Mathematical models of protein trafficking in
neurons
Paul C Bressloff
Department of Mathematics, University of Utah
December 7, 2008
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Protein transport
neurodegeneration
learning and
memory
protein
trafficking
synaptic plasticity
cortical
development
synaptogenesis
I
I
I
Proteins must be transported long distances along dendrites
and axons
Protein transport crucial in synapse formation and plasticity
Certain neurodegenerative diseases involve a breakdown in
protein transport
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Outline
I
Protein receptor trafficking and synaptic plasticity
I
Cable theory of receptor trafficking
I
Motor transport of mRNA
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Dendritic spines
I
I
Most excitatory synapses are located in small mushroom–like
appendages called dendritic spines which are rich in actin.
Associated neurotransmitter is glutamate (Glu).
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Glutamate (Glu) receptors
I
Two major types of Glu receptor: AMPA and NMDA
I
Binding of Glu receptors opens AMPA ion channels leading to
depolarization
I
Binding of Glu + sufficient depolarization opens NMDA ion
channels leading to influx of calcium
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Long-term potentiation/depression
I
High frequency (100 Hz) stimulus induces LTP
I
Low frequency (1Hz) stimulus induces LTD
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Separation of time-scales
I
I
I
Induction: Calcium signal via NMDA induces action of
kinases and phosphotases
Expression: (De)Phosphorylation of AMPAR receptors
regulates their trafficking and their conductance state
Maintenance: Persistent changes require protein synthesis
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Regulation of AMPAR trafficking
I
A major expression mechanism for LTP/LTD is a
Ca2+ -induced change in the number of synaptic AMPARs
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Various mechanisms of receptor trafficking
4
1
2
3
5
1.
2.
3.
4.
Newly synthesized proteins inserted in cell surface
Lateral surface diffusion along dendrite
Surface entry into spine
Recycling with intracellular pools/binding to scaffolding
proteins
5. Motor transport of mRNA and local synthesis of AMPAR
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Single-particle tracking
(Choquet and Triller, Nat Rev. Neurosci. 2003)
I
Sub-µm latex bead is bound to a receptor using ligands
(antibodies or scaffolding proteins) and imaged using lasers
I
GFP tags reveal regions of high receptor concentration coincide with confinement domains (red)
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Inactivation and recovery of surface receptors
Passafaro et al (Nature 2001)
I
Transfect AMPAR with an immunoflourescent tag (HA/T)
I
Reduce temperature to temporarily stop trafficking
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Treat neuron with thrombin to eliminate surface staining of
HA/T
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Washout thrombin and return to normal temperatures:
surface expression of HA/T recovers in 30 min.
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Outline
I
Protein receptor trafficking and synaptic plasticity
I
Cable theory of receptor trafficking
I
Motor transport of mRNA
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Single–spine model
B
A
Q
END
EXO
DEG
ω+
ω−
β
α
k
σdeg
R
C
σrec
synaptic membrane
σ
intracellular pool
AMPA receptor
scaffolding protein
R, Q: free and bound receptor concentrations in synapse
C: number of intracellular receptors
k, σrec, σdeg, σ: rates of endo/exocytosis, deg/syn
ω±: hopping rates in and out of synapse
α, β: rates of binding/unbinding to scaffolding proteins
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Kinetic equations
dR
= ω+ U − ω− R − kR + σ rec C − α[Z − Q]R + βQ
dt
dQ
a
= α[Z − Q]R − βQ
dt
dC
= −σ rec C − σ deg C + kR + σ
dt
a
I
U is dendritic receptor concentration at spine boundary
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Z is concentration of scaffolding proteins
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a is area of synapse
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Adiabatic approximation
Q(t) =
I
Reduced single-spine model
a
I
αZR(t)
αR(t) + β
dR
= ω+ U − ω− R − kR + σ rec C
dt
dC
= −σ rec C − σ deg C + kR + σ
dt
Strength of a synapse
W = a[gR R + gQ Q]
where gR , gQ are conductances of free and bound AMPARs
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Multi-spine diffusion–trapping model3
receptor
scaffolding
protein
EXO
END
DEG
Js
x=0
spine
x=L
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Consider a dendritic cable of circumference l and length L.
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Somatic flux Js at x = 0. Reflecting BC at x = L
3
Bressloff et. al. PRE 2007, SIAM J. Appl. Math. 2008
Paul C Bressloff
Mathematical models of protein trafficking in neurons
1D Continuum model4
I
Assume uniform concentration around circumference of cable
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Represent fluxes between spines and cable using Dirac delta
functions:
∂2U
∂U
= D 2 − ρ(x)[ω+ U(x, t) − ω− R(x, t)] ,
∂t
∂x
with
ρ(x) =
N
1 X
δ(x − xj )
2πl
j=1
I
4
Continuum approximation: take ρ(x) to be a smooth function
so that we obtain a 1D cable equation.
Earnshaw and Bressloff J. Comp Neurosci 2008, Bressloff 2009
Paul C Bressloff
Mathematical models of protein trafficking in neurons
A.
J(x,t)
U(x,t)
JS
x=0
x=L
B.
σdeg
k
J(x,t)
C(x,t)
R(x,t)
σrec
synaptic membrane
I
σ
intracellular pool
Transverse current
J(x, t) = ω+ U(x, t) − ω− R(x, t)
I
Basal parameter values
k = 10−3 µm2 s −1 , σ rec = 10−3 s −1 , σ rec = 10−5 s −1 , D = 0.1µm2 s −1
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Steady–state cable equation
I
Uniform spine density ρ0 = N/Ll:
ρ0 ω
d 2U
−
[U − R] = 0
2
dx
D
with
ω=
I
ω+ k(1 − Λ)
,
ω− + k(1 − Λ)
R=
ω− Λσ
,
ω+ k(1 − Λ)
Λ=
σ rec
.
σ rec + σ deg
Solution is
U(x) = ZJS
cosh(γ[x − L])
+R
sinh(γL)
where Z is characteristic impedance of cable and ξ ≡ γ −1 is a
diffusive space constant:
r
ρ0 ω
1
, Z=
.
γ=
D
lDγ
Paul C Bressloff
Mathematical models of protein trafficking in neurons
350
ω± = ω =10−3
180
160
140
120
300
250
σ = 10-3 s-1
γ−1[µm]
concentration [µm-2]
200
100
80
60
200
150
100
40
50
σ=0
20
0
D
σdeg
ω
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
distance from soma [mm]
0
0.1
1
10
parameter value [x baseline]
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Diffusion from soma is insufficient to supply distal synapses
with receptors
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Source at soma favors proximal synapses. Synaptic democracy
requires non-uniform spines
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Given solution U(x) to cable equation, the free synaptic
receptor concentration is
R(x) =
I
ω+ U(x) + Λσ
ω− + k(1 − Λ)
The steady–state synaptic weight is then
α(x)Z (x)R(x)
W (x) = a(x) gR R(x) + gQ
α(x)R(x) + β(x)
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Heterosynaptic interactions
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Spine parameters can be classified according to whether or
not they have a nonlocal effect on steady-state receptor
numbers mediated by diffusion
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Synaptic parameters are purely local: rates of
binding/unbinding, number of binding sites, area of synapse
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Parameters of constitutive recycling are nonlocal: rates of
exo/endocytosis, local production and degradation
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If the expression of LTP/LTD involves changes in number of
binding sites, then receptor diffusion unlikely to mediate
heterosynaptic plasticity
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Local increase in σ
receptor number or
concentration [µm-2]
Local increase in σdeg
80
240
70
200
60
160
120
80
50
intracellular
spine
dendrite
40
30
20
40
10
0
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
I
Increasing local production rate σ for spines between 90µm
and 110µm leads to a global increase in receptor numbers.
I
Increasing degradation rate σ deg leads to a global reduction in
receptor numbers
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Local decrease in σrec
Local increase in k
receptor number or
concentration [µm-2]
120
100
120
intracellular
spine
dendrite
80
60
60
40
40
20
20
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
I
100
80
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
Increasing rate of endocytosis k or decreasing rate of recycling
σ rec for spines between 90µm and 110µm leads to a global
decrease in receptor numbers.
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Green’s functions
I
Response to a time–dependent surface current source I(x, t)
switched on at t = 0:
Z tZ L
A(x, t) =
GA (x, y ; t − t 0 )I(y , t 0 )dydt 0
0
0
for A = U, R, C and A(x, 0) = 0.
I
Green’s function GA (x, x0 ; t) is probability density that at time
t a single labeled receptor is at position x and in state A given
that at t = 0 it was injected into dendritic surface at x0
I
Can be generalized to a time–dependent modification of
constitutive recycling induced by synaptic activity.
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Plots of Green’s functions for x0 = L/2:
0.08
18
receptors [µm-1]
x 10-3
t = 200 s
t = 1000 s
t = 2000 s
t = 6000 s
0.04
0.02
10
12
10
8
8
6
6
4
4
2
2
0
0
0
20 40 60 80 100 120 140 160 180 200
-3
GC
12
14
0.06
0
0
distance from soma [µm]
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
Dependence of GU on diffusive coupling ω.
0.08
ω = 10-3
0.06
0.05
0.04
0.03
ω = 10-4
0.09
0.07
receptors [µm-1]
I
14 x 10
GR
16
GU
t = 200 s
t = 1000 s
t = 2000 s
t = 6000 s
0.02
0.01
0.08
0.07
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0
0
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
ω=0
0.01
0.01
0
0.09
0.08
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
Paul C Bressloff
0
20 40 60 80 100 120 140 160 180 200
distance from soma [µm]
Mathematical models of protein trafficking in neurons
Let PA (t) denote probability to be in state A at time t:
L
Z
PA (t) =
GA (x, L/2; t)dx
0
σdeg = 10−5 s−1
1
σdeg = 10−3 s−1
1
1
U
0.7
0.6
0.5
0.8
0.5
0
0.4
0.7
10
20
time [103 sec]
30
R
0.3
C
0.2
0
0.4
0.1
0
3000
4000
time [sec]
5000
Paul C Bressloff
6000
0
10
20
time [103sec]
30
R
0.3
0.2
2000
0.5
0.5
0
1000
U
0.6
0.1
0
1
0.9
probability
0.8
probability
probability
0.9
probability
I
C
0
1000
2000
3000
4000
time [sec]
5000
6000
Mathematical models of protein trafficking in neurons
“Sum–over–trips” on dendritic trees
dendrites
B
soma
T
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Each branch is uniform
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All terminal nodes are closed
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Continuity of receptor concentration at branch nodes
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Conservation of current at branch nodes
Paul C Bressloff
Mathematical models of protein trafficking in neurons
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Use Laplace transforms on jth branch
ej
e
d 2U
ej (x, s) = − Ij (x, s)
− Ξj (s)U
2
dx
D
I
Perform the rescalings
x → X = γj (s)x, Lj → Lj (s) = γj (s)Lj
p
with γj (s) = Ξj (s) so that
ej
e
d 2U
ej (X , s) = − Ij (X , s)
−
U
dX 2
D
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
At each branch node
ei (0, s) = U
ej (0, s)
U
for all pairs (i, j) radiating from the node and
ej X
∂U
= 0, zj (s) = lj Dγj (s),
zj (s)
∂X j
X =0
where the sum is over all branches j connected to the node.
I
At all terminal modes
ej ∂U
∂X Paul C Bressloff
= 0.
X =Lj
Mathematical models of protein trafficking in neurons
I
General solution is of the form
X Z Lj (s)
e
Ui (X , s) =
Gij (X , Y ; s)Iej (Y , s)dY
j∈T
I
0
The Green’s function Gij (X , Y ; s) satisfies the homogeneous
equation
d 2 Gij (X , Y ; s)
1
− Gij (X , Y ; s) = − δi,j δ(X − Y ),
2
dX
D
ei (X , s) for fixed j, Y .
with the same boundary conditions as U
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Using the “sum–over–trips” method5 it can be shown that
X
Gij (X , Y ; s) =
Atrip (s)G∞ (Ltrip (i, j, X , Y , s)),
trips
where G∞ (X ) is Green’s function for an infinite cable:
G∞ (X ) =
e−|X |
,
2D
and Ltrip (i, j, X , Y , s) is the length (in rescaled coordinates)
of a path along the tree starting at the point X on branch i
and ending at the point Y on branch j.
I
Sum restricted to a special class of paths or trips
5
Abbott et. al. 1991; Bressloff et. al. 1996; Coombes et. al. 2007,
Bressloff 2009
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Definition:
A trip from (X , i) to (Y , j) may start out in either direction along
branch i but it can subsequently change direction only at a branch
or terminal node. A trip is always reflected back at a terminal
node, whereas at a branch node it may be transmitted to another
branch or reflected back. A trip may pass through the points (X , i)
and (Y , j) an arbitrary number of times as long as it starts at
(X , i) and ends at (Y , j).
n
A.
2pn
m
2pm−1
B.
±1
Paul C Bressloff
Mathematical models of protein trafficking in neurons
For each trip, the associated amplitude Atrip is calculated
according to the following rules:
1. Initially take Atrip (s) = 1.
2. For every branch node at which the trip passes from an initial
segment m to a different segment n, n 6= m, multiply Atrip (s)
by a factor 2pn (s)
3. For every branch node at which the trip is reflected back along
the same segment m, multiply Atrip (s) by a factor 2pm (s) − 1
4. For every closed (open) terminal node, multiply Atrip (s) by a
factor +1 (−1)
The factor pm is given by
zm (s)
,
pm (s) = P
n zn (s)
where sum is over all branches n radiating from branch node.
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Future directions
1. How is receptor trafficking regulated following the induction
of synaptic plasticity?
2. How are stable synaptic memories maintained given rapid
protein turnover? Roles of protein synthesis and actin
cytoskeleton?
3. What is the effect of intrinsic noise (small receptor numbers)
on receptor trafficking and plasticity?
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Outline
I
Protein receptor trafficking and synaptic plasticity
I
Cable theory of receptor trafficking
I
Motor transport of mRNA
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Protein synthesis required for persistent LTP6
I
I
I
6
(A) Single train of tetanic stimulation produces decremental
potentiation
Early-phase LTP insensitive to translational inhibition
(anisomycin) and transcriptional inhibition (actinomyosin-D)
(B) Repeated stimulation induces persistent potentiation that
is sensitive to both translational and transcriptional inhibition
Kelleher et. al. Neuron 2004
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Motor transport of mRNA7
I
I
7
Newly transcribed mRNA granules are transported into the
dendrite by kinesin motors on microtubules.
Following synaptic activation, mRNA is localized to spines by
the actin-based myosin
Bramham and Wells Nat. Rev. Neurosci. 2007
Paul C Bressloff
Mathematical models of protein trafficking in neurons
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Motile mRNA particles in cultured hippocampal cells exist in 3
states: oscillatory, anterograde and retrograde
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KCL-induced depolarization increases anterograde motion
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Intermittent search8
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Many problems in nature require search for a randomly hidden
target: foraging animals, a protein searching for a specific
target site on DNA, microtubular transport of mRNA
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An efficient stochastic search strategy is to alternate between
(A) a slow motion (diffusive), intensive search phase and (B)
a fast (motor-assisted) ballistic non-search phase
8
Loverdo et. al. Nat. Phys. 2008
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Stochastic model of directed intermittent search9
Target
βv0
α
α β+
v+
x
X
L
I
Particle injected into dendrite at x = 0 and t = 0. Fixed
hidden target (dendritic spine) at x = X .
I
Particle exists in 3 states: anterograde with velocity v+ ,
retrograde with velocity −v− , and stationary.
I
In stationary state particle can be absorbed at a rate k if
within a distance a of target
I
Absorbing boundary at x = L
9
Bressloff and Newby 2009
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
I
Let pm (x, t) denote probability density for anterograde
(m = +), retrograde (m = −) and stationary (m = 0) states
Master equation of the form
∂p+
∂p+
= −v+
+ αp0 − β+ p+
∂t
∂x
∂p−
∂p−
= v−
+ αp0 − β− p−
∂t
∂x
∂p0
= −2αp0 + (β+ p+ + β− p− ) − kχ(x − X )p0
∂t
I
where χ(x) = 1 if |x| < a and is zero otherwise.
Initial and boundary conditions:
v− p− (0, t) = v+ p+ (0, t),
I
p− (L, t) = 0,
pm (x, 0) = δ(x)δm,+
Bias in anterograde direction
β+ /v+ < β− /v−
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Optimization problem
I
Probability of finding target after time t is
Z
∞ Z X +a
p0 (x, τ )dxdτ
γ(t) = k
t
X −a
I
Define hitting probability Π and conditional MFPT T
according to
Z ∞
γ(t)
dt
Π = γ(0), T =
γ(0)
0
I
Determine hitting probability Π and MFPT T by solving
backwards equation or using Laplace transforms
I
Optimization problem: find parameter values that minimize
search time and maximize hitting probability?
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Plots of Π and T for β− = 1s −1 (solid black), β− = 2.5s −1
(dashed) and unidirectional (gray)
Hitting Probability
MFPT
5000
1
4000
0.9
3000
Π 0.8
T
2000
0.7
0.
0
1
0.9
0.8
0.7
0.6
Π 0.5
0.4
0.3
0.2
0.1
0
0
1000
β+ = 1s-1
0.2
0.4 0.6
α (sec-1)
0.8
1
0 0
0.2
0.4 0.6
α (sec-1)
0.8
1
0.4 0.6
β+(sec-1)
0.8
1
1200
1000
800
T 600
400
α = 0.5 s-1
0.2
0.4 0.6
β+(sec-1)
0.8
200
1
Paul C Bressloff
0 0
0.2
Mathematical models of protein trafficking in neurons
I
Determine minimal MFPT for a given hitting probability
I
For finite β− (partially biased) there exists a minimum of T as
a function of α
I
Unidirectional transport gives smaller MFPT than
bidirectional transport
1400
1000
1000
Π = 0.8
800
increasing β−
T(sec)
T (sec)
1200
800
600
400
increasing β−
200
400
200
600
0
0.2
0.4
0.6
α(sec-1)
0.8
Paul C Bressloff
1
0
0
0.2
0.4
Π
0.6
0.8
1
Mathematical models of protein trafficking in neurons
Multiple targets
First Target
Second Target
β α
v
0
I
I
X1
X2
x
Unidirectional search and n + 1 identical targets
Hitting probability of finding most downstream target is
b = (1 − Π)n Π
Π
I
b is maximized when
Π
I
ln 1 + n1 v
(α + k).
βmax (α) =
2ka
The maximum hitting probability is
b max =
Π
Paul C Bressloff
nn
.
(n + 1)n+1
Mathematical models of protein trafficking in neurons
ˆ
Π
0.25
(a)
Π̂max
0.25
(b)
0.2
0.2
n =1
n =3
0.1
0.1
n =5
n =7
0
0
2 α
1
0
1
2
3
4
5
6
7
8
9 n
βmax
α
k
βmax
< 1
α
k < k*
(1, α* )
1
βmax
> 1
α
k > k*
k*
(c)
0
0
k*
α
Paul C Bressloff
(d)
0
0
α
Mathematical models of protein trafficking in neurons
Quasi–steady–state approximation
I
Fix units of space and time: x = 1mm and t = 104 s
I
Then v± = O(1) and transition rates α, β± 1
I
Introduce small parameter
ε=
1
1
1
+
+
β− β+ α
and set a = εα, b± = εβ±
I
Have linear reaction-hyperbolic system
1
(∂t + v+ ∂x )p+ = (−b+ p+ + ap0 )
1
(∂t − v− ∂x )p− = (−b− p− + ap0 )
1
(∂t + k(x))p0 = (b+ p+ + b− p− − 2ap0 )
Paul C Bressloff
Mathematical models of protein trafficking in neurons
I
Using projection methods one can show that
p0 (x, t) ∼
q(x, t)
,
a
p± (x, t) ∼
q(x, t)
b±
where q satisfies advection–diffusion equation
∂q
∂2q
∂q
k(x)
=−
q − veff
+ Deff 2
∂t
a
∂x
∂x
where
veff = γ− ≡
Deff = 1
1
−
b+ b−
1 − γ− 1 + γ−
+
2
2
b+
b−
Paul C Bressloff
.
Mathematical models of protein trafficking in neurons
I
If k ≡ 0 then one can write the solution in the form
x − vt
√ ,t
pm (x, t) = ηm Qε
ε
I
Can prove that10
Qε (s, t) → q(s, t) as ε → ∞
where q(s, t) is solution of heat equation.
10
cf. Reed et. al. 1990, Friedman and Hu 2007
Paul C Bressloff
Mathematical models of protein trafficking in neurons
Acknowledgements
I
AMPAR trafficking: Berton Earnshaw (Utah), Michael Ward
(UBC)
I
mRNA transport: Jay Newby (Utah)
I
Movies: Mike Ehlers (Duke University)
I
National Science Foundation
Paul C Bressloff
Mathematical models of protein trafficking in neurons
References
1. B. A. Earnshaw and P. C. Bressloff. J. Neurosci. 26 12362
(2006).
2. P. C. Bressloff and B. A. Earnshaw. Phys. Rev. E. 75 041916
(2007)
3. P. C. Bressloff, B. A. Earnshaw and M. J. Ward. SIAM J.
Appl. Math. 68 1223-1246 (2008)
4. B. A. Earnshaw and P. C. Bressloff. J. Comput. Neurosci. 25
366-389 (2008)
5. P. C. Bressloff and B. A. Earnshaw. Biophys. J. In press.
6. P. C. Bressloff and J. Newby. Phys. Rev. Lett. Submitted.
Paul C Bressloff
Mathematical models of protein trafficking in neurons