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Parikh vectors
7NNG_1 1TIJ_1 9EMZ_1 Letter Amino acid
29 2 8 I Isoleucine
34 7 6 K Lycine
24 5 12 F Phenylalanine
34 4 4 Y Tyrosine
30 8 7 R Arginine
30 5 4 N Asparagine
32 8 11 G Glycine
40 8 11 S Serine
3 1 9 W Tryptophan
53 10 18 V Valine
46 10 20 A Alanine
29 7 9 D Aspartic acid
26 4 0 C Cysteine
50 8 24 L Leucine
30 8 4 P Proline
50 7 5 T Threonine
18 7 6 Q Glutamine
23 5 3 E Glutamic acid
12 3 3 H Histidine
8 3 5 M Methionine 

7NNG_1|Chains A, B|SARS-CoV-2 helicase NSP13|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1TIJ_1|Chains A, B|Cystatin C|Homo sapiens (9606)
>9EMZ_1|Chains A, B, C, D|Lipoprotein signal peptidase|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
7NNG , Knot 238 601 0.84 40 268 568 
AVGACVLCNSQTSLRCGACIRRPFLCCKCCYDHVISTSHKLVLSVNPYVCNAPGCDVTDVTQLYLGGMSYYCKSHKPPISFPLCANGQVFGLYKNTCVGSDNVTDFNAIATCDWTNAGDYILANTCTERLKLFAAETLKATEETFKLSYGIATVREVLSDRELHLSWEVGKPRPPLNRNYVFTGYRVTKNSKVQIGEYTFEKGDYGDAVVYRGTTTYKLNVGDYFVLTSHTVMPLSAPTLVPQEHYVRITGLYPTLNISDEFSSNVANYQKVGMQKYSTLQGPPGTGKSHFAIGLALYYPSARIVYTACSHAAVDALCEKALKYLPIDKCSRIIPARARVECFDKFKVNSTLEQYVFCTVNALPETTADIVVFDEISMATNYDLSVVNARLRAKHYVYIGDPAQLPAPRTLLTKGTLEPEYFNSVCRLMKTIGPDMFLGTCRRCPAEIVDTVSALVYDNKLKAHKDKSAQCFKMFYKGVITHDVSSAINRPQIGVVREFLTRNPAWRKAVFISPYNSQNAVASKILGLPTQTVDSSQGSEYDYVIFTQTTETAHSCNVNRFNVAITRAKVGILCIMSDRDLYDKLQFTSLEIPRRNVATLQ
1TIJ , Knot 63 120 0.83 40 101 117 
SSPGKPPRLVGGPMDASVEEEGVRRALDFAVGEYNKASNDMYHSRALQVVRARKQIVAGVNYFLDVELGRTTCTKTQPNLDNCPFHDQPHLKRKAFCSFQIYAVPWQGTMTLSKSTCQDA
9EMZ , Knot 76 169 0.77 38 115 161 
MPDVDRFGRLPWLWITVLVFVLDQVSKAFFQAELSMYQQIVVIPDLFSWTLAYNTGAAFSFLADSSGWQRWLFALIAIVVSASLVVWLKRLKKGETWLAIALALVLGGALGNLYDRMVLGHVVDFILVHWQNRWYFPAFNLADSAITVGAVMLALDMFRSKKSGEAAHG

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 \(c\).

\(|P_{f(7NNG_1)}(2) \setminus P_{f(1TIJ_1)}(2)|=194\), \(|P_{f(1TIJ_1)}(2) \setminus P_{f(7NNG_1)}(2)|=27\). 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 1:1111011000000100110100111000000001100000111010101001110010010010111100000000111011101010111100000110001001011100010011001110000001011110010100001010011101001100001010101101011100001101001000001011000100100101110010000010110011100001111011011100001010110101010001000110000111000001011110100011111110010101100100011101100011001110000011110101001001010001000110010111000101111001011000010110101010001011011011110011001010100100100110011101111000001101100101110000101000001001011001110001001100101111001100011100111101000001110011111000100001000001110000001000010010111001011110110000100010100101100011010
Pair \(Z_2\) Length of longest common subsequence
7NNG_1,1TIJ_1 221 3
7NNG_1,9EMZ_1 227 3
1TIJ_1,9EMZ_1 138 3

Newick tree

 
[
	7NNG_1:12.05,
	[
		1TIJ_1:69,9EMZ_1:69
	]:54.05
]

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 \((0.85)(0.8)(\frac{721 }{\log_{20} 721}-\frac{120}{\log_{20}120})=172.\)
Status Protein1 Protein2 d d1/2
Query variables 7NNG_1 1TIJ_1 220 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]