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Parikh vectors
2FKZ_1 4DFX_1 3PZO_1 Letter Amino acid
8 23 18 A Alanine
3 15 18 R Arginine
5 6 11 M Methionine
2 13 15 P Proline
13 34 22 K Lycine
2 6 13 W Tryptophan
6 14 25 Y Tyrosine
13 19 25 D Aspartic acid
20 26 32 E Glutamic acid
10 22 39 G Glycine
7 8 16 H Histidine
14 21 23 I Isoleucine
2 3 2 C Cysteine
5 17 24 S Serine
2 15 10 T Threonine
2 20 24 V Valine
5 16 21 N Asparagine
7 15 3 Q Glutamine
27 32 25 L Leucine
2 25 17 F Phenylalanine 

2FKZ_1|Chains A, B, C, D, E, F, G, H|Bacterioferritin|Azotobacter vinelandii (354)
>4DFX_1|Chain A[auth E]|cAMP-dependent protein kinase catalytic subunit alpha|Mus musculus (10090)
>3PZO_1|Chain A|Mannan endo-1,4-beta-mannosidase. Glycosyl Hydrolase family 5|Thermotoga petrophila RKU-1 (390874)
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 \(w=f(c)\)
2FKZ , Knot 73 155 0.79 40 104 148 
MKGDKIVIQHLNKILGNELIAINQYFLHARMYEDWGLEKLGKHEYHESIDEMKHADKLIKRILFLEGLPNLQELGKLLIGEHTKEMLECDLKLEQAGLPDLKAAIAYCESVGDYASRELLEDILESEEDHIDWLETQLDLIDKIGLENYLQSQMD
4DFX , Knot 153 350 0.85 40 211 337 
GNAAAACKGSEQESVKEFLAKAKEDFLKKWETPSQNTAQLDQFDRIKTLGTGSFGRVMLVKHKESGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVAGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF
3PZO , Knot 160 383 0.82 40 218 363 
MGSSHHHHHHSSGLVPRGSHMLNGKEFRFIGSNNYYMHYKSNRMIDSVLESARDMGIKVLRIWGFLDGESYCRDKNTYMHPEPGVFGVPEGISNAQNGFERLDYTIAKAKELGIKLIIVLVNNWDDFGGMNQYVRWFGGTHHDDFYRDERIKEEYKKYVSFLINHVNVYTGVPYREEPTIMAWELANELRCETDKSGNTLVEWVKEMSSYIKSLDPNHLVAVGDEGFFSNYEGFKPYGGEAEWAYNGWSGVDWKKLLSIETVDFGTFHLYPSHWGVSPENYAQWGAKWIEDHIKIAKEIGKPVVLEEYGIPKSAPVNRTAIYRLWNDLVYDLGGDGAMFWMLAGIGEGSDRDERGYYPDYDGFRIVNDDSPEAELIREYAKLF

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 \(c\).

\(|P_{f(2FKZ_1)}(2) \setminus P_{f(4DFX_1)}(2)|=35\), \(|P_{f(4DFX_1)}(2) \setminus P_{f(2FKZ_1)}(2)|=142\). 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 1:10100111001001110011110001101010001110011000000010010010011001111011101001101111000001100010100111101011110000110010001100110000001011000101100111000100010
Pair \(Z_2\) Length of longest common subsequence
2FKZ_1,4DFX_1 177 4
2FKZ_1,3PZO_1 198 4
4DFX_1,3PZO_1 167 5

Newick tree

 
[
	2FKZ_1:97.11,
	[
		4DFX_1:83.5,3PZO_1:83.5
	]:13.61
]

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 \((0.85)(0.8)(\frac{505 }{\log_{20} 505}-\frac{155}{\log_{20}155})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 2FKZ_1 4DFX_1 131 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]