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Parikh vectors
8CZN_1 4IPF_1 6PBM_1 Letter Amino acid
5 4 12 I Isoleucine
17 4 54 A Alanine
14 7 16 Q Glutamine
3 4 16 H Histidine
11 3 15 F Phenylalanine
2 0 7 W Tryptophan
8 4 14 Y Tyrosine
21 6 31 V Valine
2 1 9 C Cysteine
12 7 24 E Glutamic acid
13 4 34 G Glycine
12 12 60 L Leucine
6 3 6 M Methionine
9 3 13 T Threonine
7 2 10 N Asparagine
12 2 23 D Aspartic acid
7 3 29 P Proline
8 6 29 S Serine
3 3 42 R Arginine
17 7 5 K Lycine 

8CZN_1|Chains A, B|Thiol:disulfide interchange protein DsbA|Escherichia coli K-12 (83333)
>4IPF_1|Chain A|E3 ubiquitin-protein ligase Mdm2|Xenopus laevis (8355)
>6PBM_1|Chains A, B|Pseudopaline Dehydrogenase|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (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 \(w=f(c)\)
8CZN , Knot 86 189 0.79 40 134 185 
AQYEDGKQYTTLEKPVAGAPQVLEFFSFFCPHCYQFEEVLHISDNVKKKLPEGVKMTKYHVNFMGGDLGKDLTQAWAVAMALGVEDKVTVPLFEGVQKTQTIRSASDIRDVFINAGIKGEEYDAAWNSFVVKSLVAQQEKAAADVQLRGVPAMFVNGKYQLNPQGMDTSNMDVFVQQYADTVKYLSEKK
4IPF , Knot 45 85 0.78 38 73 83 
EKLVQPTPLLLSLLKSAGAQKETFTMKEVLYHLGQYIMAKQLYDEKQQHIVHCSNDPLGELFGVQEFSVKEHRRIYAMISRNLVS
6PBM , Knot 179 449 0.81 40 214 411 
HHHHHHSSGLVPRGSHMNAADESLGNVLLVGLGAVAIQVALDLRRHGAGRLGALNHPGRRSQRIAEALARGACLQLEGQGQHRWLSGNAALDVFHQDPAELRDDWQTLVLCVPADSYLDVVRGLPWERLGGVRTLLLVSAFIGANLLVRSALPAGCQATVLSLSSYYAATKVIDETQPLRALTKAVKRRVYLGSSRPDCPARETWRRVLAGSGVEVVPLATPEAAEGRNVTTYVHSPFFLGEFALARILSEQGPPGFMYKLYPEGPITPGAIGAMRRLWCELSELLRRMGAEPLNLLRFLNDDNYPVHETMLPRASIDGFAEAGAERQEYLLFVRYAALLVDPFSPADEQGRHFDFSAVPFRRVSRDEDGLWRLPRVPLEDYRKLALIVALAAHFDLAMPQARSLLASYENAVSRFIDCQGASQCHPSLYPIDSRPAADAIYRQWCSTC

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 \(c\).

\(|P_{f(8CZN_1)}(2) \setminus P_{f(4IPF_1)}(2)|=98\), \(|P_{f(4IPF_1)}(2) \setminus P_{f(8CZN_1)}(2)|=37\). 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 1:100001000001001111110110110110100001001101000100011011010000101111011001001111111111000101111011000001001001001110111010000111001110011100001110101011111110100010101100001011100010010010000
Pair \(Z_2\) Length of longest common subsequence
8CZN_1,4IPF_1 135 3
8CZN_1,6PBM_1 198 3
4IPF_1,6PBM_1 199 3

Newick tree

 
[
	6PBM_1:10.77,
	[
		8CZN_1:67.5,4IPF_1:67.5
	]:40.27
]

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 \((0.85)(0.8)(\frac{274 }{\log_{20} 274}-\frac{85}{\log_{20}85})=60.4\)
Status Protein1 Protein2 d d1/2
Query variables 8CZN_1 4IPF_1 75 54.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]