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Parikh vectors
2GJD_1 6HQF_1 3FWR_1 Letter Amino acid
10 63 6 A Alanine
6 41 14 I Isoleucine
8 45 19 V Valine
6 30 7 R Arginine
7 39 4 N Asparagine
9 55 8 E Glutamic acid
16 32 4 P Proline
8 16 2 Y Tyrosine
8 27 12 D Aspartic acid
9 23 7 Q Glutamine
12 73 17 L Leucine
2 22 6 M Methionine
5 26 4 F Phenylalanine
4 42 12 T Threonine
2 7 2 C Cysteine
10 61 7 G Glycine
2 16 9 H Histidine
17 40 12 K Lycine
11 54 7 S Serine
5 4 0 W Tryptophan 

2GJD_1|Chains A, B, C, D|Ubiquitin-conjugating enzyme E2-18 kDa|Saccharomyces cerevisiae (4932)
>6HQF_1|Chains A, B|Phenylalanine ammonia-lyase 1|Petroselinum crispum (4043)
>3FWR_1|Chains A, B|YqzB protein|Bacillus subtilis (1423)
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 \(w=f(c)\)
2GJD , Knot 75 157 0.80 40 111 151 
MSSLCLQRLQEERKKWRKDHPFGFYAKPVKKADGSMDLQKWEAGIPGKEGTNWAGGVYPITVEYPNEYPSKPPKVKFPAGFYHPNVYPSGTICLSILNEDQDWRPAITLKQIVLGVQDLLDSPNPNSPAQEPAWRSFSRNKAEYDKKVLLQAKQYSK
6HQF , Knot 277 716 0.84 40 284 662 
MENGNGATTNGHVNGNGMDFCMKTEDPLYWGIAAEAMTGSHLDEVKKMVAEYRKPVVKLGGETLTISQVAAISARDGSGVTVELSEAARAGVKASSDWVMDSMNKGTDSYGVTTGFGATSHRRTKQGGALQKELIRFLNAGIFGNGSDNTLPHSATRAAMLVRINTLLQGYSGIRFEILEAITKFLNQNITPCLPLRGTITASGDLVPLSYIAGLLTGRPNSKAVGPTGVILSPEEAFKLAGVEGGFFELQPKEGLALVNGTAVGSGMASMVLFEANILAVLAEVMSAIFAEVMQGKPEFTDHLTHKLKHHPGQIEAAAIMEHILDGSAYVKAAQKLHEMDPLQKPKQDRYALRTSPQWLGPQIEVIRSSTKMIEREINSVNDNPLIDVSRNKAIHGGNFQGTPIGVSMDNTRLAIAAIGKLMFAQFSELVNDFYNNGLPSNLSGGRNPSLDYGFKGAEIAMASYCSELQFLANPVTNHVQSAEQHNQDVNSLGLISSRKTSEAVEILKLMSTTFLVGLCQAIDLRHLEENLKSTVKNTVSSVAKRVLTMGVNGELHPSRFCEKDLLRVVDREYIFAYIDDPCSATYPLMQKLRQTLVEHALKNGDNERNLSTSIFQKIATFEDELKALLPKEVESARAALESGNPAIPNRIEECRSYPLYKFVRKELGTEYLTGEKVTSPGEEFEKVFIAMSKGEIIDPLLESLESWNGAPLPIS
3FWR , Knot 80 159 0.85 38 123 154 
MTGKTGTQLLADKLKKLQVKDFQSIPVVIHENVSVYDAICTMFLEDVGTLFVVDRDAVLVGVLSRKDLLRASIGQQELTSVPVHIIMTRMPNITVCRREDYVMDIAKHLIEKQIDALPVIKDTDKGFEVIGRVTKTNMTKILVSLSENEILLQHHHHHH

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 \(c\).

\(|P_{f(2GJD_1)}(2) \setminus P_{f(6HQF_1)}(2)|=23\), \(|P_{f(6HQF_1)}(2) \setminus P_{f(2GJD_1)}(2)|=196\). 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 1:1001010010000001000011110101100101010100101111100100111110110100100010011010111110010101010101011000001011101001111100110010100110011100100001000001110100000
Pair \(Z_2\) Length of longest common subsequence
2GJD_1,6HQF_1 219 5
2GJD_1,3FWR_1 156 3
6HQF_1,3FWR_1 201 5

Newick tree

 
[
	6HQF_1:11.68,
	[
		2GJD_1:78,3FWR_1:78
	]:34.68
]

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 \((0.85)(0.8)(\frac{873 }{\log_{20} 873}-\frac{157}{\log_{20}157})=199.\)
Status Protein1 Protein2 d d1/2
Query variables 2GJD_1 6HQF_1 254 152.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]