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Parikh vectors
5ZRL_1 8OTK_1 9HNC_1 Letter Amino acid
14 10 16 R Arginine
1 6 12 N Asparagine
8 16 25 E Glutamic acid
16 18 33 G Glycine
3 10 20 I Isoleucine
1 2 5 Y Tyrosine
20 13 40 A Alanine
2 8 14 K Lycine
5 7 25 T Threonine
14 9 26 V Valine
2 8 9 Q Glutamine
10 5 6 H Histidine
3 4 9 M Methionine
0 3 6 F Phenylalanine
8 3 13 P Proline
10 9 16 S Serine
4 0 2 W Tryptophan
0 0 6 C Cysteine
18 22 24 L Leucine
11 2 15 D Aspartic acid 

5ZRL_1|Chains A, B|Putative mutator protein MutT2/NUDIX hydrolase|Mycobacterium smegmatis (strain ATCC 700084 / mc(2)155) (246196)
>8OTK_1|Chain A|ATP-dependent Clp protease ATP-binding subunit ClpC / Negative regulator of tic competence clcC/mecB|Bacillus subtilis (1423)
>9HNC_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD], E[auth EEE], F[auth FFF], G[auth GGG], H[auth HHH], I[auth III], J[auth JJJ], K[auth KKK], L[auth LLL], M[auth MMM], N[auth NNN], O[auth OOO], P[auth PPP]|beta-aspartyl-peptidase|Phaseolus vulgaris (3885)
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 \(w=f(c)\)
5ZRL , Knot 66 150 0.73 36 90 139 
MGSSHHHHHHSSGLVPRGSHMTKQIVVAGALISRGTLLVAQRDRPAELAGLWELPGGKVTPGESDADALARELREELGVDVAVGERLGADVALNDAMTLRAYRVTLRSGSPHPHDHRALRWVGADEIDGLAWVPADRAWVPDLVAALSGR
8OTK , Knot 68 155 0.73 36 105 139 
IDPFTMMFGRFTERAPKVLALAQEEALRLGHNNIGTEHILLGLVREGEGIAAKALQALGLGSEKIQKEVESLIGRGQEMSQTIHYTPRAKKVIELSMDEARKLGHSYVGTEHILLGLIREGEGVAARVLNNLGVSLNKARQQVLQLLGSNETGSS
9HNC , Knot 135 322 0.80 40 187 301 
MGWAIALHGGAGDIPLSLPPERRHPREEALRHCLQIGVEALKAKLPPLDVVERVVRELENIPQFNAGKGSVLTSNGTVEMEASIMDGTTMDCGAVSGLTTVVNAISLARLVMEKTPHIYLAFDGAEEFARQQGVETLDSSHFITAENIERLKQAKEANRVQIDYTQPIQNDTKKETAIANGDSQIGTVGCVAVDGNGNLASATSTGGLVNKMVGRIGDTPLIGAGTYADARCAVSATGKGEAIIRGTVARDVAALMEFKGLSLEEAATCVVHERTPKGTLGLIAVSAKGEVAMPYNTTGMFRACATEDGYSEVAIWPSAKID

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 \(c\).

\(|P_{f(5ZRL_1)}(2) \setminus P_{f(8OTK_1)}(2)|=54\), \(|P_{f(8OTK_1)}(2) \setminus P_{f(5ZRL_1)}(2)|=69\). 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 1:110000000000111101001000111111110010111100001101111101111010110001011100100011101111001110111001101010010100101010000110111100101111111001111011111010
Pair \(Z_2\) Length of longest common subsequence
5ZRL_1,8OTK_1 123 3
5ZRL_1,9HNC_1 167 4
8OTK_1,9HNC_1 156 5

Newick tree

 
[
	9HNC_1:86.27,
	[
		5ZRL_1:61.5,8OTK_1:61.5
	]:24.77
]

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 \((0.85)(0.8)(\frac{305 }{\log_{20} 305}-\frac{150}{\log_{20}150})=47.6\)
Status Protein1 Protein2 d d1/2
Query variables 5ZRL_1 8OTK_1 57 56
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]