MODELING RECEPTOR-LIGAND INTERACTIONS FOR THE EXTRINSIC APOPTOSIS PATHWAY

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Apoptosis: an important pathway in systems biology and medicine

Apoptosis comprises a set of chemical reactions through which cells can be eliminated when exposed to intracellular or extracellular stress signaling events (Renault and Chipuk, 2014; Tait and Green, 2010). When cell damage becomes relevant and endangers the rightful genotypic transmission to the incoming cell generations the cell generally chooses to commit apoptosis in order to protect the tissue from alarming and malignant mutations that can propagate to a tissue-level. The proper control of apoptosis is thus essential for the correct development of biological tissues and maintenance of homeostasis and its deregulation is commonly associated to disease conditions (Jacobson et al., 1997; Kerr et al., 1972). Inhibition of apoptosis often correlates with immune disorders and cancer formation and its excessive activation is an underlying feature of Parkinson, Alzheimer and Huntington’s disease. (Brown and Attardi, 2005; Qi and Shuai, 2016).
Apoptosis is defined as either intrinsic or extrinsic depending on the origin of the death-signaling stimulus (Dewson and Kluck, 2009; Rehm et al., 2002). In the intrinsic apoptosis pathway the signal source is due to an intracellular event caused e.g. by a viral infection, excessive exposure to an oxidative stress or high levels of a DNA-damaging agent, such as UV radiation, that lead to the activation of a mitochondrial dependent pathway and the production of effector caspases. The proteins belonging to the effector caspase family are capable for cleaving a subset of intracellular targets, ultimately causing nuclear condensation, DNA fragmentation and consequent cell death (Brenner and Mak, 2009; Brune and Andoniou, 2017). The extrinsic apoptosis pathway is triggered instead by a death-ligand that interacts with the death-receptors on the cell membrane, activating an intra-cellular cascade that results in the production of initiator caspases. These are capable of directly activating effector caspases in the so-called type I cells. However, in some cell-lineages known as type II cells, effector caspases are inhibited by a pool of existing anti-apoptotic proteins and the initiator caspases must follow an indirect pathway and cause first the release of molecular substances from the mitochondria to inhibit the anti-apoptotic proteins present in the cell. In fact, both intrinsic and extrinsic pathways intersect at the level of the mitochondria, differing solely in their signal provenience. In the two pathways the same order is followed: initiator caspases are activated first and only afterwards there’s a rise on the effector caspases concentration inside the cell.
Although massively studied, it is still unclear which step by step conditions or molecular proportions of pro- and anti-apoptotic proteins should be achieved along the several stages of the pathway for the cell to irreversibly commit to death. This has direct implications in clinical applications, for instance in cancer. Cancer cells typically possess innate resistances to both chemotherapy and death-inducing drug agents rendering them almost unaffected to these approaches (Almendro et al., 2013; Fulda and Vucic, 2012; Juin et al., 2013; Lopes et al., 2007). In some cases, resistance arises from specific mutations at the genomic level (Holohan et al., 2013). However, in general, non-genetic factors seem to be the key mechanisms behind resistance to drug and chemotherapy treatments (Cohen et al., 2008; Kreso et al., 2013; Roesch et al., 2010; Roux et al., 2015; Sharma et al., 2010; Spencer et al., 2009). In worse case scenarios these factors can accumulate and eventually persist at the cell population level (Brock et al., 2009; Wakamoto et al., 2013).
The link between the miss-regulation of apoptosis and the resulting phenotypic impairments make it a very challenging topic in both biology and medicine. One of the major drawbacks is to understand why the phenotypic responses of clonal or sister cells are so highly heterogeneous when exposed to identical death signals (Balázsi et al., 2011). This effect, described as “fractional-response”, is a hallmark of apoptosis and is a topic that has been under intense investigation in the past decades.
Multiple modeling efforts have tried to bring to light levels of interactions and mechanisms of control that can make cells to choose between life (apoptosis escape) and death (apoptosis commitment). Along the next lines of the manuscript, the role of the most important proteins of the apoptotic system and a proposed mathematical model will be discussed in parallel as an attempt to tackle this question. In chapter 1, a deep inspection of the literature gathered a theoretical scheme of the extrinsic apoptotic reactions and a general explanation on the most relevant biological events is provided. A quick overview of previous models is also given, with a focus on five models that show how the modeling questions in apoptosis evolved in the past two decades. Chapter 1 ends with a simple description of noise in biology, its different sources and the existing strategies to model each noise source. In Chapter 2, the ARROM1 mathematical model is built from the biological information explained in Chapter 1. The signal of the available data set is briefly explained along with some methods to match the model output with the experimental signal information. Arguments are then presented on the hypothesis that the reactions defined in the literature might be incomplete and an extra set of proteins might exist so that model fittings become more coherent with respect to previous parameter values available in the literature. Chapter 3 provides some examples of how the new model version, with two new proteins, reproduces known biological results, validating the model approach. In Chapter 4, the noise sources introduced in chapter 1 are simulated into the model equations and an inspection is performed to verify which of them can reproduce the heterogeneity observed in the data set. Chapter 5 continues with the fits of the entire cell population and the analysis of which resulting distributions in parameter values and initial conditions contain the largest differences between the population of resistant and sensitive cells. Hypothesis are presented for the dependencies that might exist between the new proposed proteins in Chapter 2 and their connection to FLIP or Caspase-8, the two major apoptotic intervenients at the receptor level. Finally, Chapter 6 summarizes the results of this thesis and proposes new directions in the modeling field of apoptosis.

Apoptosis signaling network

TRAIL, a death-inducing molecular agent initiating apoptosis

Tumor necrosis factor (TNF)-related apoptosis inducing ligand (TRAIL) is a cysteine polypeptide belonging to the class of death-inducing ligand molecules termed TNF-superfamily, capable of selectively triggering apoptosis in cancer cells in vitro and in vivo (Ashkenazi et al., 1999; Walczak et al., 1999). Its application in the cancer-therapy field has been especially encouraging due to its singular mode of action, not only because of its exclusive impact on cancer cells, but also because of its capability of triggering an intracellular death response independent of the p53 signaling cascade that is commonly mutated in a variety of cancer entities (Kozłowska and Puszynski, 2016; Muller and Vousden, 2013; Olivier et al., 2010).
Like the majority of the TNF-family members, TRAIL’s induction of an apoptotic response depends on its ability to recruit complementary death-receptors to the cell membrane in order to form relatively stable homodimers or heterodimers at the cell surface. In particular, two classes of death-receptors, the class TRAIL-receptor 1 (TRAIL-R1/DR4) and/or the class TRAIL-receptor 2 (TRAIL-R2/DR5) aggregate with TRAIL to form active structures capable of propagating the death signal to the interior of the cell (Hymowitz et al., 1999; Jones et al., 1999). Although a natural component of the human bloodstream, TRAIL baseline concentration is insufficient to induce an apoptotic response in natural tissues and its biological role in normal conditions is therefore unknown. (Gibellini et al., 2007; Mariani et al., 1997).
As part of an anti-cancer drug therapy TRAIL shows ulterior advantages as it doesn’t promote the usual toxicity signals derived from cross-activation of pro-survival inflammatory pathways observed in other TNF members (Ashkenazi et al., 1999; Roberts et al., 2011; Walczak et al., 1999). However, it is also long known that besides the desired apoptotic response TRAIL also unleashes the activation of the NF-KB pathway (Chaudhary et al., 1997; Degli-Esposti et al., 1997; Schneider et al., 1997; Trauzold et al., 2001) , MAPK pathway (Tran et al., 2001), and PI3K pathways (Azijli et al., 2012; Xu et al., 2010), all of these pro-survival and undesired secondary signalling pathways. The result is an overall decrease of efficiency of the treatment as at the same time it induces pro-apoptotic signals, TRAIL also unleashes parallel pro-survival signalling responses.
Up to these days TRAIL tests in clinical trials have proven rather disappointing as it was shown in a recent metastatic pancreatic cancer study where phase II-trial patients revealed no overall improvement neither in 12-month survival rate nor on overall survival rate in comparison with a gemcitabine monotherapy [gemcitabine is a common chemotherapeutic agent] (Kretz et al., 2018). Other DR agonistic antibodies including mapatumumab, drozitumab, conatumumab , lexatumumab and LBY135 , all triggering extrinsic apoptosis in a manner similar to TRAIL, have all been discontinued after revealing also unfruitful results in clinical applications (Herbst et al., 2010; Merchant et al., 2012; Rocha Lima et al., 2012; Sharma et al., 2014; Younes et al., 2010). The reasons explaining their unsuccess are still to be properly characterized. In future applications, the optimal direction strategy will pass by the development of more potent and stable drugs that can successfully stimulate the extrinsic apoptosis pathway in a selective manner, avoiding the activation of undesired pro-survival responses. For that matter it is necessary to unveil the dynamics of these interlinked pathways and to understand the different thresholds that allows the cell to activate a so called “fractional survival”, where some cells avoid the apoptotic death-fate and resist the therapy (Shlyakhtina et al., 2017).
Recent findings suggest that the interaction between the death-ligand TRAIL and the complementary death receptors is already a decisive factor that distinguishes the downstream activated pathways. Incomplete trimeric interactions with just two binding sites attached in a TRAIL-R1-TRAIL complex are sufficient for activation of a NF-kB signaling response (Morton et al 2015) and a specific oligomerization between trimeric complexes may be required for the proper activation of the apoptotic signal (Wajant, 2019). A detail understanding of these interactions is thus essential.
The time of application has also shown to be important. A pulse-like TRAIL dosage instead of a continuous treatment has demonstrated to be associated with therapy efficacy. In a study from 2016 it was shown that a 1-min pulse of TNF was more efficient in inducing a cell apoptotic response than a 30-min or 60-min pulse and it was as efficient as a continuous treatment with a 10-hours dosage. The conclusion was that the timing of cell exposure impacts cell fate decision and long applications may stimulate the death receptors in such a way that they can be more prone to activate secondary pro-survival responses. Short-pulse scenarios are then related with weaker NF-kB signaling and stronger pro-apoptotic outcomes (Lee et al., 2016).
The application of the drug, and specifically the time of successive dosages, has also shown interesting dynamics. As a consequence of the augmented levels of pro-survival proteins during the hours that follow a first TRAIL dosage, the subsequent rounds may end up ineffective if a “drug-holiday” is not considered. In this sense, and similar to other chemotherapy drug scenarios, it can be beneficial to establish neutral periods of non-administration until the pro-survival protein levels return to their basal levels (Becker et al., 2011; Das Thakur et al., 2013).
The adequate treatment may pass by the right usage and combination of multiple drugs while also considering the timing of TRAIL application so that in the end the cell system can be sensitized into a correct cell death direction.

DR4 and DR5, decoy death receptors, and clustering modes with TRAIL

TRAIL can bind to a variety of death-receptors (DR) in the cell membrane. The receptor isoforms propagating the apoptotic signal are the DR4 and DR5 that exist in a stable equilibrium of monomer and dimeric versions, being the monomeric the most common based on molecular studies of CD95 (Liesche et al., 2018). The binding affinity between monomeric receptors is low, justifying the existence of a higher proportion of monomeric receptors (Chan, 2000; Clancy et al., 2005; Neumann et al., 2014). Despite less common, not all the receptors in the cell membrane are functional and another class, termed decoy receptors (DcR), is able to interfere with the death-ligand signal and disrupt the interaction with the adequate receptors DR4/DR5. These are the TRAIL-R3 (DcR1) and the TRAIL-R4 (DcR2) receptors which not only compete for direct contact with the ligand but are also capable of sequestering free DR4/DR5 receptors and bind to them, forming non-functional complexes (LeBlanc and Ashkenazi, 2003).

DISC complex, the basic unit structure for activation of Caspase-8, and the role of FLIP as an inhibitor of apoptosis

TRAIL sequentially adds receptor molecules DR4/DR5 until a trimeric complex is formed. After the trimerization, other molecules join with the death receptors to form a functional death-inducing signalling complex (DISC), capable of propagating the signal to the remaining steps of the network. These are the FAS-associated death domain (FADD) molecule and the initiator pro-caspase 8 (pC8), added in the same order and in a sequential manner (Dickens et al., 2012). While the receptor multimerization and association with the ligand is a quick process, the addition of FADD is a limiting and delayed step, essential for the binding of pC8 to the DISC platform (Pennarun et al., 2010). FADD binds to one assembled trimeric receptor complex and acts as an adaptor molecule for the recruitment of several pro-caspases, allowing a first pC8 to be fixed and directly bind to its death effector domain (DED). From there, a second pro-caspase can bind to the free DED of the first pro-caspase and a chain of interlinked molecules is sequentially formed (Schleich et al., 2016). Once a dimerization occurs, by proximity-induced catalytic activation, a functional form of the pro-caspases is released from the receptors, in the form of caspase 8 (C8), sending the apoptotic signal from the receptors to downstream reactions occurring in the cytoplasm of the cell (de Miguel et al., 2016; Dickens et al., 2012) .
A controlling step in this process depends on a homolog molecule with a very similar structure to that of the pro-caspases, the FADD-like IL-1-converting enzyme (FLICE)-inhibitory protein (c-FLIP), that competes for a binding position with FADD suppressing the elongation chain of linked pro-caspases and obstructing their activation (Hughes et al., 2016). Its major role is to hinder the apoptotic signal setting a threshold between quantities of pro-apoptotic caspases vs. the quantity of pro-survival FLIP molecules in the DISC unit.

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Caspase-8, a threshold for cell-fate decision

The requirements for the activation of C8 is the presence of a DISC assembled structure with at least two attached pC8 chains, interlinked through their DED domains. In this configuration, the two pC8 proteins inter-activate themselves and form a single structure with catalytic activity that is next released from the DISC unit and navigates into the cytoplasm to participate in the cleavage of downstream molecules. The amount of C8 released from the receptor in this way is essential as a first step towards an apoptotic cell response but yet little is known about the actual mechanisms that control C8 numbers inside the cell.
Recently, with the scope of understanding and quantifying the C8 activity level in apoptotic committed cells, Roux and colleagues analyzed hundreds of HeLa-cells treated with TRAIL at multiple concentrations and concluded that the dynamics of the C8 protein plays a central role in cell-fate decision, unveiling a necessary relationship between the rate of C8 activity (k) and the time of maximum C8 activity ( ) for cells to properly engage into apoptosis (Roux et al., 2015). Each cell was fitted to the parametric model ∗ ( − ) and the optimal maximum derivative value, ( ∗ [ − ]), that best separated the group of sensitive cells from the group of resistant cells was defined as . The maximum derivative value assigned to can then be interpreted as a biological threshold, able to distinguish surviving cells from dying cells in the (k, ) space (red line in figure 1-C).
The work was pioneer in the field by presenting for the first time a metric to predict apoptosis propensity on a given cell. Although simplified by the usage of the lumped parameters k and , the approach was accurate enough to correctly forecast the cell fate of more than 80% of the cell population in several cell lines. This number, as a result of a model approximation that considered uniquely DISC-related events, agreed with the result of Gaudet and colleagues that mentions that the non-DISC related proteins contribute to the overall heterogeneity of the apoptotic signal in a range of 20% (Gaudet et al., 2012).
An interesting conclusion of this study was that cells can be sensitized to cross in multiple ways such as by increasing the ligand (TRAIL) concentration (by stabilizing the DISC and consequently augmenting the C8 rate of activation (k)) or by adding drugs that stabilize the C8 molecule (e.g. bortezomib, an inhibitor of protein degradation triggered in a proteasome dependent way) [increasing the time of C8 activation ( )]. In the opposite direction, cells can also turn more resistant to apoptosis by overexpressing anti-apoptotic proteins as the FLIP isoforms (specially the FLIP-s (FLIP short) configuration). All these scenarios are DISC-related events and do not change = 2.61×10−3 s−1. However, when adding a drug acting downstream of C8 activation, the value of the threshold changes in a concomitant way. This was verified by the addition of the pro-apoptotic BCL-2 inhibitor, ABT-263, that by decreasing the mitochondrial outer membrane permeabilization (MOMP) “sensitivity”, it was able to lower in those cell populations, making them more prone to die. In the end, the framework was especially relevant as it compiled information from C8 trajectories that varied importantly in signal and had (k, ) pair values changing tenfold and threefold, respectively, across the cell populations (Roux et al., 2015).
Although simple in the description of the underlying apoptotic pathway, the work still has the potential to be further explored and extended so as to include the role of other molecules also activated under an apoptotic stimulus but known to be C8-independent (Scott et al., 2017). A more complete version would be most useful to understand in better detail how to make a cell irreversibly enter in the death-region space (figure 1-C, D). The framework suggests several ways in which cells can be sensitized to cross the theta-threshold line but cells still remain close to the theta vicinity, calling for more effective co-drugging treatments.
A) A simplified scheme for the activation of C8 from the DISC and consequent cleavage of its substrate ICRP (a reporter fluorescent protein), returning the output signal of the cleaved fluorescence reporter (FR). The lumped parameter k represents the rate of C8 activation and t0 the time required to start receiving the signal from the first cleaved ICRP molecule. B) The shape of an output signal produced by an averaged surviving and dying cell. Dying cells have a more pronounced parabolic trajectory curve, a sign of higher C8 activities. The model ( ) = ( − ) describes the period after ligand stimulation, when C8 activity increases and follows a quadratic polynomial curve and is valid until the maximum value of dFR/dt, corresponding to the maximum C8 activity value at time t = . C) Dosage increase of the ligand causes an overall increase in C8 activity (k) and a small decrease on the time interval required to reach the associated maximum activity ( ). The -value of the population remained unchanged D), E) Co-treatments cause a deviation on the (k, ) values of a mean-cell of the population and on the landscape of the curve. Figure adapted from (Roux et al., 2015).

Bcl-2 like proteins, pro- and anti-apoptotic role

Once the DISC complexes are activated and functional C8 is generated, a group of proteins of the BCL-2 family group intervenes into the apoptosis cascade and holds the next decision to block the incoming signal or propagate it to the mitochondria outer membrane (MOM). This decision point, specific of type II cells, starts with the C8 cleavage of the Bcl-2 family member Bcl-2 homology domain 3 (BH3) interacting domain (Bid) into a truncated form tBid that, in this form, is translocated from the cytosol to the MOM where it can interact with other elements of the Bcl-2 family. This step is of major importance and should be a result of sustained C8 activation in order to reach an important pool of tBid molecules and surpass an irreversible pro-apoptotic threshold.
The molecular targets of tBid are the Bcl-2-associated X protein (Bax) and Bcl-2 homologous antagonist killer (Bak), both responsible for MOMP and propagation of the apoptotic signal (Green, 2004; Huang et al., 2016). The agent tBid allows the oligomerization of Bax and Bak to occur, changing them into an active form that is capable of perforating the MOM and transform it into an all-or-none MOMP signal.
A tight level of control is also set by a group of pro-survival proteins, unbalancing the signal from Bax and Bak and allowing the cell to control spurious apoptotic stimulus. Under these conflicting forces a minimum limit is then set for Bax and Bak pro-apoptotic intensity. Among the pro-survival proteins there is the B-cell lymphoma-extra-large (Bcl-XL) and the myeloid leukemia cell differentiation proteins (Mcl-1) which bind to Bax and Bak and convert them into their inactive and non-oligomerized form not capable of triggering MOMP (Llambi et al., 2011; Willis, 2005).
As a synthesis, the Bcl-2 family members can be divided into three groups: pore-forming effector proteins, Bax and Bak; BH3-like proteins such as tBid; and the pro-survival proteins such as Bcl-2, Bcl-xL and Mcl-1. The effector proteins are capable of forming mitochondrial pores but only after a conformational change induced by the BH3-like proteins (Chipuk et al., 2010). On the other hand the pro-survival proteins sequester the pro-apoptotic ones, rendering them inactive until a sufficiently high stoichiometric proportion of BH3-like proteins participates and sets the effector proteins free from the inhibitory effect of the pro-survival proteins (Sarosiek et al., 2013). This occurs by direct competition where the BH3-like proteins interact with the pro-survival Bcl-2 members and release them from the effector proteins, making them accumulate at the MOM. Lower intensity apoptotic signals are not able to summon sufficient BH3-like proteins and in these conditions effector proteins are constantly translocated by the pro-survival proteins from the MOM back to the cytosol (Edlich et al., 2011; Schellenberg et al., 2013; Todt et al., 2015).
A positive feedback loop has been proposed for the BCL-2 effector family member Bax, where already activated Bax could reinforce the apoptotic signal by contributing to the activation of still inactive Bax molecules (Cui et al., 2008; Tan et al., 2006). In this way the cell could decrease the necessary levels of tBid required to trigger MOMP or reinforce the response into an all-or-none switch. This complicated network of interactions is not completely understood and may hide other still unknown dynamics.

MOMP, an irreversible commitment to cell-death

Upon a sufficient amount of tBid accumulation and Bax / Bak oligomerization at the MOM, the MOMP event ensues and molecular agents such as cytochrome c (CyC) and the second mitochondrial activator of caspases (Smac) are released from the mitochondrial intermembrane space (IMS) into the cytoplasm (Tait and Green, 2010). Both of these agents are implicated in effector caspases activation (Galluzzi et al., 2009). CyC accumulation together with the apoptotic protease activating factor-1 (Apaf-1) create the apoptosome, a stimulation platform for caspase-9 (C9) activation. C9 positively contributes for the activation of the effector caspases -3 (C3), -6 (C6) and -7 (C7) and from this point apoptosis is irreversibly unleashed (Riedl and Salvesen, 2007). Smac contributes to effector caspases onset by causing direct binding and inhibition of X-linked inhibitor of apoptosis protein (XIAP), ensuring a fast activation of C3, C7 and C9 and a stronger pro-apoptotic response (Deng, 2002).
While the onset of MOMP is highly variable and can last up for several hours after an extrinsic or intrinsic apoptotic stimulus exposure, the time frame of a MOMP response after tBid translocation to the membrane is in the order of minutes and suggests a switch type like-response (Albeck et al., 2008; Goldstein et al., 2000; Rehm et al., 2009). The speed of MOMP event depends on the accumulation of Bax/Bak at the MOM. Since only 10% of the Bax cytosolic pool is already sufficient to cause MOM depolarization and consequent MOMP, the speed of these ending processes is fast and tightly regulated among all the molecular intervenients (Hantusch et al., 2018).

Table of contents :

1 INTRODUCTION
1.1 MOTIVATIONS
1.2 APOPTOSIS SIGNALING NETWORK
1.2.1 TRAIL, a death-inducing molecular agent initiating apoptosis
1.2.2 DR4 and DR5, decoy death receptors, and clustering modes with TRAIL
1.2.3 DISC complex, the basic unit structure for activation of Caspase-8, and the role of FLIP as an inhibitor of apoptosis
1.2.4 Caspase-8, a threshold for cell-fate decision
1.2.5 Bcl-2 like proteins, pro- and anti-apoptotic roles
1.2.6 MOMP, an irreversible commitment to cell-death
1.2.7 Global overview on the extrinsic apoptosis pathway
1.3 MODELLING IN APOPTOSIS
1.3.1 Fussenegger’s model, 2000
1.3.2 Albeck’s EARM1 model, 2008
1.3.3 Schlatter’s & Calzone’s model, two Boolean modeling approaches
1.3.4 Bertaux’s model, a stochastic protein-turnover approach 2014
1.3.5 A summary list of models of apoptosis
1.4 HETEROGENEITY IN BIOLOGY, INTRINSIC AND EXTRINSIC NOISE
2 MODELING RECEPTOR-LIGAND INTERACTIONS FOR THE EXTRINSIC APOPTOSIS* PATHWAY
2.1 A NETWORK OF REACTIONS DEFINING A MODEL STRUCTURE FOR THE RECEPTOR LAYER OF THE EXTRINSIC APOPTOSIS NETWORK
2.2 AVAILABLE EXPERIMENTAL DATA
2.3 CONVERSION OF FRET-SIGNAL INTO NUMBER OF ICRP-CLEAVED MOLECULES
2.4 ARROM1: INITIAL CONDITIONS AND PARAMETER VALUES
2.5 ARROM2: A LIGAND-RECEPTOR MODEL WITH AN EXTRA SET OF PROPOSED REACTIONS
3 VALIDATION OF ARROM2: A RECEPTOR-LIGAND MODEL IN AGREEMENT WITH EXPERIMENTAL DATA
3.1 FLIP, A STRONG ANTI-APOPTOTIC PROTEIN WITH IRREVERSIBLE BINDING AT THE DISC STRUCTURE
3.2 DIMERIC VS. TRIMERIC LIGAND VALENCY, AN UNEQUAL RECEPTOR BINDING RATE
4 SOURCES OF HETEROGENEITY IN APOPTOTIC CELL-FATE DECISION 
4.1 INTRINSIC NOISE: A COMPUTATIONAL APPROACH WITH THE GILLESPIE ALGORITHM
4.2 EXTRINSIC NOISE: A COMPUTATIONAL APPROACH WITH INITIAL CONDITION VARIATION.
4.3 PARAMETER NOISE: A COMPUTATIONAL APPROACH WITH VARIATION IN REACTION RATES
5 FITTING THE ENTIRE CELL POPULATION AND SEARCHING FOR SIGNATURES IN SENSITIVE AND RESISTANT POPULATIONS
6 DISCUSSION AND PERSPECTIVES
7 APPENDIX
7.1 APPENDIX 1: ARROM1 MODEL INPUTS
7.2 APPENDIX 2: ARROM1 FIT TO THE MEDIAN_CELL
7.3 APPENDIX 3: ARROM2 FIT TO THE MEDIAN_CELL
7.4 APPENDIX 4 : ARROM1 MODEL EQUATIONS
7.5 APPENDIX 5: A NEW METHOD TO QUANTIFY PARAMETER VARIABILITY AMONG DIFFERENT PARAMETER POPULATIONS
8 BIBLIOGRAPHY

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