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Parikh vectors
6JIR_1 5GHT_1 4EZS_1 Letter Amino acid
30 46 13 G Glycine
20 14 16 T Threonine
13 32 18 K Lycine
9 11 6 M Methionine
14 16 4 F Phenylalanine
24 14 6 P Proline
39 48 25 A Alanine
28 30 7 R Arginine
10 17 9 N Asparagine
10 14 6 H Histidine
37 33 12 V Valine
9 9 16 Q Glutamine
26 55 19 E Glutamic acid
37 44 20 L Leucine
0 3 0 W Tryptophan
8 12 0 Y Tyrosine
27 27 16 D Aspartic acid
1 1 0 C Cysteine
24 34 14 I Isoleucine
26 17 12 S Serine 

6JIR_1|Chains A, B|Beta sliding clamp|Caulobacter vibrioides CB15 (190650)
>5GHT_1|Chain A|SsDNA-specific exonuclease|Thermococcus kodakarensis (strain ATCC BAA-918 / JCM 12380 / KOD1) (69014)
>4EZS_1|Chain A|Chaperone protein DnaK|Escherichia coli (83333)
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 \(w=f(c)\)
6JIR , Knot 161 392 0.81 38 190 371 
MGSSHHHHHHSSGLVPGRGSMKLTIERAALLKALGHVQSVVERRNTIPILSNILLSAEGDRLSFSATDLDMEIIDEGFAQIDVPGQITAPAHTLYEIVRKLPDGADVSLSFSGDDPRLVIQAGRSRFNLPVLPAGDFPVMSSDGLSSRIAVDTNELIRLIDKTRFAISTEETRYYLNGLYVHTVNEGGETKLRAVATDGHRLALAEMPAPEGAVGIPGVIVPRKTIAEARRLMESAGETVDLQVSPQKVRFEFGAAALTSKVIDGAFPDYMRVIPRDNAKILTLDNDLFAKAVDRVATISAEKSRSVKLAVEPGRITLTVRNMEAGQAVEEVEVDYDGEPFEIGFNARYLLDVCGQIAGPQAEFRFADPASPTLVVDPVDPGVKYVLMPLRV
5GHT , Knot 190 477 0.82 40 216 429 
MDKEAFLERVREGAELIKMHIELGHTIRLISHRDADGITAGAILAKAVAREGGTFQLSIVKQVSEELIDQLAREKREIYVFSDLGSGSIELIEEKLNFATVVVADHHPPEKDSFSTDSHVLVNPVPFGANSVRDLSGSGVAYFVAREMNRKNRDMAYVAIVGAVGDMQEIDGTFHGLNLEIIEDGKELGILEVRKELRLFGRESRPLYQMLAYATNPEIPEITGDERKAIEWLRAKGFDPEMKYWQLREEEKRKLHEALLVHMIKHGAPKEAIDRLIGDVVISPLYPEGDVRHEAREFATLLNATGRLNAGTLGVAICLGDEEAYKVARKMLDDYKKEQIEARKFIIQNWNMVEEGEHAYVFYAGKNIRDTLVGIAANMAINAGLADPEKPVVVLADSDEDENLVKGSARTTEKALEKGYHLGEALKEVAEKLGGEGGGHAIAAGIRFPKNRIDEFIKLFNEALGRQVKGGGSEGEG
4EZS , Knot 95 219 0.78 34 145 212 
VLLLDVTPLSLGIETMGGVMTTLIAKNTTIPTKHSQVFSTAEDNQSAVTIHVLQGERKRAADNKSLGQFNLDGINPAPRGMPQIEVTFDIDADGILHVSAKDKNSGKEQKITIKASSGLNEDEIQKMVRDAEANAEADRKFEELVQTRNQGDHLLHSTRKQVEEAGDKLPADDKTAIESALTALETALKGEDKAAIEAKMQELAQVSQKLMEIAQQQHA

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 \(c\).

\(|P_{f(6JIR_1)}(2) \setminus P_{f(5GHT_1)}(2)|=60\), \(|P_{f(5GHT_1)}(2) \setminus P_{f(6JIR_1)}(2)|=86\). 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 1:11000000000011111010101010011110111010011000001111001110101001010100101011001110101110101110010011001101101010101001011101100010111111101111000110001110000110110000111000000001011010010011000101110010011110111101111111111100011010011001100101010100101011111100011011110010111000101101000111011001101010000010111011010101001011011001010001011011101001101010111101010110110101110110111001111101
Pair \(Z_2\) Length of longest common subsequence
6JIR_1,5GHT_1 146 4
6JIR_1,4EZS_1 135 4
5GHT_1,4EZS_1 165 5

Newick tree

 
[
	5GHT_1:81.06,
	[
		6JIR_1:67.5,4EZS_1:67.5
	]:13.56
]

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 \((0.85)(0.8)(\frac{869 }{\log_{20} 869}-\frac{392}{\log_{20}392})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 6JIR_1 5GHT_1 158 145.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]