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Parikh vectors
5ZHE_1 1FUL_1 2PCC_1 Letter Amino acid
6 0 7 W Tryptophan
27 1 17 A Alanine
20 1 10 R Arginine
15 0 20 N Asparagine
2 4 1 C Cysteine
20 2 25 G Glycine
13 0 11 I Isoleucine
7 0 16 P Proline
10 0 15 T Threonine
14 0 14 V Valine
12 2 24 D Aspartic acid
19 0 20 E Glutamic acid
8 0 6 H Histidine
6 0 6 M Methionine
11 1 18 F Phenylalanine
13 0 15 S Serine
13 0 9 Q Glutamine
24 0 25 L Leucine
9 0 23 K Lycine
4 0 14 Y Tyrosine 

5ZHE_1|Chains A, B|Ditrans,polycis-undecaprenyl-diphosphate synthase ((2E,6E)-farnesyl-diphosphate specific)|Escherichia coli K-12 (83333)
>1FUL_1|Chain A|RGD PEPTIDE ISOMER-B|null
>2PCC_1|Chains A, C|CYTOCHROME C PEROXIDASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
5ZHE , Knot 114 253 0.83 40 173 246 
MMLSATQPLSEKLPAHGCRHVAIIMDGNGRWAKKQGKIRAFGHKAGAKSVRRAVSFAANNGIEALTLYAFSSENWNRPAQEVSALMELFVWALDSEVKSLHRHNVRLRIIGDTSRFNSRLQERIRKSEALTAGNTGLTLNIAANYGGRWDIVQGVRQLAEKVQQGNLQPDQIDEEMLNQHVCMHELAPVDLVIRTGGEHRISNFLLWQIAYAELYFTDVLWPDFDEQDFEGALNAFANRERRFGGTEPGDETA
1FUL , Knot 7 11 0.50 12 9 9 
ACDCRGDCFCG
2PCC , Knot 136 296 0.87 40 195 289 
MITTPLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHISGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDAGYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGPWGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL

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 \(c\).

\(|P_{f(5ZHE_1)}(2) \setminus P_{f(1FUL_1)}(2)|=171\), \(|P_{f(1FUL_1)}(2) \setminus P_{f(5ZHE_1)}(2)|=7\). 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 1:1110100110001110100011111010101100010101110011100100110111001101101011000010011001011101111110001001000010101110000100010001000011011001101011100110101101100110010010101001000110001010011110111001100010011110110101010011110100001011101110000011100110001
Pair \(Z_2\) Length of longest common subsequence
5ZHE_1,1FUL_1 178 2
5ZHE_1,2PCC_1 176 4
1FUL_1,2PCC_1 200 2

Newick tree

 
[
	1FUL_1:96.77,
	[
		5ZHE_1:88,2PCC_1:88
	]:8.77
]

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 \((0.85)(0.8)(\frac{264 }{\log_{20} 264}-\frac{11}{\log_{20}11})=87.1\)
Status Protein1 Protein2 d d1/2
Query variables 5ZHE_1 1FUL_1 112 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]