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Parikh vectors
9DPP_1 4KTX_1 6VBG_1 Letter Amino acid
4 33 56 F Phenylalanine
9 27 29 V Valine
7 1 8 C Cysteine
10 35 12 K Lycine
7 19 12 P Proline
8 28 29 S Serine
2 2 8 W Tryptophan
2 14 12 R Arginine
2 10 11 Q Glutamine
6 27 11 E Glutamic acid
2 8 14 M Methionine
5 29 33 I Isoleucine
7 36 54 L Leucine
7 30 19 T Threonine
7 19 35 A Alanine
1 38 16 N Asparagine
6 24 6 D Aspartic acid
9 29 34 G Glycine
4 13 4 H Histidine
5 23 14 Y Tyrosine 

9DPP_1|Chain A|Growth/differentiation factor 2|Homo sapiens (9606)
>4KTX_1|Chain A|Botulinum neurotoxin A light chain|Clostridium botulinum A (441771)
>6VBG_1|Chains A, C[auth B]|Galactoside permease|Escherichia coli (562)
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 \(w=f(c)\)
9DPP , Knot 59 110 0.84 40 94 105 
SSGAGSHCQKTSLRVNFEDIGWDSWIIAPKEYEAYECKGGCFFPLADDVTPTKHAIVQTLVHLKFPTKVGKACCVPTKLSPISVLYKDDMGVPTLKYHYEGMSVAECGCR
4KTX , Knot 179 445 0.81 40 216 409 
MGSSHHHHHHSSGLVPRGSHMPFVNKQFNYKDPVNGVDIAYIKIPNAGQMQPVKAFKIHNKIWVIPERDTFTNPEEGDLNPPPEAKQVPVSYYDSTYLSTDNEKDNYLKGVTKLFERIYSTDLGRMLLTSIVRGIPFWGGSTIDTELKVIDTNSINVIQPDGSYRSEELNLVIIGPSADIIQFECKSFGHEVLNLTRNGYGSTQYIRFSPDFTFGFEESLEVDTNPLLGAGKFATDPAVTLAHELIHAGHRLYGIAINPNRVFKVNTNAYYEMSGLEVSFEELRTFGGHDAKFIDSLQENEFRLYYYNKFKDIASTLNKAKSIVGTTASLQYMKNVFKEKYLLSEDTSGKFSVDKLKFDKLYKMLTEIYTEDNFVKFFKVLNRKTYLNFDKAVFKINIVPKVNYTIYDGFNLRNTNLAANFNGQNTEINNMNFTKLKNFTGLFEF
6VBG , Knot 165 417 0.79 40 197 380 
MYYLKNTNFWMFGLFFFFYFFIMGAYFPFFPIWLHDINHISKSDTWIIFAAISLFSLLFQPLFGLLSDKLGLRKYLLWIITGMLVMFAPFFIFIFGPLLQYNILVGSIVGGIYLGFCFNAGAPAVEAFIEKVSRRSNFEFGRARMFGCVGWALCASIVGIMFTINNQFVFWLGSGCALILAVLLFFAKTDAPSSATVANAVGANHSAFSLKLALELFRQPKLWFLSLYVIGVSCTYDVFDQQFANFFTSFFATGEQGTRVFWYVTTMGELLNASIMFFAPLIINRIGGKNALLLAGTIMSVRIIGSSFATSALEVVILKTLHMFEVPFLLVGCFKYITSQFEVRFSATIYLVCFCFFKQLAMIFMSVLAGNMYESIGFQGAYLVLGLVALGFTLISVFTLSGPGPLSLLRRQVNEVA

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 \(c\).

\(|P_{f(9DPP_1)}(2) \setminus P_{f(4KTX_1)}(2)|=34\), \(|P_{f(4KTX_1)}(2) \setminus P_{f(9DPP_1)}(2)|=156\). 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 1:00111000000010101001110011111000010000110111110010100011100110101100110100110010110110000111101000001101100100
Pair \(Z_2\) Length of longest common subsequence
9DPP_1,4KTX_1 190 3
9DPP_1,6VBG_1 181 3
4KTX_1,6VBG_1 153 4

Newick tree

 
[
	9DPP_1:97.60,
	[
		6VBG_1:76.5,4KTX_1:76.5
	]:21.10
]

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 \((0.85)(0.8)(\frac{555 }{\log_{20} 555}-\frac{110}{\log_{20}110})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 9DPP_1 4KTX_1 159 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]