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Parikh vectors
3WDD_1 5EJM_1 4ZZQ_1 Letter Amino acid
1 26 33 D Aspartic acid
10 36 17 Q Glutamine
3 33 21 P Proline
4 9 19 Y Tyrosine
11 65 21 A Alanine
3 16 33 N Asparagine
11 28 15 I Isoleucine
22 68 23 L Leucine
2 8 10 M Methionine
12 35 33 V Valine
11 34 9 R Arginine
0 7 18 C Cysteine
17 31 20 E Glutamic acid
4 13 19 K Lycine
7 30 44 S Serine
0 17 7 W Tryptophan
16 40 44 G Glycine
12 19 5 H Histidine
2 14 14 F Phenylalanine
5 27 33 T Threonine 

3WDD_1|Chain A|Probable ATP-dependent Clp protease ATP-binding subunit|Mycobacterium tuberculosis (1773)
>5EJM_1|Chains A, B, C, D, E, F, G, H|2-succinyl-5-enolpyruvyl-6-hydroxy-3-cyclohexene-1-carboxylate synthase|Escherichia coli K12 (83333)
>4ZZQ_1|Chain A|CELLULOSE 1,4-BETA-CELLOBIOSIDASE|DICTYOSTELIUM DISCOIDEUM (44689)
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 \(w=f(c)\)
3WDD , Knot 63 153 0.69 36 91 129 
MYERFTDRARRVVVLAQEEARMLNHNYIGTEHILLGLIHEGEGVAAKSLESLGISLEGVRSQVEEIIGQGQQAPSGHIPFTPRAKKVLELSLREALQLGHNYIGTEHILLGLIREGEGVAAQVLVKLGAELTRVRQQVIQLLSGYLEHHHHHH
5EJM , Knot 225 556 0.85 40 250 532 
MSVSAFNRRWAAVILEALTRHGVRHICIAPGSRSTPLTLAAAENSAFIHHTHFDERGLGHLALGLAKVSKQPVAVIVTSGTAVANLYPALIEAGLTGEKLILLTADRPPELIDCGANQAIRQPGMFASHPTHSISLPRPTQDIPARWLVSTIDHALGTLHAGGVHINCPFAEPLYGEMDDTGLSWQQRLGDWWQDDKPWLREAPRLESEKQRDWFFWRQKRGVVVAGRMSAEEGKKVALWAQTLGWPLIGDVLSQTGQPLPCADLWLGNAKATSELQQAQIVVQLGSSLTGKRLLQWQASCEPEEYWIVDDIEGRLDPAHHRGRRLIANIADWLELHPAEKRQPWCVEIPRLAEQAMQAVIARRDAFGEAQLAHRICDYLPEQGQLFVGNSLVVRLIDALSQLPAGYPVYSNAGASGIDGLLSTAAGVQRASGKPTLAIVGDLSALYDLNALALLRQVSAPLVLIVVNNNGGQIFSLLPTPQSERERFYLMPQNVHFEHAAAMFELKYHRPQNWQELETAFADAWRTPTTTVIEMVVNDTDGAQTLQQLLAQVSHL
4ZZQ , Knot 185 438 0.85 40 234 417 
QKIGKLTPEVHPPMTFQKCSEGGSCETIQGEVVVDANWRWVHSAQGQNCYTGNTWNPTICPDDETCAENCYLDGANYESVYGVTTSEDSVRLNFVTQSQGKNIGSRLFLMSNESNYQLFHVLGQEFTFDVDVSNLDCGLNGALYLVSMDSDGGSARFPTNEAGAKYGTGYCDAQCPRDLKFISGSANVDGWIPSTNNPNTGYGNLGSCCAEMDLWEANNMATAVTPHPCDTSSQSVCKSDSCGGAASSNRYGGICDPDGCDYNPYRMGNTSFFGPNKMIDTNSVITVVTQFITDDGSSDGKLTSIKRLYVQDGNVISQSVSTIDGVEGNEVNEEFCTNQKKVFGDEDSFTKHGGLAKMGEALKDGMVLVLSLWDDYQANMLWLDSSYPTTSSPTDPGVARGSCPTTSGVPSKVEQNYPNAYVVYSNIKVGPIDSTYKK

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 \(c\).

\(|P_{f(3WDD_1)}(2) \setminus P_{f(5EJM_1)}(2)|=16\), \(|P_{f(5EJM_1)}(2) \setminus P_{f(3WDD_1)}(2)|=175\). 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 1:100010001001111100010110000110001111110010111100100111010110001001110100110101110101001101010011011000110001111110010111101110111010010001101101010000000
Pair \(Z_2\) Length of longest common subsequence
3WDD_1,5EJM_1 191 4
3WDD_1,4ZZQ_1 213 3
5EJM_1,4ZZQ_1 172 5

Newick tree

 
[
	3WDD_1:10.71,
	[
		5EJM_1:86,4ZZQ_1:86
	]:19.71
]

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 \((0.85)(0.8)(\frac{709 }{\log_{20} 709}-\frac{153}{\log_{20}153})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 3WDD_1 5EJM_1 202 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]