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Parikh vectors
7UUX_1 1UFQ_1 1HKC_1 Letter Amino acid
12 13 51 A Alanine
20 13 59 R Arginine
37 15 62 E Glutamic acid
7 4 21 H Histidine
43 16 59 K Lycine
5 0 4 W Tryptophan
8 3 20 C Cysteine
12 19 27 Q Glutamine
40 23 91 L Leucine
18 21 71 V Valine
17 18 78 G Glycine
21 16 53 I Isoleucine
5 2 34 M Methionine
22 13 39 F Phenylalanine
25 16 49 S Serine
11 8 21 Y Tyrosine
15 10 35 N Asparagine
15 17 62 D Aspartic acid
16 11 21 P Proline
15 14 60 T Threonine 

7UUX_1|Chains A, B[auth C]|Cyclic GMP-AMP synthase|Mus musculus (10090)
>1UFQ_1|Chains A, B, C, D|Uridine-cytidine kinase 2|Homo sapiens (9606)
>1HKC_1|Chain A|D-GLUCOSE 6-PHOSPHOTRANSFERASE|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 \(w=f(c)\)
7UUX , Knot 156 364 0.84 40 200 344 
GTGPDKLKKVLDKLRLKRKDISEAAETVNKVVERLLRRMQKRESEFKGVEQLNTGSYYEHVKISAPNQFNVMFKLEVPRIELQEYYETGAFYLVKFKRIPRGNPLSHFLEGEVLSATKMLSKFRKIIKEEVKEIKDIDVSVEKEKPGSPAVTLLIRNPEEISVDIILALESKGSWPISTKEGLPIQGWLGTKVRTNLRREPFYLVPKNAKDGNSFQGETWRLSFSHTEKYILNNHGIEKTCCESSGAKCCRKECLKLMKYLLEQLKKEFQELDAFCSYHVKTAIFHMWTQDPQDSQWDPRNLSSCFDKLLAFFLECLRTEKLDHYFIPKFNLFSQELIDRKSKEFLSKKIEYERNNGFPIFDKL
1UFQ , Knot 113 252 0.82 38 166 240 
PGMAGDSEQTLQNHQQPNGGEPFLIGVSGGTASGKSSVCAKIVQLLGQNEVDYRQKQVVILSQDSFYRVLTSEQKAKALKGQFNFDHPDAFDNELILKTLKEITEGKTVQIPVYDFVSHSRKEETVTVYPADVVLFEGILAFYSQEVRDLFQMKLFVDTDADTRLSRRVLRDISERGRDLEQILSQYITFVKPAFEEFCLPTKKYADVIIPRGADNLVAINLIVQHIQDILNGGPSKRQTNGCLNGYTPSRK
1HKC , Knot 321 917 0.79 40 296 754 
MIAAQLLAYYFTELKDDQVKKIDKYLYAMRLSDETLIDIMTRFRKEMKNGLSRDFNPTATVKMLPTFVRSIPDGSEKGDFIALDLGGSSFRILRVQVNHEKNQNVHMESEVYDTPENIVHGSGSQLFDHVAECLGDFMEKRKIKDKKLPVGFTFSFPCQQSKIDEAILITWTKRFKASGVEGADVVKLLNKAIKKRGDYDANIVAVVNDTVGTMMTCGYDDQHCEVGLIIGTGTNACYMEELRHIDLVEGDEGRMCINTEWGAFGDDGSLEDIRTEFDREIDRGSLNPGKQLFEKMVSGMYLGELVRLILVKMAKEGLLFEGRITPELLTRGKFNTSDVSAIEKNKEGLHNAKEILTRLGVEPSDDDCVSVQHVCTIVSFRSANLVAATLGAILNRLRDNKGTPRLRTTVGVDGSLYKTHPQYSRRFHKTLRRLVPDSDVRFLLSESGSGKGAAMVTAVAYRLAEQHRQIEETLAHFHLTKDMLLEVKKRMRAEMELGLRKQTHNNAVVKMLPSFVRRTPDGTENGDFLALDLGGTNFRVLLVKIRSGKKRTVEMHNKIYAIPIEIMQGTGEELFDHIVSCISDFLDYMGIKGPRMPLGFTFSFPCQQTSLDAGILITWTKGFKATDCVGHDVVTLLRDAIKRREEFDLDVVAVVNDTVGTMMTCAYEEPTCEVGLIVGTGSNACYMEEMKNVEMVEGDQGQMCINMEWGAFGDNGCLDDIRTHYDRLVNEYSLNAGKQRYEKMISGMYLGEIVRNILIDFTKKGFLFRGQISETLKTRGIFETKFLSQIESDRLALLQVRAILQQLGLNSTCDDSILVKTVCGVVSRRAAQLCGAGMAAVVDKIRENRGLDRLNVTVGVDGTLYKLHPHFSRIMHQTVKELSPKCNVSFLLSEDGSGKGAALITAVGVRLRTEASS

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 \(c\).

\(|P_{f(7UUX_1)}(2) \setminus P_{f(1UFQ_1)}(2)|=99\), \(|P_{f(1UFQ_1)}(2) \setminus P_{f(7UUX_1)}(2)|=65\). 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 1:1011001001100101000010011001001100110010000001011001001000001010110010111010110101000000111011010011010110011010110100110010011000100100101010000110111011100100101011111000101110000111101111001000100011011100100100101001010100000011000110000000011000000010110011001000100101100001001110110001000010100100010011111100100001000111010110001100000011000100000011111001
Pair \(Z_2\) Length of longest common subsequence
7UUX_1,1UFQ_1 164 4
7UUX_1,1HKC_1 178 4
1UFQ_1,1HKC_1 172 4

Newick tree

 
[
	1HKC_1:89.27,
	[
		7UUX_1:82,1UFQ_1:82
	]:7.27
]

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 \((0.85)(0.8)(\frac{616 }{\log_{20} 616}-\frac{252}{\log_{20}252})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 7UUX_1 1UFQ_1 128 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]