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Parikh vectors
4HQX_1 4UGK_1 7TMF_1 Letter Amino acid
11 18 8 V Valine
8 36 14 E Glutamic acid
1 12 8 H Histidine
8 16 8 T Threonine
1 11 3 W Tryptophan
6 16 12 P Proline
10 20 17 R Arginine
5 9 9 Q Glutamine
7 28 9 I Isoleucine
3 12 9 F Phenylalanine
0 15 1 Y Tyrosine
6 31 26 A Alanine
2 23 9 D Aspartic acid
2 22 14 G Glycine
7 26 6 K Lycine
4 17 7 S Serine
5 15 4 N Asparagine
8 4 3 C Cysteine
7 27 18 L Leucine
1 5 6 M Methionine 

4HQX_1|Chain A|Platelet-derived growth factor subunit B|Homo sapiens (9606)
>4UGK_1|Chain A|NITRIC OXIDE SYNTHASE OXYGENASE|BACILLUS SUBTILIS (224308)
>7TMF_1|Chains A, B|NAD(P)H nitroreductase|Klebsiella pneumoniae subsp. pneumoniae HS11286 (1125630)
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 \(w=f(c)\)
4HQX , Knot 51 102 0.77 38 85 98 
SLGSLTIAEPAMIAECKTRTEVFEISRRLIDRTNANFLVWPPCVEVQRCSGCCNNRNVQCRPTQVQLRPVQVRKIEIVRKKPIFKKATVTLEDHLACKCETV
4UGK , Knot 153 363 0.82 40 216 348 
MEEKEILWNEAKAFIAACYQELGKAAEVKDRLADIKSEIDLTGSYVHTKEELEHGAKMAWRNSNRCIGRLFWNSLNVIDRRDVRTKEEVRDALFHHIETATNNGKIRPTITIFPPEEKGEKQVEIWNHQLIRYAGYESDGERIGDPASCSLTAACEELGWRGERTDFDLLPLIFRMKGDEQPVWYELPRSLVIEVPITHPDIEAFSDLELKWYGVPIISDMKLEVGGIHYNAAPFNGWYMGTEIGARNLADEKRYDKLKKVASVIGIAADYNTDLWKDQALVELNKAVLHSYKKQGVSIVDHHTAASQFKRFEEQAEEAGRKLTGDWTWLIPPISPAATHIFHRSYDNSIVKPNYFYQDKPYE
7TMF , Knot 86 191 0.78 40 134 181 
MAHHHHHHMDALDLLLNRRSASRLAEPAPAGEQLENILRAGLRAPDHGTLQPWRFFIIADDGRERFSQLLEKGAREAGQDEKGIEKARNAPFRAPLIITVVARCEDHPKVPRWEQEMSAGCAVMAMQMAALAQGFNGIWRSGALTESPVVRAGFECREQDKIVGFLYLGTPQLKASTTIALPDTAPFVTRF

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 \(c\).

\(|P_{f(4HQX_1)}(2) \setminus P_{f(4UGK_1)}(2)|=31\), \(|P_{f(4UGK_1)}(2) \setminus P_{f(4HQX_1)}(2)|=162\). 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 1:011010110111110000000110100011000010111111010100001000000100010010101101001011000111001010100011000001
Pair \(Z_2\) Length of longest common subsequence
4HQX_1,4UGK_1 193 3
4HQX_1,7TMF_1 139 4
4UGK_1,7TMF_1 180 3

Newick tree

 
[
	4UGK_1:99.99,
	[
		4HQX_1:69.5,7TMF_1:69.5
	]:30.49
]

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 \((0.85)(0.8)(\frac{465 }{\log_{20} 465}-\frac{102}{\log_{20}102})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 4HQX_1 4UGK_1 139 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]