Bistability, Stochasticity, and Oscillations in the Mitogen-Activated Protein Kinase Cascade

Article (PDF Available)inBiophysical Journal 90(6):1961-78 · April 2006with36 Reads
DOI: 10.1529/biophysj.105.073874 · Source: PubMed
Abstract
Signaling pathways respond to stimuli in a variety of ways, depending on the magnitude of the input and the physiological status of the cell. For instance, yeast can respond to pheromone stimulation in either a binary or graded fashion. Here we present single cell transcription data indicating that a transient binary response in which all cells eventually become activated is typical. Stochastic modeling of the biochemical steps that regulate activation of the mitogen-activated protein kinase Fus3 reveals that this portion of the pathway can account for the graded-to-binary conversion. To test the validity of the model, genetic approaches are used to alter expression levels of Msg5 and Ste7, two of the proteins that negatively and positively regulate Fus3, respectively. Single cell measurements of the genetically altered cells are shown to be consistent with predictions of the model. Finally, computational modeling is used to investigate the effects of protein turnover on the response of the pathway. We demonstrate that the inclusion of protein turnover can lead to sustained oscillations of protein concentrations in the absence of feedback regulation. Thus, protein turnover can profoundly influence the output of a signaling pathway.
Bistability, Stochasticity, and Oscillations in the Mitogen-Activated
Protein Ki nase Cascade
Xiao Wang,* Nan Hao,
y
Henrik G. Dohlman,
y
and Timothy C. Elston
z
*Department of Statistics and Operations Research,
y
Department of Biochemistry and Biophysics, and
z
Department of Pharmacology,
University of North Carolina at Chapel Hill, Chapel Hill, North Caroli na
ABSTRACT Signaling pathways respond to stimuli in a variety of ways, depending on the magnitude of the input and the
physiological status of the cell. For instance, yeast can respond to pheromone stimulation in either a binary or graded fashion.
Here we present single cell transcription data indicating that a transient binary response in which all cells eventually become
activated is typical. Stochastic modeling of the biochemical steps that regulate activation of the mitogen-activated protein kinase
Fus3 reveals that this portion of the pathway can account for the graded-to-binary conversion. To test the validity of the model,
genetic approaches are used to alter expression levels of Msg5 and Ste7, two of the proteins that negatively and positively
regulate Fus3, respectively. Single cell measurements of the genetically altered cells are shown to be consistent with
predictions of the model. Finally, computational modeling is used to investigate the effects of protein turnover on the response of
the pathway. We demonstrate that the inclusion of protein turnover can lead to sustained oscillations of protein concentrations
in the absence of feedback regulation. Thus, protein turnover can profoundly influence the output of a signaling pathway.
INTRODUCTION
The pheromone response of yeast is one of the best-
characterized signaling pathways (1). Much is known about
the proteins that transmit the pheromone signal, as well as
about the mechanisms by which events at the cell surface are
linked to subsequent biochemical changes in the cytoplasm
and nucleus. A diagram of the pathway is given in Fig. 1 A.
Activation of the pathway is initiated by binding of mating
pheromones to specific cell surfa ce receptors, and ends with
the fusion of a- and a-haploid cell types to form an a/a-
diploid (mating). In the a-cell type, signaling is initiated
by binding of the pheromone a-factor to its receptor Ste2.
Receptor activation in turn leads to the exchange of GDP for
GTP on the G-protein a -subunit Gpa1, and subsequent dis-
sociation from the G-protein bg-subunit dimer composed of
Ste4 and Ste18. The signal is then transmitted and amplified
through effector proteins that bind to Gbg. A major target of
the Gbg-subunits is a cascade of four protein kinases that
begins with Ste20 and ends with the mitogen-activated pro-
tein kinase (MAPK) Fus3. Ste20 phosphorylates and acti-
vates Ste11, which phosphorylates and activates Ste7, which
in turn phosphorylates and activates the MAPK Fus3 (on
Tyr-182 and Thr-180) . Fus3 has a numbe r of substrates
including the transcription factor Ste12, which is responsible
for induction of genes required for mating. Inactivation of
signaling requires that Fus3 is dephosphorylated on both Tyr
and Thr. Both sites are recognized by the dual-specificity
phosphatase Msg5. Tyr is also dephosphorylated by at least
two other phosphatases, Ptp2 and Ptp3. In the absence of
Fus3, however, a closely related MAP kinase Kss1 can carry
out most of the functions of Fus3 (1).
Most of the components of the pheromone response path-
way have been identified genetically, through the isolation of
mating-defective or sterile gene mutations. Further genetic,
biochemical, and molecular biological analysis of the path-
way revealed the order of each signaling event, and has
established many basic principles of G-protein and MAPK
signaling relevant to all eukaryotes (2). Having now deter-
mined the essential components and events in G-protein-
coupled receptor signaling, an emer ging goal is to develop
mathematical models that describe their behavior over time.
Cell signaling pathways can vary their response to a
stimulus in a variety of different ways. At the receptor level,
pathway activation is determined by the number of liganded
receptors and therefore is proportional to the concentration of
the input stimulus. This graded binding event can propagate
through the pathway and produce a graded transcriptional
response that is proportional to the input signal. However,
many pathways convert a graded input instead into a binary
transcriptional response (3–5). A binary response refers to an
all-or-none situation in which the probability of an individual
cell responding to a stimulus is proportional to the strength of
the signal. Binary responses are often attributed to multiple
steady states that arise from feedback regulation (4,6). How-
ever, graded and binary outputs are not the only time-depen-
dent responses of which signaling pathways are capable.
Sustained oscillations and more complicated dynamics are
also possible.
In yeast, pheromone signaling can produce either a graded
or binary transcriptional response depending on the dose
of pheromone, the time of treatment, and the intracellular
activation event being measured. A binary response may be
appropriate in some physiological situations but not in
Submitted September 6, 2005, and accepted for publication November 21,
2005.
Address reprint requests to Timothy C. Elston, Dept. of Pharmacology,
University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7365.
Tel.: 919-962-8655; E-mail: telston@amath.unc.edu.
Ó 2006 by the Biophysical Society
0006-3495/06/03/1961/18 $2.00
doi: 10.1529/biophysj.105.073874
Biophysical Journal Volume 90 March 2006 1961–1978 1961
others. For instance in yeast, pheromones initiate a process
leading to mating, an inherently irreversible process where
an all-or-none decision is appropriate. Binary outputs are
also appropriate durin g cell division, cell differentiation, and
cellular apoptosis. Thus, establishing the mechanisms by
which the graded-to-binary conversion is accomplished is a
fundamental problem in cell biology. Here we seek to iden-
tify components of the pheromone response pathway that
mediate the graded-to-binary conversion and to uncover the
mechanism by which this conversion is accomplished. We
employ an approach that combines experimental analysis
with computational modeling. Data from fluorescence-based
transcriptional induction assays in single cells are used as the
experimental basis for a stochastic model of the biochemical
steps that regulate MAPK activation. The mathematical
model is motivated by the theoretical work of Markevich
et al. (7) on multisite phosphorylation of protein kinases.
Computational analysis is used to validate the model and
generate testable hypotheses, which are in turn confirmed
experimentally.
Our stochastic model ing reveals that small changes in
protein abundances can have large effects on MAPK acti-
vation. Additionally, data on protein turnover suggests that
protein degradation plays an important role in regulating the
pheromone pathway. Pheromone-stimulated degradation has
been documented previously for the receptor (Ste2) (8), a
regulator of G-protein signaling (Sst2) (9), components of
the effector kinase cascade (Ste11, Ste7) (9–11), and the
transcription factor (Ste12) (12). However, the functional
consequences of accelerating protein turnover have not been
well characterized. These observations motivate extending
the computational model to include protein turnover. We
demonstrate that the qualitative behavior of the pathway
depends on the mechanism through which protein degrada-
tion occurs. In particular, we show that protein degradation
can generate sustained oscillations in protein concentrations.
Finally, stochastic modeling is used to demonstrate that
biochemical fluctuations increase the param eter range over
which the oscillations occur, and that the oscillations can
generate an apparent binary response in flow cytometry
experiments.
MATERIALS AND METHODS
Model description
In this study, we focus on the MAPK portion of the pheromone response
pathway. The biochemical steps involved in the regulation of the MAP
kinase Fus3 are shown in Fig. 1 B. Transmission of the intracellular signal
involves phosphorylation of Fus3 by Ste7. Pathway activation can also occur
via the MAPK Kss1. However, in this study we do not distinguish between
Fus3 and Kss1. Ste7 is a dual-specificity kinase that modifies Fus3 at Thr-
180 and Tyr-182 sites and stimulates its catalytic activity. Msg5 is a dual-
specificity phosphatase that inactivates Fus3. Ptp2 and Ptp3 are tyrosine
phosphatases that also inactivate Fus3. For simplicity, we do not distinguish
between these three proteins. As shown in Fig. 1 B, we assume a distributive
kinetic mechanism for the dual phosphorylation and dephosphorylation
reactions (7,13). A distributive mechanism refers to one in which the kinase
and phosphatase release the monophosphorylated substrate intermediate and
a second interaction is required to generate the final product. We use [KK],
[K], and [P] to denote the concentrations of Ste7 (MAPK kinase), Fus3
(MAPK), and Msg5 (phosphatase), respectively. [Kp] and [Kpp] denote the
concentrations of the singly phosphorylated and doubly phosphorylated
forms of Fus3, respectively. Protein/protein complexes are denoted with a
dot (). For example, [KKK] denotes the concentration of the Fus3/Ste7
complex. The biochemical reactions used in the stochastic and rate equation
models are given in Appendix A. Where available, we use experimentally
measured values for the model parameters. For the parameters that have not
been measured, we use biologically realistic values. A summary of the
model parameters and the values used in the simulations is given in Table 1.
Computational methods
Stochastic and deterministic modeling
Biochemical reactions are inherently random processes. The Gillespie algo-
rithm can be used to generate single realizations of biochemical networks
(14). We have implemented an efficient version of the Gillespie algorithm in
our software package BioNetS (15). Most of the stochastic modeling was
done using BioNetS. The BioNetS scripts used to generate the results are
available upon request.
FIGURE 1 (A) The pheromone signaling pathway. Shown are the
pheromone receptor (Ste2), G-protein a-, b-, and g-subunits (Gpa1, Ste4,
Ste18), the Regulator of G-protein signaling (RGS, Sst2) which accelerates
Gpa1 GTPase activity, effector kinases including the MAPKKK kinase
(Ste20), MAPKK kinase (Ste11), MAPK kinase (Ste7), MAP kinases (Kss1,
Fus3), and nuclear transcription factor (Ste12). Three of the kinases bind to a
kinase scaffold protein (Ste5). Several other effectors and regulatory
components are not shown, for clarity. Pheromone-dependent transcriptional
induction of SST2 represents a negative feedback loop. (B) A schematic
diagram of Fus3 regulation. Fus3 activation and deactivation are assumed to
occur through a distributive kinetic mechanism in which two collisions with
Ste7 (MAPK kinase) and Msg5 (MAPK phosphatase) are required. The
MAPK Kss1 also activates the pheromone response pathway, and the
phosphatases Ptp2 and Ptp3 also inactivate kinase activity. However, to
simplify the models the kinase and phosphatase activities of these proteins
are not distinguished from those of Fus3 and Msg5.
1962 Wang et al.
Biop hysical Journal 90(6) 1961–1978
When protein abundances are sufficiently large, the law of mass action
can be used to construct rate equations for the concentrations of the various
chemical species. Here we present a very simple example to illustrate the
connection between the stochastic models and the rate equations. Consider
the following two biochemical reactions:
B
E
*
g
K
d
K
K; (1)
K 1 KK
E
*
k
1
k
K KK: (2)
When Eq. 1 proceeds in the forward direction, a new molecule of Fus3 (K)is
synthesized. The reverse reaction represents the degradation of a single Fus3
molecule. Eq. 2 represents the binding and dissociation of a Fus3 molecule
(K) with a Ste7 molecule (KK). In the stochastic models the system is
described in terms of molecule numbers. Therefore the rate constants, d
k
, g
k
,
k
1
, and k
, all have units of inverse time. The second-order rate constant k
1
is inversely proportional to the effective volume V. That is, k
1
;k9
1
=V,
where the constant k9
1
is independent of the volume and has units of
[concentration]
1
[time]
1
. The effective volume might be that of a yeast cell
or smaller if the proteins are spatially restricted in their localization. Let
N
K
(t), N
KK
(t), and N
KKK
(t) denote the number of molecules of the proteins
Fus3, Ste7, and the complex Fus3Ste7, respectively, at time t. Note that in
the example given by Eqs. 1 and 2, the total number of Ste7 molecules,
N
KK
1N
KKK
, remains constant in time and that Fus3 cannot be degraded
when it is in a complex with Ste7. Concentrations are formed by dividing the
number of molecules by the volume. To convert to molar concentration, the
molecule number also must be divided by Avogadro’s number N
A
. For
example, [K] ¼ N
K
/(VN
A
) is the Fus3 concentration. Using the law of mass
action, we can write rate equations for the concentrations:
d½K
dt
¼
g
K
VN
A
d
K
½K 1 k
½K KKk
1
VN
A
½K½KK; (3)
d½KK
dt
¼ k
½K KKk
1
VN
A
½K½KK: (4)
The conservation of total Ste7 number can now be written as d/dt([KK] 1
[KKK]) ¼ 0. Therefore, once the total Ste7 concentration, [KK]
T
, has been
specified, [KKK] can be found from the relation [KKK] ¼ [KK]
T
[KK].
Equations 3 and 4 represent a macroscopic description of the process,
because they ignore biochemical fluctuations. The macroscopic limit is
reached as the molecule numbers and volume become large with their ratios
(concentrations) remaining finite. Fluctuations in concentration typically
scale like 1/V
1/2
, so that Eqs. 3 and 4 are valid in limit of large volume. When
investigating the range of validity of these equations through comparisons
with stochastic simulations, the synthesis rates must be scaled with the
volume, whereas the second-order rate constants must scale inversely with
volume. This scaling ensures that Eqs. 3 and 4 remain unchanged as the
volume is increased. The numerical simulations of the rate equations were
carried out in MatLab (The MathWorks, Natick, MA) and the bifurcation
analysis was done in MatLab and XPPAUT (16).
Protein degradation and synthesis
We investigate several different models of protein degradation and
synthesis.
Case I—conservation of enzyme concentrations. The simplest model
assumes that proteins are neither degraded nor synthesized. In this case the
total concentration of all three enzymes is constant in time. That is,
d½K
T
dt
¼
d
dt
ð½K 1 ½Kp 1 ½Kpp 1 ½K KK 1 ½Kp KK
1 ½Kp P 1 ½ðKp PÞ 1 ½Kpp P 1 ½K PÞ ¼ 0;
(5)
TABLE 1 Model parameters and the values used in the simulations
Parameter Description Value (Cases I, II, III) Reference
k
1
Association rate constant K/KK binding 0.00275, 0.0011, 0.0198 (7)
k
1
Dissociation rate constant for KKK 2.5, 1, 1 (7)
k
2
k
cat
for first phosphorylation event 0.025, 0.01,0.01 (7)
k
3
Association rate constant for Kp /KK binding 0.00445, 0.00178, 0.03204 (7)
k
3
Dissociation rate constant for KpKK 2.5, 1, 1 (7)
k
4
k
cat
for second phosphorylation event 37.5, 15, 15 (7)
h
1
Association rate constant Kpp/P binding 0.00625, 0.0025, 0.045 (7)
h
1
Dissociation rate constant for KppP 2.5, 1, 1 (7)
h
2
Rate constant for phosphate release of the phosphotyrosine 0.23, 0.092, 0.092 (7)
h
3
Dissociation rate constant for KpP 2.5, 1, 1 (7)
h
3
Association rate constant for Kp/P 0.0014, 5.6e-4, 0.0099 (7)
h
4
Association rate constant for Kp/P binding 0.0014, 5.6e-4, 0.0099 (7)
h
4
Dissociation rate constant for (KpP) 2.5, 1,1 (7)
h
5
Rate constant for phosphate release of the phosphothreonine 1.25, 0.5,0.5 (7)
h
6
Dissociation rate constant for KP 0.215, 0.086, 0.086 (7)
h
6
Association rate constant for K/P binding 1.525e-4, 6.1e-5, 0.099 (7)
g
k
Synthesis rate of Fus3 N/A, 0.9, 0.095 Determined from abundance and degradation rate
g
kk
Synthesis rate of Ste7 N/A, 0.33504, 0.016 Determined from abundance and degradation rate
g
p
Synthesis rate of Msg5 N/A, 0.828, 0.023 Determined from abundance and degradation rate
d
k
Degradation rate of Fus3 N/A,1e-4, 1e-4 Experiment
d
kk
Degradation rate of Ste7 N/A, 3.2e-4, 3.2e-4 (9)
d
p
Degradation rate of Msg5 N/A, 4.6e-4, 4.6e-4 Experiment
N
KT
Molecular abundance of Fus3 9000, N/A, N/A (18)
N
KKT
Molecular abundance of Ste7 900, N/A, N/A (18)
N
PT
Molecular abundance of Msg5 1800, N/A, N/A (7,18,22)
All rate constants have units of s
1
. K denotes Fus3 (MAPK), KK denotes Ste7 (MAPKK), and P denotes Msg5 (phosphatase).
Computational Analysis of MAPK 1963
Biophysical Journal 90(6) 1961–1978
d½KK
T
dt
¼
d
dt
ð½KK 1 ½K KK 1 ½Kp KKÞ ¼ 0; (6)
d½P
T
dt
¼
d
dt
ð½P 1 ½Kp P 1 ½Kpp P 1 ½ðKp PÞ
1 ½K PÞ ¼ 0;
(7)
where the subscript T stands for total. The chemical species (KpP) in Eq. 5
results from the assumption that the phosphotyrosine is dephosphorylated
first (7,17). That is, the chemical species (KpP) indicates the phosphatase
attacking the phosphothreonine, whereas KpP is the product after
phosphotyrosine has been dephosphorylated. The assumption of conserved
enzyme concentrations is generally justified by the observation that protein
synthesis and degradation occur on timescales that are considerably longer
than phosphorylation/dephosphorylation reactions. However, it was recently
shown for the pheromone response pathway that the degradation rate of Ste7
is large and increases upon exposure to pheromone (9). Below we present
experimental results that show the degradation rate of Msg5 is also large, and
slightly increases with pheromone stimulation. The fact that steady-state
levels of these proteins do not change appreciably after exposure to pher-
omone implies that protein synthesis must also increase (data not shown).
There are many possibilities for how to incorporate protein degradation into
the model. We will investigate two cases that illustrate that the dynamics of
the system depends critically on this choice.
Case II—no protection from degradation. In addition to Eq. 1 for Fus3,
the reactions for the synthesis and degradation of free Ste7 and Msg5 are
B
E
*
g
KK
d
KK
½KK; (8)
B
E
*
g
P
d
P
½P: (9)
In this model, we also allow proteins to be degraded regardless of their
chemical state. That is, all reactions of the form
½KK
!
d
KK
B; (10)
½K KK
!
d
KK
½K; (11)
½K KK
!
d
K
½KK; (12)
are included. The total concentration of each protein species is no longer
conserved, and Eqs. 5–7 become
d½K
T
dt
¼ g
K
d
K
½K
T
; (13)
d½KK
T
dt
¼ g
KK
d
KK
½KK
T
; (14)
d½P
T
dt
¼ g
P
d
P
½P
T
: (15)
At steady state, the synthesis and degradation rates balance, and the steady-
state values of the total protein concentrations are ½K
ss
T
¼ g
K
=d
K
; ½KK
ss
T
¼
g
KK
=d
KK
; and ½P
ss
T
¼ g
P
=d
P
; where the superscript ss denotes steady state.
Case III—degradation of free protein only. In this scenario, only the free
inactive form of the protein is degraded. That is, protein degradation only
occurs via the reactions given in Eqs. 1, 8, and 9, so that proteins are
protected against degradation when they are in a protein-protein complex or
phosphorylated. In this case, Eqs. 13–15 become
d½K
T
dt
¼ g
K
d
K
½K; (16)
d½KK
T
dt
¼ g
KK
d
KK
½KK; (17)
d½P
T
dt
¼ g
P
d
P
½P: (18)
At steady state, the free enzyme concentrations are given by [K]
ss
¼ g
K
/d
K
,
[KK]
ss
¼ g
KK
/d
KK
,and[P]
ss
¼ g
P
/d
P
. Below we show that this model can
produce sustained oscillations in concentration levels. These oscillations persist
if the phosphorylated species of Fus3 are not protected from degradation.
For simplicity, we will focus on the three cases described here. However,
many other possibilities exist. For example, it is possible that degradation
rates are different for free proteins versus those in complexes or those that
are phosphorylated.
Experimental methods
Strains and plasmids
The S. cerevisiae strains used in this study were BY4741 (MATa leu2D
met15D his3D ura3D), BY4741-derived strains containing the tandem-
affinity purification (TAP) tag fused to FUS3 or MSG5 (18), or BY4741-
derived deletion mutants lacking SST2 or MSG5 (Research Genetics, Huntsville,
AL). The single copy SST2 expression plasmid pRS316-SST2 (2X SST2)
was described previously (19). The single-copy MSG5 plasmid was con-
structed by PCR amplification of the MSG5 gene, using flanking PCR
primers that anneal ;600-bp upstream (GAGGATCCGACGATGATGAC-
GATGATGATG) or ;600-bp downstream (GAGGATCCTGCAGCAA-
CACCTTTGG) of the open reading frame. The PCR product was then
subcloned into pRS316 (American Type Culture Collection, Manassas, VA)
by BamHI digestion and ligation to yield pRS316-MSG5. The STE7 over-
expression plasmid was constructed by PCR amplification (forward primer:
ATGTTTCAACGAAAGACTTTA; reverse primer: AATGGGTTGATCT-
TTCCGATTG) of the STE7 gene and then ligated into pYES2.1/V5-His-
TOPO (Invitrogen, Carlsbad, CA) to yield pGAL-STE7. The FUS1-green
fluorescent protein (GFP) reporter (containing destabilized PEST-domain
containing variant of GFP) was subcloned from pDS30 (20) into pRS303
(American Type Culture Collection) using EcoRI and NotI, and then
linearized by XcmI to drive genomic integration at the FUS1 locus.
Protein degradation time course and immunoblot detection
Cells were treated with 3 mM a-factor for indicated times. To monitor the
degradation of proteins over time, midlog cell cultures were treated with
cycloheximide (10 mg/ml in 0.1% ethanol, final concentrations) for up to 90
min before harvesting. The cell growth and treatment was stopped by the
addition of 10 mM NaN
3
and transfer to an ice bath. Cells were washed and
resuspended directly in boiling SDS-PAGE sample buffer for 10 min,
disrupted by glass-bead homogenization, and clarified by microcentrifuga-
tion. After SDS-PAGE and transfer to nitrocellulose, the membrane was
probed with antibodies to Sst2 or protein A (Sigma-Aldrich, St. Louis, MO).
Immunoreactive species were visualized by enhanced chemiluminescence
detection (Pierce Biotechnology, Rockford, IL) of horseradish peroxidase-
conjugated anti-rabbit IgG (Bio-Rad, Hercules, CA).
Fluorescence-activated cell sorting
The measuremen t of GFP in individual yeast cells was described previously
(19,21). Briefly, cells containing the integrated FUS1-GFP reporter were treated
with indicated concentrations of a-fac tor for indicated times. The cell growth and
a-factor treatment were stopped by the addition of 10 mM NaN
3
and transfer to
an ice bath. The resulting fluorescence in each cell was monitored by cell sorting.
RESULTS
The pheromone response can be binary or graded
Transcriptional induction can occur in a graded or binary
fashion (5). A binary response can exist transiently with all
1964 Wang et al.
Biop hysical Journal 90(6) 1961–1978
the cells eventually becoming activated or can be permanent
with a persistent subpopulation of cells in the inactivate state.
To investigate the temporal response of the pheromone path-
way, we conducted single cell fluorescence measurements of
transcription. Fig. 2 presents flow cytometry data using the
destabilized green fluorescent protein (GFP) as a reporter for
the transcription induction activity of individual cells after
exposure to pheromone. The distributions shown as solid
lines in the first column of Fig. 2 A are results at different
time-points for wild-type cells exposed to 1 mM pheromone.
The cell s display a transient binary response with all cells
eventually responding after 120 min. The distributions shown
as solid lines in the top row of Fig. 2 B are flow cytom etry
data for wild-type cells at various pheromone doses taken
60 min after exposure to pheromone. At this time-point, the
binary response is most pronounced at 1 mM of pheromone.
Sst2 is a regulator of G-protein signaling that accelerates
G-protein-catalyzed GTP hydrolysis, and in this way inhibits
pathway activation. Sst2 transcription and synthesis is in-
duced by pheromone producing a negative feedback loop.
We recently demonstrated that Sst2 degradation is also in-
creased upon exposure to pheromone (19). Computational
modeling revealed that this positive feedback mechanism
can counteract the negative effects of Sst2 and generate a
binary response. A prediction of the computational model is
that the deletion of Sst2 should remove the transient binary
response resulting in a graded temporal response. To test this
possibility we conducted single cell fluorescence measure-
ments on cells lacking Sst2 (sst2D). These cells exhibited a
graded response for all doses and time-points tested. The
distributions shown as dotted lines in Fig. 2, A and B, are a
subset of these results. Again the columns of Fig. 2 A are the
results for different time-points using a pheromone concen-
tration of 1 mM and the rows of Fig. 2 B are the results for
different doses at 60 min after exposure. Because the sst2D
mutant always produced a grade response, we have included
these results on all the plots as a reference. To further inves-
tigate how Sst2 attenuates the pheromone response, we per-
formed fluorescence measurement on cells containing an extra
copy of the SST2 gene (2XSst2). The results are shown in the
second column of Fig. 2 A and second row of Fig. 2 B. These
data indicate that the 2XSst2 strain also exhibits a transient
binary response and at pheromone doses of 0.3, 1, and 3 mM,
this response lags that of wild-type cells (compare column
1 with 2 in Fig. 2 A and row 1 with 2 in Fig. 2 B). Fig. 2 B
also indicates that the binary response is more pronounced at
higher pheromone concentrations than in wild-type cells.
The results presented in Fig. 2 demonstrate that the
graded-to-binary conversion is regulated by the G-protein;
however, these data do not establish that the conversion is
mediated by the G-protein itself or by another downstream
signaling component. Previous results suggest that it is the
MAPK that mediates the switch. Evidence for this comes
from the work of Poritz et al. (21). Using sst2D mutants, they
demonstrated over a range of pheromone concentrations that
inhibition of the pathway downstream of the G-protein
converts a graded response to a binary one. Their measure-
ments were made 4–6 h after pheromone treatment, indicat-
ing that the binary response is permanent. This suggests that
inhibiting the pathway downstream of the G-protein gener-
ates bistability. Recent theoretical work of Markevich et al.
(7) also suggests that regulation of MAPK is sufficient to
generate bistability, and thus might account for the graded-
to-binary conversion observed by Poritz et al. (21). To test if
regulation of MAPK is also sufficient to explain the temporal
response of the pathway described here, we constructed a
stochastic model of Fus3 activation. Stochastic models treat
biochemical reactions as random processes and therefore pro-
vide information about the effects of concentration fluctu-
ations on the response of the pathway.
Stochastic modeling, bistability, and the transient
binary response
A diagram of the MAPK portion of the pheromone response
pathway is shown in Fig. 1 B. Ste7 is a dual-specificity ki-
nase that phosphorylates Fus3 at both threonine and tyrosine
residues and stimulates its catalytic activity. Msg5 is a dual-
specificity phosphatase that inactivates Fus3. As shown in
Fig. 1 B, we assume a distributive kinetic mechanism for the
dual phosphorylation and dephosphorylation reactions. That
is, two collisions between Fus3 and Ste7 are required for the
dual phosphorylation of Fus3. Evidence for a distributive
phosphorylation mechanism comes from work on MAPKK-
1 phosphorylation of p42 MAPK in Xenopus (13), where it
was shown that during the phosphorylation process the
amount of monophosphorylated MAPK exceeded the amount
of dually phosphorylated MAPK. This result is only possible
if two collisions between the MAPKK and MAPK are
required for full phosphorylation. We also assume that two col-
lisions between Fus3 and Msg5 are required to convert the
doubly phosphorylated Fus3 back to the unphosphorylated
state (7). The biochemical reactions used in the computa-
tional model are given in Appendix A. Where available, we
use experimentally measured values for the model param-
eters. For the parameters that have not been measured, we
use biologically realistic values. A summary of the model
parameters and the values used in the simulations is given in
Table 1.
A surprising property of the reaction scheme illustrated
in Fig. 1 B is that the system can exhibit bistability in the
absence of feedback regulation (7). Bistability refers to the
situation in which the system posses ses two stable steady
states and is generally attributed to feedback regulation.
Fig. 3 shows a plot of the steady-state values of the doubly-
phosphorylated (active) Fus3 concentration [Kpp]
ss
as a
function of the total (phosphorylated and unphosphorylated)
Ste7 concentration [KK]
T
. This type of graph, often referred
to as a bifurcation diagram, illustrates where qualitative changes
to the steady-state values of [Kpp]
ss
occur. If an effective cell
Computational Analysis of MAPK 1965
Biophysical Journal 90(6) 1961–1978
FIGURE 2 Pheromone-dependent transcriptional induction was measured for four different strains of cells: wild-type cells (WT), an sst2 gene deletion
mutant (sst2D), cells transformed with a single copy plasmid (pRS316) containing genomic clone of SST2 (2XSST2) and sst2D containing a single copy
plasmid (pRS316) containing genomic clone of MSG5 (2XMSG5). An integrated pheromone-responsive FUS1 promoter-GFP reporter was used to monitor
expression. Cells were then treated with various concentrations of a-factor (0.3, 1, 3, and 10 mM) and the resulting fluorescence in each cell was monitored by
cell sorting at 45, 60, 75, 90, and 120 min. All experiments were repeated at least twice with similar results. Fluorescence measurements are reported on a log
scale (x axis). (A) The rows in this figure correspond to fluorescence measurements made at different time-points after treatment with 1 mM of pheromone. In
each column the distributions shown as dotted lines are data for the sst2D mutant that responds in a graded fashion. In contrast the distributions shown as solid
lines are for the WT (first column), 2XSST (second column) and 2XMSG5 in sst2D (third column) strains and show a transient binary response. The shaded
dashed line in the first two columns is a guide for the eye to help illustrate the delayed response of the 2XSST2 strain relative to the WT case. (B) The columns
in this figure correspond to fluorescence measurements made at different pheromone concentrations 60 min after exposure. Again the distributions shown as
dotted lines are data for the sst2D mutant and the distributions shown as solid lines are for the WT (first row), 2XSST2 (second row), and 2XMSG5 in sst2D
(third row) strains. The delayed response of the 2XSST2 strain can again be observed for a-factor concentrations of 0.3, 1, and 3 mM.
1966 Wang et al.
Biop hysical Journal 90(6) 1961–1978
volume of 30 mm
3
is assumed, then the concentrations of
Fus3, Ste7, and Msg5 used to generate Fig. 3 correspond to
total molecular abundances of 9000, 700; 1200, and 1800,
respectively, which are similar to experimentally determined
values (18,22). The solid portions of the curve indicate stable
steady states and the dashed portion indicates unstable steady
states. As is typical of bistable systems, in the bistable region
there exist three steady states for each value of the total Ste7
concentration [KK]
T
, two stable and one unstable. Because
the unstable steady state is unstable agains t all perturbations,
it is not experimentally observable.
The effects of random fluctuations in protein concentra-
tions can be investigated by considering a stochastic model
of the system. Concentration fluctuations enable the system
to undergo random transitions between the two stable states
in the b istable region. Therefore, a histogram of protein con-
centration taken from a population of cells exhibiting bi-
stability would be bimodal with a subpopulation of cells in
an activated state and another subpopulation in the inactive
state. Equivalently, histograms generated from a sufficiently
long time-series from a single cell would also be bimodal.
However, stochastic modeling revealed that when realistic
protein abundances are used, the average time for sponta-
neous transitions between stable steady states to occur is
much longer than all biologically relevant timescales. This is
illustrated in Fig. 4. This figure shows simulated time-se ries
of the activated Fus3 molecule number for cases near the
bifurcation point at 58.4 nM (1051 Ste7 molecules/cell).
Proceeding from top to bottom, each panel of this figure
corresponds to the addition of one molecule of Ste7. In each
panel, two different sets of initial conditions were used; one
set was chosen to be near the steady state corresponding to
low level s of activated Fus3 and the other set corresponds to
high levels of activated Fus3. In each graph only a single
time-series started near the high state is shown, because
FIGURE 2 Continued
FIGURE 3 A bifurcation diagram illustrating bistability in the reactions
shown in Fig. 1 B. The steady-state values of the activated Fus3 concen-
tration [Kpp]
ss
are plotted as a function of the total Ste7 concentration [KK]
T
.
The solid portion of the curve indicates stable steady states and the dashed
portion unstable steady states. The bistable region occurs between 44.6 nM
and 58.4 nM. Assuming a cell volume of 30 mm
3
, this corresponds to 802
and 1051 Ste7 molecules, respectively. The values of the model parameters
used to produce this plot are given in Table 1.
Computational Analysis of MAPK 1967
Biophysical Journal 90(6) 1961–1978
multiple runs did not generate any transitions to the low
state. This indicates that the activated state is very stable
against fluctuations in protein concentration. Each graph in
Fig. 4 shows 10 time-series started near the low state. The
top panel corresponds to a case in which two stable stead y
states exist, and the transitions from low to high occur
infrequently. The bottom panel corresponds to a case with a
unique steady state. In this case, the system transitions from
low to high levels of activated Fus3 relatively quickly. That
is, the average transition time is noticeably shorter even
though the top and bottom panels only differ by two Ste7
molecules. Although this sensitivity to molecular abundance
is surprising and of theoretical interest, it seems unlikely to
have biological significance and does not underlie any of the
results presented here. Note, however, that there is consid-
erable variability in the transition time even when a single
steady state exists (Fig. 4, middle and bottom). This vari-
ability continues for Ste7 concentrations well beyond the
region of bistability and is sufficient to generate a binary
response. The model shows a similar sensitivity on the deac-
tivation time to Msg5 levels (data not shown).
The solid line s shown in the bottom and middle panels of
Fig. 4 are time-series from the rate equations for the protein
concentrations (see Materials and Methods and Appendix
A). Notice that these trajectories show a long time delay
before the system moves from the low to high state. This
time lag is referred to as a bottleneck (23). The bottleneck
occurs for values of the total Ste7 concentration just beyond
the bistable region shown in Fig. 3, because at low activated
Fus3 concentrations Fus3 phosphorylat ion and dephospho-
rylation rates nearly balance. However, the dephosphoryla-
tion rate is slightly smaller and not able to maintain Fus3 in a
deactivated state.
The time delay in Fus3 activation produced by the model
might underlie the delay in Sst2 induction observed exper-
imentally in cells overexpressing Sst2 (19). We next inves-
tigated if a transient binary response could be generated even
for values of the total Ste7 concentration that do not generate
bistability. In Fig. 5, we present results from stochastic
simulations using a total Ste7 molecule number of 1062. For
this value, the system is still close enough to the transition to
bistability so that the rate equations produce a time lag (black
solid line, Fig. 5 A). To simulate pathway activation the
system was started near the low activated Fus3 steady state.
Next 1250 sample paths were generated, 10 of which are
shown in Fig. 5 A. The dashed line shown in this figure is the
result of averaging the sample paths. This curve agrees fairly
well with the rate equation result. However, the fluctuations
cause the response to become less sharp. Next the sample
paths were used to generate histograms of the activated Fus3
at several different times. The results are shown in Fig. 5 B.
As can be seen, the system exhibits a transient binary
response. That is, at intermediate times the distribution of
activated Fus3 is bimodal, whereas at long times the
distribution is centered at the high state, indicating that all
FIGURE 4 Time-series of the number of activated Fus3 molecules from
stochastic simulations. Moving down the column corresponds to adding one
molecule of Ste7 to the system. The plots illustrate pathway activation as the
system moves out of the bistable regime through the bifurcation at 58.4 nM
of Ste7 (1051 molecules/cell). Each graph shows 10 time-series started with
identical initial conditions near the low state and one time-series started near
the high state. As can be seen, the transition time from the low to high state is
very sensitive to Ste7 molecule number. Also shown in the two lower panels
are the results from the rate equation for [Kpp] (solid black lines). To convert
to molecule number, the concentration was multiplied by the volume and
Avogadro’s number. In the top panel, the system is in the bistable regime so
the solution to the rate equation (not shown) remains near the low state.
1968 Wang et al.
Biop hysical Journal 90(6) 1961–1978
the cells have become activated. This transient binary re-
sponse can be observed for Ste7 concentrations well beyond
the bistable region; however, the time lag becomes less
pronounced.
We note that a transient binary response is also observed if
the pathway is operating within the bistable regime where the
activated state is sufficiently stable so that transitions back to
the deactivated state do not occur on biologically relevant
timescales (,2 h). In this case, the time-series of Fus3
activation does not show the S-shape seen in Fig. 5 A. That
is, the initial lag phase leading up to Fus3 activation is not
present. Also, unless the system is very close to the transition
to bistability at 44.6 nm, the lower stable steady state is
sufficiently stable that pathway activation does not occur on
timescales consi stent with the experimental results.
The stochastic model can now be used to interpret the
results of the single cell fluorescence experiments shown in
Fig. 2. We postulate that under normal conditions in wild-
type cells, the pathway is not bistable, but operating suffi-
ciently close to a bistable region to generate a transient binary
response. For the sst2D strain, the pathway is operating well
beyond the bistable regime and therefore always responds in
a graded fashion. For the 2XSst2 strain at moder ate pher-
omone concentrations (1–3 mM), the pathway is just to the
right of the bistable regime. This produces the transient
binary response as well as the previously documented delay
in Sst2 induction (19). For 2XSst2 strain, low pheromone
concentrations (,0.3 mM) are insufficient to generate a
response (data not shown). This indicates that the pathway is
operating in the bistable region or to the left of it. If this
model is correct, and if the binary-to-graded response is truly
mediated at the level of Fus3, it should be possible to pre-
dict the outcome of experiments in which the proteins that
regulate Fus3 are perturbed through deletion or twofold
overexpression of the gene.
Experimental analysis of model predictions
Although the model described above lacks many biological
details, it does make predictions that can be tested experi-
mentally. Sst2 is a negative regulator of the pheromone
response pathway. Deletion of Sst2 resulted in a temporal ly
graded response for each concentration of pheromone we
tested. Within the context of the computational model, dele-
tion of Sst2 increases the amount of active Ste7 and moves
the pathway away from the bistable regime shown in Fig. 3.
Msg5 is a negative regulator of the pathway that dephos-
phorylates Fus3. Therefore, increasing the activity or ex-
pression of Msg5 should counteract the effect of deleting
Sst2 and restore the binary response. To test this prediction,
we engineered cells lacking Sst2 to express twice the normal
amount of Msg5 (sst2D/2XMsg5). We then performed single
cell fluorescence measurements of transcription at various
pheromone concentrations. Whereas the parent sst2D strain
shows the typical graded response, the sst2D/2XMsg5 strain
exhibits a transient binar y response for all pheromone con-
centrations tested. The results for the sst2D/2XMsg5 strain
are shown in the third column of Fig. 2 A and the third row of
Fig. 2 B. These results indicate that modest overexpression
of Msg5 can restore the transient binary response in cells
lacking Sst2, supporting the hypothesis that the graded-to-
binary conversion occurs at the level of the MAPK Fus3.
Sst2 synthesis is induced upon exposure to pheromone
(24). Induction of SST2 requires the transcription factor
Ste12, which is itself phosphorylated and activated by
Fus3 (25). Conversely, the activity of Fus3 is attenuated by
Sst2 acting on the G-protein. We recently reported that cells
engineered to overexpress Sst2 show a time delay in tran-
scriptional induction (19). Computational modeling sug-
gested that alterations in Fus3 activity could account for the
observed delay in Sst2 induction. The model predicts that the
delay results from a bottleneck that occurs near the transition
to bistability. That is, for the 2XSst2 strain at 3 mMof
FIGURE 5 (A) Simulation time-series for the activated Fus3 molecule
number N
Kpp
. In this figure, the total Ste7 molecule number is 1062 (59 nM)
The solid line is the result from the rate equations and the dashed line is the
result from averaging 1250 realizations of the process. (B) Histograms from
the stochastic simulations at times 50, 65, 69, and 80 min. Panels A and B
illustrate that the Fus3 activation can account for the delay in the Sst2
dynamics and the transient binary response that are observed experimentally.
Computational Analysis of MAPK 1969
Biophysical Journal 90(6) 1961–1978
pheromone, the pathway operates just to the right of
bistable region shown in Fig. 3. If this mechanism is correct,
then increasing the expression of a positive regulator of Fus3
should remove the time delay by moving the system further
away from the bistable region. To test this possibility, we
engineered cells to overexpress the MAPKK Ste7 in 2XSst2
strains (2XSst2/GAL-STE7). This was accomplished by in-
serting a plasmid containing the STE7 gene under the control
of the galactose-inducible GAL1/10 promoter. Fig. 6, A and
B, shows time-series for Sst2 induction in the 2XSst2/GAL-
STE7 and 2XSst2 strains grown in galactose. In agreement
with the model, the 2XSst2 strain shows a delayed response,
whereas in the 2XSst2/GAL-STE7 strain the delay is absent.
These results are in agreement with our model in which the
biochemical steps that regulate MAPK activation mediate
the graded-to-binary conversion.
Protein synthesis and degradation
The modeling results above indicate that small changes in
Ste7 abundance can have profound effects on Fus3 phos-
phorylation. A similar sensitivity on protein abundance was
found for the dephosphorylation of Fus3 by Msg5. It was
recently demonstrated that exposure to pherom one increases
the degradation rate of Ste7 (9). To further investigate the
role of protein turnover in the regulation of the pheromone
response pathway, we monitored the effects of pheromone
on the degradation of Fus3 and Msg5. After 1 h of growth in
the absence or presence of pherom one the cells were treated
with cycloheximide to block new protein synthesis. Steady-
state levels of Fus3 and Msg5 remaining were then monitored
by immunob lotting. As shown in Fig. 7, Msg5 abundance
declined quickly, and this decline was marginally faster
when the cells wer e pretreated with pheromone. A different
pattern of degradation was o bserved for Fus3. In this case
degradation was fairly slow, and was slowed even further
after pheromone treatment. These results are in contrast to
Ste7, which is degraded more slowly than Msg5 and more
rapidly than Fus3, and degradation is accelerated by pher-
omone treatment.
Based on these experimental results and the sensitivity of
Fus3 activation to Ste7 and Msg5 levels observed in the sto-
chastic simulations, we expanded the computational model to
include protein synthesis and degradation. Modeling protein
degradation requires assumptions about when proteins are
susceptible to proteolysis. We lim it our investigations to the
Case II and Case III scenarios discussed in the Materials and
Methods. In Case I, protein synthesis and degradation are
ignored. In Case II, each protein is degraded at a rate that is
independent of its chemical state, and in Case III, each pro-
tein is protected from degradation when it forms a multimeric
complex or is phosphorylated. Clearly, there are other possi-
bilities for modeling protein degradation, and the mechanisms
for protein stabilization and destabilization are not fully
understood. However , these two scenarios represent extreme
cases, and serve to illustrate the important effects protein syn-
thesis and degradation have on the dynamics of the system.
We start with Case II, in which each protein is degraded at
a rate that is independent of its chemical state. The data
shown in Fig. 7 indicate that, in the presence of pheromone,
Fus3 is stable for the duration of the experiment. Therefore,
we only consider the degradation and synthesis of Ste7
and Msg5. At steady state, the total concentrations of Ste7
ð½KK
ss
T
Þ and Msg5ð½P
ss
T
Þ are given by the ratio of their
synthesis rate to their degradation rate (see Materials and
Methods). The degradation rate d
P
of Msg5 was estimat ed
from the data presented in Fig. 7 and the data of Wang and
Dohlman (9) was used to estimate the degradation rate d
KK
of Ste7 (see Table 1). Fig. 8 A is a bifurcation diagram for
this system as a function of the Ste7 synthesis rate g
KK
.
In this figure, the synthesis rate g
P
of Msg5 was chosen to
produce molecular abundances similar to those measured
FIGURE 6 (A) Whole-cell extracts were prepared from wild-type cells
transformed with a single copy plasmid (pRS316) containing genomic SST2
(2XSST2) and either an empty vector (pYES) or the same vector containing
STE7 under control of the galactose-inducible GAL1/10 promoter (GAL-
STE7). Cells treated with 3 mM a-factor for the indicated times, collected,
resolved by 7.5% SDS-PAGE and immunoblotting, and probed using anti-
Sst2 polyclonal antiserum as indicated (IB). The specificity of each antibody
was confirmed using gene deletion or diploid cells lacking the indicated gene
product (19). (B) To estimate the difference in protein expression the Sst2
band was analyzed by densitometric scanning.
1970 Wang et al.
Biop hysical Journal 90(6) 1961–1978
experimentally. The bifurcation diagram is very similar to
the one shown in Fig. 3, except that the model parameter
being varied in this case is the Ste7 synthesis rate rather than
the Ste7 abundance. We performed stochastic simulations
to investigate if a binary response is possible in this case.
Initially, the protein levels were taken to be at their steady-
state values for a Ste7 synthesis rate of g
KK
¼ 0.224 s
1
. The
model does not consider upstream elements of the pathway
such as the MAPK kinase kinase Ste11. Therefore, to simu-
late pathway activation the Ste7 synthesis rate was increased
to g
KK
¼ 0.339 s
1
at t ¼ 0. Because our model does not take
into account the phosphorylation of Ste7, mathematically
FIGURE 8 (A) A bifurcation diagram showing the steady-state concen-
tration of activated Fus3 concentration [Kpp] as a function of the Ste7
synthesis rate g
KK
for Case II (no protection from degradation). (B)
Simulation time-series for the activated Fus3 molecule number N
Kpp
. In this
figure the Ste7 synthesis rate is 0.339 s
1
. The solid line is the result from the
rate equations and the dashed line is an average over 500 realizations of the
process. The discrepancy between these two curves is due to the large effects
of the fluctuations in the activated Fus3 concentration. These fluctuations
eliminate the time lag in Fus3 activation. (C) Histograms of activated Fus3
molecule number from the stochastic simulations at times 150, 250, 300, and
350 min. Even though the system no longer shows a lag in Fus3 activation,
the transient binary response is still produced.
FIGURE 7 Cells containing an integrated TAP-tagged form of (A) Fus3
or (B) Msg5 were treated with 3 mM a-factor for 60 min, and then treated
with the protein synthesis inhibitor cycloheximide (CHX) for the indicated
times. Cell extracts were analyzed by immunoblotting with anti-protein
A antibodies. To estimate the difference in protein half-life, the intensity
of each band was analyzed by densitometric scanning and expressed as
a percentage of the amount of protein at the beginning of cycloheximide
treatment.
Computational Analysis of MAPK 1971
Biophysical Journal 90(6) 1961–1978
increasing the synthesis rate of Ste7 is similar to modeling
the pheromone-induced activation of Ste7 by Ste11. In-
creasing the Ste7 synthesis rate moves the pathway just
beyond the bistable regime (see Fig. 8 A). Fig. 8 B shows a
simulated time-series for Fus3 activation. Because the total
molecule numbers of Ste7 and Msg5 now fluctuate, there is
considerably more noise in the system (compare with Fig.
5 A). This increase in fluctuations allows the system to
transition between the two steady states more rapidly. This
can be seen by comparing the rate equation result (solid line)
and the result from averaging 500 realizations of the process
(dashed line). The increased fluctuat ions are almost suffi-
cient to eliminate the time lag. However, Fig. 8 C shows that
the system still produces a transient binary response. Notice
that the timescale for the pathway to respond is now con-
siderably longer than that for the activation of Fus3 shown
in Fig. 5 A. This is due to the fact that we have simulated
pathway activation by increasing the synthesis rate as a
surrogate for increasing the phosphorylation rate. Phospho-
rylation of Ste7 by Ste11 occurs on a faster timescale than
protein synthesis. In fact, in Fig. 5 it was assumed to occur
instantaneously. However, as long as Ste7 activation is not
rate-limiting, we expect an extended model that includes
Ste7 regul ation to respond on a timescale similar to that ob-
served experimentally. Case II represents an extreme scenario
in which proteins can be degraded regardless of the chemical
state. Therefore, this case maximizes the effects of biochem-
ical fluctuations. One consequ ence of the large fluctuations is
that the time lag produced by the rate equations is almost
eliminated. In contrast, Case I minimizes the effects of fluc-
tuations, because variability in concentration levels that arises
from protein synthesis and degradation events is not present
in this model.
We now move to an intermediate case in which protein-
protein interactions (such as those leading to protein di-
merization) and phosphorylation protect proteins from
degradation. It is commonly observed that proteins in a func-
tional complex are more stable than when expressed alone.
One familiar example is the Gb- and Gg-subunits, neither of
which can be stably expressed or puried in the absence of the
other. For simplicity, we start with the case in which only Ste7
is degraded and synthesized, with the levels of total Fus3 and
Msg5 remaining constant. That is, d[K]
T
/dt ¼ d[P]
T
/dt ¼
0 and [KK]
T
satisfies Eq. 17, above. The rate equations for
the rest of the chemical species remain unchanged except for
the equation for the Ste7 concentration [KK], which now
includes the same synthesis and degradation terms as in Eq.
17. Equation 17 implies that at steady state the free Ste7
concentration is given by the synthesis rate g
KK
divided by
the degradation rate d
KK
. Fig. 9 A shows a plot of the total
Ste7 concentration ½KK
ss
T
as function of the synthesis rate
g
KK
. Surprisingly this plot is not monotonic, but goes
through a local maximum when g
KK
is ;0.7 s
1
. This effect
is a result of the protein complexes being protected from
degradation and the dual phosphorylation reaction. At low
levels of Ste7, most of the Fus3 is in the unphosphorylated or
singly phosphorylated state. Ste7 interacts strongly with
these two states and is then protected from degradation. As
Ste7 levels increase, the reactions in Fig. 1 B favor the dually
phosphorylated state. Therefore, there is less substrate for
Ste7 to interact with and to protect from degradation. As the
synthesis rate is increased further, the total amount of Ste7
again begins to increase. To illustrate the counterintuitive
observation that increasing synthesis rate of Ste7 can lead to a
decrease in its total concentration, the degradation rate of Ste7
was artificially increased to d
KK
¼ 4 3 10
3
s
1
. If the
experimentally measured value d
KK
¼ 3.2 3 10
4
s
1
is
FIGURE 9 (A) The steady-state total Ste7 concentration ½KK
ss
T
as a
function of its synthesis rate g
KK
. The total concentration goes through a
local maximum near g
KK
¼ 0.7 s
1
. This results from protection from
degradation and the dual phosphorylation reactions. In this figure the
degradation rate was artificially increased to d
KK
¼ 4 3 10
3
s
1
to illustrate
the nonmonotonic behavior. (B) When the experimentally determined value
d
KK
¼ 3.2 3 10
4
s
1
is used, the steady-state total Ste7 concentration
½KK
ss
T
again shows nonmonotonic behavior. However, in the case the
steady-state is not stable for the part of the curve with negative slope (dashed
line). In this region the system undergoes sustained oscillations. The
minimum and maximum values of [KK]
T
are plotted as circles. (Inset)
A time-series [KK]
T
of illustrating the periodic behavior. In this figure,
g
KK
¼ 0.1 s
1
.
1972 Wang et al.
Biop hysical Journal 90(6) 1961–1978
used, the results shown in Fig. 9 B are produced. As can be
seen, the stead y value of the total Ste7 concentration still
goes through a local maximum. However, where the
concentration declines, the steady state is unstable (dashed
line). In this region, the system produces sustained oscilla-
tions. An example of these oscillations is shown in the inset.
The circles shown in Fig. 9 B indicate the minimum and
maximum values of the total Ste7 concentration. This model
can also exhibit bistability (see Appendix B). However, the
parameter range over which two stable steady states exist
appears to be limited.
We now consi der the case in which the uncomplexed/free
forms of Fus3 and Msg5 also are degraded. In this case, the
total concentrations satisfy Eqs. 16–18. To estimate the de-
gradation rate d
K
of Fus3, we used the experimental results
in absence of pheromone presented in Fig. 7. The rationale
for this decision is that, in the absence of pheromone, most
Fus3 molecules are in the inactive state and available for
degradation, whereas, upon pheromone stimulation, phos-
phorylation of Fus3 leads to kinase/substrate interactions that
protect the protein from degradation. Therefore, the data in
the absence of pheromone provi de a good estimate for the
degradation rate of free inactive Fus3. If this reasoning is
correct, then it implies that the pheromone-induced increase
in the degradation rate of Ste7 is actually larger than that
reported in Table 1, because these estimates did not take into
account protection against degradation through protein/
protein interactions. However, for simplicity we assume
that the rates estimated from the data for these two proteins
are the rates at which free protein is degraded. The synthesis
rates were chosen to produce protein numbers consistent
with experimental measurements. In this case, the steady-
state concentrations of free Fus3, Ste7, and Msg5 are given
by the ratio of their synthesis rates and degradation rates (i.e.,
[K]
ss
¼ g
K
/d
K
, [KK]
ss
¼ g
KK
/d
KK
, and [P]
ss
¼ g
P
/d
P
), and it
is straightforward to show that the system has a unique
steady state. That is, this system cannot be bistable. Fig. 10 A
is a bifurcation diagram for the system as a function of the
Ste7 synthesis rate g
KK
. At small values of g
KK
, the system
always approaches steady state. However, at values greater
than g
KK
¼ 0.013 s
1
, the system undergoes sustained
oscillations in concentration. The circles again indicate the
maximum and minimum values of the activated Fus3 con-
centration [Kpp]. The periodic behavior is illustrated in Fig.
10 B, which shows time-series of the total Fus3 concentra-
tion [K]
T
(solid line), the activated Fus3 concentration [Kpp]
(shaded line), and the unphosphorylated Fus3 concentration
[K] (dashed line).
The quali tative explanation for the origin of the oscilla-
tions is that protein degradation acts as a negative feedback
on the bistable system. The combination of bistability and
negative feedback often leads to periodic behavior referred to
as hysteresis oscillations (6). This effect is illustrated in Fig.
10 C, which is a plot of the free Ste7 concentration [KK]
versus the total Ste7 concentration [KK]
T
. Also drawn is
FIGURE 10 (A) A bifurcation diagram showing the activated Fus3
concentration [Kpp] as a function of its synthesis rate g
KK
. For small values
of g
KK
there is a single stable steady state (solid line). At g
KK
¼ 0.013 s
1
,
this steady state becomes unstable (dashed line) and the system undergoes
sustained oscillations (circles). (B) Time-series of the total Fus3 concentra-
tion [K]
T
(solid line), the activated Fus3 concentration [Kpp] (shaded line),
and the inactive Fus3 concentration [K] (dashed line). In this figure, g
KK
¼
0.016 s
1
.(C) A plot of the activated the free Ste7 concentration [KK]
versus the total Ste7 concentration [KK]
T
(dotted line) using the same value
of g
KK
as in B. Also drawn on this figure is the bifurcation diagram
generated by using the time-averaged values of [K]
T
and [P]
T
. The periodic
trajectory closely follows the upper and lower branches of this curve and
rapidly transition between the two branches near the bifurcation points.
Computational Analysis of MAPK 1973
Biophysical Journal 90(6) 1961–1978
the bifurcation curve computed under the assumption of
constant enzyme concentrations (i.e., no synthesis or degra-
dation). Therefore, to draw the bifurcation curve the time-
averaged values of the total Fus3 and Msg5 concentrations
were used. As can be seen, the periodic trajectory closely
follows the upper and lower branches of the curve in the re-
gion of bistability, rapidly jumping between the two
branches at the bifurcation points.
Stochastic modeling and oscillations
We next performed stochastic simulations to determine how
random fluctuations in protein abundance affect the oscilla-
tory dynamics described above. One important result of these
investigations is that the fluctuations increase the parameter
range over which the syst em exhibits oscillatory behavior.
Fig. 11 A shows a time-series from a stochastic simulation. In
this figure, the Ste7 synthesis rate is g
KK
¼ 0.01 s
1
. For this
value, the rate equations predict a stable steady state (see Fig.
10 A). However, the biochemical fluctuations are sufficiently
strong to destabilize the steady state and generate behavior
that appears periodic. Su ch behavior is typical of noisy
systems near a bifurcation and is referred to as noise-induced
oscillations. To investigate the periodicity of the oscillations,
we computed the power spectrum of the time-se ries gener-
ated from the stochastic model (Fig. 11 B). A clear peak is
seen in the spectrum at a frequency of 8 3 10
4
min
1
,
indicating that the noise-induced oscillations are indeed
periodic. The inset in Fig. 11 B shows a histogram of the
interbeat interval (i.e., time betw een successive peaks in con-
centration) and provides a measure of the amount of vari-
ability in the period of the oscillations. The fact that stochastic
effects enlarge the parameter range over which the protein
concentrations oscillate makes it more likely that this peri-
odic behavior has biological significance.
The explanation of the transient binary response required
that the pathway operate near a bistable region. However,
when synthesis and degradation of all three enzymes are
included in the model, the system can no longer exhibit
bistability. To investigate if a binary response is possible in
this scenario, we performed stochastic simulations of path-
way activati on. Histograms from the simulations are shown
in Fig. 12. To generate the histograms, the stochastic simu-
lations were started near the stable steady state that exists for
a Ste7 synthesis rate o f g
KK
¼ 0.005 s
1
. Once again, to
model pathway activation, the Ste7 synthesis rate was
increased at t ¼ 0. In this case, the increase in synthesis rate
to g
KK
¼ 0.016 s
1
moves the system into a regime where
oscillations occur (see Fig. 10 A). The results presented in
Fig. 12 illustrate that the strength of the biochemical
fluctuations is sufficient to generate significant variability
in the activation time of individual cells and produce a clear
binary response. The slow response of the pathway again
FIGURE 11 Noise-induced oscillations. (A) A time-series from a sto-
chastic simulation with g
KK
¼ 0.01 s
1
. For this value of the Ste7 synthesis
rate the rate equations predict a stable steady state. However, biochemical
fluctuations are sufficient to destabilize the steady state leading to sustained
oscillations. (B) The power spectrum computed from stochastic simulations
using the same parameter values as in A. A clear peak is seen at a frequency
of 8 3 10
4
min
1
.(Inset) A histogram of the in interbeat interval.
FIGURE 12 Histograms from the stochastic model when protein synthe-
sis and degradation is considered. These results illustrate that a binary
response is possible when the system is undergoing sustained oscillations.
1974 Wang et al.
Biop hysical Journal 90(6) 1961–1978
results from simulating pathway activation by increasing the
Ste7 synthesis rate rather than modeling the phosphorylation
of Ste7. If larger values of the synthesis rate are used, the
response becomes graded (data not shown).
The oscillatory behavior presented above represents a
macroscopic phenomenon in the sense that it emerges from
the stochastic dynamics only if suffici ently large volumes
and protein abundances are considered. To investigate the
onset of oscillations, we performed stochastic simulations at
various volumes. The synthesis rate and second-order rate
constant were scaled appropriately to ensure that the rate
equations remained unchanged as the volume was increased
(see Materials and Methods). The results are summarized in
Figs. 13 and 14. Fig. 13 shows time-series of the active Fus3
molecule number for various volumes. Fig. 13 A corresponds
to a volume of 0.6 mm
3
. For this volume and protein abun-
dance, the time-series is dominated by fluctuations and no
periodic behavior is discernable. The steady-state distribu-
tion (two-dimensional histogram) for the activated Fus3 and
total Ste7 molecule numbers is shown in Fig. 14 A. In this
figure, red indicates regions where the system spends large
amounts of time and blue indicates regions that are visited
infrequently. For this small volume, the steady-state distri-
bution has very little structure. Figs. 13 B and 14 B
correspond to a volume of 6.0 mm
3
. Periodic behavior is
beginning to appear in the time-series. However, the steady-
state distribution indicates that the system is still dominated
by fluctuations. Figs. 13 C and 14 C are for a volume of 30
mm
3
, which is similar to the volume of a yeast cell. Here, the
periodic behavior is clear. The steady-state distribution is
clearly structured and lies mostly along the deterministic
trajectory (compare Fig. 14, C and D). These investigations
reveal that at volumes and protein abundances typical of yeast
cells, the qualitative behavior of the system is accurately
captured by the rate equations. However, these findings
highlight the importance of stochastic simulations to account
for the variability observed in single cell measurements.
DISCUSSION
The mating response system in yeast is arguably the best-
characterized signaling pathway of any eukaryote, and it has
long served as a prototype for hormone, neurotransmitter,
and sensory response systems in humans. At the receptor
level, signal transduction occurs in a graded fashion, with
pathway activation proportional to the agonist concentration.
However, downstream components of the pathway can
convert the graded response to a binary one. A common
mechanism for achieving this conversion is through positive
feedback (6). For example, we recently demonstrated that
pheromone promotes transcriptional induction as wel l as
ubiquitin-mediated degradation of Sst2, a negative regulator
of the pheromone pathway. These pheromone-regulated
changes in expression are likely to be functionally important,
since twofold overexp ression of Sst2 converts the normally
graded response into a binary response (19). Moreover,
induction of Sst2 expression by pheromone is delayed when
Sst2 is overexpressed (2XSst2 strain). These results led us to
suggest a model in which pheromone-induced degradation of
Sst2 acts as a positive feedback mechanism to counteract the
negative effects of Sst2 on G-protein signaling. The model of
Sst2 regulation also suggested that the binary response exists
only transiently, with all cells eventually becoming activated.
FIGURE 13 Simulation time-series of the activated Fus3 molecule
number N
Kpp
.(A) At small volumes (0.6 mm
3
) fluctuations dominate the
dynamics, and no periodic behavior is observable. (B) As the volume
increases (6 mm
3
), the fluctuations become less significant and periodic
behavior is starting to emerge. (C) At biological realistic volumes (30 mm
3
)
periodic behavior is clearly observable. (D) The result from the rate
equations.
FIGURE 14 (AC) The corresponding steady-state distribution for the
activated Fus3 molecule number N
Kpp
and total Ste7 molecule number N
KKT
for the cases shown in Fig. 13. In these figures, red corresponds to regions
where the system spends large amounts of time and blue regions are visited
infrequently. (D) Plot of N
Kpp
versus N
KKT
computed from the rate
equations.
Computational Analysis of MAPK 1975
Biophysical Journal 90(6) 1961–1978
Although Sst2 acts at the G-protein level, other steps of
the pathway may likewise regulate the graded- to-binary
transformation. Poritz et al. (21) demonstrated, using sst2D
mutants, that inhibiting the pathway downstream of the
G-protein also converts a graded response to a binary one. In
this case, the binary response was long-lived, existing for
several hours. To further investigate pathway activation and
attenuation, we performed single cell fluorescence-based
transcription measurements on sst2D, wild-type, and 2XSst2
strains. The wild-type and 2XSst2 cells were both found to
exhibit a transient binary response, with all the cells be-
coming activated within 2 h. The sst2D mutant strain, in
contrast, showed a graded response. These new findings
coupled with our previous work and the results of Poritz et al.
(21) led us to investigate the regulation of the protein kinase
Fus3 as a potential mediator of the graded-to-binary re-
sponse. We focused on Fus3, because previous theoretical
results of Markevich et al. (7) demonstrated that a distrib-
utive kinetic mecha nism for the dual phosphorylation of
protein kinases is sufficient to generate bistability. Our
stochastic modeling revealed that this mechanism, applied to
Fus3 regulation, can account for all the experimental ob-
servations outlined above. To test the validity of the model,
we genetically altered expression of proteins that normally
regulate Fus3. As predicted by the model, increasing ex-
pression of Msg5, a negative regulator of the pathway, coun-
teracted the positive effects of deleting Sst2 and restored the
binary response. Conversely, to counteract the increased neg-
ative effects of Sst2 in the 2XSst2 strain, we engineered cells
to overexpress Ste7. In full agreement with model predic-
tions, increased levels of Ste7 had the effect of removing the
time delay in Sst2 induction.
Our computational and experimental analysis suggests
that Fus3 regulation is responsible for the graded-to-binary
conversion in the yeast pheromone response. However, the
possibility that this conversion takes place downstream of
Fus3 cannot be ruled out. For example, Blake et al. (26) used
a stochastic model of transcription initiation to show that
pulsatile mRNA production, through reinitiation of tran-
scription, could account for the binary transcriptional re-
sponse observed for the yeast GAL1 promoter. However, our
ability to predict the results of altering the expression of
proteins that regulate Fus3 provides strong evidence that this
step of the pathway mediates the graded-to-binary conver-
sion in the pheromone response pathway.
Protein kinase cascades are a reoccurring feature of signal
transduction pathways and are found in all eukaryotic cells.
For this reason, many recent theoretical investigations have
focused on understanding their behavior (7,27–33). Protein
kinase cascades have been shown to exhibit ultrasensitivity
(28) and to lead to bistability and sust ained oscillations of
concentration levels in the presence of feedback regulation
(30,31,33). Here we have demonstrated that a MAPK
cascade can generate sustained oscillations in the absence
of feedback regul ation. This result builds on the work of
Markevich et al. (7), who demonstrated that multisite phos-
phorylation in protein kinase cascades is sufficient to
generate bistability. We have also shown that periodic
behavior occurs when protein synthesis and degradation are
included in the model of MAPK regulation. Thus, protein
turnover, which is often overlooked in MAPK signaling, can
profoundly influence the response of a signaling pathway
and may provide an important regulatory mechanism. In our
model, protein degradation acts as a negative feedback on a
bistable system and produces periodic behavior through
hysteresis oscillations. The oscillations occur when physical
modifications such as protein oligomerization and phos-
phorylation protect enzymes against degradation. The period
of the oscillations is determined mainly by the protein
degradation rate. Taken together, these findings highlight the
importance of considering protein stability and degradation
in generating models of biological processes.
One measure of our understanding of biological systems is
our ability to predict their behavior in detail. Intracellular
signaling pathways are highly nonlinear and often regulated
by multiple feedback loops. Additionally, these networks are
subject to stochastic fluctuations that arise from the random
nature of biochemical reactions. These features make pre-
dicting a pathway’s response to an external stimulus difficult,
if not impossible, without the aid of mathematical modeling.
Our stochastic modeling of the biochemical steps that reg-
ulate the MAPK Fus3 reveal that very small changes in the
abundance of the MAPKK Ste7 have a significant effect on
pathway activation. Such a high sensitivity to Ste7 levels
might underlie the cell’s ability to convert a small external
signal into a strongly amplified response. Any computational
model necessarily includes biological assumptions and math-
ematical simplifications. For example, many models of sig-
naling pathways ignore protein synthesis and degradation.
However, our analysis reveals that including protein turnover
can lead to sustained oscillations in protein concentrations that
are likely to have biological significance. These findings
demonstrate that mathematical modeling, when combined
with experimental analysis, provides a powerful tool for under-
standing the complex behavior of signaling pathways.
APPENDIX A: BIOCHEMICAL REAC TIONS
In this Appendix, we list the biochemical reactions used in the stochastic
simulations and rate equations. Case I, in which proteins are not synthesized
or degraded, consists of the following reactions:
K 1 KK
E
*
k
1
k
1
K KK; (19)
K KK
!
k
2
Kp 1 KK; (20)
Kp 1 KK
E
*
k
3
k
3
Kp KK; (21)
Kp KK
!
k
4
Kpp 1 KK; (22)
1976 Wang et al.
Biop hysical Journal 90(6) 1961–1978
Kpp 1 P
E
*
h
1
h
1
Kpp P; (23)
Kpp P
!
h
2
Kp P; (24)
Kp P
E
*
h
3
h
3
Kp 1 P; (25)
Kp 1 P
E
*
h
4
h
4
ðKp PÞ; (26)
ðKp PÞ
E
*
k
5
k
5
K P; (27)
K P
E
*
h
6
h
6
K 1 P: (28)
The above reactions are identical to the ones considered by Markevich et al.
(7). Both phosphorylation events follow standard Michaelis-Menten kinet-
ics, Eqs. 19–22. Dephosphorylation occurs in two chemical steps. First the
phosphate is released, Eqs. 24 and 27, and then the kinase and phosphatase
dissociate, Eqs. 25 and 28. The dissociation of the kinase and phosphatase is
assumed to be reversible. The backward rate constants, h
3
and h
6
, for this
process can be taken to be zero without significantly changing the results. In
this case, the kinetics is essentially Michaelis-Menten. The chemical species
(KpP) in Eqs. 26 and 27 results from the assumption that the phosphotyrosine
is dephosphorylated first (7,17). That is, the chemical species (KpP)
indicates the phosphatase attacking the phosphothreonine, whereas KpPis
the product after phosphotyrosine has been dephosphorylated.In addition to
Eqs. 19–28, Case II consists of the following reactions:
f
E
*
g
KK
d
KK
KK; (29)
f
E
*
g
K
d
K
K; (30)
f
E
*
g
P
d
P
P; (31)
Kp
!
d
K
f; (32)
Kpp
!
d
K
f; (33)
K KK
!
d
KK
K; (34)
K KK
!
d
K
KK; (35)
Kp KK
!
d
KK
Kp; (36)
Kp KK
!
d
K
KK; (37)
Kpp P
!
d
K
P; (38)
Kpp P
!
d
P
Kpp; (39)
Kp P
!
d
K
P; (40)
Kp P
!
d
P
Kp; (41)
ðKp PÞ
!
d
K
P; (42)
ðKp PÞ
!
d
P
Kp; (43)
K P
!
d
K
P; (44)
K P
!
d
P
K: (45)
Case III consists of Eqs. 19–31.
APPENDIX B: DISCUSSION ON BISTABILITY
FOR CASE II
It was shown that the inclusion of protein synthesis and degradation of Ste7
in the mathematical description of Fus3 regulation destroyed bistability in
this model for the parameter values listed in Table 1. The addition of protein
degradation and synthesis forces the steady-state Ste7 concentration to take
the value g
KK
/d
KK
, and a plot of the active Fus3 concentration [Kpp]
ss
versus the synthesis rate g
KK
, does not have an S-shape similar to Fig. 3.
However, if [Kpp]
ss
is plotted against the total Ste7 concentration ½KK
ss
T
, the
resulting curve is identical to Fig. 3. The multiple values of [Kpp]
ss
for a
single value of ½KK
ss
T
results from the nonmonotonic behavior of ½KK
ss
T
shown in Fig. 9, A and B. The reason why the functional dependence of
[Kpp]
ss
on ½KK
ss
T
is the same in this case as it is in Case I (no synthesis or
degradation) is as follows. The concentrations of all the chemical species
satisfy exactly the same equations as in Case I, except for the equation for the
Ste7 concentration [KK], which contains the two additional terms g
KK
d
KK
[KK]. In Case I, when the system is in the bistable region, the steady-state
equations have three solutions. However, Eq. 17 for the total Ste7 concen-
tration imposes an additional constraint on the steady state of the system,
[KK]
ss
¼ g
KK
/d
KK
. This constraint uniquely selects one of the three possible
solutions. This argument does not rule out the possibility of bistability in this
model for other parameter values. In fact, it can be shown that the system can
be bistable. For this to occur, the steady-state equations for the concentra-
tions possess three solutions with identical values of [KK]
ss
. We have
assumed that only the free form of Ste7 is degraded. If we relax this
assumption and allow other chemical forms of Ste7 to be degraded, then the
constraint [KK]
ss
¼ g
KK
/d
KK
no longer applies, and it seems likely that the
region of parameter values that show bistability would increase.
We thank Yuqi Wang and Christine Fraser for providing the protein
turnover data.
This work was supported by National Institutes of Health grants No.
GM059167 (to H.G.D.) and No. GM073180 (to H.G.D. and T.C.E.), and
DARPA grant No. F30602-01-2-0579 (to T.C.E.).
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    • "The majority of the models had shown that negative feedback loops are chiefly responsible for the emergence of the oscillatory behaviour. Some models also propose that the interplay between positive and negative feedback is fundamental to generate signals that code for specific responses [23][24][25][26]. These oscillatory behaviours are suggested to be responsible for allowing the cell to choose to proliferate, go into senescence or differentiate. "
    [Show abstract] [Hide abstract] ABSTRACT: Signal transduction through the Mitogen Activated Protein Kinase (MAPK) pathways is evolutionarily highly conserved. Many cells use these pathways to interpret changes to their environment and respond accordingly. The pathways are central to triggering diverse cellular responses such as survival, apoptosis, differentiation and proliferation. Though the interactions between the different MAPK pathways are complex, nevertheless, they maintain a high level of fidelity and specificity to the original signal. There are numerous theories explaining how fidelity and specificity arise within this complex context; spatio-temporal regulation of the pathways and feedback loops are thought to be very important. This paper presents an agent based computational model addressing multi-compartmentalisation and how this influences the dynamics of MAPK cascade activation. The model suggests that multi-compartmentalisation coupled with periodic MAPK kinase (MAPKK) activation may be critical factors for the emergence of oscillation and ultrasensitivity in the system. Finally, the model also establishes a link between the spatial arrangements of the cascade components and temporal activation mechanisms, and how both contribute to fidelity and specificity of MAPK mediated signalling.
    Full-text · Article · May 2016
    • "It is worth emphasizing that if one is interested in the details of the dynamics at the level of a single cell, a different approach has to be taken for the small number of molecules. Gillespie [9] has considered in detail how simulations of such situations could be performed and used by many others[16, 26, 14, 18, 28]. "
    [Show abstract] [Hide abstract] ABSTRACT: Cells maintain cellular homeostasis employing different regulatory mechanisms to respond external stimuli. We study two groups of signal-dependent transcriptional reg- ulatory mechanisms. In the first group, we assume that repressor and activator proteins compete for binding to the same regulatory site on DNA (competitive mechanisms). In the second group, they can bind to different regulatory regions in a noncompeti- tive fashion (noncompetitive mechanisms). For both competitive and noncompetitive mechanisms, we studied the gene expression dynamics by increasing the repressor or decreasing the activator abundance (inhibition mechanisms), or by decreasing the re- pressor or increasing the activator abundance (activation mechanisms). We employed delay differential equation models. Our simulation results show that the competitive and noncompetitive inhibition mechanisms exhibit comparable repression effective- ness. However, response time is fastest in the noncompetitive inhibition mechanism due to increased repressor abundance, and slowest in the competitive inhibition mech- anism by increased repressor level. The competitive and noncompetitive inhibition mechanisms through decreased activator abundance show comparable and moderate response times, while the competitive and noncompetitive activation mechanisms by increased activator protein level display more effective and faster response. Our anal- ysis exemplify the importance of mathematical modeling and computer simulation in the analysis of gene expression dynamics.
    Article · May 2016
    • "Many signaling pathways and regulatory networks have been described with sets of ordinary differential equations (ODE). These models enable a representation of their general wiring and of the kinetics of individual reactions, and can be simulated to follow the dynamics of the investigated system (examples for the yeast MAPK pathways are, amongst many others, described in1234567). A frequently used framework to describe the dynamics of gene regulatory networks is Boolean modeling8910. "
    [Show abstract] [Hide abstract] ABSTRACT: Cellular decision-making is governed by molecular networks that are highly complex. An integrative understanding of these networks on a genome wide level is essential to understand cellular health and disease. In most cases however, such an understanding is beyond human comprehension and requires computational modeling. Mathematical modeling of biological networks at the level of biochemical details has hitherto relied on state transition models. These are typically based on enumeration of all relevant model states, and hence become very complex unless severely - and often arbitrarily - reduced. Furthermore, the parameters required for genome wide networks will remain underdetermined for the conceivable future. Alternatively, networks can be simulated by Boolean models, although these typically sacrifice molecular detail as well as distinction between different levels or modes of activity. However, the modeling community still lacks methods that can simulate genome scale networks on the level of biochemical reaction detail in a quantitative or semi quantitative manner. Here, we present a probabilistic bipartite Boolean modeling method that addresses these issues. The method is based on the reaction-contingency formalism, and enables fast simulation of large networks. We demonstrate its scalability by applying it to the yeast mitogen-activated protein kinase (MAPK) network consisting of 140 proteins and 608 nodes. The probabilistic Boolean model can be generated and parameterized automatically from a rxncon network description, using only two global parameters, and its qualitative behavior is robust against order of magnitude variation in these parameters. Our method can hence be used to simulate the outcome of large signal transduction network reconstruction, with little or no overhead in model creation or parameterization.
    Full-text · Article · Dec 2015
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