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Parikh vectors
2CUK_1 1COI_1 6GIR_1 Letter Amino acid
10 1 7 H Histidine
6 0 9 M Methionine
27 1 26 G Glycine
2 1 24 Q Glutamine
25 7 44 E Glutamic acid
51 4 41 L Leucine
29 0 24 R Arginine
8 0 25 I Isoleucine
21 0 15 P Proline
19 0 20 T Threonine
2 0 5 W Tryptophan
5 0 13 Y Tyrosine
13 0 26 D Aspartic acid
5 0 20 N Asparagine
1 0 8 C Cysteine
10 5 42 K Lycine
5 0 24 F Phenylalanine
4 1 21 S Serine
32 4 24 V Valine
36 5 38 A Alanine 

2CUK_1|Chains A, B, C, D|glycerate dehydrogenase/glyoxylate reductase|Thermus thermophilus (300852)
>1COI_1|Chain A|COIL-VALD|synthetic construct (32630)
>6GIR_1|Chain A|Serine--tRNA ligase, cytoplasmic|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
2CUK , Knot 124 311 0.76 40 148 282 
MRVLVTRTLPGKALDRLRERGLEVEVHRGLFLPKAELLKRVEGAVGLIPTVEDRIDAEVMDRAKGLKVIACYSVGVDHVDLEAARERGIRVTHTPGVLTEATADLTLALLLAVARRVVEGAAYARDGLWKAWHPELLLGLDLQGLTLGLVGMGRIGQAVAKRALAFGMRVVYHARTPKPLPYPFLSLEELLKEADVVSLHTPLTPETHRLLNRERLFAMKRGAILLNTARGALVDTEALVEALRGHLFGAGLDVTDPEPLPPGHPLYALPNAVITPHIGSAGRTTRERMAEVAVENLLAVLEGREPPNPVV
1COI , Knot 14 31 0.51 20 18 21 
XEVEALEKKVAALESKVQALEKKVEALEHGX
6GIR , Knot 188 456 0.84 40 234 422 
MVDINLFREEKGNNPEIIRESQRRRFASVEIVDEIIKLDKEWRQRQFEVDSFRKEFNKLNKQVAQLKIKKEDASEIIQQTEKNKQDSTAKEAEVREAYAALKAKLEQVGNLVHDSVPVDKDEANNLVIKLWGEKRFSTPGLKLKNHVDLVELLGIADTKRGAEIAGARGFFLKGDGLMLNQALINFGLTFLKKRGFTGLQPPFFMRKDVMAKCAQLAQFDEELYKVTGEGDDKYLIATAEQPLCAYHIDEWIHPTELPLRYAGYSSCFRKEAGSHGRDTLGIFRVHQFEKIEQFCITGPNENASWEMLDEMMKNSEDFYQALKLPYQIVSIVSGALNDAAAKKYDLEAWFPSSETFRELVSCSNCTDYQARRLEIRYGQKKSNEQTKQYVHMLNSTLTATERTICCILENYQREDGVDIPEVLQPFMGGETFLPFKAKPVVADTKGKKSKAAAALE

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 \(c\).

\(|P_{f(2CUK_1)}(2) \setminus P_{f(1COI_1)}(2)|=139\), \(|P_{f(1COI_1)}(2) \setminus P_{f(2CUK_1)}(2)|=9\). 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 1:10111000111011001000110101001111101011001011111110100010101100101101110001110010101100011010001111001010101111111100110111010011101101011111010110111111101101110011111101100100101110111010011001011010011010000110000111100111110010111100011101101011111101001011111011011101110101101100000011011100111110100110111
Pair \(Z_2\) Length of longest common subsequence
2CUK_1,1COI_1 148 4
2CUK_1,6GIR_1 162 4
1COI_1,6GIR_1 224 4

Newick tree

 
[
	6GIR_1:10.45,
	[
		2CUK_1:74,1COI_1:74
	]:30.45
]

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 \((0.85)(0.8)(\frac{342 }{\log_{20} 342}-\frac{31}{\log_{20}31})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 2CUK_1 1COI_1 117 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]