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Parikh vectors
3SEN_1 6KOX_1 1IGO_1 Letter Amino acid
7 5 8 D Aspartic acid
7 6 7 Q Glutamine
9 6 24 G Glycine
11 1 3 H Histidine
4 0 6 W Tryptophan
8 7 28 T Threonine
15 2 9 A Alanine
3 4 5 R Arginine
4 2 15 N Asparagine
12 7 11 I Isoleucine
5 1 4 M Methionine
10 6 7 E Glutamic acid
14 7 13 K Lycine
7 3 7 P Proline
6 4 9 V Valine
0 0 1 C Cysteine
17 9 10 L Leucine
3 2 7 F Phenylalanine
8 3 18 S Serine
7 1 13 Y Tyrosine 

3SEN_1|Chains A, B, C, D|Caskin-1|Homo sapiens (9606)
>6KOX_1|Chain A|Polyubiquitin-B|Homo sapiens (9606)
>1IGO_1|Chains A, B|family 11 xylanase|Bacillus subtilis (1423)
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 \(w=f(c)\)
3SEN , Knot 72 157 0.77 38 111 148 
MKHHHHHHHMHAKEGKSSEAVSQWLTAFQLQLYAPNFISAGYDLPTISRMTPEDLTAIGVTKPGHRKKIAAEISGLSIPDWLPEHKPANLAVWLSMIGLAQYYKVLVDNGYENIDFITDITWEDLQEIGITKLGHQKKLMLAVRKLAELQKAEYAKY
6KOX , Knot 40 76 0.76 36 61 74 
MQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGG
1IGO , Knot 92 205 0.79 40 139 198 
ATTITSNQTGTHDGYDYELWKDSGNTSMTLNSGGAFSAQWSNIGNALFRKGKKFDSTKTHSQLGNISINYNATFNPGGNSYLCVYGWTKDPLTEYYIVDNWGTYRPTGTPKGTFTVDGGTYDIYETTRINQPSIIGIATFKQYWSVRQTKRTSGTVSVSEHFKKWESLGMPMGKMYETALTVEGYQSNGSANVTANVLTIGGKPL

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 \(c\).

\(|P_{f(3SEN_1)}(2) \setminus P_{f(6KOX_1)}(2)|=90\), \(|P_{f(6KOX_1)}(2) \setminus P_{f(3SEN_1)}(2)|=40\). 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 1:1000000001010010000110011011010101101101100110100101001011110011000011101011011011100011011111011111000011100100010110010100100111001100001111100110100100100
Pair \(Z_2\) Length of longest common subsequence
3SEN_1,6KOX_1 130 3
3SEN_1,1IGO_1 154 3
6KOX_1,1IGO_1 154 3

Newick tree

 
[
	1IGO_1:80.60,
	[
		3SEN_1:65,6KOX_1:65
	]:15.60
]

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 \((0.85)(0.8)(\frac{233 }{\log_{20} 233}-\frac{76}{\log_{20}76})=51.3\)
Status Protein1 Protein2 d d1/2
Query variables 3SEN_1 6KOX_1 63 47.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]