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Parikh vectors
4PTG_1 7DTJ_1 6INY_1 Letter Amino acid
6 3 15 M Methionine
30 11 28 T Threonine
34 7 41 A Alanine
18 11 16 N Asparagine
38 17 57 L Leucine
19 8 21 F Phenylalanine
27 16 40 R Arginine
22 7 33 D Aspartic acid
20 10 27 E Glutamic acid
15 13 16 H Histidine
20 15 12 I Isoleucine
29 12 22 V Valine
9 5 2 C Cysteine
19 12 23 Q Glutamine
31 4 22 P Proline
36 6 35 S Serine
2 0 12 W Tryptophan
19 8 17 Y Tyrosine
24 9 35 G Glycine
23 8 8 K Lycine 

4PTG_1|Chains A, B|Glycogen synthase kinase-3 beta|Homo sapiens (9606)
>7DTJ_1|Chain A|ATP-dependent RNA helicase cgh-1|Caenorhabditis elegans (6239)
>6INY_1|Chain A|UPF0061 protein YdiU|Escherichia coli (562)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
4PTG , Knot 180 441 0.82 40 237 409 
MHSSHHHHHHSSGENLYFQGHMSGRPRTTSFAESCKPVQQPSAFGSMKVSRDKDGSKVTTVVATPGQGPDRPQEVSYTDTKVIGNGSFGVVYQAKLCDSGELVAIKKVLQDKRFKNRELQIMRKLDHCNIVRLRYFFYSSGEKKDEVYLNLVLDYVPETVYRVARHYSRAKQTLPVIYVKLYMYQLFRSLAYIHSFGICHRDIKPQNLLLDPDTAVLKLCDFGSAKQLVRGEPNVSYICSRYYRAPELIFGATDYTSSIDVWSAGCVLAELLLGQPIFPGDSGVDQLVEIIKVLGTPTREQIREMNPNYTEFKFPQIKAHPWTKVFRPRTPPEAIALCSRLLEYTPTARLTPLEACAHSFFDELRDPNVKLPNGRDTPALFNFTTQELSSNPPLATILIPPHARIQAAASTPTNATAASDANTGDRGQTNNAASASASNST
7DTJ , Knot 84 182 0.80 38 143 175 
MGHHHHHHMEELTLLGVTQYYAFVQEKQKVHCLNTLFRKLQINQSIIFCNSTQRVELLAKKITEIGYSCYYIHSKMAQNHRNRVFHDFRQGNCRNLVCSDLLTRGIDIQAVNVVINFDFPRNAETYLHRIGRSGRFGHLGVAINLITYEDRHTLRRIEQELRTRIEPIPKTVDPKLYVADQQ
6INY , Knot 195 482 0.83 40 241 450 
GPGSMTLSFITRWRDELPATYTTLSPTPLNNARLIWHNAELANTLGIPSSLFKNGAGVWGGETLLPGMSPLAQVYSGHQFGVWAGQLGDGRGILLGEQRLADGTTMDWHLKGAGLTPYSRMGDGRAVLRSTIRESLASEAMHYLGIPTTRALSIVTSDSPVYRETVEPGAMLMRVAPSHLRFGHFEHFYYRREPEKVRQLADFAIRHYWSHLADDEDKYRLWFTDVVARTASLIAQWQTVGFAHGVMNTDNMSLLGLTLDYGPFGFLDDYEPGFICNHSDHQGRYSFDNQPAVALWNLQRLAQTLSPFVAVDALNEALDSYQQVLLTHYGQRMRQKLGFMTEQKEDNALLNELFSLMARERSDYTRTFRMLSLTEQHSAASPLRDEFIDRAAFDDWFARYRGRLQQDEVSDSERQQLMQSVNPALVLRNWLAQRAIEAAEKGDMTELHRLHEALRNPFSDRDDDYVSRPPDWGKRLEVSCSS

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(4PTG_1)}(2) \setminus P_{f(7DTJ_1)}(2)|=136\), \(|P_{f(7DTJ_1)}(2) \setminus P_{f(4PTG_1)}(2)|=42\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:100000000000100101010101010000110000110010111010100000100100111011011001001000000111010111100101000101111001100001000010110010000110100110001000001010111001100100110000010001111010101001100110100111000010100111010011101001101001101010100100000011011111000000101101101110111101111100110011011011101000010010100001011010101100110100110111100011000101010110101001100100101011010001111010000100011110111110101011100100101100100100100001101010000
Pair \(Z_2\) Length of longest common subsequence
4PTG_1,7DTJ_1 178 6
4PTG_1,6INY_1 160 3
7DTJ_1,6INY_1 212 4

Newick tree

 
[
	7DTJ_1:10.14,
	[
		4PTG_1:80,6INY_1:80
	]:23.14
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{623 }{\log_{20} 623}-\frac{182}{\log_{20}182})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 4PTG_1 7DTJ_1 159 111.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]