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Parikh vectors
6KRL_1 3ZRM_1 4QYE_1 Letter Amino acid
51 24 17 A Alanine
17 22 22 K Lycine
11 2 6 W Tryptophan
24 28 18 P Proline
31 13 14 N Asparagine
43 19 20 G Glycine
27 20 17 I Isoleucine
33 37 28 L Leucine
29 17 9 F Phenylalanine
6 8 6 C Cysteine
6 7 15 H Histidine
58 23 12 S Serine
39 24 9 T Threonine
21 18 10 Y Tyrosine
39 28 22 V Valine
16 24 17 R Arginine
25 20 18 D Aspartic acid
21 15 7 Q Glutamine
38 18 24 E Glutamic acid
10 4 7 M Methionine 

6KRL_1|Chain A|Mating factor alpha,GH30 Xylanase B|Saccharomyces uvarum (230603)
>3ZRM_1|Chains A, B|GLYCOGEN SYNTHASE KINASE-3 BETA|HOMO SAPIENS (9606)
>4QYE_1|Chain A|Serine/threonine-protein kinase Chk1|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 \(w=f(c)\)
6KRL , Knot 219 545 0.84 40 253 515 
MRFPSIFTAVLFAASSALAAPVNTTTEDETAQIPAEAVIGYSDLEGDFDVAVLPFSNSTNNGLLFINTTIASIAAKEEGVSLEKREAEAYVEFQINVDLQARYQSVDGFGCSQAFQRAEDIFGKYGLSPKNQSYVLDLMYSEERGAGFTILRNGIGSSNSSTSNLMNSIEPFSPGSPSSTPNYTWDHYNSGQFPLSQQARARGLPYIYADAWSAPGYMKTNQDENWSGFLCGIEGETCPSGDWRQAYADYLVQYVKFYAESGVPVTHLGFLNEPQEVVSYASMGSNGTQAAEFVKILGQTLEREGIDIELTCCDGVGWSEQEAMIPGLQVVGPDGKSAEDYLSVVTGHGYSSAPTFPLSTKRRTWLTEWTDLSGAFTPYTFFADGGAGEGMTWANHIQTAFVNANVSAFIYWIGAENSTTNSGMINLINDEVIPSKRFWSMASFSKFVRPNAQRVKATSSDASVTVSAFENTNGVVAIQVINNGTSAASLTIDLGKTHKEVKKVVPWVTSNDYDLEEMSEIDVKHNSFLASVPARSLTSFVTECE
3ZRM , Knot 157 371 0.83 40 213 351 
FGSMKVSRDKDGSKVTTVVATPGQGPDRPQEVSYTDTKVIGNGSFGVVYQAKLCDSGELVAIKKVLQDKRFKNRELQIMRKLDHCNIVRLRYFFYSSGEKKDEVYLNLVLDYVPETVYRVARHYSRAKQTLPVIYVKLYMYQLFRSLAYIHSFGICHRDIKPQNLLLDPDTAVLKLCDFGSAKQLVRGEPNVSYICSRYYRAPELIFGATDYTSSIDVWSAGCVLAELLLGQPIFPGDSGVDQLVEIIKVLGTPTREQIREMNPNYTEFKFPQIKAHPWTKVFRPRTPPEAIALCSRLLEYTPTARLTPLEACAHSFFDELRDPNVKLPNGRDTPALFNFTTQELSSNPPLATILIPPHARIQAAASTPTN
4QYE , Knot 129 298 0.82 40 180 280 
MAVPFVEDWDLVQTLGEGAYGEVQLAVNRVTEEAVAVKIVDMKRAVDCPENIKKEICINKMLNHENVVKFYGHRREGNIQYLFLEYCSGGELFDRIEPDIGMPEPDAQRFFHQLMAGVVYLHGIGITHRDIKPENLLLDERDNLKISDFGLATVFRYNNRERLLNKMCGTLPYVAPELLKRREFHAEPVDVWSCGIVLTAMLAGELPWDQPSDSCQEYSDWKEKKTYLNPWKKIDSAPLALLHKILVENPSARITIPDIKKDRWYNKPLKKGAKRPRVTSGGVSESPSGGHHHHHHHH

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 \(c\).

\(|P_{f(6KRL_1)}(2) \setminus P_{f(3ZRM_1)}(2)|=102\), \(|P_{f(3ZRM_1)}(2) \setminus P_{f(6KRL_1)}(2)|=62\). 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 1:10110110111111001111110000000010111011110001010101111110000001111100011011100011010000101010101010101000010111000110010011100110100000110110000011110110011100000000110010110110100010001000001011100010101110101011011101000000010111011010001010100101001100101010011110011110010011001011001001101101110010001101010000111100001111110111101001000101101010001101110000001100100101110100111011110110110010011101010111011110000000111011000111000110110100110101001010000101010110000111110110010011010101100000100111110000001001001010000111011100100110000
Pair \(Z_2\) Length of longest common subsequence
6KRL_1,3ZRM_1 164 4
6KRL_1,4QYE_1 175 4
3ZRM_1,4QYE_1 171 6

Newick tree

 
[
	4QYE_1:87.95,
	[
		6KRL_1:82,3ZRM_1:82
	]:5.95
]

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 \((0.85)(0.8)(\frac{916 }{\log_{20} 916}-\frac{371}{\log_{20}371})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 6KRL_1 3ZRM_1 185 155
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]