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Parikh vectors
5DDS_1 8AHO_1 3ZGC_1 Letter Amino acid
27 15 11 I Isoleucine
11 10 2 K Lycine
9 7 9 M Methionine
20 20 14 P Proline
10 11 17 Y Tyrosine
12 21 15 N Asparagine
35 24 16 D Aspartic acid
3 1 8 C Cysteine
3 10 6 W Tryptophan
13 9 6 Q Glutamine
16 8 16 H Histidine
58 18 22 L Leucine
54 31 22 A Alanine
30 21 24 S Serine
31 32 18 T Threonine
19 8 4 F Phenylalanine
40 41 28 V Valine
45 16 23 R Arginine
40 13 11 E Glutamic acid
47 39 38 G Glycine 

5DDS_1|Chains A, B, C, D|CrmG|Actinoalloteichus sp. WH1-2216-6 (1074250)
>8AHO_1|Chains A, B|Transpeptidase,Lipoprotein LppS|Mycobacterium tuberculosis (1773)
>3ZGC_1|Chains A, B|KELCH-LIKE ECH-ASSOCIATED PROTEIN 1|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 \(w=f(c)\)
5DDS , Knot 209 523 0.83 40 225 488 
MTHPSGEPVYADAVLNGWLTSMGLGVEYVRAEGNTVYYLDDEGREVPVLDHACGFGSLIFGHNHPEIIAHAKAALDAGTVVHAQLSRQPRANQISRILNDIMRRETGRDDRYNAIFANSGAEANEICMKHAELERQERITALFAEIDAELDTAREALTTGTATLDTASLPLVGGGAGDVDGVIADIHRHNDERRAERPLFLTLDGSFHGKLVGSIQLTQNEPWRTPFTALSSPARFLPADEPELIGKIVEDERRSVLTLSLDKDTVRVVERDFPVVAAIFVEPVRGGSGMKTVTPELAEELHRLRDTLGCPLVVDEVQTGIGRTGAFFGSALLGIRGDYYTLAKAIGGGIVKNSVALIRQDRFLPAMEVIHSSTFAKDGLSASIALKVLEMVEADGGRVYQRVRERGQRLEAMLESVRADHSDVVSAVWGTGLMLALELRDQSNATSQAIREKAAHGFLGYVLAGFLLREHHIRVLPAGPRSGFLRFSPSLYITDEEIDRTETALRSLFTALRDQDGDRLVLS
8AHO , Knot 151 355 0.83 40 206 340 
QSDLLVPKLTASVTDGAVGVTVDAPVSVTAADGVLAAVTMVNDNGRPVAGRLSPDGLRWSTTEQLGYNRRYTLNATALGLGGAATRQLTFQTSSPAHLTMPYVMPGDGEVVGVGEPVAIRFDENIADRGAAEKAIKITTNPPVEGAFYWLNNREVRWRPEHFWKPGTAVDVAVNTYGVDLGEGMFGEDNVQTHFTIGDEVIATADDNTKILTVRVNGEVVKSMPTSMGKDSTPTANGIYIVGSRYKHIIMDSSTYGVPVNSPNGYRTDVDWATQISYSGVFVHSAPWSVGAQGHTNTSHGCLNVSPSNAQWFYDHVKRGDIVEVVNTVGGTLPGIDGLGDWNIPWDQWRAGNAKA
3ZGC , Knot 122 310 0.75 40 162 260 
MGSSHHHHHHSSGLVPRGSHMAPKVGRLIYTAGGYFRQSLSYLEAYNPSDGTWLRLADLQVPRSGLAGCVVGGLLYAVGGRNNSPDGNTDSSALDCYNPMTNQWSPCAPMSVPRNRIGVGVIDGHIYAVGGSHGCIHHNSVERYEPERDEWHLVAPMLTRRIGVGVAVLNRLLYAVGGFDGTNRLNSAECYYPERNEWRMITAMNTIRSGAGVCVLHNCIYAAGGYDGQDQLNSVERYDVATATWTFVAPMKHRRSALGITVHQGRIYVLGGYDGHTFLDSVECYDPDTDTWSEVTRMTSGRSGVGVAVT

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 \(c\).

\(|P_{f(5DDS_1)}(2) \setminus P_{f(8AHO_1)}(2)|=82\), \(|P_{f(8AHO_1)}(2) \setminus P_{f(5DDS_1)}(2)|=63\). 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 1:1001010110101110111001111100101010010010001001111001011101111000101110101110110110101000101001001100110000100000011110011010010100101000001011110101010010011001010100101111111110101111010000000010011110101010101110101000011001101100110111100101110110000001101010000101100011111111101101101100101011001001000110111100100111001111101111101000011011111110001111000011111011000011001101011101101101011010001000100101110010100001101111011111101000001000110001101111011111110000101111110011101010101000010000011001101100001001110
Pair \(Z_2\) Length of longest common subsequence
5DDS_1,8AHO_1 145 4
5DDS_1,3ZGC_1 167 4
8AHO_1,3ZGC_1 168 4

Newick tree

 
[
	3ZGC_1:87.17,
	[
		5DDS_1:72.5,8AHO_1:72.5
	]:14.67
]

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 \((0.85)(0.8)(\frac{878 }{\log_{20} 878}-\frac{355}{\log_{20}355})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 5DDS_1 8AHO_1 173 147
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]