📝 Abstract

Positive feedback regulation is ubiquitous in cell signaling networks, often leading to binary outcomes in response to graded stimuli. However, the role of such feedbacks in clustering, and in spatial spreading of activated molecules, has come to be appreciated only recently. We focus on the latter, using a simple model developed in the context of Ras activation with competing negative and positive feedback mechanisms. We find that positive feedback, in the presence of slow diffusion, results in clustering of activated molecules on the plasma membrane, and rapid spatial spreading as the front of the cluster propagates with a constant velocity (dependent on the feedback strength). The advancing fronts of the clusters of the activated species are rough, with scaling consistent with the Kardar-Parisi-Zhang (KPZ) equation in one dimension. Our minimal model is general enough to describe signal transduction in a wide variety of biological networks where activity in the membrane-proximal region is subject to feedback regulation.

💡 Analysis

Positive feedback regulation is ubiquitous in cell signaling networks, often leading to binary outcomes in response to graded stimuli. However, the role of such feedbacks in clustering, and in spatial spreading of activated molecules, has come to be appreciated only recently. We focus on the latter, using a simple model developed in the context of Ras activation with competing negative and positive feedback mechanisms. We find that positive feedback, in the presence of slow diffusion, results in clustering of activated molecules on the plasma membrane, and rapid spatial spreading as the front of the cluster propagates with a constant velocity (dependent on the feedback strength). The advancing fronts of the clusters of the activated species are rough, with scaling consistent with the Kardar-Parisi-Zhang (KPZ) equation in one dimension. Our minimal model is general enough to describe signal transduction in a wide variety of biological networks where activity in the membrane-proximal region is subject to feedback regulation.

📄 Content

1 Positive Feedback Regulation Results in Spatial Clustering and Fast Spreading of Active Signaling Molecules on a Cell Membrane

Jayajit Das1#, Mehran Kardar4, Arup K. Chakraborty1-3,* Departments of 1Chemical Engineering, 2Chemistry, 3Biological Engineering, & 4Physics, Massachusetts Institute of Technology, Cambridge, MA 02139.

• address correspondence to arupc@mit.edu

ABSTRACT

Positive feedback regulation is ubiquitous in cell signaling networks, often leading to binary outcomes in response to graded stimuli. However, the role of such feedbacks in clustering, and in spatial spreading of activated molecules, has come to be appreciated only recently. We focus on the latter, using a simple model developed in the context of Ras activation with competing negative and positive feedback mechanisms. We find that positive feedback, in the presence of slow diffusion, results in clustering of activated molecules on the plasma membrane, and rapid spatial spreading as the front of the cluster propagates with a constant velocity (dependent on the feedback strength). The advancing fronts of the clusters of the activated species are rough, with scaling consistent with the Kardar-Parisi-Zhang (KPZ) equation in one dimension. Our minimal model is general enough to describe signal transduction in a wide variety of biological networks where activity in the membrane-proximal region is subject to feedback regulation.

Present Address: Battelle Center for Mathematical Medicine, The

Research Institute at the Nationwide Children’s Hospital, Departments of Pediatrics, Physics and Biophysics Graduate Program, Ohio State University, 700 Children’s Drive, Columbus, OH 43205

2 INTRODUCTION

Cell signaling networks often generate binary (on or off) responses in presence of a diverse set of stimuli in the local environment. A common element present in many of these networks is a positive feedback loop 1,2, which can give rise to discrete decisions 1,3,4. A positive feedback loop can arise when an activated signaling molecule creates a mediatory molecule that in turn enhances the activation of the signaling molecule. When the timescales of the biochemical reactions involved are slower, or of the same order, as that of the diffusion of the molecules participating in the reactions, nonlinearities associated with positive feedback could couple with diffusion to result in spatial clustering, and in spreading of activated molecules at a rate much faster than diffusion. Various models describing growth of advantageous alleles in the area of population biology5, kinetics of reaction fronts in autocatalytic systems6 7, spreading of bacterial colonies 8, and pattern formation during embryonic development 9, have incorporated the coupling of nonlinearities with diffusion in their dynamics. It was also found that auto catalytic reactions in the presence microscopic discreteness could give rise to spatial clustering of active species5,6. These models are naturally constructed to explore the behavior of systems at the length and time scales relevant to the population, organism, or cellular level. The role of positive feedback in spatial dynamics of sub-cellular processes, relevant for signal transduction in cell signaling networks, has begun to be appreciated only very recently10,11. In this paper, we study the spatial-temporal evolution of cell signaling dynamics subject to feedback regulation when diffusive processes occur on time scales similar to the signaling reactions. The diffusion of molecules is usually much slower (about ~100 times 12) in the

3 plasma membrane compared to the cytosol, thus the effects we described are most relevant for the molecules in the plasma membrane participating in membrane proximal cell signaling.

MODEL

As a prototype of signaling events on a two-dimensional cell membrane that involves positive feedback regulation, we consider the activation of the membrane associated Ras family of proteins. Ras can be activated by a Guanine exchange factor protein, Son of Sevenless (SOS). Specifically SOS catalytically converts GDP bound Ras to its GTP bound activated form. It was discovered recently 13,14 that catalysis of Ras-GDP to Ras-GTP aided by SOS becomes even faster (~75 fold) when a membrane associated SOS molecule is bound to Ras-GTP at an allosteric site. This mechanism introduces positive feedback regulation of Ras activation15. Activated Ras has been observed to form clusters, and the diffusion coefficient in these clusters can be small.

We study the following simplified set of three reactions that aim to mimic positive feedback, as in Ras activation:

Z + Y k1 ⎯ → ⎯ X + Y, Z + X + Y k2 ⎯ → ⎯ 2X + Y, Y k3 ⎯ → ⎯ ∅ .

(1)

In the above reactions, the Y species can be thought of as representing SOS, while Z and X are analogous to Ras-GDP and Ras-GTP respectively. The first two reactions in Eq.(1) co

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