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Parikh vectors
9JBM_1 5XHW_1 4QZG_1 Letter Amino acid
49 4 16 Q Glutamine
57 26 15 I Isoleucine
60 4 25 R Arginine
30 14 10 Y Tyrosine
95 15 23 V Valine
57 13 21 F Phenylalanine
102 8 26 A Alanine
27 7 10 N Asparagine
46 22 24 D Aspartic acid
27 4 7 C Cysteine
65 23 25 G Glycine
27 10 15 H Histidine
22 4 12 M Methionine
66 8 15 P Proline
13 0 5 W Tryptophan
67 13 32 E Glutamic acid
151 17 40 L Leucine
58 21 28 K Lycine
82 17 31 S Serine
62 7 20 T Threonine 

9JBM_1|Chain A|Endoplasmic reticulum transmembrane helix translocase|Homo sapiens (9606)
>5XHW_1|Chain A|Putative 6-deoxy-D-mannoheptose pathway protein|Yersinia pseudotuberculosis (633)
>4QZG_1|Chain A|DNA nucleotidylexotransferase|Mus musculus (10090)
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 \(w=f(c)\)
9JBM , Knot 415 1163 0.84 40 335 996 
LAAGPALIANGDELVAAVWPYRRLALLRRLTVLPFAGLLYPAWLGAAAAGCWGWGSSWVQIPEAALLVLATICLAHALTVLSGHWSVHAHCALTCTPEYDPSKATFVKVVPTPNNGSTELVALHRNEGEDGLEVLSFEFQKIKYSYDALEKKQFLPVAFPVGNAFSYYQSNRGFQEDSEIRAAEKKFGSNKAEMVVPDFSELFKERATAPFFVFQVFCVGLWCLDEYWYYSVFTLSMLVAFEASLVQQQMRNMSEIRKMGNKPHMIQVYRSRKWRPIASDEIVPGDIVSIGRSPQENLVPCDVLLLRGRCIVDEAMLTGESVPQMKEPIEDLSPDRVLDLQADSRLHVIFGGTKVVQHIPPQKATTGLKPVDSGCVAYVLRTGFNTSQGKLLRTILFGVKRVTANNLETFIFILFLLVFAIAAAAYVWIEGTKDPSRNRYKLFLECTLILTSVVPPELPIELSLAVNTSLIALAKLYMYCTEPFRIPFAGKVEVCCFDKTGTLTSDSLVVRGVAGLRDGKEVTPVSSIPVETHRALASCHSLMQLDDGTLVGDPLEKAMLTAVDWTLTKDEKVFPRSIKTQGLKIHQRFHFASALKRMSVLASYEKLGSTDLCYIAAVKGAPETLHSMFSQCPPDYHHIHTEISREGARVLALGYKELGHLTHQQAREVKREALECSLKFVGFIVVSCPLKADSKAVIREIQNASHRVVMITGDNPLTACHVAQELHFIEKAHTLILQPPSEKGRQCEWRSIDGSIVLPLARGSPKALALEYALCLTGDGLAHLQATDPQQLLRLIPHVQVFARVAPKQKEFVITSLKELGYVTLMCGDGTNDVGALKHADVGVALLANAPERVVERRRRPRDSPTLSNSGIRATSRTAKQRSGLPPSEEQPTSQRDRLSQVLRDLEDESTPIVKLGDASIAAPFTSKLSSIQCICHVIKQGRCTLVTTLQMFKILALNALILAYSQSVLYLEGVKFSDFQATLQGLLLAGCFLFISRSKPLKTLSRERPLPNIFNLYTILTVMLQFFVHFLSLVYLYREAQARSPEKQEQFVDLYKEFEPSLVNSTVYIMAMAMQMATFAINYKGPPFMESLPENKPLVWSLAVSLLAIIGLLLGSSPDFNSQFGLVDIPVEFKLVIAQVLLLDFCLALLADRVLQFFLGTP
5XHW , Knot 102 237 0.78 38 142 219 
MHHHHHHGGGGGMYDVVILAGGLGTRLKSVSGDIPKPMVDISGKPFLYRLMEYLEDQGVQRIILSLSYKADYIIDSLIKDNPVNSKVDFIIENEPLGTGGAIKNACKYIEASKFIVINGDTYCELDYKKFIESSLDSDLQISGVEVDDVCRYGSLDIDHDFHVNSIVEKGHQGTGIINSGTYILSKKIIEDYPHDKFSFELDFLPKFTGVFKAFVTKAYFIDIGIPEDYYKACEKFK
4QZG , Knot 169 400 0.84 40 224 376 
MGSSHHHHHHSSGLVPRGSHMSPSPVPGSQNVPAPAVKKISQYACQRRTTLNNYNQLFTDALDILAENDELRENEGSCLAFMRASSVLKSLPFPITSMKDTEGIPCLGDKVKSIIEGIIEDGESSEAKAVLNDERYKSFKLFTSVFGVGLKTAEKWFRMGFRTLSKIQSDKSLRFTQMQKAGFLYYEDLVSCVNRPEAEAVSMLVKEAVVTFLPDALVTMTGGFRRGKMTGHDVDFLITSPEATEDEEQQLLHKVTDFWKQQGLLLYCDILESTFEKFKQPSRKVDALDHAQKCFLILKLDHGRVHSEKSGQQEGKGWKAIRVDLVMCPYDRRAFALLGWTGSRQFERDLRRYATHERKMMLDNHALYDRTKRVFLEAESEEEIFAHLGLDYIEPWERNA

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 \(c\).

\(|P_{f(9JBM_1)}(2) \setminus P_{f(5XHW_1)}(2)|=209\), \(|P_{f(5XHW_1)}(2) \setminus P_{f(9JBM_1)}(2)|=16\). 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 1:11111111101001111111000111100101111111101111111111011110011011011111110101101101101010101001100010001001011011101001000111100001001101101010010000011000011111111101100000001100000101100011000101111010011000101111110110111101000100011010111110101100010010010011001011010000010111000111101101100100011100111101001100111010011010011001010011010100010111110011001110010011011001011011001100001011001111100101001001111111111111111011101000100000011100011100111101110101110001111101010000110111110101001000101000011101111100100101100111000011100001101001011101100111011010100000111001000110100010110110010111000011000100111101110010011000110000100010001101111100011010000100100011000101111111001101000111001001000111101001101001100101100100111011000100001001010111111010101111001101010111010100100110111010111011100001110010011010110101000111100101111111011001100000100010100011010000100001111000010000001001100100000111011010111110001001001001100100011001011011110111110000110101101001010101111110111100001100100001110110100110111011101101101000101001000001101000101011000101111110110111000111110011000111101110111111111100101000111101110101111011110101111100110111101
Pair \(Z_2\) Length of longest common subsequence
9JBM_1,5XHW_1 225 3
9JBM_1,4QZG_1 157 5
5XHW_1,4QZG_1 192 6

Newick tree

 
[
	5XHW_1:11.92,
	[
		9JBM_1:78.5,4QZG_1:78.5
	]:33.42
]

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 \((0.85)(0.8)(\frac{1400 }{\log_{20} 1400}-\frac{237}{\log_{20}237})=305.\)
Status Protein1 Protein2 d d1/2
Query variables 9JBM_1 5XHW_1 385 229.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]