Condensation transition and forced unravelling of DNA-histone H1 toroids: a multi-state free energy landscape

A flow cell was incubated for 1 h with an antidigoxigenin containing buffer of 100 mM NaCl, 100 mM NaP and 20 µg mL⁻¹ antidoxigenin (Sigma Aldrich). The antidigoxigenin non-specifically binds to the coverslip. One milliliter of wash buffer (100 mM NaCl, 100 mM NaP, 0.1% Tween, 1 mg mL⁻¹ non-fat dry milk, ∼0.05 mg mL⁻¹ Sodium Azide) was flowed into the flow-cell and incubated for one hour. This process coats the surface with casein to prevent bead sticking. A bead and DNA solution of streptavidin-coated, 1.26 μ m-diameter microspheres (Spherotech) at 20 fM and 10 pM 4200 basepair DNA with biotin and digoxigenin labels at opposite ends in wash buffer is then incubated for one to 20 h. Before use, the samples were rinsed by flowing 2 mL of wash buffer to remove unattached beads. Video microscopy with particle tracking identified beads singly-tethered to the flow-cell's coverslip by observation of their Brownian motion.

The pet-11d vector containing wild-type mouse H1⁰ was obtained from Professor Jeff Hansen. The H1 containing vector was transformed into E. coli BL21 cells and induced with 0.6 M IPTG (American Bioanalytical) at an OD of 0.6. The induced cells were grown for an additional 4 hours, then harvested and centrifuged. The pellet was resuspended in a lysis buffer (25 mM Tris HCl, 2.5 mM EDTA, 1 M NaCl) and sonicated. Cellular debris was centrifuged while H1 remained soluble in the supernatant. The supernatant was diluted to 300 mM NaCl using a no salt buffer (25 mM Tris HCl, 2.5 mM EDTA). This causes a precipitant to form containing DNA and histone H1. The precipitant was centrifuged and resuspended in 8 M urea. The H1 dissolves but the DNA does not and the DNA is centrifuged away. The supernatant containing H1 was placed on a HiTrap SP HP column. This was washed with 8 M urea plus 300 mM NaCl and eluted with 8 M urea plus 600 mM NaCl. Elution was then dialyzed into PBS (0.01 M phosphate buffer, 0.0027 M KCl and 0.137 M NaCl) pH 7.3.

Fluid flow into the observation flow cell was driven at a constant rate by a computer controlled syringe pump (Harvard Apparatus, Micro-Liter OEM Syringe Pump Modules), inducing a viscous force on the bead proportional to the flow velocity [35] and consequently extending the DNA molecule [36, 37]. The dead-time of our microfluidics flow cell, namely the time between flowing in buffer and the arrival time of buffer at the viewing location, was calibrated using food coloring, the presence of which strongly affects the total light intensity viewed in the microscope. Fluid mechanics dictates a higher flow rate at the center of the channel than at the coverslip surface and hence uncertainty about when liquid arrives at the tether. To minimize this uncertainty, high flow rates were used initially to compress all delays as much as possible. At the flow rates used in these measurements, when a new buffer was introduced, the total intensity of the microscope image transitioned quickly (in less than 1 s), permitting us to approximate the arrival of a H1 as essentially instantaneous. Twenty seconds after the introduction of the protein, the flow speed was reduced to a rate corresponding to a viscous force of 0.3 pN, applied to the bead and thus the DNA. This force was chosen to minimally perturb the DNA while at the same time providing readout of the DNA's length more simply and more rapidly than tethered particle motion without flow. Throughout the period of flow, the bead's position was monitored via video microscopy and tracked in software [38].

All force-extension measurements were carried out in the presence of 1.5 µM histone H1. Prior to the introduction of H1, a singly tethered bead was centered in the optical trap by means of a piezo-stage (NanoMAX 311, Thorlabs) and held in a non-condensed state by means of a force in the axial direction of about ∼1 pN [39, 40]. Using an axial pulling geometry minimizes the possibility of bead-coverslip or DNA-coverslip interactions in condensed states. The calibration was performed with a previously published method [40], where power spectrum measurements are taken over the range of coverslip displacements used during the measurement and the background signal dependence on coverslip displacement is measured and accounted for. Histone H1 at 1.5 µM was introduce as described above. After other beads on the screen were observed to condense, the trapped bead, still held uncondensed by our optical tweezers trapping beam, was re-centered above its tether attachment point. Force-versus-extension measurements were then performed repeatedly by moving the piezoelectric stage back and forth in the axial direction at a constant speed of 500 nm s⁻¹.

Images from an example DNA condensation measurement with matching cartoon interpretations are shown in figure 2. Before the introduction of H1, the bead performs Brownian motion constrained by its tethering DNA, shown in figure 2(a) with the cartoon depicting the many positions available for the bead. Once the flow is introduced, the DNA is extended by the flow force on the bead, as shown in figure 2(b). After some time, in spite of the flow pushing the bead to the right, the bead spontaneously moves to the left, seen in figure 2(c). A red marker at the same position in figures 2(b) and (c) highlights the difference in bead position between these two images. To quantify the condensation behavior, we tracked the bead's position in each frame of a movie of the condensation measurement, such as those shown in figure 2.

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Figure 3 shows several example tracking measurements during DNA condensation obtained for H1 concentrations from 100 nM to 2 µM. The bead coordinate is the tracked position of the bead in the x or left-to-right direction. Focusing on the 200 nM example trace, shown in purple in figure 3, it can be seen that at early times the bead performs Brownian motion over a range of ∼2 µm. At −40 s, a high flow is introduced to quickly deliver the new histone H1-containing buffer. The vertical, blue, dashed line at 0 s represents the time at which the new buffer was calibrated to arrive at the tether. The flow is reduced at 20 s, resulting in a small decrease in bead-coordinate and a noticeable increase in the variance of the bead coordinate. Because the bead is now under less force, it is less constrained and Brownian motions occur over a greater range of extensions. As the bead incubates in H1-containing buffer, there is no noticeable change in the the bead coordinate's mean or variance until the DNA suddenly condenses at ∼700 s for the 400 nm H1 case.

Similarly, for all concentrations greater than 100 nM, after a concentration dependent lag time, there is a sharp decrease in the bead coordinate, as each bead studied returns to approximately its mean position before the introduction of flow. This observation implies that the length of the DNA is drastically reduced in a condensed state, as shown in the cartoon of figure 2(c). The lag time is taken as the time from when the H1 reached the tether (0 s) to the time of the condensation event, with the latter time defined to be the first tracked bead position that lies outside the preceding, measured position distribution by more than seven standard deviations.

These condensation transitions occur either as a single abrupt event, over a time of ∼0.5 s—a 'sharp' transition—or as a series of similarly abrupt, but smaller, steps—a 'stepped' transition. Examples of both types of transition can be seen in figure 4, where the bead coordinates of three condensation events are shown, measured in the same movie at an H1 concentration of 400 nM.

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We now investigate this structure with force-versus-extension measurements using optical tweezers. A representative force-versus-extension trace, obtained for naked DNA, is shown in figure 6(a). This trace is consistent with the worm-like chain (WLC) model [22, 26–28], illustrated as the solid black line, which shows forces less than 0.4 pN to extensions of 1000 nm. The fluctuations in figure 6(a) are the result of Brownian motion within the optical trap. However, in the presence of 1.5 µM histone H1 (figures 6(b)–(e)), the DNA-H1 complex shows a non-zero force of about 1.2 pN, from zero extension up to about 1100 nm, previously referred to as the plateau region [26], even though the force is not strictly constant. Beyond 1100 nm, the force-versus-extension of DNA-H1 is indistinguishable from that of naked DNA at comparable extensions. A key feature of the plateau region is the presence of a number of sudden jumps in force-extension, in addition to Brownian fluctuations. Similar features were reported in earlier studies of the force-versus-extension of DNA in the presence of other multivalent cations [22, 26, 27], but not fully understood.

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In individual traces, jumps occur at slightly different extensions and forces and are of slightly different lengths. However, a regular pattern emerges from a two-dimensional, false-color histogram of the number of times a particular force-extension pair is realized in the ensemble of measurements, as shown in figure 7(a). Figures 7(b) and (c) show the histogram for pulling only and relaxing only, respectively. It is especially clear from figures 7(b) and (c) that the force-extension distribution shows six distinct regions of high probability—six states—both for pulling and for relaxing. Six states can also be seen in the summed distribution of figure 7(a), but, because of overlaps, somewhat less distinctly. To quantify the force-versus-extension for each putative state, we fit a quadratic polynomial to the collected force-versus-extension traces in the neighborhood of the state in question. The best fit curves, including the transitions between states, determined as described below, are shown as the black curves in each of figures 7(a)–(c) and provide a good description of the most probable force-versus-extension.

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To investigate whether individual traces can be described in terms of states, we implemented a simple algorithm that assigns each point on an experimental trace to the state corresponding to the nearest quadratic fit. The results of this procedure are shown as the purple lines in figures 6(b)–(e). Although Brownian fluctuations add noise, individual experimental force-versus-extension traces are generally consistent with the state model. All together, our results support the hypothesis that there are six states sampled sequentially during each pulling and relaxing cycle. Back-stepping, or returning to the previous state for a brief time, is observed, such as in figure 6(c) at 1100 nm extension, further bolstering the claim of reversible states.

We have developed a simple model, that permits us to interpret the observed states and to describe the experimental force-versus-extension curves semi-quantitatively. Our starting point, based on the previous observation of H1 toroids [41], is the expression, adapted from reference [42], for the free energy (F) of a histone DNA-H1 complex, partially wound into a helical toroid, while the remaining, unwound DNA-H1 is stretched to an extension x:

$$\begin{eqnarray} F =&&2R \varepsilon_0 (\alpha-\pi) + \frac{\kappa \alpha}{R} + \int^x_0 f {\rm d}x \nonumber\\ &&-2 fR \left(\cos \beta \sin \alpha -\frac{H \alpha \sin \beta}{2 \pi R} \right) \nonumber\\ &&+4\sqrt{f \kappa} \left(2- \sqrt{2+\frac{2 R \cos \alpha \cos \beta}{P}+\frac{H \sin \beta}{\pi P}}\right), \nonumber\\ \end{eqnarray} \tag{ 5 }$$

where L_C is the DNA-H1 contour length, κ is the bending modulus of DNA-H1, R is the helical toroid's radius, H is its helical pitch, $P = \sqrt{R^2+(H/2 \pi)^2}$, α is one half of the winding angle of the DNA-H1 in the toroid, β is the toroid tilting angle [42], f is the force-versus-extension of unwound DNA-H1 of length L_C−2αR and ε₀ is the DNA-H1 self interaction energy per unit length, which is the only way that H1 enters the model. The periodic terms in equation (5) lead to minima in F spaced in α by integer multiples of π [42]. An analysis of equation (5) reveals that, for a range of realistic parameter values (κ ≃ 200 pN nm⁻¹, f ≃ 1.2 pN, R ≃ 30 nm, R ≫ H/2π), minima in F occur near $\alpha = \pi (n +\frac{1}{3})$ where n is an integer. Accordingly, in order to achieve an analytically tractable result, we approximate equation (5) as

$$\begin{equation} F = F_n= 2\pi R \varepsilon_0 \left(n-\frac{2}{3}\right) + \frac{\pi \kappa (n+\frac{1}{3})}{R} + \int_0^x f {\rm d}x. \end{equation} \tag{ 6 }$$

Equation (6) explains the states observed experimentally: different states correspond to different, discrete values of n. To predict the extensions at which each state is realized and the corresponding force-versus-extension curve, we determined the value of R that minimizes equation (6) for each value of n. Between jumps, the superhelix shrinks in radius, thus releasing DNA, permitting the extension to increase. This minimum free energy (F_n) determines the force-versus-extension for the $(n+\frac{1}{3})$-turn state via f_n = −dF_n/dx. The forces, f_n are shown in figure 7(a) as the white curves, which correspond to n = 6, 5, 4, 3 and 2 from top to bottom. The value of f_n for the state with the lowest free-energy is highlighted in red. Finally, the rightmost red curve shows the WLC result for completely unwound DNA-H1 (n = 0). For these and a range of nearby parameter values, the state corresponding to n = 1 is not the lowest free energy state at any extension and therefore we do not plot its force-versus-extension. This means that when the DNA condenses from the fully extended state in these measurements, the DNA condenses directly to the $(2+\frac{1}{3})$-turn state, not the $(1+\frac{1}{3})$-turn state.

There are important features in common between the experimental (black) and theoretical (white and red) force-versus-extension curves in figure 7: Both show six states and the transitions between them. The theoretical curves (white) also overlap with the high probability regions of the experimental force-extension histograms. On this basis, we identify the sequence of states realized experimentally for increasing extension as n = 6, 5, 4, 3, 2 and 0, as given by theory. For toroids in which the DNA packs hexagonally across the toroid's cross-section, the force required to unravel one turn is predicted to increase about threefold as the number of turns in the toroid increases from two to seven, as a result of the increasing number of strand-strand contacts per turn in this range [22]. However, our measurements show that the unwinding force is approximately constant. We conclude that 4200 base-pair DNA-H1 is not hexagonal in cross-section, but rather forms a superhelix with only one additional contact for each additional turn.

One important difference between experiment and theory, however, is that the slopes of the force-versus-extension curves are predicted to be smaller than observed, except for the $\frac{7}{3}$-turn state. As the number of turns (n) increases and the extension decreases, the model diverges further and further from experiment. At low extension and high n, the model predicts that the DNA is almost entirely in the form of a toroid. Increasing extension results in the toroid shrinking in radius and releasing DNA. The experimental data suggests that the toroid is more resistant to changing radius under force than our model predicts. Possibly, the linear free energy reduction with increasing DNA in the toroid (the first term of equation (6)) is too simplified; one might expect higher order interactions than simply nearest neighbor for bound DNAs. Secondly, recent measurements of DNA bending have found that high bending angles occur at a much higher frequency than the worm-like chain model would predict [43, 44]. Another difference between experiment and theory is that each state is predicted to be stable at a smaller extension than is found experimentally, suggesting that the toroid free energies are actually lower relative to the free energy of the 0-turn state than given by equation (6). Most likely, the energy of bending the DNA into a toroid is overestimated by the worm-like chain model [43, 44], the second term of equation (6). Because the bending free energy of DNA and the interaction free energy of multiple DNAs is unknown, we use the simplest models available and achieve semi-quantitative agreement.

Recently, methods have been developed to determine free energy differences on the basis of force-versus-extension measurements [31–34]. For the distribution of the work (W) required to go from one state to another, Crooks Fluctuation Theorem (CFT) [31] informs us that

$$\begin{equation} \frac{P_{\rm F}\left(W\right)}{P_{\rm R}\left(-W\right)}={\rm exp}\left(\frac{W-\it{\Delta} F}{k_{\rm B}T}\right), \end{equation} \tag{ 7 }$$

where ΔF is the free energy difference between the two states, P_F(W) is the probability distribution of the work in the forward direction and P_R(W) is the probability distribution of the work in the reverse direction. It follows that ΔF is given by the value of W at which P_F(W) is equal to P_R(−W).

To measure the work involved in transitioning from one state m to the next state m − 1 for each trial, we first designated a fiducial extension for each state, corresponding to the extension of maximum state occupancy. Thus, these extensions correspond closely to the peaks in the histogram of figure 7(a). The following analysis determines the free energy difference between neighboring states at the extensions of the fiducial markers. We obtained the work by properly integrating the area under each experimental force trace that starts in state m at its fiducial extension and finishes in state m − 1 at its fiducial extension. (The work that must be used in applying the CFT is correctly determined by integrating under the experimental force-versus-stage-extension curve [34].) The work for the reverse transition was obtained similarly, but starting in state m − 1 and finishing in state m. The integration may be carried out either by integrating the data directly (green traces in figure 6) or by integrating the state fits (purple traces in figure 6) (in hope of reducing the effect of Brownian noise). In fact, both procedures yield similar results for the probability distributions of the work.

Figures 8(a)–(e) shows the work distribution for each of the five transitions. Shown in green is the distribution obtained on pulling [P_F(W)]; in red is that for relaxing [P_R(−W)]. In every case, the two distributions overlap. For convenience, we fitted the distributions to Gaussian functions, shown as the solid black lines in figure 8. We then identified the work at the crossing point as the free energy difference between the two states in question. Thus, application of CFT determines experimentally the free energy differences: F₅ − F₆ = 331 pN nm. F₄ − F₅ = 243 pN nm, F₃ − F₄ = 255 pN nm, F₂ − F₃ = 230 pN nm and F₀ − F₂ = 314 pN nm. Given a value for F₀, the free energy of each state at its fiducial extension follows immediately. These results are summarized in figure 9, assuming F₀ = 224 pN nm at 1249 nm, marked as the open square in the figure.

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The solid and open circles in figure 9 correspond to the free energy determined from the state fits and data traces, respectively. Clearly, both free energy estimates agree.

With the free energy of each state known at its fiducial extension, the free energy at any extension can be found by integrating the state's quadratic fit from the fiducial extension to the extension of interest. The minimum free energy curve so-obtained is shown as the solid line in figure 9. The red, magenta, green, blue, cyan and purple sections correspond to the range of extensions within which the zero, $\frac{7}{3}$, $\frac{10}{3}$, $\frac{13}{3}$, $\frac{16}{3}$ and $\frac{19}{3}$-turn state, respectively, exhibits the lowest free energy. This curve is the helical toroid's complete, experimentally-determined, free energy landscape. The steps in the black curve of figure 7 correspond to the extensions at which the lowest free energy state transitions from one value of n to another. Shown in figure 9 as the dotted lines is the theoretical minimum free energy, according to equation (6). Again, the red, magenta, green, blue, cyan and purple sections of this curve correspond to the range of extensions within which the zero, $\frac{7}{3}$, $\frac{10}{3}$, $\frac{13}{3}$, $\frac{16}{3}$ and $\frac{19}{3}$-turn state, respectively, shows the lowest free energy theoretically. Although we chose the value of the DNA-H1 self interaction energy per unit length (ε₀ = −0.9 pN) to reproduce the experimental plateau force, the corresponding theoretical free energy (using the same value of ε₀) may be seen to be significantly higher than our experimentally determined free energy.