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Parikh vectors
6NMJ_1 1KWU_1 3KGF_1 Letter Amino acid
73 10 47 L Leucine
10 4 13 M Methionine
43 13 55 A Alanine
39 4 34 R Arginine
9 4 5 C Cysteine
40 14 29 E Glutamic acid
19 4 28 P Proline
22 10 21 S Serine
1 2 5 W Tryptophan
10 10 15 N Asparagine
16 3 21 Q Glutamine
20 10 33 G Glycine
13 4 16 H Histidine
17 14 9 K Lycine
17 7 12 F Phenylalanine
23 11 27 T Threonine
9 2 9 Y Tyrosine
27 6 30 D Aspartic acid
11 7 19 I Isoleucine
34 10 36 V Valine 

6NMJ_1|Chains A, B|Resistance to inhibitors of cholinesterase 8 homolog A (C. elegans)|Rattus norvegicus (10116)
>1KWU_1|Chains A, B, C|MANNOSE-BINDING PROTEIN A|Rattus norvegicus (10116)
>3KGF_1|Chains A, B|Probable 3-deoxy-D-arabino-heptulosonate 7-phosphate synthase AroG|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
6NMJ , Knot 180 453 0.81 40 214 411 
GMEPRAVADALETGEEDAVTEALRSFNREHSQSFTFDDAQQEDRKRLAKLLVSVLEQGLSPKHRVTWLQTIRILSRDRSCLDSFASRQSLHALACYADIAISEEPIPQPPDMDVLLESLKCLCNLVLSSPTAQMLAAEARLVVRLAERVGLYRKRSYPHEVQFFDLRLLFLLTALRTDVRQQLFQELHGVRLLTDALELTLGVAPKENPLVILPAQETERAMEILKVLFNITFDSVKREVDEEDAALYRYLGTLLRHCVMADAAGDRTEEFHGHTVNLLGNLPLKCLDVLLALELHEGSLEFMGVNMDVINALLAFLEKRLHQTHRLKECVAPVLSVLTECARMHRPARKFLKAQVLPPLRDVRTRPEVGDLLRNKLVRLMTHLDTDVKRVAAEFLFVLCSESVPRFIKYTGYGNAAGLLAARGLMAGGRPEGQYSEDEDTDTEEYREAKASI
1KWU , Knot 72 149 0.80 40 121 146 
AIEVKLANMEAEINTLKSKLELTNKLHAFSMGKKSGKKFFVTNHERMPFSKVKALCSELRGTVAIPRNAEENKAIQEVAKTSAFLGITDEVTEGQFMYVTGGRLTYSNWKKDEPNDHGSGEDCVTIVDNGLWNDISCQASHTAVCEFPA
3KGF , Knot 192 464 0.84 40 235 437 
GAMNWTVDIPIDQLPSLPPLPTDLRTRLDAALAKPAAQQPTWPADQALAMRTVLESVPPVTVPSEIVRLQEQLAQVAKGEAFLLQGGDCAETFMDNTEPHIRGNVRALLQMAVVLTYGASMPVVKVARIAGQYAKPRSADIDALGLRSYRGDMINGFAPDAAAREHDPSRLVRAYANASAAMNLVRALTSSGLASLHLVHDWNREFVRTSPAGARYEALATEIDRGLRFMSACGVADRNLQTAEIYASHEALVLDYERAMLRLSDGDDGEPQLFDLSAHTVWIGERTRQIDGAHIAFAQVIANPVGVKLGPNMTPELAVEYVERLDPHNKPGRLTLVSRMGNHKVRDLLPPIVEKVQATGHQVIWQCDPMHGNTHESSTGFKTRHFDRIVDEVQGFFEVHRALGTHPGGIHVEITGENVTECLGGAQDISETDLAGRYETACDPRLNTQQSLELAFLVAEMLRD

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 \(c\).

\(|P_{f(6NMJ_1)}(2) \setminus P_{f(1KWU_1)}(2)|=134\), \(|P_{f(1KWU_1)}(2) \setminus P_{f(6NMJ_1)}(2)|=41\). 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 1:110101110110010001100110010000000101001000000011011101100110100010110010110000001001100001011100101110001110110101110010010011100101011110101110110011100000010010110101111101100010001100101101100110101111100011111110000011011011101010010001000011100011011000111011100000101001011101110010111110100101011110101101111110001000001000111110110001010011001101011111001000101101100011011001000100111011111000011011000101011111110111111010100000000000000010101
Pair \(Z_2\) Length of longest common subsequence
6NMJ_1,1KWU_1 175 4
6NMJ_1,3KGF_1 129 4
1KWU_1,3KGF_1 188 4

Newick tree

 
[
	1KWU_1:98.02,
	[
		6NMJ_1:64.5,3KGF_1:64.5
	]:33.52
]

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 \((0.85)(0.8)(\frac{602 }{\log_{20} 602}-\frac{149}{\log_{20}149})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 6NMJ_1 1KWU_1 162 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]