First-passage time to clear the way forreceptor-ligand binding in a crowded

environment

Jay Newby∗ Jun Allard†

March 8, 2016

AbstractCertain biological reactions, such as receptor-ligand binding at cell-

cell interfaces and macromolecules binding to biopolymers, require manysmaller molecules crowding a reaction site to be cleared. Examples includethe T cell interface, a key player in immunological information processing.Diffusion sets a limit for such cavitation to occur spontaneously, therebydefining a timescale below which active mechanisms must take over. Weconsider N independent diffusing particles in a closed domain, contain-ing a sub-region with N0 particles, on average. We investigate the timeuntil the sub-region is empty, allowing a subsequent reaction to proceed.The first passage time is computed using an efficient exact simulationalgorithm and an asymptotic approximation in the limit that cavitationis rare. In this limit, we find that the mean first passage time is sub-exponential, T ∝ eN0/N2

0 . For the case of T cell receptors, we find thatstochastic cavitation is exceedingly slow, 109 seconds at physiological den-sities, however can be accelerated to occur within 5 second with only afour-fold dilution.

Diffusion drives many biological processes, both positively, by deliveringcargo to a target, and negatively, by removal of cargo from a region of in-terest (ROI). While the temporal dynamics of diffusional delivery have beenextensively studied [5, 4, 24, 12], diffusion-driven removal has been less charac-terized experimentally or theoretically [3]. Removal is of particular interest inthe crowded environment of cells, where large biomolecules and cellular struc-tures require the displacement of smaller molecules, a phenomenon we termstochastic cavitation.

A specific example arises in the study of cell-cell interfaces including theT-cell/antigen-presenting-cell interface [22, 2, 29, 8] (see Fig. 1). A fundamen-tal question for all cell-cell interfaces is how receptors and ligands come into∗Department of Mathematics, University of North Carolina, Chapel Hill, 329 Phillips Hall,

Chapel Hill, NC 27599†Department of Mathematics, University of California, Irvine, 340 Rowland Hall, Irvine,

CA 92697

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contact, despite being separated by large molecules, the extracellular fluid, andother structures in the glycocalyx. On either cell surface, large molecules suchas CD45 and LFA-1 undergo 2D diffusion in the cell membrane with a diffusioncoefficient of D ∼ 0.1µm2/s [27, 6]. These large molecules impair interactionsbetween smaller pairs of molecules, such as the T cell receptor and its ligand—akey step in immunological information processing and decision-making. It hasbeen estimated that a region of radius ∼ 100nm, devoid of large molecules, isnecessary for spontaneous T cell receptor interaction [2], which is occupied byon average ∼ 30 particles at equilibrium. A natural question is whether thisempty region can form spontaneously in a biologically relevant time. Under-standing contact formation will address cell-cell interactions in the crowded,heterogeneous environment inside organisms and what machinery is necessaryfor general cell-cell interactions [15, 31].

Examples of diffusional cavitation in biology also arise in other dimension-alities. In 1D, microtubules (inflexible polymers of the protein tubulin) aredecorated by hundreds of microtubule-associated proteins [1, 28]. These pro-teins exhibit significant crowding [11] and lateral diffusion along the microtubulelattice [13, 17]. Large microtubule-binding molecules may therefore have to waitfor a region to be clear before binding. What is the mean time for such clearance,and is it the rate-limiting step in microtubule binding? A similar situation oc-curs for DNA and the myriad of DNA-binding molecules, some of which undergolateral diffusion across base pairs [16, 25]. A significant waiting-time for largeDNA-binding molecules has potential implications for the study of the chemicalmodification of DNA and RNA, all of which require an enzyme to attach to thepolymer.

For some of the above scenarios, it has been hypothesized that clearance ofthe target region requires an active process [2, 18]. To address the feasibilityof passive diffusion-driven cavitation, a theoretical assessment of the timescalesinvolved is needed. In other words, can diffusion-driven cavitation reliably oc-cur on biologically relevant timescales? To address this question, we considerN independent particles undergoing simple diffusion in either the 1D domain

Figure 1: Cell-cell interface formation between a T-cell and an antigen-presenting cell. CD45 molecules (black) block the receptor-ligand (purple) bondfrom forming while they inhabit the ROI (orange).

2

(−L,L) or the 2D domain (−L,L)2. In each case, we study the first-passagetime until a smaller region, a disk of radius L0, is empty.

The cavitation event can be rare (i.e., the first passage time can be verylarge compared to the diffusion timescale L2/D) under certain circumstances.To understand this, consider the 1D domain with N particles. At equilibrium,each particle has a uniformly distributed postion within the domain. In thelimit L → ∞ and N → ∞ with the average particle density ϕ = N/(2L)fixed, the equilibrium probability of finding a region of radius L0 containing noparticles is small, P = e−N0 , where N0 = 2Lϕ. Therefore, when N0 1 weexpect cavitation to be a rare event. Although we might expect the 1D meanfirst-passage time (MFPT) to scale as1 T ∝ P−1 ∼ eN0 , we instead we find anasymptotic scaling of T ∝ eN0/N2

0 .In this Letter, we develop a simulation algorithm to efficiently generate ex-

act realizations of the first passage time, based on Green’s function reactiondynamics [26]. For situations where cavitation is a rare event and computa-tion becomes unfeasible, i.e., when N is very large or the ROI occupies most ofthe explorable area, we develop an asymptotic approximation of the mean firstpassage time.

Consider N independent random walkers Yn(t), with n = 1, · · · , N , that areconfined to the interval −L < y < L. The ROI is the inner domain centered atthe origin with radius L0 < L. The event we wish to characterize is the first timeat which the ROI is empty (i.e., minnYn(t) = L0). We first nondimensionalizethe problem using the space scale L and the time scale L2/D, where D is thediffusion coefficient. We define the nondimensional distances Rn = |Yn| /L,ε = 1 − L0/L, and l0 = L0/L. Then, a given particle is inside the ROI if0 < Rn < l0.

By formulating a simulation algorithm, we can generate exact samples of thefirst passage time. We take advantage of explicit formulas for the probability dis-tributions that govern single particle Brownian motion in a closed domain. Notethat even though we focus on the 1D and 2D cavitation problem in this letter,exact distributions are also known for 3D Brownian motion [7]. The algorithmproceeds as follows. Given a set of random starting positions Rn(t0)1≤n≤N ,select a particle that is inside the ROI and closest to the origin. That is, selectRm = minRn < l0. The first step is to compute the first time τ at whichthe selected particle leaves the ROI (i.e., Rm(t0 + τ) = l0). Once τ has beencomputed, set t′ = t0 + τ . The cavitation event cannot have occurred beforetime t′ because we are certain that Rm(t) < l0 for all t0 < t < t′. Therefore,the position of the other particles between time t0 and time t′ is irrelevant, weneed only generate the random position for each of the remaining particles attime t′. Once all positions have been updated, select a new Rm = minRn(t′).We know that the cavitation event has occurred if Rm ≥ l0. If Rm < l0, then

1This problem is equivalent to the diffusion of a particle in ND dimensions (the productof number of particles and dimensionality of space). Since ND 2, this Brownian motionis not recurrent, so we might naively expect the system to be well-mixed in ND-dimensionalphase space, and the rate of first passage would be the attempt rate times the probability ofbeing in the target state.

3

set t0 = t′ and repeat the above procedure.At each step, the jump times τ can be sampled from the exact distribution

f(τ |r0), obtained from the fundamental solution to the diffusion equation witha reflecting boundary at r = 0 and an absorbing boundary at r = l0. Therandom positions can be sampled from the distribution p(r|r0, τ), satisfyingthe diffusion equation with reflecting boundaries at r = 0 and r = 1. Anefficient way of sampling from p is to use a rejection method, similar to theone described in [26]. For the jump time sampled from f , we found that therejection method could not easily be adapted to our situation. Instead, wesample the jump time using a root finding algorithm. Additional details areprovided in Supplementary Material. The simulation algorithm is maximallyfast in the sense that only the (average) slowest particle determines the nextevent time, allowing us to efficiently access densities around φL0 ∼ 13. We findthis is sufficiently high to validate our asymptotic approximations.

To obtain a complete picture of cavitation in the rare event limits, we developan asymptotic approximation for the MFPT, T . The approximation is derivedfor 1D cavitation, and based on simulations, we observe that in the limit L→∞with a fixed particle density, the approximation is also surprisingly accurate for2D cavitation. We first state the main results (Eqs. 46-37) and then summarizetheir derivation.

For fixed N , the first term in the asymptotic approximation for 0 < ε 1of the MFPT, averaged over a uniformly distributed initial position for each ofthe N particles, is given by

T ∼ 2NAN(CN ε)N−2

+O(1), N ≥ 3 (1)

where

AN =Γ(N2 )

2πN2 (N − 2)

. (2)

The constant CN is the Newtonian capacitance of a hypercube in RN ; as ex-plained below, it determines the far field behavior of certain solutions to Laplace’sequation [32]. An explicit formula for the Newtonian capacitance of a cube forN > 2 is unknown. However, a good approximation for N = 3 is C3 ≈ 1.3214[19]. The ε 1 approximation (solid line) is compared to simulations (symbols)in Fig. 2. For N = 3, we find good agreement between simulation and the inde-pendently derived estimate for C3 from [19]. From physical arguments detailedat the end of this letter, we have determined an expansion of the Newtoniancapacitance for large N given by

CN ∼√

2N

πe

(1 +

3 logN

2N+α2

N+O(N−2)

). (3)

The unknown constant in the above expansion is independent of all parame-ters. Using the exact simulation algorithm, we obtain the numerical estimate,α2 ≈ −1.67. Our MFPT calculation thus provides an approximation for thecapacitance CN , which otherwise remains challenging to compute [19].

4

0.5 0.6 0.7 0.8 0.9 1.0

L0/L

10-1

100

101

102

MFP

T T

N=5

N=4

N=3

Figure 2: The MFPT (in nondimensional units) as a function of L0/L = 1− ε.The small ε approximation (solid lines) is compared to simulations (symbols),using C3 = 1.3214, C4 = 1.44, and C5 = 1.55.

For fixed 0 < ε < 1, an asymptotic expansion for N 1 is given by

T ∼ κ1D

N2εN−2, N 1, (4)

where κ1D ≈ 2.2 depends only on α2 (via Eq. 19). The N 1 MFPT approxi-mation is compared to simulations in Fig. 3.

5 10 15 20 25 30 35 40 45

N

10-2

10-1

100

101

102

103

MFP

T T

L0/L=0.4

L0/L=0.3

L0/L=0.2

Figure 3: The large N MFPT approximation (solid lines) compared to simula-tions (symbols).

Finally, we consider the case the radius of the ROI L0 is fixed and L → ∞with a fixed number of particles per unit length ϕ = N/(2L). Let N0 be theaverage number of particles in the ROI. The L→∞ MFPT approximation (indimensional units) is

T∞ ∼κ1DL

20eN0

N20D

, N0 1. (5)

5

The MFPT is shown in Fig. 4 as functions of N0, for different values of L.The approximation (37) matches closely with the L/L0 = 33.3 simulations for

6 7 8 9 10 11 12 13

N0

101

102

103

104

MFP

T (L/L

0)2T

L/L0 =4.17

L/L0 =8.33

L/L0 =25

L/L0 =33.3

L/L0 =∞

Figure 4: The MFPT vs N0, the average number of particles in the ROI. Thesymbols indicate 103 averaged simulations; the 1D simulations are shown as cir-cles and the 2D simulations are shown as diamonds. Also shown is the L→∞approximation (dashed curve) for both 1D and 2D. Note that time is nondimen-sional using the L2

0/D timescale.

N0 > 10. Hence, the effect of a small domain size compared to the ROI isto increase the MFPT, making the cavitation event more rare. This providesa quantitative measure of when the domain size L no longer influences thecavitation event, which is relevant when, for example, considering cavitation ona relatively long strand of DNA compared to a shorter plasmid. Microtubulefilaments also vary in length.

Simulations of 2D cavitation are also shown in Fig. 4 as diamond symbols.Although the asymptotic approximation (37) is derived for 1D, we find that it isa remarkably good fit to the simulation data after changing a single parameter:the prefactor κ2D ≈ 0.7. We therefore infer that cavitation is roughly threetimes faster in 2D than in 1D. Heuristically, this speed-up occurs because themean time for a random walker to escape a spherical region decreases withdimensionality.

The asymptotic approximations (46)-(37) for 1D cavitation are derived asfollows. Because all of the N walkers are independent, the problem can bereformulated as the first passage time of a single random walker in a N di-mensional domain. Define the domain Ω ≡ (0, 1)N , and let Ωε ≡ (1 − ε, 1)N

be the small target domain. The random process R(t) ∈ Ω \ Ωε representsthe original process with R(t) = (R1(t), · · · , RN (t)). Define the MFPT as

6

T ≡ 〈inft > 0 : min1≤n≤N Rn(t) = l0〉; it satisfies

N∑n=1

∂2T

∂r2n

= −1, r ∈ Ω, (6)

∂ηT (r) = 0, r ∈ ∂Ω, (7)T (r) = 0, r ∈ ∂Ωε. (8)

An approximate solution to (6) can be obtained using the method of matchedasymptotics [33, 10, 30, 9, 20]. We split the solution into two parts: an inner andouter solution. The inner solution satisfies the absorbing boundary conditionon ∂Ωε and ignores the reflecting boundary. The outer solution satisfies thereflecting boundary on ∂Ω and is singular as r→ (1, · · · , 1). The two solutionsare then matched to obtain a uniformly accurate approximation using the Van–Dyke matching principle [23].

Define the inner coordinates z = r−rbε , and let z = ‖z‖. The inner solution

satisfies∆zw = 0, w(z ∈ ∂ZN ) = 0, (9)

where ZN is the unit hypercube. The exact solution to the inner problem forarbitrary N is unknown. However, from electrostatics [21], for large z, the innersolution has the two term expansion,

w ∼ BN (ε)[(z/CN )

2−N − 1], N ≥ 3. (10)

where BN is a constant determined by matching to the outer solution. Theconstant CN , called the Newtonian capacitance, is a boundary dependent termdiscussed below.

Up to an unknown constant T , the outer solution is

Tout ∼ −GN (r, rb) + T , (11)

where the Green’s function GN satisfies,

N∑n=1

∂2GN∂r2n

= 1− δ(r− r′), r ∈ Ω, (12)

∂ηGN (r, r′) = 0, r ∈ ∂Ω, (13)∫Ω

GN (r, r′)dr = 0. (14)

By integrating (11) over Ω using (36), we find that T is the MFPT averagedover a uniformly distributed set of initial positions, i.e., T = T . Again fromelectrostatics [21], in the limit rn → 1 with r′n = 1 and ‖r − r′‖ = εz, theGreen’s function scales like

GN ∼ 2NAN (εz)2−N +O(1), N ≥ 3, (15)

7

where AN is given by (45).Matching the inner and outer solutions we find that the z dependent terms

match provided that BN (ε) = −2NAN ε2−N for N ≥ 3. The remaining unknown

term T yields the approximation (46).In order to access the rare event limit where both N 1 and ε 1, we

must find how the Newtonian capacitance CN scales with N . This problem hasno known exact solution for N > 2 [19].

If the cuboid boundary ∂Ωε were replaced by a spheroid with the samehypervolume, then the Newtonian capacitance is known for general N ,

CN ≈2√π

Γ

(1 +

N

2

)1/N

∼√

2N

πe. (16)

We therefore propose a general expansion of CN (for the present case of cuboidboundary) having the same form as the large-N expansion of (16),

CN ∼√

2N

πe

(1 +

α1 logN

N+α2

N+O(N−2)

). (17)

Note that (16) and (17) have the same leading-order term.To elucidate how the unknown constants α1,2 affect the large N MFPT

approximation, we use Stirling’s formula, leading to

2NAN

CN−2N

∼ κ1D

Nβ, N 1, (18)

whereβ = α1 + 1/2, κ1D =

2√πeα2+1

. (19)

In dimensional units, the MFPT approximation is

T ∼ L2κ1D

NβD

(1− L0

L

)2−N

. (20)

We determine the value of α1 by exploiting a physical constraint as follows. AsL → ∞ with the density of particles ϕ = N/(2L) held constant, the MFPTmust converge to a finite value. Substituting L = N/(2ϕ) and N0 = 2L0ϕ into(20) yields

T ∼ N2−βκ1D

4ϕ2D

(1− N0

N

)2−N

. (21)

Since limN→∞(1− N0

N

)2−N= eN0 , we must have that β = 2 (and therefore

α1 = 3/2) in order for (44) to converge to a finite, nonzero value in the limit(L,N)→∞. We also find that β = 2 is supported by numerical simulations (seeSupplementary Material). The limiting result is the approximation Eq. (37).

While the approximation matches well with simulations in 2D, a more sys-tematic asymptotic analysis for the 2D case should be feasible. For small ε and

8

finite L, the leading order in (46) holds in 2D. A notable feature of our 1D caseis that there are no terms in the expansion between the leading order term andthe O(1) term, making our 1D approximation converge particularly fast. Thisfeature is lacking in 2D, where there are other terms singular in ε, therefore weexpect this approximation to converge more slowly. In 2D for large L at con-stant density, a different scaling between N and L prevents the approximationin (44) from converging, necessitating an alternative strategy that will be thesubject of future research.

Returning to the specific question of cell-cell contact at T cell interfaces,large diffusing molecules such as CD45 disfavor proximity between receptorsand ligands on apposing cells. These molecules have diffusion coefficients ofD ≈ 0.1µm2/s [27] and density such that on average there areN0 = 30 moleculesin the 100-nanometer ROI [2]. The approximation (37), using the prefactorκ2D = 0.7 from the numerical fit to simulations, yields an estimate of T ≈109 seconds. In contrast, the MFPT for a single particle to escape a circulardomain is T = L2

0/(4D) = 0.025 seconds. Since T cell receptor triggeringoccurs within seconds [14], the above calculation predicts that receptor-ligandbinding must involve a mechanism faster than passive diffusion. We thereforesuggest the alternative hypothesis that an active force drives receptor-ligandproximity [2]. To obtain an empty ROI spontaneously in less than five seconds,we would require N0 ≤ 7, corresponding to a four-fold dilution, which could beexperimentally accessible. The biological system is complicated by interactionsof large molecules within and between molecular species, lipid heterogeneity,and transient immobilization, all of which could be exploited to dynamicallytune the rate of ligand binding and will be studied by expanding the presentframework.

1 AcknowledgmentsJN was supported by a NSF-funded postdoctoral fellowship (NSF DMS-1100281,DMS-1462992). JA was supported by a NSF CAREER award (DMS-1454739).

A Simulation algorithmThe exact simulation algorithm makes use of two solutions to the 1D diffusionequation. Let pa,r(x, x0, t) be solutions to

∂p

∂t=∂2p

∂x2, 0 < x, x0 < xa,r (22)

∂p

∂x= 0, x = 0 (23)

p(x, x0, 0) = δ(x− x0), (24)

with two different right boundary conditions. Let pa be the solution with anabsorbing BC at xa = l0 = 1−ε. This solution is used to derive f , the jump time

9

distribution. Let pr be the solution with a reflecting BC at xr = 1. This solutionis used to generate the random position of each particle given a jump time. Inboth cases, the solution is represented as an infinite series. Two different seriesrepresentations are derived for each solution: one that converges quickly forshort times and one for long times.

Letan =

π

l0(n− 1/2), bn = nπ. (25)

For large times, we have

pa(x, x0, t) =2

l0

∞∑n=1

cos(anx) cos(anx0)e−a2nt, (26)

and

pr(x | x0, t) = 1 + 2

∞∑n=1

cos(bnx) cos(bnx0)e−b2nt. (27)

For short times we have

pa(x, x0, t) =1√4πt

∞∑n=0

(−1)n(e−

(2l0n+(x+x0))2

4t − e−(2l0(n+1)−(x+x0))2

4t

+e−(2l0n+|x−x0|)

2

4t − e−(2l0(n+1)−|x−x0|)

2

4t

)(28)

and

pr(x, x0, t) =1√4πt

∞∑n=0

(e−

(2n+(x+x0))2

4t + e−(2(n+1)−(x+x0))2

4t

+e−(2n+|x−x0|)

2

4t + e−(2(n+1)−|x−x0|)

2

4t

)(29)

For short times, the first passage time density is

f(t | x0) = − ∂

∂xpa(l0, x0, t)

=4π

(4πt)3/2

∞∑n=0

(−1)n

(l0(2n+ 1) + x0)e−(l0(2n+1)+x0)2

4t

+(l0(2n+ 1)− x0)e−(l0(2n+1)−x0)2

4t

(30)

with the cumulative distribution,

F (t | x0) = 1 +

∞∑n=0

(−1)n

erf(l0(2n+ 1) + x0√

4t) + erf(

l0(2n+ 1)− x0√4t

)

(31)

10

For long times, the first passage time density and cumulative distribution are

f(t | x0) = − 2

l0

∞∑n=1

(−1)nan cos(anx0)e−a2nt, (32)

F (t | x0) = 1 +2

l0

∞∑n=1

(−1)n cos(anx0)e−a

2nt

an. (33)

The jump time is sampled using a standard root finding algorithm. Given auniform random variable U , the jump time is the unique solution to

F (t | x0)− U = 0. (34)

The distribution pr can be sampled using a rejection method as follows. Amajoring function C(x) must be chosen such that C(x) > p(x | x0, t) for allx ∈ (0, 1). A tentative value X is sampled from the distribution

P (x) =C(x)∫ 1

0C(x)dx

. (35)

A second random variable is drawn according to Y = C(X)U , where U is a unituniform random variable. If Y > pr(X | x0, t), then the sample X is rejected.The procedure is repeated until a sample is accepted.

For the long time expansion (27) we selectX to be a uniform random variablein (0, 1) and set

Y =

(1 + e−π

2t

1− e−π2t

)U. (36)

For the short time expansion (29) we select X to be a normal random variablewith mean x0 and variance

√2t. Note that care must be taken to ensure that

0 < X < 1. In this case,

Y = e−(X−x0)2/(4t) U√πt. (37)

A.1 2D simulationsFor 2D cavitation, the outer boundary is a square of side length 2L. Thisgeometry allows us to reuse the jump propagator from the 1D algorithm toupdate positions. The x and y coordinate of each particle are updated fromseparate samples of the 1D propagator pr as described in the previous section.The jump times are generated from the 2D distribution of first passage timesto the boundary of a circle.

The 2D first passage time problem is

∂

∂tp(r, t | r0) =

1

r

∂

∂r

(r∂p

∂r

), (38)

p(r, 0 | r0) =δ(r − r0)

2πr0, (39)

p(l0, t | r0) = 0. (40)

11

The solution can be written as an expansion in Bessel functions. The solutionis

p(r, t | r0) =2

l0

∞∑j=1

J0(rβn)J0(r0βn)

J1(l0βn)2e−β

2nt, (41)

where αn are the roots of J0(αn) = 0, and βn = αn/l0. The jump time densityfunction is

f(t | r0) = − ∂

∂rp(l, t | r0) =

2

l0

∞∑j=1

βnJ0(r0βn)

J1(l0βn)e−β

2nt, (42)

and the cumulative distribution is

F (t | r0) = 1− 2

l0

∞∑j=1

J0(r0βn)

βnJ1(l0βn)e−β

2nt. (43)

We use a root finding method to sample the jump time.

B Parameter estimationWe use maximum likelihood to estimate parameter values in the large N expan-sion of the Newtonian capacitance of a hypercube. We exploit the one to onecorrespondance between α1 and β and between α2 and κ. The parameters βand κ are estimated using realizations of the first passage time. The likelihoodfunction is computed by assuming that the first passage time is an exponentiallydistributed random variable with mean

T ∼ κ

NβεN−2. (44)

This assumption is valid asymptotically as N →∞ when the first passage timeis a rare event. The likelihood function for β and κ from n iid samples τk,k = 1, · · · , n is given by

P (τk | β, κ) = exp

[−n(TnTβ,κ

+ log Tβ,κ

)], (45)

where

Tn =1

n

n∑k=1

τk. (46)

Two data sets were generated for N = 20, 25, 30, · · · , 115, 120 with ε = 10−5/N

and ε = 10−6/N . A value of Tn was generated for each parameter set using104 samples of the first passage time from the exact simulation algorithm. Wenumerically computed the maximum of the product of the likelihood functionsfrom all parameter values. The likelihood functions were computed on a 500×500 grid for 1.5 < β < 2.5 and 1.5 < κ < 3. The resulting maximizers were

12

0.00 0.02 0.04 0.06 0.08 0.10

1/N

101

102

103T

ε=10−5/N

ε=10−6/N

0.00 0.02 0.04 0.06 0.08 0.10

1/N

10-5

10-4

10-3

10-2

10-1

100

rela

tive e

rror

Figure 5: The MFPT approximation compared to Monte Carlo simulation esti-mates. Each symbol shows the sample mean of 104 simulations.

β ≈ 2.00 and κ ≈ 2.19. As shown in Fig. 5, T and Tn are in good agreement withthese parameter values. The corresponding parameter values in the capacitanceexpansion are α1 ≈ 3/2 and α2 ≈ −1.67. The capacitance approximation isshown in Fig. 6 compared to numerical estimates. From the expansion, weexpect the error (given by the absolute difference divided by

√N) to scale like

1/N2 as N →∞. We find good agreement between the error and 15/N2.

20 40 60 80 100 120

N

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

CN

asymptotic

numerical

20 40 60 80 100 120

N

10-4

10-3

10-2

10-1

error

15/N2

Figure 6: The Newtonian capacitance of a hypercube.

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