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Parikh vectors
4ERQ_1 7AKT_1 6IQY_1 Letter Amino acid
10 1 10 W Tryptophan
21 6 47 V Valine
20 9 38 G Glycine
8 2 16 H Histidine
0 2 12 M Methionine
11 5 40 P Proline
37 9 26 S Serine
17 1 28 N Asparagine
6 6 20 Q Glutamine
11 3 30 F Phenylalanine
27 7 33 L Leucine
33 5 11 K Lycine
21 9 39 T Threonine
9 5 28 Y Tyrosine
22 9 35 D Aspartic acid
10 6 1 C Cysteine
19 4 23 I Isoleucine
19 11 43 A Alanine
3 2 28 R Arginine
8 7 30 E Glutamic acid 

4ERQ_1|Chains A, B, C|WD repeat-containing protein 5|Homo sapiens (9606)
>7AKT_1|Chain A|Ferredoxin, chloroplastic|Chlamydomonas reinhardtii (3055)
>6IQY_1|Chains A, B|Bilirubin oxidase|Myrothecium verrucaria (1859699)
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 \(w=f(c)\)
4ERQ , Knot 122 312 0.74 38 162 268 
ATQSKPTPVKPNYALKFTLAGHTKAVSSVKFSPNGEWLASSSADKLIKIWGAYDGKFEKTISGHKLGISDVAWSSDSNLLVSASDDKTLKIWDVSSGKCLKTLKGHSNYVFCCNFNPQSNLIVSGSFDESVRIWDVKTGKCLKTLPAHSDPVSAVHFNRDGSLIVSSSYDGLCRIWDTASGQCLKTLIDDDNPPVSFVKFSPNGKYILAATLDNTLKLWDYSKGKCLKTYTGHKNEKYCIFANFSVTGGKWIVSGSEDNLVYIWNLQTKEIVQKLQGHTDVVISTACHPTENIIASAALENDKTIKLWKSDC
7AKT , Knot 57 109 0.81 40 91 105 
MASWSHPQFEKIEGRYKVTLKTPSGDKTIECPADTYILDAAEEAGLDLPYSCRGGVCSSCAGKVAAGTVDQSDQSFLDDAQMGNGFVLTCVAYPTSDCTIQTHQEEALY
6IQY , Knot 215 538 0.83 40 256 507 
YVEFVAQISPQYPMFTVPLPIPPVKQPRLTVTNPVNGQEIWYYEVEIKPFTHQVYPDLGSADLVGYDGMSPGPTFQVPRGVETVVRFINNAEAPNSVHLHGSFSRAAFDGWAEDITEPGSFKDYYYPNRQSARTLWYHDHAMHITAENAYRGQAGLYMLTDPAEDALNLPSGYGEFDIPMILTSKQYTANGNLVTTNGELNSFWGDVIHVNGQPWPFKNVEPRKYRFRFLDAAVSRSFGLYFADTDAIDTRLPFKVIASDSGLLEHPADTSLLYISMAERYEVVFDFSDYAGKTIELRNLGGSIGGIGTDTDYDNTDKVMRFVVADDTTQPDTSVVPANLRDVPFPSPTTNTPRQFRFGRTGPTWTINGVAFADVQNRLLANVPVGTVERWELINAGNGWTHPIHIHLVDFKVISRTSGNNARTVMPYESGLKDVVWLGRRETVVVEAHYAPFPGVYMFHCHNLIHEDHDQMAAFNATVLPDYGYNATVFVDPMEELWQARPYELGEFQAQSGQFSVQAVTERIQTMAEYRPYAAADE

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 \(c\).

\(|P_{f(4ERQ_1)}(2) \setminus P_{f(7AKT_1)}(2)|=119\), \(|P_{f(7AKT_1)}(2) \setminus P_{f(4ERQ_1)}(2)|=48\). 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 1:100001011010011010111000110010101010111000100110111100101000101001110011100000111010000010110100100100101000011000101000111010100010110100100100111000110110100010111000001100110010100100110000111011010101001111010001011000010010000100000001110101011011101000011011010000110010100011100100100011101110000010110000
Pair \(Z_2\) Length of longest common subsequence
4ERQ_1,7AKT_1 167 4
4ERQ_1,6IQY_1 202 3
7AKT_1,6IQY_1 213 4

Newick tree

 
[
	6IQY_1:10.71,
	[
		4ERQ_1:83.5,7AKT_1:83.5
	]:26.21
]

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 \((0.85)(0.8)(\frac{421 }{\log_{20} 421}-\frac{109}{\log_{20}109})=94.5\)
Status Protein1 Protein2 d d1/2
Query variables 4ERQ_1 7AKT_1 108 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]