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Parikh vectors
9ILW_1 9DUD_1 2DVE_1 Letter Amino acid
7 4 3 W Tryptophan
20 16 11 R Arginine
0 0 0 C Cysteine
37 14 21 I Isoleucine
10 5 5 M Methionine
32 6 8 P Proline
37 13 14 S Serine
31 10 9 T Threonine
18 7 3 Q Glutamine
16 12 1 H Histidine
55 20 31 L Leucine
35 19 7 A Alanine
38 6 13 N Asparagine
23 9 14 D Aspartic acid
31 25 14 E Glutamic acid
27 6 19 K Lycine
26 6 5 Y Tyrosine
18 16 24 G Glycine
18 7 9 F Phenylalanine
34 17 24 V Valine 

9ILW_1|Chains A, B|Acid-sensing system histidine kinase EvgS|Escherichia coli (562)
>9DUD_1|Chains A, B|ADP-ribose pyrophosphatase|Klebsiella pneumoniae (573)
>2DVE_1|Chains A, B|Biotin Protein Ligase|Pyrococcus horikoshii (53953)
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 \(w=f(c)\)
9ILW , Knot 204 513 0.82 38 242 478 
YIEYRGISSNNRVTLDPLRLSNKELRWLASKKNLVIAVHKSQTATLLHTDSQQRVRGINADYLNLLKRALNIKLTLREYADHQKAMDALAEGEVDIVLSHLVTSPPLNNDIAATKPLIITFPALVTTLHDSMRPLTSPKPVNIARVANYPPDEVIHQSFPKATIISFTNLYQALASVSAGHNDYFIGSNIITSSMISRYFTHSLNVVKYYNSPRQYNFFLTRKESVILNEVLNRFVDALTNEVRYEVSQNWLDTGNLAFLNKPLELTEHEKQWIKQHPNLKVLENPYSPPYSMTDENGSVRGVMGDILNIITLQTGLNFSPITVSHNIHAGTQLSPGGWDIIPGAIYSEDRENNVLFAEAFITTPYVFVMQKAPDSEQTLKKGMKVAIPYYYELHSQLKEMYPEVEWIQVDNASAAFHKVKEGELDALVATQLNSRYMIDHYYPNELYHFLIPGVPNASLSFAFPRGEPELKDIINKALNAIPPSEVLRLTEKWIKMPNVTIDTWDLYSEQFY
9DUD , Knot 98 218 0.80 38 148 210 
MAHHHHHHMSKPTQQGITFSKNDVEIIARETLYRGFFSLDLYRFRHRLFNGGMSGEITREIFERGHAAVLLPFDPVRDEVVLVEQIRIAAYDTSESPWLLEMVAGMIEAGETVEDVARREALEEAGLEVGRTKPILSYLASPGGTSERLSILVGEVDASTAKGIHGLAEENEDIRVHVVSREQAYQWVEEGKIDNAASVIALQWLQLHYHNLRNEWTK
2DVE , Knot 104 235 0.80 38 146 225 
MLGLKTSIIGRRVIYFQEITSTNEFAKTSYLEEGTVIVADKQTMGHGRLNAKWESPEGGLWLSIVLSPKVPQKDLPKIVFLGAVGVVETLKEFSIDGRIKWPNDVLVNYKKIAGVLVEGKGDKIVLGIGLNVNNKVPNGATSMKLELGSEVPLLSVFRSLITNLDRLYLNFLKNPMDILNLVRDNMILGVRVKILGDGSFEGIAEDIDDFGRLIIRLDSGEVKKVIYGDVSLRFL

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 \(c\).

\(|P_{f(9ILW_1)}(2) \setminus P_{f(9DUD_1)}(2)|=140\), \(|P_{f(9DUD_1)}(2) \setminus P_{f(9ILW_1)}(2)|=46\). 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 1:010001100000101011010000101110000111110000010110000000101101001011001101010100010000110111010101110011001110001110011110111110010001011001011011011001100110001101011010010011101011000011100110001100010001011000001000011100000111001100110110001000100011001011110011010000001100010101100100110010000101011110110110100110101101000101100101111011111100000000111101110010111100110000010011011110000100010010101011010010111001001010111100100001100001001001111111010101111010101001100110111100110100011011010100101000010
Pair \(Z_2\) Length of longest common subsequence
9ILW_1,9DUD_1 186 4
9ILW_1,2DVE_1 180 4
9DUD_1,2DVE_1 140 4

Newick tree

 
[
	9ILW_1:97.63,
	[
		2DVE_1:70,9DUD_1:70
	]:27.63
]

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 \((0.85)(0.8)(\frac{731 }{\log_{20} 731}-\frac{218}{\log_{20}218})=143.\)
Status Protein1 Protein2 d d1/2
Query variables 9ILW_1 9DUD_1 179 127
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]