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