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Parikh vectors
6PQE_1 3BGT_1 4PJH_1 Letter Amino acid
2 10 12 I Isoleucine
5 7 11 K Lycine
2 7 7 M Methionine
7 23 14 P Proline
6 13 16 T Threonine
1 2 9 W Tryptophan
6 17 14 R Arginine
5 15 21 E Glutamic acid
2 10 15 Y Tyrosine
8 19 16 V Valine
18 18 17 G Glycine
1 12 10 F Phenylalanine
4 12 15 S Serine
3 13 14 D Aspartic acid
7 27 25 L Leucine
0 2 4 C Cysteine
3 8 16 Q Glutamine
0 6 13 H Histidine
9 23 13 A Alanine
0 2 9 N Asparagine 

6PQE_1|Chain A|Alkaline phosphatase,Chaperone protein DnaJ 2 fusion|Escherichia coli (strain K12) (83333)
>3BGT_1|Chains A, B, C, D|Probable acetoacetate decarboxylase|Chromobacterium violaceum (536)
>4PJH_1|Chains A, C|Major histocompatibility complex class I-related gene protein|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 \(w=f(c)\)
6PQE , Knot 42 89 0.70 34 61 78 
METATAGEWQGKGSGGSGGSGGSQDLYATLDVPAPIAVVGGKVRAMTLEGPVEVAVPPRTQAGRKLRLKGKGFPGPAGRGDLYLEVRIT
3BGT , Knot 113 246 0.84 40 166 240 
MKQQEVRQRAFAMPLTSPAFPPGPYRFVNREYMIITYRTDPAAIEAVLPEPLQMAEPVVRYEFIRMPDSTGFGDYSESGQVIPVTFRGERGSYTLAMFLDDQPPLAGGRELWGFPKKAGKPRLEVHQDTLVGSLDFGPVRIATGTMGYKYEALDRSALLASLAEPNFLLKIIPHVDGSPRICELVRYHTTDVAIKGAWSAPGSLELHPHALAPVAALPVLEVLSARHFVCDLTLDLGTVVFDYLRQ
4PJH , Knot 127 271 0.87 40 195 261 
MRTHSLRYFRLGVSDPIHGVPEFISVGYVDSHPITTYDSVTRQKEPRAPWMAENLAPDHWERYTQLLRGWQQMFKVELKRLQRHYNHSGSHTYQRMIGCELLEDGSTTGFLQYAYDGQDFLIFNKDTLSWLAVDNVAHTIKQAWEANQHELLYQKNWLEEECIAWLKRFLEYGKDTLQRTEPPLVRVNRKETFPGVTALFCKAHGFYPPEIYMTWMKNGEEIVQEIDYGDILPSGDGTYQAWASIELDPQSSNLYSCHVEHSGVHMVLQVP

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 \(c\).

\(|P_{f(6PQE_1)}(2) \setminus P_{f(3BGT_1)}(2)|=19\), \(|P_{f(3BGT_1)}(2) \setminus P_{f(6PQE_1)}(2)|=124\). 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 1:10010110101010110110110001010101111111111010110101110111110001100101010111111101010101010
Pair \(Z_2\) Length of longest common subsequence
6PQE_1,3BGT_1 143 3
6PQE_1,4PJH_1 176 3
3BGT_1,4PJH_1 169 3

Newick tree

 
[
	4PJH_1:90.65,
	[
		6PQE_1:71.5,3BGT_1:71.5
	]:19.15
]

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 \((0.85)(0.8)(\frac{335 }{\log_{20} 335}-\frac{89}{\log_{20}89})=76.9\)
Status Protein1 Protein2 d d1/2
Query variables 6PQE_1 3BGT_1 96 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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