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Parikh vectors
2QRL_1 7UJQ_1 2AHO_1 Letter Amino acid
14 6 14 N Asparagine
4 5 4 C Cysteine
9 10 14 Q Glutamine
26 17 41 I Isoleucine
17 10 11 F Phenylalanine
28 13 21 P Proline
19 11 26 S Serine
25 9 22 T Threonine
24 17 41 V Valine
37 24 24 A Alanine
22 13 16 D Aspartic acid
23 12 6 H Histidine
35 30 34 L Leucine
5 2 8 M Methionine
9 11 8 Y Tyrosine
23 14 18 R Arginine
19 23 32 E Glutamic acid
24 17 34 G Glycine
26 19 33 K Lycine
5 5 7 W Tryptophan 

2QRL_1|Chain A|Saccharopine dehydrogenase, NAD+, L-lysine-forming|Saccharomyces cerevisiae (4932)
>7UJQ_1|Chains A, B|Calcium/calmodulin-dependent protein kinase type II subunit alpha|Homo sapiens (9606)
>2AHO_1|Chain A|Translation initiation factor 2 gamma subunit|Sulfolobus solfataricus (2287)
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 \(w=f(c)\)
2QRL , Knot 167 394 0.84 40 228 367 
MGHHHHHHHHHHSSGHIEGRHMAAVTLHLRAETKPLEARAALTPTTVKKLIAKGFKIYVEDSPQSTFNINEYRQAGAIIVPAGSWKTAPRDRIIIGLKEMPETDTFPLVHEHIQFAHCYKDQAGWQNVLMRFIKGHGTLYDLEFLENDQGRRVAAFGFYAGFAGAALGVRDWAFKQTHSDDEDLPAVSPYPNEKALVKDVTKDYKEALATGARKPTVLIIGALGRCGSGAIDLLHKVGIPDANILKWDIKETSRGGPFDEIPQADIFINCIYLSKPIAPFTNMEKLNNPNRRLRTVVDVSADTTNPHNPIPIYTVATVFNKPTVLVPTTAGPKLSVISIDHLPSLLPREASEFFSHDLLPSLELLPQRKTAPVWVRAKKLFDRHCARVKRSSRL
7UJQ , Knot 122 268 0.84 40 176 261 
TRFTEEYQLFEELGKGAFSVVRRCVKVLAGQEYAAKIINTKKLSARDHQKLEREARICRLLKHPNIVRLHDSISEEGHHYLIFDLVTGGELFEDIVAREYYSEADASHCIQQILEAVLHCHQMGVVHRNLKPENLLLASKLKGAAVKLADFGLAIEVEGEQQAWFGFAGTPGYLSPEVLRKDPYGKPVDLWACGVILYILLVGYPPFWDEDQHRLYKQIKAGAYDFPSPEWDTVTPEAKDLINKMLTINPSKRITAAEALKHPWISHR
2AHO , Knot 171 414 0.83 40 226 390 
AWPKVQPEVNIGVVGHVDHGKTTLVQAITGIWTSKHSEELKRGMTIKLGYAETNIGVCESCKKPEAYVTEPSCKSCGSDDEPKFLRRISFIDAPGHEVLMATMLSGAALMDGAILVVAANEPFPQPQTREHFVALGIIGVKNLIIVQNKVDVVSKEEALSQYRQIKQFTKGTWAENVPIIPVSALHKINIDSLIEGIEEYIKTPYRDLSQKPVMLVIRSFDVNKPGTQFNELKGGVIGGSIIQGLFKVDQEIKVLPGLRVEKQGKVSYEPIFTKISSIRFGDEEFKEAKPGGLVAIGTYLDPSLTKADNLLGSIITLADAEVPVLWNIRIKYNLLERVVGAKEMLKVDPIRAKETLMLSVGSSTTLGIVTSVKKDEIEVELRRPVAVWSNNIRTVISRQIAGRWRMIGWGLVEI

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 \(c\).

\(|P_{f(2QRL_1)}(2) \setminus P_{f(7UJQ_1)}(2)|=104\), \(|P_{f(7UJQ_1)}(2) \setminus P_{f(2QRL_1)}(2)|=52\). 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 1:1100000000000010101001111010101000110101110100100111011010100010001010000011111111101001100011111001100001111000101100000011100111011010101001011000010011111101111111111001110000000001111010100011100100000011101100101111111100101110110011110101101010000011110011010111001010011111001001001000100110101000010011110011011001011110011101011010011011100100110001110101110000111110100110000101000001
Pair \(Z_2\) Length of longest common subsequence
2QRL_1,7UJQ_1 156 4
2QRL_1,2AHO_1 138 4
7UJQ_1,2AHO_1 170 4

Newick tree

 
[
	7UJQ_1:85.35,
	[
		2QRL_1:69,2AHO_1:69
	]:16.35
]

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 \((0.85)(0.8)(\frac{662 }{\log_{20} 662}-\frac{268}{\log_{20}268})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 2QRL_1 7UJQ_1 136 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]