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Fussenegger’s model, 2000
Models of Apoptosis date as far back as the beginning of the century when Fussenegger and colleagues proposed a model for apoptosis with both extrinsic and intrinsic associated pathways (Fussenegger et al., 2000). The model included some of the knowledge of the time such as the receptor role in the activation of functional C8 and the importance of mitochondria in intersecting and transducing both exterior and stress-induced signals. Assembling those facts, the authors were capable of describing known phenomena like the ability to inhibit the apoptotic cascade by overexpressing inhibitors of the apoptotic proteins (IAPs) and to define quantitative proportions of anti-apoptotic BCL-2 molecules necessary to reduce the amount of produced effector caspases in the system. Also, supported by cancer-like scenarios with an overexpression of BCL2-anti-apoptotic proteins, the model permitted the calculation of compensation ratios of pro-apoptotic proteins that could reverse the resistant phenotype and a relevant role was given to the reactions happening at the receptor level in what concerns their impact on the overall response of the system (figure 3-C) ).
Although not entirely innovative in its conclusions it was a pioneer work that launched the usage of ODE’s in the mathematical field of apoptosis in a manner that agreed with the biological description of the known chemical reactions. In what concerns the underlying mathematical approach, the model did not follow standard mass-action description and the expression for the rates laws are somehow unclear. Same examples are provided in table 1.
Schlatter’s & Calzone’s model, two Boolean modeling approaches
The transduction and processing of intracellular signals often results from the contribution of multiple chemical agents and signaling pathways whose exact dynamics are frequently unknown. This lack of information greatly compromises the usage of continuous-time modelling approaches, which depend on precise model parameters, and can eventually bias model predictions. In this sense a simpler but useful alternative to study the interactions in a reaction network is to consider a Boolean or logical modeling framework to qualitatively assess the interactions and dependencies between the considered molecular species. The concentration of every element is replaced by a binary variable {0, 1}, in either an “on” or “off” state, and the collection of interactions is embed in an oriented graph that can integrate several types of dependencies, including activation and inhibition effects and positive and negative feedback loops.
In the field of apoptosis two studies based on Boolean formalisms stand out by the complexity and relevance of their conclusions, the work of (Schlatter et al., 2009) and (Calzone et al., 2010). In the former, a large network of reactions describing the intrinsic and extrinsic apoptosis and a myriad of associated pathways was proposed to analyze the effect of an input of Fas ligand, TNF-a, UV-B irradiation, interleukin-1b and insulin into the phenotype outcome of the system. The complexity of the reported interactions made the authors choose for a multi-value node representation where each variable could assume multiple states, instead of the common “1” or “0” and all-or-none definition, to account for “low-active amount” and “high-active amount” and establish higher-valued states where one variable could surpass the inhibitory role of another or instead reinforce the inhibition of a given substrate. The inclusion of more detailed timescale dynamics, with subsets of reactions being active only at certain time points, proved also essential to reproduce threshold dependencies and reaction delays that are known to be apoptotic signatures. The phenotypic outcome of the system was found to depend considerably on the feedback loops and highly connected nodes which included crosstalks with the survival and insulin pathways. A non-reported negative feedback loop from Smac to RIP (a central molecule in the necroptosis pathway inhibiting DISC and C8 activation) was suggested as a mechanism to enhance the stability of the DISC structures and lead to more effective apoptotic responses (Schlatter et al., 2009). Although hypothetical, the possibility of uncovering a new level of regulation in an already complex network is definitely one of the potential benefits of using modeling approaches and overall strengthen the results of this model.
The model of (Calzone et al., 2010) tried to establish a functional relationship between the NF-kB survival pathway and the cellular decision for either apoptosis or RIP1-dependent necrosis, after activation of the death-receptors on the cell membrane. By assembling a large network of reactions with the most representative elements of these three signaling pathways, the C3 activation, a significant drop in ATP levels and emergence of NF-kB were set as representative hallmarks of apoptosis, necrosis and survival decisions, respectively. A Boolean variable was assigned to each node and rules were imposed to define multiple events, such as the activation of a protein. For instance, C8 was considered to effectively change into an active state after direct stimulation by either DISC-TNF or DISC-FAS but only in the absence of FLIP. In this case, the associated logical rule was defined as (DISC-TNF OR DISC-FAS) AND NOT FLIP. By analyzing the multiple steady-states of their system, the authors could propose novel insights about cellular decisions towards different cell-death modalities. Even in the presence of C8-mediated cleavage of RIP1 the system was shown to contain attractors eventually converging into a necrotic phenotype, suggesting that the presence of C8 per se does not imply a cellular apoptotic response. For cells in which important apoptotic elements were mutated (APAF1, BAX, C8, FADD deletions and z-VAD treatment), simulations uncovered the existence of an optimal TNF treatment coincident with maximum proportions of obtained necrotic cells. The authors also found the role of the positive feedback loop from C3 to C8 to be non-essential when TNF or FAS levels were constant in the cell medium. Oppositely, when the same ligands are to be administrated in pulses the feedback loop ensures the persistence of the apoptotic signal (Calzone et al., 2010). This result reassures the importance of positive feedback loops in natural tissues where the cell receives non-sustained death-signals from its surrounding environment. The work still has the potential to uncover, with the availability of more data, the paradigms of cell decision towards necrosis, necroptosis or apoptosis and enlighten how we can force the cell into a specific type of death.
In what concerns the modeling framework, many formulations are available (among Boolean, ODE’s or PDE’s) and the modeler can choose for a more qualitative or quantitative approach according to the data at hand and the desired analysis or conclusions to extract. Some authors chose to build hybrid models combining both Boolean and ODE components and some are available in the field of apoptosis, such as the model for the NF-kB module proposed by (Chaves et al., 2009).
Bertaux’s model, a stochastic protein-turnover approach 2014
It is long known that identical sister cells often respond in different ways when exposed to equivalent stimulus. In apoptosis these differences have been attributed to inequalities in protein numbers that, when summed up for the plethora of intra-signaling reactions of a network, can influence the phenotype decision. In an effort to study these effects, Bertaux and colleagues used the EARM1 model, previously proposed by Albeck, and added a second contribution in the form of a gene stochastic layer (Bertaux et al., 2014). Each protein of the network interacted with other proteins of the system and also received an extra-input signal resulting from the fluctuations of the corresponding gene expression (figure 5). The goal was to follow the impact of random fluctuations in time in the absolute protein quantities, instead of the traditional sampling of variable initial conditions at the beginning of the simulation that is a common tool used to reproduce the behavior of non-identical cells. This analysis allowed for the simulation of a population of sister cells whose proteome decorrelate in time due to the inherent fluctuations of the underlying gene expression, as observed experimentally (Spencer et al., 2009).
Heterogeneity in biology, intrinsic and extrinsic noise
Signals in biology are noisy. Even when an adequate degree of confidence lies in the technique used for signal acquisition and the experimentally-associated errors are minimized, cells have nonetheless their own inherent stochastic fluctuations. These fluctuations are caused by inter-cellular differences in genetic and non-genetic expression levels that contribute for unique cell signatures in time (Chabot et al., 2007; Elowitz, 2002; Newman et al., 2006; Ozbudak et al., 2002; Raj et al., 2006; Raser, 2004; Stewart-Ornstein et al., 2012). The exact role and nature of signal fluctuations in biology is not entirely understood but some defend that cell diversification might be essential for the evolution of life. The presence of non-homogenous cells among a cell population allows both adaptation and robustness of the organism to different stresses and a high-level tissue response might depend on the inclusion of heterogeneous single-cell outputs as a whole (Bódi et al., 2017; Dueck et al., 2016; Lehner and Kaneko, 2011; Pujadas and Feinberg, 2012). Although positive and advantageous in the sense of evolution it can also be extremely deleterious when it promotes drug-resistant phenotypes, such as is often the case in cancer cells.
The current literature summarizes heterogeneity in biology as the result of intrinsic and extrinsic noise contributions. Intrinsic noise is derived directly from the reaction kinetics, a natural consequence of the thermodynamics of every chemical process that causes the reaction times to be stochastic (Gillespie, 1976). This impacts the absolute number of reaction products at a given time point, such as the total quantity of expressed RNA and proteins, causing even isogenic cells to decorrelate importantly in a matter of few hours (Spencer et al., 2009) . This phenomenon is highly dependent on the quantity of the intervening molecules and is increasingly relevant for systems with low number of particles. It is usually introduced in studies of gene expression and RNA dynamics where the reactants are not abundant but its influence on large scale signaling networks, where protein quantities are commonly high, has been confirmed to be minimal (Iwamoto et al., 2016; Labavić et al., 2019). The inclusion of intrinsic noise in model simulations usually follows the Gillespie algorithm, which returns an exact solution for the associated master equation (Gillespie, 1977). For gene expression models, where intrinsic noise is more notable, ON-OFF “ telegraph-models” are well established and are able to explain gene noise levels and its impact in RNA and protein amounts (Blake et al., 2006, 2003; Golding et al., 2005; Harper et al., 2011; Lionnet and Singer, 2012; Raj et al., 2006; Raj and van Oudenaarden, 2008; Raser, 2004; Suter et al., 2011).
Table of contents :
ABSTRACT
RESUME
ACKNOWLEDGMENTS
LIST OF TABLES
GLOSSARY
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
