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Parikh vectors
6ENV_1 1VBL_1 6WCB_1 Letter Amino acid
12 32 19 D Aspartic acid
28 24 77 L Leucine
2 20 34 P Proline
6 25 10 Y Tyrosine
9 24 7 K Lycine
10 26 32 S Serine
11 34 26 G Glycine
4 28 24 I Isoleucine
3 5 12 M Methionine
15 21 52 A Alanine
6 30 3 N Asparagine
11 14 22 Q Glutamine
15 22 17 E Glutamic acid
6 23 30 T Threonine
1 9 5 W Tryptophan
8 30 41 V Valine
11 15 30 R Arginine
2 0 13 C Cysteine
6 15 16 H Histidine
8 19 34 F Phenylalanine 

6ENV_1|Chain A|Ferritin light chain|Equus caballus (9796)
>1VBL_1|Chain A|pectate lyase 47|Bacillus sp. TS-47 (132676)
>6WCB_1|Chains A, B|Endosomal/lysosomal potassium channel TMEM175|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 \(w=f(c)\)
6ENV , Knot 83 174 0.82 40 127 170 
SSQIRQNYSTEVEAAVNRLVNLYLRASYTYLSLGFYFDRDDVALEGVCHFFRELAEEKREGAERLLKMQNQRGGRALFQDLQKPSQDEWGTTLDAMKAAIVLEKSLNQALLDLHALGSAQADPHLCDFLESHFLDEEVKLIKKMGDHLTNIQRLVGSQAGLGEYLFERLTLKHD
1VBL , Knot 177 416 0.85 38 226 395 
KELGHEVLKPYDGWAAYGEGTTGGAMASPQNVFVVTNRTELIQALGGNNHTNQYNSVPKIIYVKGTIDLNVDDNNQPVGPDFYKDPHFDFEAYLREYDPATWGKKEVEGPLEEARVRSQKKQKDRIMVYVGSNTSIIGVGKDAKIKGGGFLIKNVDNVIIRNIEFEAPLDYFPEWDPTDGTLGEWNSEYDSISIEGSSHIWIDHNTFTDGDHPDRSLGTYFGRPFQQHDGALDIKNSSDFITISYNVFTNHDKVTLIGASDSRMADSGHLRVTLHHNYYKNVTQRLPRVRFGQVHIYNNYYEFSNLADYDFQYAWGVGVFSQIYAQNNYFSFDWDIDPSLIIKVWSKNEESMYETGTIVDLPNGRRYIDLVASYNESNTLQLKKEVTWKPMFYHVIHPTPSVPALVKAKAGAGNLH
6WCB , Knot 196 504 0.80 40 233 453 
MSQPRTPEQALDTPGDCPPGRRDEDAGEGIQCSQRMLSFSDALLSIIATVMILPVTHTEISPEQQFDRSVQRLLATRIAVYLMTFLIVTVAWAAHTRLFQVVGKTDDTLALLNLACMMTITFLPYTFSLMVTFPDVPLGIFLFCVCVIAIGVVQALIVGYAFHFPHLLSPQIQRSAHRALYRRHVLGIVLQGPALCFAAAIFSLFFVPLSYLLMVTVILLPYVSKVTGWCRDRLLGHREPSAHPVEVFSFDLHEPLSKERVEAFSDGVYAIVATLLILDICEDNVPDPKDVKERFSGSLVAALSATGPRFLAYFGSFATVGLLWFAHHSLFLHVRKATRAMGLLNTLSLAFVGGLPLAYQQTSAFARQPRDELERVRVSCTIIFLASIFQLAMWTTALLHQAETLQPSVWFGGREHVLMFAKLALYPCASLLAFASTCLLSRFSVGIFHLMQIAVPCAFLLLRLLVGLALATLRVLRGLARPEHPPPAPTGQDDPQSQLLPAPC

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 \(c\).

\(|P_{f(6ENV_1)}(2) \setminus P_{f(1VBL_1)}(2)|=38\), \(|P_{f(1VBL_1)}(2) \setminus P_{f(6ENV_1)}(2)|=137\). 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 1:000100000001011100110101010000101110100001110110011001100000110011010000110111001001000011001011011111000100111010111010101010011000110001011001100100100111001111001100101000
Pair \(Z_2\) Length of longest common subsequence
6ENV_1,1VBL_1 175 3
6ENV_1,6WCB_1 192 3
1VBL_1,6WCB_1 175 4

Newick tree

 
[
	6WCB_1:93.25,
	[
		6ENV_1:87.5,1VBL_1:87.5
	]:5.75
]

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 \((0.85)(0.8)(\frac{590 }{\log_{20} 590}-\frac{174}{\log_{20}174})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 6ENV_1 1VBL_1 150 106
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]