- Research article
- Open Access
Modeling heterogeneous responsiveness of intrinsic apoptosis pathway
- Hsu Kiang Ooi^{1} and
- Lan Ma^{1}Email author
https://doi.org/10.1186/1752-0509-7-65
© Ooi and Ma; licensee BioMed Central Ltd. 2013
- Received: 19 December 2012
- Accepted: 19 July 2013
- Published: 23 July 2013
Abstract
Background
Apoptosis is a cell suicide mechanism that enables multicellular organisms to maintain homeostasis and to eliminate individual cells that threaten the organism’s survival. Dependent on the type of stimulus, apoptosis can be propagated by extrinsic pathway or intrinsic pathway. The comprehensive understanding of the molecular mechanism of apoptotic signaling allows for development of mathematical models, aiming to elucidate dynamical and systems properties of apoptotic signaling networks. There have been extensive efforts in modeling deterministic apoptosis network accounting for average behavior of a population of cells. Cellular networks, however, are inherently stochastic and significant cell-to-cell variability in apoptosis response has been observed at single cell level.
Results
To address the inevitable randomness in the intrinsic apoptosis mechanism, we develop a theoretical and computational modeling framework of intrinsic apoptosis pathway at single-cell level, accounting for both deterministic and stochastic behavior. Our deterministic model, adapted from the well-accepted Fussenegger model, shows that an additional positive feedback between the executioner caspase and the initiator caspase plays a fundamental role in yielding the desired property of bistability. We then examine the impact of intrinsic fluctuations of biochemical reactions, viewed as intrinsic noise, and natural variation of protein concentrations, viewed as extrinsic noise, on behavior of the intrinsic apoptosis network. Histograms of the steady-state output at varying input levels show that the intrinsic noise could elicit a wider region of bistability over that of the deterministic model. However, the system stochasticity due to intrinsic fluctuations, such as the noise of steady-state response and the randomness of response delay, shows that the intrinsic noise in general is insufficient to produce significant cell-to-cell variations at physiologically relevant level of molecular numbers. Furthermore, the extrinsic noise represented by random variations of two key apoptotic proteins, namely Cytochrome C and inhibitor of apoptosis proteins (IAP), is modeled separately or in combination with intrinsic noise. The resultant stochasticity in the timing of intrinsic apoptosis response shows that the fluctuating protein variations can induce cell-to-cell stochastic variability at a quantitative level agreeing with experiments. Finally, simulations illustrate that the mean abundance of fluctuating IAP protein is positively correlated with the degree of cellular stochasticity of the intrinsic apoptosis pathway.
Conclusions
Our theoretical and computational study shows that the pronounced non-genetic heterogeneity in intrinsic apoptosis responses among individual cells plausibly arises from extrinsic rather than intrinsic origin of fluctuations. In addition, it predicts that the IAP protein could serve as a potential therapeutic target for suppression of the cell-to-cell variation in the intrinsic apoptosis responsiveness.
Keywords
- Intrinsic apoptosis pathway
- Stochastic model
- Intrinsic noise
- Extrinsic noise
Background
Apoptosis, the major form of programmed cell death, is a conserved cell suicide process critical for the health and survival of multicellular organisms [1–3]. Apoptosis plays a fundamental role in animal development, by sculpting tissues and structures, as well as in tissue homeostasis, by regulating and maintaining balanced cell number [4–6]. Dysregulation of apoptosis is associated with various human diseases, ranging from developmental disorders, neurodegeneration to cancer [7, 8].
Since the key constituents and molecular interactions of apoptosis pathways have been experimentally identified, the approach of mathematical modeling and computer simulations have been employed extensively to help elucidate the complicated regulatory network and dynamic responsiveness related to apoptosis at average cellular population level [14, 15, 17–19]. Nevertheless, recent experiments at single-cell resolution have discovered noisy phenotypic diversity of apoptosis activity in that significant cell-to-cell heterogeneity of the dynamic apoptosis responses exist across a genetically-identical cell population [16]. Toward understanding such single-cell variability in apoptosis response, some theoretical efforts have been taken recently to model the stochastic response of receptor-mediated apoptotic pathway. The stochastic behavior of intrinsic apoptosis pathway, on the other hand, has been the subject of relatively little mathematical modeling to date. In this work, we will focus on addressing the intrinsic apoptosis pathway under stochastic perturbations by developing theoretical and computational models at single-cell level. The models will be exploited to investigate the heterogeneous behavior of intrinsic apoptosis network among individual cells.
Deterministic model based on ordinary differential equations (ODEs) is the most widely used mathematical approach to describe the molecular kinetics during cell death signaling. Fussenegger et al. developed a well-accepted ODE model that integrates components of the extrinsic as well as the intrinsic apoptosis pathways [20]. Qualitatively the Fussenegger model compares reasonably well with published experimental kinetics of caspase activation at average cell population level. Nevertheless, there is lack of understanding of the nonlinear stability and systems properties of this model, which hinders deeper understanding of the system behavior. For instance, studies have suggested that bistability is a key system feature for apoptotic signaling networks [15, 16, 21–23], which could achieve the all-or-none responses and in addition confer robustness to the apoptosis system [18, 24, 25]. It is unclear whether the Fussenegger model presents the property of bistability. Since then, there have been considerable theoretical efforts on modeling and systems analysis using ODE models of death-receptor mediated apoptosis [17, 18, 26, 27], mitochondria-mediated apoptosis [28, 29], or integrated extrinsic and intrinsic apoptosis pathways [15, 30–34].
The past few years have seen increasing efforts in stochastic modeling to address the heterogeneous apoptosis responses at single-cell level. Specifically, these efforts incorporate cellular noise perturbations into the apoptosis framework. Cellular noise is defined as stochastic fluctuations of biomolecular processes within and between cells. It can be divided into intrinsic noise and extrinsic noise [35, 36]. Intrinsic noise in genetically identical cells refers to random deviation of the molecular processes from their average deterministic kinetics within a cell, mostly due to probabilistic biochemical reactions associated with low copy number of molecular quantities [35, 37]. Extrinsic noise arises from global perturbation factors such as cellular environment and organelle distribution, which results in cell-to-cell variation in rate constants of biochemical reactions, expression levels of genes and proteins, and other parameters of biochemical processes [35, 38, 39]. Towards the analysis of cellular noise, several statistical measures of noise have been proposed to quantify the level of stochastic fluctuations of biomolecular processes [40–43]. Two measures of noise are commonly used to characterize the stationary averages and variances of random cellular components. In particular, noise strength can be quantified by Fano factor, which is defined as the steady-state variance over average and has a value of 1 for Poisson process. The Fano factor of an arbitrary stochastic system reveals deviations from Poissonian behavior [44–46]. A more standard and frequently used measure of cellular noise is the dimensionless coefficient of variation, which is defined as standard deviation divided by mean. It measures the inverse signal-to-noise ratio and has been widely employed to characterize intrinsic and extrinsic noises of gene and protein expression and their determining factors with respect to cellular network organization [36, 43, 47–52]. For the latter measure, the coefficient of variation squared may be alternatively used [53]. In this work, we use the coefficient of variation to quantify the noise of the random distribution of molecular components and stochastic cellular response time. In the aspect of mathematical modeling of noise in apoptosis pathway, several previous stochastic apoptosis models have taken into account of the intrinsic noise by either applying Gillespie’s stochastic simulation algorithm to the ODE models or constructing Monte Carlo models from first principles [22–24, 54, 55]. With regard to modeling the impact of extrinsic noise on apoptosis pathway, there have been a few studies notably only on receptor-mediated apoptosis pathway [16, 56].
In this study, we attempt to develop mathematical and computational models of the intrinsic apoptosis pathway at single-cell level, and to identify plausible sources of non-genetic heterogeneity of apoptosis dynamics observed in a cell population using stochastic simulations. We start with a deterministic ODE model of intrinsic apoptosis pathway adapted from the Fussenegger model and find that bistability is missing. By adding positive feedback regulations that are supported by previous experimental evidences, we develop a model of intrinsic apoptosis pathway that functions as a bistable switch. We are particularly interested in understanding the stochastic behavior of this apoptosis switch under perturbation of intrinsic noise and/or extrinsic noise. Stochastic modeling and simulations of intrinsic apoptosis pathway indicate that noise could enhance robustness of the bistable switch. In addition, we show that intrinsic noise is not sufficient to induce the observed level of cell-to-cell variability of apoptosis response at biologically relevant level of molecular numbers, while the extrinsic noise of protein variations is plausibly the main source giving rise to the degree of heterogeneous responses of intrinsic apoptosis pathway between single cells.
Results and discussion
Deterministic model of intrinsic apoptosis pathway
Summary of abbreviation terms
Abbreviation | Full name or description |
---|---|
Apaf-1 | Apoptotic protease activating factor 1 |
a1cc | Apoptosome complex formed by Cytochrome C and Apaf-1 |
CC | Cytochrome C |
CEA | Executioner caspase 3 |
CV | Coefficient of variation |
c3p | Executioner procaspase-3 |
c9a | Initiator caspase 9 |
c9p | Initiator procaspase-9 |
IAP | Inhibitor of apoptosis proteins |
ODE | Ordinary differential equation |
SSA | Stochastic simulation algorithm |
T _{ d } | Caspase 3 response delay time |
Stochastic model of intrinsic apoptosis pathway under intrinsic noise perturbation
The deterministic model of the intrinsic apoptosis pathway has allowed us to analyze nonlinear properties of the system and quantify region of robust behavior. Nevertheless, the ODE modeling approach accounts for average cellular dynamics while ignoring the inevitable unpredictability embedded in biomolecular reactions and in intra- and extra-cellular environments. It has been observed by several different experimental techniques that the apoptosis response at single-cell resolution is subject to inherent stochastic perturbations, giving rise to pronounced cell-to-cell variability even in genetically identical cell population [13, 14, 16]. Therefore, it is necessary for us to develop a stochastic modeling framework of intrinsic apoptosis pathway, which can be used to explore the plausible origin of the stochasticity underlying the phenotypic heterogeneity.
To illustrate the cellular variability in the stochastic response due to intrinsic noise, histograms of the steady-state CEA copy numbers in the same 150 cells is plotted in Figure 4B against different input CC levels. The histogram shows that if the CC molecule is above 15 copy number the distribution of steady-state CEA is bimodal, with a low mean steady-state value of ∼50 number of CEA molecules and a high mean steady-state value of ∼960 CEA molecules. Such bimodal distribution of CEA response indicates that the stochastic response of intrinsic apoptosis pathway subject to intrinsic noise under this particular c9p/c3p condition is bistable [67]. The bistability behavior persists even when the copy number of CC increases to ∼800 molecules, showing that the corresponding fold change of bistability domain under intrinsic noise is above five times that of the deterministic model, where the bistability region of CC is [0.08, 0.83] (μ M) as shown in Figure 2D. Such phenomenon of enhanced robustness induced by intrinsic noise is in agreement with previous computational work which suggests that stochastic signaling networks may perform more robustly than their deterministic counterpart [68, 69]. It is noteworthy that the existence and range of bistable CEA response are dependent on the abundances of c9p and c3p in that the bistability boundary shrinks as the copy numbers of procaspases increase, and the bistable range becomes almost undetectable at significantly high amount of procaspases (10^{4}) (Additional file 2: Table S1). This result implies that the bistability property of apoptosis might only be observable at proper condition of molecular abundance.
The above stochastic simulations show that the cell-to-cell variability of intrinsic apoptosis due to intrinsic noise is especially evident when the number of c3p is relatively small (below 1000), where the CV of T_{ d } persist at the level of non-genetic noise of apoptosis observed by experiments (CV ∼[0.2, 0.3]) [3, 16] (Figure 6C). Moreover, the abundances of c9p and c3p inversely regulate the cellular variability among the eight conditions. Specifically, the lowest c9p/c3p copy-number condition leads to a CV slightly above 0.3 while the highest c9p/c3p copy-number condition leads to a CV slightly below 0.15 (Figure 6C), suggesting that the wide copy-number range of procaspases, especially procaspase-3, under the sub-1000 condition is likely a source of cell-to-cell variability under the perturbation of intrinsic noise. Previously, there has been study of apoptosis pathway implying that the molecular numbers of participating biochemical species seem to reside in a regime much higher than 1,000 [16]. Therefore, our stochastic model with intrinsic noise is further explored under the various copy numbers of c9p and c3p with sufficiently strong CC input. As shown in Figure 6D, under all the eight conditions the CV of T_{ d } drops dramatically when the number of CC molecules increases above 1000. In particular, when the copy numbers of c9p, c3p and CC molecules are all raised to 1000 and above, the CV of T_{ d } tends to an almost negligible level of ∼0.01. Such low degree of CV of T_{ d } at physiologically more plausible condition indicates that the perturbation by intrinsic noise alone seems insufficient to induce the observed degree of cell-to-cell stochasticity of apoptosis dynamics (that is, CV ∼[0.2, 0.3]), and other sources of uncertainty needs to be taken into account.
Stochastic model of intrinsic apoptosis pathway under extrinsic noise perturbation
Recently, natural protein variations across cell population has been identified as a major source of extrinsic fluctuations for apoptosis pathway [16]. Experiments have suggested that the concentration of a protein naturally varies among different cells following a log-normal distribution with a typical CV value of 0.2 to 0.3 [16, 71]. To address the impact of this kind of extrinsic noise on intrinsic apoptosis pathway, we assume that in the deterministic single-cell model of intrinsic apoptosis pathway described above certain protein(s) of interest has log-normally distributed concentration with CV equal to 0.25.
It is interesting to find out that when the 2D heat map is superimposed with the boundary of the two-parameter bistability diagram of the stochastic model, which is estimated using the histograms of CEA response (black curves in Figure 8A), the CV of T_{ d } with relatively high values (0.2–0.45) is located between the low-threshold curve (broken black line) and the high-threshold curve (solid black line). Such behavior is confirmed under systematic perturbations of the parameter K_{ c } (Additional file 3: Figure S2) and the other three parameters selected for the sensitivity analysis in Figure 3 (data not shown). Compared to the threshold lines for the bistability domain of the corresponding deterministic model (gray curves), the extrinsic noise in CC seems not significantly affect the area of bistability domain, albeit the thresholds of CC shift toward smaller values. This trend also holds under different values of K_{ c } (Additional file 3: Figure S2).
As a further exploration of the impact of extrinsic noise, we allow the concentrations of both the CC and IAP proteins to be log-normally distributed random numbers with CV equal to 0.25. The resulting 2D heat map of the CV of T_{ d } in Figure 8B shows that the level of stochasticity across cells is in general higher than the case under individual extrinsic perturbation of CC protein or IAP protein. In most of the area in the 2-parameter region of Figure 8B, the CV of T_{ d } achieves a value above 0.2. Similar to Figure 8A, the CV of T_{ d } with double extrinsic noises attains highest values (up to 0.8) between the low-threshold and high-threshold curves estimated for the stochastic bistability diagram, which is almost twice the CV value in the scenario under the extrinsic noise in IAP protein only. Moreover, the activation of CEA response can now be elicited even in the non-responsive region of Figure 8A due to the additional degree of fluctuation in the input signal. Collectively, the 2D heat-map in Figure 8B suggests that the extrinsic noises in CC and IAP proteins are sufficient to yield the experimentally observed degree of cell-to-cell variability in the apoptosis response.
The bistability domains under the nominal value of K_{ c } in Figure 8B and those under systematically perturbed values of K_{ c } in Additional file 4: Figure S3 demonstrate the same trend of behavior as those in Figure 8A and Additional file 3: Figure S2. Close comparison between Figure 8A/Additional file 3: Figure S2 and Figures 8B/Additional file 4: Figure S3 indicate that the area of bistability domain on average is slightly larger in the latter group. Therefore, additional degree of extrinsic noise may induce extra robustness for bistability of apoptosis.
Stochastic model of intrinsic apoptosis pathway under combined intrinsic and extrinsic noise perturbations
We have so far analyzed the model of intrinsic apoptosis pathway subject to either intrinsic noise or extrinsic noise independently. Additionally, we would like to find out if both types of noises are present in the intrinsic apoptosis pathway, how the cell-to-cell variability of the delay time of CEA response is influenced. To simulate a model under the perturbation of combined intrinsic and extrinsic noises, we implement the stochastic simulation algorithm of the apoptosis model as described above to mimic the intrinsic noise, and simultaneously allow certain protein concentrations to be log-normally distributed random variables to represent the extrinsic noise. In order to make comparison with the results of the intrinsic-noise only case, we again use the eight conditions of molecule numbers at c9p = {5×10^{3},10^{4}} in combination with c3p = { 10,10^{2},10^{3},10^{4}}.
First, the impact of extrinsic fluctuation in the amount of CC protein on top of the intrinsic noise is simulated. The resulting CV of T_{ d } in response to the abundance of CC less than 600 is shown in the inset of Figure 7B. Note that the unit of CC protein is now copy number of molecules rather than μ M as a requirement by the implementation of SSA. Comparing it to Figure 6C, we find that the behavior of cell-to-cell variability due to the combined types of noises is almost the same as that under intrinsic noise alone. That is, under all the eight molecule-number conditions, the CV of T_{ d } monotonically decreases in a near exponentially-decaying fashion as the number of CC molecules increases, and it falls into similar range of value (∼[0.15, 0.3]). And similar to Figure 6C, the variability value is inversely correlated with the abundances of c9p/c3p, achieving CV of T_{ d } above 0.2 if c9p/c3p have sub-1000 copy numbers. This result indicates that at the limit of low number of molecules (sub-1000), the intrinsic noise seems to make the dominating contribution to the cell-to-cell stochasticity of intrinsic apoptosis response.
We subsequently run the same stochastic model under the perturbation of intrinsic noise plus extrinsic noise in CC, while increasing the mean abundance of CC molecules up to 10,000, a more plausible reacting scale for apoptosis pathway. We find that the stochasticity curves under different c9p/c3p abundance conditions relatively converge at CC equal to 10,000 copy number (Figure 7B), due to diminished intrinsic noise. The comparison between Figure 7B and Figure 6D enables us to see the relative contributions of intrinsic and extrinsic noises: except for the case of almost undetectable abundance of c3p (= 10), which induce severe intrinsic noise, the CV of T_{ d } due to combined noises is significantly higher than that due to intrinsic-only noise. Specifically, the contrast between the case of combined noises and the case of intrinsic-only noise is highest (CV ∼0.3 versus CV <0.02) when the copy numbers of c9p, c3p and CC molecules are all above 1000. Therefore, at higher molecular numbers, the cellular stochasticity seems to be majorly contributed by the source of extrinsic noise.
Simulations under the eight conditions of c9p and c3p abundances demonstrate that the overall cell-to-cell variability is, the same as what we see earlier, inversely correlated with the copy numbers of molecules, where the CV of T_{ d } on average declines through row 1 to row 4 as the abundance of c3p increases from 10 to 10^{4}, while the CV of T_{ d } on average slightly declines from column 1 to column 2 as the abundance of c9p increases from 5×10^{3} to 10^{4}. This is consistent with the aforementioned observation of the evident variability induced by the procaspase-3 under sub-1000 copy-number condition through the channel of intrinsic noise.
It is noteworthy that the CV of T_{ d } in Figure 9 is generally larger at higher mean value of IAP molecular number, agreeing with the inference from Figure 8A that the abundance of IAP protein is capable of promoting the stochasticity of apoptosis pathway. It highlights the interesting role of IAP protein in controlling the cell-to-cell variability of intrinsic apoptosis response, and suggests that in treating diseases exploiting the apoptosis mechanism, such as cancer, the IAP protein offers a potential therapeutic target not only for effective modulation of apoptosis [74] but also for eliminating the undesired cell-to-cell heterogeneity, a major obstacle to effective cancer treatment and personalized medicine [75]. This finding is in line with the current view of the therapeutic function of IAP based on the study of apoptosis pathway at cell population level [17, 76].
Conclusions
The recently observed heterogeneous apoptosis phenotypes at single cell level have drawn increasing attention from researchers. Mathematical modeling and computer simulation provide an efficient approach to gain deep insight into the dynamic behavior of apoptosis network. This paper develops a theoretical and computational framework for single-cell stochastic modeling of the intrinsic apoptosis pathway. Using this modeling framework we explore the stochastic behavior of the intrinsic apoptosis response at single-cell level and seek to understand the plausible sources underlying the experimentally observed cell-to-cell variability of apoptosis response. We show that in the presence of noise, the bistable response of intrinsic apoptosis pathway can be more robust than its deterministic behavior. The coefficient of variation (CV) of the delayed timing of the activity of executioner caspase is utilized to quantify the stochasticity in the apoptosis dynamics. We find that the intrinsic noise can introduce significant cell-to-cell variability if the abundances of reacting biomolecules are relatively low. The level of cellular stochasticity solely due to intrinsic noise decreases dramatically to a negligible level of CV equal to ∼0.01 when the copy number of Cytochrome C is raised to 10,000, which is the amount suggested by a previous study. In addition, the extrinsic noise caused by the natural variations in protein concentrations of two key components in the intrinsic apoptosis pathway, Cytochrome C and the inhibitor of apoptosis (IAP) proteins, is also accounted for without or with intrinsic noise. Series of simulations indicate that the extrinsic noise is plausibly the major source of the cell-to-cell variability of intrinsic apoptosis response at high number of biomolecules. Furthermore, we find that the mean abundance of fluctuating IAP is positively correlated with the degree of cell-to-cell variability, thus making IAP a potential target for therapeutically suppressing the stochasticity of intrinsic apoptosis response across cell population in treating diseases such as cancer. In summary, this study based on our theoretical and computational models characterizes the behavior of the intrinsic apoptosis pathway under complex stochastic perturbations, and suggests that certain deterministic features, such as system bistability and IAP as potential therapy target, still remain in the presence of noise. Altogether, the work can enable us to gain deeper understanding toward the experimentally observed uncertainty in cellular decision making.
Methods
Deterministic ODE model
A deterministic model accounting for the average dynamics of intrinsic apoptosis pathway at single-cell level is developed. The model is adapted and modified based on the previous Fussenegger model [20]. Its reaction diagram and our modification method are given in the Results and discussion section. The model is formulated by 5-dimensional interconnected ODEs. The biochemical processes underlying each of the five ODEs are explained in detail as follows. Note that all the binding and unbinding processes are compactly represented by Michaelis-Menten kinetics under the quasi-steady-state assumption as previously described [20], and thus are not explained separately below.
Nominal parameter values of the deterministic model, which are proposed based on the Fussenegger et al model [20]
Deterministic model parameter values | |||
---|---|---|---|
Parameter | Description | Unit | Value |
k _{f 1} | Binding rate constant for Cytochrome C and Apaf-1 | m i n ^{−1} | 0.12 |
k _{f 2} | Catalysis rate constant for c9a | μ M^{−1}·m i n^{−1} | 0.3 |
k _{f 3} | Catalysis rate constant for CEA | μ M^{−1}·m i n^{−1} | 0.015 |
k _{r 1} | Unbinding rate constant for Cytochrome C and Apaf-1 | m i n ^{−1} | 0.4 |
K _{ H } | Equilibrium rate constant for binding of Cytochrome C and Apaf-1 | μ M ^{−1} | 1 |
K _{ K } | Equilibrium constant for binding of c9p and Apoptosome | μ M ^{−1} | 0.5 |
K _{ L } | Equilibrium constant for binding of c9p and Apoptosome | μ M ^{−1} | 0.3 |
K _{ P } | Equilibrium constant for binding of c9a and c3p | μ M ^{−1} | 0.1 |
K _{ U } | Equilibrium constant for binding of IAP and CEA | μ M ^{−1} | 0.1 |
k _{ u } | Inhibition rate constant of CEA by IAP | μ M^{−1}·m i n^{−1} | 0.03 |
I A P | Concentration of free IAP protein | μ M | 0.018 |
K _{ c } | Threshold concentration for catalysis of c9p by CEA | μ M | 0.1 |
μ _{1} | Degradation rate constant for the complex of Cytochrome C and Apaf-1 | m i n ^{−1} | 0.005 |
μ _{2} | Degradation rate constant for c9p | m i n ^{−1} | 0.005 |
μ _{3} | Degradation rate constant for c9a | m i n ^{−1} | 0.005 |
μ _{4} | Degradation rate constant for c3p | m i n ^{−1} | 0.005 |
μ _{5} | Degradation rate constant for CEA | m i n ^{−1} | 0.005 |
Ω_{ E Z } | Basal synthesis rate for c3p | μ M·m i n^{−1} | 0.05 |
Ω_{9} | Basal synthesis rate for c9p | μ M·m i n^{−1} | 0.05 |
Stochastic model with intrinsic noise
When the reacting species of a cellular signaling network have low molecular numbers, the inherent fluctuations of biochemical reactions become prominent. As a consequence, the deterministic formulation is no longer accurate to account for such effect of intrinsic noise, while probabilistic kinetic model is necessary. Toward this end, we refer to the Gillespie stochastic simulation algorithm (SSA) for simulating the stochastic biochemical kinetics with intrinsic noise. It is known that the standard SSA accounts for exact stochasticity of each molecule and every reaction event. To be applied to our ODE model, it requires expanding each term of the Michaelis-Menten kinetics into corresponding elementary reactions, which will add to significant computational demand especially when we intend to simulate more than one hundred single cells. Various recent studies have shown that using the lumped Hill functions for Michaelis-Menten kinetics in SSA actually yielded similar results as using the fully decomposed elementary biomolecular reactions model, thus validating the approach of applying quasi-steady-state assumption to reduce the complexity of stochastic models [78–81]. This modified Gillespie SSA is employed here, where the Michaelis-Menten kinetics is not expanded. Specifically, each binding reactions implemented by the Michaelis Menten kinetics in ODE is now treated as single reaction step, and the corresponding Hill function is incorporated directly as propensity function.
Elementary reactions of the stochastic model denoted by their corresponding propensity functions (PF)
Rn | Elementary reaction (PF) | Rn | Elementary reaction (PF) |
---|---|---|---|
Rn 1 | $\frac{{k}_{f1}\left[\mathit{\text{CC}}\right]}{1+{K}_{H}\left[\mathit{\text{CC}}\right]}$ | Rn 7 | μ_{ 3 }[ c 9a] |
Rn 2 | k_{r 1}[ a 1c c] | Rn 8 | Ω_{ E Z } |
Rn 3 | μ_{ 1 }[ a 1c c] | Rn 9 | ${k}_{f3}\xb7{\left[\phantom{\rule{0.3em}{0ex}}c9a\right]}^{n}\xb7\frac{\left[c3p\right]}{\frac{1}{{K}_{P}}+\left[c3p\right]}$ |
Rn 4 | Ω_{ 9 } | Rn 10 | μ_{ 4 }[ c 3p] |
Rn 5 | $\frac{{k}_{f2}\left[\mathit{\text{CEA}}\right]}{\left[\mathit{\text{CEA}}\right]+{K}_{c}}\xb7\frac{\left[c9a\right]\xb7\left[a1\mathit{\text{cc}}\right]\xb7{\left[c9p\right]}^{2}}{\frac{1}{{K}_{K}\xb7{K}_{L}}+\frac{\left[c9p\right]}{{K}_{L}}+{\left[c9p\right]}^{2}}$ | Rn 11 | μ_{ 5 }[ C E A] |
Rn 6 | μ_{ 2 }[ c 9p] | Rn 12 | $\frac{{k}_{u}\xb7\mathit{\text{IAP}}\xb7\left[\mathit{\text{CEA}}\right]}{1+\mathit{\text{IAP}}\xb7{K}_{U}}$ |
Stochastic model with extrinsic noise
As described in the Results and discussion section, the natural variation of protein concentration from cell to cell is considered as extrinsic noise in the apoptosis reactions. Our model of intrinsic apoptosis pathway subject to extrinsic noises in the Cytochrome C and IAP proteins is established using the above deterministic ODE model as the average single-cell model, with randomly selected parameter values as the varying protein concentrations. For instance, in each run of simulation the concentration of CC can be assumed to be a random number generated based on a log-normal distribution around its mean concentration with a CV of 0.25. Again, 150 independent runs of the stochastic model are generated to represent a sample size of 150 cells. To simulate the responses at different levels of CC signal, different mean concentrations of CC are used. Similarly, we can generate and simulate the stochastic model of intrinsic apoptosis pathway with extrinsic noise only in IAP protein, or with extrinsic noises in both CC and IAP proteins. The above algorithm is written in MATLAB program (Additional file 7: Model Script 3).
Stochastic model with combined intrinsic and extrinsic noise
To establish a stochastic model with sources of both intrinsic and extrinsic noises, we employ the stochastic model with intrinsic noise, implemented by modified Gillespie SSA method as described above, while generating random abundances of CC protein and/or IAP protein. Again, the random protein is assumed to be log-normally distributed with a CV of 0.25. The simulation of stochastic models with intrinsic noise plus the extrinsic noises in CC and/or IAP proteins across a cell population is performed by a sample size of 150 cells. The above algorithm is written in MATLAB program (Additional file 8: Model Script 4).
The MATLAB codes for stochastic models are distributed to a high-performance computer cluster consisting of one master node and 96 slave nodes to achieve parallel computation that simulates responses in multiple single cells simultaneously.
Declarations
Acknowledgements
The authors thank Dr. Zhenyu Xuan and Donald Moore for their helpful assistance in the use of the computer cluster at the Center for Systems Biology, and are grateful for the Start-up Fund from the University of Texas at Dallas.
Authors’ Affiliations
References
- Vousden K, Lu X: Live or Let Die: The Cell’s Response to p53. Nat Rev Cancer. 2002, 2: 594-604. 10.1038/nrc864.PubMedView ArticleGoogle Scholar
- Taylor R, Cullen S, Martin S: Apoptosis: controlled demolition at the cellular level. Nat Rev Mol Cell Biol. 2008,, 9: 231-41.PubMedView ArticleGoogle Scholar
- Spencer S, Sorger P: Measuring and Modeling Apoptosis in Single Cells. Cell. 2011, 144: 926-939. 10.1016/j.cell.2011.03.002.PubMedPubMed CentralView ArticleGoogle Scholar
- Xu G, Shi Y: Apoptosis signaling pathways and lymphocyte homeostasis. Cell Res. 2007, 17: 759-771. 10.1038/cr.2007.52.PubMedView ArticleGoogle Scholar
- Fuchs Y, Steller H: Programmed cell death in animal development and disease. Cell. 2011, 147 (4): 742-758. 10.1016/j.cell.2011.10.033.PubMedPubMed CentralView ArticleGoogle Scholar
- Elmore S: Apoptosis: a review of programmed cell death. Toxicol Pathol. 2007, 35 (4): 495-516. 10.1080/01926230701320337.PubMedPubMed CentralView ArticleGoogle Scholar
- Wang J, Zheng L, Lobito A, Chan F, Dale J: Inherited human Caspase 10 mutations underlie defective lymphocyte and dendritic cell apoptosis in autoimmune lymphoproliferative syndrome type II. Cell. 1999, 98: 47-48. 10.1016/S0092-8674(00)80605-4.PubMedView ArticleGoogle Scholar
- Green D: A matter of life and death. Cancer Cell. 2002, 1: 19-30. 10.1016/S1535-6108(02)00024-7.PubMedView ArticleGoogle Scholar
- Chipuk J, Green D: Dissecting p53-dependent Apoptosis. Nat Rev Cell Death Differ. 2006, 13: 994-1002. 10.1038/sj.cdd.4401908.View ArticleGoogle Scholar
- Riedl S, Salvesen G: The apoptosome: signaling platform of cell death. Nat Rev Mol Cell Biol. 2007, 8: 405-413.PubMedView ArticleGoogle Scholar
- Fuentes-Prior P, Salvesen G: The protein structures that shape caspase activity, specificity, activation and inhibition. Biochem J. 2004, 384: 201-232. 10.1042/BJ20041142.PubMedPubMed CentralView ArticleGoogle Scholar
- Jiang S, Chow S, Nicotera P, Orrenius S: Intracellular Ca2+ signals activate apoptosis in thymocytes: studies using the Ca2+−ATPase inhibitor thapsigargin. Exp Cell Res. 1994, 212: 84-92. 10.1006/excr.1994.1121.PubMedView ArticleGoogle Scholar
- Rehm M, Dussmann H, Janicke R, Tavare J, Kogel D, Prehn J: Single-cell fluorescence resonance energy transfer analysis demonstrates that caspase activation during apoptosis is a rapid process. Role of caspase-3. J Biol Chem. 2002, 277 (27): 24506-24514. 10.1074/jbc.M110789200.PubMedView ArticleGoogle Scholar
- Albeck J, Burke J, Aldridge B, Zhang M, Lauffenburger D, Sorger P: Quantitative analysis of pathways controlling extrinsic apoptosis in single cells. Mol Cell. 2008, 30: 11-25. 10.1016/j.molcel.2008.02.012.PubMedPubMed CentralView ArticleGoogle Scholar
- Albeck J, Burke J, Spencer S, Lauffenburger D, Sorger P: Modeling a snap-action, variable-delay switch controlling extrinsic cell death. PLoS Biol. 2008, 6 (12): 2831-2852.PubMedView ArticleGoogle Scholar
- Spencer S, Gaudet S, Albeck J, Burke J, Sorger P: Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis. Nature. 2009, 459: 428-433. 10.1038/nature08012.PubMedPubMed CentralView ArticleGoogle Scholar
- Bentele M, Lavrik I, Ulrich M, Stober S, Heermann D, Kalthoff H, Krammer P, Eils R: Mathematical modeling reveals threshold mechanism in CD95-induced apoptosis. J Cell Biol. 2004, 166: 839-851. 10.1083/jcb.200404158.PubMedPubMed CentralView ArticleGoogle Scholar
- Eissing T, Conzelmann H, Gilles E, Allogowert F, Bullinger E, Scheurich P: Bistability analyses of a caspase activation model for receptor-induced apoptosis. J Biol Chem. 2004, 279 (35): 36892-36897. 10.1074/jbc.M404893200.PubMedView ArticleGoogle Scholar
- Huber H, Duessmann H, Wenus J, Kilbride S, Prehn J: Mathematical modelling of the mitochondrial apoptosis pathway. Biochimica et Biophysica Acta. 2011, 1813 (4): 608-615. 10.1016/j.bbamcr.2010.10.004.PubMedView ArticleGoogle Scholar
- Fussenegger M, Bailey J, Varner J: A mathematical model of caspase function in apoptosis. Nat Biotechnol. 2000, 18: 768-774. 10.1038/77589.PubMedView ArticleGoogle Scholar
- Nair V, Yuen T, Olanow C, Sealfon S: Early single cell bifurcation of pro- and antiapoptotic states during oxidative stress. J Biol Chem. 2004, 279 (26): 27494-27501. 10.1074/jbc.M312135200.PubMedView ArticleGoogle Scholar
- Skommer J, Brittain T, Raychaudhuri S: Bcl-2 inhibits apoptosis by increasing the time-to-death and intrinsic cell-to-cell variations in the mitochondrial pathway of cell death. Apoptosis. 2010, 15: 1223-1233. 10.1007/s10495-010-0515-7.PubMedPubMed CentralView ArticleGoogle Scholar
- Raychaudhuri S: A minimal model of signaling network elucidates cell-to-cell stochastic variability in apoptosis. PLoS ONE. 2010, 5 (8): 1-7.View ArticleGoogle Scholar
- Eissing T, Allogowert F, Bullinger E: Robustness properties of apoptosis models with respect to parameter variations and intrinsic noise. IEEE Proc Syst Biol. 2005, 152 (4): 221-228. 10.1049/ip-syb:20050046.View ArticleGoogle Scholar
- Huber H, Bullinger E, Rehm M: System biology approaches to the study of apoptosis. Essentials of Apoptosis,. Edited by: Yin XM, Dong Z. 2009, New York: Humana Press, 283-297.View ArticleGoogle Scholar
- Aldridge B, Haller G, Sorger P, Lauffenburger D: Direct Lyapunov exponent analysis enables parametric study of transient signalling governing cell behaviour. Syst Biol (Stevenage). 2006, 153 (6): 425-432. 10.1049/ip-syb:20050065.View ArticleGoogle Scholar
- Hua F, Cornejo M, Cardone M, Stokes C, Lauffenburger D: Effects of Bcl-2 levels on Fas signaling-induced caspase-3 activation: molecular genetic tests of computational model predictions. J Immunol. 2005, 175: 985-995.PubMedView ArticleGoogle Scholar
- Legewie S, Bluthgen N, Herzel H: Mathematical modeling identifies inhibitors of apoptosis as mediators of positive feedback and bistability. PLoS Comput Biol. 2006, 2 (9): 1061-1073.View ArticleGoogle Scholar
- Zhang T, Brazhnik P, Tyson J: Computational analysis of dynamical responses to the intrinsic pathway of programmed cell death. Biophys J. 2009, 97: 415-434. 10.1016/j.bpj.2009.04.053.PubMedPubMed CentralView ArticleGoogle Scholar
- Bagci E, Vodovotz Y, Billiar T, Ermentrout G, Bahar I: Bistability in apoptosis: Roles of Bax, Bcl-2, and mitochondrial permeability transition pores. Biophys J. 2006, 90 (5): 1546-1559. 10.1529/biophysj.105.068122.PubMedPubMed CentralView ArticleGoogle Scholar
- Stucki J, Simon H: Mathematical modeling of the regulation of caspase-3 activation and degradation. J Theor Biol. 2005, 234: 123-131. 10.1016/j.jtbi.2004.11.011.PubMedView ArticleGoogle Scholar
- Nakabayashi J, Sasaki A: A mathematical model for apoptosome assembly: the optimal cytochrome c / Apaf-1 ratio. J Theor Biol. 2006, 242 (2): 280-287. 10.1016/j.jtbi.2006.02.022.PubMedView ArticleGoogle Scholar
- Harrington H, Ho K, Ghosh S, Tung K: Construction and analysis of a modular model of caspase activation in apoptosis. Theor Biol Med Model. 2008, 5 (26): 1-15.Google Scholar
- Kutumova E, Zinovyev A, Sharipov R, Kolpakov F: Modeling composition through model reduction: a combined model of CD95 and NF-kB signaling pathways. BMC Syst Biol. 2013, 7: 13-10.1186/1752-0509-7-13.PubMedPubMed CentralView ArticleGoogle Scholar
- Elowitz M, Levine A, Siggia E, Swain P: Stochastic gene expression in a single cell. Science. 2002, 297: 1183-1186. 10.1126/science.1070919.PubMedView ArticleGoogle Scholar
- Swain P, Elowitz M, Siggia E: Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc Natl Acad Sci USA. 2002, 99: 12795-12800. 10.1073/pnas.162041399.PubMedPubMed CentralView ArticleGoogle Scholar
- Samoilov M, Arkin A: Deviant effects in molecular reaction pathways. Nat Biotech. 2006, 24: 1235-1240. 10.1038/nbt1253.View ArticleGoogle Scholar
- Raser J, O’Shea E: Control of stochasticity in eukaryotic gene expression. Science. 2004, 304: 1811-1814. 10.1126/science.1098641.PubMedPubMed CentralView ArticleGoogle Scholar
- Johnston I: Mitochondrial variability as a source of extrinsic cellular noise. PLoS Comput Biol. 2012, 8 (3): 1-14.View ArticleGoogle Scholar
- J P: Summing up the noise in gene networks. Nature. 2004, 427 (6973): 415-418. 10.1038/nature02257.View ArticleGoogle Scholar
- J P: Models of stochastic gene expression. Physics Life Rev. 2005, 2: 157-175. 10.1016/j.plrev.2005.03.003.View ArticleGoogle Scholar
- Chalancon G, Ravarani C, Balaji S, Martinez-Arias A, Aravind L, Jothi R, Babu M: Interplay between gene expression noise and regulatory network architecture. Trends Genet. 2012, 28 (5): 221-232. 10.1016/j.tig.2012.01.006.PubMedPubMed CentralView ArticleGoogle Scholar
- EA A: Determining biological noise via single cell analysis. Anal Bioanal Chem. 2009, 393: 73-80. 10.1007/s00216-008-2431-z.View ArticleGoogle Scholar
- Thattai M, Oudenaarden A: Intrinsic noise in gene regulatory networks. Proc Natl Acad Sci U S A. 2001, 98 (15): 8614-8619. 10.1073/pnas.151588598.PubMedPubMed CentralView ArticleGoogle Scholar
- Ozbudak E, Thattai M, Kurtser I, Grossman A, van Oudenaarden A: Regulation of noise in the expression of a single gene. Nat Genet. 2002, 31: 69-73. 10.1038/ng869.PubMedView ArticleGoogle Scholar
- Tao Y: Intrinsic and external noise in an auto-regulatory genetic network. J Theor Biol. 2004, 229 (2): 147-156. 10.1016/j.jtbi.2004.03.011.PubMedView ArticleGoogle Scholar
- Rosenfeld N, Young J, Alon U, Swain P, Elowitz M: Gene Regulation at the Single-Cell Level. Science. 2005, 307 (5717): 1962-1965. 10.1126/science.1106914.PubMedView ArticleGoogle Scholar
- Newman J, Ghaemmaghami S, Ihmels J, Breslow D, Noble M, DeRisi J, Weissman J: Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise. Nature. 2006, 441 (7095): 840-846. 10.1038/nature04785.PubMedView ArticleGoogle Scholar
- Shahrezaei V, Ollivier J, Swain P: Colored extrinsic fluctuations and stochastic gene expression. Mol Syst Biol. 2008, 4 (196):PubMedPubMed CentralView ArticleGoogle Scholar
- Stekel D, Jenkins D: Strong negative self regulation of prokaryotic transcription factors increases the intrinsic noise of protein expression. BMC Syst Biol. 2008, 2: 6-10.1186/1752-0509-2-6.PubMedPubMed CentralView ArticleGoogle Scholar
- Kar S, Baumann W, Paul M, Tyson J: Exploring the roles of noise in the eukaryotic cell cycle. Proc Natl Acad Sci U S A. 2009, 106 (16): 6471-6476. 10.1073/pnas.0810034106.PubMedPubMed CentralView ArticleGoogle Scholar
- Thomas P, Matuschek H, Grima R: Intrinsic intrinsic noise analyzer: a software package for the exploration of stochastic biochemical kinetics using the system size expansion. PLoS ONE. 2012, 7 (6): e38518-10.1371/journal.pone.0038518.PubMedPubMed CentralView ArticleGoogle Scholar
- Singh A, Razooky B, Dar R, Weinberger L: Dynamics of protein noise can distinguish between alternate sources of gene-expression variability. Mol Syst Biol. 2012, 8: 607-PubMedPubMed CentralView ArticleGoogle Scholar
- Raychaudhuri S, Willgohs E, Nguyen T, Khan E, Goldkorn T: Monte Carlo simulation of cell death signaling predicts large cell-to-cell stochastic fluctuations through the type 2 pathway of apoptosis. Biophys J. 2008, 95: 3559-3562. 10.1529/biophysj.108.135483.PubMedPubMed CentralView ArticleGoogle Scholar
- Raychaudhuri S: How can we kill cancer cells: Insights from the computational models of apoptosis. World J Clin Oncol. 2010, 1: 24-28. 10.5306/wjco.v1.i1.24.PubMedPubMed CentralView ArticleGoogle Scholar
- Calzolari D, Paternostro G, Harrington PJ, Piermarocchi C, Duxbury P: Selective control of the apoptosis signaling network in heterogeneous cell populations. PLoS ONE. 2007, 2 (6): e547-10.1371/journal.pone.0000547.PubMedPubMed CentralView ArticleGoogle Scholar
- Goldbeter A, Koshland D: An amplified sensitivity arising from covalent modification in biological systems. Proc Natl Acad Sci U S A. 1981, 78: 6840-6844. 10.1073/pnas.78.11.6840.PubMedPubMed CentralView ArticleGoogle Scholar
- Shah N, Sarkar C: Robust network topologies for generating switch-like cellular responses. PLoS Comput Biol. 2011, 7 (6): e1002085-10.1371/journal.pcbi.1002085.PubMedPubMed CentralView ArticleGoogle Scholar
- Srinivasula S, Ahmad M, Fernandes-Alnemri T, Alnemri E: Autoactivation of Procaspase-9 by Apaf-1-Mediated Oligomerization. Mol Cell. 1998, 1: 949-957. 10.1016/S1097-2765(00)80095-7.PubMedView ArticleGoogle Scholar
- Creagh E, Martin S: Caspase: cellular demolition experts. Biochem Soc Trans. 2001, 29 (6): 696-702. 10.1042/BST0290696.PubMedView ArticleGoogle Scholar
- Budihardjo I, Oliver H, Lutter M, Luo X, Wang X: Biochemical pathways of caspase activation during apoptosis. Annu Rev Cell Dev Biol. 1999, 15: 269-290. 10.1146/annurev.cellbio.15.1.269.PubMedView ArticleGoogle Scholar
- Lancker JLV: Apoptosis, Genomic Integrity, and Cancer. 2006, Massachusetts: Jones and Bartlett PublishersGoogle Scholar
- Hill M, Adrain C, Duriez P, Creagh E, Martin S: Analysis of the composition, assembly kinetics and activity of native Apaf-1 apoptosomes. Eur Mol Biol Organ J. 2004, 23: 2134-2145. 10.1038/sj.emboj.7600210.View ArticleGoogle Scholar
- Gillespie D: Exact stochastic simulation of coupled chemical reactions. J Phys Chem. 1977, 81: 2340-61. 10.1021/j100540a008.View ArticleGoogle Scholar
- Gillespie D: Stochastic simulation of chemical kinetics. Annu Rev Phys Chem. 2007, 58: 35-55. 10.1146/annurev.physchem.58.032806.104637.PubMedView ArticleGoogle Scholar
- Svingen P, Loegering D, Rodriquez J, Meng X, Mesner PJ, Holbeck S, Monks A, Krajewski S, Scudiero D, Sausville E, Reed J, Lazebnik Y, Kaufmann S: Components of the cell death machine and drug sensitivity of the national cancer institute cell line panel. Clin Cancer Res. 2004, 10 (20): 6807-6820. 10.1158/1078-0432.CCR-0778-02.PubMedView ArticleGoogle Scholar
- Song C, Phenix H, Abedi V, Scott M, Ingalls B, Kaern M, Perkins T: Estimating the stochastic bifurcation structure of cellular networks. PLoS Comput Biol. 2010, 6 (3): e1000699-10.1371/journal.pcbi.1000699.PubMedPubMed CentralView ArticleGoogle Scholar
- Kim J, Heslop-Harrison P, Postlethwaite I, Bates D: Stochastic noise and synchronisation during dictyostelium aggregation make cAMP oscillations robust. PLoS Comput Biol. 2007, 3 (11): e218-10.1371/journal.pcbi.0030218.PubMedPubMed CentralView ArticleGoogle Scholar
- Kim D, Debusschere B, Najm H: Spectral methods for parametric sensitivity in stochastic dynamical systems. Biophys J. 2007, 92 (2): 379-393. 10.1529/biophysj.106.085084.PubMedPubMed CentralView ArticleGoogle Scholar
- Niepel M, Spencer S, Sorger P: Non-genetic cell-to-cell variability and the consequences for pharmacology. Curr Opin Chem Biol. 2009, 13 (5–6): 556-561.PubMedPubMed CentralView ArticleGoogle Scholar
- Sigal A, Milo R, Cohen A, Klein Y, Liron Y, Rosenfeld N, Danon T, Perzov N, Alon U, Geva-Zatorsky N: Variability and memory of protein levels in human cells. Nature. 2006, 444 (30): 643-646.PubMedView ArticleGoogle Scholar
- Potts P, Singh S, Knezek M, Thompson C, Deshmukh M: Critical function of endogenous XIAP in regulating caspase activation during sympathetic neuronal apoptosis. J Cell Biol. 2003, 163: 789-799. 10.1083/jcb.200307130.PubMedPubMed CentralView ArticleGoogle Scholar
- Hu Y, Cherton-Horvat G, Dragowska V, Baird S, Korneluk R: Antisense oligonucleotides targeting XIAP induce apoptosis and enhance chemotherapeutic activity against human lung cancer cells in vitro and in vivo. Clin Cancer Res. 2003, 9: 2826-2836.PubMedGoogle Scholar
- Fulda S, Vucic D: Targeting IAP proteins for therapeutic intervention in cancer. Nat Rev Drug Discov. 2012, 11 (2): 109-124. 10.1038/nrd3627.PubMedView ArticleGoogle Scholar
- Almendro V, Marusyk A, Polyak K: Cellular heterogeneity and molecular evolution in cancer. Annu Rev Pathol Mech Dis. 2012, 8: 277-302.View ArticleGoogle Scholar
- Bagci E, Sen S, Camurdan M: Analysis of a mathematical model of apoptosis: individual differences and malfunction in programmed cell death. J Clin Monit Comput. 2013, 27 (4): 465-479. 10.1007/s10877-013-9468-z.PubMedView ArticleGoogle Scholar
- Ermentrout B: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. 2002, Philadelphia: Society for Industrial and Applied MathematicsView ArticleGoogle Scholar
- Gonze D, Halloy J, Goldbeter A: Deterministic versus stochastic models for circadian rhythms. J Biol Phys. 2002, 28: 637-653. 10.1023/A:1021286607354.PubMedPubMed CentralView ArticleGoogle Scholar
- Gonze D, Halloy J, Jean-Christophe L, Goldbeter A: Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour. C R Biol. 2003, 326: 189—203-PubMedView ArticleGoogle Scholar
- Kim H, Gelenbe E: Stochastic gene expression modeling with hill function for switch-like gene responses. IEEE/ACM Trans Comput Biol Bioinformatics. 2012, 9: 973-979.View ArticleGoogle Scholar
- Smolen P, Baxter D, Byrne J: Interlinked dual-time feedback loops can enhance robustness to stochasticity and persistence of memory. Phys Rev. 2009, 79: 031902-1–11.Google Scholar
- Arnoult D, Gaume B, Karbowski M, Sharpe J, Cecconi F, Youle R: Mitochondrial release of AIF and EndoG requires caspase activation downstream of Bax/Bak-mediated permeabilization. EMBO J. 2003, 22 (17): 4385-4399. 10.1093/emboj/cdg423.PubMedPubMed CentralView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.