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Parikh vectors
5XOX_1 1TNM_1 1JPU_1 Letter Amino acid
17 3 31 I Isoleucine
23 6 29 K Lycine
20 3 10 F Phenylalanine
10 5 26 G Glycine
11 7 9 H Histidine
23 4 29 L Leucine
13 3 12 Y Tyrosine
10 3 18 D Aspartic acid
6 1 2 C Cysteine
7 5 6 Q Glutamine
17 9 33 E Glutamic acid
10 9 32 V Valine
3 1 2 W Tryptophan
7 5 10 R Arginine
3 2 2 M Methionine
14 13 16 S Serine
12 12 22 T Threonine
11 4 59 A Alanine
18 2 13 N Asparagine
8 3 9 P Proline 

5XOX_1|Chains A, B, C, D, E, F|tRNA(His) guanylyltransferase|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>1TNM_1|Chain A|TITIN MODULE M5|Homo sapiens (9606)
>1JPU_1|Chain A|glycerol dehydrogenase|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
5XOX , Knot 113 243 0.85 40 161 234 
MANSKFGYVRQFETHDVILPQCYIVVRIDGKKFHEFSKFYEFAKPNDENALKLMNACAKNLVLKYKNDIILAFGESDEYSFILKSSTTLFNRRKDKLATLFGSFFTSNYVALWAKFFPEKPLNIKHLPYFDSRCVAYPNLQTIKDYLSWRYVDTHINNLYNTTFWQLIIKCGLTPQESEKKLCGTFSNEKQEILFSECGINYNNEPEMFKKGSLVTRKGEILHINVIAQIDELFEGYHHHHHH
1TNM , Knot 50 100 0.76 40 80 94 
MHHHHHHSSRILTKPRSMTVYEGESARFSCDTDGEPVPTVTWLRKGQVLSTSARHQVTTTKYKSTFEISSVQASDEGNYSVVVENSEGKQEAEFTLTIQK
1JPU , Knot 153 370 0.81 40 183 338 
MAAERVFISPAKYVQGKNVITKIANYLEGIGNKTVVIADEIVWKIAGHTIVNELKKGNIAAEEVVFSGEASRNEVERIANIARKAEAAIVIGVGGGKTLDTAKAVADELDAYIVIVPTAASTDAPTSALSVIYSDDGVFESYRFYKKNPDLVLVDTKIIANAPPRLLASGIADALATWVEARSVIKSGGKTMAGGIPTIAAEAIAEKCEQTLFKYGKLAYESVKAKVVTPALEAVVEANTLLSGLGFESGGLAAAHAIHNGFTALEGEIHHLTHGEKVAFGTLVQLALEEHSQQEIERYIELYLCLDLPVTLEDIKLKDASREDILKVAKAATAEGETIHNAFNVTADDVADAIFAADQYAKAYKEKHRK

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 \(c\).

\(|P_{f(5XOX_1)}(2) \setminus P_{f(1TNM_1)}(2)|=117\), \(|P_{f(1TNM_1)}(2) \setminus P_{f(5XOX_1)}(2)|=36\). 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 1:110001101001000011110001110101001001001001101000011011010100111000001111110000001110000011000000110111011000011111011100110100110100001101010010001010010001001000011011100110100000010101000000111000110000010110010110001011010111010011010000000
Pair \(Z_2\) Length of longest common subsequence
5XOX_1,1TNM_1 153 6
5XOX_1,1JPU_1 166 3
1TNM_1,1JPU_1 153 3

Newick tree

 
[
	1JPU_1:80.89,
	[
		5XOX_1:76.5,1TNM_1:76.5
	]:4.39
]

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 \((0.85)(0.8)(\frac{343 }{\log_{20} 343}-\frac{100}{\log_{20}100})=75.4\)
Status Protein1 Protein2 d d1/2
Query variables 5XOX_1 1TNM_1 94 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]