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Parikh vectors
5PQH_1 7YAS_1 6BYW_1 Letter Amino acid
5 17 38 T Threonine
0 5 18 W Tryptophan
14 4 28 R Arginine
4 6 59 N Asparagine
1 5 6 C Cysteine
7 17 47 G Glycine
10 8 16 H Histidine
6 13 30 F Phenylalanine
10 14 50 V Valine
13 14 56 D Aspartic acid
6 23 38 K Lycine
13 24 45 E Glutamic acid
18 30 80 L Leucine
7 3 15 M Methionine
5 12 28 Y Tyrosine
11 18 57 A Alanine
8 7 52 Q Glutamine
5 17 39 I Isoleucine
6 12 46 P Proline
7 8 68 S Serine 

5PQH_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>7YAS_1|Chain A|PROTEIN (HYDROXYNITRILE LYASE)|Hevea brasiliensis (3981)
>6BYW_1|Chains A, B, C, D|GoxA|Pseudoalteromonas luteoviolacea DSM 6061 (1365250)
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 \(w=f(c)\)
5PQH , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
7YAS , Knot 117 257 0.84 40 173 246 
MAFAHFVLIHTICHGAWIWHKLKPLLEALGHKVTALDLAASGVDPRQIEEIGSFDEYSEPLLTFLEALPPGEKVILVGESCGGLNIAIAADKYCEKIAAAVFHNSVLPDTEHCPSYVVDKLMEVFPDWKDTTYFTYTKDGKEITGLKLGFTLLRENLYTLCGPEEYELAKMLTRKGSLFQNILAKRPFFTKEGYGSIKKIYVWTDQDEIFLPEFQLWQIENYKPDKVYKVEGGDHKLQLTKTKEIAEILQEVADTYN
6BYW , Knot 311 816 0.85 40 295 739 
MSNCQYKIYPPLGIARVGNGPAIKPLSLSTPEVPWAHLYDTNVQYLVTQQELEQLLEEAFGGNVINEISQIKTKLDERKAEKFKQEEIETITGLLGLSHLVPQQQLSRSLDNLELKSTKDSDDIVQQIKGALLKVLSDHYLHAVKKQAQNFYIYKCDEQGNPVEKLKLTDGDKVTWRVEVANKKSFWYDYNNALDLSLHTQGSGNLSKNVSKHRLAPAMTAKRRNPNVITNSLRKQLVISSQGSVSSDNNTQVPLRGKFPANEPDTNNRLSDLLNLQERHNVLQGSIECDNEGVLRFYAGNGISQALSPSSLNTDFADNSNWFDDICDGRVTAVVELKNGDTFEIQDEQSSAWVATTPPDYAPQIEPIVTMYDMVSGAALKEQDLDNLTTQFSDVFPILYRLYRMQWVNQADFTDNAVNTQIRELNSELGFAQLLDNSASAKSLREGIFNQFRNPLFDQDIDVDDPGQSSNEWVSNSRIIPSKDETNIAAKPATSSLKLPFYPNDGIDYPGSPVQWFAIPPFMYQHLQNWAAGDFSVTQVEKESANTIEELGLFYSEQFKNSPNSALLCARGALDALYGGGFHPGVELTWPMRHNLIYSQNDYVSSVTPEINLLGLREFRLKQDLQGLNSPNMYQDFGHVIAVDNVTASIDPNSDAAWLWRSTPGDLTKWMGIPWQSDAASCQAVYTPEDFPIPSWWAANLPVHVLPLARYNKFKDSQSADLPEINGMTHSIAQGMSEETFEHLRLEQFSQRLDWLHTADLGFVGYHAEGGYTNGLIQMVSQWKNMAMVMARPVENPGSSGIPNVVYVAYSQADKD

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 \(c\).

\(|P_{f(5PQH_1)}(2) \setminus P_{f(7YAS_1)}(2)|=56\), \(|P_{f(7YAS_1)}(2) \setminus P_{f(5PQH_1)}(2)|=109\). 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 1:100000000110110001010010011101010010011001100100001101110110100110010010011011010001010100010010001011100010001000110011101000111110010001001110010110110011
Pair \(Z_2\) Length of longest common subsequence
5PQH_1,7YAS_1 165 4
5PQH_1,6BYW_1 219 4
7YAS_1,6BYW_1 194 4

Newick tree

 
[
	6BYW_1:10.53,
	[
		5PQH_1:82.5,7YAS_1:82.5
	]:27.03
]

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 \((0.85)(0.8)(\frac{413 }{\log_{20} 413}-\frac{156}{\log_{20}156})=76.7\)
Status Protein1 Protein2 d d1/2
Query variables 5PQH_1 7YAS_1 97 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]