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Parikh vectors
6IMK_1 4GOP_1 2GPA_1 Letter Amino acid
4 0 12 W Tryptophan
18 8 49 D Aspartic acid
19 4 36 P Proline
25 5 38 F Phenylalanine
22 12 36 T Threonine
28 5 63 A Alanine
22 6 30 Q Glutamine
40 4 48 K Lycine
19 2 63 R Arginine
22 6 46 N Asparagine
23 8 48 G Glycine
11 2 22 H Histidine
27 5 50 I Isoleucine
39 8 79 L Leucine
6 4 21 M Methionine
16 13 29 S Serine
6 1 9 C Cysteine
28 5 65 E Glutamic acid
17 1 36 Y Tyrosine
28 15 62 V Valine 

6IMK_1|Chain A|DNA ligase|African swine fever virus (10497)
>4GOP_1|Chains A, D[auth X]|Putative uncharacterized protein|Ustilago maydis (5270)
>2GPA_1|Chain A|PROTEIN (GLYCOGEN PHOSPHORYLASE)|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
6IMK , Knot 174 420 0.83 40 235 402 
SMLNQFPGQYSNNIFCFPPIESETKSGKKASWIICVQVVQHNTIIPITDEMFSTDVKDAVAEIFTKFFVEEGAVRISKMTRVTEGKNLGKKNATTVVHQAFKDALSKYNRHARQKRGAHTNRGMIPPMLVKYFNIIPKTFFEEETDPIVQRKRNGVRAVACQQGDGCILLYSRTEKEFLGLDNIKKELKQLYLFIDVRVYLDGELYLHRKPLQWIAGQANAKTDSSELHFYVFDCFWSDQLQMPSNKRQQLLTNIFKQKEDLTFIHQVENFSVKNVDEALRLKAQFIKEGYEGAIVRNANGPYEPGYNNYHSAHLAKLKPLLDAEFILVDYTQGKKGKDLGAILWVCELPNKKRFVVTPKHLTYADRYALFQKLTPALFKKHLYGKELTVEYAELSPKTGIPLQARAVGFREPINVLEII
4GOP , Knot 57 114 0.79 38 84 110 
MEKPTPLINSSMLGQYVGQTVRIVGKVHKVTGNTLLMQTSDLGNVEIAMTPDSDVSSSTFVEVTGKVSDAGSSFQANQIREFTTVDCGHDVDLTLVENVVQISAAFPNLFSDST
2GPA , Knot 321 842 0.85 40 316 773 
SRPLSDQEKRKQISVRGLAGVENVTELKKNFNRHLHFTLVKDRNVATPRDYYFALAHTVRDHLVGRWIRTQQHYYEKDPKRIYYLSLEFYMGRTLQNTMVNLALENACDEATYQLGLDMEELEEIEEDAGLGNGGLGRLAACFLDSMATLGLAAYGYGIRYEFGIFNQKICGGWQMEEADDWLRYGNPWEKARPEFTLPVHFYGRVEHTSQGAKWVDTQVVLAMPYDTPVPGYRNNVVNTMRLWSAKAPNDFNLKDFNVGGYIQAVLDRNLAENISRVLYPNDNFFEGKELRLKQEYFVVAATLQDIIRRFKSSKFGCRDPVRTNFDAFPDKVAIQLNDTHPSLAIPELMRVLVDLERLDWDKAWEVTVKTCAYTNHTVLPEALERWPVHLLETLLPRHLQIIYEINQRFLNRVAAAFPGDVDRLRRMSLVEEGAVKRINMAHLCIAGSHAVNGVARIHSEILKKTIFKDFYELEPHKFQNKTNGITPRRWLVLCNPGLAEIIAERIGEEYISDLDQLRKLLSYVDDEAFIRDVAKVKQENKLKFAAYLEREYKVHINPNSLFDVQVKRIHEYKRQLLNCLHVITLYNRIKKEPNKFVVPRTVMIGGKAAPGYHMAKMIIKLITAIGDVVNHDPVVGDRLRVIFLENYRVSLAEKVIPAADLSEQISTAGTEASGTGNMKFMLNGALTIGTMDGANVEMAEEAGEENFFIFGMRVEDVDRLDQRGYNAQEYYDRIPELRQIIEQLSSGFFSPKQPDLFKDIVNMLMHHDRFKVFADYEEYVKCQERVSALYKNPREWTRMVIRNIATSGKFSSDRTIAQYAREIWGVEPSRQRLPAPDEKIP

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 \(c\).

\(|P_{f(6IMK_1)}(2) \setminus P_{f(4GOP_1)}(2)|=175\), \(|P_{f(4GOP_1)}(2) \setminus P_{f(6IMK_1)}(2)|=24\). 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 1:011001110000011011110000001001011101011000011110001100010011101100111001110100100100100110001001100110011000000100001100001111111100101110011000001110000011011100010101110000000111100100010010111010101010101000110111101010000001010110011000101100000011001100000101100100101001001101010110010011110010110011000000101101011101011110000100100111111100110000111010010010001110010111100010100101001010100111101011110011011011
Pair \(Z_2\) Length of longest common subsequence
6IMK_1,4GOP_1 199 3
6IMK_1,2GPA_1 149 6
4GOP_1,2GPA_1 254 4

Newick tree

 
[
	4GOP_1:12.51,
	[
		6IMK_1:74.5,2GPA_1:74.5
	]:50.01
]

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 \((0.85)(0.8)(\frac{534 }{\log_{20} 534}-\frac{114}{\log_{20}114})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 6IMK_1 4GOP_1 162 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]