Sequence alignment using Markov Model I am learning about applying Markov model to sequence alignment. The prof says that the transition probabilities from a gap-residue alignment to a residue-gap alignment and vice versa are both 0. Is there any biological/mathematical reason behind this statement? Why are the (X,Y) and (Y,X) cell 0? This is a lecture slide of lecture 1, week 4 of the "Bioinformatics: Introdcution and Methods" course on coursera.
If I understand your question correctly, then I think for case of pairwise alignment, there is a simple explanation.

I believe the key insight is that: a mismatch should always score better than a gap.*

This follows biologically since the insertion/deletion (indel) rate is roughly 1/10th that of the substitution rate (i.e. the occurrence of single nucleotide changes), at least in vertebrates. (This varies across the tree of life but I think the substitution rate virtually always exceeds the indel rate.)

To understand why this matters, consider an example:

ATG-AG
ATGT-G


This is an 'impossible alignment' under the probabilities you gave since here we have a transition from a gap-residue alignment to a residue-gap.

However, under our assumption that mismatches are more likely biologically than indels, the correct alignment should be:

ATGAG
ATGTG


Indeed, the latter does look like a better alignment.

This also follows for more complex examples, so this:

ATG--AAG
ATGTT-AG


Becomes this:

ATG-AAG
ATGTTAG


(Or this:

ATGA-AG
ATGTTAG


)

* Strictly, I mean a substitution should score better than an indel (with the associated gap opening and extension penalties). In fact, for the assumption to always be true, a run of mismatches should still score worse than a single indel. This may not always be a correct assumption, consider this example below, is the true alignment case 1) or 2) or something else? Or is in fact a global alignment bad here and this should be split into 2 local alignments? Is there a likely biological mutational event that could explain this? I ask these questions just to point out it is not black-and-white, I don't have clear answers

1)

CGTACGTAGAGGAATGCCCCCCCCC--------AGCAACGTAGCAT     
CGTACGTAGAGGAATG---------TTTTTTTTAGCAACGTAGCAT


2)

CGTACGTAGAGGAATGCCCCCCCCCAGCAACGTAGCAT     
CGTACGTAGAGGAATGTTTTTTTT-AGCAACGTAGCAT