1 Introduction
Sharp bending of double-stranded DNA on the nanoscale occurs in many biological processes, such as gene regulation,[1-2] transcription factor binding,[3-4] and double-stranded DNA (dsDNA) packaging.[5-6] Recent experiments[7-8] found that the cyclization rates of short dsDNA are considerably larger than that predicted by the worm-like chain model, which has attracted successive studies on the mechanism underlying the greater flexibility of DNA at small length scales.[9-12] Although local hinges in the form of bubbles or kinks, suggested by experiments and theoretical studies,[13-15] should be responsible for the increase in dsDNA flexibility, the elasticity of the shorter dsDNA is not well understood due to some technical difficulties in the direct measurement of forces, such as the lack of good tools to measure the internal forces that actually bent the short dsDNA and the non-triviality of the synthesis of dsDNA loops at small-length scales.
To overcome the above-mentioned technical difficulties, a novel force sensor with optical readout was developed by Shroff and co-workers[16] to measure the forces inside the short, strained loops composed of both dsDNA and ssDNA. However, their results are much lower than those of the theoretical prediction from the worm-like chain model of dsDNA. The discrepancy may be due to a bend softening of the tightly bent dsDNA section in their novel sensor or the fork structure at the junctions between ssDNA and dsDNA sections; until now, no consistency regarding the mechanism has been reached. In Shroff and co-workers' further investigation,[17] they supposed that dsDNA did not appear to melt with high probability under sharp bending and excluded transient melting (microsecond to millisecond duration) as a mechanism for the relief of compressive forces in the similar force sensor by fluorescence correlation spectroscopy. However, recent single-molecule FRET experiments were carried out using similar force-sensor constructs, and it was identified that strong bending induces two types of local melting, namely, a kink in the dsDNA section and forks at the junctions between ssDNA and dsDNA, and, further, that the two types of deformed DNA structures dynamically inter-convert on a millisecond timescale.[18-19]
The ssDNA in the force sensor exerts not only compression forces but also shear forces on the dsDNA, and the shear forces must contribute to the melting of base pairs of dsDNA at the junction between ssDNA and dsDNA (shown in Fig. 1(c)) and then contribute to the fork generation. In Shroff and co-workers' theoretical analysis on the force-sensor results,[17] the shear forces are thought to be not large enough to contribute to the fork generation in the force sensor according to their estimates of the expected magnitude of the shear forces. The fork structure has obvious effects on the measurement results of the force sensor, i.e., it will alter the contour length of ssDNA in the force sensor and the force measurement, and, at the same time, it also determines whether the twist energy of dsDNA in the force sensor should be considered in simulation. Our simulation indicated that the fork generation at both junctions in the force sensor has equivalent probability and the generation at one junction will alter the twist of dsDNA clockwise or anti-clockwise, which will be offset by the twist in opposite direction caused by fork generation at the other junction, so that the twist energy of dsDNA in the force sensor should not be considered.
Fig. 1 (Color online) Schematic depiction of force sensor, fork generation, and the transition between fork state and kink state. (a) Force exerted on dsDNA and its components of shear force and compressive force. (b) Fork-structure generation unzipped by shearing force at the junction of dsDNA and ssDNA; $\Delta x$ is length of unzipped dsDNA corresponding to every fork generation. (c) Force sensor in fork state and kink state, the transition between the force sensor in fork state and the one in kink state. |
In the current work, the force-sensor model is designed according to the experimental constructs, in which dsDNA is separately described by a worm-like chain with or without kink structure and ssDNA is represented by a free-joint chain model. Monte Carlo simulations are first performed to identify the existence of a fork structure at the junction in the force sensor under the conditions with or without a kink structure in dsDNA, and then the forces that ssDNA exerts on the dsDNA at different conditions are extracted and compares with experimental results. Finally, a phase diagram of the fork and kink in the space of the nearest-neighbor (NN) stacking interaction energy and excited energy is provided to understand the transition between the fork and kink states.
2 Model and Simulation Method
2.1 Force-Sensor Model
The common designation of the force sensor in recent single-molecule FRET experiments[16-19] is to prepare an ssDNA ring of various lengths and then hybridize the ring partially with a complementary ssDNA, finally producing D-shaped DNA, as shown by Fig. 1(a). In the current simulation, the conformation of dsDNA in the force sensor was undertaken by a succession of segments with length $l_{0}$, and the elastic energies were carried by its vertices. $\hat{t}_{i+1}$ and $\hat{t}_{i}$ are the tangent vectors of the successive segments at the $i$-th vertex, and its elastic energy of dsDNA under the condition without kink structure can be expressed as: $E_{i}=({\beta}/{2})(\hat{t}_{i+1}-\hat{t}_{i})^{2}$, where $\beta$ is a dimensionless quantity and satisfies $l_{0}\beta=\alpha$, where $\alpha$ is the persistence length of dsDNA.[20] A sharp bend at the $i$-th vertex (namely the kink state shown in Fig. 1(c)) will reduce the bending rigidity from $\beta$ to $\beta'$, and the elastic energy of dsDNA at this vertex can be expressed as: $E_{i}=({\beta'}/{2})(\hat{t}_{i+1}-\hat{t}_{i})^{2}$. Considering $\beta>\beta'$ and the excited energy $\mu>0$, the transition from a sufficient small bend to a sharp bend occurs when the angle between $\hat{t}_{i+1}$ and $\hat{t}_{i}$ becomes larger than the critical one $\theta_{c}$, which is determined by the relation: $({\beta}/{2})(\hat{t}_{i+1}-\hat{t}_{i})^{2}- ({\beta'}/{2})(\hat{t}_{i+1}-\hat{t}_{i})^{2}\approx \mu$; namely, $\cos\theta_{c}\approx 1-{\mu}/({\beta-\beta'})$.[21]
The ssDNA in the force sensor was modelled as a modified freely jointed chain with the length per nucleotide $b=0.63$ nm and its Kuhn length $l_{c}=1.0$ nm; these parameters are consistent with the established experimental results. Thus, the force exerted on the dsDNA can be expressed as:
$f=\frac{k_{B}T}{l_{c}}L^{-1}\Bigl(\frac{x}{l_{ss}}\Bigr),$
where $L(s)=\coth(s)-{1}/{s}$ is the Langevin function and the analytical approximation of its inverse with appropriate accuracy can be reached by $L^{-1}(s)\approx s[({3-s^{2}})/({1-s^{2}})]$.[22] By integration, the elastic energy is then obtained as: $E_{ss}=({k_{B}T}/{l_{c}})\{{x^{2}}/{2l_{ss}}-l_{ss}\ln[1-({x}/{l_{ss}})^{2}]\}$, where $x$ is the end-to-end distance of dsDNA in the force sensor.[23-24]
Finally, the total energies of this force sensor with a kink structure in dsDNA can be determined to be:
$\frac{E}{k_{B}T}= \frac{1}{l_{c}}\Bigl\{\frac{x^{2}}{2l_{ss}}-l_{ss}\ln \Bigl[1-\bigg(\frac{x}{l_{ss}}\bigg)^{2}\Bigr]\Bigr\} +\sum^{N-1}_{i=1}\frac{\delta_{n_{i},0}\beta+\delta_{n_{i},1}\beta'}{2} \\ \times(\hat{t}_{i+1}-\hat{t}_{i})^{2} +\delta_{n_{i},1}\mu+\sum^{N}_{1}\frac{1}{2}k'(l_{i}-l_{0})^{2} +\frac{\beta'}{2}(\hat{r}-\hat{t}_{1})^{2} \\ +\frac{\beta'}{2}(\hat{r}- \hat{t}_{N})^{2},$
where $n_{i}$ are two-state variables, indicating whether the $i$-th vertex is either in double-helix form ($n_{i}=0$) or contains a hinge defect ($n_{i}=1$), and $\hat{r}$ is a tangential vector along the end-to-end vector; $({\beta'}/{2})(\hat{r}-\hat{t}_{1})^{2}$ and $({\beta'}/{2})(\hat{r}-\hat{t}_{N})^{2}$ are the elastic energies of the vertices at the junction between ssDNA and dsDNA. $k'$ is the stretching modulus of dsDNA scaled by $k_{B}T$, approximately 292.7 nm$^{-2}$.
2.2 Simulation Method
Monte Carlo simulation was used to generate the new conformation of the force sensor. Due to the limited freedoms of this system, perturbation was used to generate the new trial conformation; namely, all of the vertices in dsDNA fluctuate around their original position along different coordinate directions and the movement of all of the vertices can be controlled by the perturbation strength $\Delta$ and an average random number $R$ ranging from 0 to 1. To guarantee the ergodicity of simulation, the net displacement in every direction relative to their original position fell in the range $(-\Delta,\Delta)$. The Metropolis criterion was applied to decide whether the new trial conformation is accepted.
To assess the feasibility of this method, we first applied this method to identify the Euler critical force of dsDNA and then compared the simulation results with the theoretical Euler critical force of dsDNA. The simulation of the Euler critical force of dsDNA was the same as in our previous work.[24] During the simulation process, the segment lengths of dsDNA, $l_{0}= 0.34$ nm, 0.68 nm, and 1.0 nm, were used sequentially to identify the effects of segment length on the Euler critical force. The Euler critical forces obtained by simulation are consistent with the theoretical Euler critical forces and not obviously subject to the segment length $l_{0}$. To save the simulation time, the segment lengths $l_{0}=1.0$ nm and $l_{0}=0.68$ nm were used in the subsequent simulations.
3 Results
As shown by Fig. 1(a), the force exerted on dsDNA by ssDNA as shown by Eq. (1) has the components of compressive force and shear force, and the shear force can be expressed as: $f\sin\theta=({k_{B}T}/{l_{c}})L^{-1}({x}/{l_{ss}}) \sin\theta$. The fork generation is dependent on whether the work done by the shear force could overcome the NN stacking interaction energy $\Delta\xi$, namely, $\int^{\Delta x}_{0}({k_{B}T}/{l_{c}})L^{-1}({x}/{l_{ss}}) \sin\theta {\rm d} x\geq \chi\Delta\xi$, which, in our simulation, it is roughly expressed as:
$\frac{1}{l_{c}}L^{-1}\Bigl(\frac{x}{l_{ss}}\Bigr)\sin\theta\Delta x\geq \frac{\chi\Delta\xi}{k_{B}T},$
as shown by Fig. 1(b), where $\Delta x$ is the length of unzipped dsDNA and taken to be 0.68 nm or 1.0 nm in the current simulation and $\chi$ is the number of base pairs per segment. The NN stacking interaction $\Delta\xi$ plays a crucial role in dsDNA stabilization and depends on the sequence composition of neighboring base pairs. The data outlined in Refs. [25-26] have provided the enthalpy and entropy changes for each base-pair step during DNA melting, the NN stacking interaction $\Delta\xi$ ranges from $1.1k_{B}T$ to $2.7k_{B}T$ for $A-T$ pairs and from $2.7k_{B}T$ to $3.3k_{B}T$ for $G-C$ pairs, which were used in the current simulation to estimate the stability of DNA at different temperatures and ionic strengths. This force sensor model can identify the generation of fork at junctions, but can be only applied to some given sequences consisted of $A-T$ or $G-C$, for the force sensor model is coarse-grained without considering the detailed information of nucleotide at atomic level.
3.1 Generation of Forks at Junctions in Force Sensor
Based on the criterion in Eq. (3), the forks were first identified at the junction of dsDNA and ssDNA in the force sensor with $\Delta\xi=1.3k_{B}T$, and its generation does not depend on whether there are kink structures in dsDNA. As indicated by Fig. 2, the conditions without kink structure in dsDNA were first considered. The ssDNA loop in the force sensors with respective contour lengths 37 bp, 57 bp, 81 bp, and 101 bp is complemented by ssDNA to form dsDNA with different contour lengths. When the contour length of dsDNA in every sensor exceeds its critical value, forks begin to occur, and its frequency depends on the contour length of dsDNA proportionally. The critical values corresponding to Figs. 2(a)-2(d) are approximately 20 bp, 33 bp, 51 bp, and 63 bp, respectively.
Fig. 2 (Color online) Fork generation at junction of dsDNA and ssDNA under conditions without kink generation in dsDNA. Loop lengths of the sensor in Figs. 2(a)-2(d) are 37 bp, 57 bp, 81 bp, and 101 bp, respectively, in which the loop is complemented by ssDNA with different lengths to form dsDNA; generation frequency of forks pertains to contour lengths of dsDNA in sensor proportionally. A critical contour length of dsDNA exists in every sensor and their values are 20 bp, 33 bp, 51 bp, and 63 bp, respectively. Only when the contour length of dsDNA in sensor becomes larger than the critical length does the fork structure begins to be generated. |
Fig. 3 (Color online) Fork generation at the junction of dsDNA and ssDNA under conditions with kink structure in dsDNA. Loop length of sensor in Figs. 3(a)-3(d) is 37 bp, 57 bp, 81 bp, and 101 bp, respectively, in which the loop is complemented by ssDNA with different lengths to form dsDNA; generation frequency of the fork pertains to contour length of dsDNA in the sensor proportionally. A critical contour length of dsDNA exists in every sensor and their values are approximately 20 bp, 39 bp, 60 bp, and 78 bp, respectively. Fork structure begins to be generated when the contour length of dsDNA in sensor becomes larger than the critical length. |
Similar to the results without a kink in dsDNA, there exists a critical contour length of dsDNA in every sensor, which are 20 bp, 39 bp, 60 bp, and 78 bp, respectively, but the critical contour length is larger than the corresponding one in Fig. 2 and decreases with the increasing excited energy $\mu$. In a recent FRET experiment, Kim and co-authors hybridized different-length ssDNA rings with complementary ssDNA with constant length and found that the fork and kink in the force sensor began to inter-convert on a millisecond timescale when the contour length of ssDNA was reduced gradually to a definite value and less,[18] which provided direct support of our simulation results.
3.2 Forces Measured in Force Sensor
As reviewed in the Introduction, the measuring of the force that develops in dsDNA in the force sensor as it is mechanically deformed is the central problem to be solved in current experimental and theoretical work. An empirical three-parameter formula in the FRET experiment[16] was taken to fit the FRET data and its inverse, $f=f_{\rm char}\ln(({E_{0}-E_{\infty}})/({E_{\rm FRET}-E_{\infty}}))$, was used to obtain the force in the force sensor, where $f_{\rm char}$, $E_{\infty}$, and $E_{0}$ are free parameters and correspond to the characteristic force, the efficiency at infinite force, and the efficiency at zero force, respectively. The best-fitting values for these parameters in the FRET experiment are $f_{\rm char}=4.0$ pN, $E_{\infty}=0.34$, and $E_{0}=0.73$, respectively.
At this point, the maximum and minimum of the FRET data measured in the force sensor with different loop length were extracted, and then the force calculated by $f=f_{\rm char}\ln[({E_{0}-E_{\infty}})/({E_{\rm FRET}-E_{\infty}})]$, and finally compared with the force obtained in our simulations. $E^{\max}_{\rm FRET}$ and $E^{\min}_{\rm FRET}$ extracted from the FRET data for the force sensor with loop length 101 bp are 0.61 and 0.52, respectively, and their corresponding forces calculated based on the best-fitting parameters and $f=f_{\rm char}\ln[({E_{0}-E_{\infty}})/({E_{\rm FRET}-E_{\infty}})]$ range from $f_{\min}=1.47$ pN to $f_{\max}=3.09$ pN. Following the same routines, the force inferred from the FRET data for the force sensor with loop length 81 bp ranges from $f_{\min}=1.94$ pN to $f_{\max}=5.05$ pN. For the force sensor with loop length 57 bp, $f_{\min}$ inferred from the FRET data is 4.1 pN and $f_{\max}$ cannot be inferred from the FRET data because its corresponding $E^{\min}_{\rm FRET}$ was below the calibration range of the force sensor. In addition, as for the force sensor with loop length 37 bp, a force of approximately 6 pN was measured with the loop upon hybridization.[15] In addition to the forces mentioned above, the forces ranging from 15 pN to 20 pN or even larger forces ($> 20$ pN) were detected in their small loops with loop lengths of 37 bp and 57 bp.[17]
The forces inferred from our simulations are demonstrated in Figs. 4(a)-4(d), which were obtained under three different conditions, namely, the condition with kink structure in dsDNA and a fork at the junction between ssDNA and dsDNA (condition I), the condition with kink structure in dsDNA and without a fork at the junction between ssDNA and dsDNA (condition II), and the condition without kink structure in dsDNA but with a fork at the junction between ssDNA and dsDNA (condition III). The loop size of the force sensor used in our simulation is the same as that used in the FRET experiments, namely, 37 bp, 57 bp, 81 bp, and 101 bp, respectively. With the increasing contour length of dsDNA in every sensor, the forces in every sensor measured under different conditions can be grouped into two different states. One state corresponds to the forces measured under conditions I and II, which are close to each other and consistent with the experimental results extracted from FRET data;[17] the other state represents the forces inferred from the simulations under condition III, which are relatively larger than those measured under conditions I and II, especially for the larger contour length of dsDNA in every sensor, and are comparable with the relatively large forces in the range 15 pN-20 pN measured by Shroff and co-workers in single-molecule FRET experiments.[17]
3.3 Effects of NN Stacking Interaction $\Delta\xi$ on Force Detection
The simulation results above are obtained based on $\Delta\xi=1.3k_{B}T$, given the dependence of the criterion in Eq. (3) on the NN stacking interaction $\Delta\xi$. Simulations based on different NN stacking interactions $\Delta\xi$ were carried out to identify their effects on force detection in the sensor. Except for the NN stacking interaction $\Delta\xi$, all the following simulations were performed with the same conditions and parameters as those with the NN stacking interaction $\Delta\xi=1.3k_{B}T$. Figure 5 demonstrates forces inferred from the simulations under three conditions as a function of the contour length of dsDNA in the sensor with the NN stacking interaction $\Delta\xi=2.7k_{B}T$ (simulations with $\Delta\xi=1.7k_{B}T$ were conducted, but the results are not given here). The curves have the same features with the results indicated in Fig. 4 ($\Delta\xi=1.3k_{B}T$), namely, the forces as a function of the contour length of dsDNA in the sensor can be categorized as two states. The forces inferred from simulations under conditions I and II have no obvious difference and their values are consistent with the forces measured in the FRET experiments,[16-17] ranging from 1.94 pN to 6.0 pN.
Fig. 4 Forces extracted from simulations as function of contour length of dsDNA in sensor. Three conditions are considered in the simulations, namely, that without kink structure in dsDNA but with a fork at the junction of dsDNA and ssDNA, that with kink structure in dsDNA but without a fork at the junction of dsDNA and ssDNA, and that with kink structure in dsDNA and a fork at the junction of dsDNA and ssDNA. NN base pair stacking interaction $\Delta\xi$ and excited energy $\mu$ used in simulations are $1.3k_{B}T$ and $8k_{B}T$, respectively. |
Fig. 5 Forces extracted from simulations as function of contour length of dsDNA in sensor. Three conditions are considered in simulations, namely that without kink structure in dsDNA but with a fork at the junction of dsDNA and ssDNA, that with kink structure in dsDNA but without a fork at the junction of dsDNA and ssDNA, and that with kink structure in dsDNA and a fork at the junction of dsDNA and ssDNA. NN stacking interaction $\Delta\xi$ and the excited energy $\mu$ used in simulations are $2.7k_{B}T$ and $8k_{B}T$, respectively. |
Regarding the forces inferred from simulations under condition III, the forces as a function of the contour length of dsDNA in sensors with different $\Delta\xi$ are redrawn in Fig. 6 to highlight their dependence on the NN stacking energy $\Delta\xi$. It can be seen that the curves represent two different stages. When the contour length of dsDNA in the sensor is equal to or less than the critical length, as depicted in Fig. 2, the curves overlap and the forces increase linearly with the extension of the contour length of dsDNA in the sensor, and then transition to the nonlinear stage when the contour length of dsDNA becomes larger than the critical length. The linear dependence of the forces on the contour length of dsDNA originates from the entropy elasticity of ssDNA. Generally, the force required to hold two chain ends separated by a general vector $\vec{R}$ is linear in $\vec{R}$ as $\vec{f}=({3k_{B}T}/{Nb^{2}})\vec{R}$.[23] In the sensor, $|\vec{R}|$ should be the contour length of dsDNA and is expressed as $|\vec{R}|=x$. $N$ is the number of base pairs in ssDNA and is reduced gradually with the contour length of dsDNA being extended, so that the force $\vec{f}=({3k_{B}T}/{Nb^{2}})\vec{R}$ can be re-expressed as $f=({3k_{B}T}/{b^{2}})({x}/({N_{t}-3x}))$ ($N_{t}$ is the loop size), which is consistent with the simulation results when the contour length of dsDNA is less than the critical length shown in Fig. 2. In the nonlinear stage, the fork generation is assumed to account for the nonlinear dependence of the forces on the contour length of dsDNA in the sensor. For the sensor with loop lengths of 37 bp and 57 bp, their maximum values pertain to the NN stacking interaction $\Delta\xi$ proportionally and match the ultra-large forces ($>20$ pN) detected in FRET experiments.[17]
Fig. 6 Forces extracted from simulations under condition III as function of contour length of dsDNA in sensor, namely, that without kink structure in dsDNA but with a fork at the junctions of dsDNA and ssDNA. NN stacking interaction $\Delta\xi$ ranges from $1.3k_{B}T$ to $2.7k_{B}T$ and the excited energy $\mu$ used in simulations is $8k_{B}T$. |
Taken together, we have identified the existence of a fork at the junction of dsDNA and ssDNA in the sensor, and the results shown in Figs. 4 and 5 not only indicate that the fork generation has tiny effects on the forces inferred from the sensors with a kink in the dsDNA, but highlight that two possible states have appeared in the sensors. To further identify the two possible states, the forces achieved by annealing different-length ssDNA to the loops and their dependence on the NN stacking interaction $\Delta\xi$ are listed in Fig. 7, where each point corresponds to a distinct sensor, the empty symbols and solid symbols indicated the forces inferred from different loops under conditions II (kink state) and III (fork state), respectively. In the kink state, the forces are not obviously dependent on the NN stacking energy $\Delta\xi$ and range from 1.9 pN to 6.0 pN, which are totally consistent with the FRET experiment data. In the fork state, the forces corresponding to the sensors with loop sizes 57 bp and 37 bp are strongly dependent on increasing $\Delta\xi$ and their maximums at different $\Delta\xi$ are consistent with the relatively large forces (15 pN-20 pN) and the extra-high values ($>$20 pN) detected in single-molecule FRET experiments.[17]
Fig. 7 All of the possible forces extracted from loops annealed by different-length ssDNA. NN stacking interaction $\Delta\xi$ for every figure are $1.3k_{B}T$, $1.7k_{B}T$, and $2.7k_{B}T$. Solid and empty symbols indicate the fork state and kink state, respectively. Each point corresponds to a distinct construct. |
3.4 State Transition between Fork and Kink
The fork state and kink state were identified in the sensor and their transition in the force sensor was detected further, and the results are shown in Figs. 8(a)-8(d), where the loop lengths are 37 bp, 57 bp, 81 bp, and 101 bp and complemented by 30 bp, 48 bp, 69 bp, and 87 bp ssDNA, respectively, all of which are the same as those used in the FRET experiment.[16-17] With the excited energy $\mu$ increasing, the forks occur more frequently and the number of forks in every sensor increase gradually and approximate to their corresponding one in the loop with the largest contour length of dsDNA in Figs. 2(a)-2(d) without a kink in the dsDNA in the sensor. It can be supposed that the kink structure is difficult to excite in dsDNA for larger excited energy $\mu$, so that the dsDNA is subject to a stronger stretching force exerted by ssDNA and the generation of the fork is enhanced.
Considering the results in Fig. 8 obtained based on the NN stacking energy $\Delta\xi=1.3k_{B}T$, similar results inferred from simulations based on other NN stacking energies $\Delta\xi$ with the sensor complemented by different-length ssDNA were generalized in the phase space of the NN stacking energy $\Delta\xi$ and the excited energy $\mu$. The phase diagrams of the sensor with loop size 57 bp are presented in Fig. 9; from top to bottom, the sensor is sequentially complemented by ssDNA with lengths 39 bp, 45 bp, and 48 bp. Figures 9(a), 9(c), and 9(e) show the phase diagrams of the fork in the phase space of NN stacking energy $\Delta\xi$ and excited energy $\mu$. The phase diagrams have the common features that three different phase spaces can be differentiated in every phase diagram, namely, the phase space with an extra-high number of forks, a small number of forks, and a medium number of forks, respectively. In the complementary phase diagrams of the kink presented in Figs. 9(b), 9(d), and 9(f), the differentiation of the phase space also can be identified according to the number of kinks.
Fig. 8 (Color online) Effects of excited energy on fork structure generation. Loop lengths of sensor in Figs. 8(a)-8(d) are 37 bp, 57 bp, 81 bp, and 101 bp and complemented by 30 bp, 48 bp, 69 bp, and 87 bp ssDNA, respectively. Simulations are carried out under the condition with kink structure in dsDNA, fork at the junction between ssDNA and dsDNA, and NN stacking energy $\Delta\xi=1.3k_{B}T$. The fork in every sensor is generated more frequently with increasing excited energy. |
Fig. 9 (Color online) Phase diagram of fork ((a), (c), and (e)) and kink ((b), (d), and (f)) in the phase space of NN stacking energy $\Delta\xi$ and excited energy $\mu$. Color bar indicates the number of forks or kinks generated in the sensor with loop size 57 bp. From top to bottom, the sensor is complemented by ssDNA with lengths 39 bp, 45 bp, and 48 bp, respectively. |
The general features can be further understood by the phase diagrams in Figs. 9(a) and 9(b). The sensors corresponding to the phase space in Fig. 9(a) with an extra-low number of forks (indicated by black bars) and with a relatively high number of forks should be in the kink state and fork state, respectively, which is evidenced by its complementary phase spaces in Fig. 9(b) with an extra-high number of kinks and an extra-low number of kinks, respectively, where the detection of the transition between the kink state and fork state can not be realized easily. Furthermore, the transition between the kink state and fork state should be easily detected at the phase space with the medium number of forks in Fig. 9(a) and with the medium number of kinks in Fig. 9(b).
The single-molecule FRET experiments[18-19] not only identify the transition between the kink state and fork state, but also highlight the temperature effects on the transition. The FRET results[18] indicate that the probability of fork generation is dominant over that of kink generation in the temperature range 18$^{\circ}$C to 30$^{\circ}$C tested. It is necessary to find a temperature equivalent to the exited energy $\mu$ and to ensure that the effects of excited energy $\mu$ on the state transition are consistent with those of temperature. According to the Arrhenius equation $k\approx {\rm e}^{-{\Delta G}/{k_{B}T}}$,[27] the relation ${\rm e}^{-{\Delta G}/{k_{B}T}}={\rm e}^{-{1k_{B}T}/{k_{B}(T'-T)}}$ was used to find the equivalent temperature with different excited energy $\mu$, where $T$ is room temperature in K. The free-energy barrier $\Delta G$ corresponding to the transition between the fork state and kink state is mainly contributed by the excited energy $\mu$ and the NN stacking energy $\Delta\xi$, namely $\Delta G \approx \mu +n(\chi\Delta\xi)$ ($n$ is the number of forks). Thus, the temperature $T'$ equivalent to the excited energy $\mu$ was obtained as $T'=({1}/[{(\mu + n(\chi\Delta\xi))}/{k_{B}T}]+1)T$, which is inversely dependent on the NN stacking energy $\Delta\xi$, the excited energy $\mu$, and the number of forks. While two forks at the junctions of dsDNA and ssDNA were considered, with the excited energy $\mu$ and NN stacking energy $\Delta\xi$ increasing from $20k_{B}T$ and $1.3k_{B}T$, respectively, its equivalent temperature was reduced gradually from 35.5 $^{\circ}$C, including the temperature range tested in the FRET experiment,[18] and the generation of forks dominated that of kink states, as shown in Fig. 9, which is consistent with the FRET experiment results.[18]
4 Conclusions and Discussions
In this article, based on the criterion that the work done by the shear force should overcome the NN stacking interaction energy $\Delta\xi$, Monte Carlo simulation was used to identify the fork generation, extract the forces in sensors, and demonstrate the transition between the kink state and fork state. Fork structure was generated at the junctions between dsDNA and ssDNA in the sensor, even with kink structure in dsDNA, when the contour length of dsDNA in the sensor becomes larger than its critical length. The fork state and kink state can be differentiated from the forces inferred from simulations under three different conditions, and the forces as a function of contour length of dsDNA in the sensor are grouped into two states, namely, the fork state and kink state. In addition, fork generation has no obvious effects on forces with a kink in dsDNA.
The transition between a kink state and fork state can be identified by the comparison between the diagrams of forks and kinks in the phase space of NN stacking energy $\Delta\xi$ and excited energy $\mu$. In the phase space with an extra-low number of forks in Figs. 9(a), 9(c), and 9(e), the sensor should be in a kink state, which is evidenced by the number of kinks in their complementary phase space in Figs. 9(b), 9(d), and 9(f) with an extra-high number of kinks. On the contrary, the sensor corresponding to the phase space with the relatively high number of forks in Figs. 9(a), (c), and (e) should be in a fork state. The kink state or fork state is usually in the phase space with relatively high NN stacking energy or excited energy, respectively, so that it is difficult to detect the transition between the fork state and kink state at the phase space with the relatively low or high number of forks in Figs. 9(a), 9(c), and 9(e). This can help us understand the FRET experiment results that the forces ranging from 1.94 pN to 6.0 pN and extra large forces larger than 20 pN in the sensor with loop size 57 bp were detected, but no melting transitions on a wide range of timescales were found by the technique combining fluorescence correlation spectroscopy and fluorescence energy transfer.[28] Regarding the phase space with a medium number of forks in Figs. 9(a), 9(c), and 9(e) and with a medium number of kinks in Figs. 9(b), 9(d), and 9(f), the transition between the fork state and kink state can be detected easily, which can be used to understand the FRET experiments in which a kink state and fork state were not only identified, but dynamically transitioned on a millisecond timescale.[18] Finally, the temperature effects on the transition between the fork state and kink state are also evidenced by the phase diagrams by converting the excited energy to its equivalent temperature based on the Arrhenius plot.[27]