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Parikh vectors
9FST_1 4GGQ_1 6SHQ_1 Letter Amino acid
0 0 9 C Cysteine
6 8 13 Q Glutamine
9 8 15 F Phenylalanine
10 4 15 Y Tyrosine
18 11 31 A Alanine
12 9 21 P Proline
23 11 30 S Serine
26 15 37 L Leucine
2 2 6 W Tryptophan
16 16 37 V Valine
14 15 29 D Aspartic acid
23 26 27 G Glycine
15 11 25 I Isoleucine
3 9 10 H Histidine
18 13 19 K Lycine
4 6 14 M Methionine
19 13 12 T Threonine
9 11 32 R Arginine
6 4 19 N Asparagine
17 17 30 E Glutamic acid 

9FST_1|Chains A, O|Proteasome subunit alpha type-2|Saccharomyces cerevisiae (4932)
>4GGQ_1|Chains A, B, C, D|Ubiquitin-like protein SMT3, Peptidyl-prolyl cis-trans isomerase|Saccharomyces cerevisiae (559292)
>6SHQ_1|Chains A, B, C, D|Glucose-1-phosphate adenylyltransferase|Escherichia coli (562)
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 \(w=f(c)\)
9FST , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL
4GGQ , Knot 92 209 0.78 38 147 197 
MGHHHHHHSGEVKPEVKPETHINLKVSDGSSEIFFKIKKTTPLRRLMEAFAKRQGKEMDSLRFLYDGIRIQADQTPEDLDMEDNDIIEAHREQIGGSTVVTTESGLKYEDLTEGSGAEARAGQTVSVHYTGWLTDGQKFDSSKDRNDPFAFVLGGGMVIKGWDEGVQGMKVGGVRRLTIPPQLGYGARGAGGVIPPNATLVFEVELLDV
6SHQ , Knot 184 431 0.86 40 236 410 
MVSLEKNDHLMLARQLPLKSVALILAGGRGTRLKDLTNKRAKPAVHFGGKFRIIDFALSNCINSGIRRMGVITQYQSHTLVQHIQRGWSFFNEEMNEFVDLLPAQQRMKGENWYRGTADAVTQNLDIIRRYKAEYVVILAGDHIYKQDYSRMLIDHVEKGARCTVACMPVPIEEASAFGVMAVDENDKIIEFVEKPANPPSMPNDPSKSLASMGIYVFDADYLYELLEEDDRDENSSHDFGKDLIPKITEAGLAYAHPFPLSCVQSDPDAEPYWRDVGTLEAYWKANLDLASVVPELDMYDRNWPIRTYNESLPPAKFVQDRSGSHGMTLNSLVSGGCVISGSVVVQSVLFSRVRVNSFCNIDSAVLLPEVWVGRSCRLRRCVIDRACVIPEGMVIGENAEEDARRFYRSEEGIVLVTREMLRKLGHKQER

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 \(c\).

\(|P_{f(9FST_1)}(2) \setminus P_{f(4GGQ_1)}(2)|=77\), \(|P_{f(4GGQ_1)}(2) \setminus P_{f(9FST_1)}(2)|=65\). 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 1:1000001010010101011010011011001100111010011111000000011110001001011010111100111100011100000110000001010011001110011011001000111011110111110001011010010101001110101110101110011000100010100110111101000101010100101111100010111001110001101001000010001011
Pair \(Z_2\) Length of longest common subsequence
9FST_1,4GGQ_1 142 4
9FST_1,6SHQ_1 175 3
4GGQ_1,6SHQ_1 175 4

Newick tree

 
[
	6SHQ_1:92.34,
	[
		9FST_1:71,4GGQ_1:71
	]:21.34
]

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 \((0.85)(0.8)(\frac{459 }{\log_{20} 459}-\frac{209}{\log_{20}209})=72.8\)
Status Protein1 Protein2 d d1/2
Query variables 9FST_1 4GGQ_1 92 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]