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Parikh vectors
6KRX_1 4WMW_1 7CQN_1 Letter Amino acid
16 44 16 K Lycine
30 23 22 S Serine
4 11 9 W Tryptophan
9 17 10 Y Tyrosine
9 7 19 H Histidine
29 43 44 L Leucine
21 33 33 D Aspartic acid
7 1 5 C Cysteine
21 31 17 I Isoleucine
17 23 26 P Proline
26 29 20 T Threonine
22 33 24 V Valine
15 18 22 R Arginine
19 25 14 N Asparagine
16 37 22 E Glutamic acid
11 9 14 M Methionine
19 42 39 G Glycine
12 22 20 F Phenylalanine
21 55 63 A Alanine
16 15 11 Q Glutamine 

6KRX_1|Chain A|Protein-tyrosine-phosphatase PTP1|Arabidopsis thaliana (3702)
>4WMW_1|Chain A|MBP-MCL1 chimera protein|Escherichia coli (83333)
>7CQN_1|Chains A, B, C|Type III glutamate--ammonia ligase|Rhodovulum sp. 12E13 (2203891)
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 \(w=f(c)\)
6KRX , Knot 145 340 0.82 40 212 322 
MATGKTSSAANLFTGSTRFDLSSADSPPSKLSLSSDQLNHCHQALGVFRGKIQNPDSIAHEFTGLQANRMWPSELLLNSTVAMNSVNVEKNRYSDVVPFDKNRIVLNPCKDSSAKGYVNASLIKTSESESISQFIATQGPLPHTMEDFWEMVIQQHCPIIVMLTRLVDNNRTVKCGDYFQDEDGPREFGNISLTTKWIKTTDTSLMLRNLEVNYKETEDQPMSVLHIQYPEWPDHGVPKDTVAVREILKRLYQVPPSLGPIIVHCSAGIGRTGTYCAIHNTIQRILAGDMSALDLAKTVALFRKQRIGMVQTMDQYFFCYNAIVDELEDLTAGTNAGTSS
4WMW , Knot 207 518 0.83 40 244 476 
GKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTGSELYRQSLEIISRYLREQATGAADTAPMGASGATSRKALETLRRVGDGVQRNHETAFQGMLRKLDIKNEDDVKSLSRVMIHVFSDGVTNWGRIVTLISFGAFVAKHLKTINQESCIEPLAESITDVLVRTKRDWLVKQRGWDGFVEFFHV
7CQN , Knot 186 450 0.84 40 231 417 
MGSSHHHHHHSSGLVPRGSHMTDLASIAREKGIEFFLISFTDLLGVQRAKLVPARAIADMAVNGAGFAGFAAWLDMSPADADILAIPDPESLIQLPWKPSVGWLAADVHFEGRPFPKAPRVALKSVLARAAGKDMHLKHGVECEFFLIQPDGSAISDPADTQAKPCYDQDALMRRFDVIAEICSYMVDLGWGPYQNDHEDANGQFEMNWDYADALVTADRHAFFKFMVKSVAERHGLRATFMPKPFAHLTGNGCHTHLSMWTAAGDNLFEGDGELGLSPTAYAFLGGLIGHAKGLTAVVNPTVNSYKRLNAPVTVSGATWSPNTITYGGNNRTHMVRIPDAGRLELRLPDGAANPYLMPAAILAAGLDGIETQADPGQRLDIDMYVEGHSVEAEQLPLNLLDAVRALEADEVLAGGLGAAAAAFAKFKRAEWADYKSQLTEWERRTTLDC

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 \(c\).

\(|P_{f(6KRX_1)}(2) \setminus P_{f(4WMW_1)}(2)|=72\), \(|P_{f(4WMW_1)}(2) \setminus P_{f(6KRX_1)}(2)|=104\). 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 1:1101000011011010001010010011001010000100000111110101001001100101101001110011100011100101000000011110000111010000010101010110000000100111001111001001101110000111111001100000100100100001100110101000110000001110010100000000110110100101100111000111001100100111011111100011110010001100010011110101101100111100001111001000110001110010010110011000
Pair \(Z_2\) Length of longest common subsequence
6KRX_1,4WMW_1 176 4
6KRX_1,7CQN_1 165 4
4WMW_1,7CQN_1 163 5

Newick tree

 
[
	6KRX_1:86.52,
	[
		7CQN_1:81.5,4WMW_1:81.5
	]:5.02
]

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 \((0.85)(0.8)(\frac{858 }{\log_{20} 858}-\frac{340}{\log_{20}340})=139.\)
Status Protein1 Protein2 d d1/2
Query variables 6KRX_1 4WMW_1 178 145.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]