The energy dot plot

A nucleic acid secondary structure dot plot is a triangular plot that depicts base pairs as dots or other symbols. We shall refer to these symbols as dots. A dot in column i and row j of a triangular array, $\{ (i,j) \vert 1 \leq i \leq j \leq n \}$ represents the base pair i.j. The advantage of a dot plot is that it can display the base pairs in more than 1 folding simultaneously. It can be used to compare a few foldings, or the base pair distribution in many millions of foldings.

Mfold computes a number, $\Delta G(i,j)$ for every possible base pair, i.j. This is the minimum free energy of any folding that contains the i.j base pair. As above, we let $\Delta G$ be the overall minimum folding free energy, and $\Delta \Delta G$ a user selected free energy increment. Clearly

$$\Delta G= \min_{1 \leq i < j \leq n} \Delta G(i,j).$$

The energy increment is derived from $\Delta G$ and P. That is, $\Delta \Delta G= P \times \Delta G/ 100$. The current convention is to lower $\Delta \Delta G$ to 12 kcal/mole when it would otherwise be greater, and to raise it to 1 kcal/mole when it would otherwise be smaller. Then the energy dot plot is defined to be the collection of all base pairs i.j satisfying:

$$\Delta G(i,j) \leq \Delta G+ \Delta \Delta G.$$

This dot plot contains the superposition of all possible foldings whose folding energy is within $\Delta \Delta G$ of the minimum folding energy. Typically, $\vert\Delta \Delta G\vert$ is small compared to $\vert\Delta G\vert$, or P is a small percentage. In this case, the energy dot plot contains the superposition of all close to optimal foldings.

The energy dot plot gives an overall visual impression of how ``well-defined'' the folding is. A cluttered plot, or cluttered regions, indicate either structural plasticity (the lack of well-defined structure) or else the inability of the algorithm to predict a structure with confidence. A couple of crude measures of ``well-definedness'' have been introduced in mfold . The first is ``P-num''. $P\!-\!num(i)$ is a measure of the level of promiscuity of r<sub>i</sub> in its pairing with other bases in foldings within $\Delta \Delta G$ of $\Delta G$. It is the number of different base pairs, i.j, or k.i that can form in this set of foldings, and is simply the number of dots in the i<sup>th</sup> row and i<sup>th</sup> column of the energy dot plot . If $\delta(expression)$ is defined to be 1 when ``expression'' is true, and 0 otherwise, then P-num may be defined as:

$$P\!-\!num(i) = \sum_{k < i} \delta (\Delta G(k,i) \leq \Delta... ...{i < j} \delta (\Delta G(i,j) \leq \Delta G+ \Delta \Delta G).$$

P-num pertains to individual bases. H-num is ``well-definedness'' measure for a base pair i.j. It is the average value of the two P-num quantities, adjusted by removing the ``desirable'' i.j base pair. That is:

$$H\!-\!num(i,j) = ( P\!-\!num(i) + P\!-\!num(j) - 1 )/2.$$

A helix, already defined as a collection of two or more consecutive base pairs, may be described as a triple i,j,k, where k is the number of base pairs, and the actual base pairs are $i.j, i\!+\!1.j\!-\!1, \dots, i\!+\!k\!-\!1.j\!-\!k\!+\!1$. When k=1, the helix becomes a single base pair. With some abuse of notation, we may also write $H\!-\!num(i,j,k)$ to be the H-num value of the helix, i,j,k. This is the average value of H-num over all the base pairs in the helix.

There are 5 files associated with the energy dot plot .

`FILE_NAME.PLOT' : This is a text file that contains all the base pairs on the energy dot plot , organized into helices for which $\Delta G(i,j)$ is constant. The first record is a header, and each subsequent record describes a single helix. The records are usually sorted by $\Delta G(i,j)$, and are often filtered so that short helices or isolated base pairs (helices of length 1) in suboptimal foldings are removed. Figure 9 shows a sample plot file.

  

Figure 9: Selected records from a plot file. ``level'' refers to a free energy range that is to be plotted in the same color, where 1 is always optimal. The ``level'' parameter is obsolete in the newer plotting programs of mfold 3.0. ``istart'', ``jstart'' and ``length'' define a helix and refer to i,j,k, respectively. The ``energy'' is free energy expressed as an integer in 10^ths of a kcal/mole. Note that this is not the free energy of the helix, but the mimimum free energy of any folding that contains the helix.

   level  length istart jstart energy
      1      8    206    242   -972
      1      7    319    434   -972
      1      7    108    141   -972
      1      7     53    185   -972
      1      6    334    412   -972
      1      6    308    444   -972
      1      6    288    472   -972
      1      6    247    279   -972
     ...
      2      4      8     23   -971
      2      2     69     78   -971
      2      4      1     24   -970
      2      2     10     17   -970
      2      3    345    400   -967
      2      2    297    462   -967
     ...

`FILE_NAME.ANN' : This file contains P-num information for a particular $\Delta \Delta G$. The i^th record contains i and $P\!-\!num(i)$. This file is used for annotating plotted structures.

`FILE_NAME.H-NUM' : This file is the same as `file_name.plot', except that the ``energy'' column is replaced by an ``h-num'' column. These files are usually sorted by h-num; lowest to highest, or best determined to worst determined. Often, only helices in optimal foldings are retained. Figure 10 shows part of a sorted and filtered h-num file corresponding to the plot file in Figure 9.

  

Figure 10: The beginning and end of an h-num file sorted by h-num and filtered to include only helices in optimal foldings. As with P-num, H-num values are relative to a particular sequence and free energy increment.

   level  length istart jstart h-num
      1      4     38    194    6.8
      1      4    215    232    7.3
      1      5     31    201    8.4
      1      7     53    185    8.4
      1      2     47    189   11.0
      1      8    206    242   11.9
      1      6     61    176   13.7
      1      4     89    163   13.8
      1      3    255    271   14.0
      1      3    104    145   15.0
      1      1     68     79   16.0
      1      4    121    131   17.0
      1      6    288    472   17.3
    ...
      1      2    353    389   35.0
      1      3    364    377   38.7
      1      3    297    459   39.0

`FILE_NAME.PS' : This is a PostScript file of the energy dot plot .

`FILE_NAME.GIF' : This is an image of the energy dot plot in ``gif'' format, suitable for display on web pages.