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Parikh vectors
4KZB_1 6PBP_1 4KOA_1 Letter Amino acid
0 9 3 C Cysteine
31 60 27 L Leucine
6 6 8 M Methionine
13 7 4 W Tryptophan
11 42 20 R Arginine
16 10 11 N Asparagine
5 16 14 H Histidine
26 29 10 P Proline
22 31 31 V Valine
29 34 40 G Glycine
22 12 20 I Isoleucine
9 15 7 F Phenylalanine
18 29 16 S Serine
23 13 18 T Threonine
37 54 48 A Alanine
14 23 16 D Aspartic acid
25 16 4 Q Glutamine
15 24 24 E Glutamic acid
21 5 8 K Lycine
15 14 4 Y Tyrosine 

4KZB_1|Chains A, B|Beta-lactamase|Escherichia coli (83333)
>6PBP_1|Chains A, B|Pseudopaline Dehydrogenase|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
>4KOA_1|Chain A|1,5-anhydro-D-fructose reductase|Sinorhizobium meliloti (266834)
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 \(w=f(c)\)
4KZB , Knot 154 358 0.84 38 208 348 
APQQINDIVHRTITPLIEQQKIPGMAVAVIYQGKPYYFTWGYADIAKKQPVTQQTLFELGSVSKTFTGVLGGDAIARGEIKLSDPTTKYWPELTAKQWNGITLLHLATYTAGGLPLQVPDEVKSSSDLLRFYQNWQPAWAPGTQRLYANSSIGLFGALAVKPSGLSFEQAMQTRVFQPLKLNHTWINVPPAEEKNYAWGYREGKAVHVSPGALDAEAYGVKSTIEDMARWVQSNLKPLDINEKTLQQGIQLAQSRYWQTGDMYQGLGWEMLDWPVNPDSIINGSDNKIALAARPVKAITPPTPAVRASWVHKTGATGGFGSYVAFIPEKELGIVMLANKNYPNPARVDAAWQILNALQ
6PBP , Knot 179 449 0.81 40 214 411 
HHHHHHSSGLVPRGSHMNAADESLGNVLLVGLGAVAIQVALDLRRHGAGRLGALNHPGRRSQRIAEALARGACLQLEGQGQHRWLSGNAALDVFHQDPAELRDDWQTLVLCVPADSYLDVVRGLPWERLGGVRTLLLVSAFIGANLLVRSALPAGCQATVLSLSSYYAATKVIDETQPLRALTKAVKRRVYLGSSRPDCPARETWRRVLAGSGVEVVPLATPEAAEGRNVTTYVHSPFFLGEFALARILSEQGPPGFMYKLYPEGPITPGAIGAMRRLWCELSELLRRMGAEPLNLLRFLNDDNYPVHETMLPRASIDGFAEAGAERQEYLLFVRYAALLVDPFSPADEQGRHFDFSAVPFRRVSRDEDGLWRLPRVPLEDYRKLALIVALAAHFDLAMPQARSLLASYENAVSRFIDCQGASQCHPSLYPIDSRPAADAIYRQWCSTC
4KOA , Knot 139 333 0.80 40 175 302 
MIRWGLIGASTIAREWVIGAIRAAGGEVVSVMSSSAERGEAYAAENGIAKAVTSVDDLVGDPDVDAVYISTTNELHHGQALAAIRAGKHVLCEKPLAMNLNDGCEMVLKACEAGVVLGTNHHLRNAATHRAMREAIAAGRIGRPIAARVFHAVYLPPHLQGWRLDKPEAGGGVILDITVHDADTLRFVLNDDPIEAVAISHSAGMGKEGLEDGVMGVLRFRSGVIAQFHDAFTTKFAETGLEVHGTAGSLIGRNVMTQRPVGTVVLRNEEGESELPLDHRNLYETAIAAFHSAIGGNGRPSASGEDGVWSLATGLAVVKAAATGGAVEIETGL

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 \(c\).

\(|P_{f(4KZB_1)}(2) \setminus P_{f(6PBP_1)}(2)|=84\), \(|P_{f(6PBP_1)}(2) \setminus P_{f(4KZB_1)}(2)|=90\). 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 1:1100100110001011100001111111110010100101101011000110000110110100010111110111010101001000011010100101101101100011111101100100000110100010111111000101000111111111010110100110001101101000110111100000111000101101011110101011000100110110001011010000100110110000100101001111011011101001101000011111011011011011101011000110111100111110001111111000010110101110110110
Pair \(Z_2\) Length of longest common subsequence
4KZB_1,6PBP_1 174 4
4KZB_1,4KOA_1 161 4
6PBP_1,4KOA_1 145 4

Newick tree

 
[
	4KZB_1:87.25,
	[
		4KOA_1:72.5,6PBP_1:72.5
	]:14.75
]

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 \((0.85)(0.8)(\frac{807 }{\log_{20} 807}-\frac{358}{\log_{20}358})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 4KZB_1 6PBP_1 152 139.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]