Introduction

The stretch of covarying positions corresponds to the main stem of the RNA secondary structure. Note that the trace is interrupted by a few positions having complete sequence conservation. The respective parts of the sequence logo profile have also been displayed along the edges of the plot.

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Discussion of Information Content

Two issues on information content for sequence/structure alignments are discussed here. First the contribution to the computation of information content when gaps are included in the alignment, and secondly the sample size robustness of mutual information for RNA secondary structure co-variance analysis. For background and details consult the mentioned references.

• Calculating the information content of an alignment that have gaps.

  The conceptual problem in computing the information content or relative entropy of an alignment that has gaps is the background probablities. The background probabilities are typically found by counting base (or amino acid) frequencies in the genomes (or data set) from which the aligned sequences originate. The gaps, however, do not appear before the alignment is made, so a ``gap background probability'' or a ``gap expectation value'' cannot be dealt with in the same way. It only makes sense to talk about this when the alignment has been performed.

  Calculation of the information content of an alignment containing gaps has been derived by Hertz and Stormo, 1995 (In Lim & Cantor (eds), Proc. of the Third Int. Conf. on Bioinformatics and genome Res., pp. 201-216). They derived the expression

  where and are the fraction of gaps and symbol k at position i in the alignment, and where k is a symbol in the alphabet A. The fraction is the background probability of symbol k, so . Note that if only a few sequences contain a gap () or almost all sequences have a gap () at some position i, the ``gap'' term does not contribute to the sum. Since the gap background probability is one in this expression, then all the background probabilities sum to 2 rather than one, which is the formal claim to define information content or relative entropy. This is resolved as follows. Hertz and Stormo derived a large-devation rate function given by and as they write, it normalizes all background probabilities with a factor of 1/2. This corresponds to adding to the information content on each of L positions in the alignment. Clearly, the difference between I and , is where to put the baseline in a sequence logo profile plot. The normalizing factor of Hertz and Stormo was found by considering the minimum of . When using the normalizing factor, one interpretation is that the a priori expectation to gaps is probability 0.5, however, another interpretation is that one has two models, one for which the expectation values for all the symbols sum to one, and one model dealing only with gaps and the expectation to gaps is one. When combining the two models a (re-)normalization of all the parameters can be introduced.

  The considerations for any gap background probability can in general be extended by subtracting from the standard form

  When using the normalization , it is easily found that . Clearly . Hence, the normalizing factor and the a priori expectation of gap frequency is directly related and the same when . Using this relation one can readily write so when the difference is equal to , and the same as above is obtained. The result becomes independent of . The interpretation of a gap a priori probability of 0.5 also makes sense. If no prior knowledge is known about the alignment it makes sense to expect gaps to appear randomly, i.e. having an a priori expectation of 0.5. So a ``renormalization'' of the two sets of probabilities, the set of base frequencies, and the gap probability, ensures that the background probabilities sum to one and fulfill the formal claim to define information content.

  The current version of Inform computes the information content according to , but MatrixPlot contains an option ``neg'' for which the baseline can be moved as described (when neg=n). Otherwise (when neg=y) the baseline is placed so that ``negative information content'' can show up on the profile. This can be useful in identifying positions which neither contains many or few gaps, but really is degenerated.

• Mutual information only for basepairs
  As has been discussed by Gorodkin et al. (Comput. Appl. Biosci., 13:583-586, 1997) and used in the display of structure logos the mutual information content for RNA sequences can be limited to include only pairs of bases that are complementary. In that way a more relevant measure is constructed. This is done by defining a complementary matrix (Gorodkin et al. Nucl. Acids. Res. 25:3724-3732,1997) which lists the bases that are complementary. The complementary matrix lists all bases against themselves, and a number is assigned to indicate the degree of ``belief'' in the complementarity. The matrix is clearly symmetric and most positions in the matrix hold the value zero. The mutual information between position i and j in the alignment using the complementary matrix is given by (A) where , A the alphabet, and where and . The element of the complementary matrix lists the degree of complementarity between base k and l. The fraction of pairs of base k at position i and base l at position j is and is the fraction of base k at position i. Gaps is dealt with by having for any base k. There are in particular two qualitative differences between this measure and the standard form of mutual information given by (B) First in (A) the observed and expected values are computed independently for two positions i and j, and the the overall covariance is computed. In contrast in (B) the observed and expected values are compared for each combination of the involved symbols. Secondly, in (A) only symbols for which a basepair rule is defined are included in the sum. In (B) all symbols are included (also gaps) even if it is already known that they do not (for RNA secondary structure) contribute to the covariance. This fact makes (A) much more robust to sampling size noise as illustrated below by two examples. In the Inform program measure (A) is referred to as mtype 2, and (B) to mtype 1. The upper triangle is mutual information content computed by type 2, and the lower triangle is the mutual information content computed by using type 1.
  
  Sample size corrections for plain sequence information content (e.g. Schneider et al. J. Mol. Biol. 188:415-431, 1986; Basharin, Theory Probability Appl. 4:333-336, 1959), and for the standard form of mutual information and correlation functions (e.g. Weiss and Herzel, J. Theor. Biol. 190:341-353, 1998) has previously been studied.

Acknowledgements

Thanks to Anders Krogh for his contribution in the discussion of information content. Thanks to Claus A. Andersen, Lars J. Jensen, and Christopher Workman for suggestions and feedback.

Reference

MatrixPlot: visualizing sequence constraints. J. Gorodkin, H. H. Stærfeldt, O. Lund, and S. Brunak. Bioinformatics. 15:769-770, 1999. (http://www.cbs.dtu.dk/services/MatrixPlot/)