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Parikh vectors
4RHB_1 1NGW_1 5PTL_1 Letter Amino acid
37 11 11 A Alanine
26 10 13 E Glutamic acid
22 9 6 F Phenylalanine
41 21 5 T Threonine
53 10 5 Y Tyrosine
37 13 7 G Glycine
27 5 5 I Isoleucine
25 11 6 P Proline
47 28 7 S Serine
45 10 13 D Aspartic acid
32 14 8 Q Glutamine
16 2 10 H Histidine
14 3 7 M Methionine
14 2 0 W Tryptophan
35 17 10 V Valine
36 6 14 R Arginine
43 9 4 N Asparagine
2 4 1 C Cysteine
42 14 18 L Leucine
20 14 6 K Lycine 

4RHB_1|Chains A, C|LPS-assembly protein LptD|Escherichia coli (83333)
>1NGW_1|Chains A, C[auth L]|Mature Metal Chelatase Catalytic Antibody, Light chain|Mus musculus (10090)
>5PTL_1|Chains A, B|Bromodomain-containing protein 1|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 \(w=f(c)\)
4RHB , Knot 240 614 0.83 40 265 582 
MKKRIPTLLATMIATALYSQQGLAFKVGPVPIFYSPYLQLPVGDKRRSGFLIPNAKYTTTNYFEFYLPYYWNIAPNMDATITPHYMHRRGNIMWENEFRYLSQAGAGLMELDYLPSDKVYEDEHPNDDSSRRWLFYWNHSGVMDQVWRFNVDYTKVSDPSYFNDFDNKYGSSTDGYATQKFSVGYAVQNFNATVSTKQFQVFSEQNTSSYSAEPQLDVNYYQNDVGPFDTRIYGQAVHFVNTRDDMPEATRVHLEPTINLPLSNNWGSINTEAKLLATHYQQTNLDWYNSRNTTKLDESVNRVMPQFKVDGKMVFERDMEMLAPGYTQTLEPRAQYLYVPYRDQSDIYNYDSSLLQSDYSGLFRDRTYGGLDRIASANQVTTGVTSRIYDDAAVERFNISVGQIYYFTESRTGDDNITWENDDKTGSLVWAGDTYWRISERWGLRGGIQYDTRLDNVATSNSSIEYRRDEDRLVQLNYRYASPEYIQATLPKYYSTAEQYKNGISQVGAVASWPIADRWSIVGAYYYDTNANKQADSMLGVQYSSCCYAIRVGYERKLNGWDNDKQHAVYDNAIGFNIELRGLSSNYGLGTQEMLRSNILPYQNTLHHHHHHHH
1NGW , Knot 98 213 0.82 40 156 206 
ELVMTQTPKFMSTTVGDRVSITCKASQNVGTPVAWYQQKPGQSPKLLIYSASNRYTGVPDRFTGSGSGTDFTLTISNMQSEDLADYFCQQYSSYPLTFGGGTKVEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRNE
5PTL , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA

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 \(c\).

\(|P_{f(4RHB_1)}(2) \setminus P_{f(1NGW_1)}(2)|=150\), \(|P_{f(1NGW_1)}(2) \setminus P_{f(4RHB_1)}(2)|=41\). 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 1:10001101110111011000011110111111100101011110000011111010000000101011001011101010101001000101110001001001111110100110001000001000000011101000111001101010000100100100100001000010100010110110010101000010110000000001010101000000111100010101101100000110100101010101110001101000101110000000101000000001000100111010101011100010111110000101010010110000001000000110000011100000111001101001001100010001110010101101001000001000101000000101111100010100011101110000010011000001000000001101000010100101011000001000001100111110111100101111000000100010011110000000110110000101100000011000111101010110000111000110001110000100000000
Pair \(Z_2\) Length of longest common subsequence
4RHB_1,1NGW_1 191 4
4RHB_1,5PTL_1 209 6
1NGW_1,5PTL_1 178 3

Newick tree

 
[
	4RHB_1:10.53,
	[
		1NGW_1:89,5PTL_1:89
	]:14.53
]

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 \((0.85)(0.8)(\frac{827 }{\log_{20} 827}-\frac{213}{\log_{20}213})=169.\)
Status Protein1 Protein2 d d1/2
Query variables 4RHB_1 1NGW_1 214 142.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]