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Steve Andrews:
Beyond particle-based simulation: rule-based and filament modeling
Pavol Bauer: Parameter sweeps and sensitivity in stochastic
simulations of chemical kinetics
Essentially all current biochemical modeling software focuses on the
chemical reaction networks that animate living cells. These networks
are important, of course. For example, metabolism, cellular signaling,
and cell cycling all depend on complex reaction networks. Recent
algorithm and software advances now enable researchers to explore
the roles of intracellular stochasticity and spatial organization
in biochemical reaction networks. For example, my Smoldyn software
represents proteins as point-like particles that diffuse, react,
and interact with membranes in 3-dimensional continuous space, much
as proteins do in real cells. However, this focus on modeling reaction
networks misses a large fraction of what really happens in real cells.
Many, and perhaps most, real proteins don't act as individuals, but
perform their work as components of transient multi-protein complexes.
A special type of extended complex is the intracellular filament,
such as actin and microtubules. I will briefly survey the state
of the art in multimeric complex and filament modeling and I will
present my recent work on these topics.
Ruth Baker:
Models of cellular migration for cells of different shapes and sizes
Continuum, partial differential equation models are often used to describe
the collective motion of cell populations, with various types of motility
represented by the choice of diffusion coefficient, and cell proliferation
captured by the source terms. Previously, the choice of diffusion
coefficient has been largely arbitrary, with the decision to choose a
particular linear or nonlinear form generally based on calibration arguments
rather than making any physical connection with the underlying
individual-level properties of the cell motility mechanism. In this talk I
will discuss a series of individual-level models, which account for
important cell properties such as varying cell shape and volume exclusion,
and their corresponding population-level partial differential equation
formulations. I will demonstrate the ability of these models to predict the
population-level response of a cell spreading problem for both proliferative
and non-proliferative cases. I will also discuss the potential of the models
to predict long time travelling wave invasion rates.
Thomas Bartol: How to build a synapse from molecules, membranes,
and Monte Carlo methods
Biochemical signaling pathways are integral to the information storage,
transmission, and transformation roles played by neurons in the nervous system.
Far from behaving as well-mixed bags of biochemical soup, the intra- and
inter-cellular environments in and around neurons are highly organized
reaction-diffusion systems, with some subcellular specializations consisting of
just a few copies each of the various molecular species they contain. For
example, glutamtergic synapses at dendritic spines in area CA1 hippocampal
pyramidal cells contain perhaps 100 AMPA receptors, 20 NMDA receptors, 10
CaMKII complexes, and 5 free Ca++ ions in the spine head. Much experimental
data has been gathered about the neuronal signaling pathways involved in
processes such as synaptic plasticity, especially recently, thanks to new
molecular probes and advanced imaging techniques. Yet, fitting these
observations into a clear and consistent picture that is more than just a
cartoon but rather can provide biophysically accurate predictions of function
has proven difficult due to the complexity of the interacting pieces and their
relationships. Gone are the days when one could do a simple thought experiment
based on the known quantities and imagine the possibilities with any degree of
accuracy. This is especially true of biological reaction-diffusion systems
where the number of discrete interacting particles is small, the spatial
relationships are highly organized, and the reaction pathways are non-linear
and stochastic. Here I will present how biophysically accurate computational
experiments performed on cell signaling pathways can be a powerful way to study
such systems and can help formulate and test new hypotheses in conjunction with
bench experiments. MCell is a Monte Carlo simulator designed for the purpose
of simulating exactly these sorts of cell signaling systems. I will introduce
fundamental concepts of cell signaling processes in the organized and compact
spaces of synapses, and the insights that can be gained through building
realistic models of neurotransmission.
Jon Chapman: Diffusion of finite size particles
Short range interactions such as volume exclusion or contact
inhibition processes can play an important role in determining
transport properties for diffusing particles. Often such interactions
are modelled at a mean-field level using a phenomenological guess at a
nonlinear diffusion coefficient, or via some closure assumption.
Here we use matched asymptotic expansions to determine the effective
diffusion coefficient for a system of small particles
undergoing Brownian diffusion. The resulting equations compare
favourably with stochastic simulations, and outperform both point
particles and closure. The difference between self and collective
diffusion is highlighted.
David Fange: From micro to macro using spatially discrete stochastic
reaction-diffusion models
Spatially discrete stochastic reaction-diffusion models, based on
the reaction-diffusion master equation (RDME) has been shown to give
diverging results with increasing spatial resolution. The underlying
reason is that RDME commonly uses macroscopic rate constants, which
includes time for diffusional transport between reactions. However,
for bi-molecular reactions, in a high spatial resolution RDME, the
diffusional transport is modelled explicitly and should not be
included in the rate constant.
Recently we have sought to resolve this issue by deriving rate
constants based on a microscopic hard-sphere model.
Here the amount of diffusional mixing within each lattice point
depend on the spatial resolution of the simulations,
and thus the reaction rate constants become scale dependent in
a microscopic consistent manner.
By implementing the scale dependent reaction rate constants into
MesoRD, we have now made it easy to perform physically consistent
stochastic reaction-diffusion simulations down to a spatial resolution
of the size of molecules while still retaining the speed gained
when only low spatial resolution is needed.
By using simple examples we have shown how strikingly important
it is to choose a correct model framework for the problem at hand.
For example, spatial correlations between reactive substrates can
change the overall properties of the system, especially for molecules
in two dimensional geometries such as cell membranes.
Mark Flegg: Matching stochastic reaction-diffusion simulation
strategies over an interface
Traditionally reaction-diffusion phenomena have been modelled with
the use of continuous distributions of chemicals in the form of
partial differential equations (PDEs). In biology, there are
cases where a continuous deterministic description, like PDEs, is
appropriate. However, biological phenomena that are highly sensitive
to small concentrations or critical processes are dependent and/or
controlled by fluctuations in local concentrations. There are two
fundamental classifications to the modelling of stochastic
reaction-diffusion processes; on-lattice and off-lattice. Whilst
both of these modelling processes are similar in that they
describe a discrete number of molecules with the same stochastic
macroscopic physical kinetics, neither approach has emerged as
dominant amongst modellers. This is because the method that is
most appropriate, whether this be a deterministic continuous PDE
method or one of the two stochastic methods, depends on the
particular biology that is being modelled. Often the biological
phenomena occur on multiple scales for which a macroscopic (PDE),
a mesoscopic (stochastic on-lattice), or microscopic (stochastic
off-lattice) model is appropriate in different regions of the domain.
In this talk I will discuss the ways of correctly coupling these
regions. In particular I will discuss how an off-lattice model of
a stochastic reaction-diffusion process should be coupled to an
adjoining discrete (on-lattice) subdomain and/or to a subdomain
that describes the molecules in an expected distribution sense
and governed by a set of PDEs.
Martin Howard: Dissecting the noisy pom1p intracellular concentration
gradient in fission yeast
Chemical gradients can generate pattern formation in biological systems. In
the fission yeast Schizosaccharomyces pombe, a cortical gradient of the
kinase pom1p functions to position sites of cytokinesis and cell polarity,
and to control cell length. Here, using quantitative imaging, fluorescence
correlation spectroscopy and mathematical modelling, we study how its
gradient distribution is formed. Pom1p gradients exhibit large cell-to-cell
variability as well as dynamic fluctuations in each individual gradient.
Our data lead to a two-state model for gradient formation where pom1p
molecules associate with the plasma membrane at cell tips, and then diffuse
on the membrane while aggregating into and fragmenting from clusters, before
disassociating from the membrane. In contrast to a classical one-component
gradient, this two-state gradient buffers against cell-to-cell variations in
protein concentration. This buffering mechanism, together with
time-averaging to reduce intrinsic noise, allows the pom1p gradient to
specify positional information in a robust manner.
Samuel Isaacson: Stochastic reaction diffusion methods for studying
the influence of subcellular structure on the dynamics of biochemical
processes
We will discuss our work developing stochastic reaction-drift-diffusion
methods and their application to the study of cellular processes. We will
first describe our recent work constructing numerical methods for solving
both spatially-discrete and spatially-continuous models. The methods we
develop will then be applied to study how volume exclusion due to cellular
substructure influences the dynamics of cellular processes. Our detailed
three-dimensional simulations will be constructed using several different
types of high-resolution imaging of the ultrastructure within mammalian
cells.
Karen Lipkow: Towards a complete spatial model of bacterial
chemotaxis - and the yeast nucleus
The chemotaxis system of Escherichia coli, which helps the bacteria to swim
towards the most favourable environment, is arguably the best studied
biological signal transduction pathway and as such a favoured subject of
Systems Biology. We have pioneered a spatially detailed model using the
particle-based software Smoldyn and studied the effects of (dynamic)
molecule positioning on signalling properties such as speed, fidelity and
robustness. While early models were restricted to the cytoplasm, we have
recently added transmembrane reactions and complex interactions in the polar
chemoreceptor cluster. This had led to surprising insights which are only
possible to gain in a three-dimensional model of this resolution. In
parallel, we developed a 2-dimensional Java-based model at slightly higher
detail. We are currently testing our major modelling predictions in vivo.
Time allowing, I will also present some of our Smoldyn simulations on
transcription factor movement and genome organisation in the nucleus of
Saccharomyces cerevisiae.
Per Lotstedt: Multiscale simulation of stochastic processes
in systems biology
In living cells, molecules are transported actively or by diffusion
and react with each other when they are close. The molecules move
and react in three space dimensions (3D), on two dimensional (2D)
surfaces and on one dimensional (1D) structures. The surfaces can
be the cell membrane or the nuclear membrane and the DNA and the
microtubules are examples of one dimensional structures embedded
in 3D. A molecule diffuses in 3D, attaches to the lower dimensional
manifold, diffuses on the manifold and is released from it.
The cell domain is partitioned into voxels or compartments in 1D,
2D, and 3D in a mesoscopic model. The diffusion of the molecules
is simulated by jumps between the voxels and these jumps and the
reactions occur with a certain probability. At a finer, microscopic
level, each individual molecule is tracked, it moves by Brownian
motion and reacts with other molecules according to the
Smoluchowski equation.
Algorithms for simulations with the mesoscopic, microscopic and
meso-micro models in 1D, 2D, and 3D will be described and applied
to systems in molecular biology.
Alan McKane: Stochastic pattern formation: the linear noise
approximation and beyond
Models of biological systems are typically studied by analysing deterministic
differential equations or by carrying out computer simulations. It is less
common to analyse individual based models mathematically, although the tools
required to do this frequently exist and several interesting and important
mechanisms can be understood through their application. In this talk I will
briefly review some aspects of the formalism that can be used to investigate
stochastic effects in individual based models and then go on to apply these
to study stochastic Turing patterns and stochastic travelling waves. These
patterns exist outside of the region in which similar patterns are found in
the corresponding deterministic system, and often occur for a wider range of
parameter values. I will end by briefly discussing other instances of
stochastic pattern formation which can only be understood by going beyond the
previously used linear noise approximation.
Hans Othmer: A stochastic model of actin waves in Dictyostelium discoideum
In the absence of directional signals, neutrophils and Dictyostelium
cells explore their environment randomly, and thus the signaling
networks that control the cytoskeleton are tuned to produce this
random movement. The balance between the RhoA and Rac pathways
determines whether dendritic network formation or bundling of F-actin
and myo-II-controlled contraction dominates, and the competition
between them leads to complex patterns of traveling actin waves in
the cortex in both cell types. Treatment of Dd with latrunculin, which
acts as an F-actin depolymerizing agent, leads to dissolution of
the cortex, and a variety of actin waves emerge during its
reconstruction. First static spots of actin network arise, then some
of these evolve into traveling spots, and some in turn give rise
to full-fledged traveling waves. These waves propagate by treadmilling,
as shown by actin recovery after bleaching. In this talk we
discuss results from a detailed stochastic model of actin polymerization
at the membrane that describes cortical re-construction and wave
propagation, and predicts how different processes control
the various behaviors.
Linda Petzold: Spatial stochastic amplification in cell polarization
Polarization is an essential behavior of living cells, yet the
dynamics of this symmetry-breaking are not fully understood.
Previously, noise was thought to interfere with this process; however,
we show that stochastic dynamics play an essential role in robust cell
polarization and the dynamic response to changing cues. We describe a
spatial stochastic model of polarisome formation in mating yeast. The
model is built on simple mechanistic components, but is able to
achieve a highly polarized phenotype with a relatively shallow input
gradient, and to track movement in the gradient. The spatial
stochastic simulations are able to reproduce experimental observations
to an extent that is not possible with deterministic simulation.
Spatial stochastic simulation is a challenging computational problem.
We report on our progress to date on the development of accurate and
efficient algorithms.
Wouter-Jan Rappel: Eukaryotic chemotaxis: modeling cell
motion and gradient sensing
Understanding chemotaxis, during which cells move in the presence of
chemical gradients, is a challenging and fascinating problem.
Chemotaxis plays a crucial role in a number of biological processes,
including neuronal patterning, wound healing, embryogenesis, and cancer
metastasis. Even though many of the key biochemical components
involved in chemotaxis are known, it remains a poorly understood
process. In this talk, I will present our recent modeling efforts
aimed towards a better quantitative and mechanistic understanding
of chemotaxis.
Koichi Takahashi: Simulating cellular environments with E-Cell
System Version 4
Predictive modeling of cellular behaviors bottom up from molecular
interactions is a holy grail in computational systems
biology. Although 'rigorous' chemical master equations exist
since the midst of the 20th century, biochemical network
simulations has been suffering from lack of quantitative bases.
I argue that part of this lack of predictive power in
systems biology is due to the negligence of the non-idealistic
intracellular environments, including molecular crowding,
spatial heterogeneity of proteins including localization
and clustering in microdomains, and microscopic spatio-temporal
correlations. Such an environment is beyond the scopes
of classical theories such as Einstein-Smoluchowski theory of
molecular diffusion and Michaelis-Menten's enzymatic
reaction kinetics. The advent of advanced laser micro-/spectro-scopy
that can directly observe macromolecules in living cells
made us realize the need of modeling at the level of individual
molecules and their internal states, instead of electrical
circuit-like reaction networks. To fulfill this computational
need, we have been developing high-performance and rule-based
biochemical reaction network simulation platform E-Cell System
Version 4 for supercomputing architectures that can potentially
handle the enormous amount of computation required to
reproduce macroscopic physiological responses of cellular systems
from microscopic molecular events. I will introduce a few
types of single-molecular simulation methods implemented
on E-Cell 4, including (1) the enhanced Green's Function Reaction
Dynamics, which is an exact and high-performance method for
particle reaction-diffusion problems, (2) Spatiocyte, a
massively parallel microscopic lattice method, and
(3) a reaction Brownian Dynamics (BD) method. I will discuss
some applications of these methods such as (a) the role of
spatio-temporal correlations and molecular crowding on the MAP
Kinase double-phosphorylation cycle using the eGFRD and BD,
(b) PI3K/PTEN self-organized waves in Dictyostelium discoideum
cAMP response pathway using Spatiocyte, and (c) a mechanism to
generate cell-to-cell variations in signal response in human
epidermal growth factor pathway using Spatiocyte.
Abstracts of Posters
Spatiotemporal stochastic models in the reaction-diffusion master
equation formalism can be simulated by resolving the geometry
using two or three dimensional unstructured meshes as in our
modular software framework URDME.
Thanks to its flexible structure and well-defined interfaces, new
solvers may be developed in an independent manner and connected
directly to the underlying layers of the simulation environment.
We have developed a solver for stochastic sensitivity analysis
allowing for a path-wise control of the processes occurring
during the simulation of complex kinetic networks. This method
allows us to compare single trajectories under arbitrary
perturbations and calculate the difference and even derivative
in the expected value to a reference simulation run.
This new functionality in URDME enables outstanding possibilities
to compare and optimize models under different configurations,
including an accurate estimation of a system's response.
Tommaso Biancalani: Stochastic patterns in population models
A common starting point to understand pattern formation is formulating
the problem in terms of reaction-diffusion partial differential
equations; the emergence of patterns can then often be explained via
a linear instability of an homogeneous state, as proposed by Turing
in the 50s. However, using partial differential equations requires
that every population is considered as continuous, which is not
appropriate when the number of degrees of freedom is not macroscopic.
To overcome this, stochastic models have been introduced: the
finiteness of the degrees of freedom translates into an
"intrinsic stochasticity" which can be successfully treated
using techniques from the theory of stochastic processes. The research
reported in this poster has been devoted to study the role of
stochasticity in pattern forming systems. Specifically,
we show that in stochastic models the parameter region for which
patterns arise is greatly enlarged compared to the deterministic
counterpart, and it does not require a separation of diffusivities.
We carried out this analysis in a simple 2-species reaction-diffusion
model which exhibits steady patterns and travelling waves. The stochastic
patterns have been observed in numerical simulations and analytically
characterised using a linear noise approximation.
Pascal Bochet:
A representation of microtubules in lymphocytes using Smoldyn
Helper lymphocytes regulate the immune responses and are the
targets of the HIV virus. One key regulators of the multiplication and
differentiation of helper lymphocytes is interleukin-7 (IL-7).
IL-7 signaling in helper lymphocytes is mediated by STAT5 which is
phosphorylated by the IL-7 receptor signaling complex at the cell
membrane and transported into the nucleus along microtubules.
Smoldyn is a widely-used program which simulates the diffusion
and the reactions of individual molecules at the cellular scale.
However, so far, Smoldyn has no method for the representation
of microtubules in simulations. Here we describe an implementation
in Smoldyn of hundreds of microtubules bridging cytoplasmic and
nuclear membranes and a simulation for the
transport of several thousands signaling molecules carried by
unidirectional molecular motors moving along those microtubules.
With this simulation we represented signal transduction in helper
lymphocytes. This simulation helps us to interpret experimental
data on signal transduction in these cells. They showed that the
transport of STAT5 assisted by molecular-motors is slower than simple
Brownian diffusion. This leads to new hypothesis about the role of
microtubules in intracellular signaling in this system.
This is a joint work with Blanche Tamarit and Thierry Rose.
Katarina Bodova: The role of randomness in the collective food
finding behavior and trail formation in ant colonies
Ants are able to form narrow paths between the nest and the food source
using multiple pheromones to mark the trail. We designed a mathematical
model based on a release of two different signalling pheromones where the
motion of ants is governed by two main components - random direction change
that improves food finding ability, and the systematic component, based on
pheromone concentration, important for trail signalling. Numerical
simulations show two phases of the trail formation: initial food search and
trail refining. The randomness is crucial not only for finding the shortest
trail but also for an ability of ants to simultaneously search for multiple
food sources. This is a joint work with my student Miriam Malickova
(Comenius University).
Maria Bruna: Excluded-volume effects in the diffusion of hard spheres
Excluded-volume effects can play an important role in determining transport
properties in diffusion of particles. Here, the diffusion of finite-sized
hard-core interacting particles in two or three dimensions is considered
systematically using the method of matched asymptotic expansions. The result
is a nonlinear diffusion equation for the one-particle distribution
function, with excluded-volume effects enhancing the overall collective
diffusion rate. An expression for the effective (collective) diffusion
coefficient is obtained. Stochastic simulations of the full particle system
are shown to compare well with the solution of this equation for two
examples.
Niall Deakin: Stochastic multiscale models of cancer invasion
There are several important steps involved in the growth and spread
of cancer. Local invasion of tissue is one of these steps or "hallmarks".
The current work will focus on the local invasion of the host tissue
achieved by the secretion of enzymes involved in proteolysis (tissue
degradation). We present a mathematical model of cancer cell invasion
of host tissue at both the macro- (tissue) and micro-scale (cell).
The model considers cancer cells and a number of different matrix-degrading
enzymes from the MMP family (matrix metalloproteinases) and their
interaction with, and effect on, the extracellular matrix (ECM)
in a 2D domain. While the macro-scale incorporates partial
differential equations, the micro-scale operates over a smaller
spatial and temporal scale, and as such stochastic properties can
be useful in accurately modelling this scale of invasion. Also
included will be work that focuses solely on the activation
system of a specific MMP (MMP-2) at the cell level with a guide
to its actions surrounding the individual cell.
Omer Dushek: Ultrasensitivity in multisite phosphorylation
of membrane-anchored proteins
Cellular signaling is initially confined to the plasma membrane,
where the cytoplasmic tails of surface receptors and other
membrane-anchored proteins are phosphorylated in response to
ligand binding. These proteins often contain multiple
phosphorylation sites that are regulated by membrane-confined
enzymes. Phosphorylation of these proteins is thought to be
tightly regulated, because they initiate and regulate signaling
cascades leading to cellular activation, yet how their
phosphorylation is regulated is poorly understood. Ultrasensitive
or switchlike responses in their phosphorylation state are not
expected because the modifying enzymes are in excess. Here, we
describe a novel mechanism of ultrasensitivity exhibited by
multisite membrane-anchored proteins, but not cytosolic proteins,
even when enzymes are in excess. The mechanism underlying this
concentration-independent ultrasensitivity is the local
saturation of a single enzyme by multiple sites on the substrate.
Local saturation is a passive process arising from slow membrane
diffusion, steric hindrances, and multiple sites, and therefore
may be widely applicable. Critical to this ultrasensitivity is
the brief enzymatic inactivation that follows substrate
modification. Computations are presented using ordinary
differential equations and stochastic spatial simulations. We
propose a new role, to our knowledge, for multisite
membrane-anchored proteins, discuss experiments that can be used
to probe the model, and relate our findings to previous
theoretical work.
Benjamin Franz: Multiscale modelling: Coupling Brownian dynamics
with mean-field PDEs
We discuss two different situations where coupling Brownian
dynamics simulations with mean-field PDEs can be advantageous.
In many biological applications, a strongly inhomogeneous domain
leads to the necessity of multiscale methods that allow the use
of mean-field approaches in regions of high copy numbers, whilst
exact particle dynamics need to be considered in other parts
of the domain. We present an algorithm that achieves this coupling
in a way that statistical properties are preserved. The algorithm
provides exact particle tracking data in parts of the domain,
whilst using quick continuum approaches everywhere else. In the
second situation, we consider multi-species systems that have
strong between-species differences in copy numbers, so that
different approaches need to be applied for different species.
We present how the coupling can be achieved and demonstrate its
functionality on the example of bacterial chemotaxis.
Andreas Hellander: Multiscale simulation of reaction-diffusion
kinetics in the URDME software framework
URDME is a flexible software framework for stochastic
reaction-diffusion simulation. Using unstructured meshes, URDME
is capable of handling realistic geometries. The framework is
designed to combine an advanced modeling environment and an
efficient built-in core simulation algorithm with the support for
easy extensions of the software. Mesoscopic simulation based on
the reaction-diffusion master equation (RDME) offers many
computational challenges. Well resolved geometries lead to
expensive simulations due to the large number of diffusion
events. Furthermore, in the highly diffusion limited regime the
RDME can give inaccurate results for fine meshes, as compared to
microscopic particle tracking methods. We have recently shown
that no local modification of the mesoscopic reaction rates in
the classical RDME can rectify this problem. In this light we
show how a microscopic-mesoscopic hybrid method that can achieve
the accuracy of the microscopic modeling level at a much reduced
cost was implemented in the URDME framework. In addition, URDME
has been used to develop models of active transport, a
mesoscopic-macroscopic hybrid simulation algorithm and
approximate algorithms. We also report on recent work to extend
URDME to allow for simulation in distributed computing
environments such as Grids and Clouds and discuss the challenges
involved in managing and handling the large amounts of simulation
data that is generated by e.g. parameter sweeps.
Stefan Hellander: Multiscale simulation of reaction-diffusion
processes
We have developed a multiscale method that makes it possible to
simulate parts of the domain and subsets of the molecules at a
microscopic, particle-tracking, level and the rest of the domain
and the molecules at a mesoscopic level. For some problems we
have been able to reduce the number of molecules simulated at the
microscopic level by an order of magnitude without an appreciable
loss of accuracy. In addition to this we are also able to
represent general surfaces in an efficient and flexible manner,
and are able to apply the multiscale method also to systems with
reactions on membranes and microtubules.
The multiscale method is based on splitting the mesoscopic and
the microscopic variables during a global time step. It is
crucial to choose this time step correctly to balance all errors.
We will show some recent results concerning how to choose this
time step optimally for a simple model problem.
Max Hoffmann:
Spatial stochastic simulations of focal adhesions
Focal adhesions are cell-matrix contacts which transduce and
integrate mechanical as well as biochemical cues from the
environment. They consist of more than 170 proteins with
more than 700 interactions (collectively known as the "adhesome").
Due to their association with the plasma membrane, focal
adhesions are spatially organized in a layer-like manner.
The layer of transmembrane receptors from the integrin family
recruits a layer of cytoplasmic components like talin, paxillin
and vinculin, which in turn connect to the actin cytoskeleton,
which close to focal adhesions is also organized in a
mainly horizontal manner.
In recent years, many aspects of this assembly process and the subsequent
hierarchical structure have been investigated in detail. High throughput
RNAi screens have revealed the effect of knocking down different fractions
of the adhesome. However, to further understand these effects, it is
important to place it in the context of the spatial organization of focal
adhesions.
We present a spatial simulation of the focal adhesion assembly process,
incorporating the main known features of focal adhesions. We especially
focus on the maturation of focal complexes into focal adhesions and the
signaling feedback loops that govern this process.
Hye-Won Kang: The effect of the signaling scheme on the robustness
of boundary formation in development
Boundary formation between different cell types
involves spatially-distributed signals called morphogens that
influence the phenotypic identity of cells. Usually different
cell types are spatially segregated, and the boundary between
them may be determined by a threshold value of some state
variable. Our question is how sensitive the boundary location is
to variations in parameter values in the system, such as reaction
and diffusion rates, size of the system, and input flux. In the
poster, we analyze both deterministic and stochastic
reaction-diffusion models to understand how the signaling scheme
affects the variability of boundary determination between cell
types.
Heinrich Klein:
Brownian dynamics of direct and hierarchical virus assembly
The assembly of proteins into supra-molecular complexes is at the
heart of many biological processes and the dynamic interplay of
the different components ensures biological functionality. Their
size ranges from the nanometer-scale (for example for
bi-molecular complexes) through tens of nanometers (for example
for viruses, which typically consist of multiples of 60
components) up to the micrometer scale (for example for the actin
filament networks of the cytoskeleton). Even in steady state most
biological complexes remain highly dynamic, with dissociation
events being balanced by association events.
Studying bi-molecular association reaction has a long tradition
in chemistry, physics and mathematics. Early approaches relied
mainly on analytical solution, e.g. using the Smoluchowski
equation. The advances in computational sciences now make it
possible to address complex questions beyond those analytically
tractable.
Here we present a computational approach to study the assembly of
viruses based upon the overdamped Langevin equation (Brownian
dynamics). As fundamental building blocks we use "patchy
particles": hard spheres partly covered by reactive patches.
The particles associate stochastically upon diffusional overlap
of two patches, thus implementing the concept of an encounter
complex. Similarly the dissociation of bound structures is
treated stochastically based on intrinsic dissociation rates. We
first study direct assembly from single particles for simple
virus geometries and determine values for the association and
dissociation rates that are optimal for fast and complete
assembly. We then compare to the results for hierarchical
assembly, when the rates change depending on the completed
assembly of sub-complexes.
Richard Kollar:
Mathematical modeling of macrophage motion in diffusive environment
We design and implement a mathematical model describing an immune
reaction: motion of a macrophage and phagocytosis in a diffusive
environment. The motion of a macrophage is modeled by a stochastic
cellular Potts model. We capture both an internal diffusive dynamics
of actin activators inside the macrophage and external dynamics of
chemoattractant in the environment along with a chemotactic motion
of bacteria. The key parameters influencing performance efficiency
of macrophage are numerically studied. This is a joint work with my
student Peter Dizo (Comenius University).
Anel Mahmutovic:
Microscopically constrained reaction-diffusion processes
There exist a number of modeling frameworks for describing intracellular
reaction processes, where each framework has its advantages and
disadvantages. The key is knowing when it is appropriate to choose one
framework over the other or when a certain framework breaks down. In order
to show the drastic impacts that may arise if the spatial and stochastic
aspects are not taken into account, we have analyzed three small, but
instructive, reaction networks. Simulations were carried out using the
MesoRD software, which has recently implemented microscopically consistent
reaction kinetics. In the first example a spatial stochastic model is
required to correctly describe local saturation of degradation enzymes,
which lead to an increased number of substrate molecules when compared to a
non-stochastic spatial model. In the following two examples, we show how
spatial correlation between reactants in bimolecular reactions give changes
in the overall model behavior when microscopic associations and
dissociations are accounted for, as compared to when they are not. In 3D
this effect is shown by reactant degradation affecting microscopic
rebinding. In 2D we describe a system where long-range spatial correlations
make the enzyme molecules apparently more effective in consuming its
substrate. Consequently, using these simple examples, we have shown that it
may be imperative to take into account both the stochastic and spatial
aspects in a physically consistent manner when modeling intracellular
processes.
Partha Sarathi Mandal:
Noise regulated spatio-temporal pattern formation in a ratio-dependent
prey-predator model
Spatio-temporal models of prey-predator interaction in presence
of environmental fluctuations are receiving significant attention
from researchers as it is evident that noise have ability to
alter the resulting pattern significantly. Existence of
spatio-temporal chaos for deterministic model are examined by
some researchers but whether it will appear for the parameter
values within Turing-Hopf domain or as a non-Turing pattern
remain as a debatable question. In [Banerjee and Petrovskii,
Theoretical Ecology 4, pp. 37-53, 2011], it was established that
spatio-temporal chaos can be observed for parameter values within
the Turing-Hopf domain for a ratio-dependent prey-predator model
where both the species are distributed over two dimensional
landscapes. Apart from spatio-temporal chaos, other two types of
stationary patterns (cold spot and labyrinthine) were observed.
In the present work, we have investigated the same model in presence
of temporally correlated and spatially uncorrelated coloured noise
terms. The noise terms are introduced to take care of the variability
of intrinsic birth rate for the prey population and intrinsic death
rate for the predators. Exhaustive numerical simulation reveals
the role of environmental noise to modulate the abundance of both
the species and they are capable to suppress the spatio-temporal
chaos. Further, the strength and correlation time of noise terms
are responsible for the irregular distribution of population over
the two dimensional landscape but the magnitude of deterministic
parameters regulate the chance of extinction of either species.
Lina Meinecke: Diffusion simulation on unstructured meshes
via first exit times
Molecules move in biological cells by diffusion. For simulation
of such diffusion, the cell is partitioned into compartments or
voxels in a mesoscopic model. The number of molecules in a voxel
is recorded and the molecules can jump between the voxels to
model diffusion. In order to accurately represent the geometry of
the cell it is helpful to use unstructured meshes for the voxels.
The probability for a molecule to jump is determined by the
diffusion coefficient and the geometry of the voxel and its
neighbors. Algorithms simulating diffusion in discrete space
based on finite element or finite volume methods sometimes
encounter problems on these unstructured meshes in 3D. If the
mesh is of poor quality the jump coefficients may be negative.
We present a new approach to diffusion simulation using first
exit times that for unstructured meshes guarantees positive jump
coefficients. These first exit times can be sampled from the
survival probability for molecules within a voxel. It will be
shown that this approach yields similar results as the SSA on
cartesian meshes.
Iwona Mroz: The lattice model of an evolving population
An individual-based lattice model of an evolving population is
presented [Mroz, Pekalski and Sznajd-Weron, Phys Rev Lett 76,
pp. 3025-3028, 1996; Mroz and Pekalski, Eur Phys J B 10,
pp. 181-186, 1999; Mroz, Physica A 323, pp. 569-577, 2003].
The population is composed of individuals
characterised by genotypes, phenotypes and ages. The individuals
move over habitats represented by square lattices, mate, produce
offsprings and die. The habitats are spatially separated. Each of
the habitats is characterised by a "model phenotype"
describing the phenotype of an individual that is fully adapted
to it. Using computer simulations based on the standard Monte Carlo
technique we investigate the conditions under which a model
population can survive in a given habitat and is able to colonise
a new habitat. The individual's probability of survival is
related to the individual's adaptation to a given habitat. The
individual's adaptation is defined basing on the individual's
similarity to the "model phenotype": linear and power
functions are used. The process of colonisation is observed in
time and space. In particular we investigate formation of a
hybrid zone [Mroz and Pekalski, Eur Phys J B 10, pp. 181-186, 1999]
that can occur between populations living in the neighbouring
habitats.
Andrew Mugler:
Divide and conquer: the signaling benefit of spatial partitioning
Spatial heterogeneity is a hallmark of living systems, even at the
molecular scale in individual cells. A key example is the spatial
partitioning of membrane-bound proteins, either via lipid domain
formation or cytoskeleton-induced corralling. Here we show that such
partitioning can lead to improved signal propagation. We exactly solve
a stochastic model describing a ubiquitous motif in membrane signaling:
a species that switches between an inactive and an active state and drives
the switching of a second species. The solution reveals that
partitioning can reduce the noise in the signal by suppressing
correlations between molecules. Moreover, an optimal partition
density arises from the tradeoff between suppressing correlations
and avoiding near-empty partitions. The predicted density agrees
quantitatively with experimentally observed systems.
Tim Rogers:
Demographic noise leads to the spontaneous formation of species
When a collection of phenotypically diverse organisms compete
with each other for limited resources, the population can evolve
into tightly localised clusters. Past studies have neglected the
effects of demographic noise and studied the population on a
macroscopic scale, where cluster formation is found to depend on
the shape of the curve describing the decline of competition
strength with phenotypic distance. I will show how including the
effects of demographic noise leads to a radically different
conclusion. Two situations are identified: a weak-noise regime in
which the population exhibits patterns of fluctuation around the
macroscopic description, and a strong-noise regime where clusters
(species) appear spontaneously even in the case that all
organisms have equal fitness.
Alexander Rush: Validity of a two-step model alternative to full
spatial simulation of diffusion limited reactions
We explore the usefulness of a two-step diffusion limited
differential and stochastic model as an alternative to expensive
spatial simulations in modelling diffusion limited reactions,
with applications to modelling diffusion limited processes on
cell membranes. We use a Gillespie algorithm implementation of
the two step differential model and compare the ordinary
differential and stochastic results to those yielded by Smoldyn,
a full spatial stochastic simulation using Smoldyn, looking in
detail at the approach to equilibrium and the time series at
steady state, via auto-correlation. We observe the two-step model
is valid over a large range of diffusion constants and
concentrations regimes. We discuss parameter regimes where the
two step model begins to fail in predicting the behaviour up to
and at equilibrium, predicting a different steady state
altogether. With the scope of the model established, we use it to
study the MAPK pathway and demonstrate both the processive and
distributive behaviour established by recent, more expensive
spatial simulations. Finally, we discuss the importance of these
results in allowing for efficient computations that captures the
spatial-temporal correlations that exists in diffusion limited
reactions.
Michael Sadovsky:
Smart diffusion in spatially distributed biological communities
Presented is a study of the dynamics of spatially distributed
populations and communities. The study requires that each individual
has detailed knowledge of the environmental conditions everywhere
in order to simulate or model the spatial behaviour of each
coupled population to determine the survivability of each
population and the dynamics of the system as a whole.
This process of collective reaction-diffusion behaviour is called
smart diffusion. The high level of detail needed in this model
requires the simulation of individuals. The work aims to implement
a general model of such system. A community occupies a discrete
space (lattice), so that migration is a transfer of N_{i,j}
individuals out the node (i,j) to the nearest ones, or,
reciprocally, a transfer of some number of individuals
into the node. In practice, the direction a population is
biased towards migrating in is determined by the relation
between the net reproduction functions observed in the nodes;
the transfer flux depends on the knowledge of the beings towards
the environmental conditions in the nodes of immigration. We
are interested to see how these migratory strategies can aid
the survival of a population by comparing it to a pure
reaction-diffusion model, whereby no migration strategy is
chosen. This reaction-diffusion model is therefore simulated
using the spatial Gillespie algorithm.
Matthew Simpson:
Mean field descriptions of collective migration with strong adhesion
Random walk models based on an exclusion process with contact
effects are often used to represent collective migration where individual
agents are affected by agent-to-agent adhesion. Traditional mean field
representations of these processes take the form of a nonlinear diffusion
equation which, for strong adhesion, becomes ill-posed and does not predict
the averaged discrete behaviour. Alternatively, we show that collective
migration with strong adhesion can be accurately represented using a moment
closure approach. The moment closure results are extremely robust. We obtain
accurate results by incorporating nearest neighbour correlation effects
meaning that the additional computational requirement of the moment closure
approach is small. This is joint work with Dr Ruth Baker (Oxford) and Mr
Stuart Johnston (Queensland University of Technology).
Thomas Sokolowski: Modelling Cell Polarization with eGFRD
Many intracellular processes such as cell polarization and
locomotion require the establishment of localized concentrations
of signaling proteins. Such accumulations typically arise due to
an interplay between passive transport, i.e. cytosolic and
membrane diffusion, and one-dimensional active transport on
linear filaments such as microtubules. The underlying mechanisms
and symmetry breaking effects however remain to be elucidated.
To study how distinct modes of intracellular transport can give
rise to polarized intracellular patterns we combine experiments
in vivo and in microfabricated chambers with stochastic
simulations. For our in silico approach we employ eGFRD (enhanced
Green's Function Reaction Dynamics), a spatially resolved
stochastic simulation technique which is both exact and
efficient, to set up a whole-cell model. eGFRD partitions the
simulation volume into simple sub-volumes and makes use of exact
analytical solutions of the reaction-diffusion problem to
implement an event-driven scheme with single-particle resolution.
This technique is superior to brute-force Brownian Dynamics
within a wide regime of biologically relevant system parameters.
Here we present the results of our recent work which supplements
the classical eGFRD framework by 2D-diffusion and 1D-diffusion
with drift on confined structures as well as particle-structure
interactions in a way that retains exacticity. These new features
allow us to study the interplay between various stochastic
transport modes and simulate reaction-diffusion systems in
complex geometries with high efficiency. We apply the enhanced
framework to model the localization of growth factors in fission
yeast.
Marc Sturrock:
Stochastic spatial modelling of the Hes1 pathway
Many intracellular signalling pathways exist where the low copy
number and spatial distribution of molecular species (e.g. mRNA,
proteins) cannot be neglected. These pathways often contain
negative feedback loops and can exhibit oscillatory dynamics in
space and time. One such pathway that embodies these properties
is the Hes1 pathway, which is involved in the regulation of
somitogenesis and is also implicated in cancer. We present a
stochastic spatial model of the Hes1 negative feedback system and
show that the observed experimental data can be more faithfully
captured when such a model is used rather than a system of
partial differential equations.
Bartosz Szczesny:
Spiral waves and nonlinear diffusion in bacterial metapopulations
Stochastic metapopulation lattice model of three
species cyclic dominance is enriched by adding two different
mobility rates resulting in nonlinear diffusive terms. The model
exhibits spiral waves which affect the biological landscape.
Following the van Kampen expansion, the model can be approximated
by simple PDEs. Then, instead of the usual mapping treatment,
weakly nonlinear asymptotic analysis of the PDEs is performed to
yield the Complex Ginzburg-Landau equation (CGLE). This well
known amplitude equation predicts spiral wave patterns for
certain parameter ranges and a connection is made to the
biological parameters in the original stochastic model.
Thomas Woolley: Stochastic reaction and diffusion on growing
domains: Understanding the breakdown of robust pattern formation
Biological patterns, from population densities to animal coat
markings, can be thought of as heterogeneous spatiotemporal
distributions of mobile agents. Many mathematical models have
been proposed to account for the emergence of this complexity,
but, in general, they have consisted of deterministic systems of
differential equations, which do not take into account the
stochastic nature of population interactions. One particular,
pertinent criticism of these deterministic systems is that the
exhibited patterns can often be highly sensitive to changes in
initial conditions, domain geometry, parameter values, etc. Due
to this sensitivity, we seek to understand the effects of
stochasticity and growth on paradigm biological patterning
models. Through Fourier analysis we are able to suggest a reason
behind this lack of robustness and identify possible mechanisms
by which to reclaim it.
Kit Yates:
From microscopic to macroscopic descriptions of cell migration on growing
domains
Cell migration and growth are essential components of the
development of multicellular organisms. The role of various cues
in directing cell migration is widespread, in particular, the role
of signals in the environment in the control of cell motility and
directional guidance. In many cases, especially in developmental
biology, growth of the domain also plays a large role in the
distribution of cells and, in some cases, cell or signal
distribution may actually drive domain growth. There is
a near-ubiquitous use of partial differential equations (PDEs) for
modelling the time evolution of cellular density and environmental
cues. In the last twenty years, a lot of attention has been devoted
to connecting macroscopic PDEs with more detailed microscopic models
of cellular motility, including models of
directional sensing and signal transduction pathways. However,
domain growth is largely omitted in the literature. In this poster,
individual-based models describing cell movement and domain
growth are studied, and correspondence with a macroscopic-level PDE
describing the evolution of cell density is demonstrated.
The individual-based models are formulated in terms of random
walkers on a lattice. Domain growth provides an extra mathematical
challenge by making the lattice size
variable over time. A reaction-diffusion master equation
formalism is generalised to the case of growing lattices and
used in the derivation of the macroscopic PDEs.