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Parikh vectors
3PBI_1 3ZJQ_1 5AEJ_1 Letter Amino acid
12 14 11 T Threonine
29 13 1 A Alanine
12 9 11 R Arginine
34 10 11 G Glycine
2 5 9 H Histidine
7 8 8 I Isoleucine
1 0 9 C Cysteine
6 9 6 F Phenylalanine
16 10 7 P Proline
16 12 6 V Valine
7 17 4 Y Tyrosine
5 18 10 E Glutamic acid
16 16 9 L Leucine
4 13 12 K Lycine
16 8 14 S Serine
2 5 1 W Tryptophan
6 7 7 N Asparagine
8 10 3 D Aspartic acid
11 7 8 Q Glutamine
4 4 5 M Methionine 

3PBI_1|Chain A|INVASION PROTEIN|Mycobacterium tuberculosis (1773)
>3ZJQ_1|Chains A, B|PROTOGLOBIN|METHANOSARCINA ACETIVORANS (2214)
>5AEJ_1|Chains A, B, C, D|GREMLIN-1|HOMO SAPIENS (9606)
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 \(w=f(c)\)
3PBI , Knot 95 214 0.79 40 133 205 
SMDAEPGQWDPTLPALVSAGAPGDPLAVANASLQATAQATQTTLDLGRQFLGGLGINLGGPAASAPSAATTGASRIPRANARQAVEYVIRRAGSQMGVPYSWGGGSLQGPSKGVDSGANTVGFDCSGLVRYAFAGVGVLIPRFSGDQYNAGRHVPPAEAKRGDLIFYGPGGGQHVTLYLGNGQMLEASGSAGKVTVSPVRKAGMTPFVTRIIEY
3ZJQ , Knot 92 195 0.83 38 148 188 
MSVEKIPGYTYGETENRAPFNLEDLKLLKEAVMFTAEDEEYIQKAGEVLEDQVEEILDTWYGFVGSHPHLLYYFTSPDGTPNEKYLAAVRKRFSRWILDTSNRSYDQAWLDYQYEIGLRHHRTKKNQTDNVESVPNIGYRYLVAFIYPITATMKPFLARKGHTPEEVEKMYQAWFKATTLQVALWSYPYVKYGDF
5AEJ , Knot 73 152 0.80 40 112 139 
MNTIHHHHHHNTSGSGGGGGRLVPRGSMSENLYFQGSMSEVLESSQEALHVTERKYLKRDWCKTQPLKQTIHEEGCNSRTIINRFCYGQCNSFYIPRHIRKEEGSFQSCSFCKPKKFTTMMVTLNCPELQPPTKKKRVTRVKQCRCISIDLD

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 \(c\).

\(|P_{f(3PBI_1)}(2) \setminus P_{f(3ZJQ_1)}(2)|=76\), \(|P_{f(3ZJQ_1)}(2) \setminus P_{f(3PBI_1)}(2)|=91\). 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 1:0101011010101111101111101111101010101010000101100111111101111110110110011001101010011001100110011110011110101100110011001110001110011111111110101000011001111010010111011111001010110101101010110101011001110111001100
Pair \(Z_2\) Length of longest common subsequence
3PBI_1,3ZJQ_1 167 3
3PBI_1,5AEJ_1 155 3
3ZJQ_1,5AEJ_1 170 4

Newick tree

 
[
	3ZJQ_1:86.38,
	[
		3PBI_1:77.5,5AEJ_1:77.5
	]:8.88
]

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 \((0.85)(0.8)(\frac{409 }{\log_{20} 409}-\frac{195}{\log_{20}195})=63.2\)
Status Protein1 Protein2 d d1/2
Query variables 3PBI_1 3ZJQ_1 80 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]