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Parikh vectors
4ZQW_1 6QLT_1 3SEZ_1 Letter Amino acid
11 6 74 A Alanine
10 8 58 G Glycine
20 13 70 L Leucine
11 8 14 K Lycine
4 2 11 M Methionine
9 5 32 T Threonine
6 3 20 Y Tyrosine
10 9 52 R Arginine
1 1 4 C Cysteine
4 4 21 Q Glutamine
7 10 36 P Proline
3 1 10 W Tryptophan
6 14 16 N Asparagine
13 7 44 D Aspartic acid
8 4 20 H Histidine
8 10 32 I Isoleucine
13 6 43 E Glutamic acid
8 8 30 F Phenylalanine
15 6 44 S Serine
10 13 49 V Valine 

4ZQW_1|Chains A[auth B], C[auth D]|Immunity protein CdiI-o11|Escherichia coli O157:H7 (strain EC869) (478008)
>6QLT_1|Chain A|Galectin-3|Homo sapiens (9606)
>3SEZ_1|Chains A, B, C, D|Glutamine-dependent NAD(+) synthetase|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
4ZQW , Knot 86 177 0.83 40 139 170 
MAFNKDQDYWANIFVTPDFLSVETYSGLGMTGRDPLFSPRLLQPDVDDKSLGEEILQALSDSRTLDVLEERVAFFDLEKSKEQYAAWIATLMEKYGYRTKRALFKNMKKVGIHLVNDVITIRPSFHEKLEAWSGNRINESDYVVLPADSSPTEIGSGLRLALSRCKGTSLEHHHHHH
6QLT , Knot 68 138 0.81 40 112 136 
PLIVPYNLPLPGGVVPRMLITILGTVKPNANRIALDFQRGNDVAFHFNPRFNENNRRVIVCNTKLDNNWGREERQSVFPFESGKPFKIQVLVEPDHFKVAVNDAHLLQYNHRVKKLNEISKLGISGDIDLTSASYTMI
3SEZ , Knot 259 680 0.82 40 262 605 
SMNFYSAYQHGFVRVAACTHHTTIGDPAANAASVLDMARACHDDGAALAVFPELTLSGYSIEDVLLQDSLLDAVEDALLDLVTESADLLPVLVVGAPLRHRHRIYNTAVVIHRGAVLGVVPKSYLPTYREFYERRQMAPGDGERGTIRIGGADVAFGTDLLFAASDLPGFVLHVEIAEDMFVPMPPSAEAALAGATVLANLSGSPITIGRAEDRRLLARSASARCLAAYVYAAAGEGESTTDLAWDGQTMIWENGALLAESERFPKGVRRSVADVDTELLRSERLRMGTFDDNRRHHRELTESFRRIDFALDPPAGDIGLLREVERFPFVPADPQRLQQDCYEAYNIQVSGLEQRLRALDYPKVVIGVSGGLDSTHALIVATHAMDREGRPRSDILAFALPGFATGEHTKNNAIKLARALGVTFSEIDIGDTARLMLHTIGHPYSVGEKVYDVTFENVQAGLRTDYLFRIANQRGGIVLGTGDLSELALGWSTYGVGDQMSHYNVNAGVPKTLIQHLIRWVISAGEFGEKVGEVLQSVLDTEITPELIPTGEEELQSSEAKVGPFALQDFSLFQVLRYGFRPSKIAFLAWHAWNDAERGNWPPGFPKSERPSYSLAEIRHWLQIFVQRFYSFSQFKRSALPNGPKVSHGGALSPRGDWRAPSDMSARIWLDQIDREVPKG

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 \(c\).

\(|P_{f(4ZQW_1)}(2) \setminus P_{f(6QLT_1)}(2)|=88\), \(|P_{f(6QLT_1)}(2) \setminus P_{f(4ZQW_1)}(2)|=61\). 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 1:111000000110111010110100001111010011101011010100001100110110000010110001111010000000111110110001000001110010011101100110101010001011010010000011111000100110110111000010010000000
Pair \(Z_2\) Length of longest common subsequence
4ZQW_1,6QLT_1 149 3
4ZQW_1,3SEZ_1 179 3
6QLT_1,3SEZ_1 192 4

Newick tree

 
[
	3SEZ_1:98.15,
	[
		4ZQW_1:74.5,6QLT_1:74.5
	]:23.65
]

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 \((0.85)(0.8)(\frac{315 }{\log_{20} 315}-\frac{138}{\log_{20}138})=54.4\)
Status Protein1 Protein2 d d1/2
Query variables 4ZQW_1 6QLT_1 68 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]