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Parikh vectors
4XRX_1 5WST_1 8XJM_1 Letter Amino acid
5 0 4 C Cysteine
18 3 7 Q Glutamine
33 1 17 G Glycine
10 1 4 H Histidine
30 0 17 I Isoleucine
14 3 5 M Methionine
31 10 16 A Alanine
17 0 13 N Asparagine
17 1 14 F Phenylalanine
24 0 14 T Threonine
17 0 6 Y Tyrosine
14 10 20 R Arginine
29 20 16 E Glutamic acid
7 0 3 W Tryptophan
25 1 20 D Aspartic acid
12 0 6 P Proline
22 2 15 S Serine
25 0 14 V Valine
27 8 18 L Leucine
37 11 17 K Lycine 

4XRX_1|Chains A, B|Isocitrate dehydrogenase [NADP] cytoplasmic|Homo sapiens (9606)
>5WST_1|Chains A, B|Unconventional myosin-VIIa|Mus musculus (10090)
>8XJM_1|Chain A|Engineered miniGq|synthetic construct (32630)
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 \(w=f(c)\)
4XRX , Knot 176 414 0.85 40 234 401 
MSKKISGGSVVEMQGDEMTRIIWELIKEKLIFPYVELDLHSYDLGIENRDATNDQVTKDAAEAIKKHNVGVKCATITPDEKRVEEFKLKQMWKSPNGTIRNILGGTVFREAIICKNIPRLVSGWVKPIIIGHHAYGDQYRATDFVVPGPGKVEITYTPSDGTQKVTYLVHNFEEGGGVAMGMYNQDKSIEDFAHSSFQMALSKGWPLYLSTKNTILKKYDGRFKDIFQEIYDKQYKSQFEAQKIWYEHRLIDDMVAQAMKSEGGFIWACKNYDGDVQSDSVAQGYGSLGMMTSVLVCPDGKTVEAEAAHGTVTRHYRMYQKGQETSTNPIASIFAWTRGLAHRAKLDNNKELAFFANALEEVSIETIEAGFMTKDLAACIKGLPNVQRSDYLNTFEFMDKLGENLKIKLAQAKL
5WST , Knot 34 71 0.68 24 40 63 
GSEFRLEAERMRLAEEEKLRKEMSAKKAKEEAERKHQERLAQLAREDAERELKEKEEARRKKELLEQMEKA
8XJM , Knot 109 246 0.81 40 167 234 
MGSTVSAEDKAAAERSKMIDKNLREDGEKARRTLRLLLLGADNSGKSTIVKQMRILHGGSGGSGGTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRKEFVDISTASGDGRHICYPHFTCAVDTENARRIFNDCKDIILQMNLREYNLV

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 \(c\).

\(|P_{f(4XRX_1)}(2) \setminus P_{f(5WST_1)}(2)|=207\), \(|P_{f(5WST_1)}(2) \setminus P_{f(4XRX_1)}(2)|=13\). 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 1:100010110110101001001110110001111010101000011100001000010001101100001110010101000010010100110010101001111011001110001101101110111110010100001001111111010100010010001001100100111111110000001001100010111001111010000011000010100110010000000010100110000110011101100011111100000101000011010101111001110101001010110101000001000100000011101111001110010100000111110110010100101111000111010111010000010010110011001010110101
Pair \(Z_2\) Length of longest common subsequence
4XRX_1,5WST_1 220 3
4XRX_1,8XJM_1 181 4
5WST_1,8XJM_1 161 3

Newick tree

 
[
	4XRX_1:10.61,
	[
		8XJM_1:80.5,5WST_1:80.5
	]:26.11
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{71}{\log_{20}71})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 4XRX_1 5WST_1 167 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]