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Parikh vectors
4IQB_1 7ALQ_1 5YNQ_1 Letter Amino acid
11 7 24 N Asparagine
12 9 9 M Methionine
13 10 10 P Proline
5 2 6 W Tryptophan
22 18 21 V Valine
14 16 19 A Alanine
15 17 11 E Glutamic acid
25 12 19 G Glycine
9 5 5 H Histidine
29 19 36 L Leucine
19 12 16 K Lycine
11 7 12 F Phenylalanine
17 11 20 T Threonine
7 2 6 C Cysteine
13 4 10 Y Tyrosine
16 14 24 S Serine
16 13 9 R Arginine
17 10 11 Q Glutamine
18 11 19 I Isoleucine
26 10 16 D Aspartic acid 

4IQB_1|Chains A, B|Thymidylate synthase|Caenorhabditis elegans (6239)
>7ALQ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T|GTP cyclohydrolase 1|Homo sapiens (9606)
>5YNQ_1|Chain A|nsp16 protein|Human betacoronavirus 2c EMC/2012 (1235996)
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 \(w=f(c)\)
4IQB , Knot 140 315 0.85 40 203 302 
MEVMNKENIIADAPSDVVKTVQQQVHLNQDEYKYLKQVEQILREGTRRDDRTGTGTISIFGMQSKYCLRNGTIPLLTTKRVYWKGVLEELLWFISGSTDGKLLMEKNVKIWEKNGDRAFLDNLGFTSREEGDLGPVYGFQWRHFGAKYVDCHTDYSGQGVDQLAEVIRQIKEQPDSRRIIMSAWNPSDLGQMVLPPCHTMCQFYVDNGELSCQLYQRSGDMGLGVPFNLASYGLLTHMIAKVCGLKPGTLVHTLGDAHVYSNHVDALKIQLDREPYAFPKIRFTRDVASIDDFTSDMIALDDYKCHPKIPMDMAV
7ALQ , Knot 99 209 0.84 40 159 206 
PEAKSAQPADGWKGERPRSEEDNELNLPNLAAAYSSILSSLGENPQRQGLLKTPWRAASAMQFFTKGYQETISDVLNDAIFDEDHDEMVIVKDIDMFSMCEHHLVPFVGKVHIGYLPNKQVLGLSKLARIVEIYSRRLQVQERLTKQIAVAITEALRPAGVGVVVEATHMCMVMRGVQKMNSKTVTSTMLGVFREDPKTREEFLTLIRS
5YNQ , Knot 136 303 0.85 40 193 295 
ASADWKPGHAMPSLFKVQNVNLERCELANYKQSIPMPRGVHMNIAKYMQLCQYLNTCTLAVPANMRVIHFGAGSDKGIAPGTSVLRQWLPTDAIIIDNDLNEFVSDADITLFGDCVTVRVGQQVDLVISDMYDPTTKNVTGSNESKALFFTYLCNLINNNLALGGSVAIKITEHSWSVELYELMGKFAWWTVFCTNANASSSEGFLLGINYLGTIKENIDGGAMHANYIFWRNSTPMNLSTYSLFDLSKFQLKLKGTPVLQLKESQINELVISLLSQGKLLIRDNDTLSVSTDVLVNTYRKLR

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 \(c\).

\(|P_{f(4IQB_1)}(2) \setminus P_{f(7ALQ_1)}(2)|=110\), \(|P_{f(7ALQ_1)}(2) \setminus P_{f(4IQB_1)}(2)|=66\). 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 1:101100001110110011001000101000000010010011001000000010101011110000010010111100001010111001111101000101110001011000100111001110000010111101101001110010000000101100110110010001000011101101001101111100010010100101000100001011111110110011100111010110110110011010100001011010100010111010100011010010001111000000101110111
Pair \(Z_2\) Length of longest common subsequence
4IQB_1,7ALQ_1 176 3
4IQB_1,5YNQ_1 178 4
7ALQ_1,5YNQ_1 166 3

Newick tree

 
[
	4IQB_1:90.26,
	[
		7ALQ_1:83,5YNQ_1:83
	]:7.26
]

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 \((0.85)(0.8)(\frac{524 }{\log_{20} 524}-\frac{209}{\log_{20}209})=90.7\)
Status Protein1 Protein2 d d1/2
Query variables 4IQB_1 7ALQ_1 116 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]