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Parikh vectors
3BRJ_1 5UCF_1 5ZSG_1 Letter Amino acid
6 0 37 R Arginine
11 3 70 N Asparagine
1 0 19 C Cysteine
10 1 37 Q Glutamine
10 6 51 K Lycine
7 0 38 P Proline
12 0 78 S Serine
2 11 42 T Threonine
7 6 24 A Alanine
17 5 39 D Aspartic acid
23 2 134 L Leucine
3 1 11 M Methionine
0 1 8 W Tryptophan
12 5 40 E Glutamic acid
10 4 25 G Glycine
3 0 19 H Histidine
18 2 42 I Isoleucine
8 4 44 F Phenylalanine
3 2 27 Y Tyrosine
12 3 38 V Valine 

3BRJ_1|Chains A, B, C, D|Mannitol operon repressor|Vibrio parahaemolyticus RIMD 2210633 (223926)
>5UCF_1|Chain A|Immunoglobulin G-binding protein G|Streptococcus sp. GX7805 (1325)
>5ZSG_1|Chains A, B|Toll-like receptor 7|Macaca mulatta (9544)
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 \(w=f(c)\)
3BRJ , Knot 80 175 0.78 38 122 165 
SNAMADNINESEIIERLNSAPSVRGFFIATVDVFNESIDGLIQRIFRKDNFAVQSVVGPLLQDSGPLGDLSVRLKLLFGLGVLPDDIYHDIEDIIKLKNHLNSDASDYEFTDPNILEPIKKLHLVKKMGMVQLEVNEPDDDIDLEFYQLQLQRQQQIIKSGLSLAIVEICNELGK
5UCF , Knot 33 56 0.79 30 46 54 
MTFKAIINGKTLKGETTTEAVDAATAEKVFKQYFNDNGLDGEWTYDDATKTFTVTE
5ZSG , Knot 299 823 0.81 40 278 687 
RSPWARWFPKTLPCDVTLDVSKNHVIVDCTDKHLTEIPGGIPTNTTNLTLTINHIPDISPASFHRLVHLVEIDFRCNCVPIRLGSKSNMCPRRLQIKPRSFSGLTYLKSLYLDGNQLLEIPQGLPPSLQLLSLEANNIFSIRKEQLTELANIEILYLGQNCYYRNPCYVSYSIEKDAFLNLTKLKVLSLKDNNVTTVPTVLPSTLTELYLYNNMIAEIQEDDFNNLNQLQILDLSGNCPRCYNAPFPCTPCKNNSPLQIPVNAFDALTELKVLRLHSNSLQHVPPRWFKNINNLQELDLSQNFLAKEIGDAKFLHFLPNLIQLDLSFNFELQVYRASMNLSQAFSSLKSLKILRIRGYVFKELKSFQLSPLHNLQNLEVLDLGTNFIKIANLSMFKQFKRLKVIDLSVNKISPSGDSLVPRGSSNARTSVESYEPQVLEQLYYFRYDKYARSCRFKNKEASFTSVQESCYKYGQTLDLSKNSIFFIKSSDFQHLSFLKCLNLSGNLISQTLNGSEFQPLAELRYLDFSNNRLDLLHSTAFEELRKLEVLDISSNSHYFQSEGITHMLNFTKNLKVLQKLMMNDNDISSSTSRTMESESLRTLEFRGNHLDVLWRDGDNRYLQLFKNLLKLEELDISKNSLSFLPSGVFDGMPPNLKNLSLAKNGLKSFIWEKLRYLKNLETLDLSHNQLTTVPERLSNCSRSLKNLILKNNQIRSLTKYFLQDAFQLRYLDLSSNKIQMIQKTSFPENVLNNLKMLLLHHNRFLCTCDAVWFVWWVQHTEVTIPYLATDVTCVGPGAHKGQSVISLDLYTCELDLTNEFLVPR

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 \(c\).

\(|P_{f(3BRJ_1)}(2) \setminus P_{f(5UCF_1)}(2)|=105\), \(|P_{f(5UCF_1)}(2) \setminus P_{f(3BRJ_1)}(2)|=29\). 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 1:0011100100001100100110101111101011000101110011000011100111111000111101010101111111110010001001101000100010000100101101100101100111101010010001010100101000001100110111101000110
Pair \(Z_2\) Length of longest common subsequence
3BRJ_1,5UCF_1 134 2
3BRJ_1,5ZSG_1 210 4
5UCF_1,5ZSG_1 254 3

Newick tree

 
[
	5ZSG_1:12.86,
	[
		3BRJ_1:67,5UCF_1:67
	]:61.86
]

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 \((0.85)(0.8)(\frac{231 }{\log_{20} 231}-\frac{56}{\log_{20}56})=58.1\)
Status Protein1 Protein2 d d1/2
Query variables 3BRJ_1 5UCF_1 73 48.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]