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Parikh vectors
4MLG_1 3QXI_1 1GZA_1 Letter Amino acid
5 1 8 C Cysteine
7 3 14 Q Glutamine
23 28 28 L Leucine
15 11 38 S Serine
9 2 2 W Tryptophan
17 16 19 V Valine
15 18 15 R Arginine
11 1 4 H Histidine
10 6 7 M Methionine
23 14 29 P Proline
23 2 3 Y Tyrosine
12 9 13 N Asparagine
21 18 14 E Glutamic acid
15 17 17 I Isoleucine
20 10 4 K Lycine
15 8 18 F Phenylalanine
12 14 20 T Threonine
15 45 33 A Alanine
28 13 20 D Aspartic acid
28 29 38 G Glycine 

4MLG_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Beta-xylosidase|uncultured organism (155900)
>3QXI_1|Chains A, B, C|Enoyl-CoA hydratase EchA1|Mycobacterium marinum (216594)
>1GZA_1|Chain A|PEROXIDASE|'Arthromyces ramosus' (5451)
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 \(w=f(c)\)
4MLG , Knot 141 324 0.83 40 216 304 
MKEPRYLVPGDYMADPAAHVFNDKLYIYPSHDWESGIPENDNGDHFNMKDYHVFSMDDVEQGEVTDHGVVLRTEDIPWAGRQLWDSDVAFRNGKYYMYFPLKDQNDIFRIGVAISDRPEGPFIPQENPIKGSYSMDPCIWPDKDGEYYMYFGGLWGGQLQRYRNNKALECALLPEGDEPALCPKVVRLREDMLEFAEEPRDLMILDEKGKLLSAGDTKRRFFEASWMHYYNGKYYFSYSTGDTHLICYATGDNPYGPFTYRGVILTPVVGWTTHHSIVEFKGKWYLFHHDCVPSKGKTWLRSLKVAELKYNPDGSIQPIKGTAE
3QXI , Knot 113 265 0.79 40 136 240 
GPGSMVSELNGDTEPEVLVEQRDRILIITINRPKAKNSVNAAVSRALADAMDRLDADAGLSVGILTGAGGSFCAGMDLKAFARGENVVVEGRGLGFTERPPAKPLIAAVEGYALAGGTELALATDLIVAARDSAFGIPEVKRGLVAGGGGLLRLPERIPYAIAMELALTGDNLSAERAHALGMVNVLAEPGAALDAAIALAEKITANGPLAVAATKRIITESRGWSLDTRFAQQMKILFPIFTSNDAKEGAIAFAEKRPPRWTGT
1GZA , Knot 141 344 0.79 40 189 315 
QGPGGGGGSVTCPGGQSTSNSQCCVWFDVLDDLQTNFYQGSKCESPVRKILRIVFHDAIGFSPALTAAGQFGGGGADGSIIAHSNIELAFPANGGLTDTIEALRAVGINHGVSFGDLIQFATAVGMSNCPGSPRLEFLTGRSNSSQPSPPSLIPGPGNTVTAILDRMGDAGFSPDEVVDLLAAHSLASQEGLNSAIFRSPLDSTPQVFDTQFYIETLLKGTTQPGPSLGFAEELSPFPGEFRMRSDALLARDSRTACRWQSMTSSNEVMGQRYRAAMAKMSVLGFDRNALTDCSDVIPSAVSNNAAPVIPGGLTVDDIEVSCPSEPFPEIATASGPLPSLAPAP

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 \(c\).

\(|P_{f(4MLG_1)}(2) \setminus P_{f(3QXI_1)}(2)|=128\), \(|P_{f(3QXI_1)}(2) \setminus P_{f(4MLG_1)}(2)|=48\). 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 1:100100111100110111011000101010001001110000100101000011010010010100011110000111110011000111001000101110000011011111000101111100011010001010111000100010111111101000000011001111010011101011010001101100100111100010110110000011010110000100010000100011001010010111000111101111100000110101010110000110010011001011010001010101101010
Pair \(Z_2\) Length of longest common subsequence
4MLG_1,3QXI_1 176 4
4MLG_1,1GZA_1 179 4
3QXI_1,1GZA_1 145 4

Newick tree

 
[
	4MLG_1:93.54,
	[
		3QXI_1:72.5,1GZA_1:72.5
	]:21.04
]

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 \((0.85)(0.8)(\frac{589 }{\log_{20} 589}-\frac{265}{\log_{20}265})=91.3\)
Status Protein1 Protein2 d d1/2
Query variables 4MLG_1 3QXI_1 119 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]