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Parikh vectors
2VJL_1 1KSW_1 4JWP_1 Letter Amino acid
18 20 8 Q Glutamine
42 32 18 G Glycine
18 11 4 M Methionine
19 17 7 F Phenylalanine
15 26 8 S Serine
18 15 8 N Asparagine
28 39 17 E Glutamic acid
9 7 11 H Histidine
8 9 3 W Tryptophan
35 29 19 A Alanine
17 27 13 R Arginine
30 18 5 D Aspartic acid
6 9 1 C Cysteine
28 26 4 K Lycine
22 22 5 P Proline
22 16 9 I Isoleucine
33 46 22 L Leucine
22 32 8 T Threonine
9 23 4 Y Tyrosine
29 28 11 V Valine 

2VJL_1|Chains A, B|FORMYL-COENZYME A TRANSFERASE|OXALOBACTER FORMIGENES (847)
>1KSW_1|Chain A|PROTO-ONCOGENE TYROSINE-PROTEIN KINASE SRC|Homo sapiens (9606)
>4JWP_1|Chains A, B|GCN5-related N-acetyltransferase|Brucella abortus (359391)
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 \(w=f(c)\)
2VJL , Knot 183 428 0.86 40 241 412 
MTKPLDGINVLDFTHVQAGPACTQMMGFLGANVIKIERRGSGDMTRGWLQDKPNVDSLYFTMFNCNKRSIELDMKTPEGKELLEQMIKKADVMVENFGPGALDRMGFTWEYIQELNPRVILASVKGYAEGHANEHLKVYENVAQCSGGAAATTGFWDGPPTVSGAALGDSNSGMHLMIGILAALEIRHKTGRGQKVAVAMQDAVLNLVRIKLRDQQRLERTGILAEYPQAQPNFAFDRDGNPLSFDNITSVPRGGNAGGGGQPGWMLKCKGWETDADSYVYFTIAANMWPQICDMIDKPEWKDDPAYNTFEGRVDKLMDIFSFIETKFADKDKFEVTEWAAQYGIPCGPVMSMKELAHDPSLQKVGTVVEVVDEIRGNHLTVGAPFKFSGFQPEITRAPLLGEHTDEVLKELGLDDAKIKELHAKQVV
1KSW , Knot 189 452 0.85 40 249 429 
MVTTFVALYDYESRTETDLSFKKGERLQIVNNTEGDWWLAHSLSTGQTGYIPSNYVAPSDSIQAEEWYFGKITRRESERLLLNAENPRGTFLVRESETTKGAYCLSVSDFDNAKGLNVKHYKIRKLDSGGFYITSRTQFNSLQQLVAYYSKHADGLCHRLTTVCPTSKPQTQGLAKDAWEIPRESLRLEVKLGQGCFGEVWMGTWNGTTRVAIKTLKPGTMSPEAFLQEAQVMKKLRHEKLVQLYAVVSEEPIYIVGEYMSKGSLLDFLKGETGKYLRLPQLVDMAAQIASGMAYVERMNYVHRDLRAANILVGENLVCKVADFGLARLIEDNEYTARQGAKFPIKWTAPEAALYGRFTIKSDVWSFGILLTELTTKGRVPYPGMVNREVLDQVERGYRMPCPPECPESLHDLMCQCWRKEPEERPTFEYLQAFLEDYFTSTEPQYQPGENL
4JWP , Knot 85 185 0.80 40 121 174 
MAHHHHHHMGTLEAQTQGPGSMTLLIRHATEADLPALLAIYNDAVENTLAIWNETLVDLENRHQWLENRNRDGFPVLVAEREGQVVGYASYGPFRPFEGFRHSSELSVYVASNARGGGIGRTLLAELIEEARERKVHVLIAGIEAGNAASIALHRSQGFEECGTLKQVGQKFGRWLDLLFMQKIL

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 \(c\).

\(|P_{f(2VJL_1)}(2) \setminus P_{f(1KSW_1)}(2)|=75\), \(|P_{f(1KSW_1)}(2) \setminus P_{f(2VJL_1)}(2)|=83\). 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 1:10011011011010010111100011111110110100010101001110001010010101100000010101001010011001100101110011111100111010010010101111010101010100010100011000111110011101110101111100001101111111110100001010011111001110110101000001000111100101010111000101101001001101101111101111100011000100010101110111010011001010001100010101001101101100011000010100111001110111101001100101001101101100101001011111010110101001111100000110011100101001010011
Pair \(Z_2\) Length of longest common subsequence
2VJL_1,1KSW_1 158 3
2VJL_1,4JWP_1 192 3
1KSW_1,4JWP_1 192 4

Newick tree

 
[
	4JWP_1:10.03,
	[
		2VJL_1:79,1KSW_1:79
	]:22.03
]

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 \((0.85)(0.8)(\frac{880 }{\log_{20} 880}-\frac{428}{\log_{20}428})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 2VJL_1 1KSW_1 156 150.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]