Figure 1.
A model to study dynamical properties of a scaffold mediated signaling cascade.
(A) Schematic of the events considered in the scaffold mediated signaling cascade. Each kinase, if activated, can activate its downstream substrate when the two proteins are in close proximity. Kinases can bind and unbind to the scaffold and phosphatases can, upon encountering an active kinase, deactivate it. Activation potentially occurs both in solution and on a scaffold. Each forward and backward reaction is modeled as a elementary reactive collision with an energy barrier, E. Energy barriers, E, were taken to be zero so that all kinetics are diffusion limited (B) Key variables and the main quantities computed are shown. ζ is the dimensionless scaffold concentration. The concentration of scaffold proteins [Scaffold] is scaled to the density of the first kinase in the cascade [MAP3K]0 (). The survival probability
, where σ(t) is zero is a kinase has become active and one otherwise (brackets denote an ensemble average); S(t) is the probability that the final kinase has been activated given that it was inactive at t = 0.
, where fA is the fraction of active kinases at steady state; R(t) is the integrated reactive flux of kinase activation. The characteristic time scale of signal propagation τ is defined by the relation S(t = τ) = e−1.
Table 1.
Notation and parameters used.
Figure 2.
The concentration of scaffold proteins sets time scales for signal propagation.
(A) S(t) as a function of time for different values of relative scaffold concentration, ζ (). ζ ranges from 0.005 to 2 times the optimal value. (B) The characteristic time scale τ (S(t = τ) = e−1) is extracted from the curves in (A), and its variation with ζ is shown. Two regimes are observed and are separated by an inflection point. (C) Integrated signal flux R(t) for different values of ζ.
Figure 3.
Scaffold proteins allow for signals to be distributed over many time scales.
Variation of the survival probability with time scaled to characteristic time scales, τ. (A) S(t/τ) (on a semi-logarithmic scale) for different values of ζ. Values of ζ are given in the legend. Large deviations of exponential decay are observed near the optimal value of ζ (ζ = 1, red). (B) Survival probabilities were fit to a stretched exponential function (). Values of stretching exponent β as a function of scaffold density ζ are shown. Two cases are considered: (1) kinases can be activated only while bound to a scaffold (red) and (2) kinases can be activated while in solution and bound to a scaffold (blue). β deviates most from a purely an exponential (β = 1) at the optimal value of scaffold density (ζ = 1) in both cases.
Figure 4.
Dynamics can be characterized by a multi-state kinetic mechanism.
Important time scales in scaffold mediated signaling. (A) Graph of multi-state kinetic model whose dynamics are governed by 8 transitions. Each kinase can transition between four states denoted with four subscripts: in solution (S), bound to a signaling competent complex (C), bound to a signaling incompetent complex (I), and activated (A). (B) Diagram depicting how the various processes occurring at fast time scales couple with scaffold concentration ζ to give rise to collective behavior occurring at slower time scales.
Figure 5.
Power spectra for kinase activation at high, low, and optimal scaffold concentrations.
Plots of where
, are considered. Three cases are considered: low concentration (ζ = 0.005, τ = 1.9×105 mcsteps, β = 1.03), high concentration (ζ = 3.5, τ = 2.2×107 mcsteps, β = 0.97), and optimal concentration (ζ = 1.0, τ = 3.0×106 mcsteps, β = 0.60). On the x-axis, frequency is reported in units that are scaled to the characteristic time for the ζ = 1.0 case, τopt = 3.0×106 mcsteps. The y-axis contains values of P(ωτopt)/(τopt)2.
Figure 6.
Relation between first passage time statistics and signal duration.
(A) Trajectories of x(t) on a semi-log plot. The abscissa represents time scaled by the characteristics time τ. Trajectories for three values of β are shown: β = 0.06 (blue), β = 0.8 (green), and β = 1.0 (red). kφ = 5 for each curve. Smaller values of β result in larger values of x(t) at longer times. (B) Values of signal duration as a function of β for different choices of threshold, T (defined in text); values of T are provided in the legend. For small values of T, β<1 (i.e., the presence of scaffolds) markedly increases signal duration.