#  In silico model of infection of a CD4(+) T-cell by a human immunodeficiency
type 1 virus, and a mini-review on its molecular pathophysiology.

Vivanco-Lira  Department of Medicine and Nutrition, University of Guanajuato,
Leon, Mexico.  Department of Exact Sciences and Engineering, Open and Distance
Learning University of Mexico, Mexico City, Mexico.  A.
<EMAIL_ADDRESS> Department of Medicine and Nutrition, University of
Guanajuato, Leon, Mexico.  Department of Exact Sciences and Engineering, Open
and Distance Learning University of Mexico, Mexico City, Mexico.  Nieto-
Saucedo  Department of Medicine and Nutrition, University of Guanajuato, Leon,
Mexico.  J.R  Department of Medicine and Nutrition, University of Guanajuato,
Leon, Mexico.

(Submitted on February 2021)

###### Abstract

Introduction. Infection by the Human Immunodeficiency Virus can be defined as
a chronic viral infection which mainly affects T-cells; this virus displays a
set of both structural and regulatory proteins that aid in its survival within
the host’s cell, through these proteins, can the virus alter the host’s gene
expression pattern and with it, signaling involved in cell cycle control,
cytokine response, differentiation, metabolism, and others; therefore by
hijacking the host’s genetic machinery there exists a promotion in viral
fitness, and the ground is cemented for changes in the host’s cell
differentiation status to occur. Methods. We will consider two stochastic
Markov chain models, one which will describe the T-helper cell differentiation
process, and another one describing that process of infection of the T-helper
cell by the virus; in these Markov chains, we will consider a set of states
$\\{X_{t}\\}$ comprised of those proteins involved in each of the processes
and their interactions (either differentiation or infection of the cell), such
that we will obtain two stochastic transition matrices ($A,B$), one for each
process; afterwards, the computation of their eigenvalues shall be performed,
in which, should the eigenvalue $\lambda_{i}=1$ exist, the computation for the
equilibrium distribution $\pi^{n}$ will be obtained for each of the matrices,
which will inform us on the trends of interactions amongst the proteins in the
long-term. Results. The stochastic processes considered possess an equilibrium
distribution, when reaching their equilibrium distribution, there exists an
increase in their informational entropy, and their log-rank distributions can
be modeled as discrete beta generalized distributions (DGBD). Discussion. The
equilibrium distributions of both process can be regarded as states in which
the cell is well-differentiated, ergo there exists an induction of a novel
HIV-dependent differentiated state in the T-cell; these processes due to their
DGBD distribution can be considered complex processes; due to the increasing
entropy, the equilibrium states are stable ones. Conclusion. The HIV virus can
promote a novel differentiated state in the T-cell, which can give account for
clinical features seen in patients (decrease in naïve T cell counts, and the
general T-cell count decline); this model, notwithstanding does not give
account of YES/NO logical switches involved in the regulatory networks.

###### Contents

  1. 1 Introduction
    1. 1.1 Biology of HIV-1.
    2. 1.2 Insights into molecular pathophysiology: viral proteins.
      1. 1.2.1 gp120: organizing the foundations for infection from membrane to nucleus.
      2. 1.2.2 gp41: fusion, metabolism, and T-cell function suppression.
      3. 1.2.3 p7 (nucleoprotein, $NC\in gag$): viral genome package and facilitator.
      4. 1.2.4 p6 ($p6\in gag$): incorporating vpr into salient viral particles.
      5. 1.2.5 p24 (capsid protein, $CA\in gag$): protector of the viral genome.
      6. 1.2.6 p17 (matrix protein, $MA\in gag$): multifunctional protein.
      7. 1.2.7 p10 (protease, $PR\in POL$): catalyzer of viral and host’s proteins breakdown.
      8. 1.2.8 p51 (reverse transcriptase, $RT\in POL$): loader of nucleic acids and structural support.
      9. 1.2.9 p15 (RNase H, $(p15\vee p66\in pol$): converter of RNA into dsDNA.
      10. 1.2.10 p32 (Integrase, $IN\in pol$): immersing viral DNA into the host’s genome.
      11. 1.2.11 vpr (virus protein r): nuclear importer of viral genome.
      12. 1.2.12 vpu (virus protein unique, p16): CD4 downregulator.
      13. 1.2.13 nef (negative regulating factor, p27): downregulator of host’s proteins.
      14. 1.2.14 tat (transactivator protein, p14): promoting viral genome expression.
      15. 1.2.15 rev (RNA splicing regulator, p19): exporter of viral RNA.
      16. 1.2.16 vif (viral infectivity protein, p23): evading APOBEC3G response.
      17. 1.2.17 tev (tat/env/rev protein, p26): fusion protein.
    3. 1.3 T-cell differentiation.
    4. 1.4 Role of TOX in thymocyte differentiation
  2. 2 Materials and methods.
    1. 2.1 Computational requirements.
    2. 2.2 Data acquisition.
    3. 2.3 Stochastic model.
  3. 3 Results.
    1. 3.1 Equilibrium distribution.
    2. 3.2 The logarithmic equilibrium distribution follows a discrete beta generalized distribution.
    3. 3.3 The stochastic processes can be coupled with both in vivo processes of T-cell differentiation, and HIV-cellular infection.
    4. 3.4 Evolution of distributions until equilibrium.
    5. 3.5 The Th-cell differentiation and the HIV-infection processes tend towards an increasing entropy, $S_{1}\leq S_{2}\leq...\leq S_{n}$ when reaching its equilibrium distribution.
  4. 4 Discussion.
    1. 4.1 Th-cell fully differentiated state as an equilibrium distribution, $\pi^{n}$.
    2. 4.2 Th-differentiation, and HIV-infection processes as martingales.
    3. 4.3 Th-cell differentiation, and HIV-infection equilibrium distributions as discrete generalized beta ones.
  5. 5 Conclusions & Limitations.
  6. 6 Acknowledgements.

##  1 Introduction

Human Immunodeficiency Virus infection and the spectrum of related disorders
are relatively new to the human being, with the first known displays of this
disease occurring in the decade of the 1980s, however, efforts were made to
determine a more precise temporal origin of the virus, with some estimates
computing that it appeared as early as in the 1920s decade [311]; while it
became clearer a bit later that HIV may have originated by means of zoonoses
from primates in Africa [252]. During the first-ever recorded outbreak in the
1980s, the epidemic was said to be limited to the population of the 4 Hs:
homosexuals, haemophiliacs, haitians, heroin addicts [88] nevertheless, was
this soon proved to be a far too narrow assumption, and that the actual
boundaries of the infection were not these, for the set of susceptible
population was indeed a larger one. HIV was later discovered to exhibit two
basic genotypes: HIV-1 and HIV-2 with differences described both in the
clinical outcome and course of the disease [236]. HIV is known to exhibit
affinity for the receptors in T cells CD4, CCR5, CXCR4, as well as other
cofactors necessary for entry into the cell [302], then using these cells for
replication, inducing later apoptosis in these ones, this apoptosis in the CD4
(+) T cells (and as well in CD8(+) T cells) is responsible for the
immunodeficiency seen in the HIV(+) patients, in which, when untreated, may
result in the development of opportunistic infections, including infection by
human herpesvirus type 8, being able this virus to promote the establishment
of a malignancy, say Kaposi’s sarcoma.

###  1.1 Biology of HIV-1.

Both HIV-1 and HIV-2 are species belonging to the Lentivirus genus which
contains other notable viruses, such as Simian immunodeficiency virus, Feline
immunodeficiency virus and others; this genus belongs itself to the subfamily
Orthoretrovirinae, part of the Retroviridae family which also contains the
subfamily Spumaretrovirinae in which we may find those endogenous retroviruses
which have incorporated into the host’s genome and may take part in the
genetic diversification. The Retroviridae family is part of those reverse
transcribing RNA and DNA viruses [262]. The viral particle is about 100 nm in
diameter, with an outer membrane as envelope, this envelope contains 72 knobs,
each knob comprised by trimers of Env proteins, the Env protein possesses in
its structure the gp120 and the gp41 (gp160) proteins in its stead, where the
gp41 anchors to the lipid membrane and gp120 binds to gp41 finally protruding
from the membrane. In its envelope, the viral particle also contains MHC
pertaining to the host. Within the particle, we find another membrane,
comprised by the matrix (p17) and the protease (p10) proteins; attached to the
matrix protein membrane we find the lateral bodies, which are polarized bodies
inside the particle. Finally, the capsid lies in the core of the viral
particle, the shell being composed of the p24 protein (capsid protein) and
attached to the matrix protein by means of the p6 protein (link protein). In
the innards of the capsid lies the genome, where the reverse transcriptase
complex is found attached to the genome, along with the nucleic acid binding
protein (p7) [1]. The viral particle’s bilayer is enriched in specific lipids:
aminophospholipids, dyhydrosphingomyelin, plasmenyl phosphoethanolamine,
phosphatidylserine, phosphatidylethanolamine and phosphoinositides; being the
lipid composition bearers of importance in HIV fusion and infectivity (chol-
chelating compounds inhibit these actions), interaction with T-cell
immunoglobulin and mucin domain proteins block release of HIV from infected
cells, interaction with bavituximab (which targets phosphatidylserine)
suppresses productive HIV infection [111]. The clues leading to the thought of
HIV being a retrovirus came when in the 1980s outbreak, HIV-2 was shown to
also cause AIDS in patients, HIV-2 was related to HIV-1 but also to a simian
virus which caused immunodeficiency in macaques. It has been determined that
the SIV from the Sooty mangabey originated the HIV-2 (groups A-H), while the
SIV from the chimpanzee originated both the HIV-1 (groups M, N and probably O)
as well as the SIV from Western gorillas, which in turn gave place to HIV-1
(groups P and probably O) [252]; it has been proposed as well that HIV
originated from SIV by means of several genetic mutation transitions and that
the use of unsterile injections may have promoted the increase in transmission
[161], other mechanisms have been proposed, such as: population growth,
changing sexual practices, migration, increased hunting and deforestation in
post-colonial Africa [196]; however, retroviruses have been computed to appear
as far back as 460 to 550 million years ago (during the Palaeozoic Era), along
with the radiation of Vertebrata, nevertheless it remains unclear whether
Retroviridae originated from Metaviridae or whether Retroviridae gave place to
Metaviridae [106]. HIV-1 displays three major groups: major (M) with a
transmission event occurring between 1912 and 1941, outlier (O), and nonmajor
and nonoutlier (N); the virus may have commenced with the HIV-1 O group in
gorillas, then spread among humans from the Congo River into Kinshasa, Zaire,
with the earliest documented case of HIV-1 infection (M group) in humans
dating from 1959. The M group is the predominant circulating group, and it has
been divided into subtypes: A1, A2, A3, A4, B, C, D, F1, F2, F, H, J, K;
further recognition of subtypes by means of full-genome sequencing has been
made (CRFs and URFs) [274]. Meanwhile, HIV-2 displays A-H grups, with the A
group possessing two subdivisions: A1 and A2 [1]. The life of those patients
infected with HIV gave a very important turn with the advent of the
antiretroviral drugs, and has approached the amount of years to be lived at
age 20 years two thirds that of the general population [49], other
measurements have been made estimating expected average ages of death in those
patients receiving antiretroviral therapy, finding 67.6 years for men and 67.9
years for women [50]. In addition, the mutation rate of the HIV-1 genome in
vivo has been shown to be rather high, i.e., $4.1\pm 1.7\times 10^{-3}$ per
base per cell, this rate is contributed 2% by the reverse transcriptase and
98% by the host’s cytidine deaminases of the A3 family [56].

###  1.2 Insights into molecular pathophysiology: viral proteins.

####  1.2.1 gp120: organizing the foundations for infection from membrane to
nucleus.

gp120 owes its name to its molecular weight ranging from 110 to 120 kDa [55],
being this protein a portion of the gp160 complexes (due to the fact that
gp120 matures from a larger, gp160 peptide), the gp120 associates with the
gp41 molecule assemblying into a heterodimer, which will trimerize to form the
mature Env protein, in its turn bound to the viral bilayer membrane. The
protein is comprised by five variable regions and five conserved regions,
folding into a globular structure with an inner and an outer domain bound by a
bridging sheet. gp120 is glycated mostly by high-mannose glycans, and complex
glycans fucosylated and containing multiple antennas, along with a variable
amount of sialic acid ; the high-mannose oligosaccharides were found to be
$(GlcNAc)_{2}(Man)_{5}$ and $(GlcNAc)_{2}(Man)_{9}$ [218]; deficiencies in
glycosylation (as they have been induced in bacteria-produced gp120) result in
unsuccessful binding of gp120 to CD4 [78]. gp120 binds to CD4 by means of a
pocket found above the bridging sheet [313], this pocket is known to be
hydrophobic in gp120 and when bound to CD4 a phenylalanine residue caps this
pocket, this residue has proved essential for a successful binding, the whole
complex (gp120 hydrophobic pocket + phenylalanine residue) is termed the Phe43
cavity [65]; moreover, the affinity with which CD4 interacts with gp120 has
been determined to be $K_{d}=4\times 10^{-9}M$ [275] while the entropy of this
process has been seen to be $\Delta S=220\pm 13kJmol^{-1}$ and an enthalpy of
$\Delta H=-259\pm 13kJmol^{-1}$ (at 310 K) (in which we find that this binding
of CD4 to gp120 is an irreversible (due to $\Delta S>0$) and spontaneous
(because $\Delta G<0$, considering $\Delta G$ as the free energy) process
[157]) with an intermolecular hydrogen-bond network of 166 atoms (for CD4) and
130 atoms (for gp120) [109]. The binding of gp120 to CD4 induces
conformational changes in gp120, and promotes the creation of a high-affinity
binding site for CCR5, as well as the exposure of gp41 in order to induce
membrane fusion; furthermore, has it been shown that the binding of the
soluble form of CD4 (sCD4) to gp120 promotes the dissociation of gp120 from
gp41, and some variable loops (V1, V2, V3) change their conformation or become
more more exposed; the exposure of these loops (V1 and V2) could be
potentially recognized by the monoclonal antibodies 17b and 48d (this
nomenclature is due to the epitopes exposed when the conformational changes
are induced) [264], where the monoclonal antibody 17b is an anti-HIV-1 gp120
monoclonal antibody which is obtained from EBV transformation of B cells (let
us recall that these transformed B cells are those infected by EBV in which
EBNA1, EBER1 and EBER2 are expressed, inducing the translocation of MYC into
the immunoglobulin loci, activating permanently this transcription factor
[134]) of an asymptomatic HIV-1 infected individual [226]; however, due to
steric hindrance and conformational masking, these epitopes remain obscured or
unaccesible to the antibodies directed against gp120, therefore, by means of a
chimeric protein comprised of a soluble CD4 bound to the monoclonal antibody
directed towards 17b, the necessary conformational change occurs and the
epitopes are exposed, method through which, the monoclonal antibody may then
bind to the epitope [136]; nevertheless, there exist also entropic barriers in
the gp120 core which may decrease the potential binding of antibodies [217].
The binding of CD4 to the heterohexamer ($(gp120)_{3}(gp41)_{3}$) happens in a
stoichiometry of $(CD4)_{3}(gp120)_{3}(gp41)_{3}$ this because two CD4
molecules may be as close as $19.8nm$ in the cell membrane context, that is,
the root of the CD4 molecules for these molecules protrude from the cell
membrane and be closer in the extracellular space, the maximum distance at
which they can be approached one from the other has been measured to be
$3.5nm$ in the extracellular space and without steric interference [135].
After the binding of CD4 to gp120, we have described that there exist some
conformational changes which promote the interaction of CCR5 with gp120 (used
in early infection) and with CXCR4 (in late infection), it has been displayed
that in the absence or changes in sequence of the loops V1 and V2, there could
be a loss in the ability of gp120 of binding to CD4, then being able to infect
CD4(-) cells with CCR5(+) or CXCR4(+); the conformational changes of gp120 by
means of CD4 diminish the entropy of the protein [217]; this binding of gp120
to the chemokines happens through the V3 loop which is predominantly composed
of 35 residues, connected by a disulfide bridge between residues 1 and 35,
being those residues essential for the binding with the coreceptors those from
13 to 21, additionally, gp120 binds with near equal energetic properties to
CCR5 and CXCR4 [269]; the binding of CD4 to gp120 induces a diminishment in
the distance of the trimer to the target cell membrane by $2nm$ [135]. We
previously commented on the observed fact that the HIV-1 in its early
infection phase exhibits tropism towards CCR5, while during late infection,
this tropism shifts towards CXCR4 [223] and there have been some reasons of
this shift implicating gp120: loss of N-linked glycosylation site in V2 in
late stages, increase in positive-charged residues in V1/V2 and V4/V5,
addition of N-linked glycosylation site near the CD4-binding site (N362);
however the implications of this change in tropism are viral-fitness-
dependent, that is, whether this switch is convenient for the virus to occur
or not, in which, given the case, the switch shall not occur [174]; this may
be challenged (or sustained) by the fact that CCR5 is expressed in memory T
cells, while CXCR4 is in change expressed in naïve T cells, in which the
memory T cells have shown to possess a higher division rate than naïve T
cells, which promotes an advantage in the fitness of the virus during early
infection, nevertheless when time passes and CD4(+) T cell depletion occurs,
both memory and naïve T cell numbers are consequentially diminished, but the
fraction of naïve T cells in division has been seen to divide more rapidly in
the late stage of disease than memory T cells, those augmenting the selection
of those viruses with CXCR4 tropism or the switch from CCR5 to CXCR4 in late
stages of disease [223]. It has been shown that gp120 folding occurs at the
rough endoplasmic reticulum (and not in the Golgi apparatus) and this folding
is essential to the binding of gp120 to the CD4 molecule; the folding time of
this molecule has been shown to be more lengthy than usual, this due perhaps
to the formation of disulfide bonds within gp120, probably aided by disulfide
isomerase (disulfide bond-formation are dependent on several enzymes, those
thiol-disulphide oxidoreductases, such as disulphide isomerase (this one is
yet to know whether it catalyzes de novo formation of disulfide bonds or
shuffles pre-existing bonds), as well as other proteins, such as ERO1 which is
a glycosylated ER protein which induces oxidizing equivalents into the ER
lumen [249]); gp120 is constrained to proper folding for its exit from the
rough endoplasmic reticulum [78]. gp120 not only exerts fusion-related
functions, but also contributes to the pathogenesis of HIV by activating
complement, inducing polyclonal B cell activation, binding to immune
complexes, participates in influencing T cell function (promoting inefficiency
in T cell response), augments production of TGFB [313]. The gp120-dependent
activation of complement occurs via a dettachment of gp120 from the viral
envelope either spontaneously or after gp120 binding to the CD4 molecule, then
gp120 may circulate in the patient’s blood and may attach in a soluble form to
CD4(+) T cells, promotion of opsonization and elimination of CD4(+)gp120(+) T
cells by means of the reticuloendothelial system; the complement is activated
in its classical pathway, and gp120 has been seen to bind to C4, C3, C5, C9
complement proteins [266]; thus ending with the membrane atttack complex (MAC)
formation in the gp120-dependent complement activation directed towards CD4(+)
T cells, contributing to the depletion of these cells; this membrane attack
complex is assembled by means of the cleavage of C5 to C5b and C5a, C5b
recruits C6 (C5b6) through C9 and forms the MAC pore [319]; abnormal
complement activation has been observed in HIV patients, associating this
infection with certain autoimmune disorders, such as: immune thrombocytopenic
purpura, inflammatory myositis, Guillain-Barré syndrome, sarcoidosis,
myasthenia gravis, Graves’ disease, Hashimoto thyroiditis, autoimmune
hemolytic anemia, autoimmune hepatitis, rheumatoid arthritis, systemic lupus
erythematosus [287]. Abnormalities of the humoral response have been observed
in HIV patients (which relates to abnormal B cell function), such as:
polyclonal hypergammaglobulinemia, B-cell hyperplasia, circulating immune
complexes, spontaneous Ig secretion, proliferation with elevated
autoantibodies; that has been shown to be induced by gp120 due to an increase
in TNFA, which increases B cell proliferation, this increase in TNFA not only
stimulates cellular proliferation, but also IgM and IgG secretion;
furthermore, it has been seen that B cells display binding to the gp120
protein (experimentally, 2.9% to 4.3% of the B cell CD20(+) population binds
to gp120, therefore we may think of this ratio of binding as a probability p
for B cell to bind to gp120 in a range of $p\in[0.029,0.043]$); in B cells,
gp120 also promotes cAMP generation [199] which is known to suppress the
activity of NFKB by means of BCR and TLR4 signaling blocking, possibly through
a protein kinase A [169], NFKB is also a lymphopoietic regulator in B cells,
by maintaining TNFA production at appropriate levels (then we see an increase
in TNFA activity, and suppression of NFKB, which may contribute to a
malignancy development in HIV patients, even in those patients with
antiretroviral treatment, there exists the possibility of developing, for
instance, Hodgkin’s lymphoma [119]), and plays also a role in the expression
of $Ig\lambda$ in immature B cells; mature B cell survival and population
restoration is dependent on the NFKB canonical pathway [241]. gp120 also
exerts actions down- and upregulating cell cycle- or transcriptional
regulation-related genes in T cells (as seen when T cells are treated with the
V3 loop of gp120), some of these upregulated genes are: NOC2L (inhibitor of a
histone acetyltransferase independent of HDAC, which is seen upregulated)
[173], SEPT9 (member of a family of GTP-binding proteins, which exhibit roles
in cytokinesis, cytoskeleton, cell cycle control; whose hypermethylation is
related to several cancers [254]), IFI6, SPIN, HNRNPM; while some other
downregulated genes are: ABCG1, PGPEP1, PTPLA, SPATA21; overall, the V3 loop
of gp120 affects the following sets of genes within T cells: cell cycle (62
genes), cellular development (function, maintenance, compromise, morphology)
(54 genes), aminoacid or lipid metabolism (36 genes), gene expression (65
genes), DNA metabolism (replication, recombination, repair) (27 genes),
cellular assembly and organization (30 genes), cellular movement (25 genes),
hematopoiesis (25 genes), immune response (32 genes), cellular death (growth
and proliferation) (100 genes), infection mechanism (34 genes); therefore we
witness a hijacking of the genetic machinery in T cells which may compromise
by all means their further function and proliferation [173]. Variois gp120
inhibitors have been described, such as: BMS-378806, BMS-488043, BMS-626529,
BMS-663068, NBD-556, JRC-II-191, and 18A [150], BMS-378806 does not display
action against HIV-2, SIV, and other viruses, whose metabolism is cythochrome-
dependent [310] and thus major drug-drug interactions are expected [294].
BMS-488043 is an analog of the BMS-378006 compound and has been seen to
display in vivo efficacy against HIV-1 in a monotherapy regime [150].
BMS-626529 or temsavir is administered as a prodrug (BMS-663068), a methyl
phosphate prodrug (fostemsavir) which is hydrolized by an esterase, whose
metabolism is equally contributed by the cytochromes [328], the recorded
adverse effects include: headache, rash, and micturition urgency [180];
additionally, this prodrug has shown a better performance in the inhibition of
dually tropic viruses [192]; fostemsavir has very recently been suggested to
be used as a drug in multidrug-resistant HIV-1 infection [130].

####  1.2.2 gp41: fusion, metabolism, and T-cell function suppression.

This protein mediates fusion with the host’s cell membrane [39], portion of
the env protein pertaining to HIV; this protein displays C-terminal helices
which back around N-terminal helices, forming a six-helix bundle [166], this
structure organizes into an extracellular ectodomain, anchoring gp120 to the
cell surface, a transmembrane domain (promoting further anchoring to the lipid
bilayer), and a cytoplasmic tail, this tail contains a highly conserved
endocytosis motif, and three alpha helical motifs known as lentiviral lytic
peptides [80]; where the transmembrane region of gp41 is followed by a region
of high hydrophilicity (the cytoplasmic domain), containing a highly
immunogenic region and a C-terminal region with two amphipathic segments (the
lentiviral lytic peptides , LLPs), as well as a leucine zipper motif between
LLP2 and LLP1; the cytoplasmic domain of gp41 has been suggested to confer
conformational stability to the Env protein, but it has been determined that
the cytoplasmic domain interacts with host proteins: AP1, AP2, CAM, CTNNA,
luman, MA, TP115, perilipin-3, plasma membrane, PRA1, prohibitin 1/2, TAK1.
Furthermore, the cytoplasmic domain has been proved essential for both
replication and incorporation into several cell lines, such as: monocyte-
derived macrophages, peripheral-blood mononuclear cells, B cells, epithelial
carcinoma-derived cells; however, the replication and incorporation processes
have proved independence of the cytoplasmic domain of gp41 in other cell
lines, mostly CD4(+) T cells [208]. Furthermore, gp41 fusion protein works as
an inhibitor of T cell activation by different mechanisms. The
immunosupressive (ISU) sequence impairs T cell activation through the
interaction with the T cell receptor (TCR) complex and the direct inhibition
of protein kinase C mediated phosphorylation; gp41 fusion peptide inhibits
antigen-specific T cell activation by binding to the TCR$\alpha$ transmembrane
domain; gp41 transmembrane domain can enhance the overall immunosupressive
effect through an interaction with the fusion peptide; and the recent
described immunosupressive loop-associated determinant (ISLAD) is another
inhibitor of antigen-specific T cell proliferation and proinflammatory
cytokine release by interacting with the TCR$\alpha$ [12].

Peptides derived from gp41 N-terminal heptad repeat (NHR) and C-terminal
heptad repeat (CHR) sequences can inhibit HIV-1 infection by interaction with
their counterparts in gp41. Some of these peptide fusion inhibitors are:
DP178, later named T20 (generic name: enfuvirtide) was the first fusion
inhibitor approved by the U.S. FDA., but the high cost and inconvenience of
twice daily injection, prevents it from being considered as a regular drug
[34]; FB006M (generic name: albuvirtide) was approved in 2018 by the Chinese
FDA.; SFT (generic name: sifuvirtide) is a novel and potent gp41 inhibitor
that has shown promising results; other gp41 fusion inhibitors under
development and study are: T1249, 2F5, 4E10, C52L, VIR-576, among others
[210].

####  1.2.3 p7 (nucleoprotein, $NC\in gag$): viral genome package and
facilitator.

This protein derives from the Gag precursor which is cleaved into p17, p24,
and p7 proteins [164]. Nucleocapsid protein 7 (NCp7) is the major internal
component of the HIV virion core, it has been seen to be highly conserved
(however not so in spumaretroviruses), displays RNA-binding properties,
containing two zinc fingers (ZF) [62], reminding us that these domains are
maintained by a zinc ion, coordinating a cysteine and a histidine molecule,
these motifs have two $\beta$-sheets and one $\alpha$-helix; it has been seen
that these proteins bind to DNA but may as well bind to RNA [38]. p7 possesses
functions involved in the selection and packaging of the viral genome, as well
as others which play a role in viral replication [164]; NCp7 interacts with
the viral RNA and is required for its dimerization, encapsidation, and
initiation of its reverse transcription, where NCp7 enhances the reverse
transcriptase processivity and RNase H activity [143]; p7 additionally binds
to proviral DNA (by means of its basic residues), and protects it from
nuclease digestion [140]. Interestingly, zinc ejection or mutations affecting
the zinc finger folding and conformation of the nucleocapsid hydrophobic
plateau, lead to non-infectious viral particles [94]. The importance of its
conserved structure is the low probability of mutations found in treatment-
resistant strains [127]. Thus, NCp7 represents a promising therapeutic target
for an effective next-generation antiretroviral therapy. NCp7 inhibitors are
divided into covalent and non-covalent inhibitors. Covalent inhibitors, also
referred to as irreversible inhibitors or zinc ejectors, which can recognise
NCp7 among cellular proteins containing ZFs, some examples of them are
3-Nitrosobenzamides (NOBAs); disulfide-substituted benzamides (DIBAs);
thioesters and pyridinioalkanoyl derivatized thioesters (PATEs);
benzisothiazolone (BITAs); azodicarbonamide (ADA); thiocarbamates (TICAs);
S-acyl 2-mercaptobenzamides (SAMTs); transition metal complexes;
diselenobisbenzamides (DISeBAs); and more recently, thioether prodrugs. All of
these covalent inhibitors are structurally characterised by a weak
electrophilic group that is attached by the distal ZF domain, after this
covalent complex has been formed, the zinc is ejected, causing the loss of the
tertiary protein structure and consequently, all of its functions [282, 238].
Non-covalent inhibitors and nucleic acid (NA) binders are another therapeutic
target against NCp7. These compounds have weaker antiviral potency when
compared with covalent binders, and they have not been approved for clinical
trials. Several interactions in the NA/NCp7 complexes are involved with the
W37 hydrophobic plateau residue, offering a chance to develop competitive
inhibitors. Some of these are pseudodinucleotides, HTS-derived small
molecules, thiadiazoles, and thiazolidinones. All of these molecules are
characterized by a $\pi$-rich area capable of interacting with W37, seeking to
avoid the interaction of NCp7 with NAs. On the other hand, NA-binding NCp7
inhibitors include stem loop structure-binders and anthraquinones, which block
the NCp7-RNA-DNA complex formation, but most of them are unable to disrupt a
preformed complex [117].

####  1.2.4 p6 ($p6\in gag$): incorporating vpr into salient viral particles.

This protein, also a byproduct of the Gag precursor cleavage, promotes virus
particle budding, and the incorporation of the vpr protein into the viral
particles [122]; inclusively, p6 is a factor involved in the capsid maturation
and virus core formation processes, this by means of its phosphoprotein
features; these functions are performed due to the close Euclidean distance
which is encountered between p6 and the plasma membrane of the cells, this
protein may be adsorbed onto the inner surface of the plasma membrane and
promote, for instance, the incorporation of vpr into the viral particles. This
protein possesses two $\alpha$-helical domains which are connected by a
flexible region, this structure is more pronounced in hydrophobic conditions;
its C-terminal region contains vpr-binding residues; the Ser40 residue has
been seen to be a potential protein kinase C phosphorylation site [194].
TSG101 (tumor susceptibility gene 101) is a key cellular protein as part of
the endosomal sorting complexes required for transport (ESCRT), which is
recruited to viral assembly sites via p6, where the Pro-Thr-Ala-Pro (PTAP)
motif in p6 acts as a docking site for TSG101. This process is critical for
HIV release. The duplication of this PTAP motif has shown an enhanced
replication advantage of HIV-1 subtype C by engaging TSG101 with a higher
affinity [250]. Furthermore, the deletion of the YPxnL motif, which binds to
ESCRT component ALG-2-interacting protein X (ALIX), is associated with a
decrease in virus release from infected cells (as seen in HIV-1 subtype C),
and conversely, PYxE motif insertion can reconstitute the p6 binding to ALIX
and consequently, viral budding mediated through the ESCRT pathway [68].
Depletion of the ESCRT components have shown a powerful block to HIV particle
release [259], for this reason, therapeutic targets are principally focused on
disrupting the p6-TSG101 interface, such as peptoid hydrazones, cyclic peptide
11, F15 (esomeprazole) and N16 (tenatoprazole), these last two are able to
join to the ubiquitin E2 variant domain of TSG101, highlighting the
possibility of interfering with previously unknown therapeutic targets and
expanding the future perspectives of TSG101 inhibitors [66].

####  1.2.5 p24 (capsid protein, $CA\in gag$): protector of the viral genome.

Capsid protein (CA) is member of the subset of proteins derived from the Gag
polyprotein cleavage, a $[24,25]kDa$ protein which may be detected before
seroconversion, this protein may assemble into a protective shell around the
viral RNA [99] which is a spontaneous process [35], this has been described as
a ”fullerene cone”, hexamers of p24 link into an hexagonal surface lattice,
with 12 capsid-protein pentamers, finally possessing around 1500 p24 monomers.
This protein is comprised by seven $\alpha$-helices, a $\beta$-hairpin
(N-terminal region), a C-terminal region with four $\alpha$-helices, and a
flexible linker [321], when assembled into the cone, the N-terminal domain is
located on the outer surface of the cone and the C-terminal domain is oriented
towards its interior; inside this cone, the RNA genome, and the POL proteins
are located (that is the integrase, protease, reverse transcriptase and
others); the cone may act in order to shield the genetic content from a host
response [35], such as those induced by STING, DDX14 or IFI16 acting as
foreign DNA sensors [289]. Several models have been proposed in regards to the
viral uncoating, that is, the dissociation of the capsid cone into monomers or
simpler polymers: immediate uncoating, cytoplasmic uncoating and nuclear pore
complex uncoating; there exist studies which support each of these models, in
the first case, the uncoating occurs rapidly after the entry into the host’s
cell has been performed, however, this model has lost its predictive power due
to the fact that the core provides protection against the host’s foreign-DNA
sensors, furthermore, the capsid possesses a pocket to which the reverse
transcriptase complex binds to and allows it to be imported into the nucleus.
The second model, the one of cytoplasmic uncoating proposes the disassembly of
the capsid after a certain time interval has been spanned within the host’s
cell, this model may be nevertheless challenged as well by the factors which
we have already commented (foreign DNA host’s response), moreover some
additional cellular factors may protect the viral’s genome from the host’s
response, it may occur by means of the HMGA1 protein (high mobility group AT-
hook, also called HMGIY), which has been recollected from the preintegration
complexes, that is the reverse transcriptase complex before they integrate
into the host’s genome [75]. The third model refers to the nuclear pore
complex (NPC) uncoating, in which when the intact capsid reaches the nuclear
pores, and disassembles in situ, while the reverse transcriptase complex is
imported into the nucleus; this model displays another range of problems which
are not compatible entirely with the experimental data, ergo a more suitable
model should be provided in following years. Both viral and host’s factors
take part in the uncoating process: PPIA (or CYPA, a peptidylpropyl isomerase
A, with a native function of accelerating the folding of proteins and
isomerize the proline imidic bonds [188]), dynein (cytoplasmic trafficking),
CPSF6 (nuclear import, which is a cleavage and polyadenylation specific
factor, interacts with RNA [186]), TNPO3 (nuclear import, this protein is
member of the importin-$\beta$ family of proteins, has been seen to bind to
the viral integrase, and as well interacts with the capsid protein), NUP358,
NUP153 (nuclear import) [35]. Maturation inhibitors are a novel class of
antiretroviral drugs targeting the cleavage site between the C-terminal
portion of CA and the spacer peptide 1 (SP1), this cleavage site usually
triggers a conformational switch that destabilizes the immature Gag and the
mature core formation. Maturation inhibitors cause an accumulation of CA-SP1
precursor, which eventually leads to the loss of viral infectivity [66]. The
first reported maturation inhibitor was Bevirimat (BVM), which causes an
abnormal virion morphology and inhibition of viral replication, but failed in
the phase IIb trial due to resistance mutations in CA-SP1. A second compound
named PF-46396 was identified, it shows structural differences compared with
BVM, but it induced resistance mutations at different locations, this lead to
the identification of second-generation maturation inhibitors, such as
GSK3532795, which successfully overcame the inconvenience of drug resistance,
but showed a high rate of adverse gastrointestinal events and frequency of
treatment-emergent nucleoside reverse transcriptase inhibitor (NRTI)
resistance, reason why its evaluation was interrupted [172]. However, the
promising results obtained support the continued development of drugs against
this therapeutic target. Small molecules and peptide-based antivirals designed
to disrupt CA-CA interactions in the immature Gag lattice, the mature core, or
both have been studied during the last years. CAP-1 was the first small
molecule developed to target the CA protein, it produced abnormal core
morphologies, and consequently noninfectious particles. Besides small
molecules compounds, 12-mer peptide CA inhibitor (CAI), binds in a hydrophobic
CA dimerization interface, but it can not penetrate cell membranes, limiting
its clinical use. Some more stable $\alpha$-helical peptides, such as NYAD-1
and NYAD-13, showed a stronger affinity for the binding site than CAI. Other
classes of CA inhibitors are benzodiazepienes and benzimidazoles. PF74 and the
pyrrolopyrazolones BI-1 and BI-2, seem to compete with CPSF6 and Nup153 for CA
binding, disrupting nuclear import. The main problem of the compounds
described so far, is the low clinical relevance due to their pharmacological
characteristics. Recently, a new type of CA inhibitors has been described,
this group includes GS-CA1 and its derivative, GS-6207, are promising drugs
that have showed higher potency than PF74 [66].

####  1.2.6 p17 (matrix protein, $MA\in gag$): multifunctional protein.

The HIV-1 p17 protein (matrix protein, MA) is associated with the inner
surface of the viral envelope, and may primarily function as an anchor of the
gp41 protein on the virion surface. When in solution it is mainly encountered
in a monomeric form, while in a solid state it trimerizes [163]; it displays
five $\alpha$-helices, and a three-strand $\beta$-sheet, whereby the
C-terminal domain, exposes carboxyl-terminal residues which aid in the early
states of HIV infection, and basic residues promote membrane binding and
nuclear localization (the residues which take part in this process are located
in a cationic loop connecting $\beta$-strands one and two). This protein
additionally functions in RNA targeting to the plasma membrane, incorporation
of the envelope into virions and particle assembly, and aids in the transport
of the reverse transcriptase complex through the nuclear pores [162], it also
acts as a viral cytokine, by binding to a cellular receptor, namely p17R [83],
this receptor has been seen to be expressed in Raji B cells, and activates
AP-1, as well as ERK1/2 and downregulates AKT, this by means of maintaining
PTEN in its active state through the serine/threonin kinase ROCK [45]; the
identity of this p17R protein is a bit obscured in the literature, nonetheless
through a non-exhaustive search in regards to Raji cells’ receptors we came up
with a list of potential proteins which may play such role, bearing in mind as
well that the H9 cell line lacks this p17R protein in its surface, if we
consider that the surface proteins of the Raji cells are members of the set
$R=\\{p_{1},...,p_{n}\\}$ and those of the H9 cell line pertain to the set
$H=\\{r_{1},...,r_{m}\\}$, therefore $\exists p_{i}\vee r_{i}:p_{i}\in
R,p_{i}\notin H$ [231]; those proteins in the set $R\cap H$ are: PVRL1,
LILRB1, DRD4, CRLF3, and ADGRD2; such that these receptors may not be the p17R
protein; now, those proteins which are exclusively expressed in Raji cells and
take part in PI3K/AKT/PTEN/AP1/GPC (G-protein coupling) are: NTSR2, TACR3,
FOLR2, MC5R, OR2J3, TNFRS13B, PTAFR, CCR8, GPR173; this sets are shown in
Figure 1; the activation of this pathway (by means of this p17R) may lead to a
promotion in proliferation and release of proinflammatory cytokines from
T-cells [83]; but if p17R is also expressed in B cells, it ought indeed to
promote B cell growth and tumorigenesis [45]. MA protein has a fundamental
role in virion assembly, because of its highly conserved PI[4,5]P2/nucleic
acid binding site, it has become an attractive site for the development of new
antiretroviral drugs, nevertheless, other drugs have been described targeting
the nuclear localization signal of MA or the MA-RNA interaction. Thiadiazolane
based compounds where first described, they target the MA-RNA interaction, but
showed significant levels of toxicity. In contrast, PI[4,5]P2 binding site
inhibitors were not associated with cytotoxic effects, but work is actively
ongoing in optimizing the affinity/potency of this type of chemotypes. New
targeting sites are highly desirable, one of them could be the involvement of
MA in Env incorporation, where MA trimerization is important for the
recognition of Env cytoplasmic tail (gp41-CT) and virus assembly [66].

Figure 1: Venn diagram of the sets representing the receptors expressed in
Raji cells (Raji) and H9 cells (H9), where we regard at their cardinalities,
that is, $|R|=28$, $|H|=68$, and $R\cap H=6$; in this case the p17R protein
lies in the $R-H$ set.

####  1.2.7 p10 (protease, $PR\in POL$): catalyzer of viral and host’s
proteins breakdown.

This protease is encoded by the pol gene being an aspartic protease, and as in
other retroviruses, displays its function when in its homodimeric form; each
monomer is an aspartic peptidase with four elements: two hairpin loops, wide
catalytic aspartic acid loop, and an $\alpha$ helix [70], the dimeric
interface comprises eight N- and C-terminal residues of each chain, some of
these are exposed to the solvent, and some others (the hydrophobic ones) are
oriented towards the interior of the enzyme [320]. This protein displays two
states (when in the homodimer form), either open or closed, depending on the
presence of a ligand (open when in the free state, and closed when in the
bound state) [308] where the open form is more stable than the closed one
(considering the free energy of the process $\Delta G$) [320]. The HIV
protease cleaves the polyproteins Gag and Gag-Pol creating protein subunits
[167]; all the same the viral protease displays proteolytic abilities of the
host’s proteins, such as those belonging to: the cytoskeleton: vimentin,
desmin, GFAP [256], actin, troponin [277], laminin [182]; immune system:
proIL1B; and transmembrane proteins: APP [277]; cytosolic proteins: BCL2,
CASP8 [182]. Vimentin plays a role in the diapedesis process of T cells,
whereby this protein, T cells migrate across the endothelium, and in genotypes
$vim^{-/-}$, there is a diminished capacity of T cells to home to mesenteric
lymph nodes and spleen, this may be mediated by changes in expression or
distribution of ICAM1 and VCAM1 [183]; while actin polymerizes or
depolymerizes when the T cell becomes active, this T cell activation leads
towards the formation of a distal-pole complex which is an actin-rich
structure; disruptions in the cytoskeleton might as well induce changes in the
organization of the supramolecular activation clusters in T cells (SMACs) [21]
which takes a central part in the immunological synapse, being the SMAC a
nanocluster comprised of an actin mesh, transmembrane proteins (members of the
TCR complex), and cholesterol-enriched nanodomains [89]; these disruption in
the cytoskeletal proteins might be added to the one induced by the gp120
protein, as this protein increases ICAM1 expression [221, 273], inducing the
formation of a type of SMAC but in this case gp120-dependent (and likely
protease-aided), in which gp120 clusters in the centre, and LFA1 and ICAM1
concentrate in the periphery, this type of SMAC is termed the virological
synapse, throughout which the virus may spread between T cells without the
need of being externalized from the cell (no need of viral budding) [159];
this SMAC recruits the host’s proteins which we may as well find in the
immunological synapse, such as: TCRZ, ZAP70, LAT, SLP76, ITK, PLCG, and only a
weak recruitment of: CD3E, and ABTCR; however when this protein nanocluster
was assembled, there was an abnormal signaling (distinct to the one induced by
the immunological synapse), where no PKC recruitment, no calcium mobilization
nor CD69 upregulation were observed [284]. By means of the cleavage of both
BCL2 and CASP8 can the protease induce cell death [182, 142]. HIV-1 protease
inhibitors (PI) play a key role in the antiretroviral treatment. First
generation of PIs were based on hydroxyethylene and hydroxyethylamine
isosteresstarted. Saquinavir was the first PI approved by the FDA in 1995,
since then many other PIs have been developed. Ritonavir was found to be a
potent inhibitor of cytochrome P450 3A, a major metabolic enzyme for PIs
[133], due to this finding, ritonavir is more frequently used as a PI
pharmacokinetic booster than as a PI itself. Other first generation
antiretrovirals are indinavir, nelfinavir, and amprenavir. Due to the major
problems of first-generation PIs (high metabolic clearance, low half-life,
poor oral bioavailability, gastrointestinal distress, and the emergence of
drug-resistance strains), second-generation PIs were developed. The balance of
hydrophobicity and hydrophilicity allowed for a longer half-life. Some of
these drugs are lopinavir, atazanavir, tipranavir (reserved as a salvage
therapy), and darunavir (high potency against multidrug-resistance strains)
[92]. Despite the progress in PIs therapy, some drug-resistance variants have
emerged, they are classified in two groups: primary mutations, which involve
changes in residues directly involved in substrate binding and manifest
themselves in the active site of the enzyme; while secondary mutations are
located away from the active site and are usually compensatory mutations to
mitigate the deleterious effects of primary mutations [171]. Even, secondary
mutations may not manifest in the protease itself but instead in the protein
cleavage site on the Gag-Pol and Gag substrates [202]. One of the main
strategies to avoid drug resistance, is the design of PIs by promoting
hydrogen bonding interactions with the backbone atoms in the HIV-1 protease
active site, since mutations that cause drug resistance cannot significantly
alter protease active site backbone conformation, as an example we have
darunavir, that showed extensive binding interactions with backbone atoms and
maintained a potent antiviral activity against panels of clinically relevant
multidrug-resistant HIV-1 variants [92, 93]. Recently, a new pharmacokinetic
enhancer, cobicistat, which does not have anti-HIV activity, has been
developed, offering the advantage over ritonavir, where cobicistat will not
contribute toward the emergence of drug-resistant HIV-1 variants [305]. Many
other new classes of PIs with innovative ligands are under preclinical
development for the next generation, molecular design efforts have focused on
the synthesis of P2-ligands promoting enhanced backbone binding and non-
peptide PIs containing different structural scaffolds distinct from
hydroxyethylsulfonamide isosteres [92].

####  1.2.8 p51 (reverse transcriptase, $RT\in POL$): loader of nucleic acids
and structural support.

The reverse transcriptase’s main function is to convert the viral RNA which
has just entered the host’s cell after membrane fusion into double-stranded
DNA, this occurs in the host’s cell cytoplasm, this DNA will later be
translocated towards the nucleus in order for it to be finally integrated into
the host’s genome. It is a heterodimer comprised of p66 (or p15 or RNase H)
and p51, both of which derive from a common Gag-Pol polyprotein, cleaved by
the viral protease (p10, PR); these subunits (p66 and p51) share a common
amino terminus; the details of p15 will be addressed in the following section
1.2.9. p51 possesses 4 subdomains (which are homonymous to those of p15 or
p66): fingers, palm, thumb, and connection [240]. The heterodimerization
process is dependent on factors such as: mutations (e.g., the L289K mutation,
the L289 residue in the p51 subunit makes various hydrophobic contacts with
p66 but the mutation does not yield towards an ill-heterodimerization or no
heterodimerization at all; contrarywise, the same mutation but occurring at
the protein p66 yields towards a lack of dimerization), nucleotide substrates,
temperature, magnesium, and solution conditions [324]. It has been noted that
the catalytic site of the reverse transcriptase lies in the p15 (p66) domain,
however p51 may take part in: structural support and facilitating the loading
of nucleic acids onto p66, which would explain why the N348I mutation to p51
may confer resistance to reverse transcriptase inhibitors (both nucleoside and
non-nucleoside), as well, p51 has been seen to contribute to the architecture
of the RNase H primer grip/phosphate binding pocket, finally, deletions to the
C-terminal sequence of p51 may alter changes in the RNase H activity [44]. The
p51 subunit in itself may undergo homodimerization under certain conditions,
this was observed by small-angle X-ray scattering (SAXS), this phenomenon
occurs dependent on the concentration of the p51 monomer, and may coexist with
the homodimer p51/p51 in equilibrium, furthermore it has been noted that p51
may exist in two forms: p51E and p51C, the latter is the form in which it may
be encountered when bound to the p66 in the heterodimer, and the former is an
extended form in which it may resemble the structure of p66, the
homodimerization may occur with one p51E species and one p51C species
assemblying into p51E/p51C, this could happen due to the stabilizing effect of
p51C on the structure of p51E. p51 interacts with non-nucleoside reverse
transcriptase inhibitors (NNRTIs) forming two different species: p51E-NNRTI
and p51E/p51C-NNRTI, this due to the fact that p51E is in possession of a
p66-like structure [324]; the monomeric form of p51 is favored by low
concentration, low salt, and low temperature conditions [325]. Reverse
transcriptase inhibitors are divided into nucleoside/nucleotide and non-
nucleoside reverse transcriptase inhibitors, NRTIs and NNRTIs, respectively.
NRTIs inhibit viral replication through competition with purines and
pyrimidines, avoiding adding new nucleotides and consequently, finishing viral
DNA replication. In contrast, NNRTIs bind to an hydrophobic pocket in p66,
this binding leads to a stereo-chemical change in the protein, preventing the
addition of new nucleosides and blocks the cDNA elongation. Typically, NRTIs
constitute the backbone of the antiretroviral therapy, among them we can find
tenofovir disoproxil fumarate (TDF) and tenofovir alafenamide (TAF), which are
adenosine-derivated; emtricitabine (FTC) and lamivudine as cytosine analogs;
abacavir as guanosine analog; zidovudine and stavudine as thymidine analogs;
and didanosine as an inosine derived. An advantage of the current NRTIs is the
low clinically significant drug-drug interactions but they have the
disadvantage of significant side effect profiles and problems with resistant
HIV-1 variants [216]. All of them can be affected by selected resistance
mutations, either by mutations in the N-terminal polymerase domain of the
enzyme, where the most common are K65R, L74V, Q151M, and M184V; or by
thymidine analog mutations, where are included M41L, D67N, K70R, L210W,
T215Y/F, and K219Q/E [216, 155, 11]. Among NNRTIs, we can find the first
generation (efavirenz and nevirapine) and second generation (rilpivirine and
etravirine). Resistance mutations against NNRTIs are based on the inhibited
drug interaction with the NNRTI binding pocket (as seen with K103N and K101E),
disruption between the drug and NNRTI binding domain residues (Y181C and
Y188L), or changes in the conformation or size of the NNRTI binding pocket
(Y188L and G190E)[4, 61, 60, 220]. Unlike NRTIs, NNRTIs have a higher plasma
half-life, so they can act as perpetrators of drug interactions additionally
to their significant side-effect profile [216]. New NRTIs and NNRTIs are
focused on overcoming resistances and reduce side effects [215]. Islatravir
(MK-8591) is a nucleoside nonobligate chain terminator that inhibits the
enzyme by preventing its translocation, and therefore it has been categorized
as a nucleoside reverse transcriptase translocation inhibitor (NRTTI),
demostrating a robust antiviral activity in vivo against HIV-1 [156].
Doravirine (MK-1439), a novel NNRTI, has showed its potency at low doses
against common resistant strains and has less potential drug interactions than
its counterparts because it does not have appreciable inductive or inhibitory
effects on CYP enzymes [23]. Elsulfavirine is the prodrug of the active
compound VM-1500A, it has shown a high genetic barrier to the development of
resistant drug mutations [318]. A study showed that elsulfavirine was not
inferior to efavirenz combined with TDF/FTC in virologic control, and in
addition, elsulfavirine was better tolerated [176].

####  1.2.9 p15 (RNase H, $(p15\vee p66\in pol$): converter of RNA into dsDNA.

We have previously commented that the reverse transcriptase protein is
comprised of two subunits: p51 and p66 (or p15), where the enzymatic activity
of the protein is performed by the p15 subunit, this enzymatic activity is
performed by both the polymerase and by the RNase H, the polymerase domain
contains 4 subdomains: fingers, palm, thumb, and connection [240]. The RNase H
belongs to a nucleotidyl-transferase superfamily, this family includes enzymes
such as: transposase, retroviral integrase, Holliday junction resolvase, and
RISC nuclease Argonaute. In retroviruses, this RNase H converts an ssRNA into
dsDNA, removes the template RNA after the DNA synthesis and produces a
polypurine primer for the second-DNA-strand synthesis. In reactions catalyzed
by the RNase H, the nucleophile is derived from a water molecule or a 3’-OH
from the nucleic acid; the active site of these enzymes is dependent on
magnesium or manganese to bind to the substrate and catalyze the reactions,
these reactions occur in a one-step bimolecular nucleophilic substitution; two
metallic ions are required for the reactions, in which one metal ion activates
the hydroxyl nucleophile and the other one stabilizes the pentacovalent
intermediate. Various RNases display a conserved $\sim 100$ residue core
structure: five stranded $\beta$ sheet and three $\alpha$ helices (A,B,D).
Specificity for the RNA strand is determined by contacts between the RNase H
and five consecutive 2’-OH groups, and when the RNA complexes (around 6 bp)
with the enzyme, around 100 nm of the enzymatic surface becomes buried, a
groove which protrudes near the site of the buried surface is the active site,
where catalytic carboxylates (E109 and D132) make hydrogen bonds with 2’-OH
groups. The DNA binding site is located in the alpha helices (polymerase),
which fold into a groove, a phosphate binding pocket accompanies this binding
site (comprised of the amino acids T104, N106, S147, T148); the pocket when
not in enzymatic activity is occupied by a sulfate ion, which is replaced by
the DNA’s phosphate [193]. It has been noted that the manganese may
specifically promote a specific hydrolytic activity (hydrolysis of dsRNA) to
the enzyme, rather than the more general endonuclease and directional
processing activities, divalent metal binding may be mediated by the p66
residue E478 [46]. The initial RNAse H inhibitors were described almost 30
years ago, however, the first inhibitor with a relevant effect was N-(4-tert-
butylbenzoyl)-2-hydroxy-1-naphthaldehyde hydrazine (BBNH), although it was
also an inhibitor of the RT DNA polymerase [26]. After, a derivative,
(E)-3,4-dihydroxy-N’-((2-methoxynaphthalen-1-yl)methylene)benzohydrazide
(DHBNH), which has its binding site between the RT polymerase active site and
the polymerase primer grip, alters the trajectory of the template-primer, so
the RNAse H cannot cleave the RNA strand in RNA/DNA complexes [108];
metalchelating RNase H active-site inhibitors, such as 1-hydroxy-
pyridopyrimidinone, pyridopyrimidinone, dihydroxycoumarin, diketo acids,
$\beta$-keto acids, 3-hydroxypyrimidine-2,4-diones and
hydroxypyridonecarboxylic acids, might sequester Mg+2 ions required for RNAse
H activity [279, 293]. An alternative to metalchelating inhibitors are
allosteric inhibitors that interfere with the RNA-DNA binding and induce
changes in the RNAse H active site, examples of these are the
cycloheptathiophene-3-carboxamide (cHTC) derivatives [293]. Several other
inhibitors have been proposed and developed such as the dual inhibitors
RDS1643 (co-targeting HIV-1 IN and RNAse H), EMAC2005 (co-targeting HIV-1 RNA-
dependent DNA polymerase and RNAse H), and RMNC6 (co-targeting RT and RNAse H)
[293]. However, despite extensive research and improvement in RNAse H
inhibitors, the efficacy of these molecules remains low [279].

####  1.2.10 p32 (Integrase, $IN\in pol$): immersing viral DNA into the host’s
genome.

After the dsDNA has been produced in the reverse transcriptase enzyme, this
DNA is part of a large nucleoprotein complex, the reverse transcription
complex [54], this DNA shall later transite from the reverse transcriptase
towards the pre-integration complex (PIC) which contains the integrase enzyme,
as well as the viral proteins: vpr, capsid, and matrix, and the host proteins:
INI1 and PML [116], moreover, it has been noted that vpr-defective viral
particles fail for their PICs to be effectively translocated to the nucleus,
whereas matrix-defective viral particles are only partially effective in the
transport process, this could be related to the finding that vpr binds to KPNA
(karyopherin alpha), the ideal concentration of the vpr protein in order for
this to occur is $<5nM$, for when concentrations of the vpr protein such as
$>42.8nM$ happen, the transport mechanism is inhibited [207]; this pre-
integration complex will afterwards be transported towards the nucleus by
active mechanisms (due to the size impediment of the PIC, being this complex
around 28 nm wide, versus 9 nm as the limit of passive transport towards the
nucleus), the active transport mechanism is dependent on the capsid protein
(CA) from the HIV-1 particle, as well as various nuclear transport proteins:
NUP153, NUP358, CPSF6, TNPO3, and ADAM10 [73]. The integration of the viral
genome into that of the host’s seems partially non-random, due to the fact
that the viral genome has been seen to prefer transcriptionally active regions
of the host’s genome; the integrase enzyme also displays preference towards
chromatinised DNA substrates rather than naked ones, i.e., the viral genome
integrating process can be described by means of a stochastic process
$\\{S_{x}\\}$, where the x represents the host’s DNA fragment, the probability
of the viral DNA being bound to the genome in a position near the nucleosome
increases 4-fold, and when this position shifts towards naked DNA, the
probability distribution becomes a uniform one. The efficiency of integration
has also been computed to increase when the chromatin is found in an open
fashion (euchromatin) and not in its closed one (heterochromatin) [168];
furthermore, not only does the viral DNA prefers transcriptionally active
sites, but as well as intronic genetic ones. The viral DNA integration is
aided partially by the host, through a protein product of the PSIP1 gene, say
the LEDGF/p75 protein which belongs to the hepatoma-derived growth factor-
related proteins, which contains a conserved N-terminal PWWP (this domain is a
member of the Royal superfamily, functions as a chromatin methylation reader,
by recognizing methylated DNA or histones, through its high content of basic
residues, thus raising its isoelectric point and facilitating DNA interactions
[228]) and A/T hook domains (this is a DNA binding peptide motif, with a Gly-
Arg-Pro tripeptide, where we may find in its sides basic amino acids, these
DNA binding motifs, that might as well bind to RNA [82]) [5]. Available
integrase inhibitors include bictegravir, cabotegravir, dolutegravir,
elvitegravir, and raltegravir, that may be used in combination with other
antiretrovirals (cobicistat, emtricitabine, tenofovir, rilpivirine, and
others) [243]; bictegravir tends to be used along with emtricitabine and
tenofovir alafenamide, (BET) in which case bictegravir displays a high genetic
barrier in HIV-1 resistance development, this combination of antiretrovirals
was similar or superior to the one of dolutegravir, abacavir, and lamivudine
(DAL); where in naïve adults, BET established virological suppression in
treatment during 96 weeks [63]; bictegravir inhibits the integrase with an
$IC_{50}=7.5\pm 0.3nM$ [280]. Cabotegravir, dolutegravir, and bictegravir have
been seen to most efficiently inhibit B HIV-1 subtypes, compared to non-B
subtypes [179].

####  1.2.11 vpr (virus protein r): nuclear importer of viral genome.

This protein is a 96 amino acid, 14 kDa protein, which is conserved among
other retroviruses [128]. The structure of the vpr protein is featured by
three $\alpha$-helices, with the N- and C-terminal domains surrounding these
helices [95], this protein has been seen to exists as an oligomer being the
amino acids 36-76 of essence for this polymerization to occur [322], in vivo
both dimers and hexamers have been observed, where in the dimeric state only
helices II and III of the protein’s structure are involved, while in the
hexameric state, helix I also takes part [285]. Its functions include: the
nuclear import of the preintegration complex [281], induction of G2 cell cycle
arrest, modulation of T-cell apoptosis, transcriptional activation of both
viral and host genes, regulation of NFKB activity [128]; these functions are
regulated by distinct domains of the vpr protein, the N-terminal domain may
take part in nuclear localization and virion packaging, the LR domain is
involved in the nuclear localization of vpr, and the C-terminal domain
regulates cell cycle arrest [153]. Cell cycle arrest (in G2-M phase) is
described as one of the most dramatic functions of vpr, process occurring with
the aid of some host’s proteins, for instance: DCAF1, CRL4, which includes
CUL4A, RBX1, and DDB1 [303]; DCAF1 can promote ubiquitylation of proteins
through CRL4 and another E3 ligase, assemblying later with another protein
into a complex, DCAF1-DDB1-DYRK2-EDD; it has been determined that DCAF1
regulates cell-cycle progression and apoptosis either with TP53 or without it,
where DCAF1 negatively regulates TP53, as well as some TP53 target genes
(TP21, TP27), while another E3 ligase, MDM2 can promote TP53’s ubiquitylation,
inducing then its degradation; DCAF1 not only acts directly on TP53 but can
additionally alter its epigenetic landscape through an interaction with HDAC1,
then recruiting this deacetylase towards TP53-dependent promoters [244];
nonetheless TP53 is not the only protein whose expression is affected by
DCAF1, for MUS81 and MCM10 are also affected by it [303], where MUS81 is a
Holliday junction resolvase, regulating cell growth and meiotic recombination
[102], and MCM10 is a protein essential for the initiation of chromosome
replication containing a single-stranded DNA binding domain, this protein
complexes with CDC45 and GINS proteins which togetherly display helicase
activity [297] (ATP-dependent DNA unwinding enzyme [214]). Interestingly, vpr
can alter the host’s transcriptome, this done through ubiquitination of HDAC1,
HDAC2, HDAC3, and HDAC8, being HDAC3 the most affected one; this has a two-
fold response: enhance the viral genome expression and that of the host [227],
some of the host’s proteins whose expression has been seen to become altered
by vpr are: ANT, VDAC, PMCA, GLUD2 (mitochondrial), HK1 (glucose metabolism),
G6PDH (pentose phosphate pathway), GAPDH (glycolysis), ATR (stress response),
DCAF1 (proteasome), NFAT, NFKB, CEBP, AP1, HIF1A, HDAC1, GR, SP1
(transcription), SLX4, HLTF, and UNG2 (DNA metabolism) [95]; therefore, vpr
possesses the potential to radically modify the host’s T-cell transcriptome.
Some naturally occurring compounds have been recorded to exhibit inhibitory
activity against vpr, such as isopimarane diterpenoids and picrasane
quassinoids [300]; while coumarin compounds in in silico experiments have
yielded information that these compounds may bind to the hydrophobic pocket in
vpr and could inhibit the protein [42]; a novel hematoxylin derivative was
developed, this compound binds to vpr and can inhibit HIV-1 replication, this
hematoxylin derivative can be produced by reacting the hematoxylin molecule
with 2,2-dimethoxypropane in acetone with p-toluene sulphonic acid and
phosphorus pentoxide, with an intermediate molecule being produced, the
acetonide-protected hematoxylin, this is reacted with allyl bromide with
$K_{2}CO_{3}$ in dimethylformamide, where we might then obtain the stable
hematoxylin derivative [103].

####  1.2.12 vpu (virus protein unique, p16): CD4 downregulator.

This protein is an 81-amino acid type I transmembrane protein, pertains to the
viroporin family of proteins; this protein displays certain functions:
downregulates CD4 receptor expression, promotes release of virions from
infected cells; this protein is translated from the gene which also encodes
for the env complex [97], and it has been suggested that the expression of env
and vpu are coordinated; the vpu gene lies between the first exon of the tat
and env genes, this gene is absent in HIV-2. Vpu may assemble into
homooligomers, and the monomers exhibits an N-terminal domain (luminal), a
single transmembrane domain, and C-terminal hydrophilic domain (into the
cytoplasm); where this transmembrane domain appears essential for the
oligomerization of Vpu, and a stoichiometry of $(Vpu)_{n},n=5$ is required for
the formation of the pore [69]; there have been experimental suggestions in
regards to the ion channel activity of this protein, such as that result of a
selective conductance for cations of this protein (where vpu has been
expressed in frog oocytes), this protein then has been compared with the M2
protein of the influenza virus, because vpu possesses a similar transmembrane
topology, an uncleaved transmembrane domain, similar length, and casein kinase
II phosphorylation sites; features whereby the M2 protein has been seen to
transport protons, in this protein a motif within the transmembrane domain has
been proved essential for its ion channel activity, the His-X-X-X-Trp motif,
in this case, His is the proton sensor and Trp is the pore; when drugs which
targeted the transmembrane domain of vpu, the release of virions could be
decreased, additionally, the vpu protein displays an Ala-X-X-X-Trp motif in a
very similar position as the previously mentioned motif does in the M2
protein; nonetheless, these discoveries while they indeed proof the necessity
of this protein by the virus in releasing the virions mainly through its
transmembrane domain, its role as an ion channel has not yet been fully proved
[234], but this protein may alter potassium transport, with a possible role as
an ion channel and additionally with an increased TASK1/KCNK3 protein
degradation (KCNK3 is a member of the superfamily of potassium channel
proteins, it is activated by halothane and isoflurane [187]) which can lead to
membrane depolarization and viral release [97]. The mechanism through which
vpu promotes CD4 downregulation is by means of targeting newly synthesized CD4
molecules for proteasomal degradation, promoting the binding of CD4 to the
SKP1-CUL1 E3 ubiquitin ligase which allows for the ubiquitination of CD4 on
lysine, serine, and threonine residues, then enabling the retention of the CD4
protein in the endoplasmic reticulum, being afterwards recruited the VCP-
UFD1-NPL4 dislocase complex (NPL4 tends to recognize Lys48-linked
polyubiquitinylated substrates [242], VCP extracts ubiquitinated proteins from
lipid membranes [189]), finally leading to degradation by means of the
proteasome. Downregulation of CD4 is not accomplished purely by the action of
the vpu protein, but instead it is a constructive result of vpu, nef, and env
proteins [232]. If the viral particle uses CD4 as a main receptor for its
entry into the host’s cell, why would it downregulate this molecule? Because
the unlimited binding of the viral particle to the CD4 molecules would lead to
superinfection (acquisition of different viral strains infecting a single
T-cell [20]) of a T-cell, and superinfection of T-cells leads to increased
apoptotic rate, by diminishing the available CD4 molecules for the viral
particle to bind to, this superinfection is in its stead decreased, and
consequently, the likelihood of increased T-cell apoptosis, which can lead to
a more everlasting viral infection [298]. Downregulation of the CD4 molecule
expression on the membrane surface could also serve the virus in promoting the
virion release from the cell surface (due to the finding that CD4 may bind to
the env protein in virions, as well as CD4 prevents the env protein from
inserting itself into the virion surface) [270]. In vpu-defective virions,
where the viral release was still allowed, when treated with type 1
interferons, these viral particles were no longer released, and were tethered
onto the host’s cell surface; it was later observed that tetherin (BST2) was
one of the proteins involved in this process [69], whose expression is
promoted by type I interferons [209]. Vpu aids virion release and avoidance of
virion tethering onto the cellular surface by ubiquitinating BST2 at the
cytoplasmic lysine residues 18 and 21, likely promoting subsequently
proteasomal degradation, nevertheless, lysosomal degradation is also possible
and has been docummmented (because lysosomal inhibitors have been seen to
inhibit vpu-mediated BST2 degradation) [69]. Tetherin displays this
restraining activity in regards to viral release from the host’s cell not only
in HIV1, but as well in: HIV2, SIV (simian immunodefficiency virus), alpha-,
beta-, and gammaretroviruses; filoviruses (Marburg and ebola viruses),
arenaviruses (Lassa virus), herpesvirus (type 8 or associated with Kaposi’s
sarcoma); thereof can tetherin be regarded as an antiviral molecule [131]. A
molecule BIT225 (2-naphthalenecarboxamide,
N-(aminoiminomethyl)-5-(1-methyl-1H-pyrazol-4-yl)) has been described as a vpu
inhibitor regarding its viroporin function, and can also work as a hepatitis C
virus protein 7 inhibitor [132], SM111
(1-(2-(azepan-1-yl))nicotinoyl)guanidine) has also been described as a
potential vpu inhibitor, for HIV-1 replication diminishes when SM111 comes
into play even with protease, reverse transcriptase, and integrase inhibitors-
resistant strains [177]; another class of molecules which show promise in
inhibiting vpu function, are phlorotannins (brown-seaweed derived molecules,
with a structural unit of polyphenols [286]), where these molecules have been
tested in silico, one advantage of these molecules is the finding that they
can as well inhibit the reverse transcriptase and the protease proteins,
reducing the likelihood of HIV replication within the cell, one point,
nonetheless, to be considered is the hydrophyllicity of these molecules which
may reduce oral bioavailability [138].

####  1.2.13 nef (negative regulating factor, p27): downregulator of host’s
proteins.

Nef is a 27 to 35 kDa N-myristoylated protein, in which the myristoylation
process is mediated by the N-myristoyl transferase (NM1), and is further
cleaved by the viral protease, in a site near a hydrophobic groove, which
stabilizes a PxxP loop [91]; structurally, this protein is highly plastic, and
many conformations have been determined [123]. Through its N-myristoylation,
is nef able to latch onto the inner surface of the host’s cell membrane and
can therefore interact with the clathrin-coated vesicle machinery, redirecting
some transmembrane proteins from the cell surface. Nef can interact with AP2
(heterotetramer comprising the monomers $\alpha$, $\beta 2$, $\mu 2$, and
$\sigma 2$) which is a bridge between a membrane protein substrate (which will
be later cargoed onto the clathrin-mediated endocytosis process towards its
degradation) and clathrin, throughout this AP2-Nef interaction the latter
protein can downregulate the host’s proteins: CD4, CD8, CD28, CD3, SERINC3,
SERINC5, CXCR4, CCR5; Nef also interacts with AP1 (a heterotetrameric adaptin,
which mediates the recruitment of clathrin and recognition of sorting signals
of transmembrane receptors [184]) being able subsequently to downregulate both
tetherin and MHCI [31]. We can witness the impact of the Nef-induced
downregulation of proteins, such as CD3, essential for TCR signaling; and CD28
with vital roles in T-cell proliferation, and Th2 cell development [185]. The
small molecule 2c
(2,4-dihydroxy-5-(1-methoxy-2-methylpropyl)benzene-1,3-dialdehyde [67]) has
been seen to inhibit the downregulation of MHC-1 mediated by the action of nef
through disrupting the interaction of nef with a Src family kinase [64];
4-amino substituted diphenylfuropyrimidines have also been described as nef
inhibitors, and subsequently inhibiting the nef-dependent HCK activation, the
values of the inhibitory concentration $IC_{50}$ range the micromolar order;
one of these compounds relates to the tyrosine kinase type of inhibitors
(which are built around a 5,6-biarylsulfo[2,3-d]pyrimidine pharmacophore [72].

####  1.2.14 tat (transactivator protein, p14): promoting viral genome
expression.

This protein has a weight which may be found within the interval of
$w\in[14,16]kDa$ [211], which uses the alternative splicing machinery of the
host in producing this set of possible tat translated isoforms [248]. It has
been observed that the tat protein displays what could be considered as
antagonistic properties to other viral proteins, such as vpu or nef, in
regards to the fact that tat promotes CD4 surface expression, as well as it
promotes cellular survival (through BCL2) [211], however it has been observed
that the interplay between the proteins tat and nef, which would seem
initially antagonistic, instead provide optimal viral infection, which by
binding nef to tat, induction of transcription occurs, leading to a specific
cellular fate determined by the viral fitness [124]. This protein promotes
gene expression (both viral and those of the host cell) by binding to a
nascent RNA structure termed transactivator response region (TAR) which
results in the transactivation of the long terminal repeat (LTR) promoter; tat
can also recruit histone acetyltransferases, inducing histone acetylation in
those genes whose expression is mediated by the LTR; these LTR-induced
expressed genes may not only be promoted by means of TAR, but as well tat may
induce these genes’ expression by binding to the NFKB enhancer [59].
Furthermore, tat regulates a set of cellular genes by various mechanisms: by
means of binding to TAR-like sequences, binding to promoter region or
interaction with transcription factors; these mechanisms regulate then the
expression of genes such as: IL6, TNFB, MAP2K6, IRF7, PTEN, PPP2R1B, PPP2R5E,
REL, IL2, IL2RA, OGG1, LMP2, CD69, VAV3, ADCYAP1, and FAM46C [47]; as it is
possible for us to regard, these genes are related to cytokine signaling, cell
cycle, TCR activation, amongst other processes. Tat can be inhibited by
various compounds: peptide-based, oligonucleotide-based, and small molecules.
Peptide-based compounds are arginine-enriched and can assemble into complexes
with the Transactivation response element RNA (TAR RNA), both of which can
compete for binding to the tat protein; peptoids are another type of peptide-
based compounds which have been seen to inhibit the assembly of the tat-TAR
RNA complex, peptoids are peptide isomers, in which the branching amino acids
are all bound to nitrogens in the backbone. In regards to oligonucleotide-
based compounds we can find TAR decoys, which resemble the TAR RNA sequence
and can bind to the tat protein, however with a limited impact on HIV-1
replication; antisense sequences to that of the TAR RNA one can also inhibit
tat activity; small interfering RNAs can also be used in inhibiting tat
activity by promoting the degradation of the target mRNA sequence.
Interestingly, in regards to the small molecule compounds, the quinolones have
displayed anti-tat activity at the nanomolar range by means of chelating
$Mg^{+2}$ in the TAR RNA sequence, one specific quinolone compound, WM5 has
been tested
(6-amino-1-methyl-4-oxo-7-[4-(2-pyridyl)piperazin-1-yl]quinoline-3-carboxylic
acid) [224], the use of quinolones in tat inhibition can aid in the treatment
of bacterial coinfections, although by all means can also may give rise to
quinolone-resistant bacteria, which poses a challenge in its clinical use.

####  1.2.15 rev (RNA splicing regulator, p19): exporter of viral RNA.

This protein is a 18 kDa, 116 amino acid phosphoprotein, and binds to rev-
responsive elements, assemblying into multimers [205]; rev is phosphorylated
by the kinases CSNK2A1 and MAPK, and it has been seen that the residues Ser8
as well as Ser 5 are phosphorylated by CSNK2A1, and CSNK2A1-induced
phosphorylation may induce downregulation of the rev protein [165]. These
multimers, when in solution, display a hollow fiber-like configuration (with
around 20 nm in diameter), and when interacting with RNA, the RNA molecule
might be engulfed into the hollow volume in order to protect it from nucleic
acid metabolism, promoting its cytoplasmic translation [301]. HIV-1 mRNA
nuclear export has been seen to be one of the main functions of the rev
protein, but it may as well promote the translation of rev-responsive mRNAs
(RRE), and is able also to downregulate its own expression as well as that of
other viral genes [205]; the RRE is found within env-coding region, it spans
around $\sim 350$ nucleotides in length, and when transcribed, this region
binds to rev, where the complex rev-mRNA (ribonucleoproteic in nature) then
promotes the recruitment of XPO1 and RAN [79], where this multimeric
ribonucleoproteic complex shall subsequently be exported from the nucleus,
XPO1 is an exportin (also termed CRM1 or chromosome region maintenance 1),
member of the importin beta superfamily of nuclear transport receptors,
required for the export of many RNAs (but is enabled to export proteins by all
means), CRM1 interacts with a nuclear export signal (NES) which is leucine-
enriched (rev possesses a NES domain [17]), but can additionally be a
hydrophobic domain; thus, the rev-mRNA-CRM1-RAN-GTP complex found in the
nucleus binds to the RAN binding protein 3 (RANBP3) which will then attach to
the nucleoporin complex comprised by the NUP214 and NUP88 proteins (found in
the nuclear membrane), hatched to the NUP358 protein with its cytoplasmic
ending bound to the RAN GTPase activating protein 1; resulting in the export
of the rev-mRNA-CRM1-RAN-GDP complex and release onto the cytoplasm [112].
Cellular splicing tends to occur within the nucleus [81, 37] (however there do
exist reports of cytoplasmic intronic unspliced mRNA [268], some of these
unspliced mRNAs are: APP, ATF4, CACNA1H, CAMK2B, FMR1, HP, IL1B, and others
which may promote in increasing the variability of the transcribed message
[29]), and export of mRNAs tends to be that of spliced mRNAs, nevertheless,
rev promotes the export towards the cytoplasm of both spliced and unspliced
viral RNA [76]. A small molecule PKF050-638 (Ethyl
(Z)-3-[5-(2-amino-5-chlorophenyl)-1,2,4-triazol-1-yl]prop-2-enoate) is
recorded to disrupt the XPO1-NES interaction, thereof being able to inhibit
rev-mediated nuclear export, this compounds binds to the Cys539 residue of
XPO1, and exhibits similar effects to those induced by the XPO1 inhibitor,
leptomycin B [58]. Three alternate compounds were found to suppress HIV-1 gene
expression of tat and rev, these compounds are a pyrimidin-7-amine, a
diphenylcarboxamide, and a benzohydrazide [15].

####  1.2.16 vif (viral infectivity protein, p23): evading APOBEC3G response.

This is a basic $\sim 23kDa$ phosphoprotein, required by the HIV-1 particle in
order to replicate in nonpermissive cells, such as lymphocytes, macrophages
and leukemic T-cells; but vif is redundant in permissive cells (T-cells)
[230]. Vif’s phosphorylation occurs within the C-terminus: Ser144, Thr155, and
Thr188, which may occur through protein kinase C or cGMP-dependent kinases
[309] (serine-threonine protein kinase family of proteins). Interestingly,
vif-defective viral particles when infecting a permissive cell, may continue
to replicate, and produce infectios vif-defective viral particles which may
then infect nonpermissive cells; when these vif-defective viral particles
derived from nonpermissive cells infect either a permissive cell or a
nonpermissive cell, viral replication ceases, this is due to negative
imprinting mechanisms applied onto vif [230]. It has been observed that the
permisiveness feature of the cell relies on the expression pattern of the
APOBEC3G protein which can be found in T-cells, B-cells, macrophages, and
myeloid cells; in T-cells, APOBEC3G’s expression is promoted by T-cell
activation [129], this protein is member of a family of cytosine deaminases
which play a role in innate immunity by restricting viral replication through
inducing deamination and mutation of viral genes, these enzymes display one or
two zinc-binding motifs, one of the members of this family is essential for
antigen-driven B-cell differentiation (AICDA); the APOBEC3 genes are clustered
in chromosome 22; and the expression of the gene APOBEC3G in monocytes has
been seen to be linked to IFNA, IFNB, IFNG, TNFA, and IL4 [53]. APOBEC3G is
able to deaminate ssDNA cytosines to uracils, whose ability is not constrained
merely to HIV1, but as well to other retroviruses, and inclusively
retrotransposons which effectively reduces the viral capability of replicating
in the absence of vif [195], due to vif being able to induce ubiquitination
and degradation ov APOBEC3G through complexing with CUL5, and KITLG (vif-
CUL5-KITLG) [316]. How can vif-defective viral particles derived from
permissive cells infect nonpermissive cells which indeed contain APOBEC3G?
This could be evaded by action of both the viral core and the nucleocapsid
proteins which can then esterically protect reverse transcription from the
deleterious actions of APOBEC3G [230, 30]. The compound RN-18 which is a small
molecule containing three benzene rings, has been reported to diminish vif
activity, and increasing APOBEC3G levels [178]; another small molecule
compound (IMB-301) inhibits vif-mediated APOBEC3G by binding to APOBEC3G and
inhibiting the binding of vif to APOBEC3G [151], another study reported that
the small molecule compound, Redoxal, inhibits the DHODH enzyme
(dihydroorotate dehydrogenase), thereof diminishing the pool of available
pyrimidine molecules, which has been seen to increase APOBEC3G protein
stability [203]; these three molecules, RN-18, IMB-301, and Redoxal, aid then
in decreasing HIV-1 replication rate.

####  1.2.17 tev (tat/env/rev protein, p26): fusion protein.

This is one of the fusion proteins present in the HIV1 proteome, comprised of
portions from three genes: env, rev, and tat, displaying a molecular weight of
$26kDa$; this protein is able to transactivate viral transcription (through
its tat portion) at an activity of 70% relative to that of the tat protein,
however its rev activity is weak or nonexistant. This protein localizes to the
nucleolus, and may be phosphorylated [139].

###  1.3 T-cell differentiation.

Based on the one cell multiple fate theory, several T cell subsets can be
differentiated from a common T cell precursor (either a naïve T-cell or
already primed T-cell) [87]. Each subset of T helper cells is involved in
immunity against specific pathogens. It is well known that the cytokine-
mediated signal traducers and activators of transcription (STATs) activation
is critical for Th-cell-fate determination [327].

For Th1 differentiation, T-cell receptors (TCRs) are stimulated by IL12
through the STAT4 pathway, promoting TBX21 expression, which promotes the
transcription of the IFNG gene, and its own via STAT1 activation. TBX21
inhibits GATA3, the principal regulator of Th2 cells. Th1 cells mainly express
CXCR3, INFG, IL2, and TNFA [87, 327, 245].

Th2 cells are essential in the control of helminth infections. Their effector
cytokines are IL4, IL5, and IL13, while IL2 and IL4 mediate their
differentiation. These cells express CCR4 and STAT6 (mediated by IL4), which
induce GATA3 expression, the master transcriptional regulator of Th2 cells
differentiation, it directly binds to the Il4/Il13 gene locus and through this
binding, it induces IL5 and IL13 transcription. Retroviral expression of GATA3
is sufficient to induce endogenous GATA3 [327]. In addition to IL4, IL2
activates STAT5, critical for Th2 differentiation, inducing IL4RA and
maintaining GATA3 expression. Other proposed transcriptional regulators are
NLRP3 and IL25 [87, 245].

Th17 cells are principally located in the gastrointestinal tract. Their
primary function is to respond to extracellular bacteria or fungi infections.
Their differentiation is mediated by TGFB, IL1B, IL6, and IL23. Through
IL6-mediated STAT3 activation, TGFB induces IL23R and RORC (the principal
regulator of Th17 differentiation), leading to the production of both IL17A,
and IL17F. IL21 and IL23 have a similar function as IL6, inducing STAT3
activation. Th17 cells also can produce IL22 and TNFA. IL2 activates STAT5,
which inhibits STAT3 and consequently reduces IL17A expression. FOSL2, MINA,
FAS, and POU2AF1 promote Th17 differentiation, against TSC22D3, which
negatively regulates it. NFIL3 (induced by melatonin) blocks RORC [87, 327].

FOXP3(+) Treg cells are essential to regulate auto-reactive T cells. They are
characterized by the constitutive expression of the IL2 receptor alpha chain
(CD25). These cells can be generated within the thymus (tTreg) or in
peripheral lymphoid organs (pTreg). Their master transcriptional regulator is
FOXP3. IL2 and TGFB1 are their main differentiation factors, through the
IL2-mediated STAT5 activation, which also suppresses Th17-cell production. In
the absence of IL2, IL2RA, or IL2RB, Treg cells are downregulated [87, 327].

Transcriptional factors usually cross-regulate the expression of factors
involved in the development of other lineages. For example, TBX21 suppresses
GATA3 through direct protein-protein interaction; GATA3 suppresses the
expression of STAT4 and the production of IFNG; RORC and FOXP3 antagonize each
other through a protein-protein interaction [327]. Interestingly, all T-cell
subsets have demonstrated some degree of plasticity, adapting their phenotype
to the other T-cell subsets under certain conditions [87].

###  1.4 Role of TOX in thymocyte differentiation

Thymocyte selection-associated high-mobility group box (TOX) is a DNA-binding
factor that regulates transcription by producing specific changes in DNA
structure and allowing the formation of multi-protein complexes [152].

TOX was initially identified as a thymic transcript; it is transiently
upregulated during $\beta$-selection and positive selection of developing
thymocytes. This upregulation is mediated by TCR-mediated calcineurin
signaling. Expression of TOX induces the factor RUNX3, resulting in CD4
downregulation and CD8 single positive cell formation, but TOX stimulus alone
is insufficient to substitute the complete TCR signaling during positive
selection. Mice deficient in TOX revealed a requirement for TOX in CD4 T cell
lineage development. RUNX3 and ZBTB7B are key nuclear factors for CD8 and CD4
T cells differentiation in the thymus, respectively. It has been suggested
that TOX is an upstream regulator of ZBTB7B. TOX is required also for the
development of NK, LTi and CD4(+) cells, probably by modulating E protein
activity thought the upregulation of ID2, an E protein inhibitor [6].

##  2 Materials and methods.

###  2.1 Computational requirements.

The stochastic model, further computations, and graphs were performed by means
of RStudio: Integrated Development Environment for R [233].

###  2.2 Data acquisition.

The data was acquired from two main databases: The BioGrid [22], and STRING
[2], as well as from various papers: [120, 208, 34, 306, 326, 118, 304, 276,
41, 213, 239, 40, 107, 267, 261, 204, 323, 74, 201, 31, 212, 158, 8, 110, 290,
237, 292, 113, 14, 260, 121, 154, 90, 219, 57, 148, 257, 251, 258, 206, 175,
197, 314, 7, 125, 235, 288, 115, 255, 291, 315, 33, 32, 149, 271, 52, 86, 35,
18, 265, 246, 36, 191, 317, 3, 307, 141, 144, 27, 296, 295, 278, 272, 137,
104, 263, 312, 48, 126, 139, 229, 145, 10, 9, 19, 105, 165, 96, 100, 101].

###  2.3 Stochastic model.

The interactions pertaining to the T-cell differentiation molecular network
were assessed, 562 proteins were included, a Markov chain was then associated
to these interactions, the stochastic process $\\{X_{t}\\}$ displays a space
of states $S$, such that $S=\\{p_{1},p_{2},...,p_{n}\\}$ where $n=562$ and
$p_{i}$ stands for the protein or gene that the system encounters itself in;
the space of the temporal parameter is $T=\\{1,2,...m\\}$, the units of this
temporal parameter shall be defined subsequently. The space of states of the
stochastic process $\\{X_{t}\\}$ may be divided, in order to analyse the
selection of the molecules taking part of it, into two subsets, the TOX-
related process subset $A\subset S$, and the Th-cell-related process subset
$B\subset S$, such that, $A+B-(A\cap B)=S$ due to the fact that some molecules
in each subset are repeated, $A\cap B\neq\emptyset$. The TOX-related process
is comprised of three major networks, $A=\\{C,D,E\\}$, where the C network
stands for one of the roots of the process: the TOX protein; the D network
stands for those proteins which interact with TOX and amongst themselves
(KRTAP12-1, KRTAP26-1, DMRTB1, ZDHHC17, TOX2, BANP, GRAP, FUS); the E network
stands for those proteins which interact with TOX-related proteins and amongst
themselves. The Th-cell-related process subset is in its turn comprised of ten
networks, $B=\\{K,L,M,N,O,P,Q,R,S,T\\}$, the K network stands for those
proteins involved in the Tfh-differentiation process, the L network stands for
the proteins involved in the Th1-differentiation process, the M network stands
for those proteins involved in the Th2-differentiation process, the N network
stands for those proteins involved in the Th17-differentiation process, the O
network is composed of those proteins involved in the Treg-differentiation
process; we may regards that the set $U\subset B$, $U=\\{K,L,M,N,O\\}$ is a
set of roots of the process (TBX21, GATA3, STAT4, STAT5, STAT6, RUNX3, IRF1,
HLX, EOMES, EST1, FI1, ASCL2, TCF7, BCL6, IRF4, CMAF, JUNB, DEC2, IKAROS,
RORC, FOXP3, BATF1, RORA, RUNX1, HELIOS, FOXO). The P network stands for those
proteins interacting with the Tfh-related ones, the Q network is comprised of
proteins interacting with the Th1-related ones, the R network stands for those
proteins interacting with the Th2-related ones, the S network stands for those
proteins interacting with the Th17-related ones, and finally the T network is
comprised of proteins which interact with the Treg-related ones; the set
$v=\\{P,Q,R,S,T\\}$ is thus a secondary network. The adjacency matrix and
corresponding graph summarising this data is shown in 2. The states comprising
the processes forecommented are not mutually exclusive, i.e., one protein may
be involved in various networks.

(a) The subsets belonging to the space of states, and their interactions are
shown by means of their adjacency matrix, colour key: shell/beige:
nonexistence of interaction between networks, brown/dark orange: interaction
between networks.

(b) Graph displaying the adjacency matrix interactions between the TOX and Th-
related networks. C=TOX root (TOX), D=TOX secondary network, E=TOX tertiary
network, K=Tfh root network, L=Th1 root network, M=Th2 root network, E=TOX
tertiary network, O=Treg root network, P=Tfh secondary network, Q=Th1
secondary network, R=Th2 secondary network, S=Th17 secondary network, T=Treg
secondary network.

Figure 2: Adjacency matrix and corresponding graph depicting the interactions
between the TOX and the Th networks.

We may describe the probabilities of going from one state to another in a
Markov chain by means of a transition probability matrix $A=(a_{ij})$ where
$a_{ij}$ is the probability of transiting from the i-th state to the j-th
state, the existence of a probability between two molecules in this model is
given by the following condition,

$\exists R\mid p_{i}Rp_{j}\rightarrow\exists a_{ij}$ (1)

Should the relationship R between two elements of the space of states ($p_{i}$
and $p_{j}$) exist, the probability of transiting from the i-th state to the
j-th state exists ($a_{ij}$), the probability is computed by the amount of
states the process can transite to from a given initial state; say we start
our process at the $p_{1}$ molecule, this molecule interacts with $q$
molecules, these molecules in the vector $Q=(p_{2},...p_{q})$, $Q\subset S$,
then the probability of transiting from $p_{1}$ to $p_{q}$ shall be,

$(a_{1q})=\frac{1}{q}$ (2)

in this case, we have considered that all molecules involved interact with
themselves, i.e., $a_{ii}>0$. And the condition for the stochastic matrix is
thus fulfilled,

$\sum_{i=1}^{k}(a_{ik})=1$ (3)

The row is normalised. The transition matrix corresponding to the molecules
interacting in the T-cell differentiation process is shown in 3.

Figure 3: T-cell differentiation molecular network (TCDMN) transition matrix,
represented as a heatplot, the minimum value of this matrix is $a_{ij}\in
P,min\\{a_{ij}\\}=0$, that is where there exists no interaction between two
molecules, and the maximum possible value is $a_{ij}\in
P,max\\{a_{ij}\\}=\frac{1}{2}$, due to the fact that every molecule interacts
at least with itself (as we can see that all the elements of the main diagonal
of the matrix, $\forall a_{ii}\in P,a_{ii}>0$ and the ”root” molecule.

The eigenvalues were also computed in order to obtain a matrix $\Lambda$,
where $\lambda\in\Lambda$ is an eigenvalue of P; since
$\lambda_{1}=1\in\Lambda$, we may intuit that the equilibrium distribution
$\pi^{n}$ exists, such that,

$\pi^{n}=\pi P^{n},n\rightarrow\infty$ (4)

The equilibrium distribution has been computed taking in consideration the
following linear system,

$\begin{split}\pi(1)(p_{11}-1)+p_{21}\pi(2)+...+p_{n1}\pi(n)=0\\\
p_{12}\pi(1)+\pi(2)(p_{22}-1)+...+p_{n2}\pi(n)=0\\\ \vdots\\\
p_{1n}\pi(1)+p_{2n}\pi(2)+...+\pi(n)(p_{nn}-1)=0\\\
\sum_{i=1}^{n}\pi_{i}(n)=1\end{split}$ (5)

Where $\pi(i)$ stands for the i-th state, $p_{ii}$ stands for the probability
of transiting from the i-th state to the i-th state when the transition matrix
is $P^{1}$. The eigenvalues’ matrix $\Lambda$ is shown in section 3, as is as
well displayed the equilibrium distribution.

In regards to the T-cell differentiation network when the HIV particle enters
the cell, the network was extended to 2874 molecules (including those that
were previously in the T-cell differentiation network). The most connected
node in this network was the one of the TP53 molecule with an adjacency
$k_{TP53}$ of $k_{TP53}=169$, the transition matrix is shown in figure 4.

Figure 4: Transition matrix corresponding to the interaction of one viral
particle with a T-cell, 2874 molecules are included, displaying thus 8’259,876
interactions in total.

##  3 Results.

###  3.1 Equilibrium distribution.

The eigenvalues for the T-cell differentiation process matrix were obtained,
further assembled into a set $\Lambda$, this set contained two eigenvalues of
special interest, the presence of $\lambda_{1}=1$ which, as has been
commented, permits us to intuit the existence of an equilibrium distribution
$\pi^{n}$ [225], which has been computed; the computation of this equilibrium
distribution has been performed by means of a numerical method yielding
results $\hat{f}(\pi)$ where $\hat{f}$ is the approximation method which
differ from the theoretical ones by a factor
$\epsilon_{a}=|f(\pi)-\hat{f}(\pi)|$ where $\epsilon_{a}$ is the absolute
error from the process; considering that the events are subject to a
$\sigma$-algebra $\mathcal{F}$ and that there exists a measurement $P$ of the
probability over $\mathcal{F}$, we may consider that $\sum\pi=1$. In this
case, $\epsilon_{a}=5.32039697\times 10^{-15}$ and we are able to correct our
probabilities, therefore we performed a mapping $g:\hat{f}(\pi)\rightarrow
f(\pi)$. These corrected probabilities were then the argument of the base-10
logarithm (in order to more vividly represent the computed probabilities,
where some probabilities were $p<<0$) that is, $h(\pi)=\log_{10}(p)$, this is
shown in 5(a); from this we can regard that $2.588919\times 10^{-12}$ of the
probability is contained in 555 molecules, while only $3$ molecules span
$p=1-2.588919\times 10^{-12}$ of the probability, should we exclude for a
moment these 3 molecules, the logarithmic probability would be seen as is
displayed in 5(b). The corresponding eigenvalues for the HIV+Th-cell
differentiation network were computed, and poured into a matrix $\Lambda_{1}$,
where we come to find $\lambda_{i}\in\Lambda_{1}:\lambda_{i}=1$, therefore
should we start intuiting about the existence of an equilibrium distribution
for our compound network; the second largest eigenvalue in the network is
$\lambda_{j}\neq\lambda_{i}:\mod(\lambda_{j})=0.999654150505886$; our
intuition serves well for the HIV process indeed possesses an equilibrium
distribution $\pi_{HIV}^{n}$, this is shown in figure 6.

(a) Equilibrium distribution of the T-cell differentiation molecular network
process, the x-axis represents the index of the molecule, where there exists a
bijection between the space of states $S$ and the set of the index; the y-axis
represents the base-10 logarithm of the corrected probability.

(b) Equilibrium distribution of the T-cell differentiation molecular network
process, in this case, we have excluded the extreme cases in which the
probability was accumulated, here we display those molecules in which
$p=2.588919e^{-12}$ of the probability is condensed.

(c) Probability vector when 50 iterations have been performed, $\pi^{50}=\pi
P^{50}$.

(d) Probability vector when 100 iterations have been performed, $\pi^{100}=\pi
P^{100}$.

(e) Probability vector when 500 iterations have been performed, $\pi^{500}=\pi
P^{500}$.

(f) Probability vector when 1148 iterations have been performed,
$\pi^{1148}=\pi P^{1148}$, this number has not been randomly chosen, it is
$f(\lfloor T\rfloor)$, that is the floor function of the rate towards
convergence, $T$.

Figure 5: Rank-logarithmic graphs of the equilibrium distribution pertaining
to the T-cell differentiation molecular network process. Figure 6: Equilibrium
distribution of the HIV molecular network process, the x-axis represents the
index of the molecule, where there exists a bijection between the space of
states $S$ and the set of the index; the y-axis represents the base-10
logarithm of the corrected probability.

###  3.2 The logarithmic equilibrium distribution follows a discrete beta
generalized distribution.

Graph 5(b) may be considered a rank-ordered relation, where the rank is the
molecule and the probabilities are logarithmic frequencies, Zipf showed that
when plotted, the logarithmic frequencies fall on a slope $m=-1$, which may
suggest a power law, we may try and fit the data by means of the following
function,

$\begin{split}x(r)=C\frac{(N+1-r)^{b}}{r^{a}}\\\ \equiv
log_{10}(p)\end{split}$ (6)

Where $r$ is the rank value, $N$ is the maximum value, $C$ is a normalization
constant and $a,b\in\mathbb{R}$ are two constants to be determined [160],
furthermore, it has been observed that many types of data follow this
distribution: ranks of articles and citations in journals, linguistic data,
citation profile, English, and Spanish letter frequency distribution, and
more; this distribution, depending on the value of the constants $a,b$, may
give rise to other common probability distributions, that is, when $a=b=0$, it
yields a constant random variable, when $a=0,b=1$ we may obtain a uniform
distribution, when $b=0$ it becomes a Pareto distribution, when $a=b$ it
yields a Lavalette distribution [85]. Here we will determine the constants
$a,b$ for the equilibrium distributions of both the T-cell differentiation and
the HIV infection processes, as well as the $R^{2}$ fitting values for said
functions. This we may do by means of firstly obtaining the logarithm of the
function $X(r)$ (in order to ease the computations),

$\ln(x(r))=\ln(C)+b\ln(N+1-r)-a\ln(r)$ (7)

Then, the first moment of the logarithmic function is
($E[\ln(x(r)]\equiv\break E[\ln(\log_{10}(p))]$),

$E[g(r)]=\int^{\infty}_{-\infty}g(r)dF_{X}(x)$ $\rightarrow
E[\ln(x(r))]=\int^{1}_{0}\ln(x(r))\cdot\frac{dr}{dx}dx$ (8)
$=\int^{1}_{0}\ln(x(r))dr$

We can substitute the function $\ln(x(r))$,

$E[\ln(x(r))]=\int^{1}_{0}[\ln(C)+b\ln(N+1-r)-a\ln(r)]dr$
$=[r\ln(C)+b(N+1-r)(1-\ln(N+1-r))-ar(\ln(r)-r)]^{1}_{0}$ (9)
$=\ln(C)+a+b[N(1-\ln(N))-(N+1)(1-\ln(N+1))]$

When $N=0$,

$E[\ln(x(r))]=\ln(C)+a-b$ (10)

Which goes in line with [85]; now, when we plug in the values from the
simulation ($N=2.70546008=\max\\{\ln(\log_{10}(p))\\}$), we obtain,

$E[\ln(x(r))]=\ln(C)+a+1.60703188b$ (11)

Let us compute the variance now,

$Var[\ln(x(r))]=E\big{[}[\ln(x(r))]^{2}]\big{]}-E^{2}[\ln(x(r))]$ (12)

We have already computed the expression for the first moment $E[\ln(x(r))]$,
we are missing now the computation of $E[[\ln(x(r))]^{2}]$,

$\ln^{2}(x(r))=(\ln(C)+b\ln(N+1-r)-a\ln(r))^{2}$
$=\ln^{2}(C)+2\ln(C)(\ln(N+1-r)-a\ln(r))+(b\ln(N+1-r)-a\ln(r))^{2}$ (13)

The integral that we are looking for then is,

$I=\ln^{2}(C)\int^{1}_{0}1\cdot
dr+2\ln(C)\int^{1}_{0}\ln(N+1-r)dr-2a\ln(C)\int^{1}_{0}\ln(r)dr+\\\
b^{2}\int^{1}_{0}\ln^{2}(N+1-r)dr-2ab\int^{1}_{0}\ln(N+1-r)\ln(r)dr+a^{2}\int^{1}_{0}\ln^{2}(r)dr$
(14)

Which equals,

$I=\ln^{2}(C)+2\ln(C)[(N+1)\ln(N+1)-N(\ln(N)-1)]+2a\ln(C)+\\\
b^{2}[(N+1)\ln(N+1)(\ln(N+1)-2)-N\ln(N)(\ln(N)-2)+2]-\\\
2ab\bigg{[}(N+1)\bigg{(}\ln(-N)-\ln(-N-1)-Li_{2}\bigg{(}\frac{1}{N+1}\bigg{)}\bigg{)}-\ln(N)+2\bigg{]}+2a^{2}$
(15)

Where $Li_{2}(.)$ is the polylogarithm function. The variance equals (we make
$\ln(C)=\phi$),

$Var(\ln(x(r)))=\phi^{2}+2\phi[\ln(N+1)(N+1)-N(l\ln(N)-1)]+2a\phi+b^{2}[(N+1)\\\
\ln(N+1)(\ln(N+1)-2)-N\ln(N)(\ln(N)-2)+2]-2ab\bigg{[}N\bigg{(}\ln(-N)\\\
-\ln(-N-1)-Li_{2}\bigg{(}\frac{1}{N+1}\bigg{)}-\ln(N)+2\bigg{)}+1\bigg{]}+2a^{2}-(c+a+\\\
b[N(1-\ln(N))-N(1-\ln(N+1))-1])^{2}$ (16)

In the case of $N=2.70546008$,

$Var[\ln(\log_{10}(x(r)))]=a^{2}-9.475027627b^{2}-0.025582926ab+7.385337236b\phi\\\
-2a\phi+9.732461206\phi$ (17)

The median is where $r=0.5$,

$Med(\ln(x(r)))=\phi+b\ln(N+1/2)-a\ln(1/2)$ (18)
$=\phi+1.164855631b+0.6931471806a$ (19)

the values of the first moment, variance, and median are,

$E[\ln(\log_{10}(p))]=\overline{Z}=2.64148234\times 10^{0}$
$Var[\ln(\log_{10}(p))]=Var(Z)=6.1653831\times 10^{-2}$ (20)
$Med(\ln(x(r)))=Med(Z)=2.659718516\times 10^{0}$

Therefore we possess a system of three variables and three equations,

$\overline{Z}=\phi+a+1.60703188b=2.64148234$
$Var(Z)=a^{2}-9.475027627b^{2}-0.025582926ab+7.385337236b\phi\\\
-2a\phi+9.732461206\phi$ (21)
$Med(Z)=\phi+1.164855631b+0.06931471806a=2.659718516$

We have consequently an underdetermined linear system,

$\Xi=\begin{pmatrix}1&1&1.60703188&2.64148234\\\
1&0.6931471806&1.164855631&2.659718516\end{pmatrix}$ (22)

Which yields the partial solution,

$a=3.22596-1.2164\phi$ $b=0.134657\phi-0.363698$ (23)

By plugging these solutions into the quadratic equation, we obtain the
expression for $\phi$,

$2.306500645\phi^{2}+3.323241286\phi-23.48052236=0$ (24)

the corresponding solutions are then,
$\Phi=(\phi_{1},\phi_{2})=(2.550545919,-3.991361162)$, and the constants for
each $\phi_{i}\in\Phi$,

$\phi_{1},\begin{pmatrix}\hat{a}_{1}\\\
\hat{b}_{1}\end{pmatrix}=\begin{pmatrix}8.0810517\\\ -0.9011627\end{pmatrix}$
$\phi_{2},\begin{pmatrix}\hat{a}_{2}\\\
\hat{b}_{2}\end{pmatrix}=\begin{pmatrix}3.2034876\\\ -0.3612103\end{pmatrix}$
(25)

Whose graphs are shown in figure 7.

Figure 7: Discrete generalized beta distributions modeling the logarithmic
equilibrium distribution corresponding to the Th-cell differentiation process,
where $\hat{a}_{1}=8.0810517,\hat{b}_{1}=-0.9011627$,
$\hat{a}_{2}=3.2034876,\hat{b}_{2}=-0.3612103$.

###  3.3 The stochastic processes can be coupled with both in vivo processes
of T-cell differentiation, and HIV-cellular infection.

The second eigenvalue of interest is one which allows us to compute the rate
of convergence, should it exist, and in this case, we have already commented
on its existence; this rate is dependent on the second maximum eigenvalue
$\mu$ [283],

$\begin{split}\mu=max\\{\lambda_{i}\in\Lambda\ :\mu\neq\lambda_{1}=1\\}\\\
\mu=0.99799714\end{split}$ (26)

We thus can compute the rate to convergence $T$, that is, the necessary steps
for the process to achieve equilibrium,

$\begin{split}T_{1}=\frac{1}{log(\mu^{-1})}\\\ =1148.49753u\end{split}$ (27)

The process, in this case a T-cell, may reach equilibrium after $1148.49753$
arbitrary temporal units of time, these units may be then converted to an
international system unit, taking in account the in vivo or in vitro data
collected in the past, this achievement of equilibrium may be regarded as the
complete lifespan of a T-cell, and has been reported to be as high as 17 years
[114] and as low as 6 days [25, 98], thus this unit may range in the following
interval,

$1u\in[4.51372324\times 10^{4},4.66985088\times 10^{5}]s$ (28)

In regards to the HIV infection process, we may regard that the coupled set of
viral particle and T-cell reach equilibrium when there begins to exist a
decline in the Th-cell count, moment in which we can start to perceive that
the genetic/proteic machinery of the virus has effectively hijacked that of
the cell towards another more deleterious, that of more massive apoptosis,
this is more generally seen after five years of infection [77], then the time
to reach equilibrium $T_{2}$, will depend on the second maximum eigenvalue
$\nu$,

$\begin{split}T_{2}=\frac{1}{log(\nu^{-1})}\\\ =6656.62441u\end{split}$ (29)

Now, $6656.62441u\equiv 5y\rightarrow 1331.32488u/y$.

###  3.4 Evolution of distributions until equilibrium.

The probability distributions $\pi_{i}$ for $i\in[1,T]$ where $T$ is the time
when the process has converged towards equilibrium $\pi^{n}$ have been
computed and are shown in 8.

Figure 8: Selected probability distributions of seven molecules pertaining to
the Th-cell differentiation process, where the probability for every time step
in the interval $[1,T]$ is shown, in this plot, 1=TOX, 2=EOMES, 3=PFDN1,
4=NOTCH2NL, 5=TP53, 6=HDAC1, 7=EP300.

###  3.5 The Th-cell differentiation and the HIV-infection processes tend
towards an increasing entropy, $S_{1}\leq S_{2}\leq...\leq S_{n}$ when
reaching its equilibrium distribution.

Entropy in a thermodynamic setting may relate to the concepts of reversibility
or order in a system, more specifically, should we have a protein with a
thermodynamic state
$\pi_{0}(x)=exp\big{[}-\frac{\beta\mathcal{H}_{0}(x)}{Z_{0}}\big{]}$ where
$\mathcal{H}_{0}(x)$ is the Hamiltonian ($\mathcal{H}=T+V$, T is the kinetic
energy and V is the potential energy), $\beta=\frac{1}{kT}$ and $Z_{0}$ is the
partition function, we can thus compute the entropy $S(\pi)$ of the system by
means of,

$H(\pi)=-k\sum_{i=1}^{N}\pi(x)\ln\pi(x)$ (30)

[198] this could be therefore considered as the summation over all the
particles of each of their stochastic entropies $s_{t}$,

$s_{t}=-\ln\pi(x_{t},t)$ (31)

The probability is to be computed by means of solving the Fokker-Planck
equation [247], which should we recall, describes the evolution of the
probabilities $\pi$ from an ensemble of particles when this particles start
their trajectories in $t=0$, we would call this distribution the initial
vector $\pi^{0}$ [16] that we have previously commented on, thus we may regard
the solution to this Fokker-Planck equation in our case as a discrete one
(which comes as no surprise, due to the fact that the continuous probability
considered in the Fokker-Planck equation is a continuous Markovian one).
Should we divide the expression 30 by $1/k$, we would obtain,

$S(\pi)=-\sum_{i=1}^{N}\pi(x)\ln\pi(x)$ (32)

This is the Shannon entropy of the system [198], which will help us in the
understanding of the changes of the microscopical states $\pi$ in our system,
these states are then the thermodynamic states pertaining to each particle in
the regarded system. We have thereof computed $S(\pi_{i})$ for $i\in[1,N]$
where $N=1148$ in this case, due to this being the time of convergence for the
Th-differentiation process, which may be seen in 9, the entropy $S(\pi_{HIV})$
was also computed for the HIV-infection process, and is seen in 10

Figure 9: Entropy computations for each time step in the Th-cell
differentiation process, until this process reaches its equilibrium
distribution in $t=1148u$, we may regard the increasing values of entropy with
time, $S_{1}<S_{2}<...<S_{n}$. Figure 10: Shannon’s entropy for the HIV-
infection process, in which it is as clear as for the previous process that
the values of entropy $S_{i}$ display increases, $S_{1}\leq S_{2}\leq...\leq
S_{n}$ for $n\rightarrow\infty$.

We may regard that there exists a feature for each $S_{i}$ computed,

$S_{1}<S_{2}<...<S_{n}$ (33)

Therefore, the process tends towards an increasing entropy value $S_{n}$,
which according to Boltzmann, systems tend towards increases in entropy when
they reach equilibrium due to this being the state in which there exist the
most possible configurations [13]. However we may notice two aspects in this
entropy computation:

  1. 1.

$\Delta S_{1}>\Delta S_{2}>...>\Delta S_{n-1}\rightarrow(\Delta
S_{n}\rightarrow 0)$, therefore the system tends towards steadiness in its
equilibrium distribution.

Interestingly, considering the series in 33, if we regard the entropy $S_{t}$
as a measurement of the information or the number of states in the system at
any time $t\in T$, therefore, the set of states $Z_{s}^{t}$ in the system
accounted for by the entropy are defined by,

$t=1,Z^{1}_{s}=\\{Z_{1}^{1},Z_{2}^{1},...,Z_{m}^{1}\\}$ (34)

Then $Z_{s}$ is a measurement of events $\pi\in\Omega$ (some molecules may
have a repeated value of probability) over $P=[0,1]$, so $Z_{s}$ is a
$\sigma$-algebra; now, let us establish a relationship between the
cardinalities of the sets of states,

$|Z_{s}^{1}|<|Z_{s}^{2}|<...<|Z_{s}^{n}|$ (35)

And,

$\begin{split}\forall Z^{1}_{m}\in Z^{1}_{s},Z^{1}_{m}\in Z^{2}_{s}\\\ \forall
Z^{2}_{m+i}\in Z^{2}_{s},Z^{2}_{m+i}\notin Z_{s}^{1}\\\ \rightarrow
Z^{1}_{s}\subseteq Z^{2}_{s}\wedge Z^{2}_{s}\supset Z^{1}_{s}\end{split}$ (36)

In our simulation we can state thus,

$Z_{s}^{1}\subseteq Z_{s}^{2}\subseteq...\subseteq Z_{s}^{n}$ (37)

And we may say that there exists a collection of $\sigma$-algebras for every
step $t=1,2,...,N$ in the process, this collection is then a filtration
$\\{Z_{s}^{n}\\}_{n\geq 1}$ such that the following is fulfilled,

$Z^{n}_{s}\subseteq Z^{n+1}_{s}$ (38)

The process $\\{X_{n}\\}$ can be $Z$-measurable, and we even may couple the
entropy with the filtration $\\{Z_{s}^{n}\\}_{n\geq 1}$ and, the acquisition
of information throughout the Th-cell differentiation process may be
considered a martingale.

  2. 2.

The Euclidean distance $\delta_{1}\in\mathbb{R}$ between the initial entropy
$S_{1}$ and the final entropy $S_{f}$ is $\delta(S)=S_{f}-S_{1}>0$, therefore
this model describes a possible process, and moreover an irreversible one:
When reaching equilibrium the change in entropy, $\delta(S)>0$ tells us the
differentiated status is a committed one in the cell. It has been previously
commented that natural systems tend to display an increase in entropy in time
[299, 28], therefore this simulation is a phenomenologically accurate one.

##  4 Discussion.

HIV-1 transmission results from viral exposure at mucosal surfaces or from
percutaneous inoculation. Since direct analysis of these mechanisms is
inaccessible, understanding HIV-1 infection has been done by indirect methods
[253].

The initial period between the moment when the first cell is infected and the
moment when the virus can be detected in blood, is called the eclipse phase
(estimated at 7-21 days). During this phase, clinically silent, HIV-1 is
propagated. Once virus becomes detectable in blood plasma, it increases
exponentially [253, 222].

Most dynamic models of HIV infection do not usually take into account the
earliest stage of infection, where stochastic models play a key role [200].

###  4.1 Th-cell fully differentiated state as an equilibrium distribution,
$\pi^{n}$.

One of the motivating questions for this paper was the question: can the HIV
change the differentiation status of a T-cell? In order to answer such
conundrum one would ask oneself how would the modeling for the differentiation
process in the T-cell ever occur, we have addressed this problem in a two-fold
fashion, one in a biological manner, i.e., a differentiated state in a cell
may be understood as a committed state in the cell, one in which the cell
cannot return towards a totipotent or pluripotent cell and has a narrower,
more defined set of functions [170]; and another in a more physical manner,
i.e., this differentiated state as a state which is in equilibrium, or that it
would require energy from its environment in order to dedifferentiate and
return to a less differentiated state (as it may occur in certain processes,
such as neoplasms [190]), this energy $E$ is not available readily for such
endeavours for any cell, requiring a special effort for it to happen, this
equilibrium state can be regarded as the equilibrium reached for a Markov
process, a state which is independent of any initial state of the system for
it will be reached notwithstanding the initial conditions; a Markov process
can model as well the behaviour of the differentiating cell due to the
stochastic nature of interactions between signaling, structural, and energy-
related molecules. Through this simulation we have observed that in both
stochastic processes (Th-cell differentiation, and HIV-infection) there exist
equilibrium distributions, providing us with in silico evidence that these
processes indeed lead to differentiated states of the cell, the Th-cell
differentiation may lead to the various phenotypes (Th1, Th2, Th17, Treg,
etc), and the HIV-infection-dependent differentiation may lead to a novel
differentiated state, $Th_{HIV}$ 11, this state may be featured by an altered
immunological synapse, and promotion of induction of the virological synapse,
expression of antiapoptotic factors in early phases of infection, but lately
would it be overrun with promotion of proapoptotic factors leading to the
decline of T-cell count in a chronic phase of the infection. This has been
seen to occur in an in vivo setting, where CD45RA(+) (PTPRC(+)) T cells
decline over time, with a more rapid decrease in the naïve phenotype type of T
cells than those which bear a T-cell memory phenotype [51], this molecule was
included in our simulation, and displayed $-\log_{10}(p)=13.24022$ interacting
with other 6 molecules, here can we witness ways in which this simulations can
be embedded onto the in vivo or clinical settings, for we can track the
decline in the CD45RA(+) T-cells in the course of the HIV-1 infection, and
establish a relation with the adimensional temporal units in this model.

Figure 11: Commitment and differentiation of T-helper cells towards their
known phenotypes Th1, Th2, Th17, and Treg, where we may find that the
stochastic molecular network which is involved in the regulation of the
differentiation process displays an equilibrium distribution, then can we
intuit these states are stable ones, where the cell should necessitate an
amount of energy in order to leave such state, this perturbation in the energy
distribution might be induced by chronic infection of HIV-1, and then promote
the differentiation of a T-cell towards a particular Th-HIV cell, where the
genetic machinery has been hijacked by the virus, and can also be regarded as
a stable cellular state. Created with BioRender.com

###  4.2 Th-differentiation, and HIV-infection processes as martingales.

Here, we have commented on a method through which one may be able to prove an
stochastic process $\\{X_{t}\\}$ is a martingale by a non-conventional path,
for we have proposed that a martingale can be regarded as such set of random
variables which are coupled with filtrations, these filtrations are steps in
the Markov chain in which by each step, the process gains information,
therefore increases its informational Shannon entropy. Both stochastic
processes considered here have been proved to gain entropy when
$t\rightarrow\infty$, this increase in entropy can also be seen as a way in
which the systems tend towards equilibrium, both the T-cell differentiation
process, and the HIV-infection process. A connection between Shannon’s entropy
and the more thermodynamic sense of it can be performed should we remind
ourselves that the increase in the states in which the system may be
encountered (the increase in information) are molecules within such system,
and probabilities the system may find itself in given molecule for a given
amount of time. Interestingly, our simulation has yielded results, in which
the probabilities of the system are very unequally divided between the members
of the system, that is, there is a small set of molecules
$\\{S\\}\subset\\{X\\}:|S|<<|X|$, while
$\sum_{i}p^{S}_{i}>>\sum_{j}p^{X}_{j}$, these molecules contain most of the
probability within a very small set of them, this has been observed in other
simulations which do not pertain to the biological realm, but to the
economical one, in which in simulations of continuous bids within a given
population, a very small amount of people possess most of the wealth, and the
majority of the population end up with a very small amount of wealth [84, 24],
in the yard-sale model trying to give account for wealth distribution
inequality and the formation of oligarchies, it is also of interest to note
that should we consider $\\{X_{n}\\}$ as the wealth of a given player in the
model in the n-th moment, the conditional expectancy equals,

$E[(X_{n+1}^{i}-X_{n}^{i})^{2}|X_{n}^{i}]=a^{2}\min{(X_{n}^{i},1-X_{n}^{i})^{2}}$
(39)

where $X_{\infty}\sim Ber(X_{0})$, that is the stochastic process
$\\{X_{n}\\}$ is also a non-negative martingale which converges to the random
variable $X^{i}_{\infty}$ [43]. In our model, we can consider that there
exists the formation of molecular ”oligarchies” which possess most of the
wealth (time spent on those molecules) which can play central roles in both
stochastic processes considered.

###  4.3 Th-cell differentiation, and HIV-infection equilibrium distributions
as discrete generalized beta ones.

One additional result which astounded the authors of this paper is that of the
log-rank equilibrium distribution of the stochastic processes, for they
display the morphology of the discrete generalized beta distributions, this
distribution has been encountered in various processes: distribution of impact
factor in journals [71], letter frequency distribution in political speeches
[147], k-mer distributions in the human genome [146], traffic networks [181],
however it extends to art, and genetic regulatory networks [160]; all these
phenomena might be regarded as complex networks, which is the case we display
here, and its importance of an HIV-infection towards a single cell (as well as
the T-cell differentiation process) as complex ones, that is, that the gross
phenomena which arise from it might differ from the individual products of its
constituents, i.e., while we might not be able to explain the clinical course
of the HIV infection by means of tracking the course of a single molecule
included in the process, should we regard the network and the behaviour it
displays as a whole, then can we establish those mappings from the molecular
realm towards the clinical and more macroscopic one.

##  5 Conclusions & Limitations.

Three main findings may be exposed: the T-cell differentiation process tends
towards an equilibrium state which can ensure its differentiated state (for it
could be required an energetic stimulus in order to exit from it), and the
HIV-infection of a T-cell leads to another, novel T-cell differentiated state,
which can be antiapoptotic at first, but when time passes by and equilibrium
is reached, there is a tendency towards a proapoptototic state. These
processes lead toward ”oligarchic” states, that is, states in which only a few
molecules possess the wealth of the process (time spent on those molecules),
this can be seen as how some molecules can play central roles in the process.
Finally, these two processes can be considered complex ones, their
phenomenology corresponds to that in which the behavior in whole cannot be
directly explained by the individual behaviour of its constituents. This model
can serve as scaffold for future simulations of pathogen/cell infections, the
interaction of this systems with additional pathogens can be considered, for
instance, what occurs when an opportunistic virus such as herpes virus type 8
infects the body? What is the nature of the interaction of this virus with the
T-cells and their novel HIV-induced differentiated state? Other questions
which can be addressed are those which correspond to the treatment: what
happens when we block gp120? Can the path towards equilibrium continue? If so,
how does this path look?

One of the clearest limitations of this study is the lack of the logical gates
YES and NO, that is, whether a protein with a higher probability to be
interacted with represents an inhibition or a promotion of its function, this
can be addressed in further studies; another limitation is the one of the
heterogeneity in the nature of the information treated with, that is, the
levels of evidence included in the interactions, The BioGrid database already
displays a system of levels of evidence regarding the interactions of every
protein, however in the data-acquisition process, we have not only considered
interactions derived from this database, for we have considered interactions
recorded in individual papers, as well as in the STRING database, all of them
with distinct levels of evidence. Additionally, the interactions considered do
not consider the very nature of the interaction, it does not distinguish
whether the interaction is genetic, physical, or else. Finally, this model
does not take into account mutations which may lead to dysfunctional protein
products of both the host and the virus, as well as it does not consider the
compound effect of the infection in various cells, but merely in one cell.

##  6 Acknowledgements.

To my friend and colleague Luz Fernanda Jiménez Muñoz for her input in regards
to section 3.5.

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