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Parikh vectors
5FDS_1 4GTU_1 8AIX_1 Letter Amino acid
7 8 17 T Threonine
4 9 11 N Asparagine
18 10 22 G Glycine
3 5 8 H Histidine
3 13 1 F Phenylalanine
6 12 23 S Serine
7 26 52 L Leucine
6 16 13 K Lycine
7 16 32 D Aspartic acid
2 4 0 C Cysteine
8 6 35 Q Glutamine
8 16 50 E Glutamic acid
13 12 21 I Isoleucine
9 12 90 A Alanine
7 11 3 P Proline
2 4 0 W Tryptophan
6 12 1 Y Tyrosine
2 11 53 R Arginine
6 6 6 M Methionine
7 8 19 V Valine 

5FDS_1|Chain A|Profilin-2|Hevea brasiliensis (3981)
>4GTU_1|Chains A, B, C, D, E, F, G, H|GLUTATHIONE S-TRANSFERASE|Homo sapiens (9606)
>8AIX_1|Chains A, B, D[auth E], E[auth F], F[auth G], G[auth H], H[auth K], I[auth L], M, N, P[auth Q], Q[auth R], R[auth U], S[auth V], T[auth W], U[auth X]|Crescentin|Caulobacter vibrioides (155892)
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 \(w=f(c)\)
5FDS , Knot 66 131 0.81 40 107 127 
MSWQAYVDDHLMCEIEGNHLSAAAIIGQDGSVWAQSANFPQFKSEEITGIMSDFHEPGTLAPTGLYIGGTKYMVIQGEPGAVIRGKKGPGGVTVKKTNQALIIGIYDEPMTPGQCNMIVERLGDYLIDQGY
4GTU , Knot 101 217 0.83 40 145 209 
SMTLGYWDIRGLAHAIRLLLEYTDSSYEEKKYTMGDAPDYDRSQWLNEKFKLGLDFPNLPYLIDGAHKITQSNAILCYIARKHNLCGETEEEKIRVDILENQAMDVSNQLARVCYSPDFEKLKPEYLEELPTMMQHFSQFLGKRPWFVGDKITFVDFLAYDVLDLHRIFEPNCLDAFPNLKDFISRFEGLEKISAYMKSSRFLPKPLYTRVAVWGNK
8AIX , Knot 167 457 0.74 36 162 375 
MRLLSKNSRETKNGKPTVLGDEARAEAMQHQIESTQAIGQRYETIHGGLDSIGRVMEHLKAIEPLIAEIRGPVSQEFEARRAEHAELIAVRANLDQAQRQIALIQAEEREVSARLAAAETALGESDARRQTQDAALEDNALEIDRLRNALLQSDLKVSSLDASLRDATARIEHLVQDVEGLRVQAQDIDARRGDAEAALARANQDNALLGEEAATLKKRVDQAGLDLARLSRIETDLEAQLAAERARVQAVENALAAHQADSGRTIRGLESQVEANRAEISALQTRLETATGRADKLEEMNGQISARLADSSAQQKAVERRAGDLNVALERALDRIRALEEEADGLRQRHAGVDTARATAIERADQLAKSAVAQEKALKRAEERAQQLRARLDAMQEAQDQVRRDHEAKIAELQATIERLTSEAALAEGALEAARRDRSRLQMALLGASDGDVAASA

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 \(c\).

\(|P_{f(5FDS_1)}(2) \setminus P_{f(4GTU_1)}(2)|=62\), \(|P_{f(4GTU_1)}(2) \setminus P_{f(5FDS_1)}(2)|=100\). 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 1:10101010001100101001011111100101110010110100001011100100110111011011100011101011111010011111010000011111100011011000111001100110010
Pair \(Z_2\) Length of longest common subsequence
5FDS_1,4GTU_1 162 4
5FDS_1,8AIX_1 163 3
4GTU_1,8AIX_1 161 4

Newick tree

 
[
	5FDS_1:81.49,
	[
		4GTU_1:80.5,8AIX_1:80.5
	]:0.99
]

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 \((0.85)(0.8)(\frac{348 }{\log_{20} 348}-\frac{131}{\log_{20}131})=66.3\)
Status Protein1 Protein2 d d1/2
Query variables 5FDS_1 4GTU_1 85 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]