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Parikh vectors
1LXA_1 5KVQ_1 5IKL_1 Letter Amino acid
16 17 23 T Threonine
6 9 12 Y Tyrosine
12 26 8 H Histidine
26 14 24 I Isoleucine
10 5 27 K Lycine
10 10 17 F Phenylalanine
25 32 68 A Alanine
15 27 31 R Arginine
13 6 18 N Asparagine
6 19 21 Q Glutamine
11 29 25 S Serine
0 8 4 W Tryptophan
9 27 28 P Proline
25 21 36 V Valine
11 19 24 D Aspartic acid
30 28 60 G Glycine
12 52 56 L Leucine
4 7 11 M Methionine
5 10 6 C Cysteine
16 19 38 E Glutamic acid 

1LXA_1|Chain A|UDP N-ACETYLGLUCOSAMINE O-ACYLTRANSFERASE|Escherichia coli (83333)
>5KVQ_1|Chains A, B|Irp3 protein|Yersinia enterocolitica (630)
>5IKL_1|Chains A[auth B], B[auth D], C[auth F]|Geranyl-CoA carboxylase, beta-subunit|Pseudomonas aeruginosa (strain ATCC 15692 / PAO1 / 1C / PRS 101 / LMG 12228) (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 \(w=f(c)\)
1LXA , Knot 114 262 0.80 38 160 247 
MIDKSAFVHPTAIVEEGASIGANAHIGPFCIVGPHVEIGEGTVLKSHVVVNGHTKIGRDNEIYQFASIGEVNQDLKYAGEPTRVEIGDRNRIRESVTIHRGTVQGGGLTKVGSDNLLMINAHIAHDCTVGNRCILANNATLAGHVSVDDFAIIGGMTAVHQFCIIGAHVMVGGCSGVAQDVPPYVIAQGNHATPFGVNIEGLKRRGFSREAITAIRNAYKLIYRSGKTLDEVKPEIAELAETYPEVKAFTDFFARSTRGLIR
5KVQ , Knot 162 385 0.83 40 204 359 
MGSSHHHHHHSSGLVPRGSHMPSASPKQRVLIVGAKFGEMYLNAFMQPPEGLELVGLLAQGSARSRELAHAFGIPLYTSPEQITGMPDIACIVVRSTVAGGAGTQLARHFLARGVHVIQEHPLHPDDISSLQTLAQEQGCCYWINTFYPHTRAGRTWLRDAQQLRRCLAKTPPVVHATTSRQLLYSTLDLLLLALGVDTAAVECDVVGSFSDFHCLRLFWPEGEACLLLQRYLDPDDPDMHSLIMHRLLLGWPEGHLSLEASYGPVIWSSSLFVADHQENAHSLYRRPEILRDPPGLTRSAAPLSWRDCCETVGPEGVSWLLHQLRSHLAGEHPPVACQNVHQIALSRLWQQILRKTGNAEIRRLTPPHHDRLAGFYNDDDKEAL
5IKL , Knot 216 537 0.84 40 236 493 
PAIQSELDVNGEDFARNREAMLAAVAGFRELEQKVLDKAAEARPKFEKRGQLLPRERLALLLDPGAPFLELSSLAGYKLHDDKDGTQAGGGIIAGIGYIAGVRCLVSASNSAIKGGTISPTGLKKTLRLQQIAMENKLPVVTLTESGGANLNYAAEIFVEGARGFANQARISAMGIPQVTVVHGSSTAGGAYQPGLSDYVVVVRGKAKMFLAGPPLLKAATGEIASDEELGGAELHAQVAGTAEYLAENDADGVRLAREIVGMLPWNAQLPARPARSWREPLYPVEELLGVVPADPKKPYDVREIVARIADGSEFLDFKNEFDGQTVCGHLRIEGHACGLIGNNGPITPQGAAKAAQFIQLCEQSNTPLLFLHNTTGFMVGTESERQGVIKHGSKMIQAVANARVPKLTLVVGGSYGAGNYAMCGRGLDPRFIFAWPNSRTAVMGGAQAGKVLRIVTEEKHAKEGKEADPKMLEMLETVTAQKLDSQSTALYGTASLWDDGLVDPRDSRRLLGYLLDICAEAEARPLKGNSFGVARF

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 \(c\).

\(|P_{f(1LXA_1)}(2) \setminus P_{f(5KVQ_1)}(2)|=67\), \(|P_{f(5KVQ_1)}(2) \setminus P_{f(1LXA_1)}(2)|=111\). 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 1:1100011101011100110111010111101111010110101100011101000110000100110110100010011010010110000100010100101011110011000111101011000011000111001011101010011111110110010111101111100111001110111010010111101011000110001101100100110001001001010110110001010110011100001110
Pair \(Z_2\) Length of longest common subsequence
1LXA_1,5KVQ_1 178 3
1LXA_1,5IKL_1 172 3
5KVQ_1,5IKL_1 166 4

Newick tree

 
[
	1LXA_1:88.96,
	[
		5IKL_1:83,5KVQ_1:83
	]:5.96
]

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 \((0.85)(0.8)(\frac{647 }{\log_{20} 647}-\frac{262}{\log_{20}262})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 1LXA_1 5KVQ_1 137 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]