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Parikh vectors
3VSH_1 8GMK_1 7PTZ_1 Letter Amino acid
17 23 16 D Aspartic acid
17 23 5 E Glutamic acid
26 33 18 G Glycine
13 26 20 P Proline
15 17 20 S Serine
32 55 22 A Alanine
15 33 7 R Arginine
10 16 16 I Isoleucine
8 11 6 F Phenylalanine
8 8 10 Y Tyrosine
6 12 8 H Histidine
13 16 16 T Threonine
6 7 6 W Tryptophan
8 5 17 N Asparagine
3 3 2 C Cysteine
6 10 8 Q Glutamine
27 28 11 L Leucine
11 5 3 K Lycine
4 11 1 M Methionine
26 37 23 V Valine 

3VSH_1|Chains A, C|2-amino-5-chlorophenol 1,6-dioxygenase alpha subunit|Comamonas testosteroni (543891)
>8GMK_1|Chains A, B|citrate synthase|Mycobacterium tuberculosis H37Rv (83332)
>7PTZ_1|Chain A[auth AAA]|Auxiliary activity 9|Lentinus similis (292560)
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 \(w=f(c)\)
3VSH , Knot 121 271 0.83 40 177 259 
MTVVSAFLVPGTPLPQLKPEVPSWGQLAAATERAGKALAASRPDVVLVYSTQWLAVLDQQWLTRPRSEGVHVDENWYEFGDLAYDIRADTALAEACVTSSPLHGVHARGVNYDGFPIDTGTITACTLMGIGTDAFPLVVGSNNLYHSGEITEKLAALAVDCAKDQNKRVAVVGVGGLSGSLFREEIDPREDRIANEEDDKWNRRVLKLIEAGDVSALREAMPVYAKEARVDMGFKHLHWILGALKGKFSGANVLGYGPSYGSGAAVIEFRL
8GMK , Knot 157 379 0.82 40 193 352 
MTVVPENFVPGLDGVVAFTTEIAEPDKDGGALRYRGVDIEDLVSQRVTFGDVWALLVDGNFGSGLPPAEPFPLPIHSGDVRVDVQAGLAMLAPIWGYAPLLDIDDATARQQLARASVMALSYVAQSARGIYQPAVPQRIIDECSTVTARFMTRWQGEPDPRHIEAIDAYWVSAAEHGMNASTFTARVIASTGADVAAALSGAIGAMSGPLHGGAPARVLPMLDEVERAGDARSVVKGILDRGEKLMGFGHRVYRAEDPRARVLRAAAERLGAPRYEVAVAVEQAALSELRERRPDRAIETNVEFWAAVVLDFARVPANMMPAMFTCGRTAGWCAHILEQKRLGKLVRPSAIYVGPGPRSPESVDGWERVLTTAHHHHHH
7PTZ , Knot 105 235 0.81 40 154 226 
HTLVWGVWVNGVDQGDGRNIYIRSPPNNNPVKNLTSPDMTCNVDNRVVPKSVPVNAGDTLTFEWYHNTRDDDIIASSHHGPIAVYIAPAASNGQGNVWVKLFEDAYNVTNSTWAVDRLITAHGQHSVVVPHVAPGDYLFRAEIIALHEADSLYSQNPIRGAQFYISCAQITINSSDDSTPLPAGVPFPGAYTDSTPGIQFNIYTTPATSYVAPPPSVWSGALGGSIAQVGDASLE

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 \(c\).

\(|P_{f(3VSH_1)}(2) \setminus P_{f(8GMK_1)}(2)|=63\), \(|P_{f(8GMK_1)}(2) \setminus P_{f(3VSH_1)}(2)|=79\). 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 1:1011011111101110101011011011110001101111001011110000111110001100100011010001001101100101001110101000110110101100011110010101001111100111111100010001010001111110010000001111111110101100010100001100000010001101101101011001111010010101110010111111010101101110110010111110101
Pair \(Z_2\) Length of longest common subsequence
3VSH_1,8GMK_1 142 4
3VSH_1,7PTZ_1 153 4
8GMK_1,7PTZ_1 175 4

Newick tree

 
[
	7PTZ_1:85.58,
	[
		3VSH_1:71,8GMK_1:71
	]:14.58
]

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 \((0.85)(0.8)(\frac{650 }{\log_{20} 650}-\frac{271}{\log_{20}271})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 3VSH_1 8GMK_1 126 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]