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Parikh vectors
4PIN_1 1QMQ_1 5AIP_1 Letter Amino acid
13 18 3 F Phenylalanine
11 26 12 I Isoleucine
24 33 15 E Glutamic acid
6 13 1 H Histidine
39 42 24 L Leucine
7 12 8 K Lycine
20 20 7 T Threonine
23 24 5 V Valine
6 23 9 Q Glutamine
5 5 1 W Tryptophan
6 9 2 Y Tyrosine
23 25 7 G Glycine
26 26 11 R Arginine
7 13 7 N Asparagine
26 22 8 D Aspartic acid
2 7 1 C Cysteine
5 10 3 M Methionine
12 30 4 P Proline
16 24 5 S Serine
46 32 13 A Alanine

4PIN_1|Chains A, B|Histidine-specific methyltransferase EgtD|Mycobacterium smegmatis (1772)
>1QMQ_1|Chain A|CYTOCHROME P450|PSEUDOMONAS PUTIDA (303)
>5AIP_1|Chains A, B|TRANSCRIPTIONAL REGULATOR, MARR FAMILY|NEISSERIA MENINGITIDIS SEROGROUP B (491)
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 \(w=f(c)\)
4PIN , Knot 133 323 0.79 40 168 300
GHMALSLANYLAADSAAEALRRDVRAGLTATQKSLPPKWFYDAVGSDLFDQITRLPEYYPTRTEAQILRTRSAEIISAAGADTLVELGSGTSEKTRMLLDAMRDAELLRRFIPFDVDAGVLRSAGAAIGAEYPGIEIDAVCGDFEEHLGKIPHVGRRLVVFLGSTIGNLTPAPRAEFLSTLADTLQPGDSLLLGTDLVKDTGRLVRAYDDAAGVTAAFNRNVLAVVNRELSADFDLDAFEHVAKWNSDEERIEMWLRARTAQHVRVAALDLEVDFAAGEEMLTEVSCKFRPENVVAELAEAGLRQTHWWTDPAGDFGLSLAVR
1QMQ , Knot 172 414 0.83 40 218 395
TTETIQSNANLAPLPPHVPEHLVFDFDMYNPSNLSAGVQEAWAVLQESNVPDLVWTRCNGGHWIATRGQLIREAYEDYRHFSSECPFIPREAGEAYDFIPTSMDPPEQRQFRALANQVVGMPVVDKLENRIQELACSLIESLRPQGQCNFTEDYAEPFPIRIFMLLAGLPEEDIPHLKYLTDQMTRPDGSMTFAEAKEALYDYLIPIIEQRRQKPGTDAISIVANGQVNGRPITSDEAKRMCGLLLVGGLDTVVNFLSFSMEFLAKSPEHRQELIERPERIPAACEELLRRFSLVADGRILTSDYEFHGVQLKKGDQILLPQMLSGLDERENAAPMHVDFSRQKVSHTTFGHGSHLCLGQHLARREIIVTLKEWLTRIPDFSIAPGAQIQHKSGIVSGVQALPLVWDPATTKAV
5AIP , Knot 72 146 0.82 40 105 140
MPTQSKHASINIGLIQAREALMTQFRPILNQANITDQQWRIIRLLAENGTLDFQDLANQACILRPSLTGILTRLEKAGLVVRLKPSNDQRRVFLKLTAEGEKLYEEIGEEVDERYDAIEEVLGREKMLLLKDLLAELAKIEDALNS

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 \(c\).

\(|P_{f(4PIN_1)}(2) \setminus P_{f(1QMQ_1)}(2)|=54\), \(|P_{f(1QMQ_1)}(2) \setminus P_{f(4PIN_1)}(2)|=104\). 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 1:10111011001110011011000101110100001110110011100110010011000100001011000010110111100110110100000011101100101100111101011110011111110011101011010100011011011001111110011010111010110011001011001111001100010110100011110111000111110001010101011001101000000101110100100101111010101111001100100010100111011011100001100111011101110
Pair \(Z_2\) Length of longest common subsequence
4PIN_1,1QMQ_1 158 6
4PIN_1,5AIP_1 145 4
1QMQ_1,5AIP_1 173 4

Newick tree

 
[
	1QMQ_1:86.00,
	[
		4PIN_1:72.5,5AIP_1:72.5
	]:13.50
]

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 \((0.85)(0.8)(\frac{737 }{\log_{20} 737}-\frac{323}{\log_{20}323})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 4PIN_1 1QMQ_1 145 126
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]