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Parikh vectors
8GVL_1 4ZYL_1 7ARK_1 Letter Amino acid
20 16 40 A Alanine
5 0 0 C Cysteine
27 9 18 E Glutamic acid
13 6 34 G Glycine
14 11 16 T Threonine
14 12 14 D Aspartic acid
27 3 25 Q Glutamine
6 10 1 H Histidine
19 4 10 K Lycine
16 8 8 Y Tyrosine
11 9 27 I Isoleucine
5 3 20 M Methionine
8 6 21 P Proline
17 9 23 S Serine
22 12 36 V Valine
13 14 21 R Arginine
10 6 16 N Asparagine
33 18 53 L Leucine
16 5 10 F Phenylalanine
5 2 6 W Tryptophan 

8GVL_1|Chains A, B|Tyrosine-protein phosphatase non-receptor type 21|Homo sapiens (9606)
>4ZYL_1|Chains A, B|RphyB protein|Rhodopseudomonas palustris CGA009 (258594)
>7ARK_1|Chain A[auth C]|Lipoprotein-releasing ABC transporter permease subunit LolC|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
8GVL , Knot 135 301 0.85 40 199 291 
SSKSCLVARIQLLNNEFVEFTLSVESTGQESLEAVAQRLELREVTYFSLWYYNKQNQRRWVDLEKPLKKQLDKYALEPTVYFGVVFYVPSVSQLQQEITRYQYYLQLKKDILEGSIPCTLEQAIQLAGLAVQADFGDFDQYESQDFLQKFALFPVGWLQDEKVLEEATQKVALLHQKYRGLTAPDAEMLYMQEVERMDGYGEESYPAKDSQGSDISIGACLEGIFVKHKNGRHPVVFRWHDIANMSHNKSFFALELANKEETIQFQTEDMETAKYIWRLCVARHKFYRLNQCNLQTQTVTV
4ZYL , Knot 78 163 0.81 38 118 155 
MNRQRTLPTVLVAEDHDYDKLILTEVFARASISADLRFVSDGEQTLDYIYGRNRFADRGDAPYPAIVLLDLNMPRLDGRKVVRLLRQDETVRHLVVIALSTSESAKHITEAYSIGFNAYLVKPANIADYVEAIRSLWHFWMNTASLPTTEAYRTGLEHHHHHH
7ARK , Knot 162 399 0.81 38 200 361 
MYQPVALFIGLRYMRGRAADRFGRFVSWLSTIGITLGVMALVTVLSVMNGFERELQNNILGLMPQAILSSEHGSLNPQQLPETAVKLDGVNRVAPITTGDVVLQSARSVAVGVMLGIDPAQKDPLTPYLVNVKQTDLEPGKYNVILGEQLASQLGVNRGDQIRVMVPSASQFTPMGRIPSQRLFNVIGTFAANSEVDGYEMLVNIEDASRLMRYPAGNITGWRLWLDEPLKVDSLSQQKLPEGSKWQDWRDRKGELFQAVRMEKNMMGLLLSLIVAVAAFNIITSLGLMVMEKQGEVAILQTQGLTPRQIMMVFMVQGASAGIIGAILGAALGALLASQLNNLMPIIGVLLDGAALPVAIEPLQVIVIALVAMAIALLSTLYPSWRAAATQPAEALRYE

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 \(c\).

\(|P_{f(8GVL_1)}(2) \setminus P_{f(4ZYL_1)}(2)|=124\), \(|P_{f(4ZYL_1)}(2) \setminus P_{f(8GVL_1)}(2)|=43\). 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 1:0000011101011000110101010001000101110010100100101100000000011010011000100011010101111101101001000100000010100011010110010011011111101011010000000110011111111100001100100011110000011011010110100100101010000110000100101110101111000010011110100110100000111101100000101000010010011010110001001000010000101
Pair \(Z_2\) Length of longest common subsequence
8GVL_1,4ZYL_1 167 3
8GVL_1,7ARK_1 151 4
4ZYL_1,7ARK_1 150 4

Newick tree

 
[
	8GVL_1:81.07,
	[
		7ARK_1:75,4ZYL_1:75
	]:6.07
]

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 \((0.85)(0.8)(\frac{464 }{\log_{20} 464}-\frac{163}{\log_{20}163})=88.7\)
Status Protein1 Protein2 d d1/2
Query variables 8GVL_1 4ZYL_1 115 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]