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Parikh vectors
8RUO_1 6QTU_1 1QFX_1 Letter Amino acid
10 18 18 I Isoleucine
14 20 32 T Threonine
2 6 6 W Tryptophan
16 30 24 V Valine
14 22 40 A Alanine
12 22 25 E Glutamic acid
21 14 36 G Glycine
1 13 6 H Histidine
15 18 36 L Leucine
23 17 12 K Lycine
11 9 26 P Proline
10 21 25 D Aspartic acid
10 5 14 Q Glutamine
7 13 20 F Phenylalanine
3 17 14 R Arginine
6 8 8 M Methionine
14 39 35 S Serine
5 12 35 Y Tyrosine
11 17 38 N Asparagine
0 9 10 C Cysteine 

8RUO_1|Chain A|Outer membrane protein MIP|Legionella pneumophila (446)
>6QTU_1|Chain A|E3 ubiquitin-protein ligase COP1|Arabidopsis thaliana (3702)
>1QFX_1|Chains A, B|PROTEIN (PH 2.5 ACID PHOSPHATASE)|Aspergillus niger (5061)
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 \(w=f(c)\)
8RUO , Knot 91 205 0.78 38 139 196 
TDKDKLSYSIGADLGKNFKNQGIDVNPEAMAKGMQEAMSGAQLALTEQQMKDVLNKFQKDLMAKRTAEFNKKADENKVKGEAFLTENKNKPGVVVLPSGLQYKVINSGNGVKPGKSDTVTVEYTGRLIDGTVFDSTEKTGKPATFQVSQVIPGWTEALQLMPAGSTWEIYVPSGLAYGPRSVGGPIGPNETLIFKIHLISVKKSS
6QTU , Knot 145 330 0.85 40 214 320 
GAMTFTRYSRLRVIAEIRHGDIFHSANIVSSIEFDRDDELFATAGVSRCIKVFDFSSVVNEPADMQCPIVEMSTRSKLSCLSWNKHEKNHIASSDYEGIVTVWDVTTRQSLMEYEEHEKRAWSVDFSRTEPSMLVSGSDDCKVKVWCTRQEASVINIDMKANICCVKYNPGSSNYIAVGSADHHIHYYDLRNISQPLHVFSGHKKAVSYVKFLSNNELASASTDSTLRLWDVKDNLPVRTFRGHTNEKNFVGLTVNSEYLACGSETNEVYVYHKEITRPVTSHRFGSPDMDDAEEEAGSYFISAVCWKSDSPTMLTANSQGTIKVLVLAA
1QFX , Knot 188 460 0.83 40 234 434 
FSYGAAIPQSTQEKQFSQEFRDGYSILKHYGGNGPYSERVSYGIARDPPTSCEVDQVIMVKRHGERYPSPSAGKDIEEALAKVYSINTTEYKGDLAFLNDWTYYVPNECYYNAETTSGPYAGLLDAYNHGNDYKARYGHLWNGETVVPFFSSGYGRVIETARKFGEGFFGYNYSTNAALNIISESEVMGADSLTPTCDTDNDQTTCDNLTYQLPQFKVAAARLNSQNPGMNLTASDVYNLMVMASFELNARPFSNWINAFTQDEWVSFGYVEDLNYYYCAGPGDKNMAAVGAVYANASLTLLNQGPKEAGSLFFNFAHDTNITPILAALGVLIPNEDLPLDRVAFGNPYSIGNIVPMGGHLTIERLSCQATALSDEGTYVRLVLNEAVLPFNDCTSGPGYSCPLANYTSILNKNLPDYTTTCNVSASYPQYLSFWWNYNTTTELNYRSSPIACQEGDAMD

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 \(c\).

\(|P_{f(8RUO_1)}(2) \setminus P_{f(6QTU_1)}(2)|=54\), \(|P_{f(6QTU_1)}(2) \setminus P_{f(8RUO_1)}(2)|=129\). 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 1:0000010001110110010001101010111011001101101110000100110010001110001010001000010101110000001111111011000110010110110000101000101101011000000101101010011111001101111100101011011101100111111100011101011010000
Pair \(Z_2\) Length of longest common subsequence
8RUO_1,6QTU_1 183 3
8RUO_1,1QFX_1 169 4
6QTU_1,1QFX_1 182 4

Newick tree

 
[
	6QTU_1:93.39,
	[
		8RUO_1:84.5,1QFX_1:84.5
	]:8.89
]

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 \((0.85)(0.8)(\frac{535 }{\log_{20} 535}-\frac{205}{\log_{20}205})=95.0\)
Status Protein1 Protein2 d d1/2
Query variables 8RUO_1 6QTU_1 124 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]