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Parikh vectors
3HUO_1 3RJG_1 5UGJ_1 Letter Amino acid
23 22 34 G Glycine
27 26 26 L Leucine
4 14 10 F Phenylalanine
13 14 11 P Proline
3 1 1 W Tryptophan
15 19 14 R Arginine
11 12 12 N Asparagine
1 3 2 C Cysteine
10 36 11 K Lycine
5 7 6 M Methionine
29 19 23 T Threonine
5 12 5 Y Tyrosine
36 18 33 A Alanine
2 9 17 H Histidine
11 23 20 I Isoleucine
14 18 25 V Valine
13 21 21 D Aspartic acid
18 13 6 Q Glutamine
12 21 11 S Serine
11 27 14 E Glutamic acid 

3HUO_1|Chains A, B|CTX-M-9 extended-spectrum beta-lactamase|Escherichia coli (562)
>3RJG_1|Chain A|DNA polymerase beta|Homo sapiens (9606)
>5UGJ_1|Chains A, B, C, D|4-hydroxy-tetrahydrodipicolinate reductase|Neisseria meningitidis serogroup B (strain MC58) (122586)
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 \(w=f(c)\)
3HUO , Knot 117 263 0.82 40 160 250 
QTSAVQQKLAALEKSSGGRLGVALIDTADNTQVLYRGDERFPMCGTSKVMAAAAVLKQSETQKQLLNQPVEIKPADLVNYNPIAEKHVNGTMTLAELSAAALQYSDNTAMNKLIAQLGGPGGVTAFARAIGDETFRLDRTEPTLNTAIPGDPRDTTTPRAMAQTLRQLTLGHALGETQRAQLVTWLKGNTTGAASIRAGLPTSWTAGDKTGSGDYGTTNDIAVIWPQGRAPLVLVTYFTQPQQNAESRRDVLASAARIIAEGL
3RJG , Knot 146 335 0.84 40 199 321 
MSKRKAPQETLNGGITDMLTELANFEKNVSQAIHKYNAYRKAASVIAKYPHKIKSGAEAKKLPGVGTKIAEKIDEFLATGKLRKLEKIRQDDTSSSINFLTRVSGIGPSAARKFVDEGIKTLEDLRKNEDKLNHHQRIGLKYFGDFEKRIPREEMLQMQDIVLNEVKKVDSEYIATVCGSFRRGAESSGDMDVLLTHPSFTSESTKQPKLLHQVVEQLQKVHFITDTLSKGETKFMGVCQLPSKNDEKEYPHRRIDIRLIPKDQYYCGVLYFTGSDIFNKNMRAHALEKGFTINEYTIRPLGVTGVAGEPLPVDSEKDIFDYIQWKYREPKDRSE
5UGJ , Knot 126 302 0.79 40 169 274 
MHHHHHHGLPIPNPLLGLDSTENLYFQGIDPFTMTPLKIAIAGANGRMGRVLVEAVNNHPDTVLSGALEHSGSEALGLDAGYAVGLKTGIAISDDVDAVLAQSDVLIDFTRPEPTLKHLQKCVEKQVNIIIGTTGFDDTGKAAIHTAAEKTGIVFAANFSVGVNLTFHILDTVARVLNEGYDIEIIEGHHRHKVDAPSGTALRMGEVIAGALGRDLKQCAVYGREGHTGPRDPSTIGFATVRAGDIVGDHTALFATDGERVEITHKASSRMTFAAGAVRAAVWVNGKTGLYDMQDVLGLNNR

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 \(c\).

\(|P_{f(3HUO_1)}(2) \setminus P_{f(3RJG_1)}(2)|=60\), \(|P_{f(3RJG_1)}(2) \setminus P_{f(3HUO_1)}(2)|=99\). 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 1:00011000111100001101111110010000110010001110100011111111000000001100110101101100011100010101011010111100000011001110111111101110111000101000010100111101000001011100100101101110000101101101000111010111100101100010100100001111110101111110010010001000001110110111011
Pair \(Z_2\) Length of longest common subsequence
3HUO_1,3RJG_1 159 4
3HUO_1,5UGJ_1 151 4
3RJG_1,5UGJ_1 164 4

Newick tree

 
[
	3RJG_1:82.43,
	[
		3HUO_1:75.5,5UGJ_1:75.5
	]:6.93
]

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 \((0.85)(0.8)(\frac{598 }{\log_{20} 598}-\frac{263}{\log_{20}263})=94.3\)
Status Protein1 Protein2 d d1/2
Query variables 3HUO_1 3RJG_1 119 105
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]