biophysical models of ampa receptor trafﬁcking in dendrites

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


Neurons communicate via synapses


Synaptic transmission

Kandel et al., Principles of Neural Science (2000)


Synaptic plasticity

Collingridge et al., Nat. Rev. Neurosci. (2004)


Outline

1. AMPA receptor trafficking

2. Single-spine model

3. 2D multi-spine model

4. 1D continuum multi-spine model


AMPA receptors

Huganir & Song, Nat. Rev. Neurosci. (2002)

Fast synaptic transmission

Complexes with other proteins −→ trafficking


Long-range receptor trafficking

Groc & Choquet, Cell Tissue Res. (2006)

Vesicle transport along microtubules

Diffusion from soma to synapse?


Dendritic spines

Matus, Science (2000)

Excitatory synapses located on surface ofmushroom-like protrusions of the dendritic membranecalled spines


Receptor trafficking at spines

Sheng & Kim, Science (2002)

Constitutive recycling

Crosslink to scaffolding in PSD

Lateral diffusion in membrane


Expression of LTP/LTD

Scannevin & Huganir, Nat. Rev. Neurosci. (2000)


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


Outline



3. 2D mutli-spine model



Single-spine model

DEG

END

EXO

EXO

PSD

AMPA receptor

scaffolding protein

Earnshaw & Bressloff, J. Neurosci. (2006)


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


Single-spine results: Block exo/endocytosis

Luscher et al., Neuron (1999)


Single-spine results: LTPA

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t [min]

nu

mb

er

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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PS

D

TotalBound GluR1/2Free GluR1/2Bound GluR2/3Free GluR2/3Scaffolding


Single-spine results: LTD

0 5 10 15 20 25 300

5

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40

t [min]

nu

mb

er o

f re

cep

tors

in P

SD

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t [min]

nu

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SD

TotalBound GluR1/2Free GluR1/2Bound GluR2/3/GRIPFree GluR2/3/GRIPBound GluR2/3/PICKFree GluR2/3/PICKScaffolding

0 30 60 90 120 150 180 2100

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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)


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)


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)


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)


Conclusions





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)


Conclusions





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)


Outline






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.


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


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


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

]


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


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 |)


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 |.


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.


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


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


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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URS

Distance from soma [μm]

Re

ce

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rs

j

j

j

D


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)


Effect of ε

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1

1.2

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2.2

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1.2

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2

2.2

2.4


Outline






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


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


Delivery of synaptic receptorsFast recycling, production

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IntracellularPSDBound

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Slow recycling, no production

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Slow recycling, production

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Fast recycling, no production

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distance from soma

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Heterosynaptic effect of constit. recyclingreduced σrec in gray

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Intracellular

PSD

Bound

increased k in gray

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increased δ in gray

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increased σdeg in gray

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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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IntracellularESMScaffoldingPSDBound

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LTP at x = 100 μm

ScaffoldingTotalBoundFree

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LTP at x = 100 μm


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


Conclusions

1. Surface trafficking of AMPA receptors is effectivelyone-dimensional with spines treated as pointsources/sinks (or density if there are enough spines)


Conclusions


2. Somatic synthesis + lateral diffusion are not sufficient tosupply distal synapses with receptors


Conclusions



3. Local changes to constitutive recycling producenon-local changes in synaptic receptor numbers


Conclusions



3. Local changes to constitutive recycling producenon-local changes in synaptic receptor numbers

4. Lateral diffusion of AMPA receptors not responsible forheterosynaptic LTP/LTD


Thank you!


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


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


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.


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).


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


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


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


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


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

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