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Parikh vectors
4LKC_1 9DNZ_1 5PTS_1 Letter Amino acid
18 80 18 L Leucine
34 56 7 S Serine
7 38 14 R Arginine
6 27 4 N Asparagine
9 39 13 D Aspartic acid
15 14 8 Q Glutamine
10 45 13 E Glutamic acid
13 63 7 G Glycine
13 54 10 V Valine
5 18 1 C Cysteine
2 16 10 H Histidine
12 35 6 K Lycine
9 44 6 F Phenylalanine
9 30 5 Y Tyrosine
8 64 5 I Isoleucine
0 23 7 M Methionine
10 35 6 P Proline
15 69 11 A Alanine
17 49 5 T Threonine
2 19 0 W Tryptophan 

4LKC_1|Chain A|Fab fragment of PCSK9 antibody|Homo sapiens (9606)
>9DNZ_1|Chains A, B[auth C]|H(+)/Cl(-) exchange transporter 3|Homo sapiens (9606)
>5PTS_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
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 \(w=f(c)\)
4LKC , Knot 101 214 0.84 38 148 202 
AIQLTQSPSSLSASVGDRVTITCRASQGISSALAWYQQKPGKAPKLLIYDASSLESGVPSRFSGSGSGTDFTLTISSLQPEDFATYYCQQFNSYLITFGQGTRLEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC
9DNZ , Knot 311 818 0.85 40 304 751 
MESEQLFHRGYYRNSYNSITSASSDEELLDGAGVIMDFQTSEDDNLLDGDTAVGTHYTMTNGGSINSSTHLLDLLDEPIPGVGTYDDFHTIDWVREKCKDRERHRRINSKKKESAWEMTKSLYDAWSGWLVVTLTGLASGALAGLIDIAADWMTDLKEGICLSALWYNHEQCCWGSNETTFEERDKCPQWKTWAELIIGQAEGPGSYIMNYIMYIFWALSFAFLAVSLVKVFAPYACGSGIPEIKTILSGFIIRGYLGKWTLMIKTITLVLAVASGLSLGKEGPLVHVACCCGNIFSYLFPKYSTNEAKKREVLSAASAAGVSVAFGAPIGGVLFSLEEVSYYFPLKTLWRSFFAALVAAFVLRSINPFGNSRLVLFYVEYHTPWYLFELFPFILLGVFGGLWGAFFIRANIAWCRRRKSTKFGKYPVLEVIIVAAITAVIAFPNPYTRLNTSELIKELFTDCGPLESSSLCDYRNDMNASKIVDDIPDRPAGIGVYSAIWQLCLALIFKIIMTVFTFGIKVPSGLFIPSMAIGAIAGRIVGIAVEQLAYYHHDWFIFKEWCEVGADCITPGLYAMVGAAACLGGVTRMTVSLVVIVFELTGGLEYIVPLMAAVMTSKWVGDAFGREGIYEAHIRLNGYPFLDAKEEFTHTTLAADVMRPRRNDPPLAVLTQDNMTVDDIENMINETSYNGFPVIMSKESQRLVGFALRRDLTIAIESARKKQEGIVGSSRVCFAQHTPSLPAESPRPLKLRSILDMSPFTVTDHTPMEIVVDIFRKLGLRQCLVTHNGRLLGIITKKDILRHMAQTANQDPASIMFN
5PTS , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA

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 \(c\).

\(|P_{f(4LKC_1)}(2) \setminus P_{f(9DNZ_1)}(2)|=29\), \(|P_{f(9DNZ_1)}(2) \setminus P_{f(4LKC_1)}(2)|=185\). 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 1:1101000100101011001010001001100111100001101101110010010011100101010100101010010100110000001000110110100101000111101111110000100101011011001010010101010011001000001000000000001000101001000000101001000110011000100100
Pair \(Z_2\) Length of longest common subsequence
4LKC_1,9DNZ_1 214 3
4LKC_1,5PTS_1 166 4
9DNZ_1,5PTS_1 226 3

Newick tree

 
[
	9DNZ_1:11.68,
	[
		4LKC_1:83,5PTS_1:83
	]:34.68
]

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 \((0.85)(0.8)(\frac{1032 }{\log_{20} 1032}-\frac{214}{\log_{20}214})=221.\)
Status Protein1 Protein2 d d1/2
Query variables 4LKC_1 9DNZ_1 279 174.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]