Which Two Proteins of the Bcl-2 Family Stimulate the Formation of the Apoptosome?

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The Mathematical Model of the Bcl-2 Family Mediated MOMP Regulation Can Perform a Not-Trivial Design Recognition

• Jozef Ulicny

The Mathematical Model of the Bcl-2 Family Mediated MOMP Regulation Tin Perform a Non-Niggling Pattern Recognition

• Tomas Tokar,
• Jozef Ulicny

x

• Published: December 26, 2013
• https://doi.org/10.1371/periodical.pone.0081861

Figures

Abstract

Interactions between individual members of the B-cell lymphoma 2 (Bcl-2) family of proteins form a regulatory network governing mitochondrial outer membrane permeabilization (MOMP). Bcl-2 family initiated MOMP causes release of the inter-membrane pro-apoptotic proteins to cytosol and creates a cytosolic surround suitable for the executionary stage of apoptosis. Nosotros designed the mathematical model of this regulatory network where the synthesis rates of the Bcl-2 family unit members served equally the independent inputs. Using computational simulations, nosotros have then analyzed the response of the model to up-/downregulation of the Bcl-ii proteins. Nether several assumptions, and using estimated reaction parameters, a not-linear stimulus-response emerged, whose characteristics are associated with bistability and switch-like behavior. Interestingly, using the principal component analysis (PCA) we have shown that the given model of the Bcl-2 family interactions classifies the random combinations of inputs into two distinct classes, and responds to these past 1 of the two qualitatively singled-out outputs. As nosotros showed, the emergence of this behavior requires specific arrangement of the interactions betwixt detail Bcl-2 proteins.

Commendation: Tokar T, Ulicny J (2013) The Mathematical Model of the Bcl-2 Family Mediated MOMP Regulation Tin can Perform a Non-Lilliputian Pattern Recognition. PLoS I eight(12): e81861. https://doi.org/10.1371/journal.pone.0081861

Editor: Georg Häcker, University Freiburg, Federal republic of germany

Received: May 14, 2013; Accepted: October 21, 2013; Published: Dec 26, 2013

Copyright: © 2013 Tokar, Ulicny. This is an open-access commodity distributed under the terms of the Creative Eatables Attribution License, which permits unrestricted utilise, distribution, and reproduction in whatsoever medium, provided the original author and source are credited.

Funding: This work was funded by Slovak Research and Development Bureau, grant no. APVV-0242-eleven, from the projection SEPO-II, grant no. ITMS 26220120039, project Technicom, grant no. ITMS 26220220182, FP7-REGPOT-2012-2013-1 grant No. 316310 - CELIM and from the Scientific Grant Agency of the Ministry building of Education of the Slovak Republic, grant no. VEGA-1/4019/07. The funders had no role in written report design, data collection and assay, decision to publish, or preparation of the manuscript.

Competing interests: The authors have alleged that no competing interests exist.

Introduction

Programmed cell death (PCD), often denoted as a cellular suicide, plays an of import role in the homeostasis of every multi-cellular organism [one]. One of the main forms of PCD is called apoptosis [1]–[3], a process which is well distinguished by its characteristic morphology [4]. Defects in apoptosis regulation may crusade a multifariousness of serious diseases, including the neurodegenerative disorders [5], autoimmune diseases [6], or cancer [vii]–[9]. Apoptosis can be initiated by either extracellular stimuli or by signals originating from a prison cell's internal space [9], [x]. Signals initiating apoptosis so proceed through the apoptotic signaling pathways, which contain several control points [9], [10]. 1 of the about important of such points, integrating a multitude of incoming apoptotic (and antiapoptotic) signals is formed by a family of Bcl-two (B-cell lymphoma 2) proteins [11], [12]. The Bcl-2 family controls mitochondrial outer membrane permeabilization (MOMP) [13], [14], the crucial outcome of apoptosis.

MOMP allows the release of cardinal apoptotic players - Smac/DIABLO and cytochrome c, from a mitochondrial intermembrane infinite to cytosol [xiii], [14]. In the presence of ATP, released cytochrome c binds to a cytosolic protein Apaf-1, causing Apaf-1 oligomerization and recruitment of an inactive pro-caspase-9, leading to germination of a multi-protein complex known as an apoptosome [15]–[17]. Inside the apoptosome, pro-caspase-nine subsequently undergoes processing and activation [15]–[17]. The active caspase-ix proteolytically activates caspase-3 [xviii]. Smac/DIABLO, once released to the cytosol, inhibits XIAP (Ten-linked inhibitor of apoptosis) - the near prominent suppressor of caspases-3 and -9 [xix]. Caspase-3 and other effector caspases (caspases-6 and -7) are the primary executioners of the apoptosis [9], [20]. Activation of effector caspases signifies the signal of no-return, after which apoptosis irreversibly occurs [21].

Bcl-ii family members are functionally classified as either antiapoptotic, or proapoptotic. Structurally, Bcl-2 proteins tin can be categorized co-ordinate to the number of Bcl-two homology domains (BH) in their α-helical regions [ix], [22]. Antiapoptotic members (Mcl-1, A1, Bcl-xL, Bcl-2, Bcl-w and Bcl-B) are characterized by the presence of four BH domains (BH1-4) [23], [24]. Their role is to prevent MOMP by inhibition of proapoptotic family unit members [23], [24]. Proapoptotic members can be divided to BH3-only proteins and multidomain proteins - effectors [ix]. BH3-but proteins can be further subdivided based on their role in apoptotic signaling. BH3-but subgroup members, termed sensitizers, or enablers (Noxa, Bad, Puma, Hrk, Bmf and Bik) can only demark to antiapoptotic Bcl-2 proteins, forming inactive dimers [22]. Members of another BH3 subgroup, termed activators (Bim and Bid), tin can act in the same way [22], but in improver, activators tin directly actuate effectors [23], [25]. Effectors, once activated, undergo oligomerization and form pores – mitochondrial apoptosis channels (MAC) in the mitochondrial outer membrane (MOM), leading eventually to MOMP. [xiv], [26]. Therefore, effectors are the chief target of inhibition past their antiapoptotic relatives [25].

Individual Bcl-2 family members are regulated by wide-diversity of factors, e.g. growth factor impecuniousness, cytokine withdrawal, heat shock, DNA damage, hypoxia, death receptors stimulation and many others [23]. Mechanisms of regulation include transcription command and/or post-translational modifications by phosporylation, or proteolytic cleavage [23]. Bcl-2 family unit thus integrates a multitude of converging signals to decide whether to commit MOMP or not. This determination is carried in an all or nothing manner, giving no possibility of intermediate MOMP. This interesting behavior has made the Bcl-2 family an attractive field of study of mathematical modeling and computer simulations.

There are several works regarding modeling and simulation of the Bcl-two family and the control of MOMP, revealing and examining a diverseness of non-linear organisation behaviors such every bit robustness, stimulus-response ultrasensitivity [27] and steady-state bistability [28]–[30]. Besides these, the Bcl-two family was involved in several other, more general models of apoptosis signaling [31]–[33].

In the above mentioned works, the authors reduce the complexity of their model by grouping of several functionally similar species together. Usually, the Bcl-ii family'southward members are assigned into iv groups co-ordinate to their structural and functional classification. The most prominent group'south fellow member is taken as the model'due south representation of the whole group of species. Previous models of the Bcl-2 regulatory network differ by level of details, nevertheless they all adopt such simplification. Although, such simplification provides an attractive trade-off between the model's complexity and plausibility, functional specificities of Bcl-ii family'due south individuals are being omitted.

In the proposed piece of work nosotros provide a literature-based mathematical model in which interactions between individual Bcl-two family members are distinguished. Our goal was to investigate the behavior of the detailed model, in particular to testify/disprove the switching properties obtained by models based on functional grouping. In the process we obtained additional insight into the decision machinery of the Bcl-two control of MOMP. The not-little pattern-recognition emerged as a consequence of functional specificities of Bcl-2 family individuals. In addition, an explicit model of pair interactions allowed united states to probe the pro- and anti-apoptotic potency of individual members of the Bcl-two family and to rank them co-ordinate to their ability to promote or prevent the MOMP event.

Results

Bistability of the Bcl-ii family unit regulation of MOMP

Several previous works [28]–[30] accept been focused on the analysis of bistable behavior of the Bcl-2 family mediated regulation of MOMP. Similarly, we performed a variation of the single stimulus parameter of the model to analyze its steady-land stimulus-response dependence. The product charge per unit of the tBid ( ) was considered as the stimulus and the steady-land concentration of the MAC ( ) was the measure of response. The stimulus – was varied through the chosen range of values, while the other parameters of the model remained unchanged.

Using the default parameter setup of the model (come across section Model and its biological relevance) nosotros accept obtained the stimulus-response dependence depicted in the Figure ane. The obtained steady-state stimulus-response forms hysteresis, the typical authentication of bistability in dynamic system. The two thresholds (Fig. i), marked by the left and correct vertical dashed lines) enclose the bistable region. Inside the bistable region, the arrangement under the aforementioned value of the stimulus may occur in one of two stable steady-states, depending on the initial conditions. These steady-states are separated by the unstable steady-states (marked by dashed curve). Such systems are often described as the mechanism of a "toggle switch" [34].

Figure one. Steady-country concentration of the MAC ( ) is plotted equally a function of the product rate of tBid (kptBid).

is increasing with increasing value of . The remains at very low levels (pro-survival – the blue solid bend), until the production charge per unit exceeds the threshold (correct vertical dashed line). Exceeding the threshold value causes sudden increase of the (onset of MOMP – crimson solid bend). The subsequent decrease of the production of tBid cause simply slow decrease of , until the another threshold is crossed (left vertical dashed line). And then the suddenly drops back to very low levels. Vertical dashed lines are enclosing the bistable region. Within this region organization tin persist in 1 of the two stable steady-states (solid curves), which are separated by unstable steady-states (dashed curve).

https://doi.org/10.1371/periodical.pone.0081861.g001

Previous works limited themselves to the response of the Bcl-2 family induced by activators tBid/Bim solely. In the presented work we have explored the model's response to variation of all components, including both activators, all the anti-apoptotic proteins, enablers and effectors. Utilizing individual production rates as the input stimuli, we performed ready of analyses analogous to the previous one.

In addition to tBid, we have observed that steady-country stimulus-response hysteresis resulted also from stimulation by the second of activators – Bim (information non shown), as well as by the the anti-apoptotic proteins. Surprisingly, even a variation of the production rates of enablers Puma and Bad & Bmf yields stimulus-response hysteresis (encounter Fig. 2).

Variation of the Hrk & Bik productions yields hysteresis as well, but the obtained range of bistability is extremely narrow, close to ultrasensitive sigmoid curve. Similarly to bistability, a sigmoid curve indicates that the modeled organization is insensitive to low levels of the given input, simply that it tin reply significantly high levels of the given input. In contrast to bistability, sigmoidal response is not discretized, every bit the response is continuously increasing/decreasing with the growing/reducing input strength. While the bistability tin can be compared to mechanism of "toggle switch" sigmoid stimulus-response is often compared to the performance of the "push-button" [34].

Robust hysteresis with a broad bistable region was yielded by the variation of production rate of Bax and less pronounced hysteresis was obtained past variation of production of Bak. This shows that "toggle" switching of the Bcl-2 response tin as well exist achieved past significant upregulation of the effector proteins.

Interestingly, variation of the production rate of the Noxa produced a hyperbolic response. Hyperbolic dependence, in dissimilarity to hysteresis and sigmoid response, indicates sensitivity to even a minor increase of input stimuli. Stimulation of the model through the changing production of Noxa, can therefore exist viewed as "tuning" of the model'due south sensitivity to other incoming stimuli (come across Fig. 3).

Monte-Carlo variation of production rates results in bimodal distribution of steady-state affluence of MAC

In the previous section, we analyzed the dependence of the steady-land concentration of MAC on the production of individual proteins. In what follows, we performed Monte-Carlo analysis of the dependence of the steady-state concentration of MAC on the simultaneous variation of multiple product rates.

In each single iteration, the values of all the production rates (kpMcl1, kpA1, kpBclXl, kpBcl2, kpBclw, kpBclB, kpHrk, kpBik, kptBid, kpPuma, kpBim, kpBad, kpBmf, kpNoxa, kpBax, kpBak) of the model were simultaneously varied according to following dominion: (one) where kp is the variated production rate, is its default value. and q is the uniformly distributed real number, randomly chosen at each iteration, and for each production rate, from interval . Other parameters of the model were kept at their default values. Like to the analysis of bistability, the steady-land concentration of MAC, was used every bit the output.

We have plotted the distribution of the model's response obtained for ten⁴ iterations. The results (run across Fig. 4) evidence clear bimodal distribution of the model'south response. The bimodal distribution of the response proves that the model of Bcl-2 regulatory network can turn random combinations of inputs into two qualitatively distinct outputs [29], [35]–[37].

Figure 4. Distribution of steady-state concentrations of MAC produced by 10⁴ random variations of the model'due south production rates.

Vertical dashed line denotes the minimum betwixt ii local maxima, defining the threshold value of the steady-state concentration of the MAC, distinguishing pro-survival and pro-MOMP responses.

https://doi.org/x.1371/journal.pone.0081861.g004

The minimum located betwixt the local maxima of the response distribution (Fig. 4, marked by vertical dashed line) was considered equally the threshold value of steady-state concentration of MAC, separating the pro-survival (colored blue) and MOMP (colored crimson) responses. Futurity, the steady-state concentrations of the MAC below this threshold are considered as pro–survival, while the concentrations to a higher place the threshold are considered as MOMP initiating (pro–MOMP). It is worth mentioning, that the value of the threshold intersects all the hysteresis curves (Fig. i and 2), inside the bistable range. The value of this threshold – ∼350, assumes that MOMP occurs once the number of effector dimers in mitochondria surrounding surroundings exceeds 350. This is remarkably good agreement with experimental estimations fabricated by Martinez-Caballero et al. [38].

Bcl–2 family performs non-piddling pattern recognition

Using the aforementioned sets of the product rates every bit were used in the previous analysis we arranged the matrix of stimuli. Each row of this matrix – stimuli vector corresponds to 1 iteration of the Monte-Carlo analysis from the previous department and each of its columns corresponds to the one of the production rates, defining the size of the stimuli matrix to . Each column was normalized to its mean. So we performed the principal component analysis (PCA) of the matrix of stimuli and plotted the input vectors within the plane defined by chief components.

The results (meet Fig. 5, top) show that when the random input stimuli vectors are plotted in the PCA–divers plane, they form a randomly scattered deject. But, when the each vector is colored based on the associated response, information technology appears that the stimuli associated with the given response are clustered. This shows, that the model of Bcl-ii regulatory network is capable of taking a wide range of random combinations of incoming signals and classifying them into two sharply defined responses of distinct biological relevance. Such functionality defines what is in the field of machine learning and neural networks known every bit non-trivial pattern recognition [39]–[42].

Figure five. Scatter plot of all input combinations, plotted in the plane defined by the main components analysis.

Inputs associated with pro-MOMP response are colored red, remaining inputs are colored blue. The clusterization of inputs according to response quality is apparent for reference model (top), but plain absent when altering the model'southward topology (bottom).

https://doi.org/10.1371/periodical.pone.0081861.g005

In following, we created ten alternative models of the Bcl-ii family unit, by mutating the topology of the default model. By mutation nosotros mean random addition of non-existing or deletion of existing inhibitory interactions between anti-apoptotic and pro-apoptotic proteins and/or activations of effectors by activators. Such mutations permit alteration of the Bcl-2 family model on its detailed level, but preserve the interactions betwixt the functional groups consistent with the default model.

For each of the alternative models we have performed a total of five of such mutations. Nosotros then performed Monte-Carlo assay of these models by generating the 10⁴ random combinations of input stimuli. For each culling model nosotros so identified the threshold value of the effectors activity and subsequently performed the PCA of the stimuli matrix.

As a event we have found that all the alternative models produced bimodal distribution of response, merely none of them clustered input stimuli similarly to the default model (see Fig. five, lesser). This point that, while the bistability tin can emerge from alternative topologies of the Bcl-2 family interaction network, the pattern recognition is strictly associated with this particular topology of this regulatory network.

Pivotal pro- and anti-apoptotic Bcl-two family members

In the following we wanted to answer the naturally arising question: Which of the production rates are pivotal regarding the determination of the model'south response?

The model's response varies over several orders of magnitude. However, all the response values below/above the threshold, regardless of the value itself, are considered to be qualitatively equal – pro–survival/pro–MOMP, providing the same biological consequences. Therefore, we were interested in correlation of the values of the production rates with the response quality (pro–survival/pro–MOMP), instead of the correlation with the response quantity.

We utilized the point-biserial correlation coefficient (PBCC) - as the the measure of the correlation betwixt the value of the production rate and quality of the model's response (pro–survival/pro–MOMP). PBCC is frequently used to measure the correlation between the ii variables, one of which is dichotomous (either naturally, or artificially dichotomized) [43]. PBCC for each production rate was calculated as follows: (2) where the M and Southward are the ways of values of the given product rate respective to pro–MOMP and pro–survival responses respectively. and are the number of values of the given production rate, respective to pro–MOMP and pro-survival responses respectively. n is the full number of the values of the given production rate and σ is its standard deviation.

The results we obtained (Fig. half-dozen) prove the role of the tBid and Bim – the only activators of effectors – as the main pro-apoptotic proteins. Similarly, the Puma is the most efficient MOMP promoter among the enabler proteins within the model. On the other hand, as model predicts, the well-nigh efficient MOMP preventers are proteins A1 and Bcl-Xl, followed by Mcl-1 and Bcl-westward.

Word

The Bcl-2 family of proteins consists of xvi (excluding putative tissue-specific Bcl-ii protein known as Bcl-2 ovarian killer – Bok) proteins that differ in their upstream regulation every bit well every bit in their interactions with other family members. We translated electric current biological noesis of these interactions into mathematical model, which was utilized to study the regulation of the MOMP – crucial apoptotic consequence, which is maintained by the Bcl-2 family. Under given assumptions, that accept been fabricated to construct the model (see Sec. Model and its biological relevance), and using estimated parameters, interesting behaviors have been found to emerge from Bcl-2 protein interactions.

Assay of the model's response to variation of productions of private proteins, revealed that the organisation property called steady-state bistability emerges as a robust feature of the modeled system. Our model predicts that the well-nigh of the Bcl-2 proteins tin can potentially serve similarly to bistable "toggles", up- or downregulation of these cause "switching on" the MOMP. Steady-state bistability is currently being, the favorite framework for thinking nigh the switch between life and death" [44]. Therefore, the above-mentioned results were expected. Still, we have too found that other Bcl-2 proteins tin can potentially act as a "push button-buttons" – culling switching mechanisms with very narrow or missing bistable range and one of them every bit a "tuner" of the model's sensitivity – upregulation of which results in hyperbolic model's response.

We have shown that the model is able to process random combinations of inputs to produce output that has bimodal distribution. This strongly suggests that orchestration of these "toggles", "push-buttons" and "tuners" can constitute a molecular device whose function is to integrate multitude of incoming, continuous inputs into binary outcomes. Bimodal distribution was previously observed by Sun et al [29] in the menstruum cytometry of Bax activation as the response to staurosporine treatment of the HeLa cells population, supporting our finding.

Moreover, our model of this molecular device shows power to perform pattern recognition – which is non-petty functionality, often associated with neural networks and machine learning algorithms. This functionality is impaired by deletion of a relatively small number of reactions, as well every bit past addition of artificial interactions, even for interactions which are consequent with the relationships between the functional groups of the family.

Measurements of the correlation between the individual signals and the model's response predict that the nearly potent inputs of this network are associated with the regulation of activators tBid and Bim, enabler poly peptide Puma and the anti-apoptotic sentinels A1, Bcl-Twoscore, Bcl-w and Mcl-1.

Finally, to outline the directions in which the proposed piece of work could be extended, we would like to point out the necessity of utilization of more sophisticated, machine-learning based approaches to better clarify the synergy of the Bcl-2 family regulation of MOMP. Nosotros believe that its understanding can be crucial in evolution of novel anti-cancer drugs and/or handling of the serious neurodegenerative diseases.

Model and its biological relevance

In the proposed work, we modeled the MOMP regulatory network formed by the interactions of the members of the Bcl-2 family of proteins (see Tabular array i). Our model of the Bcl-2 family is divers by the extensive set of reactions which are listed in Table ii and the respective set up of reaction rates (Table 3). In post-obit we describe of import features of our model, giving special emphasize to the assumptions and simplification which we have fabricated toward the model feasibility and simplicity.

In each simulation, the initial abundance of each protein was either gear up to zero for a protein that is not synthesized from external sources (reactions 46–61), otherwise divers by following rule: (3) setting the initial affluence to balance the production (reactions 46–61) and degradation of the given protein at the initial conditions. Bcl-2 proteins' productions, as appear in our model, involve de-novo synthesis of the particular protein and may also involve the postal service-translational activation of this protein by external cellular signals (as is the case of the Bid which is truncated by caspase-eight to active tBid). Although synthesis of the Bcl-ii proteins could potentially by regulated by activity of other Bcl-2 family relatives on the mail-transcriptional level, we are not aware whether any mechanistic details of such kind of intra-familiar regulation are currently known, and therefore we assume the production rates to be fully independent of the Bcl-two proteins activity. Similar assumption is often made in similar models, equally was too made in the previous models of the Bcl-two apoptotic switch (e.g. Cui et al. [28]). Protein productions can thus serve equally an independent, multivariate input, model's response to which nosotros study by computational simulations.

Degradation half-life times of the Bcl-2 proteins were reported to range between xv–45 minutes (Mcl-ane) and from 12 to more than 24 hours (Bax) [45]. However, to reduce number of parameters and reduce the complexity of the model, deposition charge per unit of all proteins was modeled by catch-all reaction (catch-all reaction 62) and prepare to resemble the ∼10 hours deposition half-life time. Although this simplification was establish to bear on the obtained results but quantitatively (data no shown), information technology needs to be highlighted every bit a difference from biological reality.

The abundance of the mitochondrial apoptosis channels – MACs of the model was calculated equally the sum of the abundances of the aBax∼aBax, aBak∼aBak, aBax∼aBak dimers. This is an important simplification, as MACs are reported to comprise normally several to several tens of monomeric units [38]. Similar simplification is, however, often adopted by modeling works, and was too used in several previous models of the Bcl-2 apoptotic switch (e.g. Cui et al. [28] and Sun et al. [29]).

Default values of the reaction rates (listed in the Table 3) were estimated co-ordinate to typical rates of biomolecular interactions of similar type, and in accord with the previously published models of the Bcl-2 family [27]–[30]. Information technology has to be noted that in that location has been no analysis of the model'due south parametric robustness performed nonetheless, and thus the obtained results must exist interpreted as valid merely under the given parametrization and on the basis of given assumptions.

Protein interactions were modeled using mass action kinetics, translated into a set of ordinary differential equations. The model was implemented and all the analyses were performed within Python programming language, by using the PySCes (Python Simulator for Cellular Systems) [46] module.

Author Contributions

Conceived and designed the experiments: TT. Performed the experiments: TT. Analyzed the data: TT. Contributed reagents/materials/analysis tools: TT. Wrote the paper: TT JU. Proposed the model of Bcl-2 family, implemented the model within the Python environment and performed the simulations: TT. Processed and analyzed the obtained data, prepared all the illustrations and major part of the manuscript: TT. Substantively contributed to this work, by extensive revision of the manuscript: JU.

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