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Pause Point Spectra in DNA Constant-Force Unzipping

Under constant applied force, the separation of double-stranded DNA into two single strands is known to proceed through a series of pauses and jumps. Given experimental traces of constant-force unzipping, we present a method whereby the locations of pause points can be extracted in the form of a pause point spectrum. A simple theoretical model of DNA constant-force unzipping is demonstrated to produce good agreement with the experimental pause point spectrum of lambda phage DNA. The locations of peaks in the experimental and theoretical pause point spectra are found to be nearly coincident below 6000 bp. The model only requires the sequence, temperature and a set of empirical base pair binding and stacking energy parameters, and the good agreement with experiment suggests that pause points are primarily determined by the DNA sequence. The model is also used to predict pause point spectra for the BacterioPhage PhiX174 genome. The algorithm for extracting the pause point spectrum might also be useful for studying related systems which exhibit pausing behavior such as molecular motors.


I. INTRODUCTION
The unbinding of double-stranded DNA (dsDNA) into single-stranded DNA (ssDNA) is a ubiquitous event central to many cellular processes. Much research has focused on understanding the thermal unbinding of dsDNA (Wartell and Benight, 1985). These studies have revealed quantitative aspects of the thermal unbinding transition through the extraction of sequence-dependent free energy differences between bound and unbound DNA (Blossey and Carlon, 2003;Rouzina and Bloomfield, 1999a,b;SantaLucia et. al., 1996). In living cells, however, the unbinding of dsDNA is typically achieved using molecular motors which utilize chemical energy and exert forces to pull apart the strands of dsDNA. To have a quantitative understanding of these processes it is important to first study the simpler case of unbinding of dsDNA by a constant external force. This process is typically referred to as 'unzipping' of dsDNA. For early experiments which unzip lambda phage DNA with a constant velocity and a fluctuating force, see .
Recently, single-molecule experiments have allowed study of this process (see Fig. 1 for a schematic illustration of the experiment). Both theory (Bhattacharjee, 2000;Cocco et. al., 2001Cocco et. al., , 2002Kafri et. al., 2002;Nelson, 2000, 2002;Marenduzzo et. al., 2002;Nelson, 2003;Sebastian, 2000) and experiments (Danilowicz et al., 2003b) show that at a given temperature the dsDNA separates into ssDNA when the applied force exceeds a critical value F c . Moreover, for forces near F c , the dynamics of the unzipping process is highly irregular (Danilowicz et al., 2003a), as displayed in the time evolution of the junction between the separated ssDNA and the bound dsDNA. This junction is referred to as the unzipping fork. Rather than a smooth time evolution, the position of the unzipping fork progresses through a series of long pauses separated by rapid bursts of unzipping.
Pauses and jumps in constant-force unzipping can have several origins. For temperatures near the dsDNA melting transition, portions of the dsDNA can unbind and form transient 'bubbles' below the unzipping fork. In addition, because of the helical nature of the dsDNA structure, a natural twist can be accumulated during the force-induced unzipping process. If the unzipping fork encounters a thermal bubble in the course of its progress, then we would expect a jump in its position. Furthermore, if the unzipping occurs on time scales much faster than the time scales associated with untwisting, then one would expect pauses when the DNA has to unravel accumulated twist (Thomen et. al., 2002). Moreover, since AT and CG base pairs (bp) have different interaction strengths (and associated base pair stacking energies -see Table I), pauses and jumps could also be due to effects associated with the particular sequence of the DNA.
Experiments on multiple identical copies of the same DNA have shown that locations of pauses are highly conserved from one strand to another, (Danilowicz et al., 2003a). Hence, it seems likely that in these experiments at least, transient bubbles and accumulated twist play only a minor role in determining the jumps and pauses. In this work we study the location of the pause points both experimentally and theoretically. To facilitate this study, we introduce a pause point spectrum which is a function of the number of base pairs unzipped. The locations of peaks in the pause point spectrum signify the location of pause points in unzipping, and peak areas can be used as a measure of the strength of pause points. We predict pause point locations by adapting a model of the dynamics of the unzipping fork in a constant-force unzipping experiment on heterogeneous DNA (Lubensky and Nelson, 2002). The only input information into the analysis is the DNA sequence, free-energy differences between dsDNA and ssDNA obtained using F   000  000  111  111  000  000  111  111  000 111   00 00 11 11  00 00 11 11   000 111   0 0   0  1 1   1   00 00 11 11  00 00 11 11  00 11   000  000  111  111  000  000  111  111   00 11   00 00   00  11 11 11 m ( t ) FIG. 1 Schematic diagram of the DNA constant-force unzipping experiment. One strand of the dsDNA is attached to a fixed support, typically via a linker DNA strand (not shown). The other strand is pulled with a constant force, F , via a magnetic bead (not to scale) attached to the strand. If the force is large enough, the dsDNA separates into two ssDNA strands. The position of this separation, measured in base pairs opened m(t), locates the unzipping fork. See Figure 2 for a more detailed description of the experimental setup used in this paper. melting experiments, and temperature. Both thermal bubbles and build up of twist are ignored within our treatment. We find that we can predict most experimentally observed pause points, thus confirming that pause point locations are primarily a function of sequence. Our algorithm might also prove useful for analyzing pause points arising in other single molecule experiments, such as RNA polymerase and exonuclease (Davenport et. al., 2000;Neuman et. al., 2003;Perkins et. al., 2003;Wang et. al., 1998).
The paper is organized as follows: In section II, we describe constant-force unzipping experiments performed on lambda phage DNA. In section III, we present an algorithm for constructing a pause point spectrum from experimental traces of unzipping fork position versus time. Section IV.A describes a theoretical model of DNA constant-force unzipping which defines a free energy landscape as a function of the number of bases unzipped, m, used to describe the unzipping process. In section IV.B, this free energy landscape is used as a surface on which to perform Monte Carlo simulations mimicking the unzipping experiments. In the same way as performed for experimental unzipping trajectories, these trajectories are combined to form theoretical pause point spectra, which are compared with experiment in section V.

II. EXPERIMENTAL METHOD
The experimental procedure has been discussed previously in (Assi et. al., 2002;Danilowicz et al., 2003a). As shown in Figure 2, our setup consisted of two pieces of lambda phage DNA, covalently bound to each other. One strand of DNA was used as a spacer between the glass capillary and the other strand, which was to be unzipped. The spacer strand of DNA was attached to the capillary with a digoxigenin/anti-digoxigenin antibody bond. The capillary was coated with digoxigenin antibody while the spacer strand of DNA was hybridized with a digoxigenin labeled oligonucleotide. One end of the strand of DNA to be unzipped was hybridized and ligated with a hairpin to prevent the complete separation of the unzipped DNA molecule. The other end of the strand was hybridized and ligated with a biotinylated oligonucleotide which specifically bound to a streptavidin-coated super-paramagnetic bead. When a magnetic field was applied, the force induced on the bead slowly unzipped the DNA. The unzipping direction (order of nucleotides unzipped) was controlled by the selection of oligonucleotides. Figure 2b shows the round, antibody coated capillary inside an uncoated square microcell. The round capillary was 0.5 mm in diameter, while the square microcell was 0.8 mm across, leaving a space for a solution of DNA, beads and buffer inside the microcell but outside the sealed, empty, round capillary. The capillary was incubated in a solution containing digoxigenin antibody at 5 o C for at least two days. The DNA solution and bead suspension were also individually kept at 5 o C prior to the experiment. We inserted a digoxigenin antibody coated capillary and the DNA and bead suspension into the microcell and then incubated it at 37 o C for 45 minutes, allowing the DNA to bind to the capillary via the antigen-antibody bond. Finally, we rotated the microcell so the beads that had settled on top of the round capillary were hanging off of its side. We focused using a microscope objective on the beads that were attached to the outermost point on the capillary so that we could accurately measure the distance between the beads FIG. 2 Molecular construction and square cell. (A) Schematic of the DNA binding to the inner glass capillary and the magnetic bead such that pulling the bead away from the surface will cause the dsDNA shown on the right side of the diagram to be separated into two single DNA strands. Note that the figure is not to scale, considering that lambda DNA contains 48,502 bp. (B) Schematic of the side view of the square capillary containing the round glass capillary to which the DNA molecules are bound. The magnetic tweezer apparatus exerts the controlled force on the magnetic beads, a microscope is used for observation, and two thermoelectric coolers are used to control the temperature of the sample during the initial incubation. The magnetic beads are pulled to the right in a direction parallel to the bottom and top surfaces of the square capillary, and perpendicular to the surface of the round capillary at a height equal to the radius of the round capillary, where we focus the microscope. This design allows us to view DNA molecules that are offset from the surfaces of the square capillary, and to infer the number of separated base pairs (bp) by measuring the separation between the magnetic bead and the surface of the round capillary.
in our field of view and the capillary.
We applied a magnetic force by bringing a stack of small magnets mounted on an xyz translation stage near the microcell. The magnets could be approximated as a solenoid with its long axis in the z-direction, so the field gradient acted in the z-direction only and was essentially uniform (Assi et. al., 2002) over our field of view, which was much smaller than the solenoid radius.
We measured the distance the DNA molecules had unzipped by tracking their attached beads. We took still digital photographs of the field of view through a 10x objective lens once every 10 seconds. An image processing program found the coordinates of each bead in each frame. Figure 3 shows part of our field of view at two different times. In Figure 3(a), we had just applied the magnetic field. Figure 3(b) shows the same beads 28 frames, or just over 3 minutes later.
In each experiment, we saw approximately 50 beads in our field of view. Approximately 10 beads unzipped slowly over the course of the experiment, pausing at various points. An individual bead paused at fixed extension until, through thermal fluctuations, the unzipping proceeded. After overcoming an energy barrier which we attribute to sequence heterogeneity, the strand unzipped up to the next pause point, where the same process repeated. Pause points seemed very reproducible in experiment from bead to bead (each attached to a genetically identical DNA), even when force and temperature varied considerably. This statement will be quantified below.
In order to compare simulation results to experimental data, we converted microns unzipped to the numbers of base pairs unzipped. The centers of beads attached to fully zipped strands of lambda phage DNA under a force near 15 pN were observed 16.5µm from the round capillary in experiments. The centers of beads attached to fully unzipped strands of lambda phage DNA under a force of 15 pN were observed 77.4µm from the round capillary in experiments. Thus DNA strands being unzipped were stretched to a length of 60.9µm. Lambda phage DNA is 48, 502 base pairs long, so to convert from µm to base pairs, we use the conversion factor 48, 502bp/60.9µm ≈ 800bp/µm. Since two strands of ssDNA are produced during unzipping, the monomer spacing along a ssDNA strand is found from the inverse of this factor divided by two to be a ≈ 0.6nm. Such a linear interpolation seems reasonable given the fairly large forces (∼ 15 − 20pN ) acting on the unzipped 'handles'. Figure 5(b-c) displays several experimental unzipping trajectories. As can be seen in the trajectories, the unzipping fork progresses through long pauses at specific locations, separated by rapid transitions between these pauses. Moreover, a sample of trajectories from identical DNA sequences display a uniformity in the locations at which the DNA unzipping pauses. Note also that pauses at certain locations seem consistently longer than others. From these considerations, we are motivated to develop a method for combining many trajectories to form a distribution reflecting the location and relative strengths of pause points.

III. PAUSE POINT ALGORITHM
A pause point 'spectrum' can be computed as follows (see Figure 4 for an example): 1. Create a histogram (area normalized to 1) based on the position of the unzipping fork during the time duration of the experiment for all trajectories using the highest resolution possible.
2. In order to smooth this histogram according to the real experimental resolution, define a window centered around each position of the histogram, with a width equal to the experimental resolution.
3. Compute the average histogram peak height within this window, and assign the value of this average to the position of the center of the window in the pause point spectrum.
The location of the peaks in a pause point spectrum correspond to the distances at which the trajectories paused, and thus correspond to pause points. In addition, the peak area is proportional to the amount of time trajectories spent at the peak locations. Hence, relative peak area can be used as a measure of the relative strength of pause points. As can be seen from direct comparison between the spectrum and trajectories ( Figure 5), this method of analysis provides an excellent summary of pauses and jumps observed in experiment. From this intuition, we expect that higher experimental spatial resolution would allow us to resolve further peaks in the spectrum. The experimental pause point spectrum was computed using 15 unzipping trajectories of which 10 are shown in Figure 5(b-c). In addition to applying the above algorithm, the following steps were taken: Trajectories were first The grey spectrum is created by sliding a window of size 400 bp along the x-axis, and assigning a y-value to the midpoint of the window equal to the average of the high resolution histogram within that window. Note that highly localized pause points appear as broad peaks according to a much lower resolution in the window average. individually shifted so that the starting position of each trajectory was zero µm. This was done in order to subtract the linker length from the measured distance for each bead. In order to collect as many trajectories as possible for better statistics, the experimental resolution was 0.5µm. The spectrum was thus created with a window of 0.5µm, corresponding to about 400 bp. In each of the experimental trajectories, there was a period in the beginning that was noisy due to transient bead adjustments in the turning on of the magnetic field ( Figure 5). These regions were not included in the experimental spectra. To convert from µm to bp, we multiplied by (48502 bp/ 60.9 µm), which represents the appropriate bp/µm factor for lambda phage under these experimental conditions (Section II).

A. Defining the Free Energy Landscape
We consider the DNA unzipping experiment ( Figure 1) as a chemical reaction from dsDNA → 2 ssDNA. This system possesses a natural one-dimensional reaction coordinate, namely the number of base pairs (bp) unzipped m, or equivalently the spatial location of the unzipping fork. Theoretical descriptions can then be naturally reduced from a complicated, three-dimensional system to a one-dimensional description with some of the interactions renormalized, expressing the three-dimensional nature of the problem.
A very simple one-dimensional effective model Nelson, 2000, 2002) can then be written down based on the above picture. We define a free energy as a function of the number of bases unzipped, E(m), which represents the difference in free energy between the states with m base pairs unzipped and the fully zipped states (E(m = 0) = 0). There is a contribution to E(m) − E(m − 1) from the free energy difference between the bound and unbound m th base pair, ∆G bp = k B T η(m), as well as a contribution due to adding two additional monomers to the free ssDNA strands under a tension, F , denoted as 2k B T g(F ). We can write these contributions as The DNA sequence information is stored in the function η(m). In principle, this function might depend on time due to transient bubbles or twists in the DNA. At forces low enough (F ≤ 5pN ) such that hairpin formation in the unzipped handles is possible (Dessinges et. al., 2002;Montanari and Mézard, 2001), g(F ) could also be sequence dependent. These compilations are neglected here. If we iterate equation (1) until we reach m = 0, we have Here we are setting η(0) = 0, which ensures E(0) = 0. The long DNA sequences we are considering should be insensitive to such edge effects. Thermodynamically we expect in equilibrium that unzipping of a dsDNA molecule of M bases will occur when E(M ) < E(0), with the transition region between ds and ss DNA occurring when E(M ) = E(0). If we define the shifted function η(n) = η(n) − η, where η is the average of η(n) over the sequence, we can rewrite (2) in the form The parameter f is a reduced-force, which defines the overall tilt of the free energy landscape (FEL), with the particular base sequence overlaid on this tilt with the function m n=0 η(n). With this definition, the critical reduced force is defined by the equation f = 0. Values of f > 0 represent forces too low to unzip, while values f < 0 represent forces where full equilibrium unzipping is thermodynamically favorable. In all of the above, we have assumed that thermal bubbles do not form under the experimental unzipping conditions. This simple model can be defined in a more rigorous fashion by integrating out three-dimensional degrees of freedom in a statistical mechanical microscopic definition of the system. Effects due to bubbles can be incorporated into a coarse grained model, with renormalized parameters (Lubensky and Nelson, 2002).
It is interesting to note that even for this simple model, non-trivial phenomena can occur due to the buildup of free energies naturally present in (3). Indeed, for the case of a completely random base sequence of length M , a sum over the independent random variables in (3) would give an energy barrier ∼ k B T √ M (Lubensky and Nelson, 2002). Since GC base pairs are ∼ k B T stronger than AT pairs at room temperature (Table I), we expect large peaks to appear due to the presence of long GC-rich regions. For a sequence of length M = 48, 000, one expects barriers of the order of 200 k B T .
In practice, FEL's are computed for a particular genome sequence using the experimentally determined free energies of base quartet formation (SantaLucia et. al., 1996). There are 10 distinct base quartets, where m now represents the m th base quartet, while η(m) represents this base quartet's free energy (Table I). These free energy parameters are determined through thermal denaturation studies on short dsDNA fragments, and were found for temperatures of around 310 K. By using base quartet free energies, base stacking interactions are included, which are thought to be more important for overall dsDNA stability than the hydrogen bonds in between base pairs (Blossey and Carlon, 2003;Grosberg and Khokhlov, 1994). To compute FEL's for different temperatures, the free energies for a given quartet were calculated from k B T η(m) = ∆G qt = ∆H qt − T ∆S qt . Here ∆H qt (∆S qt ) is the enthalpy (entropy) difference between the bound and unbound DNA base quartet. Once temperature and f are specified, the FEL is computed with equation (1) or (3).
The case of lambda phage DNA is particularly interesting since it is known that this genome consists of a GC-rich half connected to an AT-rich half 1 . Using the free energy parameters of (SantaLucia et. al., 1996), one finds that   A closer view is given in the inset. Because the lambda phage genome splits into a GC-rich 'front end', followed by an AT-rich region, the effective critical unzipping force is the one shown here, which produces an approximately flat energy landscape for the first ∼ 20, 000 base pairs. Note that there are still energy barriers ∼ 20kB T in this region due to sequence heterogeneity. the large GC-rich region creates a peak of approximately 3000k B T at f = 0 and T = 298K (25 o C), representing an insurmountable barrier to unzipping, and which is much larger than that expected for a random sequence. For the lambda phage genome, we thus define an operational critical reduced force of unzipping as the value of f such that E(0) = E(m GC ), where m GC is the boundary of the GC-rich region. As can be seen in Figure 6, this operational critical reduced force corresponds to approximately f = −0.15. Forces greater than this should allow easier unzipping since the AT-rich portion of the FEL has a negative slope for these forces. However, even at these large forces, there are barriers on the order of 20k B T to unzip (Figure 6 inset.) Using a Freely-Jointed Chain (FJC) model for the single strands, with monomer spacing, a = 0.6nm (corresponding to the bp/µm conversion factor in section II), and Kuhn length, b = 1.9nm (Smith, Cui and Bustamante, 1996), a reduced force value of f = −0.15 corresponds to an experimental force value of F ≈ 16pN.

B. Dynamics
To study the location of the pause points one has to study the dynamics related to moving along the chemical reaction coordinate, m. Macroscopic unzipping occurs only if f < 0, when the equilibrium state of the system is unbound. In this case, the experimental traces of DNA unzipping represent the approach of the system toward its equilibrium single-stranded state.
As outlined in (Lubensky and Nelson, 2002), there are four dynamical time scales associated with DNA constantforce unzipping in the setup shown in Figure 1: τ end and τ bulk represent base pairing and unpairing at the end of the strand and in the bulk, respectively; τ ss (m) represents the relaxation time of the liberated single strands; and τ rot (m) represents the relaxation time of twist built up in the zipped portion of the strand due to the helical nature of the DNA. The latter two time scales vary as a function of m. The dynamics of unzipping are determined by the slowest of these time scales. Here we assume that this time scale, for any value of m, is related to the unbinding of base pairs.
In the analysis below, we assume that the slowest timescale is m-independent. Furthermore, it can be argued that bubble formation is suppressed in DNA for the relevant experimental conditions because of strong base stacking interactions (Blossey and Carlon, 2003).
We will be interested in the unzipping dynamics for f < 0, that is for forces above the critical force of unzipping where it is thermodynamically favorable to unzip. However, even under these conditions, the approach to thermodynamic equilibrium is far from simple. Smooth progress of the unzipping fork is hampered by very large energy barriers in E(m) that can be caused by the buildup of positive η(m). As mentioned above, for random DNA sequences of length M , these barriers can grow as √ M . Forces slightly above the critical force are unable to remove these barriers through tilting the landscape, and we expect to observe difficulty in traversing these barriers. Since the barriers are sequence dependent, it is possible that the dynamics of unzipping display characteristic signatures of the sequence.
To study the behavior of a particular DNA sequence, and to make direct contact with experiments, it is useful to have a dynamical model that closely mimics the experiment. This can be achieved most simply through Monte Carlo (MC) simulations of a random walker on the one-dimensional FEL for the specific DNA sequence under study, at the specified reduced force and temperature conditions. The position of the walker on the FEL represents the position of the unzipping fork in experiments. The walker moves from position m to a nearest neighbor position m ± 1 with the rate Details of the algorithm are outlined in Appendix A. A simulation consists of specifying the FEL (DNA sequence, temperature and reduced force), and propagating the MC algorithm for a specified number of steps. The initial condition is such that the walker starts at m = 0, representing the experimental circumstance of tracking DNA's that begin as fully zipped. What results is trajectory data, m(t), which contains the same information obtained in experiments. Sample theoretical trajectories are shown in Figure 7. There are clear pauses and jumps of the trajectories for reduced forces much higher than the critical force (−0.39 ≤ f ≤ −0.5). Only extremely large reduced forces (f = −5) are sufficient to remove all barriers and allow smooth unzipping.
It must be stressed that there is a certain freedom in choosing particular MC algorithms (Appendix A). Algorithms can differ in how long it takes random walkers to traverse energy barriers, and thus we do not expect to be able to compare timescale information between experiment and theory, quantitatively. Although pause point locations can be predicted, we do not expect pause point strengths to match between experiment and theory.
To calculate the theoretical pause point spectra, simulations were performed at a variety of reduced forces, f , and on FEL's at the same temperature of the experiments. Each simulated trajectory consisted of 10 7 Monte Carlo steps, and 300 trajectories were used to create the theoretical histograms. Corresponding to the 0.5µm experimental resolution, the window averages were taken over 400 base pairs with a length per base pair for lambda phage DNA under these conditions of 60.9µm/48502 bp as discussed in Section II. For f values in which some of the trajectories reached the fully unzipped state (m = 48, 502 bp), the trajectories were cutoff past 48, 000 base pairs before being included in the spectra.

V. DISCUSSION
An examination of a few experimental unzipping trajectories ( Figure 5) reveals that the pausing locations are often encountered by multiple copies of the same DNA. These copies are subject to different realizations of thermal noise, but share the same base sequence, an indication that sequence is a strong factor in governing pause point locations.
Experimental and theoretical pause point locations can be compared by examining the peak positions in the corresponding pause point spectra (Table II). There is strong agreement between experimental and theoretical pause point locations at distances less than 6000 bp. In addition, theory predicts a gap in the pause point spectra of ∼ 9000 bp starting at 5400 bp, which is similar in size and location to that observed in experiment. The fact that the positions of the pause points and gaps match to such a high degree between experiment and theory are evidence that for these experimental conditions, the approximations inherent in the concept of dynamics on the FEL representing DNA constant-force unzipping as a model for the experiments are sound. Since the theoretical model only requires the base sequence and thermodynamic parameters for base quartet formation, the agreement is proof that pause point locations are strongly governed by base sequence. The good comparison also shows that neglect of bubble formation for these temperatures is appropriate, as is also found by other means (Blossey and Carlon, 2003). The pause point spectra contain much more information than the pause point locations. Theoretical and experimental pause point spectra are shown in Figure 8. The values of f used in the simulations lie in the range −0.25 ≤ f ≤ −0.5. Recall that f = −0.15 corresponds roughly to a flat average FEL in the GC-rich region of the lambda phage DNA (Figure 6). A value of F ≈ 17pN corresponds to f = −0.37 under these conditions, which is within this range. We have compared experimental and theoretical pause points in the front half of the unzipping process. The much steeper energy landscape in the back half (see Figure 6) eliminates most pause points. As the values of f are gradually decreased from f = −0.25, the theoretical pause point spectra grow into more peaks at larger distances, although low base pair peaks are still preserved. Thus the locations of pause points are fairly robust with respect to f values. Once the forces are high enough to allow exploration of the whole FEL, the location of the peaks in the spectra do not change, and peak areas are adjusted reflecting a changing of the strength of the pause points.
There are some noticeable disagreements in pause point location between experiment and theory. In particular, the experimental doublet peak at 7100 bp was not observed in any theoretical spectra for a variety of parameters, including longer simulation times. The pair of peaks centered around 14000 bp in the experimental spectra are also not picked up in the theoretical spectra, rather a single peak lying in the middle of the experimental peaks is found. This most likely does not represent an averaging of the two pause point locations in the theoretical spectrum because we would expect a broad peak covering the two locations in this scenario. The doublet of experimental peaks represents data from two separate runs, and thus a slight miscalibration in the 800bp/µm conversion factor (Section II) for those particular runs could cause the two peaks to separate, since a miscalibration has a larger effect for longer distance pause points. For strong enough forces, theoretical pause point spectra also display many more peaks than present in the experimental spectra. It could be that more experimental trajectories need to be included to observe these peaks. While the peak positions in the experimental and theoretical pause point spectra coincide quite well, the peak areas noticeably differ. One source of this discrepancy is due to the time scales of the DNA unzipping. Each step in the Monte Carlo propagation of the unzipping can be thought to occur on the microscopic time scale governing the DNA unzipping, which is estimated to be ∼ 10 −7 s (Danilowicz et al., 2003a), with a large error in the exponent (Mathé et. al., 2004). The accuracy of this figure is not high enough to allow direct comparison with theoretical and experimental time scales. As mentioned above, the particular choice of Monte Carlo algorithm can change the characteristic unzipping times of simulations and could account for the discrepancy between theoretical and experimental time scales. Figure 9 shows pause point spectra obtained for simulations of 10 8 steps. A reduced force of f = −0.39 is not strong enough to allow DNA's to unzip under this length of time. Comparing to the 10 7 step simulations (Figure 8), we see that longer times in this case allow peaks at slightly higher base pair to be observed, but mainly result in a change in peak area. For f = −0.50, which allows for some fraction of unzipping even with 10 7 step runs, longer runs only serve to change peak areas (compare with Figure 8). Longer simulation times do not give spectra that approach the experimental pause point spectrum. We thus expect that even longer times will not provide quantitative agreement of pause point strength with the current MC algorithm. The choice of the Metropolis MC algorithm is an efficient choice to satisfy the detailed balance condition (A1), but it is not the only choice for MC algorithm. Other choices for algorithms can give different pause times at pause points which result in different peak areas in pause point spectra.
The discrepancy of unzipping timescales between experiment and theory is also reflected in the fact that multiple values of f needed to be used to obtain the theoretical pause point information. At low f values, theoretical simulations could not pass certain barriers of the FEL, as is seen in the abrupt cutoff of peaks in Figure 8(a-c), which necessitated further tilting of the landscape. The technique of increasing the force is also used in experiment to probe farther out regions of the unzipping landscape ( Figure 5). However, the range of reduced forces used in the theoretical simulations corresponds roughly to 15-17pN, while the experimental range is roughly 15-30pN (Danilowicz et al., 2003b).
We might expect discrepancies between experimental and theoretical pause point locations to be due to large Arich regions in the genome, since these are more susceptible to bubble formation due to the weaker base pairing and stacking interactions (Table I). Thus further investigation into these pause point discrepancies can lead to interesting genomic information. Experiments involving higher temperatures and different ionic conditions will help elucidate these discrepancies (Danilowicz et al., 2003b).
This procedure to investigate pause points in DNA unzipping is easily extended to the study of other genomes since all that is required is the base sequence and temperature of interest. As an example, we theoretically investigated the pause point unzipping spectrum for the microvirus Bacteriophage Phi-X174 (BP-φX174) (M = 5386) 2 . The FEL for BP-φX174 has barriers that are on the order of √ M , and is a good example of a landscape which can be approximated by an integrated random walk ( Figure 10). Figure 11 plots several sample simulation trajectories alongside a segment of the BP-φX174 FEL for f = −0.25, and figure 12 plots pause point spectra for several values of f , all at T = 298K (25 o C). Once again we see that for low values of f , the spectra grow into peaks at higher base pair as the value of f is increased. A value of f = −0.45 is large enough to cause unzipping, and we can see that the spectrum at this value has contributions from the whole surface. For BP-φX174, f = −0.45 corresponds to F ≈ 17pN at 298K (25 o C).

VI. CONCLUSION
We have presented experimental evidence that the dynamics of DNA constant force unzipping are not smooth, but rather display characteristic pauses and jumps. Furthermore, we have given strong evidence that the locations of these pauses are primarily governed by the DNA sequence at room temperature (25 o C). We have also presented a general scheme for computing pause point spectra for any DNA sequence, with the only inputs being the sequence, temperature, and a set of ten empirical parameters representing DNA duplex stability.
The ideas presented above can be applied to any system in which the concept of a pause point can be well defined, or in which a 'spectrum' representation of trajectory data can be useful in other ways. We can then enumerate the steps involved in constructing a theoretical representation of the system in order to facilitate comparison with experiments: