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Parikh vectors
6BPR_1 5WGZ_1 1GDV_1 Letter Amino acid
11 49 7 G Glycine
6 10 3 M Methionine
11 58 12 A Alanine
9 31 1 R Arginine
8 23 3 Q Glutamine
5 30 2 Y Tyrosine
4 16 2 C Cysteine
16 40 5 E Glutamic acid
17 61 5 L Leucine
12 33 6 K Lycine
9 34 2 F Phenylalanine
7 41 2 P Proline
14 27 9 N Asparagine
10 36 8 D Aspartic acid
11 11 1 H Histidine
8 32 4 I Isoleucine
14 44 4 S Serine
13 36 3 T Threonine
3 17 1 W Tryptophan
11 38 5 V Valine 

6BPR_1|Chain A|Cysteine dioxygenase type 1|Homo sapiens (9606)
>5WGZ_1|Chain A|Flavin-dependent halogenase|Malbranchea aurantiaca (78605)
>1GDV_1|Chain A|CYTOCHROME C6|Porphyra yezoensis (2788)
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 \(w=f(c)\)
6BPR , Knot 99 200 0.87 42 160 194 
SEQTEVLKPRTLADLIRILHQLFAGDEVNVEEVQAIMEAYESDPTEWAMYAKFDQYRYTRNLVDQGNGKFNLMILCWGEGHGSSIHDHTNSHCFLKMLQGNLKETLFAWPDKKSNEMVKKSERVLRENQCAYINDSVGLHRVENISHTEPAVSLHLXSPPFDTCHAFDQRTGHKNKVTMTFHSKFGIRTPNATSGSLENN
5WGZ , Knot 266 667 0.86 40 303 630 
MAPTPKYTFTERAAAGNLSDAEILNSNNPTGSELPDESDVVVGGAGIHGLIYALHASKYKPNNLKISVIEKNTRPGYKIGESTLPIFYTWCKLHGISAAYLLRLFGLKDGLCFYFLDRENQGQYTDFCSVGAPGLVLASLQIERPMSELLFTILAQRNGVNVYHGREVDFKSTVVQGGGQGNKIAVSRGKYDSTPKTIDSALFVDATGRFRQFCSKKAPRHRFDGWNCNAFWGYFTAPKDESKIPFDLYEGDHTNHLCFPEGWVWVIRLPSWEGSPIANLMDMVTYILECADAGVPGDELPSSEELARMFGLKFQWVTSIGFAVRNDVKYPEDLSAYGTREAEQKFNYFVQKYELLQQFMSNFELIENLYGPGTTWFIRKTLAYQSPVVSGPGWLAIGDACGFTNPLYSPGINVGMSTSTWAAQLSHPIVEIGKSAPADAAESSIRKLLVPYDDYCKSLVPALEQMNRFNYVCYRDTRLGPQVACLWQFFAGIERYLSDVNIETFAHYAIKWVWGAMVPEYQQVAQKCIEHIETVPLDERLPDAMVDELLAFSNRIKSAAVAADDFSLRWDAILRSFDRSLNFVEGKTSRDIYTRQCSGCGAWLQLRPDWKKCHSCGLLGTEPQTAVTFDPPLTAEEEALLYAAWNTAPKYDPSKELKLPTPTRPAA
1GDV , Knot 48 85 0.83 40 72 83 
ADLDNGEKVFSANCAACHAGGNNAIMPDKTLKKDVLEANSMNTIDAITYQVQNGKNAMPAFGGRLVDEDIEDAANYVLSQSEKGW

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 \(c\).

\(|P_{f(6BPR_1)}(2) \setminus P_{f(5WGZ_1)}(2)|=29\), \(|P_{f(5WGZ_1)}(2) \setminus P_{f(6BPR_1)}(2)|=172\). 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 1:00000110100110110110011110010100101110100001001110101000000001100101010111101101010010000000011011010100011111000000110000011000001010001110010010000111010100111000011000010000101010001110010100101000
Pair \(Z_2\) Length of longest common subsequence
6BPR_1,5WGZ_1 201 4
6BPR_1,1GDV_1 164 3
5WGZ_1,1GDV_1 253 3

Newick tree

 
[
	5WGZ_1:12.12,
	[
		6BPR_1:82,1GDV_1:82
	]:41.12
]

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 \((0.85)(0.8)(\frac{867 }{\log_{20} 867}-\frac{200}{\log_{20}200})=184.\)
Status Protein1 Protein2 d d1/2
Query variables 6BPR_1 5WGZ_1 236 150.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]