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Parikh vectors
6KVS_1 5TQP_1 2XFD_1 Letter Amino acid
28 56 7 A Alanine
30 46 4 D Aspartic acid
1 4 2 C Cysteine
14 15 2 M Methionine
6 51 0 P Proline
19 52 13 V Valine
10 37 5 R Arginine
15 56 4 E Glutamic acid
19 46 14 T Threonine
9 27 9 Q Glutamine
33 54 13 G Glycine
4 25 1 H Histidine
4 14 3 W Tryptophan
10 40 11 N Asparagine
27 53 4 I Isoleucine
20 86 4 L Leucine
23 49 0 K Lycine
10 30 2 F Phenylalanine
22 61 11 S Serine
9 37 3 Y Tyrosine 

6KVS_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 3|Staphylococcus aureus (strain Mu3 / ATCC 700698) (418127)
>5TQP_1|Chains A, B|Seed linoleate 13S-lipoxygenase-1|Glycine max (3847)
>2XFD_1|Chain A|CARBOHYDRATE BINDING MODULE|ESCHERICHIA COLI (562)
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 \(w=f(c)\)
6KVS , Knot 135 313 0.82 40 179 296 
MNVGIKGFGAYAPEKIIDNAYFEQFLDTSDEWISKMTGIKERHWADDDQDTSDLAYEASVKAIADAGIQPEDIDMIIVATATGDMPFPTVANMLQERLGTGKVASMDQLAACSGFMYSMITAKQYVQSGDYHNILVVGADKLSKITDLTDRSTAVLFGDGAGAVIIGEVSEGRGIISYEMGSDGTGGKHLYLDKDTGKLKMNGREVFKFAVRIMGDASTRVVEKANLTSDDIDLFIPHQANIRIMESARERLGISKDKMSVSVNKYGNTSAASIPLSIDQELKNGKLKDDDTIVLVGFGGGLTWGAMTIKWGK
5TQP , Knot 319 839 0.85 40 305 778 
MFSAGHKIKGTVVLMPKNELEVNPDGSAVDNLNAFLGRSVSLQLISATKADAHGKGKVGKDTFLEGINTSLPTLGAGESAFNIHFEWDGSMGIPGAFYIKNYMQVEFFLKSLTLEAISNQGTIRFVCNSWVYNTKLYKSVRIFFANHTYVPSETPAPLVEYREEELKSLRGNGTGERKEYDRIYDYDVYNDLGNPDKSEKLARPVLGGSSTFPYPRRGRTGRGPTVTDPNTEKQGEVFYVPRDENLGHLKSKDALEIGTKSLSQIVQPAFESAFDLKSTPIEFHSFQDVHDLYEGGIKLPRDVISTIIPLPVIKELYRTDGQHILKFPQPHVVQVSQSAWMTDEEFAREMIAGVNPCVIRGLEEFPPKSNLDPAIYGDQSSKITADSLDLDGYTMDEALGSRRLFMLDYHDIFMPYVRQINQLNSAKTYATRTILFLREDGTLKPVAIELSLPHSAGDLSAAVSQVVLPAKEGVESTIWLLAKAYVIVNDSCYHQLMSHWLNTHAAMEPFVIATHRHLSVLHPIYKLLTPHYRNNMNINALARQSLINANGIGETTFLPSKYSVEMSSAVYKNWVFTDQALPADLIKRGVAIKDPSTPHGVRLLIEDYPYAADGLEIWAAIKTWVQEYVPLYYARDDDVKNDSELQHWWKEAVEKGHGDLKDKPWWPKLQTLEDLVEVCLIIIWIASALHAAVNFGQYPYGGLIMNRPTASRRLLPEKGTPEYEEMINNHEKAYLRTITSKLPTLISLSVIEILSTHASDEVYLGQRDNPHWTSDSKALQAFQKFGNKLKEIEEKLVRRNNDPSLQGNRLGPVQLPYTLLYPSSEEGLTFRGIPNSISI
2XFD , Knot 57 112 0.80 36 87 108 
MSNSITVRARGVNGQESVSLQVGGTTVQTWTLTTAMQDYTASTSLTGEIRVAFTNDATGRDVQVDYIVVNGQTRQAENQSVNTGVWANNQCGGSGNSEWLHCNGYISFGNVS

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 \(c\).

\(|P_{f(6KVS_1)}(2) \setminus P_{f(5TQP_1)}(2)|=21\), \(|P_{f(5TQP_1)}(2) \setminus P_{f(6KVS_1)}(2)|=147\). 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 1:1011101111011001100101001100000110010110000110000000011001010111011101001011111010101111011011000110101101001110011100110100010010000111111001001001000001111101111111101001011100011001011001010000101010100110111011101000110010100001011110010101100100011100001010100010001101110100010010100000111111111101111010110
Pair \(Z_2\) Length of longest common subsequence
6KVS_1,5TQP_1 168 4
6KVS_1,2XFD_1 164 3
5TQP_1,2XFD_1 238 5

Newick tree

 
[
	5TQP_1:10.10,
	[
		6KVS_1:82,2XFD_1:82
	]:27.10
]

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 \((0.85)(0.8)(\frac{1152 }{\log_{20} 1152}-\frac{313}{\log_{20}313})=221.\)
Status Protein1 Protein2 d d1/2
Query variables 6KVS_1 5TQP_1 285 192.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]