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Parikh vectors
5YHP_1 4PKE_1 9LPT_1 Letter Amino acid
19 20 14 T Threonine
14 10 21 Y Tyrosine
16 10 4 H Histidine
26 18 32 E Glutamic acid
26 26 46 L Leucine
9 9 10 M Methionine
18 7 13 F Phenylalanine
6 5 1 W Tryptophan
36 16 12 A Alanine
2 0 2 C Cysteine
10 13 6 Q Glutamine
31 15 4 G Glycine
16 25 27 I Isoleucine
17 12 48 K Lycine
21 19 6 P Proline
5 19 39 N Asparagine
21 15 31 D Aspartic acid
22 28 21 S Serine
20 9 17 V Valine
20 15 15 R Arginine 

5YHP_1|Chains A, B|Cold active proline iminopeptidase|Glaciozyma antarctica (105987)
>4PKE_1|Chain A|Protein C10C5.1, isoform i|Caenorhabditis elegans (6239)
>9LPT_1|Chain A|Surface protein P113|Plasmodium falciparum 3D7 (36329)
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 \(w=f(c)\)
5YHP , Knot 158 355 0.87 40 213 341 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDHPFTMTSLFAAIQPYKTHLLRVSPLHRLSIKEYGNPQGKPVVFLHGGPGGGASDSDARRFNPTTYRIVLFDQRGSGESTPASCLEDNTTQALVEDIEKIREFLQVGAAWHVFGGSWGSTLALAYAQAHPARVKSLTLRGIFTLRKKELDFFYQGPGSSFVFPEYWEEYLDPIPVAERGDMVKAYYERLTGSDEKVRAEAGRAWSRWEMATSRLHVDPDYISKADAPGFADAFARIESHYFVNGGFMPEGELLKPENIAKISHIPAVIVQGRYDMVCPITTAYELTKLWPEAKFVVIPDAGHSAIEAGTEKALVEATEEFAKLA
4PKE , Knot 128 291 0.83 38 184 278 
MSLLNQIGTISMPEKVTLRISIEGYPPLYEMEAQGSNHDNAELGMIKPDQLASLNQALTDSYTTRDTNSILRSRMSVSYLKGYTYEDILIVRFRPESEIYWPISQDSRNAMIDKLSRNTSVNFEVSLEFTRPYDPNENAALKHSKSWLVPISLDMTIRAKIQSALRGDPGHPILIPQSIPAFIQVPNQGELTLPTSIGNTIINDGNPRINTTGMEKSDEARAWFDSLTLNLEQGKSQNEKMWIATSEHPGDQNAKLWIKTANTTYSGRPYLQVVGFIDRAFPSLEHHHHHH
9LPT , Knot 146 369 0.78 40 176 336 
SSNDVNNNTDTLVNNKENKEFVLKEKSSLTSKINKELAHRTALFNKLADNISLLLNKKYDSFEIKDVLEDRYNEMKRDANPDVYYIYLMDTLDIEKIEDINLEEVKMSLLASLKETMNKIDTIEKKIEEFKNKYISLYNKVKTTMPELFDLNEDLVLLYNDFPFDNGMISSDIFFKYNPSENIMDHQEMVKKGSITEDELRIVNDLEPLDNYRRRKRITELRKILVEKLRILYLEKNNLFNTQASCIKSYCYKNPLNLKTLEVLLKKNYYRLKENKDYDVVSSIIQHLDNVDANKKKKWLTHERILKKLQVLIAEGYKRINEKEKDIDRRMAVYNALYEKAQSYNLQKLFNDSNDFLKKYAIMGNSFDD

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 \(c\).

\(|P_{f(5YHP_1)}(2) \setminus P_{f(4PKE_1)}(2)|=98\), \(|P_{f(4PKE_1)}(2) \setminus P_{f(5YHP_1)}(2)|=69\). 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 1:1010000000111010110011001000000001101001111101000011010110010100010101011111011111110000100101000011110001010001100100000011100100100110111110111101100111101010110100101011101000010110011100111100100010111110010110100001010000101011011001011000101010010010111110111010000110111110101101001101001111110100011011001001001110101111101100110110001110100011011
Pair \(Z_2\) Length of longest common subsequence
5YHP_1,4PKE_1 167 6
5YHP_1,9LPT_1 157 5
4PKE_1,9LPT_1 166 4

Newick tree

 
[
	4PKE_1:84.77,
	[
		5YHP_1:78.5,9LPT_1:78.5
	]:6.27
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{291}{\log_{20}291})=98.8\)
Status Protein1 Protein2 d d1/2
Query variables 5YHP_1 4PKE_1 125 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]