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Parikh vectors
7XYE_1 3TKC_1 3LLI_1 Letter Amino acid
19 18 9 I Isoleucine
13 8 17 F Phenylalanine
28 14 17 P Proline
29 10 4 T Threonine
9 1 8 W Tryptophan
43 15 22 V Valine
32 17 13 E Glutamic acid
26 9 11 H Histidine
3 6 6 C Cysteine
7 14 16 K Lycine
60 12 29 A Alanine
36 13 17 R Arginine
8 8 6 Y Tyrosine
52 11 11 G Glycine
12 8 3 M Methionine
11 14 10 Q Glutamine
52 34 25 L Leucine
32 24 16 S Serine
9 7 11 N Asparagine
31 16 10 D Aspartic acid 

7XYE_1|Chains A, B, C, D, E, F, G, H|N-formimidoyl fortimicin A synthase|Streptomyces luteocolor (285500)
>3TKC_1|Chain A|Vitamin D3 receptor|Homo sapiens (9606)
>3LLI_1|Chain A|Sulfhydryl oxidase 1|Homo sapiens (9606)
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 \(w=f(c)\)
7XYE , Knot 205 512 0.83 40 233 475 
MGSSHHHHHHSSGLVPRGSHMMSKTADAPGQTDVIVVGNGVLGLSVGVEIARTRPDVRVTLLGKPARQYGATPAAGAMLGAFGEVTAHALASEHGRKKHALAVQAQRLWPEWIESLEATGTAADGRIKTADDTVVLLNTVGHSALDDANFAAVLTALKEANAPHEEIAVESVDWIDPDPNSRPLRALHIEGEGSVDSGILLAALERSFLQAGGRLHPVDATEIRASHGRVEGVVTDDGDFLPAGHVVVAAGARSQRLVAALPGLAHRIPRIYDGVGVSALVDTWDGSGPATVLRTSNRAFACGLHLVPRAGGSVYIGATNAVCLEPRGAASIEETVFLFNCATHQLHRGLNGSELRKVQVGSRPAPIDGFPLIGGTSVEGLWMLSGTYRDGLHMSPLLARHVVSLMDGGTGVDGLREFRPERDLISAWSREEILDDVVRHTMATGYEFPWRLPLEWPHMMETFLQGPFAELADRLSDTYTPPADLMTAIMFSEREQQDELIAYYADVHREWH
3TKC , Knot 114 259 0.81 40 173 247 
DSLRPKLSEEQQRIIAILLDAHHKTYDPTYSDFCQFRPPVRVNDGGGSVTLELSQLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTSEDQIVLLKSSAIEVIMLRSNESFTMDDMSWTCGNQDYKYRVSDVTKAGHSLELIEPLIKFQVGLKKLNLHEEEHVLLMAICIVSPDRPGVQDAALIEAIQDRLSNTLQTYIRCRHPPPGSHLLYAKMIQKLADLRSLNEEHSKQYRCLSFQPECSMKLTPLVLEVFGNEIS
3LLI , Knot 118 261 0.83 40 171 254 
IAPTVWKLADRSKIYMADLESALHYILRIEVGRFPVLEGQRLVALKKFVAVLAKYFPGRPLVQNFLHSVNEWLKRQKRNKIPYSFFKTALDDRKEGAVLAKKVNWIGCQGSEPHFRGFPCSLWVLFHFLTVQAARQNVDHSQEAAKAKEVLPAIRGYVHYFFGCRDCASHFEQMAAASMHRVGSPNAAVLWLWSSHNRVNARLAGAPSEDPQFPKVQWPPRELCSACHNERLDVPVWDVEATLNFLKAHFSPSNIILDFPA

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 \(c\).

\(|P_{f(7XYE_1)}(2) \setminus P_{f(3TKC_1)}(2)|=107\), \(|P_{f(3TKC_1)}(2) \setminus P_{f(7XYE_1)}(2)|=47\). 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 1:11000000000011110100110001011100011111011111011101100010101011101100011011111111111010101110001000011110100111011001010101101010010001111001100110010111110110010110001110010110101000110110101010100111111100011011101011010010100101011100010111110111111100001111111110011010011110111001010111011000001110110111011101011100110101011101000111100100010011010010010110011110111111100101111101000011010111100110110110110110010100011011000011001100011010011101110110110011011110110010000011101101111000000001110010100010
Pair \(Z_2\) Length of longest common subsequence
7XYE_1,3TKC_1 154 4
7XYE_1,3LLI_1 172 5
3TKC_1,3LLI_1 166 4

Newick tree

 
[
	3LLI_1:86.87,
	[
		7XYE_1:77,3TKC_1:77
	]:9.87
]

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 \((0.85)(0.8)(\frac{771 }{\log_{20} 771}-\frac{259}{\log_{20}259})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 7XYE_1 3TKC_1 178 134
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]