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Parikh vectors
1FWE_1 1RMP_1 3HDH_1 Letter Amino acid
5 19 31 K Lycine
3 12 15 F Phenylalanine
5 15 23 T Threonine
0 0 1 W Tryptophan
7 6 7 R Arginine
2 9 7 H Histidine
6 12 14 I Isoleucine
12 19 28 L Leucine
1 10 6 Y Tyrosine
4 17 16 D Aspartic acid
6 21 20 G Glycine
5 4 7 M Methionine
9 17 26 V Valine
2 13 9 N Asparagine
2 8 7 Q Glutamine
5 10 25 S Serine
9 8 28 A Alanine
0 2 1 C Cysteine
11 15 19 E Glutamic acid
6 10 12 P Proline 

1FWE_1|Chain A|UREASE|Klebsiella aerogenes (548)
>1RMP_1|Chain A|SIGF1-GFP fusion protein|Aequorea victoria (6100)
>3HDH_1|Chains A, B, C|PROTEIN (L-3-HYDROXYACYL COA DEHYDROGENASE)|Sus scrofa (9823)
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 \(w=f(c)\)
1FWE , Knot 51 100 0.78 36 79 96 
MELTPREKDKLLLFTAALVAERRLARGLKLNYPESVALISAFIMEGARDGKSVASLMEEGRHVLTREQVMEGVPEMIPDIQVEATFPDGSKLVTVHNPII
1RMP , Knot 106 228 0.84 40 152 215 
SKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPXPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYIMADKQKNGIKVNFKIRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSALSKDPNEKRDHMVLLEFVTAAGI
3HDH , Knot 130 302 0.82 40 181 285 
SSSSTAAASAKKILVKHVTVIGGGLMGAGIAQVAAATGHTVVLVDQTEDILAKSKKGIEESLRKVAKKKFAENPKAGDEFVEKTLSSISTSTDAASVVHSTDLVVEAIVENLKVKSELFKRLDKFAAEHTIFASNTSSLQITSLANATTRQDRFAGLHFFNPVPLMKLVEVVKTPMTSQKTLESLVDFSKTLGKHPVSCKDTPGFIVNRLLVPYLIEAVRLYERGDASKEDIDTAMKLGAGYPMGPFELLDYVGLDTTKFIIDGWHEMDSQNPLFQPSPAMNKLVAENKFGKKTGEGFYKYK

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 \(c\).

\(|P_{f(1FWE_1)}(2) \setminus P_{f(1RMP_1)}(2)|=35\), \(|P_{f(1RMP_1)}(2) \setminus P_{f(1FWE_1)}(2)|=108\). 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 1:1010100000111101111100011011010010011110111101100100110110010011000011011101110101010110100110100111
Pair \(Z_2\) Length of longest common subsequence
1FWE_1,1RMP_1 143 3
1FWE_1,3HDH_1 158 4
1RMP_1,3HDH_1 165 3

Newick tree

 
[
	3HDH_1:83.63,
	[
		1FWE_1:71.5,1RMP_1:71.5
	]:12.13
]

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 \((0.85)(0.8)(\frac{328 }{\log_{20} 328}-\frac{100}{\log_{20}100})=71.1\)
Status Protein1 Protein2 d d1/2
Query variables 1FWE_1 1RMP_1 89 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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