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Parikh vectors
1KGU_1 5DCD_1 1GPZ_1 Letter Amino acid
32 21 14 S Serine
17 3 8 W Tryptophan
21 10 18 Y Tyrosine
42 12 22 N Asparagine
34 21 22 D Aspartic acid
25 36 28 L Leucine
22 20 21 K Lycine
23 14 22 P Proline
36 19 24 V Valine
28 33 20 A Alanine
27 19 22 R Arginine
14 11 26 Q Glutamine
28 24 20 I Isoleucine
21 17 21 T Threonine
19 24 23 E Glutamic acid
9 10 10 M Methionine
11 5 14 C Cysteine
50 29 37 G Glycine
11 17 11 H Histidine
26 6 16 F Phenylalanine 

1KGU_1|Chain A|ALPHA-AMYLASE, PANCREATIC|Homo sapiens (9606)
>5DCD_1|Chains A, B, C, D|Phospho-2-dehydro-3-deoxyheptonate aldolase|Neisseria meningitidis serogroup B (strain MC58) (122586)
>1GPZ_1|Chains A, B|COMPLEMENT C1R COMPONENT|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 \(w=f(c)\)
1KGU , Knot 212 496 0.88 40 259 473 
QYSPNTQQGRTSIVHLFEWRWVDIALECERYLAPKGFGGVQVSPPNENVAIYNPFRPWWERYQPVSYKLCTRSGNEDEFRNMVTRCNNVGVRIYVDAVINHMCGNAVSAGTSSTCGSYFNPGSRDFPAVPYSGWDFNDGKCKTGSGDIENYNDATQVRDCRLTGLLDLALEKDYVRSKIAEYMNHLIDIGVAGFRLDASKHMWPGDIKAILDKLHNLNSNWFPAGSKPFIYQEVIDLGGEPIKSSDYFGNGRVTEFKYGAKLGTVIRKWNGEKMSYLKNWGEGWGFVPSDRALVFVDNHDNQRGHGAGGASILTFWDARLYKMAVGFMLAHPYGFTAVMSSYRWPRQFQNGNDVNDWVGPPNNNGVIKEVTINPDTTCGNDWVCEHRWRQIRNMVIFRNVVDGQPFTNWYDNGSNQVAFGRGNRGFIVFNNDDWSFSLTLQTGLPAGTYCDVISGDKINGNCTGIKIYVSDDGKAHFSISNSAEDPFIAIHAESKL
5DCD , Knot 147 351 0.81 40 203 335 
MTHHYPTDDIKIKEVKELLPPIAHLYELPISKEASGLVHRTRQEISDLVHGRDKRLLVIIGPCSIHDPKAALEYAERLLKLRKQYENELLIVMRVYFEKPRTTVGWKGLINDPHLDGTFDINFGLRQARSLLLSLNNMGMPASTEFLDMITPQYYADLISWGAIGARTTESQVHRELASGLSCPVGFKNGTDGNLKIAIDAIGAASHSHHFLSVTKAGHSAIVHTGGNPDCHVILRGGKEPNYDAEHVSEAAEQLRAAGVTDKLMIDCSHANSRKDYTRQMEVAQDIAAQLEQDGGNIMGVMVESHLVEGRQDKPEVYGKSITDACIGWGATEELLALLAGANKKRMARAS
1GPZ , Knot 175 399 0.87 40 250 389 
IKCPQPKTLDEFTIIQNLQPQYQFRDYFIATCKQGYQLIEGNQVLHSFTAVCQDDGTWHRAMPRCKIKDCGQPRNLPNGDFRYTTTMGVNTYKARIQYYCHEPYYKMQTRAGSRESEQGVYTCTAQGIWKNEQKGEKIPRCLPVCGKPVNPVEQRQQIIGGQKAKMGNFPWQVFTNIHGRGGGALLGDRWILTAAHTLYPKEHEAQSNASLDVFLGHTNVEELMKLGNHPIRRVSVHPDYRQDESYNFEGDIALLELENSVTLGPNLLPICLPDNDTFYDLGLMGYVSGFGVMEEKIAHDLRFVRLPVANPQACENWLRGKNRMDVFSQNMFCAGHPSLKQDACQGDSGGVFAVRDPNTDRWVATGIVSWGIGCSRGYGFYTKVLNYVDWIKKEMEEED

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 \(c\).

\(|P_{f(1KGU_1)}(2) \setminus P_{f(5DCD_1)}(2)|=127\), \(|P_{f(5DCD_1)}(2) \setminus P_{f(1KGU_1)}(2)|=71\). 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 1:0001000010001101101011011100000111011111010110001110011011100001100010000100001001100000111010101110010101101100000100101100011111001101001000010101000001001000010111011100001000110010011011111101010001111010111001001000111110011100011011101100000110101001001101101100101001001001101111110001111100000001011111011011010100111111110101101110000110010010010011111000111001010100001001100001001001111001101011001000100011110100111110000101010100111110000110100101000110101000101010100010011111010001
Pair \(Z_2\) Length of longest common subsequence
1KGU_1,5DCD_1 198 4
1KGU_1,1GPZ_1 157 4
5DCD_1,1GPZ_1 175 3

Newick tree

 
[
	5DCD_1:97.89,
	[
		1KGU_1:78.5,1GPZ_1:78.5
	]:19.39
]

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 \((0.85)(0.8)(\frac{847 }{\log_{20} 847}-\frac{351}{\log_{20}351})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 1KGU_1 5DCD_1 177 149.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]