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Parikh vectors
3UHL_1 2VAK_1 4AXN_1 Letter Amino acid
7 6 21 K Lycine
0 38 13 S Serine
2 13 7 W Tryptophan
1 15 19 Y Tyrosine
7 27 11 P Proline
11 22 18 V Valine
4 44 40 A Alanine
4 24 8 R Arginine
4 26 27 N Asparagine
4 29 13 Q Glutamine
9 30 17 T Threonine
13 27 27 G Glycine
1 3 3 H Histidine
2 14 8 M Methionine
4 18 13 F Phenylalanine
8 40 25 L Leucine
5 17 22 D Aspartic acid
0 5 0 C Cysteine
4 10 13 E Glutamic acid
9 15 23 I Isoleucine 

3UHL_1|Chains A, B, C, D|Protease|Human immunodeficiency virus 1 (11676)
>2VAK_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|SIGMA A|AVIAN ORTHOREOVIRUS (38170)
>4AXN_1|Chains A, B|CHITINASE C1|SERRATIA MARCESCENS (615)
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 \(w=f(c)\)
3UHL , Knot 52 99 0.80 36 80 95 
PQITLWKRPFVTVKVGGQLKEALLDTGADNTIFEDINLPGRWKPKMVGGIGGFLKVREYDQVPIEIAGHKVIGTVLVGPTPVNVIGRDTMTQIGATLNF
2VAK , Knot 175 423 0.83 40 228 398 
ISEFGSTMARAIYDFFSTPFGNRGLATNRTQLSSLLSSSNSPWQRFLSSMTPLTAPGIVSTPEAPYPGSLMYQESMLHSATVPGVLGSRDAWRTFNVFGLSWTDEGLSGLVAAQDPPPAAPYQPASAQWSDLLNYPRWANRRRELQSKYPLLLRSTLLSAMRAGPVLYVETWPNMISGRLADWFMSQYGNNFVDMCARLTQSCSNMPVEPDGNYDQQMRALISLWLLSYIGVVNQTNTISGFYFSSKTRGQALDSWTLFYTTNTNRVQITQRHFAYVCARSPDWNVDKSWIAAANLTAIVMACRQPPVFANQGVINQAQNRPGFSMNGGTPVHELNLLTTAQECIRQWVMAGLVSAAKGQALTQEANDFSNLIQADLGQIKAQDDALYNQQPGYARRIKPFVNGDWTPGMTAQALAVLATFTA
4AXN , Knot 144 328 0.84 38 194 317 
MSTNNTINAVAADDAAIMPSIANKKILMGFWHNWAAGASDGYQQGQFANMNLTDIPTEYNVVAVAFMKGQGIPTFKPYNLSDTEFRRQVGVLNSQGRAVLISLGGADAHIELKTGDEDKLKDEIIRLVEVYGFDGLDIDLEQAAIGAANNKTVLPAALKKVKDHYAAQGKNFIISMAPEFPYLRTNGTYLDYINALEGYYDFIAPQYYNQGGDGIWVDELNAWITQNNDAMKEDFLYYLTESLVTGTRGYAKIPAAKFVIGLPSNNDAAATGYVVNKQAVYNAFSRLDAKNLSIKGLMTWSINWDNGKSKAGVAYNWEFKTRYAPLIQ

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 \(c\).

\(|P_{f(3UHL_1)}(2) \setminus P_{f(2VAK_1)}(2)|=30\), \(|P_{f(2VAK_1)}(2) \setminus P_{f(3UHL_1)}(2)|=178\). 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 1:101011001110101110100111001100011001011101010111111111010000011101110011101111101101110001001110101
Pair \(Z_2\) Length of longest common subsequence
3UHL_1,2VAK_1 208 3
3UHL_1,4AXN_1 172 3
2VAK_1,4AXN_1 166 4

Newick tree

 
[
	3UHL_1:99.22,
	[
		4AXN_1:83,2VAK_1:83
	]:16.22
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{99}{\log_{20}99})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 3UHL_1 2VAK_1 160 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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