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Parikh vectors
4GMK_1 5WOG_1 7QTB_1 Letter Amino acid
4 2 2 M Methionine
7 7 9 P Proline
0 1 5 C Cysteine
6 2 3 R Arginine
10 4 14 N Asparagine
7 1 8 Q Glutamine
26 7 20 G Glycine
21 18 20 L Leucine
6 7 6 F Phenylalanine
10 11 26 S Serine
9 21 26 A Alanine
2 1 6 W Tryptophan
11 9 20 T Threonine
17 10 10 K Lycine
19 4 14 E Glutamic acid
4 10 7 H Histidine
20 0 18 I Isoleucine
5 2 16 Y Tyrosine
23 12 12 V Valine
21 8 25 D Aspartic acid 

4GMK_1|Chains A, B|Ribose-5-phosphate isomerase A|Lactobacillus salivarius (362948)
>5WOG_1|Chains A, B|Hemoglobin subunit alpha|Homo sapiens (9606)
>7QTB_1|Chains A, B|Nuclease S1|Aspergillus oryzae RIB40 (510516)
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 \(w=f(c)\)
4GMK , Knot 99 228 0.78 38 149 217 
GPNQDELKQLVGTKAVEWIKDGMIVGLGTGSTVKYMVDALGKRVNEEGLDIVGVTTSIRTAEQAKSLGIVIKDIDEVDHIDLTIDGADEISSDFQGIKGGGAALLYEKIVATKSNKNMWIVDESKMVDDLGQFPLPVEVIPYGSGTVFKRFEEKGLNPEFRKNEDGSLLHTDSDNYIIDLHLGKIENPKELGDYLINQVGVVEHGLFLDIVNTVIVGRQDGPEVLEAR
5WOG , Knot 66 137 0.79 38 96 133 
LSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTS
7QTB , Knot 119 267 0.83 40 169 258 
WGNLGHETVAYIAQSFVASSTESFCQNILGDDSTSYLANVATWADTYKYTDAGEFSKPYHFIDAQDNPPQSCGVDYDRDCGSAGCSISAIQNYTNILLESPNGSEALNALKFVVHIIGDIHQPLHDENLEAGGNGIDVTYDGETTNLHHIWDTNMPEEAAGGYSLSVAKTYADLLTERIKTGTYSSKKDSWTDGIDIKDPVSTSMIWAADANTYVCSTVLDDGLAYINSTDLSGEYYDKSQPVFEELIAKAGYRLAAWLDLIASQPS

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 \(c\).

\(|P_{f(4GMK_1)}(2) \setminus P_{f(5WOG_1)}(2)|=103\), \(|P_{f(5WOG_1)}(2) \setminus P_{f(4GMK_1)}(2)|=50\). 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 1:110000100111001101100111111101001001101110010001101111000100100100111110010010010101011001000101101111111000111000000111100001100110111110111010101100100011010100000101100000001101011010010011001100111100111101100111100011011010
Pair \(Z_2\) Length of longest common subsequence
4GMK_1,5WOG_1 153 3
4GMK_1,7QTB_1 164 3
5WOG_1,7QTB_1 157 4

Newick tree

 
[
	7QTB_1:81.48,
	[
		4GMK_1:76.5,5WOG_1:76.5
	]:4.98
]

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 \((0.85)(0.8)(\frac{365 }{\log_{20} 365}-\frac{137}{\log_{20}137})=69.3\)
Status Protein1 Protein2 d d1/2
Query variables 4GMK_1 5WOG_1 88 69.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]