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Parikh vectors
3FOT_1 5MJK_1 3NWX_1 Letter Amino acid
14 12 1 Y Tyrosine
17 11 5 Q Glutamine
20 13 2 F Phenylalanine
49 15 0 S Serine
9 2 2 W Tryptophan
22 37 13 G Glycine
25 19 6 K Lycine
7 11 0 M Methionine
24 19 8 T Threonine
45 25 5 A Alanine
30 14 4 R Arginine
31 19 4 D Aspartic acid
38 30 4 E Glutamic acid
23 9 4 N Asparagine
9 2 1 C Cysteine
34 29 12 I Isoleucine
56 15 12 L Leucine
11 11 1 H Histidine
24 10 7 P Proline
31 24 7 V Valine 

3FOT_1|Chain A|15-O-acetyltransferase|Fusarium sporotrichioides (5514)
>5MJK_1|Chains A, B, C, D|Thioredoxin reductase|Lactococcus lactis subsp. cremoris (1359)
>3NWX_1|Chains A, B|protease|null
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 \(w=f(c)\)
3FOT , Knot 207 519 0.83 40 247 485 
MSASSSSALPPLVPALYRWKSTGSSGRQVQRRCVGAEAIVGLEEKNRRALYDLYIATSLRNIAPASTLLTLQNLKEMFELALLDARFEHPECACTVSWDDEVPAIITYESPESNESARDWARGCIHVQPTAKSALDLWSEMEEGRAAANDNTPSKSIELFLLSDVSTDSTPIPQDATVEILFHSNHLFWDGIGCRKFVGDLFRLVGSYIGRSDSREMKKIQWGQEIKNLSPPVVDSLKLDINTLGSEFDDKCTEYTSALVANYKSRGMKFQPGLALPRCVIHKLSADESIDIVKAVKTRLGPGFTISHLTQAAIVLALLDHLKPNDLSDDEVFISPTSVDGRRWLREDIASNFYAMCQTAAVVRIENLKSITVSHKDEKELQVRALESACRNIKKSYRQWLENPFLQALGLRVHNFEASYLHAKPIPFEGEANPLFISDGINERFIPHEIKQTATGENVLSVESIDFVVNQSLPYLAIRLDSWRDASTLNIIYNDANYTEAEVQKYLQSIVEFMLAFRL
5MJK , Knot 140 327 0.82 40 188 308 
GSSHHHHHHSSGLVPRGSHMTEKKYDVVIIGSGPAGMTAAMYTARSEMKTLLLERGVPGGQMNNTAEIENYPGYETIMGPELSMKMAEPLEGLGVENAYGFVTGIEDHGDYKKIITEDDEFITKSIIIATGANHRKLEIPGEEEYGARGVSYCAVCDGAFFRNQEILVIGGGDSAVEEALYLTRFGQSVTIMHRRDKLRAQEIIQQRAFKEEKINFIWDSVPMEIKGDDKKIQSVVYKNVKTGEVTEKAFGGIFIYVGLDPVAEFVSDLGITDEAGWIITDDHMRTNIPGIFAVGDVRQKDFRQITTAVGDGAQAAQEAYKFVVELG
3NWX , Knot 52 99 0.80 38 80 96 
PQITLWKRPLVTIRIGGQLKEALLDTGADDTVIEELNLPGCWKPKLIGGXGGFIKVRQYDQIPVEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF

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 \(c\).

\(|P_{f(3FOT_1)}(2) \setminus P_{f(5MJK_1)}(2)|=110\), \(|P_{f(5MJK_1)}(2) \setminus P_{f(3FOT_1)}(2)|=51\). 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 1:101000011111111100100010010010000111011111000000110010110010011110011010010011011110101001001001010001111100001000001001101010101010011011001001011100001000101111001000001110010101110000111011100011101101110011000000100101100100101111001010100110010000000001111000001101011111100110010100010110110001111101001001111111100101001000011101001010011000110010110001111010010010100000001010110010001000000110011101111010010100101011110101011110011000111001000101001101001011100011011101001001001011000100001010001001101111101
Pair \(Z_2\) Length of longest common subsequence
3FOT_1,5MJK_1 161 4
3FOT_1,3NWX_1 203 4
5MJK_1,3NWX_1 178 3

Newick tree

 
[
	3NWX_1:99.94,
	[
		3FOT_1:80.5,5MJK_1:80.5
	]:19.44
]

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 \((0.85)(0.8)(\frac{846 }{\log_{20} 846}-\frac{327}{\log_{20}327})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 3FOT_1 5MJK_1 179 143.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]