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Parikh vectors
6YCA_1 5WGL_1 4CUP_1 Letter Amino acid
33 27 4 G Glycine
11 21 2 H Histidine
19 21 4 I Isoleucine
25 18 6 P Proline
9 4 2 W Tryptophan
0 7 2 C Cysteine
31 16 7 T Threonine
12 11 3 Y Tyrosine
60 32 4 A Alanine
38 22 5 R Arginine
18 21 8 E Glutamic acid
39 37 11 L Leucine
12 14 7 F Phenylalanine
24 35 9 S Serine
5 17 5 N Asparagine
27 17 11 D Aspartic acid
13 8 1 Q Glutamine
2 8 14 K Lycine
1 11 5 M Methionine
36 17 7 V Valine 

6YCA_1|Chains A, B, C, D, E, F|Uncharacterized N-acetyltransferase D2E76_00625|Mycobacteroides abscessus (36809)
>5WGL_1|Chains A, B|Hdac6 protein|Danio rerio (7955)
>4CUP_1|Chain A|BROMODOMAIN ADJACENT TO ZINC FINGER DOMAIN PROTEIN 2B|HOMO SAPIENS (9606)
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 \(w=f(c)\)
6YCA , Knot 161 415 0.78 38 185 368 
GAFTAQPQWPGASRRNNTGQEAVTDIRFLQSRAEHERAFTVFWRAMVGLPALGAVAADELLELGRYLGAFVQGELIGGADSYTSWLTVPGGSRVPHAAVTHIGVLPTHTRRGILTALVTRQLTDIAGRGEIVASLRASEAVIYRRFGYGIATSSATYRIQRRRAAPLRPIDTGAIALLDAAASPEGLAAIYERAAWTGSVARPPQWWRLHELFDAADPVKPYVVTHPDGYVRYRPQDTAEWFSSSARTISVDDLVAHSDEAYRALVGHLLDLDLVDVIELGPRPIDDPLPHLVTDPRAVAVAGIRDETWLRLVDVEAALAARTYTDGAPVVIEVQDTLLPHNAARFSVSSDKVRRTQHTPDISVDVAALGSVYLGGNTWTRLERAGLVSAQSPGAIRAADALFSTGTQPFAGTNF
5WGL , Knot 156 364 0.84 40 210 351 
SNAGGSSPITGLVYDQRMMLHHNMWDSHHPELPQRISRIFSRHEELRLLSRCHRIPARLATEEELALCHSSKHISIIKSSEHMKPRDLNRLGDEYNSIFISNESYTCALLAAGSCFNSAQAILTGQVRNAVAIVRPPGHHAEKDTACGFCFFNTAALTARYAQSITRESLRVLIVDWDVHHGNGTQHIFEEDDSVLYISLHRYEDGAFFPNSEDANYDKVGLGKGRGYNVNIPWNGGKMGDPEYMAAFHHLVMPIAREFAPELVLVSAGFDAARGDPLGGFQVTPEGYAHLTHQLMSLAAGRVLIILEGGYNLTSISESMSMCTSMLLGDSPPSLDHLTPLKTSATVSINNVLRAHAPFWSSLR
4CUP , Knot 59 117 0.80 40 98 112 
SMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS

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 \(c\).

\(|P_{f(6YCA_1)}(2) \setminus P_{f(5WGL_1)}(2)|=71\), \(|P_{f(5WGL_1)}(2) \setminus P_{f(6YCA_1)}(2)|=96\). 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 1:1110101011110000001001100101100010000110111011111111111110011011001111101011111000001101111001101110011111000001110111000100111010111010100111000110111000100010000111101100111111011101011111000111010110110110100110110110101100101010001000101100010010100111000010011110110101101101110110011101100101111111000011011010111110000011111101000111001101010000100000010101011111010111001001001111010011110110111001001111001
Pair \(Z_2\) Length of longest common subsequence
6YCA_1,5WGL_1 167 4
6YCA_1,4CUP_1 175 3
5WGL_1,4CUP_1 196 4

Newick tree

 
[
	4CUP_1:95.82,
	[
		6YCA_1:83.5,5WGL_1:83.5
	]:12.32
]

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 \((0.85)(0.8)(\frac{779 }{\log_{20} 779}-\frac{364}{\log_{20}364})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 6YCA_1 5WGL_1 136 132.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]