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Parikh vectors
5NVL_1 6LPX_1 7ADE_1 Letter Amino acid
2 10 1 C Cysteine
15 63 44 L Leucine
9 16 16 F Phenylalanine
10 25 17 P Proline
12 32 21 S Serine
9 34 31 A Alanine
15 28 30 R Arginine
12 26 36 E Glutamic acid
13 16 31 I Isoleucine
6 3 1 W Tryptophan
9 15 11 Q Glutamine
14 50 29 G Glycine
6 23 8 H Histidine
7 9 16 M Methionine
12 25 23 T Threonine
8 13 21 N Asparagine
8 21 23 D Aspartic acid
9 17 28 K Lycine
6 13 7 Y Tyrosine
7 42 25 V Valine 

5NVL_1|Chains A, C|Eukaryotic translation initiation factor 4E type 2|Homo sapiens (9606)
>6LPX_1|Chains A, B|D-2-hydroxyglutarate dehydrogenase, mitochondrial|Homo sapiens (9606)
>7ADE_1|Chains A[auth a], B[auth b], C[auth c], D[auth d], E[auth e], F[auth f]|Transcription termination factor Rho|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 \(w=f(c)\)
5NVL , Knot 94 189 0.87 40 148 185 
GPHMLEAEHPLQYNYTFWYSRRTPGRPTSSQSYEQNIKQIGTFASVEQFWRFYSHMVRPGDLTGHSDFHLFKEGIKPMWEDDANKNGGKWIIRLRKGLASRCWENLILAMLGEQFMVGEEICGAVVSVRFQEDIISIWNKTASDQATTARIRDTLRRVLNLPPNTIMEYKTHTDSIKMPGRLGPQRLLF
6LPX , Knot 192 481 0.82 40 228 450 
GPGSRYPVRRLPFSTVSKQDLAAFERIVPGGVVTDPEALQAPNVDWLRTLRGCSKVLLRPRTSEEVSHILRHCHERNLAVNPQGGNTGMVGGSVPVFDEIILSTARMNRVLSFHSVSGILVCQAGCVLEELSRYVEERDFIMPLDLGAKGSCHIGGNVATNAGGLRFLRYGSLHGTVLGLEVVLADGTVLDCLTSLRKDNTGYDLKQLFIGSEGTLGIITTVSILCPPKPRAVNVAFLGCPGFAEVLQTFSTCKGMLGEILSAFEFMDAVCMQLVGRHLHLASPVQESPFYVLIETSGSNAGHDAEKLGHFLEHALGSGLVTDGTMATDQRKVKMLWALRERITEALSRDGYVYKYDLSLPVERLYDIVTDLRARLGPHAKHVVGYGHLGDGNLHLNVTAEAFSPSLLAALEPHVYEWTAGQQGSVSAEHGVGFRKRDVLGYSKPPGALQLMQQLKALLDPKGILNPYKTLPSQAHHHHHH
7ADE , Knot 177 419 0.85 40 219 405 
MNLTELKNTPVSELITLGENMGLENLARMRKQDIIFAILKQHAKSGEDIFGDGVLEILQDGFGFLRSADSSYLAGPDDIYVSPSQIRRFNLRTGDTISGKIRPPKEGERYFALLKVNEVNFDKPENARNKILFENLTPLHANSRLRMERGNGSTEDLTARVLDLASPIGRGQRGLIVAPPKAGKTMLLQNIAQSIAYNHPDCVLMVLLIDERPEEVTEMQRLVKGEVVASTFDEPASRHVQVAEMVIEKAKRLVEHKKDVIILLDSITRLARAYNTVVPASGKVLTGGVDANALHRPKRFFGAARNVEEGGSLTIIATALIDTGSKMDEVIYEEFKGTGNMELHLSRKIAEKRVFPAIDYNRSGTRKEELLTTQEELQKMWILRKIIHPMGEIDAMEFLINKLAMTKTNDDFFEMMKRS

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 \(c\).

\(|P_{f(5NVL_1)}(2) \setminus P_{f(6LPX_1)}(2)|=53\), \(|P_{f(6LPX_1)}(2) \setminus P_{f(5NVL_1)}(2)|=133\). 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 1:110110100110000011000001101000000000100110110100110100011011010100010110011011100010001101110100111000100111111100111100101111010100011011000100010010100010011011100110000000010111011100111
Pair \(Z_2\) Length of longest common subsequence
5NVL_1,6LPX_1 186 4
5NVL_1,7ADE_1 175 3
6LPX_1,7ADE_1 157 4

Newick tree

 
[
	5NVL_1:93.89,
	[
		7ADE_1:78.5,6LPX_1:78.5
	]:15.39
]

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 \((0.85)(0.8)(\frac{670 }{\log_{20} 670}-\frac{189}{\log_{20}189})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 5NVL_1 6LPX_1 170 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]