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Parikh vectors
7KQN_1 3ADD_1 5KAJ_1 Letter Amino acid
13 0 2 C Cysteine
69 45 9 K Lycine
46 9 13 F Phenylalanine
32 14 16 S Serine
6 3 5 W Tryptophan
32 10 46 R Arginine
20 0 11 Q Glutamine
23 23 31 E Glutamic acid
46 29 14 I Isoleucine
36 15 34 V Valine
27 7 57 A Alanine
31 19 6 N Asparagine
28 19 32 D Aspartic acid
17 6 35 G Glycine
57 22 59 L Leucine
9 5 10 M Methionine
26 6 24 P Proline
29 7 15 T Threonine
17 7 12 H Histidine
32 13 7 Y Tyrosine 

7KQN_1|Chains A, D|Telomerase reverse transcriptase|Tribolium castaneum (7070)
>3ADD_1|Chains A, B|L-seryl-tRNA(Sec) kinase|Methanocaldococcus jannaschii (243232)
>5KAJ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|(3,5-dihydroxyphenyl)acetyl-CoA 1,2-dioxygenase|Streptomyces toyocaensis (55952)
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 \(w=f(c)\)
7KQN , Knot 233 596 0.83 40 265 560 
MVHYYRLSLKSRQKAPKIVNSKYNSILNIALKNFRLCKKHKTKKPVQILALLQEIIPKSYFGTTTNLKRFYKVVEKILTQSSFECIHLSVLHKCYDYDAIPWLQNVEPNLRPKLLLKHNLFLLDNIVKPIIAFYYKPIKTLNGHEIKFIRKEEYISFESKVFHKLKKMKYLVEVQDEVKPRGVLNIIPKQDNFRAIVSIFPDSARKPFFKLLTSKIYKVLEEKYKTSGSLYTCWSEFTQKTQGQIYGIKVDIRDAYGNVKIPVLCKLIQSIPTHLLDSEKKNFIVDHISNQFVAFRRKIYKWNHGLLQGDPLSGCLCELYMAFMDRLYFSNLDKDAFIHRTVDDYFFCSPHPHKVYDFELLIKGVYQVNPTKTRTNLPTHRHPQDEIPYCGKIFNLTTRQVRTLYKLPPNYEIRHKFKLWNFNNQISDDNPARFLQKAMDFPFICNSFTKFEFNTVFNDQRTVFANFYDAMICVAYKFDAAMMALRTSFLVNDFGFIWLVLSSTVRAYASRAFKKIVTYKGGKYRKVTFQCLKSIAWRAFLAVLKRRTEIYKGLIDRIKSREKLTMKFHDGEVDASYFCKLPEKFRFVKINRKASI
3ADD , Knot 111 259 0.79 36 145 231 
MNHKVHHHHHHMLIILTGLPGVGKSTFSKNLAKILSKNNIDVIVLGSDLIRESFPVWKEKYEEFIKKSTYRLIDSALKNYWVIVDDTNYYNSMRRDLINIAKKYNKNYAIIYLKASLDVLIRRNIERGEKIPNEVIKKMYEKFDEPGKKYKWDEPFLIIDTTKDIDFNEIAKKLIEKSKEIPKFYVLEENKNKNNNISDKIDKETRKIVSEYIKSKKLDKDKIKEVVELRKEFLKKIKKMEEVDADRVLKEFKDLLNSY
5KAJ , Knot 177 438 0.82 40 203 405 
MTTVLPPLEDTDGLWAALTEAAASVEKLLATLPEHGARSSAERAEIAAAHDAARALRVRFLDTHADAVYDRLTDHRRVHLRLAELVEAAATAFPGLVPTQQQLAVERSLPQAAKEGHEIDQGIFLRAVLRSPLAGPHLLDAMLRPTPRALELLPEFVRTGEVEMEAVHLERRDGVARLTMCRDDRLNAEDGQQVDDMETAVDLALLDPGVRVGLLRGGVMSHPRYRGKRVFSAGINLKYLSQGGISLVDFLMRRELGYIHKLVRGVLTNDDRPGWWHSPRIEKPWVAAVDGFAIGGGAQLLLVFDRVLASSDAYFSLPCAKEGIIPGAANLRLGRFAGPRVSRQVILEGRRIWAKEPEARLLVDEVVEPDELDAAIERSLTRLDGDAVLANRRMLNLADESPDGFRAYMAEFALMQALRLYGHDVIDKVGRFGGRPPA

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 \(c\).

\(|P_{f(7KQN_1)}(2) \setminus P_{f(3ADD_1)}(2)|=152\), \(|P_{f(3ADD_1)}(2) \setminus P_{f(7KQN_1)}(2)|=32\). 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 1:11000010100000110110000001101110010100000000110111110011100011000010010011001100001001010110000000111110010101010111000111100110111110001100101001011000001010001100100100110100010101110111000010111011100100111011000100110000000101000100100000101011010100101010111100110011001100000011100100011110001001001110101101010010111100101001000111000100011001010010010111011001010000001100001000110010110100001001001110001000101101000100001101100110111100010010100110000011101001110110010111111000111001111111100010101001100110001100001010010011101111110000010011100100000101010010101001001100101101000101
Pair \(Z_2\) Length of longest common subsequence
7KQN_1,3ADD_1 184 4
7KQN_1,5KAJ_1 186 4
3ADD_1,5KAJ_1 178 4

Newick tree

 
[
	7KQN_1:93.63,
	[
		3ADD_1:89,5KAJ_1:89
	]:4.63
]

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 \((0.85)(0.8)(\frac{855 }{\log_{20} 855}-\frac{259}{\log_{20}259})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 7KQN_1 3ADD_1 208 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]