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Parikh vectors
6DGZ_1 5MVD_1 3LKB_1 Letter Amino acid
1 7 14 Y Tyrosine
6 34 32 V Valine
4 31 22 R Arginine
2 4 6 M Methionine
8 18 20 T Threonine
13 40 34 G Glycine
1 8 14 H Histidine
13 12 21 I Isoleucine
2 1 3 W Tryptophan
3 29 43 A Alanine
4 9 18 N Asparagine
4 22 28 E Glutamic acid
6 19 21 P Proline
4 20 15 D Aspartic acid
2 0 2 C Cysteine
5 16 16 Q Glutamine
0 19 14 S Serine
12 45 30 L Leucine
7 18 20 K Lycine
2 11 19 F Phenylalanine 

6DGZ_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
>5MVD_1|Chain A|Dihydroorotate dehydrogenase (quinone), mitochondrial|Homo sapiens (9606)
>3LKB_1|Chains A, B|Probable branched-chain amino acid ABC transporter, amino acid binding protein|Thermus thermophilus (300852)
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 \(w=f(c)\)
6DGZ , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF
5MVD , Knot 150 363 0.81 38 189 338 
DERFYAEHLMPTLQGLLDPESAHRLAVRFTSLGLLPRARFQDSDMLEVRVLGHKFRNPVGIAAGFDKHGEAVDGLYKMGFGFVEIGSVTPKPQEGNPRPRVFRLPEDQAVINRYGFNSHGLSVVEHRLRARQQKQAKLTEDGLPLGVNLGKNKTSVDAAEDYAEGVRVLGPLADYLVVNVSSPNTAGLRSLQGKAELRRLLTKVLQERDGLRRVHRPAVLVKIAPDLTSQDKEDIASVVKELGIDGLIVTNTTVSRPAGLQGALRSETGGLSGKPLRDLSTQTIREMYALTQGRVPIIGVGGVSSGQDALEKIRAGASLVQLYTALTFWGPPVVGKVKRELEALLKEQGFGGVTDAIGADHRR
3LKB , Knot 169 392 0.85 40 220 375 
MSLGQQQVTLFWSGAITGPTSDAGAPYGAAVEDYCKWANERKLVPGVVFNCVVRDDQYNNANTQRFFEEAVDRFKIPVFLSYATGANLQLKPLIQELRIPTIPASMHIELIDPPNNDYIFLPTTSYSEQVVALLEYIAREKKGAKVALVVHPSPFGRAPVEDARKAARELGLQIVDVQEVGSGNLDNTALLKRFEQAGVEYVVHQNVAGPVANILKDAKRLGLKMRHLGAHYTGGPDLIALAGDAAEGFLWATSFYMAHEDTPGIRLQKEIGRKYGRPENFIESVNYTNGMLAAAIAVEAIRRAQERFKRITNETVYQAIVGMNGPNAFKPGFAVSTKQGVEIDFTKSEHTGAEGLRILEAKGGRFVPVTEPFTSALFRKVHYGEGHHHHHH

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 \(c\).

\(|P_{f(6DGZ_1)}(2) \setminus P_{f(5MVD_1)}(2)|=25\), \(|P_{f(5MVD_1)}(2) \setminus P_{f(6DGZ_1)}(2)|=135\). 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 1:101011001110101110100111001100011001011101010111111111010000011101010011101111101101110011001100101
Pair \(Z_2\) Length of longest common subsequence
6DGZ_1,5MVD_1 160 4
6DGZ_1,3LKB_1 183 3
5MVD_1,3LKB_1 139 4

Newick tree

 
[
	6DGZ_1:90.76,
	[
		5MVD_1:69.5,3LKB_1:69.5
	]:21.26
]

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 \((0.85)(0.8)(\frac{462 }{\log_{20} 462}-\frac{99}{\log_{20}99})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 6DGZ_1 5MVD_1 134 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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