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Parikh vectors
7OSI_1 5FLK_1 1SXP_1 Letter Amino acid
10 6 0 M Methionine
17 31 0 P Proline
35 10 0 S Serine
16 8 0 Y Tyrosine
28 18 0 R Arginine
2 4 1 C Cysteine
8 17 0 H Histidine
15 14 0 F Phenylalanine
7 9 0 W Tryptophan
28 21 6 A Alanine
26 19 0 D Aspartic acid
23 9 3 T Threonine
43 20 3 G Glycine
24 19 0 I Isoleucine
22 24 0 E Glutamic acid
37 29 0 L Leucine
9 8 0 K Lycine
30 21 0 V Valine
30 9 0 N Asparagine
26 4 0 Q Glutamine 

7OSI_1|Chain A|Probable ABC transporter binding protein NosD|Pseudomonas stutzeri ATCC 14405 = CCUG 16156 (32042)
>5FLK_1|Chain A|DHAA101|RHODOCOCCUS RHODOCHROUS (1829)
>1SXP_1|Chain A[auth C]|5'-D(*A*AP*TP*AP*CP*TP*AP*AP*GP*AP*TP*AP*G)-3'|
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 \(w=f(c)\)
7OSI , Knot 178 436 0.82 40 218 411 
MFKAQATFSRYSAAVSLLLLFSGAAQAAPQSITTLPLQPDGENRWRLPAGEYQGQFTIEQPMQLRCEPGAVIQSQGQGSSLLISAPDVLVEGCTLYEWGSDLTAMDSAVFILPAAERAQISNNRMRGPGFGVFVDGTRDVQVIGNEIDGDAGVRSQDRGNGIHLFAVSGARVLHNHVRNARDGIYIDTSNGNHLEGNVIEDVRYGVHYMFANENSLIDNVTRRTRTGYALMQSRKLTVTGNRSEQDQNYGILMNYITYSTITGNFVSDVQRGDTGGDSMISGGEGKALFIYNSLFNTIENNHFEKSSLGIHLTAGSEDNRISGNAFVGNQQQVKYVASRTQEWSVDGRGNYWSDYLGWDRNNDGLGDIAYEPNDNVDRLLWLYPQVRLLMNSPSIEVLRWVQRAFPVIKSPGVQDSHPLMKLPTEKLLTEKQEPTS
5FLK , Knot 131 300 0.83 40 184 282 
MSEIGTGFPFDPHYVEVLGSRMHYVDVGPRDGTPVLFLHGNPTSSYLWRNIIPHVAPSHRCIAPDLIGMGKSDKPDLDYRFDDHVRYLDAFIEALGLEEVVLVIHDWGSALGFHWAKRNPERVKGIACMEFIRPIPTWDEWPEFARETFQAFRTPDVGRELIIDQNAFIEGALPKCVVRPLTEVEMDHYREPFLKPVDREPLWRFPNELPIAGEPANIVALVEAYMNWLHQSPVPKLLFWGTPGVLIPPAEAARLAESLPNCKTVDIGPGLHYLQEDNPDLIGSEIARWLPALEHHHHHH
1SXP , Knot 6 13 0.39 8 7 9 
GATACTAAGATAG

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 \(c\).

\(|P_{f(7OSI_1)}(2) \setminus P_{f(5FLK_1)}(2)|=105\), \(|P_{f(5FLK_1)}(2) \setminus P_{f(7OSI_1)}(2)|=71\). 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 1:1101010100001110111110111011100100111010100010111100010101001101000111110001010011101101110100100110010110011111111001010000101111111101000101110010101110000010110111101101100010010011010000100101011001001100111000011001000000101110000101010000000001111001000010101100100100110011011010111100011001000010000111010110000010101111000010011000001010101001000111000001110110010001001111010101110010101101100111110011100001110110001100000100
Pair \(Z_2\) Length of longest common subsequence
7OSI_1,5FLK_1 176 3
7OSI_1,1SXP_1 213 3
5FLK_1,1SXP_1 183 2

Newick tree

 
[
	1SXP_1:10.76,
	[
		7OSI_1:88,5FLK_1:88
	]:14.76
]

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 \((0.85)(0.8)(\frac{736 }{\log_{20} 736}-\frac{300}{\log_{20}300})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 7OSI_1 5FLK_1 155 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]