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Parikh vectors
4CIZ_1 9EMD_1 4DAT_1 Letter Amino acid
24 9 5 F Phenylalanine
10 8 10 Y Tyrosine
20 30 13 R Arginine
11 6 7 N Asparagine
19 33 13 G Glycine
30 50 24 L Leucine
23 28 23 A Alanine
5 10 3 H Histidine
15 21 19 S Serine
15 14 10 T Threonine
2 7 2 W Tryptophan
20 28 12 V Valine
14 16 14 D Aspartic acid
22 19 7 Q Glutamine
7 2 9 M Methionine
13 21 5 P Proline
4 4 2 C Cysteine
35 25 29 E Glutamic acid
13 11 10 I Isoleucine
15 10 17 K Lycine 

4CIZ_1|Chain A|RETINALDEHYDE-BINDING PROTEIN 1|HOMO SAPIENS (9606)
>9EMD_1|Chain A|Probable N-acetyltransferase 16|Homo sapiens (9606)
>4DAT_1|Chain A|14-3-3 protein sigma|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 \(w=f(c)\)
4CIZ , Knot 135 317 0.81 40 179 296 
MSEGVGTFRMVPEEEQELRAQLEQLTTKDHGPVFGPCSQLPRHTLQKAKDELNEREETREEAVRELQEMVQAQAASGEELAVAVAERVQEKDSGFFLRFIRARKFNVGRAYELLRGYVNFRLQYPELFDSLSPEAVRCTIEAGYPGVLSSRDKYGRVVMLFNIENWQSQEITFDEILQAYCFILEKLLENEETQINGFCIIENFKGFTMQQAASLRTSDLRKMVDMLQDSFPARFKAIHFIHQPWYFTTTYNVVKPFLKSKLLERVFVHGDDLSGFYQEIDENILPSDFGGTLPKYDGKAVAEQLFGPQAQAENTAF
9EMD , Knot 143 352 0.79 40 188 325 
MKLESSETRPQEVEAEPRSGSGPEAEAEPLDFVVATEREFEEVLAISGGIYGGLDYLPSRYHSWLRDPDRTVVLAKRNGGVIALESVNVIDAGETVLVEGLRVAPWERGKGVAGLLQRFCSQLVKRQHPGVKVARLTRDDQLGPRELKKYRLITKQGILLVRFNASALLAGLGARLAALRTSGTFSPLPTEAVSEAGGDVARLLLSPSVQRDVLPGGTIIQDWQPYRPSESNLRLLAAKGLEWRVDSRARPRVLTLCTRPFPIPHGGDGTWRYLNIDAFGSDGAQVQSQLLWHLQRQAPRLVGLNVMCQLFLEPQLWSQLADFCQVGLGLELVKGYTEQYLLEADIHHHHHH
4DAT , Knot 108 234 0.84 40 164 227 
MGSMERASLIQKAKLAEQAERYEDMAAFMKGAVEKGEELSCEERNLLSVAYKNVVGGQRAAWRVLSSIEQKSNEEGSEEKGPEVREYREKVETELQGVCDTVLGLLDSHLIKEAGDAESRVFYLKMKGDYYRYLAEVATGDDKKRIIDSARSAYQEAMDISKKEMPPTNPIRLGLALNFSVFHYEIANSPEEAISLAKTTFDEAMADLHTLSEDSYKDSTLIMQLLRDNLTLWT

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 \(c\).

\(|P_{f(4CIZ_1)}(2) \setminus P_{f(9EMD_1)}(2)|=72\), \(|P_{f(9EMD_1)}(2) \setminus P_{f(4CIZ_1)}(2)|=81\). 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 1:10011101011100000101010010000011111100011000100100010000000001100100110101101001111110010000011110110100101101001101010101001011001010110001011011110000001011111010010000101001101001110011000000101101100101101001101000010011011000111010110110011010000011011100011001110100101100010001110011101100010111001111010100011
Pair \(Z_2\) Length of longest common subsequence
4CIZ_1,9EMD_1 153 4
4CIZ_1,4DAT_1 177 4
9EMD_1,4DAT_1 158 3

Newick tree

 
[
	4DAT_1:86.20,
	[
		4CIZ_1:76.5,9EMD_1:76.5
	]:9.70
]

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 \((0.85)(0.8)(\frac{669 }{\log_{20} 669}-\frac{317}{\log_{20}317})=97.3\)
Status Protein1 Protein2 d d1/2
Query variables 4CIZ_1 9EMD_1 121 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]