Gene circuit designs for noisy excitable dynamics

arXiv:1102.4026v1 [q-bio.MN] 19 Feb 2011 Gene circuit designs for noisy excitable dynamics Pau Ru´ea, Jordi Garcia-Ojalvoa,∗ aDepartament de F´ısica i Enginyeria Nuclear, Universitat Polit`ecnica de Catalunya, Edifici GAIA, Rambla de Sant Nebridi s/n, Terrassa 08222,Barcelona, Spain Abstract Certain cellular processes take the form of activity pulses that can be interpreted in terms of noise-driven excitable dynamics. Here we present an overview of different gene circuit architectures that exhibit excitable pulses of protein expression, when subject to molecular noise. Different types of excitable dynamics can occur depending on the bifurcation structure leading to the specific excitable phase-space topology. The bifurcation structure is not, however, linked to a particular circuit architecture. Thus a given gene circuit design can sustain different classes of excitable dynamics depending on the system parameters. Keywords: excitable dynamics, noise, activator-repressor circuits, SNIC bifurcation, saddle-homoclinic bifurcation, Hopf bifurcation PACS: 87.18.Vf, 87.18.Tt, 87.10.Ed 1. Introduction Dynamical behavior is ubiquitous in gene regulatory processes. While many of the dynamical phenomena ex- hibited by genetic circuits take the form of periodic os- cillations, in certain cases the behavior is governed by randomly occurring pulses of protein expression. Math- ematically, this type of dynamics can be understood as an instance of excitability, by which a dynamical system with a stable fixed point is forced to undergo, when subject to a relatively small perturbation, a large excursion in phase space before relaxing back to the fixed point [1] (top panel in Fig. 1). This feature arises when the system is close to a bifurcation point beyond which the dynamics has the form of a limit cycle (bottom panel in Fig. 1). Functionally, excitability provides cells with a mecha- nism to amplify molecular noise and transform it into a macroscopic cellular response. Biochemical activity pulses have been reported to exist in cAMP signaling in amoebae [2], response to DNA damage in human cells [3], compe- tence in bacteria [4, 5], and differentiation priming in em- bryonic stem cells [6]. In a different (but still biological) context, electrical activity pulses in a specific cell type, namely neurons, have long been associated with excitable dynamics [7]. Within that context, early work by Hodgkin [8] identified two main different types of excitability on the basis of the frequency response of the neuron across a bifurcation leading from excitable to oscillatory behav- ior (Fig. 2). Type I excitability corresponds to a situation in which the limit cycle is born at the bifurcation with a ∗Corresponding author Email address: jordi.g.ojalvo@upc.edu (Jordi Garcia-Ojalvo) frequency equal to zero, i.e. the oscillatory activity pulses become infinitely sparse as the bifurcation is approached (left panels in Fig. 2). In type II excitability, on the other hand, the limit cycle is born with a non-zero frequency (and thus with a finite period), as shown in the right panel of Fig. 2. The distinction is not purely academic: given that neurons encode information mainly in the timing be- tween pulses, the two types of excitability correspond to two fundamentally different modes of information trans- mission [9]. Additionally, they exhibit distinct statistical features in their response to noise [9]. !"#$%& #'(')&*+*#%& ,& Figure 1: Schematic representation of excitable dynamics (top) and its associated limit-cycle behavior (bottom). The solid circle in the top plot represents the stable fixed point of the system. In this regime, large enough perturbations (starting at the empty circle in top panel) result in a large excursion through phase space before coming back to the stable state. Representation adapted from [7]. Preprint submitted to Elsevier November 16, 2018 Figure 2: Frequency response to the increase of a given control pa- rameter, for the two types of excitable behavior discussed in the text, as a bifurcation from excitability to oscillations is crossed. The dif- ferent bifurcation types underlying the different scenarios are listed in top of each plot. The two types of excitability shown in Fig. 2 are as- sociated with different classes of bifurcation to oscillatory behavior. Two kinds of bifurcation lead to type I excitabil- ity. The first is a saddle-node on an invariant circle (SNIC) bifurcation, in which a stable node and a saddle point ap- pear together on a limit cycle. In that regime, the tra- jectory along the remnant of the limit cycle delimits the excitable pulse. The second bifurcation leading to type I excitability is a saddle-homoclinic bifurcation, which oc- curs when a stable limit cycle collides with a saddle point, being transformed into a homoclinic orbit at the bifurca- tion point. Again this orbit delimits the excitable pulse. In both cases the existence of a saddle provides the system with a well defined excitability thre

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