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Parikh vectors
5PYZ_1 3FJJ_1 4YPC_1 Letter Amino acid
5 7 7 D Aspartic acid
6 13 0 G Glycine
6 11 1 H Histidine
13 5 2 F Phenylalanine
13 10 3 S Serine
8 6 5 V Valine
11 6 8 R Arginine
11 8 4 N Asparagine
13 11 5 K Lycine
5 1 2 M Methionine
8 8 0 P Proline
12 2 0 C Cysteine
12 6 7 Q Glutamine
3 1 0 W Tryptophan
5 8 0 Y Tyrosine
19 9 16 E Glutamic acid
6 8 2 T Threonine
11 17 12 L Leucine
3 4 6 A Alanine
10 5 3 I Isoleucine 

5PYZ_1|Chains A, B|Nuclear autoantigen Sp-100|Homo sapiens (9606)
>3FJJ_1|Chains A, B|Heparin-binding growth factor 1|Homo sapiens (9606)
>4YPC_1|Chain A|Vimentin|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 \(w=f(c)\)
5PYZ , Knot 87 180 0.83 40 137 176 
ENSNICEVCNKWGRLFCCDTCPRSFHEHCHIPSVEANKNPWSCIFCRIKTIQERCPESQSGHQESEVLMRQMLPEEQLKCEFLLLKVYCDSKSCFFASEPYYNREGSQGPQKPMWLNKVKTSLNEQMYTRVEGFVQDMRLIFHNHKEFYREDKFTRLGIQVQDIFEKNFRNIFAIQETSK
3FJJ , Knot 69 146 0.78 40 111 138 
HHHHHHFNLPPGNYKKPKLLYCSNGGHFLRILPDGTVDGTRDRSDQHIQLQLSAESVGEVYIKSTETGQYLAMDTDGLLYGSQTPNEEVLFLERLEENHYNTYISKKHAEKNWFVGLKKNGSCKRGPRTHYGQKAILFLPLPVSSD
4YPC , Knot 41 83 0.72 30 61 78 
VDQLTNDKARVEVERDNLAEDIMRLREKLQEEMLQREEAENTLQSFRQDVDNASLARLDLERKVESLQEEIAFLKKLHEEEIQ

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 \(c\).

\(|P_{f(5PYZ_1)}(2) \setminus P_{f(3FJJ_1)}(2)|=95\), \(|P_{f(3FJJ_1)}(2) \setminus P_{f(5PYZ_1)}(2)|=69\). 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 1:000010010001101100000100100000110101000110011001001000010000100000111001110001000111101000000011100100000100110011110010001000100010111001011100000100000100111010011000100111100000
Pair \(Z_2\) Length of longest common subsequence
5PYZ_1,3FJJ_1 164 3
5PYZ_1,4YPC_1 138 3
3FJJ_1,4YPC_1 124 3

Newick tree

 
[
	5PYZ_1:79.84,
	[
		4YPC_1:62,3FJJ_1:62
	]:17.84
]

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 \((0.85)(0.8)(\frac{326 }{\log_{20} 326}-\frac{146}{\log_{20}146})=55.0\)
Status Protein1 Protein2 d d1/2
Query variables 5PYZ_1 3FJJ_1 71 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]