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Parikh vectors
8ZDI_1 9EIV_1 3PZH_1 Letter Amino acid
23 29 21 R Arginine
18 39 19 I Isoleucine
30 51 36 L Leucine
15 42 28 K Lycine
38 23 9 T Threonine
27 26 25 D Aspartic acid
27 55 16 G Glycine
4 19 8 M Methionine
18 32 11 S Serine
4 16 5 W Tryptophan
10 14 22 Y Tyrosine
54 48 16 A Alanine
1 13 4 C Cysteine
6 13 12 H Histidine
12 28 12 F Phenylalanine
16 44 13 P Proline
40 33 25 V Valine
9 27 17 N Asparagine
13 18 12 Q Glutamine
17 52 21 E Glutamic acid 

8ZDI_1|Chains A, B, C, D, E, F, G, H, I|Major Capsid Protein (gp8)|Mycolicibacterium smegmatis MC2 155 (246196)
>9EIV_1|Chain A|Phosphoenolpyruvate carboxykinase, cytosolic [GTP]|Rattus norvegicus (10116)
>3PZH_1|Chain A|Casein kinase II subunit alpha|Zea mays (4577)
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 \(w=f(c)\)
8ZDI , Knot 158 382 0.82 40 187 349 
MAHIFVKPELVAEIGVKQLQREIVLPGLVWTNPLTDFGGSKNDTITVRVPAITTANRRDLRDPDRTVIASELVEHSFGVTLDKHVYAALKFTDEQRTLDIRDYTKQVLMPQVSAVAYELEDYIAELIEGAPYEETILIDPADTVPAFITADQRMGEANVPTDSRRLVVGSAVAAALAKDKQFRHADWSGDQANAALREAHVGRLAGMNVIRSNAIAPDKAYLWHRTAFILAYRTPVVPEGAKAGASFSANGVALRWLADYDYSQLGDRTLLDVFTGRKVVTEVDGSFVRAVELQLQASSITIVGGAFALATTTGTKQLKVRDDNGTDVTARCTFASSAGTKATVSAAGLVTGVAAGTADITASYVPPQGGTAKTATVTVTVP
9EIV , Knot 254 622 0.87 40 281 577 
MPPQLHNGLDFSAKVIQGSLDSLPQEVRKFVEGNAQLCQPEYIHICDGSEEEYGRLLAHMQEEGVIRKLKKYDNCWLALTDPRDVARIESKTVIITQEQRDTVPIPKSGQSQLGRWMSEEDFEKAFNARFPGCMKGRTMYVIPFSMGPLGSPLAKIGIELTDSPYVVASMRIMTRMGTSVLEALGDGEFIKCLHSVGCPLPLKKPLVNNWACNPELTLIAHLPDRREIISFGSGYGGNSLLGKKCFALRIASRLAKEEGWLAEHMLILGITNPEGKKKYLAAAFPSACGKTNLAMMNPTLPGWKVECVGDDIAWMKFDAQGNLRAINPENGFFGVAPGTSVKTNPNAIKTIQKNTIFTNVAETSDGGVYWEGIDEPLAPGVTITSWKNKEWRPQDEEPCAHPNSRFCTPASQCPIIDPAWESPEGVPIEGIIFGGRRPAGVPLVYEALSWQHGVFVGAAMRSEATAAAEHKGKVIMHDPFAMRPFFGYNFGKYLAHWLSMAHRPAAKLPKIFHVNWFRKDKNGKFLWPGFGENSRVLEWMFGRIEGEDSAKLTPIGYVPKEDALNLKGLGDVNVEELFGISKEFWEKEVEEIDKYLEDQVNADLPYEIERELRALKQRISQM
3PZH , Knot 143 332 0.83 40 204 317 
MSKARVYADVNVLRPKEYWDYEALTVQWGEQDDYEVVRKVGRGKYSEVFEGINVNNNEKCIIKILKPVKKKKIKREIKILQNLCGGPNIVKLLDIVRDQHSKTPSLIFEYVNNTDFKVLYPTLTDYDIRYYIYELLKALDYCHSQGIMHRDVKPHNVMIDHELRKLRLIDWGLAEFYHPGKEYNVRVASRYFKGPELLVDLQDYDYSLDMWSLGCMFAGMIFRKEPFFYGHDNHDQLVKIAKVLGTDGLNVYLNKYRIELDPQLEALVGRHSRKPWLKFMNADNQHLVSPEAIDFLDKLLRYDHQERLTALEAMTHPYFQQVRAAENSRTRA

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 \(c\).

\(|P_{f(8ZDI_1)}(2) \setminus P_{f(9EIV_1)}(2)|=37\), \(|P_{f(9EIV_1)}(2) \setminus P_{f(8ZDI_1)}(2)|=131\). 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 1:1101110101110111001000111111110011001110000010101111001000010010001110011000111010001011101000000101000000111101011100100011011011100001110110011111010001101011000001111011111110000100101010010111001011011110110001111001011000111110001111011011101010111101110000001100011011010011001010110110101010010111111111000100010100001001010001100110010101111101111101010100111011010010101011
Pair \(Z_2\) Length of longest common subsequence
8ZDI_1,9EIV_1 168 4
8ZDI_1,3PZH_1 173 5
9EIV_1,3PZH_1 177 4

Newick tree

 
[
	3PZH_1:88.64,
	[
		8ZDI_1:84,9EIV_1:84
	]:4.64
]

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 \((0.85)(0.8)(\frac{1004 }{\log_{20} 1004}-\frac{382}{\log_{20}382})=165.\)
Status Protein1 Protein2 d d1/2
Query variables 8ZDI_1 9EIV_1 214 169
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]