Stochastic Models of Signaling Complexes

This research project is performed in close cooperation with the Smith lab. We jointly work on modeling techniques and tools to support the modeling of signaling complexes.

__For students: __

This project offers a multitude of interesting research questions that may serve as a topic for a Phd, Master or MasterÕs project. Specific topics are provided upon request. Note that for interested Computer Science students, we do not(!) expect any previous expertise in biological modeling.

__A brief motivation of the topic from a biology point of
view:__

The stochastic gating of voltage- and ligand-gated ion
channels in biological membranes that is observed by single channel recording
techniques is often modeled using discrete-state continuous-time Markov chains
(CTMCs). While these single channel models can be relatively simple (e.g., two physicochemically
distinct states) or complex (hundreds of states), most include only two
conductance levels (closed and open). For example, a transition state diagram
for a three-state calcium (Ca^{2+})-regulated channel activated by
sequential binding of two Ca^{2+}ions is given by

where k^{+}_{i} c and k^{−}_{i}
with i ∈ {a, b} are transition rates with
units of reciprocal time, k^{+}_{i} is an association rate
constant with units of conc^{−1} time^{−1}, and c
is the [Ca^{2+}] near the channel. If this local [Ca^{2+}] is
specified, the transition-state diagram of the channel (1) defines a CTMC that
takes on values in the state-space (C1, C2, O1). The experimentally observable
conductance of this stochastically gating channel is the aggregated process of
transitions between the closed and open classes of states: C = {C1, C2} and O =
{O1}. The scientific literature developing stochastic models for the behavior
of ion channels is largely focused on single channels or populations of
independent channels. Notable exceptions are the work of Ball and colleagues
analyzing interacting aggregated CTMC models of membrane patches containing
several ion channels and the work by Smith et al. A second example are
simulations of clusters of intracellular Ca^{2+}-regulated Ca^{2+}
channels inositol 1,4,5-trisphosphate receptors (IP3Rs) and ryanodine receptors
(RyRs)—located on the surface of the endoplasmic reticulum or
sarcoplasmic reticulum membrane—that give rise to localized intracellular
[Ca^{2+}] elevations known as Ca^{2+} puffs and sparks. When
Markov chain models of Ca^{2+}-regulated Ca^{2+} channels such
as (1) are coupled via a mathematical representation of buffered diffusion of
intracellular Ca^{2+}, simulated Ca^{2+} release sites may
exhibit the phenomenon of Òstochastic Ca^{2+} excitabilityÓ where the
IP3Rs or RyRs open and close in a concerted fashion (see Fig. 1 for
representative simulation) .

Such models are stochastic automata networks (SANs) that
involve a large number of functional transitions, that is, the transition
probabilities of one automata (i.e., an individual channel) may depend on the
local [Ca^{2+}] and thus the state of the other channels. The
experimentally observable quantity is either the local [Ca^{2+}] or the
number of channels in the open class of states, N_{O}(t) (see Fig. 1,
middle). The relationship between single channel kinetics of Ca^{2+}-regulated
channels and the emergent phenomenon of Ca^{2+} puffs and sparks is not
well understood. However, if each release site configuration is known, several
informative response measures can be determined from the steady-state
probability distribution. For example, the so-called puff/spark Score given by
Var[f_{O}]/E[f_{O}] is the index of dispersion of the
steady-state fraction of open channels, f_{O} = N_{O}/N (see
Fig. 1, right). This response measure takes values between 0 and 1, and a
puff/spark Score of greater than approximately 0.3 indicates the presence of Ca^{2+}
excitability. However, Ca^{2+} release sites are composed of
5–250 channels and this leads to a state-space explosion that makes
numerical calculation of the stationary and transient distributions of model Ca^{2+}
release sites challenging.

__Publications__

H. DeRimigio, P. Kemper, M.D. LaMar, G. Smith.

Markov chain models of coupled intracellular Calcium channels: Kronecker structured representations and benchmark stationary distribution calculations.

In: Proc. Pacific Symposium on Biocomputing 13:354-365(2008) (online proceedings),

a preliminary version appeared as Technical Report WM-CS-2007-06, College of William and Mary, June 2007.