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Parikh vectors
3NEO_1 8WFY_1 3KLZ_1 Letter Amino acid
10 30 15 S Serine
12 19 28 A Alanine
4 31 4 R Arginine
0 24 9 Q Glutamine
10 44 8 E Glutamic acid
4 19 6 H Histidine
7 40 33 L Leucine
7 19 7 P Proline
5 20 8 Y Tyrosine
12 43 25 V Valine
5 30 6 D Aspartic acid
1 8 4 C Cysteine
7 39 26 G Glycine
6 41 12 K Lycine
5 16 20 F Phenylalanine
2 7 6 W Tryptophan
3 25 12 N Asparagine
10 33 14 T Threonine
5 25 23 I Isoleucine
1 12 14 M Methionine 

3NEO_1|Chains A, B|Transthyretin|Homo sapiens (9606)
>8WFY_1|Chains A, B|Tyrosine-protein phosphatase non-receptor type 11|Homo sapiens (9606)
>3KLZ_1|Chains A, B, C, D, E|Putative formate transporter 1|Vibrio cholerae (666)
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 \(w=f(c)\)
3NEO , Knot 62 116 0.84 38 94 114 
CPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNP
8WFY , Knot 212 525 0.84 40 258 482 
MKSRRWFHPNITGVEAENLLLTRGVDGSFLARPSKSNPGDLTLSVRRNGAVTHIKIQNTGDYYDLYGGEKFATLAELVQYYMEHHGQLKEKNGDVIELKYPLNCADPTSERWFHGHLSGKEAEKLLTEKGKHGSFLVRESQSHPGDFVLSVRTGDDKGESNDGKSKVTHVMIRCQELKYDVGGGERFDSLTDLVEHYKKNPMVETLGTVLQLKQPLNTTRINAAEIESRVRELSKLAETTDKVKQGFWEEFETLQQQECKLLYSRKEGQRQENKNKNRYKNILPFDHTRVVLHDGDPNEPVSDYINANIIMPEFETKCNNSKPKKSYIATQGCLQNTVNDFWRMVFQENSRVIVMTTKEVERGKSKCVKYWPDEYALKEYGVMRVRNVKESAAHDYTLRELKLSKVGQGNTERTVWQYHFRTWPDHGVPSDPGGVLDFLEEVHHKQESIMDAGPVVVHCSAGIGRTGTFIVIDILIDIIREKGVDCDIDVPKTIQMVRSQRSGMVQTEAQYRSIYMAVQHYIETL
3KLZ , Knot 121 280 0.81 40 180 264 
MEHNQFDSLLPPQMAERAAITGEGKAKKAAYKSFLLAISAGIQIGIAFVFYTVVTTGAHDMPYGVTKLLGGLAFSLGLILVVITGGELFTSSVLILVAKASGKISWKELVRNWTVVYFGNLCGSIILVFIMLATRQFMEDGGQLGLNAMAISQHKLHHTFLQAFALGLMCNILVCLAVWMTFSARSLTDKVMVLILPVAMFVSSGFEHCIANMFQVPMAIGIKYFAPESFWAMTGANIAQYADLNFVNFIVNNLIPVTLGNIVGGGVFVGMWYWLIYLKD

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 \(c\).

\(|P_{f(3NEO_1)}(2) \setminus P_{f(8WFY_1)}(2)|=20\), \(|P_{f(8WFY_1)}(2) \setminus P_{f(3NEO_1)}(2)|=184\). 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 1:01111011011010111011101100110001011101000001010110000011011001010000010111101100010111010001100001111101000000111001
Pair \(Z_2\) Length of longest common subsequence
3NEO_1,8WFY_1 204 3
3NEO_1,3KLZ_1 178 4
8WFY_1,3KLZ_1 186 4

Newick tree

 
[
	8WFY_1:10.30,
	[
		3NEO_1:89,3KLZ_1:89
	]:11.30
]

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 \((0.85)(0.8)(\frac{641 }{\log_{20} 641}-\frac{116}{\log_{20}116})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 3NEO_1 8WFY_1 193 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]