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Parikh vectors
3GIJ_1 5WKY_1 3DMD_1 Letter Amino acid
24 27 35 A Alanine
4 18 6 Q Glutamine
18 28 24 G Glycine
37 24 30 I Isoleucine
12 26 12 F Phenylalanine
11 20 5 P Proline
19 19 12 S Serine
15 25 5 Y Tyrosine
27 44 27 L Leucine
38 25 33 K Lycine
5 10 5 M Methionine
30 23 30 V Valine
0 3 2 W Tryptophan
24 19 18 R Arginine
13 18 9 N Asparagine
19 25 22 D Aspartic acid
1 17 0 C Cysteine
32 35 37 E Glutamic acid
2 8 5 H Histidine
10 25 11 T Threonine 

3GIJ_1|Chains A, D[auth B]|DNA polymerase IV|Sulfolobus solfataricus P2 (273057)
>5WKY_1|Chains A, B, C|Acid-sensing ion channel 1|Gallus gallus (9031)
>3DMD_1|Chains A, B, C|Signal recognition particle receptor|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
3GIJ , Knot 145 341 0.82 38 180 322 
GIVLFVDFDYFYAQVEEVLNPSLKGKPVVVCVFSGRFEDSGAVATANYEARKFGVKAGIPIVEAKKILPNAVYLPMRKEVYQQVSSRIMNLLREYSEKIEIASIDEAYLDISDKVRDYREAYNLGLEIKNKILEKEKITVTVGISKNKVFAKIAADMAKPNGIKVIDDEEVKRLIRELDIADVPGIGNITAEKLKKLGINKLVDTLSIEFDKLKGMIGEAKAKYLISLARDEYNEPIRTRVRKSIGRIVTMKRNSRNLEEIKPYLFRAIEESYYKLDKRIPKAIHVVAVTEDLDIVSRGRTFPHGISKETAYSESVKLLQKILEEDERKIRRIGVRFSKFI
5WKY , Knot 185 439 0.85 40 255 415 
SSTLHGISHIFSYERLSLKRVVWALCFMGSLALLALVCTNRIQYYFLYPHVTKLDEVAATRLTFPAVTFCNLNEFRFSRVTKNDLYHAGELLALLNNRYEIPDTQTADEKQLEILQDKANFRNFKPKPFNMLEFYDRAGHDIREMLLSCFFRGEQCSPEDFKVVFTRYGKCYTFNAGQDGKPRLITMKGGTGNGLEIMLDIQQDEYLPVWGETDETSFEAGIKVQIHSQDEPPLIDQLGFGVAPGFQTFVSCQEQRLIYLPPPWGDCKATTGDSEFYDTYSITACRIDCETRYLVENCNCRMVHMPGDAPYCTPEQYKECADPALDFLVEKDNEYCVCEMPCNVTRYGKELSMVKIPSKASAKYLAKKYNKSEQYIGENILVLDIFFEALNYETIEQKKAYEVAGLLGDIGGQMGLFIGASILTVLELFDYAYEVIKHR
3DMD , Knot 137 328 0.80 38 170 305 
GSHMASMFGKLREKLKSFVKRVEEEVEKEEEEVEKKGLLDRILTVEIKEKDVDKALDELEIDLLEADVALEVVDALREKIKQKLVGKKVRIGTDKGKIIEEAVKEAVSEILETSRRIDLIEEIRKAEKPYVIMFVGFNGSGKTTTIAKLANWLKNHGFSVVIAASDTFRAGAIEQLEEHAKRIGVKVIKHSYGADPAAVAYDAIQHAKARGIDVVLIDTAGRSETNRNLMDEMKKIARVTKPNLVIFVGDALAGNAIVEQARQFNEAVKIDGIILTKLDADARGGAALSISYVIDAPILFVGVGQGYDDLRPFEKEWFLERIFGEENA

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 \(c\).

\(|P_{f(3GIJ_1)}(2) \setminus P_{f(5WKY_1)}(2)|=42\), \(|P_{f(5WKY_1)}(2) \setminus P_{f(3GIJ_1)}(2)|=117\). 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 1:11111101001010100110101010111101101010001111010001001110111111010011101101110001000100011011000000101101001010100010000010011101000110000101011100001110111011010110110000100110010110111110101001001110011001010100101111010100110110000001100010001101101000000100101011011000000100011011011110001011001001101100001000010110011000000100111010011
Pair \(Z_2\) Length of longest common subsequence
3GIJ_1,5WKY_1 159 4
3GIJ_1,3DMD_1 132 4
5WKY_1,3DMD_1 167 4

Newick tree

 
[
	5WKY_1:86.07,
	[
		3GIJ_1:66,3DMD_1:66
	]:20.07
]

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 \((0.85)(0.8)(\frac{780 }{\log_{20} 780}-\frac{341}{\log_{20}341})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 3GIJ_1 5WKY_1 151 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]