for [NaCl] = 0.1, 0.2, 0.3, 0.4 M from top to bottom, respectively.

Figure 4. Cholesteric phase data fits.

Colors, symbols, and [NaCl] are the same as in Fig. 3. Dashed lines are the bare interaction pressures, Π₀(d_int), calculated using the parameters extracted from LH phase data fittings (see Fig. 3). Solid lines show the fits of the cholesteric phase data to Π_cho(d_int). Each data set for each [NaCl] is fitted individually, and the prefactor c is extracted from the fits as given on top right for each [NaCl]. It is independent of the ionic strength, i.e., c ≈ 3. Horizontal lines show the transitions from cholesteric phase to LH phase.

Figure 5. The application of the standard Maxwell equal-area construction in order to extract the effective attractive component Π_ea in the total osmotic pressure.

Colors, symbols, and [NaCl] are the same as in Fig. 3. Top panel: Solid lines show the net repulsion with the fluctuation part, i.e., fits of the cholesteric phase data to Π_cho(d_int). They converge at high pressures. In the LH phase, there are no fluctuation effects and the net repulsion is equal to Π₀(d_int). Brown dashed line shows Π₀(d_int) for [NaCl] = 0.4 M in LH phase. Bottom panel: Thick colored lines show Π_ea(d_int). Each line ends at due to the transition to the cholesteric phase, where Π_att = 0. From the simultaneous fits of Π_ea vs. d_int data to (thin black dashed lines), we extract the decay length ratios, f = f_h = f_e ≈ 2.4 (see SI for the details).

Figure 6. Calculated free energy per length, (W/L), vs. log[NaCl] in different regions of the phase diagram.

[NaCl] is in mM concentration units. (a): Cholesteric phase. (b): LH phase. (c): Cholesteric → LH transition. (d): From R → ∞ to R = R₀. Black dashed curves are quadratic (a and d) and line (b and c) fits.

Figure 7. Osmotic pressure-induced transitions for different [CoHex] at [NaCl] = 0.3 M.

Horizontal dashed lines show the cholesteric → LH transitions. Horizontal solid lines show the LH → orthorhombic transitions. For [CoHex] = 3 mM (blue left-facing triangles) and [CoHex] = 12 mM (brown right-facing triangles), data are shown up to the pressure where d_int is approximately 0.1Å bigger than the interaxial distance measured when [CoHex] = [CoHex]* = 28 mM (purple inverted triangles); above that they superimpose with [CoHex]* data. Black dashed curve (with jump at the transition) is the fit of [CoHex] = 0 (green trirangles) data to the total osmotic pressure. At less than [CoHex]*, the dependence of osmotic pressure on d_int is slightly sensitive to temperature at low pressures () and [CoHex] = [CoHex]* data is shown only for T = 20°C. No detectable temperature dependence for other [CoHex] (see SI and ref. for details) and the transitions can be measured with high accuracy using temperature variations. Insets: Typical x-ray images of oriented DNA bundles in LH phase with DNA helical axis parallel and perpendicular to the x-ray beam in (a) and (b), respectively. The sixfold symmetry in (a) shows the long-range bond orientational order in the LH phase. This symmetry does not exist in the x-ray images of DNA samples in the cholesteric phase. The twofold symmetry in (b) shows the parallel alignment of DNA chains. In the oriented sample preparations, we align the bundles of DNA chains in the same direction in order to make macroscopically oriented samples so that 2D ordering of DNA chains can be seen in the x-ray images. The angular widths of the arcs are due to the mosaic spread in our samples.