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Parikh vectors
4LLU_1 8UPB_1 9CPN_1 Letter Amino acid
6 11 37 E Glutamic acid
3 0 7 H Histidine
2 0 9 M Methionine
13 1 23 P Proline
13 4 55 A Alanine
9 3 33 D Aspartic acid
12 3 43 K Lycine
9 4 22 F Phenylalanine
31 1 23 S Serine
20 0 29 T Threonine
10 2 25 N Asparagine
5 0 1 C Cysteine
8 2 15 Q Glutamine
2 3 31 I Isoleucine
19 12 43 L Leucine
4 1 11 W Tryptophan
22 6 33 V Valine
7 9 19 R Arginine
10 1 17 Y Tyrosine
22 1 41 G Glycine 

4LLU_1|Chains A, C|PERTUZUMAB FAB Heavy chain|Homo sapiens (9606)
>8UPB_1|Chain A|De novo designed IL-6 mimetic|synthetic construct (32630)
>9CPN_1|Chains A, B, C, D|Induced myeloid leukemia cell differentiation protein Mcl-1|synthetic construct (32630)
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 \(w=f(c)\)
4LLU , Knot 101 227 0.80 40 142 214 
EVQLVESGGGLVQPGGSLRLSCAASGFTFTDYTMDWVRQAPGKGLEWVADVNPNSGGSIYNQRFKGRFTLSVDRSKNTLYLQMNSLRAEDTAVYYCARNLGPSFYFDYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDKTHT
8UPB , Knot 33 64 0.71 32 48 57 
ANLYFQSDREEFQWLVEEFIRVLERGDVEKAREILRLLKEVAEKVNDPLLRLLFRIARRLVEEL
9CPN , Knot 207 517 0.83 40 245 476 
AIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTGSELYRQSLEIISRYLREQATGAADTAPMGRSGATSRKALETLRRVGDGVQRNHETAFQGMLRKLDIKNEDDVKSLSRVMIHVFSDGVTNWGRIVTLISFGAFVAKHLKTINQESCIEPLAESITDVLVRTKRDWLVKQRGWDGFVEFFHV

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 \(c\).

\(|P_{f(4LLU_1)}(2) \setminus P_{f(8UPB_1)}(2)|=123\), \(|P_{f(8UPB_1)}(2) \setminus P_{f(4LLU_1)}(2)|=29\). 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 1:01011001111101110101001101101000010110011101101110101001101000010101010100000010101001010001100010011101010011010110100100011011111100000011011110110001101101010011100110011111000110010011011000110000100100010000100010100000000
Pair \(Z_2\) Length of longest common subsequence
4LLU_1,8UPB_1 152 3
4LLU_1,9CPN_1 187 4
8UPB_1,9CPN_1 219 3

Newick tree

 
[
	9CPN_1:10.07,
	[
		4LLU_1:76,8UPB_1:76
	]:33.07
]

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 \((0.85)(0.8)(\frac{291 }{\log_{20} 291}-\frac{64}{\log_{20}64})=73.1\)
Status Protein1 Protein2 d d1/2
Query variables 4LLU_1 8UPB_1 91 57.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]