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Parikh vectors
8EEC_1 6GOL_1 1MOE_1 Letter Amino acid
9 5 12 D Aspartic acid
29 2 28 G Glycine
45 10 16 L Leucine
23 4 20 T Threonine
6 1 4 W Tryptophan
18 3 7 R Arginine
9 10 6 N Asparagine
12 7 10 K Lycine
26 0 12 P Proline
25 2 23 S Serine
4 3 14 Y Tyrosine
31 5 16 V Valine
10 2 2 H Histidine
11 2 11 I Isoleucine
22 3 18 A Alanine
20 7 11 E Glutamic acid
10 1 3 M Methionine
11 3 8 F Phenylalanine
8 0 4 C Cysteine
14 2 15 Q Glutamine 

8EEC_1|Chain A|Isoform 2 of Mitogen-activated protein kinase kinase kinase kinase 1|Homo sapiens (9606)
>6GOL_1|Chain A|Hemagglutinin tri-stalk|Influenza A virus (A/California/VRDL69/2009(H1N1)) (705461)
>1MOE_1|Chains A, B|anti-CEA mAb T84.66|Mus musculus (10090)
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 \(w=f(c)\)
8EEC , Knot 146 343 0.82 40 199 321 
GSRKGCALLVKLFNGCPLRIHSTAAWTHPSTKDQHLLLGAEEGIFILNRNDQEATLEMLFPSRTTWVYSINNVLMSLSGKTPHLYSHSILGLLERKETRAGNPIAHISPHRLLAAKNMVSTKIQDTKGCRACCVAEGASSGGPFLCGALETSVVLLQWYQPMNKFLLVRQVLFPLPTPLSVFALLTGPGSELPAVCIGVSPGRPGKSVLFHTVRFGALSCWLGEMSTEHRGPVQVTQVEEDMVMVLMDGSVKLVTPEGSPVRGLRTPEIPMTEAVEAVAMVGGQLQAFWKHGVQVWALGSDQLLQELRDPTLTFRLLGSPRPVVVETRPVDDPTAPSNLYIQE
6GOL , Knot 42 72 0.83 36 62 70 
MNTQFTAVGKEFNHLEKRIENLNKKVDDGFLDIWTYNAELLVLLENERTLDYHDSNVKNLYEKVRSQLKNNA
1MOE , Knot 111 240 0.84 40 158 232 
DIVLTQSPASLAVSLGQRATMSCRAGESVDIFGVGFLHWYQQKPGQPPKLLIYRASNLESGIPVRFSGTGSRTDFTLIIDPVEADDVATYYCQQTNEDPYTFGGGTKLEIKGGGSGGGGEVQLQQSGAELVEPGASVKLSCTASGFNIKDTYMHWVKQRPEQGLEWIGRIDPANGNSKYVPKFQGKATITADTSSNTAYLQLTSLTSEDTAVYYCAPFGYYVSDYAMAYWGQGTSVTVSS

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 \(c\).

\(|P_{f(8EEC_1)}(2) \setminus P_{f(6GOL_1)}(2)|=164\), \(|P_{f(6GOL_1)}(2) \setminus P_{f(8EEC_1)}(2)|=27\). 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 1:1000101111011010110100011100100000011111001111100000010101111000011001001110101001010000111110000001101110101001111001100010000100100110110011111011100011110100110011110011111101101111101110011110111011011001110010111100111010000011101001000111111010101101010110110010111001101111111010111001101111100011001001010101110101111000110010110010100
Pair \(Z_2\) Length of longest common subsequence
8EEC_1,6GOL_1 191 3
8EEC_1,1MOE_1 173 4
6GOL_1,1MOE_1 160 3

Newick tree

 
[
	8EEC_1:94.52,
	[
		1MOE_1:80,6GOL_1:80
	]:14.52
]

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 \((0.85)(0.8)(\frac{415 }{\log_{20} 415}-\frac{72}{\log_{20}72})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 8EEC_1 6GOL_1 134 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]