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Parikh vectors

8URQ_1	3RDN_1	6XOL_1	Letter	Amino acid
5	5	23	G	Glycine
15	7	38	L	Leucine
22	2	17	S	Serine
8	8	15	V	Valine
9	2	17	Q	Glutamine
12	4	23	E	Glutamic acid
12	5	23	I	Isoleucine
4	7	12	M	Methionine
13	7	22	T	Threonine
8	7	11	P	Proline
1	0	8	W	Tryptophan
7	1	12	Y	Tyrosine
7	6	31	A	Alanine
8	1	19	R	Arginine
13	6	23	N	Asparagine
13	2	28	D	Aspartic acid
4	0	3	C	Cysteine
1	0	9	H	Histidine
10	4	31	K	Lycine
10	0	21	F	Phenylalanine

8URQ_1|Chain A[auth F]|Meiotic recombination protein REC104|Saccharomyces cerevisiae S288C (559292)
>3RDN_1|Chain A|ANTIFREEZE PROTEIN RD3 TYPE III|Lycodichthys dearborni (8201)
>6XOL_1|Chain A|Lysozyme, DCN1-like protein 1 chimera|Enterobacteria phage T4 (10665)

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 $w=f(c)$
8URQ , Knot	85	182	0.81	40	130	178	MSIEEEDTNKITCTQDFLHQYFVTERVSIQFGLNNKTVKRINKDEFDKAVNCIMSWTNYPKPGLKRTASTYLLSNSFKKSATVSLPFILGDPVCMPKRVESNNNDTCLLYSDTLYDDPLIQRNDQAGDEIEDEFSFTLLRSEVNEIRPISSSSTAQILQSDYSALMYERQASNGSIFQFSSP
3RDN , Knot	40	74	0.77	32	64	71	MNKASVVANQLIPINTALTLIMMKAEVVTPMGIPAEEIPNLVGMQVNRAVPLGTTLMPDMVKNYEDGTTSPGLK
6XOL , Knot	162	386	0.83	40	216	363	MHHHHHHSSGVDLGTENLYFQSNAMNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWAEAAVNLAKSRWYNQTPNRTKRVITTFATGTWDAYKNLRKKLEQLYNRYKDPQDENKIGIDGIQQFCDDLALDPASISVLIIAWKFRAATQCEFSKQEFMDGMTELGCDSIEKLKAQIPKMEQELKEPGRFKDFYQFTFNFAKNPGQKGLDLEMAIAYWNLVLNGRFKFLDLWNKFLLEHHKRSIPKDTWNLLLDFSTMIADDMSNYDEEGAWPVLIDDFVEFARPQIAGTKSTTV

Protein code

c

$c$

LZ-complexity

L Z (w)

$\mathrm{LZ}(w)$

Length

n = | w |

$n=|w|$

\frac{L Z (w)}{n / \log_{20} n}

$\frac{\mathrm{LZ}(w)}{n /\log_{20} n}$

p_{w} (1)

$p_w(1)$

p_{w} (2)

$p_w(2)$

p_{w} (3)

$p_w(3)$

Sequence

w = f (c)

$w=f(c)$

8URQ , Knot

182

0.81

130

178

MSIEEEDTNKITCTQDFLHQYFVTERVSIQFGLNNKTVKRINKDEFDKAVNCIMSWTNYPKPGLKRTASTYLLSNSFKKSATVSLPFILGDPVCMPKRVESNNNDTCLLYSDTLYDDPLIQRNDQAGDEIEDEFSFTLLRSEVNEIRPISSSSTAQILQSDYSALMYERQASNGSIFQFSSP

3RDN , Knot

0.77

MNKASVVANQLIPINTALTLIMMKAEVVTPMGIPAEEIPNLVGMQVNRAVPLGTTLMPDMVKNYEDGTTSPGLK

6XOL , Knot

162

386

0.83

216

363

MHHHHHHSSGVDLGTENLYFQSNAMNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWAEAAVNLAKSRWYNQTPNRTKRVITTFATGTWDAYKNLRKKLEQLYNRYKDPQDENKIGIDGIQQFCDDLALDPASISVLIIAWKFRAATQCEFSKQEFMDGMTELGCDSIEKLKAQIPKMEQELKEPGRFKDFYQFTFNFAKNPGQKGLDLEMAIAYWNLVLNGRFKFLDLWNKFLLEHHKRSIPKDTWNLLLDFSTMIADDMSNYDEEGAWPVLIDDFVEFARPQIAGTKSTTV

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 (8 U R Q_{1})} (2) ∖ P_{f (3 R D N_{1})} (2) | = 102

$|P_{f(8URQ_1)}(2) \setminus P_{f(3RDN_1)}(2)|=102$ ,

| P_{f (3 R D N_{1})} (2) ∖ P_{f (8 U R Q_{1})} (2) | = 36

$|P_{f(3RDN_1)}(2) \setminus P_{f(8URQ_1)}(2)|=36$ . Let

Z_{k} (x, y) = | P_{x} (k) ∖ P_{y} (k) | + | P_{y} (k) ∖ P_{x} (k) |

$Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)|$ be a LZ76 style (set of subwords) Jaccard distance numerator for

P (k)

$P(k)$ .Hydrophobic-polar version of Sequence 1:10100000001000001100011000101011100001001000010011001101000101110001000110001000101011111101101100100000000110000100011100000110010001010110001001011000001011000001110000100101101001

Pair	$Z_2$	Length of longest common subsequence
8URQ_1,3RDN_1	138	4
8URQ_1,6XOL_1	194	3
3RDN_1,6XOL_1	200	3

Newick tree

 
[
	6XOL_1:10.54,
	[
		8URQ_1:69,3RDN_1:69
	]:37.54
]

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{256}{\log_{20} 256} - \frac{74}{\log_{20} 74}) = 59.0

$(0.85)(0.8)(\frac{256 }{\log_{20} 256}-\frac{74}{\log_{20}74})=59.0$

Status	Protein1	Protein2	d	d₁/2
Query variables	8URQ_1	3RDN_1	75	52

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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