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Parikh vectors
3JWS_1 1BXB_1 4AID_1 Letter Amino acid
25 13 37 T Threonine
18 9 13 N Asparagine
25 37 52 E Glutamic acid
25 30 71 G Glycine
14 9 15 H Histidine
23 7 49 I Isoleucine
36 43 55 L Leucine
19 22 27 F Phenylalanine
13 6 4 W Tryptophan
11 1 2 C Cysteine
18 9 22 Q Glutamine
21 33 30 R Arginine
23 22 30 P Proline
24 24 63 V Valine
21 46 77 A Alanine
26 26 55 D Aspartic acid
27 17 54 K Lycine
11 8 24 M Methionine
26 9 36 S Serine
16 16 10 Y Tyrosine 

3JWS_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
>1BXB_1|Chains A, B, C, D|XYLOSE ISOMERASE|Thermus thermophilus (300852)
>4AID_1|Chains A, B, C|POLYRIBONUCLEOTIDE NUCLEOTIDYLTRANSFERASE|CAULOBACTER VIBRIOIDES (190650)
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 \(w=f(c)\)
3JWS , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVATKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
1BXB , Knot 161 387 0.82 40 194 352 
MYEPKPEHRFTFGLWTVGNVGRDPFGDAVRERLDPVYVVHKLAELGAYGVNLHDEDLIPRGTPPQERDQIVRRFKKALDETGLKVPMVTANLFSDPAFKDGAFTSPDPWVRAYALRKSLETMDLGAELGAEIYVVWPGREGAEVEATGKARKVWDWVREALNFMAAYAEDQGYGYRFALEPKPNEPRGDIYFATVGSMLAFIHTLDRPERFGLNPEFAHETMAGLNFVHAVAQALDAGKLFHIDLNDQRMSRFDQDLRFGSENLKAAFFLVDLLESSGYQGPRHFDAHALRTEDEEGVWAFARGCMRTYLILKERAEAFREDPEVKELLAAYYQEDPAALALLGPYSREKAEALKRAELPLEAKRRRGYALERLDQLAVEYLLGVRG
4AID , Knot 270 726 0.81 40 263 655 
MGSSHHHHHHSQDPMFDIKRKTIEWGGKTLVLETGRIARQADGAVLATMGETVVLATAVFAKSQKPGQDFFPLTVNYQEKTFAAGKIPGGFFKREGRPSEKETLVSRLIDRPIRPLFVKGFKNEVQVVVTVLQHDLENDPDILGMVAASAALCLSGAPFMGPIGAARVGWVDGAYVLNPTLDEMKESKMDLVVAGTADAVMMVESEIQELSEEIVLGGVNFAHQQMQAVIDAIIDLAEHAAKEPFAFEPEDTDAIKAKMKDLVGADIAAAYKIQKKQDRYEAVGAAKKKAIAALGLSDENPTGYDPLKLGAIFKELEADVVRRGILDTGLRIDGRDVKTVRPILGEVGILPRTHGSALFTRGETQAIVVATLGTGDDEQFIDALEGTYKESFLLHYNFPPYSVGETGRMGSPGRREIGHGKLAWRALRPMLPTKEDFPYTIRLVSEITESNGSSSMATVCGSSLAMMDAGVPLVRPVSGIAMGLILEQDGFAVLSDILGDEDHLGDMDFKVAGTSEGLTSLQMDIKIAGITPAIMEQALAQAKEGRAHILGEMNKAMDAPRADVGDFAPKIETINIPTDKIREVIGSGGKVIREIVATTGAKVDINDDGVVKVSASDGAKIKAAIDWIKSITDEAEVGKIYDGKVVKVVDFGAFVNFFGAKDGLVHVSQISNERVAKPSDVLKEGQMVKVKLLGFDDRGKTKLSMKVVDQETGEDLSKKEAAAEEA

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 \(c\).

\(|P_{f(3JWS_1)}(2) \setminus P_{f(1BXB_1)}(2)|=121\), \(|P_{f(1BXB_1)}(2) \setminus P_{f(3JWS_1)}(2)|=53\). 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 1:01011010010001110001010001001000010110111100000010011000011111001100000010011001010010010001000000010000110110011001000110101001011010000010111001000100100010100110111000010001011000110011000101001101101010010100110110101011111101010010110111011101110010101100111010111110011101111010101101101100111000000000011001100101010000011000111010111100100001011000010001100100000001101101111111101010111000110001010100010110001101
Pair \(Z_2\) Length of longest common subsequence
3JWS_1,1BXB_1 174 4
3JWS_1,4AID_1 145 4
1BXB_1,4AID_1 151 3

Newick tree

 
[
	1BXB_1:84.22,
	[
		3JWS_1:72.5,4AID_1:72.5
	]:11.72
]

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 \((0.85)(0.8)(\frac{809 }{\log_{20} 809}-\frac{387}{\log_{20}387})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 3JWS_1 1BXB_1 151 139.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]