node5.html

mfold

<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\delta \delta G_{H} = \delta \delta G_{H}^{1} + \delta \delta G_{H}^{2} + \delta \delta G_{H}^{3}+ \delta \delta G_{H}^{4},
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="DDGH">&#160;</A><IMG
 WIDTH="283" HEIGHT="28"
 SRC="img36.gif"
 ALT="\begin{displaymath}
\delta \delta G_{H} = \delta \delta G_{H}^{1} + \delta \delta G_{H}^{2} + \delta \delta G_{H}^{3}+ \delta \delta G_{H}^{4},
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where
<DL COMPACT>
<DT>1.
<DD>
<!-- MATH: $\delta \delta G_{H}^{1}$ -->
<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img37.gif"
 ALT="$\delta \delta G_{H}^{1}$">
is the size dependent contribution from the <I>loop</I>
file, or from equation <A HREF="node5.html#DDGLOG2">2</A> for sizes &gt; 30,
<DT>2.
<DD>
<!-- MATH: $\delta \delta G_{H}^{2}$ -->
<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img38.gif"
 ALT="$\delta \delta G_{H}^{2}$">
is the terminal mismatch stacking free energy, taken
from the <I>tstackh</I> file (0 for hairpin loops of size 3),
<DT>3.
<DD>
<!-- MATH: $\delta \delta G_{H}^{3}$ -->
<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img39.gif"
 ALT="$\delta \delta G_{H}^{3}$">
is the bonus free energy for triloops or
tetraloops listed in the  <SMALL>TRILOOP</SMALL> or  <SMALL>TLOOP</SMALL> files. This value is
0 for loops not listed in the  <SMALL>TRILOOP</SMALL> or  <SMALL>TLOOP</SMALL> files and for
loop sizes &gt; 4,
<DT>4.
<DD>
<!-- MATH: $\delta \delta G_{H}^{4}$ -->
<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img40.gif"
 ALT="$\delta \delta G_{H}^{4}$">
is the bonus or penalty free energy 
for special cases not covered by the above.
</DL>
<P>
A 2-loop, <B>L</B>
is closed by a base pair <I>i</I>.<I>j</I> and contains a
single base pair, <I>i</I>'.<I>j</I>', satisfying 
<I>i</I> &lt; <I>i</I>' &lt; <I>j</I>' &lt; <I>j</I>. In this case,
the loop size, l<SUB>s</SUB>(<B>L</B>),
can be written as:
<BR><P></P>
<DIV ALIGN="CENTER">
<I>l</I><SUB>s</SUB>(<B>L</B>) = <I>l</I><SUB>s</SUB><SUP>1</SUP>(<B>L</B>) + <I>l</I><SUB>s</SUB><SUP>2</SUP>(<B>L</B>),
</DIV><BR>

where

<I>l</I><SUB>s</SUB><SUP>1</SUP>(<B>L</B>) = <I>i'-i-1</I>
and  
<I>l</I><SUB>s</SUB><SUP>2</SUP>(<B>L</B>) = <I>j-j'-1</I>.

<P>
A 2-loop of size 0 is called a <I>stacked pair</I>. This refers to
the stacking between the <I>i</I>.<I>j</I> and immediately adjacent 
<!-- MATH: $i\!+\!1.j\!-\!1$ -->
<IMG
 WIDTH="65" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="img45.gif"
 ALT="$i\!+\!1.j\!-\!1$">
base
pair contained in the loop. Free energies for these loops are stored
in a file named <I>stack.dg</I>, or <I>stack.TC</I>, where <I>TC</I> is a
temperature, as defined above. The layout is the same as for the
<I>tstackh</I> file. A portion of such a file is given in Figure
<A HREF="node5.html#STK">4</A>. A group of 2 or more consecutive base pairs is called a
<I>helix</I>. The first and last are the closing base pairs of the
helix. They may be written as <I>i</I>.<I>j</I> and <I>i</I>'.<I>j</I>', where 
<!-- MATH: $i < i' < j' <
j$ -->
<I>i</I> &lt; <I>i</I>' &lt; <I>j</I>' &lt;
<I>j</I>. Then <I>i</I>.<I>j</I> is called the external closing base pair and <I>i</I>'.<I>j</I>' is
called the internal closing base pair. This nomenclature is used for
circular RNA as well, even though it depends on the choice of origin.

<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="STK">&#160;</A><A NAME="815">&#160;</A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 4:</STRONG>
Sample free energies in kcal/mole for CG base
pairs stacked over all possible base pairs, XY.  X refers to row and
Y refers to column, in the order A, C, G and U respectively. Entries
denoted by an isolated period, `.', are undefined, and may be
considered as <IMG
 WIDTH="32" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.gif"
 ALT="$+\infty $">.</CAPTION>
<TR><TD><PRE>
          5' --> 3'     
             CX         
             GY         
          3' <-- 5'     
   Y:  A    C    G    U 
  ----------------------
X:A |  .    .    .   -2.1
  C |  .    .   -3.3  . 
  G |  .   -2.4  .   -1.4
  U | -2.1  .   -2.1  . 
</PRE>
</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
Only Watson-Crick and wobble GU pairs are allowed as <I>bona fide</I>
base pairs, even though the software is written to allow for any base
pairs. The reason is that nearest neighbor rules break down for
non-canonical, even GU base pairs, and that mismatches must instead be
treated as small, symmetric interior loops. Note that the stacks

<!-- MATH: $\begin{array}{ccc}
5' & --&gt; & 3'  \\
   &     WX      \\
   &     ZY      \\
     3' & <-- & 5'
\end{array}
and
\begin{array}{ccc}
     5'& --&gt;& 3'  \\
   &     YZ      \\
   &     XW      \\
     3' & <-- & 5'  
\end{array}$ -->
<IMG
 WIDTH="258" HEIGHT="92" ALIGN="MIDDLE" BORDER="0"
 SRC="img47.gif"
 ALT="$\begin{array}{ccc}
5' & --&gt; & 3' \\
& WX \\
& ZY \\
3' & <-- & 5'
\end{...
...in{array}{ccc}
5'& --&gt;& 3' \\
& YZ \\
& XW \\
3' & <-- & 5'
\end{array}$">are identical, and yet formally different for <IMG
 WIDTH="55" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img48.gif"
 ALT="$W \neq Y$">
and <IMG
 WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img49.gif"
 ALT="$X \neq
Z$">.
These stacked pairs are stored twice in the file, and
the <I>mfold</I> software checks for symmetry. This is an example of
built in redundancy as a check on precision.

<P>
A 2-loop, L of size &gt; 0 is called a <I>bulge loop</I> if
<I>l</I><SUB>s</SUB><SUP>1</SUP>(<B>L</B>) = 0
or  
<I>l</I><SUB>s</SUB><SUP>2</SUP>(<B>L</B>) = 0
and an interior loop
if <B>both</B> 
<I>l</I><SUB>s</SUB><SUP>1</SUP>(<B>L</B>) = 0
and 
<I>l</I><SUB>s</SUB><SUP>2</SUP>(<B>L</B>) = 0.

<P>
Bulge loops up to size 30 are assigned free energies from the <I>loop</I> file (See Figure <A HREF="node5.html#LOOP">1</A>). For larger bulge loops, equation
<A HREF="node5.html#DDGLOG2">2</A> is used. When a bulge loop has size 1, the stacking
free energy for base pairs <I>i</I>.<I>j</I> and <I>i</I>'.<I>j</I>' are used (from the <I>stack</I> file).

<P>
Interior loops have size <IMG
 WIDTH="28" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="img54.gif"
 ALT="$\geq 2$">.
If <I>l</I><SUB>s</SUB><SUP>1</SUP>(<B>L</B>) = 
<I>l</I><SUB>s</SUB><SUP>2</SUP>(<B>L</B>), 
the loop is called <I>symmetric</I>; otherwise, it is <I>asymmetric</I>, 
or lopsided. The asymmetry of an interior loop, a(<B>L</B>) is defined by:
<BR><P></P>
<DIV ALIGN="CENTER">
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="ASYM">&#160;</A>
a(<B>L</B>) = <B>|</B> <I>l</I><SUB>s</SUB><SUP>1</SUP>(<B>L</B>) - <I>l</I><SUB>s</SUB><SUP>2</SUP>(<B>L</B>) <B>|</B>
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
<P>
The free energy, 
<!-- MATH: $\delta \delta G_{I}$ -->
<IMG
 WIDTH="39" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img57.gif"
 ALT="$\delta \delta G_{I}$">,
of an interior loop is the sum of 4components:
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{equation}
\delta \delta G_{I} = \delta \delta G_{I}^{1} + \delta \delta G_{I}^{2} + \delta \delta G_{I}^{3} + \delta \delta G_{I}^{4}.
\end{equation} -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="DDGI">&#160;</A><IMG
 WIDTH="259" HEIGHT="28"
 SRC="img58.gif"
 ALT="\begin{displaymath}
\delta \delta G_{I} = \delta \delta G_{I}^{1} + \delta \delta G_{I}^{2} + \delta \delta G_{I}^{3} + \delta \delta G_{I}^{4}.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
<P>
<DL COMPACT>
<DT>1.
<DD>
<!-- MATH: $\delta \delta G_{I}^{1}$ -->
<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img59.gif"
 ALT="$\delta \delta G_{I}^{1}$">
is the size dependent contribution from the <I>loop</I>
file, or from equation <A HREF="node5.html#DDGLOG2">2</A> for sizes &gt; 30.
<DT>2.
<DD>
<!-- MATH: $\delta \delta G_{I}^{2}$ -->
<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img60.gif"
 ALT="$\delta \delta G_{I}^{2}$">
and 
<!-- MATH: $\delta \delta G_{I}^{3}$ -->
<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img61.gif"
 ALT="$\delta \delta G_{I}^{3}$">
are terminal mismatch stacking
free energies, taken from the <I>tstacki</I> file. The format of this
file is identical to the format of the <I>tstackh</I> file. There are 2
terms because of the terminal stacking of both <I>r</I><SUB><I>i</I>+1</SUB> and <I>r</I><SUB><I>j</I>-1</SUB>on the <I>i</I>.<I>j</I> base pair, and of both <I>r</I><SUB><I>i</I>'-1</SUB> and <I>r</I><SUB><I>j</I>'+1</SUB> on the
<I>i</I>'.<I>j</I>' base pair. This may be visualized as
<BR>
<DIV ALIGN="CENTER">

<!-- MATH: \begin{eqnarray}
\begin{array}{ccccc}
{\rm 5'}-&r_{i}&-&r_{i+1}&-{\rm 3'} \\
&    \bullet  & & \circ          \\
{\rm 3'}-&r_{j}&-&r_{j-1}&-{\rm 5'}
\end{array}  
\: \: {\rm and} \: \:
\begin{array}{ccccc}
{\rm 5'}-&r_{j'}&-&r_{j'+1}&-{\rm 3'} \\
           &   \bullet &  &  \circ           \\
{\rm 3'}-&r_{i'}&-&r_{i'-1}&-{\rm 5'},
\end{array} \nonumber

\end{eqnarray} -->
 <IMG  WIDTH="427" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"  SRC="img62.gif"
 ALT="$\displaystyle \begin{array}{ccccc}
{\rm 5'}-&r_{i}&-&r_{i+1}&-{\rm 3'} \\
& \b...
... \\
& \bullet & & \circ \\
{\rm 3'}-&r_{i'}&-&r_{i'-1}&-{\rm 5'},
\end{array}$">
</DIV>
<BR CLEAR="ALL"><P></P>
where <IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img63.gif"
 ALT="$\bullet$">
denotes a base pair and <IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img64.gif"
 ALT="$\circ$">
denotes a mismatched pair.
<DT>3.
<DD>
<!-- MATH: $\delta \delta G_{I}^{4}$ -->
<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img65.gif"
 ALT="$\delta \delta G_{I}^{4}$">
is the asymmetry penalty, and is a function of
a(<B>L</B>) defined in equation <A HREF="node5.html#ASYM">4</A>. The penalty is 0 for
symmetric interior loops. The asymmetric penalty free energies come
from the <I>miscloop.dg</I> or <I>miscloop.TC</I> file.
</DL>
<P>
Equation <A HREF="node5.html#DDGI">5</A> is now used only for loops of size &gt; 4 or of
asymmetry &gt; 1. This means that special rules apply to 
1 &#215; 1, 1 &#215; 2 and  2 &#215; 2
interior loops. Free energies for these
symmetric and almost symmetric interior loops are stored in files <I>sint2.dg</I>, <I>asint1x2.dg</I> and <I>sint4.dg</I>, respectively. As
above, the suffix <I>TC</I> is used in place of <I>dg</I> when explicit
attention is paid to temperature. These files list all possible values
of the single stranded bases, and all possible Watson-Crick and GU
base pair closings. The <I>sint2</I> file comprises a 
6 &#215; 6 array of 4 &#215; 4 tables. There is a table for all possible 
6 &#215; 6 closing base pairs. The free energy values for each choice
of closing base pairs are arranged in 4 &#215; 4 tables. The term
``closing base pairs'' refers to the closing base pair of the loop and
the contained base pair of the loop, as in the strict definition of a
loop. An example of such a table is given in Figure <A HREF="node5.html#SINT2AND12">5</A>.

<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="SINT2AND12">&#160;</A><A NAME="817">&#160;</A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 5:</STRONG>
Left: Free energies for all 1 &#215; 1
interior loops in DNA closed by a CG and an AT base pair.  Right: Free
energies for all 1 &#215; 2 interior loops in RNA closed by a CG and
an AU base pair, with a single stranded U 3' to the double stranded
U. As in similar Figures, X refers to row and Y to column.</CAPTION>
<TR><TD><PRE>
          5' --> 3'                  5' --> 3'        
              X                          X            
             C A                        C  A
             G T                        G  U           
              Y                          YA            
          3' <-- 5'                  3' <-- 5'        
   Y:   A    C    G    T      Y:   A    C    G    U
   ---------------------      ---------------------   
X:A | 1.1  2.1  0.8  1.0   X:A | 3.2  3.0  2.4  4.8   
  C | 1.7  1.8  1.0  1.4     C | 3.1  3.0  4.8  3.0   
  G | 0.5  1.0  0.3  2.0     G | 2.5  4.8  1.6  4.8   
  T | 1.0  1.4  2.0  0.6     U | 4.8  4.8  4.8  4.8   
</PRE>
</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
The <I>asint1x2</I> file comprises a 24 row by 6 column array of 
4 &#215; 4 tables. There is a 4 &#215; 4 table for all possible 6 &#215; 6
closing base pairs and choice of one of the single
stranded bases. The free energy values for each choice of closing base
pairs and a single stranded base are arranged in 4 &#215; 4
tables. An example of these tables is given in Figure <A HREF="node5.html#SINT2AND12">5</A>.

<P>
Finally, the <I>sint4</I> file contains 36 16 &#215; 16
tables, 1 for each pair of closing base pairs. A 2 &#215; 2 interior loop can
have 4<SUP>4</SUP> combinations of single stranded bases. If, for example,
the loop is closed by a GC base pair and an AU base pair, we can write
it as:
<PRE>
  5' ------&gt; 3'
   G \/ \_/ A
   C /\  |  U
  3' &lt;------ 5'
</PRE>Both the large `X' and large `Y' refer to an unmatched pair of