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Parikh vectors
4ZXG_1 5PSK_1 7PQY_1 Letter Amino acid
2 1 8 C Cysteine
1 7 0 M Methionine
9 5 12 T Threonine
1 0 2 W Tryptophan
8 10 16 V Valine
10 11 12 A Alanine
3 10 7 H Histidine
21 18 13 L Leucine
20 6 10 K Lycine
17 6 14 F Phenylalanine
5 14 9 R Arginine
24 4 17 N Asparagine
13 13 9 D Aspartic acid
13 13 7 E Glutamic acid
15 5 7 I Isoleucine
7 7 15 S Serine
6 8 6 Q Glutamine
9 7 16 G Glycine
6 6 10 P Proline
15 5 15 Y Tyrosine 

4ZXG_1|Chains A, B|Glutathione S-transferase|Plasmodium falciparum (5833)
>5PSK_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>7PQY_1|Chains A, B[auth E]|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
4ZXG , Knot 95 205 0.82 40 144 196 
DNIVLYYFDARGKAELIRLIFAYLGIEYTDKRFGVNGDAFVEFKNFKKEKDTPFEQVPILQIGDLILAQSQAIVRYLSKKYNICGESELNEFYADMIFCGVQDIHYKFNNTNLFKQNETTFLNEDLPKWSGYFEKLLKKNHTNNNNDKYYFVGNNLTYADLAVFNLYDDIETKYPSSLKNFPLLKAHNEFISNLPNIKNYITNRK
5PSK , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
7PQY , Knot 94 205 0.81 38 142 195 
ETGHHHHHHTNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGKK

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 \(c\).

\(|P_{f(4ZXG_1)}(2) \setminus P_{f(5PSK_1)}(2)|=93\), \(|P_{f(5PSK_1)}(2) \setminus P_{f(4ZXG_1)}(2)|=69\). 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 1:0011100101010101101111011100000011101011101001000000110011110110111100011100100000101000100101011101100100010000110000001100011010101001100000000000001110010010111101000100001001001111010001100110100010000
Pair \(Z_2\) Length of longest common subsequence
4ZXG_1,5PSK_1 162 3
4ZXG_1,7PQY_1 156 3
5PSK_1,7PQY_1 158 6

Newick tree

 
[
	5PSK_1:80.66,
	[
		4ZXG_1:78,7PQY_1:78
	]:2.66
]

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 \((0.85)(0.8)(\frac{361 }{\log_{20} 361}-\frac{156}{\log_{20}156})=61.9\)
Status Protein1 Protein2 d d1/2
Query variables 4ZXG_1 5PSK_1 78 68.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]