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Parikh vectors
2HPC_1 8UMH_1 6XAR_1 Letter Amino acid
8 9 13 Q Glutamine
7 23 19 E Glutamic acid
1 5 15 H Histidine
6 20 19 S Serine
1 7 11 Y Tyrosine
9 22 11 V Valine
11 31 39 L Leucine
9 36 25 K Lycine
1 13 21 P Proline
4 23 16 A Alanine
1 17 13 N Asparagine
2 10 5 C Cysteine
1 10 10 M Methionine
0 1 6 W Tryptophan
10 16 17 R Arginine
8 21 15 D Aspartic acid
1 23 20 G Glycine
7 26 18 I Isoleucine
0 9 22 F Phenylalanine
0 23 20 T Threonine 

2HPC_1|Chains A, D, G, J|Fibrinogen alpha chain|Homo sapiens (9606)
>8UMH_1|Chain A[auth M]|Transcription initiation factor IIB|Saccharomyces cerevisiae (4932)
>6XAR_1|Chains A, B|E3 ubiquitin-protein ligase CBL|Homo sapiens (9606)
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 \(w=f(c)\)
2HPC , Knot 43 87 0.73 34 62 84 
VSEDLRSRIEVLKRKVIEKVQHIQLLQKNVRAQLVDMKRLEVDIDIKIRSCRGSCSRALAREVDLKDYEDQQKQLEQVIAKDLLPSR
8UMH , Knot 146 345 0.82 40 208 324 
MMTRESIDKRAGRRGPNLNIVLTCPECKVYPPKIVERFSEGDVVCALCGLVLSDKLVDTRSEWRTFSNDDHNGDDPSRVGEASNPLLDGNNLSTRIGKGETTDMRFTKELNKAQGKNVMDKKDNEVQAAFAKITMLCDAAELPKIVKDCAKEAYKLCHDEKTLKGKSMESIMAASILIGCRRAEVARTFKEIQSLIHVKTKEFGKTLNIMKNILRGKSEDGFLKIDTDNMSGAQNLTYIPRFCSHLGLPMQVTTSAEYTAKKCKEIKEIAGKSPITIAVVSIYLNILLFQIPITAAKVGQTLQVTEGTIKSGYKILYEHRDKLVDPQLIANGVVSLDNLPGVEKK
6XAR , Knot 146 335 0.84 40 200 316 
GSLIGLMKDAFQPHHHHHHHLSPHPPGTVDKKMVEKCWKLMDKVVRLCQNPKLALKNSPPYILDLLPDTYQHLRTILSRYEGKMETLGENEYFRVFMENLMKKTKQTISLFKEGKERMYEENSQPRRNLTKLSLIFSHMLAELKGIFPSGLFQGDTFRITKADAAEFWRKAFGEKTIVPWKSFRQALHEVHPISSGLEAMALKSTIDLTCNDYISVFEFDIFTRLFQPWSSLLRNWNSLAVTHPGYMAFLTYDEVKARLQKFIHKPGSYIFRLSCTRLGQWAIGYVTADGNILQTIPHNKPLFQALIDGFREGFYLFPDGRNQNPDLTGLCEPTP

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 \(c\).

\(|P_{f(2HPC_1)}(2) \setminus P_{f(8UMH_1)}(2)|=20\), \(|P_{f(8UMH_1)}(2) \setminus P_{f(2HPC_1)}(2)|=166\). 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 1:100010001011000110010010110001010110100101010101000010000111001010000000001001110011100
Pair \(Z_2\) Length of longest common subsequence
2HPC_1,8UMH_1 186 3
2HPC_1,6XAR_1 196 4
8UMH_1,6XAR_1 172 4

Newick tree

 
[
	2HPC_1:98.50,
	[
		8UMH_1:86,6XAR_1:86
	]:12.50
]

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 \((0.85)(0.8)(\frac{432 }{\log_{20} 432}-\frac{87}{\log_{20}87})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 2HPC_1 8UMH_1 132 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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