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Parikh vectors
5SFJ_1 9EKD_1 8QWD_1 Letter Amino acid
21 4 49 A Alanine
12 6 4 C Cysteine
16 4 52 G Glycine
12 2 14 M Methionine
15 2 74 F Phenylalanine
7 0 12 W Tryptophan
14 4 110 N Asparagine
24 12 139 E Glutamic acid
13 9 25 H Histidine
24 7 62 S Serine
22 3 53 T Threonine
12 7 65 Y Tyrosine
16 7 48 V Valine
14 2 31 R Arginine
36 3 114 L Leucine
18 7 172 K Lycine
14 2 33 P Proline
15 11 90 D Aspartic acid
17 3 39 Q Glutamine
21 9 75 I Isoleucine 

5SFJ_1|Chains A, B, C, D|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|Homo sapiens (9606)
>9EKD_1|Chains A, C[auth B]|Salivary anti-complement protein|Lutzomyia longipalpis (7200)
>8QWD_1|Chain A|ReChb|synthetic construct (32630)
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 \(w=f(c)\)
5SFJ , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED
9EKD , Knot 52 104 0.77 38 81 94 
SEDCENIFHDNAYLLKLDCEAGRVDPVEYDDISDEEIYEITVDVGVSSEDQEKVAKIIRECIAQVSTQDCTKFSEIYDCYMKKKICNYYPENMGSGHHHHHHHH
8QWD , Knot 439 1261 0.82 40 302 1007 
GPKMFSNFTNQYPLSKTLRFELKPVGKTLEHIEKKGLLEQDEKRAEDYKKVKKIIDEYHKDFIEEALNNVKLNGEGLEEYYELYFKKNKDDKDKKKKEFEKIQDNLRKQIVEAFKNHEKYKNLFKKELIKEDLPNWLKNSEDTGEEDKETVEKFKNFTTYFTGFHENRKNMYSDEEKSTAIAYRLIHENLPKFLDNMKVFEKIKEKHPEAEQLEKTLKDLGEEEVLGGNNVEDIFSLDYFNHTLTQSGIDIYNTIIGGKTEEDGNKKIQGLNEYINLYRQKNNEKNRKLPKLKPLYKQILSDRESLSFIPEAFENDEELLEAIEEFYENLNFSNNNEATNVLEKLKELLSNLADYDLNKIYIRNDTSLTDISQKIFGDWSVIKDALNAHYDQTYPKKKKKKSKEKLEEKREKWLKKQKYFSIAELQEALDSYCKESDESKEQKENSIADYFKTLAQTKNETDKKTDLIENIKSKYQYPNDKKLAQDKEFKDVEKIKAFLDSIMNLQHFVKPLHLVKGGSAGAEMEKDEAFYSEFEALYEELSQVIPLYNKVRNYLTQKPYSTEKIKLNFENSTLLDGWDVNKETDNTSVLLRKDGLYYLGIMNKKHNKVFENIPESNENDKCYEKMDYKLLPGANKMLPKVFFSNKNIDYFNPSAEILEIYENGTHKKSGDNFNLDDCHKLIDFFKESINKHEDWKKFGFKFSPTSSYEDISGFYREVEQQGYKISFKNISESYIDELVDEGKLYLFQIYNKDFSPYSKGKPNLHTLYWKALFDEENLKDVVYKLNGEAEVFYRKASINETIVHKANEPIKNKNPLNPKKQSTFEYDIIKDRRYTVDKFQFHVPITMNFKAEGNSNINDEVNEFLKGNAPDVNIIGIDRGERHLLYLTLIDQKGKIVEQDSLNTITNEHNETDYHALLDDKEKERDKARKSWGTIENIKELKEGYLSQVVHKIAKLMVEHNAIVVMEDLNFGFKRGRFKVEKQVYQKFEKMLIDKLNYLVDKDKEPNEPGGLLNAYQLTNKFESFQKMGKQSGFLFYVPAWNTSKIDPTTGFVNLFHPRYENVEKAKEFFNKFDSIRYNSEKDYFEFAFDYNNFTEKAEGTKWTVCTYGERIKTYRNADKNNQWDSKEVNVTEEFKNLFDEYNIDYKNGNDLKEAILSQDDADFFKSLLHLLRLTLQMRNSITGTEIDYIISPVANENGEFFDSRKADESLPKDADANGAYHIARKGLWVLEQIKQTDDLKKVNLAISNKEWLEFVQERKN

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 \(c\).

\(|P_{f(5SFJ_1)}(2) \setminus P_{f(9EKD_1)}(2)|=170\), \(|P_{f(9EKD_1)}(2) \setminus P_{f(5SFJ_1)}(2)|=30\). 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 1:1001000001011101011101000101101011110011111110110000100010100100111010000001100010011011001011100000110010001111101000100011000010010011111000001000010001011010100110010000000110110011110011101100001001000101010000000011111101001001001111001010010101110100100111011111000000011010111001111100001001111001110100001001001101000101100101100111000
Pair \(Z_2\) Length of longest common subsequence
5SFJ_1,9EKD_1 200 3
5SFJ_1,8QWD_1 171 4
9EKD_1,8QWD_1 245 4

Newick tree

 
[
	9EKD_1:11.30,
	[
		5SFJ_1:85.5,8QWD_1:85.5
	]:33.80
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{104}{\log_{20}104})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 5SFJ_1 9EKD_1 136 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]