Nonlinear signaling on biological networks: the role of stochasticity and spectral clustering

Signal transduction within biological cells is governed by networks of interacting proteins. Communication between these proteins is mediated by signaling molecules which bind to receptors and induce stochastic transitions between different conformational states. Signaling is typically a cooperative process which requires the occurrence of multiple binding events so that reaction rates have a nonlinear dependence on the amount of signaling molecule. It is this nonlinearity that endows biological signaling networks with robust switch-like properties which are critical to their biological function. In this study, we investigate how the properties of these signaling systems depend on the network architecture. Our main result is that these nonlinear networks exhibit bistability where the network activity can switch between states that correspond to a low and high activity level. We show that this bistable regime emerges at a critical coupling strength that is determined by the spectral structure of the network. In particular, the set of nodes that correspond to large components of the leading eigenvector of the adjacency matrix determines the onset of bistability. Above this transition, the eigenvectors of the adjacency matrix determine a hierarchy of clusters, defined by its spectral properties, which are activated sequentially with increasing network activity. We argue further that the onset of bistability occurs either continuously or discontinuously depending upon whether the leading eigenvector is localized or delocalized. Finally, we show that at low network coupling stochastic transitions to the active branch are also driven by the set of nodes that contribute more strongly to the leading eigenvector.Comment: 30 pages, 12 figure

The Timing Statistics of Spontaneous Calcium Release in Cardiac Myocytes - Figure 3

<p>(A) Plot of vs the RyR single channel current . The curves shown are numerical solution using the exact stochastic algorithm (black circles), exact solution (black line) according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e198" target="_blank">Eq. (11)</a>, asymptotic solution valid in regime II <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e224" target="_blank">Eq. (12)</a> (red line), high excitability limit using <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e226" target="_blank">Eq. 13</a> (blue line), and finally the MFPT for a single channel opening given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e229" target="_blank">Eq. (14)</a> (green line). Here, we fix and . (B) Plot of vs for small for the same parameter choice as <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone-0062967-g003" target="_blank">Fig. (3A)</a>. Black circles are numerical simulation results, black line is the exact solution using <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e198" target="_blank">Eq. (11)</a> and the red line is the asymptotic solution given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e224" target="_blank">Eq. (12)</a>. (C) Plot of vs using and . To speed up simulations we have used . Black circles are the numerical simulation results, the red line is computed using the summation shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e314" target="_blank">Eq. (24)</a>. Black line is the asymptotic approximation evaluated via <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e320" target="_blank">Eq. (27)</a> and the horizontal blue line is the high excitability limit given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e312" target="_blank">Eq. (23)</a>. The vertical dashed line indicates the current where .</p

Quantum metric spaces as a model for pregeometry

A new arena for the dynamics of spacetime is proposed, in which the basic quantum variable is the two-point distance on a metric space. The scaling dimension (that is, the Kolmogorov capacity) in the neighborhood of each point then defines in a natural way a local concept of dimension. We study our model in the region of parameter space in which the resulting spacetime is not too different from a smooth manifold

The Timing Statistics of Spontaneous Calcium Release in Cardiac Myocytes - Figure 1

<p>(A) Schematic illustration of the calcium release unit (CRU) showing a cluster of RyR channels on the SR in the vicinity of a few LCC channels on the membrane. (B) Four state Markovian scheme describing the RyR channel. (C) Birth-death process describing the closed to open transitions of RyR channels in the cluster.</p

The Timing Statistics of Spontaneous Calcium Release in Cardiac Myocytes - Figure 4

<p>(A) Schematic illustration of the voltage time course, SR Ca concentration, and the FPD, following an AP. (B) Plot of (black line) computed using the exact stochastic algorithm with time dependent excitability according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e382" target="_blank">Eq. (29)</a>. Probability distribution is computed by binning the first passage time of independent samples. The parameters used are . Red line corresponds to a plot of using <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062967#pone.0062967.e320" target="_blank">Eq. (27)</a>. The units of is indicated on the right y-axis. Late time behavior of is fitted using an exponential with decay rate (green line). (C) Schematic illustration showing cell-to-cell variations of the FPD .</p

Model Parameters.

<p>Model Parameters.</p

Quantum metric spaces as a model for pregeometry

A new arena for the dynamics of spacetime is proposed, in which the basic quantum variable is the two-point distance on a metric space. The scaling dimension (that is, the Kolmogorov capacity) in the neighborhood of each point then defines in a natural way a local concept of dimension. We study our model in the region of parameter space in which the resulting spacetime is not too different from a smooth manifold

Quantum metric spaces as a model for pregeometry

A new arena for the dynamics of spacetime is proposed, in which the basic quantum variable is the two-point distance on a metric space. The scaling dimension (that is, the Kolmogorov capacity) in the neighborhood of each point then defines in a natural way a local concept of dimension. We study our model in the region of parameter space in which the resulting spacetime is not too different from a smooth manifold