Increased accuracy of ligand sensing by receptor diffusion on cell surface
aa r X i v : . [ q - b i o . S C ] A ug Increased accuracy of ligand sensing by receptor diffusion on cell surface
Gerardo Aquino and Robert G. Endres
Division of Molecular Biosciences and Centre for Integrated Systems Biology at Imperial College,Imperial College London, SW7 2AZ, London, UK (Dated: September 30, 2018)The physical limit with which a cell senses external ligand concentration corresponds to the perfectabsorber, where all ligand particles are absorbed and overcounting of same ligand particles doesnot occur. Here we analyze how the lateral diffusion of receptors on the cell membrane affects theaccuracy of sensing ligand concentration. Specifically, we connect our modeling to neurotransmissionin neural synapses where the diffusion of glutamate receptors is already known to refresh synapticconnections. We find that receptor diffusion indeed increases the accuracy of sensing for both theglutamate AMPA and NDMA receptors, although the NMDA receptor is overall much noiser. Wepropose that the difference in accuracy of sensing of the two receptors can be linked to their differentroles in neurotransmission. Specifically, the high accuracy in sensing glutamate is essential for theAMPA receptor to start membrane depolarization, while the NMDA receptor is believed to work ina second stage as a coincidence detector, involved in long-term potentiation and memory.
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I. INTRODUCTION
Biological cells live in very noisy environments fromwhich they receive many different stimuli. The survivalof a cell highly depends on its ability to respond to suchstimuli and to adapt to changes in the environment. Afundamental role in this task is played by membrane re-ceptors, proteins on the cell surface which are able tobind external ligand molecules and trigger signal trans-duction pathways. Although the precision with whicha cell can measure the concentration of a specific lig-and is negatively affected by many sources of noise [1–4],several examples exist in which such measurements areperformed with surprisingly high accuracy. In bacterialchemotaxis for instance, the bacterium
Escherichia coli can respond to changes in concentration as low as 3.2 nM[5], corresponding to only three molecules in the volumeof a cell. High accuracy is observed also in spatial sensingby single cell eukaryotic organisms as e.g. in the slimemold
Dictyostelium discoideum , which is able to sense aconcentration difference of only 1 −
5% across the celldiameter [6], and in
Saccharomyces cerevisiae (buddingyeast), which is able to orient growth in a gradient of α -pheromone mating factor down to estimated 1% receptoroccupancy difference across the cell [7]. Spatial sensingis also efficiently performed by growing neurons, lympho-cytes, neutrophils and other cells of the immune system.There has been substantial progress in our theoreticalunderstanding of the accuracy of concentration sensing.In 1977, Berg and Purcell were the first to point outthat ligand sensing is limited by ligand diffusion, pro-ducing a “counting noise” at the receptors [8]. Bialekand Setayeshgar [9] (and later others [10]) used the Fluc-tuation Dissipation Theorem (FDT) to separate the con-tributions from random binding/unbinding events fromrebinding due to diffusion of ligand molecules. Subse-quently, Wingreen and Endres showed that the perfect dendrite axon FIG. 1: (Color online) Synaptic junction during neurotrans-mission. A vesicle containing a neurotransmitter (e.g. glu-tamate for excitatory synapses), releases its content into thesynaptic cleft. Receptors on the post-synaptic surface bindthe neurotransmitter, increasing the likelihood of the propa-gation of the electric impulse (action potential). Red arrowindicates the incoming potential, yellow arrow the set of reac-tions leading to propagation of potential, small green arrowsindicate receptor mobility. absorber is the absolute fundamental limit of ligand sens-ing accuracy; it does not suffer from ligand unbinding andrebinding noise [10, 11].In previous work we analyzed the contribution of endo-cytosis, i.e. the internalization of cell-surface receptors,to the accuracy of external ligand concentration sensing[12]. We showed with a simple model that internalization,by making the cell act as an absorber of ligand, increasessensing accuracy by reducing the noise from rebindingof already measured ligand molecules. In this paper weconsider the effect of lateral diffusion of receptors on theaccuracy of ligand concentration sensing.In a situation in which ligand sensing is concentratedin a specific region on the cell membrane (signalinghotspot), receptors bind ligand molecules locally but mayrelease them remotely from the region of interest, thuspreventing those particles from rebinding locally. Noisefrom overcounting could therefore be reduced, potentiallyincreasing the accuracy of sensing.A potential implementation of such localized sensingoccurs in neural synapses, responsible for transmissionof action potentials and short-term synaptic plasticity(see Fig. 1 for a schematic description). At such crit-ical places in the central nervous system, the accuracyof sensing ought to be important [13]. At a synapse thepropagation of an action potential arriving from the pre-synaptic neuron causes the opening of ion channels onthe pre-synaptic membrane which increases calcium-ioninflow [14]. Calcium ions trigger a biochemical cascadewhich results in the formation of vesicles containing neu-rotransmitter, e.g. glutamate, in the case of excitatorysynapses. One or more vesicles eventually fuse with themembrane at a point on the pre-synaptic surface to forma fusion pore. Glutamate is subsequently released anddiffuses into the synaptic cleft, reaching the receptors onthe post-synaptic surface on the opposite side of the cleft.These receptors, mainly the AMPA and NMDA receptors(named after the agonists α -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid and N-methyl-D-aspartic acidrespectively), cause ion channels on the post-synapticmembrane to open, facilitating the propagation of theaction potential.The AMPA receptor (AMPAR) has a tetrameric struc-ture with four binding sites for glutamate. Binding glu-tamate on two sites leads to conformational change andopening of a pore. More occupied binding sites imply ahigher current through the channel, mainly of calcium,sodium and potassium ions. The NMDA receptor (NM-DAR) has a tetrameric structure as well, but its chan-nel needs coincidental binding of glutamate and glycinemolecules on dedicated sites for it to open. A small mem-brane depolarization is furthermore necessary to clear outthe magnesium ions blocking the channel. This thirdcondition makes the NMDA receptor a type of coinci-dence detector for membrane depolarization and synap-tic transmission, playing an important role in memoryand learning. Interestingly, AMPA receptors diffuse onthe post-synaptic surface unusually fast, especially whenligand is bound [15]. In addition to refreshing the synap-tic plasticity [16], such diffusion may also increase theaccuracy of sensing in neurotransmission.This paper is organized as follows: in Sec. II we re-view the case of a single immobile receptor; in Sec. III weconsider receptor diffusion on the membrane and derivethe dynamical equations determining the spatial concen-tration of ligand and the occupancy of the receptor. InSec. IV we derive the stationary solution for the receptoroccupancy, and in Sec. V, using a non-equilibrium ap- proach based on the effective temperature, we derive theaccuracy of sensing. In Section VI we describe how ourtheoretical results connect to neurotransmission in neu-ral synapses. The final section is devoted to an overalldiscussion including cases of further biological relevance.Technical details are provided in two appendices. II. REVIEW OF THE SINGLE RECEPTOR
In this section we review previous results for a single,immobile receptor. As depicted in Fig. 2, such a recep-tor can bind and release ligand with rates k + ¯ c and k − ,respectively. The kinetics for the occupancy n ( t ) of thereceptor are therefore given by dn ( t ) dt = k + ¯ c [1 − n ( t )] − k − n ( t ) , (1)where the concentration of ligand, ¯ c , is assumed uniformand constant. The steady-state solution for the receptoroccupancy is given by¯ n = ¯ c ¯ c + K D (2)with K D = k − /k + the ligand dissociation constant. Therates of binding and unbinding are related to the (nega-tive) free energy F of binding through the detailed bal-ance k + ¯ ck − = e FT (3)with T the temperature in energy units. In the limit ofvery fast ligand diffusion, i.e. when a ligand molecule isimmediately removed from the receptor after unbinding,the receptor dynamics are effectively decoupled from thediffusion of ligand molecules and hence, diffusion doesnot need to be included explicitly.Following Bialek and Setayeshgar [9], the accuracy ofsensing is obtained by applying the FDT [17], which re-lates the spectrum of fluctuations in occupancy to thelinear response of the receptor occupancy to a pertur-bation in the receptor binding energy. Furthermore, atequilibrium the fluctuations in occupancy can be directlyrelated to the uncertainty in ligand concentration usingEq. (2). After time-averaging over a duration τ muchlarger than the correlation time of the binding and un-binding events, the normalized uncertainty of sensing isgiven by [9, 18] h ( δc ) i τ ¯ c = 2 k + ¯ c (1 − ¯ n ) τ → πD ¯ csτ , (4)where the right-hand side is obtained for diffusion-limitedbinding [18], i.e. when k + ¯ c (1 − ¯ n ) → π ¯ cD s , with D the diffusion constant and s the dimension of the (spher-ical) receptor. Eq. (4) shows that the uncertainty islimited by the random binding and unbinding of ligand. k c + k − x O FIG. 2: (Color online) Single receptor, immobile at position ~x , binds and unbinds ligand with rates k + ¯ c and k − , respec-tively. The accuracy of sensing is defined as the inverse of theuncertainty.In the case where diffusion of ligand is slow, ligandbinding to the receptor is perturbed [9]. The kineticsof receptor occupancy and ligand concentration are de-scribed by dn ( t ) dt = k + c ( ~x , t )[1 − n ( t )] − k − n ( t ) (5a) ∂c ( ~x, t ) ∂t = D ∇ c ( ~x, t ) − δ ( ~x − ~x ) dn ( t ) dt , (5b)where ~x indicates the position of the receptor. In thelast term in the second equation, the Dirac delta func-tion δ ( x − ~x ) describes a sink or source of ligand at ~x corresponding to ligand binding or unbinding from thereceptor, respectively.Following a similar procedure as in the previous case,the uncertainty of sensing is given by [9, 18] h ( δc ) i τ ¯ c = 2 k + ¯ c (1 − ¯ n ) τ + 1 πD ¯ csτ (6a) → πD ¯ csτ , (6b)where the first term on the right-hand side of Eq. (6a)is the same as in Eq. (4), while the second term is theincrease in uncertainty due to diffusion. This term ac-counts for the additional measurement uncertainty fromrebinding of previously bound ligand to the receptor. Fordiffusion-limited binding, one obtains Eq. (6b) [18].Comparison of Eqs. (4) and (6) shows that removal ofpreviously bound ligand by fast diffusion increases theaccuracy of sensing, since the same ligand molecule isnever measured more than once.The uncertainty in Eq. (4) can be further reducedby a factor of two, consequently increasing the accuracyof sensing by the same factor, by considering the fun-damental limit. In this limit, ligand particles are ab-sorbed, and hence only binding noise contributes. Fordiffusion-limited binding, the fundamental limit is givenby [8, 10, 11] h ( δc ) i τ ¯ c = 14 πD ¯ csτ , (7) D (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) D D x ∆ A k c + O k k − FIG. 3: (Color online) Ligand-receptor binding by mobile re-ceptors characterized by diffusion coefficient D (green ar-rows). Ligand (red dots) is concentrated only in a small area∆ A around ~x due to e.g. local release of glutamate from vesi-cle fusion pore with rate k ∞ . Ligand can bind to receptors at ~x , but unbind everywhere on the membrane. Parameter D describes the diffusion coefficient of the ligand. calculated from the diffusive flux to a small sphere ofradius s , which represents the receptor. III. RECEPTOR DIFFUSION MODEL
In this model, depicted in Fig. 3, receptors are freeto diffuse on the 2-dimensional (2D) membrane surfacewith diffusion coefficient D . A receptor located in thelimited area ∆ A centered at ~x , can bind and unbindligand of average concentration ¯ c with rates k + ¯ c and k − ,respectively. When a receptor leaves this area and dif-fuses to the remaining large region of negligible ligandconcentration, it can either release ligand or diffuse backin the area.We consider a situation in which receptor diffusion hasreached a stationary condition resulting in a uniform den-sity ρ of receptors on the membrane, and this conditionis not changed by interaction with ligand molecules. Lig-and binding and unbinding only affects the number ofoccupied and unoccupied receptors with respective den-sities ρ b and ρ u , given by ρ = ρ b ( ~r ) + ρ u ( ~r ) , (8)where ~r = ( x, y ) is a position on the 2D membrane sur-face. For purposes of comparison with the single immo-bile receptor model, we also assume that there is alwaysexactly one receptor in the area ∆ A around ~x = ( ~r , z ),i.e. ρ = 1 / ∆.The dynamics of such a system is captured by the fol-lowing set of coupled differential equations ∂ρ b ( ~r, t ) ∂t = D ∇ ρ b ( ~r, t ) − k − ρ b ( ~r, t )+ (9a) k + [ ρ − ρ b ( ~r, t )] c ( ~r, z , t ) ∆ A δ ( ~r − ~r ) ∂c ( ~x, t ) ∂t = D ∇ c ( ~x, t ) − δ ( z − z ) (cid:20) ∂ρ b ( ~r, t ) ∂t − D ∇ ρ b ( ~r, t ) (cid:21) + k ∞ δ ( ~x − ~x ∞ ) , (9b) where rate constant k ∞ ensures the replenishment of lig-and molecules, ~x = ( x, y, z ) = ( ~r, z ) indicates a locationin 3D space. Specifically, ρ b ( ~r, t ) is the density of boundreceptors and c ( ~x, t ) is the ligand concentration, both de-pending on space and time. An additional equation for ρ u ( ~r, t ) is not necessary due to receptor conservation inEq. (8).After normalization by the average density, using n ( ~r, t ) = ρ b ( ~r, t ) /ρ in Eq. (9), we obtain ∂n ( ~r, t ) ∂t = D ∇ n ( ~r, t ) + k + [1 − n ( ~r, t )] c ( ~r, z = z , t ) ∆ A δ ( ~r − ~r ) − k − n ( ~r, t ) (10a) ∂c ( ~x, t ) ∂t = D ∇ c ( ~x, t ) − ρ δ ( z − z ) (cid:20) ∂n ( ~r, t ) ∂t − D ∇ n ( ~r, t ) (cid:21) + k ∞ δ ( ~x − ~x ∞ ) . (10b)These equations describe the coupled dynamics ofthe diffusing receptors and their occupancy with ligand.In summary, receptors can bind ligand molecules onlywithin area ∆ A but can release them anywhere on themembrane. The solution is provided in the following sec-tions. IV. NON-EQUILIBRIUM STATIONARYSOLUTION FOR RECEPTOR OCCUPANCY
It is possible to extract an exact analytical expressionfor the stationary solution ¯ n ( ~r ) for the receptor occu-pancy from Eq. (10a). Setting the right-hand side tozero and Fourier Transforming both sides leads toˆ¯ n ( ~q ) = k + [1 − ¯ n ( ~r )]¯ c ( ~x )∆ AD q + k − , (11)where ~q is the wave vector of magnitude q = | ~q | . Theinverse Fourier Transform, upon introducing a cut-offΛ ∼ /s due to the receptor size, leads to the follow-ing solution¯ n ( ~r ) = k + [1 − ¯ n ( ~r )]¯ c ( ~x ) ∆ A πD log (cid:18) D Λ k − (cid:19) . (12)This stationary solution corresponds to a non-equilibrium condition since ligand is continuously takenaway from area ∆ A . Setting ∆ A = 4 π/ Λ ∼ s andevaluating Eq. (12) in ~r leads therefore to the followingvalue for the occupancy at ~r ¯ n ( ~r ) = 11 + h k + ¯ c ( ~x ) ∆ A πD log (cid:16) πD ∆ Ak − (cid:17)i − , (13)which is plotted in Fig. 4 and, in the limit D →
0, leadsto the recovery of the immobile receptor solution Eq. (2), n ¯ ˜ n H - n L ¬ A M P A R H Μ m s - L ¬ N M DA R D k - s - L H m FIG. 4: (Color online) Receptor occupancy at ~r . Eq. (13)for ¯ n is plotted as a function of lateral diffusion coefficient D and unbinding rate k − for a range of experimentally relevantvalues in neurotransmission ( top surface ). Specifically, the k − values for AMPA and NMDA receptors are indicated (seeTable I). Corresponding plot for ¯ n (1 − ¯ n ) is also shown ( bottomsurface ). namely ¯ n = k + ¯ c/ ( k − + k + ¯ c ). In order to quantify thedeviation from equilibrium we apply Fick’s first law tocalculate the flux ~J of occupied receptors ~J = D ~ ∇ ¯ n ( ~r, t ) (14)out of region ∆ A [19]. Integrating Eq. (14) over the bor-der of ∆ A allows us to define an effective “internalizationrate” for removal of ligand in analogy to [12], via k ei ¯ n = − D ∆ A I ∇ ¯ n ( ~r ) d~r = − D ∆ A Z ∆ A ∇ ¯ n ( ~r ) d ~r = 1∆ A Z ∆ A d ~r [ k + ¯ c (1 − ¯ n )∆ Aδ ( ~r ) − k − ¯ n ] , (15)where the second equality follows from integration byparts and the last equality results from the stationarycondition for Eq. (10a). Carrying out the last integrationleads to the following final expression for the effectiveinternalization rate k ei = k + ¯ c (cid:18) − ¯ n ¯ n (cid:19) − k − = 4 πD ∆ A log (cid:16) πD ∆ Ak − (cid:17) − k − , (16)where we used Eq. (13) to express ¯ n in terms of the re-ceptor diffusion coefficient. Using condition ∆ A = 4 π/ Λ we can define a modified unbinding rate κ − = k − + k ei = k + ¯ c (1 − ¯ n )¯ n = D Λ log (cid:16) D Λ k − (cid:17) (17)and extend the equilibrium condition Eq. (3) to the sta-tionary non-equilibrium condition via an effective tem- perature T e defined by [12] k + ¯ cκ − = e FTe . (18)This effective temperature allows us to generalize the or-dinary FDT to a stationary non-equilibrium conditionand to derive the receptor accuracy of sensing in the nextsection. V. ACCURACY OF SENSING
Here we we derive the accuracy of sensing for diffus-ing receptors. Starting from Eq. (10) we consider anexpansion to first order around the stationary solutions n ( ~r, t ) = ¯ n + δn ( ~r, t ) (19a) c ( ~x, t ) = ¯ c + δc ( ~x, t ) , (19b)where ¯ n and ¯ c are the stationary solutions of Eq. (10),which, in this case, have a spatial dependence, i.e. ¯ n =¯ n ( ~r ) and ¯ c = ¯ c ( ~x ). Hence, the linearized equation for thereceptor occupancy and ligand concentration are givenby ∂ [ δn ( ~r, t )] ∂t = D ∇ δn ( ~r, t ) + (cid:20) (1 − ¯ n ) δc ( ~r, z , t ) − ¯ cδn ( ~r, t ) + (1 − ¯ n )¯ c δk + ( t ) k + − ¯ n δk − ( t ) k + (cid:21) k + ∆ A δ ( ~r − ~r ) − k − δn ( ~r, t )(20a) ∂ [ δc ( ~x, t )] ∂t = D ∇ δc ( ~x, t ) − δ ( z − z ) ρ ( ∂ [ δn ( ~r, t )] ∂t − D ∇ δn ( ~r, t ) ) , (20b)where we assumed that fluctuations in the bind-ing/unbinding rate constants occur as well. This math-ematical trick allows us to introduce fluctuations in thereceptor-binding free energy and so to apply the FDT[9, 17]. Using the stationary non-equilibrium conditionEq. (18), we obtain the fluctuations in the rates andbinding free energy δk + k + − δk − κ − = δFT e . (21)Using this equation allows us to replace fluctuations inthe rate constants in Eq. (20a) with fluctuations in the binding free energy, resulting in ∂ [ δn ( ~r, t )] ∂t = D ∇ δn ( ~r, t ) − k − δn ( ~r, t ) + (cid:20) k + ¯ c (1 − ¯ n ) δFT e + k + (1 − ¯ n ) δc ( ~r, z , t ) − k + ¯ cδn ( ~r, t ) (cid:21) ∆ A δ ( ~r − ~r ) . (22)Fourier Transforming Eq. (22) and setting q = | ~q | , wesolve for δ ˆ n ( ~q, ω ) and obtain δ ˆ n ( ~q, ω ) = G ( ω, ~r , z ) e i~q · ~r D q + k − − iω , (23)where, for convenience of calculation, we have defined thefollowing function G ( ω, ~r , z ) = (cid:20) k + (1 − ¯ n ) δ ˆ c ( ~r , z , ω ) − k + ¯ c δ ˆ n ( ~r , ω )+(1 − ¯ n )¯ c δF ( ω ) T e (cid:21) ∆ A. (24)Inverse Fourier Transforming Eq. (23) leads to the fol-lowing expression for the spectrum of the fluctuations inoccupancy δ ˆ n ( ~r , ω ) = Z d~q (2 π ) e − i~q · ~r δ ˆ n ( ~q, ω ) (25)= G ( ω, ~r , z ) Z d~q (2 π ) D q + k − − iω = G ( ω, ~r , z )Σ ( ω ) , where we definedΣ ( ω ) = Z d~q (2 π ) D q + k − − iω (26)with the integral provided in Appendix A. In order toremove the ligand concentration in Eq. (24), we FourierTransform Eq. (20b), leading to δ ˆ c ( ~q, ω ) = ρ iω − D q D ( q + q ⊥ ) − iω δ ˆ n ( ~q, ω ) e iq ⊥ z . (27)Inserting Eq. (23) in Eq. (27) and inverse Fourier Trans-forming leads to the following expression for the spectrumof the fluctuations in ligand concentration δ ˆ c ( ~r , z , ω ) = Z dq ⊥ π e − iq ⊥ z Z d~q (2 π ) e − i~q · ~r δ ˆ c ( ~q, ω )= G ( ω, ~r , z ) ρ Σ ( ω ) (28)withΣ ( ω ) = Z dq ⊥ π Z d~q (2 π ) iω/D − q D ( q + q ⊥ ) − iω e − i~q · ~r q + k − D − iωD (29)provided in Appendix B. Inserting Eqs. (25) and (28)in Eq. (24) leads to a closed equation for the function G ( ω, ~r , z ) G ( ω, ~r , z ) = (cid:20) k + (1 − ¯ n ) ρ Σ ( ω ) G ( ω, ~r , z )+ (30) − k − ¯ c Σ ( ω ) G ( ω, ~r , z ) + k + ¯ c (1 − ¯ n ) δ ˆ F ( ω ) T e ∆ A, which can be solved for G ( ω, ~r , z ). This leads to G ( ω, ~r , z ) = k + ¯ c (1 − ¯ n )∆ A δ ˆ F ( ω ) /T e − k + (1 − ¯ n )Σ ( ω ) + k + ¯ c Σ ( ω )∆ A , (31) O cc up a n c y f l u c t u a ti on s Exact solutionEff. intern. rate
AMPAR a ( un it s o f - τ ) Diffusion strength O cc up a n c y f l u c t u a ti on s Exact solutionEff. intern. rate
NMDAR b η .51 ( un it s o f - τ ) FIG. 5: Receptor-occupancy fluctuations h ( δn ) i τ in units of τ as a function of η = 16 D / ( s k − ) for AMPAR (a) andNMDAR (b). Shown are ( solid line ) plot of h ( δn ) i τ as givenby Eqs. (34) and (36), ( dashed line ) plot of Eq. (37) using theeffective internalization rate Eq. (16), and ( vertical thin line )experimental value of η for each receptor type (see Table Ifor all parameters value). Glutamate concentration was set to¯ c =0.1mM, here and in Fig. 6. In order to produce a significanteffect from ligand rebinding, we used D = 0 . µm /s here andin subsequent plots, as may result from binding of glutamateto neuroligins, neurexins and other cleft proteins. where condition ρ = 1 / ∆ A was applied. Using Eqs. (25)and (31), a final expression for the response is obtained δ ˆ n ( ~r , ω ) δ ˆ F ( ω ) = G ( ω, ~r , z )Σ ( ω ) δ ˆ F ( ω ) (32)= 1 T e k + ¯ c (1 − ¯ n )∆ A Σ ( ω )1 − k + (1 − ¯ n )Σ ( ω ) + k + ¯ c Σ ( ω )∆ A .
Applying the generalized FDT we derive the spectrumof the fluctuations in receptor occupancy δ ˆ n ( ~r , ω ) fromthe response, namely S ( ω ) = 2 kT e ω Im " δ ˆ n ( ~r , ω ) δ ˆ F ( ω ) . (33)By replacing the expressions for Σ ( ω ) and Σ ( ω ) in Eq.(32), taking the imaginary part and setting ω ≃
0, weobtain for the zero-frequency limit of the power spectrum S η (0) = k + ¯ c (1 − ¯ n )∆ A Λ πk − (1 + η ) h k + ¯ ck − L η + k + (1 − ¯ n )Λ8 πD (1 − A η ) i × (cid:20) k + (1 − ¯ n )Λ16 πD (cid:18) D Λ − ηk − D Λ / (1 + η ) A η + 1 − A η L η / (cid:19) L η (cid:21) , (34)where we conveniently defined the following functions ofthe dimensionless parameter η = D Λ /k − L η ≡ log (1 + η ) η , A η ≡ arctan √ η √ η . (35)Note that in Eq. (34), occupancy ¯ n also depends on η via Eq. (13). The zero-frequency spectrum is relatedto the receptor occupancy fluctuations, averaged over atime τ ≫ k − − , ( k + ¯ c ) − , by the following relation: h ( δn ) i τ ≃ S η (0) /τ. (36)In Fig. 5 we plot the occupancy fluctuations, given byEq. (36) as a function of η , i.e. the effective diffusionstrength. This is done for the two main receptors in-volved in neurotransmission, namely AMPAR and NM-DAR, with parameters taken from Table I. For AMPAR,the effect of diffusion is that of decreasing the fluctuationsin occupancy due to reduced overcounting of previouslybound molecules (Fig. 5a). For small unbinding rates k − though, such as occurs for the NMDA receptor, anactual increase in the noise is observed for a range ofphysiologically relevant diffusion strengths. GLUTAMATE D µ m /s(acqueous solution)[20]10 µ m /s (synaptic cleft)[21, 22]¯ c τ D µ m /s [27, 28] 0.021 µ m /s [27, 28] k + × M − s − [20, 29] 5-8.4 × M − s − [29, 30] k − × s − [20, 29] 5-80 s − [29, 30]TABLE I: Summary of experimentally determined parametersfor glutamate ligand (top), AMPA receptor (bottom left) andNMDA receptor (bottom right). In order to make the origin of this effect more transpar-ent, we also plot the occupancy fluctuations in Fig. 5 ascalculated for the case of receptor internalization [12] h ( δn ) i τ ≃ n (1 − ¯ n ) k + ¯ cτ , (37)where ¯ n = k + ¯ c/ ( k + ¯ c + k − + k ei ), with k ei defined in Eq.(16). The experimental unbinding rate of NMDAR leadsto values for ¯ n such that starting from D = 0, the func-tion ¯ n (1 − ¯ n ) increases, reaches a maximum and thendecreases again. Such a maximum is avoided by AM-PAR whose spectrum monotonically decreases when D increases (see Fig. 4). The comparison between recep-tor diffusion and receptor internalization in Fig. 5 showsqualitatively similar results, confirming the effective lig-and removal by receptor diffusion. A. Equilibrium limit for vanishing D Here we show that Eq. (34) leads to the expressionfor the single receptor obtained in Refs. [9, 18] in thelimit for D →
0, i.e. η →
0. Taking the limit and using∆ A = 4 π/ Λ , we indeed obtainlim η → S η (0) = 2 k + ¯ c (1 − ¯ n )( k − + k + ¯ c ) (cid:20) k + (1 − ¯ n ) Λ8 πD (cid:21) , (38)which is identical to the result for the single receptor forΛ = 4 /s . B. Near-equilibrium result for small D We derive here the exact result, Eq. (34), in the limitof slow receptor diffusion on the membrane, using η = D / ( s k − ) as a small parameter. Specifically, we areinterested in the accuracy of sensing, which is derivedfrom the normalized variance for the ligand concentrationof ligand as in [9] h ( δc ) i τ ¯ c = h ( δn ) i τ ¯ n (1 − ¯ n ) ≃ n (1 − ¯ n ) S η (0) τ , (39)where the right-hand term follows from Eq. (36). Ex-pansion to first order in parameter η leads to h ( δc ) i τ ¯ c = 2 k + ¯ c (1 − ¯ n ) τ (1 − n η ) + 1 πsD ¯ cτ (cid:20) − (cid:18) k − ¯ n πk + ¯ c D s + 8¯ n k − − k + ¯ c k + ¯ c + k − D s (cid:19) η (cid:21) . (40)Comparison with Eq. (6a) shows that the correctiondue to lateral receptor diffusion is twofold to first or- der: (i) the first term is reduced due to a reduction in ¯ n , U n ce r t a i n t y Exact solution1st term onlyLinear approx.Fund. limit
AMPAR a ( un it s o f - τ ) U n ce r t a i n t y Exact solution1st term onlyLinear approx.Fund. limit
NMDAR b η ( un it s o f - τ ) FIG. 6: Uncertainty h ( δc ) i τ / ¯ c as a function of η =16 D / ( s k − ) for the AMPA receptor (a) and the NMDA re-ceptor (b) in units of τ . Shown are ( solid line ) plot of Eq.(39) as obtained from the exact solution Eq. (34), ( dashedline ) linear approximation for small η , and ( dotted line ) con-tribution to the accuracy of sensing from only the first term inEq. (40). Also shown are ( vertical thin line ) the experimen-tal value of η for each receptor (see Table I) and ( horizontaldot-dashed line ) the fundamental physical limit Eq. (7) (ii) the second term is reduced for most parameter val-ues due to a reduction in rebinding of already measuredligand molecules.In Fig. 6 we plot the uncertainty in sensing for bothAMPAR and NMDAR. In both cases the total effect oflateral diffusion is that of decreasing the uncertainty inthe concentration and therefore of increasing the accu-racy of sensing. Receptor diffusion for AMPAR (Fig.6a) mainly reduces the rebinding term, while for NM-DAR (Fig. 6b) the first term from binding and unbind-ing is reduced. In the latter case, the maximum in theoccupancy fluctuations (see Fig. 5) is removed through¯ n (1 − ¯ n ) in the denominator of Eq. (39). For large val-ues of η , the occupancy fluctuations and hence the accu-racy of sensing become unphysical (zero, i.e. below thephysical limit, or diverge) as the effective temperaturebreaks down far from equilibrium. (For large η , recep-tors remove ligand molecules efficiently, which introduceslarge non-equilibrium ligand fluxes. These cannot be rep- EPSC (pA) H i s t og r a m ExperimentFit with 3 Gaussians a current I (pA) D i s t r i bu ti on Diffusing receptorsImmobile receptorsIndividual Gaussians b FIG. 7: Quantal transmission. (a) Histogram of experimen-tal EPSC (excitatory post-synaptic current) data taken fromRef. [34] and fitted by three Gaussian distributions. (b) Dis-tribution of predicted EPSCs. Shown are individual Gaus-sian distributions corresponding to one, two and three vesi-cles (from left to right; thin lines), as well as envelope func-tions (sums with equal weighting; thick lines). Calculation isdone for AMPA receptors using Eqs. (41, 42), Eq. (39) forthe variances of diffusing (thin solid lines) and Eq. (6a) forimmobile (thin dashed lines) receptors, assuming glutamateconcentration ¯ c n = c b + nc v for n = 1 , , c b = 0 . mM , vesicle concentration c v = 0 . mM and N = 100 receptors [44, 45]. Furthermore, Eq. (41) withparameters from Refs. [34, 35] was used. resented by an effective temperature.)In summary, lateral receptor diffusion affects the accu-racy of sensing in two ways: (i) by reducing the station-ary value for the receptor occupancy, and (ii) by reduc-ing the noise from rebinding of already measured ligandmolecules. VI. QUANTAL TRANSMISSION
A remarkable property of synaptic transmission is thatthe amplitude of the synaptic response, i. e. the excita-tory post-synaptic current (EPSC), varies by an integralmultiple of a quantum. This property, first unveiled bythe pioneering work of Katz [31], was traced to the forma-tion of synaptic vesicles and subsequent pore formation.The latter leads to a release of glutamate into the synap-tic cleft [32]. This quantization is considered a way tominimize the variability of the response from each con-tributing synapse and therefore to allow efficient neuralcomputation. This variability is quantified by the coeffi-cient of variation (CV) defined as the standard deviationdevided by the mean value. The reason for a large CVis controversial (cf. Fig. 7a), but can be attributed to avariation in the size of the vesicles carrying glutamate, orto the fact that large EPSCs are generated by multiplevesicles [13, 46]. The experimental detection of EPSCswith an amplitude distribution of more than one peak isconsistent with the latter explanation [33].In Fig. 7a, the experimental data taken from Ref. [34]show EPSC peaks as a result of repetitive suprathresh-old stimulation of granule cell somata in CA3 pyramidalcells. Some stimuli did not evoke an EPSC (termed fail-ures). The histogram of successful events can be fittedwith three Gaussian distributions, showing the quantalnature of the EPSC. The peaks of the Gaussians corre-spond to the discrete multiple values 7 ,
14, and 21 pA,respectively.We can compare this distribution to the result pre-dicted by the accuracy of sensing. Specifically, we wouldlike to know if fast diffusion helps resolve the EPSCpeaks. A simple model based on the structural propertiesof AMPAR was previously used to derive the dynamicsof glutamate binding and unbinding, as well as channelopening and closing [34]. Based on this model the fol-lowing relation connecting the current through the ionchannel of AMPAR and the concentration of glutamatewas deduced: I = I
11 + ( λ/c ) n , (41)where I , λ , and the Hill coefficient n are parametersprovided in Ref. [34, 35]. Based on Eq. (41) the followingrelation connecting the width in the distribution of EPSCcurrents to the fluctuations in glutamate concentrationcan be derived: δI = I nλ ( λ/c ) n − [1 + ( λ/c ) n ] δc. (42)We further assume that a number N of receptors inde-pendently contribute to the measurement, i. e. h ( δc ) i τ,N ¯ c = h ( δc ) i τ N ¯ c , (43)with the uncertainty of a single receptor given by Eq.(39). Averaging over N , therefore, further reduces theuncertainty in Eq. (43) and hence increases the accuracyof sensing.Figure 7b shows that the distributions of predicted EP-SCs is in qualitative agreement with the experiments. From our calculations we found that the effect of lateralreceptor diffusion is to increase the resolution of peaksand so also the ability of the synapse to count the num-ber of vesicles released into the synaptic cleft. Note thatin order to resolve the EPSC peaks experimentally, theprobability of vesicle release from the presynaptic sidewas drastically reduced. This was achieved by modulat-ing the ratio of Ca + /Mg + ions in the extracellular so-lution, resulting in a 50-60% reduction of peak currents[34]. VII. DISCUSSION AND CONCLUSIONS
Our previous work on the fundamenal physical limitof ligand sensing [12] led us to conclude that receptordiffusion, by removing bound ligand from a region ofinterest, may reduce the local overcounting of ligandmolecules, therefore potentially increasing the local ac-curacy of sensing. In this paper we set out to investigatethis possibility by constructing a mathematical model,that contains all the necessary ingredients to describethe role of receptor diffusion in ligand sensing. We haveconsidered a receptor, that can bind and unbind ligandmolecules as well as diffuse on a 2D membrane. Lig-and is allowed to diffuse in 3D space. Using this modelwe derived the fluctuations in receptor occupancy in thearea of interest via a generalization of the FDT with theintroduction of an effective temperature [12]. This ulti-mately allowed us to derive an equation for the accuracyof sensing.We applied our model to the biologically relevant caseof glutamate receptors, which are responsible for trans-mitting action potentials at neural synapses. We foundthat for AMPAR and NMDAR, our model shows that thelateral diffusion of receptors increases their accuracy ofsensing and hence may allow synapses to count the num-ber of released vesicles. However, the local occupancyfluctuations are decreased only for AMPAR; for NMDARthese fluctuations are increased. This difference is due tothe smaller unbinding rate of the latter receptor. Conse-quently, the accuracy of sensing for NMDAR is an orderof magnitude smaller than for AMPAR. It is importantto remark that these results are consistent with the differ-ent roles these receptors play. In fact, the transmissionof an excitatory potential through a synapse occurs inseveral steps. Initially, it is AMPAR which senses gluta-mate and, by letting in ions, starts a small depolarizationof the membrane. This in turn allows NMDAR to openits channel and start an even bigger influx of ions (es-pecially Ca + ). This stimulates the production of moreAMPARs and so increases the strength and sensitivityof the synapses. Sensitivity to glutamate is importantfor AMPAR, while the roles of NMDAR are coincidencedetection and amplification, important for long-term po-tentiation and memory.In order to derive the accuracy of sensing for diffusingreceptors, we made a number of symplifying assumptions.0We neglected fluctuations in the total receptor densityand vesicle size, assumed to be constant [13, 46]. Fur-thermore, we assumed a constant ligand concentrationin the hotspot area ∆ A during the receptor measure-ment time, while in reality the concentration profile willbroaden out. Additionally, we made the simplifying as-sumption that multiple vesicles are released in the samespot, so that released glutamate concentrations add up,and that other effects such as spillover from neighboringcells can be neglected [42, 43]. Finally, we introducedthe effective temperature T e to generalize the FDT tonon-equilibrum processes [12, 36–39]. The approximatenature of T e , for which we neglected any potential timeor frequency dependence, limits the quantitative valid-ity of our model to small deviations from equilibrium.In fact, for large receptor diffusion coefficients, the occu-pancy fluctuations and hence the accuracy of sensing be-come unphysical. Nevertheless, the mapping of receptordiffusion to an effective internalization process, alreadystudied in Ref. [12], provides confidence in our method.In conclusion, in this paper we highlighted the roleof diffusion in increasing the accuracy of sensing. Thismight be important for synaptic counting of glutamatevesicles. This hypothesis can be tested experimentally byreducing the mobility of receptors. This can be achievedby cross-linking or addition of cholesterol. Alternatively,the glutamate diffusion costant can be reduced by addingdextran. Such perturbations should lead to a broadeningof the EPSC peaks, although signaling may be affectedas well. Similarly to AMPAR [15], an increase in re-ceptor diffusion upon ligand binding was recently alsoobserved in the pseudopods of migrating Dictyosteliumdiscoideum cells [40]. This phenomenon was attributedto signal amplification. However, since cells may senselocally in pseudopods, at least in shallow gradients [41],receptor diffusion may also be responsible for increasingthe accuracy of sensing.
Acknowledgments
We would like to thank Aldo Faisal and Peter Jonasfor helpful comments. We acknowledge financial supportfrom Biotechnological and Biological Sciences ResearchCouncil grant BB/G000131/1 and the Centre for Inte-grated Systems Biology at Imperial College.
Appendix A: Calculation of Σ ( ω ) We devote this Appendix to the calculation of the in-tegral Σ ( ω ) in Eq. (26). Σ ( ω ) is given by:Σ ( ω ) = Z d~q (2 π ) /D q + k − D − iωD = Z ∞ dq π q/D q + k − D − iωD , (A1)which is divergent. Hence, we introduce a cut-off Λ ∼ /s to account for the finite dimension s of the receptor. Thisprocedure leads to the final result:Σ ( ω ) = Z Λ0 dq π q/D q + k − D − iωD = log (cid:16) D Λ k − − iω (cid:17) πD . (A2) Appendix B: Calculation of Σ ( ω ) In this appendix, we calculate Σ ( ω ) in Eq. (29).Without loss of generality, we set ( x , y , z ) = (0 , , ( ω ) = Z ∞ dq ⊥ π Z ∞ qdq π iω/D − q ( q + q ⊥ ) − iωD /D q + k − D − iωD = Z ∞ dq ⊥ π Z ∞ dq π q/D ( q + q ⊥ ) − iωD k − /D q + k − D − iωD − ! = I ( ω ) − I ( ω ) , (B1)where I ( ω ) = Z ∞ dq ⊥ π Z ∞ qdq π /D ( q + q ⊥ ) − iωD k − /D q + k − D − iωD = Z Λ0 qdq π i/D p iω/D − q · k − /D q + k − D − i ωD (B2)= k − / (8 πD ) q k − D − iω ( D − D ) " arctan s Λ − iω/D k − D − iω ( D − D ) ! − arctan s − iω/D k − D − iω ( D − D ) ! and I ( ω ) = Z ∞ dq ⊥ π Z ∞ qdq π − iω + D ( q + q ⊥ ) (B3)= 1 D Z Λ0 qdq π i p − q + iω/D = 18 πD r − iωD − r Λ − iωD ! using a cut off to account for the finite size of the receptor.Taken together, the integral Σ ( ω ) is given by:Σ ( ω ) = I ( ω ) − I ( ω ) = ( r Λ − iωD − r − iωD ! ++ k − /D q k − D − iω ( D − D ) " arctan s Λ − iω k − D − iω ( D − D ) ! − arctan s − iω k − D − iω ( D − D ) ! πD . (B4)1 [1] A. Raj and A. van Oudenaarden, Cell. , 216 (2008).[2] R. C. Yu et al. , Nature , 755 (2008).[3] M. S. Samoilov1, G. Price, and A. P. Arkin, Sci. STKE , re17 (2006).[4] T. Gregor et al. , Cell , 153 (2007).[5] H. Mao, P.S. Cremer, M. D. Manson, Proc. Natl. Acad.Sci. USA , 5449 (2003).[6] R. A. Arkowitz Trends Cell. Biol. , 20 (1999).[7] J. E. Segall, Proc. Natl. Acad. Sci. USA , 8332 (1993).[8] H. C. Berg and E. M. Purcell, Biophys. J., , 193 (1977)[9] W. Bialek and S. Setayeshgar, Proc. Natl. Acad. Sci. USA , 10040 (2005).[10] R. G. Endres and N. S. Wingreen, Proc. Natl. Acad. Sci.USA , 15749 (2008).[11] R. G. Endres and N. S. Wingreen, Phys. Rev. Lett. ,158101 (2009).[12] G. Aquino and R. G. Endres, Phys. Rev. E , 021909(2010).[13] A.A. Faisal, L.P.J. Selen and D.M. Wolvert, Nature ,292 (2008).[14] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Robertsand Peter Walter, Molecular Biology of the Cell , GarlandPublishing Inc. US, fifth edition (2007).[15] M. Renner et al. , Curr. Opin. Neurobio. , 532 (2008).[16] M. Heine et al. , Science , 201 (2008).[17] R. Kubo, Rep. Prog. Phys. , 255 (1966).[18] R. G. Endres and N.S. Wingreen, Progr. Biophys. Mol.Biol. , 33 (2009).[19] H. C. Berg, Random Walks in Biology , Princeton Uni-versity Press (1993).[20] M. Postlethwaite et al. , J. Physiol. , 69 (2007).[21] T. A. Nielsen, D. A. Di Gregorio and R. A. Silver, Neuron , 757 (2004); E. ´E. Saftenku, J. Theor. Biol. , 363(2005).[22] D. Choquet and A. Triller, Nat. Rev. Neurosci. , 251(2003); D. Holcman and A. Triller, Biophys. J. , 2405 (2006).[23] F. Ventriglia and V. Di Maio, Biosys. , 67 (2000).[24] P. Jonas, News Physiol. Sci. , 83 (2000).[25] Molecular Biology of the Cell M. J. T. Fitzgerald, G.Gruener and E. Mtui, Clinical Neuroanatomy and Neu-roscience , Saunders Elsevier Ed., fifth edition (2007).[26] T. Nakagawa, Y. Cheng, M. Sheng and T. Walz, Biol.Chem. , 179 (2006).[27] L. Groc et al. , Nature Neurosci. , 695 (2004); L. Groc et al. , Proc. Nat. Acad. of Sci. USA , 18769 (2006).[28] A. J. Borgdorff and D. Choquet, Nature , 649 (2002).[29] D. Attwell and A. Gibb, Neuroscience , 841 (2005).[30] K. Erreger and S. F. Traynelis, J. Physiol. , 381(2005).[31] B. Katz, The Release of Neural Neurotransmitter Sub-stances , Liverpool University Press, Liverpool (1969).[32] J. E. Heuser, J. Cell Biol. , 275 (1979).[33] M. J. Wall and M. M. Usowicz, Nature Neurosci. , 675(1998).[34] P. Jonas and B.Sakmann, J. Physiol. , 615 (1993)[35] P. Jonas and B.Sakmann, J. Physiol. , 143 (1992),[36] L. F. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev.E , 3898 (1997).[37] Th. M. Nieuwenhuizen, Phys. Rev. Lett. , 5580 (1998).[38] A. Crisanti and F. Ritort, J. Phys. A , R181 (2003).[39] L. Leuzzi, J. of Non-Cryst. Solids , 686 (2009).[40] S. de Keijzer et al. , J. Cell. Sci. , 1750 (2008).[41] N. Andrew and R. H. Insall, Nat. Cell. Biol. , 193 (2007).[42] J.J. Lawrence et al. , J. of Phys. , 175 (2003).[43] P. B. Sargent et al , J. of Neurosci. , 8173 (2005).[44] D. M. Kullmann, M-Y Min, F. Asztely, D.A. Rusakov,Phil. Trans. R. Soc. Lond. B , 395 (1999).[45] J. R. Cottrell et al. , J. Neurophysiol. , 1573 (2000).[46] K. M. Franks, C.F. Stevens, T. K. Sejnowski, J. Neurosci.23