Citation: Planat, M.; Amaral, M.M.;
Fang, F.; Chester, D.; Aschheim, R.;
Irwin, K. Group Theory of Syntactical
Freedom in DNA Transcription and
Genome Decoding. Curr. Issues Mol.
Biol. 2022, 44, 1417–1433. https://
doi.org/10.3390/cimb44040095
Academic Editors: Chiara Rosso and
Antonio Francavilla
Received: 21 February 2022
Accepted: 20 March 2022
Published: 22 March 2022
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Article
Group Theory of Syntactical Freedom in DNA Transcription
and Genome Decoding
Michel Planat 1,*,†
, Marcelo M. Amaral 2,†
, Fang Fang 2,†
, David Chester 2,†
, Raymond Aschheim 2,†
and Klee Irwin 2,†
1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne-Franche-Comté, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; marcelo@QuantumGravityResearch.org (M.M.A.);
fang@QuantumGravityResearch.org (F.F.); davidc@QuantumGravityResearch.org (D.C.);
raymond@QuantumGravityResearch.org (R.A.); klee@QuantumGravityResearch.org (K.I.)
* Correspondence: michel.planat@femto-st.fr
†
These authors contributed equally to this work.
Abstract: Transcription factors (TFs) are proteins that recognize specific DNA fragments in order
to decode the genome and ensure its optimal functioning. TFs work at the local and global scales
by specifying cell type, cell growth and death, cell migration, organization and timely tasks. We
investigate the structure of DNA-binding motifs with the theory of finitely generated groups. The
DNA ‘word’ in the binding domain—the motif—may be seen as the generator of a finitely generated
group Fdna on four letters, the bases A, T, G and C. It is shown that, most of the time, the DNA-binding
motifs have subgroup structures close to free groups of rank three or less, a property that we call
‘syntactical freedom’. Such a property is associated with the aperiodicity of the motif when it is seen
as a substitution sequence. Examples are provided for the major families of TFs, such as leucine
zipper factors, zinc finger factors, homeo-domain factors, etc. We also discuss the exceptions to the
existence of such DNA syntactical rules and their functional roles. This includes the TATA box in the
promoter region of some genes, the single-nucleotide markers (SNP) and the motifs of some genes of
ubiquitous roles in transcription and regulation.
Keywords: DNA transcription factors; finitely generated groups; aperiodic order
1. Introduction
In his recent paper, Klee Irwin writes “Reality would be non-deterministic, not because it
is random, but because it is a code—a finite set of irreducible symbols and syntactical rules. 0ur
definition of information is meaning conveyed by symbolism. And expressions of code or language
are strings of symbols allowed by syntax—ordering rules with syntactical freedom.” [1].
Our last papers focused on the relevance of free groups in the encoding of the sec-
ondary structure of proteins [2] and in the encoding of tonal music and poetry [3].
In this present contribution, the concept of syntactical freedom is associated with that
of a free group in the encoding of strings of symbols—the motifs of DNA transcription. It
is shown for the first time that most transcription factors, but not all, have motifs where
the DNA letters in the motif form a finitely generated group whose structure is close to a
free group. The exceptions rely either on a specific functional role of the DNA sequence
under investigation or a potential dysfunction in the transcription of the gene, resulting
in disease.
A few definitions in the domain of genetics that we use in the paper are as follows:
Sequence motif An amino-acid sequence pattern that is related to a biological function
or a gene. The motif is sometimes called a ‘consensus sequence’.
DNA-binding domain A folded protein domain that contains a structural motif that
recognizes double- or single-stranded DNA.
Curr. Issues Mol. Biol. 2022, 44, 1417–1433. https://doi.org/10.3390/cimb44040095
https://www.mdpi.com/journal/cimb
Curr. Issues Mol. Biol. 2022, 44
1418
Transcription factor A sequence-specific DNA-binding factor, or transcription factor,
is a protein that controls the rate of transcription of a gene from DNA to messenger RNA
by binding to a specific DNA sequence. There are approximately 1600 binding domains in
the human genome that function as transcription factors. Classes of DNA-binding domains
of transcription factors also exist. The most common are zinc-coordinating DNA-binding
domains, helix-loop-helix or helix-turn-helix motifs, basic leucine zipper domains and
homeobox domains (playing critical roles in the regulation of development). A classification
of human transcription factors and their structural motifs is in References [4–6].
Exon A part of a gene that encodes a part of the mature RNA produced by that gene
after removing all introns (the non-coding regions of RNA transcript) by RNA splicing.
Promoter A sequence of DNA in which proteins initiate the transcription of a single
RNA from the DNA downstream of it. The TATA box is a sequence found in the core
promoter region of some genes in archaea and eukaryotes.
Zinc finger A small protein structural motif containing one or more zinc ions in order
to stabilize the protein fold.
Protein isoform A set of highly similar proteins may originate from a single gene. This
process is regulated by the alternative splicing of mRNA. In this process, particular exons
of a gene may be included within or excluded from the final, processed messenger RNA
(mRNA) produced from that gene. Alternative splicing and the multi-exonic genes are a
common feature in eukaryotes.
2. Materials and Methods
2.1. Finitely Generated Groups
In the domain of algebra, a group is a set equipped with an operation between every
pair of elements of the set that is associative, has an identity element, and every element
has an inverse. The elements of a group may be numbers or other structures.
One familiar group is the set Z of relative integers, together with the addition as the
group law. Notice that Z has infinite elements. The addition in Z is commutative, not a
general property for other groups.
The groups under consideration in this paper have elements that are words on four
letters A, T, G and C—the bases in DNA—and the group law is the product. In the context
of transcription factors (TFs), a motif fully describes the group. For example, taking only
two letters/bases A and T and a relation/motif rel(A,T) = TATA, the group is of the finitely
generated type and the mathematical notation is fp = 〈A, T|TATA = I〉, in which the
motif equals the identity element I. Infinite elements exist in f p. The group f p is non-
commutative and contains a wealth of symmetries. One useful method for investigating
them consists of two important concepts.
The first one is the concept of a free group on two generators F2 = 〈A, T〉 with no
explicit relation (but not forgetting the relations following from the axioms of a group:
AA−1 = TT−1 = I). The second concept is that of a conjugacy class of subgroups. Two
elements a and b belong to the same class if they are conjugate, meaning that a and
b = g−1ag belong to the same class for every g ∈ f p.
The finitely generated group with the relation rel(A,T) = TATA may be character-
ized by the number of conjugacy classes at each index d ∈ [1, 2, 3, . . .] with the sequence
ηd = [1, 3, 3, 10, 15, 56, 131, 462, . . .]. A signature of the group is also shared by the group of
‘dessins d’enfants’ in [7]—which the reader may consult for an introduction to the field of
finitely generated groups and the related graphs, topology and geometry.
2.2. Free Groups and Their Conjugacy Classes
Let Fr be the free group on r generators. Following a theorem derived by Hall in
1949 [8], the number Nd,r of subgroups of index d in Fr is
Nd,r = d(d!)r−1 −
d−1
∑
i=1
[(d− i)!]r−1Ni,r
Curr. Issues Mol. Biol. 2022, 44
1419
leading to the number Isoc(X; d) of connected d-fold coverings of a graph X (i.e., the
number of conjugacy classes of subgroups in the fundamental group of X) is as follows [9]
(Theorem 3.2, p. 84),
Isoc(X; d) =
1
d ∑
m|d
Nm,r ∑
l| dm
µ
(
d
ml
)
l(r−1)m+1,
where µ denotes the number-theoretic Möbius function.
Table 1 provides the values of Isoc(X; d) for small values of r and d, see [2] and
([9] [Table 3.2]).
Table 1. The number of conjugacy classes of subgroups of index d in the free group of rank r [9].
r
d = 1 d = 2 d = 3
d = 4
d = 5
d = 6
d = 7
1
1
1
1
1
1
1
1
2
1
3
7
26
97
624
4163
3
1
7
41
604
13,753
504,243
24,824,785
4
1
15
235
14,120
1,712,845
371,515,454
127,635,996,839
5
1
31
1361
334,576 207,009,649 268,530,771,271 644,969,015,852,641
We are interested in the cardinality sequence (card seq) of conjugacy classes for
subgroups of a finitely generated group f p with a relation (rel) given by the sequence
motif. Most of the time, the DNA motif in the transcription factor is close to that of a
free group Fr, with r + 1 being the number of distinct bases involved in the motif. How-
ever, the finitely generated group fp = 〈x1, x2|rel(x1, x2)〉, or fp = 〈x1, x2, x3|rel(x1, x2, x3)〉,
or fp = 〈x1, x2, x3, x4|rel(x1, x2, x3, x4)〉 (where the xi are taken in the four bases A, T, G,
C and rel is the motif) is not the free group F1 = 〈x1, x2|x1x2〉, or F2 = 〈x1, x2, x3|x1x2x3〉,
or F3 = 〈x1, x2, x3, x4|x1x2x3x4〉. The closeness of fp to Fr can be checked in the finite range
of indices of the card seq.
2.3. Content of the Paper
The structure of the TATA box in the core promoter region of many eukaryotes is not
close to that of a free group. Remarkably, the card seq of fp for the TATA box is close to
that of the Hecke group Hq =
〈
x1, x2|x21 = x
q
2
〉
[10]. The case q = 3 corresponds to the
modular group PSL(2,Z), which is the fundamental group of the trefoil knot manifold
K3a1 = 31. The Hecke group Hq, with q ≥ 3, is the discrete group generated by z→ −1/z,
z → z + λq, where λq = 2 cos(π/q) with λ3 = 1, λ4 =
√
2, λ5 = (1 +
√
5)/2, λ6 =
√
3,
etc. In Section 3.1, it is shown that the card seq for motifs in the standard TATA box
corresponds to H3 or H4 and that, in the case of a Gilbert’s syndrome, it is only approximate
or corresponds to Hq, q > 4.
In the same section, we investigate single-nucleotide polymorphisms (SNPs) of some
genes. In the case of SNP markers involving 3 bases, the fit of the card seq to that of the
free group F2 is obtained, or not. The fit of the card seq of the selected SNP to that of F2 is
well correlated to a lower risk of disease.
In Section 3.4, we analyze the binding domains and the card seq associated with motifs
of the immediately early genes Fos, EGR1 and Myc. In such cases, the card seq of the group
f p, taken with the relation as the motif, is that of a free group Fr (in the finite range of
indices). Most of the time, the motif of a transcription factor for a gene leads to the card
seq of a Fr. This statement follows from an almost exhaustive investigation of transcription
factors found in the databases defined in References [4,6].
However, it is also important to investigate the transcription factors with a group
structure away from a free group. This is done in Section 3.5 with the claim that the lack of
syntactical freedom (i.e., that the card seq of the gene is not that of a free group) is a marker
of potential dysfunction of the gene through mutations or isoforms.
Curr. Issues Mol. Biol. 2022, 44
1420
In Section 4, we show that group theoretical freedom correlates to the aperiodicity of
motifs when the latter are seen as substitution sequences.
In the Conclusions, we offer some roads of progress in the connection of group theory
to genetics.
3. Results
3.1. The TATA Box, the Hecke Groups and More
The TATA box (also called the Goldberg–Hogness box) is a DNA sequence located in
the core promoter region of genes in many eukaryotes, as well in archaea [11]. The TATA
box is a non-coding sequence whose name comes from the fact that it contains a consensus
sequence with repeating T and A base pairs [12,13]. The TATA box is a component of
eukaryotic promoters in which it initiates the transcription of TATA-containing genes.
The TATA box binds to the TATA-binding protein (TBP) and some other transcription
factors. TBP binds to the minor groove of the TATA box via a region of antiparallel β sheets
in the protein.
The regulation of gene transcription by transcription factors depends on the gene and
is governed by the RNA polymerase II (PolII) transcription complex. In the core promoter
of a typical PolII, they are key elements such as a TATA box.
Mutations such as insertions, deletions and point mutations to this consensus sequence
can result in phenotypic changes. These changes can then be related to diseases such as
gastric cancer, blindness, immunosuppression, Gilbert’s syndrome, etc. [12].
3.2. Gilbert’s Syndrome
Gilbert’s syndrome is a genetic polymorphism associated with the gene UGTIA1,
a phase II drug-metabolizing enzyme, which is essential in the metabolism of bilirubin
and other drugs. The core promoter in UGTIA1 contains a TATA box located at position
−28 with respect to the transcriptional start site [14]. A polymorphism with AT(TA)lTAA
(with l = 5. . . 8) is common in all ethnic populations with l = 6 as the major allele. Minor
alleles with l > 6 have less UGTIA1 transcription efficiency, leading to Gilbert’s syndrome,
neonatal jaundice and toxicity in cancer chemotherapy [14].
In Table 2, we look at the finitely generated groups f p = 〈A, T|rel〉, where rel is the
consensus sequence in the TATA box. The first two rows are for a standard TATA box. For
this case, the group f p is found to have the same cardinality structure of cc of subgroups as
the group H3 (the modular group) or the Hecke group H4. Rows 3 and 4 are for the TATA
box in the core promoter of the UGTIA1 gene for normal subjects, while rows 7 and 8 are
for subjects with a Gilbert’s syndrome. In the former case, the group f p has a cardinality
structure of cc of subgroups corresponding to the Hecke groups H6 and H7, while in the
latter case, the cardinality structure of cc of subgroups fits that of the Hecke groups H4
and H3 only up to index 8. Thus, we find that Gilbert’s syndrome is associated with an
imperfect fit of the group G to a Hecke group.
Table 2. Group structure of a TATA box. Column 1 is for the selected consensus sequence (rows 4 to 6
are for the TATA box in the core promoter of UGTIA1 gene). Column 2 is for the cardinality sequence
(card seq) of conjugacy classes (cc) of subgroups in the finitely generated group whose relation (rel) is
the consensus sequence (cons seq). Column 3 identifies the Hecke group Hq =
〈
A, T|A2T−q
〉
, which
is close to the group under consideration (based on its card seq of subgroups). Column 4 refers to
some references in the literature. Bold digits feature the fit to a Hecke group.
Rel: Cons Seq
Card Struct of cc of Subgroups
Group
Literature
TATAAAA
[1, 1, 2, 3, 2, 8, 7, 10, 18, 28, · · · ]
H3
[6] (MA0108.1)
TATAAAAA
[1, 3, 2, 8, 6, 19, 16, 69, 83, 238, · · · ]
H4
[13]
A(TA)5TAA
[1, 3, 3, 7, 6, 34, 42, 123, 319, 706, · · · ]
H6
[14]
A(TA)6TAA
[1, 1, 1, 1, 1, 1, 34, 77, 79, 51, · · · ]
H7
.
A(TA)7TAA
[1, 3, 2, 8, 6, 19, 16, 171, 315, 1022 · · · ]
≈H4
.
A(TA)8TAA
[1, 1, 2, 3, 2, 8, 7, 10, 308, 792 · · · ]
≈H3
.
Curr. Issues Mol. Biol. 2022, 44
1421
3.3. Single Nucleotide Polymorphism
The canonical form of TBP-binding sites, the TATA box, is the best-studied regulatory
element among human gene promoters. Tables identifying single-nucleotide polymor-
phisms (SNPs) in the (gene-dependent) TATA box have been collected in Reference [15].
At present, there are approximately 108 stored SNP markers that have been identified
in the human genome and approximately 1010 potentially possible markers. Most of them
are neutral and do not affect health in any way. Markers in protein-coding regions of genes
may damage proteins but are uncorrectable by treatment or lifestyle changes. However, a
large number of the variants that have been identified are located in non-protein-coding
regions and are presumed to affect gene expression regulation [16]. Regulatory SNPs in
the TATA regions have biomedical usefulness and are correctable by medication and/or
lifestyle. Ref. [15] collects 126 known SNP markers in 7 tables. We made use of these
tables to compute the finitely generated group f p whose relation (rel) is the marker; see
Table 3. For simplicity, we only took SNP markers built from 3 bases (and the exceptional
SNP marker with 2 bases). We made the cardinality sequence of cc of subgroups (card
seq) explicit. The computed closeness of the finitely generated group to the free group F2
correlates to a lower risk of illness. On the contrary, markers leading to a card seq away
from that of F2 indicate a potential higher risk of illness. The symbol * corresponds to
the only two-base SNP marker in the table. In this case, the card seq is the same as the
sequence for the fundamental group of 3-manifold m002 = net0200000. The latter manifold
is the smallest volume closed 3-manifold and is non orientable [17].
As a way of example, we take SNP markers in the first section of Table 3 that corre-
spond to potential tumors in reproductive organs. Five of them show a card seq away from
that of the free group F2, and they also correspond to a potential higher risk of disease.
The last two markers in the same section, whose card seq is close to that of F2, are expected
to produce a lower risk of breast cancer. Similar conclusions are valid for the SNP markers
in other sections of Table 3.
Table 3. Group analysis of a few known and candidate SNP markers (taken from [15]) Column 1 is
for the selected gene. Column 2 is for the SNP marker. Column 3 is for the card seq for the finitely
generated group f p whose relation (rel) is the marker. Column 4 is for the reference paper and the
letter indicates the heuristic confidence level of the candidate SNP marker (in alphabetical order
from the best (A) to the worst (E)). The computed closeness of the finitely generated group to the
free group F2, most of time, correlates to a lower risk of illness, as described in [15]. The symbol *
corresponds to the only two-base SNP marker in the table. The card seq is the same as the sequence
for the fundamental group of 3-manifold m002. The latter manifold is the smallest volume closed
3-manifold and is non-orientable [17].
Gene
Rel: Marker
Card Seq of cc of Subgroups
Literature
ESR2
TTAAAAGGAA
[1, 7, 17, 114, 423, 4526, 30364, 293306 · · · ] Table 1 in [15], B
HSD17B1 AGCCCAGAGC
[1, 3, 7, 26, 217, 124, 18443, 219870 · · · ]
., A
.
CAAGCCCAGA
[1, 7, 14, 109, 396, 3347, 19758, 188940 · · · ]
., A
PGR
AAAGGAGCCG [1, 7, 17, 142, 475, 4125, 23509, 225871 · · · ]
., A
GSTM3 GGGTATAAAG
[1, 7, 14, 109, 396, 3347, 19758, 188940 · · · ]
., E
.
CCCCTCCCGC
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
., C
.
CCCTCCCGCT
.
., C
IL1B
AAAACAGCGA
[1, 7, 14, 89, 224, 1842, 10191, 86701 · · · ] Table 2 in [15], A
CYP2A6 AAAGGCAAC
[1, 7, 17, 134, 683, 7077, 64225 · · · ]
., A
DHFR GGGACGAGGG
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
., A
.
GGACGAGGGG
.
., A
Curr. Issues Mol. Biol. 2022, 44
1422
Table 3. Cont.
Gene
Rel: Marker
Card Seq of cc of Subgroups
Literature
LEP
GGGGCGGGA
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
Table 3 in [15], C
GCG
TGCGCCTTGG
[1, 3, 7, 26, 119, 816, 4865, 40489 · · · ]
., B
GH1
TATAAAAAGG
[1, 7, 14, 109, 396, 3347, 19758, 188940]
., E
.
GTATAAAAAG
.
., D
.
GGTATAAAAA
.
., E
.
AGGGCCCACA
[1, 3, 7, 26, 127, 860, 5661, 45710 · · · ]
., A
.
AAAGGGCCCC
[1, 3, 10, 67, 266, 3458, 30653, 312237 · · · ]
., A
.
AAAGGGCCA
.
., A
NOS2
TCTTGGCTGC
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
Table 4 in [15], A
TPI1
ATATAAGTGG
[1, 3, 7,30, 125, 856, 4832, 40246 · · · ]
., B
GJA5
TATTAAACAC
[1, 3,10, 35, 140, 921, 5778, 47238 · · · ]
., E
HBD AAAAGGCAGG
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
Table 5 in [15], A
F2
AACCCAGAGG
[1, 3, 7, 26,127, 860, 5661, 45710 · · · ]
., A
F8
GGAAGAGGGA
[1, 3,2, 7, 4, 18, 9, 27, 36, 68 · · · ] *
Table 6 in [15], A
F3
GCGCGGGGCA
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
., A
F11
TTTTTAGTAA
.
., D
.
TTTTTAGTAA
[1,7, 17, 114, 423, 4526, 30364, 293306 · · · ]
., A
.
AAGGAAATTT
[1, 3, 7, 26,195, 1692, 11803, 73192 · · · ]
., A
AR
GTGGAAGATT
[1, 3, 7,34, 139, 931, 5208, 43867 · · · ]
Table 7 in [15], A
.
CCACGACCCG
[1,7, 20, 167, 754, 7232, 60860, 683597 · · · ]
., D
MTHFR
TCCCTCCCA
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
., A
DMNT1 TGTGTGGCCCG
.
., A
.
GTGTGTGCCC
.
., A
.
GACGAGCCCA
[1, 3, 7,42, 131, 912, 6011, 47322 · · · ]
., A
NR5A1 ACAAGAGAAA
[1, 3, 7, 26, 97, 624, 4163, 34470 · · · ]
., A
.
GGTGTGAGAG
[1,7, 14, 89, 264, 1987, 11086, 93086 · · · ]
., A
3.4. A Few DNA/Protein Complexes and Their Transcription Factors
As mentioned in the Introduction, most transcription factors have a binding domain
whose finitely generated group f p has a subgroup signature equal to that of a free group
Fr of rank r, with r ≤ 3 corresponding to r + 1 ≤ 4 distinct letters. An almost exhaustive
search has been performed by using the catalogs found in [4,6]. We first give details
of our calculations for a few immediate early genes. Then, in Section 3.5, most found
counterexamples are summarized.
3.4.1. Immediate Early Genes and Their Motifs
Immediate early genes (IEGs) are genes that are activated transiently and rapidly
in response to a wide variety of cellular stimuli. They represent a standing response
mechanism that is activated at the transcription level in the first round of response to
stimuli, before any new proteins are synthesized. The earliest known and best characterized
include c-fos, c-myc and c-jun, genes that were found to be homologous to retroviral
oncogenes. IEGs are well known as early regulators of cell growth and differentiation
signals. However, other findings suggest roles for IEGs in many other cellular processes as
‘gateways to genomic response’. Many IEG products are natural transcription factors or
other DNA-binding proteins. Important classes of IEG products include secreted proteins,
cytoskeletal proteins and receptor subunits.
Some IEGs such as ZNF268 and Arc have been implicated in learning, memory
and long-term potentiation. Neuronal IEGs are used prevalently as a marker to track
brain activities in the context of memory formation and the development of psychiatric
disorders [18].
Curr. Issues Mol. Biol. 2022, 44
1423
The group structure of the motifs of some IEGs in the Fos, EGR and Myc classes is
summarized in Table 4.
Table 4. Group structure of motifs for transcription factors of immediately early genes Fos, EGR
and Myc. Most of the time, the card seq of the group defined with the relation/motif is the free
group F2 (for a 3 letter motif) or F3 (for a 4 letter motif). There are two exceptions for the EGR1 gene,
depending on the selected motif, where the card seq corresponds to the modular group H3 or the
Baumslag–Solitar group BS(−1, 1), which is the fundamental group of the Klein bottle. The card seq
for H3 is in Table 2. The card seq for BS(−1, 1) is [1, 3, 2, 5, 2, 7, 2, 8, 3, 8, 2, 13, 2, 9, 4, · · · ].
Gene
Rel: Motif
Card Seq
Literature
Fos
TGAGTCA
F3
[19]
.
TGACTCA
F3
[6], MA MA0099.2
EGR1
GCGTGGGCG
F2
[6], MA0162.1
.
CCGCCCCCG
H3
., MA0162.2
.
CCGCCCCCGC
BS(−1, 1)
., .
.
ACGCCCACGCA
F2
., MA0162.3
.
GGCCCACGC
.
., MA0162.4
EGR2
CCGCCCACGC
.
., MA0472.1
.
ACGCCCACGCA
.
., MA0472.2
EGR3,EGR4
ACGCCCACGCA
.
., [ MA0732.1, MA0733.1]
Myc
CACGTG
F3
[19]
.
CGCACGTGGT
.
[6], MA0147.1
.
CCCACGTGCTT
.
., MA0147.2
.
CCACGTGC
.
., MA0147.3
Mycn, Max::Myc, etc GACCACGTGGT, etc.
.
., [MA0104.1, etc.]
3.4.2. The DNA-Binding Domain Fos
The Fos family (as well as the Jun family) are eukaryotic transcription factors that
heterodimerize to form complexes binding elements such as 5′-TGAGTCA-3′ DNA ele-
ments [19]. The X-ray crystal structure was determined and the bZIP region (with 62 aa)
of the c-Fos protein bound to DNA is available in the protein data bank as PDB: 1FOS.
The protein secondary structure of this subunit of c-Fos protein consists of two alpha helices,
as shown in Figure 1.
Let us consider the group
f p = 〈A,T,G,C|rel)〉 on 4 letters with the relation
rel = bind = TGAGTCA. The card seq of fp up to index 6 is that of the free group
F3 = 〈A,T,G,C|AGTC)〉 of rank 3. One can use the coset enumeration (with the
Todd–Coxeter procedure) to check that, up to index 6, the permutation groups organizing
the cosets in the cc of groups fp and F3 are the same. This shows that both groups are
close, at least in the finite range of subgroups. However, f p and F3 are not the same group.
Incidentally, the group f p′ = 〈A,T,G,C|AGTC, bind)〉, with the joint relations of f p and
F3, is close to the free group F2 = 〈x, y, z|xyz〉 on two generators in the sense that the
cardinality sequence of the cc of subgroups is that of F2, up to the higher index 9 that we
could reach in our calculations.
Similarly, the finitely generated groups f p = 〈A,T,G,C|rel〉, where the relation is with
the whole DNA chains rel = AATGGATGAGTCATAGGAGA (1FOS_1) or rel = TTCTC-
CTATGACTCATCCAT (1FOS_2) involved in the DNA/protein Fos complex (PDB:1F0S),
have the same card seq as F3 up to index 6.
Curr. Issues Mol. Biol. 2022, 44
1424
Figure 1. The DNA-binding domain of the immediate early gene Fos. The name in the protein data
bank is 1FOS.
3.4.3. The DNA-Binding Domain EGR1
The DNA-binding domain EGR1 (for early growth response protein 1) is a mammalian
transcription factor also called ZNF268 (the zinc finger protein 268). This is because the
protein encoded by the EGR1 gene has the Cys2His2-like fold structure of a zinc finger, as
shown in Figure 2. It binds to the motif 5′-bind-3′ [20], with bind = GCG(T/G)GGGCG.
Figure 2. (Left) Cartoon representation of the Cys2His2 zinc finger motif, consisting of an α-helix
and an antiparallel β-sheet. The zinc ion (green) is coordinated by two histidine residues and two
cysteine residues. (Right) Cartoon representation of the protein ZNF268 (blue) containing three zinc
fingers in complex with DNA (orange). The coordinating amino acid residues and zinc ions (green)
are highlighted. The name of the DNA-binding domain in the protein data bank is 4R2A.
The protein in the DNA-binding domain EGR1 is a nuclear protein and functions as
a transcriptional regulator. The products of the target genes that it activates are required
for differentiation and mitogenesis. When located in the brain, it has an essential role
in memory formation and in brain neuron epigenetic reprogramming. An X-ray crystal
structure is available in the protein data bank as PDB: 4R2A. In such a EGR1 DNA-binding
domain, the DNA chains are rel = AGCGTGGGCGT and rel = TACGCCCACGC.
Curr. Issues Mol. Biol. 2022, 44
1425
As for the Fos domain above, let us consider the group f p = 〈A,T,G,C|rel)〉 on 4 letters
with the relation bind or rel. The card seq of f p up to index 6 is similar to that of the free
group F3 = 〈A,T,G,C|ATGC)〉 of rank 3. One can use the coset enumeration (with the
Todd–Coxeter procedure) to check that, up to index 6, the permutation groups organizing
the cosets in the cc of subgroups of fp and F3 are the same. This shows that both groups
are close, at least in the finite range of subgroups. However, f p and F3 are not the same
group. Again, the groups built from the joint relations of f p and F3 are of rank 2, but the
cardinality structure of cc of subgroups is not that of F2.
The group f p′ = 〈A,T,G,C|ATGC, bind)〉, with the joint relations of f p and F3, is close
to the free group F2 = 〈x, y, z|xyz〉 on two generators in the sense that the card seq is that
of F2, up to the higher index 9 that we could reach in our calculations.
The early growth response protein 1 contains the chain of amino acids
GPLGS ERPYACPVESCDRRFSRSDELTRHIRIHTG QKPFQCRICMRNFSRSDHLT-
THIRTHTG EKPFACDICGRKFARSDERKRHTKIHLR QKD.
The central portion of the protein contains 86 = 30 + 28 + 28 aa decomposed into
3 zinc fingers with the following secondary structure (letter H is for the α-helix segment,
letter E is for the β-sheet segment and letter C is for the random coil segment)
CCCEECCCCCCCCEECHHHHHHHHHHHHHH CCCEECCCCCCEECHHHHH-
HHHHHHHHH CCCEECCCCCCEECHHHHHHHHHHHHHC
Taking the former 3-letter chain as the relation of a finitely generated group on
3 letters (and rank 2), we get the cardinality sequence for the cc of its subgroups as
[1, 3, 7, 26, 112, 717, · · · ], which fits the cardinality sequence of cc of subgroups of the free
group F2 only up to the index 4.
3.4.4. The DNA-Binding Domain Myc
Myc proto-oncogene is a transcription factor encoding a nuclear phosphoprotein that
plays a role in cell cycle progression, apoptosis and cellular transformation [21]. The protein
contains a basic helix-loop-helix zipper (bHLHZ) structural motif. The encoded protein
forms a heterodimer with the related transcription factor Max, as shown in Figure 3. Am-
plification of this gene is frequently observed in numerous human cancers. Translocations
involving this gene are associated with Burkitt lymphoma and multiple myeloma in hu-
man patients.
The bHLHZ domain of Myc-Max binds to the common DNA (palindromic) target
5′-CACGTG-3′. In the protein data bank, the reference of the complex is PDB: 1NKP.
The whole DNA chain is
rel = CGAGTAGCACGTGCTACTC (1NKP_1).
Let us consider the group f p = 〈A,T,G,C|rel)〉 on 4 letters with the relation rel =
bind = CACGTG. The conjugacy classes of f p up to index 6 have card seq equal to
that of the free group F3 = 〈A,T,G,C|GACT)〉 of rank 3. Again, one can use the coset
enumeration (with the Todd–Coxeter procedure) to check that, up to index 6, the per-
mutation groups organizing the cosets in the cc of groups fp and F3 are the same. This
shows that both groups are close, at least in the finite range of subgroups. However, f p
and F3 are not the same group. The group f p′ = 〈A,T,G,C|GACT, bind)〉, with the joint
relations of f p and F3, is close to the group π2 defined in Figure 3 (Down). The group
π2 = 〈x, y, z|(x, (y, z)) = z〉 is the fundamental group of the union of two links A and B,
which are not splittable. The proof is in Refs. [22] and ([23] p. 90) and follows from the fact
that π2 is not a free group. The group f p′ is close to π2 in the sense that the cardinality
sequence [1, 3, 10, 51, 164, 1365, 9422, 81594, 721305, · · · ] of the cc of subgroups is that of π2,
up to the higher index 9 that we could reach in our calculations.
The non-closeness of f p′ to F2 and the fact that π2 is not free are distinguishing features of
the Myc domain. It is tempting to associate such features with a potential abnormal replication.
Curr. Issues Mol. Biol. 2022, 44
1426
Figure 3. (Up) Crystal structure of Myc and Max in complex with DNA. (Down) The link L = A ∪ B
(which is supposed to control the binding domain Myc) is attached to the plane R2 in the half-
space R3+.
It is not splittable. This can be proven by checking that the fundamental group
π = π2(L) is not free [22] and ([23] p. 90). One gets π2 = 〈x, y, z|(x, (y, z)) = z〉, where (.,.)
means the group theoretical commutator. The cardinality sequence of cc of subgroups of π2 is
[1, 3, 10, 51, 164, 1365, 9422, 81594, 721305, · · · ].
3.5. Genes Whose Transcription Factors Have a Group Structure Away from a Free Group
We analyzed the group structure of motifs for some transcription factors that do not
lead to free groups. This is shown in Table 5. A short account of the function or dysfunction
of the corresponding genes is given in Table 6. It is observed that several transcription
factors whose group structure is away from a free group have the same card seq. We
conjecture that it is indicative of a related 3-dimensional structure of the corresponding
domain, although these families do not fit the standard classification [6].
Table 5.
Group structure of motifs for some transcription factors that are not lead-
ing to free groups.
The card seq for π1
is
[1, 4, 1, 2, 4, 2, 1, 7, 2, 2, 4, 2, 2, 8, 1, 2, 7, 2, 3, · · · ];
for π′1
it
is
[1, 1, 1, 2, 1, 3, 3, 1, 2, 2, 1, 1, 9, 2, 14, 2, 1, · · · ].
The card seq for π2
is already
in Figure 3 as
[1, 3, 10, 51, 164, 1365, 9422, 81594, 721305, · · · ].
The card seq for π3
is
[1, 7, 14, 89, 264, 1987, 11086, 93086, · · · ]; for π′3, it is [1, 7, 50, 867, 15906, 570528, · · · ]; for π′′3 , it is
[1, 7, 50, 739, 15234, 548439, · · · ]; for π(3)
3
, it is [1, 7, 41, 668, 14969, 550675]. The card seq for π4 is
[15, 82, 1583, 30242 · · · ]. The index i in πi refers to the rank of the group under examination. The three
sections are for motifs on 2, 3 and 4 letters, respectively.
Gene
Rel: Motif
Card Seq
Literature
NKX6-2
TAATTAA
H3
[6], [MA0675.1, MA0675.2]
HoxA1, HoxA2
TAATTA
π1
[6], [MA1495.1, MA0900.1]
POU6F1, Vax
.
.
., [MAO628.1, MA0722.1]
RUNX1
TGTGGT
.
., MA0511.1
Curr. Issues Mol. Biol. 2022, 44
1427
Table 5. Cont.
Gene
Rel: Motif
Card Seq
Literature
RUNX1
TGTGGTT
π′1
[6], MA0002.2
EHF
CCTTCCTC
.
., MA0598.1
POU6F1
TAATGAG
π2
[6] MA1549.1
PITX2
TAATCCC
.
., [MA1547.1, MA1547.2]
ELK4
CTTCCGG
.
., MA0076.2
OTX2, Dmbx1
GGATTA
π3
[6], [MA0712.2, MA0883.1]
PitX1, PitX2, PitX3, OTX1
TAATCC
.
.,[MA0682.1, MA0711.1]
N-box
TTCCGG
.
[24]
p53
CACATGTCCA
π′3
[25]
GZF1
TGCGCGTCTATA
.
[4]
NF-kappa-B
GGGAATTTCC
.
[6], [MA0107.1, MA1911.1]
STAT1
TTTCCCGGAA
.
., MA0137.2
.
TTCCAGGAA
.
., MA0137.3
STAT4
TTCCAGGAAA
.
., MA0518.1
FOSL1::Jun
ATGACGTCAT
π′′3
[6], MA1129.1
USF2
GTCATGTGACC
.
. , MA0626.1
PAX1
CGTCACGCATGA
.
. , MA0779.1
STAT2
TTCCAGGAAG
.
. , MA0144.1
FOS
GATGACGTCATCA
π
(3)
3
[6], MA1951.1
MAFA, MAFF,MAFK
TGCTGAGTCAGCA
.
., [MA1521.1, MA0495.2, MA0946.2]
CREB
TGACGTCA
π4
[6], [MA0018.2, MA018.3]
USF2
GGTCACGTGACC
.
., MA0526.4
SMAD3, SMAD5
GTCTAGAC
.
., [MA0795.1, MA1557.1], [26]
The DNA-Binding Domain of p53
Tumor protein p53 (also called tumor suppressor p53) has been called the guardian
of the genome. The main reason behind this status is the critical role that p53 plays in
preventing cancer development. p53’s role in tumor suppression is due to its ability to
induce the apoptosis, cell cycle arrest and senescence of pre-cancerous cells. However, it
also regulates other genes involved in metabolism.
According to [25], a motif for p53 is the DNA sequence CACATGTCCA. In our Table 5,
the attached card seq in the finite range of indices is that of a group π′3. However, there
are motifs leading to a card seq associated with the free group F3 or with other non-free
groups that are not of type π′3. This may be due to the fact that p53 has many isoforms to
fill its role.
In Figure 4, we draw the crystal structure of the p53 domain for the binding domain
of the PDB sequence 4HJE [27]. The p53 domain forms a tetramer, but other symmetries of
the binding domain of p53 may be found.
Table 6. A short account of the function or dysfunction (through mutations or isoforms) of genes
associated with transcription factors and sections in Table 5.
Gene
Type
Function
Dysfunction
NKX6-2
homeobox
central nervous system,
pancreas
spastic ataxia
HoxA1
homeobox
embryonic devt of face and
heart
autism
HoxA2
.
.
cleft palate
Curr. Issues Mol. Biol. 2022, 44
1428
Table 6. Cont.
Gene
Type
Function
Dysfunction
Pou6F1
.
neuroendocrine system
clear cell adenocarcinoma
Vax
.
forebrain development
craniofacial malform.
RunX1
Runt-related
cell differentiation, pain
neurons
myeloid leukemia
EHF
homeobox
epithelial expression
carcinogenesis, asthma
PitX2
.
eye, tooth, abdominal organs
Axenfeld–Rieger syndrome
ELK4
Ets-related
serum response for c-Fos
OTX1,OTX2
homeobox
brain and sensory organ devt
medulloblastomas
Dmbx1
.
.
farsightedness and strabismus
PitX1
.
organ devt, left–right
asymmetry
autism, club foot
PitX3
.
lens formation in eye
congenital cataracts
N-box
Ets-related
synaptic expression
drug sensitivity
p53
p53 domain
‘Guardian of the genome’
cancers
GZF1
Zinc fingers
protein coding
short stature, myopia
NF-kappa-B
.
DNA transcription, cytokines
apoptosis
STAT1
Stat family
signal activator of
transcription
immunodeficiency 31
STAT4
Stat family
signal activator of
transcription
rheumatoid arthritis
FOSL1::Jun
leucine zipper
cellular proliferation
marker of cancer
USF2
helix-loop-helix
transcription activator
PAX1
paired box
fetal development
Klippel–Feil syndrome
FOS
leucine zipper
cellular proliferation
cancers
Maf
.
pancreatic development
congenital cerulean cataract
CREB
bZIP
neuronal plasticity
Alzheimer’s disease
USF2
helix-loop-helix
transcription activator
SMAD
homeo domain
cell development and growth
Alzheimer’s disease
Figure 4. Crystal structure of p53 binding domain. The reference number in the protein data bank
is 4HJE.
Curr. Issues Mol. Biol. 2022, 44
1429
4. Discussion
According to Reference [1], aperiodicity is the correlate of syntactical freedom of
ordering rules. How can we check this statement in the realm of transcription factors?
First, we introduce the concept of a general substitution rule in the context of free
groups. A general substitution rule ρ on a finite alphabetAr on r letters is an endomorphism
of the corresponding free group Fr [28] (Definition 4.1). The endomorphism property means
the two relations ρ(uv) = ρ(u)ρ(v) and ρ(u−1) = ρ−1(u), for any u, v ∈ Fr.
A special role is played by the subgroup Aut(Fr) of automorphisms of Fr. We introduce
the map α : Fr → Zr from Fr to the Abelian group Zr in order to investigate the substitution
rule ρ with the tools of matrix algebra.
The map α induces a homomorphism M : End(Fr) → Mat(r,Z). Under M, Aut(Fr)
maps to the general linear group of matrices with integer entries GL(r,Z). Given ρ, there is
a unique mapping M(ρ) that makes the map diagram commutative [28] (p. 68). The substi-
tution matrix M(ρ) of ρ may be specified by its elements at row i and column j as follows
(M(ρ))i,j = card(ρai (aj)).
We will apply this approach to binding motifs of transcription factors. The binding mo-
tif rel in the finitely presented group f p = 〈A, T, G, C|rel(A,T,G,C)〉 is split into appropriate
segments so that rel = relArelTrelGrelC with the substitution rules A → relA, T → relT ,
G → relG, C → relC.
We are interested in the sequence of finitely generated groups
f (l)
p = 〈A, T, G, C|rel(rel(rel · · · (A, T, G, C)))〉 (with rel applied l times)
whose card seq is the same at each step l and equal to the card seq of the free group Fr (in
the finite range of indices that it is possible to check with the computer).
Under these conditions, we will see that (group) syntactical freedom correlates to the
aperiodicity of sequences.
4.1. Aperiodicity of Substitutions
There is no definitive classification of aperiodic order, the intermediate between
crystalline order and strong disorder, but in the context of substitution rules, some criteria
can be found. We need a few definitions.
A non-negative matrix M ∈ Mat(d,R) is one whose entries are non-negative numbers.
A positive matrix M (denoted M > 0) has at least one positive entry. A strictly positive
matrix (denoted M >> 0) has all positive entries. An irreducible matrix M = (Mij)1≤i,j≤d
is one for which there exists a non-negative integer k with (Mk)ij > 0 for each pair (i, j). A
primitive matrix M is one such that Mk is a strictly positive matrix for some k.
A Perron–Frobenius (PF for short) eigenvector v of an irreducible non-negative matrix
is the only one whose entries are positive: v > 0. The corresponding eigenvalue is called
the PF eigenvalue.
We will use the following criterion [28] (Corollary 4.3). A primitive substitution rule ρ
of substitution matrix M(ρ) with an irrational PF-eigenvalue is aperiodic.
A well-studied primitive substitution rule is the Fibonacci rule ρ = ρF : a→ ab, b→
a of substitution matrix MF =
(
1 1
1 0
)
and PF eigenvalue equal to the golden ratio
λPF = τ = (
√
5 + 1)/2 [28] (Example 4.6). As expected, the irrationality of λ corresponds
to the aperiodicity of the Fibonacci sequence.
The sequence of Fibonacci words is as follows:
a, b, ab, aba, abaab, abaababa, abaababaabaab, · · ·
The words have lengths equal to the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, · · ·
Curr. Issues Mol. Biol. 2022, 44
1430
It is straightforward to check that all finitely generated groups f (l)
p whose relations
rel(a, b) = ab, aba, abaab, abaababa, · · · have a card seq whose elements are 1’s, as for the
card seq of the free group F1. The Fibonacci sequence is our first example where group
syntactical freedom correlates to aperiodicity.
We will now provide examples taken for transcription factors involving 2, 3 or 4 letters.
4.1.1. A Two-Letter Sequence for the Transcription Factor of Gene DBX in Drosophila
Melanogaster
Let us consider the motif rel = TTTATTA for the gene DBX in drosophila melanogaster
(fruit flies) [6] (MA0174.1). The roles of the DBX gene include neuronal specification
and differentiation.
We split rel into two segments so that rel = relArelT with the substitution maps
A→ relA = TTTA, T → relT = TTA to produce the substitution sequence
A, T, AT, TTTATTA, TTATTATTATTTATTATTATTTA · · ·
The substitution matrix for this sequence is M =
(
1 3
1 2
)
; it is a primitive matrix of
PF eigenvalue λPF = (3 +
√
13)/2 so that the sequence associated with the DBX factor
is aperiodic.
Similarly to the Fibonacci generator rules, all finitely generated groups f (l)
p whose re-
lations are rel(A, T) = AT, TTTATTA, TTATTATTATTTATTATTATTTA · · · have a card
seq whose elements are 1’s as for the card seq of the free group F1.
The sequence for the DBX transcription factor is our second example where group
syntactical freedom correlates to aperiodicity.
4.1.2. A Three-Letter Sequence for the Transcription Factor of Gene EGR1
The transcription factor of gene EGR1 was investigated in Section 3.4.1. The selected
motif is rel = GCGTGGGCG. We split rel into two segments so that rel = relCrelGrelT with
the substitution maps C → relC = G, G → relG = CGT, T → relT = GGGCG to produce
the substitution sequence:
C, G, T, CGT, GCGTGGGCG, CGTGCGTGGGCGCGTCGTCGTGCGT · · ·
The substitution matrix for this sequence is M =
0 1 1
1 1 4
0 1 0
; it is a primitive matrix
(since M2 >> 0) whose eigenvalues follow from the vanishing of the polynomial −λ3 +
λ2 + 5λ + 1 = 0. There are three real irrational roots λ1 ≈ 2.86619, λ2 ≈ −0.21075 a,d
λ3 ≈ −1.65544. The PF eigenvalue is λPF = λ1 with an eigenvector of (positive) entries
(1, λ1/(λ21 − λ1 − 4), 1/(λ21 − λ1 − 4)T ≈ (1, 2.12485, 0.74134)T .
It follows that the selected sequence for the EGR1 gene is aperiodic.
All finitely generated groups f (l)
p whose relations are rel(C, G, T) = CGT, GCGTG
GGCG,CGTGCGTGGGCGCGTCGTCGTGCGT · · · have a card seq whose elements are
those of the free group F2.
The sequence for the EGR1 transcription factor is our third example where group
syntactical freedom correlates to aperiodicity.
4.1.3. A Four-Letter Sequence for the Transcription Factor of the Fos Gene
The transcription factor of gene Fos was investigated in Section 3.4.1. The selected
motif is rel = TGAGTCA.
We split rel into two segments so that rel = relArelTrelGrelC with the substitution
maps A → relA = T, T → relT = G, G → relG = AGTC, C → relC = A, to produce the
substitution sequence
Curr. Issues Mol. Biol. 2022, 44
1431
A, T, G, C, ATGC, TGAGTCA, GAGTCTAGTCGAT · · ·
The substitution matrix for this sequence is M =

0 0 1 1
1 0 1 0
0 1 1 0
0 0 1 0
.
It is a primi-
tive matrix (M4 >> 0) whose eigenvalues follow from the vanishing of the polynomial
λ4 − λ3 − λ2 − λ− 1. There are two real eigenvalues λ1 ≈ 1.92756 and λ2 ≈ −0.77480, as
well as two complex conjugate eigenvalues λ3,4 ≈ −0.07637± 0.81470i.
The PF eigenvalue is λPF = λ1 with an eigenvector of (positive) entries ≈ (1, 0.37298,
0.40211, 0.20861 )T .
It follows that the selected sequence for the Fos gene is aperiodic.
All the finitely generated groups f (l)
p whose relations are:
rel(A,C,G,T) = ATGC, TGAGTCA, GAGTCTAGTCGAT · · ·
have a card seq whose elements are 1, 7, 41, 604, 13753, 504243 · · · .
which is the card seq of the free group F3.
The sequence for the Fos transcription factor is our fourth example where group
syntactical freedom correlates to aperiodicity.
5. Conclusions
We made use of group theory for investigating transcription factors in genetics. Finite
group theory plays a major role in the attempts to model the genetic code; see [29,30]
and other references therein. Finitely generated groups (whose cardinality is infinite)
are necessary to deal with the secondary structures of proteins [2]. It was already noted
that many structures for the protein secondary codes tend to be close to free groups.
The card seq for such codes is model-dependent. In the map from amino acids to proteins,
the transcription factors play a critical role. The study of the group theoretical structure
of TFs has been our goal in the present paper. The DNA motifs that serve as a relation for
the corresponding f p groups are, in general, short sequences with around 10 amino acids.
Taking random sequences instead of the gene-specific DNA sequences in TFs also leads
to a majority of cases where the card seq of fp is close to a free group Fr and less frequent
cases where the card seq is away. However, motifs in TFs are codes with a particular
meaning—the specific gene function or dysfunction. In this sense, we found it appropriate
to use the concept of ‘syntactical freedom’ to qualify most TFs and to associate the lack of
syntactical freedom with a potential source of illness. In our context, syntactical freedom
means free groups and aperiodicity.
Potentially, there are many potential applications of our group theoretical method of
TFs for characterizing genes with problematic mutations, such as cancer and Alzheimer’s
disease. More work will be performed in the future.
Another interesting line of research is about the neurogenetic correlation of human
consciousness and the related TFs. If the reader is interested in studying this research
further, we recommend [24,31–33].
As a final note, we refer to the paper [34,35] in the domain of quantum gravity, where
the ordering rules with syntactical freedom are those of quasicrystals instead of those of
the biological crystal structures.
Author Contributions: Conceptualization, M.P., F.F. and K.I.; methodology, M.P., D.C. and R.A.; soft-
ware, M.P.; validation, R.A., F.F., D.C. and M.M.A.; formal analysis, M.P. and M.M.A.; investigation,
M.P., D.C., F.F. and M.M.A.; writing—original draft preparation, M.P.; writing—review and editing,
M.P.; visualization, F.F. and R.A.; supervision, M.P. and K.I.; project administration, K.I.; funding
acquisition, K.I. All authors have read and agreed to the published version of the manuscript.
Funding: Funding was obtained from Quantum Gravity Research in Los Angeles, CA, USA.
Curr. Issues Mol. Biol. 2022, 44
1432
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Computational data are available from the authors.
Conflicts of Interest: The authors declare no conflict of interest.
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