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Parikh vectors
4YVI_1 1WVQ_1 4PEJ_1 Letter Amino acid
26 10 13 G Glycine
13 15 7 I Isoleucine
14 13 2 K Lycine
8 2 5 M Methionine
13 4 8 S Serine
13 6 4 T Threonine
3 2 0 C Cysteine
22 16 11 E Glutamic acid
14 8 8 P Proline
5 3 3 Y Tyrosine
16 23 12 V Valine
4 8 4 N Asparagine
11 2 10 Q Glutamine
5 1 4 W Tryptophan
16 10 10 R Arginine
14 7 6 D Aspartic acid
30 12 10 L Leucine
7 7 4 F Phenylalanine
18 14 10 A Alanine
14 4 10 H Histidine 

4YVI_1|Chains A, B|tRNA (guanine-N(1)-)-methyltransferase|Haemophilus influenzae (strain ATCC 51907 / DSM 11121 / KW20 / Rd) (71421)
>1WVQ_1|Chains A, B, C|hypothetical protein PAE2307|Pyrobaculum aerophilum (13773)
>4PEJ_1|Chains A, B|Retro-aldolase|ARTIFICIAL GENE (32630)
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 \(w=f(c)\)
4YVI , Knot 119 266 0.83 40 178 254 
MGSSHHHHHHSSGLVPRGSHMWIGVISLFPEMFKAITEFGVTGRAVKHNLLKVECWNPRDFTFDKHKTVDDRPYGGGPGMLMMVQPLRDAIHTAKAAAGEGAKVIYLSPQGRKLDQGGVTELAQNQKLILVCGRYEGIDERLIQTEIDEEWSIGDYVLTGGELPAMTLIDAVARFIPGVLGKQASAEEDSFADGLLDCPHYTRPEVLEGLTVPPVLMSGHHEEIRKWRLKQSLQRTWLRRPELLEGLALTDEQRKLLKEAQAEHNS
1WVQ , Knot 81 167 0.82 40 123 163 
MTDMSIKFELIDVPIPQGTNVIIGQAHFIKTVEDLYEALVTSVPGVKFGIAFCEASGKRLVRHEANDEELRNLAIDLCKKIAAGHVFVIYIRNAWPINVLNAIKNVPEVVRIFAATANPLKVIVAEVEPERRGVVGVVDGHSPLGVETEKDREERKKFLREVVKYKL
4PEJ , Knot 70 141 0.82 38 116 134 
MNLPTAQEVQGLMARFIELVDVGDIEAIVQMYADDATVESPFGQPPIHGREQIAAHYRQWLGGGKFRVSLTGPVRASHNGSGAMPLRKEWVWNGQPSALDVILVMRFDEHGRIQTEQRYWSEVNLSVREPQGSLEHHHHHH

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 \(c\).

\(|P_{f(4YVI_1)}(2) \setminus P_{f(1WVQ_1)}(2)|=110\), \(|P_{f(1WVQ_1)}(2) \setminus P_{f(4YVI_1)}(2)|=55\). 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 1:11000000000011110100111111011101101100111010110001101001010010100000100010111111111101100110010111101101101010100100111001100001111010001100011000100010110011011011110110111011111110010100001101110010000101101101111110100001001010001000110010110111100000011001010000
Pair \(Z_2\) Length of longest common subsequence
4YVI_1,1WVQ_1 165 3
4YVI_1,4PEJ_1 148 6
1WVQ_1,4PEJ_1 157 3

Newick tree

 
[
	1WVQ_1:82.58,
	[
		4YVI_1:74,4PEJ_1:74
	]:8.58
]

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 \((0.85)(0.8)(\frac{433 }{\log_{20} 433}-\frac{167}{\log_{20}167})=78.8\)
Status Protein1 Protein2 d d1/2
Query variables 4YVI_1 1WVQ_1 99 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]