(A) Fraction of parameter sets resulting from 10,000 algorithm iterations that generated hysteretic (n = 4,541), biphasic (n = 4,878) or dual dynamics (Dual(H/B); n = 1,290). Orange region within results for hysteresis and biphasic are the subset that support dual behavior. Hysteresis is measured by calculating the path difference 24 hours after an increase in serum from 0.01% or after a decrease from 10%. Biphasic response is measured 36 hours after an increase in MYC synthesis (parameter ke_MC; Text S1). (B) Network state for biphasic behavior. Median value of parameters that support biphasic dose-response using hysteresis as the baseline (i.e. zero value). Related parameters were grouped into individual modules. (C) (Left) Distribution of solution module strengths (product over all parameters) for solutions to hysteresis and biphasic behavior. Also indicated are median values for hysteresis (square), biphasic response (triangle), and Dual(H/B) (circle) dynamics. (Right) Coordination of hysteresis and biphasic behavior. Relative module strengths are indicated by line thickness. I – growth inputs from serum and MYC; E – E2F. (D) Simulations of a representative Dual(H/B) solution in the vicinity of the median. R module strength was modulated by replacing it with median value from solutions for hysteresis (−R) or biphasic response (+R). E2F is expressed in µM. (E) Resilience of individual solutions. Boxplots summarize the resilience of 50 solutions for hysteresis, biphasic, and Dual (H/B). Solutions were selected by identifying the smallest box (values of sensor and R) centered on the median containing 50 solutions. Resiliency is defined as the ability to maintain at least 10% of their objective score following parameter perturbation. Y-axis shows the fraction of 10,000 repeated perturbations to a particular solution that are resilient. Circle indicates median; Medians are significantly different at the 5% significance level if there is no overlap between intervals defined by their triangular notches.

(A) Extended model includes duplicated E2F species (E2F′) with independent regulation of synthesis and degradation. E2F dynamics are the sum of E2F and E2F′. All other interactions described in Figure 1B are maintained but omitted here for clarity. (B) Fraction of parameter sets resulting from 10,000 algorithm iterations that support hysteresis (n = 2,508), adaptation (n = 3,704) or dual dynamics (n = 182). (C) Network configuration for adaptation relative to hysteresis. Parameters specific to E2F′ are shown by open bars and all others are shared between the two E2F species. (D) Resilience of solutions in the vicinity of the median for hysteresis, adaptation, and Dual(H/A) data. Each boxplot summarize the results from 50 solutions, each perturbed 10,000 times.

(A) (Left) A generic pathway consisting of system inputs (I); upstream and downstream mediators (black bars); a core network; and outputs (O). (Middle) The mitogen-activated protein kinase (MAPK) cascade mediates a myriad of growth factor signals and elicits, sometimes opposing responses in a cell-type specific manner. (Right) The p53 network responds to a variety of cell stresses and elicits an appropriate course of gene expression that mitigates uncontrolled proliferation. (B) The RB-E2F switch, partitioned into four modules: Sensor (cyan edges); repression (R, red); negative feedback (NFB, purple); and positive feedback (PFB, green). This pathway performs at least three dynamic tasks in response to growth signals. (C) Algorithmic approach to search parameter space. (D) Calculating tension. Given a network, solution sets of parameters able to generate each dynamic (A and B) are identified. Tension between tasks (T_AB) is defined as the weighted sum of the log-ratio of median parameter values.

(A) Fraction of parameter sets resulting from 10,000 algorithm iterations that generated hysteresis (n = 4,541 solutions), adaptation (n = 3,353) or dual dynamics (n = 105). Temporal adaptation occurs in response to a shift in serum from 0.01% to 10%. (B) Network state for adaptation. (C) (Left) Distribution of module strengths for solutions to hysteresis and adaptation. Also indicated are median values for solutions to hysteresis (square), adaptation (triangle), and dual (circle) dynamics. (Right) Coordination of hysteresis and adaptation. (D) Simulations of a representative Dual(H/A) solution in the vicinity of the median (circle in Figure 3C). The NFB strength was modulated by replacing it with median values from hysteresis (−NFB) or adaptation (+NFB). For serum response, solid lines represent levels 24 hours after an increase from 0.01% serum; dotted lines are levels when initial conditions were 10% serum. E2F is expressed in µM. (E) Resiliency of solutions in the vicinity of the median for hysteresis, adaptation, and Dual (H/A) data. Each boxplot summarize the results from 50 solutions, each perturbed 10,000 times.

(A) Relationship between tension and the accessibility of dual solutions for each pair of dynamics examined. Abbreviations: H/B – hysteresis paired with biphasic; H/A – hysteresis paired with adaptation; 1×E2F – single E2F model; 2×E2F – double E2F model. (B) Relationship between tension and resiliency to parameter perturbation. Boxplots span inter-quartile range and medians are indicated with a circle for 50 dual solutions in each category. Two medians are significantly different at the 5% level if the interval between triangular notches do not overlap. Data taken from results presented in Figure 2,3, and 4.