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Parikh vectors
4FHC_1 8XYX_1 1ADB_1 Letter Amino acid
25 44 30 K Lycine
16 13 20 P Proline
15 29 26 S Serine
17 27 17 D Aspartic acid
17 6 7 H Histidine
28 28 21 E Glutamic acid
20 27 24 I Isoleucine
7 8 9 M Methionine
21 12 18 F Phenylalanine
26 11 4 Y Tyrosine
18 17 39 V Valine
8 27 8 N Asparagine
4 3 14 C Cysteine
12 30 8 Q Glutamine
26 21 38 G Glycine
28 28 25 L Leucine
24 7 24 T Threonine
2 5 2 W Tryptophan
25 21 28 A Alanine
29 14 12 R Arginine 

4FHC_1|Chain A|Spore photoproduct lyase|Geobacillus thermodenitrificans (420246)
>8XYX_1|Chain A|MT-a70 family protein|Tetrahymena thermophila SB210 (312017)
>1ADB_1|Chains A, B|ALCOHOL DEHYDROGENASE|Equus caballus (9796)
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 \(w=f(c)\)
4FHC , Knot 153 368 0.81 40 215 351 
MKHHHHHHPMSDYDIPTTENLYFQGAMGKPFVPKLVYFEPEALSYPLGKELYEKFTQMGIKIRETTSHNQVRGIPGETELARYRNAKSTLVVGVRRTLKFDSSKPSAEYAIPLATGCMGHCHYCYLQTTLGSKPYIRVYVNLDDIFAQAQKYINERAPEITRFEAACTSDIVGIDHLTHSLKKAIEFIGATDYGRLRFVTKYEHVDHLLDARHNGKTRFRFSINSRYVINHFEPGTSSFDGRLAAARKVAGAGYKLGFVVAPIYRHEGWERGYFELFQELARQLEGMDLSDLTFELIQHRFTKPAKRVIEQRYPKTRLDLDETKRKYKWGRYGIGKYVYRDEEAKELEDTMRRYIEQFFPGAYVQYFT
8XYX , Knot 160 378 0.83 40 207 363 
GPEFKLMSKAVNKKGLRPRKSDSILDHIKNKLDQEFLEDNENGEQSDEDYDQKSLNKAKKPYKKRQTQNGSELVISQQKTKAKASANNKKSAKNSQKLDEEEKIVEEEDLSPQKNGAVSEDDQQQEASTQEDDYLDRLPKSKKGLQGLLQDIEKRILHYKQLFFKEQNEIANGKRSMVPDNSIPICSDVTKLNFQALIDAQMRHAGKMFDVIMMAPPWQLSSSQPSRGVAIAYDSLSDEKIQNMPIQSLQQDGFIFVWAINAKYRVTIKMIENWGYKLVDEITWVKKTVNGKIAKGHGFYLQHAKESCLIGVKGDVDNGRFKKNIASDVIFSERRGQSQKPEEIYQYINQLCPNGNYLEIFARRNNLHDNWVSIGNEL
1ADB , Knot 158 374 0.83 40 210 358 
STAGKVIKCKAAVLWEEKKPFSIEEVEVAPPKAHEVRIKMVATGICRSDDHVVSGTLVTPLPVIAGHEAAGIVESIGEGVTTVRPGDKVIPLFTPQCGKCRVCKHPEGNFCLKNDLSMPRGTMQDGTSRFTCRGKPIHHFLGTSTFSQYTVVDEISVAKIDAASPLEKVCLIGCGFSTGYGSAVKVAKVTQGSTCAVFGLGGVGLSVIMGCKAAGAARIIGVDINKDKFAKAKEVGATECVNPQDYKKPIQEVLTEMSNGGVDFSFEVIGRLDTMVTALSCCQEAYGVSVIVGVPPDSQNLSMNPMLLLSGRTWKGAIFGGFKSKDSVPKLVADFMAKKFALDPLITHVLPFEKINEGFDLLRSGESIRTILTF

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 \(c\).

\(|P_{f(4FHC_1)}(2) \setminus P_{f(8XYX_1)}(2)|=88\), \(|P_{f(8XYX_1)}(2) \setminus P_{f(4FHC_1)}(2)|=80\). 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 1:10000000110000110000101011110111101101010110011100100010011101000000001011110001100001000111110001010000101001111101011000000100011001010101010011101000100011010010110000111100100010011011110001010110000010011010001000101010000110010110001010111100111110011111111000011001010110011001011010010101100010011001100001000101000000001100111001000001001000100010011111010010
Pair \(Z_2\) Length of longest common subsequence
4FHC_1,8XYX_1 168 4
4FHC_1,1ADB_1 173 5
8XYX_1,1ADB_1 163 4

Newick tree

 
[
	4FHC_1:86.47,
	[
		8XYX_1:81.5,1ADB_1:81.5
	]:4.97
]

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 \((0.85)(0.8)(\frac{746 }{\log_{20} 746}-\frac{368}{\log_{20}368})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 4FHC_1 8XYX_1 133 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]