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Parikh vectors
4LAM_1 5WKV_1 5EJT_1 Letter Amino acid
14 25 16 T Threonine
3 3 7 W Tryptophan
31 23 14 V Valine
27 20 16 P Proline
3 17 1 C Cysteine
19 24 9 I Isoleucine
14 25 23 K Lycine
17 26 18 F Phenylalanine
21 25 25 D Aspartic acid
18 35 20 E Glutamic acid
26 28 24 G Glycine
14 8 6 H Histidine
43 44 25 L Leucine
9 25 14 Y Tyrosine
47 27 17 A Alanine
9 18 19 N Asparagine
9 18 9 Q Glutamine
12 10 5 M Methionine
23 19 15 S Serine
17 19 11 R Arginine 

4LAM_1|Chains A, B|Uracil-5-carboxylate decarboxylase|Cordyceps militaris (983644)
>5WKV_1|Chains A, B, C|Acid-sensing ion channel 1|Gallus gallus (9031)
>5EJT_1|Chain A|Cytochrome c peroxidase, mitochondrial|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
4LAM , Knot 162 376 0.85 40 211 359 
MAASTPVVVDIHTHMYPPSYIAMLEKRQTIPLVRTFPQADEPRLILLSSELAALDAALADPAAKLPGRPLSTHFASLAQKMHFMDTNGIRVSVISLANPWFDFLAPDEAPGIADAVNAEFSDMCAQHVGRLFFFAALPLSAPVDAVKASIERVKNLKYCRGIILGTSGLGKGLDDPHLLPVFEAVADAKLLVFLHPHYGLPNEVYGPRSEEYGHVLPLALGFPMETTIAVARMYMAGVFDHVRNLQMLLAHSGGTLPFLAGRIESCIVHDGHLVKTGKVPKDRRTIWTVLKEQIYLDAVIYSEVGLQAAIASSGADRLMFGTNHPFFPPIEEDVQGPWDSSRLNAQAVIKAVGEGSSDAAAVMGLNAVRVLSLKAE
5WKV , Knot 185 439 0.85 40 255 415 
SSTLHGISHIFSYERLSLKRVVWALCFMGSLALLALVCTNRIQYYFLYPHVTKLDEVAATRLTFPAVTFCNLNEFRFSRVTKNDLYHAGELLALLNNRYEIPDTQTADEKQLEILQDKANFRNFKPKPFNMLEFYDRAGHDIREMLLSCFFRGEQCSPEDFKVVFTRYGKCYTFNAGQDGKPRLITMKGGTGNGLEIMLDIQQDEYLPVWGETDETSFEAGIKVQIHSQDEPPLIDQLGFGVAPGFQTFVSCQEQRLIYLPPPWGDCKATTGDSEFYDTYSITACRIDCETRYLVENCNCRMVHMPGDAPYCTPEQYKECADPALDFLVEKDNEYCVCEMPCNVTRYGKELSMVKIPSKASAKYLAKKYNKSEQYIGENILVLDIFFEALNYETIEQKKAYEVAGLLGDIGGQMGLFIGASILTVLELFDYAYEVIKHR
5EJT , Knot 134 294 0.86 40 195 286 
TTPLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHTSGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDADYVRTFFQRLNMNDREVVALMGAHALGKTHLKRSGYEGPWGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL

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 \(c\).

\(|P_{f(4LAM_1)}(2) \setminus P_{f(5WKV_1)}(2)|=65\), \(|P_{f(5WKV_1)}(2) \setminus P_{f(4LAM_1)}(2)|=109\). 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 1:1110011110100010110011110000011110011010010111100011110111101110111011000110110010110001101011011011101111001111101101010010100110111111111011101101010010010000111110011101100101111101110101111101001110010110000010111111111100011110101111100100101111001101111110100011001011001011000001101100010101110001110111100110011110001111110001011100001010111011101000111111101101101010
Pair \(Z_2\) Length of longest common subsequence
4LAM_1,5WKV_1 174 4
4LAM_1,5EJT_1 156 4
5WKV_1,5EJT_1 158 3

Newick tree

 
[
	5WKV_1:84.72,
	[
		4LAM_1:78,5EJT_1:78
	]:6.72
]

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 \((0.85)(0.8)(\frac{815 }{\log_{20} 815}-\frac{376}{\log_{20}376})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 4LAM_1 5WKV_1 153 140
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]