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Parikh vectors
4GUT_1 9IME_1 6TJW_1 Letter Amino acid
42 12 15 P Proline
49 12 26 S Serine
31 3 8 Y Tyrosine
50 10 17 V Valine
60 30 17 A Alanine
25 2 9 C Cysteine
38 4 12 Q Glutamine
52 3 19 K Lycine
45 5 15 E Glutamic acid
16 1 7 M Methionine
43 10 28 T Threonine
34 8 15 R Arginine
42 6 20 I Isoleucine
65 10 25 L Leucine
33 3 8 F Phenylalanine
14 1 5 W Tryptophan
24 4 26 N Asparagine
39 8 10 D Aspartic acid
54 15 32 G Glycine
20 0 10 H Histidine 

4GUT_1|Chain A|Lysine-specific histone demethylase 1B|Homo sapiens (9606)
>9IME_1|Chains A, B|Pilin (Bacterial filament) subfamily|Burkholderia thailandensis E264 (271848)
>6TJW_1|Chains A, C, E|Hemagglutinin HA1|Influenza A virus (A/harbour seal/Germany/1/2014(H10N7)) (1572188)
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 \(w=f(c)\)
4GUT , Knot 299 776 0.85 40 312 715 
PLGSRKCEKAGCTATCPVCFASASERCAKNGYTSRWYHLSCGEHFCNECFDHYYRSHKDGYDKYTTWKKIWTSNGKTEPSPKAFMADQQLPYWVQCTKPECRKWRQLTKEIQLTPQIAKTYRCGMKPNTAIKPETSDHCSLPEDLRVLEVSNHWWYSMLILPPLLKDSVAAPLLSAYYPDCVGMSPSCTSTNRAAATGNASPGKLEHSKAALSVHVPGMNRYFQPFYQPNECGKALCVRPDVMELDELYEFPEYSRDPTMYLALRNLILALWYTNCKEALTPQKCIPHIIVRGLVRIRCVQEVERILYFMTRKGLINTGVLSVGADQYLLPKDYHNKSVIIIGAGPAGLAAARQLHNFGIKVTVLEAKDRIGGRVWDDKSFKGVTVGRGAQIVNGCINNPVALMCEQLGISMHKFGERCDLIQEGGRITDPTIDKRMDFHFNALLDVVSEWRKDKTQLQDVPLGEKIEEIYKAFIKESGIQFSELEGQVLQFHLSNLEYACGSNLHQVSARSWDHNEFFAQFAGDHTLLTPGYSVIIEKLAEGLDIQLKSPVQCIDYSGDEVQVTTTDGTGYSAQKVLVTVPLALLQKGAIQFNPPLSEKKMKAINSLGAGIIEKIALQFPYRFWDSKVQGADFFGHVPPSASKRGLFAVFYDMDPQKKHSVLMSVIAGEAVASVRTLDDKQVLQQCMATLRELFKEQEVPDPTKYFVTRWSTDPWIQMAYSFVKTGGSGEAYDIIAEDIQGTVFFAGEATNRHFPQTVTGAYLSGVREASKIAAF
9IME , Knot 69 147 0.78 38 105 141 
GSAYQDYLARSRVGEGLALAASARLAVAENAASGNGFSGGYVSPPATRNVESIRIDDDTGQIAIAFTARVAAAGANTLVLVPSVPDQADTPTARVALSKGVIQAGTITWECFAGDKASSSLPAPGAGPMPTDAPTLAGKLAPPECRA
6TJW , Knot 144 324 0.85 40 202 312 
DPDKICLGHHAVANGTIVKTLTNEQEEVTNATETVESTSLNRLCMKGRNHKDLGNCHPIGMLIGTPACDLHLTGTWDTLIERKNAIAYCYPGATVNEEALRQKIMESGGISKINTGFTYGSSINSAGTTKACMRNGGNSFYAELKWLVSKNKGQNFPQTTNTYRNADTAEHLIMWGIHHPSSTQEKNDLYGTQSLSISVGSSTYKNNFVPVVGARPQVNGLSRIDFHWTLVQPGDKITFSHNGGLIAPSRVSKLIGRGLGIQSEAPIDNSCESKCFWRGGSINTRLPFQNLSPRTVGQCPKYVNKKSLMLATGMRNVPELVQGR

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 \(c\).

\(|P_{f(4GUT_1)}(2) \setminus P_{f(9IME_1)}(2)|=222\), \(|P_{f(9IME_1)}(2) \setminus P_{f(4GUT_1)}(2)|=15\). 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 1:11100000011001001101101000010010000100100100100001000000000100000010011000100010101111000110110000100001001000101010110000011010011010000000110010110100011001111111100011111101001001110100000001110101011010000111010111100010110010001011010101101001001100000101011100111111000000110100011011101110100100100110110001110011101110001110000000111111111111110010011101011010001110110000101101101101101010011111000111010011000011001101001010001010101110110010000001001111001001001110001101001010110101001001010010010100100001110111000110110011100110110101001100100010010100001010010011101111110011101011100001011001111110011101100110001011011101110100011111100101000001110111101110100100001100011010011000011010001100100011101100110011010100111001010111110100001100101101011001001111
Pair \(Z_2\) Length of longest common subsequence
4GUT_1,9IME_1 237 4
4GUT_1,6TJW_1 174 4
9IME_1,6TJW_1 163 3

Newick tree

 
[
	4GUT_1:11.42,
	[
		6TJW_1:81.5,9IME_1:81.5
	]:28.92
]

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 \((0.85)(0.8)(\frac{923 }{\log_{20} 923}-\frac{147}{\log_{20}147})=215.\)
Status Protein1 Protein2 d d1/2
Query variables 4GUT_1 9IME_1 273 159.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]