Stochastic fire-diffuse-fire model with realistic cluster dynamics
Ana Calabrese,
1,2
Daniel Fraiman,
3
Daniel Zysman,
4
and Silvina Ponce Dawson
1
1
Departamento de Física, FCEN-UBA, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
2
Center for Theoretical Neuroscience, Columbia University, 1051 Riverside Drive, New York, New York 10032, USA
3
Departamento de Matemática y Ciencias, Universidad de San Andrés, Buenos Aires, Argentina
4
Department of Biology and Centre for Neural Dynamics, University of Ottawa, 30 Marie-Curie, Ottawa, Ontario, Canada K1N 6N5
Received 7 March 2010; revised manuscript received 19 July 2010; published 22 September 2010
Living organisms use waves that propagate through excitable media to transport information. Ca
2+
waves are
a paradigmatic example of this type of processes. A large hierarchy of Ca
2+
signals that range from localized
release events to global waves has been observed in Xenopus laevis oocytes. In these cells, Ca
2+
release occurs
trough inositol 1,4,5-trisphosphate receptors IP
3
Rs which are organized in clusters of channels located on the
membrane of the endoplasmic reticulum. In this article we construct a stochastic model for a cluster of IP
3
R’s
that replicates the experimental observations reported in D. Fraiman et al., Biophys. J. 90, 3897 2006. We
then couple this phenomenological cluster model with a reaction-diffusion equation, so as to have a discrete
stochastic model for calcium dynamics. The model we propose describes the transition regimes between
isolated release and steadily propagating waves as the IP
3
concentration is increased.
DOI: 10.1103/PhysRevE.82.031910 PACS numbers: 87.16.Xa, 87.16.A-, 87.10.Mn
I. INTRODUCTION
Oscillations and waves in the concentration of free intra-
cellular calcium Ca
2+
are seen in a variety of cells and are
known to be an important intra and intercellular signaling
system 1. It is thus of interest to determine the mechanisms
underlying such complex dynamic behavior. In many cell
types, a key component of this signaling pathway is the
inositol triphosphate receptor IP
3
R, which is also a Ca
2+
channel. The spatiotemporal properties of signals arising
through IP
3
R’s have been extensively characterized by opti-
cal imaging in Xenopus laevis oocytes 2. In these cells,
Ca
2+
imaging techniques have revealed that the cytoplasm
does not act as a continuous, homogeneous excitable me-
dium. Instead, Ca
2+
liberation occurs at discrete functional
release sites spaced a few micrometers apart, composed of
several clustered IP
3
R’s 2–8. The open probability of
IP
3
R’s depends on both the IP
3
and cytosolic Ca
2+
concen-
trations 9,10. A key feature, is the well-established biphasic
action of Ca
2+
in both facilitating and inhibiting the opening
of IP
3
R’s, through which Ca
2+
is liberated into the cytosol.
For relatively low Ca
2+
, the Ca
2+
released by one channel
increases the open probability of neighboring channels,
whereas at high Ca
2+
, it inhibits the channels and termi-
nates the release 11–15. This dependence of the open prob-
ability of the release channels on cytosolic Ca
2+
creates com-
munication between channels.
As a result of the combination of the channels spatial
organization and of the process of Ca
2+
-induced Ca
2+
release
CICR16–18, cytosolic Ca
2+
signals in oocytes display a
hierarchical spatio-temporal organization spanning over six
orders of magnitude, which include Ca
2+
“blips” that repre-
sent the release of Ca
2+
through a single or a few IP
3
R’s
4,19,20, “puffs” that involve the concerted opening of sev-
eral IP
3
R’s in a cluster 4,20–22 and Ca
2+
waves that propa-
gate globally across the cell by successive cycles of CICR
and Ca
2+
diffusion between clusters 21–23. Ca
2+
puffs re-
flect the dynamics of IP
3
R’s within a cluster. This dynamics
is ruled by the kinetics of each channel and by the interaction
among them due to the spatiotemporal Ca
2+
distribution on
the nanometer scale.
Several mathematical models have been proposed to de-
scribe Ca
2+
release through clustered IP
3
R’s. The approaches
vary depending on the spatial and time scale that they try to
resolve. The dynamics of localized signals such as puffs has
been simulated with models that include a detailed stochastic
description of the channels in the cluster and which resolve
distances on the nanometer scale 5,7. Using this fine spatial
resolution and the characteristic time scale of single channel
transitions ms to describe Ca
2+
waves which travel mil-
limeter distances and last for hundreds of seconds is compu-
tationally expensive 5,24. For this reason, mathematical
models of this type of global signals involve different ap-
proximations. For some time most such models were deter-
ministic see, e.g., 25 while stochastic models were left to
account for local signals such as puffs 5,18,26–28. It is
currently clear, however, that stochastic effects are not only
relevant for local release events but are a fundamental aspect
of the Ca
2+
dynamics for the full range of observed signals,
including waves 29–33. Efficient modeling strategies are
necessary to include the intrinsic stochasticity of Ca
2+
global
signals describing, at the same time, the largest scales in-
volved. Some of the approaches presented in the literature
assumed that clustered IP
3
R’s are in such close contact that
Ca
2+
could be considered homogeneous throughout the
cluster 26,34 –38. This type of models can be further sim-
plified as done in 39,40, where it is assumed that Ca
2+
at
one cluster depends only on the number of nearest release
sites where there are open channels and that each neighbor-
ing active site adds a time-independent contribution to
Ca
2+
. Some years ago a finite element hybrid scheme was
introduced to resolve the spatial gradients close to a channel
and simulate calcium signals efficiently 41. The algorithm
was used to simulate local signals. More recently, we intro-
duced a model that resolves the dynamics of Ca
2+
in the
intracluster region using a fine grid and some simplifications
42. The dynamics in the region outside the cluster is de-
PHYSICAL REVIEW E 82, 031910 2010
1539-3755/2010/823/03191012 ©2010 The American Physical Society 031910-1