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Parikh vectors
4PGA_1 5UKM_1 4GCP_1 Letter Amino acid
0 15 0 C Cysteine
17 32 13 Q Glutamine
8 24 3 M Methionine
9 31 4 P Proline
38 38 29 A Alanine
19 23 21 T Threonine
33 37 23 V Valine
13 42 11 R Arginine
27 40 49 G Glycine
26 77 20 L Leucine
23 56 18 K Lycine
4 7 2 W Tryptophan
8 22 29 Y Tyrosine
21 17 30 N Asparagine
21 40 27 D Aspartic acid
16 60 14 E Glutamic acid
5 27 1 H Histidine
19 32 12 I Isoleucine
6 36 19 F Phenylalanine
24 39 16 S Serine 

4PGA_1|Chains A, B|GLUTAMINASE-ASPARAGINASE|Pseudomonas sp. 7A (65406)
>5UKM_1|Chain A|Beta-adrenergic receptor kinase 1|Bos taurus (9913)
>4GCP_1|Chains A, B|Outer membrane protein F|Escherichia coli (83333)
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 \(w=f(c)\)
4PGA , Knot 144 337 0.83 38 197 323 
KEVENQQKLANVVILATGGTIAGAGASAANSATYQAAKVGVDKLIAGVPELADLANVRGEQVMQIASESITNDDLLKLGKRVAELADSNDVDGIVITHGTDTLEETAYFLNLVQKTDKPIVVVGSMRPGTAMSADGMLNLYNAVAVASNKDSRGKGVLVTMNDEIQSGRDVSKSINIKTEAFKSAWGPLGMVVEGKSYWFRLPAKRHTVNSEFDIKQISSLPQVDIAYSYGNVTDTAYKALAQNGAKALIHAGTGNGSVSSRVVPALQQLRKNGTQIIRSSHVNQGGFVLRNAEQPDDKNDWVVAHDLNPEKARILAMVAMTKTQDSKELQRIFWEY
5UKM , Knot 266 695 0.83 40 286 643 
MADLEAVLADVSYLMAMEKSKATPAARASKKILLPEPSIRSVMQKYLEDRGEVTFEKIFSQKLGYLLFRDFCLKHLEEAKPLVEFYEEIKKYEKLETEEERLVCSREIFDTYIMKELLACSHPFSKSAIEHVQGHLVKKQVPPDLFQPYIEEICQNLRGDVFQKFIESDKFTRFCQWKNVELNIHLTMNDFSVHRIIGRGGFGEVYGCRKADTGKMYAMKCLDKKRIKMKQGETLALNERIMLSLVSTGDCPFIVCMSYAFHTPDKLSFILDLMNGGDLHYHLSQHGVFSEADMRFYAAEIILGLEHMHNRFVVYRDLKPANILLDEHGHVRISDLGLACDFSKKKPHASVGTHGYMAPEVLQKGVAYDSSADWFSLGCMLFKLLRGHSPFRQHKTKDKHEIDRMTLTMAVELPDSFSPELRSLLEGLLQRDVNRRLGCLGRGAQEVKESPFFRSLDWQMVFLQKYPPPLIPPRGEVNAADAFDIGSFDEEDTKGIKLLDSDQELYRNFPLTISERWQQEVAETVFDTINAETDRLEARKKTKNKQLGHEEDYALGKDCIMHGYMSKMGNPFLTQWQRRYFYLFPNRLEWRGEGEAPQSLLTMEEIQSVEETQIKERKCLLLKIRGGKQFVLQCDSDPELVQWKKELRDAYREAQQLVQRVPKMKNKPRAPVVELSKVPLIQRGSANGLHHHHHH
4GCP , Knot 144 341 0.82 38 175 317 
GAEIYNKDGNKVDLYGKAVGLHYFSKGNGENSYGGNGDMTYARLGFKGETQINSDLTGYGQWEYNFQGNNSEGADAQTGNKTRLAFAGLKYADVGSFDYGRNYGVVYDALGYTDMLPEFGGDTAYSDDFFVGRVGGVATYRNSNFFGLVDGLNFAVQYLGKNERDTARRSNGDGVGGSISYEYEGFGIVGAYGAADRTNLQEAQPLGNGKKAEQWATGLKYDANNIYLAANYGETRNATPITNKFTNTSGFANKTQDVLLVAQYQFDFGLRPSIAYTKSKAKDVEGIGDVDLVNYFEVGATYYFNKNMSTYVDYIINQIDSDNKLGVGSDDTVAVGIVYQF

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 \(c\).

\(|P_{f(4PGA_1)}(2) \setminus P_{f(5UKM_1)}(2)|=37\), \(|P_{f(5UKM_1)}(2) \setminus P_{f(4PGA_1)}(2)|=126\). 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 1:0010000011011111011011111101100100011011100111111011011010100110110001000011011001101100001011110010001000101101100000111111010110110101110100111110000001011110100010010010001010001100111111111010001101110000100010100100110101100010100010011100110111011010101000111110010001001100001001111100100100000111100101001011111110000000010011100
Pair \(Z_2\) Length of longest common subsequence
4PGA_1,5UKM_1 163 4
4PGA_1,4GCP_1 140 4
5UKM_1,4GCP_1 189 4

Newick tree

 
[
	5UKM_1:93.53,
	[
		4PGA_1:70,4GCP_1:70
	]:23.53
]

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 \((0.85)(0.8)(\frac{1032 }{\log_{20} 1032}-\frac{337}{\log_{20}337})=185.\)
Status Protein1 Protein2 d d1/2
Query variables 4PGA_1 5UKM_1 233 171.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]