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Parikh vectors
2XXW_1 3SXY_1 7BBT_1 Letter Amino acid
14 15 3 R Arginine
9 1 2 Q Glutamine
17 10 4 S Serine
4 1 1 W Tryptophan
8 9 7 N Asparagine
2 5 4 H Histidine
9 2 4 P Proline
18 8 8 T Threonine
14 17 3 V Valine
12 8 8 A Alanine
3 1 2 C Cysteine
18 31 7 E Glutamic acid
22 11 12 G Glycine
20 20 4 I Isoleucine
12 6 5 Y Tyrosine
16 13 4 D Aspartic acid
23 26 8 L Leucine
21 21 16 K Lycine
9 5 2 M Methionine
10 8 4 F Phenylalanine 

2XXW_1|Chains A, B|GLUTAMATE RECEPTOR, IONOTROPIC KAINATE 2|RATTUS NORVEGICUS (10116)
>3SXY_1|Chains A, B|Transcriptional regulator, GntR family|Thermotoga maritima (2336)
>7BBT_1|Chains A, B, C, D|Cytochrome c iso-1|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
2XXW , Knot 116 261 0.82 40 172 244 
GSNRSLIVTTILEEPYVLFKKSDKPLYGNDRFEGYCIDLLRELSTILGFTYEIRLVEDGKYGAQDDVNGQWNGMVRELIDHKADLAVAPLAITYVREKVIDFSKPFMTLGISILYRKGTPIDSADDLAKQTKIEYGAVEDGATMTFFKKSKISTYDKMWAFMSSRRQSVLVKSNEEGIQRVLTSDYAFLMESTTIEFVTQRNCNLTQIGGLIDSKGYGVGTPMGSPYRKKITIAILQLQEEGKLHMMKEKWWRGNGCPEPR
3SXY , Knot 99 218 0.81 40 136 206 
GAMGMKKIEVDLVRTKVYNLLKEMILNHELKLGEKLNVRELSEKLGISFTPVRDALLQLATEGLVKVVPRVGFFVTDVDEKFIRETIETRIMMEVFCLENYFDKIAGSEELLEIKGEIDDVEKSAKREIFDDSDERLHKLFIRASGNELIISLYEKIWDRIDLVRHLNERYVVSNREHKELIERIISGDKEGAIEKLKEHLKNVEAETIKNLYTYERS
7BBT , Knot 58 108 0.83 40 92 105 
AEFKAGSAKKGATLFKTRCLQCHTVEKGGPHKVGPNLHGIFGRHSGQAEGYSYTDANIKKNVLWDENNMSEYLTNPKKYIPGTKMAFGGLKKEKDRNDLITYLKKATE

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 \(c\).

\(|P_{f(2XXW_1)}(2) \setminus P_{f(3SXY_1)}(2)|=95\), \(|P_{f(3SXY_1)}(2) \setminus P_{f(2XXW_1)}(2)|=59\). 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 1:100001110011001011100000110100010100101100100111100010110010011000101010111001100010111111110010001101001110111011000101100100110000100111001101011000010000011111000000111000001100110000111100001011000000100111110001011101110100001011110100010101100011010101010
Pair \(Z_2\) Length of longest common subsequence
2XXW_1,3SXY_1 154 4
2XXW_1,7BBT_1 178 3
3SXY_1,7BBT_1 140 3

Newick tree

 
[
	2XXW_1:87.17,
	[
		3SXY_1:70,7BBT_1:70
	]:17.17
]

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 \((0.85)(0.8)(\frac{479 }{\log_{20} 479}-\frac{218}{\log_{20}218})=75.6\)
Status Protein1 Protein2 d d1/2
Query variables 2XXW_1 3SXY_1 95 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]