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Parikh vectors
9ATM_1 1YWN_1 6JUO_1 Letter Amino acid
2 10 4 H Histidine
18 14 30 T Threonine
16 11 6 Y Tyrosine
7 8 2 N Asparagine
5 9 3 C Cysteine
7 8 9 Q Glutamine
6 23 21 E Glutamic acid
12 18 25 P Proline
4 5 7 W Tryptophan
9 21 27 D Aspartic acid
19 36 35 L Leucine
12 21 11 K Lycine
6 15 8 F Phenylalanine
14 21 39 A Alanine
33 14 19 S Serine
22 18 38 V Valine
6 20 28 R Arginine
21 20 27 G Glycine
5 15 9 I Isoleucine
2 9 8 M Methionine 

9ATM_1|Chain A[auth H]|VIR-7229 Fab heavy chain|Homo sapiens (9606)
>1YWN_1|Chain A|Vascular endothelial growth factor receptor 2|Homo sapiens (9606)
>6JUO_1|Chains A, D[auth F]|DNA polymerase IV|Mycobacterium smegmatis (strain ATCC 700084 / mc(2)155) (246196)
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 \(w=f(c)\)
9ATM , Knot 106 226 0.84 40 146 216 
EVQLVESGGSVVQPGRALRLSCAASGITFSSFGMYWVRQAPGKGLEWLGYIAYDGSDTYYADSVKGRFTISRDNSKNTLYLQMNSLRIEDTAVYYCAKDYYPLLSYYYGLDVWGQGTTVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSC
1YWN , Knot 138 316 0.83 40 201 296 
MDPDELPLDEHCERLPYDASKWEFPRDRLKLGKPLGRGAFGQVIEADAFGIDKTATCRTVAVKMLKEGATHSEHRALMSELKILIHIGHHLNVVNLLGACTKPGGPLMVIVEFCKFGNLSTYLRSKRNEFVPYKVAPEDLYKDFLTLEHLICYSFQVAKGMEFLASRKCIHRDLAARNILLSEKNVVKICDFGLARDIYKDPDYVRKGDARLPLKWMAPETIFDRVYTIQSDVWSFGVLLWEIFSLGASPYPGVKIDEEFCRRLKEGTRMRAPDYTTPEMYQTMLDCWHGEPSQRPTFSELVEHLGNLLQANAQQD
6JUO , Knot 146 356 0.80 40 175 331 
MTKWVLHVDLDQFYASVELRRRPDLRGQPVIVGGSGDPSEPRKVVTCASYEAREFGVHAGMPLRAAARRCPDATFLPSDPAAYDEASEQVMGLLRDLGHPLEVWGWDEAYLGADLPDESDPVEVAERIRTVVAAETGLSCSVGISDNKQRAKVATGFAKPAGIYVLTEANWMTVMGDRPPDALWGVGPKTTKKLAAMGITTVADLAVTDPSVLTTAFGPSTGLWLLLLAKGGGDTEVSSEPWVPRSRSHVVTFPQDLTERREMDSAVRDLALQTLAEIVEQGRIVTRVAVTVRTSTFYTRTKIRKLPAPSTDAGQIVDTALAVLDQFELDRPVRLLGVRLELAMDDVPRPAVTAGT

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 \(c\).

\(|P_{f(9ATM_1)}(2) \setminus P_{f(1YWN_1)}(2)|=63\), \(|P_{f(1YWN_1)}(2) \setminus P_{f(9ATM_1)}(2)|=118\). 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 1:0101100110110110110100110110100111011001110110111011001000001001010101000000001010100101000110001000011100001101110100101001000110111111000000110111101100011011010100111001100111110001100100110110001100001001000100001000101000
Pair \(Z_2\) Length of longest common subsequence
9ATM_1,1YWN_1 181 3
9ATM_1,6JUO_1 145 3
1YWN_1,6JUO_1 168 4

Newick tree

 
[
	1YWN_1:91.71,
	[
		9ATM_1:72.5,6JUO_1:72.5
	]:19.21
]

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 \((0.85)(0.8)(\frac{542 }{\log_{20} 542}-\frac{226}{\log_{20}226})=90.4\)
Status Protein1 Protein2 d d1/2
Query variables 9ATM_1 1YWN_1 119 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]