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Parikh vectors
1QCY_1 8JLK_1 3SCN_1 Letter Amino acid
13 30 46 A Alanine
14 27 35 L Leucine
14 22 27 K Lycine
12 20 23 S Serine
15 18 25 T Threonine
8 18 24 F Phenylalanine
1 4 13 W Tryptophan
4 16 35 Y Tyrosine
10 16 31 N Asparagine
11 28 26 D Aspartic acid
16 22 14 E Glutamic acid
4 6 11 H Histidine
3 7 10 M Methionine
10 31 22 R Arginine
1 7 4 C Cysteine
8 19 25 Q Glutamine
3 8 23 P Proline
13 16 39 G Glycine
15 25 19 I Isoleucine
18 21 29 V Valine 

1QCY_1|Chain A|I-DOMAIN OF INTEGRIN ALPHA1BETA1|Homo sapiens (9606)
>8JLK_1|Chain A|mGs|Homo sapiens (9606)
>3SCN_1|Chains A, B|Beta-glucosidase 7|Oryza sativa Japonica Group (39947)
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 \(w=f(c)\)
1QCY , Knot 90 193 0.81 40 135 185 
TQLDIVIVLDGSNSIYPWDSVTAFLNDLLKRMDIGPKQTQVGIVQYGENVTHEFNLNKYSSTEEVLVAAKKIVQRGGAQTMTALGTDTARKEAFTEARGARRGVKKVMVIVTDGESHDNHRLKKVIQDCEDENIQRFSIAILGSYNRGNLSTEKFVEEIKSIASEPTEKHFFNVSDELALVTIVKTLGERIFA
8JLK , Knot 157 361 0.85 40 219 345 
MGCTLSAEDKAAVERSKMIEKQLQKDKQVYRATHRLLLLGADNSGKSTIVKQMRIYHVNGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCSVDTENARRIFNDCRDIIQRMHLRQYELL
3SCN , Knot 198 481 0.84 40 258 459 
AMADVVPKPNWLGGLSRAAFPKRFVFGTVTSAYQVEGMAASGGRGPSIWDAFAHTPGNVAGNQNGDVATDQYHRYKEDVNLMKSLNFDAYRFSISWSRIFPDGEGRVNQEGVAYYNNLINYLLQKGITPYVNLYHYDLPLALEKKYGGWLNAKMADLFTEYADFCFKTFGNRVKHWFTFNEPRIVALLGYDQGTNPPKRCTKCAAGGNSATEPYIVAHNFLLSHAAAVARYRTKYQAAQQGKVGIVLDFNWYEALSNSTEDQAAAQRARDFHIGWYLDPLINGHYPQIMQDLVKDRLPKFTPEQARLVKGSADYIGINQYTASYMKGQQLMQQTPTSYSADWQVTYVFAKNGKPIGPQANSNWLYIVPWGMYGCVNYIKQKYGNPTVVITGNGMDQPANLSRDQYLRDTTRVHFYRSYLTQLKKAIDEGANVAGYFAWSLLDNFEWLSGYTSKFGIVYVDFNTLERHPKASAYWFRDMLKH

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 \(c\).

\(|P_{f(1QCY_1)}(2) \setminus P_{f(8JLK_1)}(2)|=39\), \(|P_{f(8JLK_1)}(2) \setminus P_{f(1QCY_1)}(2)|=123\). 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 1:0010111110100010110010111001100101110000111100100100010100000000111110011001110010111000100011001011001100111110010000000100110000000100101111100001010000110010011001000011010001111011001100111
Pair \(Z_2\) Length of longest common subsequence
1QCY_1,8JLK_1 162 4
1QCY_1,3SCN_1 199 4
8JLK_1,3SCN_1 169 3

Newick tree

 
[
	3SCN_1:95.77,
	[
		1QCY_1:81,8JLK_1:81
	]:14.77
]

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 \((0.85)(0.8)(\frac{554 }{\log_{20} 554}-\frac{193}{\log_{20}193})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 1QCY_1 8JLK_1 130 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]