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bases that are juxtaposed. They can each take on 16 different values,
from `AA',`AC', <IMG
 WIDTH="22" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img71.gif"
 ALT="$\dots$">,
to `UU', or 1 to 16, respectively. 
The number in row `X' and column `Y' of the table is the free
energy of the 2 &#215; 2 interior loop with the indicated single
stranded bases. Figure <A HREF="node5.html#SINT4">6</A> shows the full table for the CG and
AU closing base pairs.

<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="SINT4">&#160;</A><A NAME="819">&#160;</A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 6:</STRONG>
Free energies for all interior loops in RNA
closed by a CG and an AU base pair. Values of `X' or `Y' that
correspond to bases that could form Watson-Crick pairs have been
removed for brevity.</CAPTION>
<TR><TD><PRE>
                        5' ------> 3'
                         C \/ \_/ A
                         G /\  |  U
                       3' <------ 5'
 Y:   A    A    A    C    C    C    G    G    G    U    U    U  
      A    C    G    A    C    U    A    G    U    C    G    U  
 --------------------------------------------------------------
  AA 2.0  1.6  1.0  2.0  2.6  2.6  1.0  1.4  0.2  2.3  1.5  2.2
  AC 2.4  1.9  1.3  2.4  2.4  2.4  1.3  1.7 -0.4  2.1  0.8  1.5
  AG 0.9  0.4 -0.1  0.9  1.9  1.9 -0.1  0.2 -0.1  1.6  1.2  1.8
  CA 1.9  1.5  0.9  1.9  1.9  1.9  0.9  1.3 -0.9  1.6  0.4  1.1
  CC 2.8  1.8  2.2  2.2  2.2  2.2  2.2  2.2  0.4  1.9  1.7  1.4
X CU 2.7  1.6  2.0  2.1  2.1  2.1  2.0  2.0  0.3  1.8  1.5  1.2
  GA 1.0  0.6  0.0  1.0  2.0  2.0  0.0  0.4  0.0  1.7  1.3  2.0
  GG 1.8  1.3  0.7  1.8  2.4  2.4  0.7  1.1  0.0  2.1  1.2  1.9
  GU 1.8  0.4  1.6  0.8  1.8  1.8  1.6  1.2 -2.0  1.5 -0.7  1.8
  UC 2.7  1.6  2.0  2.1  2.1  2.1  2.0  2.0  0.3  1.8  1.5  1.2
  UG 0.3 -1.1  0.1  0.7  0.3  0.3  0.1  0.3 -3.5  0.0 -2.2  0.3
  UU 2.2  0.7  1.9  1.2  1.2  1.2  1.9  1.5  0.2  0.9  1.5  0.3
</PRE>
</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
Some special rules apply to 2-loops. A stacked pair that occurs at
the end of a helix has a different free energy than if it were in the
middle of a helix. Because of the availablility and precision of data,
we distinguish between GC closing and non-GC closing base pairs. In
particular, a penalty (terminal AU penalty) is assigned to each non-GC
closing base pair in a helix. The value of this penalty is stored in
the  <SMALL>MISCLOOP</SMALL> file. 

<P>
Because free energies are assigned to loops, and not to helices, there
is no <I>a priori</I> way of knowing whether or not a stacked pair will
be terminal or not. For this reason, the terminal AU penalty is built
into the  <SMALL>TSTACKH</SMALL> and  <SMALL>TSTACKI</SMALL> tables. For bulge,
multi-branch and exterior loops, the penalty is applied explicitly. In
all of these cases, the penalty is <I>formally</I> assigned to the
adjacent loop, although it really belongs to the helix.

<P>
A ``Grossly Asymmetry Interior Loop (GAIL)'' is an interior loop that
is 1 &#215; n,
where <I>n</I>&gt;2. The special ``GAIL'' rule that is used
in this case substitutes AA mismatches next to both closing base
pairs of the loop for use in assigning terminal stacking free energies
from the  <SMALL>TSTACKI</SMALL> file.

<P>
A <I>k</I>-loop, <IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img41.gif"
 ALT="${\bf L}$">,
where <I>k</I> &gt; 2, is called a <I>multi-branch</I>
loop. It contains <I>k</I>-1 base pairs, and is closed by a <I>k</I><SUP><I>th</I></SUP> base
pair. Thus there are <I>k</I> stems radiating out from this loop.
Because so little is known about the effects of multi-branch loops on
RNA stability, we assign free energies in a way that makes the
computations easy. This is the justification for the use of an <I>affine</I> free energy penalty for multi-branch loops. The free energy,

<!-- MATH: $\delta \delta G({\bf L})$ -->
<IMG
 WIDTH="55" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img74.gif"
 ALT="$\delta \delta G({\bf L})$">,
is given by:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\delta \delta G({\bf L}) = a + b \times l_{s}({\bf L}) + c \times l_{d}({\bf L})
+ \delta \delta G_{stack},
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="DDGM1">&#160;</A><IMG
 WIDTH="325" HEIGHT="28"
 SRC="img75.gif"
 ALT="\begin{displaymath}
\delta \delta G({\bf L}) = a + b \times l_{s}({\bf L}) + c \times l_{d}({\bf L})
+ \delta \delta G_{stack},
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(6)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <I>a</I>, <I>b</I> and <I>c</I> are constants that are stored in the <I>miscloop</I> file and 
<!-- MATH: $\delta \delta G_{stack}$ -->
<IMG
 WIDTH="62" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.gif"
 ALT="$\delta \delta G_{stack}$">
includes stacking interactions that
will be explained below.  This simple energy function allows the
dynamic programming algorithm used by <I>mfold</I> to find optimal
multi-branch loops in time proportional to <I>n</I><SUP>3</SUP>. It would take
exponentially increasing time (with sequence length) to use a more
appropriate energy function derived from Jacobson-Stockmeyer theory
[<A
 HREF="node19.html#JACH5001">30</A>] that grows logarithmically with 
<!-- MATH: $l_{s}({\bf L})$ -->
<IMG
 WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img25.gif"
 ALT="$l_{s}({\bf L})$">.
In
the <I>efn2</I> program that recalculates folding free energies using
more realistic rules (defined below), equation <A HREF="node5.html#DDGM1">6</A> is replaced
by:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\delta \delta G({\bf L}) = a + 6b + 1.75 \times RT \times \ln (l_{s}({\bf L})/6) +
c \times l_{d}({\bf L}) + \delta \delta G_{stack}.
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="DDGM2">&#160;</A><IMG
 WIDTH="465" HEIGHT="28"
 SRC="img77.gif"
 ALT="\begin{displaymath}
\delta \delta G({\bf L}) = a + 6b + 1.75 \times RT \times \l...
...bf L})/6) +
c \times l_{d}({\bf L}) + \delta \delta G_{stack}.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(7)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
That is, the linear dependence on <I>l</I><SUB><I>s</I></SUB> changes to a logarithmic
dependence for more than 6 single stranded bases in a multi-branch
loop. 

<P>
Stacking free energies, 
<!-- MATH: $\delta \delta G_{stack}$ -->
<IMG
 WIDTH="62" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.gif"
 ALT="$\delta \delta G_{stack}$">
are computed for multi-branch
and exterior loops. In the folding algorithm these are single strand
stacking free energies, also known as <I>dangling base</I> free
energies, because they are applied to single stranded bases adjacent
to a base pair that is either in the loop, or closes the loop.  This
single stranded base may ``dangle'' from the 5' or 3' end of the
base pair.  These parameters are stored in a file named <I>dangle.dg</I> or <I>dangle.TC</I>, as above. 

<P>
Figure <A HREF="node5.html#DANGLE">7</A> shows some single strand stacking
free energies.

<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="DANGLE">&#160;</A><A NAME="821">&#160;</A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 7:</STRONG>
Free energies for all possible single stranded
bases that are adjacent to a CG base pair. `X' refers to column. Note
that the 3' dangling free energies are larger in magnitude than the
5' dangling free energies.</CAPTION>
<TR><TD><PRE>
         X                           X          
 ------------------          ------------------ 
  A    C    G    U            A    C    G    U  
 ------------------          ------------------ 
     5' --> 3'                   5' --> 3'      
        CX                          C           
        G                           GX          
     3' <-- 5'                   3' <-- 5'      
 -1.7 -0.8 -1.7 -1.2         -0.2 -0.3  0.0  0.0
</PRE>
</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
If <I>i</I>.<I>j</I> and <IMG
 WIDTH="46" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img79.gif"
 ALT="$j\!+\!2.k$">
are 2 base pairs, then <I>r</I><SUB><I>j</I>+1</SUB> can interact
with both of them. In this case, the stacking is assigned to only 1 of
the 2 base pairs, whichever has a lower free energy (usually the 3'stack). If <I>k</I>.<I>l</I> is a base pair and both <I>r</I><SUB><I>k</I>-1</SUB> and <I>r</I><SUB><I>l</I>+1</SUB> are
single stranded, then both the 5' and 3' stacking are
permitted. The value of 
<!-- MATH: $\delta \delta G_{stack}$ -->
<IMG
 WIDTH="62" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.gif"
 ALT="$\delta \delta G_{stack}$">
is then the sum of all the
single base stacking free energies associated with the base pairs and
closing base pair of the loop.

<P>
It has been evident for some time that to make the free energy rules
more realistic for multi-branch and exterior loops, and to improve
folding predictions, we would be compelled to take into account the
stacking interactions between adjacent helices. Two helices, <IMG
 WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.gif"
 ALT="$\bf
H_{1}$">
and <IMG
 WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img81.gif"
 ALT="$\bf H_{2}$">
in a multi-branch or exterior loop are adjacent
if there are 2 base pairs <I>i</I>.<I>j</I> and <IMG
 WIDTH="46" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img82.gif"
 ALT="$j\!+\!1.k$">,
<I>i</I>.<I>j</I> and <IMG
 WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img83.gif"
 ALT="$i\!+\!1.k$">
or
<I>i</I>.<I>j</I> and <IMG
 WIDTH="46" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img84.gif"
 ALT="$k.j\!-\!1$">
that close <IMG
 WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.gif"
 ALT="$\bf
H_{1}$">
and <IMG
 WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img81.gif"
 ALT="$\bf H_{2}$">,
respectively. The last 2 cases can only occur in a multi-branch loop.
In addition, we define <I>almost adjacent</I> helices as 2 helices
where the addition of a single base pair (usually non-canonical),
results in an adjacent pair. The concept of adjacent helices is
important, since they are often coaxial in 3 dimensions, with a
stacking interaction between the adjacent closing base pairs. The
concept of almost adjacent comes from tRNA where, in many cases, the
addition of a GA base pair at the base of the anti-codon stem creates
a helix that is adjacent to, and stacks on, the D-loop stem.

<P>
<I>Mfold</I> does not yet take into account coaxial stacking of adjacent or
almost adjacent helices. The <I>efn2</I> program that re-evaluates
folding energies based on our best estimates does take this into
account. It is not a trivial matter to decide which combination of
coaxial stacking and single base stacking gives the lowest free energy
in a multi-branch or external loop, and a recursive algorithm is
employed to find this optimal combination. For example, coaxial
stacking excludes single base stacking adjacent to the stacked
helices. Free energies for the stacking of adjacent helices are stored
in a file called <I>coaxial.dg</I>. The format is the same as for <I>stack.dg</I>. When 2 helices are almost adjacent, then 2 files, named
<I>coaxstack.dg</I> and <I>tstackcoax.dg</I> are used. The format is the
same as for stacking free energies. The use of these 2 files is
explained with the aid of Figure <A HREF="node5.html#COAX">8</A>.
Thus, in the <I>efn2</I> program, 
<!-- MATH: $\delta \delta G_{stack}$ -->
<IMG
 WIDTH="62" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.gif"
 ALT="$\delta \delta G_{stack}$">
is a combination of
single base stacking and coaxial stacking, depending on the loop.

<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="COAX">&#160;</A><A NAME="822">&#160;</A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 8:</STRONG>
The helices closed by G<SUP>3</SUP>-C<SUP>18</SUP> and
C<SUP>20</SUP>-G<SUP>36</SUP> are almost adjacent. Their stacking is mediated by
a non-canonical G<SUP>19</SUP>-A<SUP>37</SUP> base pair. The free energy for the 
G<SUP>19</SUP>-A<SUP>37</SUP> to C<SUP>20</SUP>-G<SUP>36</SUP> comes from the <I>
tstackcoax.dg</I> file. This is used where the phosphate backbone is
unbroken, since there are 2 covalent links. The C<SUP>18</SUP>-G<SUP>3</SUP> to
G<SUP>19</SUP>-A<SUP>37</SUP> stacking free energy comes from the <I>
coaxstack.dg</I>, which is used for stacking where the backbone is
broken. In this case, G<SUP>3</SUP> and A<SUP>37</SUP> are not linked.</CAPTION>
<TR><TD><IMG
 WIDTH="461" HEIGHT="188"
 SRC="img85.gif"
 ALT="\begin{figure}\centering
\epsfig{file=figure.2dps,width=0.9\textwidth}
\end{figure}"></TD></TR>
</TABLE>
</DIV>
<BR>
<P>
In the case of circular RNA, the choice of origin is
arbitrary. However, once it is made, what would be the exterior loop 
in linear RNA becomes equivalent to a  hairpin, bulge, interior or
multi-branch loop, or a stacked pair.

<P>
The Turner parameters for RNA folding have been published and
summarized a number of times. The most significant older publications
are [<A
 HREF="node19.html#FRES8601">31</A>,<A
 HREF="node19.html#TURD8701">32</A>,<A
 HREF="node19.html#TURD8801">33</A>], and <I>mfold</I> was originally used
these results alone.  Version 1 of <I>mfold</I> had no <I>tloop.dg</I> file,
and there was a single terminal stacking free energy file, <I>tstack.dg</I>. Tetraloop bonus free energies were added in version
2.0. The <I>tstack</I> file was split into 2 files in version 2.2. Version
3.0 introduces triloop bonus energies, and both tetraloop and triloop
bonus energies now depend on the closing base pair. Special rules for
small interior loops are also new. For example, the 2 &#215; 2
interior loop rules have evolved from [<A
 HREF="node19.html#WUM9501">34</A>].  Coaxial
stacking was also introduced in version 3.0, although it's importance
was realized eariler [<A
 HREF="node19.html#WALA9401">35</A>].

<P>
A complete set of DNA folding parameters have recently become
available [<A
 HREF="node19.html#SANJ9801">36</A>]. These are based on measuremments for
stacking and mismatches, and on the literature for loop and other
effects. Parameters are also available for predicting the formation of
RNA/DNA duplexes [<A
 HREF="node19.html#SUGN9501">37</A>].

<P>
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<ADDRESS>
<TABLE><TR><TD><IMG SRC=img/shield16.gif HSPACE=20></TD><TD><I>Michael Zuker <BR>Institute for Biomedical Computing<BR>
Washington University in St. Louis<BR>
1998-12-05</I></TD></TR></TABLE>
</ADDRESS>
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