text: user3797: I would like to ask how BLOSUM matrix is constructed and calculated ? I read the wikipedia and I still do not understand it. I do not understand the mathematical calculations as I have low knowledge about logarithms. 
  Maybe could someone give a simple example and then a real-scenario example of how the BLOSUM matrix can be used and calculated, maybe in relation to the Blast algorithm where it uses the BLOSUM scoring matrix to determine high scoring words for each word in the query sequence ?
  Also, why are they different types of BLOSUM matrices like BLOSUM62 and so on ?
  
  user3848: Amino acid matrices are contructed by maximum likelihood and easily performed using PAML. You present a known tree and an amino acid alignment and ask what amino acid matrix would maximise the likelihood. Hence all amino acid matrices using this old school approach are dependent on the alignment composition. JTT was generally regarded as obtaining the highest likelihood. However, I reitereate that things have moved on a lot since this 2-D reversible amino acid substitution matrix was devised.
  
  user1769: Sean Eddy wrote an excellent explanation of the BLOSUM Matrix in his Nature Biotechnology paper Where did the BLOSUM62 alignment score matrix come from? He also gives real examples of how one would calculate the score for different amino acid pairs and where the numbers come from.  Further references given at the end of the paper might also be useful reading, specifically: Amino acid substitution matrices from an information theoretic perspective, Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes, and Amino acid substitution matrices from protein blocks.
  
  user3939: BLOSUM stands for BLOcks SUbstitution Matrices (Henikoff & Henikoff, 1992), and were created by observing substitution frequencies in local ungapped multiple sequence alignments. The score reflects the chance (log-odds) one amino acid is substituted for another in a set of protein multiple sequence alignments. BLOSUM scoring matrices are normally followed by a number eg BLOSUM62. The number is % sequence identity between the sequences in the multiple sequence alignment (MSA) used to create the score matrix. eg BLOSUM62 matrices were created from multiple sequence alignments with blocks that shared 62% identity. Note because these are ungapped alignments, the multiple sequence alignments were generated from conserved protein families.  With the older PAM matrices, the number is opposite, a higher number is greater distance. PAM matrices reflect evolutionarily distant in global (aka gapped) alignments of the full-length sequences that include both conserved and diverged regions.  See Wikipedia https://en.wikipedia.org/wiki/BLOSUM. There are several newer substitution matrices.  The R package SubVis allows you to compare/visualize different substitution matrices. 
  
original_text: 3797: <p>I would like to ask how BLOSUM matrix is constructed and calculated ? I read the wikipedia and I still do not understand it. I do not understand the mathematical calculations as I have low knowledge about logarithms. </p>
  
  <p>Maybe could someone give a simple example and then a real-scenario example of how the BLOSUM matrix can be used and calculated, maybe in relation to the Blast algorithm where it uses the BLOSUM scoring matrix to determine high scoring words for each word in the query sequence ?</p>
  
  <p>Also, why are they different types of BLOSUM matrices like BLOSUM62 and so on ?</p>
   3848: <p>Amino acid matrices are contructed by maximum likelihood and easily performed using PAML. You present a known tree and an amino acid alignment and ask what amino acid matrix would maximise the likelihood. Hence all amino acid matrices using this old school approach are dependent on the alignment composition. JTT was generally regarded as obtaining the highest likelihood. However, I reitereate that things have moved on a lot since this 2-D reversible amino acid substitution matrix was devised.</p>
   1769: <p>Sean Eddy wrote an excellent explanation of the BLOSUM Matrix in his Nature Biotechnology paper <a href="https://www.nature.com/articles/nbt0804-1035" rel="nofollow noreferrer">Where did the BLOSUM62 alignment score matrix come from?</a> He also gives real examples of how one would calculate the score for different amino acid pairs and where the numbers come from.  Further references given at the end of the paper might also be useful reading, specifically: <a href="https://www.sciencedirect.com/science/article/pii/002228369190193A?via%3Dihub" rel="nofollow noreferrer">Amino acid substitution matrices from an information theoretic perspective</a>, <a href="https://www.pnas.org/content/87/6/2264.long" rel="nofollow noreferrer">Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes</a>, and <a href="https://www.pnas.org/content/89/22/10915.long" rel="nofollow noreferrer">Amino acid substitution matrices from protein blocks</a>.</p>
   3939: <p>BLOSUM stands for BLOcks SUbstitution Matrices (Henikoff &amp; Henikoff, 1992), and were created by observing substitution frequencies in local ungapped multiple sequence alignments. The score reflects the chance (log-odds) one amino acid is substituted for another in a set of protein multiple sequence alignments. BLOSUM scoring matrices are normally followed by a number eg BLOSUM62. The number is % sequence identity between the sequences in the multiple sequence alignment (MSA) used to create the score matrix. eg BLOSUM62 matrices were created from multiple sequence alignments with blocks that shared 62% identity. Note because these are ungapped alignments, the multiple sequence alignments were generated from conserved protein families.  With the older PAM matrices, the number is opposite, a higher number is greater distance. PAM matrices reflect evolutionarily distant in global (aka gapped) alignments of the full-length sequences that include both conserved and diverged regions.  See Wikipedia <a href="https://en.wikipedia.org/wiki/BLOSUM" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/BLOSUM</a>. There are several newer substitution matrices.  The R package SubVis allows you to compare/visualize different substitution matrices. </p>
  