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Parikh vectors
5NZQ_1 8FJX_1 7XKK_1 Letter Amino acid
9 3 6 M Methionine
2 2 3 W Tryptophan
11 8 24 R Arginine
8 3 10 C Cysteine
10 8 16 Q Glutamine
3 11 5 H Histidine
12 16 23 I Isoleucine
15 10 18 T Threonine
1 1 13 Y Tyrosine
18 21 16 A Alanine
14 14 22 E Glutamic acid
23 15 27 G Glycine
11 12 15 K Lycine
7 7 16 F Phenylalanine
8 9 12 N Asparagine
22 19 29 L Leucine
13 14 22 S Serine
16 21 31 V Valine
9 9 10 D Aspartic acid
11 7 14 P Proline 

5NZQ_1|Chains A, B|D-3-phosphoglycerate dehydrogenase|Homo sapiens (9606)
>8FJX_1|Chain A|Trifunctional purine biosynthetic protein adenosine-3|Homo sapiens (9606)
>7XKK_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Gap junction delta-2 protein|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 \(w=f(c)\)
5NZQ , Knot 106 223 0.85 40 162 218 
MLVMNTPNGNSLSAAELTCGMIMCLARQIPQATASMKDGKWERKKFMGTELNGKTLGILGLGRIGREVATRMQSFGMKTIGYDPIISPEVSASFGVQQLPLEEIWPLCDFITVHTPLLPSTTGLLNDNTFAQCKKGVRVVNCARGGIVDEGALLRALQSGQCAGAALDVFTEEPPRDRALVDHENVISCPHLGASTKEAQSRCGEEIAVQFVDMVKGKSLTGV
8FJX , Knot 93 210 0.79 40 139 199 
MARVAVLISGTGSNLQALIDSTREPNSSAQIDIVISNKAAVAGLDKAERAGIPTRVINHKLYKNRVEFDSAIDLVLEEFSIDIVCLAGFMRILSGPFVQKWNGKMLNIHPSLLPSFKGSNAHEQALETGVTVTGCTVHFVAEDVDAGQIILQEAVPVKRGDTVATLSERVKLAEHKIFPAALQLVASGTVQLGENGKICWVKEEHHHHHH
7XKK , Knot 145 332 0.84 40 213 318 
MGEWTILERLLEAAVQQHSTMIGRILLTVVVIFRILIVAIVGETVYDDEQTMFVCNTLQPGCNQACYDRAFPISHIRYWVFQIIMVCTPSLCFITYSVHQSAKQRERRYSTVFLALDRDPPESIGGPGGTGGGGSGGGKREDKKLQNAIVNGVLQNTENTSKETEPDCLEVKELTPHPSGLRTASKSKLRRQEGISRFYIIQVVFRNALEIGFLVGQYFLYGFSVPGLYECNRYPCIKEVECYVSRPTEKTVFLVFMFAVSGICVVLNLAELNHLGWRKIKLAVRGAQAKRKSIYEIRNKDLPRVSVPNFGRTQSSDSAYVSRGDMLEVLFQ

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 \(c\).

\(|P_{f(5NZQ_1)}(2) \setminus P_{f(8FJX_1)}(2)|=90\), \(|P_{f(8FJX_1)}(2) \setminus P_{f(5NZQ_1)}(2)|=67\). 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 1:1111001010010110100111101100110101010010100001110010100111111101100110010011100110011101010101110011100111100110100111100011100001100001101100101111001111011001001111101100011000111000011001011100001000010011101101101001011
Pair \(Z_2\) Length of longest common subsequence
5NZQ_1,8FJX_1 157 4
5NZQ_1,7XKK_1 161 3
8FJX_1,7XKK_1 164 3

Newick tree

 
[
	7XKK_1:82.15,
	[
		5NZQ_1:78.5,8FJX_1:78.5
	]:3.65
]

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 \((0.85)(0.8)(\frac{433 }{\log_{20} 433}-\frac{210}{\log_{20}210})=65.2\)
Status Protein1 Protein2 d d1/2
Query variables 5NZQ_1 8FJX_1 84 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]