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Parikh vectors
7JPE_1 4IIB_1 2AHG_1 Letter Amino acid
22 54 13 N Asparagine
5 7 3 C Cysteine
20 89 38 G Glycine
5 15 8 H Histidine
7 20 13 W Tryptophan
18 31 18 I Isoleucine
32 65 33 L Leucine
12 11 3 M Methionine
11 28 14 F Phenylalanine
12 37 20 Y Tyrosine
9 32 35 R Arginine
18 51 33 D Aspartic acid
12 32 11 Q Glutamine
10 45 11 P Proline
24 59 21 S Serine
18 49 18 T Threonine
21 75 35 A Alanine
8 41 25 E Glutamic acid
18 33 10 K Lycine
19 67 15 V Valine 

7JPE_1|Chain A|2'-O-methyltransferase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4IIB_1|Chains A, B|Beta-glucosidase 1|Aspergillus aculeatus (5053)
>2AHG_1|Chains A, B|unsaturated glucuronyl hydrolase|Bacillus sp. (84635)
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 \(w=f(c)\)
7JPE , Knot 137 301 0.86 40 200 290 
SNASSQAWQPGVAMPNLYKMQRMLLEKCDLQNYGDSATLPKGIMMNVAKYTQLCQYLNTLTLAVPYNMRVIHFGAGSDKGVAPGTAVLRQWLPTGTLLVDSDLNDFVSDADSTLIGDCATVHTANKWDLIISDMYDPKTKNVTKENDSKEGFFTYICGFIQQKLALGGSVAIKITEHSWNADLYKLMGHFAWWTAFVTNVNASSSEAFLIGCNYLGKPREQIDGYVMHANYIFWRNTNPIQLSSYSLFDMSKFPLKLRGTAVMSLKEGQINDMILSLLSKGRLIIRENNRVVISSDVLVNN
4IIB , Knot 315 841 0.84 40 295 775 
DELAFSPPFYPSPWANGQGEWAEAYQRAVAIVSQMTLDEKVNLTTGTGWELEKCVGQTGGVPRLNIGGMCLQDSPLGIRDSDYNSAFPAGVNVAATWDKNLAYLRGQAMGQEFSDKGIDVQLGPAAGPLGRSPDGGRNWEGFSPDPALTGVLFAETIKGIQDAGVVATAKHYILNEQEHFRQVAEAAGYGFNISDTISSNVDDKTIHEMYLWPFADAVRAGVGAIMCSYNQINNSYGCQNSYTLNKLLKAELGFQGFVMSDWGAHHSGVGSALAGLDMSMPGDITFDSATSFWGTNLTIAVLNGTVPQWRVDDMAVRIMAAYYKVGRDRLYQPPNFSSWTRDEYGFKYFYPQEGPYEKVNHFVNVQRNHSEVIRKLGADSTVLLKNNNALPLTGKERKVAILGEDAGSNSYGANGCSDRGCDNGTLAMAWGSGTAEFPYLVTPEQAIQAEVLKHKGSVYAITDNWALSQVETLAKQASVSLVFVNSDAGEGYISVDGNEGDRNNLTLWKNGDNLIKAAANNCNNTIVVIHSVGPVLVDEWYDHPNVTAILWAGLPGQESGNSLADVLYGRVNPGAKSPFTWGKTREAYGDYLVRELNNGNGAPQDDFSEGVFIDYRGFDKRNETPIYEFGHGLSYTTFNYSGLHIQVLNASSNAQVATETGAAPTFGQVGNASDYVYPEGLTRISKFIYPWLNSTDLKASSGDPYYGVDTAEHVPEGATDGSPQPVLPAGGGSGGNPRLYDELIRVSVTVKNTGRVAGDAVPQLYVSLGGPNEPKVVLRKFDRLTLKPSEETVWTTTLTRRDLSNWDVAAQDWVITSYPKKVHVGSSSRQLPLHAALPKVQ
2AHG , Knot 156 377 0.81 40 211 351 
MWQQAIGDALGITARNLKKFGDRFPHVSDGSNKYVLNDNTDWTDGFWSGILWLCYEYTGDEQYREGAVRTVASFRERLDRFENLDHHNIGFLYSLSAKAQWIVEKDESARKLALDAADVLMRRWRADAGIIQAWGPKGDPENGGRIIIDCLLNLPLLLWAGEQTGDPEYRRVAEAHALKSRRFLVRGDDSSYHTFYFDPENGNAIRGGTHQGNTDGSTWTRGQAWGIYGFALNSRYLGNADLLETAKRMARHFLARVPEDGVVYWDFEVPQEPSSYRDSSASAITACGLLEIASQLDESDPERQRFIDAAKTTVTALRDGYAERDDGEAEGFIRRGSYHVRGGISPDDYTIWGDYYYLEALLRLERGVTGYWYERGR

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 \(c\).

\(|P_{f(7JPE_1)}(2) \setminus P_{f(4IIB_1)}(2)|=35\), \(|P_{f(4IIB_1)}(2) \setminus P_{f(7JPE_1)}(2)|=130\). 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 1:0010001101111110100100111000010001001011011110110000100010010111100101101111000111110111001110101110001001100100011100101001001011100100100001000000001110010111000111110111010000101010011101111011100101000011111000110100010101101001110000110100001101001110101011101001010011101100101110000011100011100
Pair \(Z_2\) Length of longest common subsequence
7JPE_1,4IIB_1 165 4
7JPE_1,2AHG_1 177 4
4IIB_1,2AHG_1 154 4

Newick tree

 
[
	7JPE_1:88.21,
	[
		4IIB_1:77,2AHG_1:77
	]:11.21
]

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 \((0.85)(0.8)(\frac{1142 }{\log_{20} 1142}-\frac{301}{\log_{20}301})=222.\)
Status Protein1 Protein2 d d1/2
Query variables 7JPE_1 4IIB_1 279 188
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]