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Parikh vectors
7AAW_1 2WOL_1 5TRS_1 Letter Amino acid
12 54 21 R Arginine
12 13 7 N Asparagine
14 53 13 D Aspartic acid
34 26 19 E Glutamic acid
19 65 20 L Leucine
8 6 3 M Methionine
11 16 8 F Phenylalanine
13 16 10 Y Tyrosine
29 43 17 V Valine
18 39 13 T Threonine
2 1 1 C Cysteine
7 13 10 Q Glutamine
24 14 9 I Isoleucine
26 5 7 K Lycine
8 51 8 P Proline
27 58 35 A Alanine
35 46 21 G Glycine
4 19 2 H Histidine
17 15 16 S Serine
1 9 0 W Tryptophan 

7AAW_1|Chains A, B|Thioredoxin reductase|Bacillus cereus (strain ATCC 14579 / DSM 31 / JCM 2152 / NBRC 15305 / NCIMB 9373 / NRRL B-3711) (226900)
>2WOL_1|Chain A|CLAVULANIC ACID BIOSYNTHESIS OLIGOPEPTIDE BINDING PROTEIN 2|STREPTOMYCES CLAVULIGERUS (1901)
>5TRS_1|Chains A, B, C, D, E, F, G, O, P, Q, R, S, T, U|Proteasome subunit alpha|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 \(w=f(c)\)
7AAW , Knot 137 321 0.82 40 180 302 
MNSVSEEKIYDVVIIGAGPAGMTAAVYTSRANLSTLMLERGIPGGQMANTEDVENYPGYESILGPDLSNKMFEHAKKFGAEYAYGDVKEVIDGKEYKTIIAGKKEYKARAIIVASGAEYKKIGVPGETELGGRGVSYCAVCDGAFFKGKELIVIGGGDSAVEEGVFLTRFASKVTIVHRRDTLRAQKILQDRAFQNEKVDFIWNHTIKEINEASGKVGSVTLVDVNSGEEKEVKTDGVFVYIGMLPLSKPFVELGITNENGYLETNERMETKIPGIFAAGDVREKMLRQIVTATGDGSIAAQSAQHYVEELLEELKTVSEK
2WOL , Knot 207 562 0.77 40 221 506 
MTTAARRPAPTTAGAGWDAGVGALVNPSRRRGGTLRLVSSADVDSLDPARTYYVWVWLLQRLLNRTLMAYPTDPGPAGLVPAPDLAEGPGEVSDGGRTWTYRLRRGLRYDDGTPITSDDVRHAVQRVFAQDVLPGGPTYLIPLLDDPERPYPGPYRTDEPLRSVLTPDEHTIVFRLTRPFSDFDHLMAQPCAAPVPRRSDTGADYGRDPRSSGPYRVARHEPDTLLHLERNPHWDRATDPIRPALPDRVELTIGLDVDVLDARLIAGEFDINLEGRGLQHAAQRRATADEVLRSHTDNPRTSFLHFVAMQPHIPPFDNVHVRRAVQYAADKILLQDARGGPVNGGDLTTALFPPTLPAHQDLDLYPTGPDLRGDLDAARAELAAAGLPDGFRAVIGTQRGKFRLVADAVVESLARVGIELTVKELDVATYFSLGAGHPETVREHGLGLLVTDWGADFPTEYGFLAPLVDGRQIKRNGGNWNLPELDDPEVNALIDETLHTTDPAARAELWRAVERRVMEHAVLLPLVHDKTLHFRNPWVTNVYVHPAFGLYDIQAMGLAEED
5TRS , Knot 107 240 0.81 38 148 225 
MEQAMRERSELARKGIARAKSVVALAYAGGVLFVAENPSRSLQKISELYDRVGFAAAGKFNEFDNLRRGGIQFADTRGYAYDRRDVTGRQLANVYAQTLGTIFTEQAKPYEVELCVAEVAHYGETKRPELYRITYDGSIADEPHFVVMGGTTEPIANALKESYAENASLTDALRIAVAALRAGSADTSGGDQPTLGVASLEVAVLDANRPRRAFRRITGSALQALLVDQESPQSDGESSG

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 \(c\).

\(|P_{f(7AAW_1)}(2) \setminus P_{f(2WOL_1)}(2)|=60\), \(|P_{f(2WOL_1)}(2) \setminus P_{f(7AAW_1)}(2)|=101\). 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 1:100100001001111111111101110000101001110011111011000010001100011110100011001001110010101001101000001111000001011111011000011111000111011000110011110100111111100110011110011001011000001010011000110000101110001001001010110101101001000010001111011111100111011100001010000010001111111101000110011010101011100100010011001001000
Pair \(Z_2\) Length of longest common subsequence
7AAW_1,2WOL_1 161 4
7AAW_1,5TRS_1 156 5
2WOL_1,5TRS_1 145 4

Newick tree

 
[
	7AAW_1:81.38,
	[
		5TRS_1:72.5,2WOL_1:72.5
	]:8.88
]

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 \((0.85)(0.8)(\frac{883 }{\log_{20} 883}-\frac{321}{\log_{20}321})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 7AAW_1 2WOL_1 186 146
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]