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Parikh vectors
1MTW_1 4YGQ_1 2IPA_1 Letter Amino acid
14 10 8 A Alanine
34 8 6 S Serine
14 16 10 K Lycine
3 9 3 F Phenylalanine
10 13 1 Y Tyrosine
6 20 7 D Aspartic acid
4 17 11 E Glutamic acid
14 31 10 L Leucine
3 8 1 H Histidine
15 14 5 I Isoleucine
2 5 4 M Methionine
8 9 5 P Proline
17 14 12 V Valine
2 15 0 R Arginine
16 5 2 N Asparagine
12 0 1 C Cysteine
4 4 2 W Tryptophan
10 4 4 Q Glutamine
25 16 7 G Glycine
10 11 5 T Threonine 

1MTW_1|Chain A|TRYPSIN|Bos taurus (9913)
>4YGQ_1|Chain A|Hydrolase|Thermococcus onnurineus (strain NA1) (523850)
>2IPA_1|Chain A|Thioredoxin|Bacillus subtilis (1423)
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 \(w=f(c)\)
1MTW , Knot 101 223 0.81 40 140 216 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
4YGQ , Knot 102 229 0.80 38 150 217 
MKAVLFDIDGTILTEEPLIMLFLPQVYDKLSRKLGISKDEARERFLSEILGRRDSYDWHDWNFFFKLFDLDLKYEELLERYPHKLQVYPDTIPTLEWLRDTGYKLGIVTSGPKYQRLKLKLTGLLDYFDVVITRDDVNAIKPEPKIFLYTIERLGVEPGEAVMVGDSLSQDVYGAKSVGMTAVWINRNGDRGYNMADYEIRTLYELRKILGGERVDKLAAALEHHHHHH
2IPA , Knot 54 104 0.80 38 86 100 
MAIVKATDQSFSAETSEGVVLADFWAPWCGPSKMIAPVLEELDQEMGDKLKIVKIDVDENQETAGKYGVMSIPTLLVLKDGEVVETSVGFKPKEALQELVNKHL

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 \(c\).

\(|P_{f(1MTW_1)}(2) \setminus P_{f(4YGQ_1)}(2)|=80\), \(|P_{f(4YGQ_1)}(2) \setminus P_{f(1MTW_1)}(2)|=90\). 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 1:1111000110011001010010010110110001110110000011010110001011010001101000110100000010001111010011010001101011000101100011011100000100010110010111100000001011010001101101011000001001111100101011101101010000111000100010110001100
Pair \(Z_2\) Length of longest common subsequence
1MTW_1,4YGQ_1 170 3
1MTW_1,2IPA_1 162 3
4YGQ_1,2IPA_1 142 4

Newick tree

 
[
	1MTW_1:86.66,
	[
		2IPA_1:71,4YGQ_1:71
	]:15.66
]

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 \((0.85)(0.8)(\frac{452 }{\log_{20} 452}-\frac{223}{\log_{20}223})=66.5\)
Status Protein1 Protein2 d d1/2
Query variables 1MTW_1 4YGQ_1 86 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]