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Parikh vectors
4RNI_1 8SFY_1 4HXR_1 Letter Amino acid
10 21 5 F Phenylalanine
19 15 3 R Arginine
24 30 2 G Glycine
15 8 9 Q Glutamine
14 32 7 I Isoleucine
39 55 13 L Leucine
9 12 5 M Methionine
19 29 1 S Serine
5 14 8 T Threonine
4 23 12 N Asparagine
14 22 6 D Aspartic acid
13 28 12 P Proline
28 27 5 A Alanine
3 4 2 C Cysteine
7 34 12 K Lycine
3 3 3 W Tryptophan
6 13 7 Y Tyrosine
18 22 5 V Valine
23 32 7 E Glutamic acid
13 13 1 H Histidine 

4RNI_1|Chains A, B|Motility regulator|Pseudomonas aeruginosa PAO1 (208964)
>8SFY_1|Chain A|UDP-glycosyltransferase 202A2|Tetranychus urticae (32264)
>4HXR_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
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 \(w=f(c)\)
4RNI , Knot 122 286 0.80 40 166 264 
MGSSHHHHHHSSGLVPRGSHMAEMNARALERLELESDLRRALELGEFVLHYQPQFTGDGRRLTGAEALLRWQHPRRGLVPPSEFIPVLEEIGLVAQVGDWLLAEACKQLRSWHKAKVRVPKVSVNLSARQFADGQLGERIAAILYETGIPPACLELELTESILMSDVAEAMQILSGLKRLGLAIAVDDFGTGYSSLNYLKQFPIDVLKIDRSFVDGLPHGEQDAQIARAIIAMAHSLNLMVIAEGVESQAQLDFLREHGCDEVQGYLFGRPMPAEQFGMLYASDVL
8SFY , Knot 176 437 0.81 40 223 411 
MNPETMNENGKSLKILFTALFGPGHLNACLGIGSLLRKRGHQIYFAHFPRHRATIEKHGFLFISLLDYAEPEFPIVDMLPDIGIIAKFAFERMHKLTPLELFRHASGKHTFAGMVNGSKGENYAMMKIVKEYKPDVCLADYLFNMPWMFTVDCPVIPVKSVNPIELYNGPPALTGCSIHDPPSVREEIEQLARKSELELESELEKLFAHFNVPLVSYNYAQQLGIYIYPGPLDYKELGSPKENWVRLDSSIRSTEISNFELPEKLKDKPGKLIYVSMGSLASAVTELLTMILTPLANSPHRFIVSTGPNGDSIKLYDNMWGDKFINQVALLPKVDLFITHGGSNSLIEGLTAGKPLIAIPQFGDQLDNAQRIADLGLGVRLNLHEFSGEKLLKAIEDVLNDEKINANVARVSEELKKSDSKDKVISLIEKLARDKKL
4HXR , Knot 65 125 0.83 40 108 121 
MNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTE

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 \(c\).

\(|P_{f(4RNI_1)}(2) \setminus P_{f(8SFY_1)}(2)|=54\), \(|P_{f(8SFY_1)}(2) \setminus P_{f(4RNI_1)}(2)|=111\). 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 1:1100000000001111010011010101100101000100110110111000101010100101101110100100111110011111001111101101111010001001001010110101010100110101100111110001111101010100011100110110110110011111110011010001001001110110100011011101000101101111110010111110110001010110001000101011101111001111010011
Pair \(Z_2\) Length of longest common subsequence
4RNI_1,8SFY_1 165 5
4RNI_1,4HXR_1 180 4
8SFY_1,4HXR_1 195 5

Newick tree

 
[
	4HXR_1:97.30,
	[
		4RNI_1:82.5,8SFY_1:82.5
	]:14.80
]

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 \((0.85)(0.8)(\frac{723 }{\log_{20} 723}-\frac{286}{\log_{20}286})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 4RNI_1 8SFY_1 150 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]