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Parikh vectors
7RLF_1 7KIT_1 9CKY_1 Letter Amino acid
63 28 6 D Aspartic acid
59 30 12 I Isoleucine
3 7 5 W Tryptophan
26 26 10 Q Glutamine
67 26 10 E Glutamic acid
63 56 28 G Glycine
69 50 11 L Leucine
40 28 13 S Serine
58 41 12 R Arginine
32 22 13 N Asparagine
12 1 7 C Cysteine
13 17 19 Y Tyrosine
53 55 18 V Valine
10 9 2 H Histidine
20 14 0 M Methionine
32 28 8 T Threonine
43 33 10 P Proline
65 63 14 A Alanine
47 27 10 K Lycine
31 18 4 F Phenylalanine 

7RLF_1|Chains A, B, C, D, E, F|Transitional endoplasmic reticulum ATPase|Homo sapiens (9606)
>7KIT_1|Chain A|Peptidoglycan D,D-transpeptidase FtsI|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
>9CKY_1|Chain A|Papain|Carica papaya (3649)
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 \(w=f(c)\)
7RLF , Knot 295 806 0.81 40 279 706 
MASGADSKGDDLSTAILKQKNRPNRLIVDEAINEDNSVVSLSQPKMDELQLFRGDTVLLKGKKRREAVCIVLSDDTCSDEKIRMNRVVRNNLRVRLGDVISIQPCPDVKYGKRIHVLPIDDTVEGITGNLFEVYLKPYFLEAYRPIRKGDIFLVRGGMRAVEFKVVETDPSPYCIVAPDTVIHCEGEPIKREDEEESLNEVGYDDIGGCRKQLAQIKEMVELPLRHPALFKAIGVKPPRGILLYGPPGTGKTLIARAVANETGAFFFLINGPEIMSKLAGESESNLRKAFEEAEKNAPAIIFIDELDAIAPKREKTHGEVERRIVSQLLTLMDGLKQRAHVIVMAATNRPNSIDPALRRFGRFDREVDIGIPDATGRLEILQIHTKNMKLADDVDLEQVANETHGHVGADLAALCSEAALQAIRKKMDLIDLEDETIDAEVMNSLAVTMDDFRWALSQSNPSALRETVVDVPQVTWEDIGGLEDVKRELQELVQYPVEHPDKFLKFGMTPSKGVLFYGPPGCGKTLLAKAIANECQANFISIKGPELLTMWFGESEANVREIFDKARQAAPCVLFFDELDSIAKARGGNIGDGGGAADRVINQILTEMDGMSTKKNVFIIGATNRPDIIDPAILRPGRLDQLIYIPLPDEKSRVAILKANLRKSPVAKDVDLEFLAKMTNGFSGADLTEICQRACKLAIRESIESEIRRERERQTNPSAMEVEEDDPVPEIRRDHFEEAMRFARRSVSDNDIRKYEMFAQTLQQSRGFGSFRFPSGNQGGAGPSQGSGGGTGGSVYTEDNDDDLYG
7KIT , Knot 236 579 0.86 40 248 542 
MKLNYFQGALYPWRFCVIVGLLLAMVGAIVWRIVDLHVIDHDFLKGQGDARSVRHIAIPAHRGLITDRNGEPLAVSTPVTTLWANPKELMTAKERWPQLAAALGQDTKLFADRIEQNAEREFIYLVRGLTPEQGEGVIALKVPGVYSIEEFRRFYPAGEVVAHAVGFTDVDDRGREGIELAFDEWLAGVPGKRQVLKDRRGRVIKDVQVTKNAKPGKTLALSIDLRLQYLAHRELRNALLENGAKAGSLVIMDVKTGEILAMTNQPTYNPNNRRNLQPAAMRNRAMIDVFEPGSTVKPFSMSAALASGRWKPSDIVDVYPGTLQIGRYTIRDVSRNSRQLDLTGILIKSSNVGISKIAFDIGAESIYSVMQQVGLGQDTGLGFPGERVGNLPNHRKWPKAETATLAYGYGLSVTAIQLAHAYAALANDGKSVPLSMTRVDRVPDGVQVISPEVASTVQGMLQQVVEAQGGVFRAQVPGYHAAGKSGTARKVSVGTKGYRENAYRSLFAGFAPATDPRIAMVVVIDEPSKAGYFGGLVSAPVFSKVMAGALRLMNVPPDNLPTATEQQQVNAAPAKGGRG
9CKY , Knot 99 212 0.83 38 143 204 
IPEYVDWRQKGAVTPVKNQGSCGSCWAFSAVVTIEGIIKIRTGNLNEYSEQELLDCDRRSYGCNGGYPWSALQLVAQYGIHYRNTYPYEGVQRYCRSREKGPYAAKTDGVRQVQPYNEGALLYSIANQPVSVVLEAAGKDFQLYRGGIFVGPCGNKVDHAVAAVGYGPNYILIKNSWGTGWGENGYIRIKRGTGNSYGVCGLYTSSFYPVKN

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 \(c\).

\(|P_{f(7RLF_1)}(2) \setminus P_{f(7KIT_1)}(2)|=70\), \(|P_{f(7KIT_1)}(2) \setminus P_{f(7RLF_1)}(2)|=39\). 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 1:11011000100100111000001001110011000001101001010010110100111010000011011100000000010100110001010110110101010100100101111000101101011010101011010011001011110111011010110001010011110011000101100000000100110001110000110100110111001111011110110111101111010011101110001111111011011001110000010011001000111111100101111000000101000110011011011000101111110001001011100110100010111101010101101000010110010100110000101110111100011101100010110100001010110011101001011100001011000110110101001111001000100110011001001101110100111101111010011101110000101101011011011110001010011001001110111100100110101101101111100110011001011000001111110001011011110110100110111100000111101010001110010101110100110110100100010011100010001000000000101101000011101000010011011000100001000011100100001110101101001111100101110110100000000101
Pair \(Z_2\) Length of longest common subsequence
7RLF_1,7KIT_1 109 4
7RLF_1,9CKY_1 198 4
7KIT_1,9CKY_1 183 4

Newick tree

 
[
	9CKY_1:10.47,
	[
		7RLF_1:54.5,7KIT_1:54.5
	]:50.97
]

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 \((0.85)(0.8)(\frac{1385 }{\log_{20} 1385}-\frac{579}{\log_{20}579})=204.\)
Status Protein1 Protein2 d d1/2
Query variables 7RLF_1 7KIT_1 252 220
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]