Dose-Response: An International Journal
Volume 8 | Issue 3
Article 8
9-2010
EMPLOYING A MECHANISTIC MODEL
FOR THE MAPK PATHWAY TO EXAMINE
THE IMPACT OF CELLULAR ALL OR NONE
BEHAVIOR ON OVERALL TISSUE
RESPONSE
Nicholas S Luke
North Carolina Agricultural and Technical State University, Greensboro, NC
Michael J DeVito
National Institute of Environmental Health Sciences, Research Triangle Park, NC
Christopher J Portier
National Institute of Environmental Health Sciences, Research Triangle Park, NC
Hisham A El-Masri
U.S. Environmental Protection Agency, Research Triangle Park, NC
Follow this and additional works at: http://scholarworks.umass.edu/dose_response
Recommended Citation
Luke, Nicholas S; DeVito, Michael J; Portier, Christopher J; and El-Masri, Hisham A (2010) "EMPLOYING A MECHANISTIC
MODEL FOR THE MAPK PATHWAY TO EXAMINE THE IMPACT OF CELLULAR ALL OR NONE BEHAVIOR ON
OVERALL TISSUE RESPONSE," Dose-Response: An International Journal: Vol. 8 : Iss. 3 , Article 8.
Available at: http://scholarworks.umass.edu/dose_response/vol8/iss3/8
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Luke et al.: Modeling all-or-none MAPK activation
Dose-Response, 8:347–367, 2010
Formerly Nonlinearity in Biology, Toxicology, and Medicine
Copyright © 2010 University of Massachusetts
ISSN: 1559-3258
DOI: 10.2203/dose-response.09-017.Luke
EMPLOYING A MECHANISTIC MODEL FOR THE MAPK PATHWAY TO
EXAMINE THE IMPACT OF CELLULAR ALL OR NONE BEHAVIOR ON
OVERALL TISSUE RESPONSE
Nicholas S. Luke 䊐 Department of Mathematics, North Carolina Agricultural
and Technical State University, Greensboro, NC 27411
Michael J. DeVito 䊐 National Toxicology Program, National Institute of
Environmental Health Sciences, Research Triangle Park, NC 27709
Christopher J. Portier 䊐 Environmental Systems Biology Group, Laboratory of
Molecular Toxicology, National Institute of Environmental Health Sciences,
Research Triangle Park, NC 27709
Hisham A. El-Masri 䊐 Integrated Systems Toxicology Division, National Health
and Environmental Effects Laboratory, Office of Research and Development, U.S.
Environmental Protection Agency, Research Triangle Park, NC 27711
䊐 The mitogen activated protein kinase (MAPK) cascade is a three-tiered phosphorylation cascade that is ubiquitously expressed among eukaryotic cells. Its primary function is
to propagate signals from cell surface receptors to various cytosolic and nuclear targets.
Recent studies have demonstrated that the MAPK cascade exhibits an all-or-none response
to graded stimuli. This study quantitatively investigates MAPK activation in Xenopus
oocytes using both empirical and biologically-based mechanistic models. Empirical models can represent overall tissue MAPK activation in the oocytes. However, these models lack
description of key biological processes and therefore give no insight into whether the cellular response occurs in a graded or all-or-none fashion. To examine the propagation of
cellular MAPK all-or-none activation to overall tissue response, mechanistic models in conjunction with Monte Carlo simulations are employed. An adequate description of the dose
response relationship of MAPK activation in Xenopus oocytes is achieved. Furthermore,
application of these mechanistic models revealed that the initial receptor-ligand binding
rate contributes to the cells’ ability to exhibit an all-or-none MAPK activation response,
while downstream activation parameters contribute more to the magnitude of activation.
These mechanistic models enable us to identify key biological events which quantitatively
impact the shape of the dose response curve, especially at low environmentally relevant
doses.
1 INTRODUCTION
The mitogen-activated protein kinase (MAPK) cascades are an integral part of signal transduction in eukaryotic cells. These cascades are
well-known, ubiquitous, and conserved in cells from yeast to mammals
(Lewis et al., 1998; Widmann et al., 1999). In mammalian cells, there are
Address correspondence to Nicholas S. Luke, Department of Mathematics, North
Carolina Agricultural and Technical State University, 1601 East Market Street, Greensboro, NC
27411. Phone: (336) 334-7766 ext. 4015, Fax: (336) 256-0876, [email protected].
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Upstream Activation
MAPKKK
RAF
MEKKs
MEKKs
MAPKK
MEK
JNKK
MKK3
MAPK
ERK
JNK
P38
Downstream − Gene expression
FIGURE 1: Schematic of the three-tiered MAPK cascades (Adapted from Luster et al. (2004).
three distinct, parallel cascades which have been well studied. These are
the extracellular-signal-regulated-kinase (ERK) cascade, the c-Jun N-terminal kinase (JNK) cascade, and the p38 MAPK cascade. Each cascade is
structured in a three-tier design, where the first level is a MAPK kinase
kinase (MAPKKK), the second level is a MAPK kinase (MAPKK), and the
final level is the MAPK (ERK, JNK or p38). A schematic of these cascades
is depicted in Figure 1. Each kinase in the cascade phosphorylates and
activates the immediate downstream kinase. The final component of the
cascade, the MAPK, upon activation, phosphorylates various cytosolic targets and translocates to the nucleus to phosphorylate transcription factors affecting gene expression.
Studies have shown that the MAPK pathways can be activated by a wide
variety of stimulants. Arsenic (Dong, 2002; Cooper et al., 2004), asbestos,
cigarette smoke (Mossman et al., 2006), adenosine triphosphate (ATP)
(Tai et al., 2001), copper (Mattie and Freedman, 2004), and numerous
other compounds have been shown to be activators of the MAPK pathway.
Similarly, activation of the MAPK pathway has been shown to lead to different end results. Transcriptional factors, such as AP-1, are activated
(Shaulian and Karin, 2001; Zhong et al., 2005) and genes, such as c-fos and
c-jun, are induced (Owuor and Kong, 2002). Additionally, depending on
the amplitude and duration of the stimulus, MAPK activation can lead to
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Luke et al.: Modeling all-or-none MAPK activation
Modeling all-or-none MAPK activation
A
B
Increasing Stimulus
FIGURE 2: Schematic of an all-or-none response (A) versus a graded response (B) in the presence
of increasing stimulus.
cell proliferation or apoptosis (Ramos-Nino et al., 2002; Apati et al., 2003)
in conjunction with other signaling cascades. Because the MAPK cascade
can be activated by a variety of stimulants and plays such an important role
in cell fate, it would be beneficial to mathematically model the signal transduction that occurs within the cascade.
Models of different aspects of the MAPK cascade have been presented throughout the literature. For example, El-Masri & Portier (1999)
examined a mechanistic model of the MAPK pathway in relation to the
cell cycle. It is well accepted that the MAPK pathway is governed by positive and negative feedback loops, and models incorporating the importance of this feedback have been described by Kholodenko (2007),
Mutalik & Venkatesh (2006), and Salazar & Hofer (2006). These feedback loops have led to system models that exhibit bistability (Bhalla et al.,
2002; Markevich et al., 2004; Ortega et al., 2006), ultrasensitivity
(Kholodenko, 2000; Kolch et al., 2005), and oscillatory behavior
(Chickarmane et al., 2007). A spatial model of the MAPK pathway was presented by Markevich et al. (2006), and Schwacke & Voit (2007) examined
a model that accounted for interaction between parallel MAPK cascades.
The aforementioned models are but a few examples; for a more detailed
review of MAPK modeling efforts, see Orton et al. (2005) or Vayttanden
et al. (2004).
Ferrell & Machleder (1998) and Mackeigan et al. (2005) postulate
that MAPK phosphorylation occurs in an all-or-none, step-wise manner.
From an initial glance at the dose response results exhibited by MAPK in
the presence of an increasing stimulus, one might assume that the
response is graded. That is, as the stimulus increases, the response
increases uniformly. However, a graded tissue response could mean that
each individual cell had a graded response, or it could be interpreted that
each cell had a perfectly switch-like (all-or-none) response. Figure 2 displays a schematic for each of these possibilities. If the individual respons349
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es are graded, then each cell should have an increasing, intermediate
level of MAPK phosphorylation as the stimulus increases. Under the
switch like assumption, each cell can exhibit either a steady-state (none)
level of MAPK phosphorylation, or a level which is elevated above the
steady state level (all). Ferrell & Machleder (1998) present data that suggest that the latter, all-or-none response, is what occurs in the MAPK cascade. Studies suggest that the all-or-none behavior is also exhibited in
gene induction and protein expression (Pirone and Elston, 2004; Zhang
et al., 2006).
Biologically based mathematical models are increasingly used to predict the shape of the dose response curve, specifically at low environmentally-relevant exposure levels. Hence, the ability to accurately model
the all-or-none response becomes important in distinguishing between
cell level and tissue level models. In this study, we investigate the all-ornone response using a mechanistic model that describes MAPK phosphorylation.
1.1 MATERIALS AND METHODS
All simulations were executed utilizing MATLAB® R2006b (The
Mathworks, Inc., Natick, MA). The system of equations was integrated
using MATLAB®’s ode15s command, a variable order method for solving
a system of stiff differential equations. We examined MAPK phosphorylation mathematically using both empirical and mechanistic models. The
empirical models utilized a Hill equation or a log normal cumulative distribution function. The mechanistic models were based on previously
published models by El-Masri & Portier (1999) and Bhalla et al. (2002).
Each modeling effort was compared to MAPK phosphorylation data
presented by Ferrell & Machleder (1998). The data were produced during an experiment in which Xenopus oocytes were exposed to varying
amounts (ranging from 0 µM to 8 µM) of progesterone. The phosphorylation fold levels of MAPK were examined in 190 individual progesteronetreated oocytes, and 19 individual untreated oocytes. Each data point represents a sample of 11 to 39 oocytes. Data were digitized using the digitization software, TechDig v2.0 (Jones, R.B., Mudelein, Illinois, 1998).
All optimal parameters were identified using MATLAB®’s
fminsearchbnd command (available at www.matlabcentral.com). This command is an implementation of the Nelder-Meade simplex algorithm
which allows the user to specify lower and upper bounds for the possible
parameter space. fminsearchbnd was used to find the parameters which
minimize the following ordinary least squares (OLS) cost function
(Sheiner and Beal, 1985),
Σ | f (d ;q ) – z | ,
N
J=
2
i
i
(1)
i =1
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100
90
Ferrell Data
lognormal c.d.f.
Hill function
% − MAPK phosphorylation
80
70
60
50
40
30
20
10
0
0
10
2
10
progesterone (nM)
4
10
FIGURE 3: Comparison of a hill equation and a lognormal cumulative distribution function approximation of MAPK phosphorylation.
where f (di ;q ) is the model approximation at dose di , for the parameter
set q, and zi represents the data corresponding to dose di . Minimization
of the cost function effectively minimizes the discrepancy between the
model prediction and the available data.
2 EMPIRICAL MODELS FOR MAPK ACTIVATION
Ferrell & Machleder (1998) note that their data are approximated
well by utilizing a Hill function with Hill coefficient nH = 1. The Hill function is given by the equation (Ferrell and Machleder, 1998)
CHill (x ) =
xnH
k + xnH
nH
(1)
where y represents the percentage of MAPK phosphorylation, x represents the amount of progesterone, nH is the Hill coefficient, and k represents the concentration of x at which the oocytes respond half-maximally. This Hill equation fits the data with a least squares cost function of
1519, and its approximation to the data is depicted in Figure 3. This Hill
equation approximates MAPK activation at a tissue level, and thus, displays a graded response.
Under the all-or-none MAPK activation assumption, the level of stimulus required to fully phosphorylate each cell varies (i.e. one cell may
exhibit full MAPK phosphorylation at a different dose than another). We
further investigated the impact of the distribution of activation doses on
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the dose response curve. Studies have shown that the uptake of radioactivity is log-normally distributed among a group of cells (Neti and Howell,
2006). If we consider a similar assumption for MAPK activation, that is,
that the amount of stimulus required to cause full phosphorylation is distributed log-normally among the cells, then the percentage of cells which
have become activated for any given dose can be represented by the
cumulative distribution function for a lognormal distribution. This cumulative distribution function takes the form
1 + 1 erf
2
2
CLog-norm(x ) =
冤
ln(x) – µ
σ√2
冥,
(2)
where x is the level of stimulus, µ is the logarithmic mean of the activation dose, σ is the logarithmic standard deviation, and erf is the error
function, given by
erf (x ) =
x
兰e
2
√π
–t 2
dt ,
(3)
0
Using the fminsearch optimization routine, we identify optimal values for
the parameters µ and σ. The best fit to the data was produced when µ =
−3.43 (µM) and σ = 2.78 (µM). Note that since we are looking at the log
normal mean and standard deviation, the optimized value of µ does not
reflect a negative value, but rather a value on the order of 0.001. This
approximation results in a least squares cost function of 821, and is shown
in comparison to the Hill approximation and the data in Figure 3. Similar
to the Hill equation approximation, this lognormal cumulative distribution function represents MAPK activation at the tissue level and exhibits
a graded response.
While both the Hill equation and the lognormal cumulative distribution function give reasonable fits to the Ferrell & Machleder (1998) data,
they both are representative of a tissue level graded response. However,
experimental evidence has shown that on the cellular level an all-or-none
response is exhibited (Ferrell and Machleder, 1998; Bagowski and Ferrell,
2001). To adequately investigate the role of MAPK in Xenopus oocytes, it
would be beneficial to have the ability to focus on both tissue level and
cellular level responses. The empirical models presented here lack
description of key biological processes at the cellular level and therefore
give no insight into whether the cellular response occurs in a graded or
all-or-none fashion. In this paper, we employ mechanistically-based models of MAPK phosphorylation to investigate the propagation from cellular to tissue response.
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Luke et al.: Modeling all-or-none MAPK activation
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3 EMPLOYING A MECHANISTIC MAPK MODEL
3.1 Model Structure
We consider a model that accounts for the biological structure of the
MAPK signaling cascade. The mechanistic model which we initially consider is based upon a model of the MAPK cascade developed by El-Masri
& Portier (1999). This model, whose end result predicts fold levels of AP1 production, consists of three sections. The first section, dealing with the
activation of protein kinase C (PKC), focuses on the initial interpretation
of the external signal. The second section, which involves the activation
of MAPK kinases, models the translation of the growth signal from the
cytoplasm to DNA. The final section deals with the nuclear events leading
to the expression of AP-1. The primary objective of the study by El-Masri
and Portier (1999) was to model the effect of MAPK on cellular replication, hence the inclusion of the transcripton factor AP-1. Because the
main focus of our study deals with MAPK, only the first two sections of the
model will be utilized. In the following, we give a brief detailed description of the activities occurring during each section of the model. For a
more in-depth description, we refer readers to El-Masri & Portier (1999).
Model equations are presented in Appendix A, and parameter values are
reported in Tables I and II.
The first section of the model has been adapted from a previously
published calcium oscillator model (Cuthbertson and Chay, 1991). The
signaling cascade is activated when growth factors bind to cellular receptors. This binding initiates a build up of activated guanosine triphosphate
(GTP) binding proteins, which, in turn, leads to the activation of phospholipid lipase C (PLC). This causes the hydrolysis of phosphatidylinositol (4,5)-bisphosphate (PIP2), producing inositol (1,4,5) triphosphate
(InsP3) and diacylglycerol (DAG). InsP3 facilitates the release of Ca2+
from the endoplasmic reticulum (ER). This free Ca2+, along with DAG,
activates PKC. Activated PKC is then responsible for degrading active
GTP (a negative feedback process), as well as initiating the next portion
of the model.
In the second section of the model, activation of PKC is transferred
into the nucleus by multiple steps involving phosphorylation and dephosphorylation of MAPK (Yamaguchi et al., 1995). Initially, PKC activates Raf,
which then activates MAP kinase kinase (MAPKK). Activation of MAPKK
ultimately increases nuclear MAPK activity. A negative feedback process is
present, as MAPK deactivates Raf, essentially regulating the activation of
the cascade.
The model is used to generate time course data, as well as a dose
response curve. Time course and dose response results generated using
the parameter values listed in Tables I and II are displayed in Figures 4
and 5, respectively. The dose response results are illustrated by the solid
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TABLE I: Parameter values (Cuthbertson and Chay, 1991) for the PKC activation portion of the
mechanistic model
Parameter
Function
Value
Units
KGF
hg
Kd
Kg
n
m
KDAG
hd
ld
Ks
hc
lc
KP
KC
1st order constant of stimulus binding to receptor
rate constant of GTP inactivation
Hill binding coefficient of DAG to PLC
Hill binding coefficient of GTP to PLC
Hill exponent – DAG binding to PLC
Hill exponent – GTP binding to PLC
rate of DAG formation by PLC
rate constant for breakdown of DAG
rate of background release of DAG
Hill binding coefficient of InsP3 to calcium membrane
rate constant of calcium uptake
background production rate of calcium
Binding coefficient for DAG to PKC
Binding coefficient for calcium to PKC
1
300
5
50
2
4
30000
240
18
5
24
4800
40
100
min–1
min–1
nM
nM
unitless
unitless
nM/min
min–1
nM/min
nM
min–1
nM/min
nM
nM
TABLE II: Parameter values (El-Masri and Portier, 1999) for the MAPK activation section of the
mechanistic model. Values denoted with a (*) have been modified from the original sources.
Parameter
Function
Value
units
RPKCi
Vraf
Kdc
Vmapkk
Vdmapkk
Vmapk
Vdmapk
α1
α2
α3
background activation of PKC
Phosphorylation rate constant of Raf
rate constant for deactivation of Raf via RMAPK
rate constant for phosphorylation of MAPKK
rate constant for MAPKK* deactivation
rate constant for phosphorylation of MAPK
rate constant for MAPK* deactivation
ratio used to determine Raf steady state
ratio used to determine MAPKK steady state
ratio used to determine MAPK steady state
0.006
1.6
0.04
0.02
2
0.01
1.1
0.2247*
0.0099*
0.009*
unitless
min–1
min–1
min–1
min–1
min–1
min–1
unitless
unitless
unitless
blue line in Figure 5. The system was studied under different stimulus levels to examine the presence of an all-or-none activity. Figure 4 was produced by running the model with a constant stimulus (of 50 nM or 800
nM) present from time t = 1000 min to t =1200 min. The first 1000 simulated minutes were run stimulus-free in order to attain steady state
response folds of unity prior to the introduction of the stimulus. Note
that a dose of 50 nM leads to negligible activation above background (less
than 1.01 fold induction), while the fold level of phosphorylated MAPK
elevates in the presence of the 800 nM stimulus. We believe that the 50
nM level does not elicit an elevation of phosphorylated MAPK because
there is not enough stimulus present to overcome the negative feedback
built into the model. This negative feedback, inherent to the system,
keeps the low dose exposure response close to the steady state levels.
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3
800 nM stimulus
50 nM stimulus
Fold increase − MAPK−P
2.5
2
1.5
1
0.5
0
0
500
1000
1500
time (min)
FIGURE 4: Time course results for MAPK fold induction, with stimulus present from t=1000 to
t=1200 minutes. Note that the model was run for 1000 simulated minutes (stimulus free) to establish
a steady state.
4.5
Fold increase − MAPK−P
4
3.5
Vmapk=.005
Vmapk=0.01
(original value)
Vmapk=0.02
3
2.5
2
1.5
1
0
10
2
4
10
10
stimulus concentration nM
6
10
FIGURE 5: Sensitivity comparison for the parameter Vmapk , displayed as MAPK fold induction at
t=1100 min. The solid blue line represents model results using the original parameter value.
The dose response data given in Figure 5 was generated by running
the model with varying stimulus levels, and plotting the fold induction
level for a given time (t = 1100) at each level. As the stimulus level increases, the fold induction level reaches a higher steady state. Notice that fold
levels are close to the unstimulated steady state at doses less than 100 nM.
Again, this may be attributed to the negative feedback in the system.
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4.5
Fold increase − MAPK−P
4
Kgf=1
3.5
Kgf=2
3
2.5
Kgf=4
Kgf=0.5
2
1.5
Kgf=0.1
1
0
10
2
4
10
10
stimulus concentration nM
6
10
FIGURE 6: Sensitivity analysis for KGF , displayed as MAPK fold induction at t=1100 min. KGF = 1 represents the model results using the original parameter value.
In order to better understand the role that the parameters play in the
model, a visual sensitivity analysis was conducted. Parameter values were
varied by a factor of 2, and the ensuing changes in the dose response
curve of phosphorylated MAPK was examined. Figures 5 and 6 display
the sensitivity analysis results for a few selected parameters. Vmapk (in
Figure 5) represents the rate constant for MAPK activation. Examination
of this figure shows that the parameter (analogous to other activation
parameters) has a marked effect on the maximum fold induction level of
phosphorylated MAPK reached. The results shown in Figure 5 are representative of the results for the other activation parameters (Vmapk k, Vraf ,
etc.). The parameter whose results are shown in Figure 6, KGF , represents
the first order rate constant which governs the rate at which the stimulus
binds to the receptor, thus activating the cascade. We notice that changes
in the value of KGF shifts the dose response curve horizontally. These horizontal shifts impact the lowest levels of stimulus at which MAPK activation increases over background. It should be noted that this is the only
parameter in the system which exhibits such an effect in the sensitivity
analyses. Thus, variations of the value of KGF reflect an increased or
decreased sensitivity of the cell to the stimulus.
In this study, we consider the percent activation of MAPK in tissue.
Our mechanistic model predicts fold induction of activated MAPK in
cells, which is converted to percent activation. Then, the steady state level
of unity will correspond to zero percent activation, while the maximum
fold induction corresponds to 100% activation. This will normalize the
fold induction results. Thus, the variation in the maximum amplitude
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attained in Figure 5 would be rendered insignificant, as each maximum
would correspond to 100% activation. Therefore, the various activation
parameters are of little concern to us, allowing us to concentrate our
efforts on the KGF parameter.
3.2 Monte Carlo Simulation
The results produced by the mechanistic model can be considered
the MAPK fold level for a single cell, or it can represent the tissue-level
response exhibited under the assumption that each cell demonstrates a
graded response. In order to implement this model with the all-or-none
assumption (with single cells either responding or not responding), we
assume that the model results are representative of a single cell, and we
employ a Monte Carlo simulation, as follows.
We first assume that the KGF parameter varies from cell-to-cell and is
log-normally distributed (with mean µK and standard deviation σK ).
GF
GF
Assigning variability to the parameter KGF can be supported by several
mechanisms. First, it is possible that the number of receptors varies from
cell to cell, causing some cells to be more sensitive than others. Perhaps
there is a structural difference in the receptors, allowing some receptors to
have a higher affinity to the stimulus. Finally, there could be a difference in
the concentration of stimulus reaching the cells. While these factors are
speculative, they do provide motivation for varying KGF in the model.
The model is evaluated for a sampling of KGF in the distribution. If
the fold level of phosphorylated MAPK is elevated (as illustrated by the
example of the 800 nM results in Figure 4), then the cell is considered to
be on. If the fold level remains equal to 1, then the cell is off. The percent of activated MAPK is defined to be the percentage of activated cells
(number of active cells ÷ total number of cells × 100).
The Monte Carlo simulation of the MAPK model was run with an
arbitrary number of n=100 and optimized to find the logarithmic mean
and standard deviation for KGF , using the data from Ferrell & Machleder
(1998). We found optimal parameters µK = 0.77 (1/min) and σK = 2.35
GF
GF
(1/min), resulting in a least squares cost function value of 832. The
resulting MAPK activation is illustrated by the solid blue line in Figure 7
in comparison to the data.
Upon examination of these results in Figure 7, we notice that the low
dose (< 1 nM) results are not in accord with our expectations. As evidenced by the data, we would expect that low doses would have little
effect on MAPK activation. These initial results, however, suggest that at
very low doses, 8% of the cells exhibit MAPK activation. By assuming that
the parameter KGF has a log normal distribution, we subject our model to
outliers which have the potential to generate a few values that are much
larger or much smaller than the majority of the others. We believe that
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100
MC MAPK approx (unbounded)
MC MAPK approx (bounded)
data
90
80
70
60
50
40
30
20
10
0
−1
10
0
10
1
2
10
10
3
10
4
10
FIGURE 7: Approximation of MAPK activation computed using the Monte Carlo simulation with
unbounded (solid blue line) and bounded (dashed green line), log normally distributed parameter
KGF compared to data reported by Ferrell & Machleder (1998).
this is the cause for the discrepancy between simulated results and data at
low doses.
If we instead consider a log normal distribution (using the optimal
parameters µK = 0.77 (1/min) and σK = 2.35 (1/min)) that is boundGF
GF
ed above and below (i.e. KGF 僆 [0.05, 40]), then our model produces the
results depicted by the dashed green line in Figure 7. The corresponding
cost function value is now J = 701, implying that the fit is improved, especially when considering the low dose data. The need to limit the possible
values of KGF is not surprising, as physiological parameters need lie in a
range that is biologically feasible.
3.3 Further Investigation
Bhalla et al. (2002) present a more complex model of the MAPK cascade. Their model is comprised of 105 binding and enzyme reactions,
modeled by a system of 90 differential equations. The model was developed to examine the role of MAP kinase phosphatase on the MAPK cascade. Phosphorylation of the MAPKs occurs as a dual step process, and
forward and backward feedback loops are inherent in the model.
While there are many differences between the Bhalla et al. (2002)
model and our MAPK model, there are also several similarities. An end
result of both models is the prediction of phosphorylated MAPK. Also,
activation of the cascade is initiated by a binding process in each model.
That is, there is a binding coefficient in the Bhalla et al. (2002) model that
is analogous to the KGF parameter of our model. A brief sensitivity analy358
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0.18
K=0.5
0.16
Phosphorolated MAPK
0.14
0.12
K=1
0.1
0.08
K=0.1
0.06
K=2
0.04
K=0.05
0.02
0
−1
10
0
10
1
2
3
10
10
10
stimulus concentration (nM)
4
10
5
10
FIGURE 8: Sensitivity analysis for binding parameter in the Bhalla et al. (2002) model.
sis was conducted, varying the value of the initial binding parameter, and
the results are presented in Figure 8. These results appear to be similar to
those presented in Figure 6 for the El-Masri and Portier (1999) Model.
To further examine our method for incorporating the all-or-none
response into a mechanistic model, we employ the previously described
Monte Carlo method in conjunction with the Bhalla et al. (2002) model.
We assume that the binding coefficient from the Bhalla et al. (2002)
model has the same distribution that we found to be optimal for our
model. Thus, the binding coefficient is log-normally distributed with a
mean of 0.77 (1/min) and a standard deviation of 2.35 (1/min), and is
contained in the interval [0.05, 40]. The Bhalla et al. (2002) Model is evaluated with each value in a sampling (n = 100) of the distribution. If the
level of phosphorylated MAPK increases above background, then the cell
is considered to be “on”. The percent of activated MAPK is again defined
to be the percentage of activated cells. Results generated by the Bhalla et
al. (2002) model are shown in comparison to the results generated by our
model, as well as the data from Ferrell & Machleder (1998), in Figure 9.
The least squares cost function value associated with the Bhalla et al.
model is 869. Examining Figure 9, we see that the use of our optimal distribution parameters with the Bhalla et al. (2002) model produces similar
results (which could be improved by finding optimal parameters for it,
rather than using parameters optimized for the El-Masri & Portier (1999)
model). This suggests that our Monte Carlo method for incorporating
the all-or-none response can be employed in conjunction with different
mechanistic models.
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100
90
Bhalla model
El−Masri model
Ferrell data
% − MAPK phosphorylation
80
70
60
50
40
30
20
10
0
0
10
2
10
progesterone (nM)
4
10
FIGURE 9: Approximation of MAPK activation under the assumption of a cellular all or none
response, computed using the Monte Carlo simulation in conjunction with the Bhalla et al. (2002).
Model compared to results from the El-Masri model (El-Masri and Portier, 1999) and data reported
by Ferrell & Machleder (1998).
TABLE III: Optimal parameters and corresponding cost values
Model Description
Optimal Parameters
Hill Modela
Lognormal distribution
El-Masri, All-or-None, unboundedb
El-Masri, All-or-None, boundedb
K = 31.25
µ = –3.43, σ = 2.78
µKGF = 0.77, σKGF = 2.35
µKGF = 0.77, σKGF = 2.35,
KGF 僆 [0.05 40]
µK = 0.77, σK = 2.35,
K 僆 [0.05 40]
KGF = 23.98
K = 5.66
Bhalla, All-or-None, boundedc,*
El-Masri, Gradedb
Bhalla, Gradedc
OLS Cost Value
1519
821
832
701
869
2580
3127
a
model from Ferrell and Machleder (1999); bmodel from El-Masri and Portier (1999); cmodel
from Bhalla et al. (2002); *parameters for the Bhalla, All-or-None model were not optimized
A comparison of the least squares cost function values, presented in
Table III, shows that the mechanistic models, modified to incorporate the
all-or-none MAPK response, provide a better fit to the data than the
empirical models do. The benefit of using these mechanistic models is
that they can incorporate more information about the biological processes. They also help to provide an understanding of key biological events
contributing to the shape of the dose response curve. In this example, we
found that the initial receptor-ligand binding rate contributes to the cell’s
ability to exhibit an all-or-none MAPK activation response, while down360
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Luke et al.: Modeling all-or-none MAPK activation
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100
90
Bhalla graded
El−Masri graded
Ferrell Data
% − MAPK phosphorylation
80
70
60
50
40
30
20
10
0
0
10
2
10
Progesterone (nM)
4
10
FIGURE 10: MAPK activation results predicted by the Bhalla et al. (2002) Model and the El-Masri &
Portier (1999) model for a graded cellular response compared to data reported by Ferrell &
Machleder (1998).
stream activation parameters contribute more to the magnitude of activation (see Figure 5).
3.4 All-or-None Cellular Response vs. Graded Cellular Response
When coupled with the Monte Carlo simulation, the El-Masri &
Portier (1999) and Bhalla et al. (2002) MAPK models can predict MAPK
activation, assuming an all-or-none cellular response. These models, without the Monte Carlo simulation, can also be used to predict overall MAPK
activation if we assume that the cellular response is graded. Under the
assumptions that each cell exhibits an intermediate graded response to
increasing stimulus and that the tissue level response is an average of the
cellular responses, then the results generated by these models (without
the Monte Carlo simulation) would represent tissue level MAPK activation with graded cellular responses.
Again using the data reported in Ferrell & Machleder (1998), we wish
to compare the accuracy of the graded response model with our implementation of the all-or-none model. Since the El-Masri & Portier (1999)
and Bhalla et al. (2002) models predict fold induction and actual units of
MAPK activation, respectively, the results of both models are linearly
scaled so that the maximum induction relates to a 100% activation level
and basal levels become 0% activation. Results generated using the models with a graded cellular response are shown in Figure 10. Optimization
of the graded models to the Ferrell & Machleder (1998) data yields a
binding parameter of 5.66 sec−1 for the Bhalla et al. (2002) model and a
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KGF value of 23.98 min−1 for the El-Masri & Portier (1999) model. The optimal parameter values produce cost function values of 3127 and 2580,
respectively. These cost function values are considerably higher than those
computed using the models with an all-or-none approach, suggesting that
the ability of the models to predict MAPK activation is enhanced by assuming that cellular MAPK responses occur in an all-or-none fashion.
4. DISCUSSION
Many signaling biological processes (nuclear transcriptional receptors, kinase or phosphatase cascades, G-coupled protein receptors, etc.)
are mediated through cascading processes leading to cellular events (e.g.,
commitment to cell division, cell differentiation, and phenotypic alterations) that eventually translate to tissue and/or systemic responses.
These processes have been typically examined and modeled at a tissue
level, but technology has facilitated the examination of these events at the
cellular level.
Experimental evidence of either a tissue-based graded or cellular
based all-or-none response was investigated using Xenopus oocyte maturation in the presence of progesterone (Ferrell and Machleder, 1998). For
populations of Xenopus oocytes, the progesterone concentrations leading
to maturation were distributed fairly broadly. Data for the tissue response
were modeled empirically, and it was found that the data were fit well by
a Hill equation with Hill coefficient of approximately 1, implying a graded response. However, Hill coefficients for individual oocytes were found
to be between 20 and 30 (Ferrell and Machleder, 1998). These high values indicate an all-or-none switch for maturation in any individual oocyte.
While several models of varying levels of complexity have been developed for the MAPK signaling cascade (see Orton et al. (2005) for a
detailed review), few, if any, of the published models account for the allor-none response. In this paper, we utilized biological knowledge of
underlying mechanisms for individual oocyte maturation using a detailed
description of the MAPK pathway to investigate the impact of all-or-none
behavior of individual cells on the overall oocyte population doseresponse relationship. We have presented a Monte Carlo simulation
approach which can be used to incorporate the all-or-none behavior into
MAPK modeling efforts. Using this approach with our MAPK model (ElMasri and Portier, 1999), we recapitulated experimental data presented
by Ferrell & Machleder (1998).We further investigated our method with
a more complicated MAPK model developed by Bhalla et al. (2002). Both
models, in conjunction with the Monte Carlo simulation, generated
results which are consistent with the Ferrell &Machleder (1998) data.
Furthermore, in both models, it was determined that the critical parameter for incorporating the all-or-none response was the initial binding
rate parameter. Additionally, we compared model results generated with
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the assumption that cellular responses occur in a graded manner with
results compiled under the assumption that an all-or-none cellular
response is present. We found that the model incorporating the all-ornone response provides a better fit to MAPK activation data, then our
implementation of the graded response model.
The examined importance of the initial binding rate parameter suggested that receptor binding was the driving mechanism for the presence
of a threshold-like level of the ligand to increase the activity of MAPK
leading to oocyte maturation. As noted earlier, allowing the binding rate
to vary across cells can potentially be explained by several factors including availability of the ligand, affinity of the ligand to the receptors, structure of the receptor and quantity of available receptor sites. Our modeldriven investigation provides an example where mechanistic information
relating to the MAPK pathway’s role in cell maturation was organized
based on biological representation of important processes to investigate
dose-response at the cellular level. This mechanistic description of the
cellular-response can be introduced into a larger computational model
where receptor binding via the MAPK pathway is part of a general mode
of action for toxicological effects (such as cellular proliferation) to
describe an overall biologically based dose-response (BBDR) model.
Examples of signal-mediated cascading biological motifs are plentiful.
These signaling pathways have been described experimentally and computationally either as “all-or-none” phenomena or as graded responses.
For example, neuronal cell firing in response to neurotransmitters is an
all-or-none phenomenon, where as biochemical changes in response to
activation of the steroid receptor superfamily, such as the progesterone
receptor, have been described as graded responses. As technology has
advanced, many of the observed graded responses appear due to an
underlying all or none response at the cellular level. For example, the
transcriptional changes due to activation of hepatic nuclear receptors,
such as the Ah receptor or the Constitutive Androstane Receptor, once
thought to yield graded responses at the tissue level, are due to dose
dependent changes in the number of fully activated cells (Buhler et al.,
1992; Tritscher et al., 1992; Fox et al., 1993). In many cases, these cascading biological motifs have nonlinear dose response behaviors for endogenous ligands and for exogenous toxicants, acting as switches with “all-ornone” responses over a narrow range of concentrations (Andersen et al.,
2002). In addition, these induction responses appear to occur at the level
of individual cells, thus a 50% response of the liver for induction does not
represent 50% induction in each individual cell. Instead, half of the cells
are fully induced and half are unaffected, while cells “switch” from one
phenotypic state to another (Andersen, 2002). The difference between a
graded response and a switch-like behavior may have significant implications for understanding dose response relationships for endogenous hormones or exogenous toxicants.
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Biologically based dose response models are used to provide simulations that link molecular events to tissue and organ level responses such
as morphogenesis and carcinogenesis. These models, such as the one we
used for MAPK activation, accounting for the biological switches, would
improve our understanding of these important biological processes.
Similarly, BBDR models may prove useful in linking molecular, cellular,
tissue and organ responses which will predict adverse health effects.
These BBDR models must account for normal control of the signaling
motifs and for the perturbations by toxic compounds (Hoel et al., 1983;
Andersen, 2002). As the biological basis of these switches becomes unraveled and incorporated into computational models, these BBDR models
should eventually serve to improve risk assessment with a variety of toxicants exhibiting receptor-based modes of action.
ACKNOWLEDGEMENTS
This manuscript has been reviewed in accordance with the policy of
the National Health and Environmental Effects Research Laboratory,
U.S. Environmental Protection Agency, and approved for publication.
Approval does not signify that the contents necessarily reflect the views
and policies of the Agency, nor does mention of trade names or commercial products constitute endorsement or recommendation for use.
The authors would like to thank Dr. Karen Yokley for valuable insight
provided during the completion of this project. Additionally, the authors
express appreciation towards Dr. Marina Evans, Dr. Rory Conolly, and Dr.
Linda Birnbaum for helpful comments during the preparation of this
manuscript.
APPENDIX A
A.1 Model Equations
Equations for PKC activation (Cuthbertson and Chay, 1991)
dGTP = K · GF – GTP · hg · RPKC
GF
dt
RPLC =
(GTP/Kg)m
(DAG/Kd)n
·
(DAG/Kd)n + 1 (GTP/Kg)m + 1
dDAG = K
· RPLC – hd · DAG + ld
DAG
dt
dCa = (InsP3/Ks)n – hc · Ca + lc
dt
(InsP3/Ks)n + 1
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RPKC =
DAG ·
Ca
+ RPKCi
KP + DAG
Ca + KC
Equations for MAPK activation (El-Masri and Portier, 1999)
dRaf * = V · RPKC · Raf – K · Raf * · RMAPK
raf
dc
dt
dRaf = – dRaf *
dt
dt
dMAPKK * = V
· MAPKK · RRaf – Vdmapkk · MAPKK *
mapkk
dt
dMAPKK = – dMAPKK *
dt
dt
dMAPK * = V
· MAPK · RMAPKK – Vdmapk · MAPK *
mapk
dt
dMAPK = – dMAPK *
dt
dt
RRaf =
RMAPKK =
RMAPK =
Raf *
α1 · Rafi
MAPKK *
α2 · MAPKKi
MAPK *
α3 · MAPKi
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