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Parikh vectors
3GVQ_1 1MKR_1 1DVV_1 Letter Amino acid
6 1 2 C Cysteine
30 24 7 G Glycine
17 18 0 F Phenylalanine
26 16 6 P Proline
11 15 3 S Serine
5 7 2 W Tryptophan
11 15 3 Y Tyrosine
6 20 3 N Asparagine
17 25 5 D Aspartic acid
22 9 5 Q Glutamine
16 16 2 T Threonine
39 17 14 A Alanine
26 20 4 E Glutamic acid
9 5 1 H Histidine
11 9 4 I Isoleucine
43 25 4 L Leucine
10 5 3 M Methionine
26 14 5 V Valine
24 10 1 R Arginine
12 23 8 K Lycine 

3GVQ_1|Chain A|Uroporphyrinogen decarboxylase|Homo sapiens (9606)
>1MKR_1|Chain A|Cytochrome c Peroxidase|Saccharomyces cerevisiae (4932)
>1DVV_1|Chain A|CYTOCHROME C551|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
3GVQ , Knot 153 367 0.82 40 200 341 
MEANGLGPQGFPELKNDTFLRAAWGEETDYTPVWCMRQAGRYLPEFRETRAAQDFFSTCRSPEACCELTLQPLRRFPLDAAIIFSDILVVPQALGMEVTMVPGKGPSFPEPLREEQDLERLRDPEVVASELGYVFQAITLTRQRLAGRVPLIGFAGAPWTLMTYMVEGGGSSTMAQAKRWLYQRPQASHQLLRILTDALVPYLVGQVVAGAQALQLFESHAGHLGPQLFNKFALPYIRDVAKQVKARLREAGLAPVPMIIFAKDGHFALEELAQAGYEVVGLDWTVAPKKARECVGKTVTLQGNLDPCALYASEEEIGQLVKQMLDDFGPHRYIANLGHGLYPDMDPEHVGAFVDAVHKHSRLLRQN
1MKR , Knot 134 294 0.86 40 195 286 
TTPLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWYTSGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDADYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGPWGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL
1DVV , Knot 45 82 0.80 38 70 79 
EDPEVLAKNKGCMACHAIDTKMVGPAYKDVAAKYAGQAGAEAYLAQRIKNGSQGVWGPIPMPPNAVSDDEAQTLAKWILSQK

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 \(c\).

\(|P_{f(3GVQ_1)}(2) \setminus P_{f(1MKR_1)}(2)|=89\), \(|P_{f(1MKR_1)}(2) \setminus P_{f(3GVQ_1)}(2)|=84\). 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 1:1010111101110100001101111000000111010011001101000011001100000101000101011001110111110011111011110101111011011011000001001001011100110110110100001110111111111110110011011100011010011000101000110110011110111011111011011000110111011001111010011001010100111111111111001011100110110011110101110010001100101010101011010000110110011001110001101101101010100111110110000011000
Pair \(Z_2\) Length of longest common subsequence
3GVQ_1,1MKR_1 173 4
3GVQ_1,1DVV_1 186 4
1MKR_1,1DVV_1 193 3

Newick tree

 
[
	1DVV_1:97.36,
	[
		3GVQ_1:86.5,1MKR_1:86.5
	]:10.86
]

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 \((0.85)(0.8)(\frac{661 }{\log_{20} 661}-\frac{294}{\log_{20}294})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 3GVQ_1 1MKR_1 129 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]