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Parikh vectors
1ZEB_1 6YMR_1 4FZB_1 Letter Amino acid
13 2 6 M Methionine
4 3 5 W Tryptophan
17 28 7 Y Tyrosine
53 28 21 A Alanine
16 20 10 N Asparagine
31 24 7 D Aspartic acid
38 16 16 L Leucine
20 11 14 K Lycine
44 36 7 G Glycine
14 8 14 H Histidine
17 10 9 F Phenylalanine
33 25 17 T Threonine
31 22 9 V Valine
19 14 11 Q Glutamine
18 18 16 I Isoleucine
27 26 16 S Serine
29 10 12 R Arginine
5 0 3 C Cysteine
29 7 18 E Glutamic acid
26 8 9 P Proline 

1ZEB_1|Chain A|Alkaline phosphatase|Homo sapiens (9606)
>6YMR_1|Chain A[auth E]|Thermolysin|Geobacillus stearothermophilus (1422)
>4FZB_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Probable thymidylate synthase|Paramecium bursaria Chlorella virus 1 (10506)
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 \(w=f(c)\)
1ZEB , Knot 200 484 0.85 40 253 463 
IIPVEEENPDFWNREAAEALGAAKKLQPAQTAAKNLIIFLGDGMGVSTVTAARILKGQKKDKLGPEIPLAMDRFPYVALSKTYNVDKHVPDSGATATAYLCGVKGNFQTIGLSAAARFNQCNTTRGNEVISVMNRAKKAGKSVGVVTTTRVQHASPAGTYAHTVNRNWYSDADVPASARQEGCQDIATQLISNMDIDVILGGGRKYMFRMGTPDPEYPDDYSQGGTRLDGKNLVQEWLAKRQGARYVWNRTELMQASLDPSVTHLMGLFEPGDMKYEIHRDSTLDPSLMEMTEAALRLLSRNPRGFFLFVEGGRIDHGHHESRAYRALTETIMFDDAIERAGQLTSEEDTLSLVTADHSHVFSFGGYPLRGSSIFGLAPGKARDRKAYTVLLYGNGPGYVLKDGARPDVTESESGSPEYRQQSAVPLDEETHAGEDVAVFARGPQAHLVHGVQEQTFIAHVMAFAACLEPYTACDLAPPAGTTD
6YMR , Knot 132 316 0.80 38 179 302 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGNGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSQMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK
4FZB , Knot 106 227 0.84 40 164 218 
MASMTGSAKLISVTKPVVEGVNTAEELIAYAARVSNPENQINNKTASGLLKYCIRHKHWSIFETAFMTLELKTSRGIAAQVLRHRSFHFQEFSQRYASVMETPPPHQARFQDHKNRQNSLDTVPEDDQTWWATEQEKLYAQSMELYNKALEKGIAKECARFILPLSTPTTIYMSGTIRDWIHYIELRTSNGTQREHIDLANACKEIFIKEFPSIAKALDWVHHHHHH

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 \(c\).

\(|P_{f(1ZEB_1)}(2) \setminus P_{f(6YMR_1)}(2)|=116\), \(|P_{f(6YMR_1)}(2) \setminus P_{f(1ZEB_1)}(2)|=42\). 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 1:1111000010110001101111100101100110011111101111001011011010000011101111100110111000001000110011010101011010100111011101000000010011011001001100111100001001011100100100010001011101000100011001100101011111100011011010100100000110010100110011100011001100001101010101001111101101000100000101011010011101100010111111011010010000010011000111001100110100000010110100001101110110100111111101000010011101011101100110101000001010000001111000001100111110110101101100001110111111010100100111111000
Pair \(Z_2\) Length of longest common subsequence
1ZEB_1,6YMR_1 158 5
1ZEB_1,4FZB_1 189 3
6YMR_1,4FZB_1 167 3

Newick tree

 
[
	4FZB_1:92.31,
	[
		1ZEB_1:79,6YMR_1:79
	]:13.31
]

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 \((0.85)(0.8)(\frac{800 }{\log_{20} 800}-\frac{316}{\log_{20}316})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 1ZEB_1 6YMR_1 171 136.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]