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Parikh vectors
4GSS_1 9ISS_1 9QEA_1 Letter Amino acid
8 19 45 R Arginine
7 22 27 I Isoleucine
14 25 40 V Valine
11 12 13 Y Tyrosine
12 36 13 K Lycine
2 14 10 M Methionine
11 27 27 P Proline
10 17 30 S Serine
2 6 5 W Tryptophan
8 19 7 N Asparagine
13 22 19 Q Glutamine
2 13 17 H Histidine
32 50 53 L Leucine
8 25 17 F Phenylalanine
9 21 32 T Threonine
15 32 54 A Alanine
13 33 39 D Aspartic acid
4 3 4 C Cysteine
10 36 35 E Glutamic acid
18 23 39 G Glycine 

4GSS_1|Chains A, B|GLUTATHIONE S-TRANSFERASE|Homo sapiens (9606)
>9ISS_1|Chains A, B|Bifunctional cytochrome P450/NADPH--P450 reductase|Priestia megaterium (1404)
>9QEA_1|Chain A|Lysine--tRNA ligase 1|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
4GSS , Knot 97 209 0.82 40 138 200 
PPYTVVYFPVRGRCAALRMLLADQGQSWKEEVVTVETWQEGSLKASCLYGQLPKFQDGDLTLYQSNTILRHLGRTLGLYGKDQQEAALVDMVNDGVEDLRCKYISLIFTNYEAGKDDYVKALPGQLKPFETLLSQNQGGKTFIVGDQISFADYNLLDLLLIHEVLAPGCLDAFPLLSAYVGRLSARPKLKAFLASPEYVNLPINGNGKQ
9ISS , Knot 188 455 0.84 40 239 438 
TIKEMPQPKTFGELKNLPLLNTDKPVQALMKIADELGEIFKFEAPGWVLRYLSSQRLIKEACDESRFDKNLSQVLKFVRDFAGDGLFTSWTHEKNWKKAHNILLPSFSQQAMKGYHAMMVDIAVQLVQKWERLNADEHIEVPEDMTRLTLDTIGLCGFNYRFNSFYRDQPHPFITSMVRALDEAMNKPQRANPDDPAYDENKRQFQEDIKVMNDLVDKIIADRKASGEQSDDLLTHMLNGKDPETGEPLDDENIRYQIITFLIAGHETTSGLLSFALYFLVKNPHVLQKAAEEAARVLVDPVPSYKQVKQLKYVGMVLNEALRLWPTAPAFSLYAKEDTVLGGEYPLEKGDELMVLIPQLHRDKTIWGDDVEEFRPERFENPSAIPQHAFKPFGNGQRACIGQQFALHEATLVLGMMLKHFDFEDHTNYELDIKETLTLKPEGFVVKAKSKKIPL
9QEA , Knot 203 526 0.80 40 236 475 
MGSSHHHHHHSSGENLYFQGHMSAADTAEDLPEQFRIRRDKRARLLAQGRDPYPVAVPRTHTLAEVRAAHPDLPIDTATEDIVGVAGRVIFARNSGKLCFATLQDGDGTQLQVMISLDKVGQAALDAWKADVDLGDIVYVHGAVISSRRGELSVLADCWRIAAKSLRPLPVAHKEMSEESRVRQRYVDLIVRPEARAVARLRIAVVRAIRTALQRRGFLEVETPVLQTLAGGAAARPFATHSNALDIDLYLRIAPELFLKRCIVGGFDKVFELNRVFRNEGADSTHSPEFSMLETYQTYGTYDDSAVVTRELIQEVADEAIGTRQLPLPDGSVYDIDGEWATIQMYPSLSVALGEEITPQTTVDRLRGIADSLGLEKDPAIHDNRGFGHGKLIEELWERTVGKSLSAPTFVKDFPVQTTPLTRQHRSIPGVTEKWDLYLRGIELATGYSELSDPVVQRERFADQARAAAAGDDEAMVLDEDFLAALEYGMPPCTGTGMGIDRLLMSLTGLSIRETVLFPIVRPHSN

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 \(c\).

\(|P_{f(4GSS_1)}(2) \setminus P_{f(9ISS_1)}(2)|=41\), \(|P_{f(9ISS_1)}(2) \setminus P_{f(4GSS_1)}(2)|=142\). 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 1:11001101110100111011110010010001101001001010100101011010010101000001100110011101000001111011001100100001011100001100001011110101100110000110011110010110001101111001111101011111010110101010101111010010111010100
Pair \(Z_2\) Length of longest common subsequence
4GSS_1,9ISS_1 183 4
4GSS_1,9QEA_1 190 4
9ISS_1,9QEA_1 149 4

Newick tree

 
[
	4GSS_1:98.73,
	[
		9ISS_1:74.5,9QEA_1:74.5
	]:24.23
]

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 \((0.85)(0.8)(\frac{664 }{\log_{20} 664}-\frac{209}{\log_{20}209})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 4GSS_1 9ISS_1 165 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]