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Parikh vectors
5GIV_1 7LCQ_1 9EZL_1 Letter Amino acid
48 26 46 G Glycine
18 13 40 P Proline
63 17 41 A Alanine
43 12 39 R Arginine
27 14 17 Q Glutamine
14 8 20 H Histidine
11 12 18 I Isoleucine
26 24 34 V Valine
0 12 5 C Cysteine
37 9 34 E Glutamic acid
50 30 62 L Leucine
24 16 29 F Phenylalanine
23 16 26 S Serine
9 19 21 N Asparagine
33 17 45 D Aspartic acid
8 11 14 K Lycine
7 10 12 M Methionine
29 26 28 T Threonine
11 3 14 W Tryptophan
22 11 26 Y Tyrosine 

5GIV_1|Chains A, B, C, D, E, F|Carboxypeptidase 1|Deinococcus radiodurans str. R1 (243230)
>7LCQ_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus (694009)
>9EZL_1|Chains A, B, C, D, E, F, G, H|maltose alpha-D-glucosyltransferase|Deinococcus radiodurans (1299)
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 \(w=f(c)\)
5GIV , Knot 198 503 0.81 38 217 461 
MTTTRQDTQWQQLTEHWQELADFGGIEALLGWDQSTFLPAGAAEDRARQQSLLAGLRHARATDAGYGKLLDAASSRSDLSPEQARMVQVARQDFEKATRIPAEFVREFSGHVGQSYSAWTEARPANDFGRMVPYLEKTLDLSLQAASYFPEFGDPLDYYINESDEGMTAEQVGQVFAELRAALVPLADAVIAAGAPRTDFLGRGFAQERQLAFGERVIRDYGYDFRRGRQDLTHHPFMTRLGGHDVRITTRVKEQDPTDALYSTLHEAGHALYEQGVDAAFLGTPLGGGVSAGVHESQSRLWENLVGRSRAFWAAYFGDWRDTFPEQLAGVTEEEMYRAVNTVSRSLIRTDADELTYNLHVITRFELEREMLAGKLAVRDLADAWHAAYEQNLGLRAPSDVDGALQDVHWYFGPIGGSFQGYTIGNVLSAQFYAAAEAANPGLEADFARKDFSRLHGWLRENVYRHGRRWTPGELIERATGQALTAGPYLKYLRGKYGELYGV
7LCQ , Knot 133 306 0.83 40 193 288 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDTVYCPRHVICTAEDMLNPNYEDLLIRKSNHSFLVQAGNVQLRVIGHSMQNCLLRLKVDTSNPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNHTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGKFYGPFVDRQTAQAAGTDTTITLNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCAALKELLQNGMNGRTILGSTILEDEFTPFDVVRQCSGVTFQ
9EZL , Knot 235 571 0.87 40 260 531 
MVPTQAHPEWYKSAVFYELSVRTFQDGNGDGKGDFPGLTSRLDYLKNLGVDCLWLLPWFPSPLRDDGYDVADYRGIHPDLGTLDDFKVFLREAHARGLWVIGDLVTNHTSSDHPWFQAARRGPTLPDGSPNEYHDYYVWSDEGKEYADTRIIFTDTEVSNWTLDEQAGKYYWHRFFASQPDLNYDNPKVVEELHGAARFWLDLGLDGFRVDAVPYLIEREGTSCENLPETHEILKGFRAMVDREYPGRLLLAEAHQWPEEVVEYFGTEAEPEFHMCFNFPVMPRLYMSLKREDTSSIREIMGRLPKIPSFGQWCIFLRNHDELTLEMVTDDERAFMYAAYAPDARMKINVGIRRRLAPLLDNDRRRIELLNTVLLALPGSPVLYYGDEIGMGDDLGLPDRNGVRTPMQWNAGTSGGFSTAQPSDCFFPPIQDPVYGFGRVNVQSQLQDPSSLLKWTARQLELRRAHPAFAHGDLTFIETGNPAILAFTRQYDGETLLIVSNFAGNAQAGLLDLAPFVGRAPVTLSGASPLPVVTGNGQYPVVMGKYDYYWLRLNSRVDKLAAALEHHHHHH

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 \(c\).

\(|P_{f(5GIV_1)}(2) \setminus P_{f(7LCQ_1)}(2)|=92\), \(|P_{f(7LCQ_1)}(2) \setminus P_{f(5GIV_1)}(2)|=68\). 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 1:10000000010010001001101111011111000011111110001000011111001010011010110110000010100101101100010010011101100101011000011001011001101110100010101011001101101100010000011010011011101011111110111111110001110111000011110011000100100100010001110011100101000100001001100010011011000110111110111111011100000011001110001111101101000110011110000100110010001100010010001011001010001111011100110110110000111011001011100101011111101010011011010101110110111010110001001011100010001001011011001010110111010010100101011
Pair \(Z_2\) Length of longest common subsequence
5GIV_1,7LCQ_1 160 4
5GIV_1,9EZL_1 133 4
7LCQ_1,9EZL_1 169 3

Newick tree

 
[
	7LCQ_1:86.90,
	[
		5GIV_1:66.5,9EZL_1:66.5
	]:20.40
]

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 \((0.85)(0.8)(\frac{809 }{\log_{20} 809}-\frac{306}{\log_{20}306})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 5GIV_1 7LCQ_1 172 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]