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Parikh vectors
6YZK_1 7CIA_1 4APC_1 Letter Amino acid
19 20 30 S Serine
13 23 19 A Alanine
8 18 27 I Isoleucine
11 18 16 F Phenylalanine
17 10 15 P Proline
13 16 9 T Threonine
7 19 18 R Arginine
9 10 15 N Asparagine
19 21 14 D Aspartic acid
12 8 15 Q Glutamine
22 23 22 G Glycine
23 16 32 K Lycine
2 6 7 M Methionine
7 1 2 W Tryptophan
8 6 15 Y Tyrosine
18 30 20 V Valine
1 0 7 C Cysteine
13 23 23 E Glutamic acid
12 5 12 H Histidine
26 26 32 L Leucine 

6YZK_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>7CIA_1|Chain A|UDP-3-O-acyl-N-acetylglucosamine deacetylase|Pseudomonas aeruginosa PAO1 (208964)
>4APC_1|Chains A, B|SERINE/THREONINE-PROTEIN KINASE NEK1|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 \(w=f(c)\)
6YZK , Knot 112 260 0.79 40 171 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHSFQVTFDDSQDKAVLKGGPLDGTYRLLQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDVGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTEGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLAECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
7CIA , Knot 129 299 0.82 38 180 285 
MIKQRTLKNIIRATGVGLHSGEKVYLTLKPAPVDTGIVFSRTDLDPVVEIPARAENVGETTMSTTLVKGDVKVDTVEHLLSAMAGLGIDNAYVELSASEVPIMDGSAGPFVFLIQSAGLQEQEAAKKFIRIKREVSVEEGDKRAVFVPFDGFKVSFEIDFDHPVFRGRTQQASVDFSSTSFVKEVSRARTFGFMRDIEYLRSQNLALGGSVENAIVVDENRVLNEDGLRYEDEFVKHKILDAIGDLYLLGNSLIGEFRGFKSGHALNNQLLRTLIADKDAWEVVTFEDARTAPISYMRP
4APC , Knot 149 350 0.83 40 208 334 
MHHHHHHSSGVDLGTENLYFQSMEKYVRLQKIGEGSFGKAILVKSTEDGRQYVIKEINISRMSSKEREESRREVAVLANMKHPNIVQYRESFEENGSLYIVMDYCEGGDLFKRINAQKGVLFQEDQILDWFVQICLALKHVHDRKILHRDIKSQNIFLTKDGTVQLGDFGIARVLNSTVELARACIGTPYYLSPEICENKPYNNKSDIWALGCVLYELCTLKHAFEAGSMKNLVLKIISGSFPPVSLHYSYDLRSLVSQLFKRNPRDRPSVNSILEKGFIAKRIEKFLSPQLIAEEFCLKTFSKFGSQPIPAKRPASGQNSISVMPAQKITKPAAKYGIPLAYKKYGDKK

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 \(c\).

\(|P_{f(6YZK_1)}(2) \setminus P_{f(7CIA_1)}(2)|=77\), \(|P_{f(7CIA_1)}(2) \setminus P_{f(6YZK_1)}(2)|=86\). 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 1:10001101000110010001111010000110100001000101011010000100101100100101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
6YZK_1,7CIA_1 163 3
6YZK_1,4APC_1 175 4
7CIA_1,4APC_1 154 4

Newick tree

 
[
	6YZK_1:86.92,
	[
		7CIA_1:77,4APC_1:77
	]:9.92
]

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 \((0.85)(0.8)(\frac{559 }{\log_{20} 559}-\frac{260}{\log_{20}260})=84.7\)
Status Protein1 Protein2 d d1/2
Query variables 6YZK_1 7CIA_1 106 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]