# The Tension on dsDNA Bound to ssDNA-RecA Filaments May Play an Important
Role in Driving Efficient and Accurate Homology Recognition and Strand
Exchange

Julea Vlassakis  Efraim Feinstein  Darren Yang  Antoine Tilloy  Dominic
Weiller  Julian Kates-Harbeck  Vincent Coljee  Mara Prentiss  Harvard
University, Department of Physics, Cambridge, MA, 02138
<EMAIL_ADDRESS>
###### Abstract

It is well known that during homology recognition and strand exchange the
double stranded DNA (dsDNA) in DNA/RecA filaments is highly extended, but the
functional role of the extension has been unclear. We present an analytical
model that calculates the distribution of tension in the extended dsDNA during
strand exchange. The model suggests that the binding of additional dsDNA base
pairs to the DNA/RecA filament alters the tension in dsDNA that was already
bound to the filament, resulting in a non-linear increase in the mechanical
energy as a function of the number of bound base pairs. This collective
mechanical response may promote homology stringency and underlie unexplained
experimental results.

###### pacs:

87.15.ad, 87.15.La

##  I Introduction

Sexual reproduction and DNA damage repair often include homologous
recombination facilitated by RecA family proteins Roca and Cox (1990);
Kowalczykowski and Eggleston (1994). In homologous recombination, a single
stranded DNA molecule (ssDNA) locates and pairs with a sequence matched
double-stranded DNA molecule (dsDNA). In the first step of the process, the
incoming ssDNA binds to site I in RecA monomers, resulting in a helical ssDNA-
RecA filament with 3 base pairs/monomer and $\sim 6$ monomers/helical turn
Chen et al. (2008). This helical ssDNA-RecA filament then searches dsDNA
molecules for homologous sequences by rapidly binding and unbinding dsDNA to
site II in RecA Chen et al. (2008). Thus, the sequence of the ssDNA in the
searching filament is fixed, and the system then searches through the
available dsDNA to find a sequence match for that ssDNA. The binding of dsDNA
to site II is very unstable, so if the dsDNA is not homologous to the ssDNA
bound to site I, the dsDNA rapidly unbinds from the ssDNA-RecA filament. If
the dsDNA is homologous, strand exchange should occur, probably via base-
flipping that transfers the Watson-Crick pairing of the complementary strand
from the outgoing DNA strand bound to site II to the incoming DNA strand bound
to site I Folta-Stogniew et al. (2004). This strand exchange reduces the
unbinding rate for the dsDNA Xiao et al. (2006). RecA is an ATPase, but in
vitro homology recognition and strand exchange can occur without ATP
hydrolysis Mazin and Kowalczykowski (1996); Menetski et al. (1990); Rosselli
and Stasiak (1990). Thus, each step in the homology search/strand exchange
process is fully reversible.

During the homology search and strand exchange process, dsDNA bound to RecA is
extended significantly beyond the B-form length Stasiak et al. (1981). Recent
theoretical work proposed that the free energy penalty associated with
extension may promote rapid unbinding of non-homologous sequences, but the
free energy penalty was assumed to be a linear function of the number of bound
triplets and the kinetic trapping due to near homologs was not considered
Savir and Tlusty (2010).Earlier work had also suggested that the dsDNA
extension promotes base-flipping Mazin and Kowalczykowski (1999) and reduces
kinetic trapping since the lattice mismatch between extended dsDNA and B-form
dsDNA presents a steric barrier to interactions between unbound dsDNA and
bound ssDNA which implies that the dsDNA must bind to the filament in order to
interact with the ssDNA Klapstein et al. (2004). These studies assumed the
dsDNA in the DNA/RecA filament is uniformly extended; however, the X-ray
crystal structures of the dsDNA in the final post-strand exchange state and
the ssDNA in the homology searching state both consist of base pair triplets
in a nearly B-form conformation separated by large rises as illustrated in
Figure 1a. The rises occur at the interfaces between adjoining RecA monomers
Chen et al. (2008), as illustrated in Fig. 2. The functional role of the non-
uniform extension has been unclear.

In this paper we present a simple model that calculates the extension of each
base pair triplet in a dsDNA. Using this model, we calculate the free energy
changes associated with progression through the homology recognition/strand
exchange process. The results of that calculation suggest a resolution to the
long standing question of why strand exchange is free energetically favorable
even though the Watson-Crick pairing in the initial and final states is the
same and the DNA/protein contacts in the ssDNA-RecA filament and final post-
strand exchange state are nearly the same.Chen et al. (2008) The model also
makes several significant qualitative predictions, the most significant being
the suggestion that the collective behavior of the triplets due to their
attachment to the phosphate backbones leads to a free energy that is a non-
linear function of the number of consecutive bound triplets. As a result of
this non-linearity, total binding energy has a minimum as a function of the
number bound triplets in a given conformation. After that minimum is reached,
adding more triplets that given conformation becomes free energetically
unfavorable.

Such a change in sign in the binding energy as a function of the number of
bound triplets can never occur in a theory where the energy is a purely linear
function of the number of bound triplets since the binding of any base pair
anywhere in the system is equally likely regardless of the state of any of the
other triplets in the system. In a system with a binding energy that is a
linear function of the number of correctly paired bound triplets, if
homologous triplets can initially bind to the system, then additional
homologous triplets will always continue to bind. Thus, binding will readily
progress across a non-homologous triplet. As we will discuss in detail in this
work, in a system with a linear energy and more than $\sim 4$ binding sites,
either homologs will be too unstable or near homologs will be too stable.

In contrast, the non-linearity may provide more rapid and accurate homology
recognition than is available for systems using linear energies because the
non-linearity requires that dsDNA binding to the ssDNA-RecA proceed
iteratively triplet by triplet through a series of checkpoints which inhibit
the progression of strand exchange past a non-homologous triplet. At each
checkpoint, the progression of strand exchange to more stable binding
conformations is only free energetically favorable if a sufficient number of
contiguous homologous base pair triplets are bound to the ssDNA/RecA filament
in the appropriate conformations. The general qualitative features of the
homology recognition based on the non-linear energy follow from basic
properties of the simple model and are insensitive to the parameters chosen.
These features include the following: 1. that homology recognition will
proceed iteratively through consecutive triplets 2. that strand exchange
reversal is much more favorable at the ends of the filament than at the center
3. that there will be two checkpoints that cannot be passed unless the bound
dsDNA contains a sufficient number of contiguous homologous base pairs in the
appropriate conformations. Though the general features of the model are very
robust, the exact number of contiguous homologous bp required to progress past
a particular checkpoint depends strongly on the choice of model parameters.
Analytical modeling and numerical simulations suggest that there are a small
range of parameters that allow the free energies predicted by the model
presented in this paper to provide homology recognition which is both fast and
accurate Kates-Harbeck et al. ; Feinstein and Prentiss .

Simulation results suggest that though the initial binding of $\sim 9$ base
pairs (bp) is free energetically accessible, adding more bound triplets is not
favorable, and adding more than 15 bp is enormously unlikely unless the first
checkpoint is passed. Feinstein and Prentiss  The first major checkpoint
requires that $\sim 9$ of the $\sim 15$ bp that initially bind to the filament
are contiguous and homologous. If the initial $\sim 15$ bp do not contain
$\sim 9$ contiguous homologous base pairs, the dsDNA cannot make a transition
to the more stably bound intermediate state; therefore, the weakly bound dsDNA
will almost immediately unbind from the filament. This checkpoint rapidly
rejects all but 50 of the $\sim$ 10,000,000 possible binding positions in a
bacterial genome. If the initial $\sim 15$ bp do include $\sim 9$ contiguous
homologous base pairs, the system can make a transition to a metastable
intermediate state, which allows more base pairs to be added to the filament.
The second major checkpoint occurs when $\sim 18$ contiguous bp are bound to
the filament in the metastable intermediate state. If all of the bp are
contiguous and homologous, the system can make a transition to the final post-
strand exchange state. Otherwise, the long regions of accidental homology will
slowly reverse strand exchange and unbind. Given the statistics of bacterial
genomes, passing the homology requirement for the second checkpoint would
guarantee that the correct match had been found. These predictions are in good
agreement with known experimental results that measure the stability of strand
exchange products as a function of the number of contiguous bound base pairs
Hsieh et al. (1992).

In this work, we will not attempt to optimize the parameters of the model in
order to provide rapid and accurate homology recognition. Rather, we will
consider why homology recognition systems in which the energies are a linear
function of the number of bound base pairs can either provide rapid unbinding
of non-homologs or stable binding of complete homologs, but not both if the
number of binding sites in the system is $>\sim 4$ . We will then discuss how
qualitative features of the non-linear free energy predicted by the model
allow strand exchange to avoid kinetic trapping in near mismatches while also
permitting homologs to progress completely to strand exchange in systems where
the number of binding sites is $>4$.

###  I.1 General Issues in Self-Assembly Based on the Pairing of Arrays of
Matching Binding Sites

In efficient self-assembly/recognition systems that create correct assembly by
matching linear arrays of binding sites, correctly paired arrays of binding
sites must remain stably bound, whereas incorrectly paired arrays must rapidly
unbind even if the incorrect pairing contains only one single mismatched
binding site. For a system in thermodynamic equilibrium, the populations in
different binding configurations are determined simply by their binding
energies. Thus, in a system where every array consists of $N$ binding sites if
$U(m,N)$ is the binding energy when $m$ of the binding sites are correctly
paired, accurate recognition requires that $\exp[-(U(m,N)-U(N,N))/(kT)]\ll
1\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ m<N$, where $k$ is Boltzmann’s constant and $T$ is
the Kelvin temperature. Furthermore, in a system with a temperature $T$, the
requirement that mismatched pairings rapidly unbind implies that
$U(m,N)>-kT\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ m<N$, whereas if the correctly bound ones are to
remain stably bound $U(N,N)$ must be $\ll-kT$. If the energy is a linear
function of the number bound and a mismatch contributes zero free energy then
$U(N,N)=\leavevmode\nobreak\ -\leavevmode\nobreak\ N\epsilon kT$ and
$U(N-1,N)=\leavevmode\nobreak\ -\leavevmode\nobreak\ (N-1)\epsilon kT$, so the
condition $\exp[-(U(N-1,N)-U(N,N))/(kT)]\ll 1$ requires $\epsilon\gg 1$, which
also implies that the homolog will remain stably bound. For simplicity
consider $\epsilon=3$ , which implies $U(N,N)=-N\epsilon kT=-3NkT$ .
Substitution into the requirement for the unbinding of near homolog requires
that and $U(N-1,N)=-(3N-3)kT>-kT\leavevmode\nobreak\ \implies N<4/3$ Thus, the
requirements for rapid and accurate recognition can only be met if $N=1$. The
requirements become more stringent if the specificity ratio is more strict
than 1/20. As we will discuss below, accurate homology recognition in a
bacterial genome requires accurate recognition over a length of more than 12
bp, which implies $N>4$ since 12 base pairs is 4 triplets.

Some of the problems with realizing accurate recognition in systems at
thermodynamic equilibrium were recognized by John Hopfield in the 1970’s,
inducing him to propose a kinetic proofreading system that requires an
irreversible process. Hopfield (1974) In such systems, the energy of the bound
state can be very deep without making the energy of the searching state deep
because of the irreversible step that transfers the system from the searching
state to the bound state. Even in Hopfield’s system there is a tradeoff
between search speed and accuracy since greater sequence discrimination
requires greater unbinding probabilities for homologs. The increased binding
probability for homologs increases the searching time because the correct
binding site must be revisited many times before the homolog makes the
irreversible transition to the bound state. Earlier work had proposed that
RecA based homology recognition could proceed via kinetic proofreading, Xiao
et al. (2006), but homology recognition in vitro is known to proceed without
an irreversible step. Mazin and Kowalczykowski (1996); Menetski et al. (1990);
Rosselli and Stasiak (1990) In this work, we will consider the non-linearities
in the free energy as a function of the number of contiguous bound base pairs
that arise as a function of the differential extension of dsDNA bound to RecA
and how those non-linearities can promote rapid and accurate homology
recognition without requiring irreversibility by making transitions between
successive bound states contingent on the state and relative position of the
bound base pairs.

###  I.2 Homology Recognition and Strand Exchange are Likely to Proceed in
Units of Base pair Triplets

Previous experimental results suggest that homology recognition occurs via
base flipping of the complementary strand bases between their initial pairing
with the outgoing strand and their final pairing with incoming strand. Xiao et
al. (2006); Bazemore et al. (1997); Peacock-Villada et al. (2012).
Experimental results have already shown that strand exchange does progress in
triplets. Ragunathan et al. (2011) We propose that within each triplet the
nearly B-form structure preserves sufficient stacking to allow homology
discrimination to exploit the energy difference between the Watson-Crick
pairing of homologs and heterologs, while the large rises between triplets
result in mechanical stress that plays several important functional roles in
homology recognition and strand exchange. Furthermore, if the stacking makes
it free energetically favorable for the triplets to flip as a group, the lost
of Watson-Crick pairing due to a single mismatch in a triplet may make strand
exchange unfavorable for the entire triplet, as would be the case if Watson-
Crick pairing alone determined the free energy of the strand exchanged state.
If a single mismatch makes strand exchange of a triplet unfavorable, then
testing in triplets implies that in a random sequence the possibility of an
accidental mismatch is 1/64, whereas if the search were done by comparing
single base pairs the probability of an accidental match would be 1/4.
Homology stringency and searching speed would be greatly reduced if the search
were not done in triplets. Once a sufficient number of base pair triplets have
undergone strand exchange, the complementary strand backbone relocates to the
position shown in the X-ray structure of the final state Xiao et al. (2006);
Peacock-Villada et al. (2012), as illustrated in Figures 1c and 2.

###  I.3 General Model for dsDNA Bound to a RecA Filament

Certain underlying assumptions and qualitative features are important to the
model proposed in this paper: 1. that in the absence of hydrolysis the
extension of the protein filament is unaffected by dsDNA binding Karplus  2\.
that homology recognition and strand exchange occur in quantized units of base
pair triplets Chen et al. (2008); Ragunathan et al. (2011) 3\. that the
incoming and outgoing strands consist of nearly B-form triplets separated by
large rises where the strand is bound to the filament due to strong charged
interactions between the backbone phosphates and positive residues in the
protein Chen et al. (2008); Danilowicz et al. (2011); Peacock-Villada et al.
(2012) 4\. that the complementary strand is bound to the filament dominantly
via Watson-Crick pairing, resulting in large stress on the base pairs unless
the dsDNA is in the final post-strand exchanges state where the L1 and L2
loops provide significant mechanical support. Chen et al. (2008); van der
Heijden et al. (2008)

The model utilizes work presented by deGennes that calculated the force
required to shear dsDNA Gennes (2001). We extend deGennes’ model to the
triplet structures in the initial, intermediate, and final dsDNA
conformations. In the model the actual three dimensional helical structure is
converted into a one dimensional system. In the simple one dimensional model,
$L_{R}$ is rise between the triplets in the incoming and outgoing strands and
a single variable, $\gamma$, characterizes the equilibrium spacing between
phosphates when the complementary strand is in a particular state. Thus, for a
particular state, the difference between the equilibrium spacing is given by
$(1-\gamma)L_{R}$ .

###  I.4 Predicted Extension and Energy

As shown in Fig. 3, the extensions of rises between base pair triplets in a
strand bound directly to the protein are given by $v_{N,i}$ and for a system
with $N$ triplets bound RecA,

$v_{N,i}=iL_{R}.$ (1)

The $u_{N,i}$ specify the extensions of the rises in the complementary strand.
At equilibrium, the net force on each $u_{N,i}$ must be zero; therefore, for
$j=2$ to $N$

$Q((u_{N,j+1}-u{N,j}-\gamma L_{R})-(u_{N,j}-u_{N,j-1}-\gamma
L_{R}))+R(u_{N,j}-v_{N,j})=0,$ (2)

where $R$ and $Q$ are the spring constants for the base pairs and the
backbones, respectively. These values for $R$ and $Q$ may be substantially
different from those for individual dsDNA base pairs when the dsDNA is not
bound to RecA because of the interactions between the charged phosphates and
the protein and because dsDNA is grouped in triplets where the stacking
between the triplets is strongly disrupted; however, it is still likely that
$R\ll Q$ as it is in naked dsDNA since the interactions between the bases on
opposite strand is significantly weaker than the interaction between the
phosophates in the backbone of the same strand.

The boundary condition on the last triplet $u_{N,1}$ requires that

$+Q(u_{N,2}-u_{N,1}-\gamma L_{R})+R(u_{N,1}-v_{N,1})=0$ (3)

The values of $u_{N,i}$ can be found using Equations 1-3. The angle between
base pairs and the DNA helical axis is shown as $\theta_{bp}$ in Fig. 3.

In the continuous limit where the discrete subscript $i$ is replaced by a
continuous variable $x$, the equations have an analytical solution

$v_{N,x}=x+A_{N}(x)\sinh(\chi x/L_{R})$ (4)

where $\chi=Sqrt[R/(2Q)]$ is the deGennes length for RecA bound dsDNA and the
constant $A$ is found by using the boundary conditions for the ends, yielding

$A_{N}(x)\Big{[}R\sinh(\chi x/L_{R})+Q\chi\cosh(\chi
x/L_{R})\Big{]}=Q(\gamma-1)L_{R}.$ (5)

In the limit where $1/(2\chi)\gg 1$

$A_{N}(x)={(\gamma-1)L_{R}\over{\cosh(\chi N/2)}}.$ (6)

These assumptions and features lead to a nuanced picture of the distribution
of tension during strand exchange. The lattice mismatch between the
complementary strand and its pairing partners is largest at the ends of the
filament as shown in Fig. 4; therefore, the base pair tension is largest at
the end of the filament. Furthermore, the lattice mismatch at the ends
increases significantly with the number of bound triplets as shown in Fig. 4
and 5. The angle $\theta_{bp}$ is greatest at the ends of the molecule. When
fewer than $\sim 30$ bp are bound to the filament, the tension on the base
pairs at the ends increases rapidly as more RecA bound triplets are added, as
shown in Fig. 4 and 5; however, $\theta_{bp}$ and the tension on the base
pairs near the center decreases as more triplets are added, as shown in Fig. 4
and 5. In contrast with the tension on the base pairs, which is largest at the
ends of the complementary strand, the tension on the rises in the
complementary strand is largest at the center and smallest at the ends. These
general qualitative features are not sensitive to $R$, $Q$ or $\gamma$ as long
as $R\ll Q$.

Given the values of $u_{N,i}$ the mechanical energy of the system, which is of
the form $1/2k(x-x_{0})^{2}$, may be calculated from:

$E_{mech}(N)=[\sum\limits_{i=1}^{N}\frac{1}{2}R(u_{N,i}-v_{N,i})^{2}+\sum\limits_{i=2}^{N}\frac{1}{2}Q(u_{N,i}-u_{N,i-1}-\gamma
L_{R})^{2}]$ (7)

When $R\ll Q$ this expression simplifies to:

$E_{mech}(2)=2[\frac{1}{2}R\frac{(L_{R}(1-\gamma)}{2}^{2}]$ (8)

$E_{mech}(3)=2[\frac{1}{2}R(L_{R}(1-\gamma)^{2}]$ (9)

These energy terms represent the stress on base pairs at the ends of the
molecule due to the lattice mismatch. When $R\ll Q$, for moderate numbers of
bound base pairs, this energy is stored primarily in the extended base pairs.
This idea is particularly important because it implies that an increase in the
extension of the rises in the complementary strand can reduce the free energy
by reducing the tension on the base pairs even if the increase in extension of
the rises requires some energy. Similarly, in the RecA structure the
transition from the intermediate state to the final state may reduce the
tension on the base pairs because the interactions between the base pairs and
the L1 and L2 loops may increase the equilibrium extension of the rises in the
complementary strand.

Using the continuous limit allows us to generate scaling laws for the
extension as a function of the total number of bound base pairs. In the
continuous limit, the non-linear contribution to the free energy is given by

$E_{non-
linear}(N)=\frac{1}{2}\frac{(1-\gamma)^{2}L_{R}^{2}}{\chi^{2}\cosh^{2}[\chi
N/2]}\sum_{i=-N/2}^{i=N/2}\sinh^{2}[\chi i].$ (10)

Thus, when $\chi\ll 1$ the non linear energy term has the following scaling

$E_{non-linear}[N]\alpha(1-\gamma)^{2}L_{R}^{2}\frac{\sinh(\chi N)-\chi
N}{1+\cosh^{2}[\chi N/2]}.$ (11)

In the limit where the number of base pairs bound is much less than the
deGennes length so $\chi N\ll 1$ the energy the non-linear energy scaling is

$E_{non-linear}[N]\alpha(1-\gamma)^{2}L_{R}^{2}\chi^{2}N^{2}.$ (12)

Thus, when $N$ is small the energy increases as the square of the number of
bound triplets, consistent with the exact results for the discrete case given
in equations 7,8 and 9. In contrast, when $N$ is larger than the deGennes
length, the non-linear energy term approaches zero and the mechanical energy
increases linearly with increasing $N$. In this limit, the base pairs at the
center of the filament are no longer under tension. Thus, adding a triplet to
the end of the filament effectively adds another triplet to the unstressed
center rather than increasing the stress on all of the bound triplets.

The total energy of the system includes the mechanical energy calculated above
and the non-mechanical binding energy per RecA monomer, $E_{bind}$. Assuming
the free energy of an unbound dsDNA is zero and the free energy gain upon
binding a triplet is independent of $N$ and $i$ then the non-mechanical
contribution to the binding energy for $N$ triplets is $NE_{bind}$. When the
first RecA monomer binds, $E_{total}[1]=E_{bind}$, which is a constant
negative value. In contrast, when $N>1$, the stress on the molecule yields a
total energy of

$E_{total}[N]=E_{mech}[N]+NE_{bind}$ (13)

which changes in sign and magnitude depending on $L_{R}$, $R$, $Q$ and
$\gamma$.

###  I.5 Modeling the Homology Search and Strand Exchange Process

Recent experimental results suggest that the homology recognition/strand
exchange process uses four major dsDNA conformations: 1. B-form dsDNA, which
is the structure of the dsDNA when it is not bound to the protein 2. an
initial sequence independent searching state with the dsDNA bound to the RecA
where the complementary strand bases paired with outgoing strand 3. an
intermediate sequence dependent strand exchanged state in which the
complementary strand bases are paired with the incoming strand where the
complementary strand backbone is in a position where its bases can flip
between pairing with the incoming and complementary strands 4. the final state
known from the X-ray structure where the complementary strand bases are paired
with the incoming strand bases and the phosphate backbone is in the position
shown in the x-Ray structure. Peacock-Villada et al. (2012)

Consistent with strand exchange proceeding through base flipping of triplets,
recent experimental results have suggested that outgoing strand bases are
arranged in B-form triplets separated by large rises such that the
complementary strand bases can readily rotate between pairing with the
outgoing strand and pairing with the incoming strand. Danilowicz et al.
(2011). The crystal structure shows that the incoming strand is located near
the center of the helical DNA/protein structure whereas the residues
associated with the binding of the outgoing strand are much farther away from
the center, as illustrated in Fig. 1; however, if strand exchange occurs via
the base flipping of base pair triplets, the spacing within triplets must be
approximately the same for all three strands. Thus, given that the total
extension of the outgoing strand backbone is be much larger than the total
extension of the incoming strand backbone, the rises in the outgoing strand
must be much larger than the rises in the incoming strand, as illustrated in
Fig. 2.

In the one dimensional model considered here where $\gamma$ is the only
parameter characterizing each dsDNA conformation, $\gamma$ is smallest for the
initial searching state, where the complementary strand is paired with the
very highly extended outgoing strand, consistent with experimental results
that the dsDNA in the initial bound state has a large differential extension
between the outgoing and complementary strands that prevents more base pairs
from binding to the filament unless the dsDNA undergoes strand exchange.
Danilowicz et al. (2011) The $\gamma$ for the intermediate state is slightly
larger because the complementary strand is paired with the less extended
incoming strand. This value is consistent with experimental results that
suggest that the strand exchange of homologous triplets is favorable because
it reduces the differential tension on the dsDNA. Peacock-Villada et al.
(2012) The $\gamma$ for the final state is almost 1, because of the support
provided for the rises by the L1 and L2 loops of the protein that occupy the
rises in the complementary strand, consistent with the known experimental
result that the binding of at least $\sim 80$ bp in the final state is free
energetically favorable. van der Heijden et al. (2008) In order to probe the
role of tension during homology recognition strand exchange we apply the model
to the various stages of strand exchange using the following parameters:
$\gamma={0.4,0.75,0.8,0.9875}$ and $E_{bind}$ equal to $0,-0.75kT$, $-0.8kT$
and $-0.125kT$ per homologous triplet in the unbound, initial bound,
intermediate and final states, respectively. These $\gamma$ values are
inspired by X-ray structure DNA RecA filaments, the known properties of B-form
dsDNA, experimental results on the stability of strand exchange products Hsieh
et al. (1992), and results of numerical simulations that optimize homology
recognition. Feinstein and Prentiss

##  II Results and Discussion

###  II.1 Changes in Mechanical Tension Allows Strand Exchange to be Free
Energetically Favorable if and only if 6 Contiguous Homologous bp Undergo
Strand Exchange Insensitive to Model Parameters

Early RecA recognition systems assumed that strand exchange is always free
energetically favorable, just as Watson-Crick pairing of unpaired ssDNA is
always favorable, but this assumption is not correct. In self-assembly based
on ssDNA/ssDNA pairing, the pairing of matched bases is free energetically
favorable and the pairing of mismatched bases is approximately neutral. Thus,
in ssDNA/ssDNA pairing systems, such as DNA origami, assembly of matching
sites is always free energetically favorable because the correct Watson-Crick
pairing reduces the free energy below that for the system where no bases were
bound. In contrast, if only Watson-Crick pairing is considered, self-assembly
based on strand exchange is free energetically neutral for matched bases and
free energetically unfavorable for mismatched bases because the system begins
in a state with all of the bases in the complementary strand correctly paired
with the corresponding bases in the outgoing strand. Thus, previous models of
strand exchange that considered only the Watson-Crick pairing were faced with
a paradox: at thermal equilibrium strand exchange is only free energetically
neutral, so at best only $50\%$ of correct pairings would end up in the strand
exchanged state. The other $50\%$ would end up paired with their initial
partners; however, in vivo strand exchange proceeds to completion. Including
protein contacts did not solve the problem since the protein contacts in the
initial ssDNA-RecA filament are almost identical to the contacts in the final
post-strand exchange state Chen et al. (2008); however, the model presented in
this paper resolves the problem, as discussed below.

In the model considered here, the coupling along the dsDNA backbone makes any
change in the dsDNA conformation (such as adding or base-flipping a triplet)
alter the positions and extensions of all other bound triplets. This effect is
illustrated in Fig. 4, which shows the rises and extensions of the base pairs
calculated from the model when an additional triplet binds in site II of the
protein when all of the bound triplets are bound in site II. The triplet
highlighted with the arrow experiences increasing tension as triplets are
added to the end of the sequence. Furthermore, as shown in Fig. 5, the lattice
mismatch at the ends of the filament increases monotonically as more triplets
are added. In contrast, the lattice mismatch for the first triplet out from
the center of the sequence decreases as base pairs are added (solid black
line) because the added triplets extend the inner rises. As the number of
bound triplets approaches the deGennes length for RecA bound dsDNA, the
lattice mismatch at the ends of the filament approaches a constant and the
central rises in the complementary strand approach $L_{R}$.

The mechanical stress model considered here suggests strand exchange is free
energetically favorable if and only if strand exchange transfers a rise
because the reduction in free energy is the result of the reduction in
mechanical stress due to a reduction in the stress on the base pairs due to
decrease in the lattice mismatch between the complementary strand and its
Watson-Crick pairing partner. Thus, when the dsDNA first binds to the
searching filament and all of the triplets are in the initial state, base
flipping of a single triplet does not lower the energy of the system since a
rise is not transferred even if that triplet is perfectly homologous; however,
once one homologous triplet is strand exchanged, it becomes favorable to
transfer a contiguous homologous triplet. Thus, the transfer of the Watson-
Crick pairing of the complementary strand from the outgoing strand to the
incoming strand is not free energetically favorable except in sequence regions
containing at least six base pairs of contiguous homology. This minimum number
of contiguous homologous bases required to pass the first checkpoint is a
basic qualitative feature of the model that is extremely insensitive to the
model parameters chosen. This first checkpoint can provide very rapid
unbinding of almost all initial dsDNA pairings, which is highly advantageous
for rapid searching. Though the model specifies that 6 contiguous bases is the
minimum number required to pass the checkpoint insensitive to the parameters
used in the model,the actual number of base pairs required to pass the first
checkpoint is sensitive to parameters.

Experimental results suggest that strand exchange is only marginally stable
when $\sim 9$ contiguous homologous bp undergo strand exchange,Hsieh et al.
(1992) suggesting that rapid unbinding will occur for all tested sequences
except for $1/4^{9}\sim 2\times 10^{-6}$ which represents only $\sim$ 50
possible positions for a bacterial genome with a length of 10,000,000 bp. All
other sequences will rapidly unbind from the RecA filament because the binding
energy for sequence independent searching state is very weak when only $\sim
9$ bp are bound and adding more base pairs to the searching state would
increase rather than decrease the binding energy.

###  II.2 Calculations of the Free Energy as Function of the Number of Bound
Triplets in Each dsDNA Conformation

In order to understand the progression through homology recognition and strand
exchange, it is important to consider the free energies for all of the dsDNA
conformations involved, not just the initial bound state and the intermediate
state. Of course the process is kinetic, so the energies of the transition
states play a vital role; however, for simplicity we will only show the
energies of the various bound conformations using $\gamma$ values based on
numerical simulations. Feinstein and Prentiss  Fig. 6 shows $E_{total}$ as a
function of number of base pairs in the initial bound state, the intermediate
state and the final post-strand exchange state when all of the triplets are in
the same conformation. Except for the initial binding of $\sim 9$ to 15 bp,
all of the triplets are rarely in the same conformation. Thus, the free energy
curves for states with all of the triplets same conformation rarely represent
the free energy of the system; however, it is clear that reductions in the
mechanical energy of the bound dsDNA can drive strand exchange for homologs,
as we discuss below.

###  II.3 Non-linearities in the Free Energy Allows Homologs to Progress to
Complete Strand exchange

While it is energetically favorable to add base pairs to the initial bound
state for small numbers of base pairs, the quadratic term of $E_{total}$ from
Equation 7 rapidly increases as a function of increasing number of base pairs,
making the binding of a large number of base pairs in the initial state
unfavorable. This is because of the significant tension due to the lattice
mismatch between the complementary strand and the outgoing strand;
consequently, for the parameters based on simulation, no more than $\sim 15$
bp can bind to the filament in the initial searching state because the energy
required greatly exceeds $kT$. Thus, once $\sim$ 15 bp are bound to the
filament, the system is in a highly free energetically unfavorable state which
will force it to choose between the following: 1. unbinding from the filament
2. strand exchange, which is unfavorable for non-homologs. Again, the
prediction that there will be an checkpoint in the progression of strand
exchange from the initial bound state to the intermediate state is insensitive
to the model parameters. The parameters only determine whether more than 6
contiguous homologous base pairs are required to progress past the checkpoint.

Homologs can rapidly progress to complete strand exchange if the weak initial
binding holds long enough for $\sim 9$ homologous contiguous base pairs to
undergo strand exchange, which stabilizes the binding for those homologous bp
and allows strand exchange to progress. Kates-Harbeck et al. ; Feinstein and
Prentiss  The non-linearity in the free energy makes the strand exchange of
consecutive homologous triplets increasingly favorable as long as the number
of bound base pairs is $<\sim$ 30; consequently, the non-linearity in the free
energy makes strand exchange reversal more improbable as a more contiguous
homologous base pairs are strand exchanged. Furthermore, the non-linearity
makes strand exchange at the center of the filament increasingly unfavorable
as the number of bound triplets increases as shown in Fig. 8, while still
allowing strand exchange reversal to remain possible at the ends of the
filament. Again, these qualitative features are basic properties of the model
that are highly insensitive to the model parameters. These qualitative
features allow true homologs to progress to complete strand exchange even
though non-homologs readily unbind. In contrast, for a system with a linear
free the probability that strand exchange will be reversed for a given triplet
is independent of the number of other triplets bound and of the position of
the particular triplet in the filament. As a result, such systems either
suffer rapid unbinding of homologs or strong kinetic trapping in near
homologs, as discussed above for the general case of a system with a linear
binding energy and greater than 4 binding sites.

###  II.4 dsDNA Tension Drives the Transition from the Intermediate State to
the Final State

The free energy penalty due to binding in the intermediate state is so large
that even a perfect homolog cannot bind more than $\sim 18$ bp in the
intermediate state. The only way to continue to add base pairs to the filament
is to make a transition to the final post-strand exchange state where the L1
and L2 loops provide significant mechanical support for the rises. This
transition reduces the tension sufficiently to allow more triplets to bind in
the initial bounds state. If the transition does not occur, strand exchange
will reverse because the intermediate state energy is unfavorable. Thus, the
collectivity in the behavior of the dsDNA can have a significant effect in
enforcing homology stringency at this final step in the strand exchange
process by enforcing the following: 1. that the number of base pairs that can
be bound in the intermediate state is limited 2. that transfer of a single
triplet from the intermediate state to the final state is never favorable 3.
that transferring non- contiguous triplets from the intermediate state to the
final state is unfavorable even if they are homologs 4. that the transfer of
pairs of triplets is not favorable until $\sim 18$ bp are bound in the
intermediate state. Properties 1-3 are shared with the transition from the
initial bound state to the intermediate state, but the fourth property is
different. It may arise from some combination of two features: 1. the linear
energy in the final state may be less favorable than the linear energy in the
intermediate state 2. there is a significant boundary penalty associated with
the deformation of the backbone that occurs when the system is partially in
the intermediate state and partially in the final state. In either case, then
the transition from the intermediate state to the final state will only become
free energetically favorable when the favorable non-linear term becomes
dominant over the other terms. If, the final state only becomes free
energetically favorable when $\sim 15$ contiguous base pairs are present in
the final state, then the non-linearity in the energies of the intermediate
and final states in the energy may provide additional discrimination against
regions of accidental homology since for a particular given searching ssDNA
sequence the odds of this occurring with a given searching sequence are $\sim
1.4x10^{-11}$/base pair. Thus, even in a 10 million bp genome, the probability
that such an accidental homology is present in a given bacterial genome is
$\sim 1/10^{4}$. We have considered a few bacterial genomes and found that the
sequences are indeed random in this sense, with the exception of repeated
genes. Thus, many bacterial genomes contain no accidental mismatches
consisting of 18 contiguous bp; therefore, no further homology checking is
required if 18 contiguous bp exactly match. The important statistical property
is consistent with experimental results that strand exchange products to do
not become stable until more than $\sim 18$ bp have undergone strand exchange.

###  II.5 For dsDNA in the final state the free energy has a minimum as a
function of $N$

If the final state has a much higher $\gamma$ than the intermediate state,
then adding base pairs will remain free energetically favorable for much
larger $N$; however, as long as the mechanical contribution remains a non-
linear function of $N$, the free energy as a function of $N$ will achieve a
minimum for some $N$, after which adding more base pairs becomes free
energetically unfavorable. Eventually, as $N$ approaches the deGennes length,
the mechanical energy will become a linear function of $N$. In this case, the
energy cost of adding an additional triplet remains constant as a function of
$N$. If this cost is small in comparison with $kT$ and unbinding is forbidden,
the model suggests that the length of the strand exchange product can increase
without limit.

###  II.6 The Effect of the Non-Linearity on the Strand Exchange of a
Mismatched Triplet

It has previously been assumed that the free energy penalty for strand
exchange of a triplet is approximately equal to the loss of Watson-Crick
pairing for that triplet, with a possible additional factor due to the effect
of the mismatch on the pairing of the two neighboring bases which ranges from
$\sim 1.5$ to $\sim 4kT$ . SantaLucia, Jr. (1998) In contrast, for a system
with the non-linearity considered here if the initially bound base pairs
contain a single mismatch, then strand exchange may be significantly more
unfavorable because the unfavorable free energy contribution due to this
mismatch must include not only the Watson-Crick pairing energy for that base
pair and its neighbors, but also the the increased mechanical stress on the
two matched base pairs. This stress not only makes a direct contribution to
the free energy penalty, but it can also increase the stacking penalty by
distorting the bonds between the two homologous base pairs which lowers their
Watson-Crick pairing energy. A detailed structural calculation would be
required to correctly assess all of these factors. In what follows, we will
assume that the free energy penalty for the strand exchange of a mismatched
base is approximately equal to the Watson-Crick pairing loss as long as the
number of $<18$ bp are bound to the filament.

###  II.7 dsDNA Tension Inhibits Progression of Strand Exchange Past a
Mismatch

In a system with a linear free energy as a function of the number of bound
homologous triplets, adding more homologous triplets is always favorable even
if the last triplet added were non-homologous, resulting in enormous kinetic
trapping. In contrast, the non-linear energy inhibits binding of additional
base pairs after a non-homologous triplet has bound, as illustrated in Fig. 7.
The dashed black line shows the curve for a perfect homolog adding a triplet
to the initial bound state if all of the other bound triplets have undergone
strand exchange. For up to $\sim 18$ bp thermal energy is sufficient to bind
additional base pairs. The dashed gray line shows the free energy penalty for
adding a homologous triplet if the last triplet added was non-homologous. The
free energy penalty is only slightly larger than the penalty for a homolog;
however, the solid gray line shows that the penalty for adding a second
triplet is very large, even though both triplets added after the non-homolog
were in fact homologous. For comparison, the solid black line shows the
energetic favorability of strand exchange of a homologous triplet from the
initial binding state. This graph shows that the non-linearity makes adding
additional triplets to the initial state is unfavorable once a mismatched
triplet has bound, even when the subsequent base pairs are homologous.

###  II.8 Possible Explanations of Biological Results

We have already discussed the proposal that the energetic non-linearity
explains why strand exchange is free energetically favorable even though the
sequences of the incoming and outgoing strands are the same and the protein
contacts in the initial searching ssDNA-RecA filament are similar to those in
the final post-strand exchange state.

In addition, experimental results have shown that a rapid initial interaction
incorporating $\sim 15$ base pairs is followed by a slower progression of
strand exchange that occurs in triplets Ragunathan et al. (2011). Figure 6
suggests that the binding of dsDNA to site II is favorable for fewer than 9
bases and requires only a few $kT$ of energy for fewer than 15 bases, whereas
for more bases the binding is highly free energetically unfavorable.

Furthermore, FRET based studies indicate that homology recognition may be
accurate for short sequences, but inaccurate for longer sequences Bazemore et
al. (1997). A separate study showed that strand exchange pauses at sequence
mismatches Lee et al. (2006); D. Sagi and? T. Tlusty (2006), and we argue that
such pauses lead to the unbinding of shorter non-homologous sequences because
the binding of the dsDNA to the filament occurs sequentially In the model
presented here the pause in strand exchange at a mismatch results from the
free energy cost of transferring the non-homologous triplet to the
intermediate state as well as the cost of progressing past a mismatched
triplet. Spontaneous unbinding of the entire strand exchange product becomes
unlikely as the sequence lengthens because so many free energetically
unfavorable transitions are required. If the strand exchange product becomes
too long, the unbinding time exceeds the recognition time available to the
organism; however, as discussed above, accidental mismatches that extend
beyond 18 bp rarely exist in vivo. In vivo, strand exchange does progress
through regions of non-homology once a sufficiently long stand exchange
product is formed, but ATP hydrolysis is required. Rosselli and Stasiak
(1991); Kim et al. (1992)

Finally, it is also well known that in the presence of ATP hydrolysis the size
of the strand exchange product increases monotonically until it reaches a
limit of $M\sim 80$ bp van der Heijden et al. (2008), where $M$ is the number
of bound dsDNA base pairs. Strand exchange then continues to progress, but $M$
remains constant because the heteroduplex dsDNA unbinds from the lagging edge
of the filament at the same average rate that new dsDNA binds to site II van
der Heijden et al. (2008); Fan et al. (2011). Since the dsDNA can freely
unbind from the filament, free energy minimization implies $M$ will remain
$\sim M_{freemin}$. Additional effects associated with dynamics may explain
why the strand exchange window moves along the dsDNA with $M\sim M_{freemin}$
rather than remaining stationary Conover et al. (2011). In contrast with the
experimental results obtained in the presence of hydrolysis, experimental
results obtained in the absence of hydrolysis show that the length of the
strand exchange window can continue without bound.van der Heijden et al.
(2008) In the model presented here, if the number of base pairs bound is small
the mechanical energy penalty associated with adding base pairs is a quadratic
function of energy for small numbers already bound; however, the base pairs
redistribute the stress between the backbones, so that eventually the base
pairs in the center of the filament are not under stress. In this case, the
penalty for adding another base pair triplet to the end becomes a linear
function of the number bound rather than a quadratic function. Thus, when a
sufficient number of base pairs have undergone strand exchange, the energy
required to add an additional triplet to the filament is constant, independent
of the number bound. If, as suggested by this model, the constant energy
decrease due to additional DNA protein contacts is approximately equal to the
constant energy increase associated with the added mechanical stress, then the
filament can extend forever because both energies are independent of the
number of base pairs already bound.

###  II.9 Additional Features in Three Dimensions

In the real RecA system steric factors are associated with the mismatch
between the 150 bp persistence length of dsDNA and the strong bending of dsDNA
in the 18 bp/turn helical RecA filament. The local rigidity of the dsDNA may
play a role in limiting the initial binding length to $\sim 9$ bp since that
many base pairs can interact with the ssDNA-RecA without significant bending.
After some dsDNA triplets are bound, the rigidity may also play a role in
preventing non-contiguous triplets from being added to the filament. The
nearest unbound triplet is already very near to the filament because it is
attached to the bound triplets by the phosphate backbones which cannot extend
much more than 0.5 nm/bp. Thus, the phosphates are in a position to interact
strongly with positively charged residues on the protein which can provide
sufficient free energy for the required bending. In contrast, the second
neighboring triplet will be separated by a larger distance which reduces the
interaction with the protein and requires more bending. A detailed structural
calculation would be required to correctly evaluate these effects, but both
effects would further support rapid and accurate homology recognition. In the
simple one dimensional model discussed here, the free energy effects of the
bending can be included in the $\gamma$ for the initial bound state, but the
additional degrees of freedom would alter the coupling between the initial
bound state $\gamma$ and the $\gamma$ for the intermediate state.

In addition, in the final post-strand exchange state interactions with the L1
and L2 loops may be more favorable for homologous triplets than non-homologous
triplets due to steric factors. Thus, the final state could have a sequence
dependent linear contribution to the free energy that was not considered in
this model, but may provide additional homology stringency.

##  III Conclusion

We have proposed a simple mechanical model for the stress distribution on
dsDNA bound to ssDNA-RecA filaments. The model suggests that a change in the
conformation of one bound triplet can change the conformation of all of the
other bound triplets; consequently, the total energy is a non-linear function
of the number of bound base pairs. The model makes several significant
qualitative and quantitative predictions. The most important qualitative
prediction is that a change in the configuration of one bound triplet changes
configuration of all the other bound triplets. The collective behavior of the
triplets leads to a free energy that is a non-linear function of the number of
consecutive bound triplets, where the binding of additional triplets becomes
increasingly unfavorable as the number of bound triplets increases. This non-
linearity is important because in systems with more than $\sim 4$ binding
sites, neither thermodynamic equilibrium nor kinetic proofreading can combine
accurate and efficient homology recognition when the energy is a linear
function of the number of correct pairings. In contrast, an unfavorable non-
linear energy combined with the a favorable linear energy due to DNA/protein
contacts can promote rapid and accurate homology recognition by making initial
sequence independent binding interactions favorable for up to $\sim 9$ base
pairs, while preventing any additional base pairs from binding unless the
bound base pairs include 9 contiguous homologous base pairs. If the initially
bound base pairs do not contain 9 contiguous homologous base pairs, adding
more base pairs to the filament is highly improbable regardless of whether or
not the additional base pairs are homologous. This effect combined with the
statistics of the sequence distribution of bacterial genomes implies that of
all but the $\sim$ 50 out of 10,000,000 possible pairings will rapidly unbind.

In addition , the non-linearity forces the addition of triplets to the
filament to proceed sequentially from the initially binding, where adding more
than two base pair triplets after a mismatch is highly unlikely, even if the
additional base pairs are sequence matched. Furthermore, the model suggests
true homologs can proceed to complete strand exchange because the strand
exchange of contiguous homologous base pair triplets reduces the tension on
the dsDNA. The tension reduction associated with the strand exchange of the
initial $\sim 9$ contiguous homologous base pairs allows more base pairs to
bind to the filament due to two effects: 1. the reduction in dsDNA tension
reduces the free energy penalty for adding more triplets to the initial bound
state 2. the decrease in the energy of the bound dsDNA due to strand exchange
reduces the unbinding rate of the dsDNA by making unbinding more unfavorable.
As strand exchange progresses from the $\sim 9$ initial contiguous base pairs,
the binding of the strand exchanged state still remains weak enough to be
reversed unless $\sim 18$ contiguous bp make the transition to the final post-
strand exchange state, a transition which is not favorable for $<\sim 18$ bp.
These effects could provide exact sequence recognition for bacterial genomes
except for repeated genes. The actual speed and accuracy of homology
recognition depends on detailed values of parameters as well as additional
factors associated with the three dimensional geometry of the filament, so we
do not provide detailed estimates here; however, the simple model presented
here provides mechanisms for overcoming fundamental limitations encountered in
systems with more than 4 binding sites where the binding energy is a linear
function of the number of correctly paired binding sites.

In sum, the energy non-linearity produces three crucial advantages that are
unavailable to systems with a strictly linear energy: 1. the initial
interaction is limited to $\sim 15$ bp beyond which binding of dsDNA triplets
to the filament cannot progress without a sequence dependent transition of 9
contiguous homologous base pairs to the strand exchanged state 2. nearly
immediate unbinding of any sequence that does not contain at least 9
contiguous homologous base pairs 3. a large free energy penalty that prevents
strand exchange from progressing past a sequence mismatch even if the mismatch
is followed by homologous triplets 4. a large free energy penalty that makes
the transition from the intermediate state to the final state unfavorable
until $\sim 18$ contiguous base pairs make the transition to the final state.
These features provide much more rapid and accurate homology recognition than
systems using linear energies: in systems with linear energies addition and
strand exchange of a homologous triplet is always favorable; therefore, in
systems with linear energies even short regions of accidental homology can
produce substantial trapping times, as demonstrated by both analytical
modeling and numerical simulations. Feinstein and Prentiss  Qualitative
features of the model provide possible explanations for well known but
previously unexplained features for homology recognition and strand exchange
and suggest that the bond rotations that appear in the overstretching of naked
dsDNA may have a role in strand exchange.

###### Acknowledgements.

We would like to thank Douglas Bishop, Yuen-Ling Chan, and Chantal Prévost for
helpful conversations about the X-ray structure of DNA bound to RecA. We would
also like to thank Chantal Prévost for the PDB file of a complete RecA helix.

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Figure 1: (Color Online) a. Side view of the X-ray structure of the dsDNA
post-strand exchange (final) state with complementary (on the right at the
arrow) and incoming strands shown as green and red stick renderings. The arrow
points to a few rises between two triplets in the dsDNA structure. The VMD
(32) renderings of RecA crystal structure 3CMX (8) show site II residues
Arg226 (pink), Lys227 (cyan), Arg243 (yellow), and Lys245 (magenta) with
charged nitrogen atoms (blue). The cyan triangle indicates the approximate
position of an outgoing strand phosphate

b. top view of the same structure with circles indicating the radii occupied
by the incoming (green), outgoing (blue), and complementary strands (red for
final state and gray for intermediate state). c. top view showing the base
pairing in the homology recognition/strand exchange process superimposed on
the actual X-ray structure.

Figure 2: (Color Online) Schematic of Interactions between dsDNA and the
ssDNA-RecA filament as strand exchange progresses for a homolog with side
views shown in the central part of each panel and top views shown in the right
part of each panel where the grey region indicates the space occupied by the
protein DNA/protein and the circles indicate the radii occupied by the three
strands in their final post-strand exchange positions. In the schematic, the
filament consisting of three RecA proteins, showing site I (pale red), site II
(pale orange), and the support for the rises provided by the L1 and L2 loops
(cyan), with (i) unbound ssDNA ( (ii) ssDNA bound in site I and unbound B-form
dsDNA, (iii) ssDNA bound in site I and dsDNA bound in site II; The outgoing
strand (far left ), complementary strand (bound to the outgoing strand in
purple ), and incoming strand (bound to the protein on the right) are shown in
blue, red, and green, respectively. (iv) central triplet has undergone strand
exchange, resulting in a decrease in lattice mismatch and a decrease in the
stress on the bp (v) all three triplets shown have undergone strand exchange
(vi) ssDNA bound in site II and dsDNA bound in site I in the final assembled
state which has even less bp stress than the strand exchanged state due to
increase mechanical support for the rises. Figure 3: Schematic of dsDNA bound
to site II of the protein showing the model parameters. Figure 4: Effects of
adding triplets to the initial bound state as calculated by the model. (A)
Three triplets shown in the bound state with an arrow pointing to the second
one. (B) Adding a triplet to the initial state changes the tension in the
other triplets. (C) Again tension on the triplets change as a fifth triplet is
added to the initial state. Figure 5: The dashed, solid gray, and solid black
lines show the absolute value of $v_{i}-u_{i}$ for the outer triplet, central
triplet, and the first triplet out from the center respectively when the
number of bound triplets is odd. Figure 6: Total energy as a function of bound
base pairs all in the initial bound state (dashed line), the intermediate
state (solid gray line) and the final post-strand exchange state (solid black
line). Figure 7: The free energy penalty for adding another triplet in the
initial bound state as a function of the number of base pairs in the strand
exchanged state. The dashed black line shows the penalty when all of the
triplets are homologous. The dashed gray line shows the penalty for adding an
additional triplet to the initial bound state after a triplet that is non-
homologous. The solid gray line shows the penalty for adding a second triplet
after a non-homolog where the first triplet after the non-homolog was
homologous. The solid black line shows the energetic favorability of strand
exchanging a homologous triplet in the initial bound state. Figure 8: For a
completely homologous dsDNA with all triplets in the intermediate strand
exchanged state, the free energy penalty for a single triplet reversing strand
exchange by making a transition from the intermediate strand exchanged state
to the initial bound state as a function of then number of base pairs when the
base flipping triplet is in the center of the filament (solid line) or at the
end of the filament (dashed line)