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Parikh vectors
4YAB_1 2FUH_1 6DTS_1 Letter Amino acid
18 3 26 E Glutamic acid
3 3 14 M Methionine
14 15 22 P Proline
11 11 16 S Serine
7 8 13 T Threonine
4 5 12 Q Glutamine
20 12 26 L Leucine
10 6 17 F Phenylalanine
2 3 4 W Tryptophan
5 8 5 R Arginine
10 7 15 N Asparagine
7 12 30 I Isoleucine
16 8 32 K Lycine
8 6 18 Y Tyrosine
7 9 37 A Alanine
12 12 23 D Aspartic acid
12 4 1 C Cysteine
5 6 35 G Glycine
4 3 7 H Histidine
9 5 30 V Valine 

4YAB_1|Chains A, B|Transcription intermediary factor 1-alpha|Homo sapiens (9606)
>2FUH_1|Chain A|Ubiquitin-conjugating enzyme E2 D3|Homo sapiens (9606)
>6DTS_1|Chains A, B|maltose-binding protein MalE2|Thermotoga maritima (243274)
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 \(w=f(c)\)
4YAB , Knot 92 184 0.87 40 140 178 
SPNEDWCAVCQNGGELLCCEKCPKVFHLSCHVPTLTNFPSGEWICTFCRDLSKPEVEYDCDAPSHNSEKKKTEGLVKLTPIDKRKCERLLLFLYCHEMSLAFQDPVPLTVPDYYKIIKNPMDLSTIKKRLQEDYSMYSKPEDFVADFRLIFQNCAEFNEPDSEVANAGIKLENYFEELLKNLYP
2FUH , Knot 75 146 0.85 40 116 141 
ALKRINKELSDLARDPPAQCSAGPVGDDMFHWQATIMGPNDSPYQGGVFFLTIHFPTDYPFKPPKVAFTTRIYHPNINSNGSICLDILRSQWSPALTISKVLLSICSLLCDPNPDDPLVPEIARIYKTDRDKYNRISREWTQKYAM
6DTS , Knot 158 383 0.81 40 191 351 
MQTKLTIWCSEKQVDILQKLGEEFKAKYGIPVEVQYVDFGSIKSKFLTAAPQGQGADIIVGAHDWVGELAVNGLIEPIPNFSDLKNFYDTALKAFSYGGKLYGVPYAMEAVALIYNKDYVDSVPKTMDELIEKAKQIDEEYGGEVRGFIYDVANFYFSAPFILGYGGYVFKETPQGLDVTDIGLANEGAVKGAKLIKRMIDEGVLTPGDNYGTMDSMFKEGLAAMIINGLWAIKSYKDAGINYGVAPIPELEPGVPAKPFVGVQGFMINAKSPNKVIAMEFLTNFIARKETMYKIYLADPRLPARKDVLELVKDNPDVVAFTQSASMGTPMPNVPEMAPVWSAMGDALSIIINGQASVEDALKEAVEKIKAQIEKGSHHHHHH

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 \(c\).

\(|P_{f(4YAB_1)}(2) \setminus P_{f(2FUH_1)}(2)|=94\), \(|P_{f(2FUH_1)}(2) \setminus P_{f(4YAB_1)}(2)|=70\). 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 1:0100010110001101100000101101000110100110101100100010010100000110000000000111010110000000111110000101110011110110000110011010010001000001000100111010111000101001000110111010001001100101
Pair \(Z_2\) Length of longest common subsequence
4YAB_1,2FUH_1 164 3
4YAB_1,6DTS_1 167 4
2FUH_1,6DTS_1 169 4

Newick tree

 
[
	6DTS_1:84.65,
	[
		4YAB_1:82,2FUH_1:82
	]:2.65
]

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 \((0.85)(0.8)(\frac{330 }{\log_{20} 330}-\frac{146}{\log_{20}146})=56.2\)
Status Protein1 Protein2 d d1/2
Query variables 4YAB_1 2FUH_1 71 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]