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Parikh vectors
2QKD_1 8QWP_1 6NVK_1 Letter Amino acid
7 8 8 Y Tyrosine
42 13 19 E Glutamic acid
21 8 14 K Lycine
16 5 11 F Phenylalanine
21 18 22 P Proline
1 1 5 W Tryptophan
20 19 21 R Arginine
9 11 7 C Cysteine
12 5 10 M Methionine
33 17 36 L Leucine
20 14 25 V Valine
25 15 21 G Glycine
12 9 14 H Histidine
25 6 10 I Isoleucine
21 7 8 Q Glutamine
23 20 16 S Serine
29 16 7 T Threonine
23 9 22 A Alanine
22 9 7 N Asparagine
22 9 17 D Aspartic acid 

2QKD_1|Chain A|Zinc finger protein ZPR1|Mus musculus (10090)
>8QWP_1|Chains A, B|Cellular tumor antigen p53|Homo sapiens (9606)
>6NVK_1|Chain A|Fibroblast growth factor receptor 4|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 \(w=f(c)\)
2QKD , Knot 169 404 0.83 40 211 376 
MGHHHHHHGSIESLCMNCYRNGTTRLLLTKIPFFREIIVSSFSCEHCGWNNTEIQSAGRIQDQGVRYTLTVRSQEDMNREVVKTDSATTRIPELDFEIPAFSQKGALTTVEGLISRAISGLEQDQPTRRAVEGAIAERIDEFIGKLKDLKQMASPFTLVIDDPSGNSFVENPHAPQKDNALVITYYDRTPQQAEMLGLQAEAPEEKAEEEDLRNEVLQFNTNCPECNAPAQTNMKLVQIPHFKEVIIMATNCENCGHRTNEVKSGGAVEPLGTRITLHITDPSDMTRDLLKSETCSVEIPELEFELGMAVLGGKFTTLEGLLKDIRELVTKNPFTLGDSSNPDQSEKLQEFSQKLGQIIEGKMKAHFIMNDPAGNSYLQNVYAPEDDPEMKVERYKRTFDQNEE
8QWP , Knot 102 219 0.83 40 149 211 
SSSVPSQKTYQGSYGFRLGFLHSGTAKSVTCTYSPALNKLFCQLAKTCPVQLWVDSTPPPGTRVRAMAIYKQSQHMTEVVRRCPHHERCSDSDGLAPPQHLIRVEGNLRAEYLDDRNTFRHSVVVPYEPPEVGSDCTTIHYNCMCYSSCMGGMNRRPILTIITLEDSSGNLLGRDSFEVRVCACPGRDRRTEEENLRKKGEPHHELPPGSTKRALPNNT
6NVK , Knot 133 300 0.84 40 192 284 
MVSLDLPLDPLWEFPRDRLVLGKPLGEGCFGQVVRAEAFGMDPARPDQASTVAVKMLKDNASDKDLADLVSEMEVMKLIGRHKNIINLLGVCTQEGPLYVIVECAAKGNLREFLRARRPPGSSEGPLSFPVLVSCAYQVARGMQYLESRKCIHRDLAARNVLVTEDNVMKIADFGLARGVHHIDYYKKTSNGRLPVKWMAPEALFDRVYTHQSDVWSFGILLWEIFTLGGSPYPGIPVEELFSLLREGHRMDRPPHCPPELYGLMRECWHAAPSQRPTFKQLVEALDKVLLAVSHHHHHH

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 \(c\).

\(|P_{f(2QKD_1)}(2) \setminus P_{f(8QWP_1)}(2)|=118\), \(|P_{f(8QWP_1)}(2) \setminus P_{f(2QKD_1)}(2)|=56\). 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 1:11000000101001010000010001110011110011100100000110000100110100011000101000001000110000100011010101111000111001011100110110000100011011110010011101001001101101110010100110010110000111100000010010111101011000100001000110100001000111000101101101001111100000010000010011110111001010100100100011000000101101010111111110100101110010011000110110000100000100100011011010101011100111000100101100010101000000100000
Pair \(Z_2\) Length of longest common subsequence
2QKD_1,8QWP_1 174 3
2QKD_1,6NVK_1 173 6
8QWP_1,6NVK_1 181 4

Newick tree

 
[
	8QWP_1:89.51,
	[
		2QKD_1:86.5,6NVK_1:86.5
	]:3.01
]

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 \((0.85)(0.8)(\frac{623 }{\log_{20} 623}-\frac{219}{\log_{20}219})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 2QKD_1 8QWP_1 147 112
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]