Spatial Stochastic Simulators

Kim “Avrama” Blackwell

George Mason University

Krasnow Institute of Advanced Studies

Diverse Numbers of MoleculesSpatially Inhomogeneous

Kotaleski and Blackwell 2010

Glutamate receptors 1 M 60 molecules

molecular interactions occur stochastically

G protein coupled receptorsDiffusion required for signal interaction

Small number of molecules in spinesLarge 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

b

Association DissociationDiffusion

Membrane

MCell• Geometry from volumetric imaging data

using Blender (www.blender.org)– Mesh elements may be reflective,

transparent, or absorptive

• Surface or volume diffusion– Ray tracing determines whether molecules

would have collided during (fixed) time step

• Reaction rules depend on order of reaction, and whether surface or volume molecules involved (Kerr et al. SIAM J Sci Comput 2008)

Transparent

Reflective

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 voltage-dependent 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

VDCC

NMDAR

Pumps(Membrane)

Calcium bindingProteins(cytosol)

Pre-synapticTerminal

Spine Head

Spine Neck

Dendrite

UnpairedStimuli

• Calcium differs due to channel distribution

Keller et al. PLoS One 2008

Paired Stimuli• Calcium

depends on timing of AP versus glutamate release

EPSP-AP AP-EPSP

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

• Tau leap – non-spatial– Allow multiple reaction events, Kj, to occur for

each reaction at each time step, , according to Poisson:

Ke a

kj

aj

kj

( )

!

• Spatial Gillespie, e.g. Fange et al. 2010, PNAS

– Morphology is subdivided into small compartments

– Propensity of diffusion calculated from diffusion coefficient, ad D Nd

– Diffusion considered as another reaction

a1

a2

ad1

ad3

ad2

ad4

+

+

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, non-

critical 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

STEPS-Cerebellar LTD

calcium

PKC

ArachidonicAcid

cPLA2

ProteinPhosphatase 2A

ERK

MEK

Raf Raf-act

MapKinasePhosphatase 1

ProteinPhosphatase 1

ProteinPhosphatase 5

AMPAReceptor

CalciumBuffers

CalciumPumps

Inactivation, dephosphorylationActivation, phosphorylation

PositiveFeedbackLoop

Single Spine Model

• Average of multiple simulations reveals graded induction of LTD

• Single runs reveals bistability at intermediate calcium

Time (min)Antunes et al. J Neurosci 2012

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

• 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

P N jN

N j jp pj N j( , )

!

( ) ! !( ) ( )

1

p k N N tr r 1 2 p D t xm 2 2 /

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 system10

8

6

4

2

0

Mol

ecul

es

2000150010005000Time (msec)

Distance 0.5 8.5NeuroRD Smoldyn Determ

350

300

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150

100

50

0

Mol

ecul

es

12008004000Time (msec)

Distance 0.5 2.5 8.5 NeuroRD Smoldyn Determ

Oliveira et al. 2010, PLoS One

Molecule A

Molecule B

NeuroRD

• NeuroRD is up to 60 times faster than Smoldyn• Computations increase linearly with number of

compartments, but not moleculesNeuroRD Smoldyn

Simulation # initial molecules

# injected Time (h:mm:ss)

Memory (kb) Time (h:mm:ss)

Memory (kb)

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

NeuroRD DevelopmentBiochemical 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 OscillatorInactive mRNA in nucleus, activated by binding in cytosol compartmentVary number of compartments, and translation compartment

500

400

300

200

100

0

Mo

lecu

les

200150100500Time (hours)

Diffusion=10, 4 comptranslation in cytosol 2

R, cytosol 2 A, cytosol 2 A, cytosol 1 A, nucleus

1400

1200

1000

800

600

400

200

0

Mo

lecu

les

200150100500Time (hours)

Diffusion=10, 3 comptranslation in cytosol 1

R, cytosol 1 A, cytosol 1 A, nucleus

1400

1200

1000

800

600

400

200

0

Mo

lecu

les

200150100500Time (hours)

Diffusion=10, 4 comptranslation in cytosol 1

R, cytosol 1 A, cytosol 2 A, cytosol 1 A, nucleus

mRNA production is faster when A binds to DNAmRNA production and degradation are faster for A than RProtein synthesis and degradation are faster for A than RR degrades A (at same catalytic rate that A spontaneously degrades)

Protein quantity

Spatial Biochemical Oscillator

5

4

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1

0

Mo

lecu

les

200150100500Time (Hours)

Diffusion=10, 3 comp inactive activemRNA A, cyt 1 mRNA R, cyt 1

5

4

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2

1

0

Mo

lecu

les

200150100500Time (Hours)

Diffusion=10, 4 comp inactive activemRNA A, cyt 1 mRNA R, cyt 1 10

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6

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2

0

Mo

lecu

les

200150100500Time (Hours)

Diffusion=10, 4comp inactive active mRNA R, cyt 1 mRNA R, cyt 2 mRNA A, cyt 2

DNA

mRNA

NeuroRD

• Model specification allows good experimental design, with separate files for– Reactions– Spatial morphology– Initial conditions– Stimulation– Output specification– Top level file which specifies reactions,

morphology, initial conditions, output specs, time step and spatial grid, random seed

Tissue

Experiment

Simulation control

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" kdiffunit = "mu2/s"/>

• 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