Spatial Stochastic Simulators
Kim “Avrama” Blackwell
George Mason University
Krasnow Institute of Advanced Studies
Diverse Numbers of Molecules
Spatially Inhomogeneous
G protein coupled
receptors
Diffusion required
for signal interaction
Glutamate receptors
1 M  60 molecules
molecular
interactions occur
stochastically
Kotaleski and Blackwell 2010
Small number of molecules in spines
Large number of molecules in system
Spatial Stochastic Simulators
• Particle based
– Smoldyn, MCell, CDS
– Individual molecules are represented as point-based
particles, which diffuse random distance and random
direction at each time step
– If two reacting molecules pass near each other they
may react
– Computations increase with number of molecules
Diffusion
Association
sb
Membrane
Dissociation
MCell
• Geometry from volumetric imaging data
using Blender (www.blender.org)
Transparent
– Mesh elements may be reflective,
transparent, or absorptive
• Surface or volume diffusion
q
q q
– Ray tracing determines whether molecules
would have collided during (fixed) time step
Reflective
• Reaction rules depend on order of
reaction, and whether surface or volume
molecules involved (Kerr et al. SIAM J Sci Comput 2008)
MCell Diffusion
• Diffusion distance from probability density:
• Radial distance from uniformly distributed
random variable X:
• Speed computations by storing values of X
in look-up table
• Direction: uniformly distributed random
variable [0,2π)
Mcell – STDP example
• Calmodulin activation versus spike timing
– Do NMDA receptors and VDCC produce
different calmodulin profiles?
• Neuron model to determine voltagedependent open probability of VDCC and
NMDA
• MCell model with calmodulin, calbindin,
NCX and PMCA
•
Model: Keller et al. PLoS One 2008, tutorial: http://www.mcell.org/tutorials/
MCell Model
NMDAR
Pre-synaptic
Terminal
Spine Head
Spine Neck
Dendrite
Pumps
(Membrane)
VDCC
Calcium binding
Proteins
(cytosol)
Unpaired
Stimuli
• Calcium
differs due to
channel
distribution
Keller et al. PLoS One 2008
EPSP-AP
AP-EPSP
Paired Stimuli
• Calcium
depends on
timing of AP
versus glutamate
release
Keller et al. PLoS One 2008
CDS
• Particle based simulator with event driven
algorithm
– All possible collisions are detected during
short dt
– If collision detected, the exact collision time is
calculated
– Earliest collision (or reaction events) are
simulated one-by-one until dt
• Particles have volume, thus can simulate
crowding and volume exclusion
•
http://nba.uth.tmc.edu/cds/content/download.htm
CDS Example
• Morphology from
triangular meshes
• CaMKII diffusion
out of spine
depends on
morphology (b)
and also binding
targets and F-actin
•
Byrne et al. J Comput Neuro
2011
Stochastic (non-spatial)
Simulators
• Gillespie (Exact Stochastic Simulation Algorithm)
Propensity of reaction aj  Kf  Np
– Propensity of any reaction, a0 =  aj
– Next reaction occurs with exponential distribution with
mean a0:
– Identity of reaction selected randomly, based on
propensity
– Computations increase with number of molecules
Extensions to Gillespie Algorithms
• Spatial Gillespie, e.g. Fange et al. 2010,
PNAS
ad2
a2
– Morphology is subdivided into small
+
ad1
compartments
ad4
a1
– Propensity of diffusion calculated from
+
ad3
diffusion coefficient, ad  D  Nd
– Diffusion considered as another reaction
• Tau leap – non-spatial
– Allow multiple reaction events, Kj, to occur for
each reaction at each time step, t, according to
Poisson:
 a jt
k
Kj

e
(a j t )
k!
Hybrid Models
• Partition the reaction-diffusion space into
two or more sets of reactions (and
diffusion)
• Each set is simulated differently
– Diffusion – deterministic, reactions –
stochastic
– Fast reactions - deterministic, slow reactions
– stochastic
– “Critical” reactions - exact stochastic, noncritical reactions – tau leap
STEPS
• Spatial extension of exact stochastic
simulation algorithm
– Tetrahedral meshes allows realistic
geometries
– Diffusion constant can vary between
compartments
– Simulations are specified in python, witih
morphology, reactions and simulations
specified independently (for ideal control of
simulation experiments)
– http://steps.sourceforge.net/STEPS/Home.html
STEPSCerebellar LTD
Calcium
Buffers
AMPA
Receptor
Protein
Phosphatase 5
Protein
Phosphatase 2A
MapKinase
Phosphatase 1
Raf
MEK
Raf-act
Positive
Feedback
Loop
ERK
Inactivation, dephosphorylation
Activation, phosphorylation
Calcium
Pumps
calcium
PKC
Arachidonic
Acid
cPLA2
Protein
Phosphatase 1
Single Spine Model
• Average of multiple
simulations reveals
graded induction of
LTD
• Single runs reveals
bistability at
intermediate calcium
Antunes et al. J Neurosci 2012
Time (min)
Model Limitations
• All these model have either small volume
(single spine) or small number of reactions
(calmodulin+CaMKII)
• Only MCell model uses voltage to
determine calcium influx
• Smoldyn
– Particle simulation algorithm incorporated into
Moose (Genesis 3) and VCell
– No neuroscience examples yet
NeuroRD
• Spatial extension to Gillespie tau leap
– Multiple reaction events and diffusion events can
occur during each time step
– Morphology is subdivided into small compartments
• Cuboidal meshes
and cylindrical
meshes possible
NeuroRD – Mesoscopic
• Subdivide dendrites and spines into sub-volumes
• Pre-calculate the probability that one molecule leaves
the compartment or reacts
pr  k r  N1  N 2   t
pm  2  D   t /  x 2
N!
P( N , j ) 
p j (1  p ) ( N  j )
( N  j )! j !
• Look-up tables store the probability that j out of N
molecules leave a compartment or react
• At each time step, for each molecule, choose a random
number to determine the number, j, molecules out of N
leaving or reacting
NeuroRD
Calculate number of molecules
Calculate j reacting
or k moving using
Poisson distribution
Determine destinations for diffusing
molecules
NeuroRD - Validation
• An approximation, to allow large scale simulations
• Agrees with Smoldyn, and deterministic solution for
reaction-diffusion system
10
350
Distance 0.5
NeuroRD
Smoldyn
Determ
Molecules
250
2.5
200
150
Molecule A
8.5
8
Molecules
300
Distance 0.5
NeuroRD
Smoldyn
Determ
8.5
6
4
100
2
50
Molecule B
0
0
0
400
800
1200
Time (msec)
0
500
Oliveira et al. 2010, PLoS One
1000
1500
Time (msec)
2000
NeuroRD
• NeuroRD is up to 60 times faster than Smoldyn
• Computations increase linearly with number of
compartments, but not molecules
NeuroRD
# injected
Time
(h:mm:ss)
Memory (kb)
Smoldyn
Simulation
# initial
molecules
Diffusion
0
2000
0:00:02.86
1608
0:00:07.04
2344
Reaction
28853
0
0:00:05.97
1764
0:08:03.53
26524
Reaction &
Diffusion I
662
4000
0:00:04.51
1764
0:02:48.90
22168
Reaction &
Diffusion II
6619
40000
0:00:07.58
1772
2:19:58.00
23760
Oliveira et al. 2010, PLoS One
Time
(h:mm:ss)
Memory (kb)
NeuroRD Development
Biochemical Oscillator
Srivastava et al., J Chem Phys
Spatial Gene Oscillator
• mRNA is inactive in the nucleus, diffuses into cytosol
• A diffuses to nucleus, binds to DNA
• Effect of diffusion constant (2 cytosol compartment)
Spatial Biochemical Oscillator
Inactive mRNA in nucleus, activated by binding in cytosol compartment
Vary number of compartments, and translation compartment
Protein quantity
1000
1200
Molecules
Molecules
1200
800
600
1000
800
600
400
400
200
200
0
0
0
50
100 150
Time (hours)
Diffusion=10, 4 comp
translation in cytosol 1
R, cytosol 1
A, cytosol 2
A, cytosol 1
A, nucleus
1400
200
Diffusion=10, 4 comp
translation in cytosol 2
R, cytosol 2
A, cytosol 2
A, cytosol 1
A, nucleus
500
400
Molecules
Diffusion=10, 3 comp
translation in cytosol 1
R, cytosol 1
A, cytosol 1
A, nucleus
1400
300
200
100
0
0
50
100 150
Time (hours)
200
0
50
100
150
Time (hours)
mRNA production is faster when A binds to DNA
mRNA production and degradation are faster for A than R
Protein synthesis and degradation are faster for A than R
R degrades A (at same catalytic rate that A spontaneously degrades)
200
Spatial Biochemical Oscillator
Diffusion=10, 4 comp
inactive
mRNA A, cyt 1
mRNA R, cyt 1
5
Molecules
4
mRNA
active
10
Diffusion=10, 4comp
inactive active
mRNA R, cyt 1
mRNA R, cyt 2
mRNA A, cyt 2
3
2
8
1
0
Diffusion=10, 3 comp
inactive
mRNA A, cyt 1
mRNA R, cyt 1
5
4
Molecules
50
100
Time (Hours)
150
active
Molecules
0
200
6
4
3
2
2
1
0
0
0
DNA
50
100
Time (Hours)
150
200
0
50
100
Time (Hours)
150
200
NeuroRD
• Model specification allows good
experimental design, with separate files for
– Reactions
Tissue
– Spatial morphology
– Initial conditions
Experiment
– Stimulation
Simulation control
– Output specification
– Top level file which specifies reactions,
morphology, initial conditions, output specs,
time step and spatial grid, random seed
NeuroRD – Morphology File
• Specify start and end of each segment
• Specification includes id, region type, location (x,y,z),
radius, and optional label
<Segment id="seg1" region="dendrite">
<start x="1.0" y="1.0" z="0.0" r="0.5" />
<end x="1.0" y="2.0" z="0.0" r="0.5" label="pointA"/>
</Segment>
• Additional segments start on a previous segment
• Branching is possible – see branching.tar
NeuroRD – Reaction File
• Define each species that has either a reaction pool or
conservepool
• Include diffusion constant, which can be 0
<Specie name="mGluR" id="mGluR" kdiff="0"
"mu2/s"/>
kdiffunit =
• Specify Reactions
• First order – single reactant and product
• Second order – two reactants or two products
NeuroRD – Reaction File
• Include forward and backward rate constants
<Reaction name = "glu+mGluR--glu-mGluR reac"
id="glu+mGluR--glumGluR_id">
<Reactant specieID="glu"
/>
<Reactant specieID="mGluR" />
<Product specieID="glu-mGluR" />
<forwardRate> 5e-03 </forwardRate>
<reverseRate> 50e-03 </reverseRate>
<Q10>
0.2
</Q10>
</Reaction>
NeuroRD – Initial Condition File
• Four types of initial conditions
1. General concentration of molecule in entire
morphology, or
2. Region specific concentration
• Overrides general concentration
3. Surface Density of membrane molecules
• Overrides concentration specifications
4. Surface Density of Membrane molecules in
specific region
• Overrides general surface density
NeuroRD – Initial Condition File
• General concentration of each molecule should be
specified (zero otherwise)
<NanoMolarity specieID="mGluR" value="5e3" />
• Surface density if molecule is membrane bound
<PicoSD specieID="PLC" value="2.5" />
• Initial conditions for different parts of morphology
<ConcentrationSet region="PSD" >
followed by <NanoMolarity specieID=“IP3" value=“30" />
NeuroRD – Stimulation File
• Stimulation used to inject molecules
• Temporary fix until software is integrated with
software for simulating neuron electrical activity and
ion channels
• Specify molecule and injection site
<InjectionStim specieID="Ca" injectionSite="pointA">
• Repetitive trains can be created
• Specify onset time, duration, rate (amplitude)
• period and end used for train
• InterTrain Interval to repeat train (e.g. For LTP)
NeuroRD – Output Specification
• Specify dt for output, species and compartment
<OutputSet filename = "dt1" region="dendrite" dt="1.0">
<OutputSpecie name="glu" />
<OutputSpecie name="IP3" />
</OutputSet>
• Multiple outputSets can be specified
•
Sample slowly changing molecules less frequently
•
Sample glutamate receptors from PSD only
NeuroRD – Model file
• Specify all the other files
<reactionSchemeFile>Purkreactions</reactionSchemeFile>
<morphologyFile>Purkmorph</morphologyFile>
<stimulationFile>Purkstim</stimulationFile>
<initialConditionsFile>Purkic</initialConditionsFile>
<outputSchemeFile>Purkio</outputSchemeFile>
• Specify some other parameters, such as algorithm
variations and random seed
• Indicate total simulation time, time step and largest
compartment size
NeuroRD – running simulation
• Java -jar stochdiff.jar Purkmodel.xml
• Morphology output file
•
Purkmodel.out-mesh.txt
• Ascii output file
•
•
name of model file -- .out –output set name - conc.txt
Purkmodel.out-dt1-conc.txt