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Parikh vectors
4UIJ_1 4EZP_1 6NVO_1 Letter Amino acid
6 16 11 D Aspartic acid
1 0 2 C Cysteine
8 6 12 H Histidine
5 18 7 K Lycine
3 4 6 F Phenylalanine
5 6 17 P Proline
7 0 4 Y Tyrosine
6 25 27 A Alanine
8 7 19 R Arginine
4 9 4 N Asparagine
6 16 9 Q Glutamine
22 20 28 L Leucine
9 12 5 S Serine
11 16 8 T Threonine
8 19 12 E Glutamic acid
16 13 13 G Glycine
3 14 9 I Isoleucine
4 6 1 M Methionine
1 0 6 W Tryptophan
8 12 16 V Valine 

4UIJ_1|Chains A, B, C, D|BTB/POZ DOMAIN-CONTAINING ADAPTER FOR CUL3-MEDIATED RHOA DEGRADATION PROTEIN 1|HOMO SAPIENS (9606)
>4EZP_1|Chains A, B|Chaperone protein DnaK|Escherichia coli (83333)
>6NVO_1|Chain A|Nuclease MPE|Pseudomonas putida (strain ATCC 47054 / DSM 6125 / NCIMB 11950 / KT2440) (160488)
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 \(w=f(c)\)
4UIJ , Knot 69 141 0.80 40 113 134 
MHHHHHHSSGVDLGTENLYFQSMGPAAYGLKPLTPNSKYVKLNVGGSLHYTTLRTLTGQDTMLKAMFSGRVEVLTDAGGWVLIDRSGRHFGTILNYLRDGSVPLPESTRELGELLGEARYYLVQGLIEDCQLALQQKRETL
4EZP , Knot 95 219 0.78 34 145 212 
VLLLDVTPLSLGIETMGGVMTTLIAKNTTIPTKHSQVFSTAEDNQSAVTIHVLQGERKRAADNKSLGQFNLDGINPAPRGMPQIEVTFDIDADGILHVSAKDKNSGKEQKITIKASSGLNEDEIQKMVRDAEANAEADRKFEELVQTRNQGDHLLHSTRKQVEEAGDKLPADDKTAIESALTALETALKGEDKAAIEAKMQELAQVSQKLMEIAQQQHA
6NVO , Knot 101 216 0.83 40 146 206 
MNYQPIEHRGQALWLLADKAIYWPARRALLVADVHIGKAASYRALHQPVPRGTTEATLARLDRLLAEHDCEQLIILGDFLHARTARAPATLAKVEDWRKRHKNLKVVLIRGNHDRNAGDPPASLDIQVVDEPWVLEPFALQHEPQPHGTHPVLAGHVHPVFVLRGKARQRLRLPCFVIDEQVSLLPAFGEFTGGWEITPASASRLYLAGAERVWPL

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 \(c\).

\(|P_{f(4UIJ_1)}(2) \setminus P_{f(4EZP_1)}(2)|=54\), \(|P_{f(4EZP_1)}(2) \setminus P_{f(4UIJ_1)}(2)|=86\). 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 1:100000000110110001010011111011011010000101011101000010010100011011101010110011111110001001101100100101111000001101110100011011100001110000001
Pair \(Z_2\) Length of longest common subsequence
4UIJ_1,4EZP_1 140 3
4UIJ_1,6NVO_1 159 3
4EZP_1,6NVO_1 157 3

Newick tree

 
[
	6NVO_1:81.78,
	[
		4UIJ_1:70,4EZP_1:70
	]:11.78
]

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 \((0.85)(0.8)(\frac{360 }{\log_{20} 360}-\frac{141}{\log_{20}141})=66.5\)
Status Protein1 Protein2 d d1/2
Query variables 4UIJ_1 4EZP_1 82 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]