Figure 1.
Cell-to-cell variability in a minimal model of a gradual kinase cascade.
A Schematic representation of a five-step kinase cascade (S…extracellular stimulus; and
…active and inactive kinases, respectively;
…phosphatases;
and
…phosphorylation and dephosphorylation rate constants, respectively). B Cell-to-cell variability simulations confirm strong heterogeneity in the gradual kinase cascade. Nine signaling protein concentrations (5 kinases, 4 phosphatases) were sampled from log-normal distributions (
; coefficient of variation =
), and the dose-response curve was simulated using Eqs. 3 and 4 for a set of 1000 sampled protein concentrations. Low phosphatase activities were chosen to model a low activation resistance:
(Supplemental Table S1). The blue and orange areas are enclosed by the dose-response curves which yielded the minimal/maximal
and
, respectively. Box plots at the top and right side represent the distributions of
and
, respectively (normalized by the population medians). These box plots indicate the median (middle of box), the first and third quartile (box edges), the data points that lie within a distance of 1.5 interquartile ranges from the lower and higher quartiles (whiskers) and extreme outliers (crosses). C The variabilities of
and
respond inversely to changes in kinetic parameter values. Cell-to-cell variability simulations (similar to panel B) were repeated for various activation resistances in the cascade which were tuned by simultaneously changing the phosphatase rate constants
(x-axis). The variabilities of
and
were analyzed for each parameter configuration (y-axis) and expressed as inter-quartile ratios (IQRatio =
= ratio of the third quartile and the first quartile; related to the width of the box plots shown in B). High inter-quartile ratios imply high cell-to-cell variability while an IQRatio of 1 corresponds to no variability. Similar results are obtained when using the coefficient of variation as a measure of variability (Figure S1). D Upstream signaling protein fluctuations determine the pathway sensitivity (
) while downstream fluctuations control the maximal pathway activation (
).
and
were calculated for each simulation run in panel B and related to the concentrations of the first and the terminal kinase in the same simulation. Each dot represents a simulation of a single cell, and the solid lines are linear fits to all points.
Figure 2.
Cell-to-cell variability of kinase cascades with negative feedback.
A Schematic representation of the five-step cascade with negative feedback acting upstream (red, solid) or downstream (red, dashed). either activates the phosphatase of the second or the fifth level. B Cell-to-cell variability simulations confirm that negative feedback eliminates the variability of the pathway sensitivity (concepts similar to Figure 1B). Strong feedback was assumed and simulations were performed using Eq. 9 (parameters same as in Figure 1B; Supplemental Table S1). Colored box plots represent the
and
distribution of the feedback model, while gray box plots show the behavior of the reference feedback-less cascade (cf. Figure 1B). The inset shows that increasing the feedback cooperativity parameter
(Eq. 8) decreases
variability, measured as IQRatio (cf. Figure 1C). C–D Negative feedback abrogates the trade-off in
and
invariance. Cell-to-cell variability simulations (similar to panel B) were repeated for various parameter configurations for models with upstream feedback (C) or downstream feedback (D): activation resistances in the cascade were tuned by simultaneously changing the phosphatase rate constants
(x-axis). The variabilities of
and
were analyzed using the IQRatio as in Figure 1C, and similar results are obtained using the coefficient of variation (Figure S2).
was defined as the stimulus for a half-maximal pathway activation. The behavior of a feedback model with limited feedback strength (
; thick, solid lines) is compared to a feedback-less model (
; thin, dashed lines) and to a model with very strong feedback
; thin, solid lines). Simulations for moderate feedback strength (thick lines) were performed by numerically integrating the ODE systems (Eqs. 8 and 12), while the strong feedback calculations (thin solid lines) were done using analytical approximations (Eqs. 9 and 13).
Figure 3.
Cell-to-cell variability of kinase cascades with distributed ultrasensitive switching.
A Simulations of a cascade with distributed ultrasensitive switching and low activation resistance shows a steep response with little variability in (defined as the stimulus for a half-maximal pathway activation). The simulations of the five-step cascade were performed by iteratively applying the Hill equation describing the steady state of each level (similar to Eq. 14). The concepts and parameter values correspond to Figure 1B, with a Hill coefficient
(Supplemental Table S1). Colored box plots represent the
and
distribution of the ultrasensitive model, while gray box plots show the behavior of the reference gradual cascade (cf. Figure 1B). B
is strongly controlled by the first kinase concentration, whereas
primarily responds to fluctuations in the terminal kinase (concept similar to Figure 1D). C Simulations of a cascade with distributed ultrasensitive switching show that the threshold variability can be reduced by coregulating the first level kinase (
) and second level phosphatase (
) concentrations. Correlation was modeled by introducing a proportional relationship between both concentrations. D–E The variabilities of
and
were analyzed using the IQRatio as in Figure 1C, but plotted against changes in the kinetic parameter value for only the second level phosphatase (
). Similar results are obtained using the coefficient of variation as a measure of variability (Figure S3). The markers 1–3 correspond to the respective dose-response density plots shown in E. A high density (red) corresponds to a high number of cells showing a particular stimulus-response relationship. Three modes of variability are visible in E: 1) for low resistance values, the variability in
is low and all cells are able to respond to stimulation; 2) the variability increases at intermediate resistance levels, because only a fraction of the cells respond while the remaining cells do not even for high stimulus values; 3) in case of a high activation resistance no cell is able to respond.
Figure 4.
Cell-to-cell variability of cascades with a localized switch at the terminal level.
A Simulations show that pronounced variability for both (defined as the stimulus for a half-maximal pathway activation) and
. The concepts and parameter values correspond to Figure 1B, and the simulations were performed by iteratively applying Eqs. 16 and 17 with a Hill coefficient
(Supplemental Table S1). Colored box plots represent the
and
distribution of the ultrasensitive model, while gray box plots show the behavior of the reference gradual cascade (cf. Figure 1B). B The variabilities of a cascade with a localized switch at the terminal level were analyzed using the IQRatio, and the activation resistance was tuned by varying several phosphatase rate constants (
–
, thick, solid lines), and compared to a gradual model (thin, dashed lines). In contrast to a cascade with distributed ultrasensitivity (Figure 3D), homogeneous switching of all cells at a defined stimulus value is not possible even for low activation resistances. Similar results are obtained using the coefficient of variation as a measure of variability (Figure S4).
Figure 5.
Cell-to-cell variability of cascades with a localized switch at the terminal level and basal transcriptional feedback.
A Schematic representation of the five-step cascade with an ultrasensitive terminal step and basal transcriptional feedback. Assuming fast pathway dynamics and slow expression dynamics (time-scale separation), the system can be considered to exist in two states: at basal levels of stimulus, induces the expression of the second level phosphatase
(Eq. 18). Upon acute stimulation the pathway responds rapidly but the expression kinetics of the phosphatase are too slow to establish a significant feedback regulation (Eq. 19). B Simulations of a cascade with a localized switch at the terminal level and basal transcriptional feedback show a reduced variability when compared to the ultrasensitive model without basal transcriptional feedback shown in Figure 4A. The concepts and parameter values correspond to Figure 1B, and the simulations were performed numerically integrating the ODE system given by Eqs. 16–19, with a Hill coefficient
, a basal stimulus of
, a basal synthesis rate
, an
-induced synthesis rate constant
, and a degradation rate
(Supplemental Table S1). Colored box plots represent the
and
distribution of the basal transcriptional feedback model, while gray box plots show the behavior of the reference gradual cascade (cf. Figure 1B). C Variabilities of
(defined as the stimulus for a half-maximal pathway activation) and
were analyzed using the IQRatio, and the activation resistance was tuned by varying several phosphatase rate constants (
–
, thick, solid lines). The variability of the gradual model is shown for comparison (thin, dashed lines). The variant with basal transcriptional feedback is able to strongly reduce the variability in
for low activation resistance values when compared to the single-switch model without feedback (cf. Figure 4B). Similar results are obtained using the coefficient of variation as a measure of variability (Figure S5).
Figure 6.
Cell-to-cell variability of cascades with coherent feedforward regulation.
A Schematic representation of the five-step cascade with a coherent feedforward loop: the -mediated phosphorylation of
is positively regulated by the kinase
(see main text). B Simulations of a cascade with a coherent feedforward loop show reduced variability when compared to the single-switch model without feedforward regulation (Figure 4A). The concepts and parameter values correspond to Figure 1B, and the simulations were performed by iteratively applying Eqs. 16 and 20, with a Hill coefficient
and
(see Supplemental Table S1). Colored box plots represent the
and
distribution of the feedforward model, while gray box plots show the behavior of the reference gradual cascade (cf. Figure 1B). C The variabilities of
(defined as the stimulus for a half-maximal pathway activation) and
were analyzed as a function of the activation resistance by varying several phosphatase rate constants (
–
, thick, solid lines), and compared to a gradual model (thin dashed lines). Feedforward regulation plays no role at low activation resistances (point 1), but reduces the variability at intermediate activation resistances (point 2; see main text). High variability arises at high resistances, because not all cells reach the threshold for full
activation (point 3). Similar results are obtained using the coefficient of variation as a measure of variability (Figure S6).
Table 1.
Determinants of cell-to-cell variability in gradual and ultrasensitive signal transduction.