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Parikh vectors
1QDB_1 4NEA_1 9HBE_1 Letter Amino acid
15 18 3 R Arginine
33 46 11 G Glycine
17 15 1 H Histidine
18 37 4 I Isoleucine
31 38 1 L Leucine
10 12 0 M Methionine
48 42 8 A Alanine
18 26 6 N Asparagine
34 32 3 D Aspartic acid
10 3 2 C Cysteine
17 18 1 Q Glutamine
56 35 3 K Lycine
22 31 15 T Threonine
19 12 4 Y Tyrosine
15 21 1 F Phenylalanine
17 18 2 P Proline
11 7 7 W Tryptophan
24 36 7 V Valine
34 45 3 E Glutamic acid
24 28 8 S Serine 

1QDB_1|Chains A, B, C|CYTOCHROME C NITRITE REDUCTASE|Sulfurospirillum deleyianum (65553)
>4NEA_1|Chains A, B, C, D|Betaine aldehyde dehydrogenase|Staphylococcus aureus subsp. aureus (93062)
>9HBE_1|Chains A, B|Fucose-binding lectin protein|Ralstonia solanacearum (305)
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 \(w=f(c)\)
1QDB , Knot 192 473 0.83 40 240 448 
GIAGKEKSEEWAKYYPRQFDSWKKTKEYDSFTDMLAKDPALVIAWSGYAFSKDYNSPRGHYYALQDNVNSLRTGAPVDAKTGPLPTACWTCKSPDVPRLIEEDGELEYFTGKWAKYGSQIVNVIGCANCHDDKTAELKVRVPHLNRGLQAAGLKTFEESTHQDKRTLVCAQCHVEYYFKKTEWKDAKGADKTAMVVTLPWANGVGKDGNAGVEGMIKYYDEINFSDWTHNISKTPMLKAQHPGFEFWKSGIHGQKGVSCADCHMPYTQEGSVKYSDHQVKENPLDSMDQSCMNCHRESESKLRGIVHQKYERKEFLNKVAFDNIGKAHLETGKAIEAGASDEELKEVRKLIRHGQFKADMAIAAHGNYFHAPEETLRLLAAGSDDAQKARLLLVKILAKHGVMDYIAPDFDTKDKAQKLAKVDIAALAAEKMKFKQTLEQEWKKEAKAKGRANPELYKDVDTINDGKSSWNKK
4NEA , Knot 208 520 0.83 40 251 483 
MHHHHHHSSGVDLGTENLYFQSNAMELLKHLSQRQYIDGEWVESANKNTRDIINPYNQEVIFTVSEGTKEDAERAILAARRAFESGEWSQETAETRGKKVRAIADKIKEHREALARLETLDTGKTLEESYADMDDIHNVFMYFAGLADKDGGEMIDSPIPDTESKIVKEPVGVVTQITPWNYPLLQASWKIAPALATGCSLVMKPSEITPLTTIRVFELMEEVGFPKGTINLILGAGSEVGDVMSGHKEVDLVSFTGGIETGKHIMKNAANNVTNIALELGGKNPNIIFDDADFELAVDQALNGGYFHAGQVCSAGSRILVQNSIKDKFEQALIDRVKKIKLGNGFDADTEMGPVISTEHRNKIESYMDVAKAEGATIAVGGKRPDRDDLKDGLFFEPTVITNCDTSMRIVQEEVFGPVVTVEGFETEQEAIQLANDSIYGLAGAVFSKDIGKAQRVANKLKLGTVWINDFHPYFAQAPWGGYKQSGIGRELGKEGLEEYLVSKHILTNTNPQLVNWFSK
9HBE , Knot 43 90 0.71 38 65 81 
SSVQTAATSWGTVPSIRVYTANNGKITERCWDGKGWYTGAFNEPGDNVSVTSWLVGSAIHIRVYASTGTTTTEWCWDGNGWTKGAYTATN

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 \(c\).

\(|P_{f(1QDB_1)}(2) \setminus P_{f(4NEA_1)}(2)|=70\), \(|P_{f(4NEA_1)}(2) \setminus P_{f(1QDB_1)}(2)|=81\). 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 1:11110000001100010010010000000010011100111111101011000000101000110001001001111010011110101000010110110001010010101100100110111010000000101010110100110111100100000000001101000100010000100101100011110111101110010111011100000101001000100011101001110110011010011001000110000101000000100011001000010000000001011100000000110011100110101001011011100001001001100101010111110100101100010111110001001011110111001110011101000001001101011111100101000100010001010101010100010010010001000
Pair \(Z_2\) Length of longest common subsequence
1QDB_1,4NEA_1 151 4
1QDB_1,9HBE_1 227 3
4NEA_1,9HBE_1 222 3

Newick tree

 
[
	9HBE_1:12.07,
	[
		1QDB_1:75.5,4NEA_1:75.5
	]:46.57
]

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 \((0.85)(0.8)(\frac{993 }{\log_{20} 993}-\frac{473}{\log_{20}473})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 1QDB_1 4NEA_1 173 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]