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Parikh vectors
4RHG_1 3WTW_1 2MMR_1 Letter Amino acid
15 16 3 A Alanine
13 11 1 N Asparagine
6 2 3 C Cysteine
19 13 0 Q Glutamine
19 7 0 E Glutamic acid
14 6 0 H Histidine
12 5 0 I Isoleucine
12 4 0 M Methionine
23 16 3 T Threonine
14 5 0 Y Tyrosine
27 13 0 V Valine
17 10 0 G Glycine
12 7 0 K Lycine
14 19 0 P Proline
14 18 0 R Arginine
16 10 0 D Aspartic acid
30 15 0 L Leucine
8 8 0 F Phenylalanine
17 17 0 S Serine
4 2 0 W Tryptophan 

4RHG_1|Chain A|Tyrosine-protein phosphatase non-receptor type 3|Homo sapiens (9606)
>3WTW_1|Chains A, D[auth F]|Runt-related transcription factor 1|Mus musculus (10090)
>2MMR_1|Chain A|DNA_(5'-D(*CP*TP*AP*AP*(FAG)P*CP*TP*TP*CP*A)-3')|
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 \(w=f(c)\)
4RHG , Knot 135 306 0.84 40 207 294 
MHHHHHHSSGVDLGTENLYFQSNADTLEGSMAQLKKGLESGTVLIQFEQLYRKKPGLAITFAKLPQNLDKNRYKDVLPYDTTRVLLQGNEDYINASYVNMEIPAANLVNKYIATQGPLPHTCAQFWQVVWDQKLSLIVMLTTLTERGRTKCHQYWPDPPDVMNHGGFHIQCQSEDCTIAYVSREMLVTNTQTGEEHTVTHLQYVAWPEHGVPDDSSDFLEFVNYVRSLRVDSEPVLVHSSAGIGRTGVLVTMETAMCLTERNLPIYPLDIVRKMRDQRAMMVQTSSQYKFVCEAILRVYEEGLVQM
3WTW , Knot 100 204 0.87 40 150 200 
GELVRTDSPNFLCSVLPTHWRCNKTLPIAFKVVAKGDVPDGTLVTVMAGNDENYSAELRNATAAMKNQVARFNDLRFVGRSGRGKSFTLTITVFTNPPQVATYHRAIAITVDGPREPRRHRQKLDDQTKPGSLSFSERLSELEQLRRTAMRVSPHHPAPTPNPRASLNHSTAFNPQPQSQMQDARQIQPSPPWSYDQSYQYLGS
2MMR , Knot 6 10 0.46 8 8 8 
CTAANCTTCA

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 \(c\).

\(|P_{f(4RHG_1)}(2) \setminus P_{f(3WTW_1)}(2)|=116\), \(|P_{f(3WTW_1)}(2) \setminus P_{f(4RHG_1)}(2)|=59\). 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 1:100000000110110001010001001010110100110010111010010000111110110110010000000111000001110100001010010101111011000110011110001011011100010111110010001000000011011011001110100000000110100011100000100001001001111001110000011011001001010001111000111100111101001101000011101101100100001111000000011001110100011101
Pair \(Z_2\) Length of longest common subsequence
4RHG_1,3WTW_1 175 3
4RHG_1,2MMR_1 201 3
3WTW_1,2MMR_1 154 3

Newick tree

 
[
	4RHG_1:99.30,
	[
		3WTW_1:77,2MMR_1:77
	]:22.30
]

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 \((0.85)(0.8)(\frac{510 }{\log_{20} 510}-\frac{204}{\log_{20}204})=88.5\)
Status Protein1 Protein2 d d1/2
Query variables 4RHG_1 3WTW_1 113 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]