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WIREs Syst Biol Med
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Four‐dimensional dynamics of MAPK information‐processing systems

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Abstract Mitogen‐activated protein kinase (MAPK) cascades process a myriad of stimuli received by cell‐surface receptors and generate precise spatiotemporal guidance for multiple target proteins, dictating receptor‐specific cellular outcomes. Computational modeling reveals that the intrinsic topology of MAPK cascades enables them to amplify signal sensitivity and amplitude, reduce noise, and display intricate dynamic properties, which include toggle switches, excitation pulses, and oscillations. Specificity of signaling responses can be brought about by signal‐induced feedback and feedforward wiring imposed on the MAPK cascade backbone. Intracellular gradients of protein activities arise from the spatial separation of opposing reactions in kinase‐phosphatase cycles. The membrane confinement of the initiating kinase in MAPK cascades and cytosolic localization of phosphatases can result in precipitous gradients of phosphorylated signal‐transducers if they spread solely by diffusion. Endocytotic trafficking of active kinases driven by molecular motors and traveling waves of protein phosphorylation can propagate phosphorylation signals from the plasma membrane to the nucleus, especially in large cells, such as Xenopus eggs. Copyright © 2009 John Wiley & Sons, Inc. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models

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Graphical illustration of different MAPK responses to typical stimuli. (a) A sustained, or step input, characterized by the magnitude of the stimulus S. (b) An exponential decay input, characterized by the initial strength of the stimulus S0, and a half‐life, τ1/2; . (c) A rectangular pulse, or pulse‐chase input, characterized by the signal magnitude S, and the signal duration SD. (d) A transient or adaptive response. The peak of the signal is described by the amplitude A and the peak time τp, signal duration TD is related to how long the response lasts, the signaling time τ is the time‐averaged concentration, and the integrated signal I is the area under the curve. Different mathematical forms have been used to quantify these signaling characteristics; for example, Heinrich et al. proposed , and , where K* is activated MAPK.37 (e) A sustained response. The parameters A, τp, and I have similar meaning as for the transient response, whereas the steady‐state is described by a magnitude, Ass, and a time to reach 99% of the steady‐state value, τss. (f) A damped oscillatory response. The steady‐state and peak are characterized similarly as to the sustained response, while the duration of the initial oscillatory period is described by P. Damped oscillations do not always show a constant period. (g) A sustained oscillatory response. After an initial transient period, oscillations are steady with a constant amplitude A and period P.

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Effect of endocytosis on spatial signal propagation. (a) Schematic of spherically symmetric, endocytosis‐enhanced signal propagation. Signals are generated at the PM and in a small region of space where the endosomes reside, and are terminated by phosphatases everywhere else. Signals can diffuse throughout the region between the plasma and NMs. (b, c) Steady‐state spatial profiles of Cp as a function of the distance from the plasma membrane d = RPMr. Dash‐dot lines correspond to cases where half the kinase activity localized at the endosomes (ϕ = 0.5), while solid lines denote cases where all kinase activity is located at the PM. Panel (b) corresponds to a cell with dimensions RPM = 9 µm, Rnuc = 3.5 µm, m, = 6.2 µm. The endosome width, 0.3 µm , was taken to be three times a typical endosome diameter (∼100 nm), since endosomes are not well packed. Panel (c) corresponds to a large cell, such as a Xenopus oocyte, with dimensions RPM = 1000 µm, Rnuc = 390 µm, = 650 µm, and = 649 µm. (d, e) Endocytosis‐enhanced signal amplification at the NM as a function of α and ϕ. Signal amplification is defined as . Cell dimensions used for panels (d) and (e) are the same as for panels (b) and (c), respectively.

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The inter‐scaffold signaling scenario. Signals cause activation of a membrane component which recruits the MAPK cascade scaffold. The scaffold can be empty, or contain any combination of MAPK cascade components, active or inactive. The role of the scaffold is to concentrate the MAPK cascade components into a small volume, where they can effectively interact and propagate the signal. Green color denotes signaling activity, while red color denotes inactivity.

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Summary of the effects of various MAPK cascade features on the steady‐state, input/output relationships. Curve a (black) is a classical Michaelian response, which is the least sensitive, but responds to the largest range of input signals. Curves b (blue) and c (red) denote progressively more sensitive input/output relationships, which can be approximated by a Hill equation , where n is the Hill coefficient and S50 is the half‐maximal dose. Higher Hill coefficients give more sigmoidal responses, and therefore more sensitive responses. Curve d (green) represents a bistable response that shows hysteresis. The solid green lines denote stable steady‐states, whereas the dashed green line denotes unstable steady‐states. As the input magnitude is increased slowly from zero, the systems follow the lower branch of the curve d until the dotted line is reached, at which point the system jumps to the high branch. If the input is then decreased slowly back to zero, the upper branch of curve d is followed until the dotted line is reached, at which point the system jumps back to the low branch.

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Schematic representations of MAPK cascade signal propagation reactions. In all schemes, ATP is assumed to be in excess and rapidly binding. Throughout the literature, different, yet experimentally relevant assumptions have been made regarding rate constants, relative enzyme concentrations and saturability in diverse cell systems, and this has resulted in a variety of different rate laws being used to describe the reaction kinetics. Readers interested in these details are referred to the original papers where these different MAPK cascade models are analyzed. (a) Simple linear cascade, where double phosphorylations are lumped into a single reaction. (b) Typical cascade where MAP2K and MAPK activation consist of double phosphorylation steps. For the ERK1/2 cascade, MEK (MAP2K) activation is processive, whereas ERK (MAPK) activation is distributive. (c) The ‘reactor model’ of scaffold‐mediated MAPK cascade activation. Each kinase in the cascade is sequentially activated without dissociating from the scaffold complex.

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