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Parikh vectors
6ERF_1 5PEM_1 7UHT_1 Letter Amino acid
49 9 10 E Glutamic acid
8 8 10 H Histidine
63 13 29 L Leucine
30 5 15 R Arginine
4 2 2 C Cysteine
26 6 23 G Glycine
2 2 5 W Tryptophan
39 11 13 S Serine
23 8 20 T Threonine
49 14 8 K Lycine
15 6 6 M Methionine
28 8 5 F Phenylalanine
28 6 18 P Proline
27 4 8 I Isoleucine
19 4 8 Y Tyrosine
31 8 14 V Valine
26 4 46 A Alanine
18 6 8 N Asparagine
37 12 18 D Aspartic acid
22 2 9 Q Glutamine 

6ERF_1|Chains A, C, E, G|X-ray repair cross-complementing protein 6|Homo sapiens (9606)
>5PEM_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|Homo sapiens (9606)
>7UHT_1|Chain A|Putative metallo-beta-lactamase l1 (Beta-lactamase type ii) (Ec 3.5.2.6) (Penicillinase)|Stenotrophomonas maltophilia K279a (522373)
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 \(w=f(c)\)
6ERF , Knot 213 544 0.82 40 242 506 
MSGWESYYKTEGDEEAEEEQEENLEASGDYKYSGRDSLIFLVDASKAMFESQSEDELTPFDMSIQCIQSVYISKIISSDRDLLAVVFYGTEKDKNSVNFKNIYVLQELDNPGAKRILELDQFKGQQGQKRFQDMMGHGSDYSLSEVLWVCANLFSDVQFKMSHKRIMLFTNEDNPHGNDSAKASRARTKAGDLRDTGIFLDLMHLKKPGGFDISLFYRDIISIAEDEDLRVHFEESSKLEDLLRKVRAKETRKRALSRLKLKLNKDIVISVGIYNLVQKALKPPPIKLYRETNEPVKTKTRTFNTSTGGLLLPSDTKRSQIYGSRQIILEKEETEELKRFDDPGLMLMGFKPLVLLKKHHYLRPSLFVYPEESLVIGSSTLFSALLIKCLEKEVAALCRYTPRRNIPPYFVALVPQEEELDDQKIQVTPPGFQLVFLPFADDKRKMPFTEKIMATPEQVGKMKAIVEKLRFTYRSDSFENPVLQQHFRNLEALALDLMEPEQAVDLTLPKVEAMNKRLGSLVDEFKELVYPPDYNPEGKVTKRK
5PEM , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS
7UHT , Knot 115 275 0.78 40 157 259 
SNAASAAEAPLPQLRAYTVDASWLQPMAPLQVADHTWQIGTEDLTALLVQTAEGAVLLDGGMPQMAGHLLDNMKLRGVAPQDLRLILLSHAHADHAGPVAELKRRTGAHVAANAETAVLLARGGSNDLHFGDGITYPPASADRIIMDGEVVTVGGIAFTAHFMPGHTPGSTAWTWTDTRDGKPVRIAYADSLSAPGYQLKGNPRYPRLIEDYKRSFATVRALPCDLLLTPHPGASNWNYAVGSKASAEALTCNAYADAAEKKFDAQLARETAGTR

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 \(c\).

\(|P_{f(6ERF_1)}(2) \setminus P_{f(5PEM_1)}(2)|=160\), \(|P_{f(5PEM_1)}(2) \setminus P_{f(6ERF_1)}(2)|=27\). 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 1:1011000000010001000000010101000001000111110100111000000010110101001001010011000001111110100000001010010110010011100110100101001000100111010000100111101011001010100001111000001010001010010001101000111101101001111010110001101100001010100000100110010100000011001010100011101110011001101111010000001100000010000111111000000010100011100000001001001111111101111100000101011101000111100011011110010001111000010001110111111000010000101011110111111100000111000111010011010111001010000001001110001001011110110100110101101011000110110010011011000101010000
Pair \(Z_2\) Length of longest common subsequence
6ERF_1,5PEM_1 187 3
6ERF_1,7UHT_1 185 4
5PEM_1,7UHT_1 168 4

Newick tree

 
[
	6ERF_1:95.81,
	[
		7UHT_1:84,5PEM_1:84
	]:11.81
]

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 \((0.85)(0.8)(\frac{682 }{\log_{20} 682}-\frac{138}{\log_{20}138})=155.\)
Status Protein1 Protein2 d d1/2
Query variables 6ERF_1 5PEM_1 196 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]