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Parikh vectors
1XUZ_1 1JAK_1 9MHS_1 Letter Amino acid
13 31 26 T Threonine
19 39 29 V Valine
6 11 18 Q Glutamine
29 23 37 E Glutamic acid
32 16 40 I Isoleucine
25 41 67 L Leucine
11 29 20 Y Tyrosine
13 36 21 R Arginine
29 53 20 G Glycine
15 40 22 P Proline
1 13 6 W Tryptophan
30 60 24 A Alanine
22 40 35 D Aspartic acid
7 2 7 C Cysteine
10 6 10 M Methionine
20 21 34 S Serine
20 4 29 N Asparagine
8 20 12 H Histidine
27 18 35 K Lycine
12 9 34 F Phenylalanine 

1XUZ_1|Chain A|polysialic acid capsule biosynthesis protein SiaC|Neisseria meningitidis (487)
>1JAK_1|Chain A|Beta-N-acetylhexosaminidase|Streptomyces plicatus (1922)
>9MHS_1|Chains A, B|Drimenol Synthase|Aquimarina spongiae (570521)
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 \(w=f(c)\)
1XUZ , Knot 149 349 0.83 40 202 334 
MQNNNEFKIGNRSVGYNHEPLIICEIGINHEGSLKTAFEMVDAAYNAGAEVVKHQTHIVEDEMSDEAKQVIPGNADVSIYEIMERCALNEEDEIKLKEYVESKGMIFISTPFSRAAALRLQRMDIPAYKIGSGECNNYPLIKLVASFGKPIILSTGMNSIESIKKSVEIIREAGVPYALLHCTNIYPTPYEDVRLGGMNDLSEAFPDAIIGLSDHTLDNYACLGAVALGGSILERHFTDRMDRPGPDIVCSMNPDTFKELKQGAHALKLARGGKKDTIIAGEKPTKDFAFASVVADKDIKKGELLSGDNLWVKRPGNGDFSVNEYETLFGKVAACNIRKGAQIKKTDIE
1JAK , Knot 197 512 0.80 40 215 464 
MHHHHHHHTGAAPDRKAPVRPTPLDRVIPAPASVDPGGAPYRITRGTHIRVDDSREARRVGDYLADLLRPATGYRLPVTAHGHGGIRLRLAGGPYGDEGYRLDSGPAGVTITARKAAGLFHGVQTLRQLLPPAVEKDSAQPGPWLVAGGTIEDTPRYAWRSAMLDVSRHFFGVDEVKRYIDRVARYKYNKLHLHLSDDQGWRIAIDSWPRLATYGGSTEVGGGPGGYYTKAEYKEIVRYAASRHLEVVPEIDMPGHTNAALASYAELNCDGVAPPLYTGTKVGFSSLCVDKDVTYDFVDDVIGELAALTPGRYLHIGGDEAHSTPKADFVAFMKRVQPIVAKYGKTVVGWHQLAGAEPVEGALVQYWGLDRTGDAEKAEVAEAARNGTGLILSPADRTYLDMKYTKDTPLGLSWAGYVEVQRSYDWDPAGYLPGAPADAVRGVEAPLWTETLSDPDQLDYMAFPRLPGVAELGWSPASTHDWDTYKVRLAAQAPYWEAAGIDFYRSPQVPWT
9MHS , Knot 213 526 0.84 40 239 489 
NQNNMKEVLTPELITSLQELSQILDRYDTIVFDLGDVLLHWDSVHFTSETKGIDDVRKMVKHPVWQDLEKGLINQEFALTALSCELETPCSKLKEMLELSIASLQVNPLMVEVLRVLHKKDKQIYCLSNVDLESFSYLYKQFDFWKYFDGIYVSALLQLRKPNPDIFQYLISSASINTKSTIFIDDKSENLQEAANFGISTLKYNKDNFEYTAIEGGWPIPLQNMTPEIHKKRTLGEDYLNLRLRKFPFCKSFVSNNVELIGGEDFSKEIFSTAVILHSYTSLPDDIIASMCHEILNHDGQNKLRWCFYKNEARPDNFPDDLDTTSMVLSFLLNHNKLTIEKIIPVAEQMIANRNEEGIIQVYFDDNRPRIDAIVAINVLYLMHQIGYGERKELKETEAFVFDFLISKEYLKGTRYYPAPDVFLFFLSRLVVDFPDQFEKFHKPLTEMLITRVNCSTFPLERALRIIALKKLGIVNRVDFLKLLDTQLADGGWPVYGLFIAPRSNTYFGSRELSTAFALEALHILS

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 \(c\).

\(|P_{f(1XUZ_1)}(2) \setminus P_{f(1JAK_1)}(2)|=79\), \(|P_{f(1JAK_1)}(2) \setminus P_{f(1XUZ_1)}(2)|=92\). 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 1:1000001011000110000111100111000101001101101100111011000001100010001001111010101001100011000001010001000111110011001111010010111001101000001110111011011110011001001000101100111101110000101010001011110010011101111100001000101111111101100010001001110110010100100100110110110110000111100100011110111000100101101001110011010101000001110111001001101000010
Pair \(Z_2\) Length of longest common subsequence
1XUZ_1,1JAK_1 171 4
1XUZ_1,9MHS_1 163 4
1JAK_1,9MHS_1 178 4

Newick tree

 
[
	1JAK_1:89.10,
	[
		1XUZ_1:81.5,9MHS_1:81.5
	]:7.60
]

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 \((0.85)(0.8)(\frac{861 }{\log_{20} 861}-\frac{349}{\log_{20}349})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 1XUZ_1 1JAK_1 173 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]