Stochastic Petri Net Models of Ca2+ Signaling Complexes
and Their Analysis
Peter Kemper
Received: April 2, 2008/ Accepted: –
Abstract Mathematical models of Ca2+ release sites derived from Markov chain mod-
els of intracellular Ca2+ channels exhibit collective gating reminiscent of the experi-
mentally observed phenomenon of stochastic Ca2+ excitability (i.e., puffs and sparks).
Ca2+ release site models are structured into a number of individual channel models
whose dynamic behavior depends on the local Ca2+ concentration which is jointly in-
fluenced by the state of all channels. We consider this application area to illustrate
how stochastic Petri nets and in particular stochastic activity networks can be used to
model problems in cell biology. We highlight how state-sharing composition operations
as supported by the Mobius framework represent the common mean-field coupling as-
sumption, or, alternatively, spatial aspects, in a natural manner. We investigate to what
extent state-of-the-art techniques for the numerical and simulative analysis of Markov
chains that are associated with Stochastic Petri nets scale with research problems in
modeling Ca2+ signaling complexes.
1 Introduction
The Monte Carlo Markov Chain (MCMC) simulation and Gillespie’s method are fre-
quently applied in the analysis of stochastic models of biophysical and biochemical
phenomena, in particular for those aspects that are of rather discrete nature and where
traditional approaches based on differential equations find their limits. MCMC simula-
tion implies that the underlying stochastic model is a Markov chain. Most challenges in
working with Markov chains are related to largeness, be it largeness of model descrip-
tions or largeness of state spaces. While largeness of descriptions is mainly a challenge
for a human-computer interface and asks for a modeling formalism that is expressive,
R. Lamprecht · P. Kemper Department of Computer Science, College of William and Mary, Williamsburg, VA 23187, USA E-mail: rlampy,kemper@cs.wm.edu
G. D. Smith Department of Applied Sciences, College of William and Mary, Williamsburg, VA 23187, USA E-mail: greg@as.wm.edu
2
intuitive to use, and easy to communicate, largeness of state spaces relates to the
amount of computation time and memory necessary to compute the required results,
i.e., the properties of interest for a given model.
Many modeling formalisms that allow a modeler to work with Markov chains at a
higher level of abstraction have been developed and discussed in the last decades. Those
can be seen as almost canonical extensions of an originally untimed formalism that
describes a state transition system and then adds transition rates to those elements of
the formalism that perform changes in the system’s state. Examples for such endeavors
are (generalized) stochastic Petri nets (GSPNs) (Marsan et al 1995) as an extension of
Petri nets, Markovian process algebras like PEPA (Hillston 2005) and process algebras
like Milner’s calculus of communicating systems (CCS), pi calculus and stochastic pi
calculus (Priami 1995), state charts and stocharts (Jansen and Hermanns 2004).
GSPNs have a number of positive aspects for modeling biological systems: they
provide a clear yet simple separation of states and transitions with easy to grasp and
formally well-defined semantics; and they come with a graphical representation (with
the usual limits of scalability for any graphical formalism). GSPNs are particularly suit-
able to describe entities that act asynchronously yet occasionally synchronize. GSPNs
allow a modeler to describe extremely large continuous time Markov chains (CTMCs)
with a few powerful constructs and in a concise manner. As for Petri nets, GSPNs
can have associated CTMCs that have exponentially larger state spaces (if finite at
all), a phenomenon that is called state space explosion. Although special cases like
CTMCs that are amenable to matrix geometric solutions exist, in general, numerical
analysis techniques to compute transient or steady-state distributions are restricted to
finite CTMCs, i.e., GSPNs that have a finite set of reachable states. Albeit existing
numerical techniques (Stewart 1994), the model-based evaluation of biophysical and
biochemical phenomena often takes place by MCMC simulation and not by numerical
solution of a CTMC. The efficiency of simulation depends on the level of detail con-
sidered by a reward measure, the required precision of the results (width of confidence
intervals), and the frequency with which relevant events can be observed (rare events
are a well-known borderline case for efficient simulation). In many cases, the desired
results could, in principle, also be derived through analysis of the underlying CTMCs;
however, in practice, the size of the systems of equations that needs to be solved is
prohibitive. This “largeness problem” has motivated much research in the construction
and numerical solution of CTMCs.
Thus, we face a quest for scaleability on both ends, for a graphical model descrip-
tion as well as for scaleability of numerical analysis techniques to compute steady-state
or transient distributions. Among possible ways to make Petri net descriptions scale,
some approaches just help for scalability of descriptions, while others provide structural
information that is valuable for a subsequent analysis. The classical approach of place
and transition refinement only serves the first purpose and helps to describe large Petri
nets in an organized manner. However, compositional operators like synchronization
of transitions as for superposed GSPNs (Donatelli 1994), sharing places (state vari-
ables) in Rep/Join and Graph composition operations as supported in Mobius (Clark
et al 2001), as well as GSPN extensions towards stochastic well-formed nets (SWN)
(Chiola et al 1993) provide ways to structure large models as well as give rise to a
model decomposition that gives the potential to reduce the underlying CTMC based
on lumpability. We consider the latter approaches more valuable since they provide
a modeler with means to describe large Markov models in a well-structured manner,
while simultaneously allowing advanced analysis techniques to apply.
3
In what follows, we first discuss existing work on the use of Petri nets and, in
particular, timed and stochastic Petri nets in modeling aspects of biological systems.
In Section 3, we briefly introduce Markovian models of Ca2+ signaling complexes as a
particular application area. In Section 4, we recall stochastic Petri nets and composition
operators in order to cope with largeness of descriptions. We focus in particular on
operations implemented in the Rep/Join and Graph composer functionality of Mobius.
Section 5 is devoted to a survey of common analysis techniques for Markovian models.
In Section 6, we show different ways to model Ca2+ signaling complexes with stochastic
Petri nets when the mean-field coupling assumption applies. Finally, in Section 7, we
include spatial considerations into our Ca2+ signaling complexes model and make use
of the Graph composer to do so in a natural manner. We conclude in Section 8.
2 Related Work
Petri nets have been used in many areas including software design and performance
analysis, data analysis, reliability engineering, and biological systems. Our focus here
is on biological applications of Petri nets.
The majority of the previous work on Petri nets and biological systems was on how
and when to use Petri nets to model biological systems and behaviors. Several have dis-
cussions of how to construct and use Petri nets to solve computational biology systems
(Chaouiya 2007; Schaub et al 2007). Others relate Petri net formalisms to the model-
ings, simulations, and analysis of biological pathways (Matsuno et al 2006; Hardy and
Robillard 2007). Peleg et al (2002) developed a model of biological processes solved by
combining a Workflow/Petri Net and a biological concept model. Steggles et al (2007)
describe a technique to construct and analyze qualitative models of genetic regulatory
networks by using a Petri net formalism. Heiner and Koch (2004) discuss examples
on how and why to do model validation of qualitative models of biological pathways.
Srivastava et al (2002) give a comparison of the results of implementing a model of in-
tracellular kinetics of a generic virus both deterministically and stochastically. Griffith
et al (2006) present an algorithm to do a hybrid simulation of a well-mixed chemical
system by dynamically partitioning fast and slow reactions into discrete and continuous
reactions, respectively.
There has also been some previous work on using different tools to model and
solve Petri nets. One such work was a study and comparison of different Petri net
tools (including Mobius) based on the mathematical capabilities and the ability to
model specific features of biological systems and answer a set of biological questions
(Peleg et al 2005). The test cases used for this study involved malaria invasion, tRNA
mutations, and drug pathways in a human cell. The work in Materi and Wishart (2007)
is a summary and assessment of several techniques used to model complex temporal and
spatiotemporal processes, with a focus on using these methods for drug discovery and
development. Another gives an approach to defining molecular interaction networks
as stochastic Petri nets, with results from models solved with UltraSAN (Goss and
Peccoud 1998), the predecessor of Mobius. A brief overview of the features of Mobius
in terms of stochastic Petri nets can be found in Peccoud et al (2007). In contrast to
existing work, we particularly address scalability issues that evolve in the use of GSPNs
for stochastic models of Ca2+ signaling complexes (Nguyen et al 2005).
4
0
5
10
1 2 3 4 5 6 7 8 9 10
0
0.5
1
Fig. 1 From DeRemigio et al (2008): a) Local [Ca2+] near 3 × 3µm ER membrane with Ca2+ channels modeled as 0.05 pA point sources with positions randomly chosen from a uniform distribution on a disc radius of 2 µm. Buffered Ca2+ diffusion is modeled as in Nguyen et al (2005). b) Stochastic Ca2+ excitability reminiscent of Ca2+ puffs/sparks. c) Probability distribution of the number of open channels leading to a puff/spark Score of 0.39.
3 Markov Models of Ca2+ Release Sites
Cell signal transduction is often mediated by molecular assemblies composed of mul-
tiple interacting transmembrane receptors and systems biologists investigate the rules
that govern the emergent behavior of these signaling complexes (Gomperts et al 2002;
Krauss 2003). Examples for such problems include coliform bacteria where the high
sensitivity of lattices of plasma membrane proteins to chemoattractants is due to al-
losteric protein-protein interactions which leads to conformational spread through the
receptor array (Bray 1998; Duke and Bray 1999; Duke et al 2001). Signaling com-
plexes are not restricted to the plasma membrane, but are also found on mitochondrial
and nuclear membranes and on the endoplasmic or sarcoplasmic reticulum (Berridge
1997; Cho 2006; Berridge 2006). Clusters of 5–50 IP3Rs on both the cortical endo-
plasmic reticulum (Yao et al 1995; Sun et al 1998) and outer nuclear membrane (Mak
and Foskett 1997) of immature Xenopus laevis oocytes exhibit coordinated gating that
gives rise to spatially localized Ca2+ elevations known as “Ca2+ puffs.” During cardiac
excitation-contraction coupling, plasma membrane depolarization leads to Ca2+ influx
via L-type Ca2+ channels that triggers “Ca2+ sparks,” i.e., the release of sarcoplasmic
reticulum Ca2+ from two-dimensional arrays of ryanodine receptors.
In this paper, we consider clusters of Ca2+ regulated channels whose stochastic
gating depends on Ca2+ . The local concentration experienced by a particular chan-
nel depends on its own state (being open or closed) and the state of other channels
(in varying degrees dependent on the distance). An open channel increases the Ca2+
concentration experienced by neighboring channels. A release site consists of multiple
individual channels with some approximate spatial arrangement. Fig. 1a shows a sam-
ple channel system with 9 randomly positioned channels. Fig. 1b displays the change
in the number of open channels as time progresses while Fig. 1c shows the probability
distribution of the number of open channels at the Ca2+ release site model.
The spatial arrangement observed in reality is often unknown, not necessarily regu-
lar and this leaves room for simplifying modeling assumptions. One common modeling
assumption is that each channel experiences the same average Ca2+ concentration
that depends only on the number of open channels (i.e., mean-field coupling). How-
ever, DeRemigio et al (2008a) showed that the spatial arrangement can have an impact
on the stochastic activity of Ca2+ release site models. For a detailed model that takes
5
location of channels on the endoplasmic reticulum membrane into account, one needs
to specify those locations, the pair-wise distances between channels, and the influence
of inter-channel communication via elevated Ca2+ concentration.
Single channel techniques such as patch clamp and planar lipid bilayer recording
reveal the stochastic gating of voltage- and ligand-gated ion channels in biological
membranes. There is a long history of modeling the stochastic gating of single channels
using CTMCs. These models vary in their complexity - from aggregated, abstract ones
with only two physicochemically distinct states to very detailed ones with hundreds
of states (Colquhoun and Hawkes 1995). Several recent publications (Nguyen et al
2005; DeRemigio et al 2008b) present CTMC models of Ca2+ release sites composed
of instantaneously coupled intracellular Ca2+.
In order to understand the structure of such Ca2+ release site models, we briefly
recall definitions related to CTMCs for clarification. A CTMC for a state space S of
n states is described by an n × n matrix Q of real values where Qij ≥ 0 for i 6= j
and Qii = −P j,j 6=i Qij such that its row sums are zero and an initial distribution
π(0) ∈ IRn with P
i πi(0) = 1. An entry Qij = λ denotes a state transition from si to
sj that has a stochastic delay given by an exponential distribution with rate λ and if
the state resides in state si it moves to state sj with probability pij = λ/|Qii|. The
value of 1/|Qii| gives the mean sojourn time the process stays in state si. For a given
CTMC, we can compute the value for the probability to be in a certain state s ∈ S at
some point of time t ≥ 0, denoted by πs(t) by solving π(t) = π(0)eQt. For t → ∞ and
an irreducible CTMC, we can compute the so-called steady-state distribution π as a
solution of πQ = 0 such that P
s∈S πs = 1. Hence we can study a CTMC model of a
Ca2+ release site by transient analysis (the specific pattern of channel openings and
closings during a given time period) and by steady-state analysis (how many channels
are open on average). An important property of CTMCs is that this analysis can
be done by either simulation or numerical analysis (see Section 5). The measures of
primary interest for a Ca2+ release site model are: 1) is the model configuration of
the kind that shows puffs and sparks? and 2) if so, what is the average amplitude and
duration of a spark? For the first aspect, the Score of a model indicates the presence of
puffs and sparks observed in the Ca2+ release site model. Let NO denote the number
of open channels, N the total number of channels. The puff/spark Score is the index
of dispersion of the fraction of open channels (fO ≡ NO/N),
Score = Var[fO ]
E[NO ] (1)
a simple measure that agrees with subjective evaluations of how puff-like simulated
Ca2+ release site dynamics appear(Nguyen et al 2005). While several different statis-
tical properties of sparks are of interest, the one we focus on here is to select an initial
state where k out of n channels are open and compute the time until all channels are
closed (i.e., the spark duration). This calculation is performed by making the state
where all channels are closed an absorbing state and evaluating the time to absorption.
In the following, we will go through a sequence of CTMC models of increasing
complexity and start off with the simplest possible model of a Ca2+regulated channel,
namely that of a single two-state model. Our presentation closely follows that in Nguyen
et al (2005).
Two-State Channel
A CTMC of a two-state single channel activated by Ca2+ is derived from the
transition diagram
(closed) C
k+cη ∞
k− O (open) (2)
where k+cη ∞ and k− are transition rates with units of reciprocal time, k+ is an associ-
ation rate constant with units of conc−ηtime−1, η is the cooperativity of Ca2+binding,
and c∞ is the background [Ca2+]. For Eq. 2, the Q-matrix is,
Q =
Four-State Channel
The four-state Ca2+-regulated Ca2+ channel considered here includes both Ca2+inactivation
(vertical) and Ca2+ activation (horizontal) and has the transition state diagram
(closed) (open)
with the inactivation and de-inactivation assumed slow compared to activation and
de-activation (k± b
, k± d
<< k±a , k±c ). Furthermore, the parameters are chosen to satisfy
the thermodynamic constraint (Hill 1977)
KcKd = KaKb
where Kη i = k−i /k+
i for i ∈ {a, b, c, d} are dissociation constants. Therefore, Eq. 4 has
eight rate constraints, but only seven are free parameters because
k−d = k+ d
Kc . (5)
The Q-matrix for Eq. 4 is defined similar to the Q-matrix for Eq. 2 (not shown).
n-State Channel
If we define u to be a M × 1 column vector, where M is the number of states in
a channel, to indicate open states, the Q-matrix for both the two-state and four-state
models can be written as
Q = K− + diag(c∞e + cdu)ηK+ (6)
7
where K− and K+ involve the dissociation and association rate constants, respectively.
For the two-state channel, u = (0, 1)T and for the four-state channel u = (0, 1, 0, 0)T ,
where the states are ordered C1, O, C2, C3.
So far, we have considered only single channel models. For multi-channel models
with dependancies among channels, the representation of Q for the overall system gets
more complicated and is discussed next.
3.2 Q-matrix Expansion for Instantaneously-Coupled Channels
For release sites composed of multiple Ca2+-regulated Ca2+ channels, interaction be-
tween channels must be accounted for. The first case to consider is the Q-matrix for
two interacting two-state channels,
1 CCA (7)
where a diamond () indicates a diagonal element leading to a zero row sum in Q. The
cij values represent the effect experienced by channel j when channel i is open under
the assumption of a single high concentration of Ca2+ buffer (Nguyen et al 2005).
Instead of considering the Q-matrix expansion for four-state channels, we skip to
the general case. The single-channel Q matrix in the form of Eq. 6 is
Q (1) = K
(1) + (8)
so the expanded Q-matrix, coupled through the interaction matrix C, for N channels
is
(N) + are as follows,
Q (N) + =
diag(c∞e(N) + Γ∗n)η(I(N−n) ⊗ K (1) + ⊗ I(n−1)) (11)
and Γ∗n is the nth column of the MN × N matrix Γ given by
Γ = UC (12)
where C is an N × N ’coupling matrix’ that summarizes the channel interactions and
U is an MN ×N matrix indicating the open channels for every state of the release site.
Further explanation can be found in Nguyen et al (2005).
Expansion on Eq. 9 gives us the Q-matrix for N channels, with M states per
channel, which results in |S| = MN states in total. However, this matrix will quickly
get large as N and M are increased. A method to reduce the largeness of the model is
discussed in the next section.
8
3.3 Mean-field Ca2+ Release Site Model
As N and/or M is increased, a common simplification is to assume mean-field coupling,
where every channel experiences an average Ca2+ concentration and, consequently,
there is no need to differentiate between channels. In this case, one need only keep
track of how many of the N channels are in each of the M states. With a single,
replicated model the interactions are reduced to a count of how many are doing a
certain activity, a linear function of the number of atomic models instead of a factorial
function. This leads to a state space reduction from MN to
„ M + N − 1
« .
A consequence of using mean-field approximation is the need to find a feasible
approximation for the Ca2+ concentration if x of the N channels are open. As discussed
in Nguyen et al (2005), using c = cinfty + NOc∗, where c∗ is the average of the off-
diagonal entries of C, often works well. The approximation used for the experiments
done in this paper is based on the work done in DeRemigio and Smith (2005), where
the c∗ value used with N channels was
c∗ = c∗opt · K
N (13)
where N ≥ K and c∗opt was the ’optimal’ value of c∗ for K channels (i.e., the value of
c∗ that maximized the puff/spark Score.
3.4 A Spatial Model and Its Coupling Matrix
If the position of individual channels is to be explicitly accounted for, we need to
consider N M -state channels, which yields a state space of MN states. Following
Nguyen et al (2005), channel coupling scales the rate of Ca2+mediated transitions and
a concise representation of that relation is formalized as a N × N coupling matrix
C where cij gives scaling factor that for instance is seen in Eq. 7. Assuming that all
channels are identical, cii = c for all i = 1, . . . , N and some c ∈ IR, independently of
the spatial relationship of the channels. If the chosen spatial arrangement is irregular,
C is nevertheless a symmetric matrix with cij = cji since the distance is undirected
and implies a symmetric relation. For an arbitrary spatial arrangement, one can select
up to N(N−1)
2 distances (for the number of ways to choose 2 out of N channels). If the
spatial arrangementt is regular, this number is reduced accordingly.
In Section 7, we consider examples with a grid arrangment and a hexagonal lattice.
With a grid arrangement, the distance between pairs of channels would match for each
level of the grid. For instance, with the four channel grid arrangement,
4 1
3 2
the pairs (1,2), (2,3), (3,4), and (4,1) would all have the same distance since they are
all neighboring pairs. The pairs (1,3) and (2,4) would both be the same distance apart
since they are each one channel removed from each other. So in this case there are only
9
two different values for the distance between any pair of channels. For any fixed release
site size, we obtain a matrix C with only three different values
C =
0 BB@
c0 c1 c2 c1 c1 c0 c1 c2 c2 c1 c0 c1 c1 c2 c1 c0
1 CCA . (14)
Study of this matrix shows that the distinct values are c0 on the diagonal for the
interaction of a channel with itself, and c1 > c2 on the off-diagonal. In Section 7, we
discuss how to make use of the symmetries imposed by regular spatial arrangementss
to achieve a reduction of the CTMC based on lumpability.
4 Modeling with Stochastic Petri Nets
In this section, we outline essential steps to transform the model characteristics given in
the previous section into a model-based study of Calcium release sites with stochastic
Petri nets and a modeling framework like Mobius.
4.1 Modeling with stochastic Petri nets
For a generalized SPN model, we use the following formal definition, where we allow
for enabling, firing and weight functions being state/marking-dependent, which exceeds
that classic definition of GSPNs in Marsan et al (1995):
Definition 1 A GSPN is a 7-tuple (P, T, m0, e, f, w, p) where P and T are non-empty
finite sets of places and transitions, m0 ∈ INP 0 is the initial state (or marking) of
places, functions e, f, w, p describe the enabling e : T × INP 0 → IB of a transition for
a transition t ∈ T in given state in INP 0 , the successor state reached by firing of a
transition f : T × INP 0 → INP
0 , the weight of that transition w : T × INP 0 → IR+
0 and its
priority of any transition P : T → IN0.
In any given state (or marking in Petri net terminology) m ∈ INP 0 , a transition t ∈ T
is enabled if e(t, m) is true. Priorities impose a constraint on the possible ways to
define e in the follwing sense: let e(t,m) = true then there is no transition t′ ∈ T
with p(t′) > p(t) for which e(t′, m) = true. Let E(m) ⊆ T be the set of transitions
that are enabled in a marking m. Any enabled transition t ∈ E(m) can fire and yield a
successor marking m′ = f(t, m). A transition with p(t) = 0 is called timed and its firing
(if enabled at m) takes place after an exponentially distributed random delay with rate
w(t, m) if p(t) = 0. A transition with p(t) > 0 is called immediate, it fires with no delay.
If several transition are enabled at a marking m, then t ∈ E(m) fires with probability
w(t, m)/ P
t′∈E(m) w(t′, m). Probabilities are calculated irrespectively of the priority
of t (which is the same for all transitions in E(m)). With the help of the firing rule,
we can explore all reachable states S for a given initial state m0. States that enable
immediate transitions are called vanishing and if present imply that the resulting state
space describes a semi-Markov and not a CTMC. The reduction towards a CTMC is a
well-known procedure called elimination of vanishing states in the literature (Marsan
et al 1995). Immediate transitions are a mean to handle models that operate with
10
different time scales, i.e., models that include extremely fast transition and very slow
transitions. If one assumes that the extremely fast transitions are irrelevant for the
fraction of time the stochastic process spends in them, one can treat them as immediate
and thus reduce their sojourn times to zero as well as their steady-state probabilities
which also justifies their elimination. We introduced priorities only for the sake of
consistency with GSPNs and the SAN variant of GSPNs supported in Mobius, we will
not make use of priorities to model Ca2+ signaling complexes.
So a GSPN has a finite number of integer state variables as places and transitions
that modify a marking vector (state) by subtraction and addition of possibly marking
dependent integer vectors of appropriate dimension. To encode a single channel with
M states is straightforward if we use M places, one per state, put one token into
the place that corresponds to the initial state and leave all other places empty. Every
possible change from one state si to another sj is represented by a transition that
has the corresponding place pi as input place and pj as output place. In order to
express multiple channels, e.g., N channels, we can either generate N instances (copies)
of the single channel net or use the single channel net and assign N tokens as the
initial marking instead of 1. With the first variant, we are able to distinguish states
of individual channels and are able (at least in principle) to take spatial aspects into
account when we define transition rates. For the second variant, we cannot distinguish
among channels, such that any possible definition of transition rates can only rely on
the number of channels being in a particular state (e.g., being open). In both cases,
we need to make use of marking dependent transitions rates to take into account that
in the CTMC transition rates depend on the states of all channels. If a single channel
model includes a lot of states M or if we prefer to distinguish among instances of a single
channel model, we need some form of composition operation for GSPNs. A GSPN has
two basic concepts - places/state variables and transitions - that can serve as building
blocks for a composition operations. A composition based on transitions has two or
more separate GSPNs perform certain dedicated transitions only in a jointly manner;
such transitions fire in a synchronized manner, the synchronization is of a rendez-
vous type. Formally, this is often achieved by merging transitions of the same name
t across several GSPNs i = 1, . . . , n in that their enabling function becomes a logical
AND, e(t,M) = V
ei(ti, Mi), firing function f(t, M) = M ′ with M ′ i =
P fi(ti, Mi).
There are a number of ways to define a joint weight function, one way is w(t,M) =
wt Q
wi(ti, Mi). Note that synchronized or shared transitions are usually restricted to
timed transitions. The resulting class of nets is called superposed GSPNs in Donatelli
(1994). Synchronization by shared transitions is similar to the parallel composition
operator of process algebras like PEPA and that of stochastic automata. Transition
sharing is known to be a congruence for a performance bisimulation that matches with
lumpability for CTMCs, which means that for a model composed of multiple GSPNs,
we can first generate and minimize CTMCs associated with each individual CTMC
with the help of lumpability and then compose the reduced CTMC into a large CTMC
of the composed model.
As an alternative to sharing transitions, one can consider sharing of state vari-
ables or places in order to build large models by composing a set of smaller models.
Both types of composition are currently supported by Mobius. For the following dis-
cussion of composition based on shared variables we focus on Mobius-related concepts
since those will result in composed models that can be reduced according to lumpa-
bility. The two types of composition based on sharing variables in Mobius are called
Rep/Join composition and Graph composition. Individual basic models that are used
11
to compose large models are called atomic models. Mobius supports several formalisms
to describe atomic models; the one that extends GSPNs and that we consider here
is called stochastic activity networks (SANs) (Sanders and Meyer 2001). In addition
to common generalized SPN concepts of places and transitions with different levels of
priority, SANs provide for input and output gates which are graphically denoted by
small triangles and allow a modeler to specify state-dependent functions e, f , and w
in a very general manner. Timed activities have an activity time distribution function,
which can be a generally distributed random variable and state-dependent. Activities
also have case probabilities to represent different possible outcomes after an activity
completes.
Sharing state variables is supported by the Rep/Join composition, which particu-
larly facilitates instantiation of multiple copies of the same atomic model and results
in a tree type model structure. State variable sharing is also supported in the Graph
composition which results in a model structure of an undirected graph.
The formalism for the Rep/Join composition was first defined for SAN models in
Sanders and Meyer (1991). It uses two separate operations, Rep and Join, to achieve
composed models that have the form of a tree. Each leaf is a predefined atomic model
or a composed model, and each non-leaf is either a Join or Rep node. The Join node
is used to connect multiple submodels (composed or atomic) by merging (sharing) a
set of variables that is specified by the modeler. This implies that all children of the
Join node can read and write to a state variable they share and thus can communicate
with each other. The Rep node identifies a special case of the Join where all children
are instances of the same model. This simplifies the specification, i.e., only a single
submodel needs to be identified together with the number of instances that shall be
created. As for the Join , the replicated submodels only share those variables that are
selected as shared. The key advantage of the Rep node is that it imposes lumpability for
the associated CTMC which can be used for an automated reduction of the resulting
CMTC; this works even in a nested manner with tree-type structures of multiple Rep
and Join nodes. With both types of nodes, the modeler can specify the state variables
that are to be shared between the submodels below the node.
The second composer, Graph Composer, uses a graph structure, rather than the tree
structure of the Rep/Join Composer, and only the Join operator. Again, the Join node
allows the modeler to specify state variables that are to be shared between submodels
connected to the Join.
One motivation for looking into composition operations is to identify a natural way
to represent the fact that certain parts of a biological model result from instantiating
a certain type multiple times. The composition operations supported in Mobius and in
particular the ones based on shared variables serves this purpose very well as we will
see in Section 6. However, we would like to note that these are not the only concepts
one could discuss at this point. For instance, in stochastic well-formed nets (SWNs),
tokens are colored to exploit behavioral symmetries in the model (Chiola et al 1993).
For SWNs, reduction operations based on lumpability are also known. For object-
oriented Petri nets (OOPN), tokens are object-oriented, meaning that a token can
itself be an OOPN. Nevertheless, we decided to focus on state variable composition,
since we believe those are the most suitable to support mean-field coupled and spatial
models of Ca2+ signaling complexes.
12
4.2 Defining Rewards
Analysis of a system begins with a decision of what is to be measured. The dynamic
behavior of a stochastic model results in being in a certain state at some point of
time and performing certain transitions to move from one state to another. In terms
of the Mobius model, the state of a system is defined as the value of the places. In
close correspondence to this, rate and impulse rewards are defined. A rate reward is a
function rr : S → IR, a function of the state of the system evaluated at an instant of
time and rr(s) gives the rate at which a reward is earned while being in state s over
time. An impulse reward is a function of the state of the system and the identity of
a transition that completes, and is evaluated when a particular transition completes,
so ir : S × T → IR where T denotes the set of transitions. Computation of rewards
also needs a decision on what timing to use for reward collection. The time can be
chosen as an instant of time, a steady-state, an accumulation over an interval of time,
or an accumulation over a time-averaged interval. With Mobius, all of these options
are available plus the possibility to select the characteristics of the resulting values of
interest: mean, variance or distribution of the measure. For example, to measure the
duration of spark in a CTMC model of a signaling complex, we can choose the initial
state such that k out of n channels are open and modify the opening transitions such
that once all channels are closed the model stays in that state. In that way, we obtain
an absorbing Markov chain and can measure the duration of a spark as the time to
absorption (DeRemigio and Smith 2005). To do so, we define a rate reward rr(s) = 1
if s is a state where at least one channel is open and 0 otherwise. The accumulated
reward over an interval of time (of sufficient length) gives the time to absorption and
we are, for instance, interested in its mean and variance.
5 Analysis Techniques
In this section, we briefly outline analysis techniques for a CTMC derived from a GSPN
model. We start off with discrete event simulation that applies to stochastic models in
general and then focus on numerical methods to compute transient and steady-state
distributions, π(t) and π.
5.1 Simulation
Simulation applies to stochastic models of discrete event systems in general, including
simulation of GSPNs. The key of discrete event simulation is to exercise a given model
to observe its dynamic behavior, such that those observations result in samples for a
statistical evaluation of rewards (much in the same manner as observations of the real
system would be statistically evaluated). Simulation is a process where the behavior
of a system is represented as a sequence of events (Fishman 1978). Simulation usually
generates a set of samples from performing multiple independent simulation runs that
start from the same initial state but with different settings of values for the seed of a
random generator. This approach is straightforward to parallelize; one can distribute
computations over a network with good speedup and scalability. In case of steady-
state analysis, one can alternatively compute a single but very long simulation run and
obtain approximately independent samples from batch means. This approach saves
13
the repeated simulation of an initial transient phase that is required with replicated
simulation runs.
The efficiency of simulation depends in part on the desired measurements and the
dynamics of the model, i.e., on the ratio of events that need to be computed (simulated)
to obtain a single sample for the statistical evaluation. If this ratio is low, i.e., few
simulated events are needed to generate a sample, then this is to the advantage of a
discrete event simulation. For example, estimating the mean and variance of impulse
rewards for transitions that fire frequently requires less computation time than doing
so for transitions that occur rarely. Other examples on how this ratio is influenced are
the following. A reward defined for an early instant of time can be calculated faster
than one at a later instant. Likewise, the size of the interval can also increase the
amount of work needed to be done for a simulation to solve for the reward. Estimating
the distribution of a variable with few possible values and high probabilities per value
requires less effort than for a variable with many possible values or values that have
very small probabilities (and thus are rarely seen in a simulation run).
In addition to estimate the mean E[R], variance σ2, or distribution of a reward
R, one uses simulation to also produce confidence intervals for these estimates. Let R
denote the estimated mean reward, σ the estimated standard deviation of R, then the
confidence interval for E[R] is
[R − c1σ/ √
n]
where n is the number of samples and c1, c2 are values obtained from either a standard
normal distribution or a student t distribution for a given confidence level (Law and
Kelton 2000). Assigning an upper bound for the width of a confidence intervals for
every reward is a common way to tell a simulation framework when to stop since
confidence intervals indicate the accuracy of the computed estimates of rewards. This
information is important to be able to recognize if the amount of simulated samples
is sufficient. Note that this formula also gives rise to a classical rule in simulation: in
order to reduce confidence intervals to half of its size, one needs to generate four times
its samples (with the assumption that R and σ stay the same).
While discrete event simulation of stochastic models is common in many areas
including computer systems and networks, manufacturing and production systems,
and in the biological sciences. In the later case, Gillespie’s method is often employed to
simulate CTMCs (Gillespie 1977). This method was developed for stochastic simulation
of coupled differential equations that describe the process of changes in a molecular
population via the chemical reactions. It is an adaptation of Monte Carlo simulation
that allows for a smaller number of reactions. It can get computationally expensive
but has been adapted to get less exact results in a more reasonable time frame (Lecca
2006).
Simulation can be broadly applied because of its few constraints. The key challenge
in simulation is not its applicability but to obtain results of good quality. Due to its
random nature, there is no guarantee that a simulation visits relevant parts of a state
space of a given CTMC at all or sufficiently often to obtain accurate results.
5.2 Numerical Analysis
As alternative methods to simulation, one can use numerical analysis techniques to
compute π(t) and π for a given finite CTMC, which we briefly discuss below. As a first
14
step, CTMC analysis requires to generate the underlying CTMC from a given GSPN
model with the help of a state space exploration.
State space exploration
For a given GSPN with an initial marking s0, we can compute the overall state
space of reachable states S as the transitive closure of the reachability relation. Much
research has gone into the development of data structures and algorithms that enable
exploration of extremely large, yet finite sets S. If the GSPN models makes use of
immediate transitions (or priorities), then those give rise to so-called vanishing states
that can be eliminated such that the resulting state space (of tangible states that enable
timed transitions), and thus the dimensions of Q, becomes much smaller. Symbolic
data structures like multi-terminal binary decision diagrams (MTBDD) and matrix
diagrams (MDs) as well as Kronecker representations are suitable for representing Q
in a space-efficient manner for a subsequent numerical analysis. In particular, MD
representations of Q (Ciardo and Miner 1999) can be derived directly from many
formalisms with the help of a symbolic state-space exploration as well as from a given
sparse matrix or Kronecker representation of Q; see Miner and Parker (2004) for a
recent overview paper on symbolic representations. Generation of an MD for models
with state spaces in the order of 101000 states can be achieved, thanks to symbolic data
structures and the so-called “saturation technique” (Ciardo et al 2003). Symbolic or
Kronecker representations help to reduce the memory requirements for Q to an extent
that the bottleneck for the numerical methods described below is the space needed to
store iteraton vectors for π(t) and π.
Transient Analysis
For given π(0) and Q, the distribution π(t) at time t ≤ 0 is given by
π(t) = π(0)eQt = π(0) ∞X
P k
where α ≥ max|Q(s, s)| and P = 1 αQ + I. The right hand side of this equation is
the basis of the uniformization method (also called randomization or Jensen’s method)
which is the one that is frequently deployed for its numerical robustness and efficiency.
Uniformization computes π(t) from
π(0) otherwise (17)
where l < r are appropriate constants (determined as described in Fox and Glynn
(1988) to avoid numerical instabilities for high values of αt). Uniformization is based
on the observation that pk is the distribution after exactly k steps of a corresponding
DTMC, which is given by matrix P, and wk is the probability of observing exactly k
steps in an time interval [0, t], which turns out to be a Poisson distribution. The ad-
vantage of uniformization is that it results in an accurate computation of π(t) which is
then used to evaluate rate and impulse rewards. The fundamental and computationally
15
most expensive calculation is the sequence of vectors pk for k = 1, 2, ...r. Since this
distribution may converge towards the steady-state distribution of the DTMC for high
enough values of k, recognizing a fix point where pk+1 = pk helps to compute results
for high values of t since for pk+1 = pk the summation for π(t) becomes trivial for all
pk+i, i > 0. Given a rate reward ρ() that one evaluates for π(t) as ρt = P
s∈S ρ(s)πs(t),
it is well known that one can derive results for a series of values of t by one single com-
putation of pk based on ρt = Pr
k=l wkxk where xk = P
s∈S ρ(s)πs(t). Uniformization
gets ineffective if α is relatively high with respect to t, i.e., the uniformization rate
makes the process proceed with small steps and t is large enough such that a large
number of such steps fit into the interval from 0 to t. While the former difficulties im-
pose numerical problems and high computation times, a bottleneck in terms of space is
the representation vectors π(t), pk of lengths |S| (given that Q is represented efficently
by a symbolic or Kronecker representation). For large values of t and an irreducible
CTMC, the process may reach its steady state whose distribution, which we can com-
pute in a different manner.
Steady-state analysis
As t goes towards infinity, an irreducible CTMC will reach its steady state, which
means that there is a distribution π such that π(t) = π for all t > k some constant k.
Since π can be derived from
πQ = 0 and
i=1
πi = 1
a large number of different approaches exist (Stewart 1994). Among those approaches,
iterative methods that retain sparsity of Q and that allow for symbolic or structured
representations of Q are the ones that are best for extremely large CTMCs. Commonly
applied and simple iterative methods are the Power method, the method of Jacobi and
Gauss-Seidel. We recall the method of Jacobi to illustrate the point. Let Q = D−R be
a decomposition of Q into a diagonal matrix D, a R with 0 values on its main diagonal.
The method of Jacobi is a fix point method that computes x(k+1) = x(k)RD−1 starting
with π(0) = x(0) and terminating at some k where x(k+1) = x(k) which then gives π.
Advanced techniques include some form of preconditioning and projection methods
(Krylov subspace methods), for details we refer to the textbook of Stewart (1994). For
signaling complexes, iterative Multi-level methods (Buchholz and Dayar 2004) are the
most promising according to an experimental evaluation in DeRemigio et al (2008).
There are also many approximate techniques, and many ways to do the approxima-
tion, that can be used for analysis. One way is with approximate vector representations
(Buchholz 2004); another is with iterated fix point methods (Ciardo and Trivedi 1993).
For exact solution methods, the bottleneck for applications is the computation time
and space needed to represent iteration vectors, which directly relates to the size of S.
However, for CTMCs with certain regularities it is possible to perform a state space
reduction.
Lumping Techniques
State space lumping (e.g. Buchholz 1994; Kemeney and Snell 1960) is a well-known
approach that reduces the size of a CTMC by considering the quotient of the CTMC
with respect to an equivalence relation that yields a generator matrix eQ, preserves the
Markov property and supports the desired reward measures defined on the CTMC. By
16
solving the smaller lumped CTMC, exact results for the larger CTMC, and therefore
measures of interest for the model, can be computed. We classify the many publications
on lumping into three categories.
State-level lumping applies directly to a given generator matrix Q of a CTMC and
computes eQ of the lumped CTMC. It yields an optimal partition, i.e., the smallest pos-
sible lumped CTMC. However, the size of Q limits its application. Efficient algorithms
have been designed, e.g., see Derisavi et al (2003) for the fastest known algorithm.
Model-level lumping is used to generate eQ directly from a model description. Hence,
it is specific to a modeling formalism. The approach is based on symmetry detection
among components of a compositional model. Results are known for a variety of Petri
net related formalisms, such as stochastic well-formed networks (SWN) (Chiola et al
1993; Delamare et al 2003), stochastic activity networks (Sanders and Meyer 1991),
general state-sharing composed models (Obal II 1998), stochastic automata networks
(Benoit et al 2003), and Kronecker representations, among others. The Rep/Join and
Graph composition operations considered in Section 4 fall into this category as well
and Derisavi et al (2004) and McQuinn et al (2007) describe how to apply those for
Markov chains that are represented by a symbolic data structure like a MD. While
this approach manages to avoid processing a large matrix Q, it is limited to those
symmetries that can be identified from a given model description. Hence, in general,
it does not obtain an optimal lumping as the first approach.
Compositional lumping applies the state-level lumping approach to individual com-
ponents of a compositional model. The original components are replaced by lumped
and “equivalent” components during generation of eQ. Like model-level lumping, this
approach is formalism-dependent; specifically, it relies on properties of the composition
operators. For instance, based on the fact that lumping is a congruence with respect
to parallel composition in a number of process algebra formalisms and stochastic au-
tomata networks (SANs), compositional lumping can be used in those formalisms, e.g.,
see Hermanns (2002) and Buchholz (1995).
Note that in principle, approaches from different categories can be combined. For
example, a compositional approach may yield smaller lumped components that can
then be fed to a model-level technique for further reduction of the CTMC. Finally,
state-level lumping can be applied to obtain optimal reduction of the resulting matrix.
Finally, the approach in Derisavi et al (2005) is useful for exact and ordinary lumping
of Markov chains represented as MDs without knowledge of the modeling formalism
from which the MDs were generated. Obviously these approaches are involved and
should be well-encapsulated in a software tool such that a modeler can take advantage
of these capabilities without necessarily being aware of them.
6 Mean-field Coupling
In models of Ca2+ signaling complexes, a common simplifying assumption is to abstract
from the details of a spatial layout of individual channels and assume that a single
channel experiences an average Ca2+concentration that depends only on the number
of open channels in the overall model.
17
Fig. 2 The Mobius atomic SAN model for N two-state channels.
6.1 A Straightforward Stochastic Petri Net Model
To begin our discussion and exploration of coupled Ca2+-regulated Ca2+ channels
modeled through Mobius, we first implemented the model as a stochastic Petri net
with marking dependent rates and for the smallest possible configuration with only
two states, namely Open and Closed. Fig. 2 shows the corresponding atomic Mobius
model, which has an Open and Closed place, connected through timed activities, Open-
ing and Closing. The transition rates are as defined in Eq. 2 with the expansion for
instantaneously-coupled channels as defined in Eq. 6 and 7. With this model, we heavily
rely on the mean-field coupling assumption since we cannot distinguish among different
tokens which in turn helps us to keep the model simple. Specifically, the opening rate
is
k+(c∞ + NOc∗)ηNC
where NC ∈ {0, . . . , N} is the marking, or value, of the Closed place. The closing rate
is
k−NO
where NO ∈ {0, . . . , N} is the marking of the Open place. Note that the NC and NO
occurring as the last factor in these expressions accounts for the number of channels
that can potentially make a C → O or O → C transition, respectively. The marking of
the Closed place is initialized to the total number of channels in the system, while the
Open place has an initial marking of 0. The number of distinguishable states
|S| =
N
« (18)
for this mean field model with N channels and M = 2 states per channel, yields
|S| = N + 1 which is far less than MN states in a model where we could distinguish
each channel.
Table 1a shows numerical results for model configurations that differ in the num-
ber of channels N and the selection of c∗, using K = 19 and c∗opt(K) = 0.0637µM
(DeRemigio and Smith 2005), With the exception of the case where N = 250, the
choice of c∗ was based on Eq. 13. We found that Eq. 13 is an approximation with
a limited range, i.e., there is a NU such that for N > U , c∗(N) no longer produces
expected results and Score value. For N = 250, Eq. 13 based on K = 19 does not lead
to sparks and, through simulation, we identified 0.005262µM as a reasonable choice to
achieve a significant Score value at least for the transient case.
In Mobius, measures of interest are defined through the reward variables, which in
the Ca2+ channel problem is the number of open channels. We defined a performance
18
100
80
60
40
20
M ea
n V
al ue
Time (sec)
t = ∞
Fig. 3 As the chosen time t for the transient analysis approaches infinity, the calculated expected value for the number of channels in the open state approaches the result from the steady-state analysis. For these experiments, N = 150, c∗ = 0.008069µM, and the Score and parameters are as reported in Table 1.
variable, numopen, to return the value of the place Open at a specific moment in time.
For each configuration (N, c∗) we report the mean and variance for the number of open
channels, the resulting Score value and the wall clock time in seconds used to compute
these values with different analysis techniques. Column one reports on results for the
state of the model at time t = 2000ms. Such transient results are obtained through
Mobius with simulation or transient CTMC analysis based on randomization, i.e., the
Mobius TRS solver. However, since the only difference between the two methods was
the time taken to solve for the reward, only the numerical analysis results are included
here. The simulation times ranged from about 10 seconds for a 19 channel system up
to about 5500 seconds for the 250 channel system. For steady-state analysis (t → ∞),
column two gives results from a numerical computation of the steady-state distribu-
tion with ISS, an iterative steady-state solver performing Gauss-Seidel iterations with
successive over-relaxation (SOR).
For smaller sized systems, both transient and steady-state analysis produced similar
results. However, as the number of channels is increased, it takes more time than
t = 2000ms to reach the steady state and results begin to differ. Fig. 3 shows the
results of a transient analysis for N = 150 performed over a range of values for t. The
figure illustrates that as the value of t is increased, the expected value of the number
of channels in the open state, as calculated by the transient analysis, approaches the
value calculated by steady-state analysis. Considering computation times, we see that
for CTMCs of small size, transient and steady-state numerical solvers obtain results
much faster than a discrete event simulation. However, numerical solvers are currently
limited to CTMCs in the order of 106 − 107 states and for such large models their
performance depend on a number of factors. However, DeRemigio et al (2008) report
on numerical results that show that in particular Multi-Level solution methods for
steady-state computations are very competitive with simulation for Markovian models
of Ca2+ signaling complexes.
We observe that the stochastic Petri net model is appropriate to model mean-field
coupled channel models in a straightforward manner given that the number of states per
channel is not too large to have a clear graphical presentation. In the following Rep/Join
composition of models, we describe a way of modeling Ca2+ signaling complexes in a
19
(a) (b)
Fig. 4 The Mobius atomic SAN model for a) a two-state channel and b) a four-state channel under the REP composition when position is not considered. NumOpen is the place used to store the information of how many channels in the system are in the open state. The output gates (right pointing triangles) are used to increment and decrement the value of NumOpen.
way that results in CTMCs of same dimension as before, yet the model shows individual
channels even when working with the mean-field coupling assumption.
6.2 A Rep/Join Composed Stochastic Petri Net Model
We derive an N channel release site model by composing N appropriate single channel
models. In the previous section, we derived the same model using only the atomic
model in Mobius. Now, we use the composition methods in Mobius in order to get
a more robust model. A single channel model is described as an atomic SAN model
in Mobius as shown in Fig. 4a. The channel is in either the open or closed state, the
same as for the previous section. Because there is interaction between the channels
which now needs to be shared among different instances of the atomic model, an extra
place was added to keep a count of how many states are open, NumOpen. This place is
shared among all instances of this atomic model and used to define transition rates that
depend on the number of open channels. In SANs, transitions need not be graphically
connected with places if input and output gates are used, but we do so in order to help
visualize the movement between the states. Input and output gates allow a modeler
to define state-dependent enabling and firing functions in a very general manner. The
right-pointing arrows, OG Open and OG Close, are output gates. These are used to
change the marking of the place NumOpen which is incremented upon firing transition
Opening, decremented upon firing transition Closing. These transitions use rates from
Eq. 2 with the expansion for instantaneously-coupled channels as defined in Eq. 6 and
Eq. 7 for the rates to transition between the Open and Closed states. We consider the
two-state model as the most simple one for illustrating purposes, extensions to more
complex single channel models with more states are straightforward. Some experiments
have also been done using the four-state channel model shown in Fig. 4b. Transitions
Opening[x] and Closing[x] in Fig. 4b use transition rates from Eq. 4 with the expansion
for instantaneously-coupled channels as defined in Eq. 6 and 7. The output gates are
used in the same manner as the two-state model. For the four-state model, we identified
(N, c∗) = (19, 0.3393) from running a series of experiments for different parameter
settings for c∗. Obviously there is a limit to what the graphical Petri net notation
scales if we increase M , the number of states in a single state channel model.
When reaching a certain number of places and transitions, we are in need of some
structuring mechanism or composition operations for SANs. In addition to this, we are
20
(a) (b)
Fig. 5 a) The Mobius composed model using the REP operator. b) The Mobius composed model using the REP and JOIN operator. With the addition of the JOIN node, the modeler can study the movement and behavior of a single channel. The Submodel can be either of the atomic models shown in Fig. 4.
in need of a composition operation to construct the single channel model into a release
site model with N channels. We make use of the Mobius Rep/Join composer that
provides a Rep composition operator which instantiates N copies of a given model
according to any given value of N . All N instances can share a user-given set of
state variables/places, as it is the case for NumOpen in our model. This restricted
communication and the symmetry among the N instances of the same atomic model,
implies lumpability and Mobius leverages on this to derive a reduced lumped CTMC in
an automated manner. It is this reduction that reduces the state space from otherwise
MN states for an M -state N channel model to the size of Eq. 18 as we have seen
for the plain GSPN model discussed before. Fig. 5a shows the composed model in the
graphical notation for the Rep/Join composer. The submodel Channel is either of the
atomic models as seen in Fig. 4, but it can be easily changed to another submodel with
a different format for the channel.
To measure the number of open channels, we defined a performance variable,
numopen, to return the value of the place NumOpen at a specific moment in time. As
before, we performed a transient analysis for t = 2000ms and a steady-state analysis.
Table 1b shows results for the same type of experiments as for the plain GSPN model
we discussed in the previous section. Both sets of experiments give consistent results.
The computation times for the composed model are higher due to the automated re-
duction that is necessary to achieve a CTMC of same dimension as for the plain GSPN
model. We investigated further in this issue and exercised experiments for N = 1000
and N = 5000 and observed computation times for a numerical transient analysis of
11hrs for the smaller, 4.1 days for the larger configuration. Since the resulting lumped
CTMC are of size |S| = N + 1 only, the automated reduction induces significant over-
head if it is used to this extent. Note that also the simulator, which does not apply any
lumping but simulates the unreduced model, gives significantly higher computation
times than for the plain GSPN model, for N ≥ 100 the overhead is in the range of
hours of computation time. So the use of the Rep operator should take this effect into
account.
Note that the Rep/Join composer allows for nested, tree type structures to compose
atomic models with replicate and join operators. Hence a simple modification would
allow study of the behavior of a single channel. Reduce the number of replications for
the Rep node by 1 and use a Join node to connect this last node to the rest (see Fig.
5b). In this manner, one can collect the same data for this individual channel, as for the
rest of the group, thereby seeing the movement and behavior of a single channel. This
construction also facilitates consideration of the composition of a heterogeneous set of
21
Pr op
or tio
O C1 C2 C3
Fig. 6 Probability distribution of the state of a single 4-state channel. O is the open state and C1, C2, and C3 are the closed1, closed2, and closed3 states, respectively. The left, lighter, bar is for when the tagged channel is initialized to be in either the closed1 or open states and the right, darker, bar is for when the tagged channel is initialized to be in either the closed2 or closed3 states.
single channel models, for instance, a single complicated M state model with a large
cluster of simple 2 state models. The concept follows what is known as the ’tagged
customer’ approach in performance modeling with queuing networks, which is also
known to be tricky to implement with Petri nets. We exercised some experiments for
a homogeneous cluster of four-state models as in Fig. 4b. Fig. 6 shows the probability
distribution for the channel of interest to be in one of its four states at t = 2000ms based
on transient numerical analysis. As expected, it can be seen that a channel will spend
most of its time is either the closed1 or closed3 state, since this is the fastest transition
when starting in the closed1 state. If the channel of interest is given a different initial
condition for the starting state, the probability changes slightly between closed1/open
and closed2/closed3.
Table 2 presents results from the analysis of a four-state channel model in the
same way as 1 but only for transient analysis, so we included both the simulation
and numerical analysis results. This model results in much larger CTMCs with |S| =
(N + 3)(N + 2)(N + 1)/6 states. The result show that the simulator is faster than
the transient solver TRS, which is based on a sparse matrix representation of Q and
reaches its limit at |S| = 585, 276 for N = 150.
So far, we mainly focused on the computation of Score values for channel models
through the measurement of the number of open channels. However, there is also
interest in the probability distribution for the amplitude of puffs and sparks as well
as their duration. We can measure these values by adjusting the atomic model by
selecting an appropriate starting situation of a puff or spark, i.e, k of N channels are
open, and by changing the dynamics of the model to make all channels remain closed
once all of them are closed. With this modification, we can measure the amplitude as
the maximal value of NumOpen. In addition, we can measure the duration of a spark
as the mean time to absorption in the transformed CTMC. We chose k = 5 for N = 50
in a two-state model and added a new place to store the maximum value of the number
of open channels. While the initialization for a plain GSPN is trivial, for the replicate
composed model we need to model a randomized initialization for k out of N channels
to be open. We added input gates to ensure that the system did not start until the
initialization period was over (which took an average of 0.1 ms of simulation time) and
22
Fig. 7 The new Mobius atomic SAN model used for transient analysis. The addition of the Init place, activity and gate is used to initialize the model to have 5 channels in the open state. The new place MaxOpen is used to store the maximum value of NumOpen. Also added was two Input Gates used to stop the simulation when NumOpen gains the value of 0.
(a) (b)
Pr op
or tio
Pr op
or tio
Number of Open Channels
Fig. 8 a) Probability distribution of the number of open channels for a two-state 50 channel model leading to a Score of 0.32. b) Rescaled to see a slight increase in the proportion for N ≈ 30. Parameters used are as in Table 1.
to stop the simulation once the number of open channels reached 0. This new model
can be seen in Fig. 7. The number of channels chosen was 50, as a larger enough value
to be able to see interesting behavior, yet small enough to be easily processed.
We begin with an analysis of how much time the model spends at states with a
certain number of channels being open and make use of the simulator to compute this.
From 10,000 simulation batches, we recorded the number of open channels at time
t = 200ms with a two-state 50 channel system. The results from this experiment can
be seen in Fig. 8. The mean value for NO was 1.57, with a variance of 24.17, which gives
a Score of 0.31. The highest probability is for the smallest number of open channels,
corresponding with previous research showing a small average value for the number of
channels in the open state (Nguyen et al 2005; DeRemigio and Smith 2005). There is
a slight increse in the proportion for N ≈ 30, which occurs because of the occasional
peak that fully opens all channels.
The probability distribution of the maximum number of open channels is shown in
Fig. 9. These results came from 10,000 simulation batches and give an average value
for MaxOpen of 7.18. In Fig. 9a, it can be seen that the value with highest probability
for the maximum number of open channels is about 5, corresponding with the results
on the probability distribution of the number of open channels having the greatest
probability with a small number of channels in the open state. However, similar to the
observation that the number of open channels is a bimodal distribution, we can see
23
Pr op
or tio
Pr op
or tio
Maximal Open Channels
Fig. 9 a) Probability distribution of the maximum number of open channels for a two-state 50 channel model. b) Rescaled to show more detail. Score and parameters are as Fig. 8.
0.2
0.16
0.12
0.08
0.04
Time (ms)
Fig. 10 Probability distribution of the mean time to absorption (all channels closed) given 50 channels and an initial threshhold of 5 open channels.
a slight increase in the probability for having a maximum number of open channels
of around 45. Fig. 9b has a smaller scale for the y-axis so this activity at the values
around 45 can be more easily seen.
The probability distribution of the mean time to absorption is shown in Fig. 10.
For this experiment, a reward variable, mtta, was defined to return 1 if NumOpen
was greater than 0, for every moment from 0 ≤ t ≤ 2000, which means that mtta
was an interval-of-time variable, instead of instant-of-time like the previous reward
variables. The average simulation time to absorption is 13.39 ms, with a maximum
value of 500.56 ms and a standard deviation of 26.17 ms. Again, these results are from
10,000 simulation batches. These results shows that in almost 80% of the simulations,
all channels closed within 20ms. This again corresponds with previous research that the
average number of channels in the open state is small. This experiment was also solved
using an accumulated reward solver (ARS) which solves for transient interval-of-time
variables. ARS gives an expected value for mtta to be 12.86 ms, which is close to the
result from simulation.
7 Spatial Arrangement and Graph Composer
When considering the Ca2+ channels with position, we need to incorporate for each of
the N channels information on its distance to other channels and access to the state
24
of any other channel to formulate state-dependent transition rates. In Section 3, the
impact of distances is represented by a N × N coupling matrix C. Among the three
composers of Mobius (Rep/Join, Graph composition and Action Synchronization), the
Graph composer is the most natural fit. The Graph composer allows us to compose
instances of atomic (or composed models) with the same Join composition as in the
Rep/Join composer, however the resulting graph of composed models need not be a
tree as for the Rep/Join composer. The Graph composer is able to leverage on symme-
tries in the model composition graph to reduce the associated CTMC with the help of
lumpability. Mobius employs canonical labeling and automorphism groups to automat-
ically detect suitable symmetries (Obal II et al 2006) and to achieve a reduced CTMC.
This mitigates the state space explosion problem and helps to enable numerical meth-
ods to solve models with very large CTMCs. Using the Graph composer, a modeler can
maintain the atomic model used with a Rep/Join composer, with only minor changes.
This means that experimentation can be done with both a spatial arrangement of the
channels, as well as the mean-field coupling, with very little extra work.
For this paper, we model a cluster of four two-state channels arranged in a square,
with a minimum inter-channel distance of 0.03 µM, giving us the following C matrix:
C =
1 CCA (19)
Examination of this matrix shows there are only 3 distinct values as discussed with
Eq. 14 (c0 = 3.2664, c1 = 0.1637, and c2 = 0.1080) which we can declare as global
variables in Mobius to simplify storage and referencing. However, with a larger system,
and a different arrangement of channels, it may not be so easy and obvious which and
how many values to take from the C matrix. To help with this issue, Mobius has the
capability to allow a modeler to incorporate outside C++ code to calculate, access,
and store these values.
Since each channel is affected differently by other channels based on how far apart
they are, each channel needs to know which of the other channels are open. The atomic
model is then changed slightly from the one used under the Rep Composer, to allow for
the storage of the extra information. The challenge is to have a generic atomic model
that is then configured with respect to its location. The new atomic SAN model can
be seen in Fig 11a. It has N = 4 new places LeftStatus, RightStatus, OppStatus, and
SelfStatus to track which channel is open. During the composition this information is
shared in a manner that gives all channels the correct information. When more channels
are added to the system, more places are added to the atomic model. Fig. 11b shows
the atomic model that is used for a 19 channel system. The 19 channel system has a
spatial arrangement of a hexagonal lattice; we add three additional places: CenterStatus
is for the channel in the middle of the lattice, while NumOpen has been divided into
NumInnerOpen and NumOuterOpen to reflect that the channels are now arranged in
two tiers.
In the composition, the Join operator is used to share variables among the atomic
models – for instance, NumOpen is shared by all channels since each channel affects how
many are open. The information is shared through the Join node by carefully matching
the status places between every pair. There are two ways to proceed, the one is to use
absolute names like StatusAtOne to encode the state of the channel at the first location
and consistently use it this way throughout the model. This implies that the associated
25
(a) (b)
Fig. 11 The Mobius atomic SAN model under the GRAPH operator where position is con- sidered in a) a 4 channel system and b) a 19 channel system.
(a) (b)
Fig. 12 The Mobius composed model using the GRAPH operator with a) a four channel system in a grid arrangment and b) a 19 channel system in a hexagonal lattice.
distance and transition rates change in each instance of that atomic model, which in
turn implies that the atomic model itself is changed. The other alternative is to use
relative names like LeftStatus for the state of the neighboring channel to the left. The
latter allows us to take advantage of regular arrangements like the grid arrangment
or the hexagonal lattice and to encode the arrangement solely in the composition
operation of the Join, namely by carefully matching which variables are shared and
merged among models. For instance, as in Fig. 12a, channel 1’s RightStatus is shared
with channel 4’s SelfStatus, channel 3’s LeftStatus and channel 2’s OppStatus. This
information is then used to add (or not add) the effect, c1 or c2, of that channel to the
transition rate. If done appropriately, the symmetries of regular spatial arrangements
that imply lumpability for the underlying CTMC are carried over to symmetries in
the Graph composed model, which in turn can be detected and used by Mobius for an
automated reduction of the CTMC.
7.1 Results and Analysis of Graph composition experiments
The composed model can then be solved in Mobius either by simulation or by numer-
ical analysis where only the latter leverages on lumpability and symmetries. Table 3
shows results from some of the experiments done with the Graph composition, with a
comparison listing of the same experiments done using the Rep composition. As with
Table 1, only the results from the numerical analysis are included, since the simmu-
26
lation produces nearly matching results. The simulation times for the Rep composed
models were about 3 and 260 seconds for the steady-state analysis for N = 4 and
19, respectively. With the Graph composed models, the times were about 5 and 38
seconds, N = 4 and 19, respectively. The Graph composed models were also solved
using the symmetry detection and state-space reduction option included with Mobius.
The calculated mean and variance values for the reward variable matched the results
already reported for N = 4. For N = 19, results for a mean-field coupled model and
the spatial model differ which illustrates that the spatial arrangement may have an
impact. We do not investigate here if there is a way to calibrate both models to result
in a consistent answer, as in DeRemigio et al (2008b). The main point here is to to
show the capability of Mobius and its Graph composer for modeling the Ca2+ channel
system when position is considered. A limitation with using Mobius to solve a posi-
tional system is the increase in the amount of work needed to define the model. With
a positional system, all interactions between channels must be included in the model.
As the number of channels increases, the number of interactions increases, which also
increases the likelihood of user error in the model formation.
8 Conclusion
In this paper, we discussed ways to use stochastic Petri nets and in particular SANs
supported by Mobius to express and subsequently analyze Markovian models of Ca2+
signaling complexes. We selected this application area since it highlights two charac-
teristics of complex stochastic models in biology: 1) models are structured by their
compositional formulation, and 2) models have a spatial aspect that a modeling for-
malism should be able to take into account. We showed how to use stochastic Petri
nets and composition operations in way that naturally matches with the structure
of a model. Composition is used to make Petri net modeling scale with the size of
models but also to help a modeler express regularities that are present in a model to
the advantage of any subsequent numerical analysis. In Mobius, this means to reduce
the associated Markov chain based on lumpability that in turn can be deduced from
the compositional structure of a model in an automated manner. More concretely, we
demonstrated how to use the Mobius Rep/Join composer to model mean-field coupled
multi-channel models and the Graph composer to mode multi-channel models with a
static two dimensional spatial arrangement of channels. The models were analyzed with
respect to their ability to exhibit puffs and sparks (i.e. high Score), and the amplitude
and duration of these events. We demonstrated multiple methods of analysis using
simulation, transient and steady-state analysis of Markov chains and show trade-offs
among different modeling and analysis techniques.
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