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Parikh vectors
1UKJ_1 7XNZ_1 6IVY_1 Letter Amino acid
18 32 25 E Glutamic acid
18 13 4 H Histidine
19 29 18 P Proline
22 35 19 S Serine
23 19 16 T Threonine
1 5 4 W Tryptophan
23 53 16 V Valine
20 29 10 R Arginine
4 4 0 C Cysteine
16 20 13 I Isoleucine
9 17 22 K Lycine
14 12 6 M Methionine
18 27 15 D Aspartic acid
18 6 13 Q Glutamine
40 58 27 G Glycine
47 35 34 L Leucine
13 15 9 Y Tyrosine
10 6 11 N Asparagine
14 8 9 F Phenylalanine
51 55 36 A Alanine 

1UKJ_1|Chains A, B, C, D|Methionine gamma-lyase|Pseudomonas putida (303)
>7XNZ_1|Chains A, B, C, D|Putative cystathionine beta-synthase Rv1077|Mycobacterium tuberculosis H37Rv (83332)
>6IVY_1|Chains A, B|Periplasmic Ferric iron-binding Protein HitA|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
1UKJ , Knot 168 398 0.84 40 210 381 
MHGSNKLPGFATRAIHHGYDPQDHGGALVPPVYQTATFTFPTVEYGAACFAGEQAGHFYSRISNPTLNLLEARMASLEGGEAGLALASGMGAITSTLWTLLRPGDEVLLGNTLYGCTFAFLHHGIGEFGVKLRHVDMADLQALEAAMTPATRVIYFESPANPNMHMADIAGVAKIARKHGATVVVDNTYCTPYLQRPLELGADLVVHSATKYLSGHGDITAGIVVGSQALVDRIRLQGLKDMTGAVLSPHDAALLMRGIKTLNLRMDRHCANAQVLAEFLARQPQVELIHYPGLASFPQYTLARQQMSQPGGMIAFELKGGIGAGRRFMNALQLFSRAVSLGDAESLAQHPASMTHSSYTPEERAHYGISEGLVRLSVGLEDIDDLLADVQQALKASA
7XNZ , Knot 188 478 0.80 40 211 428 
MARIAQHISELIGGTPLVRLNSVVPDGAGTVAAKVEYLNPGGSSKDRIAVKMIEAAEASGQLKPGGTIVEPTSGNTGVGLALVAQRRGYKCVFVCPDKVSEDKRNVLIAYGAEVVVCPTAVPPHDPASYYSVSDRLVRDIDGAWKPDQYANPEGPASHYVTTGPEIWADTEGKVTHFVAGIGTGGTITGAGRYLKEVSGGRVRIVGADPEGSVYSGGAGRPYLVEGVGEDFWPAAYDPSVPDEIIAVSDSDSFDMTRRLAREEAMLVGGSCGMAVVAALKVAEEAGPDALIVVLLPDGGRGYMSKIFNDAWMSSYGFLRSRLDGSTEQSTVGDVLRRKSGALPALVHTHPSETVRDAIGILREYGVSQMPVVGAEPPVMAGEVAGSVSERELLSAVFEGRAKLADAVSAHMSPPLRMIGAGELVSAAGKALRDWDALMVVEEGKPVGVITRYDLLGFLSEGAGRRKLAAALEHHHHHH
6IVY , Knot 133 307 0.82 38 180 294 
PVTLTLYNGQHAATGIAIAKAFQDKTGIQVKIRKGGDGQLASQITEEGARSPADVLYTEESPPLIRLASAGLLAKLEPETLALVEPEHAGGNGDWIGITARTRVLAYNPKKIDEKDLPKSLMDLSDPSWSGRFGFVPTSGAFLEQVAAVIKLKGQEEAEDWLTGLKAFGSIYTNNVTAMKAVENGEVDMALINNYYWYTLKKEKGELNSRLHYFGNQDPGALVTVSGAAVLKSSKHPREAQQFVAFMLSEEGQKAILSQSAEYPMRKGMQADPALKPFAELDPPKLTPADLGEASEALSLERDVGLN

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 \(c\).

\(|P_{f(1UKJ_1)}(2) \setminus P_{f(7XNZ_1)}(2)|=73\), \(|P_{f(7XNZ_1)}(2) \setminus P_{f(1UKJ_1)}(2)|=74\). 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 1:10100011111001100100100011111111000101011010011101110011010001001010110101101011011111101111100011011011001111001010011110011101110100101101011011101100110100110101011011111011000110111000000101001101110111001000101010101111110011100101011001011110100111110110010101000010101110111001010110011110110001100010011111110101111110011011011001101101001100110100000010001001100111010111001001110100110101
Pair \(Z_2\) Length of longest common subsequence
1UKJ_1,7XNZ_1 147 5
1UKJ_1,6IVY_1 148 4
7XNZ_1,6IVY_1 149 4

Newick tree

 
[
	6IVY_1:74.49,
	[
		1UKJ_1:73.5,7XNZ_1:73.5
	]:0.99
]

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 \((0.85)(0.8)(\frac{876 }{\log_{20} 876}-\frac{398}{\log_{20}398})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 1UKJ_1 7XNZ_1 153 143
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]