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Parikh vectors
4LEB_1 3MCS_1 4QDY_1 Letter Amino acid
6 5 8 R Arginine
22 16 10 N Asparagine
8 1 0 C Cysteine
28 10 10 G Glycine
11 16 19 I Isoleucine
16 15 4 Y Tyrosine
21 16 24 V Valine
6 1 0 W Tryptophan
14 18 14 D Aspartic acid
17 14 11 L Leucine
13 26 17 K Lycine
18 10 4 F Phenylalanine
41 15 27 T Threonine
13 3 12 P Proline
30 8 31 S Serine
19 15 13 A Alanine
6 7 8 Q Glutamine
6 18 12 E Glutamic acid
2 2 0 H Histidine
3 3 3 M Methionine 

4LEB_1|Chain A|Agglutinin-like protein 3|Candida albicans (5476)
>3MCS_1|Chains A, B|Putative monooxygenase|Fusobacterium nucleatum subsp. nucleatum (190304)
>4QDY_1|Chain A|conserved hypothetical protein|Streptococcus pneumoniae (170187)
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 \(w=f(c)\)
4LEB , Knot 131 300 0.83 40 178 288 
GKTITGVFNSFNSLTWSNAATYNYKGPGTPTWNAVLGWSLDGTSASPGDTFTLNMPCVFKFTTSQTSVDLTAHGVKYATCQFQAGEEFMTFSTLTCTVSNTLTPSIKALGTVTLPLAFNVGGTGSSVDLEDSKCFTAGTNTVTFNDGGKKISINVDFERSNVDPKGYLTDSRVIPSLNKVSTLFVAPQCANGYTSGTMGFANTYGDVQIDCSNIHVGITKGLNDWNYPVSSESFSYTKTCSSNGIFITYKNVPAGYRPFVDAYISATDVNSYTLSYANEYTCAGGYWQRAPFTLRWTGYR
3MCS , Knot 100 219 0.82 40 138 208 
GMYAVPILNVYDFEVKKDKETSYKSATEDYVNKTMGVEQGVLGLFAATDERDKTTSYIVEIYNDYLAFSNHTKNQASKDFKAVIPQIAEGNLNSAEIDVQIAKDKKIEQNDNTFAVYTVIDVKPENDKEFAEIIKNIVETTFNEEGTLLVYLGTDRRNFNKWCLFEVYKDIDSYLNHRSAKYFKDYITQTKDMIAGKKRAELQVLKIENKGGLDYKKLY
4QDY , Knot 100 227 0.79 34 137 209 
GRQVKTETYTNTVTNVPIDIRYNSDKYFISGFASEVSVVLTGANRLSLASEMQESTRKFKVTADLTDAGVGTIEVPLSIEDLPNGLTAVATPQKITVKIGKKAQKDKVKIVPEIDPSQIDSRVQIENVMVSDKEVSITSDQETLDRIDKIIAVLPTSERITGNYSGSVPLQAIDRNGVVLPAVITPFDTIMKVTTKPVAPSSSTSNSSTSSSSETSSSTKATSSKTN

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 \(c\).

\(|P_{f(4LEB_1)}(2) \setminus P_{f(3MCS_1)}(2)|=101\), \(|P_{f(3MCS_1)}(2) \setminus P_{f(4LEB_1)}(2)|=61\). 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 1:100101110010010100110000011101010111110101001011001010110110100000010101011001000101100110100100010001010101110101111101110100101000001011000101001100101010100001010101000011101001001111100101000101111000101010000101110011001001100001000000000111100001111001110101010010000100100000111010011101010100
Pair \(Z_2\) Length of longest common subsequence
4LEB_1,3MCS_1 162 4
4LEB_1,4QDY_1 141 5
3MCS_1,4QDY_1 145 3

Newick tree

 
[
	3MCS_1:78.87,
	[
		4LEB_1:70.5,4QDY_1:70.5
	]:8.37
]

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 \((0.85)(0.8)(\frac{519 }{\log_{20} 519}-\frac{219}{\log_{20}219})=86.3\)
Status Protein1 Protein2 d d1/2
Query variables 4LEB_1 3MCS_1 106 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]