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Parikh vectors
8WSS_1 2ODO_1 3RHH_1 Letter Amino acid
3 6 5 W Tryptophan
18 17 33 S Serine
16 7 18 N Asparagine
10 7 1 C Cysteine
17 25 42 G Glycine
30 10 30 K Lycine
4 22 24 P Proline
19 31 40 V Valine
32 48 49 A Alanine
26 19 29 D Aspartic acid
13 14 20 Q Glutamine
27 20 36 E Glutamic acid
6 8 15 H Histidine
21 12 27 T Threonine
13 10 5 Y Tyrosine
18 25 21 R Arginine
26 17 41 I Isoleucine
27 40 47 L Leucine
9 9 8 M Methionine
19 10 14 F Phenylalanine 

8WSS_1|Chain A|Guanine nucleotide-binding protein G(i) subunit alpha-1|Homo sapiens (9606)
>2ODO_1|Chains A, B, C, D|Alanine racemase|Pseudomonas fluorescens (294)
>3RHH_1|Chains A, B, C, D|NADP-dependent glyceraldehyde-3-phosphate dehydrogenase|Bacillus halodurans (86665)
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 \(w=f(c)\)
8WSS , Knot 154 354 0.85 40 205 334 
MGCTLSAEDKAAVERSKMIDRNLREDGEKAAREVKLLLLGAGESGKNTIVKQMKIIHEAGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTTGIVETHFTFKDLHFKMFDVGAQRSERKKWIHCFEGVTAIIFCVALSDYDLVLAEDEEMNRMHASMKLFDSICNNKWFTDTSIILFLNKKDLFEEKIKKSPLTICYPEYAGSNTYEEAAAYIQCQFEDLNKRKDTKEIYTHFTCSTDTKNVQFVFDAVTDVIIKNNLKDCGLF
2ODO , Knot 151 357 0.82 40 206 337 
MRPARALIDLQALRHNYQLAREVTGAKALAVIKADAYGHGAVRCAQALEAEADGFAVACIEEALELRAAGIRAPILLLEGFFEADELPLIVEHDFWCVVHSLWQLDAIEQARLAKPIHVWLKLDSGMHRVGLHPNDYKAAYQRLQASANVAKIVLMSHFARADELDCTRSVEQVAVFLGARADLTAEISLRNSPAVLGWPRVPSDWVRPGLMLYGSSPFDEPQATASRLQPVMTLESKVICVRELPAGEPVGYGARFITPKPMRIGVVAMGYADGYPRQAPTGTPVFVDGVRSQLLGRVSMDMLCIDLTDVPQAGLGSTVELWGKNILASEVATAADTIPYQIFCNLKRVPRLYSGN
3RHH , Knot 197 505 0.81 40 230 459 
MVMALQTEQFNANILRNGEWVESRTGERISISAPASGVALGSIPALSQEEVNDAIQGAKDAQKIWKIRPIHERVDLLYAWADLLEERKEIIGELIMHEVAKPKKSAIGEVSRTADIIRHTADEALRLNGETLKGDQFKGGSSKKIALVEREPLGVVLAISPFNYPVNLAAAKIAPALVTGNTVVFKPATQGSLSGIKMVEALADAGAPEGIIQVVTGRGSVIGDHLVEHPGIDMITFTGGTTTGERISEKAKMIPVVLELGGKDPAIVLDDADLKLTASQIVSGAFSYSGQRCTAIKRVFVQDSVADQLVANIKELVEQLTVGSPEDDADITPVIDEKSAAFIQGLIDDALENGATLLSGNKRQGNLLSPTLLDDVTPAMRVAWEEPFGPVLPIIRVKDANEAISLSNQSDYGLQASIFTKDTDRAINIGKHLEVGTVHINAKTERGPDHFPFLGVKKSGLGVQGIKPSLLSMTRERVTVLNLAENLYFQSHHHHHHWSHPQFEK

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 \(c\).

\(|P_{f(8WSS_1)}(2) \setminus P_{f(2ODO_1)}(2)|=83\), \(|P_{f(2ODO_1)}(2) \setminus P_{f(8WSS_1)}(2)|=84\). 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 1:110010100011100001100010001001100101111111001000110010110011000000000011100001001111101110101011001010010011111111001110101111100110001101010000000100011001001001101001100001100010001110001010010101101110000000110010110111101110000111100001001010101100100001100001111100001100010001101001001100000011101000100100000000100010000000010111011001110001000111
Pair \(Z_2\) Length of longest common subsequence
8WSS_1,2ODO_1 167 4
8WSS_1,3RHH_1 159 3
2ODO_1,3RHH_1 146 4

Newick tree

 
[
	8WSS_1:84.17,
	[
		3RHH_1:73,2ODO_1:73
	]:11.17
]

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 \((0.85)(0.8)(\frac{711 }{\log_{20} 711}-\frac{354}{\log_{20}354})=97.6\)
Status Protein1 Protein2 d d1/2
Query variables 8WSS_1 2ODO_1 125 124.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]