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Parikh vectors
4WLQ_1 3DBV_1 6OYY_1 Letter Amino acid
16 25 1 G Glycine
4 2 0 W Tryptophan
6 8 1 Y Tyrosine
15 15 2 R Arginine
13 20 0 N Asparagine
24 23 2 K Lycine
12 19 3 T Threonine
36 23 0 E Glutamic acid
29 25 3 L Leucine
20 19 1 S Serine
21 40 2 V Valine
14 19 1 D Aspartic acid
21 20 1 I Isoleucine
23 3 0 Q Glutamine
7 9 2 H Histidine
9 7 2 M Methionine
17 5 0 F Phenylalanine
12 10 0 P Proline
24 40 2 A Alanine
5 2 1 C Cysteine 

4WLQ_1|Chain A|Ubiquitin carboxyl-terminal hydrolase isozyme L5|Mus musculus (10090)
>3DBV_1|Chains A[auth O], B[auth P], C[auth Q], D[auth R]|GLYCERALDEHYDE-3-PHOSPHATE DEHYDROGENASE|Geobacillus stearothermophilus (1422)
>6OYY_1|Chains A, C, E, G, I, K|Aspartate 1-decarboxylase beta chain|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
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 \(w=f(c)\)
4WLQ , Knot 148 328 0.87 40 207 320 
MSSNAGEWCLMESDPGVFTELIKGFGCRGAQVEEIWSLEPESFEKLKPVHGLIFLFKWQPGEEPAGSVVQDSRLETIFFAKQVINNACATQAIVSVLLNCTHQDVHLGETLSEFKEFSQSFDAAMKGLALSNSDVIRQVHNSFARQQMFEFDTKTPAKEEDAFHFVSYVPVNGRLYELDGLREGPIDLGACNQDDWITAVRPVIEKRIQKYSEGEIRFNLMAIVSDRKMIYEQKIAELQRQLAEEPMDTDQGSTVLSAIQSEVARNQMLIEEEVQKLKRYKIENIRRKHNYLPFIMELLKTLAEHQQLIPLVEKAKEKQNAKKAQETK
3DBV , Knot 138 334 0.80 40 183 317 
AVKVGINGFGRIGRNVFRAALKNPDIEVVAVNDTGGANTLAHLLKYDSVHGRLDAEVSVNGNNLVVNGKEIIVKAERDPENLAWGEIGVDIVVESTGRFTKREDAAKHLEAGAKKVIISAPAKNEDITIVMGVNQDKYDPKAHHVISNASCTTNCLAPFAKVLHEQFGIVRGMMTTVHSYTNDQRILDASHKDLRRARAAAESIIPTTTGAAKAVALVLPELKGKLNGMAMRVPTPNVSVVDLVAELEKEVTVEEVNAALKAAAEGELKGILAYSEEPLVSRDYNGSTVSSTIDALSTMVIDGKMVKVVSWYDNETGYSHRVVDLAAYIASKGL
6OYY , Knot 15 24 0.66 28 22 22 
MLRTMLKSKIHRATVTCADLHYVG

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 \(c\).

\(|P_{f(4WLQ_1)}(2) \setminus P_{f(3DBV_1)}(2)|=95\), \(|P_{f(3DBV_1)}(2) \setminus P_{f(4WLQ_1)}(2)|=71\). 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 1:1000110101100011110011011100110100110101001001011011111101011001110110000100111100110010100111011100000010110010010010001011101111000011001000110001101000011000011011001110101001011001110111000001101101110001000001010101111100001100001101000110011000010011011000110001110001001000010010000001111101100110000111110010000010010000
Pair \(Z_2\) Length of longest common subsequence
4WLQ_1,3DBV_1 166 4
4WLQ_1,6OYY_1 211 2
3DBV_1,6OYY_1 183 2

Newick tree

 
[
	6OYY_1:10.46,
	[
		4WLQ_1:83,3DBV_1:83
	]:20.46
]

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 \((0.85)(0.8)(\frac{662 }{\log_{20} 662}-\frac{328}{\log_{20}328})=92.2\)
Status Protein1 Protein2 d d1/2
Query variables 4WLQ_1 3DBV_1 116 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]