biophysical models of ampa receptor trafficking in dendrites
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Biophysical models of AMPA receptortrafficking in dendrites
Berton A. Earnshaw
Advisor: Dr. Paul C. Bressloff
Department of Mathematics, University of Utah
Salt Lake City, Utah 84112
Biophysical models of AMPA receptor trafficking in dendrites – p. 1/52
The amazing brain
Biophysical models of AMPA receptor trafficking in dendrites – p. 2/52
Neurons communicate via synapses
Biophysical models of AMPA receptor trafficking in dendrites – p. 3/52
Synaptic transmission
Kandel et al., Principles of Neural Science (2000)
Biophysical models of AMPA receptor trafficking in dendrites – p. 4/52
Synaptic plasticity
Collingridge et al., Nat. Rev. Neurosci. (2004)
Biophysical models of AMPA receptor trafficking in dendrites – p. 5/52
Outline
1. AMPA receptor trafficking
2. Single-spine model
3. 2D multi-spine model
4. 1D continuum multi-spine model
Biophysical models of AMPA receptor trafficking in dendrites – p. 6/52
AMPA receptors
Huganir & Song, Nat. Rev. Neurosci. (2002)
Fast synaptic transmission
Complexes with other proteins −→ trafficking
Biophysical models of AMPA receptor trafficking in dendrites – p. 7/52
Long-range receptor trafficking
Groc & Choquet, Cell Tissue Res. (2006)
Vesicle transport along microtubules
Diffusion from soma to synapse?
Biophysical models of AMPA receptor trafficking in dendrites – p. 8/52
Dendritic spines
Matus, Science (2000)
Excitatory synapses located on surface ofmushroom-like protrusions of the dendritic membranecalled spines
Biophysical models of AMPA receptor trafficking in dendrites – p. 9/52
Receptor trafficking at spines
Sheng & Kim, Science (2002)
Constitutive recycling
Crosslink to scaffolding in PSD
Lateral diffusion in membrane
Biophysical models of AMPA receptor trafficking in dendrites – p. 10/52
Expression of LTP/LTD
Scannevin & Huganir, Nat. Rev. Neurosci. (2000)
Biophysical models of AMPA receptor trafficking in dendrites – p. 11/52
Separation of time-scales
INDUCTION EXPRESSION MAINTENANCE
High [Ca2+] (LTP)
Low [Ca2+] (LTD)
Synaptic vesicles
AMPAR conductance Number of AMPARs
Protein synthesis
Structural changesin spine morphology
seconds minutes/hours hours/days...TIME
Ca2+ signal activates kinase/phosphotase pathways
Phosphorylation/dephosphorylation of AMPA receptorcomplexes
Regulation of AMPA receptor trafficking
Biophysical models of AMPA receptor trafficking in dendrites – p. 12/52
Outline
1. AMPA receptor trafficking
2. Single-spine model
3. 2D mutli-spine model
4. 1D continuum multi-spine model
Biophysical models of AMPA receptor trafficking in dendrites – p. 13/52
Single-spine model
DEG
END
EXO
EXO
PSD
AMPA receptor
scaffolding protein
Earnshaw & Bressloff, J. Neurosci. (2006)
Biophysical models of AMPA receptor trafficking in dendrites – p. 14/52
Single-spine model equations (GluR2/3)
dR
dt=
ω
A(U − R) −
k
AR −
h
A(R − P )
dP
dt=
h
a(R − P ) − α(Z − Q)P + βQ +
σrec(1 − f)S
adQ
dt= α(Z − Q)P − βQ
dS
dt= −σrec(1 − f)S − σdegfS + kR + δ
P = free AMPAR conc. in PSDQ = bound AMPAR conc. in PSDR = free AMPAR conc. in spine headU = free AMPAR conc. in dendriteS = # intracellular AMPARZ = scaffolding protein conc.f = fraction of S sorted for degradation
Biophysical models of AMPA receptor trafficking in dendrites – p. 15/52
Single-spine results: Block exo/endocytosis
Luscher et al., Neuron (1999)
Biophysical models of AMPA receptor trafficking in dendrites – p. 16/52
Single-spine results: LTPA
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t [min]
nu
mb
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of
rec
ep
tors
in
PS
D
O’Connor et al., PNAS (2005)
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D
TotalBound GluR1/2Free GluR1/2Bound GluR2/3Free GluR2/3Scaffolding
C
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D
TotalBound GluR1/2Free GluR1/2Bound GluR2/3Free GluR2/3Scaffolding
Biophysical models of AMPA receptor trafficking in dendrites – p. 17/52
Single-spine results: LTD
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TotalBound GluR1/2Free GluR1/2Bound GluR2/3/GRIPFree GluR2/3/GRIPBound GluR2/3/PICKFree GluR2/3/PICKScaffolding
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TotalBound GluR1/2Free GluR1/2Bound GluR2/3/GRIPFree GluR2/3/GRIPBound GluR2/3/PICKFree GluR2/3/PICKScaffolding
Dudek & Bear, J. Neurosci. (1993)
Biophysical models of AMPA receptor trafficking in dendrites – p. 18/52
Conclusions
1. Significant fraction of PSD receptors are mobile (Groc et al.,
2004; Ashby et al., 2006)
(a) Requires PSD-ESM barrier (Choquet & Triller, 2003)
(b) Required for exocytosis blockade time-course (Luscher et
al., 1999) and LTD saturation (Dudek & Bear, 1993)
Biophysical models of AMPA receptor trafficking in dendrites – p. 19/52
Conclusions
1. Significant fraction of PSD receptors are mobile (Groc et al.,
2004; Ashby et al., 2006)
(a) Requires PSD-ESM barrier (Choquet & Triller, 2003)
(b) Required for exocytosis blockade time-course (Luscher et
al., 1999) and LTD saturation (Dudek & Bear, 1993)
2. Diffusive impedance of spine neck is significant (Ashby et
al., 2006)
(a) Required for endocytosis blockade time-course (Luscher
et al., 1999) and LTP time-course (O’Connor et al., 2005)
Biophysical models of AMPA receptor trafficking in dendrites – p. 19/52
Conclusions
3. Insertion of intracellular GluR1/2 during LTP mustcombine synaptic targeting
(a) Requires increased hopping and binding rate (Schnell et
al., 2002) and scaffolding (Shi et al., 2001)
(b) Required for LTP time-course (O’Connor et al., 2005)
Biophysical models of AMPA receptor trafficking in dendrites – p. 20/52
Conclusions
3. Insertion of intracellular GluR1/2 during LTP mustcombine synaptic targeting
(a) Requires increased hopping and binding rate (Schnell et
al., 2002) and scaffolding (Shi et al., 2001)
(b) Required for LTP time-course (O’Connor et al., 2005)
4. Slow exchange of GluR1/2 with GluR2/3 after LTPrequires maintenance of additional binding sites
(a) Required for exchange time-course (McCormack et al., 2006)
Biophysical models of AMPA receptor trafficking in dendrites – p. 20/52
Conclusions
3. Insertion of intracellular GluR1/2 during LTP mustcombine synaptic targeting
(a) Requires increased hopping and binding rate (Schnell et
al., 2002) and scaffolding (Shi et al., 2001)
(b) Required for LTP time-course (O’Connor et al., 2005)
4. Slow exchange of GluR1/2 with GluR2/3 after LTPrequires maintenance of additional binding sites
(a) Required for exchange time-course (McCormack et al., 2006)
5. GRIP to PICK1 exchange during LTD must beaccompanied by loss of binding sites (Colledge et al., 2003)
(a) Required for LTD time-course (Dudek & Bear, 1992) and LTDsaturation (Dudek & Bear, 1993)
Biophysical models of AMPA receptor trafficking in dendrites – p. 20/52
Outline
1. AMPA receptor trafficking
2. Single-spine model
3. 2D mutli-spine model
4. 1D continuum multi-spine model
Biophysical models of AMPA receptor trafficking in dendrites – p. 21/52
2D multi-spine model
receptor
x = 0
spine
x = L
σ0 DEG
ENDEXO
spineneck
dendrite
l
Treat dendrite as cylinder of length L, radius l
Intersection of jth spine with dendrite as disc of radius ερ
centered at rj, j = 1, . . . ,M .
Separation of length-scales: ερ ≪ l ≪ L.
Biophysical models of AMPA receptor trafficking in dendrites – p. 22/52
2D multi-spine diffusion equation
∂tU = D∇2U, (r, t) ∈ Ωε × [0,∞)
Ωε = Ω0 \⋃M
j=1 Ωj
y = πl
x = 0 x = Ly = -πl
Ωε
Ωj
Boundary conditions:
U(x, πl, t) = U(x,−πl, t), ∂yU(x, πl, t) = ∂yU(x,−πl, t)
−D∂xU(0, y, t) =σ0
2πl, −D∂xU(L, y, t) = 0
σ0 = # AMPARs per unit time entering surface from soma
Biophysical models of AMPA receptor trafficking in dendrites – p. 23/52
2D multi-spine bcs on∂Ωj
Generalized Neumann bcs at ∂Ωj:
−D∂nU(r, t) =ωj
2περ(U(r, t) − Rj), r ∈ ∂Ωj
∂n = outward normal derivative to Ωε
ωj = spine neck hopping rate at jth spine|∂Ωj | = 2περ
Rj = free AMPAR concentration on jth spine head
Biophysical models of AMPA receptor trafficking in dendrites – p. 24/52
2D multi-spine model: Steady-state solution
Want to solve∇2U = 0, r ∈ Ωε
with all boundary conditions (called BVP1)
Assume U(r) = Uj on ∂Ωj
−D∂nU(r) =ωj
2περ(Uj − Rj), r ∈ ∂Ωj
whereωj =
ωjkj(1 − λj)
ωj + kj(1 − λj), Rj =
σrecj
k
δj
σdegj
Integrating equation over Ωε yields solvability condition:
σ0 =
M∑
j=1
ωj
[Uj − Rj
]
Biophysical models of AMPA receptor trafficking in dendrites – p. 25/52
2D multi-spine model: Steady-state solution
Solve BVP1 in two steps:1. Solve implicitly assuming Uj ’s are known (BVP2)
2. Substitute BVP1 into M bcs on ∂Ωj −→ M equationsin M + 1 unknowns Uj and χ
3. With solvability condition −→ M + 1 equations in theM + 1 unknowns Uj and χ
Solution of BVP1 requires matching solutions in M innerregions
|r − rj| = O(ε)
and outer region
|r − rj | ≫ O(ε), j = 1, . . . ,M
Biophysical models of AMPA receptor trafficking in dendrites – p. 26/52
Inner solution of BVP2
Set s = ε−1(r − rj), V (s; ε) = U(rj + εs; ε), then
∇2sV = 0, |s| > ρ
V = Uj , |s| = ρ
which has solution
V = Uj + νAj(ν) log(|s|/ρ), ν = −1
log(ερ)
Far-field behavior of inner solution is
V ∼ Uj + Aj(ν) + νAj(ν) log(|r − rj |)
Biophysical models of AMPA receptor trafficking in dendrites – p. 27/52
Outer solution of BVP2
Decompose outer solution U = U + u, where
u(r) =κ
2L(x − L)2, κ =
σ0
2πlD
Then U satisfies inhomogeneous diffusion equation
∇2U = −κ
L, r ∈ Ω0
homogeneous bcs on Ω0 (no holes)asymptotic conditions as r → rj
U ∼ −u(rj) + Uj + Aj(ν) + νAj(ν) log |r − rj |.
Biophysical models of AMPA receptor trafficking in dendrites – p. 28/52
Green’s function for BVP2
Modified Green’s function G(r, r′)
∇2G =1
|Ω0|− δ(r− r
′),
∫
Ω0
G(r; r′)dr = 0
G(x, πl; r′) = G(x,−πl; r′), ∂yG(x, πl; r′) = ∂yG(x,−πl; r′)
∂xG(0, y; r′) = 0, ∂xG(L, y; r′) = 0
G has logatrithmic singularity as r′ → r
G(r; r′) = −1
2πlog |r − r
′| + G(r; r′)
where G is regular part of G.
Biophysical models of AMPA receptor trafficking in dendrites – p. 29/52
Outer solution for BVP2 again
Replace diffusion equation and asymptotics with
∇2U = −κ
L+
M∑
j=1
2πνAj(ν)δ(r− rj)
Integrating yields
U(r) = −M∑
j=1
2πνAj(ν)G(r; rj) + χ
where χ is determined by solvability condition
Biophysical models of AMPA receptor trafficking in dendrites – p. 30/52
Outer solution for BVP1
Outer solution is
U(r) = u(r) −M∑
j=1
ωj
D[Uj − Rj ]G(r; rj) + χ
where
χ =σ0
2πνD −∑M
i,j=1 Mji(ui − Ri)∑M
i,j=1 Mji
M = (I + 2πνB)−1, Bjj =D
ωj+ Gjj , Bji = Gji, j 6= i
Biophysical models of AMPA receptor trafficking in dendrites – p. 31/52
2D multi-spine model: Numerical results2D outer solution 2D numerical solution
2D outer solution
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URS
Distance from soma [μm]
Re
ce
pto
rs
j
j
j
B
1D solution
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15
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URS
Distance from soma [μm]
Re
ce
pto
rs
j
j
j
D
Biophysical models of AMPA receptor trafficking in dendrites – p. 32/52
1D multi-spine modelRecall 2D steady-state equation for U with asymptotics
0 = ∇2U −M∑
j=1
ωj
D(Uj − Rj)δ(r− rj)
“Average over y-coordinate” to get 1D model
0 =d2U
dx2−
M∑
j=1
ωj
2πlD(Uj − Rj)δ(x − xj)
Not equal since in 2D
ωj
D(Uj − Rj) = 2πνAj(ν), ν = −
1
log(ερ)
Bressloff & Earnshaw, Phys. Rev. E (2007)
Biophysical models of AMPA receptor trafficking in dendrites – p. 33/52
Effect of ε
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1
1.2
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1.8
2
2.2
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2.4
Biophysical models of AMPA receptor trafficking in dendrites – p. 34/52
Outline
1. AMPA receptor trafficking
2. Single-spine model
3. 2D mutli-spine model
4. 1D continuum multi-spine model
Biophysical models of AMPA receptor trafficking in dendrites – p. 35/52
1D continuum multi-spine modelTreat spine distribution as density ρ:
∂U
∂t= D
∂2U
∂x2− ρω(U − R), (x, t) ∈ (0, L) × [0,∞)
−D∂U
∂x(0, t) =
σ0
2πl, −D
∂U
∂x(L, t) = 0
Biophysical models of AMPA receptor trafficking in dendrites – p. 36/52
1D cont’m multi-spine model: Steady-state
Assume uniform spine density ρ(x) = ρ0
Assume uniform effective hopping rate ω(x) = ω0 (henceω, k, σrec, σdeg uniform)
Allow production rate δ to vary
Have “cable equation” for AMPAR trafficking
Dd2U
dx2− Λ2
0(x)U = −Λ20(x)R(x)
with length constant
Λ−10 =
√D
ρ0ω0
Biophysical models of AMPA receptor trafficking in dendrites – p. 37/52
Delivery of synaptic receptorsFast recycling, production
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Biophysical models of AMPA receptor trafficking in dendrites – p. 38/52
Heterosynaptic effect of constit. recyclingreduced σrec in gray
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Intracellular
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increased k in gray
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increased σdeg in gray
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Biophysical models of AMPA receptor trafficking in dendrites – p. 39/52
Heterosynaptic effect of LTP
GluR1/2
scaffolding protein
ΙΙ
Ι
potentiated
GluR2/3
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LTP at t = 6 hrs
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ScaffoldingTotalBoundFree
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Biophysical models of AMPA receptor trafficking in dendrites – p. 40/52
Heterosynaptic effect of LTD
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LTD at t = 6 hrs
distance from soma
α β β∗
GRIP PICKµ
ν
scaffoldingprotein
AMPAR AMPAR
h*h
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ScaffoldingTotalBound AMPAR+GRIPBound AMPAR+PICKFree AMPAR+GRIPFree AMPAR+PICK
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TotalAMPAR+PICKTotal IntracellularIntra. AMPAR+GRIPIntra. AMPAR+PICKTotal in ESMAMPAR+GRIP in ESMAMPAR+PICK in ESM
Biophysical models of AMPA receptor trafficking in dendrites – p. 41/52
Conclusions
1. Surface trafficking of AMPA receptors is effectivelyone-dimensional with spines treated as pointsources/sinks (or density if there are enough spines)
Biophysical models of AMPA receptor trafficking in dendrites – p. 42/52
Conclusions
1. Surface trafficking of AMPA receptors is effectivelyone-dimensional with spines treated as pointsources/sinks (or density if there are enough spines)
2. Somatic synthesis + lateral diffusion are not sufficient tosupply distal synapses with receptors
Biophysical models of AMPA receptor trafficking in dendrites – p. 42/52
Conclusions
1. Surface trafficking of AMPA receptors is effectivelyone-dimensional with spines treated as pointsources/sinks (or density if there are enough spines)
2. Somatic synthesis + lateral diffusion are not sufficient tosupply distal synapses with receptors
3. Local changes to constitutive recycling producenon-local changes in synaptic receptor numbers
Biophysical models of AMPA receptor trafficking in dendrites – p. 42/52
Conclusions
1. Surface trafficking of AMPA receptors is effectivelyone-dimensional with spines treated as pointsources/sinks (or density if there are enough spines)
2. Somatic synthesis + lateral diffusion are not sufficient tosupply distal synapses with receptors
3. Local changes to constitutive recycling producenon-local changes in synaptic receptor numbers
4. Lateral diffusion of AMPA receptors not responsible forheterosynaptic LTP/LTD
Biophysical models of AMPA receptor trafficking in dendrites – p. 42/52
Thank you!
Biophysical models of AMPA receptor trafficking in dendrites – p. 43/52
Single-spine results: Assumptions
Let subscripts I, II denote GluR1/2, GluR2/3 respectively.
UI , UII are constant (determine self-consistently later)
SII constant (include dynamics later)dSI
dt = −σrecI SI + δI (more detailed later)
All parameters constant so concentrations approach asteady-state
Biophysical models of AMPA receptor trafficking in dendrites – p. 44/52
Single-spine results: Steady-state
Bound AMPAR concentration in PSD
QI =ρI
1 + ρI + ρIIZ, QII =
ρII
1 + ρI + ρIIZ
where
ρI =αIPI
βI, ρII =
αIIPII
βII
Free AMPAR concentration in PSD
PI = RI , PII = RII +σrec
II SII
hII
AMPAR concentration in spine head
RI =σrec
I SI + ωIUI
kI + ωI, RII =
σrecII SII + ωIIUII
kII + ωII
Biophysical models of AMPA receptor trafficking in dendrites – p. 45/52
2D mutli-spine model: Steady-state
Rj =ωjUj + λjδj
ωj + kj(1 − λj), Sj =
kjλjRj
σrecj
where
λj =σrec
j
σrecj + σdeg
j
To determine Uj, need to solve
∇2U = 0, r ∈ Ωε
with bcs.
Biophysical models of AMPA receptor trafficking in dendrites – p. 46/52
Inner behavior of outer solution
U has the near-field behavior (as r → rj)
U ∼ −2πνAj(ν)
[−
1
2πlog |r − rj| + G(rj ; rj)
]
−∑
i 6=j
2πνAi(ν)G(rj; ri) + χ
Comparison with asymptotic conditions yields the system:
(1 + 2πνGjj)Aj +∑
i 6=j
2πνGjiAi = uj − Uj + χ
where uj = u(rj), Gji = G(rj; ri) and Gjj = G(rj ; rj).
Biophysical models of AMPA receptor trafficking in dendrites – p. 47/52
Calculation of boundary concentrationsUj
Substituting inner sol. into generalized Neumann bcs gives
2πνAj(ν) =ωj
D[Uj − Rj ] ≡ Vj
Substituting into system of equations yields
Vj = 2πνM∑
i=1
Mji(ui − Ri + χ)
where M = (I + 2πνB)−1 and
Bjj =D
ωj+ Gjj , Bji = Gji, j 6= i
Biophysical models of AMPA receptor trafficking in dendrites – p. 48/52
Evaluation of Green’s function
A standard (and long) calculation shows
G(r; r′) = −1
2πln |r − r
′| + G(r; r′)
G(r; r′) =L
24πl
[h
(x − x′
L
)+ h
(x + x′
L
)]
−1
2πln
|1 − er+/l||1 − er−
/l||1 − eρ+/l||1 − eρ−
/l|
|r − r′|+ O(q)
h(θ) = 3θ2 − 6|θ| + 2
r± = −|x ± x′| + i(y − y′), ρ± = −2L + |x ± x′| + i(y − y′)
q = e−2L/l
Biophysical models of AMPA receptor trafficking in dendrites – p. 49/52
2D multi-spine model: MFPT
Mean first passage time (MFPT) for a single AMPAR totravel axial distance X < L from soma, given started atr0 = (0, y) and not degraded:
T (X|r0) =X2
2D+
NX∑
j=1
ηj
DGX(rj ; r0)
where
ηj = Aj +kj
σrecj
GX(rj ; r0) =X − xj
2πl+ O(qxj
), qxj= e−2xj/l
Biophysical models of AMPA receptor trafficking in dendrites – p. 50/52
Effective and anomalous diffusion
Large number of identical spines uniformly distributedwith spacing d (i.e., NX = X/d ≫ 1 and xj = jd for all j):
T ≈X2
2Deff, Deff = D
(1 +
A + k/σrec
2πld
)−1
Now suppose xj = d(ln(j) + 1) so NX = eX/d−1, then
T ≈X2
2Deff (X), Deff (X) = D
(
1 +A + k/σrec
2πld
eX/d−1
(X/d)2
2
)−1
Biophysical models of AMPA receptor trafficking in dendrites – p. 51/52
1D cont’m multi-spine model: Solution
Modified Green’s function for 1D Laplace equation withreflecting bcs:
G(x, x′) =cosh(Λ0[|x − x′| − L]) + cosh(Λ0[x + x − L])
2Λ0 sinh(Λ0L)
Steady-state solution is
U(x) =σ0
D
cosh(Λ0[x − L])
Λ0 sinh(Λ0L)+ Λ2
0
∫ L
0G(x, x′)R(x′)dx′
If also R(x) = R0 is uniform
U(x) =σ0
D
cosh(Λ0[x − L])
Λ0 sinh(Λ0L)+ R0
Biophysical models of AMPA receptor trafficking in dendrites – p. 52/52
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