Atomistic understanding of kinetic pathways for
single base-pair binding and unbinding in DNA
Michael F. Hagan*, Aaron R. Dinner†‡, David Chandler†§, and Arup K. Chakraborty*†§¶储
*Department of Chemical Engineering, †Department of Chemistry, and ¶Biophysics Graduate Group, University of California, Berkeley, CA 94720; and
储Physical Biosciences Division and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
Contributed by David Chandler, October 2, 2003
We combine free-energy calculations and molecular dynamics to
elucidate a mechanism for DNA base-pair binding and unbinding in
atomic detail. Specifically, transition-path sampling is used to
overcome computational limitations associated with conventional
techniques to harvest many trajectories for the flipping of a
terminal cytosine in a 3-bp oligomer in explicit water. Comparison
with free-energy projections obtained with umbrella sampling
reveals four coordinates that separate true dynamic transition
states from stable reactant and product states. Unbinding proceeds
via two qualitatively different pathways: one in which the flipping
base breaks its intramolecular hydrogen bonds before it unstacks
and another in which it ruptures both sets of interactions simultaneously. Both on- and off-pathway intermediates are observed.
The relation of the results to coarse-grained models for DNA-based
biosensors is discussed.
M
icrodevices that identify specific DNA sequences and兾or
mismatches are widely used in genomics applications.
Many of these devices operate by detecting the hybridization of
solubilized oligomers to surface-attached probe sequences (1–
6). Effective design of these device platforms would be aided by
models that describe the dynamics of DNA hybridization. There
are a number of phenomenological and coarse-grained statistical
mechanical models used in this endeavor (7–12). However, such
theories require certain assumptions regarding the microscopic
dynamics associated with hybridization, about which little is
known. In particular, there is insufficient understanding of the
mechanism underlying the fundamental event in this process, the
binding兾unbinding of an individual base pair. In this work we use
atomistic simulations to examine the free energy and dynamics
of this important reaction and thereby elucidate its mechanism.
Nucleic acids have been examined in several atomistic simulation
studies. By driving the reaction along a coordinate based on the
distance between bases, Norberg and Nilsson (13) calculated the
free energy for unstacking three nucleotides in single-stranded
RNA. The free energy for flipping an internal base in a B-form
DNA dodecamer in the presence (14) and absence (15–17) of a
methyltransferase enzyme has been investigated in a similar manner. Most closely related to the present work, Pohorille et al. (18)
determined the free energy for unbinding of an end base pair in
DNA as a function of the distance between hydrogen-bonding
groups and obtained a single 0.45-ns molecular dynamics trajectory.
Based on these data, they concluded that this coordinate alone was
not sufficient to describe the dynamics.
Boczko and Brooks (19) have argued that multiple order
parameters are required to describe complex biomolecular reactions such as those described above. Although this seems often
to be true, it is important to point out that, in studies where
reactive trajectories were compared directly with free-energy
projections (20, 21), the order parameters that described the
dynamics were found to be system-specific. Indeed, free-energy
surfaces that are functions of dynamically irrelevant variables can
be misleading in that rate-determining free-energy barriers can
be obscured. In such a case, probability densities do not evolve
on the calculated free-energy surface in a way consistent with the
corresponding projection of the microscopic dynamics. To avoid
13922–13927 兩 PNAS 兩 November 25, 2003 兩 vol. 100 兩 no. 24
this pitfall, we use transition-path sampling (TPS) (22, 23) to
overcome limitations associated with conventional simulations
to harvest many reactive trajectories. These explicit dynamic
data enable us to demonstrate the kinetic pathways by which a
base pair binds兾unbinds.
We focus on the flipping of a pyrimidine end base of a 3-bp
oligomer in explicit solvent. This represents the first step in the
melting of short DNA chains. We identify four order parameters
that provide a meaningful description of the reaction. Two
monitor the interaction energy and distance between the cytosine and guanine that unbind, and two track the interaction
energy and distances associated with intrastrand stacking (Fig.
1). The distances describe the overall reaction, whereas the
energies give a more detailed view of the initial unbinding
process. Several coordinates based on the numbers and properties of water molecules near the DNA bases were examined as
well and found to be superfluous. The analysis reveals that there
are two qualitatively different reactive pathways: one in which
the flipping base breaks its intramolecular hydrogen bonds
before it unstacks and another in which it ruptures both sets of
interactions simultaneously. In both cases, transition states are
found at the point when all CG interactions are reduced to the
magnitude of the thermal energy. These states are associated
with the rate-determining process, in which the cytosine diffuses
away from the guanine into a state with increased conformational freedom. The implications of these findings for the design
of better coarse-grained models and experiments are discussed.
Methods
Here we summarize the methods; details are given in Supporting
Methods, which is published as supporting information on
the PNAS web site. All calculations were performed with a
version of CHARMM (24) c28b2 that we modified to include TPS.
System. We model the base-pairing and -unpairing at the 5⬘ end of
a CGC DNA sequence with its complementary strand. A 3-bp
oligomer is the smallest duplex system that contains interior bases
and is thermodynamically stable at 300 K. CGC is more stable as
a duplex than a sequence containing A or T. The oligomer was
solvated in a (34 Å)3 box containing 1,345 TIP3 water molecules and
four sodium (Na⫹) ions that were included to make the system
neutral. This corresponds to a Na⫹ concentration of ⬇0.2 M.
Although salt concentration is known to influence the stability of
DNA duplexes (7, 25, 26), we chose not to include additional ions
for simplicity. We did not expect counterions to play a direct role
in the reaction, and, consistent with this idea, the included ions were
generally quite mobile in our simulations. Periodic boundary conditions were applied, and the particle mesh Ewald method (27) was
used to treat the nonbonded interactions.
Abbreviation: TPS, transition-path sampling.
‡Present address: Department of Chemistry, James Franck Institute, Institute for Biophysical
Dynamics, University of Chicago, Chicago, IL 60637.
§To
whom correspondence may be addressed. E-mail: arup@uclink.berkeley.edu or
chandler@cchem.berkeley.edu.
© 2003 by The National Academy of Sciences of the USA
www.pnas.org兾cgi兾doi兾10.1073兾pnas.2036378100
Fig. 1. Reaction being investigated, which is the flipping of C1. The nomenclature used to denote the individual bases is shown in the native state (Left). (Right)
Unfolded state.
Hagan et al.
Free Energy. As described above, order parameters are considered to provide a good description of the reaction if projection
of the atomistic dynamics onto these coordinates is consistent
with the topography of the free-energy landscape projected onto
these variables. In other words, we consider these coarse-grained
order parameters to be the significant slow variables in the
binding–unbinding kinetics. We used umbrella sampling to
obtain the free energy first as a function of E12 and E16 and then
as a function of d12 and d16. Thus, in the first set of simulations,
harmonic terms were used to constrain the system to be close to
particular values of E12 and E16 and, in the second set, constraints
were imposed in terms of d12 and d16.
Results
Overall Mechanism. In this section we project both the dynamics
and free energy of the flipping of C1 (see Fig. 1) onto the energy
(E12 and E16) and the distance (d12 and d16) coordinates.
Stacking of bases C1 and G2 is monitored by E12 or d12, whereas
the pairing of C1 and G6 is monitored by E16 or d16. It is
important to note that E16 registers contributions from stacking
interactions between C1 and G6 in nonnative conformations.
The frequency of observing values of these coordinates during
TPS dynamics is shown in Fig. 2. Fig. 2a, which shows E12 and
E16, reveals that at least two coordinates are necessary to
describe the reaction, because there are two qualitatively
different paths between the native state (E12 ⬇ ⫺6.5 and E16 ⬇
⫺16; d12 ⬇ 4 and d16 ⬇ 6) and the completely unfolded state
(E12 ⬇ 0 and E16 ⬇ 0; d12 ⬇ 13 and d16 ⬇ 16).
Along the first pathway, the hydrogen bonds between C1 and
G6 break first, and then C1 unstacks from G2. The hydrogen
bonds break mainly through twisting motions of the bases.
Presumably, twisting motions are more likely than linear separation, because the angular dependence (felt over ⬇22°) of
hydrogen bond energies is softer than the distance dependence.
Because twisting is the dominant motion along this pathway, the
distance coordinate, d16 (see Fig. 2b), does not provide as good
a description of this process as E16. The process typically begins
with rotation about the glycosidic bond, which weakens all three
hydrogen bonds, as well as a swinging motion in the original
plane of the base that causes one of the outer hydrogen bonds
to break first. This is illustrated in structure A1 (E12 ⬇ ⫺6.2 and
E16 ⬇ ⫺6.6; d12 ⬇ 4.2 and d16 ⬇ 6.6), in which the foremost
hydrogen bond has broken. The rotation proceeds until all the
hydrogen bonds are broken and C1 begins to unbind as shown
in structure A2 (E12 ⬇ ⫺4.5 and E16 ⬇ ⫺0.5; d12 ⬇ 4.2 and d16
⬇ 8.7). From this point, the path proceeds through the transition
state (TS) to the unfolded state (U) as described below. The
twisting leading to A1 often involves significant motion on the
part of G6, which enables the hydrogen bonds to break while
minimizing disruption of the stacking between C1 and G2. This
can also lead to an off-pathway stable state, A3 (E12 ⬇ ⫺6.5 and
E16 ⫽ 0; d12 ⬇ 4.8 and d16 ⬇ 15.3). In this work, we only consider
trajectories that lead to C1 unbinding.
PNAS 兩 November 25, 2003 兩 vol. 100 兩 no. 24 兩 13923
BIOPHYSICS
Obtaining Reactive Trajectories. To start TPS, it is necessary to
define reactant and product basins in terms of a set of order
parameters and generate at least one trajectory. The initial path
need not be physical but must connect the basins. Although all
our studies are conducted at 300 K, to obtain a set of such initial
trajectories we performed 43 molecular dynamics simulations,
each of which started in the native state and proceeded for 500
ps at high temperature (400 K). Unpairing events occurred in 21
of these trajectories (see Table 1, which is published as supporting information on the PNAS web site), of which 15 involved the
bases of interest, C1 and G6 (Fig. 1). In almost all these cases,
unpairing occurred by a stepwise process in which one base
unstacked significantly before the other. In the remainder of the
article, we concentrate on reactions in which C1 flips and define
the unfolded state as that in which C1 does not interact with
either G2 or G6 (see Fig. 1).
Determining a set of order parameters that reflects the
dynamics of the system is a trial-and-error procedure (23). A set
of order parameters that does not characterize the stable states
properly is indicated by dynamical trajectories that rapidly relax
from one basin to the other. As described in Supporting Methods
(see Tables 2 and 3 and Figs. 6–12, which are published as
supporting information on the PNAS web site), we tried many
sets of order parameters, a number of which characterized
properties of the solvent. We eventually found that the following
four coordinates served as suitable order parameters: the distance (d16) and the interaction energy (E16) between the unpairing bases, C1 and G6, and the distance (d12) and the
interaction energy (E12) between the unstacking bases, C1 and
G2. The energy coordinates (E12 and E16) do not include energy
associated with the backbone atoms. As explained in TPS of the
End Base-Pair Unfolding in Supporting Methods, the interaction
energies provide a good description of the reaction near the
native state, but the distances are required to describe regions in
the vicinity of the unfolded state.
The coordinates E12, E16, d12, and d16 were used as the basis for
a TPS procedure. Specifically, we harvested trajectories 400 ps in
length in which one endpoint was in the basin corresponding to the
bound state (E12 ⬍ ⫺4.0 and E16 ⬍ ⫺12.0), and there was a point
anywhere along the path in the unfolded basin (12.5 ⬍ d12 ⬍ 14.5
and 17.0 ⬍ d16 ⬍ 20.0). Throughout the article, values of E12 and
E16 are given in kcal兾mol and d12 and d16 are reported in angstroms.
As discussed in Supporting Methods (see Fig. 13, which is published
as supporting information on the PNAS web site), the length of the
trajectories was chosen to be 400 ps to ensure that they are longer
than the relaxation time across the barrier. Two types of moves were
used to generate trial paths: shifting and shooting (23). Because the
system is highly diffusive, no velocity perturbation was applied in the
shooting moves; rather, the motions of the waters far away (⬎15.0
Å) from the center of the DNA molecule were simulated with a
stochastic (Langevin) algorithm, and the path was updated in only
one direction. Shooting in only one direction at a time significantly
increased the acceptance rate, consistent with ref. 28.
Fig. 2. The frequency of observing values of the order parameters during 1,650 TPS trajectories. The structures shown next to the plots are snapshots taken
from the TPS simulations except A3, which was taken from an umbrella sampling simulation. Note that the gray scale has been cut off at 600 (a) and 980 (b) such
that regions with smaller frequencies can be seen. (a) Frequency of observing E12 and E16. (b) Frequency of observing d12 and d16.
Along the second pathway, C1 unstacks (E12 ⬇ ⫺0.5) concomitantly with a decrease in E16 to ⬇⫺6 (d12 ⬇ 7 and d16 ⬇ 6), denoted
as B1. As in the case of A1, there are two very weak hydrogen bonds
remaining in B1 but also some interactions between the ring atoms
in C1 and G6. At this point, some trajectories enter an off-pathway
kinetic intermediate (B2) in which C1 stacks on top of G6 (E12 ⬇
⫺0.5 and E16 ⬇ ⫺6; d12 ⬇ 7 and d16 ⬇ 4). Stacking interactions are
the main contribution to E16 in B2, because there are no hydrogen
bonds between C1 and G6. This structure sometimes persists for
several hundred picoseconds. A similar structure was described in
ref. 18. Further progress along this path occurs when C1 swings
away and breaks the remaining hydrogen bond (B1) or the stacking
interaction (B2).
The two pathways rejoin at the point (TS) where the magnitudes
of the interaction energies are comparable with the thermal energy
(E12 ⬇ E16 ⬇ ⫺0.1; d12 ⬇ 9 and d16 ⬇ 10). The completely unfolded
state (U) is reached when C1 entirely disengages from G6 by
diffusing to d12⬇13 and d16⬇16. As explained next, it is necessary
to use the distances to describe this process because the energy
coordinates approach zero and do not contain useful information
about configurational degrees of freedom relaxing to the entropically stabilized free-energy minimum.
13924 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.2036378100
Free Energy. Because E16 and E12 appear to describe the dynamics
near the native state well, we have projected the free energy onto
them as shown in Figs. 3 and 4. The two paths described above
correspond to low free-energy troughs that connect the two
major basins. In addition, there are local minima that correspond
to A3 and B2.
In the region of phase space near the unfolded region (E12 ⬇
0 and E16 ⬇ 0), large changes in spatial position correspond to
small changes in the interaction energies, which results in a
projection of the unfolded basin that is extremely narrow with a
high degree of curvature. Thus, in this region, E12 and E16 are no
longer slow variables compared with molecular time scales.
Rather, the slow variables in this region are d12 and d16. Thus, we
have projected the free energy onto these coordinates (see Fig.
4). On this projection, we can see that, although the unfolded
basin begins at approximately d12 ⬇ 10 and d16 ⬇ 10, C1 has to
diffuse to approximately d12 ⬇ 12.5 and d16 ⬇ 15 before the free
energy falls to more than kBT below the saddle point, indicating
that there is only a weak driving force to enter the unfolded
basin. This is because there are only a few degrees of freedom
contributing to the entropically stabilized basin when only one
base pair unbinds.
Hagan et al.
Integration of the Boltzmann weights over the basins in Fig.
4 yields a free-energy difference of ⌬G ⬇ 1.5 kcal兾mol for the
unfolded state relative to the native state. This free-energy
change is in reasonable agreement with the value one obtains
from empirical nearest-neighbor potentials, ⌬G ⬇ 1.8 (26).
However, it is important to keep in mind that empirical potentials derive from experiments in which the entire molecule melts
and are not precise estimates for the unbinding of an individual
base pair. In addition, integration of the basins corresponding to
A3 and B2 results in free-energy differences relative to the native
state of 3.3 and 3.0 kcal兾mol, respectively. To characterize the
basin corresponding to A3 fully, however, we would have to
include a coordinate that characterizes the stacking of G6.
Transition States. To determine whether the coordinates E12 and
E16 or d12 and d16 are sufficient to characterize the transition
states of the reaction in addition to the stable state, we tested
selected structures from the TPS trajectories for their probabilities of committing to the bound and unfolded basins (Pb and Pu,
respectively). Each selected structure was subjected to 10 trials,
each of which began with velocities assigned randomly from a
Maxwell–Boltzmann distribution and proceeded for 200 ps. A
trial contributed positively to Pu if its final structure was in the
unfolded basin. We counted A3, B2, and the native state (N) as
a single basin because the TPS trajectories indicate that the
rate-limiting step is going from one of these states to the fully
unfolded state. If it were a two-state reaction with strongly
attracting basins and a true reaction coordinate were identified,
we would expect the distribution of Pu for structures close to the
free-energy saddle point to be sharply peaked at 0.5.
The results of the committor calculation are overlayed on the
free-energy projection in Fig. 4. We note that in estimating Pu,
the error in an individual commitor estimation that uses 10 trials
is ⬇0.3 (23, 29). Therefore, although we labeled the structures
for which Pu ⬇ 0.5 as ‘‘transition states’’ in Fig. 4 (the white
points), we stress the statistical uncertainty in this designation.
Hagan et al.
Fig. 4. Projection of free energy onto d12 and d16 (contour lines are separated by 0.5 kcal兾mol). The points overlayed on the plot correspond to the
structures from which the committors were estimated. Color coding: red,
tends not to unfold (0 ⱕ Pu ⬍ 0.35); white, ‘‘transition state’’ (0.35 ⱕ Pu ⱕ 0.65);
black, tends to unfold (0.65 ⬍ Pu ⱕ 1.0). During the estimation, the unfolded
state was defined as d12 ⬎ 10.75 and d16 ⬎ 11. Only structures that reached a
stable state were counted in the estimation of Pu. The other stable states were
defined as the native state (d12 ⬍ 9.0; 5.5 ⬍ d16 ⬍ 8.0), G6 unfolds (d12 ⬍ 7.0;
8.0 ⬍ d16 ⬍ 18), and the stacked state (6.0 ⬍ d12 ⬍ 9.0; d16 ⬍ 5.5). The region
marked with the rectangle has been identified as the ‘‘transition state region’’
(see Fig. 5). The qualitative distribution of committor values and location of
the transition-state region are relatively insensitive to the definitions of the
stable and transition states.
We present these committor estimations not to identify specific
structures as transition states but to locate a transition-state
region in which a large fraction of structures are near Pu ⬇ 0.5.
Although statistical uncertainty in the committor estimation
precludes precise identification of structures that comprise the
transition-state ensemble, it does not prevent us from qualitatively locating a region containing the transition states.
We see in Fig. 4 that the transition states primarily fall at a
saddle point on this surface (d12 ⬇ 9, d16 ⬇ 9), although the
correspondence is not perfect. The frequency of observing
commitor values in this region is shown in Fig. 5; there is a broad
peak centered at ⬇0.5. This location can be understood as
follows. The magnitudes of the interaction energies in this area
are in the range 0.1 ⬍ E1i ⬍ 1.0 kcal兾mol, where i is 2 or 6. The
unfolded basin is entropically stabilized at large d12 and d16
because E12 ⬇ 0 and E16 ⬇ 0, which allows the greatest
conformational freedom for the C1 base. In contrast, the other
basins are energetically stabilized. The transition states thus fall
at the crossover, where the entropy is reduced by the fact that
nonzero E12 and E16 restrict the spatial location and orientations
of the base but are not significantly larger than the thermal
energy (kBT ⫽ 0.6 kcal兾mol).
Given the complexity of the system and the small number of
trial trajectories, which result in large errors in commitor estimations, the description of the transition-state region in Fig. 4 is
surprisingly good. It is still possible that additional coordinates
may be necessary to characterize the reaction in further detail.
However, we tried a number of other order parameters (as
described in Coordinates in Supporting Methods), none of which
PNAS 兩 November 25, 2003 兩 vol. 100 兩 no. 24 兩 13925
BIOPHYSICS
Fig. 3. Projection of free energy onto E12 and E16 (contour lines are separated
by 0.5 kcal兾mol). The letters overlayed on the plots correspond to the location
of the structures depicted in Fig. 2. The arrows were drawn by hand to indicate
the two predominant paths discussed in the text.
Fig. 5. The probability of observing values of Pu in the transition-state region
(boxed region in Fig. 4).
significantly improved the precision with which we are able to
identify the transition-state ensemble.
Discussion
This study integrates free-energy calculations and atomistic
dynamics to elucidate a detailed mechanism for base-pair binding and unbinding in DNA. We found four coordinates that
describe the kinetic pathway well. Two of them monitor the
interaction energies between the flipping base (C1) and its
nearest neighbors in the native structure (G2 and G6). These
parameters indicate that, close to the native state, the dynamics
is dominated by specific intramolecular hydrogen bonds and
stacking interactions. In particular, there are two pathways to
unfolding at 300 K: one in which the hydrogen bonds first break
via twisting of bases C1 and G6, followed by unstacking of C1
from G2, and one in which C1 simultaneously breaks the
stacking interactions and the hydrogen bonds with G6. The other
two coordinates that we identified are pairwise center-of-mass
distances between the flipping base and its neighbors (G2 and
G6). These parameters do not perfectly describe the initial exit
from the native state, which involves twisting out of hydrogen
bonds. However, they effectively describe the reaction outside
this region, which is the location of the transition states between
folded or misfolded states and the unfolded state. These states
occur at the crossover between the energy- and entropydominated regions, where the interaction energy is about equal
to the thermal energy. Given that E12 (E16) and d12 (d16) can be
related in some areas of phase space, it is possible that one could
reduce the number of coordinates to three or two by interpolating between the energies and distances, but we chose not to
do so because the separate coordinates are physically more
transparent.
The goal of this investigation is to understand the microscopic
details of DNA hybridization兾melting to validate or improve
existing coarse-grained models of DNA. For example, Brownian
dynamics simulations were used to study the kinetics of melting
in a model in which each nucleotide was represented by two sites
(11, 30). This model seems to capture many of the details of
melting successfully. However, it is not a priori obvious that the
simplified coordinates remaining in such a model are sufficient
to describe all the important reaction dynamics. In particular,
this model assumes that there are no important reaction coordinates associated with the solvent structure.
With these concerns in mind, we tested many coordinates
related to properties of water for their ability to describe the
1. Demers, L. M., Mirkin, C. A., Mucic, R. C., Reynolds, R. A. I., Letsinger,
R. L., Elghanian, R. & Viswanadham, G. (2000) Anal. Chem. 72, 5535–
5541.
13926 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.2036378100
reaction. These order parameters included the interaction energy between water and the hydrogen-bonding groups in the
unpairing bases, the number of waters between stacking bases,
and the hydration number of the unbinding bases. As described
in Supporting Methods, however, we did not find any combination
of these coordinates that was sufficient to describe the unfolded
state or the location of the transition states. The values of these
coordinates usually changed concomitantly with the breaking of
hydrogen bonds, which is registered in E16. Coordinates that
monitor water properties could be more important for describing
the flipping of an interior base (as in refs. 14–17) because there
is a greater change in hydration in that case.
The location of the transition states and free-energy saddle point
indicate that the outcome of a trajectory is not determined by the
breaking of the DNA–DNA hydrogen bonds or subsequent formation of DNA–water hydrogen bonds. It is furthermore not
decided by the penetration of water between stacking bases, such
as might be the case in a reaction dominated by hydrophobic forces.
Rather, unfolding requires that C1 diffuse away from G6 to a
distance at which the conformational freedom is maximized and the
DNA–DNA bonds cannot reform easily. There is only a weak bias
for the base to diffuse away due to the entropic stabilization of the
unfolded state. The fate of a transition thus is decided by a random
walk across a region that appears as a broad plateau on the
free-energy surface (Fig. 4). The entropic stabilization is limited in
this case because there is only a small increase in the number of
conformations available to the DNA after the unbinding of one
base. This can be contrasted with the cases in which an entire chain
unfolds, such as protein unfolding (31, 32), in which there is a larger
increase in available conformations and thus a much stronger
entropic driving force.
The fact that transitions are determined by structures in which
stacking and hydrogen bonding are mostly dissolved does not mean
that the rate of unfolding is independent of the strength of these
bonds. The frequency of random walks that cross the transitionstate region is directly related to the probability that a trajectory
surmounts the free-energy cliff leading to this region: the breaking
of hydrogen bonds and stacking interactions.
The fact that the dynamics are well described by coordinates
(energies and distances) that can be calculated from positions of
only the DNA atoms might imply that implicit solvent models are
adequate for describing nucleic acids. However, it is important
to consider that both the dynamics and free-energy calculations
included explicit solvent atoms, the effect of which is therefore
included in the coordinates we have used. In addition, our
description of the reaction is not perfect, and unidentified
coordinates related to the solvent could still play a role.
Conclusions
In summary, we have projected the atomistic dynamics and the
free energy of a DNA base-pair binding兾unbinding onto four
order parameters. This has enabled us to identify the kinetic
pathways by which the reaction proceeds, several kinetic intermediates, and the approximate location of the dynamical transition states. The identification of suitable order parameters
allows the determination of rate constants (23, 33, 34). These
rates can serve as the basis for testing and further development
of coarse-grained models.
We thank Arun Majumdar for fruitful discussions. Financial support was
provided by the Defense Advanced Research Projects Agency. A.R.D.
was also supported by a Burroughs Wellcome Fund Hitchings–Elion
Fellowship. D.C. is supported by the National Science Foundation.
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