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Parikh vectors
2XCL_1 1CRL_1 9CYC_1 Letter Amino acid
22 31 12 F Phenylalanine
21 45 17 S Serine
20 36 33 T Threonine
41 18 16 E Glutamic acid
10 14 8 M Methionine
13 16 18 Q Glutamine
39 56 19 G Glycine
2 5 2 W Tryptophan
42 28 20 V Valine
12 14 7 R Arginine
24 34 13 D Aspartic acid
7 5 9 H Histidine
29 48 26 L Leucine
10 20 23 Y Tyrosine
10 32 16 N Asparagine
1 5 10 C Cysteine
30 20 24 K Lycine
16 31 12 P Proline
54 47 22 A Alanine
19 29 11 I Isoleucine 

2XCL_1|Chain A|PHOSPHORIBOSYLAMINE--GLYCINE LIGASE|BACILLUS SUBTILIS (1423)
>1CRL_1|Chain A|LIPASE|Candida rugosa (5481)
>9CYC_1|Chains A, B|Papain-like protease|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
2XCL , Knot 174 422 0.83 40 206 391 
MNVLIIGKGGREHTLAWKAAQSSLVENVFAAPGNDGMAASAQLVNIEESDHAGLVSFAKQNQVGLTIVGPEVPLIEGLVDEFEKAGLHVFGPSKAAAIIEGSKQFAKDLMKKYDIPTAEYETFTSFDEAKAYVQEKGAPIVIKADGLAAGKGVTVAMTEEEAIACLHDFLEDEKFGDASASVVIEEYLSGEEFSLMAFVKGEKVYPMVIAQDHKRAFDGDKGPNTGGMGAYSPVPQISEETVRHAVETIVKPAAKAMVQEGRSFTGVLYAGLMLTENGSKVIEFNARFGDPETQVVLPRMESDLVQVLLDLLDDKEVDLRWKDTAAVSVVLASEGYPESYAKGTPIGSLAAETEQVVVFHAGTKAEGGEFVTNGGRVANVTAFDETFEAARDRVYKAVDEIFKPGLFFRKDIGARALKAAQK
1CRL , Knot 216 534 0.84 40 251 501 
APTATLANGDTITGLNAIINEAFLGIPFAEPPVGNLRFKDPVPYSGSLDGQKFTSYGPSCMQQNPEGTYEENLPKAALDLVMQSKVFEAVSPSSEDCLTINVVRPPGTKAGANLPVMLWIFGGGFEVGGTSTFPPAQMITKSIAMGKPIIHVSVNYRVSSWGFLAGDEIKAEGSANAGLKDQRLGMQWVADNIAAFGGDPTKVTIFGESAGSMSVMCHILWNDGDNTYKGKPLFRAGIMQSGAMVPSDAVDGIYGNEIFDLLASNAGCGSASDKLACLRGVSSDTLEDATNNTPGFLAYSSLRLSYLPRPDGVNITDDMYALVREGKYANIPVIIGDQNDEGTFFGTSSLNVTTDAQAREYFKQSFVHASDAEIDTLMTAYPGDITQGSPFDTGILNALTPQFKRISAVLGDLGFTLARRYFLNHYTGGTKYSFLSKQLSGLPVLGTFHSNDIVFQDYLLGSGSLIYNNAFIAFATDLDPNTAGLLVKWPEYTSSSQSGNNLMMINALGLYTGKDNFRTAGYDALFSNPPSFFV
9CYC , Knot 141 318 0.85 40 201 302 
SNAEVRTIKVFTTVDNINLHTQVVDMSMTYGQQFGPTYLDGADVTKIKPHNSHEGKTFYVLPNDDTLRVEAFEYYHTTDPSFLGRYMSALNHTKKWKYPQVNGLTSIKWADNNSYLATALLTLQQIELKFNPPALQDAYYRARAGEAANFCALILAYCNKTVGELGDVRETMSYLFQHANLDSCKRVLNVVCKTCGQQQTTLKGVEAVMYMGTLSYEQFKKGVQIPCTCGKQATKYLVQQESPFVMMSAPPAQYELKHGTFTCASEYTGNYQCGHYKHITSKETLYCIDGALLTKSSEYKGPITDVFYKENSYTTTIK

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 \(c\).

\(|P_{f(2XCL_1)}(2) \setminus P_{f(1CRL_1)}(2)|=41\), \(|P_{f(1CRL_1)}(2) \setminus P_{f(2XCL_1)}(2)|=86\). 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 1:10111110110000111011000110011111100111101011010000011110110000111011110111101110010011101111001111101000110011000011010000100100101010001111110101111101101110000111010011000011010101110001010010111110100101111100000110100110011111001110100001001100110111011100100101110111110001001101010110100011110100011011101100001010100011101111001010001010111011100001111011001011011001101101011000101100010011001101111100011101101100
Pair \(Z_2\) Length of longest common subsequence
2XCL_1,1CRL_1 127 4
2XCL_1,9CYC_1 161 4
1CRL_1,9CYC_1 156 4

Newick tree

 
[
	9CYC_1:83.85,
	[
		2XCL_1:63.5,1CRL_1:63.5
	]:20.35
]

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 \((0.85)(0.8)(\frac{956 }{\log_{20} 956}-\frac{422}{\log_{20}422})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 2XCL_1 1CRL_1 177 157.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]