2.1 Gradual protein kinase cascades show strong cell-to-cell variability.

Multi-step signaling cascades show a gradual dose-response behavior if the response of each individual cascade level is gradual as well [3,30]. A minimal model of a gradual signaling cascade can be implemented by assuming that enzyme saturation in the phosphorylation and dephosphorylation reactions at each level are negligible [4].

The following ordinary differential equation describes the temporal evolution of the active kinase :(1)

Each phosphorylation step is described as a reaction between the phosphorylated form of kinase and the non-phosphorylated form of a downstream kinase (Figure 1A). The corresponding phosphorylation rate is given by the term where is the second-order rate rate constant for phosphorylation of the -th kinase. Similarly, the dephosphorylation of the active form is described by the rate , with the dephosphorylation rate constant and total phosphatase concentration designated as and , respectively [4]. Equation 1 takes into account that the total kinase concentration at each level is constant, i.e., .

The steady state activity of each cascade level describing the activity upon long-term stimulation can be calculated by assuming that the kinase concentrations do not change over time ().(2)

This expression relates the activity of the -th kinase to that of its upstream activator , and therefore characterizes the local dose-response behavior of the cascade. It has the form of a Michaelis-Menten equation: Each cascade level may saturate if the kinase pool is fully phosphorylated () and half-maximal activation occurs when the kinase and phosphatase activities are equal .

The cellular response to stimulation is determined by the global dose-response curve which relates the activity of the terminal cascade level to the concentration of the extracellular stimulus S. By iteratively applying Eq. 2 and setting the stimulus to , one derives for the global dose-response curve of a five-step cascade ()(3)

This Michaelis-Menten-like equation increases gradually for increasing concentrations of the stimulus , confirming that the minimal cascade model shows gradual dose-response behavior. The parameter describes the maximal activation level of the pathway upon strong stimulation. equals the stimulus concentration leading to half-maximal signaling, and thus reflects the pathway sensitivity to stimulation. and are lumped parameters that can be defined as(4)

is the dissociation constant of receptor-ligand binding, and the remaining are proportional to the kinase and phosphatase concentrations in the cascade,(5)To understand the cell-to-cell variability, we need to know how and depend on the total kinase and phosphatase concentrations. We initially analyze cell-to-cell variability for the case of weak stimulation () where the pathway dose-response curve in Eq. 3 can be approximated by the following linear equation:(6)

The signaling activity upon weak stimulation is thus determined by the product of five kinase concentrations () divided by the product of four phosphatase concentrations (). This implies that a weakly stimulated cascade exhibits strong cell-to-cell variability, because the product of fluctuating species shows much greater variability than either species alone. The total variance in the signaling output upon weak stimulation () equals the sum over all signaling protein concentration variances ( for kinases and for phosphatases) (Supplemental Text S1).(7)

Two conclusions can be drawn from this equation concerning the regulation of variability: (i) the variability cannot be reduced significantly by lowering the expression noise of certain signaling proteins; instead, a simultaneous noise reduction of all species would be required: The cell-to-cell variability can be quantified using the inter-quartile ratio (IQ ratio) which expresses the difference of cells with high and low signaling activities by dividing the third and first quartiles of the distribution (see Methods). Assuming realistic protein concentration fluctuations in Eq. 6, the IQ ratio only drops from 4.1 to 3.8 if the noise of 1 out of 9 signaling protein concentrations is eliminated, implying that the cell-to-cell variability remains essentially unchanged. (ii) the variability does not depend on the choice of the kinetic parameter values . Thus, the weakly stimulated gradual signaling pathway always shows strong variability. Consistent with the expectation, we find that lesser variability may be observed upon strong stimulation where (see below).

The variability principles derived from the analytical model (Eq. 1–7) were confirmed by explicit cell-to-cell variability simulations. To this end, each of the nine protein concentrations in the cascade was sampled from a log-normal distribution with a coefficient of variation that matches the experimentally observed variability of eukaryotic protein expression [12]. Dose-response simulations were performed for each set of sampled protein concentrations, yielding cohorts of dose-response curves representing the cell population. Such a simulation is shown in Figure 1B, and cells with the highest and lowest or are highlighted by the shaded areas. These cell-to-cell variability simulations confirm that the minimal gradual protein kinase cascade generally shows pronounced variability, especially at low-level stimulation.

2.2 A trade-off in controlling the variability of maximal pathway activation and pathway sensitivity.

We investigated how the variabilities of the maximal pathway activation () and the pathway sensitivity () depend on the kinetic parameters in the cascade. and are fully described by the lumped parameters (Eq. 4). Each equals the phosphatase activity at a cascade level divided by the maximally possible kinase activity (Eq. 5). Thus, quantifies the tendency of a cascade level to be fully activated upon strong stimulation and can be considered as an activation resistance. A strong stimulus fully activates the pathway kinases only if all resistances are low ().

We tuned the activation resistances () by simultaneously changing all phosphatase activities, and performed cell-to-cell variability simulations (Figure 1C). For low phosphatase activity at each level (), we observe little variance in the maximal pathway activation, because only the concentration of the terminal kinase matters (). At the same time, the pathway sensitivity is determined by the product of multiple protein concentrations (), and therefore differs strongly between individual cells. In the opposite regime of high phosphatase activity at each level (), we find that the pathway sensitivity is completely invariant between cells. This is because the receptor level saturates before the subsequent cascade steps, implying that the dose-response curve of the terminal kinase is aligned to the half-maximal saturation point of receptor-ligand binding (). In this regime, the maximal activation level is, however, determined by all protein concentrations in the cascade and thus highly variable (). These simulations reveal that the variabilities of and are inversely related. The drop in the variability of precisely matches the parameter range where the variability of increases (Figure 1C). We show more generally in Supplemental Text S1 that and are inversely related for any parameter change in the signaling cascade. Thus, a trade-off exists in a simple gradual protein kinase cascade: either the pathway sensitivity or the maximal pathway activation can be made invariant by changing the kinetic parameter values. However, it is not possible to make and invariant at the same time.

The signaling variability could be reduced by lowering the expression noise of individual signaling proteins. We therefore investigated whether fluctuations in certain signaling protein concentrations have particularly strong impact on the variabilities of the maximal pathway activation () and the pathway sensitivity (). To this end, and were related to the signaling protein expression levels in single cells (Figure 1D). Cells with a high expression level of the terminal kinase () tend to have a higher maximal pathway output than cells harboring low levels of the terminal kinase. No such correlation is observed for the kinase concentration at the first cascade level (). Thus, the downstream species tend to exert a stronger control on the maximal pathway output than the upstream species (see also Supplemental Text S1). In contrast, the pathway sensitivity is primarily determined by upstream species in the cascade: Cells tend to respond at lower ligand concentrations the higher the expression level of the first kinase () is, and the concentrations of the downstream species play a lesser role in this respect (Figure 1D, top row; Supplemental Text S1). Taken together, we find that the maximal output and the pathway sensitivity are controlled in a very different way. We show in Supplemental Text S1 that signaling protein concentrations with strongly control over generally have lesser impact on (and vice versa). Thus, while either or can be made invariant by reducing the expression noise of certain signaling proteins, it is not possible to achieve invariance for both features at the same time.

The trade-off in the regulation of and has important ramifications for the control of intracellular signaling: Intracellular signaling regulators or pharmacological inhibitors acting upstream in the cascade primarily regulate the pathway sensitivity, whereas downstream regulators predominantly affect the maximal pathway activation. The strong and parameter-independent dose-response variability suggests that the simple gradual model is unable reflect the invariant dose-response behavior of the yeast pheromone pathway (see Rationale): all cells of the population respond upon strong stimulation if the activation resistances are low (), but then the pathway sensitivity fluctuations are high, and the most sensitive cells respond at a ∼100-fold lower stimulus concentration than the least sensitive cells (Figure 1B).

2.3 Negative feedback regulation allows for the simultaneous invariance of maximal pathway activation and pathway sensitivity.

Negative feedback regulation reduces the variability of biological systems [31–33]. In the following, we show that negative feedback resolves the above-mentioned robustness trade-off by simultaneously promoting the invariance of maximal pathway activation () or pathway sensitivity ().

In the yeast pheromone pathway, the terminal kinase promotes the deactivation of the pathway by negatively regulating the G protein level [22,34], and this negative feedback loop has been reported to control the pathway sensitivity to stimulation [34]. We implemented negative feedback in a gradual five-step protein kinase cascade by assuming that the final kinase () enhances the activity of the phosphatase at the second level (Figure 2A, solid red line). Most of the differential equations remain unchanged when compared to the basic model (Eq. 1), but the ODE for the second pathway level now reads:(8)

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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).

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The rate of deactivation is multiplied by the term to reflect that the dephosphorylation is enhanced by . The parameter determines how strongly promotes the deactivation of . The exponent indicates a possible cooperativity of negative feedback regulation (: feedback with positive cooperativity). For pronounced feedback regulation, i.e., , the steady state dose-response curve can be approximated by (Supplemental Text S1):(9)

In line with previous studies, we find that the kinase cascade with negative feedback regulation exhibits a shallower dose-response curve than the feedback-less system, because the stimulus enters as the -th root only [32,33,35]. The non-cooperative feedback system requires a ∼420-fold increase in the stimulus level to switch from 10% to 90% of the maximal activation level, while an 81-fold increase is sufficient in the corresponding feedback-less cascade (Eq. 3). Negative feedback therefore extends the gradual mode of quantitative information transmission to a large stimulus concentration range, and the effect is more pronounced for cooperative feedback regulation .

Strong negative feedback reduces the cell-to-cell variability of the dose-response curve: The half-maximal stimulus of the cascade is proportional to the half-saturation point of receptor-ligand binding (), and completely independent of fluctuations in the signaling protein concentrations (cf. Figure 2B)(10)

This protein concentration insensitivity can be explained as follows: strong negative feedback shifts the species to to very low activation levels, implying that the downstream part of the cascade does not saturate. The dose-response curve of the cascade thus follows the receptor-ligand binding isotherm, though with a more gradual shape. Such pathway alignment to the receptor dose-response curve due to negative feedback has been observed experimentally in the yeast pheromone signaling cascade [34].

The effects of strong negative feedback on the maximal pathway activation variability are less pronounced. Assuming log-normally distributed gene expression noise, the variance of can be derived from Eq. 9 and represented as a function of the signaling protein concentration variances (Supplemental Text S1)(11)

The variability is determined by the sum of all protein concentration variances, but is reduced by the feedback term . This result confirms previous observations showing that cooperative feedback suppresses noise more efficiently than linear feedback [36].

The and variabilities of the feedback system are independent of the activation resistances in the cascade and low compared to a feedback-less cascade (thin solid and thin dashed lines in Figure 2C). Thus, negative feedback allows for the simultaneous invariance of the maximal pathway activation and the pathway sensitivity, thereby resolving the robustness trade-off of the feedback-less cascade. Moreover, the negative feedback system shows the same signaling variability for low and high stimulus levels (Eq. 9), implying that quantitative information transmission is possible over a very broad stimulus concentration range. These conclusions continue to hold for an equivalent negative feedback system, where the terminal kinase inhibits the activity of , thereby controlling the phosphorylation reaction of . This can be seen in the steady state condition (Eq. 8) which can be converted to the kinase inhibition case by division with the feedback term .

We confirmed our findings concerning negative feedback regulation for more realistic feedback cascades with limited feedback strength . Figure 2C shows that the moderate feedback system shows a simultaneous invariance of maximal pathway activation () and the pathway sensitivity () over a finite range of activation resistances in the cascade (thick solid lines), and the variability tends to be lower than that of a feedback-less cascade (dashed lines). The strength of the feedback regulation primarily affects the width of the compromise range where and are simultaneously invariant: Limited feedback cannot perform any regulatory function for high activation resistances (), because is barely activated in this regime. Likewise, moderate feedback cannot efficiently counteract the strong signaling activity of a cascade with too low activation resistance ().

2.4 Negative feedback loops acting upstream and downstream in the cascade control different aspects of the dose-response curve.

Signaling cascades are often equipped with multiple negative feedback loops, some acting close to the receptor level, while others modulate the terminal cascade levels [23]. We investigated how the length of a negative feedback emanating from affects the dose-response behavior of the cascade. Consider a cascade with a short, downstream feedback, where activates its own phosphatase (Figure 2A, dashed red line). Such downstream feedback regulation occurs in the yeast pheromone pathway, as Msg5, the phosphatase acting at the terminal cascade level, is transcriptionally induced upon stimulation [22,37]. Again, most of the ODEs remain unchanged when compared to the basic cascade model (Eq. 1), but the fifth pathway level reads:(12)

The steady state condition of the upstream feedback also describes an equivalent negative feedback system, where the terminal kinase inhibits the activity of its own activator . This can be seen by dividing the steady state condition with the feedback term . We again approximate the steady state for strong feedback () and obtain (Supplemental Text S1)(13)

The steady state of is proportional to the root of the Michaelis-Menten equation, and the dose-response curve is thus as shallow as that of the system with upstream feedback (Eq. 9). In similarity to Eq. 10, the half-saturation point of Eq. 13 is proportional to the half-maximal stimulus of (i.e., ). Downstream feedback thus eliminates the impact of the terminal level on the pathway sensitivity, but any variability arising between and is transmitted. Downstream feedback suppresses the variability of the maximal pathway activation, especially for high feedback cooperativity n (Eq. 13), and achieves the same or stronger invariance when compared to the upstream feedback system (thin orange lines in Figure 2C and Figure 2D): this is because the upstream signaling protein concentrations (parameters , and ) have a lower impact in Eq. 13 than in Eq. 9 if the phosphatase activities in the cascade are low . We conclude that only upstream feedback efficiently suppresses fluctuations, while downstream feedback has the stronger impact on the variability. For both systems, increasing feedback cooperativity selectively suppresses the fluctuations.

We analyzed the downstream feedback model with limited feedback strength (thick lines in Figure 2D). Moderate downstream feedback also resolves the robustness trade-off in protein kinase signaling by allowing for a simultaneous and invariance at intermediate activation resistances. Moderate downstream feedback reduces fluctuations to a much lesser extent than upstream feedback, while having a slightly more pronounced effect on the variability (Figure 2C and Figure 2D). Taken together, upstream and downstream feedback loops differentially control the dose-response behavior also at moderate feedback strengths, although the differences are less pronounced compared to the case of strong feedback (Figure 2C and Figure 2D).

Our models predict that upstream negative feedback in the pheromone pathway may contribute to the invariant shmooing threshold, while downstream negative feedbacks may primarily ensure that all cells exhibit a similar maximal activation upon strong stimulation. One limitation of the negative feedback models is their shallow dose-response behavior which is inconsistent with the reported ultrasensitivity of the pheromone pathway [18,20,21]. We turn to ultrasensitive signaling cascades in the following to study more realistic models of yeast pheromone sensing.

3.1 Ultrasensitive cascades with distributed switching can be inherently invariant.

Multi-step signaling cascades are capable of strong ultrasensitivity amplification, implying that a combination of multiple weak switches establishes a very steep overall dose-response curve. To simplify the mathematical analysis, we initially analyze a two-step signaling cascade with ultrasensitivity at each level(14)The steady state of each cascade level is represented by the Hill equation, which has a structure analogous to the local dose-response behavior of a gradual signaling cascade (Eq. 2). The maximal activation of each cascade level equals the total concentration of the respective kinase (), and the half-saturation point is determined by the parameters and . equals the equivalence point of kinase and phosphatase activities in ultrasensitive (de)phosphorylation systems, and is thus determined by the concentration of a phosphatase [5]. Ultrasensitive, sigmoidal dose-response behavior can be observed for Hill coefficients .

We analyzed the overall dose-response curve relating the signaling output to the stimulus , and found that the ultrasensitive behavior is amplified along the cascade (Supplemental Text S1). Assuming a low phosphatase activity at the second level (), the threshold where the system switches from low to high activation is given by(15)

The activation resistances and are defined in Eq. 5. The threshold depends in a less-than linear manner on the kinase and phosphatase concentrations controlling the second level (), thus showing little variability. At the same time, the maximal activation level depends the concentration of the terminal kinase only () and shows partial invariance as well. The ultrasensitive system thus shows a less pronounced robustness trade-off when compared to the gradual system, and can simultaneously show little variability of and .

Numerical simulations were performed for a five-step signaling cascade, where each level was modeled using a Hill equation with n = 2 (similar to Eq. 14). The five-step signaling cascade exhibits very strong ultrasensitivity if low phosphatase activities are assumed for all cascade levels (Figure 3A; ). The system shows little cell-to-cell variability, as all cells respond in a switch-like manner within a ∼3-fold range of stimulus concentrations. As with the two-step cascade, the threshold in these cell-to-cell variability simulations is almost exclusively determined by the upstream signaling species: correlates well with the concentration of the first kinase , but not with signaling protein concentrations controlling subsequent cascade steps (Figure 3B).

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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.

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The inherent invariance of the ultrasensitive system can be understood intuitively by considering an extreme case scenario, where each cascade level is a very steep switch (): In this case, all downstream cascade levels simultaneously respond as soon as the first level is switched on. The system thus behaves like a chain of dominos, and the threshold of the first level sets the threshold of the whole cascade. The phenomenon is less pronounced for the case of moderate switching at each step, so that the downstream protein concentrations still matter to some extent (Eq. 15).

We conclude that the coordinated switching of the whole yeast cell population within a ∼3-fold range of pheromone concentrations could be explained based on the ultrasensitive model with distributed switching (Figure 3A). The prediction of local ultrasensitivity which is then amplified along the cascade could be tested experimentally by measuring and relating the dose-response curves of several kinases in the cascade.

3.2 Active variability regulation in ultrasensitive cascades by gene expression noise control and parameter tuning.

The threshold variability of the ultrasensitive system strongly depends on the noise of , the kinase-phosphatase ratio at the upstream cascade level (Eq. 15; Figure 3B).The noise of could be reduced by correlating the fluctuations of the respective kinase and phosphatase concentrations. Such correlated fluctuations may be realized in the yeast pheromone pathway, because the pheromone receptor Ste2 and the antagonizing G protein deactivator Sst2 are transcriptionally co-regulated by the transcription factor Ste12 [22,37]. In Figure 3C, we simulated the five-step signaling cascade with moderate switching at each level, and introduced correlated fluctuations between the first kinase concentration and the antagonizing second phosphatase concentration . We find that this system exhibits less variability when compared to the uncorrelated case, as all cells respond in a switch-like manner within a ∼2-fold range of stimulus concentrations (compare Figure 3A and Figure 3C). Experimental work supports that correlated fluctuations in upstream kinase and phosphatase concentrations reduce the variability of mammalian MAPK signaling [7]. We propose to simultaneously measure the expression levels of fluorescently labeled Ste2 and Sst2 in single-cells to confirm that a similar mechanism promotes the invariance of yeast shmooing.

Correlated fluctuations in a single kinase-phosphatase pair would also promote invariance in gradual signaling cascades, but only to a minor extent, because the remaining seven protein concentration variabilities still enter the signaling activity upon weak stimulation in an additive manner (Eqs. 6 and 7): The cell-to-cell variability of a gradual cascade with realistic protein concentration fluctuations, quantified as the inter-quartile ratio (see Methods), only drops from 4.1 to 3.5 if a perfect correlation is introduced for a single kinase-phosphatase pair. This suggests that correlations in upstream signaling protein concentrations specifically promote the robustness of ultrasensitive systems.

A way to increase the variability of the ultrasensitive cascade relative to Figure 3A is kinetic parameter tuning, e.g., by increasing the activity of certain phosphatases. Figure 3D shows the variabilities of maximal pathway activation () and the pathway sensitivity () for varying phosphatase expression at the second level. Both variabilities clearly increase for increasing phosphatase expression, and the variance of peaks at intermediate levels. Increasing phosphatase expression introduces heterogeneity because a fraction of the cell population becomes completely insensitive to stimulation. This can be seen in Figure 3E, where the dose-response curve distributions of the cell population are indicated by density plots for different phosphatase levels. For instance, at intermediate phosphatase levels, half of the cells do not respond at all to stimulation, while the remainder shows essentially complete activation of (Figure 3E, panel 2). Thus, increasing phosphatase expression introduces heterogeneity, because the system switches from a strong and synchronous response of the whole population to a strong response in only a fraction of cells. The variability peaks at intermediate phosphatase activities, because the stimulus level required for half-maximal activation is different in responding and non-responding cells (not shown). At very high phosphatase levels, the population only consists of non-responders, thus again showing less variability (Figure 3E, panel 3).

3.3 Ultrasensitive signaling cascades with switching at a single step show strong variability.

Switch-like decision making may also be established if a single cascade level shows a very steep dose-response curve. Such a localized switch has been reported for the yeast mating pathway [18]: The scaffold protein Ste5 co-localizes members of the MAPK cascade, and its activity is regulated by a multisite dephosphorylation mechanism, thereby promoting the switch-like phosphorylation of the terminal MAPK cascade member Fus3. We mimic this scenario by assuming that the terminal cascade level is phosphorylated by in a switch-like manner, whereas the upstream part of the pathway shows gradual behavior. In similarity to Eq. 3 the steady state of can be written as(16)

The switch-like dose-response at the terminal level () may be represented by the Hill equation :(17)

The threshold determines the equivalence point of kinase and phosphatase activities at the fifth level, and is proportional to the concentration of the phosphatase . We performed cell-to-cell variability simulations of this system assuming a high Hill coefficient n (Figure 4A), and investigated how the variabilities of maximal pathway activation and threshold stimulus depend on the kinetic parameter values (Figure 4B). Increasing phosphatase expression shifts the system from complete switching of the whole cell population to incomplete switching of only a fraction of cells, reminiscent of the cascade with distributed switching (Figure 3E). Cells only respond to stimulation if the maximal activation level of is larger than the Hill equation threshold . The single-switch system shows strong variability even for low phosphatase activities in the cascade, because seven signaling protein concentrations jointly determine the signaling threshold (Supplemental Text S1). The simulated signaling thresholds vary over three orders of magnitude as long as all cells of the population respond strongly to stimulation (Figure 4A), implying that the single switch system cannot explain the experimentally observed invariance of the shmooing threshold (see Rationale). We show in the following that invariance can be realized if the single-switch model is extended by feedback or feedforward loops.

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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).

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3.4 Suppression of signaling threshold variability by basal transcriptional feedback.

Negative feedback diminishes the variability of signaling cascades, though at the cost of a reduced steepness of the dose-response curve (Eq. 9). In this Section, we demonstrate that switching and invariance can be combined if the time window of variability suppression by negative feedback can be separated from the time window of switch-like decision making.

Feedback loops in mammalian signaling commonly involve transcriptional regulation, and the signaling cascades typically induce the expression of their own inhibitors [23]. In the yeast mating pathway phosphatases negative regulators like the G protein deactivator Sst2 and the phosphatase Msg5 are transcriptionally induced upon stimulation [22,37]. Transcriptional feedback requires de novo protein biosynthesis, thereby affecting signal transduction with a delay. Thus, a time window exists early after stimulation where the steepness of the dose-response curve is unaffected by transcriptional feedback. Yet, transcriptional feedback may promote robustness of the dose-response curve if it operates under basal conditions before stimulation as observed for the yeast mating pathway [34,37,40]: Basal feedback inhibitors are able to correct for the basal state variability, because their concentration reflects and tunes the basal signaling activity. This variability suppression effect can be memorized to the time window of acute stimulation, as long as the concentration of the feedback inhibitor remains stable. In the following, it will be shown that this memorization of the pre-stimulation state efficiently suppresses variability upon acute stimulation.

We model basal transcriptional feedback by implementing a negative feedback loop in the single-switch cascade. The phosphatase at the second level is transcriptionally induced by the active terminal kinase (Figure 5A) and also transcribed with a basal rate . Additionally, is subject to degradation , giving rise to the following differential equation(18)

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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).

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Basal signaling was implemented by assuming a low chronic level of the stimulus , and ligand-induced signaling was simulated by further increasing . Time scale separation was introduced by neglecting the induction of by transcriptional feedback early after stimulation. The concentration of is thus fixed to the basal level throughout the time window of acute stimulation, i.e.,(19)

We performed cell-to-cell variability simulations of this system assuming a high Hill coefficient (Figure 5B). The basal transcriptional feedback model shows a strongly reduced threshold variability when compared to a feedback-less cascade with the same kinetic parameters (compare Figure 4A and Figure 5B). The simulated signaling thresholds lie within a ∼3-fold range of stimulus concentrations (Figure 5B), which is consistent with the experimentally observed invariance of the shmooing threshold (see Rationale). Strong variability suppression is possible, because the basal signaling activity and the pathway threshold are controlled by the same combination of parameters (Supplemental Text S1): Transcriptional feedbacks correcting for fluctuations in basal signaling thereby indirectly correct the threshold variability as well. The threshold invariance of the basal feedback system is more pronounced if the dose-response curve of the signaling pathway is highly switch-like, because cooperativity promotes robustness in negative feedback circuits.

The invariance due to basal feedback is restricted to a certain range of activation resistances in the cascade, because increasing the expression of several phosphatases in the cascade shifts the system from complete switching of the whole cell population to heterogeneous switching of only a fraction of cells (Figure 5C). In Figure 5B and Figure 5C, we assumed that heterogeneity in expression solely arises from fluctuations in the upstream signaling cascade. However, basal transcriptional feedback can exert a similar variability suppression if the synthesis rate is also noisy and sampled from a log-normal distribution (not shown).

Recent experimental evidence supports that Msg5, a basal transcriptional feedback regulator of the yeast mating pathway, suppresses the signaling variability upon pheromone stimulation [26]. We propose to eliminate basal transcriptional feedback by exchanging the endogenous promoter of the Msg5 gene by a promoter that is not regulated by the pheromone, and predict a strong increase in the signaling threshold variability.

3.5 Variability suppression by coherent feedforward regulation.

Signaling networks commonly involve coherent feedforward loops where an upstream kinase controls a downstream target by two parallel pathways. Fus3, the yeast MAPK that mediates the shmooing response, is controlled by two pheromone-dependent pathways, both of which are required for full activation [18]: Fus3 is phosphorylated by the upstream MAP kinase kinase Ste7, and additionally needs to be released from an inhibitory site in the scaffold protein Ste5 for activation. The latter step involves Ste5 multisite dephosphorylation by the phosphatase Ptc1. It will be shown below that coherent feedforward regulation of Fus3 may contribute to the invariance of shmooing.

We model coherent feedforward regulation by extending the model of the five-step signaling cascade with a switch at the terminal level (Eqs. 16 and 17). The downstream signaling species and represent the MAPKK Ste7 and its target Fus3, respectively. Ptc1 is regulated by pheromone pathway intermediates upstream of Fus3 [18], but the precise molecular mechanism is not known. We assume in the model that the Ptc1 activity is directly activated by the pheromone receptor (). and that the Ptc1 pathway enhances –mediated phosphorylation of (Figure 6A). The effective kinase concentration is thus enhanced by the feedforward term and the modified steady state equation for reads(20)

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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).

https://doi.org/10.1371/journal.pcbi.1003357.g006

The threshold is proportional to the concentration of the phosphatase , and thus subject to fluctuations. The parameter determines how strongly the phosphorylation of is enhanced by feedforward regulation. We assume strong crosstalk (large ) and neglect saturation in the upstream part of the cascade (i.e., and ) to ensure that both feedforward branches jointly regulate in a stimulus-dependent manner. The latter assumption is justified if the activation resistances in the cascade are large (). Then, steady state modifies to(21)

In line with previous reports, we find that coherent feedforward regulation increases the ultrasensitivity of the dose-response curve, as the stimulus now enters with the exponent [41]. The switching threshold(22)contains the variabilities of and (determined by the ratios and , respectively). The total variability of the feedforward system is determined by the product of the individual branch variabilities, but each branch variability enters as a root only. Feedforward regulation thus reduces the threshold variability compared to the simple cascade if the feedforward branch is a shortcut and regulated by only a few signaling protein concentrations.

Explicit cell-to-cell variability simulations using Eq. 20 confirm that the threshold variability of the feedforward pathway (Figure 6B) is less than that of a simple ultrasensitive cascade with the same kinetic parameters (Figure 4A). Feedforward regulation robustly reduces the cell-to-cell variability, because partial invariance is observed over a broad range of activation resistances – in the upstream cascade (point 2 in Figure 6C). The cell-to-cell variability increases for low activation resistances (point 1 in Figure 6C), because is activated at much lower stimulus levels than , which prevents efficient feedforward regulation. For too high activation resistances, the maximal activation levels of and are too low to reliably activate , and the system shows high variability due to bimodal splitting into responding and non-responding cells (point 3 in Figure 6C).

We conclude that joint regulation of Fus3 by Ste7 and Ptc1 may reduce the cell-to-cell variability of pheromone signaling. However, the simulated signaling thresholds still span a ∼50-fold range of stimulus concentrations (Figure 6B), implying that feedforward regulation requires cooperation with other variability suppression mechanisms to bring about full robustness. The yeast pheromone pathway comprises feedforward loops other than the one comprising Ptc1, as G proteins employ several parallel pathways to activate the MAPK cascade [22]; it is possible that these multiple feedforward loops synergize to establish an invariant shmooing response.