Three level signal transduction cascades lead to reliably timed switches

Abstract

Signaling cascades proliferate signals received on the cell membrane to the nucleus. While noise filtering, ultra-sensitive switches, and signal amplification have all been shown to be features of such signaling cascades, it is not understood why cascades typically show three or four layers. Using singular perturbation theory, Michaelis-Menten type equations are derived for open enzymatic systems. Cascading these equations we demonstrate that the output signal as a function of time becomes sigmoidal with the addition of more layers. Furthermore, it is shown that the activation time will speed up to a point, after which more layers become superfluous. It is shown that three layers create a reliable sigmoidal response progress curve from a wide variety of time-dependent signaling inputs arriving at the cell membrane, suggesting the evolutionary benefit of the observed cascades.Key words: MAP-kinase network, Michaelis-Menten equations, time-dependent ODEs SignificanceIn MAP-kinase signaling networks, three-level cascades are a very common motif. We try to understand why evolution would favor such a motif by analyzing the set of differential equations describing such signaling cascades. Specifically we study how such multi-level cascades process different time dependent same sigmoidal output signal. Adding additional layers in the cascade does not change the type of output signal but delays its activation time. Introduction 10Biochemical cascades, in which upstream reaction products catalyze downstream reactions, are common patterns in eukaryotic molecular pathways. The mitogen-activated protein kinase (MAPK) system is an ancient, highly conserved example [1]. Functional and evolutionary studies of such cascades tend to emphasize their role as amplifiers of some input signal. Koshland et al. [2] 15 (see also [3]) observe that "amplification" in this context involves at least two distinct notions: amplification of the absolute size of the signal and amplification of the change in signal intensity relative to the signal background intensity, so-called "fold" amplification [4]. Classic examples of the former include bloodclotting [5, 6,7,8] and complement [9,10] among others probably evolved from 20 the same ancestral serine protease cascade [11]. The latter fold-change amplification is characteristic of signal transduction, at least for the MAPK and Wnt, β-catenin pathways [12,13]. In both general cases, amplification is generated by layering in the cascade. Layering can also produce ultrasensitive responses, in which smooth changes in input are converted to a switch-like "off-on" output 25 with a steeper signal-to-response curve than seen with traditional MichaelisMenten dynamics [14, 15,16,2,17,3].The intrinsic blood clotting mechanism of mammals was among the first biochemical cascades studied. 110We then analyze the enzyme cascade using a perturbation analysis which reduces the system to a chain of one parameter modules, allowing us to establish a Michaelis-Menten type equation for open enzymatic systems ...