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Parikh vectors
6SYE_1 6NPC_1 5FLU_1 Letter Amino acid
7 21 13 T Threonine
12 34 17 A Alanine
14 16 10 N Asparagine
1 10 2 H Histidine
8 32 9 L Leucine
6 11 12 K Lycine
2 23 5 P Proline
6 21 17 V Valine
11 27 0 R Arginine
6 15 7 I Isoleucine
2 6 1 M Methionine
3 20 8 F Phenylalanine
10 28 12 S Serine
3 12 2 Y Tyrosine
7 30 11 D Aspartic acid
8 3 2 C Cysteine
3 8 7 Q Glutamine
2 26 6 E Glutamic acid
12 34 22 G Glycine
6 6 0 W Tryptophan 

6SYE_1|Chain A|Lysozyme C|Gallus gallus (9031)
>6NPC_1|Chains A, B|TmpA, 2-trimethylaminoethylphosphonate hydroxylase|Leisingera caerulea (506591)
>5FLU_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M|PAP FIMBRIAL MAJOR PILIN PROTEIN|ESCHERICHIA COLI (562)
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 \(w=f(c)\)
6SYE , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
6NPC , Knot 160 383 0.82 40 214 362 
HHHHHHMPRSVTADASGSFLTLTFEDGSESRFHAIWLRDNALDPETRSPGNGQRLITIGDIPADTRISTALVDDGALTVTFAPEGKTVTFPGKWLKSNAYDTDQSSEVGRTSPDVETWDSSQPAPAFDWNEVQSDPKAKRDWLDAIARLGFAKLVNGPVREGALIECASMFGFVRETNYGKYFEVRTEVNPTNLAYTGLGLQAHTDNPYRDPVPSLQILYCLENSAEGGDSIVVDGFRAAERLRDEDPEGFALLAGNPARFEYKGSDGVHLRARRPMIELSPDGEMIAIRFNNRSSAPFVDIPFEKMEAYYAAYRRLGEFIDDPEMGVSFKLEPGESFIVDNTRVLHARLGYSGSGSRWLQGCYADKDGLFSTLNVLNAQLGG
5FLU , Knot 76 163 0.79 36 112 159 
APTIPQGQGKVTFNGTVVDAPCSISQKSADQSIDFGQLSKSFLEAGGVSKPMDLDIELVNCDITAFKGGNGAKKGTVKLAFTGPIVNGHSDELDTNGGTGTAIVVQGAGKNVVFDGSEGDANTLKDGENVLHYTAVVKKSSAVGAAVTEGAFSAVANFNLTYQ

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 \(c\).

\(|P_{f(6SYE_1)}(2) \setminus P_{f(6NPC_1)}(2)|=41\), \(|P_{f(6NPC_1)}(2) \setminus P_{f(6SYE_1)}(2)|=151\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
6SYE_1,6NPC_1 192 3
6SYE_1,5FLU_1 144 4
6NPC_1,5FLU_1 158 4

Newick tree

 
[
	6NPC_1:92.61,
	[
		6SYE_1:72,5FLU_1:72
	]:20.61
]

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 \((0.85)(0.8)(\frac{512 }{\log_{20} 512}-\frac{129}{\log_{20}129})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 6SYE_1 6NPC_1 142 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]