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Parikh vectors
5NFO_1 5KEZ_1 2WMN_1 Letter Amino acid
15 50 19 G Glycine
5 28 22 I Isoleucine
13 26 20 F Phenylalanine
11 21 19 T Threonine
8 17 2 W Tryptophan
17 21 20 Y Tyrosine
17 34 24 D Aspartic acid
6 42 12 N Asparagine
13 14 21 Q Glutamine
17 11 12 H Histidine
7 9 16 M Methionine
31 23 13 P Proline
32 27 28 A Alanine
8 11 7 C Cysteine
39 25 39 L Leucine
25 36 29 V Valine
16 28 28 R Arginine
10 22 34 K Lycine
23 32 20 S Serine
22 19 43 E Glutamic acid 

5NFO_1|Chains A, B|JmjC domain-containing protein 7|Homo sapiens (9606)
>5KEZ_1|Chain A|Pancreatic alpha-amylase|Homo sapiens (9606)
>2WMN_1|Chain A|DEDICATOR OF CYTOKINESIS PROTEIN 9|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 \(w=f(c)\)
5NFO , Knot 142 335 0.82 40 195 314 
GSSHHHHHHSSGLVPRGSHMAEAALEAVRSELREFPAAARELCVPLAVPYLDKPPTPLHFYRDWVCPNRPCIIRNALQHWPALQKWSLPYFRATVGSTEVSVAVTPDGYADAVRGDRFMMPAERRLPLSFVLDVLEGRAQHPGVLYVQKQCSNLPSELPQLLPDLESHVPWASEALGKMPDAVNFWLGEAAAVTSLHKDHYENLYCVVSGEKHFLFHPPSDRPFIPYELYTPATYQLTEEGTFKVVDEEAMEKVPWIPLDPLAPDLARYPSYSQAQALCCTVRAGEMLYLPALWFHHVQQSQGCIAVNFWYDMEYDLKYSYFQLLDSLTKASGLD
5KEZ , Knot 212 496 0.88 40 258 473 
QYSPNTQQGRTSIVHLFEWRWVDIALECERYLAPKGFGGVQVSPPNENVAIYNPFRPWWERYQPVSYKLCTRSGNEDEFRNMVTRCNNVGVRIYVDAVINHMCGNAVSAGTSSTCGSYFNPGSRDFPAVPYSGWDFNDGKCKTGSGDIENYNDATQVRDCRLTGLLDLALEKDYVRSKIAEYMNHLIDIGVAGFRLDASKHMWPGDIKAILDKLHNLNSNWFPAGSKPFIYQEVIDLGGEPIKSSDYFGNGRVTEFKYGAKLGTVIRKWNGEKMSYLKNWGEGWGFVPSDRALVFVDNHDNQRGHGAGGASILTFWDARLYKMAVGFMLAHPYGFTRVMSSYRWPRQFQNGNDVNDWVGPPNNNGVIKEVTINPDTTCGNDWVCEHRWRQIRNMVIFRNVVDGQPFTNWYDNGSNQVAFGRGNRGFIVFNNDDWSFSLTLQTGLPAGTYCDVISGDKINGNCTGIKIYVSDDGKAHFSISNSAEDPFIAIHAESKL
2WMN , Knot 179 428 0.84 40 238 403 
KSYASTPELRKTWLDSMARIHVKNGDLSEAAMCYVHVTALVAEYLTRKGVFRQGCTAFRVITPNIDEEASMMEDVGMQDVHFNEDVLMELLEQCADGLWKAERYELIADIYKLIIPIYEKRRDFERLAHLYDTLHRAYSKVTEVMHSGRRLLGTYFRVAFFGQGFFEDEDGKEYIYKEPKLTPLSEISQRLLKLYSDKFGSENVKMIQDSGKVNPKDLDSKYAYIQVTHVIPFFDEKELQERKTEFERSHNIRRFMFEMPFTQTGKRQGGVEEQCKRRTILTAIHCFPYVKKRIPVMYQHHTDLNPIEVAIDEMSKKVAELRQLCSSAEVDMIKLQLKLQGSVSVQVNAGPLAYARAFLDDTNTKRYPDNKVKLLKEVFRQFVEACGQALAVNERLIKEDQLEYQEEMKANYREMAKELSEIMHEQLG

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 \(c\).

\(|P_{f(5NFO_1)}(2) \setminus P_{f(5KEZ_1)}(2)|=65\), \(|P_{f(5KEZ_1)}(2) \setminus P_{f(5NFO_1)}(2)|=128\). 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 1:10000000000111101001101110110001001111100101111110100110110100011010010110011001111001011010101100010111010101011010011111000111011101101010011110100000011001101110100011110011101101101111011110010000000100110100011101100011110010011000100010101100011001111110111101100100001011000101101101111110010000101110110010001000010110010010110
Pair \(Z_2\) Length of longest common subsequence
5NFO_1,5KEZ_1 193 4
5NFO_1,2WMN_1 185 4
5KEZ_1,2WMN_1 188 3

Newick tree

 
[
	5KEZ_1:96.16,
	[
		5NFO_1:92.5,2WMN_1:92.5
	]:3.66
]

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 \((0.85)(0.8)(\frac{831 }{\log_{20} 831}-\frac{335}{\log_{20}335})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 5NFO_1 5KEZ_1 177 145.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]