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Parikh vectors
1KLQ_1 5XTK_1 3VNZ_1 Letter Amino acid
19 12 45 L Leucine
19 15 22 V Valine
1 3 16 H Histidine
17 12 13 I Isoleucine
13 8 17 K Lycine
1 4 8 M Methionine
3 4 7 W Tryptophan
7 14 68 A Alanine
9 9 23 R Arginine
17 7 21 E Glutamic acid
7 12 53 G Glycine
9 4 23 F Phenylalanine
6 8 29 P Proline
15 8 35 T Threonine
7 3 19 N Asparagine
3 1 1 C Cysteine
17 4 25 S Serine
7 5 20 Y Tyrosine
11 13 25 D Aspartic acid
9 6 18 Q Glutamine 

1KLQ_1|Chain A|MITOTIC SPINDLE ASSEMBLY CHECKPOINT PROTEIN MAD2A|Homo sapiens (9606)
>5XTK_1|Chains A, B|Xanthine phosphoribosyltransferase|Yersinia pestis (632)
>3VNZ_1|Chain A|beta-GLUCURONIDASE|Acidobacterium capsulatum (240015)
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 \(w=f(c)\)
1KLQ , Knot 90 197 0.80 40 135 192 
GSITLRGSAEIVAEFFSFGINSILYQRGIYPSETFTRVQKYGLTLLVTTDLELIKYLNNVVEQLKDWLYKCSVQKLVVVISNIESGEVLERWQFDIECDKTAKDDSAPREKSQKAIQDEIRSVIRQITATVTFLPLLEVSCSFDLLIYTDKDLVVPEKWEESGPQFITNSEEVRLRSFTTTIHKVNSMVAYKIPVND
5XTK , Knot 74 152 0.81 40 116 145 
MNEKYVVTWDMLQIHARKLAQRLLPAEQWKGIIAVSRGGLVPAGILARELGIRYVDTVCISSYDHDNQRDLKVLKRAEGDGEGFIVIDDLVDTGGTATAIREMYPKAHFVTIFAKPAGRPLVDDYVVDIPQNTWIEQPWDMAVTFVAPLSGK
3VNZ , Knot 198 488 0.83 40 239 455 
MAFARGGLAQTASQTTSSPVRVGLSVDASALGHTIPPDYTGLSYEQAQMANPNYFSGANTQLAGFLRTLGRQGVLRIGGNTSEYTFWNRHAKPTAADEHLAAGPDKGHHAAAREVITPEAVNNLSEFLDKTGWKLIYGLNLGKGTPENAADEAAYVMETIGADRLLAFQLGNEPDLFYRNGIRPASYDFAAYAGDWQRFFTAIRKRVPNAPFAGPDTAYNTKWLVPFADKFKHDVKFISSHYYAEGPPTDPSMTIERLMKPNPRLLGETAGLKQVEADTGLPFRLTETNSCYQGGKQGVSDTFAAALWAGDLMYQQAAAGSTGINFHGGGYGWYTPVAGTPEDGFIARPEYYGMLLFAQAGAGQLLGAKLTDNSAAPLLTAYALRGTDGRTRIALFNKNLDADVEVAISGVASPSGTVLRLEAPRADDTTDVTFGGAPVGASGSWSPLVQEYVPGHSGQFVLHMRKASGALLEFAKLAAALQHHHHHH

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 \(c\).

\(|P_{f(1KLQ_1)}(2) \setminus P_{f(5XTK_1)}(2)|=85\), \(|P_{f(5XTK_1)}(2) \setminus P_{f(1KLQ_1)}(2)|=66\). 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 1:10101010101110110111001100011010001001000110111000101100100110010011000010011111001001011001010100000100001100000011000100110010101011111010001011100000111100100011011000001010010001001001110011100
Pair \(Z_2\) Length of longest common subsequence
1KLQ_1,5XTK_1 151 3
1KLQ_1,3VNZ_1 190 3
5XTK_1,3VNZ_1 195 4

Newick tree

 
[
	3VNZ_1:10.24,
	[
		1KLQ_1:75.5,5XTK_1:75.5
	]:26.74
]

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 \((0.85)(0.8)(\frac{349 }{\log_{20} 349}-\frac{152}{\log_{20}152})=59.7\)
Status Protein1 Protein2 d d1/2
Query variables 1KLQ_1 5XTK_1 76 66.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]