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Parikh vectors
4YPC_1 9ELJ_1 4LEM_1 Letter Amino acid
12 100 46 L Leucine
2 77 22 F Phenylalanine
3 95 31 S Serine
8 37 37 R Arginine
4 83 15 N Asparagine
16 45 23 E Glutamic acid
1 17 22 H Histidine
3 71 28 I Isoleucine
2 93 33 T Threonine
0 52 18 Y Tyrosine
7 59 41 D Aspartic acid
0 30 2 C Cysteine
6 72 76 A Alanine
0 78 47 G Glycine
2 11 9 M Methionine
0 59 31 P Proline
5 88 43 V Valine
7 59 17 Q Glutamine
5 63 13 K Lycine
0 10 9 W Tryptophan 

4YPC_1|Chain A|Vimentin|Homo sapiens (9606)
>9ELJ_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4LEM_1|Chains A, B, C, D, E, F|1-pyrroline-5-carboxylate dehydrogenase|Mycobacterium tuberculosis (83332)
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 \(w=f(c)\)
4YPC , Knot 41 83 0.72 30 61 78 
VDQLTNDKARVEVERDNLAEDIMRLREKLQEEMLQREEAENTLQSFRQDVDNASLARLDLERKVESLQEEIAFLKKLHEEEIQ
9ELJ , Knot 431 1199 0.85 40 326 1049 
MFVFLVLLPLVSSQCVNLITTTQSYTNFTRGVYYPDKVFRSSVLHLTQDLFLPFFSNVTWFHAISGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVFIKVCEFQFCNDPFLDVYHKNNKSWMESESGVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPIIGRDFPQGFSALEPLVDLPIGINITRFQTLLALNRSYLTPGDSSSGWTAGAADYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNVTNLCPFHEVFNATRFASVYAWNRTRISNCVADYSVLYNFAPFFAFKCYGVSPTKLNDLCFTNVYADSFVIKGNEVSQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKHSGNYDYWYRSFRKSKLKPFERDISTEIYQAGNKPCKGKGPNCYFPLESYGFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTKSNKKFLPFQQFGRDIVDTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVSVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSRGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLFSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFLTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQLELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQ
4LEM , Knot 226 563 0.84 40 246 518 
MGSSHHHHHHSSGLVPRGSHMDAITQVPVPANEPVHDYAPKSPERTRLRTELASLADHPIDLPHVIGGRHRMGDGERIDVVQPHRHAARLGTLTNATHADAAAAVEAAMSAKSDWAALPFDERAAVFLRAADLLAGPWREKIAAATMLGQSKSVYQAEIDAVCELIDFWRFNVAFARQILEQQPISGPGEWNRIDYRPLDGFVYAITPFNFTSIAGNLPTAPALMGNTVIWKPSITQTLAAYLTMQLLEAAGLPPGVINLVTGDGFAVSDVALADPRLAGIHFTGSTATFGHLWQWVGTNIGRYHSYPRLVGETGGKDFVVAHASARPDVLRTALIRGAFDYQGQKCSAVSRAFIAHSVWQRMGDELLAKAAELRYGDITDLSNYGGALIDQRAFVKNVDAIERAKGAAAVTVAVGGEYDDSEGYFVRPTVLLSDDPTDESFVIEYFGPLLSVHVYPDERYEQILDVIDTGSRYALTGAVIADDRQAVLTALDRLRFAAGNFYVNDKPTGAVVGRQPFGGARGSDTNDKAGSPLNLLRWTSARSIKETFVAATDHIYPHMAVD

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 \(c\).

\(|P_{f(4YPC_1)}(2) \setminus P_{f(9ELJ_1)}(2)|=4\), \(|P_{f(9ELJ_1)}(2) \setminus P_{f(4YPC_1)}(2)|=269\). 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 1:10010000101010000110011010001000110000100010010001001011010100010010001111001000010
Pair \(Z_2\) Length of longest common subsequence
4YPC_1,9ELJ_1 273 4
4YPC_1,4LEM_1 229 5
9ELJ_1,4LEM_1 142 5

Newick tree

 
[
	4YPC_1:13.57,
	[
		4LEM_1:71,9ELJ_1:71
	]:68.57
]

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 \((0.85)(0.8)(\frac{1282 }{\log_{20} 1282}-\frac{83}{\log_{20}83})=326.\)
Status Protein1 Protein2 d d1/2
Query variables 4YPC_1 9ELJ_1 419 222.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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