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Parikh vectors
5UZP_1 8FOC_1 5WNN_1 Letter Amino acid
31 22 51 A Alanine
21 35 14 N Asparagine
18 36 14 Q Glutamine
25 47 14 E Glutamic acid
38 18 40 G Glycine
20 14 8 H Histidine
22 15 16 P Proline
19 30 4 R Arginine
21 21 19 D Aspartic acid
55 64 23 L Leucine
15 32 14 F Phenylalanine
9 5 10 W Tryptophan
19 23 10 Y Tyrosine
17 32 14 I Isoleucine
25 42 33 K Lycine
15 7 6 M Methionine
25 19 26 T Threonine
30 12 31 V Valine
3 8 0 C Cysteine
34 46 16 S Serine 

5UZP_1|Chains A, B|Citrate synthase, mitochondrial|Homo sapiens (9606)
>8FOC_1|Chain A[auth B]|DNA primase large subunit|Saccharomyces cerevisiae (4932)
>5WNN_1|Chains A, B|Phosphate-binding protein PstS|Burkholderia pseudomallei (strain K96243) (272560)
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 \(w=f(c)\)
5UZP , Knot 190 462 0.84 40 247 441 
MHHHHHHSSGVDLGTENLYFQSNASSTNLKDILADLIPKEQARIKTFRQQHGKTVVGQITVDMMYGGMRGMKGLVYETSVLDPDEGIRFRGFSIPECQKLLPKAKGGEEPLPEGLFWLLVTGHIPTEEQVSWLSKEWAKRAALPSHVVTMLDNFPTNLHPMSQLSAAVTALNSESNFARAYAQGISRTKYWELIYEDSMDLIAKLPCVAAKIYRNLYREGSGIGAIDSNLDWSHNFTNMLGYTDHQFTELTRLYLTIHSDHEGGNVSAHTSHLVGSALSDPYLSFAAAMNGLAGPLHGLANQEVLVWLTQLQKEVGKDVSDEKLRDYIWNTLNSGRVVPGYGHGVLRKTDPRYTCQREFALKHLPNDPMFKLVAQLYKIVPNVLLEQGKAKNPWPNVDAHSGVLLQYYGMTEMNYYTVLFGVSRALGVLAQLIWSRALGFPLERPKSMSTEGLMKFVDSKSG
8FOC , Knot 213 528 0.84 40 239 481 
MFRQSKRRIASRKNFSSYDDIVKSELDVGNTNAANQIILSSSSSEEEKKLYARLYESKLSFYDLPPQGEITLEQFEIWAIDRLKILLEIESCLSRNKSIKEIETIIKPQFQKLLPFNTESLEDRKKDYYSHFILRLCFCRSKELREKFVRAETFLFKIRFNMLTSTDQTKFVQSLDLPLLQFISNEEKAELSHQLYQTVSASLQFQLNLNEEHQRKQYFQQEKFIKLPFENVIELVGNRLVFLKDGYAYLPQFQQLNLLSNEFASKLNQELIKTYQYLPRLNEDDRLLPILNHLSSGYTIADFNQQKANQFSENVDDEINAQSVWSEEISSNYPLCIKNLMEGLKKNHHLRYYGRQQLSLFLKGIGLSADEALKFWSEAFTRNGNMTMEKFNKEYRYSFRHNYGLEGNRINYKPWDCHTILSKPRPGRGDYHGCPFRDWSHERLSAELRSMKLTQAQIISVLDSCQKGEYTIACTKVFEMTHNSASADLEIGEQTHIAHPNLYFERSRQLQKKQQKLEKEKLFNNGNH
5WNN , Knot 152 363 0.82 38 186 341 
MAHHHHHHMGTLEAQTQGPGSMKLMQTAFAGLAGALFAVAAHADITGAGSTFAMPIYTKWAADYQQSGGAKVNYQGIGSSGGLKQIVAKTVDFAGSDAPLKDDELAKEGLFQFPTVVGGVVPVINVPGVKAGELTLSGAVLGDIYLGKIKKWNDPAIAALNPKVKLPDTDIAVVRRADGSGTSFIWTNYLSKVNDEWKSKVGEGTTVNWPTGTGGKGNDGVAAFVQRLPGAIGYVEWAYAKKNNMVYTALKNSTGTVVEPKTETFKAAAAGANWSKSFYQILTNQPGKEAWPVVGATFVLLHAKQDKPAQGAETLKFFDWAFKNGAKAADDLDYISLPASVETEIRKQWKVKVTDAAGKAVAE

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 \(c\).

\(|P_{f(5UZP_1)}(2) \setminus P_{f(8FOC_1)}(2)|=75\), \(|P_{f(8FOC_1)}(2) \setminus P_{f(5UZP_1)}(2)|=67\). 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 1:100000000110110001010001000010011101110001010010000100111010101101110110111000011010011010110110000111010110011101111111010110000101100011001111001101100110010110010111011000001101010110000010110000101110110111010001000101111100010100010011100000100100101010000011010100001110110010101111101111110111000111110010001100100001000110010010111101011100001000000011100110011101110100111011100101001110101001111000110010000111110011111101110011111100100100011101100001
Pair \(Z_2\) Length of longest common subsequence
5UZP_1,8FOC_1 142 5
5UZP_1,5WNN_1 165 6
8FOC_1,5WNN_1 179 4

Newick tree

 
[
	5WNN_1:90.53,
	[
		5UZP_1:71,8FOC_1:71
	]:19.53
]

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 \((0.85)(0.8)(\frac{990 }{\log_{20} 990}-\frac{462}{\log_{20}462})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 5UZP_1 8FOC_1 171 163
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]