Revising Berg-Purcell for finite receptor kinetics
RRevising Berg-Purcell for finite receptor kinetics
Gregory Handy ∗† Sean D. Lawley ‡ January 18, 2021
Abstract
From nutrient uptake, to chemoreception, to synaptic transmission,many systems in cell biology depend on molecules diffusing and bindingto membrane receptors. Mathematical analysis of such systems often ne-glects the fact that receptors process molecules at finite kinetic rates. Akey example is the celebrated formula of Berg and Purcell for the ratethat cell surface receptors capture extracellular molecules. Indeed, thisinfluential result is only valid if receptors transport molecules through thecell wall at a rate much faster than molecules arrive at receptors. From amathematical perspective, ignoring receptor kinetics is convenient becauseit makes the diffusing molecules independent. In contrast, including recep-tor kinetics introduces correlations between the diffusing molecules since,for example, bound receptors may be temporarily blocked from bindingadditional molecules. In this work, we present a modeling framework forcoupling bulk diffusion to surface receptors with finite kinetic rates. Theframework uses boundary homogenization to couple the diffusion equa-tion to nonlinear ordinary differential equations on the boundary. We usethis framework to derive an explicit formula for the cellular uptake rateand show that the analysis of Berg and Purcell significantly overestimatesuptake in some typical biophysical scenarios. We confirm our analysis bynumerical simulations of a many particle stochastic system.
Introduction
Many biological systems depend on molecules diffusing and interacting withmembrane receptors. For example, cellular nutrient uptake relies on cell sur-face receptors binding and transporting diffusing molecules into the cell [1]. ∗ Departments of Neurobiology and Statistics, University of Chicago, Chicago, IL 60637USA ( [email protected] ). † Grossman Center for Quantitative Biology and Human Behavior, University of Chicago,Chicago, IL, USA. ‡ Department of Mathematics, University of Utah, Salt Lake City, UT 84112 USA( [email protected] ). The second author was supported by the National Science Founda-tion (Grant Nos. 1944574, DMS-1814832, and DMS-1148230). a r X i v : . [ q - b i o . Q M ] J a n a) (b) (c) Figure 1: (a) The blue and black molecules compete to bind to a surface re-ceptor. (b) The blue molecule binds to a receptor. The black molecule is thentemporarily blocked from binding to that receptor. (c) After some time, theblue molecule is absorbed by the receptor and that receptor is free to bind theblack molecule.Chemoreception and chemotaxis similarly depend on cell surface receptors bind-ing extracellular diffusing molecules [2]. An important part of the immune re-sponse involves antibodies binding to epitopes on the surface of a virion [3, 4].In addition, synaptic transmission requires neurotransmitter molecules releasedfrom one neuron to diffuse across the synaptic cleft and bind to receptors onthe adjacent neuron [5].In most instances, the receptors cannot bind molecules continuously, butrather binding one or more molecules temporarily hinders binding additionalmolecules (see Fig. 1 for an illustration). This could be due simply to mutualexclusion at the receptor (i.e. a receptor can only bind one molecule at a time).Alternatively, this effect could be due to the finite rate that a receptor cantransport bound molecules into the cell, as in the case of nutrient transport [6,7].Similarly, the effect could stem from a finite receptor internalization rate [8–10].In the case of synaptic transmission, a receptor that captures a molecule changesconformation, and during this time it cannot capture additional molecules [5,11, 12]. That is, the receptor must wait a transitory “recharge” time followingthe capture of any molecule before additional captures. A similar recharge timeaffects experiments where molecules are released into extracellular space in thebrain and bind to receptors on astrocytes [13]. In ecology, this notion of arecharge time is called the handling time, and it is the time a predator (the“receptor”) must wait after capturing a prey (the “molecule”) before it canhunt again.The common feature of these examples is that the receptors process moleculesat finite kinetic rates. Mathematical models often neglect receptor kinetics,which greatly simplifies the analysis since it allows the diffusing molecules to beindependent. Including receptor kinetics makes the diffusing molecules depen-dent, since the molecules may affect each other through their interactions withthe receptors. For example, if one molecule binds to a receptor, then additionalmolecules may be temporarily blocked from that receptor.2n important example of mathematical analysis that neglects receptor ki-netics is the formula of Berg and Purcell for the rate that cell surface receptorscapture extracellular molecules [2]. Assuming that the receptors occupy a smallfraction of the cell surface, they found that the uptake rate is J bp := εNεN + π J max , (1)where ε is the ratio of the receptor radius to the cell radius, N ≥ J max is the uptake rate if the entire surface is coveredby perfectly absorbing receptors [2]. In particular, J max is [14] J max := 4 πDRu , (2)where R is the cell radius, D is the diffusivity of extracellular molecules, and u is the concentration of extracellular molecules.Berg and Purcell’s formula in Eq. 1 has been very influential. Indeed, manyworks have sought to refine equation Eq. 1 to incorporate the effects of other de-tails in the problem. For example, Zwanzig used an effective medium formalismto account for the effects of interference between receptors [15]. Other workshave modified Eq. 1 to include other effects, including receptor arrangement,cell membrane curvature, and receptor motion [16–26]. In one particularly im-portant study, Wagner et al. [27] extended Eq. 1 to non-spherical geometriesand used this analysis to argue that the cylindrical morphology of cell envelopeextensions serves to increase nutrient uptake.To derive Eq. 1, Berg and Purcell [2] assumed that any molecule that touchesa receptor“is immediately (or within a time short compared to the intervalbetween arrivals) captured and transported through the cell wall,clearing the site for its next catch.”This assumption makes the diffusing molecules independent. However, it is clearthat this assumption is violated at sufficiently high extracellular concentrations.In this paper, we present a modeling framework for coupling bulk diffusion ofmolecular species to surface receptors with finite kinetics. Mathematically, thisframework uses boundary homogenization to link the diffusion equation (a par-tial differential equation (PDE)) to boundary conditions described by nonlinearordinary differential equations (ODEs). We then reduce this PDE-ODE systemto a PDE with a nonlinear boundary condition of Michaelis-Menten type. Weconfirm the predictions of this framework and analysis with detailed numericalsimulations of a full, many particle stochastic system. While the general frame-work can be applied in a variety of problems and geometries, we develop thetheory primarily in the context of the Berg-Purcell problem described above.We derive an explicit formula for the cellular uptake rate as a function of thevarious parameters in the problem, including the kinetic rates of receptors. Weshow that the classical result in Eq. 1 significantly overestimates uptake in sometypical biophysical scenarios. 3he rest of the paper is organized as follows. We first rederive the classi-cal result in Eq. 1, and then we present the PDE-ODE framework and derivea reduced Michaelis-Menten boundary condition. Next, we use the modelingframework to find explicit formulas for various steady-state quantities, includ-ing cellular uptake and the fraction of bound receptors. We then describe ournumerical methods and verify the predictions of the modeling framework bystochastic simulations. Finally, we explore some biophysical implications of ourresults in typical parameter regimes of interest. We conclude by discussingrelated work and highlighting future directions. Methods
Berg-Purcell and boundary homogenization
We begin by reviewing the model of Berg and Purcell [2] and boundary ho-mogenization [17, 19, 23, 28–30]. Consider a spherical cell in a large mediumcontaining spherical molecules of some substrate. Fixing our reference frameon the cell, the substrate concentration, u = u ( r, θ, ϕ, t ), satisfies the diffusionequation in spherical coordinates ( r, θ, ϕ ), ∂∂t u = D ∆ u, r > R, (3)where D >
R > r →∞ u = u > . (4)The cell has N (cid:29) εR with ε (cid:28)
1. Substrate molecules can be absorbedby receptors and otherwise reflect from the cell surface. We thus have mixedboundary conditions at r = R , D ∂∂r u = DR κ rec u, r = R, and ( θ, ϕ ) in a receptor ,∂∂r u = 0 , r = R, and ( θ, ϕ ) not in a receptor , (5)where κ rec ∈ (0 , ∞ ) ∪ {∞} (6)is a dimensionless parameter describing the rate that a receptor binds a substratemolecule. Berg and Purcell took κ rec = ∞ , which means that a substrate isimmediately absorbed upon contact with a receptor [2]. If κ rec = ∞ , then thefirst boundary condition in Eq. 5 means u = 0.4he method of boundary homogenization replaces the heterogeneous bound-ary conditions in Eq. 5 by a homogeneous boundary condition of the form, D ∂∂r u = DR κu, r = R, (7)for some dimensionless trapping rate κ >
0. The trapping rate κ is chosento encapsulate the effective binding properties of the heterogeneous surface inEq. 5. Notice that κ (cid:28) κ (cid:29) u = u ( r, t ) depend only on the radius r ≥ R and time t ≥
0, and not on the angularvariables ( θ, ϕ ) (assuming the initial condition is independent of ( θ, ϕ )).The idea behind boundary homogenization is that, due to diffusion in theangular variables, the surface heterogeneity only affects the substrate concentra-tion near the surface. In particular, the concentration is constant in the angularvariables outside a boundary layer, where the width of this layer depends onthe length scale of the surface heterogeneity. Many sophisticated methods havebeen developed to choose the trapping rate κ in Eq. 7 in order to incorporatethe number, size, and arrangement of receptors [17–26]. If the receptors occupya small fraction of the cell surface, then the trapping rate is linear in the numberof receptors N and is given by [16, 19, 28] κ = εNπ (cid:16) επκ rec (cid:17) − , (8)where 1 /κ rec = 0 if κ rec = ∞ .It is straightforward to solve Eqs. 3-4 and Eq. 7 at steady-state to obtainthe large-time substrate flux into the cell by integrating over the surface of thesphere of radius R > t →∞ D (cid:90) r = R ∂∂r u d S = κκ + 1 J max , (9)where J max is in Eq. 2 and is the flux in the case that the entire cell surface isabsorbing [14]. If κ rec = ∞ , then Eq. 8 and Eq. 9 yield the Berg-Purcell [2] fluxformula in Eq. 1. Including finite receptor kinetics
The model above assumes that receptors can continuously absorb substrates.That is, it assumes that there is no limit to the rate that a receptor can processsubstrate molecules. This modeling assumption is valid if receptors processmolecules much faster than molecules tend to hit receptors.When is a system in this parameter regime? If we use the following charac-teristic values (used in, for example, [2]), D = 10 µ m s − , R = 1 µ m , u = 1 µ M , J max = 7 . × s − . If we use the following characteristic values (again, see [2]) for the dimensionlessreceptor radius ε and the number of receptors N , ε = 10 − , N = 10 , then Eq. 1 implies that the arrival rate to a single receptor is J bp N = εεN + π J max = 1 . × s − . (10)Hence, in this parameter regime, the Berg-Purcell formula in Eq. 1 givesa valid estimate of cellular uptake if a single receptor can transport moleculesthrough the cell wall at a rate much faster than Eq. 10. However, the so-calledturnover rates of membrane transporters are usually in the range [31] k c ∈ [3 × , × ] s − , which is much slower than Eq. 10. Receptors modeled by ODEs
To model membrane receptors with finite kinetics, we suppose that the substratemolecules interact with the receptors via a classical substrate and enzyme reac-tion scheme, S + E k a (cid:10) k b C k c −→ P + E. (11)Here, k a > k b ≥
0, and k c ≥ S , and the receptors play the role of the enzyme E .In particular, a substrate S diffuses and binds to a receptor E when it forms thecomplex C . During this time, the receptor is unavailable to bind with anothersubstrate molecule, until it “recharges” by either producing P or releasing thesubstrate S . The product P could represent a substrate molecule that wastransported into the cell, or a modified substrate molecule that was releasedback into the extracellular bulk but can no longer interact with receptors. Notethat we allow for the possibility that k b = 0 or k c = 0.Mathematically, we replace the homogenized boundary condition in Eq. 7 atthe cell surface by D ∂∂r u ( R, t ) = k a e ( t ) u ( R, t ) − k b c ( t ) , (12)6here the free receptor and bound receptor concentrations, e ( t ) and c ( t ), satisfythe ODEs, dd t e ( t ) = − k a e ( t ) u ( R, t ) + ( k b + k c ) c ( t ) , dd t c ( t ) = k a e ( t ) u ( R, t ) − ( k b + k c ) c ( t ) . (13)Adding the equations in Eq. 13, we see that the total receptor concentration,given by the number of receptors per the surface area of the cell, e := N πR , (14)is conserved. That is, e ( t ) = e − c ( t ). Note that k a has dimension [ k a ] =(length) (time) − , whereas k b and k c are pure rates with dimension [ k b ] =[ k c ] = (time) − . Further, if all of the receptors are free (i.e. c ( t ) = 0), we choose k a so that Eq. 12 reduces to Eq. 7. In particular, we take k a = κDe R = 4 DεR (cid:16) επκ rec (cid:17) − > . (15)Summarizing, the model consists of the following PDE with nonlinear couplingto an ODE through a boundary condition, ∂∂t u = D ∆ u, r > R, t > , lim r →∞ u ( r, t ) = u > ,D ∂∂r u ( R, t ) = k a (cid:0) e − c ( t ) (cid:1) u ( R, t ) − k b c ( t ) , dd t c ( t ) = k a (cid:0) e − c ( t ) (cid:1) u ( R, t ) − ( k b + k c ) c ( t ) . (16) Michaelis-Menten boundary condition
Defining the dimensionless variables, r = rR , t = tR /D , u = uu , c = ce , Eq. 16 becomes ∂∂t u = ∆ u, r > , t > , lim r →∞ u ( r, t ) = 1 ,∂∂r u (1 , t ) = κ (cid:0) − c ( t ) (cid:1) u (1 , t ) − χ b c ( t ) ,δκ dd t c ( t ) = u (1 , t ) − (cid:16) χ b + χ c κ + u (1 , t ) (cid:17) c ( t ) , (17)7here κ is in Eq. 8 and δ := N πR u , χ b := N k b J max , χ c := N k c J max . (18)Hence, the solution to Eq. 16 depends on the four dimensionless parameters, κ , δ , χ b , and χ c .Notice that the parameter δ in Eq. 18 compares the volume concentrationsof receptors to substrates. The Briggs-Haldane analysis of Michaelis-Mentenenzyme kinetics assumes that this parameter is small [33]. In particular, if δ (cid:28) κ, (19)then we assume Eq. 17 is in quasi-steady state,0 ≈ u (1 , t ) − (cid:16) χ b + χ c κ + u (1 , t ) (cid:17) c ( t ) , and thus, c ( t ) ≈ u (1 , t )( χ b + χ c ) /κ + u (1 , t ) . In this parameter regime, the problem in Eq. 16 becomes ∂∂t u = D ∆ u, r > R, t > , lim r →∞ u ( r, t ) = u > ,D ∂∂r u ( R, t ) =
V u ( R, t ) K + u ( R, t ) , (20)where the maximum velocity and half-saturation constant in the boundary con-dition are V := e k c = N πR k c , K := k b + k c k a = N ( k b + k c )4 πDRκ . (21)That is, the ODE boundary condition in Eq. 16 is replaced by a Michaelis-Menten type boundary condition in Eq. 20. Steady-state uptake and receptor occupation
At steady-state, solving the full PDE-ODE system in Eq. 16 is equivalent tosolving the Michaelis-Menten system in Eq. 20. In either case, it is straightfor-ward to obtain u ss ( r ) := lim t →∞ u ( r, t ) = u (cid:16) − a Rr (cid:17) ,c ss := lim t →∞ c ( t ) = (cid:16) − a ( χ b + χ c ) /κ + 1 − a (cid:17) e , (22)8here a is the dimensionless parameter, a = 12 (cid:32) χ c + χ b + χ c κ + 1 − (cid:114)(cid:16) χ c + χ b + χ c κ + 1 (cid:17) − χ c (cid:33) . (23)The fraction of receptors which are bound at steady-state is c ss /e ∈ (0 , J ∗ := D (cid:90) r = R dd r u ss d S = aJ max < J bp . (24)The inequality in Eq. 24 is the desired result that the flux into the cell whenthe receptors have finite kinetics is strictly less than the flux into the cell whenthe receptors process molecules at infinite kinetic rates. To verify Eq. 24, notefirst that the case χ c = 0 is trivial since a = 0 if and only if χ c = 0. Next,suppose χ c >
0. Note that Eq. 9 means J bp = a bp J max where a bp := κ/ ( κ + 1).Hence, a bp satisfies a bp = κ (1 − a bp ) . (25)On the other hand, the boundary condition at r = R implies that a satisfies a = κ (1 − a )1 + χ b /χ c + ( κ/χ c )(1 − a ) . (26)It is clear that the solution to Eq. 25 is larger than the solution to Eq. 26 since κ and χ c are strictly positive. Therefore, a ∈ (cid:16) , κκ + 1 (cid:17) = (0 , a bp ) , which verifies Eq. 24.In addition, fixing χ b and κ and taking χ c → ∞ in Eq. 26 and comparingto Eq. 25 shows that J ∗ = aJ max → a bp J max = J bp as χ c → ∞ . (27)That is, J ∗ reduces to J bp if the receptor turnover rate k c is much faster than J max /N . Other kinetic schemes and receptor internalization
The analysis above extends to more general kinetic schemes than the standardsubstrate-enzyme reaction in Eq. 11. Indeed, alternative kinetic schemes merelyyield different systems of ODEs at the cell surface.To illustrate, suppose that receptors transport substrate molecules by en-docytosis (i.e. receptor internalization), which is often seen in eukaryotic cells[8–10]. In this case, we replace Eq. 11 by S + E k a (cid:10) k b C k c −→ P, E k −→ ∅ , ∅ k r −→ E, k c and k are the respective internalization rates for bound receptors C and free receptors E , and k r is the rate that free receptors are delivered to themembrane. In this case, the boundary condition at r = R in Eq. 12 for thesubstrate flux is unchanged and the ODEs in Eq. 13 becomedd t e ( t ) = − k a e ( t ) u ( R, t ) + k b c ( t ) − k e ( t ) + k r , dd t c ( t ) = k a e ( t ) u ( R, t ) − ( k b + k c ) c ( t ) . Numerical methods and simulations
To verify the predictions of the modeling framework developed above, we per-form numerical simulations of a stochastic, many particle system. To reducecomputational cost, the stochastic simulations are performed in a cylindricalspatial domain. We begin by extending the analysis above to this spatial do-main.
Cylindrical domain
Let the spatial domain Ω be a cylinder of radius 2 R > L > (cid:8) ( x, y, z ) ∈ R : x + y < R , z ∈ (0 , L ) (cid:9) . (28)Substrate molecules diffuse in Ω with reflecting boundaries at the top ( z = L )and the sides ( r := (cid:112) x + y = 2 R ). Hence, the substrate concentration u = u ( x, y, z, t ) satisfies ∂∂t u = D ∆ u, ( x, y, z ) ∈ Ω , t > ,∂∂r u = 0 , r = 2 R ; ∂∂z u = 0 , z = L. Analogous to the surface of the sphere in the model above, we assume that thebottom of the cylinder ( z = 0) is reflecting, except for N (cid:29) D ∂∂z u = DR κ rec u, z = 0 , and ( x, y ) in a receptor ,∂∂z u = 0 , z = 0 , and ( x, y ) not in a receptor , (29)where κ rec is as in Eq. 6. If the N receptors are roughly evenly distributed andhave common radius εR (cid:28) R , then Eq. 29 can be replaced by D ∂∂z u = DR κu, z = 0 , (30)10here κ is in Eq. 8. Hence, the problem reduces to a one-dimensional PDE for u = u ( z, t ), ∂∂t u = D ∆ u, z ∈ (0 , L ) , t > , (31) ∂∂z u = 0 , z = L, (32)with the boundary condition in Eq. 30 at z = 0.As above, we can incorporate finite receptor kinetics by replacing the bound-ary condition at z = 0 by a boundary condition that couples to an ODE. Specif-ically, D ∂∂z u (0 , t ) = k a (cid:0) e − c ( t ) (cid:1) u (0 , t ) − k b c ( t ) , (33)dd t c ( t ) = k a (cid:0) e − c ( t ) (cid:1) u (0 , t ) − ( k b + k c ) c ( t ) , (34)where c ( t ), e , k a , k b , and k c are as above (with R replaced by R in Eqs. 14 and15). Furthermore, in the parameter regime in Eq. 19, we can eliminate Eq. 34and replace Eq. 33 by a Michaelis-Menten type boundary condition with V and K given in Eq. 21, D ∂∂z u (0 , t ) = V u (0 , t ) K + u (0 , t ) . (35) Stochastic simulation method
We now describe the stochastic simulation method. We use the standard Euler-Maruyama method [34] for simulating the paths of many diffusing substratemolecules in Ω with reflecting boundary conditions on the boundaries away fromreceptors. If a molecule hits a “free” receptor, then that molecule immediatelybinds to the receptor (corresponding to κ rec = ∞ in Eq. 6). If a receptor has amolecule bound to it, then that receptor is considered “occupied” and any othermolecule that hits it simply reflects. We take k b = 0, and thus a bound moleculeis removed from the system after an exponentially distributed time with rate k c >
0. When a bound molecule is removed from the system, the correspondingreceptor changes from “occupied” back to “free” and can thus bind additionalmolecules.All stochastic simulations simulations were written in a combination of C andMATLAB [35]. The simulations were completed in the cylinder in Eq. 28 withheight L = 1 µ m and radius 2 R = 0 . µ m, with N = 500 receptors of commonradius 0 . µ m placed uniformly at random (non-overlapping) along the diskcentered at z = 0. We take D = 10 µ m s − and k c ∈ { , , , } s − .Each trial began with all receptors “free” and 10 particles placed in the domainaccording to a normal distribution with mean ( x , y , z ) = (0 , , . µ m andstandard deviation 0 . µ m in each direction. For each value of k c , 10 trialswere completed with a discrete time step of 10 − s. Additional trials and smaller11ime steps were tested on a subset of simulations and did not yield significantquantitative changes. PDE numerical solution method
We numerically solve the PDE-ODE system (Eqs. 31, 32, and 33-34) and thePDE with a Michaelis-Menten boundary condition (Eqs. 31, 32, and 35) withthe method of lines [36]. Essentially, this method approximates the PDE witha large system of ODEs by replacing spatial derivatives with finite differences.The method is fairly standard, but the nonstandard boundary conditions mustbe handled carefully.We now give the details of the method. We approximate u ( z, t ) at n (cid:29) z j := ( j + )∆ z, for j = 0 , , . . . , n − , (36)where ∆ z = L/n (cid:28) L , and denote the approximation by u j ( t ) ≈ u ( z j , t ).Replacing the Laplacian in Eq. 31 by a finite difference, u j ( t ) satisfies the ODE,dd t u j = D ( u j − − u j + u j +1 )(∆ z ) , for j = 0 , , . . . , n − . (37)Notice that the equations for dd t u and dd t u n − in Eq. 37 involve u − and u n ,which are not yet defined. To ameliorate this issue, we make use of so-calledghost points, z − := − ∆ z and z n := ( n + )∆ z , and solve for u − and u n usingthe boundary conditions. Specifically, we approximate the boundary conditionat z = L in Eq. 32 with a finite difference, u n − u n − ∆ z = 0 . (38)Hence, Eq. 38 implies u n = u n − , which we then use to solve Eq. 37 when j = n − z = 0 in Eq. 33 by D ( u − u − )∆ z = k a ( e − c ( t )) (cid:16) u + u − (cid:17) − k b c ( t ) , (39)where we have replaced u (0 , t ) by ( u + u − ) /
2. We then solve Eq. 39 for u − and use this in Eq. 37 when j = 0. In addition to the n ODEs in Eq. 37, wealso have the ODE for c ( t ) which is obtained from Eq. 34 upon replacing u (0 , t )by ( u + u − ) / D ( u − u − )∆ z = V ( u + u − ) / K + ( u + u − ) / . (40)12 a) 0 2 4 6 8 109 , , , , , , t (time) [ms] M o l ec u l e s r e m a i n i n g k c = 10 s − k c = 10 s − t (time) [ms] M o l ec u l e s r e m a i n i n g (b) 0 2 4 6 8 1002 , , , , , t (time) [ms] M o l ec u l e s r e m a i n i n g k c = 10 s − k c = 10 s −
10 0.1 0.29,50010,000 t (time) [ms] M o l ec u l e s r e m a i n i n g Figure 2: Comparison of stochastic simulations (squares and triangles) to thedeterministic PDE-ODE system (solid and dotted curves) for the cylindricaldomain for different values of the receptor turnover rate k c . The insets zoom inat early times. See the text for details.We then solve Eq. 40 for u − > j = 0.Summarizing, the PDE-ODE system is approximated by n +1 ODEs, and thePDE with a Michaelis-Menten boundary condition is approximated by n ODEs.In either case, the ODEs are solved using the ode15s function in MATLAB [35]with n = 10 spatial grid points. Results and Discussion
Analysis confirmed by stochastic simulations
In Fig. 2, we plot the results of stochastic simulations (squares and triangles) andthe solution to the deterministic PDE-ODE system (solid and dotted curves)for the cylindrical spatial domain in Eq. 28 for various choices of the receptor13urnover rate k c . Specifically, we plot the number of diffusing molecules remain-ing in the domain (molecules which are unbound and have not been absorbed)as a function of time. This plot shows that the PDE-ODE system accuratelydescribes the dynamics of the full stochastic system involving many interactingparticles.Notice in Fig. 2 that at early times ( t < . ,
000 to 9 , N = 500 receptors. This initial rapid decreaseis seen in both the PDE-ODE solution and the stochastic simulations, which isevident from the insets in Fig. 2 which zoom in at early times. Then as timeincreases, the number of remaining molecules decreases linearly with a slope of N k c , which is readily seen in Fig. 2 for both the PDE-ODE solution and thestochastic simulations.The PDE solution with a Michaelis-Menten boundary condition also pro-duces the desired linear decay with slope N k c , but it does not exhibit the initialrapid decay of N molecules binding to the N receptors (plot not shown). Thisis not surprising, since the Michaelis-Menten boundary condition was derivedassuming that there are many more diffusing molecules than receptors per somecharacteristic volume. Indeed, in such a parameter regime, the size of the initialdrop in the number of molecules (namely the number of receptors, N ) would besmall compared to the number of molecules and would thus be negligible.To reduce computational expense, the simulations were performed in a bounded,cylindrical spatial domain rather the unbounded domain exterior to a sphere inEq. 3. However, we expect that the agreement between stochastic simulationsand the PDE-ODE framework seen here extends to more general geometries,including the unbounded spherical geometry of Eq. 3. In fact, the cylindricaldomain in Eq. 28 can model a cylindrical region of height L = 1 µ m and radius2 R = 0 . µ m directly above a cell, where the base of the cylinder representsa flat patch of cell membrane with many receptors. In particular, for cells ofradius R ≥ µ m, the membrane curvature is negligible at the base of a cylinderof radius 2 R = 0 . µ m. Parameter ranges
Before discussing some biophysical implications of our analysis, we briefly dis-cuss parameter values. We do not seek precise values for any one specific appli-cation, but rather we choose ranges and orders of magnitude that are relevantacross multiple systems. Unless otherwise noted, the following “default” param-eter values are used in the figures and calculations below. The parameters aresummarized in Table 1.Cell radii range between roughly 0 . µ m for a small bacteria to 15 µ m fora large mammalian cell [31]. We set the default radius to be R = 1 µ m, whichis consistent with a bacterial cell or a small eukaryotic cell. We set the defaultdiffusion coefficient to be D = 10 µ m s − , which is the order of magnituderelevant for glucose uptake by an E. coli cell or a yeast cell [37–40] and chemo-taxis by bacterial cells and slime mold [2]. Following [2, 27], we take the radius14 arameter Default value Range of interest D = molecule diffusivity 10 µ m s − [10 , ] µ m s − R = cell radius 1 µ m [0 . , µ m ε = receptor to cell radius ratio 10 − N = number of receptors 10 [10 , ] κ rec = receptor trapping rate ∞ u = extracellular concentration 1 µ M [1 nM , k c = receptor turnover rate 10 s − [10 , ] s − k b = receptor unbinding rate 0 [0 , ] s − Table 1: Summary of parameter values and ranges. Unless otherwise noted,the “default” values are used in the figures and calculations. See the text fordetails.of each receptor to be εR = 1 nm and κ rec = ∞ , which means ε = 10 − and κ = εN/π . The number of receptors N on a cell can vary greatly [4, 41, 42], andwe take N = 10 as the default value. Extracellular concentrations of interestalso vary considerably. For example, a characteristic value in [2] is u = 1 µ M,the nutrient uptake study [27] considers u = 100 µ M, and other nutrient uptakestudies involve u on the order of 10 µ M or greater [37, 38, 40]. We follow [2]and take u = 1 µ M as the default value.Finally, the kinetic rate parameters k b and k c can also vary considerably.The typical turnover rate for sugar transporters is k c = 10 s − , with a rangeof k c ∈ [3 × s − , × s − ], though chloride-bicarbonate transporters canreach speeds on the order of k c = 10 s − [31]. We take k c = 10 s − as thedefault value. Breakup rates k b have been estimated on the order of 10 − s − [8],10 − s − [8, 43], 1 s − [43], and 10 s − [2]. Since most values satisfy k b (cid:28) k c ,we take k b = 0 as the default value for simplicity. Receptor kinetics can dominate uptake
In this subsection, we use our uptake formula ( J ∗ in Eqs. 23-24) to show thatfinite receptor kinetics play a dominant role in cellular uptake in some typicalbiophysical scenarios. In Fig. 3a, we plot the ratio J ∗ /J max as a function ofthe number of receptors N for different values of the receptor turnover rate k c .Note that J ∗ reduces to the Berg-Purcell flux, J bp , if k c is infinite (see Eq. 27).This figure shows that J ∗ is much less than J bp for typical values of the receptorturnover rate k c . For example, if k c = 10 s − and the rest of the parametersare the default values in Table 1, then J ∗ J bp ≈ . . (41)The reason for the significant discrepancy in Eq. 41 is that a large fraction ofmolecules that hit a receptor are blocked from binding because that receptor isoccupied by another molecule (this is incorporated into J ∗ but ignored by J bp ).15 a) 10 . . . . N (number of receptors) J ∗ / J m a x k c = 10 s − k c = 10 s − k c = 10 s − k c = 10 s − k c = ∞ ( J bp ) (b) 10 k c (receptor turnover rate) [s − ] N ∗ k c < ∞ k c = ∞ ( J bp ) 10 − − − f ( s u r f a ce f r a c t i o n ) Figure 3: (a) Cellular uptake as a function of the number of cell surface receptorsfor different turnover rates k c . (b) Number of receptors needed so that J ∗ = J bp ( N ∗ in Eq. 42) on the left axis as a function of k c . The right axis gives thecorresponding fraction of the cell surface covered by receptors ( f in Eq. 44).See Table 1 for parameter values.Indeed, at these default parameter values, Eq. 22 implies that more than 95%of the surface receptors are bound to a molecule at any given time. Hence, anymolecule that manages to hit a receptor has less than a 5% chance of bindingupon first contact with that receptor.It is also evident from Fig. 3a that J bp saturates at much smaller values of N compared to J ∗ . For example, increasing the receptor number from N = 10 to N = 10 increases J bp by a mere 27%, whereas this increase in the receptornumber increases J ∗ by more than 600% if k c = 10 s − . This is because inthe calculation of J bp , increasing the number of receptors merely increases thelikelihood that a single molecule hits a receptor rather than escaping to spatialinfinity. In contrast, if we include finite receptor kinetics, then increasing thenumber of receptors also increases the number of molecules that can be boundto the cell at any one time. 16o further investigate this point, in Fig. 3b we plot on the left axis thenumber of receptors, N ∗ , required for J ∗ to reach one half of J max as a functionof k c . That is, we plot N ∗ := πε + J max k c + πk b k c , (42)where the formula in Eq. 42 was found by setting J ∗ = J max / N . The corresponding number of receptors required for the Berg-Purcell flux, J bp , to reach J max / k c →∞ N ∗ = πε . (43)Using Eqs. 42-43, we see that J bp reaches one half of J max with N ∗ ≈ × receptors ( k c = ∞ ), whereas J ∗ requires N ∗ ≈ × receptors to reach onehalf of J max if k c = 10 s − .On the right axis of Fig. 3b, we plot the corresponding fraction of the cellsurface covered by receptors, f := π ( εR ) N ∗ πR = ε N ∗ ∈ (0 , . (44)Interestingly, f is very small as long as k c is not very slow. For example, f ≈ − = 0 .
1% for N ∗ = 3 × , and f = 10 − = 1% for N ∗ = 4 × .Therefore, the remarkable result of Berg and Purcell that a cell requires onlya small receptor surface fraction f in order to have uptake near the maximum J max still holds in the case of finite receptor kinetics.As mentioned in the Introduction, many previous works have sought to mod-ify and refine the Berg-Purcell formula to incorporate various details in theproblem [16–26]. It is therefore worth pointing out that the discrepancy be-tween J ∗ and J bp is much greater than some previous modifications of J bp . Forexample, Zwanzig posited the formula [15], J zw := εNεN + (1 − ε N/ π J max , in order to account for the effects of interference between receptors. However,the percentage difference between J bp and J zw is less than the dimensionlessreceptor radius ε [25], J zw − J bp J zw = N πε N ε + 4 π ≤ lim N →∞ N πε N ε + 4 π ≤ ε π < ε. That is, J zw and J bp typically differ by around one tenth of one percent, whereas J ∗ and J bp can differ by at least an order of magnitude in typical biophysicalscenarios. 17 − − − u (extracellular conc.) [ µ M] J ∗ [ s − ] k c = 10 s − k c = 10 s − k c = 10 s − k c = 10 s − k c = ∞ ( J bp ) Figure 4: Cellular uptake as a function of the extracellular concentration fordifferent turnover rates k c . The dashed horizontal lines are the maximum uptakerates for different turnover rates. See Table 1 for parameter values. Uptake is almost Michaelis-Menten
The discrepancy between J ∗ and J bp vanishes at low extracellular concentrationsand is magnified at high extracellular concentrations. In Fig. 4, we plot J ∗ and J bp as functions of the extracellular concentration u . Here, we see that J ∗ and J bp are indistinguishable at low concentrations (even for slow k c values), whereas J ∗ can be several orders of magnitude less than J bp at high concentrations.The explanation for this is straightforward. At sufficiently low concentrations,molecules arrive at receptors at a much slower rate than receptors can processmolecules, which is the assumption used to derive J bp [2]. Since the per receptorarrival rate is J bp /N and the receptor kinetic rates are k b and k c , a sufficientlylow concentration is defined by the following dimensionless number being muchless than one, ρ := J bp /Nk b + k c = κκ +1 πDRu N ( k b + k c ) (cid:28) . (45)As the concentration increases and Eq. 45 is violated, the arrival rate exceedsthe receptor kinetic rates, and thus receptor kinetics modify cellular uptake.It is intuitively clear that cellular uptake cannot increase linearly as a func-tion of u indefinitely, but rather uptake must saturate at some maximum rate.In light of this observation, a Michaelis-Menten functional form for the uptakerate is often posited [6], J mm := V max u K M + u , (46)for some maximum uptake rate V max and half-saturation constant K M . Westress that the uptake equation in Eq. 46 is not to be confused with the boundary18ondition in Eq. 20 which was derived to approximate the PDE-ODE system inEq. 16.The values of V max and K M in Eq. 46 are usually determined by matchingEq. 46 to experimental data. However, it is possible to relate V max and K M tomicroscopic biophysical parameters. First, it is clear that the maximum uptakerate must be V max = N k c . Next, if we force J mm to coincide with J bp at lowconcentrations (i.e. u (cid:28) K M ), then we must have K M = V max u J bp = N k c (1 + κ )4 πDRκ . (47)Now, it is straightforward to use Eqs. 23-24 to check that J ∗ has the desiredproperty that it saturates at N k c at high concentrations,lim u →∞ J ∗ = N k c , and J ∗ reduces to J bp at low concentrations,lim u → J ∗ /J bp = 1 . Hence, J ∗ and J mm agree at high and low concentrations. However, it is ev-ident from the formula for J ∗ in Eqs. 23-24 that J ∗ (cid:54) = J mm at intermediateconcentrations.In Fig. 5a, we plot J ∗ (solid curves) and J mm (dotted curves) as functionsof the concentration for k b = 0. While J ∗ does not have the exact Michaelis-Menten functional form as in Eq. 46, this plot shows that the J ∗ curve does havea profile very similar to the profile generated by the Michaelis-Menten functionalform (i.e. a sigmoidal curve). This is an important feature of the formula for J ∗ ,since this profile is observed in experiments measuring cellular uptake [6,37–39].Furthermore, it is straightforward to use the formula in Eqs. 23-24 for J ∗ tofind the half-saturation constant (i.e. the “apparent K M ” of J ∗ ). Indeed, solvingthe equation J ∗ = V max / u gives the half-saturationconstant, K ∗ M := N k c (1 + κ/ πDRκ + N k b πDRκ . (48)Comparing Eq. 48 with Eq. 47, we see that these two half-saturation constantsare similar if k b (cid:28) k c . Indeed, the close agreement between J ∗ and J mm inFig. 5a results from taking k b = 0. However, it follows from comparing Eq. 48and Eq. 47 that taking k b (cid:54)(cid:28) k c can make J mm saturate at much lower concen-trations than J ∗ . This is illustrated in Fig. 5b, where we plot J ∗ and J mm with k b = 10 k c . Conclusion
We have developed a framework for modeling molecular species which diffuse ina three-dimensional bulk region and interact with receptors embedded on a two-dimensional surface. The receptors bind and process molecules at finite kinetic19 a) 10 − − − . . . . u (extracellular conc.) [ µ M] J / V m a x k c = 10 s − k c = 10 s − k c = 10 s − k c = 10 s − (b) 10 − − − . . . . u (extracellular conc.) [ µ M] J / V m a x k c = 10 s − , k b = 10 k c , J ∗ k c = 10 s − , k b = 10 k c , J mm Figure 5: Cellular uptake as a function of the extracellular concentration fordifferent turnover rates k c , with breakup rate k b = 0 in panel (a) and k b = 10 k c in panel (b). The solid curves correspond to J ∗ in Eqs. 23-24 and the dottedcurves correspond to J mm in Eq. 46. See Table 1 for other parameter values.rates, which introduces significant statistical correlations between the individualdiffusing molecules. We developed the framework in the context of the Berg-Purcell cellular uptake model [2]. We found that in some typical biophysicalscenarios of interest, finite receptor kinetics can reduce cellular uptake by at leastan order of magnitude compared to the Berg-Purcell estimate. The predictionsof our analysis were confirmed by numerical simulations of a many particlestochastic system.Mathematically, the framework uses the theory of boundary homogenizationto couple a PDE (the diffusion equation) to nonlinear ODEs on a boundary. Ina certain parameter regime (or at steady-state), the boundary conditions canbe reduced to a nonlinear, Michaelis-Menten flux condition. Several interestingprior works have used PDEs with ODE boundary conditions to model reaction-diffusion systems [44–50]. These prior works have generally studied Turing20atterns and spatiotemporal oscillations. To estimate how receptor diffusion andcellular rotation influence cellular uptake, Ref. [25] used the diffusion equationwith boundary conditions described by stochastic differential equations.Previous works have employed mathematical models to study how finite re-ceptor kinetics affect diffusive uptake. Refs. [11, 12] formulated and analyzedstochastic models of diffusive interactions with receptors that must wait a tran-sitory “recharge” time between captures. In Ref. [11], it was proven that sucha recharge time can drastically reduce the number of captured molecules (thenumber of captures grows logarithmically in the number of total molecules versuslinear growth in the absence of recharge). Ref. [12] used a variety of stochasticmodels to analyze similar systems. There is a rather large literature on cellularnutrient uptake which takes the Michaelis-Menten uptake equation in Eq. 46as its starting point [6, 37–39]. The maximum uptake rate and half-saturationconstant (i.e. V max and K M ) are chosen to fit experimental uptake rates. Ouruptake formula ( J ∗ in Eqs. 23-24) does not have the Michaelis-Menten func-tional form in Eq. 46, but it nevertheless exhibits the same sigmoidal uptakecurve as a function of the extracellular concentration.While we developed the framework in the context of cellular uptake in aspherical geometry, it can be applied to other systems with potentially differentgeometries and receptor kinetic schemes. For example, synaptic transmission in-volves neurotransmitter molecules diffusing across the synaptic cleft and bindingto receptors on the adjacent neuron [5]. In this case, the shape of the synapticcleft is similar to a cylinder and the framework of the present paper could beused to investigate the effect of the finite kinetic rates of neural receptors. Asanother example amenable to the present framework, cylindrical domains withreceptors on the “sides” have been used to model catheter-based drug deliverysystems [51].Finally, we conclude by discussing the important study by Wagner et al. [27]on nutrient uptake by bacterial cells. This work used fluorescent tracing toshow that bacterial cell “stalks,” which are long and thin extensions of the cellenvelope, can bind and import nutrients from the extracellular environment.These authors then used novel mathematical analysis to generalize the Berg-Purcell model to a domain exterior to a stalk (modeled by a prolate spheroid).Based on this analysis, the authors argued that the stalk morphology increasesnutrient uptake compared to a sphere. For future work, it would be interestingto extend the analysis in the present paper to the geometry considered in [27]to investigate the effect of finite receptor kinetics.Indeed, the model in [27] assumes that receptors can absorb nutrient moleculescontinuously. As in Berg-Purcell, this assumption is valid if receptors can im-port molecules at a much faster rate than molecules tend to arrive at receptors.We now use the parameter ρ in Eq. 45 to compare these two rates in [27], where ρ (cid:28) N = 10 perfectly absorbing receptors (meaning κ rec = ∞ inour notation). The radius of each receptor was 1 nm and stalk lengths variedfrom 1 µ m to 10 µ m. The extracellular concentration used in the experimentswas u = 100 µ M. If we take these values and set R = 1 µ m, D = 10 µ m s − ,21nd k b = 0 in Eq. 45, then the receptor turnover rate k c would need to satisfy k c (cid:29) . × s − in order to have ρ (cid:28)
1. That is, a single receptor would need to import moleculesat a rate much faster than 10 s − , whereas characteristic rates are on the orderof 10 s − [31]. While this simple calculation ignores the stalk geometry, itnevertheless suggests that finite receptor kinetics may play an important role inthe uptake rate for the system studied in Ref. [27]. More broadly, the frameworkdeveloped in the present paper provides a method for investigating how receptorkinetics affect a variety of biophysical systems. Acknowledgments
The first author was supported by The Swartz Foundation. The second authorwas supported by the National Science Foundation (Grant Nos. DMS-1944574,DMS-1814832, and DMS-1148230).
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