from synapse to spiny dendrite paul bressloff berton earnshaw · march 26, 2009 bressloff,...
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Multiple spatial scales of AMPA receptor traffickingFrom synapse to spiny dendrite
Paul Bressloff Berton Earnshaw
Department of MathematicsUniversity of Utah
March 26, 2009
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 1 / 52
AMPA receptor trafficking Introduction
The amazing brain
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 2 / 52
AMPA receptor trafficking Introduction
Neurons communicate at synapses
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 3 / 52
AMPA receptor trafficking Introduction
Communication at a synapse
Kandel, Schwartz & Jessel (2000)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 4 / 52
AMPA receptor trafficking Introduction
Synapses can “learn”
Collingridge et al., Nat Rev Neurosci (2004)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 5 / 52
AMPA receptor trafficking Introduction
Synapses “learn” by regulating receptor numbers
LTDDepressed PotentiatedNaïve LTP
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 6 / 52
AMPA receptor trafficking Introduction
Synapses located in dendritic spines
Matus, Science (2000)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 7 / 52
AMPA receptor trafficking Introduction
Receptor trafficking at a synapse
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 8 / 52
AMPA receptor trafficking Introduction
Receptors diffuse laterally between synapses
Triller & Choquet, Nat. Rev. Neurosci. (2003)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 9 / 52
AMPA receptor trafficking Introduction
Long-range transport of receptors
1. somatic exocytosis
2. lateral membrane diffusion
3. surface entry into spine
4. local exo/endocytosis
12 3
4
Groc & Choquet, 2006
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 10 / 52
AMPA receptor trafficking Outline
• Single-spine model (deterministic)
• Single-synapse model (stochastic)
• 2D diffusion model
• 1D diffusion models
• Other diffusion-trapping problems
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 11 / 52
AMPA receptor trafficking Single-spine model
Model of single-spine AMPAR trafficking
Spine head:dR
dt=
1
A(µ[U − R ] − kR − h[R − P ])
PSD unbound:dP
dt=
h
a[R − P ] − α[Z − Q]P + βQ +
σEXOC
a
PSD bound:dQ
dt= α[Z − Q]P − βQ
Intracellular:dC
dt= −σEXOC − σDEGC + kR + δ
kσEXO
C
R Uµ
σDEG
hPQ
α(Z-Q)
β
δAMPA receptor
scaffolding protein
PSD Spine Head
ENDEXO
DEGDEL
BAE & Bressloff, J Neurosci (2006)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 12 / 52
AMPA receptor trafficking Single-spine model
Block exo/endocytosis
Luscher et al., Neuron (1999)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 13 / 52
AMPA receptor trafficking Single-spine model
LTP simulation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
10
20
30
40
50
60
70
80
90
100
t [min]
nu
mb
er o
f re
cep
tors
O' Connor et al (PNAS 2005)
GluR1/2
ΙΙ
Ι
GluR2/3
• Activation of GluR1/2 intracellular pool
• Rapid insertion of receptors into ESM
• AMPARs transport slot proteins into PSD
TotalBound GluR1/2Free GluR1/2Bound GluR2/3Free GluR2/3Scaffolding
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 14 / 52
AMPA receptor trafficking Single-spine model
LTD simulation
0 30 60 90 120 150 180 2100
10
20
30
40
50
60
70
t [min]
nu
mb
er o
f re
cep
tors
• Switch from AMPA-GRIP to AMPA-PICK receptor-protein complexes
• Rapid unbinding from PSD and trafficking to ESM followed by endocytosis.
• Unbound scaffolding proteins are degraded.
Dudek & Bear (1993)
PSD ESMPSD ESM ESMPSD
GRIP PICK
TotalScaffolding
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 15 / 52
AMPA receptor trafficking Single-spine model
Conclusions
1 Significant fraction of PSD receptors are mobile (Groc et al., 2004; Ashby et al.,
2006)
• Requires PSD-ESM barrier (Choquet & Triller, 2003)
2 Diffusive impedance of spine neck is significant (Ashby et al., 2006)
3 Insertion of GluR1/2 during LTP must combine synaptic targeting
• Requires increased hopping and binding rate (Schnell et al., 2002) andscaffolding (Shi et al., 2001)
4 Slow exchange of GluR1/2 with GluR2/3 after LTP requiresmaintenance of additional binding sites (McCormack et al., 2006)
5 LTD requires loss of binding sites (Colledge et al., 2003)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 16 / 52
AMPA receptor trafficking Stochastic single-synapse model
• Single-spine model (deterministic)
• Single-synapse model (stochastic)• 2D diffusion model
• 1D diffusion models
• Other diffusion-trapping problems
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 17 / 52
AMPA receptor trafficking Stochastic single-synapse model
Stochastic model of trafficking at PSDdp
dt= −α(Z − q)p + βq − µp + σ
dq
dt= α(Z − q)p − βq
DEG
ENDEXO
A
PSD
AMPA receptorscaffolding protein
B
PROD
ESM
PSD
ESM
α
βpq
(Z - q) σ
µ
Bressloff & BAE Biophys J (2009)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 18 / 52
AMPA receptor trafficking Stochastic single-synapse model
Stochastic model of trafficking at PSDdp
dt= −α(Z − q)p + βq − µp + σ
dq
dt= α(Z − q)p − βq
Pn,m(t) = Probn unbound,m bound at time tDEG
ENDEXO
A
PSD
AMPA receptorscaffolding protein
B
PROD
ESM
PSD
ESM
α
βpq
(Z - q) σ
µ
Bressloff & BAE Biophys J (2009)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 18 / 52
AMPA receptor trafficking Stochastic single-synapse model
Stochastic model of trafficking at PSDdp
dt= −α(Z − q)p + βq − µp + σ
dq
dt= α(Z − q)p − βq
Pn,m(t) = Probn unbound,m bound at time tdPn,m
dt= σPn−1,m + µ(n + 1)Pn+1,m
+ α(n + 1)[Z − (m − 1)]Pn+1,m−1
+ β(m + 1)Pn−1,m+1
− [σ + µn + αn(Z − m) + βm]Pn,m
DEG
ENDEXO
A
PSD
AMPA receptorscaffolding protein
B
PROD
ESM
PSD
ESM
α
βpq
(Z - q) σ
µ
Bressloff & BAE Biophys J (2009)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 18 / 52
AMPA receptor trafficking Stochastic single-synapse model
Stochastic model of trafficking at PSDdp
dt= −α(Z − q)p + βq − µp + σ
dq
dt= α(Z − q)p − βq
Pn,m(t) = Probn unbound,m bound at time tdPn,m
dt= σPn−1,m + µ(n + 1)Pn+1,m
+ α(n + 1)[Z − (m − 1)]Pn+1,m−1
+ β(m + 1)Pn−1,m+1
− [σ + µn + αn(Z − m) + βm]Pn,m
stochastic gate : 0 < µopen
γ−
−→
←−
γ+
µclosed = 0
σ(t) = Cµ(t) (C bath conc.)
DEG
ENDEXO
A
PSD
AMPA receptorscaffolding protein
B
PROD
ESM
PSD
ESM
α
βpq
(Z - q) σ
µ
Bressloff & BAE Biophys J (2009)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 18 / 52
AMPA receptor trafficking Stochastic single-synapse model
Analysis in two regimes: un/saturated binding sites• Can do math in two regimes:
• unsaturated binding sites: m(t) ≪ Z for all t (i.e., αn(t) ≪ β)• saturated binding sites: m(t) = Z for all t (i.e., αn(t) ≫ β)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 19 / 52
AMPA receptor trafficking Stochastic single-synapse model
Analysis in two regimes: un/saturated binding sites• Can do math in two regimes:
• unsaturated binding sites: m(t) ≪ Z for all t (i.e., αn(t) ≪ β)• saturated binding sites: m(t) = Z for all t (i.e., αn(t) ≫ β)
• Unsaturated regime: master equation is linear in n,m• Generating function
G(u, v , t) =
∞∑
n=0
∞∑
m=0
unvmPn,m(t)
satisfies first-order linear PDE
∂G
∂t+ [µ(t)(u − 1) + αZ (u − v)]
∂G
∂u− β(u − v)
∂G
∂v= σ(t)(u − 1)G
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 19 / 52
AMPA receptor trafficking Stochastic single-synapse model
Analysis in two regimes: un/saturated binding sites• Can do math in two regimes:
• unsaturated binding sites: m(t) ≪ Z for all t (i.e., αn(t) ≪ β)• saturated binding sites: m(t) = Z for all t (i.e., αn(t) ≫ β)
• Unsaturated regime: master equation is linear in n,m• Generating function
G(u, v , t) =
∞∑
n=0
∞∑
m=0
unvmPn,m(t)
satisfies first-order linear PDE
∂G
∂t+ [µ(t)(u − 1) + αZ (u − v)]
∂G
∂u− β(u − v)
∂G
∂v= σ(t)(u − 1)G
• Obtain mean, variance from derivatives of G
Eµ(n) =∂G
∂u
∣∣∣∣u=v=1
, Eµ(m) =∂G
∂v
∣∣∣∣u=v=1
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 19 / 52
AMPA receptor trafficking Stochastic single-synapse model
Averaging over realizations of µ
• Can show that
E(n) ≡ 〈Eµ(n)〉 = C + (n0 − C )〈N11〉 + (m0 − CαZ/β)〈N21〉E(m) ≡ 〈Eµ(m)〉 = CαZ/β + (n0 − C )〈N12〉 + (m0 − CαZ/β)〈N22〉
where
N (t) = exp
(−
∫ t
0M(t ′)dt ′
), M(t) =
(µ(t) + αZ −αZ
−β β
)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 20 / 52
AMPA receptor trafficking Stochastic single-synapse model
Averaging over realizations of µ
• Can show that
E(n) ≡ 〈Eµ(n)〉 = C + (n0 − C )〈N11〉 + (m0 − CαZ/β)〈N21〉E(m) ≡ 〈Eµ(m)〉 = CαZ/β + (n0 − C )〈N12〉 + (m0 − CαZ/β)〈N22〉
where
N (t) = exp
(−
∫ t
0M(t ′)dt ′
), M(t) =
(µ(t) + αZ −αZ
−β β
)
• Can derive a system of ODEs for the averages of the entries of Nusing method of Kubo and Zwanzig
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 20 / 52
AMPA receptor trafficking Stochastic single-synapse model
Saturated binding sites
• Since m(t) = Z for all t, master equation becomes
dPn
dt= µ(t) [CPn−1 + (n + 1)Pn+1(t) − (C + n)Pn]
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 21 / 52
AMPA receptor trafficking Stochastic single-synapse model
Saturated binding sites
• Since m(t) = Z for all t, master equation becomes
dPn
dt= µ(t) [CPn−1 + (n + 1)Pn+1(t) − (C + n)Pn]
• Brown et al. (Biophys J, 2000) showed that
E(n) = (n0 − C )〈w〉 + C
Var(n) = E(n) − n0〈w2〉 + (n0 − C )2(〈w2〉 − 〈w〉2
)
⟨w(t)j
⟩=
(11
)T
exp
[−t
(jµo + γ− −γ+
−γ− γ+
)](Πo
Πc
), (j = 1, 2)
Πo =γ+
γ+ + γ−, Πc =
γ−γ+ + γ−
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 21 / 52
AMPA receptor trafficking Stochastic single-synapse model
FRAP and inverse FRAP experiments
ESMPSD
PSDPSDPSD
PSDPSD
ESMESM ESM
ESMESM
FRAP
inverse FRAP
recovery
recovery
unbleached receptorbleached receptorscaffolding protein
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 22 / 52
AMPA receptor trafficking Stochastic single-synapse model
Simulations of FRAP and inverse FRAP
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
vari
ance
dynamicstatic
0 0.04 0.08 0.12 0.16 0.20
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
10
0 0.04 0.08 0.12 0.16 0.20
1
2
3
4
5
6
7
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
time [s]
vari
ance
dynamicstaticmean
0 0.04 0.08 0.12 0.16 0.20
0.4
0.8
1.2
1.6
2
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5
6
7
8
9
10
time [s]
0 0.04 0.08 0.12 0.16 0.20
2
4
6
8
10
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
14
16
18
20
0 0.04 0.08 0.12 0.16 0.20
4
8
12
16
20
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
14
16
18
20
time [s]
0 0.04 0.08 0.12 0.16 0.20
4
8
12
16
20
C
F
B
E
A
D
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 23 / 52
AMPA receptor trafficking 2D diffusion model
• Single-spine model (deterministic)
• Single-synapse model (stochastic)
• 2D diffusion model• 1D diffusion models
• Other diffusion-trapping problems
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 24 / 52
AMPA receptor trafficking 2D diffusion model
Treat dendritic membrane as cylinder with holes
x = 0
spine
x = L
Jsoma DEG
ENDEXO
receptor
scaffoldingprotein
PSD
DEL
Bressloff, BAE & Ward, SIAM J Appl Math (2008)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 25 / 52
AMPA receptor trafficking 2D diffusion model
Diffusion equation on dendritic membrane
∂U
∂t= D∇2U on Ωε
y = πl
x = 0 x = Ly = -πl
Ωε
Ωj
• U = receptor concentration
• Ωε is rectangle (0,L) × (−πl , πl) minus the holes
Ωj = r ∈ Ω0 | |r − rj | ≤ ερ, j = 1, . . . ,N
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 26 / 52
AMPA receptor trafficking 2D diffusion model
Boundary conditions
• Periodic bcs at y = ±πl
• No-flux bc at x = L, and at x = 0
−D∂U
∂x= Jsoma =
σ
2πl
• bcs at the holes:
−D∂U
∂n(r, t) =
µj
2περ(U(r, t) − Rj), r ∈ ∂Ωj
• µj = spine neck hopping rate• Rj = receptor concentration on surface of jth spine
y = πl
x = 0 x = Ly = -πl
Ωε
Ωj
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 27 / 52
AMPA receptor trafficking 2D diffusion model
Treat each spine as before
kσEXO
C
R Uµ
σDEG
hPQ
α(Z-Q)
β
δAMPA receptor
scaffolding protein
PSD Spine Head
ENDEXO
DEGDEL
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 28 / 52
AMPA receptor trafficking 2D diffusion model
Steady-state solution
• Assume concentrations in jth spine see mean value of U on ∂Ωj :
Uj =1
2περ
∫
∂Ωj
U(r)dr
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 29 / 52
AMPA receptor trafficking 2D diffusion model
Steady-state solution
• Assume concentrations in jth spine see mean value of U on ∂Ωj :
Uj =1
2περ
∫
∂Ωj
U(r)dr
• Uj ’s are determined by solving ∇2U = 0 in Ωε with boundaryconditions
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 29 / 52
AMPA receptor trafficking 2D diffusion model
Steady-state solution
• Assume concentrations in jth spine see mean value of U on ∂Ωj :
Uj =1
2περ
∫
∂Ωj
U(r)dr
• Uj ’s are determined by solving ∇2U = 0 in Ωε with boundaryconditions
• But this is hard because of boundary conditions at the holes!
−εD∂U
∂n(r) =
µj
2πρ(U(r) − Rj), r ∈ ∂Ωj
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 29 / 52
AMPA receptor trafficking 2D diffusion model
Three steps for finding approximate steady-state solution
1 Solve assuming U = Uj on the boundary of jth hole
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 30 / 52
AMPA receptor trafficking 2D diffusion model
Three steps for finding approximate steady-state solution
1 Solve assuming U = Uj on the boundary of jth hole• Singular perturbation: match logarithmic solutions in each inner
region|r − rj | = O(ε)
with Green’s function singularities in outer region
|r − rj | = O(1) for all j
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 30 / 52
AMPA receptor trafficking 2D diffusion model
Three steps for finding approximate steady-state solution
1 Solve assuming U = Uj on the boundary of jth hole• Singular perturbation: match logarithmic solutions in each inner
region|r − rj | = O(ε)
with Green’s function singularities in outer region
|r − rj | = O(1) for all j
• Solution has N + 1 unknowns: Uj ’s and integration constant
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 30 / 52
AMPA receptor trafficking 2D diffusion model
Three steps for finding approximate steady-state solution
1 Solve assuming U = Uj on the boundary of jth hole• Singular perturbation: match logarithmic solutions in each inner
region|r − rj | = O(ε)
with Green’s function singularities in outer region
|r − rj | = O(1) for all j
• Solution has N + 1 unknowns: Uj ’s and integration constant
2 Substitute this solution into N simplified bcs at holes
−εD∂U
∂n(r) =
µj
2πρ(Uj − Rj), r ∈ ∂Ωj
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 30 / 52
AMPA receptor trafficking 2D diffusion model
Three steps for finding approximate steady-state solution
1 Solve assuming U = Uj on the boundary of jth hole• Singular perturbation: match logarithmic solutions in each inner
region|r − rj | = O(ε)
with Green’s function singularities in outer region
|r − rj | = O(1) for all j
• Solution has N + 1 unknowns: Uj ’s and integration constant
2 Substitute this solution into N simplified bcs at holes
−εD∂U
∂n(r) =
µj
2πρ(Uj − Rj), r ∈ ∂Ωj
3 Conservation condition gives (N + 1)th equation
σ =∑
µj
(Uj − Rj
)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 30 / 52
AMPA receptor trafficking 2D diffusion model
Comparison of dendritic receptor concentration
perturbation solution
A
numerical solution
• Dendrite 100µm long, circumference 1µm, ǫρ = 0.1µm
• 100 identical spines spaced 1µm apart, all in a row
• Solutions are almost identical!
• Similar results if spines are not identical, not in a row
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 31 / 52
AMPA receptor trafficking 1D diffusion models
• Single-spine model (deterministic)
• Single-synapse model (stochastic)
• 2D diffusion model
• 1D diffusion models• Other diffusion-trapping problems
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 32 / 52
AMPA receptor trafficking 1D diffusion model with discrete spines
2D model well-approximated by 1D model
When the aspect ratio L/l ≫ 1, we can approximate 2D model by thefollowing 1D model
∂U
∂t= D
∂2U
∂x2−
N∑
j=1
δ(x − xj)µj(Uj − Rj)
−D∂U
∂x
∣∣∣∣x=0
= Jsoma,∂U
∂x
∣∣∣∣x=L
= 0.
Bressloff & BAE, PRE (2007)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 33 / 52
AMPA receptor trafficking 1D diffusion model with discrete spines
Comparison of models
2D model
0 20 40 60 80 10010
15
20
25
30
35
40
45
x [µm]
rece
ptor
s
Uj
Rj
Sj
B1D model
0 20 40 60 80 10010
15
20
25
30
35
40
45
x [µm]re
cept
ors
Uj
Rj
Sj
D
• 2D model as before• Dendrite 100µm long, circumference 1µm, ǫρ = 0.1µm• 100 identical spines spaced 1µm apart, all in a row
• 1D model use same parameters when relevant
• Solutions are almost identical!
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 34 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Can we make things even simpler?
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 35 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Treat spine population as continuous density
If spines are sufficiently dense, treat sum of delta functions as a density η
∂U
∂t= D
∂2U
∂x2− η(x)µ(x)(U − R)
−D∂U
∂x
∣∣∣∣x=0
= Jsoma,∂U
∂x
∣∣∣∣x=L
= 0.
BAE & Bressloff, J Comput Neurosci (2008)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 36 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Steady-state solution for identical spines: “cable” equation
• Assume all parameters are x-independent, then get “cable” equationfor receptor trafficking
d2U
dx2− Λ2U = −Λ2R
Λ =
√ηµ
Dis length-scale of diffusive coupling
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 37 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Steady-state solution for identical spines: “cable” equation
• Assume all parameters are x-independent, then get “cable” equationfor receptor trafficking
d2U
dx2− Λ2U = −Λ2R
Λ =
√ηµ
Dis length-scale of diffusive coupling
• Solve using Green’s function methods
U(x) =Jsoma
D
cosh(Λ(x − L))
Λ sinh(ΛL)+ R
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 37 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Steady-state receptor concentrations for identical spines
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
80
100
120
140
160
180
200
distance from soma [mm]
nu
mb
er o
r co
nce
ntr
atio
n [
µm-2
]
concentration in dendritenumber in PSDnumber bound in PSD
Piccini & Malinow, 2002
GluR1/2GluR2/3
• Dendrite 1 mm long
• 1,000 identical spines spaced 1µm apart
• Two sources of receptors• at soma• local intracellular delivery
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 38 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Consequences of diffusive coupling
10-fold reduction inrate of exocytosis
in gray region
0 20 40 60 80 100 120 140 160 180 2000
10
20
30
40
50
60
70
80
distance from soma [µm]
PS
D r
ecep
tors
after perturbationbefore perturbation
10-fold increase inrate of endocytosis
in gray region
0 20 40 60 80 100 120 140 160 180 2000
10
20
30
40
50
60
70
80
distance from soma [µm]
PS
D r
ecep
tors
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 39 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Steady-state is nice......but what about time-dependent phenomena?
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 40 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
AMPA receptor recycling via thrombin cleavage
Passafaro et al., Nat. Neurosci. (2001)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 41 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
AMPA receptor recycling via photoinactivation
Adesnik et al., Neuron (2005)Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 42 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Fast or slow recycling of AMPA receptors?
Passafaro et al., 2001 Adesnik et al., 2005
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 43 / 52
AMPA receptor trafficking 1D diffusion model with density of spines
Simulation of photoinactivation of AMPA receptors
0 4 8 12 16 20 240
10
20
30
40
50
60
70
80
90
reco
very
[%
bas
elin
e]
2 221814106
10 µm300 µm
time [hr]
PSD
intracellularpool
inactive AMPAR
active AMPAR
scaffolding protein
Fast insertionfrom pool
Depletion ofpool
Recovery ofpool
Replacement ofbound AMPARs
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 44 / 52
AMPA receptor trafficking Other diffusion-trapping problems
• Single-spine model (deterministic)
• Single-synapse model (stochastic)
• 2D diffusion model
• 1D diffusion models
• Other diffusion-trapping problems
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 45 / 52
AMPA receptor trafficking Other diffusion-trapping problems
CaMKII translocation waves
Rose et al., Neuron (2009)
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 46 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Diffusion-activation model of CaMKII translocation waves
primed CaMKII
activated CaMKII
diffusion-activationtranslocation
b
c
x = Lx = 0
30µm
soma dendrite
a
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 47 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Equations for diffusion-activation model
∂p
∂t= D
∂2p
∂x2− kap
∂a
∂t= D
∂2a
∂x2+ kap − ha
∂s
∂t= ha
• p = concentration of primed CaMKII in shaft
• a = concentration of activated CaMKII in shaft
• s = concentration of activated CaMKII in spines
• k, h = rate of activation, translocation
BAE & Bressloff, In prep.
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 48 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Simulation of diffusion-activation model
0
0.2
0.4
0.6
0.8
1
1.2C
aMK
II [n
orm
. co
nc.
] 10 sec 65 sec
0 30 60 90 120 1500
0.2
0.4
0.6
0.8
1
1.2
x [µm]
CaM
KII
[no
rm. c
on
c.] 140 sec
0 30 60 90 120 150
348 sec
total in spinestotal in dendriteprimed in dendriteactivated in dendrite
x [µm]
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 49 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Calculation of wave speed
• When h = 0 (no translocation), recover Fisher’s equation with speed
c = 2√
Dk
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 50 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Calculation of wave speed
• When h = 0 (no translocation), recover Fisher’s equation with speed
c = 2√
Dk
• When h 6= 0, wave speed is
c = 2√
D(k − h)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
h/k
c/√(Dk)
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
activation rate k [1/s]
wave spee
d c [
µm/s]
CaMKIIαCaMKIIβ
a b
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 50 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Calculation of wave speed
• When h = 0 (no translocation), recover Fisher’s equation with speed
c = 2√
Dk
• When h 6= 0, wave speed is
c = 2√
D(k − h)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
h/k
c/√(Dk)
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
activation rate k [1/s]
wave spee
d c [
µm/s]
CaMKIIαCaMKIIβ
a b
• Wave propagation failure when k < hBressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 50 / 52
AMPA receptor trafficking Other diffusion-trapping problems
Other projects with diffusion-trapping models
• other heterosynaptic molecules, e.g. PSD-95
• changes in spine volume during late-phase LTP• mRNA transport/capture/translation• F-actin regulation/stabilization by AMPA receptors
• protein transport/capture during synaptogenesis, e.g. NMDAreceptors
• AMPA receptor trafficking in more detailed model of PSD
• put all the pieces together!
Bressloff, Earnshaw (Utah) Multiple scales of AMPAR trafficking March 26, 2009 51 / 52
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