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Parikh vectors
2JDX_1 5PRT_1 5WQR_1 Letter Amino acid
10 8 5 Q Glutamine
23 13 4 E Glutamic acid
21 18 5 L Leucine
20 7 2 S Serine
10 0 3 W Tryptophan
16 4 6 N Asparagine
24 6 5 K Lycine
17 6 2 F Phenylalanine
23 5 4 T Threonine
15 5 1 Y Tyrosine
29 11 15 A Alanine
27 13 4 D Aspartic acid
8 1 4 C Cysteine
18 7 7 G Glycine
30 5 2 I Isoleucine
21 10 4 V Valine
23 14 2 R Arginine
11 10 1 H Histidine
10 7 1 M Methionine
29 6 6 P Proline 

2JDX_1|Chain A|PROTEIN (L-ARGININE:GLYCINE AMIDINOTRANSFERASE)|Homo sapiens (9606)
>5PRT_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>5WQR_1|Chain A|High-potential iron-sulfur protein|Thermochromatium tepidum (1050)
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 \(w=f(c)\)
2JDX , Knot 167 385 0.86 40 236 377 
STQAATASSRNSCAADDKATEPLPKDCPVSSYNEWDPLEEVIVGRAENACVPPFTIEVKANTYEKYWPFYQKQGGHYFPKDHLKKAVAEIEEMCNILKTEGVTVRRPDPIDWSLKYKTPDFESTGLYSAMPRDILIVVGNEIIEAPMAWRSRFFEYRAYRSIIKDYFHRGAKWTTAPKPTMADELYNQDYPIHSVEDRHKLAAQGKFVTTEFEPCFDAADFIRAGRDIFAQRSQVTNYLGIEWMRRHLAPDYRVHIISFKDPNPHIDATFNIIGPGIVLSNPDRPCHQIDLFKKAGWTIITPPTPIIPDDHPLWMSSKWLSMNVLMLDEKRVMVDANEVPIQKMFEKLGITTIKVNIRNANSLGGGFHCWTCDVRRRGTLQSYLD
5PRT , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
5WQR , Knot 44 83 0.78 40 72 81 
AAPANAVTADDPTAIALKYNQDATKSERVAAARPGLPPEEQHCANCQFMQANVGEGDWKGCQLFPGKLINVNGWCASWTLKAG

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 \(c\).

\(|P_{f(2JDX_1)}(2) \setminus P_{f(5PRT_1)}(2)|=154\), \(|P_{f(5PRT_1)}(2) \setminus P_{f(2JDX_1)}(2)|=38\). 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 1:0001101000000110001001110001100000101100111101001011110101010000001110000110011000100111010010011000110100101101010000101000110011100111111001101111100011000100011000100110100110101100100000110010000011101011000101010110110110011100001000111011000111000101101001010101010111111110010010001011001110110110111100011110001101011110000111010011100110011100101010010011111001000100010100010
Pair \(Z_2\) Length of longest common subsequence
2JDX_1,5PRT_1 192 4
2JDX_1,5WQR_1 218 4
5PRT_1,5WQR_1 144 3

Newick tree

 
[
	2JDX_1:11.07,
	[
		5PRT_1:72,5WQR_1:72
	]:39.07
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{156}{\log_{20}156})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 2JDX_1 5PRT_1 144 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]