Increased accuracy of ligand sensing by receptor diffusion ...

PHYSICAL REVIEW E 82, 041902 ͑2010͒

Increased accuracy of ligand sensing by receptor diffusion on cell surface

Gerardo Aquino and Robert G. Endres
Division of Molecular Biosciences and Centre for Integrative Systems Biology at Imperial College, Imperial College London,

SW7 2AZ London, United Kingdom
͑Received 8 July 2010; published 1 October 2010͒

The physical limit with which a cell senses external ligand concentration corresponds to the perfect absorber,
where all ligand particles are absorbed and overcounting of same ligand particles does not occur. Here, we
analyze how the lateral diffusion of receptors on the cell membrane affects the accuracy of sensing ligand
concentration. Specifically, we connect our modeling to neurotransmission in neural synapses where the dif-
fusion of glutamate receptors is already known to refresh synaptic connections. We find that receptor diffusion
indeed increases the accuracy of sensing for both the glutamate ␣-Amino-3-hydroxy-5-Methyl-4-isoxazole-
Propionic Acid ͑AMPA͒ and N-Methyl-D-aspartic Acid ͑NMDA͒ receptor, although the NMDA receptor is
overall much noisier. We propose that the difference in accuracy of sensing of the two receptors can be linked
to their different roles in neurotransmission. Specifically, the high accuracy in sensing glutamate is essential for
the AMPA receptor to start membrane depolarization, while the NMDA receptor is believed to work in a
second stage as a coincidence detector, involved in long-term potentiation and memory.

DOI: 10.1103/PhysRevE.82.041902 PACS number͑s͒: 87.10.Mn, 87.16.dj, 87.18.Tt

I. INTRODUCTION accuracy of external ligand concentration sensing ͓12͔. We
showed with a simple model that internalization, by making
Biological cells live in very noisy environments from the cell act as an absorber of ligand, increases sensing accu-
which they receive many different stimuli. The survival of a racy by reducing the noise from rebinding of already mea-
cell highly depends on its ability to respond to such stimuli sured ligand molecules. In this paper we consider the effect
and to adapt to changes in the environment. A fundamental of lateral diffusion of receptors on the accuracy of ligand
role in this task is played by membrane receptors, proteins on concentration sensing.
the cell surface which are able to bind external ligand mol-
ecules and trigger signal transduction pathways. Although In a situation in which ligand sensing is concentrated in a
the precision with which a cell can measure the concentra- specific region on the cell membrane ͑signaling hotspot͒, re-
tion of a specific ligand is negatively affected by many ceptors bind ligand molecules locally but may release them
sources of noise ͓1–4͔, several examples exist in which such remotely from the region of interest, thus preventing those
measurements are performed with surprisingly high accu- particles from rebinding locally. Noise from overcounting
racy. In bacterial chemotaxis, for instance, the bacterium Es- could therefore be reduced, potentially increasing the accu-
cherichia coli can respond to changes in concentration as racy of sensing.
low as 3.2 nM ͓5͔, corresponding to only three molecules in
the volume of a cell. High accuracy is observed also in spa- A potential implementation of such localized sensing oc-
tial sensing by single cell eukaryotic organisms as, e.g., in curs in neural synapses, responsible for transmission of ac-
the slime mold Dictyostelium discoideum, which is able to tion potentials and short-term synaptic plasticity ͑see Fig. 1
sense a concentration difference of only 1 – 5 % across the for a schematic description͒. At such critical places in the
cell diameter ͓6͔, and in Saccharomyces cerevisiae ͑budding central nervous system, the accuracy of sensing ought to be
yeast͒, which is able to orient growth in a gradient of important ͓13͔. At a synapse the propagation of an action
␣-pheromone mating factor down to estimated 1% receptor potential arriving from the presynaptic neuron causes the
occupancy difference across the cell ͓7͔. Spatial sensing is opening of ion channels on the presynaptic membrane which
also efficiently performed by growing neurons, lymphocytes, increases calcium-ion inflow ͓14͔. Calcium ions trigger a
neutrophils, and other cells of the immune system. biochemical cascade which results in the formation of

There has been substantial progress in our theoretical un- vesicles containing neurotransmitter, e.g., glutamate, in the
derstanding of the accuracy of concentration sensing. In
1977, Berg and Purcell were the first to point out that ligand case of excitatory synapses. One or more vesicles eventually
sensing is limited by ligand diffusion, producing a “counting
noise” at the receptors ͓8͔. Bialek and Setayeshgar ͓9͔ ͑and fuse with the membrane at a point on the presynaptic
later others ͓10͔͒ used the fluctuation dissipation theorem
͑FDT͒ to separate the contributions from random binding or surface to form a fusion pore. Glutamate is subsequently
unbinding events from rebinding due to diffusion of ligand
molecules. Subsequently, Endres and Wingreen showed that released and diffuses into the synaptic cleft, reaching
the perfect absorber is the absolute fundamental limit of
ligand sensing accuracy; it does not suffer from ligand un- the receptors on the postsynaptic surface on the opposite
binding and rebinding noises ͓10,11͔.
side of the cleft. These receptors, mainly the AMPA
In previous work we analyzed the contribution of endocy- and NMDA receptors ͑named after the main binding mol-
tosis, i.e., the internalization of cell-surface receptors, to the ecules ␣-Amino-3-hydroxy-5-Methyl-4-isoxazole-Propionic
Acid and N-Methyl-D-aspartic Acid, respectively͒, cause ion
channels on the postsynaptic membrane to open, facilitating

the propagation of the action potential.
The AMPA receptor ͑AMPAR͒ has a tetrameric structure

with four binding sites for glutamate. Binding glutamate on

two sites leads to conformational change and opening of a

pore. More occupied binding sites imply a higher current

1539-3755/2010/82͑4͒/041902͑10͒ 041902-1 ©2010 The American Physical Society

GERARDO AQUINO AND ROBERT G. ENDRES PHYSICAL REVIEW E 82, 041902 ͑2010͒

axon k c k−
+

dendrite x
0
FIG. 1. ͑Color online͒ Synaptic junction during neurotransmis-
sion. A vesicle containing a neurotransmitter ͑e.g., glutamate for O
excitatory synapses͒ releases its content into the synaptic cleft. Re-
ceptors on the postsynaptic surface bind the neurotransmitter, in- FIG. 2. ͑Color online͒ Single receptor, immobile at position xជ0,
creasing the likelihood of the propagation of the electric impulse binds and unbinds ligand with rates k+¯c and k−, respectively.
͑action potential͒. Dark ͑red͒ arrow indicates the incoming poten-
tial, light ͑yellow͒ arrow indicates the set of reactions leading to dn͑t͒ ͑1͒
propagation of potential, and small ͑green͒ arrows indicate receptor dt = k+¯c͓1 − n͑t͔͒ − k−n͑t͒,
mobility.
where the concentration of ligand, ¯c, is assumed uniform and
through the channel, mainly of calcium, sodium, and potas- constant. The steady-state solution for the receptor occu-
sium ions. The NMDA receptor ͑NMDAR͒ has a tetrameric pancy is given by
structure as well, but its channel needs coincidental binding
of glutamate and glycine molecules on dedicated sites for it ¯c ͑2͒
to open. A small membrane depolarization is furthermore ¯n = ,
necessary to clear out the magnesium ions blocking the chan-
nel. This third condition makes the NMDA receptor a type of ¯c + KD
coincidence detector for membrane depolarization and syn-
aptic transmission, playing an important role in memory and with KD = k− / k+ as the ligand dissociation constant. The rates
learning. Interestingly, AMPA receptors diffuse on the of binding and unbinding are related to the ͑negative͒ free-
postsynaptic surface unusually fast, especially when ligand is
bound ͓15͔. In addition to refreshing the synaptic plasticity energy F of binding through the detailed balance
͓16͔, such diffusion may also increase the accuracy of sens-
ing in neurotransmission. k+¯c = eF/T, ͑3͒
k−
This paper is organized as follows: in Sec. II we review
the case of a single immobile receptor; in Sec. III we con- with T as the temperature in energy units. In the limit of very
sider receptor diffusion on the membrane and derive the dy- fast ligand diffusion, i.e. when a ligand molecule is immedi-
namical equations determining the spatial concentration of ately removed from the receptor after unbinding, the receptor
ligand and the occupancy of the receptor. In Sec. IV we dynamics are effectively decoupled from the diffusion of
derive the stationary solution for the receptor occupancy, and ligand molecules and hence, diffusion does not need to be
in Sec. V, using a nonequilibrium approach based on the included explicitly.
effective temperature, we derive the accuracy of sensing. In
Sec. VI we describe how our theoretical results connect to Following Bialek and Setayeshgar ͓9͔, the accuracy of
neurotransmission in neural synapses. The final section is sensing is obtained by applying the FDT ͓17͔, which relates
devoted to an overall discussion including cases of further the spectrum of fluctuations in occupancy to the linear re-
biological relevance. Technical details are provided in Ap- sponse of the receptor occupancy to a perturbation in the
pendixes A and B. receptor-binding energy. Furthermore, at equilibrium the
fluctuations in occupancy can be directly related to the un-
II. REVIEW OF THE SINGLE RECEPTOR certainty in ligand concentration using Eq. ͑2͒. After time
averaging over a duration ␶ much larger than the correlation
In this section we review previous results for a single time of the binding and unbinding events, the normalized
immobile receptor. As depicted in Fig. 2, such a receptor can uncertainty of sensing is given by ͓9,18͔
bind and release ligand with rates k+¯c and k−, respectively.
The kinetics for the occupancy n͑t͒ of the receptor are there- ͗͑␦c͒2͘␶ = 2 ¯n͒␶ → 1 , ͑4͒
fore given by ¯c2 k+¯c͑1 − 2␲D3¯cs␶

where the right-hand side is obtained for diffusion-limited

binding ͓18͔, i.e., when k+¯c͑1 −¯n͒ → 4␲¯cD3s, with D3 as the
diffusion constant and s as the dimension of the ͑spherical͒

receptor. Equation ͑4͒ shows that the uncertainty is limited

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INCREASED ACCURACY OF LIGAND SENSING BY … PHYSICAL REVIEW E 82, 041902 ͑2010͒

by the random binding and unbinding of ligand. The accu- k 8
racy of sensing is defined as the inverse of the uncertainty. D3
D2 k+c k− D2
In the case where diffusion of ligand is slow, ligand bind-
ing to the receptor is perturbed ͓9͔. The kinetics of receptor
occupancy and ligand concentration are described by

dn͑t͒ ͑5a͒
dt = k+c͑xជ0,t͓͒1 − n͑t͔͒ − k−n͑t͒,

‫ ץ‬c͑xជ , t͒ = D3ٌ2c͑xជ , t͒ − ␦͑xជ − dn͑t͒ , ͑5b͒ x0 ∆ A
‫ץ‬t xជ0͒ dt
O

where xជ0 indicates the position of the receptor. In the last FIG. 3. ͑Color online͒ Ligand receptor binding by mobile recep-
term in the second equation, the Dirac delta function
␦͑x − xជ0͒ describes a sink or source of ligand at xជ0 corre- tors characterized by diffusion coefficient D2, light ͑green͒ arrows.
sponding to ligand binding or unbinding from the receptor, Ligand ͑small disks͒ is concentrated only in a small area ⌬A around

respectively. xជ0 due to, e.g., local release of glutamate from vesicle fusion pore
with rate kϱ. Ligand can bind to receptors at xជ0 but unbind every-
Following a similar procedure as in the previous case, the where on the membrane. Parameter D3 describes the diffusion co-
uncertainty of sensing is given by ͓9,18͔ efficient of the ligand.

͗͑␦c͒2͘␶ = 2 ¯n͒␶ + 1 ͑6a͒
¯c2 k+¯c͑1 − ␲D3¯cs␶
We consider a situation in which receptor diffusion has
3 ͑6b͒
→, reached a stationary condition, resulting in a uniform density
␳0 of receptors on the membrane, and this condition is not
2␲D3¯cs␶ changed by interaction with ligand molecules. Ligand bind-

where the first term on the right-hand side of Eq. ͑6a͒ is the ing and unbinding only affect the number of occupied and
same as in Eq. ͑4͒, while the second term is the increase in unoccupied receptors with respective densities ␳b and ␳u,
uncertainty due to diffusion. This term accounts for the ad- given by
ditional measurement uncertainty from rebinding of previ-
ously bound ligand to the receptor. For diffusion-limited ␳0 = ␳b͑rជ͒ + ␳u͑rជ͒, ͑8͒
binding, one obtains Eq. ͑6b͒ ͓18͔.
where rជ = ͑x , y͒ is a position on the 2D membrane surface.
Comparison of Eqs. ͑4͒ and ͑6͒ shows that removal of For purposes of comparison with the single immobile recep-
previously bound ligand by fast diffusion increases the accu-
racy of sensing since the same ligand molecule is never mea- tor model, we also assume that there is always exactly one
sured more than once. receptor in the area ⌬A around xជ0 = ͑rជ0 , z0͒, i.e., ␳0 = 1 / ⌬.

The uncertainty in Eq. ͑4͒ can be further reduced by a The dynamics of such a system is captured by the follow-
factor of 2, consequently increasing the accuracy of sensing
by the same factor, by considering the fundamental limit. In ing set of coupled differential equations:
this limit, ligand particles are absorbed, and hence only bind-
ing noise contributes. For diffusion-limited binding, the fun- ‫␳ ץ‬b͑rជ, t͒ = D2ٌ2␳b͑rជ, t͒ − k−␳b͑rជ, t͒
damental limit is given by ͓8,10,11͔ ‫ץ‬t

͗͑␦c͒2͘␶ 1 + k+͓␳0 − ␳b͑rជ,t͔͒c͑rជ,z0,t͒⌬A␦͑rជ − rជ0͒, ͑9a͒
¯c2 4␲D3¯cs␶
= , ͑7͒

ͫ ͬ‫ץ‬c͑xជ,t͒
‫ץ‬t
calculated from the diffusive flux to a small sphere of radius = D3ٌ2c͑xជ , t͒ − ␦͑z − z0͒ ‫␳ ץ‬b͑rជ, t͒ − D2ٌ2␳b͑rជ, t͒
s, which represents the receptor. ‫ץ‬t

+ kϱ␦͑xជ − xជϱ͒, ͑9b͒

III. RECEPTOR DIFFUSION MODEL where rate constant kϱ ensures the replenishment of ligand
molecules, and xជ = ͑x , y , z͒ = ͑rជ , z͒ indicates a location in
In this model, depicted in Fig. 3, receptors are free to three-dimensional ͑3D͒ space. Specifically, ␳b͑rជ , t͒ is the
diffuse on the two-dimensional ͑2D͒ membrane surface with density of bound receptors and c͑xជ , t͒ is the ligand concen-
diffusion coefficient D2. A receptor located in the limited
area ⌬A centered at xជ0 can bind and unbind ligand of average tration, both depending on space and time. An additional
concentration ¯c with rates k+¯c and k−, respectively. When a
receptor leaves this area and diffuses to the remaining large equation for ␳u͑rជ , t͒ is not necessary due to receptor conser-
region of negligible ligand concentration, it can either release vation in Eq. ͑8͒.
ligand or diffuse back in the area.
After normalization by the average density, using

n͑rជ , t͒ = ␳b͑rជ , t͒ / ␳0 in Eq. ͑8͒, we obtain

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GERARDO AQUINO AND ROBERT G. ENDRES PHYSICAL REVIEW E 82, 041902 ͑2010͒

‫ ץ‬n͑rជ, t͒ = D2ٌ2n͑rជ, t͒ + k+͓1 − n͑rជ, t͔͒c͑rជ, z = z0, t͒⌬A␦͑rជ − rជ0͒
‫ץ‬t

− k−n͑rជ,t͒, ͑10a͒ n

‫ ץ‬c͑xជ , t͒

ͫ ͬ‫ץ‬t
= D3ٌ2c͑xជ , t͒ − ␳0␦͑z − z0͒ ‫ ץ‬n͑rជ, t͒ − D2ٌ2n͑rជ, t͒
‫ץ‬t
AMPAR
+ kϱ␦͑xជ − xជϱ͒. ͑10b͒

6

These equations describe the coupled dynamics of the dif- 0.4 k ms 1
fusing receptors and their occupancy with ligand. In sum- 13
mary, receptors can bind ligand molecules only within area D2 Μm2 s 1 0.2
⌬A but can release them anywhere on the membrane. The n1 n
solution is provided in the following sections. 0.1 0 NMDAR

IV. NONEQUILIBRIUM STATIONARY SOLUTION FIG. 4. ͑Color online͒ Receptor occupancy at ជr0. Eq. ͑13͒ for ¯n
FOR RECEPTOR OCCUPANCY is plotted as a function of lateral diffusion coefficient D2 and un-
binding rate k− for a range of experimentally relevant values in
It is possible to extract an exact analytical expression for neurotransmission ͑top surface͒. Specifically, the k− values for
the stationary solution ¯n͑rជ͒ for the receptor occupancy from AMPA and NMDA receptors are indicated ͑see Table I͒. Corre-
Eq. ͑10a͒. Setting the right-hand side to zero and Fourier
sponding plot for ¯n͑1 −¯n͒ is also shown ͑bottom surface͒.
transforming both sides lead to

¯nˆ ͑qជ͒ = k+͓1 − ¯n͑rជ0͔͒¯c͑xជ0͒⌬A , ͑11͒
D2q2 + k−
and the last equality results from the stationary condition for
where qជ is the wave vector of magnitude q = ͉qជ͉. The inverse Eq. ͑10a͒. Carrying out the last integration leads to the fol-
lowing final expression for the effective internalization rate:
Fourier transform, upon introducing a cutoff ⌳ ϳ 1 / s due to

the receptor size, leads to the following solution: ͩ ͪ ͩ ͪkei = k+¯c
1 − ¯n 4␲D2
ͩ ͪ⌬A ¯n − k− = 1 + 4␲D2 − k−, ͑16͒

¯n͑rជ͒ = k+͓1 − ¯n͑rជ0͔͒¯c͑xជ0͒ 4␲D2 ln
1 + D2⌳2 . ͑12͒ ⌬A ln
k−
⌬Ak−

This stationary solution corresponds to a nonequilibrium where we used Eq. ͑13͒ to express ¯n in terms of the receptor
diffusion coefficient. Using condition ⌬A = 4␲ / ⌳2 we can
condition since ligand is continuously taken away from area
⌬A. Setting ⌬A = 4␲ / ⌳2 ϳ s2 and evaluating Eq. ͑12͒ in rជ0 define a modified unbinding rate
lead therefore to the following value for the occupancy at rជ0:

1 ͑13͒ kei k+¯c͑1 − ¯n͒ D2⌳2
ͫ ͩ ͪͬ¯n͑rជ0͒ = −1 , ¯n 1 + D2⌳2
1+ ͩ ͪ␬−=k− + = = ͑17͒
⌬A 1 + 4␲D2
k+¯c͑xជ0͒ 4␲D2 ln ⌬Ak− ln k−

which is plotted in Fig. 4 and, in the limit D2 → 0, leads to and extend equilibrium condition ͑3͒ to the stationary non-
the recovery of immobile receptor solution ͑2͒, namely,
equilibrium condition via an effective temperature Te defined
¯n0 = k+¯c / ͑k− + k+¯c͒. by ͓12͔
In order to quantify the deviation from equilibrium we
k+¯c = eF/Te. ͑18͒
apply Fick’s first law to calculate the flux Jជ of occupied ␬−

receptors,

Jជ = D2ٌជ ¯n͑rជ,t͒, ͑14͒

out of region ⌬A ͓19͔. Integrating Eq. ͑14͒ over the border of This effective temperature allows us to generalize the ordi-
⌬A allows us to define an effective “internalization rate” for nary FDT to a stationary nonequilibrium condition and to
removal of ligand in analogy to ͓12͔ via derive the receptor accuracy of sensing in Sec. V.

Ͷ ͵kei¯n D2 D2 ٌ2¯n͑rជ͒d2rជ V. ACCURACY OF SENSING
= − ⌬A ٌ¯n͑rជ͒drជ = − ⌬A
⌬A Here, we derive the accuracy of sensing for diffusing re-
ceptors. Starting from Eq. ͑10͒ we consider an expansion to
͵1 d2rជ͓k+¯c͑1 − ¯n͒⌬A␦͑rជ͒ − k−¯n͔, ͑15͒ first order around the stationary solutions

= ⌬A
⌬A

where the second equality follows from integration by parts n͑rជ,t͒ = ¯n + ␦n͑rជ,t͒, ͑19a͒

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INCREASED ACCURACY OF LIGAND SENSING BY … PHYSICAL REVIEW E 82, 041902 ͑2010͒

c͑xជ,t͒ = ¯c + ␦c͑xជ,t͒, ͑19b͒ ͵␦nˆ͑rជ0,␻͒ = dqជ e−iqជ·rជ0␦nˆ ͑qជ, ␻͒
͑2␲͒2
where ¯n and ¯c are the stationary solutions of Eq. ͑10͒, which,
in this case, have a spatial dependence, i.e., ¯n =¯n͑rជ͒ and ͵= G͑␻,rជ0,z0͒ dqជ 1
¯c =¯c͑xជ͒. Hence, the linearized equation for the receptor occu- ͑2␲͒2 D2q2 + k− − i␻
pancy and ligand concentration are given by
= G͑␻,rជ0,z0͒⌺1͑␻͒, ͑25͒
ͫ‫␦͓ץ‬n͑rជ,t͔͒
‫ץ‬t
= D2ٌ2␦n͑rជ, t͒ + ͑1 − ¯n͒␦c͑rជ,z0,t͒ − ¯c␦n͑rជ,t͒ where we defined

ͬ+ ͑1 − ¯n͒¯c ␦k+͑t͒ − ¯n ␦k−͑t͒ k+⌬A␦͑rជ − rជ0͒ ͵⌺1͑␻͒ = dqជ 1 ͑26͒
k+ k+ ͑2␲͒2 D2q2 + k− − i␻

− k−␦n͑rជ,t͒, ͑20a͒ with the integral provided in Appendix A. In order to remove
the ligand concentration in Eq. ͑24͒, we Fourier transform
Eq. ͑20b͒, leading to

‫␦͓ ץ‬c͑xជ , t͔͒ = D3ٌ2␦c͑xជ , t͒ ␦cˆ ͑qជ , ␻͒ = ␳0 i␻ − D2q2 ␦nˆ ͑qជ, ␻͒eiqЌz0 . ͑27͒
‫ץ‬t D3͑q2 + qЌ2 ͒ −
ͭ ͮ− ␦͑z − z0͒␳0 i␻

‫␦͓ ץ‬n͑rជ, t͔͒ − D2ٌ2␦n͑rជ, t͒ , Inserting Eq. ͑23͒ in Eq. ͑27͒ and inverse Fourier transform-
‫ץ‬t ing lead to the following expression for the spectrum of the
fluctuations in ligand concentration:
͑20b͒

͵ ͵␦cˆ͑rជ0,z0,␻͒ = dqជ
where we assumed that fluctuations in the binding or unbind- dqЌ e−iqЌz0 ͑2␲͒2 e−iqជ ·rជ0␦cˆ ͑qជ , ␻͒
ing rate constants occur as well. This mathematical trick al- 2␲
lows us to introduce fluctuations in the receptor-binding free
energy and so to apply the FDT ͓9,17͔. Using stationary = G͑␻,rជ0,z0͒␳0⌺2͑␻͒, ͑28͒
nonequilibrium condition ͑18͒, we obtain the fluctuations in
the rates and binding free energy, with

͵ ͵⌺2͑␻͒ = dqЌ dqជ i␻/D2 − q2 e−iqជ ·rជ0
2␲
␦k+ − ␦k− = ␦F . ͑21͒ ͑2␲͒2 D3͑q2 + qЌ2 ͒ − i␻ q2 + k− − i␻
k+ ␬− Te D2 D2

Using this equation allows us to replace fluctuations in the ͑29͒
rate constants in Eq. ͑20a͒ with fluctuations in the binding
free energy, resulting in provided in Appendix B. Inserting Eqs. ͑25͒ and ͑28͒ in Eq.
͑24͒ leads to a closed equation for the function G͑␻ , rជ0 , z0͒,
‫␦͓ ץ‬n͑rជ, t͔͒ ␦F
k+¯c͑1 − ¯n͒ Te ͫG͑␻,rជ0,z0͒ = k+͑1 − ¯n͒␳0⌺2͑␻͒G͑␻,rជ0,z0͒ +
ͫ‫ץ‬t
= D2ٌ2␦n͑rជ, t͒ − k−␦n͑rជ, t͒ +

ͬ+ k+͑1 − ¯n͒␦c͑rជ,z0,t͒ − k+¯c␦n͑rជ,t͒ ⌬A␦͑rជ − rជ0͒. ͬ− ␦Fˆ ͑␻͒
k−¯c⌺1͑␻͒G͑␻, rជ0, z0͒ + k+¯c͑1 − ¯n͒ Te ⌬A,

͑22͒ ͑30͒

Fourier transforming Eq. ͑22͒ and setting q = ͉qជ͉, we solve which can be solved for G͑␻ , rជ0 , z0͒. This leads to
for ␦nˆ ͑qជ , ␻͒ and obtain

␦nˆ ͑qជ, ␻͒ = G͑␻, rជ0, z0͒ eiqជ ·rជ0 , ͑23͒ G͑␻, rជ0, z0͒ = 1 − k+¯c͑1 − ¯n͒⌬A␦Fˆ ͑␻͒/Te , ͑31͒
D2q2 + k− k+͑1 − ¯n͒⌺2͑␻͒ + k+¯c⌺1͑␻͒⌬A
− i␻

where, for convenience of calculation, we have defined the where condition ␳0 = 1 / ⌬A was applied. Using Eqs. ͑25͒ and
following function: ͑31͒, a final expression for the response is obtained,

ͫG͑␻,rជ0,z0͒ = k+͑1 − ¯n͒␦cˆ͑rជ0,z0,␻͒ − k+¯c␦nˆ ͑rជ0,␻͒ ␦nˆ ͑rជ0,␻͒ = G͑␻,rជ0,z0͒⌺1͑␻͒
␦Fˆ ͑␻͒ ␦Fˆ ͑␻͒

ͬ␦F͑␻͒ 1 k+¯c͑1 − ¯n͒⌬A⌺1͑␻͒ . ͑32͒
=
+ ͑1 − ¯n͒¯c ⌬A. ͑24͒ Te 1 − k+͑1 − ¯n͒⌺2͑␻͒ + k+¯c⌺1͑␻͒⌬A
Te
Applying the generalized FDT we derive the spectrum of
Inverse Fourier transforming Eq. ͑23͒ leads to the following
expression for the spectrum of the fluctuations in occupancy: the fluctuations in receptor occupancy ␦nˆ ͑rជ0 , ␻͒ from the re-
sponse, namely,

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GERARDO AQUINO AND ROBERT G. ENDRES PHYSICAL REVIEW E 82, 041902 ͑2010͒

ͫ ͬS͑␻͒ = 2kTeIm ␦nˆ͑rជ0,␻͒ . ͑33͒ a AMPAR
␻ ␦Fˆ ͑␻͒
4
Occupancy fluctuations
By replacing the expressions for ⌺1͑␻͒ and ⌺2͑␻͒ in Eq. 2
͑32͒, taking the imaginary part, and setting ␻ Ӎ 0, we obtain (units of 10-5τ)

for the zero-frequency limit of the power spectrum,

k+¯c͑1 − ¯n͒⌬A⌳2 Exact solution
S␩͑0͒ = Eff. intern. rate
2
ͫ ͬ2␲k−2͑1 + ␩͒ 1 100
1 + k+¯c L␩ + k+͑1 − ¯n͒⌳ ͑1 − A␩͒ 0 0.01
k− 8␲D3 NMDAR
ͫ ͩϫ b1 Exact solution
1 + k+͑1 − ¯n͒⌳ D3⌳2 − ␩k− Eff. intern. rate
16␲D3 1 + D3⌳2/͑1 + ␩͒ A␩

L␩ͪ ͬ+ 1 − A␩, ͑34͒ Occupancy fluctuations .5
L␩/2
(units of 10-3τ)
where we conveniently defined the following functions of
the dimensionless parameter ␩ = D2⌳2 / k−:

ln͑1 + ␩͒ arctanͱ␩ ͑35͒ 0 0.01
L␩ ϵ ␩ , A␩ ϵ ͱ␩ . 1 100 10000
Diffusion strength η

Note that in Eq. ͑34͒, occupancy ¯n also depends on ␩ via Eq. FIG. 5. Receptor occupancy fluctuations ͗͑␦n͒2͘␶ in units of ␶ as
͑13͒. The zero-frequency spectrum is related to the receptor a function of ␩ = 16D2 / ͑s2k−͒ for ͑a͒ AMPAR and ͑b͒ NMDAR.
Shown are ͑solid line͒ plot of ͗͑␦n͒2͘␶ as given by Eqs. ͑34͒ and
occupancy fluctuations, averaged over a time ͑36͒, ͑dashed line͒ plot of Eq. ͑37͒ using the effective internaliza-
␶ ӷ k−−1 , ͑k+¯c͒−1 by the following relation: tion rate ͓Eq. ͑16͔͒, and ͑vertical thin line͒ experimental value of ␩
for each receptor type ͑see Table I for all the parameter values͒.
͗͑␦n͒2͘␶ Ӎ S␩͑0͒/␶. ͑36͒ Glutamate concentration was set to ¯c = 0.1 mM, here and in Fig. 6.

In Fig. 5 we plot the occupancy fluctuations, given by Eq. In order to produce a significant effect from ligand rebinding, we
͑36͒ as a function of ␩, i.e., the effective diffusion strength. used D3 = 0.1 ␮m2 / s here and in subsequent plots, as may result
This is done for the two main receptors involved in neu- from binding of glutamate to neuroligins, neurexins, and other cleft
rotransmission, namely, AMPAR and NMDAR, with param-
eters taken from Table I. For AMPAR, the effect of diffusion proteins.
is that of decreasing the fluctuations in occupancy due to
reduced overcounting of previously bound molecules D2 → 0, i.e., ␩ → 0. Taking the limit and using ⌬A = 4␲ / ⌳2,
͓Fig. 5͑a͔͒. For small unbinding rates k− though, such as
occurs for the NMDA receptor, an actual increase in the we indeed obtain
noise is observed for a range of physiologically relevant dif-
fusion strengths. ͫ ͬlim

In order to make the origin of this effect more transparent, ␩→0
we also plot the occupancy fluctuations in Fig. 5 as calcu- S␩͑0͒ = 2k+¯c͑1 − ¯n0͒ 1 + k+͑1 − ¯n0͒⌳ , ͑38͒
lated for the case of receptor internalization ͓12͔ ͑k− + k+¯c͒2 8␲D3

which is identical to the result for the single receptor for
⌳ = 4 / s.

͗͑␦n͒2͘␶ Ӎ 2¯n2͑1 − ¯n͒ , ͑37͒ TABLE I. Summary of experimentally determined parameters
k+¯c␶ for glutamate ligand ͑top͒, AMPA receptor ͑bottom left͒, and
NMDA receptor ͑bottom right͒.

where ¯n = k+¯c / ͑k+¯c + k− + kei ͒, with kei defined in Eq. ͑16͒. The Glutamate
experimental unbinding rate of NMDAR leads to values for
D3 300 ␮m2 / s ͑aqueous solution͒ ͓20͔
¯n such that starting from D2 = 0, the function ¯n͑1 −¯n͒ in- 10 ␮m2 / s ͑synaptic cleft͒ ͓21,22͔
creases, reaches a maximum, and then decreases again. Such
¯c 0.1–1 mM ͓20,23,24͔
a maximum is avoided by AMPAR whose spectrum mono- ␶ 1 ms ͓25͔
s 8 nm ͓14,26͔
tonically decreases when D2 increases ͑see Fig. 4͒. The com-
parison between receptor diffusion and receptor internaliza-

tion in Fig. 5 shows qualitatively similar results, confirming

the effective ligand removal by receptor diffusion. AMPAR NMDAR

A. Equilibrium limit for vanishing D2 D2 0.028– 0.1 ␮m2 / s ͓27,28͔ 0.021 ␮m2 / s ͓27,28͔

Here, we show that Eq. ͑34͒ leads to the expression for k+ ͑4 – 10͒ ϫ 106 M−1 s−1 ͓20,29͔ ͑5 – 8.4͒ ϫ 106 M−1 s−1 ͓29,30͔
the single receptor obtained in Refs. ͓9,18͔ in the limit for
k− ͑2 – 8͒ ϫ 103 s−1 ͓20,29͔ 5 – 80 s−1 ͓29,30͔

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INCREASED ACCURACY OF LIGAND SENSING BY … PHYSICAL REVIEW E 82, 041902 ͑2010͒

B. Near-equilibrium result for small D2 a3 AMPAR

We derive here the exact result ͓Eq. ͑34͔͒ in the limit of 2
slow receptor diffusion on the membrane using ␩ Uncertainty
= D2 / ͑s2k−͒ as a small parameter. Specifically, we are inter- 1 Exact solution
ested in the accuracy of sensing, which is derived from the (units of 10-3τ)
normalized variance for the ligand concentration of ligand as 1st term only
in ͓9͔
Linear approx.
͗͑␦c͒2͘␶ ͗͑␦n͒2͘␶ 1 S␩͑0͒
¯c2 ¯n2͑1 − ¯n͒2 ¯n2͑1 − ␶ Fund. limit

= Ӎ ¯n͒2 , ͑39͒ 0
0.001
0.01 0.1

where the right-hand term follows from Eq. ͑36͒. Expansion b3 NMDAR
to first order in parameter ␩ leads to

͗͑␦c͒2͘␶ = 2 ͑1 − 8¯n0␩͒ + 1 Uncertainty 2
¯c2 k+¯c͑1 − ¯n0͒␶ ␲sD3¯c␶
(units of 10-2τ)
ͫ ͩ ͪ ͬϫ 1 −
16k−2¯n20 + 8¯n0 3k− − k+¯c + k− s2 ␩ . 1
3␲k+¯c2D3s 3k+¯c 2D3
Exact solution
͑40͒ 1st term only

Comparison with Eq. ͑6a͒ shows that the correction due to Linear approx. 1
lateral receptor diffusion is twofold to first order: ͑i͒ the first
term is reduced due to a reduction in ¯n and ͑ii͒ the second Fund. limit
term is reduced for most parameter values due to a reduction
0
in rebinding of already measured ligand molecules.
0.01
In Fig. 6 we plot the uncertainty in sensing for both AM-
Diffusion strength η
PAR and NMDAR. In both cases the total effect of lateral
FIG. 6. Uncertainty ͗͑␦c͒2͘␶ /¯c2 as a function of
diffusion is that of decreasing the uncertainty in the concen- ␩ = 16D2 / ͑s2k−͒ for ͑a͒ the AMPA receptor and ͑b͒ the NMDA re-
ceptor in units of ␶. Shown are ͑solid line͒ plot of Eq. ͑39͒ as
tration and therefore of increasing the accuracy of sensing. obtained from the exact solution ͓Eq. ͑34͔͒, ͑dashed line͒ linear
Receptor diffusion for AMPAR ͓Fig. 6͑a͔͒ mainly reduces approximation for small ␩, and ͑dotted line͒ contribution to the
the rebinding term, while for NMDAR ͓Fig. 6͑b͔͒ the first accuracy of sensing from only the first term in Eq. ͑40͒. Also shown
term from binding and unbinding is reduced. In the latter are ͑vertical thin line͒ the experimental value of ␩ for each receptor
case, the maximum in the occupancy fluctuations ͑see Fig. 5͒ ͑see Table I͒ and ͑horizontal dotted-dashed line͒ the fundamental
is removed through ¯n2͑1 −¯n͒2 in the denominator of Eq. ͑39͒.
For large values of ␩, the occupancy fluctuations and hence physical limit ͓Eq. ͑7͔͒.
the accuracy of sensing become unphysical ͑zero, i.e., below
the physical limit or diverge͒ as the effective temperature tified by the coefficient of variation ͑CV͒ defined as the stan-
breaks down far from equilibrium. ͑For large ␩, receptors dard deviation divided by the mean value. The reason for a
remove ligand molecules efficiently, which introduces large large CV is controversial ͓cf. Fig. 7͑a͔͒ but can be attributed
to a variation in the size of the vesicles carrying glutamate or
nonequilibrium ligand fluxes. These cannot be represented
by an effective temperature.͒ to the fact that large EPSCs are generated by multiple
vesicles ͓13,46͔. The experimental detection of EPSCs with
In summary, lateral receptor diffusion affects the accuracy an amplitude distribution of more than one peak is consistent
of sensing in two ways: ͑i͒ by reducing the stationary value with the latter explanation ͓33͔.
for the receptor occupancy and ͑ii͒ by reducing the noise
from rebinding of already measured ligand molecules. In Fig. 7͑a͒, the experimental data taken from Ref. ͓34͔
show EPSC peaks as a result of repetitive suprathreshold
VI. QUANTAL TRANSMISSION stimulation of granule cell somata in CA3 pyramidal cells.
Some stimuli did not evoke an EPSC ͑termed failures͒. The
A remarkable property of synaptic transmission is that the histogram of successful events can be fitted with three
amplitude of the synaptic response, i.e., the excitatory Gaussian distributions, showing the quantal nature of the
postsynaptic current ͑EPSC͒, varies by an integral multiple EPSC. The peaks of the Gaussians correspond to the discrete
of a quantum. This property, first unveiled by the pioneering multiple values of 7, 14, and 21 pA.
work of Katz ͓31͔, was traced to the formation of synaptic
vesicles and subsequent pore formation. The latter leads to a We can compare this distribution to the result predicted by
release of glutamate into the synaptic cleft ͓32͔. This quan- the accuracy of sensing. Specifically, we would like to know
tization is considered a way to minimize the variability of the if fast diffusion helps resolve the EPSC peaks. A simple
response from each contributing synapse and therefore to model based on the structural properties of AMPAR was pre-
allow efficient neural computation. This variability is quan- viously used to derive the dynamics of glutamate binding
and unbinding, as well as channel opening and closing ͓34͔.
Based on this model the following relation connecting the
current through the ion channel of AMPAR and the concen-
tration of glutamate was deduced:

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GERARDO AQUINO AND ROBERT G. ENDRES PHYSICAL REVIEW E 82, 041902 ͑2010͒

Histogram a Experiment Figure 7͑b͒ shows that the distribution of predicted EP-
Fit with 3 Gaussians SCs is in qualitative agreement with the experiments. From
Distribution 60 our calculations we found that the effect of lateral receptor
50 10 20 30 40 50 diffusion increases the resolution of peaks and so also the
40 EPSC (pA) ability of the synapse to count the number of vesicles re-
30 leased into the synaptic cleft. Note that in order to resolve
20 Diffusing receptors the EPSC peaks experimentally, the probability of vesicle
10 Immobile receptors release from the presynaptic side was drastically reduced.
Individual Gaussians This was achieved by modulating the ratio of Ca+ / Mg+ ions
00 in the extracellular solution, resulting in a 50– 60 % reduc-
tion of peak currents ͓34͔.
b
VII. DISCUSSION AND CONCLUSIONS
0.25
Our previous work on the fundamental physical limit of
0.2 ligand sensing ͓12͔ led us to conclude that receptor diffusion,
by removing bound ligand from a region of interest, may
0.15 reduce the local overcounting of ligand molecules, therefore
potentially increasing the local accuracy of sensing. In this
0.1 paper, we set out to investigate this possibility by construct-
ing a mathematical model that contains all the necessary in-
0.05 gredients to describe the role of receptor diffusion in ligand
sensing. We have considered a receptor that can bind and
00 10 20 30 40 50 unbind ligand molecules as well as diffuse on a 2D mem-
brane. Ligand is allowed to diffuse in 3D space. Using this
current I (pA) model we derived the fluctuations in receptor occupancy in
the area of interest via a generalization of the FDT with the
FIG. 7. Quantal transmission. ͑a͒ Histogram of experimental introduction of an effective temperature ͓12͔. This ultimately
EPSC data taken from Ref. ͓34͔ and fitted by three Gaussian distri- allowed us to derive an equation for the accuracy of sensing.
butions. ͑b͒ Distribution of predicted EPSCs. Shown are individual
We applied our model to the biologically relevant case of
Gaussian distributions corresponding to one, two, and three vesicles glutamate receptors, which are responsible for transmitting
action potentials at neural synapses. We found that for AM-
͑from left to right; thin lines͒, as well as envelope functions ͑sums PAR and NMDAR, our model shows that the lateral diffu-
with equal weighting; thick lines͒. Calculation is done for AMPA sion of receptors increases their accuracy of sensing and
receptors using Eqs. ͑41͒ and ͑42͒, Eq. ͑39͒ for the variances of hence may allow synapses to count the number of released
diffusing ͑thin solid lines͒, and Eq. ͑6a͒ for immobile ͑thin dashed vesicles. However, the local occupancy fluctuations are de-
lines͒ receptors, assuming glutamate concentration ¯cn = cb + ncv for creased only for AMPAR; for NMDAR these fluctuations are
n = 1 , 2 , 3 vesicles, background cb = 0.02 mM, vesicle concentration increased. This difference is due to the smaller unbinding
cv = 0.04 mM, and N = 100 receptors ͓44,45͔. Furthermore, Eq. ͑41͒ rate of the latter receptor. Consequently, the accuracy of
with parameters from Refs. ͓34,35͔ was used. sensing for NMDAR is one order of magnitude smaller than
for AMPAR. It is important to remark that these results are
1 ͑41͒ consistent with the different roles these receptors play. In
I = I01 + ͑␭/c͒n , fact, the transmission of an excitatory potential through a
synapse occurs in several steps. Initially, it is AMPAR which
where I0, ␭, and the Hill coefficient n are parameters pro- senses glutamate and, by letting in ions, starts a small depo-
vided in Refs. ͓34,35͔. Based on Eq. ͑41͒ the following re- larization of the membrane. This in turn allows NMDAR to
lation connecting the width in the distribution of EPSC cur- open its channel and start an even bigger influx of ions ͑es-
pecially Ca+͒. This stimulates the production of more AM-
rents to the fluctuations in glutamate concentration can be PARs and so increases the strength and sensitivity of the
synapses. Sensitivity to glutamate is important for AMPAR,
derived: while the roles of NMDAR are coincidence detection and
amplification, important for long-term potentiation and
␦I = I0n␭ ͑␭/c͒n−1 ␦c . ͑42͒ memory.
͓1 + ͑␭/c͒n͔2
In order to derive the accuracy of sensing for diffusing
We further assume that a number N of receptors indepen- receptors, we made a number of symplifying assumptions.
dently contribute to the measurement, i.e., We neglected fluctuations in the total receptor density and
vesicle size, assumed to be constant ͓13,46͔. Furthermore,
͗͑␦c͒2͘␶,N = ͗͑␦c͒2͘␶ , ͑43͒ we assumed a constant ligand concentration in the hotspot
¯c2 N¯c2 area ⌬A during the receptor measurement time, while in re-

with the uncertainty of a single receptor given by Eq. ͑39͒.
Averaging over N, therefore, further reduces the uncertainty
in Eq. ͑43͒ and hence increases the accuracy of sensing.

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INCREASED ACCURACY OF LIGAND SENSING BY … PHYSICAL REVIEW E 82, 041902 ͑2010͒

ality the concentration profile will broaden out. Additionally, Grant No. BB/G000131/1 and the Centre for Integrative Sys-
we made the simplifying assumption that multiple vesicles tems Biology at Imperial College.
are released in the same spot, so that released glutamate con-
centrations add up and that other effects such as spillover APPENDIX A: CALCULATION OF ⌺1(␻)
from neighboring cells can be neglected ͓42,43͔. Finally, we
introduced the effective temperature Te to generalize the We devote this appendix to the calculation of the integral
FDT to nonequilibrum processes ͓12,36–39͔. The approxi-
mate nature of Te, for which we neglected any potential time ⌺1͑␻͒ in Eq. ͑26͒. ⌺1͑␻͒ is given by
or frequency dependence, limits the quantitative validity of
our model to small deviations from equilibrium. In fact, for ͵ ͵⌺1͑␻͒ = ϱ dq q/D2
large receptor diffusion coefficients, the occupancy fluctua- dqជ 1/D2 i␻ = ,
tions and hence the accuracy of sensing become unphysical. ͑2␲͒2 q2 + k− −
Nevertheless, the mapping of receptor diffusion to an effec- 0 2␲ q2 + k− − i␻
tive internalization process, already studied in Ref. ͓12͔, pro- D2 D2 D2 D2
vides confidence in our method.
͑A1͒
In conclusion, in this paper we highlighted the role of
diffusion in increasing the accuracy of sensing. This might which is divergent. Hence, we introduce a cutoff ⌳ ϳ 1 / s to
be important for synaptic counting of glutamate vesicles.
This hypothesis can be tested experimentally by reducing the account for the finite dimension s of the receptor. This pro-
mobility of receptors. This can be achieved by cross linking
or addition of cholesterol. Alternatively, the glutamate diffu- cedure leads to the final result,
sion constant can be reduced by adding dextran. Such per-
turbations should lead to a broadening of the EPSC peaks, ͵ ͩ ͪ⌺1͑␻͒ =
although signaling may be affected as well. Similarly to AM- ⌳ dq q/D2 ln 1 + D2⌳2 .
PAR ͓15͔, an increase in receptor diffusion upon ligand bind- = k− − i␻
ing was recently also observed in the pseudopods of migrat- 0 2␲ q2 + k− − i␻ 4␲D2
ing Dictyostelium discoideum cells ͓40͔. This phenomenon
was attributed to signal amplification. However, since cells D2 D2
may sense locally in pseudopods, at least in shallow gradi-
ents ͓41͔, receptor diffusion may also be responsible for in- ͑A2͒
creasing the accuracy of sensing.
APPENDIX B: CALCULATION OF ⌺2(␻)
ACKNOWLEDGMENTS
In this appendix, we calculate ⌺2͑␻͒ in Eq. ͑29͒. Without
We would like to thank Aldo Faisal and Peter Jonas for loss of generality, we set ͑x0 , y0 , z0͒ = ͑0 , 0 , 0͒ and obtain
helpful comments. We acknowledge financial support from
Biotechnological and Biological Sciences Research Council ͵ ͵⌺2͑␻͒ = ϱ qdq i␻/D2 − q2
ϱ dqЌ 1/D3
0 2␲
0 2␲ ͑q2 + qЌ2 ͒ − i␻ q2 + k− − i␻
D3 D2 D2

͵ ͵ ΂ ΃= ϱ dqЌ ϱ dq q/D3
k−/D2 − 1
2␲ 2␲ i␻ q2 + k− − i␻
0 0 ͑q2 + qЌ2 ͒ − D3 D2 D2

= I1͑␻͒ − I2͑␻͒, ͑B1͒

where

͵ ͵I1͑␻͒ = ϱ qdq
ϱ dqЌ 1/D3 k−/D2
0 2␲
0 2␲ ͑q2 + qЌ2 ͒ − i␻ q2 + k− − i␻
D3 D2 D2

͵=⌳ qdq i/D3 q2 · k−/D2 ␻
0 8␲ + k− −
ͱi␻/D3 − q2 i

D2 D2
ͱ ͩ ͪ΄ ΂ͱ ͩ ͪ΃ ΂ͱ ͩ ͪ΃΅k− −i␻
= k−/͑8␲D3͒ arctan ⌳2 − i␻/D3 − arctan − i␻/D3 ͑B2͒
͑B3͒
11 k− − i␻ 11 k− − i␻ 11
− − −
D2 D2 D3 D2 D2 D3 D2 D2 D3

and

͵ ͵ ͵ ͩͱ ͱ ͪI2͑␻͒ =ϱ qdq
ϱ dqЌ 1 1 ⌳ qdq i 1 i␻ ⌳2 − i␻
0 2␲ 0 2␲ − i␻ + D3͑q2 + qЌ2 ͒ = D3 0 8␲ ͱ− q2 + i␻/D3 = 8␲D3 −−
D3 D3

using a cutoff to account for the finite size of the receptor. Taken together, the integral ⌺2͑␻͒ is given by

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GERARDO AQUINO AND ROBERT G. ENDRES PHYSICAL REVIEW E 82, 041902 ͑2010͒

Άͩͱ ͱ ͪ ͱ ͩ ͪ΄ ΂ͱ ͩ ͪ΃⌺2͑␻͒ = I1͑␻͒ − I2͑␻͒ = ⌳2 − i␻
⌳2 − i␻ − i␻ k−/D2 arctan
D3 −+
k− − i␻ 11 k− − i␻ 11
D3 − −

D2 D2 D3 D2 D2 D3

΂ͱ ͩ ͪ΃΅·−arctan − i␻ 1 ͑B4͒
.
k− − i␻ 11
− 8␲D3

D2 D2 D3

͓1͔ A. Raj and A. van Oudenaarden, Cell 135, 216 ͑2008͒. ͓23͔ F. Ventriglia and V. Di Maio, BioSystems 58, 67 ͑2000͒.
͓2͔ R. C. Yu et al., Nature ͑London͒ 456, 755 ͑2008͒. ͓24͔ P. Jonas, News Physiol. Sci. 15, 83 ͑2000͒.
͓3͔ M. S. Samoilov, G. Price, and A. P. Arkin, Sci. STKE 2006, ͓25͔ M. J. T. Fitzgerald, G. Gruener, and E. Mtui, Clinical Neu-

re17 ͑2006͒. roanatomy and Neuroscience, 5th ed. ͑Saunders Elsevier, Lon-
͓4͔ T. Gregor et al., Cell 130, 153 ͑2007͒. don, 2007͒.
͓5͔ H. Mao, P. S. Cremer, and M. D. Manson, Proc. Natl. Acad. ͓26͔ T. Nakagawa, Y. Cheng, M. Sheng, and T. Walz, Biol. Chem.
387, 179 ͑2006͒.
Sci. U.S.A. 100, 5449 ͑2003͒. ͓27͔ L. Groc et al., Nat. Neurosci. 7, 695 ͑2004͒; Proc. Natl. Acad.
͓6͔ R. A. Arkowitz, Trends Cell Biol. 9, 20 ͑1999͒. Sci. U.S.A. 103, 18769 ͑2006͒.
͓7͔ J. E. Segall, Proc. Natl. Acad. Sci. U.S.A. 90, 8332 ͑1993͒. ͓28͔ A. J. Borgdorff and D. Choquet, Nature ͑London͒ 417, 649
͓8͔ H. C. Berg and E. M. Purcell, Biophys. J. 20, 193 ͑1977͒. ͑2002͒.
͓9͔ W. Bialek and S. Setayeshgar, Proc. Natl. Acad. Sci. U.S.A. ͓29͔ D. Attwell and A. Gibb, Nat. Rev. Neurosci. 6, 841 ͑2005͒.
͓30͔ K. Erreger and S. F. Traynelis, J. Physiol. 569, 381 ͑2005͒.
102, 10040 ͑2005͒. ͓31͔ B. Katz, The Release of Neural Neurotransmitter Substances
͓10͔ R. G. Endres and N. S. Wingreen, Proc. Natl. Acad. Sci. ͑Liverpool University Press, Liverpool, 1969͒.
͓32͔ J. E. Heuser, J. Cell Biol. 81, 275 ͑1979͒.
U.S.A. 105, 15749 ͑2008͒. ͓33͔ M. J. Wall and M. M. Usowicz, Nat. Neurosci. 1, 675 ͑1998͒.
͓11͔ R. G. Endres and N. S. Wingreen, Phys. Rev. Lett. 103, ͓34͔ P. Jonas, G. Major and B. Sakmann, J. Physiol. 472, 615
͑1993͒.
158101 ͑2009͒. ͓35͔ P. Jonas and B. Sakmann, J. Physiol. 455, 143 ͑1992͒.
͓12͔ G. Aquino and R. G. Endres, Phys. Rev. E 81, 021909 ͑2010͒. ͓36͔ L. F. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E 55,
͓13͔ A. A. Faisal, L. P. J. Selen, and D. M. Wolvert, Nat. Rev. 3898 ͑1997͒.
͓37͔ Th. M. Nieuwenhuizen, Phys. Rev. Lett. 80, 5580 ͑1998͒.
Neurosci. 9, 292 ͑2008͒. ͓38͔ A. Crisanti and F. Ritort, J. Phys. A 36, R181 ͑2003͒.
͓14͔ B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and ͓39͔ L. Leuzzi, J. Non-Cryst. Solids 355, 686 ͑2009͒.
͓40͔ S. de Keijzer et al., J. Cell. Sci. 121, 1750 ͑2008͒.
Peter Walter, 5th ed., Molecular Biology of the Cell ͑Garland ͓41͔ N. Andrew and R. H. Insall, Nat. Cell Biol. 9, 193 ͑2007͒.
Publishing, Inc., New York, 2007͒. ͓42͔ J. J. Lawrence et al., J. Physiol. ͑London͒ 554, 175 ͑2003͒.
͓15͔ M. Renner et al., Curr. Opin. Neurobiol. 18, 532 ͑2008͒. ͓43͔ P. B. Sargent et al., J. Neurosci. 25, 8173 ͑2005͒.
͓16͔ M. Heine et al., Science 320, 201 ͑2008͒. ͓44͔ D. M. Kullmann, M.-Y. Min, F. Asztely, and D. A. Rusakov,
͓17͔ R. Kubo, Rep. Prog. Phys. 29, 255 ͑1966͒. Philos. Trans. R. Soc. London, Ser. B 354, 395 ͑1999͒.
͓18͔ R. G. Endres and N. S. Wingreen, Prog. Biophys. Mol. Biol. ͓45͔ J. R. Cottrell et al., J. Neurophysiol. 84, 1573 ͑2000͒.
100, 33 ͑2009͒. ͓46͔ K. M. Franks, C. F. Stevens, and T. K. Sejnowski, J. Neurosci.
͓19͔ H. C. Berg, Random Walks in Biology ͑Princeton University 23, 3186 ͑2003͒.
Press, New Jersey, 1993͒.
͓20͔ M. Postlethwaite et al., J. Physiol. 579, 69 ͑2007͒.
͓21͔ T. A. Nielsen, D. A. Di Gregorio, and R. A. Silver, Neuron 42,
757 ͑2004͒; E. É. Saftenku, J. Theor. Biol. 234, 363 ͑2005͒.
͓22͔ D. Choquet and A. Triller, Nat. Rev. Neurosci. 4, 251 ͑2003͒;
D. Holcman and A. Triller, Biophys. J. 91, 2405 ͑2006͒.

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