 Research
 Open Access
On the modeling and analysis of the biological regulatory network of NF\({\kappa }\)B activation in HIV1 infection
 Zurah Bibi^{1},
 Jamil Ahmad^{1}Email author,
 Amjad Ali^{2},
 Amnah Siddiqa^{1},
 Shaheen Shahzad^{3},
 Samar HK Tareen^{1},
 Hussnain Ahmed Janjua^{2} and
 Shah Khusro^{4}
 Received: 6 November 2015
 Accepted: 22 December 2015
 Published: 8 January 2016
Abstract
Purpose
The complex interactions between genetic machinery of HIV1 and host immune cells mediate dynamic adaptive responses leading to Autoimmune Deficiency Syndrome. These interactions are captured as Biological Regulatory Network (BRN) which acts to maintain the viability of host cell machinery through feedback control mechanism which is a characteristic of complex adaptive systems. In this study, the BRN of immune response against HIV1 infection is modeled to investigate the role of NFκB and TNFα in disease transmission using qualitative (discrete) and hybrid modeling formalisms.
Methods
Qualitative and Hybrid modeling approaches are used to model the BRN for the dynamic analysis. The qualitative model is based on the logical parameters while the hybrid model is based on the time delay parameters.
Results
The qualitative model gives useful insights about the physiological condition observed as the homeostasis of all the entities of the BRN as well as pathophysiological behaviors representing high expression levels of NFκB, TNFα and HIV. Since the qualitative model is time abstracted, so a hybrid model is developed to analyze the behavior of the BRN by associating activation and inhibition time delays with each entity. HyTech tool synthesizes time delay constraints for the existence of homeostasis.
Conclusion
Hybrid model reveals various viability constraints that characterize the conditional existence of cyclic states (homeostasis). The resultant relations suggest larger cycle period of HIV1 than the cycle periods of the other two entities (NFκB and TNFα) to maintain a homeostatic expressions of these entities.
Keywords
 HIV
 NFκB
 TNF\(\alpha\)
 BRN
 Qualitative modeling
 Hybrid modeling
Background
Autoimmune Deficiency Syndrome (AIDS) is a major pandemic, caused by the human immunodeficiency virus (HIV). HIV targets CD4+ T lymphocytes and hinders the regulation and amplification of immune cells (Douek et al. 2002). This infection results in the persistent activation of immune system, and consequently, CD4+ T lymphocytes would greatly reduce in number resulting in body’s ability to fight against HIV infection. There is a need to understand the mechanism of immune activation and how it gets repressed by viral overload so the major elements of immune system that plays an important role are discussed below in detail.
Another important protein that plays significant role both in innate and adaptive immune response is the transcription factor NFκB. In the absence of foreign antigen(s) NFκB dimers are retained in the cytoplasm through the inhibitory action of the IκB molecules (Ghosh and Karin 2002). The proinflammatory cytokines TNFα and Interleukin1\(\beta\) (IL1\(\beta\)) induce the activation of NF\(\kappa\)B pathway accompanied by rapid degradation of I\(\kappa\)B\(\alpha\), releasing NF\(\kappa\)B dimers to the nucleus (Grey 2008), where they activate HIV transcription in infected cells (Ghosh and Karin 2002).
The immune regulators, such as cytokines, interact with the viral proteins in a feedback mechanism (see Fig. 1) to maintain homeostatic behavior of the organism. Homeostasis is one of the vital creeds of selfregulation and internal organization and is defined as the synchronized physiological reactions responsible for maintaining most of the healthy states and conditions in the body. Positive and negative feedback loops induce an on/off switching mechanism (Brandman and Meyer 2008), which stops the respective activation state after the arrival of a set point or threshold (Thomas and Thieffry 1995; Thomas et al. 1998). Such feedback mechanisms represent a characteristic feature of complex adaptive systems which facilitates the emergence of complex patterns of behaviors such as homeostasis and stable states (reference: Complex Adaptive Systems Modeling: A multidisciplinary Roadmap). These positive and negative interactions among biological entities comprise a Biological Regulatory Network (BRN) providing the basis to converge into one of the many possible outcomes. The modeling and analysis of expression dynamics of BRNs can be used to increase our understanding of the underlying adaptive mechanisms in response to various stimuli but the presence of large number of unknown biological parameters e.g., kinetic rates of reactions and the expression values of molecules etc. makes it challenging. The kinetic logic modeling formalism of Rene Thomas is one of the established methods to perform qualitative analysis of such systems (Thomas and Thieffry 1995; Gagneur and Casari 2005; De Jong et al. 2004) without any prior knowledge of kinetic parameters. The parameters of the model are computed using model checking technique based on the biological observations (see “Methods”). These parameters are then used to construct a qualitative model (state graph) which captures all the possible behaviors of the BRN as cyclic trajectories or paths diverging to the stable states. Rene Thomas has proved the similarity of differential equation models and qualitative models based on kinetic logic formalism in ([55] of rehan sepsis). However, qualitative modeling is more appropriate modeling paradigm due to its ability to infer unknown parameters ([56] of rehan sepsis). This formalism has been successfully implemented to gain insights into the pathogenesis clearance of Dengue virus and identification of the putative drug targets in Cerebral Malaria and Sepsis.
Several signaling pathways have been carefully reviewed to construct a BRN based on HIV intervention of immune system (Fig. 1). In the current study, the qualitative model developed by Bibi et al. (2011) is employed to investigate the mode of immune activation in hindering HIV1 viral proteins rapid expression. This approach facilitates in obtaining useful constraints that provides necessary conditions for the BRN to evolve and helps to analyze immune response of the body against foreign invaders. The HyTech model checker tool is used to analyze the hybrid model of the BRN (Ahmad et al. 2007) that generates delay constraints (Fromentin et al. 2010) for the existence of a particular cyclic behavior (representing homeostasis). The model findings are in line with experimental data and provides novel insights into the disease propagation mechanism which could be used for its intervention strategies.
Methods
Discrete modeling formalism
The discrete modeling formalism of René Thomas (Ahmad et al. 2012, 2007, 2009; Ahmad and Roux 2008; Thieffry and Thomas 1998; Thomas and D’Ari 1990), a type of qualitative logical modeling, has been successfully used for the construction of BRN models. This formalism uses a set of parameters that reflect the behaviors of a BRN in terms of the presence or absence of the regulatory elements (resources). Following are the formal definitions which are used to construct a discrete model of a BRN (Ahmad et al. 2012).
In a directed graph \(G = (X, A)\), where X is a set of vertices (nodes) and A is a set of edges linking the vertices, \(G^ (v)\) and \(G^+(v)\) represent the set of predeccessors and successors of a vertex \(v \in X\) respectively.
Definition 1
(Biological Regulatory Network) A BRN is a graph \(G = (X, A)\) where X represents the set of nodes representing biological entities and A is the set of edges representing interactions between biological entities. Each edge \(a \rightarrow b\) is labeled as (s\(_{ab}\), \(r_{ab}\)), where s\(_{ab}\) is a positive integer representing a threshold and \(r_{ab}\) \(\in \{+, \}\) shows the type of interactions (‘\(+\)’ for activation and ‘\(\)’ for inhibition).
To analyze the behavior of a BRN, it is necessary to know all the possible states and the transitions between them. There is a limit discrete expression value \(lm_{v}\) for each entity v of a BRN which is equal to its outgoing degree, such that \(\forall\) \(u \in G^+(v)\) each \(s_{uv}\) \(\in \{1,\ldots ,n_{v}\) \(\}\) where \(n_{v}\) \(\le\) \(lm_{v}\). Each entity carries its expression levels in the set \(Q_{v}= \{0,\ldots ,n_{v}\}.\)
Definition 2
A vector is normally used to show a qualitative state \((l_{v})_{v \in X}\), where \(l_{v}\) represents the expression level of the entity v.
A set of resources represents the activators of an entity at any instant of time.
Definition 3
From the above definition, it can be inferred that the absence of an inhibitor below threshold is considered as a resource.
The set of logical parameters assigned to each biological entity determine the dynamics of a BRN.
Definition 4
\(K_{v, R_{l_v}}\) define the target expression level towards which the expression of entity v evolves.
Definition 5

there is a unique \(v \in X\) such that \(s_{{l}_{v}} \ne s'_{l_v}\) and \(s'_{l_v} = s_{l_v} \Rsh K_{v, R_{l_v}}\);

and \(s_{l_u} = s'_{l_u} \forall u \in X \setminus {v}\).
A state graph differs from its successor state by one component only, so if a state s has n elements to be evolved then it will have at most n successor states.
The state table of genes x and y
x  y  \(R _{x}\)  \(R _{y}\)  \(K_{x,R _{x}}\)  \(K_{y,R _{y}}\) 

0  0  {}  {x}  0  1 
0  1  {x}  {y}  1  1 
0  2  {y}  {x,y}  1  2 
1  1  {y}  {}  1  0 
1  0  {}  {}  0  0 
1  2  {y}  {y}  1  2 
Hybrid modeling
In a BRN each gene is associated a clock to record evolution time of its expression. The clock is reset to zero after each transition whenever the expression passes from one level to another (Ahmad et al. 2009). In the current BRN, delays are taken as unvalued parameters to introduce the concept of Parametric Bio Linear hybrid Automaton (BioLHA) (Ahmad et al. 2007, 2009, 2012; Ahmad and Roux 2008) that could be analyzed for certain delay constraints describing the stay conditions for the behaviours of BRN.
We denote by V and \(\mathbbm {P}\), the sets of real valued variables and parameters, respectively. Let \(C =(V, \mathbbm {P})\), \(C \le (V, \mathbbm {P}))\) and \(C \ge (V, \mathbbm {P}))\) represent the set of atomic constraints using only =, ≤ and ≥, respectively.
Definition 6

L is a finite set of locations;

\(l_{0}\) is the initial location;

\(\mathbbm {P}\) is the finite set of parameters (delays);

V is a finite set of realvalued variables (clocks);

\(E\subseteq L \times C = (V, \mathbbm {P}) \times 2^{v} \times L\) is a finite set of edges. \(e = \{l ,g, \mathbbm {R}, l'\}\) \(\in E\) is the edge between the locations l and \(l'\) with the guard g and the reset set \(\mathbbm {R} \subseteq V\) contains the set of clocks which appears in g.

Inv assigns invariant condition as a conjunction of atomic constraints \(C \le (V, \mathbbm {P})) ~ and ~C \ge (V, \mathbbm {P}))\) to each location \(l \in L\);

Diff \(L \times V \rightarrow \{1, 0, 1\}\) assigns an evolution rate to each continuous variable \(v \in V\) (clocks) in each location \(l \in L\).
The semantics of a parametric BioLHA is a timed transition system. It is defined according to the time domain T. We let T* \(= T\) \\(\{0\}\).
Definition 7

Discrete transitions: \((l, v) \rightarrow (l', v')\) iff \(\exists ~\{l, g, \mathbbm {R}, l'\} \in E\) such that \(g(v)=\) true, \(v'(h)=0\) if \(h \in \mathbbm {R} ~and~ v'(h)=v(h) ~if~ h \notin \mathbbm {R}\).

Continuous transitions: For \(t \in\) T*, \((l, v) \rightarrow (l', v')\) iff \(l'= l, v'(h)= v(h) + Diff(l, h) \times t\), and for every \(t' \in [0,t], (v(h) + Diff(l, h) \times t' \models Inv(l)\).
Definition 8
(Temporal Zone) A zone in which the discrete level of entities remains the same, however the clocks of the entities keep ticking until the evolution (activation or inhibition) of an entity takes place (Ahmad et al. 2008, 2009).
Definition 9
(Temporal State Space) The temporal state of a BRN comprises of the complete set of temporal zones derived from the discrete model of the respective BRN.
In the hybrid model of a BRN, we denote \(\phi (t)\) for \(t\in \mathbb {R}_{\ge 0}\), while the sequence of points of a trajectory and the set of all points in its state pace is denoted by S (Ahmad et al. 2009). A particular trajectory is said to be viable, iff it remains within a prescribed region known as its viability domain (Ahmad and Roux 2008).
Definition 10
(Invariance kernel) An invariance kernel \(K\in S\) is the largest invariant subset of S iff for any point \(p\in K\), any trajectory starting in p is viable in K.
For the analysis of invariance kernel, the notion of Period (denoted by \(\pi )\) (Ahmad et al. 2009) is introduced.
Definition 11
(Period) It is defined as sum of all the delays that a gene goes through sequentially and successively, once through each of all its expression levels in a cycle (Ahmad et al. 2009). It is denoted as \(\pi\).
HyTech (Henzinger et al. 1997) is used to analyze the invariance kernel and convergence domain, the results of which are expressed as linear constraints that specify the conditions under which a particular behaviour exists. (Ahmad et al. 2007, 2009; Ahmad and Roux 2008).
Discrete and hybrid modeling of the NF\(\kappa\)B associated BRN
State transition table: transition from one state to the next state is given, here activators act as resources while absence of an inhibitor is also taken as its resource that decides its K Parameter (e.g. absence of TNF\(\alpha\) in Ist row act as a resource for HIV that set K Parameter for HIV as 1)
States  Resources  K Parameters  Transition to states  

HIV  TNFα  NFκB  R_{HIV}  R_{TNFα }  R_{NFκB}  K_{HIV}  K_{TNFα }  K_{NFκB}  
0  0  0  {TNF\(\alpha\)}  {}  {}  1  0  0  (1,0,0) 
0  0  1  {TNF\(\alpha\),NF\(\kappa\)B}  {}  {}  1  0  0  (1,0,1), (0,0,0) 
0  0  2  {TNF\(\alpha\),NF\(\kappa\)B}  {NF\(\kappa\)B}  {}  1  1  0  (1,0,2), (0,1,2), (0,0,1) 
0  1  0  {}  {}  {TNF\(\alpha\)}  0  0  2  (0,0,0), (0,1,1) 
0  1  1  {NF\(\kappa\)B}  {}  {TNF\(\alpha\)}  1  0  2  (1,1,1), (0,0,1), (0,1,2) 
0  1  2  {NF\(\kappa\)B}  {NF\(\kappa\)B}  {TNF\(\alpha\)}  1  1  2  (1,1,2) 
1  0  0  {TNF\(\alpha\)}  {HIV}  {HIV}  1  1  2  (1,1,0), (1,0,1) 
1  0  1  {TNF\(\alpha\),NF\(\kappa\)B}  {HIV}  {HIV}  1  1  2  (1,1,1), (1,0,2) 
1  0  2  {TNF\(\alpha\),NF\(\kappa\)B}  {HIV, NF\(\kappa\)B}  HIV  1  1  2  (1,1,2) 
1  1  0  {}  {HIV}  {HIV,TNF\(\alpha\)}  0  1  2  (0,1,0), (1,1,1) 
1  1  1  {NF\(\kappa\)B}  {HIV}  {HIV,TNF\(\alpha\)}  1  1  2  (1,1,2) 
1  1  2  {NF\(\kappa\)B}  {HIV, NF\(\kappa\)B}  {HIV,TNF\(\alpha\)}  1  1  2  ( ) 
Results
Parameter synthesis and Qualitative Model of NF\(\kappa\)B associated BRN
State graph
Realtime hybrid model
The qualitative modeling process generated all possible discrete states (Fig. 8) and their transitions which give further insights into the steady states behaviors. These steady states (cycles and stable states) are in agreement with the general hypothesis of wetlab and clinical experiments. After applying the discrete modeling formalism on NF\(\kappa\)B associated BRN, its hybrid model is obtained for the BRN with help of parametric constraints. Hybrid modeling allowed us to compute the necessary and sufficient conditions for the existence of different behaviors.
Constraints for the invariant cycle
Cycle no.  000\(\rightarrow\)100\(\rightarrow\)110\(\rightarrow\)010\(\rightarrow\)011\(\rightarrow\)001\(\rightarrow\)000 

Constraints  
1  \(\delta ^+_{NFKB010}+\delta ^_{NFKB001}+2\delta ^_{TNFA011}=\delta ^+_{HIV000} +\delta ^_{HIV110}\) 
2  \(\delta ^+_{TNFA100}\)+\(\delta ^_{TNFA011}\) \(\le \delta ^+_{HIV000}\)+\(\delta ^_{HIV110}\) 
3  \(\delta ^+_{HIV000}+\delta ^_{HIV110}+\le \delta ^+_{NFKB110}+\delta ^_{NFKB001}+2\delta ^_{TNFA011}\) 
4  \(\delta ^+_{HIV011}\ge \delta ^_{TNFA011}\) 
5  \(\delta ^+_{NFKB011}\ge \delta ^_{TNFA011}\) 
6  \(\delta ^+_{HIV001}\ge \delta ^_{NFKB001}+2\delta ^_{TNFA011}\) 
7  \(\delta ^+_{TNFA001}\ge \delta ^_{NFKB001}+\delta ^_{TNFA011}\) 
8  \(\delta ^+_{HIV000}\le \delta ^+_{NFKB000}\)+\(\delta ^_{NFKB001}\)+\(2\delta ^_{TNFA011}\) 
9  \(\delta ^+_{HIV000}\le \delta ^+_{TNFA000}\)+\(\delta ^_{TNFA011}\) 
10  \(\delta ^+_{HIV000}\ge \delta ^_{NFKB001}+2\delta ^_{TNFA011}\) 
11  \(\delta ^+_{TNFA100}\)+\(\delta ^_{TNFA 011} \le \delta ^+_{HIV000}\)+ \(\delta ^_{HIV100}\) 
12  \(\delta ^+_{TNFA100}\le \delta ^+_{NFKB100}\)+\(\delta ^_{NFKB001}\)+\(\delta ^_{TNFA011}\) 
13  \(\delta ^+_{HIV000}\le \delta ^+_{TNFA100}\)+\(\delta ^_{TNFA011}\) 
By varying time delays, i.e., the delay required by an entity to evolve or degrade, system dynamics can be accurately observed. This new model is then analyzed with linear hybrid model checker tool HyTech (Henzinger et al. 1997) in order to extract the constraints for cyclic states, the significant ones of which are discussed below:
Discussion
In this study, qualitative and hybrid models of the BRN for immune activation (upon HIV infection) were constructed based on the multivalued logical formalism of Thomas, to estimate the important dynamic properties characterized by regulation delays. On the basis of HyTech results, two important relations are drawn from the set of constraints: \(\pi (HIV)\ge \pi (NF\text{}\kappa B)\) and \(\pi (HIV) \ge \pi (TNF\text{}\alpha )\).
In order to maintain the homeostatic balance of human immune defense, the invasion period of HIV must span longer duration as compared to TNF\(\alpha\) and NF\(\kappa\)B during the initial phases of viral attack. This will tend to reduce the viral proteins ability to start its replication rapidly, giving strength to body’s innate immunity to fight against HIV1 pathogenesis. It reflects an important relationship between HIV, TNF\(\alpha\) and NF\(\kappa\)B. When the progression of HIV proteins is slow in comparison to immune defense mechanism, the virus normally takes much longer to infect and override the whole immune system of the infected individual. This phenomenon is followed by rapid generation of cytokines by cellular transcriptional machinery NF\(\kappa\)B while there is a considerably slow replication and invasion of HIV in the body. It shows frequent evolution of NF\(\kappa\)B and TNF\(\alpha\) and relatively stable behavior of HIV in comparison to the other two. However, if the fore mentioned constraint is violated, the system would get stuck in the stable steady state (1,1,2) shown in Fig. 8, where the chances of survival diminishes. This situation is generally seen during the period of acute infection. Keeping this in view, longer survival time of patients (clinical latency periods) are observed in case of longer qualitative period of HIV than that of immune defense elements (i.e., NF\(\kappa\)B and TNF\(\alpha\)). Furthermore, at initial stages of HIV infection, highlevels of CD8+ T cell frequencies have been observed to correlate with the control of viral replication (Wilson et al. 2000; Allen et al. 2000; Davenport et al. 2004). From this, it can be concluded that while designing drugs against HIV1, the delay time for HIV1 proteins’ expression must be kept greater than those of NF\(\kappa\)B and TNF\(\alpha\) by administrating greater dose of cytokines (TNF\(\alpha\)).
When the time period for HIV’s protein expression is lesser, it indicates that the system is attainting the stage of viral setpoint. At this stage, number of TNF\(\alpha\) decreases in T lymphocytes and this can be seen by the shifting of the initial steady state of TNF\(\alpha\) (cytokines) to a new equilibrium value. This new equilibrium value is lower than the preinfection value (Fauci 1996). At this stage, an exponential increase in virus load may be observed, reaching a peak and finally declining to the steady state level also called as viral setpoint (Kaufmann et al. 1998; Lindbäck et al. 2000). HIV setpoint serves as a predictor of AIDS and it gets stabilize after a period of acute HIV infection (Burg et al. 2009).
In vitro studies have shown that HIVinfected CD4+ T cells are lysed by HIV viral proteins and by mathematical modeling the primary HIV infection kinetics, it has been revealed that the viralinduced cytopathicity and susceptible T cells (also called the ’targetcell limited model’) controls the infection (Somasundaran and Robinson 1987; Muller and Bonhoeffer 2003).
By observing the different behavioral patterns of HIV, TNF\(\alpha\) and NF\(\kappa\)B with respect to time, it can be observed that it depends upon time delays taken by these key players to incite or repress a specific response. These conditions help us to understand that how the equilibrium is maintained in the body upon arrival of any invader (or infection), e.g., if the qualitative period of HIV remains slower than TNF\(\alpha\), no symptoms will appear until this condition is violated. In this way the immune cells (macrophages) can suppress the viral infected cells or hinder their binding to macrophages by speeding up the production of cytokines, chemokines and several other growth factors essential for immune stimulation. This model of HIV1 infection can be further extended to explore disease behaviour to get deeper insights. The involvement of Rel Homology domain with discriminatory domains (Rel A and Rel B) having contrasting functions make NF\(\kappa\)B a very interesting candidate for drug designing. The contrasting role of Rel B in antiinflammation as compared to Rel A (causing inflammation) revealed interesting results in initial clinical findings. Experimental data shows that suppression of Rel A domain by introducing Rel B externally (or making it active by other means) tends to slow down the disease progression in the body and helps to inhibit inflammation caused by Rel A.
By adding more entities in the BRN understudy, exact therapeutic targets can be found by docking and simulation studies of the interacting proteins. These findings would aid in the process of drug designing. Model checking techniques could help us to design a computer aided tool for rapid immunological analysis. Real time modeling and its applications would pave the way towards predicting the preventive medicines, which will give the advancement in medicine developed for the treatment of AIDS.
Conclusion
Qualitative and hybrid models of the Biological Regulatory Network (BRN) of immune response against HIV1 infection were constructed to explore the qualitative and realtime properties of the dynamics of the three entities HIV, \((NF \text{} \kappa B)\) and \(TNF \text{} \alpha\). The qualitative model predicted one stable state showing the over expression of all the interacting entities and two cycles representing a recovery mechanism of the BRN from HIV infection. The hybrid model was further analyzed to predict the delay constraints for the existence of the invariant cycle. Two important delay constraints were identified which show the oscillatory periods of HIV is greater than the oscillatory periods of \((NF \text{} \kappa B)\) and \(TNF \text{} \alpha\). These findings may help in the design of new generation drugs in the treatment of AIDS.
Declarations
Authors’ contributions
Conceived and designed the experiments:ZB JA. Performed the experiments: ZB. Analyzed the data: ZB JA AS SHKT. Contributed reagents/materials/analysis tools: ZB JA AS SHKT JA AA SS SK. Wrote the paper: ZB JA. All authors read and approved the final manuscript.
Acknowledgments
This research work was supported by Higher Education Commission (HEC) of Pakistan, National University of Sciences and Technology (NUST) and International Islamic University, Islamabad, Pakistan.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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