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Parikh vectors
2CYE_1 6UWM_1 7FYX_1 Letter Amino acid
17 15 6 R Arginine
13 22 11 G Glycine
8 12 1 P Proline
2 11 12 T Threonine
17 40 5 L Leucine
4 9 5 M Methionine
5 21 10 S Serine
3 14 2 Y Tyrosine
0 0 2 C Cysteine
16 31 17 V Valine
5 21 8 I Isoleucine
1 11 14 K Lycine
9 26 6 A Alanine
3 5 4 N Asparagine
6 9 11 D Aspartic acid
1 2 2 Q Glutamine
12 16 9 E Glutamic acid
3 9 2 H Histidine
6 10 6 F Phenylalanine
2 3 2 W Tryptophan 

2CYE_1|Chains A, B, C, D|Putative thioesterase|Thermus thermophilus (300852)
>6UWM_1|Chains A, B, C, D|Voltage-gated potassium channel|Aeropyrum pernix (56636)
>7FYX_1|Chain A|Fatty acid-binding protein, adipocyte|Homo sapiens (9606)
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 \(w=f(c)\)
2CYE , Knot 63 133 0.77 38 94 124 
MEGFPVRVRVDVRFRDLDPLGHVNNAVFLSYMELARIRYFQRISPDWLEEGHFVVARMEVDYLRPILLGDEVFVGVRTVGLGRSSLRMEHLVTANGESAAKGLGVLVWLEGGRPAPLPEAIRERIRALEGRPL
6UWM , Knot 119 287 0.78 38 162 265 
MAGGRVRNIGDVMEHPLVELGVSYAALLSVIVVVVEYTMQLSGEYLVRLYLVDLILVIILWADYAYRAYKSGDPAGYVKKTLYEIPALVPAGLLALIEGHLAGLGLFRLVRLLRFLRILLIISRGSKFLSAIADAADKIRFYHLFGAVMLTVLYGAFAIYIVEYPDPNSSIKSVFDALWWAVVTATTVGYGDVVPATPIGKVIGIAVMLTGISALTLLIGTVSNMFQKILVGEPEPSSSPAKLAEMVSSMSEEEFEEFVRTLKNLRRLENSMKLVPRGSRSHHHHHH
7FYX , Knot 69 135 0.83 40 104 129 
GSHMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA

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 \(c\).

\(|P_{f(2CYE_1)}(2) \setminus P_{f(6UWM_1)}(2)|=33\), \(|P_{f(6UWM_1)}(2) \setminus P_{f(2CYE_1)}(2)|=101\). 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 1:1011110101010100101110100111100101101001001010110010111101010010111110011111001111000101001101010011011111111011011111011000101101011
Pair \(Z_2\) Length of longest common subsequence
2CYE_1,6UWM_1 134 3
2CYE_1,7FYX_1 134 4
6UWM_1,7FYX_1 156 3

Newick tree

 
[
	7FYX_1:74.51,
	[
		2CYE_1:67,6UWM_1:67
	]:7.51
]

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 \((0.85)(0.8)(\frac{420 }{\log_{20} 420}-\frac{133}{\log_{20}133})=86.2\)
Status Protein1 Protein2 d d1/2
Query variables 2CYE_1 6UWM_1 101 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]