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Parikh vectors
8HWA_1 3BNY_1 1OWF_1 Letter Amino acid
60 16 0 D Aspartic acid
59 14 0 K Lycine
35 15 0 P Proline
53 24 0 S Serine
32 24 10 A Alanine
46 31 0 E Glutamic acid
64 12 0 I Isoleucine
15 10 0 M Methionine
46 13 0 F Phenylalanine
51 8 0 N Asparagine
4 4 0 W Tryptophan
44 22 0 V Valine
29 14 0 Y Tyrosine
43 22 0 R Arginine
19 6 6 C Cysteine
14 9 0 Q Glutamine
28 17 8 G Glycine
17 7 0 H Histidine
72 39 0 L Leucine
54 13 11 T Threonine 

8HWA_1|Chains A, B, C, D, E, F, H[auth K]|Primase D5|Monkeypox virus (10244)
>3BNY_1|Chains A, B, C, D|Aristolochene synthase|Aspergillus terreus (33178)
>1OWF_1|Chain A[auth C]|Phage lambda H' site|
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 \(w=f(c)\)
8HWA , Knot 301 785 0.85 40 291 726 
MDAAIRGNDVIFVLKTIGVPSACRQNEDPRFVEAFKCDELERYIDNNPECTLFESLRDEEAYSIVRIFMDVDLDACLDEIDYLTAIQDFIIEVSNCVARFAFTECGAIHENVIKSMRSNFSLTKSTNRDKTSFHIIFLDTYTTMDTLIAMKRTLLELSRSSENPLTRSIDTAVYRRKTTLRVVGTRKNPNCDTIHVMQPPHDNIEDYLFTYVDMNNNSYYFSLQRRLEDLVPDKLWEPGFISFEDAIKRVSKIFINSIINFNDLDENNFTTVPLVIDYVTPCALCKKRSHKHPHQLSLENGAIRIYKTGNPHSCKVKIVPLDGNKLFNIAQRILDTNSVLLTERGDHIVWINNSWKFNSEEPLITKLILSIRHQLPKEYSSELLCPRKRKTVEANIRDMLVDSVETDTYPDKLPFKNGVLDLVDGMFYSGDDAKKYTCTVSTGFKFDDTKFVEDSPEMEELMNIINDIQPLTDENKKNRELYEKTLSSCLCGATKGCLTFFFGETATGKSTTKRLLKSAIGDLFVETGQTILTDVLDKGPNPFIANMHLKRSVFCSELPDFACSGSKKIRSDNIKKLTEPCVIGRPCFSNKINNRNHATIIIDTNYKPVFDRIDNALMRRIAVVRFRTHFSQPSGREAAENNDAYDKVKLLDEGLDGKIQNNRYRFAFLYLLVKWYKKYHIPIMKLYPTPEEIPDFAFYLKIGTLLVSSSVKHIPLMTDLSKKGYILYDNVVTLPLTTFQQKISKYFNSRLFGHDIESFINRHKKFANVSDEYLQYIFIEDISSP
3BNY , Knot 138 320 0.83 40 197 306 
MKKPNGTNGASSSLEPPPSTFQPLCHPLVEEVSKEVDGYFLQHWNFPNEKARKKFVAAGFSRVTCLYFPKALDDRIHFACRLLTVLFLIDDLLEYMSFEEGSAYNEKLIPISRGDVLPDRSIPVEYIIYDLWESMRAHDREMADEILEPVFLFMRAQTDRTRARPMGLGGYLEYRERDVGKELLAALMRFSMGLKLSPSELQRVREIDANCSKHLSVVNDIYSYEKELYTSKTAHSEGGILCTSVQILAQEADVTAEAAKRVLFVMCREWELRHQLLVARLSAEGLETPGLAAYVEGLEYQMSGNELWSQTTLRYSVVVD
1OWF , Knot 15 35 0.50 8 14 24 
CGGTGCAACAAATTGATAAGCAATGCTTTTTTGGC

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 \(c\).

\(|P_{f(8HWA_1)}(2) \setminus P_{f(3BNY_1)}(2)|=124\), \(|P_{f(3BNY_1)}(2) \setminus P_{f(8HWA_1)}(2)|=30\). 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 1:10111010011111001111010000001011011000010001000100011001000010011011101010101001001011001110100011011100011100011001000101000000000010111100000100111100011010000001100010011000000101110000100001011011000100011001010000001010001001110011011110100110010011100110100100001001111100101011000000001001010011101000101000010111101001101100110000111000100111100010100001110011101000110000001101000001010100111001000001001110011101101110010010000001001101000011000101001101100101100000000010000100010110010101111001010000001100111011100100110011001101111010100011000110110010001000010010010111010100010000010111000001110010011100111101000100101001100001000101100110101000000111101110100000111101010100110111010110111000100111100100010110001101110010001000100011100100110000011010000100111001001
Pair \(Z_2\) Length of longest common subsequence
8HWA_1,3BNY_1 154 4
8HWA_1,1OWF_1 281 3
3BNY_1,1OWF_1 193 3

Newick tree

 
[
	1OWF_1:13.87,
	[
		8HWA_1:77,3BNY_1:77
	]:54.87
]

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 \((0.85)(0.8)(\frac{1105 }{\log_{20} 1105}-\frac{320}{\log_{20}320})=208.\)
Status Protein1 Protein2 d d1/2
Query variables 8HWA_1 3BNY_1 266 184
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]