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Parikh vectors
2ZHI_1 6FWX_1 7MWX_1 Letter Amino acid
26 13 13 D Aspartic acid
6 0 17 C Cysteine
33 29 17 L Leucine
21 10 6 K Lycine
23 46 17 S Serine
18 40 26 T Threonine
2 2 7 H Histidine
21 16 7 I Isoleucine
24 22 8 E Glutamic acid
10 3 4 M Methionine
1 2 8 W Tryptophan
24 19 20 R Arginine
14 30 4 Q Glutamine
26 32 20 G Glycine
10 16 11 F Phenylalanine
4 22 20 P Proline
12 10 13 Y Tyrosine
15 38 14 V Valine
22 34 12 A Alanine
10 10 15 N Asparagine 

2ZHI_1|Chain A|Annexin A4|Rattus norvegicus (10116)
>6FWX_1|Chains A, B|Titin,Titin,Titin|Homo sapiens (9606)
>7MWX_1|Chains A[auth C], B[auth G]|Core protein precursor eE2|Hepacivirus C (3052230)
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 \(w=f(c)\)
2ZHI , Knot 132 322 0.79 40 182 295 
GIPMETKGGTVKAASGFNATEDAQVLRKAMKGLGTDEDAIIGVLACRNTAQRQEIRTAYKSTIGRDLLEDLKSELSSNFEQVILGMMTPTVLYDVQELRRAMKGAGTDEGCLIEILASRNPEEIRRINQTYQQQYGRSLEEDICSDTSFMFQRVLVSLTAGGRDEGNYLDDALVKQDAQDLYEAGEKRWGTDEVKFLSILCSRNRNHLLHVFDEYKRISQKDIEQSIKSETSGSFEDALLAIVKCMRNKPAYFAERLYKSMKGLGTDDSTLIRVMVSRAEIDMLDIRANFKRLYGKSLYSFIKGDTSGDYRKVLLILCGGDD
6FWX , Knot 89 394 0.45 38 122 187 
MTTQAPTFTQPLQSVVVLEGSTATFEAHISGFPVPEVSWFRDGQVISTSTLPGVQISFSDGRAKLTIPAVTKANSGRYSLKATNGSGQATSTAELLVKAETAPPNFVQRLQSMTVRQGSQVRLQVRVTGIPTPVVKFYRDGAEIQSSLDFQISQEGDLYSLLIAEAYPEDSGTYSVNATNSVGRATSTAELLVQGETTQAPTFTQPLQSVVVLEGSTATFEAHISGFPVPEVSWFRDGQVISSGRGDSSPGVQISFSDGRAKLTIPAVTKANSGRYSLKATNGSGQATSTAELLVKAETAPPNFVQRLQSMTVRQGSQVRLQVRVTGIPTPVVKFYRDGAEIQSSLDFQISQEGDLYSLLIAEAYPEDSGTYSVNATNSVGRATSTAELLVQGE
7MWX , Knot 118 259 0.84 40 170 250 
GPRQKIQLVNTNGSWHINRTALNCNDSLHTGFIASLFYTHSFNSSGCPERMSACRSIEAFRVGWGALQYEDNVTNPEDMRPYCWHYPPRQCGVVSAKTVCGPVYCFTPSPVVVGTTDRLGAPTYTWGENETDVFLLNSTRPPLGSWFGCTWMNSSGYTKTCGAPPCRTRADFNASTDLLCPTDCFRKHPDTTYLKCGSGPWLTPRCLIDYPYRLWHYPCTVNYTIFKIRMYVGGVEHRLTAACNFTRGDRCNLEDRDRS

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 \(c\).

\(|P_{f(2ZHI_1)}(2) \setminus P_{f(6FWX_1)}(2)|=108\), \(|P_{f(6FWX_1)}(2) \setminus P_{f(2ZHI_1)}(2)|=48\). 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 1:1111000110101101101000101100110111000011111110000100001001000011001100100010001001111110101100100100110111000101101110001001001000000001001000100000111001110101110001001001110001001001100011000101101100000001101100000100001000100000101001111110010001101100100010111000001101110010101101010100101001001101000100001111101100
Pair \(Z_2\) Length of longest common subsequence
2ZHI_1,6FWX_1 156 3
2ZHI_1,7MWX_1 188 3
6FWX_1,7MWX_1 158 4

Newick tree

 
[
	7MWX_1:89.57,
	[
		2ZHI_1:78,6FWX_1:78
	]:11.57
]

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 \((0.85)(0.8)(\frac{716 }{\log_{20} 716}-\frac{322}{\log_{20}322})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZHI_1 6FWX_1 115 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]