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Parikh vectors
8YVJ_1 7HUO_1 8IMB_1 Letter Amino acid
5 9 27 D Aspartic acid
5 7 28 E Glutamic acid
4 5 18 I Isoleucine
8 12 47 L Leucine
2 3 27 F Phenylalanine
7 5 17 Q Glutamine
4 2 35 K Lycine
4 4 21 P Proline
5 8 15 Y Tyrosine
11 10 25 A Alanine
6 9 17 R Arginine
4 6 13 C Cysteine
17 14 37 G Glycine
15 14 37 S Serine
3 3 3 W Tryptophan
11 14 37 V Valine
4 5 21 N Asparagine
3 6 12 H Histidine
3 2 16 M Methionine
8 6 23 T Threonine 

8YVJ_1|Chain A|H5.2 nanobody (VHH)|Camelus dromedarius (9838)
>7HUO_1|Chain A|Protease 2A|Coxsackievirus A16 (31704)
>8IMB_1|Chains A, B, C, D|Glutaminase kidney isoform, mitochondrial|Homo sapiens (9606)
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 \(w=f(c)\)
8YVJ , Knot 67 129 0.84 40 102 123 
HMAQVTLKESGGGSVQAGGSLRLSCAGPRHIPGSVCMGWYRQVPGKEREGVALILIGDGHTNYADSVKGRFTISQDSAANTVYLQMNSLKPEDTAVYYCAAARGLSDWTCSFESGVNSWGQGTQVTVSS
7HUO , Knot 70 144 0.80 40 112 139 
SGAIYVGNYRVVNRHLATHNDWANLVWEDSSRDLLVSSTTAQGCDTIARCDCQTGVYYCSSRRKHYPVSFSKPSLIFVEASEYYPARYQSHLMLAVGHSEPGDCGGILRCQHGVVGIVSTGGNGLVGFADVRDLLWLDEEAMEQ
8IMB , Knot 198 476 0.85 40 252 450 
LVASGENKIKQGLLPSLEDLLFYTIAEGQEKIPVHKFITALKSTGLRTSDPRLKECMDMLRLTLQTTSDGVMLDKDLFKKCVQSNIVLLTQAFRRKFVIPDFMSFTSHIDELYESAKKQSGGKVADYIPQLAKFSPDLWGVSVCTVDGQRHSTGDTKVPFCLQSCVKPLKYAIAVNDLGTEYVHRYVGKEPSGLRFNKLFLNEDDKPHNPMVNAGAIVVTSLIKQGVNNAEKFDYVMQFLNKMAGNEYVGFSNATFQSERESGDRNFAIGYYLKEKKCFPEGTDMVGILDFYFQLCSIEVTCESASVMAATLANGGFCPITGERVLSPEAVRNTLSLMHSCGMYDFSGQFAFHVGLPAKSGVAGGILLVVPNVMGMMCWSPPLDKMGNSVKGIHFCHDLVSLCNFHNYDNLRHFAKKLDPRREGGDQRHSFGPLDYESLQQELALKETVWKKVSPESNEDISTTVVYRMESLGEKS

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 \(c\).

\(|P_{f(8YVJ_1)}(2) \setminus P_{f(7HUO_1)}(2)|=67\), \(|P_{f(7HUO_1)}(2) \setminus P_{f(8YVJ_1)}(2)|=77\). 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 1:011010100011101011101010011100111010111000111000011111111010000100101010100001100101010010100011000111011001000100110011010010100
Pair \(Z_2\) Length of longest common subsequence
8YVJ_1,7HUO_1 144 4
8YVJ_1,8IMB_1 202 4
7HUO_1,8IMB_1 210 4

Newick tree

 
[
	8IMB_1:11.45,
	[
		8YVJ_1:72,7HUO_1:72
	]:39.45
]

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 \((0.85)(0.8)(\frac{273 }{\log_{20} 273}-\frac{129}{\log_{20}129})=45.0\)
Status Protein1 Protein2 d d1/2
Query variables 8YVJ_1 7HUO_1 58 55
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]