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Parikh vectors
4XVK_1 2CCH_1 3MIG_1 Letter Amino acid
40 17 13 E Glutamic acid
28 19 23 P Proline
11 10 15 Y Tyrosine
50 23 27 V Valine
56 19 21 A Alanine
45 9 7 Q Glutamine
44 21 11 K Lycine
34 17 22 T Threonine
8 4 3 W Tryptophan
23 10 11 H Histidine
40 16 15 I Isoleucine
32 16 18 D Aspartic acid
27 16 16 F Phenylalanine
34 16 15 R Arginine
18 8 11 N Asparagine
74 39 18 L Leucine
11 5 16 M Methionine
43 15 18 S Serine
15 3 10 C Cysteine
33 16 24 G Glycine 

4XVK_1|Chain A|DNA polymerase nu|Homo sapiens (9606)
>2CCH_1|Chains A, C|CELL DIVISION PROTEIN KINASE 2|HOMO SAPIENS (9606)
>3MIG_1|Chain A|Peptidyl-glycine alpha-amidating monooxygenase|Rattus norvegicus (10116)
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 \(w=f(c)\)
4XVK , Knot 256 666 0.83 40 276 618 
KKHFCDIRHLDDWAKSQLIEMLKQAAALVITVMYTDGSTQLGADQTPVSSVRGIVVLVKRQAEGGHGCPDAPACGPVLEGFVSDDPCIYIQIEHSAIWDQEQEAHQQFARNVLFQTMKCKCPVICFNAKDFVRIVLQFFGNDGSWKHVADFIGLDPRIAAWLIDPSDATPSFEDLVEKYCEKSITVKVNSTYGNSSRNIVNQNVRENLKTLYRLTMDLCSKLKDYGLWQLFRTLELPLIPILAVMESHAIQVNKEEMEKTSALLGARLKELEQEAHFVAGERFLITSNNQLREILFGKLKLHLLSQRNSLPRTGLQKYPSTSEAVLNALRDLHPLPKIILEYRQVHKIKSTFVDGLLACMKKGSISSTWNQTGTVTGRLSAKHPNIQGISKHPIQITTPKNFKGKEDKILTISPRAMFVSSKGHTFLAADFSQIELRILTHLSGDPELLKLFQESERDDVFSTLTSQWKDVPVEQVTHADREQTKKVVYAVVYGAGKERLAACLGVPIQEAAQFLESFLQKYKKIKDFARAAIAQCHQTGCVVSIMGRRRPLPRIHAHDQQLRAQAERQAVNFVVQGSAADLCKLAMIHVFTAVAASHTLTARLVAQIHDELLFEVEDPQIPECAALVRRTMESLEQVQALELQLQVPLKVSLSAGRSWGHLVPLQ
2CCH , Knot 135 299 0.85 40 191 285 
SMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL
3MIG , Knot 140 314 0.85 40 207 301 
FSNECLGTIGPVTPLDASDFALDIRMPGVTPKESDTYFCMSMRLPVDEEAFVIDFKPRASMDTVHHMLLFGCNMPSSTGSYWFCDEGTCTDKANILYAWARNAPPTRLPKGVGFRVGGETGSKYFVLQVHYGDISAFRDNHKDCSGVSVHLTRVPQPLIAGMYLMMSVDTVIPPGEKVVNADISCQYKMYPMHVFAYRVHTHHLGKVVSGYRVRNGQWTLIGRQNPQLPQAFYPVEHPVDVTFGDILAARCVFTGEGRTEATHIGGTSSDEMCNLYIMYYMEAKYALSFMTCTKNVAPDMFRTIPAEANIPIPV

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 \(c\).

\(|P_{f(4XVK_1)}(2) \setminus P_{f(2CCH_1)}(2)|=124\), \(|P_{f(2CCH_1)}(2) \setminus P_{f(4XVK_1)}(2)|=39\). 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 1:000100100100110001101100111111011000100011100011001011111100010110101011101111011100010101010001110000010001100111001000011101010011011101110010100110111101011111101001010100110000000101010000100000110001000100100101010001000111011001011111111110001101000010000111110100100010111100111000001001111010101100000110011000100001110110010111011100001001000110111101001010001000101010101001010110001101001001010000110101011110001001111010010101100101010110110000000110010001001110010010000000110111011100011101111100110110011000001001101111000001011011100011101010000101010001101110101101001111011011110001010111010001110100101100111100010010010110101011101010110011011110
Pair \(Z_2\) Length of longest common subsequence
4XVK_1,2CCH_1 163 4
4XVK_1,3MIG_1 187 3
2CCH_1,3MIG_1 174 4

Newick tree

 
[
	3MIG_1:93.05,
	[
		4XVK_1:81.5,2CCH_1:81.5
	]:11.55
]

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 \((0.85)(0.8)(\frac{965 }{\log_{20} 965}-\frac{299}{\log_{20}299})=179.\)
Status Protein1 Protein2 d d1/2
Query variables 4XVK_1 2CCH_1 226 161
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]