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Parikh vectors
5VPL_1 1URH_1 4MWT_1 Letter Amino acid
16 25 25 V Valine
12 10 32 N Asparagine
7 1 9 C Cysteine
4 8 7 W Tryptophan
2 7 19 K Lycine
14 15 14 R Arginine
10 24 18 E Glutamic acid
10 28 48 L Leucine
7 7 14 H Histidine
18 14 11 I Isoleucine
4 4 10 M Methionine
9 19 24 P Proline
13 12 15 T Threonine
14 31 23 A Alanine
11 20 22 D Aspartic acid
17 6 24 Q Glutamine
17 4 27 Y Tyrosine
19 23 31 G Glycine
3 11 22 F Phenylalanine
16 11 33 S Serine 

5VPL_1|Chain A|Der f 1 variant|Dermatophagoides farinae (6954)
>1URH_1|Chains A, B|3-MERCAPTOPYRUVATE SULFURTRANSFERASE|ESCHERICHIA COLI (562)
>4MWT_1|Chains A, B|Lysosomal protective protein|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 \(w=f(c)\)
5VPL , Knot 102 223 0.82 40 157 214 
TSACRINSVNVPSELDLRSLRTVTPIRMQGGCGSCWAFSGVAATESAYLAYRNTSLDLSEQELVDCASQHGCHGDTIPRGIEYIQQNGVVEERSYPYVAREQQCRRPNSQHYGISNYCQIYPPDVKQIREALTQTHTAIAVIIGIKDLRAFQHYDGRTIIQHDNGYQPNYHAVNIVGYGSTQGVDYWIVRNSWDTTWGDSGYGYFQAGNNLMMIEQYPYVVIM
1URH , Knot 123 280 0.82 40 169 263 
STTWFVGADWLAEHIDDPEIQIIDARMASPGQEDRNVAQEYLNGHIPGAVFFDIEALSDHTSPLPHMLPRPETFAVAMRELGVNQDKHLIVYDEGNLFSAPRAWWMLRTFGVEKVSILGGGLAGWQRDDLLLEEGAVELPEGEFNAAFNPEAVVKVTDVLLASHENTAQIIDARPAARFNAEVDEPRPGLRRGHIPGALNVPWTELVREGELKTTDELDAIFFGRGVSYDKPIIVSCGSGVTAAVVLLALATLDVPNVKLYDGAWSEWGARADLPVEPVK
4MWT , Knot 179 428 0.84 40 234 403 
APDQDEIQRLPGLAKQPSFRQYSGYLKGSGSKHLHYWFVESQKDPENSPVVLWLNGGPGCSSLDGLLTEHGPFLVQPDGVTLEYNPYSWNLIANVLYLESPAGVGFSYSDDKFYATNDTEVAQSNFEALQDFFRLFPEYKNNKLFLTGESYAGIYIPTLAVLVMQDPSMNLQGLAVGNGLSSYEQNDNSLVYFAYYHGLLGNRLWSSLQTHCCSQNKCNFYDNKDLECVTNLQEVARIVGNSGLNIYNLYAPCAGGVPSHFRSGDKVRMDPPCTNTTAASTYLNNPYVRKALNIPEQLPQWDMCNFLVNLQYRRLYRSMNSQYLKLLSSQKYQILLYNGDVDMACNFMGDEWFVDSLNQKMEVQRRPWLVKYGDSGEQIAGFVKEFSHIAFLTIKGAGHMVPTDKPLAAFTMFSRFLNKQPYHHHHHH

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 \(c\).

\(|P_{f(5VPL_1)}(2) \setminus P_{f(1URH_1)}(2)|=82\), \(|P_{f(1URH_1)}(2) \setminus P_{f(5VPL_1)}(2)|=94\). 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 1:0010010010110010100100101101011010011101111000101100000101000011001000100100110110010001110000010110000000100000110000010110100100110000011111111001011000010011000010010001101110100011001110001000110010101011001111000101111
Pair \(Z_2\) Length of longest common subsequence
5VPL_1,1URH_1 176 3
5VPL_1,4MWT_1 187 4
1URH_1,4MWT_1 183 3

Newick tree

 
[
	4MWT_1:93.95,
	[
		5VPL_1:88,1URH_1:88
	]:5.95
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{223}{\log_{20}223})=80.7\)
Status Protein1 Protein2 d d1/2
Query variables 5VPL_1 1URH_1 102 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]