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Parikh vectors
1KTI_1 3RXW_1 7ODK_1 Letter Amino acid
63 41 22 A Alanine
49 14 33 D Aspartic acid
48 24 55 G Glycine
36 13 36 P Proline
36 20 21 T Threonine
12 5 6 W Tryptophan
46 9 56 N Asparagine
22 2 7 H Histidine
48 10 22 K Lycine
29 17 77 S Serine
51 8 31 I Isoleucine
38 8 22 F Phenylalanine
36 8 13 Y Tyrosine
62 17 23 V Valine
63 15 14 R Arginine
9 2 10 C Cysteine
30 9 15 Q Glutamine
65 12 35 E Glutamic acid
78 27 112 L Leucine
21 3 7 M Methionine 

1KTI_1|Chain A|GLYCOGEN PHOSPHORYLASE, MUSCLE FORM|Oryctolagus cuniculus (9986)
>3RXW_1|Chain A|Carbepenem-hydrolyzing beta-lactamase KPC|Klebsiella pneumoniae (573)
>7ODK_1|Chains A[auth AAA], B[auth BBB]|Receptor-like protein kinase HSL1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
1KTI , Knot 321 842 0.85 40 316 771 
SRPLSDQEKRKQISVRGLAGVENVTELKKNFNRHLHFTLVKDRNVATPRDYYFALAHTVRDHLVGRWIRTQQHYYEKDPKRIYYLSLEFYMGRTLQNTMVNLALENACDEATYQLGLDMEELEEIEEDAGLGNGGLGRLAACFLDSMATLGLAAYGYGIRYEFGIFNQKICGGWQMEEADDWLRYGNPWEKARPEFTLPVHFYGRVEHTSQGAKWVDTQVVLAMPYDTPVPGYRNNVVNTMRLWSAKAPNDFNLKDFNVGGYIQAVLDRNLAENISRVLYPNDNFFEGKELRLKQEYFVVAATLQDIIRRFKSSKFGCRDPVRTNFDAFPDKVAIQLNDTHPSLAIPELMRVLVDLERLDWDKAWEVTVKTCAYTNHTVIPEALERWPVHLLETLLPRHLQIIYEINQRFLNRVAAAFPGDVDRLRRMSLVEEGAVKRINMAHLCIAGSHAVNGVARIHSEILKKTIFKDFYELEPHKFQNKTNGITPRRWLVLCNPGLAEIIAERIGEEYISDLDQLRKLLSYVDDEAFIRDVAKVKQENKLKFAAYLEREYKVHINPNSLFDVQVKRIHEYKRQLLNCLHVITLYNRIKKEPNKFVVPRTVMIGGKAAPGYHMAKMIIKLITAIGDVVNHDPVVGDRLRVIFLENYRVSLAEKVIPAADLSEQISTAGTEASGTGNMKFMLNGALTIGTMDGANVEMAEEAGEENFFIFGMRVEDVDRLDQRGYNAQEYYDRIPELRQIIEQLSSGFFSPKQPDLFKDIVNMLMHHDRFKVFADYEEYVKCQERVSALYKNPREWTRMVIRNIATSGKFSSDRTIAQYAREIWGVEPSRQRLPAPDEKIP
3RXW , Knot 117 264 0.82 40 165 248 
TNLVAEPFAKLEQDFGGSIGVYAMDTGSGATVSYRAEERFPLCSSFKGFLAAAVLARSQQQAGLLDTPIRYGKNALVPWSPISEKYLTTGMTVAELSAAAVQYSDNAAANLLLKELGGPAGLTAFMRSIGDTTFRLDRWELELNSAIPGDARDTSSPRAVTESLQKLTLGSALAAPQRQQFVDWLKGNTTGNHRIRAAVPADWAVGDKTGTCGVYGTANDYAVVWPTGRAPIVLAVYTRAPNKDDKHSEAVIAAAARLALEGLG
7ODK , Knot 220 617 0.76 40 211 480 
GSSMDNQDGFILQQVKLSLDDPDSYLSSWNSNDASPCRWSGVSCAGDFSSVTSVDLSSANLAGPFPSVICRLSNLAHLSLYNNSINSTLPLNIAACKSLQTLDLSQNLLTGELPQTLADIPTLVHLDLTGNNFSGDIPASFGKFENLEVLSLVYNLLDGTIPPFLGNISTLKMLNLSYNPFSPSRIPPEFGNLTNLEVMWLTECHLVGQIPDSLGQLSKLVDLDLALNDLVGHIPPSLGGLTNVVQIELYNNSLTGEIPPELGNLKSLRLLDASMNQLTGKIPDELCRVPLESLNLYENNLEGELPASIALSPNLYEIRIFGNRLTGGLPKDLGLNSPLRWLDVSENEFSGDLPADLCAKGELEELLIIHNSFSGVIPESLADCRSLTRIRLAYNRFSGSVPTGFWGLPHVNLLELVNNSFSGEISKSIGGASNLSLLILSNNEFTGSLPEEIGSLDNLNQLSASGNKFSGSLPDSLMSLGELGTLDLHGNQFSGELTSGIKSWKKLNELNLADNEFTGKIPDEIGSLSVLNYLDLSGNMFSGKIPVSLQSLKLNQLNLSYNRLSGDLPPSLAKDMYKNSFIGNPGLCGDIKGLCGSENEAKKRGYVLEGSENLYFQ

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 \(c\).

\(|P_{f(1KTI_1)}(2) \setminus P_{f(3RXW_1)}(2)|=175\), \(|P_{f(3RXW_1)}(2) \setminus P_{f(1KTI_1)}(2)|=24\). 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 1:00110000000010101111100100100010001010110000110100001111001000111011000000000010010010101011001000110111001000100011101001001000111101111011101100110111110101100011110001011101001001100101100101010111010101000001101100011111100011110000110010110101100101001011101011100011001001101000110100101000011111010011001000011000110001011100111010000101111011011101001010011010100010000011101100111011001110010110010001100111111101001001011001110010110101110011011101000110001100100101001000001101001111001111011100110001001001001100100011100110100000101110100000101010011010100100000011001011010001000100111100111110111100110111011011101100011110010111100001011001111101000100110010101010111011101101011010110011000111111010010010001001000000110100110010011101001011001101110000101110000010000010110001001001110011001010000011001001111010000111100011
Pair \(Z_2\) Length of longest common subsequence
1KTI_1,3RXW_1 199 4
1KTI_1,7ODK_1 173 4
3RXW_1,7ODK_1 176 3

Newick tree

 
[
	3RXW_1:96.27,
	[
		1KTI_1:86.5,7ODK_1:86.5
	]:9.77
]

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 \((0.85)(0.8)(\frac{1106 }{\log_{20} 1106}-\frac{264}{\log_{20}264})=225.\)
Status Protein1 Protein2 d d1/2
Query variables 1KTI_1 3RXW_1 289 187.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]