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Parikh vectors
3LED_1 6XDK_1 3GDG_1 Letter Amino acid
22 21 15 D Aspartic acid
5 0 5 C Cysteine
23 30 16 L Leucine
16 9 17 K Lycine
12 11 16 T Threonine
13 14 8 H Histidine
29 15 13 I Isoleucine
27 21 19 S Serine
7 13 10 Y Tyrosine
45 60 34 A Alanine
23 25 8 R Arginine
22 13 8 N Asparagine
13 11 11 Q Glutamine
16 16 6 P Proline
24 18 15 E Glutamic acid
34 31 28 G Glycine
11 11 7 M Methionine
18 16 8 F Phenylalanine
4 6 3 W Tryptophan
28 28 20 V Valine 

3LED_1|Chains A, B|3-oxoacyl-acyl carrier protein synthase III|Rhodopseudomonas palustris (258594)
>6XDK_1|Chains A, B, C, D|Phosphoserine aminotransferase|Stenotrophomonas maltophilia (strain K279a) (522373)
>3GDG_1|Chains A, B, C, D|Probable NADP-dependent mannitol dehydrogenase|Cladosporium herbarum (29918)
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 \(w=f(c)\)
3LED , Knot 164 392 0.83 40 213 370 
MGSSHHHHHHSSGRENLYFQGVRPAVIAATGLYTPPDSVSNAELVEAFNTYVANFNAANKARIEAGEIEPLQPSSSEFIEKASGIKSRYVVAKPGIVDPDVMRPIIPERSNDELSILAEMAVTAAEQAIERWGKPRERIGAVLCACSNMQRAYPAMAIEVQNALGLGGFAFDMNVACSSATFGLKTAADFVGGGSVDAVLMVNPEICSGHLNFRDRDSHFIFGDVATAAIVERADDAQGGWSILGTKLKTQFSNNIRNNAGFLNRAWPEGRDKADKLFVQQGRKVFKEVVPLVSEMIIEHAREIGIDPHGLKRMWLHQANINMNEIIGRKVLGRDPTRDENVIILDDYANTSSAGSIIAFHKHQDDMAQGDLGLICSFGAGYSAGTVFVQKR
6XDK , Knot 154 369 0.82 38 198 349 
MAHHHHHHMTRAFNFSAGPATLPESVLRQAQAEMLDWHGSGASIVEMSHRGAEFMSVAAEAEADLRRLLDIPDDYAVLFLSGGATTQQALIPLNFAAPGQRADYVVSGHWGKTAVKQAGVYVDVNIAASSEANGYRELPARADWQLSRDAAYVHITANETIHGVEFRDVPDTGNVPLIADFSSSIASEPLDVRRYGVIYAGAQKNLGPVGVAVMIIRRDLLERSGQPRADIFDYRSHVARDSMLNTPPTWNWYLAGLVFKWMLAEGGVTEFAKRNAAKAALVYGAIDGSGGFYRNEVAYAARSRMNIPFFLPDAELDARFVAEAKAAGLLALKGHKVVGGIRASLYNAMPLAGAEALVAFMADFQQRHG
3GDG , Knot 117 267 0.81 40 168 256 
MPGQQATKHESLLDQLSLKGKVVVVTGASGPKGMGIEAARGCAEMGAAVAITYASRAQGAEENVKELEKTYGIKAKAYKCQVDSYESCEKLVKDVVADFGQIDAFIANAGATADSGILDGSVEAWNHVVQVDLNGTFHCAKAVGHHFKERGTGSLVITASMSGHIANFPQEQTSYNVAKAGCIHMARSLANEWRDFARVNSISPGYIDTGLSDFVPKETQQLWHSMIPMGRDGLAKELKGAYVYFASDASTYTTGADLLIDGGYTTR

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 \(c\).

\(|P_{f(3LED_1)}(2) \setminus P_{f(6XDK_1)}(2)|=79\), \(|P_{f(6XDK_1)}(2) \setminus P_{f(3LED_1)}(2)|=64\). 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 1:11000000000010001010110111111011001100100101101100011010110010101101011010000110010110000111011110101101111000000101110111011001100110100011111010001001011111010011111111101011000101110011011111010111110101001010100000011110110111100100101110111001000100010001111001110100010011100100110011111001110010011101011001110010101001110011100100000111100010000110111100000011010111100111100110111000
Pair \(Z_2\) Length of longest common subsequence
3LED_1,6XDK_1 143 6
3LED_1,3GDG_1 169 3
6XDK_1,3GDG_1 152 3

Newick tree

 
[
	3GDG_1:83.10,
	[
		3LED_1:71.5,6XDK_1:71.5
	]:11.60
]

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 \((0.85)(0.8)(\frac{761 }{\log_{20} 761}-\frac{369}{\log_{20}369})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 3LED_1 6XDK_1 132 128
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]