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Parikh vectors
5SPV_1 3CNP_1 8YEK_1 Letter Amino acid
16 29 18 N Asparagine
6 35 22 E Glutamic acid
6 18 11 H Histidine
2 12 11 M Methionine
0 8 8 W Tryptophan
15 30 40 A Alanine
13 29 19 G Glycine
13 25 29 K Lycine
6 22 19 T Threonine
2 30 20 R Arginine
3 24 14 Q Glutamine
6 31 26 I Isoleucine
19 49 44 L Leucine
5 26 30 F Phenylalanine
5 18 20 P Proline
11 36 44 S Serine
22 35 25 V Valine
9 35 30 D Aspartic acid
7 13 23 Y Tyrosine
3 11 11 C Cysteine 

5SPV_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3CNP_1|Chains A, B|fms1|Saccharomyces cerevisiae (4932)
>8YEK_1|Chain A|GtCCR2|Guillardia theta CCMP2712 (905079)
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 \(w=f(c)\)
5SPV , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
3CNP , Knot 213 516 0.86 40 262 486 
MNTVSPAKKKVIIIGAGIAGLKAASTLHQNGIQDCLVLEARDRVGGRLQTVTGYQGRKYDIGASWHHDTLTNPLFLEEAQLSLNDGRTRFVFDDDNFIYIDEERGRVDHDKELLLEIVDNEMSKFAELEFHQHLGVSDCSFFQLVMKYLLQRRQFLTNDQIRYLPQLCRYLELWHGLDWKLLSAKDTYFGHQGRNAFALNYDSVVQRIAQSFPQNWLKLSCEVKSITREPSKNVTVNCEDGTVYNADYVIITVPQSVLNLSVQPEKNLRGRIEFQPPLKPVIQDAFDKIHFGALGKVIFEFEECCWSNESSKIVTLANSTNEFVEIVRNAENLDELDSMLEREDSQKHTSVTCWSQPLFFVNLSKSTGVASFMMLMQAPLTNHIESIREDKERLFSFFQPVLNKIMKCLDSEDVIDGMRPIENIANANKPVLRNIIVSNWTRDPYSRGAYSACFPGDDPVDMVVAMSNGQDSRIRFAGEHTIMDGAGCAYGAWESGRREATRISDLLKLEHHHHHH
8YEK , Knot 193 464 0.85 40 239 432 
MKTIIALSYIFCLVFADYKDDDDAMGVASSAVITANWISFLAISASFIILLVISLRYKGPGGTESFYNGFKEQNMLTVFINLWCALAYFAKVLQSHSNDNGFAPLTVIPYVDYCTTCPLLTLDLLWCLDAPYKISSAVLVFTCLVIAVACSLAVAPFSYCWFAMGMVLFTFTYVFILSIVRQRLDFFTLCARDSNAKQSLKHLKTAVFIYFGIWLLFPLLWLLSYRAANVISNDINHIFHCILDVIAKSVYGFALLYFKMYFDKKLIESGIDEDDFAKFSKVVTTHRSEEKQKRKPAVSQSPKMYYDEAAYQDGEVESNLQSKIRKSLSRKDKTPGSPSRSPMTPRTAYSTRKSPGMHLNDHPAWGSLQASAHSWENEREGPVGDHDDEEFSEFCQSLPKKAIPPLNSQPHMYNSEDEDDGAYLERAKMAYQKAKERHDNANGQRERRDRSGSRESDFENLYFQ

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 \(c\).

\(|P_{f(5SPV_1)}(2) \setminus P_{f(3CNP_1)}(2)|=21\), \(|P_{f(3CNP_1)}(2) \setminus P_{f(5SPV_1)}(2)|=164\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5SPV_1,3CNP_1 185 4
5SPV_1,8YEK_1 194 4
3CNP_1,8YEK_1 149 4

Newick tree

 
[
	5SPV_1:10.63,
	[
		3CNP_1:74.5,8YEK_1:74.5
	]:26.13
]

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 \((0.85)(0.8)(\frac{685 }{\log_{20} 685}-\frac{169}{\log_{20}169})=146.\)
Status Protein1 Protein2 d d1/2
Query variables 5SPV_1 3CNP_1 187 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]