Noisy-threshold control of cell death
© Vilar; licensee BioMed Central Ltd. 2010
Received: 26 January 2010
Accepted: 10 November 2010
Published: 10 November 2010
Cellular responses to death-promoting stimuli typically proceed through a differentiated multistage process, involving a lag phase, extensive death, and potential adaptation. Deregulation of this chain of events is at the root of many diseases. Improper adaptation is particularly important because it allows cell sub-populations to survive even in the continuous presence of death conditions, which results, among others, in the eventual failure of many targeted anticancer therapies.
Here, I show that these typical responses arise naturally from the interplay of intracellular variability with a threshold-based control mechanism that detects cellular changes in addition to just the cellular state itself. Implementation of this mechanism in a quantitative model for T-cell apoptosis, a prototypical example of programmed cell death, captures with exceptional accuracy experimental observations for different expression levels of the oncogene Bcl-xL and directly links adaptation with noise in an ATP threshold below which cells die.
These results indicate that oncogenes like Bcl-xL, besides regulating absolute death values, can have a novel role as active controllers of cell-cell variability and the extent of adaptation.
Cells in multicellular organisms have the ability to actively control their own death and engage in an organized self-destruction process known as programmed cell death, or apoptosis. This ability is necessary in many situations, ranging from embryonic development to the maintenance of mature organisms, and its deregulation is a hallmark of cancer  and many other diseases .
The coupling of metabolism with apoptosis is remarkably important in cancer, especially because many anticancer drugs targeted against growth promoting pathways selectively induce apoptosis in cells with highly upregulated metabolism [8–10]. Extensive work has been done along these lines to decipher how different cellular components are wired to coordinate apoptotic responses [11, 12]. The connection of the single cell behavior with the collective dynamics of cell populations, however, has remained largely unexplored.
It is still not clear how the death-or-alive binary decision of single cells leads to the observed analog response of the cell population, where death increases continuously with decreasing ATP levels . This connection is particularly challenging because classic population dynamics approaches  would require the death rate to peak at an intermediate ATP level to account for adaptation at low ATP levels, which paradoxically would imply that low ATP levels strongly favor survival. Here, the single-cell and population scales are connected through a quantitative mathematical model that identifies cellular variability, or noise, as the key element.
In general, cells have different levels of ATP, of proteins that promote or prevent cytochrome c release, and of the components of the caspase cascade. The most straightforward assumption is to consider that for each individual cell there is a specific value of the ATP level, an ATP level threshold, below which apoptosis is triggered and that intercellular variability in the cellular components makes this threshold noisy. Thus, a cell will die whenever the threshold is crossed, either because its ATP level decreases below its threshold or, vice versa, because its threshold increases over its ATP level.
In this paper, the focus is on death promoting stimuli that strongly affect metabolism, such as growth factor or nutrient withdrawal. Under these conditions, intracellular ATP levels change much faster than thresholds do and it is assumed that the noise in the threshold is quenched. Explicitly, the threshold is considered to be a time-independent random variable with precise statistical properties (Figure 1B).
The connection between the threshold distribution and cell death can be expressed in mathematical terms through the survival ATP-function, S(a), which is defined as the fraction of cells with a threshold below the average intracellular ATP level, denoted here by a. The initial threshold distribution is therefore given by , where x is the ATP threshold.
which uses the chain rule for derivatives: dS (a)/dt = (dS (a)/da) (da/dt). When the ATP level increases with time, within this model, the threshold would never be crossed and the apoptotic death rate would be zero.
A far-reaching implication of the threshold crossing assumption is that the death rate is not a function of just the ATP level but also of the ATP rate of change: da/dt. Thus, a precise testable prediction of the model is that changing the ATP level from decreasing (da/dt < 0) to increasing (da/dt > 0) will bring the apoptotic death rate to zero, irrespective of the value of the ATP level itself.
This model implements that the population dynamics response to energy depletion depends on both the energetic state of the cell and its dynamics. The complexity of cell death regulation  is collapsed into the form of the function r(a, da/dt), which can be inferred from experimental data, as shown below.
This approach can be applied straightforwardly to analyze experimental data for T-cell apoptosis upon growth factor withdrawal. Following loss of receptor engagement, T-cells rapidly downregulate the glucose transporter, glut1, which leads to lower intracellular ATP levels and results in apoptosis . Experimental data is available for three levels of expression of the antiapoptotic protein Bcl-xL. For the three levels of expression of Bcl-xL, the average ATP levels follow very similar temporal dynamics, thus indicating that Bcl-xL does not affect cellular metabolism substantially.
In this case, autophagy provides the energy that allows cells to maintain a basal metabolic state in the absence of nutrient uptake .
where here x is the threshold, and μ and σ are two parameters related to the mean, , and variance, , of the distribution. The coefficient of variation is given by and can be viewed as a measure of the cell-death noise. In general, μ and σ depend on the experimental conditions, cell type, and expression levels of Bcl-xL.
The model defined by Equations (1)-(5) accurately accounts for the observed apoptotic behavior in response to growth factor withdrawal (Figure 2B) when the two parameters that characterize the threshold distribution are appropriately chosen for each cell type.
The response includes a lag phase, extensive death, and adaptation. All these features, in contrast to the intracellular ATP dynamics, strongly depend on Bcl-xL expression. For the lowest expression, the cell type "Bcl-xL" shows substantial death as soon as the ATP level starts to drop, with a very short lag phase (~ 5 hours). In this case, there is no adaptation and after 25 hours the whole cell population is dead. In the case of the cell type "Bcl-xL 1E1", the lag phase lasts 20 hours approximately and is succeeded by a period of substantial death (until a time around 75 hours) before adaptation settles in with a small subpopulation of surviving cells. With the highest expression, the cell type "Bcl-xL 4.1" is very resistant to energy depletion. It has virtually no cell death until 40 hours after growth factor withdrawal and a large fraction of cells are able to survive.
It is important to emphasize that other types of threshold distributions, besides lognormal, could also be used in the model. However, if they are able to accurately reproduce the experimental data, they will look very similar to the lognormal distributions of Figure 3 in the range where intracellular ATP changes (from 3.54 to 0.32 fmol/cell). The reason is that in this system the shape of the threshold distribution is completely determined by the death rate. This result follows straightforwardly from integration of Eq. (1), which leads to the relationship . For instance, a Gaussian distribution would be able to provide relatively accurate results for the cell type "Bcl-xL" but would fail to do so for the other two cell types. A key feature of the lognormal distribution is its ability to recapitulate the experimental observations for the three cell types.
The results of the model indicate that the increased variability of cells that overexpress Bcl-xL leads to highly heterogeneous responses, allowing a subpopulation of cells to escape death. Thus, within this model, adaptation observed in the late stages of the response to energy depletion is not the result of the apoptotic machinery adapting to low ATP levels but the result of the combined effects of high threshold variability with intracellular ATP reaching a steady state. If there were no noise, the whole population would either die or survive. When all the cells die, there is no adaptation. When all the cells survive, there is no response to the death stimulus and therefore there is no adaptation either. Thus, control of the population dynamics requires controlling not only the average values of the factors that control death but also their statistical properties.
Recently, there has been an increasing interest on the role that molecular noise can play on the cellular behavior. Noise, in the form of random fluctuations in the number of molecules has been observed in many cellular processes, such as gene expression [19–22], protein abundance regulation [23–25], intracellular oscillations [26–28], and behavioral variability [29, 30]. The results presented here indicate that the oncogene Bcl-xL exploits the inherent stochastic nature of molecular events to generate cell-death variability, which transforms a single-cell discrete event like death into a graded population response and leads to adaptation.
These two noise-induced features have far-reaching physiological consequences. On the one hand, a graded death response is essential for a smooth control of total cell numbers in multicellular organisms [13, 31, 32]. On the other hand, adaptation is a key mechanism that allows a subpopulation of cells to survive in the face of death-promoting stimuli, leading to the eventual failure of many anticancer therapies . In the case of T-cell apoptosis, these two apparently disconnected features are in fact two sides of a single control mechanism based on noisy thresholds to trigger death.
This work was supported by the MICINN under grant FIS2009-10352.
- Hanahan D, Weinberg RA: The hallmarks of cancer. Cell. 2000, 100 (1): 57-70. 10.1016/S0092-8674(00)81683-9View ArticlePubMedGoogle Scholar
- Thompson CB: Apoptosis in the pathogenesis and treatment of disease. Science. 1995, 267 (5203): 1456-1462. 10.1126/science.7878464View ArticlePubMedGoogle Scholar
- Danial NN, Korsmeyer SJ: Cell death: critical control points. Cell. 2004, 116 (2): 205-219. 10.1016/S0092-8674(04)00046-7View ArticlePubMedGoogle Scholar
- Kroemer G, Dallaporta B, Resche-Rigon M: The mitochondrial death/life regulator in apoptosis and necrosis. Annu Rev Physiol. 1998, 60: 619-642. 10.1146/annurev.physiol.60.1.619View ArticlePubMedGoogle Scholar
- Skulachev VP: Bioenergetic aspects of apoptosis, necrosis and mitoptosis. Apoptosis. 2006, 11 (4): 473-485. 10.1007/s10495-006-5881-9View ArticlePubMedGoogle Scholar
- Izyumov DS, Avetisyan AV, Pletjushkina OY, Sakharov DV, Wirtz KW, Chernyak BV, Skulachev VP: "Wages of fear": transient threefold decrease in intracellular ATP level imposes apoptosis. Biochim Biophys Acta. 2004, 1658 (1-2): 141-147. 10.1016/j.bbabio.2004.05.007View ArticlePubMedGoogle Scholar
- Hammerman PS, Fox CJ, Thompson CB: Beginnings of a signal-transduction pathway for bioenergetic control of cell survival. Trends Biochem Sci. 2004, 29 (11): 586-592. 10.1016/j.tibs.2004.09.008View ArticlePubMedGoogle Scholar
- Weinstein IB: Cancer. Addiction to oncogenes--the Achilles heal of cancer. Science. 2002, 297 (5578): 63-64. 10.1126/science.1073096View ArticlePubMedGoogle Scholar
- Pelengaris S, Khan M, Evan GI: Suppression of Myc-induced apoptosis in beta cells exposes multiple oncogenic properties of Myc and triggers carcinogenic progression. Cell. 2002, 109 (3): 321-334. 10.1016/S0092-8674(02)00738-9View ArticlePubMedGoogle Scholar
- Chin L, Tam A, Pomerantz J, Wong M, Holash J, Bardeesy N, Shen Q, O'Hagan R, Pantginis J, Zhou H, et al.: Essential role for oncogenic Ras in tumour maintenance. Nature. 1999, 400 (6743): 468-472. 10.1038/22788View ArticlePubMedGoogle Scholar
- Sharma SV, Gajowniczek P, Way IP, Lee DY, Jiang J, Yuza Y, Classon M, Haber DA, Settleman J: A common signaling cascade may underlie "addiction" to the Src, BCR-ABL, and EGF receptor oncogenes. Cancer Cell. 2006, 10 (5): 425-435. 10.1016/j.ccr.2006.09.014PubMed CentralView ArticlePubMedGoogle Scholar
- Janes KA, Albeck JG, Gaudet S, Sorger PK, Lauffenburger DA, Yaffe MB: A systems model of signaling identifies a molecular basis set for cytokine-induced apoptosis. Science. 2005, 310 (5754): 1646-1653. 10.1126/science.1116598View ArticlePubMedGoogle Scholar
- Rathmell JC: B-cell homeostasis: digital survival or analog growth?. Immunol Rev. 2004, 197: 116-128. 10.1111/j.0105-2896.2004.0096.xView ArticlePubMedGoogle Scholar
- Murray JD: Mathematical biology. 1989, Berlin; New York: Springer-VerlagView ArticleGoogle Scholar
- Rathmell JC, Vander Heiden MG, Harris MH, Frauwirth KA, Thompson CB: In the absence of extrinsic signals, nutrient utilization by lymphocytes is insufficient to maintain either cell size or viability. Mol Cell. 2000, 6 (3): 683-692. 10.1016/S1097-2765(00)00066-6View ArticlePubMedGoogle Scholar
- Lum JJ, Bauer DE, Kong M, Harris MH, Li C, Lindsten T, Thompson CB: Growth factor regulation of autophagy and cell survival in the absence of apoptosis. Cell. 2005, 120 (2): 237-248. 10.1016/j.cell.2004.11.046View ArticlePubMedGoogle Scholar
- Limpert E, Stahel WA, Abbt M: Log-normal distributions across the sciences: Keys and clues. Bioscience. 2001, 51 (5): 341-352. 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2.View ArticleGoogle Scholar
- Gaddum JH: Lognormal distributions. Nature. 1945, 156 (3964): 463-466. 10.1038/156463a0.View ArticleGoogle Scholar
- Elowitz MB, Levine AJ, Siggia ED, Swain PS: Stochastic gene expression in a single cell. Science. 2002, 297 (5584): 1183-1186. 10.1126/science.1070919View ArticlePubMedGoogle Scholar
- Ozbudak EM, Thattai M, Kurtser I, Grossman AD, van Oudenaarden A: Regulation of noise in the expression of a single gene. Nat Genet. 2002, 31 (1): 69-73. 10.1038/ng869View ArticlePubMedGoogle Scholar
- Raser JM, O'Shea EK: Control of stochasticity in eukaryotic gene expression. Science. 2004, 304 (5678): 1811-1814. 10.1126/science.1098641PubMed CentralView ArticlePubMedGoogle Scholar
- Blake WJ, Kaern M, Cantor CR, Collins JJ: Noise in eukaryotic gene expression. Nature. 2003, 422 (6932): 633-637. 10.1038/nature01546View ArticlePubMedGoogle Scholar
- Newman JR, Ghaemmaghami S, Ihmels J, Breslow DK, Noble M, DeRisi JL, Weissman JS: Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise. Nature. 2006, 441 (7095): 840-846. 10.1038/nature04785View ArticlePubMedGoogle Scholar
- Bar-Even A, Paulsson J, Maheshri N, Carmi M, O'Shea E, Pilpel Y, Barkai N: Noise in protein expression scales with natural protein abundance. Nat Genet. 2006, 38 (6): 636-643. 10.1038/ng1807View ArticlePubMedGoogle Scholar
- Sigal A, Milo R, Cohen A, Geva-Zatorsky N, Klein Y, Liron Y, Rosenfeld N, Danon T, Perzov N, Alon U: Variability and memory of protein levels in human cells. Nature. 2006, 444 (7119): 643-646. 10.1038/nature05316View ArticlePubMedGoogle Scholar
- Mihalcescu I, Hsing W, Leibler S: Resilient circadian oscillator revealed in individual cyanobacteria. Nature. 2004, 430 (6995): 81-85. 10.1038/nature02533View ArticlePubMedGoogle Scholar
- Lahav G, Rosenfeld N, Sigal A, Geva-Zatorsky N, Levine AJ, Elowitz MB, Alon U: Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat Genet. 2004, 36 (2): 147-150. 10.1038/ng1293View ArticlePubMedGoogle Scholar
- Vilar JMG, Kueh HY, Barkai N, Leibler S: Mechanisms of noise-resistance in genetic oscillators. Proc Natl Acad Sci USA. 2002, 99 (9): 5988-5992. 10.1073/pnas.092133899PubMed CentralView ArticlePubMedGoogle Scholar
- Korobkova E, Emonet T, Vilar JMG, Shimizu TS, Cluzel P: From molecular noise to behavioural variability in a single bacterium. Nature. 2004, 428 (6982): 574-578. 10.1038/nature02404View ArticlePubMedGoogle Scholar
- Pearl S, Gabay C, Kishony R, Oppenheim A, Balaban NQ: Nongenetic individuality in the host-phage interaction. PLoS Biol. 2008, 6 (5): e120- 10.1371/journal.pbio.0060120PubMed CentralView ArticlePubMedGoogle Scholar
- Arias AM, Hayward P: Filtering transcriptional noise during development: concepts and mechanisms. Nat Rev Genet. 2006, 7 (1): 34-44. 10.1038/nrg1750View ArticlePubMedGoogle Scholar
- Arias AM, Stewart A: Molecular principles of animal development. 2002, Oxford: Oxford University PressGoogle Scholar
- Hickman JA: Apoptosis and chemotherapy resistance. Eur J Cancer. 1996, 32A (6): 921-926. 10.1016/0959-8049(96)00080-9View ArticlePubMedGoogle Scholar