The most pedagogical and detailed explanation of the relationship between the various logo and matrix representations can be found in a review that we wrote in 2007 (Bussemaker et al., Annu Rev Biophys Biomol Struct. 2007;36:329-47;

https://www.ncbi.nlm.nih.gov/pubmed/17311525). I recommend that you read this paper, and some of the other relevant papers that it refers to.

A brief summary:

1. The philosophy of MatrixREDUCE is fundamentally different from that of traditional weigh matrix discovery methods such as MEME. We use a matrix representation of DNA binding specificity called position-specific affinity matrix (PSAM). The entries in the PSAM quantity the relative affinity of sequences that differ from the optimal DNA sequence by a single point mutation. Therefore, the largest value in each columns equals one, and all other values are between zero and one. The entries in a PSAM do not have to add up to one at each position.

2. The negative of the natural logarithm of the entries in the PSAM correspond to delta-delta-G values in units of RT, which are commonly used in the biophysical literature. The energy logo that we introduced in Foat et al. 2006 (

https://www.ncbi.nlm.nih.gov/pubmed/16873464) is a graphical representation in which the letter heights are directly related to these ddG/RT values. This is the logo representation that we recommend for visualizing MatrixREDUCE models.

3. It is not possible to construct a traditional information-content logo from a PSAM without making further ad hoc assumptions about the "background frequency" of each base, and therefore we do not recommend it. Traditional PWM's are statistical models of sets of aligned binding sites, in which at each position the frequency of each base is specified. These frequencies do add up to one by definition, unlike in a PSAM. While we do not recommend this, you could divide each column of the PSAM by its sum to convert the four relative affinities for each position to obtain a set of four base frequencies. You could then convert these "foreground frequencies" to relative entropies as is done to construct traditional sequence logos (for each base, divide foreground frequency by background frequency, take the logarithm base two of this ratio, then multiply this logarithm by the foreground frequency for each base, and finally sum over all four bases to get the relative entropy, which is also known as the information gain, measured in bits). The total height of the letter stack in the logo will correspond to this relative entropy, and the height of the individual letters will be proportional to the corresponding base frequency.

Finally, here is a numerical example for how a single position within the binding site could be represented:

| A | C | G | T |

relative affinity | 1.0 | 0.5 | 0.1 | 0.9 |

ddG/RT | 0.00 | 0.69 | 2.30 | 0.11 |

base frequency (not recommended) | 0.40 | 0.20 | 0.04 | 0.36 |

Hope this helps!