A Stochastic Reaction-Diffusion Active Transport Method for Studying
the Control of Gene Expression in Eukaryotic Cells.
Samuel A.
1
Isaacson ,
Charles S.
2
Peskin
1 - University of Utah, Department of Mathematics, [email protected], 2 - Courant Institute of Mathematical Sciences, [email protected]
Background
Boundary Conditions
•Want to understand how the complicated
spatial geometry of eukaryotic cells
influences gene regulatory networks.
•Also interested in stochastic effects that
arise in such networks due to small
concentrations of regulatory proteins,
mRNAs, and DNA binding sites.
•We have developed a stochastic reactiondiffusion active transport model for use in
studying spatially distributed chemical
systems where molecular noise from the
chemical reaction process is important.
•We model the nuclear membrane and
cell membrane as boundaries.
•Assume no chemicals can leave cell, so
have no-flux BC at cell membrane.
•Model nuclear pores as an effective
nuclear membrane permeability.
•Permeability is zero for most proteins/
mRNAs, as they generally require a
chemical tranport process to pass
through the membrane.
Master Equation Model
•Divide comp. domain into mesh cells
indexed by i = 1 . . . N
•Model diffusion and active transport as
jumping of particles between mesh cells,
with exponentially distributed rates.
The master equation model is
N !
N !
L
!
"
dP (m)
=
dt
i=1 j=1
+
l
kij
"
#
l
l
l
l
l
mj + 1 P (m + ej − ei ) − kji mi P (m)
κlij
"
#
l=1
N !
N !
L
!
"
#
l
l
l
l
l
mj + 1 P (m + ej − ei ) − κji mi P (m)
#
i=1 j=1 l=1
+
N !
K
!
"
aki (mi
− ν k )P (m − ei ν k ) −
aki (mi )P (m)
#
.
i=1 k=1
l
mi
Here
gives the number of the l ’th
chemical species in mesh cell i .
Denote by α ∈ {nuc, cyt} the domain of
a given mesh variable. The diffusive jump
rate from the domain α component of
cell j to the domain α component of cell
i is given by our discretization to be
α
DAij
α
kij =
.
hVjα
The trans-nuclear membrane jump rate
Bn
is
Ai ρ
αα!
ki =
.
!
α
Vi
The active transport j u m p r a t e i s
derived to be
κα
i±ek ,i
 α
%
%
A
1

i±
e

%
α
2 k %
%(v k )i± 12 ek % ,
=
Vi

 0,
if ±
α
(vk )i± 1 ek
2
mRNA Export
mRN Ai + N R–Rti → mRN A–N R–Rti
at each nuclear mesh cell
mRN A–N R–Rti ! mRN A–N R–Rt–Rbi
mRN A–N R–Rt–Rbi → mRN Ai + N Ri + RanGDPi
at each cytosolic mesh cell
“
Protein Import and Gene Regulation
Pi → ∅
Pi + N Ri → P –N Ri
P –N Ri + Rti → Pi + N R–Rti
Pi + DN A ! DN ARep
at all mesh cells
at each cytosolic mesh cell
at each nuclear mesh cell
at cell center
3D Model Results
≥ 0,
else.
Gene Expression Model
Transcription
Nuclear membrane reconstruction showing nuclear pore locations.
Spatial Jump Rates
Diffusive, trans-nuclear membrane, and
active transport jump rates can be
calculated from the discretization
weights of a Cartesian grid embedded
boundary discretization.
DN A + RN AP → DN A0
at cell center
DN A0 + n1 → DN A1
at cell center
DN Al−1 + ni → DN Al , l = 1, . . . , M − 1
at cell center
DN AM −1 + nM → mRN Ai + RN AP + DN A
at cell center
mRNA Translation and Decay
mRN Ai + Ribosomei → mRN A0i
mRN A0i + a1 → mRN A1i
“
l
mRN Al−1
+
a
→
mRN
A
i
i , l = 1, . . . , N − 1
i
−1
mRN AN
+ aN → Pi + mRN Ai
i
mRN Ai → ∅
Protein
with NLS
•The first term corresponds diffusion.
•The second term to active transport.
•The third to chemical reactions.
k
The chemical reaction propensities ai are
specified, however, the diffusive and active
transport jump rates, kil j and κlij are not.
at each cytosolic mesh cell
“
“
“
Nuclear Import Receptor
Nuclear Export Receptor
Ran!GDP
Ran!GDP
Cytosol
References
Nucleus
Ran!GTP
Protein
with NES
Ran!GTP
Cross section of spherical cell and nuclear membranes embedded in
Cartesian grid.
RanGTP import/export cycle.
•S. A. Isaacson and C. S. Peskin (2005) Incorporating diffusion in
complex geometries into stochastic chemical kinetics
simulations, SIAM J. Sci. Comp., accepted.
3D simulation movies available at:
http://www.math.nyu.edu/~isaacsas