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Parikh vectors
2LEA_1 8USC_1 1HBN_1 Letter Amino acid
15 20 49 G Glycine
9 19 43 V Valine
4 8 25 F Phenylalanine
13 14 22 S Serine
0 2 6 W Tryptophan
1 12 24 I Isoleucine
9 5 14 M Methionine
1 7 22 N Asparagine
0 2 5 C Cysteine
5 18 40 E Glutamic acid
10 4 13 H Histidine
6 20 34 L Leucine
4 14 23 K Lycine
7 23 62 A Alanine
16 11 22 R Arginine
6 13 35 T Threonine
8 10 24 P Proline
6 14 28 Y Tyrosine
12 7 34 D Aspartic acid
3 11 24 Q Glutamine 

2LEA_1|Chain A|Serine/arginine-rich splicing factor 2|Homo sapiens (9606)
>8USC_1|Chain A[auth H]|Proteasome subunit alpha type-2|Homo sapiens (9606)
>1HBN_1|Chains A, D|METHYL-COENZYME M REDUCTASE I ALPHA SUBUNIT|METHANOTHERMOBACTER THERMAUTOTROPHICUS (145262)
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 \(w=f(c)\)
2LEA , Knot 61 135 0.73 36 92 120 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMSYGRPPPDVEGMTSLKVDNLTYRTSPDTLRRVFEKYGRVGDVYIPRDRYTKESRGFAFVRFHDKRDAEDAMDAMDGAVLDGRELRVQMARYGRPPDSHHS
8USC , Knot 106 234 0.82 40 160 226 
MAERGYSFSLTTFSPSGKLVQIEYALAAVAGGAPSVGIKAANGVVLATEKKQKSILYDERSVHKVEPITKHIGLVYSGMGPDYRVLVHRARKLAQQYYLVYQEPIPTAQLVQRVASVMQEYTQSGGVRPFGVSLLICGWNEGRPYLFQSDPSGAYFAWKATAMGKNYVNGKTFLEKRYNEDLELEDAIHTAILTLKESFEGQMTEDNIEVGICNEAGFRRLTPTEVKDYLAAIA
1HBN , Knot 219 549 0.83 40 257 509 
ADKLFINALKKKFEESPEEKKTTFYTLGGWKQSERKTEFVNAGKEVAAKRGIPQYNPDIGTPLGQRVLMPYQVSTTDTYVEGDDLHFVNNAAMQQMWDDIRRTVIVGLNHAHAVIEKRLGKEVTPETITHYLETVNHAMPGAAVVQEHMVETHPALVADSYVKVFTGNDEIADEIDPAFVIDINKQFPEDQAETLKAEVGDGIWQVVRIPTIVSRTCDGATTSRWSAMQIGMSMISAYKQAAGEAATGDFAYAAKHAEVIHMGTYLPVRRARGENEPGGVPFGYLADICQSSRVNYEDPVRVSLDVVATGAMLYDQIWLGSYMSGGVGFTQYATAAYTDNILDDFTYFGKEYVEDKYGLCEAPNNMDTVLDVATEVTFYGLEQYEEYPALLEDQFGGSQRAAVVAAAAGCSTAFATGNAQTGLSGWYLSMYLHKEQHSRLGFYGYDLQDQCGASNVFSIRGDEGLPLELRGPNYPNYAMNVGHQGEYAGISQAPHAARGDAFVFNPLVKIAFADDNLVFDFTNVRGEFAKGALREFEPAGERALITPAK

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 \(c\).

\(|P_{f(2LEA_1)}(2) \setminus P_{f(8USC_1)}(2)|=40\), \(|P_{f(8USC_1)}(2) \setminus P_{f(2LEA_1)}(2)|=108\). 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 1:110000000000111101001101011001101010010111010110010100100000100100110001011010110000000001111101000001001101101111010010101100101100000
Pair \(Z_2\) Length of longest common subsequence
2LEA_1,8USC_1 148 3
2LEA_1,1HBN_1 205 3
8USC_1,1HBN_1 173 4

Newick tree

 
[
	1HBN_1:10.83,
	[
		2LEA_1:74,8USC_1:74
	]:26.83
]

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 \((0.85)(0.8)(\frac{369 }{\log_{20} 369}-\frac{135}{\log_{20}135})=71.1\)
Status Protein1 Protein2 d d1/2
Query variables 2LEA_1 8USC_1 92 69
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]