^{*}

Conceived and designed the experiments: JY TSD. Performed the experiments: JY. Analyzed the data: JY. Wrote the paper: JY TSD.

The authors have declared that no competing interests exist.

Cell growth critically depends on signalling pathways whose regulation is the focus of intense research. Without utilizing

Deregulated cell growth, a hallmark of cancer, is associated with perturbed signal transduction

The continuous advancement of high-throughout technologies in the post genomic era presents the challenge of how to interpret an ever growing amount of molecular data. Numerous experimental works have attempted to identify particular signaling molecules and their mechanisms, for example, by constructing mutants, overexpression, or reconstitutions of arrangement of genes or proteins

In this paper, we therefore apply a systems-level, multi-parametric perturbation strategy using a Monte Carlo (MC) simulation to discover molecules or reaction steps that orchestrate differential mitogen-activated protein kinase (MAPK) signaling responses. The model system is an EGF-induced signaling pathway, originally compiled by Brightman and Fell

The EGF receptor (EGFR) system implemented is based on the Brightman and Fell model (

Top-level, intermediate, and MAP kinase modules are colored in

The mechanisms in the

In the

In this

The EGFR signaling cascade system implemented here consists of 28 kinetic reactions involving 27 different protein molecules and 48 parameters. In general, these reactions follow mass action kinetics except for those catalyzed by enzymes, which follow Michaelis-Menten (MM) kinetics. The main goal of procedures detailed in this section is to calibrate key pathway elements (e.g., parameters, molecules, or reaction structures) that are chiefly responsible for processing cellular phenotypic decisions within a tolerable range. To this end, any approaches may face the following two challenges: First, even for a moderately sized cell signaling network, it is non-trivial to track the molecules' temporal behavior as kinetic parameters are often still unknown and difficult to measure experimentally. Second, even for known parameters, detailed quantitative measurements of protein activities may have been conducted under experimental condition that are far from the realities of an

We have therefore applied an approach that can avoid heavy dependence on a given set of initial parameter values. At the core is a Monte Carlo (MC) simulation that explores certain ranges of the parametric space around a given initial parameter value and generates samples of numerous parameter vectors. These vectors contain a set of multiply perturbed individual elements and are randomly generated using uniform distribution functions within known ranges of parameter uncertainty. This MC simulation is used for the entire parametric uncertainty analysis (see

Define a range for

Generate a series of independent random numbers using a uniform distribution for each parameter within defined ranges of uncertainties at Step 1. The total number of generated samples (N = 10,000) is assumed to be independent of each other and also sufficiently large in number.

Run the ordinary differential equation (ODE) model for each set of

Compare the objective function value to a threshold value. In this study, the threshold is defined as the sum of the squared errors between the active ERK profile from the unperturbed system and the average active ERK profile from all samples. Based on the threshold, each parameter set is classified into either a tolerable sample group (

Distinguish differential profiles of ERK responses using tolerable group samples only. In this study, we consider three cases of two possible differential ERK responses:

a) definitions of amplitude and duration (the ‘amplitude’ is defined as the maximum level of ERK over a time period of 60 min, and the ‘duration’ as the time period from the point of the maximum ERK level to the point of reaching 10% of the maximum within 60 min), b) transient ERK level (T) vs. sustained level (S), c) lowly transient ERK level (L-T) vs. highly transient level (H-T), and d) lowly sustained ERK level (L-S) vs. highly sustained level (H-S).

In order to efficiently classify and collect samples from the tolerable sample group for each case, we first sorted the samples with the maximum amplitude of ERK in ascending order. Then, for case 1 (T vs. S), transient samples are collected as those satisfying the criterion that the ERK level at the last time-point observation (i.e., at 60 min) is less than 10% of the maximum amplitude; sustained samples are collected according to the maximum duration, in addition to considering the maximum amplitude. For case 2 (L-T vs. H-T), L-T group samples are those below the median profile of ERK in case 1; H-T samples are those above the median. For case 3 (L-S vs. H-S), we further extracted samples with the duration of more than 30 min from the sorted samples with the maximum amplitude level in case 1. Because the maximum amplitude of ERK often occurs within the first 10 to 20 min (within the 60 min period), we assumed sustained samples would have the duration of more than 30 min; accordingly, samples of the duration of less than 30 min have been discarded. From the extracted sample list (ordered from the sample with the longest duration to that with the shortest) we have collected L-S samples from the bottom (shortest duration sample) of the list, while H-S samples have been taken from the top (longest duration sample) of the sample list. Selected were 367 samples for T and 500 samples for S in case 1, 365 samples for L-T and 367 samples for H-T in case 2, and 100 samples for both the L-S and H-S in case 3. Note that the number of samples for each group is arbitrarily chosen. During the process, our goal was that collected samples for each case have distinctively separable characteristics, so that results from the multi-parametric global sensitivity analysis can provide recognizable features for each comparison.

Evaluate parametric sensitivities by comparing the parameter distributions between two sample sets of differential ERK responses for all three cases (i.e., T vs. S, L-T vs. H-T, and L-S vs. H-S). Here, we have simply calculated cumulative frequency (CF) distributions to identify informative parameters and reactions that contribute to the difference between two differential responses. For instance, if the CF distributions between the two groups for a certain parameter are distinctively different, i.e., yielding low correlation coefficients between the two CF distributions, the parameter is classified as a sensitive, fragile, or informative factor because it contributes to the control of a particular type of differential ERK responses; otherwise, it is classified as an insensitive, robust, or uninformative factor.

In the following section, we briefly introduce overall state sensitivities (OSSs)

The OSS index is often used to capture global robustness of state variables upon parameter perturbations

With this general definition in mind, to calculate the overall state sensitivity for the individual element

Here, N_{T} represents the number of time points while N_{S} denotes the number of protein molecules or protein complexes in the system. OSS describes how robust a system is to a single parameter change while the other parameters are fixed. We perturbed at most (+/−) 50% of the original parameter value. Note that all parameter sensitivities are only valid in a local space, i.e., within the proximal space of the unperturbed parametric space.

All numerical simulations of biochemical reaction-ODEs and MC-based simulations were implemented in MATLAB (version 7.5.0, MathWorks, Natick, MA). We used

The nominal parameter set, initial conditions, and their perturbed ranges are depicted in

Parameter [unit] | Parameter Value | Test Range | |

k1 [M^{−1} min^{−1}] |
3.8e8 | 7.3e7–7.3e9 | 7.3518e7–7.2998e9 |

k2, | 0.7, | 0.1–1 | 0.1001–0.9997, |

k5, | 0.35, | 0.1002–0.9999, | |

k6, | 0.35, | 0.1000–0.9999, | |

k8 [min^{−1}] |
0.35 | 0.1001–0.9999 | |

k3 [min^{−1}] |
4.84e-2 | ||

k4 [molecule^{−1}min^{−1}] |
1.383e-3 | 1.8e-4–1.8e-2 | 1.8066e-4–1.8e-2 |

k7 [min^{−1}] |
1 | 0.1–10 | 0.1004–9.9989 |

k9 [min^{−1}] |
12 | 6.0–60 | 6.0003–59.9861 |

k11 [molecule^{−1}min^{−1}] |
2.0e-3 | 2.0e-3–2.0e-1 | 2.0e-3–2.0e-1 |

k12 [molecule^{−1}min^{−1}] |
1.63e-2 | 2.5e-5–6.0e-2 | 3.1199e-5–6.0e-2 |

k13 [min^{−1}] |
15 | 1.2–2.4e2 | 1.2045–2.3995e2 |

k14 [molecule^{−1}min^{−1}] |
5.0e-3 | 5.0e-4–5.0e-2 | 5.0141e-4–5.0e-2 |

k15 [min^{−1}] |
7.2e2 | 3.0e2–1.2e3 | 3.0015e2–1.2e3 |

k16 [molecule^{−1}min^{−1}] |
1.2e-3 | 1.2e-4–1.0e-2 | 1.2066e-4–1.0e-2 |

k17 [min^{−1}] |
27 | 0.15–2.4e2 | 0.1694–2.3994e2 |

k19, | 50, | 1.5–2.4e2 | 1.5210–2.3997e2, |

k21 [min^{−1}] |
50 | 1.5313–2.3999e2 | |

k23, | 8.3, | 1.45–2.4e2 | 1.4668–2.3998e2, |

k25 [min^{−1}] |
8.3 | 1.4829–2.3999e2 | |

k27 [min^{−1}] |
1.6 | 1.4–1.2e2 | 1.4088–1.1999e2 |

k_1 [min^{−1}] |
0.73 | 1.0e-10–1.0e-8 | 1.0052e-10–9.9991e-9 |

k_3 [min^{−1}] |
0.7 | 0.1–1 | 0.1001–0.9997 |

k_7 [min^{−1}] |
3.47e-4 | 3.47e-5–3.47e-3 | 3.5466e-5–3.5e-3 |

k_11 [min^{−1}] |
3.8 | ||

k_12 [min^{−1}] |
10 | ||

k_14 [min^{−1}] |
60 | ||

k_16 [min^{−1}] |
3 | ||

V10, | 3.0e5, | 0.6–3.0e6 | 854.6165–2.9988e6, |

V18, | 9.7e4, | 66.7873–2.9992e6, | |

V28 [molecules cell^{−1} min^{−1}] |
75 | 276.2419–2.9999e6 | |

V20, | 9.2e5, | 3.6e2–1.8e9 | 1.8663e5–1.7998e9, |

V22, | 9.2e5, | 8.9270e4–1.7995e9, | |

V24, | 2.0e5, | 4.8764e5–1.7999e9, | |

V26 [molecules cell^{−1} min^{−1}] |
4.0e5 | 2.2472e5–1.7999e9 | |

Km9, | 6.0e3, | 6.0e3–9.0e6 | 1.0976e4–8.9994e6, |

Km10, | 6.0e3, | 6.0848e3–8.9980e6, | |

Km18 [molecules cell^{−1}] |
6.0e3 | 8.0619e3–8.9990e6 | |

Km19, | 9.0e3 | 6.0e3–9.0e6 | 6.0990e3–8.9999e6, |

Km21, | 9.1468e3–8.9997e6, | ||

Km23, | 6.4348e3–8.9996e6, | ||

Km25 [molecules cell^{−1}] |
7.3475e3–8.9959e6 | ||

Km20, | 6.0e5 | 6.0e3–9.0e6 | 6.2941e3–8.9983e6, |

Km22, | 6.7470e3–9.0e6, | ||

Km24, | 7.1524e3–8.9997e6, | ||

Km26, | 6.0566e3–8.9995e6, | ||

Km27 [molecules cell^{−1}] |
9.8755e3–8.9999e6 | ||

Km28 [molecules cell^{−1}] |
2.0e4 | 6.0e3–9.0e6 | 6.4206e3–8.9995e6 |

The column of actual test range represents parameter ranges that are contained in the generated samples.

Initial Condition [unit] | Initial Value | Test Range | |

x_{0}1 [L_{0}, M] |
1.0e-7 | 2.0e-13–1.0e-7 | 2.3588e-11–9.9988e-8 |

x_{0}2 [R_{s0}, molecules cell^{−1}] |
1.11e4 | 1.0e4–1.0e6 | 1.0191e4–9.9996e5 |

x_{0}4 [R_{i0}, molecules cell^{−1}] |
3.9e3 | 1.0e3–1.0e5 | 1.0001e3–9.9992e4 |

x_{0}9 [Shc_{0}, molecules cell^{−1}] |
3.0e4 | 1.0e4–2.0e5 | 1.0009e4–1.9999e5 |

x_{0}11 [GS_{0}, molecules cell^{−1}] |
2.0e4 | 1.0e4–2.0e5 | 1.0002e4–1.9999e5 |

x_{0}14 [RasGDP_{0}, molecules cell^{−1}] |
1.98e4 | 1.0e4–1.0e6 | 1.0119e4–9.9998e5 |

x_{0}16 [RasGTP_{0}, molecules cell^{−1}] |
2.0e2 | 1.0e2–1.0e4 | 1.0049e2–9.9996e3 |

x_{0}17 [GAP_{0}, molecules cell^{−1}] |
1.5e4 | 1.0e4–2.0e5 | 1.0021e4–1.9995e5 |

x_{0}18 [Raf_{0}, molecules cell^{−1}] |
1.0e4 | 1.0e4–1.0e6 | 1.0201e4–9.9988e5 |

x_{0}22 [MEK_{0}, molecules cell^{−1}] |
3.6e5 | 1.0e4–1.0e6 | 1.0247e4–9.9994e5 |

x_{0}25 [ERK_{0}, molecules cell^{−1}] |
7.5e5 | 1.0e4–1.0e6 | 1.0024e4–9.9991e5 |

The column of actual test range represents ranges of initial values that are contained in the generated samples.

First, we investigated the differentiating pathway parameters between transient and sustained activation of ERK profiles. The cumulative frequency (CF) distributions of transient and sustained cases for each parameter are shown in

Solid lines in

The multi-parametric analysis based on the MC method identifies k14, k16, k19, and k25 as the most sensitive forward mass-action kinetic parameters while no significant effects result from reverse kinetic parameters. Furthermore, the reaction rates of V18, V20, and mildly V24 and V26 appear to be closely involved in controlling the differential ERK responses. Finally, with regards to the Michaelis constants, Km18 and Km22–Km26 are the dominating parameters. Together, this indicates that the reaction steps R14, R16, R18, R19–R20, R22, R24, and R26 are those primarily controlling differential ERK responses (i.e., a transient vs. a sustained activation profile). The corresponding protein molecules involved in those reactions include active Ras and Raf (RasGTP and Raf*), MEK, and ERK.

In taking a detailed look at the reactions, first, R14 and R16 are those that balance active Ras between an inactivated and activated state either by binding to GAP (to be finally converted to its inactive form (RasGDP)) or by binding to Raf (to further process the downstream pathway of active Raf). This activation of Raf becomes important for initiating the phosphorylation of MEK to MEKP through R19 (Note: reaction steps from R19 to R26 denote the phosphor-/dephosphorylation reactions in the MAPK module). The reverse reaction (R18) that dephosphorylates active Raf (Raf*) into its inactive form is also found to be a sensitive, controlling factor. We also find that those samples that show transient ERK activation tend to have smaller values of k14, V20, Km18, Km22 (and less so of Km25), but larger values in k16, k19, V18, and Km 24, Km26 (see

The cumulative frequency (CF) distribution informs that the higher frequencies of larger Km18 and smaller V18 in the sustained case indicate a much slower reaction rate of R18 in the sustained case than in the transient case (i.e., v_{s}<v_{t}).

To examine the impact of specific pathway modules on differential ERK activities, we divided the entire EGFR pathway into three modules and only perturbed parameters that were involved in a particular subsystem. The results follow in the next section.

For the first top-level module spanning from reaction step 1 to 8, we perturbed seven forward kinetic parameters (k1, k2, k4, k5, k6, k7, and k8) and three kinetic parameters of reverse reactions (kr1, kr3, and kr7) while fixing the other parameters at their nominal values. In examining the differentiating pathway parameters between the tolerable and intolerable group, parameter k1, i.e. the first reaction rate related to ligand-receptor binding, was found to be the most sensitive one. However, the top-level module parameter perturbations resulted in only transient responses of ERK activity. This suggests that parametric variations of the top-level module are unlikely to cause significant differences in ERK responses. A similar picture emerges for the third, i.e. MAPK module, where separate perturbation showed no sustained ERK activity pattern either. However, differential responses of ERK activity (transient or sustained behavior) were observed when the second or intermediate module was perturbed. This result was similar to those obtained with the whole pathway run (see

In parallel to the case of transient vs. sustained ERK profiles, we continually investigated the differentiation-reactions that are instrumental for controlling the amplitude of ERK activity for the transient case. The result clearly shows that R18, i.e. the dephosphorylation of active Raf (Raf*), is no longer a sensitive reaction for the amplitude-differentiation between the low-transient and high-transient level (see

We also searched for sensitive factors for the duration-differentiation pathway within sustained profiles. Here, not only did R14 and R16 turn out to be critical, but we also discovered that reactions that are specifically involved in the MAPK module became increasingly important, i.e., R19 (k19), R22(Km), R23(Km23), R24(V24, Km24), R25(Km25), R26(Km26) as shown in

First, we investigated which initial conditions are most influential in distinguishing differential ERK responses. By fixing parameters at their nominal values, a total of 11 initial conditions were perturbed at the same time. The ERK profiles all show transient behaviors with different amplitudes.

Solid lines in

Secondly, the extent of inhibitory feedback regulation of dissociation of the SOS complex, mediated by phosphorylated ERK, was varied across a range from zero feedback to 2 orders of magnitude of its original feedback strength when being multiply perturbed with the intermediate subsystems. Results showed no significant ERK response differences triggered by variations in the feedback parameter (k27, Km27) (compare

Lastly, we examined the effect of single parameter perturbation on overall state sensitivity and further compared the results with those from the multi-parametric approach. We perturbed one single parameter at a time while keeping the others fixed. Each parameter was perturbed with the maximum of (+/−) 1% and also (+/−) 50% variation around the nominal set of parameters. Note that the overall state sensitivity results were independent of the percentage variation. To identify controlling factors that contribute to transient or sustained ERK profiles, we selected two parameter sets for each case as an original parameter set, i.e., one set for transient behavior, another set for sustained ERK behavior. We replicated the perturbation simulation 100 times, ranked them for each run, and averaged their ranks by dividing the sum of ranks for each parameter with the total number of parameters (Np = 48). Thus, smaller OSS values reflect more robust parameters (

Parameter numbers from 1 to 21 denote all forward kinetic constants, k, 22 to 28 denote all reverse kinetic constants, kr, 29 to 35 denote all maximum reaction rates, V, in MM kinetics, and lastly, 36 to 48 denote all Michaelis constants, Km.

First, it is obvious that overall mass-action kinetic parameters such as k(2–21), kr (22–28) are more sensitive than MM kinetic parameters such as V (29–35) and Km (36–48). Secondly, it can be inferred that parameters that are most sensitive in orchestrating a transient ERK response have been initiated by intermediate-module reactions such as k14(13), k17(16), k19(17), k21(18), V10(29), Km9(36), Km10(37), and Km18(38). In contrast, parameters that are more sensitive for triggering a sustained ERK response are now strongly related to the MAPK module such as k23(19), k25(20), k27(21), Km20–Km22(40–42), and Km24–Km26(44–46). These are main reactions involved in phospho-/dephosphorylation of MEK and ERK by either kinases or phosphatases. Thirdly, the sensitive reactions of k23(19), k25(20), and k27(21) for the sustained case are interestingly those that are directly related to the ERK-induced inhibitory feedback loop. In addition, parameters Km27(47) and Km28(48) are less sensitive in the sustained case, implicating that R27 and R28 are more stably associated.

More mechanistic insights into key regulatory factors that control cellular phenotypic decisions are necessary to improve our understanding of cell biology in the normal and diseased state. Differential dynamics of the activation of the ERK-MAPK cascade is of importance in determining cellular responses such as cell proliferation and differentiation

In this paper, applying

We have investigated three possible cases with regards to differential patterns of ERK activation, namely

R16, R18, R19, R22, R25 | R14, R20, R23, R24, R26 | |

R14, R20, R24, R26 | R16, R19, R22, R25 | |

R14, R20, R24, R26, R28 | R16, R19, R22, R25 |

Shown is in (a) the dynamics of the transient ERK signaling response, and in (b) the dynamics of the sustained response. For the transient case in (a), the binding between RasGTP and Raf (R16) occurs faster than the binding reaction between RasGTP and GAP (R14), which further rapidly activates Raf. However, activated Raf is also quickly inactivated (R18) while it stimulates MEK phosphorylation (R19); in addition, activated MEK is also rapidly dephosphorylated (R22). These factors contribute to a faster, but short-lasting activation of ERK, thus producing a transient behavior. In contrast, for the sustained case in (b), the binding between RasGTP and GAP (R14) occurs faster, which increases the availability of RasGDP to bind to ShcGS complex so that the inactive Ras can be re-activated (R12 and R13) to continually stimulate its downstream. Meanwhile, Raf is still activated through R16 and R17. However, the inactivation of active Raf (R18) occurs much slower. In addition, the phospho-/dephospho-rylation reactions of ERK (R23, R24, and R26) are more active so that the signal can last longer within the MAPK module. Together, these factors result in a prolonged activation behavior of ERK. Note that initial protein concentrations of RasGTP, Raf, and GAP, found to be important, are depicted as

Intriguingly, our results are in good agreement with reported computational and experimental findings. Recently, the importance of Ras dynamics has been further confirmed by investigating

Technically, the main advantage of our multi-parametric approach is the ease of discovery and interpretation of informative factors contributing to differentiation-pathways between separable output observations. It also provides a mechanistic view of the key factors involved while exact kinetic information is not required. However, one caveat is that parametric ranges for each parameter need to be carefully selected to cover the range of possible values; also, it fails to capture interactive effects between distant parametric factors on the structural map. For example, no feedback effect of active ERK to SOS (on differential ERK responses) was observed through our multi-parametric global sensitivity analysis. This may suggest that this feedback effect have been buffered by other dominant, intermediate reaction factors involved. In fact, one experimental work shows that the inhibition of ERK feedback to SOS was the least active feedback loop among multiple modes of negative feedback by phosphorylated ERK (refer to

There are other EGFR-downstream pathways that function in parallel to the MAPK pathway and which deserve our attention in future work. For example, Ras may have at least one more effector other than Raf such as PI3K (Phosphatidylinositol-3-kinase)

Ultimately, we intend to integrate this powerful molecular-level pathway analysis into microscopic-macroscopic-scale

Correlation coefficients for the whole EGFR network. * T, a group of samples that show transient ERK activation; S, a group of samples that depict sustained ERK activation; L-T vs. H-T, samples of lowly transient ERK activation vs. samples of highly transient ERK activation; L-S vs. H-S, samples of lowly sustained ERK activation vs. samples of highly sustained ERK activation; Tol, samples of the tolerable group; iTol, samples of the intolerable group.

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Schematic view of the MC simulation-based multi-parametric global sensitivity analysis.

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Frequency distributions of parameters k14, V20, Km18, Km22, and Km25 in the first column, and those of k16, k19, V18, Km24, and Km26 in the second column for the whole pathway perturbation study. The red line and the blue line represent the sustained and the transient case, respectively. For instance, the frequency distributions for the transient case (blue line) in the first column show that smaller valued parameters are highly dominant whereas larger values are dominant for those parameters in the second column. These observations are exactly opposite for the sustained case (red line).

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Results of cumulative frequency distributions of the multi-parametric sensitivity analysis for the intermediate module. Solid lines in red and in blue represent the sustained and transient case, respectively.

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Results of cumulative frequency distributions between the lowly transient (L-T) and highly transient (H-T) case for the whole network. Solid lines in red and in blue represent the L-T and H-T case, respectively.

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Results of cumulative frequency distributions between the lowly sustained (L-S) and highly sustained case (H-S) for the whole network. Solid lines in red and in blue represent the L-S and H-S case, respectively.

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Results of cumulative frequency distributions of the multi-parametric sensitivity analysis for the intermediate module with the variation of feedback strength (k27, Km27). Solid lines in red and in blue represent the sustained and transient case, respectively.

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