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Parikh vectors
9BCN_1 7AAP_1 1UFR_1 Letter Amino acid
80 54 14 G Glycine
128 84 18 L Leucine
20 25 3 M Methionine
45 60 5 F Phenylalanine
16 11 1 W Tryptophan
35 59 5 Y Tyrosine
81 64 17 A Alanine
58 75 12 D Aspartic acid
37 31 1 Q Glutamine
74 34 18 E Glutamic acid
24 29 4 H Histidine
49 32 8 P Proline
54 61 9 T Threonine
74 74 19 V Valine
74 43 22 R Arginine
37 57 3 N Asparagine
17 29 0 C Cysteine
38 33 13 I Isoleucine
39 53 7 K Lycine
60 59 2 S Serine 

9BCN_1|Chains A, B|Atrial natriuretic peptide receptor 1|Homo sapiens (9606)
>7AAP_1|Chain A|Non-structural protein 12|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1UFR_1|Chains A, B, C, D|pyr mRNA-binding attenuation protein|Thermus thermophilus (274)
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 \(w=f(c)\)
9BCN , Knot 383 1040 0.85 40 320 934 
DYKDDDDKAAAGNLTVAVVLPLANTSYPWSWARVGPAVELALAQVKARPDLLPGWTVRTVLGSSENALGVCSDTAAPLAAVDLKWEHNPAVFLGPGCVYAAAPVGRFTAHWRVPLLTAGAPALGFGVKDEYALTTRAGPSYAKLGDFVAALHRRLGWERQALMLYAYRPGDEEHCFFLVEGLFMRVRDRLNITVDHLEFAEDDLSHYTRLLRTMPRKGRVIYICSSPDAFRTLMLLALEAGLCGEDYVFFHLDIFGQSLQGGQGPAPRRPWERGDGQDVSARQAFQAAKIITYKDPDNPEYLEFLKQLKHLAYEQFNFTMEDGLVNTIPASFHDGLLLYIQAVTETLAHGGTVTDGENITQRMWNRSFQGVTGYLKIDSSGDRETDFSLWDMDPENGAFRVVLNYNGTSQELVAVSGRKLNWPLGYPPPDIPKCGFDNEDPACNQDHLSTLEVLALVGSLSLLGILIVSFFIYRKMQLEKELASELWRVRWEDVEPSSLERHLRSAGSRLTLSGRGSNYGSLLTTEGQFQVFAKTAYYKGNLVAVKRVNRKRIELTRKVLFELKHMRDVQNEHLTRFVGACTDPPNICILTEYCPRGSLQDILENESITLDWMFRYSLTNDIVKGMLFLHNGAICSHGNLKSSNCVVDGRFVLKITDYGLESFRDLDPEQGHTVYAKKLWTAPELLRMASPPVRGSQAGDVYSFGIILQEIALRSGVFHVEGLDLSPKEIIERVTRGEQPPFRPSLALQSHLEELGLLMQRCWAEDPQERPPFQQIRLTLRKFNRENSSNILDNLLSRMEQYANNLEELVEERTQAYLEEKRKAEALLYQILPHSVAEQLKRGETVQAEAFDSVTIYFSDIVGFTALSAESTPMQVVTLLNDLYTCFDAVIDNFDVYKVETIGDAYMVVSGLPVRNGRLHACEVARMALALLDAVRSFRIRHRPQEQLRLRIGIHTGPVCAGVVGLKMPRYCLFGDTVNTASRMESNGEALKIHLSSETKAVLEEFGGFELELRGDVEMKGKGKVRTYWLLGERGSSTRG
7AAP , Knot 361 967 0.85 40 329 880 
SADAQSFLNRVCGVSAARLTPCGTGTSTDVVYRAFDIYNDKVAGFAKFLKTNCCRFQEKDEDDNLIDSYFVVKRHTFSNYQHEETIYNLLKDCPAVAKHDFFKFRIDGDMVPHISRQRLTKYTMADLVYALRHFDEGNCDTLKEILVTYNCCDDDYFNKKDWYDFVENPDILRVYANLGERVRQALLKTVQFCDAMRNAGIVGVLTLDNQDLNGNWYDFGDFIQTTPGSGVPVVDSYYSLLMPILTLTRALTAESHVDTDLTKPYIKWDLLKYDFTEERLKLFDRYFKYWDQTYHPNCVNCLDDRCILHCANFNVLFSTVFPPTSFGPLVRKIFVDGVPFVVSTGYHFRELGVVHNQDVNLHSSRLSFKELLVYAADPAMHAASGNLLLDKRTTCFSVAALTNNVAFQTVKPGNFNKDFYDFAVSKGFFKEGSSVELKHFFFAQDGNAAISDYDYYRYNLPTMCDIRQLLFVVEVVDKYFDCYDGGCINANQVIVNNLDKSAGFPFNKWGKARLYYDSMSYEDQDALFAYTKRNVIPTITQMNLKYAISAKNRARTVAGVSICSTMTNRQFHQKLLKSIAATRGATVVIGTSKFYGGWHNMLKTVYSDVENPHLMGWDYPKCDRAMPNMLRIMASLVLARKHTTCCSLSHRFYRLANECAQVLSEMVMCGGSLYVKPGGTSSGDATTAYANSVFNICQAVTANVNALLSTDGNKIADKYVRNLQHRLYECLYRNRDVDTDFVNEFYAYLRKHFSMMILSDDAVVCFNSTYASQGLVASIKNFKSVLYYQNNVFMSEAKCWTETDLTKGPHEFCSQHTMLVKQGDDYVYLPYPDPSRILGAGCFVDDIVKTDGTLMIERFVSLAIDAYPLTKHPNQEYADVFHLYLQYIRKLHDELTGHMLDMYSVMLTNDNTSRYWEPEFYEAMYTPHTVLQENLYFQSWSHPQFEKGGGSGGGSGGGSWSHPQFEK
1UFR , Knot 81 181 0.77 38 116 173 
MRFKAELMNAPEMRRALYRIAHEIVEANKGTEGLALVGIHTRGIPLAHRIARFIAEFEGKEVPVGVLDITLYRDDLTEIGYRPQVRETRIPFDLTGKAIVLVDDVLYTGRTARAALDALIDLGRPRRIYLAVLVDRGHRELPIRADFVGKNVPTSRSEVVKVKVEEVDGEDRVELWEREGA

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 \(c\).

\(|P_{f(9BCN_1)}(2) \setminus P_{f(7AAP_1)}(2)|=48\), \(|P_{f(7AAP_1)}(2) \setminus P_{f(9BCN_1)}(2)|=57\). 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 1:00000000111101011111111000011011011111011110101010111110100111000011110000111111101010001111111101011111101010101111011111111110000110001110010110111110001110001111010011000001111011110100010101001011000100000110011001011010001011001111110111010001110101110010110111100110010100101001101101100001001001011001001100010101001110011101001111010110001101101001001000110001011010101000100000101101010011101110001000011110100101111011101100110000110000010010111111010111111101110001010001100110101001010010001001100101010100010110001010111001000101111001000010100011101001001000010011110001101011000010101001100001010111000100011011111001110001010000011010111010001100100101001001010011011011011011101001101001111100111001110101101010011001001001110101110001001111100011001000111001010100100000001100110010001001001100000101000001011100111001100100100101011001010100111101101000110110110010001011100101001001101011101111001010100110111111011001010001000101011100111011111101100011100100100100010110101000001110011110101010101010101000111100100001
Pair \(Z_2\) Length of longest common subsequence
9BCN_1,7AAP_1 105 4
9BCN_1,1UFR_1 222 5
7AAP_1,1UFR_1 231 4

Newick tree

 
[
	1UFR_1:12.23,
	[
		9BCN_1:52.5,7AAP_1:52.5
	]:74.73
]

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 \((0.85)(0.8)(\frac{2007 }{\log_{20} 2007}-\frac{967}{\log_{20}967})=251.\)
Status Protein1 Protein2 d d1/2
Query variables 9BCN_1 7AAP_1 329 315
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]