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Parikh vectors
5VSA_1 6VTH_1 1EQY_1 Letter Amino acid
9 8 7 Y Tyrosine
22 7 8 E Glutamic acid
15 13 4 N Asparagine
1 6 2 C Cysteine
23 8 14 G Glycine
27 12 8 K Lycine
5 4 1 M Methionine
10 29 5 S Serine
2 2 2 W Tryptophan
19 12 8 A Alanine
17 13 9 F Phenylalanine
19 14 11 V Valine
24 11 13 L Leucine
24 17 7 D Aspartic acid
6 12 7 Q Glutamine
8 0 3 H Histidine
36 7 3 I Isoleucine
13 11 4 P Proline
19 11 5 T Threonine
15 8 4 R Arginine 

5VSA_1|Chains A, B, C, D|Aryldialkylphosphatase|Sulfolobus solfataricus (2287)
>6VTH_1|Chains A, C[auth D]|T-cell Receptor 12-6, Alfa chain|Homo sapiens (9606)
>1EQY_1|Chain A[auth S]|GELSOLIN|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 \(w=f(c)\)
5VSA , Knot 141 314 0.86 40 193 300 
MRIPLVGKDSIESKDIGFTLIHEHLRVFSEAVRQQWPHLYNEDEEFRNAVNEVKRAMQFGVKTIVDPTVMGLGRDIRFMEKVVKATGINLVAGTGIYIYIDLPFYFLNRSIDEIADLFIHDIKEGIQGTLNKAGFVKIAADEPGITKDVEKVIRAAAIANKETKVPIITHSNAHNNTGLEQQRILTEEGVDPGKILIGHLGDTDNIDYIKKIADKGSFIGLDRYGLDLFLPVDKRNETTLRLIKDGYSDKIMISHDYLCTFDAGTAKPEYKPKLAPRWSITLIFEDTIPFLKRNGVNEEVIATIFKENPKKFFS
6VTH , Knot 94 205 0.81 38 141 195 
MRKEVEQDPGPFNVPEGATVAFNCTYSNSASQSFFWYRQDCRKEPKLLMSVYSSGNEDGRFTAQLNRASQYISLLIRDSKLSDSATYLCVVQPGGYQKVTFGTGTKLQVIPNIQNPDPAVYQLRDSKSSDKSVCLFTDFDSQTNVSQSKDSDVYITDKCVLDMRSMDFKSNSAVAWSNKSDFACANAFNNSIIPEDTFFPSPESS
1EQY , Knot 64 125 0.82 40 101 121 
MVVEHPEFLKAGKEPGLQIWRVEKFDLVPVPTCLYGDFFTGDAYVILKTVQLRNGNLQYDLHYWLGNECSQDESGAAAIFTVQLDDYLNGRAVQHREVQGFESATFLGYFKSGLKYKKGGVASGF

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 \(c\).

\(|P_{f(5VSA_1)}(2) \setminus P_{f(6VTH_1)}(2)|=116\), \(|P_{f(6VTH_1)}(2) \setminus P_{f(5VSA_1)}(2)|=64\). 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 1:10111110001000011101100010110011000110100000010011001001101110011010111110010110011010110111101101010111011000100110111001001101010011110111001110001001101111100000111100001000011000011000110110111101100001001001100101111000110111110000000101100100001110000100101101010001011101010111000111100011000111011000100110
Pair \(Z_2\) Length of longest common subsequence
5VSA_1,6VTH_1 180 3
5VSA_1,1EQY_1 162 4
6VTH_1,1EQY_1 156 3

Newick tree

 
[
	5VSA_1:88.01,
	[
		1EQY_1:78,6VTH_1:78
	]:10.01
]

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 \((0.85)(0.8)(\frac{519 }{\log_{20} 519}-\frac{205}{\log_{20}205})=90.6\)
Status Protein1 Protein2 d d1/2
Query variables 5VSA_1 6VTH_1 116 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]