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Parikh vectors
6ZER_1 4JGD_1 1JUH_1 Letter Amino acid
15 43 30 G Glycine
8 13 12 I Isoleucine
0 6 3 M Methionine
16 34 29 V Valine
6 16 17 Q Glutamine
8 0 2 C Cysteine
6 22 12 E Glutamic acid
1 20 10 H Histidine
16 21 38 S Serine
15 7 21 Y Tyrosine
9 24 21 D Aspartic acid
19 9 14 N Asparagine
13 45 18 L Leucine
12 19 21 P Proline
12 19 39 T Threonine
2 1 6 W Tryptophan
12 30 26 A Alanine
10 19 5 K Lycine
14 11 18 F Phenylalanine
9 31 8 R Arginine 

6ZER_1|Chains A, D, G[auth E]|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4JGD_1|Chain A|Dihydroorotate dehydrogenase (quinone), mitochondrial|Homo sapiens (9606)
>1JUH_1|Chains A, B, C, D|quercetin 2,3-dioxygenase|Aspergillus japonicus (34381)
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 \(w=f(c)\)
6ZER , Knot 94 203 0.82 38 143 196 
PNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTN
4JGD , Knot 157 390 0.80 38 197 355 
MGHHHHHHHHHHSSGHIDDDDKHMATGDERFYAEHLMPTLQGLLDPESAHRLAVRFTSLGLLPRARFQDSDMLEVRVLGHKFRNPVGIAAGFDKHGEAVDGLYKMGFGFVEIGSVTPKPQEGNPRPRVFRLPEDQAVINRYGFNSHGLSVVEHRLRARQQKQAKLTEDGLPLGVNLGKNKTSVDAAEDYAEGVRVLGPLADYLVVNVSSPNTAGLRSLQGKAELRRLLTKVLQERDGLRRVHRPAVLVKIAPDLTSQDKEDIASVVKELGIDGLIVTNTTVSRPAGLQGALRSETGGLSGKPLRDLSTQTIREMYALTQGRVPIIGVGGVSSGQDALEKIRAGASLVQLYTALTFWGPPVVGKVKRELEALLKEQGFGGVTDAIGADHRR
1JUH , Knot 146 350 0.81 40 195 328 
DTSSLIVEDAPDHVRPYVIRHYSHARAVTVDTQLYRFYVTGPSSGYAFTLMGTNAPHSDALGVLPHIHQKHYENFYCNKGSFQLWAQSGNETQQTRVLSSGDYGSVPRNVTHTFQIQDPDTEMTGVIVPGGFEDLFYYLGTNATDTTHTPYIPSSSDSSSTTGPDSSTISTLQSFDVYAELSFTPRTDTVNGTAPANTVWHTGANALASTAGDPYFIANGWGPKYLNSQYGYQIVAPFVTATQAQDTNYTLSTISMSTTPSTVTVPTWSFPGACAFQVQEGRVVVQIGDYAATELGSGDVAFIPGGVEFKYYSEAYFSKVLFVSSGSDGLDQNLVNGGEEWSSVSFPADW

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 \(c\).

\(|P_{f(6ZER_1)}(2) \setminus P_{f(4JGD_1)}(2)|=57\), \(|P_{f(4JGD_1)}(2) \setminus P_{f(6ZER_1)}(2)|=111\). 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 1:10100101110110100110101100001000110001100010100100011010010010100101001110100100111100101100000110010101111000010001110000100110000101100010001001100100110110001110001101001110010011110101101110101100000
Pair \(Z_2\) Length of longest common subsequence
6ZER_1,4JGD_1 168 3
6ZER_1,1JUH_1 170 4
4JGD_1,1JUH_1 154 4

Newick tree

 
[
	6ZER_1:86.85,
	[
		4JGD_1:77,1JUH_1:77
	]:9.85
]

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 \((0.85)(0.8)(\frac{593 }{\log_{20} 593}-\frac{203}{\log_{20}203})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZER_1 4JGD_1 138 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]