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Parikh vectors
5PIY_1 7NBH_1 6LMQ_1 Letter Amino acid
18 24 8 S Serine
20 20 11 T Threonine
19 9 4 Y Tyrosine
25 33 14 A Alanine
18 10 4 R Arginine
20 13 5 N Asparagine
26 20 11 E Glutamic acid
27 22 17 G Glycine
7 6 3 C Cysteine
15 15 10 Q Glutamine
24 29 8 L Leucine
23 21 13 K Lycine
19 7 4 F Phenylalanine
11 5 3 M Methionine
14 36 14 V Valine
14 16 6 D Aspartic acid
18 12 7 H Histidine
20 25 15 I Isoleucine
18 20 5 P Proline
8 3 4 W Tryptophan 

5PIY_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
>7NBH_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|Serine racemase|Homo sapiens (9606)
>6LMQ_1|Chain A|Integrase catalytic|Human immunodeficiency virus type 1 group M subtype B (isolate NY5) (11698)
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 \(w=f(c)\)
5PIY , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR
7NBH , Knot 146 346 0.82 40 198 323 
MDAQYDISFADVEKAHINIRDSIHLTPVLTSSILNQLTGRNLFFKCELFQKTGSFKIRGALNAVRSLVPDALERKPKAVVTHSSGNHGQALTYAAKLEGIPAYIVVPQTAPDCKKLAIQAYGASIVYCEPSDESRENVAKRVTEETEGIMVHPNQEPAVIAGQGTIALEVLNQVPLVDALVVPVGGGGMLAGIAITVKALKPSVKVYAAEPSNADDCYQSKLKGKLMPNLYPPETIADGVKSSIGLNTWPIIRDLVDDIFTVTEDEIKCATQLVWERMKLLIEPTAGVGVAAVLSQHFQTVSPEVKNICIVLSGGNVDLTSSITWVKQAERPASYQSVSVHHHHHH
6LMQ , Knot 78 166 0.80 40 126 163 
GSHMHGQVDCSPGIWQLDCTHLEGKVILVAVHVASGYIEAEVIPAETGQETAYFLLKLAGRWPVKTVHTDNGSNFTSTTVKAACWWAGIKQEFGIPYNPQSQGVIESMNKELKKIIGQVRDQAEHLKTAVQMAVFIHNKKRKGGIGGYSAGERIVDIIATDIQTKE

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 \(c\).

\(|P_{f(5PIY_1)}(2) \setminus P_{f(7NBH_1)}(2)|=105\), \(|P_{f(7NBH_1)}(2) \setminus P_{f(5PIY_1)}(2)|=74\). 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 1:1000000001101100010100100100010010010001111010000100100011010001100111101111001010000001001111011001101011110000000011011000011000000011000100100001000100011011010101100000010110110100110000111101100101011110001110000101001001011010010111100100100110011110001011110001111010110001111001000110111011010011100110010110110101100101100000101010101011101101000011001000
Pair \(Z_2\) Length of longest common subsequence
5PIY_1,7NBH_1 179 6
5PIY_1,6LMQ_1 197 4
7NBH_1,6LMQ_1 168 4

Newick tree

 
[
	5PIY_1:97.24,
	[
		7NBH_1:84,6LMQ_1:84
	]:13.24
]

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 \((0.85)(0.8)(\frac{710 }{\log_{20} 710}-\frac{346}{\log_{20}346})=99.7\)
Status Protein1 Protein2 d d1/2
Query variables 5PIY_1 7NBH_1 126 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]