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Mod
elin
g &
Sim
ulat
ion
Fram
ewor
kEfficiEntly Solving thE StochaStic REaction-DiffuSion
MaStER Equation in c++ with a coMSol intERfacElukaS a. wiDMER, JoERg StElling
B. Drawert, S. Engblom, and A. Hellander, BMC Systems Biology 6, 76 (2012). [1]D. K. Samuylov, L. A. Widmer, G. Székely, and G. Paul, ISBI (2015). [2]
Department of BioSyStemS Science anD engineering, etH ZuricH, anD SwiSS inStitute of BioinformaticS, 4058 BaSel, SwitZerlanD
contact: lukas.widmer@bsse.ethz.ch
In budding yeast, we model stochastic microtubule dynamics and their regulation in 2D- and 3D geometries created with COMSOL. Our C++ simulation engine improves on state-of-the-art in performance. Its results can be used for virtual microscopy and experimental design.
csb
URDME
C++ NSM Solver
0 50 100 150 200 250 300 350 400
Time to integrate MinD example model for 900 seconds
CPU seconds
We achieved a 2-fold performance increase:
We tested our C++ Next Subvolume Method (NSM) solver against the state-of-the-art C solver URD-ME[1] on the well-known MinD model from E. coli:
Model Schematic
E. coli Mesh[1]
4.5 μm
1 μm t [s]
0 20 40 60 80 100 120 140 160 180 200
# m
olec
ules
0
500
1000
1500
2000
2500
3000Membrane-bound MinD copy number in one of the poles.
× 10-60 2.25 4.5
00.10.20.30.40.50.60.70.80.9
1Membrane-bound MinD spatiotemporal average
100 110 120 130 140 150 160 170 180 190Acquisition start time (seconds)
State samples are recorded at the acquisition times of the virtual microscope (Tint = .5s)
in s
ilico
98 99 100 101 102 103
[4]
in vivo
To accurately simulate a fluorescence microscopy ex-periment in silico, our collaborators and us developed[2] a method to enable physically-based microscopy of our simulation results:
in vivo, we often image at the resolution limit, where distinguishing hypotheses may not be trivial.
Image: Mathias Bayer
Object Space
Optics & Sampling
Pixel Space
Noise & Quantization
Image Space
Example: GFP-MinD Timelapse
in silico experiments, experimental design Reconstruction of Geometry / Photometry Benchmarking image analysis pipelinesUs
age
Scen
ario
s
LSGSZ Systems Biology
Reaction-Diffusion Model on Dynamic Subdomains (Microtubules)
Workflow for Simulation & Analysis of Stochastic Reaction-Diffusion Modelsin silico Experiment
VirtualMicroscope
in vitro / in vivo Experiment
Experimental Data
Diffusion Coefficients
Initial Conditions, Time Steps
Subvolumes
Diffusion Matrix
Model Specification
Mental Model
Mesh Generation
Geometry
Domains:• Cell Boundary• Microtubule
Mesh
MATLAB
Solver Input
HDF5
Solver Output
Model ReactionsModel Refinement
HDF5
Data Analysis
Perf
orm
ance
Eva
luat
ion:
M
inD
Osci
llatio
ns in
E. c
oli
Com
pari
son
to in
vivo
Dat
a by
Vir
tual
Mic
rosc
opy
Mic
rotu
bule
Str
uctu
re
and
Func
tion
Stoc
hast
ic R
eact
ion-
Diffu
sion
Mod
elin
g
GDP-boundαβ
GTP-boundαβ
Tubulin
Building Blocks
Plus End (+)
10811
32 1
4 5 6
71312
9
Minus End (-)
Microtubule Structure
A single microtubule consists of 13 protofilaments.
Microtubule Dynamics
GTP-tubulin subunits attach and subsequently hydrolyze to GDP-tubulin. Once GDP-tubulin reaches the growing plus tip, the microtubule rapidly starts depolymerizing, giving rise to a highly dynamic system.
Growing Microtubule
Shrinking Microtubule
Catastrophe
Rescue
Microtubule Plus-Tip GTP-tubulin attachment GTP- and GDP-tubulin dissociation Catastrophe
When 3/13 proto- filaments GDP-exposed[3]
Whole Microtubule GTP-tubulin -> GDP-tubulin (hydrolysis)
RescueOccurs as a result of
regulatory micro- tubule-associated pro-
teins in vivo.
Cytosol GTP- to GDP-tubulin Diffusion Signaling & Regulation
Microtubule Interaction with Diffusing Regulatory Molecules
We are currently step-by-step adding regu-latory molecules that modify microtubule dy-namics in budding yeast cells, testing different hypotheses about their interaction network:
[3] H. Bowne-Anderson et al, Bioessays 35, 452–61 (2013).[4] C. A. Hale et al, EMBO J. 20, 1563–1572 (2001).
We use models based on the reaction-diffusion master equation (RDME) to combine stochastic mi-crotubule dynamics with reacting & diffusing regulatory and signaling molecules:
Bik1 (CLIP-170)
Growing Microtubule
Shrinking Microtubule
-
- +
+
Stu2 (XMAP215)
CatastropheFactors
Rescue
Factors
Kip2 Kip3
Kip3
Kar3Stu1 (CLASP)Plus-Tip Tracking Proteins
Stu2 (XMAP215)
Bim1 (EB1)
Plus-End-Dire
cted
Bik1 (CLIP-170)
Kip2
Kip3
Kar3
Minus-End-Directed
Other MT ProteinsStu1 (CLASP)
Motors
In the RDME, reac-tions and diffusion events are defined on the voxels spanned by the dual mesh.
Domain Discretization
𝑗𝑗 ∈ 1, … ,𝐾𝐾 : voxel in dual mesh𝑉𝑉𝑗𝑗: Volume of voxel 𝑗𝑗
System State
Master Equation of the Probability Density Function
Reactions & Diffusion
C++ Solver:Next Subvolume
Method (NSM)
𝑖𝑖 ∈ 1, … ,𝑁𝑁 : chemical species𝑥𝑥𝑖𝑖𝑗𝑗: # molecules of chemical species 𝑖𝑖 in voxel 𝑗𝑗
𝒙𝒙 𝑡𝑡 ∈ ℕ≥0𝑁𝑁×𝐾𝐾: state (of all species in all voxels) at time 𝑡𝑡
𝑟𝑟 ∈ 1, … ,𝑅𝑅 : reaction𝒗𝒗𝑟𝑟: change vector of reaction r𝜼𝜼𝑘𝑘𝑗𝑗: change vector of diffusion from k to j
Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich
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