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Parikh vectors
5SFM_1 4QFB_1 1ERD_1 Letter Amino acid
22 15 5 T Threonine
12 8 2 Y Tyrosine
16 14 0 V Valine
16 10 4 G Glycine
36 24 2 L Leucine
12 2 6 C Cysteine
21 10 2 I Isoleucine
18 11 0 K Lycine
12 4 4 M Methionine
14 8 0 N Asparagine
15 8 2 D Aspartic acid
24 17 1 S Serine
13 3 1 H Histidine
14 14 4 P Proline
17 6 2 Q Glutamine
24 12 3 E Glutamic acid
15 12 0 F Phenylalanine
7 2 0 W Tryptophan
21 15 2 A Alanine
14 6 0 R Arginine 

5SFM_1|Chains A, B, C, D|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|Homo sapiens (9606)
>4QFB_1|Chains A, B|CT263|Chlamydia trachomatis (471472)
>1ERD_1|Chain A|PHEROMONE ER-2|Euplotes raikovi (5938)
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 \(w=f(c)\)
5SFM , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED
4QFB , Knot 94 201 0.82 40 144 195 
GSTGSMFKLLLIFADPAEAARTLSLFPFSLNKENFYTYHTENVLLDVMVLKTWGYRGVVQALSPPPSGYDLWINAGFAGAANPNIPLLKTYTITSVKELTPTTSVEEELEVTPIPRLPLAQLTSVRSPYRDGFHEHLQLVDMEGFFIAKQASLVACPCSMIKVSSDYTTREGQDFLKNNKVKLSQKLAEAIFPIYSSFIDV
1ERD , Knot 24 40 0.73 28 34 37 
DPMTCEQAMASCEHTMCGYCQGPLYMTCIGITTDPECGLP

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 \(c\).

\(|P_{f(5SFM_1)}(2) \setminus P_{f(4QFB_1)}(2)|=134\), \(|P_{f(4QFB_1)}(2) \setminus P_{f(5SFM_1)}(2)|=57\). 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 1:1001000001011101011101000101101011110011111110110000100010100100111010000001100010011011001011100000110010001111101000100011000010010011111000001000010001011010100110010000000110110011110011101100001001000101010000000011111101001001001111001010010101110100100111011111000000011010111001111100001001111001110100001001001101000101100101100111000
Pair \(Z_2\) Length of longest common subsequence
5SFM_1,4QFB_1 191 4
5SFM_1,1ERD_1 217 2
4QFB_1,1ERD_1 158 2

Newick tree

 
[
	5SFM_1:10.84,
	[
		4QFB_1:79,1ERD_1:79
	]:29.84
]

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 \((0.85)(0.8)(\frac{544 }{\log_{20} 544}-\frac{201}{\log_{20}201})=98.7\)
Status Protein1 Protein2 d d1/2
Query variables 5SFM_1 4QFB_1 129 100.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]