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Parikh vectors
6WZB_1 8PJP_1 4PPZ_1 Letter Amino acid
11 8 22 D Aspartic acid
42 10 34 G Glycine
3 2 11 H Histidine
54 10 35 L Leucine
16 2 14 F Phenylalanine
49 23 36 A Alanine
19 11 20 K Lycine
19 4 19 P Proline
16 15 19 S Serine
18 5 6 Y Tyrosine
45 14 24 V Valine
10 4 18 R Arginine
17 2 3 M Methionine
28 14 28 T Threonine
39 10 24 I Isoleucine
2 2 7 C Cysteine
12 7 13 Q Glutamine
11 6 28 E Glutamic acid
0 3 4 W Tryptophan
11 9 16 N Asparagine 

6WZB_1|Chains A, B, C|Glutamate transporter homolog|Pyrococcus horikoshii (strain ATCC 700860 / DSM 12428 / JCM 9974 / NBRC 100139 / OT-3) (70601)
>8PJP_1|Chain A|Pilin|Neisseria meningitidis 8013 (604162)
>4PPZ_1|Chain A|Succinyl-diaminopimelate desuccinylase|Neisseria meningitidis (122586)
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 \(w=f(c)\)
6WZB , Knot 169 422 0.80 38 193 385 
MGLYRKYIEYPVLQKILIGLILGAIVGLILGHYGYADAVKTYVKPFGDLFVRLLKMLVMPIVFASLVVGAASISPARLGRVGVKIVVYYLLTSAFAVTLGIIMARLFNPGAGIHLAVGGQQFQPKQAPPLVKILLDIVPTNPFGALANGQVLPTIFFAIILGIAITYLMNSENEKVRKSAETLLDAINGLAEAMYKIVNGVMQYAPIGVFALIAYCMAEQGVKVVGELAKVTAAVYVGLTLQILLVYFVLLKIYGIDPISFIKKAKDAMLTAFVTRSSSGTLPVTMRVAKEMGISEGIYSFTLPLGATINMDGTALYQGVSTFFIANALGSHLTVGQQLTIVLTAVLASIGTAGVPGAGAIMLAMVLESVGLPLTDPNVAAAYAMILCIDAILDMGRTMVNVTGDLTGTAIVAKTEGTLVPR
8PJP , Knot 77 161 0.81 40 123 156 
FTLIELMIVIAIVGILAAVALPAYQDYTARAQVSEAILLAEGQKSAVTEYYLNHGEWPGDNSSAGVATSADIKGKYVQSVTVANGVITAQMASSNVNNEIKSKKLSLWAKRQNGSVKWFCGQPVTRTTATATDVAAANGKTDDKINTKHLPSTCRDDSSAS
4PPZ , Knot 163 381 0.84 40 214 364 
MTETQSLELAKELISRPSVTPDDRDCQKLLAERLHKIGFAAEELHFGDTKNIWLRRGTKAPVVCFAGHTDVVPTGPVEKWDSPPFEPAERDGRLYGRGAADMKTSIACFVTACERFVAKHPNHQGSIALLITSDEEGDALDGTTKVVDVLKARDELIDYCIVGEPTAVDKLGDMIKNGRRGSLSGNLTVKGKQGHIAYPHLAINPVHTFAPALLELTQEVWDEGNEYFPPTSFQISNINGGTGATNVIPGELNVKFNFRFSTESTEAGLKQRVHAILDKHGVQYDLQWSCSGQPFLTQAGKLTDVARAAIAETCGIEAELSTTGGTSDGRFIKAIAQELIELGPSNATIHQINENVRLNDIPKLSAVYEGILARLLAGNAV

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 \(c\).

\(|P_{f(6WZB_1)}(2) \setminus P_{f(8PJP_1)}(2)|=114\), \(|P_{f(8PJP_1)}(2) \setminus P_{f(6WZB_1)}(2)|=44\). 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 1:11100001001110011111111111111110010101100010111011101101111111110111111010110110111011100110011110111111011011111011111001010011111011101110011111101011101111111111100110000001000100110110111011001101110011111111110011001101110110101110111010111101111010110110110010011101110000010111010110011100110010111110101010110011001111011100101100101110111101101111111111111110011111001011110111101011101100110101010101111000101110
Pair \(Z_2\) Length of longest common subsequence
6WZB_1,8PJP_1 158 4
6WZB_1,4PPZ_1 159 4
8PJP_1,4PPZ_1 171 4

Newick tree

 
[
	4PPZ_1:83.70,
	[
		6WZB_1:79,8PJP_1:79
	]:4.70
]

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 \((0.85)(0.8)(\frac{583 }{\log_{20} 583}-\frac{161}{\log_{20}161})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 6WZB_1 8PJP_1 148 101.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]