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Parikh vectors
8XJK_1 8IKU_1 8IIZ_1 Letter Amino acid
16 20 56 A Alanine
17 20 32 I Isoleucine
18 33 39 L Leucine
6 15 29 P Proline
15 17 18 S Serine
4 0 0 C Cysteine
17 12 37 G Glycine
17 34 46 K Lycine
14 15 23 F Phenylalanine
14 23 31 T Threonine
3 3 10 W Tryptophan
14 19 32 V Valine
20 4 11 R Arginine
13 12 26 N Asparagine
20 19 29 D Aspartic acid
4 5 7 H Histidine
7 17 15 Q Glutamine
16 27 37 E Glutamic acid
5 1 12 M Methionine
6 14 24 Y Tyrosine 

8XJK_1|Chain A|Engineered miniGq|synthetic construct (32630)
>8IKU_1|Chain A|NADPH-dependent alpha-keto amide reductase|Kluyveromyces marxianus (4911)
>8IIZ_1|Chain A|Maltodextrin-binding protein,YEATS domain-containing protein 4|Escherichia coli (562)
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 \(w=f(c)\)
8XJK , Knot 109 246 0.81 40 167 234 
MGSTVSAEDKAAAERSKMIDKNLREDGEKARRTLRLLLLGADNSGKSTIVKQMRILHGGSGGSGGTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRKEFVDISTASGDGRHICYPHFTCAVDTENARRIFNDCKDIILQMNLREYNLV
8IKU , Knot 135 310 0.83 38 177 295 
MTNQKFFTLSNGNKIPAVAVVGTGTKWYKAEETDATFSQELTDIVKLSLDTVPGIVHIDAAETYKTYPELGAALKETKKPREEIFITDKFSSLHKISEDPKSALETALNKLGVDYVDLYLIHSPFFDKDLNIDLETAWKQLEELYKSGKAKNIGVSNFTVEDLKKVLAIAEIKPQVNQIEFSPFLQNQTPGIVEFSQKNDILLEAYSPLGPLQKKPTDADQQPFYQYLKELSEKYNKTEAQVLLLWVYKRGILPVTTSAKIERIKQAQDIFSFDLTEEEVKKITDLGLQHEPVRLYHVDFYTKYNSEAQK
8IIZ , Knot 204 514 0.82 38 246 473 
GSMKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPAAAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYAAGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSAVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTNAAAVTIVKPIVYGNVARYFGKKREEDGHTHQWTVYVKPYRNEDMSAYVKKIQFKLHESYGNPLRVVTKPPYEITETGWGEFEIIIKIFFIDPNERPVTLYHLLKLFQSDTNAMLGKKTVVSEFYDEMIFQDPTAMMQQLLTTSR

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 \(c\).

\(|P_{f(8XJK_1)}(2) \setminus P_{f(8IKU_1)}(2)|=77\), \(|P_{f(8IKU_1)}(2) \setminus P_{f(8XJK_1)}(2)|=87\). 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 1:110010100011100001100010001001000101111110001000110010110110110110011100010100101011011100000001100100101111110000000100110010011000110010111110000111001111000100011011000010010101100101001001100011010010101001001010011000010011000001110101000011
Pair \(Z_2\) Length of longest common subsequence
8XJK_1,8IKU_1 164 4
8XJK_1,8IIZ_1 183 4
8IKU_1,8IIZ_1 153 4

Newick tree

 
[
	8XJK_1:90.07,
	[
		8IKU_1:76.5,8IIZ_1:76.5
	]:13.57
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{246}{\log_{20}246})=88.1\)
Status Protein1 Protein2 d d1/2
Query variables 8XJK_1 8IKU_1 110 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]