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Parikh vectors
7SZE_1 8BER_1 6YCM_1 Letter Amino acid
10 4 3 C Cysteine
16 35 31 G Glycine
18 10 20 I Isoleucine
15 2 6 K Lycine
22 16 32 P Proline
15 16 27 R Arginine
9 8 10 W Tryptophan
15 5 13 Y Tyrosine
30 15 30 V Valine
6 8 14 F Phenylalanine
17 12 11 N Asparagine
22 18 26 D Aspartic acid
16 2 9 Q Glutamine
28 32 35 L Leucine
9 3 8 M Methionine
14 9 42 A Alanine
17 23 37 E Glutamic acid
14 4 13 H Histidine
22 46 20 S Serine
19 22 22 T Threonine 

7SZE_1|Chains A, B, C|SxtDIOX|Microseira wollei (467598)
>8BER_1|Chains A, B, C|Core protein,Matrix protein 2,External core antigen|Hepatitis B virus (10407)
>6YCM_1|Chain A|Aromatic O-demethylase, cytochrome P450 subunit|Amycolatopsis sp. (strain ATCC 39116 / 75iv2) (385957)
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 \(w=f(c)\)
7SZE , Knot 148 334 0.85 40 217 326 
MTTADLILINNWYVVAKVEDCRPGSITTAHLLGVKLVLWRSHEQNSPIQVWQDYCPHRGVPLSMGEVANNTLVCPYHGWRYNQAGKCVQIPAHPDMVPPASAQAKTYHCQERYGLVWVCLGNPVNDIPSFPEWDDPNYHKTYTKSYLIQASPFRVMDNSIDVSHFPFIHEGILGDRNHAEVEDLEVKVDKDGLTMGKYQVHTSKFNNSTKDDSMVNWFRLSHPLCQYCSTEASEMRTVDLMVVTPIDEDNSVLRYLIMWNGSKTLESKILADYDQVIEEDIRILHSQQPTRLPLLSPKQINTQGLPQEIHVPSDRCTVAYRRWLKELGVTYGVC
8BER , Knot 93 290 0.60 40 134 186 
MDPYKEFGATVELLSFLPSDFFPSVRDLLDTASALYREALESPEHCSPHHTALRQAILCWGELMTLATWVGGNLEDGSGTSGSSGSGSGGSGSGGGGELSLLTEVETPIRNEWGSRSNDSSDDLSLLTEVETPIRNEWGSRSNDSSDDLSLLTEVETPIRNEWGSRSNDSSDDLSLLTEVETPIRNEWGSRSNDSSDDIGTSGSSGSGSGGSGSGGGGGPSRDLVVSYVNTNMGLKFRQLLWFHISCLTFGRETVIEYLVSFGVWIRTPPAYRPPNAPILSTLPETTVVC
6YCM , Knot 168 409 0.82 40 219 390 
GPMTTTERPDLAWLDEVTMTQLERNPYEVYERLRAEAPLAFVPVLGSYVASTAEVCREVATSPDFEAVITPAGGRTFGHPAIIGVNGDIHADLRSMVEPALQPAEVDRWIDDLVRPIARRYLERFENDGHAELVAQYCEPVSVRSLGDLLGLQEVDSDKLREWFAKLNRSFTNAAVDENGEFANPEGFAEGDQAKAEIRAVVDPLIDKWIEHPDDSAISHWLHDGMPPGQTRDREYIYPTIYVYLLGAMQEPGHGMASTLVGLFSRPEQLEEVVDDPTLIPRAIAEGLRWTSPIWSASARISTKPVTIAGVDLPAGTPVMLSYGSANHDTGKYEAPSQYDLHRPPLPHLAFGAGNHACAGIYFANHVMRIALEELFEAIPNLERDTREGVEFWGWGFRGPTSLHVTWEV

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 \(c\).

\(|P_{f(7SZE_1)}(2) \setminus P_{f(8BER_1)}(2)|=131\), \(|P_{f(8BER_1)}(2) \setminus P_{f(7SZE_1)}(2)|=48\). 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 1:1001011110010111010000110100101111011110000000110110000100111101101100011010011000011001011101011111010100000000011111011011001101101001000000000011010110110001010011110011110000101001010100011011000100001000000001101101001100000001001001011110110000011001111010001000111000011000101100001001111010010001110010110000011000110011100110
Pair \(Z_2\) Length of longest common subsequence
7SZE_1,8BER_1 179 3
7SZE_1,6YCM_1 168 3
8BER_1,6YCM_1 163 3

Newick tree

 
[
	7SZE_1:88.48,
	[
		6YCM_1:81.5,8BER_1:81.5
	]:6.98
]

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 \((0.85)(0.8)(\frac{624 }{\log_{20} 624}-\frac{290}{\log_{20}290})=93.3\)
Status Protein1 Protein2 d d1/2
Query variables 7SZE_1 8BER_1 130 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]