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Parikh vectors
9LYC_1 5XIP_1 7LGE_1 Letter Amino acid
7 13 1 M Methionine
14 18 10 S Serine
17 35 12 T Threonine
37 37 15 A Alanine
6 22 6 D Aspartic acid
12 9 2 C Cysteine
15 29 4 I Isoleucine
47 33 12 L Leucine
7 19 8 Q Glutamine
19 32 7 G Glycine
6 9 0 H Histidine
14 28 8 P Proline
13 15 4 Y Tyrosine
7 14 9 N Asparagine
3 47 5 E Glutamic acid
4 32 7 K Lycine
31 35 13 V Valine
8 34 7 R Arginine
11 22 3 F Phenylalanine
5 15 0 W Tryptophan 

9LYC_1|Chains A, D|G-protein coupled receptor 3|Homo sapiens (9606)
>5XIP_1|Chains A, B, C, D|Prolyl-tRNA synthetase, putative|Eimeria tenella (5802)
>7LGE_1|Chains A, B, C, D|Capsid protein|Escherichia phage Qbeta (39803)
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 \(w=f(c)\)
9LYC , Knot 121 283 0.80 40 159 259 
GPAAPLPSPKAWDVVLCISGTLVSCENALVVAIIVGTPAFRAPMFLLVGSLAVADLLAGLGLVLHFAAVFCIGSAEMSLVLVGVLAMAFTASIGSLLAITVDRYLSLYNALTYYSETTVTRTYVMLALVWGGALGLGLLPVLAWNCLDGLTTCGVVYPLSKNHLVVLAIAFFMVFGIMLQLYAQICRIVCRHAQQIALQRHLLPASHYVATRKGIATLAVVLGAFAACWLPFTVYCLLGDAHSPPLYTYLTLLPATYNSMINPIIYAFRNQDVQKVLWAVCCC
5XIP , Knot 207 498 0.86 40 255 479 
GAMGVTAKKSTNFADWYTQVIVRGELVEYYDISGCYIIRPWAYRIWEAVQKFFDDGIKRLGVENCYFPMFVSQAKLEKEKDHVEGFKPEVAWVTHYGDSELPEKVAIRPTSETIMYPAYAKWIRSHRDLPLKLNQWNNVVRWEFKQPTPFLRTREFLWQEGHTAHATEEEAYTLVLEILELYRQWYEDYLAVPVIKGEKSENEKFAGGKKTTTIEGIIPDTGRGIQAATSHLLGQNFSRMFSIEFEDEKGAKQLVHQTSWGCTTRSLGVMIMTHGDDKGLVLPPRVVAVQAVIIPIIFKETGGMEIVAKCRELERLLNAAGVRVKVDDRTNYTPGWKFNDWELKGVPLRLEIGPRDVESCQTRVVRRDTSEARNVPWAEAASTIPAMLETMQKDLFNKAKAKFEASIEQITTFDEVMPALNRRHVVLAPWCEDPETETQIKRETQRLSEIQARENADGMTGAMKPLCIPFDQPPMPEGTRCFFTGRPAKRWCLFGRSY
7LGE , Knot 67 133 0.82 36 105 129 
MAKLETVTLGNIGKDGKQTLVLNPRGVNPTNGVASLSQAGAVPALEKRVTVSVSQPSRNRKNYKVQVKIQNPTACTANGSCDPSVTRQAYADVTFSFTQYSTDEERAFVRTELAALLASPLLIDAIDQLNPAY

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 \(c\).

\(|P_{f(9LYC_1)}(2) \setminus P_{f(5XIP_1)}(2)|=54\), \(|P_{f(5XIP_1)}(2) \setminus P_{f(9LYC_1)}(2)|=150\). 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 1:1111111010110111010101100001111111110111011111111011110111111111011111011010101111111111101011011110100010100110000000100001111111111111111111110010110001110110000111111111111111101010100110001001110001111000110001110111111111101111010011101001110001011110000110111011000010011111000
Pair \(Z_2\) Length of longest common subsequence
9LYC_1,5XIP_1 204 4
9LYC_1,7LGE_1 166 3
5XIP_1,7LGE_1 204 4

Newick tree

 
[
	5XIP_1:10.59,
	[
		9LYC_1:83,7LGE_1:83
	]:24.59
]

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 \((0.85)(0.8)(\frac{781 }{\log_{20} 781}-\frac{283}{\log_{20}283})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 9LYC_1 5XIP_1 180 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]