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Virtual Cell
How to model reaction diffusion
systems
When a systems model is not enough
• Systems models ignore convection and diffusion as well as
non-uniform concentration in a cell
• Systems models also ignore stochastic events in a reaction
pathway
Live cell intracellular calcium using fluorescent Colocalization of intracellular ankyrin-B and β2biosensors
spectrin in neonatal cardiomyocytes
Different Chemical Kinetic Theories
• Fokker-Planck equation (Kolmogorov forward)
– Describes the time evolution of the probability
density function of the velocity of a particle, and
can be generalized to other observables as well
• Kolmogorov backward equation
– PDE’s that arise from the assumption of
continuous-time and continuous-state Markov
processes
Reaction Diffusion Models
[C]
source
sink
 .( DC [C])   J i
  J j   Rm
t
i
j
m
J isource  f i (C,t, x, y, z)
J sink
 f j (C,t, x, y, z)
j
[m]
Rm 
 km ([B]  [m])[C]  km [m]
t


m
B  C

where:
[C] intracellular species
concentration
DC diffusion constant for C
Jisource – source i for C
Jjsink –sink j for C
km+, km- - kinetic rates
Finite Difference Method for 1-D Heat Equation
u
 2u
D 2 0
t
x
Applying the finite difference approximations
to the 1-D heat equation:
uik 1  uik
uik1  2uik  uik1
2
4
D

O
t
,x
t
x 2
t
k- increment in time
k 1
k
 ui  ui  D 2 uik1  2uik  uik1
i- increment in space
x
This explicit (backward Euler ) scheme is stable if:
t
1
D 2 
2
x




Forward Euler Scheme
¶u
¶2u
+D
=0
2
¶t
¶x
Instead of:
Dt
k +1
k
k
k
k
ui = ui + D
u
2
u
+
u
i
i -1
Dx 2 i +1
Accuracy and stability can be
(
)
improved with an implicit scheme:
Dt
k +1
k
k +1
k +1
k +1
é
ù
ui - ui = D
u
2
u
+
u
i
i -1 û
2 ë i +1
Dx
* Virtual Cell also can convert determinist models to stochastic models using the Gibson
approximation
Defining the Biomodel
• Each reaction is defined as a function of its
flux
Defining a Geometry: 2-D
2-D Simple Shapes
2-D Image of a cell from a Microscope
Defining several different domains
Defining a Geometry
Define a geometry using mathematical
equations
Import a geometry using microscopy
images (2-D) or using a z-stack for 3-D
images
Importing a geometry into Vcell
Segmentation
Re-Segment
Import-Vcell
Smoothing
Advantages of Vcell
Can define patches of AC and PDE on a membrane
Simplistic Geometry
(still several hours to solve with
complicated reations)
Create a diffusional map in segments
Can accept Immuno-gold data in TEM’s
Defining Compartments
Defining the geometry of the compartments
with differing sizes
Creating diffusional
membranes
Creating membrane bound
reactions
Electrical Mapping
Defining the geometry of the compartments
with differing sizes
• Each membrane can have defined voltages
Creating diffusional
and capacitances
membranes
• A membrane voltage can be defined to trigger
an event (such as calcium release)
• The current or voltage of the membrane can
be clamped to better mimic experimental
conditions
Most experiments when Creating
Patch Clamping
a cellbound
either
membrane
voltage clamp or current clamp
the cell
reactions
Defining the Spatial and Temporal
Mesh
• Mesh units in the x, y, and z directions as well
as in time
• An error tolerance can also be defined which
defines the maximum allowed step change
Viewing the Results
• Can track concentration
changes in space or time
• Can graph reaction rates as
well
Virtual
Cell
•
Experiment and simulation of calcium dynamics following Bradykinin (BK) stimulation of a neuroblastoma cell. A
250 nM solution of BK was applied at time 0, and the [Ca2+]cyt is monitored with fura-2 to produce the experimental
record (left) obtained at 15 frames/sec. Representative frames are shown, and the change in calcium in the
neurite (green box) and soma (yellow box) are plotted in the inset. The Virtual Cell simulation shown in the next
column provides a good match to the experiment. The third and fourth columns display the simulation results for
[InsP3]cyt and Po, the open probability of the InsP3-sensitive calcium channel in the ER membrane (Slepchenko
BM et al. Annu Rev Biophys Biomol Struct 2002;31:423-41)
FRAP
Fluorescence Recovery After
Photobleaching
FRAP Video:
http://vcell.org/education/frap_tutorial/FRAPmovie/FRAP_0505_h264.mp4
Simple FRAP:
http://vcell.org/webstart/Rel/user_docs/Tutorial01_SimpleFRAP.pdf
FRAP with Binding:
http://vcell.org/webstart/Rel/user_docs/Tutorial02_FRAPBinding.pdf
HW
• Take a screen shot of your results for the
simple frap experiment showing diffusion into
the blocked region
• Take a screen show of your reaction diagram
for the FRAP with binding and of your final
results again showing diffusion into the
blocked region