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Parikh vectors
5LKW_1 7LNR_1 4PKF_1 Letter Amino acid
12 11 76 A Alanine
8 23 71 E Glutamic acid
12 17 61 L Leucine
9 20 59 S Serine
7 20 43 V Valine
6 31 35 N Asparagine
4 7 26 M Methionine
6 4 41 P Proline
1 6 15 W Tryptophan
0 5 13 C Cysteine
12 5 33 Q Glutamine
6 43 62 K Lycine
6 16 33 Y Tyrosine
8 7 46 R Arginine
9 21 52 D Aspartic acid
7 20 61 G Glycine
10 3 26 H Histidine
2 30 53 I Isoleucine
2 11 33 F Phenylalanine
3 12 39 T Threonine 

5LKW_1|Chains A, B|Putative OmpA family lipoprotein|Burkholderia cenocepacia (strain ATCC BAA-245 / DSM 16553 / LMG 16656 / NCTC 13227 / J2315 / CF5610) (216591)
>7LNR_1|Chain A|Beta-lactamase|Clostridioides difficile (1496)
>4PKF_1|Chain A|TutD|Thauera aromatica (59405)
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 \(w=f(c)\)
5LKW , Knot 62 130 0.77 38 100 123 
MGVDPLNDPNSPLAKRSIYFDFDSYSVKDEYQPLMQQHAQYLKSHPQRHVLIQGNTDERGTSEYNLALGQKRAEAVRRAMALLGVNDSQMEAVSLGKEKPQATGHDEASWAQNRRADLVYQQLEHHHHHH
7LNR , Knot 132 312 0.81 40 180 289 
MKRKKNFIWIAILLVGVVMVMYYAVKKHNDINQKTDKNYIKSELNKSKIKNNDKNKKESVNIVDYSDCFEGISGGAIFCNTKNKEYNIYNKELIETRRSPCSTFKIVSTLIGLEKGVINSKESVMGYDGTDYPNKNWNKNLSLEEAFKESCVWYYKKLINKVDAKSVQNILDDLKYGNCDISEWEGDLKNGKGHLNGFWLESSLQISPKEQVQTMAKIFEGDTNFKKEHINILRDIMAIDVNDANINVYGKTGTGFDEKNKRVDAWFVGMLEREGDTYYFAIKSDDSNKEITGPKVKEIAINIIKKYYSVRE
4PKF , Knot 334 878 0.86 40 320 820 
MNDIVSAKVLEYKGKKLNFTPEDPAEETIPADELHEHLQKPSTARTKRLKERCRWKHASAGEFIEKSVTAGIERMRYLTEAHKASEGKPEAIRRALGLANVLNKSTLVLQEDEFIVGYHAEDPNMFPLYPELSHMAVQDYLRSDYSPQPADEAAAINEYWKPHSLQSKCQPYFDPADLGRMYQVSSMEAPSFASGYNSIVPPYETVLEDGLLARIKLAEKHIAEAQADMSTFPWNGTKGLDNIAKIDNWKAMVIACKAVISWARRQGRLCKIVAENFETDPKRQAELLEIADICQRIPAEPCKGLKDAMQAKFFTFLICHAIERYASGYAQKEDTLLWPYYKASVVDKKFQPMSHMDAVELVEMERLKISEHGAGKSRAYREIFPGSNDLFILTVGGTNAKGEDACNDMTDAILEAAKRIRTAEPSIVFRYSKKNREKTLRWVFECIRDGLGYPSIKHDEIGTEQMKEYAKFSLNGNGATDEEAHNWVNVLCMSPGIHGRRKTQKTRSEGGGSIFPAKLLEISLNDGYDWSYADMQLGPKTGDLSSLKSFEDVWEAFRKQYQYAINLCISTKDVSRYFEQRFLQMPFVSAIDDGCMELGMDACALSEQPNGWHNPITTIVAANSLVAIKKLVFEEKKYTLEQLSQALKANWEGFEEMRVDFKRAPKWGNDDDYADGIITRFYEEIIGGEMRKITNYSGGPVMPTGQAVGLYMEVGSRTGPTPDGRFGGEAADDGGISPYMGTDKKGPTAVLRSVSKVQKNQKGNLLNQRLSVPIMRSKHGFEIWNSYIKTWHDLNIDHVQFNVVSTDEMRAAQREPEKHHDLIVRVSGYSARFVDIPTYGQNTIIARQEQDFSASDLEFLNVEISGTGSGSSHHHHHH

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 \(c\).

\(|P_{f(5LKW_1)}(2) \setminus P_{f(7LNR_1)}(2)|=48\), \(|P_{f(7LNR_1)}(2) \setminus P_{f(5LKW_1)}(2)|=128\). 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 1:1110110010011100010101000010000011100010010001000111010000010000011110001011001111111000010110110001010100010110000101100010000000
Pair \(Z_2\) Length of longest common subsequence
5LKW_1,7LNR_1 176 3
5LKW_1,4PKF_1 248 6
7LNR_1,4PKF_1 194 4

Newick tree

 
[
	4PKF_1:11.07,
	[
		5LKW_1:88,7LNR_1:88
	]:30.07
]

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 \((0.85)(0.8)(\frac{442 }{\log_{20} 442}-\frac{130}{\log_{20}130})=93.4\)
Status Protein1 Protein2 d d1/2
Query variables 5LKW_1 7LNR_1 117 82
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]