I am thinking of reconstructing an multiple sequence alignment (MSA) of protein sequences from a position-Specific Scoring Matrices (PSSM).
Is it possible? I suppose co-evolution information is lost in PSSM so maybe it is difficult?
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In short I'm afraid it isn't possible because you lose the linkage per Haplotype which is the core information and vital for any population genetics or phylogenetic analysis.
Position-specific scoring matrix e.g. A .60 C .20 G .20 T 0 I tried to produce a position scoring matrix, but it didn't work out, anyway the allele frequency for each e.g. amino acid are given for each alignment position.
You could produce a consensus sequence which, albeit be hypothetical, it might be of some use to understand the generic behaviour of all the sequences in your sample.
Here's a cartoon describing linkage and its importance,
For a multiple sequence alignment of course AB and ab are amino acid residues within the same locus. PSSM could be used to construct a matrix to analyses an established alignment, but that's a fairly complicated Phylogenetics calculation (and a separate question).