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Effect of Hydrodynamic Interactions on Reaction Rates in Membranes
Article
Effect of Hydrodynamic Interactions on Reaction Rates in Membranes Naomi Oppenheimer1,* and Howard A. Stone1,* 1
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey
ABSTRACT The Brownian motion of two particles in three dimensions serves as a model for predicting the diffusion-limited reaction rate, as first discussed by von Smoluchowski. Deutch and Felderhof extended the calculation to account for hydrodynamic interactions between the particles and the target, which results in a reduction of the rate coefficient by about half. Many chemical reactions take place in quasi-two-dimensional systems, such as on the membrane or surface of a cell. We perform a Smoluchowski-like calculation in a quasi-two-dimensional geometry, i.e., a membrane surrounded by fluid, and account for hydrodynamic interactions between the particles. We show that rate coefficients are reduced relative to the case of no interactions. The reduction is more pronounced than the three-dimensional case due to the long-range nature of two-dimensional flows.
INTRODUCTION Many chemical reactions take place on the surface of a membrane—catalysis by integral membrane proteins (1), enzymatic activation of phospholipases (2), and electron transfer aided by the diffusion of quinone (3), to name a few. It has therefore been of interest to model the kinetic rate in membranes. Kinetic rates in three dimensions (3D) were first studied by von Smoluchowski (4) and verified in many diffusionlimited experiments (see, e.g., (5–7)). Extensive generalizations have been reported since. For example, Deutch and Felderhof (8) included hydrodynamic interactions in the calculation. For other generalizations, see (9) and the numerous references within. In particular, two-dimensional (2D) diffusion was studied by, among others, Berg and Purcell (10), Keizer (11), Adam and Delbruck (12), and Linderman and colleagues (13,14). In this work, we extend the theory of kinetic rates in 2D to include hydrodynamic interactions between reactants and targets in a membrane. Membrane hydrodynamics is a well explored field. In the pioneering work of Saffman and Delbr€ uck (15), the dependence of the diffusion coefficient of membrane proteins on protein size was found to be logarithmic. Many researchers have extended the theory to account for different scenarios, a few examples of which are diffusion of large objects (16);
the effect of tension (17) or of intermonolayer friction (17–19) on membrane hydrodynamics; diffusion in membranes near a rigid surface (18,20,21), or with momentum decay (22); diffusion of needle-shaped inclusions (23,24); viscoelastic membranes (25); membranes surrounded by viscoelastic material (26); or on curved membranes (19,27,28). In unbounded 2D hydrodynamic problems, a point force (per length) perturbation decays logarithmically, much more slowly than the 1=r decay in 3D. Hydrodynamic interactions thus play a significant role in 2D, a role which will manifest itself in the kinetic rate of reactions taking place on a membrane. The next parts of this article are organized as follows. In Simple diffusion in 2D, we present the basic calculation of kinetic rates as applied to 2D. Hydrodynamic interactions introduces the formalism of including hydrodynamic interactions, whereas in Hydrodynamic interactions in membranes, we specialize to the case of 2D and show that hydrodynamics plays a central role due to the long-range nature of the response. In Constant flux from 3D, we examine a slightly different initial condition, in which reactants are coming from the 3D fluid surrounding the membrane. METHODS Simple diffusion in 2D
Submitted January 9, 2017, and accepted for publication June 7, 2017. *Correspondence: [email protected] or [email protected] Editor: Ana-Suncana Smith. http://dx.doi.org/10.1016/j.bpj.2017.06.013 Ó 2017 Biophysical Society.
440 Biophysical Journal 113, 440–447, July 25, 2017
In this section, we will re-derive the kinetic rate in a 2D system. We follow previous work by Berg and Purcell (10), Adam and Delbr€uck (12), Linderman and colleagues (13,14), and others. Since we will eventually be interested in the hydrodynamic interactions between targets and reactants, we
Reaction Rates in Membranes will treat a scenario where both targets and reactants can move. This approach will clarify the derivation presented in Hydrodynamic interactions. Let us consider a case in which the limiting step of a reaction is the time for the reactants to diffuse toward a target, i.e., here and in the rest of this work, we consider diffusion-limited reactions. We note that the discrete nature of binding/unbinding kinetics, in conjunction with random walks, was treated recently by Zimmermann and Seifert (29). Here, we assume that there is a separation of timescales that justifies the diffusion-limited assumption. The concentration of reactants is much higher than that of targets, such that targets are assumed to be independent of each other. A reactant and a target are further assumed to react instantaneously when they are within a radius r0 , the reaction radius, which here we take as the sum of the radii of the reactant and target. The reaction rate will be given by the flux of reactants toward the target taken at the reaction radius. The target therefore acts as a sink of size r0 , which provides an inner boundary condition for the diffusion problem considered below. The second boundary condition should be chosen with care, as in Constant flux from 3D we show how a choice of a different boundary condition changes the kinetic rate. For now, we adopt a strategy similar to that of Lauffenburger and Linderman (see section 4.2.3 in (14)): we assume that a reactant positioned at an equal distance to two targets will have equal probability of reaching pffiffiffiffiffiffi each target. The distance between neighboring targets is 2L ¼ 1= CT , where CT ðCR Þ is the concentration (number per area) of targets (reactants) (see Fig. 1). We assume that at L, the concentration of reactants is hardly affected by the presence of targets, therefore CR ðLÞ ¼ C0R , where C0R is the unperturbed concentration of reactants. Next, we define the joint probability of finding a target at position rT and a reactant at rR , nðrT ; rR Þ. The connection of n to the concentration deserves a word of clarification. The Fokker-Planck equation describes the evolution of the probability density, n, of a pair of particles at positions rT and rR . The diffusion equation describes the spatiotemporal evolution of concentrations CT and CR . The equations for the concentration and probability density of an ensemble often coincide, as for our case of two diffusing particles. We will use this combined picture to our benefit. The kinetic rate depends on fluxes of many particles, so it is useful to use the concentration picture at times. However, when examining hydrodynamic interactions, it is easier to consider just two particles moving toward each other in the probabilistic picture.
The diffusion equation and boundary conditions are
vt nðrR ; rT ; tÞ ¼ DT V2T nðrR ; rT ; tÞ þ DR V2R nðrR ; rT ; tÞ with
nðr ¼ r0 Þ ¼ 0
and nðr ¼ LÞ ¼ 1;
(1)
where DR ðDT Þ is the diffusion coefficient of reactants (targets), Va is the gradient operator with respect to a, and r ¼ jrT rR j . Throughout this article, Greek indices refer to a particle ðR; TÞ and Latin indices are the coordinates ðx; y; zÞ. With the specified boundary conditions, we simplified the problem to be radially symmetric. A different distribution of targets will change the results slightly, but the main effects will remain. For transport in a 2D planar layer of viscosity hm embedded in a much less viscous 3D fluid of viscosity hf , the diffusion coefficient was calculated by Saffman and Delbr€uck (15) and is given by
kB T 2 Da ¼ log g ; 4phm kaa
(2)
where aa is the radius of the target/reactant, and g 0:577 is Euler’s constant. For membranes, the cutoff distance, 1=k, which we shall call the Saffman-Delbr€uck length in the rest of the article, is determined by the ratio of the outer fluid (usually water) viscosity, hf , and the much more viscous lipid membrane, 1=k ¼ hm =ð2hf Þ, and is 1 mm. Equation 2 is valid in the limit kaa 1. Reaction rates will be determined by the relative motion of reactants toward targets. It is therefore beneficial to switch coordinates to the relative motion, r ¼ rT rR and neglect the ‘‘center-of-mass’’ motion, i.e., global gradients. The joint probability, nðrT ; rR Þ, under these assumptions is simply nðrR ; rT Þ ¼ nðrÞ, and VT ¼ Vr , and VR ¼ Vr . Equation 1 in steady state becomes
0 ¼ DV2r nðrÞ with nðr ¼ r0 Þ ¼ 0 and nðr ¼ LÞ ¼ 1: (3) The equation for n is just the diffusion equation of a reactant diffusing toward a stationary target with a diffusion coefficient equal to the mutual diffusion DhDT þ DR . In the radially symmetric case, the 2D solution for n is
n ¼
logðr=r0 Þ : logðL=r0 Þ
(4)
The rate constant, K0 , is the current (J is the flux) at the reaction radius, r0 , so
K0 ¼ 2pr0 J j r0
dn 2pD ; ¼ 2pr0 D ¼ dr r0 logðL=r0 Þ
(5)
which depends only weakly on r0 and also depends on the outer boundary condition at r ¼ L. For comparison, a similar calculation in 3D yields (9)
K3D ¼ 4pDr0 :
FIGURE 1 Schematic representation of the system. Reactants are small red dots; their radius is aR , with diffusion coefficient DR determined by Eq. 2. Targets are larger green dots of radius aT and diffusion coefficient DT . The distance between targets is 2L. We assume that the concentration of reactants exactly between two targets, i.e., at a distance L from the targets (dashed line), is fixed, such that the joint probability is nðLÞ ¼ 1. It is further assumed that the particles react when they touch, i.e., the reaction radius, r0 , is given by the sum of the radii of a target and a reactant. To see this Figure in color, go online.
(6)
Unlike that in 2D, the 3D result does not depend on the outer boundary condition. It thus does not require that the concentration be fixed at some finite distance L, i.e., in 3D, it is possible, without complications, to take the limit of L/N. In 2D, the long-range, logarithmic nature of the diffusion makes the solution more sensitive to boundary conditions in the far field.
Hydrodynamic interactions Hydrodynamic interactions are expected to reduce the kinetic rate: the motion of a reactant toward a target creates a flow that pushes the target away.
Biophysical Journal 113, 440–447, July 25, 2017 441
Oppenheimer and Stone Deutch and Felderhof (8) developed a procedure for calculating reaction rates while accounting for hydrodynamic interactions in the 3D case. In this section, we follow their procedure and the one described in (9) and present general results that apply for any fluid in any dimension. In Hydrodynamic interactions in membranes, we specialize to the 2D case of reactions on the surfaces of membranes that are surrounded by a fluid. Let us consider the hydrodynamic interaction between two particles in a fluid. In the limit of low-Reynolds-number hydrodynamics, force is linearly proportional to velocity. Lacking any coupling between the two, each particle’s velocity, va , and corresponding force, f a , acting on the particle are given by
va ¼ ma f a ðra Þ;
(7)
where ma is the scalar mobility coefficient of particle a. The motion of one particle influences that of another particle through the fluid in which they are immersed. If the particles are a distance jra rb j apart, the leading-order hydrodynamic flow generated by a particle, approximated as a point force, is given by the Green’s function, which is also known as the Oseen tensor, Gðra rb Þ (30). This tensor will differ according to the boundary conditions, dimensionality, geometry, etc. The velocity on particle a, with particle b at rb , is now
va ¼ ma f a ðra Þ þ G ðra rb Þ , f b ðrb Þ:
(8)
Next, we construct the Fokker-Planck equation for the probability distribution, n ¼ nðrR ; rT ; tÞ, of finding one particle at rR and the second at rT . In our case of diffusing particles, there is a force on particle a coming from the chemical potential, E:
f a ¼ Va E ¼ kB TVa logn:
(9)
Substituting this force in Eq. 8 gives
va ¼ Da Va logn kB TGðra rb Þ , Vbsa logn;
(10)
where Da ¼ kB Tma . Local conservation of particles means that any change in n over time for a given volume should equal the flux of particles, J, out of that volume. The fluxes are given by the probability times the velocity, Ja ¼ nva . Using Eq. 10, these steps give an equation for the joint probability, n:
vn ¼ VR , JR VT , JT vt ¼
DR V2R n
þ
DT V2T n
þ kB TVR , ½GðrT rR Þ , VT n (11)
þ kB TVT , ½GðrT rR Þ , VR n: Other particles will modify the joint probability; however, in the low-concentration limit, two-body correlations dominate, and Eq. 11 is a good approximation for the joint probability. The total probability will be a simple superposition of mutual probabilities. The probability distribution of particles is a result of forces coming from chemical potential gradients plus forces on other, distant particles as mediated by the membrane flow. As in Simple diffusion in 2D, we will switch to coordinates of the relative motion, r, and the global diffusion diffusive motion. We then assume that there are no macroscopic gradients and neglect the center-of-mass motion (8). In this limit, the joint probability is only a function of the distance between the particles, nðrR ; rT Þ ¼ nðrÞ, and also VR ¼ Vr , VT ¼ Vr . Equation 11 becomes
vnðrÞ ¼ Vr , JðrÞ ¼ Vr , ½ðD 2kB TGL ðrÞÞVr nðrÞ; vt (12)
442 Biophysical Journal 113, 440–447, July 25, 2017
where GL is the longitudinal component of the Green’s function, GL ¼ Gxx ðr ¼ rex Þ ¼ Gyy ðr ¼ rey Þ. Equation 12 is a Fokker-Planck equation with a diffusion term that depends on configuration. For the derivation of a corresponding Langevin equation, see (31). Diffusion in inhomogeneous media could also be derived from detailed balance arguments (for example, see (32)). Note however, that in our case, the inhomogeneity is not fixed but is a function of the relative distance between particles.
RESULTS AND DISCUSSION Hydrodynamic interactions in membranes We now apply the analysis of the previous section to study the effect of hydrodynamic interactions in membranes on kinetic rates. At steady state and for radial symmetry, Eq. 12 is d 2kB T dn GL ðkrÞ r D 1 ¼ 0: (13) dr D dr The Green’s function for a membrane was calculated previously (see, e.g., (15,25,33)). The calculations are valid in the limit of small particle size relative to the SaffmanDelbr€uck length, kaa 1. It is further assumed that the membrane is an infinite flat 2D viscous fluid coupled to a viscous 3D outer fluid from above and below, that both targets and reactants effectively span the thickness of the membrane, and that the particles are far enough from each other that they can be regarded as points. For more on the derivation, see (33). Under these assumptions, the Green’s function is 1 H1 ðxÞ Y0 ðxÞ Y2 ðxÞ 2 þ 2 dij Gij ðxÞ ¼ H0 ðxÞ 4hm x 2 px 1 2H1 ðxÞ 4 ri rj þ Y2 ðxÞ þ 2 ; H0 ðxÞ 4hm x px r2 (14) where x is the position vector normalized by the SaffmanDelbr€uck length, with magnitude x ¼ jx j . Also, H is the Struve function and Y is the Bessel function of the second kind. The longitudinal component of this Green’s function is 1 g ðxÞ; 4phm
(15a)
pH1 ðxÞ pðY0 ðxÞ þ Y2 ðxÞÞ 2 2: x 2 x
(15b)
GL ðxÞ ¼
with
g ðxÞh
An effective diffusion coefficient can be defined from Eqs. 13 and 15 by Deff ðrÞhD½1 dD g ðkrÞ;
(16)
where we have defined dD hkB T=ð2phm DÞ. Deff depends on the distance between the particles. Since gðxÞ is a
Reaction Rates in Membranes
monotonic function that decays to zero as x/N, the diffusion is unaffected at large distances, whereas as x/0, gðxÞ increases and hence Deff decreases. Since GL ðxÞ of Eq. 15 is positive for any x, the main effect of adding hydrodynamic interactions is to reduce the effective diffusion coefficient. These results are summarized in Fig. 2, which presents the effective diffusion normalized by D for both a membrane and a regular 3D fluid. In 3D, the effective diffusion coefficient is given by Deff ðrÞhD½1 3aR aT = ðr ðaR þ aT ÞÞ (9). In both cases, as the particles draw nearer, Deff decreases, which is a result of hydrodynamic interactions, but in 3D, we note that Deff differs from D only at much shorter distances. The kinetic rate is related to the time it takes particles to diffuse from far away up to the target. It will therefore depend on the diffusion coefficient at all distances from L up to r0 . The results in Fig. 2 indicate that the rate coefficient for membranes is hindered more than its 3D counterpart by hydrodynamic interactions. We show that this is indeed the case and discuss this result further in Conclusions. To calculate the kinetic rate in quasi-2D suspensions with hydrodynamic interactions, we must first find the joint probability, nðrÞ. Substituting Eq. 15 in Eq. 13 and solving for nðrÞ, assuming nðr0 Þ ¼ 0, gives Z r dr 0 nðrÞ ¼ AIðkrÞ; where IðkrÞ ¼ : 0 0 r0 r ð1 dD gðkr ÞÞ (17) The coefficient A is determined by the boundary conditions at r ¼ L, similar to Eq. 1: the particle density at L equals the unperturbed value, nðLÞ ¼ 1, giving A ¼
1 : IðkLÞ
(18)
From Eqs. 12 and 17, the kinetic rate constant is K ¼ 2pr0 Jðr0 Þ ¼ 2pDð1 dD gðkr0 ÞÞ
dn ¼ 2pDA: dr r0 (19)
The limit of high target concentration, kL 1
We now turn to finding the rate constant, K, analytically in different asymptotic limits. For high target concentration, the distance between two targets, 2L, is much smaller than the Saffman-Delbr€uck length, L 1=k, which is typically 1 mm for membranes. (L must still be much larger than the size of reactants or targets, 10 nm.) The hydrodynamic response in that limit is strictly 2D, i.e., no coupling to the surrounding fluid, with a cutoff given by 1=k. The Green’s function of Eq. 14 simplifies to 2 kr 1 1 gðkrÞ ! g þ log : 2 kr
(20)
In this limit, Eq. 17 can be integrated analytically. Substituting the result in Eq. 19 gives
"
kL 1
K ! hm log
kB T kL 2 þ dD 1 þ 2g þ 2log 2 kr0 2 þ dD 1 þ 2g þ 2log 2
#:
(21)
Whereas the kinetic rate constant without hydrodynamic interactions, K0 of Eq. 5, depends logarithmically on target separation, L, Eq. 21 exhibits a much weaker dependence, only through logðlogðkLÞÞ. The limit of low target concentration, kL [ 1
Let us now examine the opposite limit of large separation between targets, kL [ 1. For large distances, kr [ 1, and the hydrodynamic response is mostly 3D in nature. In this 3D point force limit, kr [ 1
gðkrÞ !
FIGURE 2 Effective diffusion coefficient, Deff, normalized by the bare diffusion, D, as a function of normalized distance, kr (see Eq. 16), for a membrane (solid line) and an unbounded 3D fluid (dashed line). Plots are for aR ¼ aT ha, and taking ka ¼ 1=1000. In both cases, the diffusion coefficient tends to D at large distances and is suppressed at shorter distances due to hydrodynamic interactions. Notice that in 3D, the deviation of Deff from D occurs at much shorter distances.
2 : kr
(22)
In general, the coefficient A in Eq. 18, and thus also the reaction rate, is determined by an integration over all distances from r0 up to L. Consequently, it is not possible to simply use the Green’s function at large distances. Even so, it is still possible to use Eq. 22 to find the dependence of the rate coefficient on L. To do so, we will rewrite the denominator in Eq. 18R as a sum of two definite integrals, L IðkLÞ ¼ IðkRÞ þ R dr 0 =ðr 0 ð1 dD gðkr 0 ÞÞÞ, where R is a number chosen such that kR [ 1. The second term only involves the flow response at large distances and therefore is
Biophysical Journal 113, 440–447, July 25, 2017 443
Oppenheimer and Stone
vn ¼ DV2 n þ q0 : vt
found analytically using the Green’s function in the limit kr [ 1 of Eq. 22. The first term, which does not depend on L, is calculated numerically. We thus have kL [ 1
K !
2pD ; logðkL 2dD Þ þ aðkRÞ
(23)
with aðkRÞ ¼ IðkRÞ log ðkR 2dD Þ. In Fig. 3, we chose R ¼ 100k (but any value >10 converges to the same result), and ka ¼ 1=1000, which produces a ¼ 25. In this limit the kinetic rate constant depends logarithmically on L, which is the same as the kinetic rate without hydrodynamics, K0 of Eq. 5, but here the cutoff of the logarithmic divergence is 1=k rather than r0 . In Fig. 3 A, we plot the reaction rate with hydrodynamic interactions, K, relative to K0 —the rate lacking hydrodynamic coupling between particles. The ‘‘’’ symbols are the result of the numerical integration of Eq. 18. The blue line corresponds to the limit of small separation between targets kL 1 (see Eq. 21) and the green line corresponds to the opposite limit of Eq. 23. Fig. 3 B is the result of the numerical integration of K=K0 for different values of the SaffmanDelbr€ uck length, while still maintaining the limit ka 1.
We will simplify the problem, as before, by assuming radial symmetry: a reactant at L has equal probability of reaching r0 or a target a distance 2L away ðL=r0 [ 1Þ. The second assumption is a simplification, as the lack of radial symmetry poses a more difficult problem to solve. We expect the nature of our results to hold even in systems lacking such symmetry, though prefactors may vary. The boundary conditions are therefore nðr0 Þ ¼ 0 and nð2LÞ ¼ 0. At steady state the source term balances the flux to the sinks. Solving Eq. 24, we find the flux of particles at the reaction radius, r0 . This gives the kinetic rate constant, 2 L r02 4 r2 K0 ¼ 2pq0 0 : (25) log ð2L=r0 Þ 2 Introducing hydrodynamic interactions, as done in Hydrodynamic interactions in membranes, we find I1 ð2kLÞ q0 r02 K ¼ 2pD (26a) Ið2kLÞ 2D and I1 ðkrÞ ¼
Constant flux from 3D For some reactions in the cell, reactants undergo a 3D diffusion before adhering to the membrane. For this physical situation, and under the assumption that reactants are well mixed in the bulk such that there are no concentration gradients, a more appropriate condition on the membrane transport equation is to include a source term, q0 , coming from the outer fluid (see Fig. 4 for a schematic representation). Working in the relative frame of the reactants and targets (see Simple diffusion in 2D) and ignoring center-of-mass motion, i.e., motion in the X direction, this problem is defined by A
(24)
q0 2Dk2
Z
kr
kr0
xdx ; ð1 dD gðxÞÞ
(26b)
where IðkrÞ is given in Eq. 17, dD is defined by Eq. 16, and gðxÞ is defined in (15). In the limit of small separation between targets, kL 1, the rate constant is 1 0 B 2 Ei½f ðkLÞ Ei½f ðkr0 =2Þ r02 C kL 1 h i K ; C !2pq0 B @k2 f ðkLÞ 2A z e log f ðkr0 =2Þ
(27a)
B
FIGURE 3 The ratio of the hydrodynamic-corrected rate constant (K) to the rate constant without hydrodynamic interactions ðK0 Þ as a function of the distance between targets, L, normalized by the Saffman-Delbr€uck length, 1=k. (A) 1=k ¼ 1000a and r0 ¼ 2a. Black crosses are given by numerical integration of Eqs. 18 and 19 divided by Eq. 5. The dashed line is the theoretical value for a small distance between targets, kL 1, as determined by Eq. 21. The solid line (gray) is the opposite limit of kL [ 1, as given by Eq. 23. (B) Numerical integration of K=K0 for four different values of the Saffman-Delbr€uck length, 1=ðkaÞ ¼ 100; 200; 500; 1000 (colors go respectively from light to dark). To see this figure in color, go online.
444 Biophysical Journal 113, 440–447, July 25, 2017
Reaction Rates in Membranes
FIGURE 4 Schematic representation of the system where there is a constant flux from 3D. Reactants are small red dots adsorbing on the membrane at a fixed rate, q0 , from the bulk, and targets are larger green dots. The distance between targets is 2L, as before. The targets act as sinks, i.e., the concentration of reactants is zero at r ¼ r0 and at r ¼ 2L. To see this figure in color, go online.
f ðxÞ ¼ z þ 2logðxÞ;
(27b)
where zh2=dD 1 þ 2g, and Ei is the exponential integral function. In the opposite limit of a low concentration of targets, kL [ 1, the kinetic rate coefficient is
K ! log ð2kL2dD ÞþaðkRÞ 4dD2 log ð2kL 2dD Þ þ 2kLðkL þ 2dD Þ þ a1 ðkRÞ pq0 r02 ; kL [ 1
pq0 =k2
(28)
where a is defined after Eq. 23 and a1 ðkRÞ ¼ I1 ðkRÞ 4dD2 log ðkR 2dD Þ R2 k2 =2 2kRdD . Here, R is chosen such that kR [ 1. We calculated IðkRÞ; I1 ðkRÞ numerically. In Fig. 5, we used kR ¼ 100, which for ka ¼ 1=1000 gives a ¼ 25 and a1 ¼ 1:3. The kinetic rate constant with hydrodynamic interactions relative to the kinetic rate without hydrodynamics, K=K0 , as calculated numerically, is displayed in Fig. 5. In addition, we provide the asymptotic limits of small and large separation between targets given by Eqs. 27 and 28. Unlike results in the previous section, the results of this section indicate a minimum with respect to target separation, L. Surprisingly, when examining the ratio K=K0 for large values of kL, i.e., for large separation between targets, the results for this and the previous sections converge (as can be seen in Fig. 5, inset), even though the rates themselves differ. Assumptions The theory presented here provides a first approximation to the reduction in the membrane kinetic rate due to hydrodynamic interactions. We have neglected several effects. First, the calculation is valid for low area fractions of reactants and targets, and we thus neglected hydrodynamic interac-
FIGURE 5 The ratio of the hydrodynamic-corrected kinetic rate constant, K, to the rate constant without hydrodynamic interactions, K0 , as a function of the distance between targets, L, normalized by the SaffmanDelbr€uck length, 1=k, in the case of a source of reactants coming from the 3D fluid. The ratio K=K0 is independent of the source term, q0 . Plots are for values of ka ¼ 1=1000 and r0 ¼ 2a. Black dots are given by numerical integration of Eq. 26. The dashed line is the theoretical value for kL 1 as given by Eq. 27. The solid line (gray) is the opposite limit of low concentration of targets, or kL [ 1, as determined by Eq. 28. The inset compares K=K0 for the current boundary condition of a source from 3D (black line) to the results in the previous section, where the source is coming from within the membrane (gray line). The two plots converge for large values of kL, i.e., in the limit of low concentration.
tions between more than two particles. Second, interactions between the reactants were neglected as well. We assumed that the distribution of targets is radially symmetric, whereas in practice it is random. Finally, the Green’s function of Eq. 14 is valid only for large distances, but to find the kinetic rate, an integration was performed up to the interaction radius. CONCLUSIONS In this work, we found that the kinetic rate constant in membranes can be reduced by almost an order of magnitude when accounting for hydrodynamic interactions between reactants and targets. The effect is stronger when the concentration of targets is high enough, such that the distance between them is smaller than the Saffman-Delbr€uck length, which is the length over which flows become 3D in nature. For typical membranes, the Saffman-Delbr€uck length is 1 mm; we therefore expect the high concentration limit to be relevant biologically. In this limit, the dependence on the outer boundary condition is practically nonexistent, Kflog ðlog ðkLÞÞ, whereas it is described by K0 flog ðL=r0 Þ when no hydrodynamic interactions are considered (compare Eqs. 5 and 21). Let us consider an example. For an area fraction of targets of just 0:01%, the distance between targets, L, is 100 nm and kL 0:1. The result in Fig. 3 indicates that the kinetic rate for these values is seven times slower than the kinetic rate without hydrodynamics. When comparing this result to the 3D case, where the reduction is by a factor of
Biophysical Journal 113, 440–447, July 25, 2017 445
Oppenheimer and Stone TABLE 1
Summary of the Results for the Kinetic Rate Constant in a Membrane in the Different Limits Summary of Membrane Kinetic Rate Constants in the Various Regimes
2D source
no H.I. H.I.
Eq. 5 kL 1
Eq. 21
K/ hm log
kL [ 1 Bulk source
no H.I.
H.I.
2pD log ðL=r0 Þ
K¼
kB T
kL 2 þ dD 1 þ 2g þ 2 log 2 kr0 2 þ dD 1 þ 2g þ 2 log 2
Eq. 23
2pD K/ logðkL 2dD Þ þ aðkRÞ
Eq. 25
K ¼ 2pq0
kL 1
Eq. 27
kL [ 1
Eq. 28
0
L2 r02 =4 r02 logð2L=r0 Þ 2 1
B 2 Ei½f ðkLÞ Ei½f ðkr0 =2Þ r02 C C ; f ðxÞ ¼ z þ 2 logðxÞ; z ¼ 2=dD 1 þ 2g K/2pq0 B @k2 f ðkLÞ 2A ez log f ðkr0 =2Þ pq0 =k2 ½4dD2 log ð2kL 2dD Þ þ 2kLðkL þ 2dD Þ þ a1 ðkRÞ pq0 r02 K/ log ð2kL 2dD Þ þ aðkRÞ
The upper three rows correspond to the initial conditions of Hydrodynamic interactions in membranes, in which there is a source of reactants within the membrane. The bottom three rows correspond to the boundary conditions of Constant flux from 3D, where reactants come from the bulk fluid. H.I., hydrodynamic interactions. The forms of the numerical factors aðkRÞ and a1 ðkRÞ are given below Eqs. 23 and 28, respectively. For ka ¼ 1=1000 and kR ¼ 100, we have a ¼ 25 and a1 ¼ 1:3.
2 (8), we see that the effect is stronger in 2D due to the long-range nature of 2D flows. For a low concentration of targets, the dependence on separation between targets, L, is logarithmic. However, the cutoff is no longer r0, the reaction radius, but 1=k, the Saffman-Delbr€ uck length. Even at a very low concentration of targets, kL [ 1, there is still a reduction of the kinetic rate: it saturates to a factor of 2 (i.e., K=K0 x1=2), comparable to the 3D case, since in this limit there are flows in the surrounding fluid. Notice, however, that this limit is not expected to be relevant biologically. Fig. 3 indicates that saturation occurs for values of kL > 104 , which is a few hundreds of microns at best, much more than the size of a typical cell. Table 1 summarizes our findings for the reaction rate constant, considering different scenarios and various limiting cases. We are hopeful that these different estimates will be useful for understanding new experiments.
REFERENCES
AUTHOR CONTRIBUTIONS
10. Berg, H. C., and E. M. Purcell. 1977. Physics of chemoreception. Biophys. J. 20:193–219.
N.O. and H.A.S. designed research, performed research, and wrote the manuscript.
11. Keizer, J. 1985. Theory of rapid bimolecular reactions in solution and membranes. Acc. Chem. Res. 18:235–241.
ACKNOWLEDGMENTS
12. Adam, G., and M. Delbruck. 1968. Reduction of dimensionality in biological diffusion processes. In Structural Chemistry and Molecular Biology. A. Rich and N. Davidson, editors. W. H. Freeman, pp. 198–215.
We thank Itay Budin, Matan Yah Ben Zion, and Haim Diamant for helpful conversations. We gratefully acknowledge the National Science Foundation for partial support of this research via grant DMS-1219366.
13. Shea, L. D., G. M. Omann, and J. J. Linderman. 1997. Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking. Biophys. J. 73:2949–2959.
1. Creighton, T. E. 1984. Proteins: Structures and Molecular Properties. W.H. Freeman, New York. 2. Vance, D. E., and J. E. Vance. 2008. Biochemistry of Lipids, Lipoproteins and Membranes. Elsevier, Amsterdam, the Netherlands. 3. Trumpower, B. L. 1982. Function of Quinones in Energy Conserving Systems. Academic Press, New York. 4. von Smoluchowski, M. 1916. Drei vortrage uber diffusion. Brownsche bewegung und koagulation von kolloidteilchen. Phys. Z. 17:557–585. 5. Periasamy, N., S. Doraiswamy, ., G. R. Fleming. 1988. Diffusion controlled reactions: experimental verification of the time-dependent rate equation. J. Chem. Phys. 89:4799–4806. 6. Arita, T., O. Kajimoto, ., Y. Kimura. 2004. Experimental verification of the Smoluchowski theory for a bimolecular diffusion-controlled reaction in liquid phase. J. Chem. Phys. 120:7071–7074. 7. Nakatani, H., and H. B. Dunford. 1979. Meaning of diffusioncontrolled association rate constants in enzymology. J. Phys. Chem. 83:2662–2665. 8. Deutch, J. M., and B. U. Felderhof. 1973. Hydrodynamic effect in diffusion controlled reaction. J. Chem. Phys. 59:1669–1671. 9. Rice, S. A. 1985. Diffusion Limited Reactions. Elsevier, New York.
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Reaction Rates in Membranes 14. Lauffenburger, D. A., and J. J. Linderman. 1993. Receptors: Models for Binding, Trafficking, and Signaling. Oxford University Press, New York.
24. Levine, A. J., T. B. Liverpool, and F. C. MacKintosh. 2004. Dynamics of rigid and flexible extended bodies in viscous films and membranes. Phys. Rev. Lett. 93:038102.
15. Saffman, P. G., and M. Delbr€uck. 1975. Brownian motion in biological membranes. Proc. Natl. Acad. Sci. USA. 72:3111–3113.
25. Levine, A. J., and F. C. MacKintosh. 2002. Dynamics of viscoelastic membranes. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66:061606.
16. Hughes, B. D., B. A. Pailthorpe, and L. R. White. 1981. The translational and rotational drag on a cylinder moving in a membrane. J. Fluid Mech. 110:349–372. 17. Fournier, J.-B. 2015. On the hydrodynamics of bilayer membranes. Int. J. Non-linear Mech. 75:67–76. 18. Camley, B. A., and F. L. H. Brown. 2013. Diffusion of complex objects embedded in free and supported lipid bilayer membranes: role of shape anisotropy and leaflet structure. Soft Matter. 9:4767–4779.
26. Komura, S., S. Ramachandran, and K. Seki. 2012. Anomalous lateral diffusion in a viscous membrane surrounded by viscoelastic media. Europhys. Lett. 97:68007.
19. Sigurdsson, J. K., and P. J. Atzberger. 2016. Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matter. 12:6685–6707. 20. Evans, E., and E. Sackmann. 1988. Translational and rotational drag coefficients for a disk moving in a liquid membrane associated with a rigid substrate. J. Fluid Mech. 194:553–561. 21. Stone, H. A., and A. Ajdari. 1998. Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth. J. Fluid Mech. 369:151–173. 22. Seki, K., and S. Komura. 1993. Brownian dynamics in a thin sheet with momentum decay. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics. 47:2377–2383. 23. Fischer, T. M. 2004. The drag on needles moving in a Langmuir monolayer. J. Fluid Mech. 498:123–137.
27. Gov, N. S. 2006. Diffusion in curved fluid membranes. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73:041918. 28. Rahimi, M., A. DeSimone, and M. Arroyo. 2013. Curved fluid membranes behave laterally as effective viscoelastic media. Soft Matter. 9:11033–11045. 29. Zimmermann, E., and U. Seifert. 2015. Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91: 022709. 30. Russel, W. B., D. A. Saville, and W. R. Schowalter. 1989. Colloidal Dispersions. Cambridge University Press, Cambridge. 31. Zwanzig, R. 1969. Langevin theory of polymer dynamics in dilute solution. Adv. Chem. Phys. 15:325–331. 32. Van Kampen, N. G. 2007. Stochastic Processes in Physics and Chemistry, Third Edition. Elsevier, Amsterdam. 33. Oppenheimer, N., and H. Diamant. 2009. Correlated diffusion of membrane proteins and their effect on membrane viscosity. Biophys. J. 96:3041–3049.
Biophysical Journal 113, 440–447, July 25, 2017 447
Effect of Hydrodynamic Interactions on Reaction Rates in Membranes Naomi Oppenheimer1,* and Howard A. Stone1,* 1
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey
ABSTRACT The Brownian motion of two particles in three dimensions serves as a model for predicting the diffusion-limited reaction rate, as first discussed by von Smoluchowski. Deutch and Felderhof extended the calculation to account for hydrodynamic interactions between the particles and the target, which results in a reduction of the rate coefficient by about half. Many chemical reactions take place in quasi-two-dimensional systems, such as on the membrane or surface of a cell. We perform a Smoluchowski-like calculation in a quasi-two-dimensional geometry, i.e., a membrane surrounded by fluid, and account for hydrodynamic interactions between the particles. We show that rate coefficients are reduced relative to the case of no interactions. The reduction is more pronounced than the three-dimensional case due to the long-range nature of two-dimensional flows.
INTRODUCTION Many chemical reactions take place on the surface of a membrane—catalysis by integral membrane proteins (1), enzymatic activation of phospholipases (2), and electron transfer aided by the diffusion of quinone (3), to name a few. It has therefore been of interest to model the kinetic rate in membranes. Kinetic rates in three dimensions (3D) were first studied by von Smoluchowski (4) and verified in many diffusionlimited experiments (see, e.g., (5–7)). Extensive generalizations have been reported since. For example, Deutch and Felderhof (8) included hydrodynamic interactions in the calculation. For other generalizations, see (9) and the numerous references within. In particular, two-dimensional (2D) diffusion was studied by, among others, Berg and Purcell (10), Keizer (11), Adam and Delbruck (12), and Linderman and colleagues (13,14). In this work, we extend the theory of kinetic rates in 2D to include hydrodynamic interactions between reactants and targets in a membrane. Membrane hydrodynamics is a well explored field. In the pioneering work of Saffman and Delbr€ uck (15), the dependence of the diffusion coefficient of membrane proteins on protein size was found to be logarithmic. Many researchers have extended the theory to account for different scenarios, a few examples of which are diffusion of large objects (16);
the effect of tension (17) or of intermonolayer friction (17–19) on membrane hydrodynamics; diffusion in membranes near a rigid surface (18,20,21), or with momentum decay (22); diffusion of needle-shaped inclusions (23,24); viscoelastic membranes (25); membranes surrounded by viscoelastic material (26); or on curved membranes (19,27,28). In unbounded 2D hydrodynamic problems, a point force (per length) perturbation decays logarithmically, much more slowly than the 1=r decay in 3D. Hydrodynamic interactions thus play a significant role in 2D, a role which will manifest itself in the kinetic rate of reactions taking place on a membrane. The next parts of this article are organized as follows. In Simple diffusion in 2D, we present the basic calculation of kinetic rates as applied to 2D. Hydrodynamic interactions introduces the formalism of including hydrodynamic interactions, whereas in Hydrodynamic interactions in membranes, we specialize to the case of 2D and show that hydrodynamics plays a central role due to the long-range nature of the response. In Constant flux from 3D, we examine a slightly different initial condition, in which reactants are coming from the 3D fluid surrounding the membrane. METHODS Simple diffusion in 2D
Submitted January 9, 2017, and accepted for publication June 7, 2017. *Correspondence: [email protected] or [email protected] Editor: Ana-Suncana Smith. http://dx.doi.org/10.1016/j.bpj.2017.06.013 Ó 2017 Biophysical Society.
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In this section, we will re-derive the kinetic rate in a 2D system. We follow previous work by Berg and Purcell (10), Adam and Delbr€uck (12), Linderman and colleagues (13,14), and others. Since we will eventually be interested in the hydrodynamic interactions between targets and reactants, we
Reaction Rates in Membranes will treat a scenario where both targets and reactants can move. This approach will clarify the derivation presented in Hydrodynamic interactions. Let us consider a case in which the limiting step of a reaction is the time for the reactants to diffuse toward a target, i.e., here and in the rest of this work, we consider diffusion-limited reactions. We note that the discrete nature of binding/unbinding kinetics, in conjunction with random walks, was treated recently by Zimmermann and Seifert (29). Here, we assume that there is a separation of timescales that justifies the diffusion-limited assumption. The concentration of reactants is much higher than that of targets, such that targets are assumed to be independent of each other. A reactant and a target are further assumed to react instantaneously when they are within a radius r0 , the reaction radius, which here we take as the sum of the radii of the reactant and target. The reaction rate will be given by the flux of reactants toward the target taken at the reaction radius. The target therefore acts as a sink of size r0 , which provides an inner boundary condition for the diffusion problem considered below. The second boundary condition should be chosen with care, as in Constant flux from 3D we show how a choice of a different boundary condition changes the kinetic rate. For now, we adopt a strategy similar to that of Lauffenburger and Linderman (see section 4.2.3 in (14)): we assume that a reactant positioned at an equal distance to two targets will have equal probability of reaching pffiffiffiffiffiffi each target. The distance between neighboring targets is 2L ¼ 1= CT , where CT ðCR Þ is the concentration (number per area) of targets (reactants) (see Fig. 1). We assume that at L, the concentration of reactants is hardly affected by the presence of targets, therefore CR ðLÞ ¼ C0R , where C0R is the unperturbed concentration of reactants. Next, we define the joint probability of finding a target at position rT and a reactant at rR , nðrT ; rR Þ. The connection of n to the concentration deserves a word of clarification. The Fokker-Planck equation describes the evolution of the probability density, n, of a pair of particles at positions rT and rR . The diffusion equation describes the spatiotemporal evolution of concentrations CT and CR . The equations for the concentration and probability density of an ensemble often coincide, as for our case of two diffusing particles. We will use this combined picture to our benefit. The kinetic rate depends on fluxes of many particles, so it is useful to use the concentration picture at times. However, when examining hydrodynamic interactions, it is easier to consider just two particles moving toward each other in the probabilistic picture.
The diffusion equation and boundary conditions are
vt nðrR ; rT ; tÞ ¼ DT V2T nðrR ; rT ; tÞ þ DR V2R nðrR ; rT ; tÞ with
nðr ¼ r0 Þ ¼ 0
and nðr ¼ LÞ ¼ 1;
(1)
where DR ðDT Þ is the diffusion coefficient of reactants (targets), Va is the gradient operator with respect to a, and r ¼ jrT rR j . Throughout this article, Greek indices refer to a particle ðR; TÞ and Latin indices are the coordinates ðx; y; zÞ. With the specified boundary conditions, we simplified the problem to be radially symmetric. A different distribution of targets will change the results slightly, but the main effects will remain. For transport in a 2D planar layer of viscosity hm embedded in a much less viscous 3D fluid of viscosity hf , the diffusion coefficient was calculated by Saffman and Delbr€uck (15) and is given by
kB T 2 Da ¼ log g ; 4phm kaa
(2)
where aa is the radius of the target/reactant, and g 0:577 is Euler’s constant. For membranes, the cutoff distance, 1=k, which we shall call the Saffman-Delbr€uck length in the rest of the article, is determined by the ratio of the outer fluid (usually water) viscosity, hf , and the much more viscous lipid membrane, 1=k ¼ hm =ð2hf Þ, and is 1 mm. Equation 2 is valid in the limit kaa 1. Reaction rates will be determined by the relative motion of reactants toward targets. It is therefore beneficial to switch coordinates to the relative motion, r ¼ rT rR and neglect the ‘‘center-of-mass’’ motion, i.e., global gradients. The joint probability, nðrT ; rR Þ, under these assumptions is simply nðrR ; rT Þ ¼ nðrÞ, and VT ¼ Vr , and VR ¼ Vr . Equation 1 in steady state becomes
0 ¼ DV2r nðrÞ with nðr ¼ r0 Þ ¼ 0 and nðr ¼ LÞ ¼ 1: (3) The equation for n is just the diffusion equation of a reactant diffusing toward a stationary target with a diffusion coefficient equal to the mutual diffusion DhDT þ DR . In the radially symmetric case, the 2D solution for n is
n ¼
logðr=r0 Þ : logðL=r0 Þ
(4)
The rate constant, K0 , is the current (J is the flux) at the reaction radius, r0 , so
K0 ¼ 2pr0 J j r0
dn 2pD ; ¼ 2pr0 D ¼ dr r0 logðL=r0 Þ
(5)
which depends only weakly on r0 and also depends on the outer boundary condition at r ¼ L. For comparison, a similar calculation in 3D yields (9)
K3D ¼ 4pDr0 :
FIGURE 1 Schematic representation of the system. Reactants are small red dots; their radius is aR , with diffusion coefficient DR determined by Eq. 2. Targets are larger green dots of radius aT and diffusion coefficient DT . The distance between targets is 2L. We assume that the concentration of reactants exactly between two targets, i.e., at a distance L from the targets (dashed line), is fixed, such that the joint probability is nðLÞ ¼ 1. It is further assumed that the particles react when they touch, i.e., the reaction radius, r0 , is given by the sum of the radii of a target and a reactant. To see this Figure in color, go online.
(6)
Unlike that in 2D, the 3D result does not depend on the outer boundary condition. It thus does not require that the concentration be fixed at some finite distance L, i.e., in 3D, it is possible, without complications, to take the limit of L/N. In 2D, the long-range, logarithmic nature of the diffusion makes the solution more sensitive to boundary conditions in the far field.
Hydrodynamic interactions Hydrodynamic interactions are expected to reduce the kinetic rate: the motion of a reactant toward a target creates a flow that pushes the target away.
Biophysical Journal 113, 440–447, July 25, 2017 441
Oppenheimer and Stone Deutch and Felderhof (8) developed a procedure for calculating reaction rates while accounting for hydrodynamic interactions in the 3D case. In this section, we follow their procedure and the one described in (9) and present general results that apply for any fluid in any dimension. In Hydrodynamic interactions in membranes, we specialize to the 2D case of reactions on the surfaces of membranes that are surrounded by a fluid. Let us consider the hydrodynamic interaction between two particles in a fluid. In the limit of low-Reynolds-number hydrodynamics, force is linearly proportional to velocity. Lacking any coupling between the two, each particle’s velocity, va , and corresponding force, f a , acting on the particle are given by
va ¼ ma f a ðra Þ;
(7)
where ma is the scalar mobility coefficient of particle a. The motion of one particle influences that of another particle through the fluid in which they are immersed. If the particles are a distance jra rb j apart, the leading-order hydrodynamic flow generated by a particle, approximated as a point force, is given by the Green’s function, which is also known as the Oseen tensor, Gðra rb Þ (30). This tensor will differ according to the boundary conditions, dimensionality, geometry, etc. The velocity on particle a, with particle b at rb , is now
va ¼ ma f a ðra Þ þ G ðra rb Þ , f b ðrb Þ:
(8)
Next, we construct the Fokker-Planck equation for the probability distribution, n ¼ nðrR ; rT ; tÞ, of finding one particle at rR and the second at rT . In our case of diffusing particles, there is a force on particle a coming from the chemical potential, E:
f a ¼ Va E ¼ kB TVa logn:
(9)
Substituting this force in Eq. 8 gives
va ¼ Da Va logn kB TGðra rb Þ , Vbsa logn;
(10)
where Da ¼ kB Tma . Local conservation of particles means that any change in n over time for a given volume should equal the flux of particles, J, out of that volume. The fluxes are given by the probability times the velocity, Ja ¼ nva . Using Eq. 10, these steps give an equation for the joint probability, n:
vn ¼ VR , JR VT , JT vt ¼
DR V2R n
þ
DT V2T n
þ kB TVR , ½GðrT rR Þ , VT n (11)
þ kB TVT , ½GðrT rR Þ , VR n: Other particles will modify the joint probability; however, in the low-concentration limit, two-body correlations dominate, and Eq. 11 is a good approximation for the joint probability. The total probability will be a simple superposition of mutual probabilities. The probability distribution of particles is a result of forces coming from chemical potential gradients plus forces on other, distant particles as mediated by the membrane flow. As in Simple diffusion in 2D, we will switch to coordinates of the relative motion, r, and the global diffusion diffusive motion. We then assume that there are no macroscopic gradients and neglect the center-of-mass motion (8). In this limit, the joint probability is only a function of the distance between the particles, nðrR ; rT Þ ¼ nðrÞ, and also VR ¼ Vr , VT ¼ Vr . Equation 11 becomes
vnðrÞ ¼ Vr , JðrÞ ¼ Vr , ½ðD 2kB TGL ðrÞÞVr nðrÞ; vt (12)
442 Biophysical Journal 113, 440–447, July 25, 2017
where GL is the longitudinal component of the Green’s function, GL ¼ Gxx ðr ¼ rex Þ ¼ Gyy ðr ¼ rey Þ. Equation 12 is a Fokker-Planck equation with a diffusion term that depends on configuration. For the derivation of a corresponding Langevin equation, see (31). Diffusion in inhomogeneous media could also be derived from detailed balance arguments (for example, see (32)). Note however, that in our case, the inhomogeneity is not fixed but is a function of the relative distance between particles.
RESULTS AND DISCUSSION Hydrodynamic interactions in membranes We now apply the analysis of the previous section to study the effect of hydrodynamic interactions in membranes on kinetic rates. At steady state and for radial symmetry, Eq. 12 is d 2kB T dn GL ðkrÞ r D 1 ¼ 0: (13) dr D dr The Green’s function for a membrane was calculated previously (see, e.g., (15,25,33)). The calculations are valid in the limit of small particle size relative to the SaffmanDelbr€uck length, kaa 1. It is further assumed that the membrane is an infinite flat 2D viscous fluid coupled to a viscous 3D outer fluid from above and below, that both targets and reactants effectively span the thickness of the membrane, and that the particles are far enough from each other that they can be regarded as points. For more on the derivation, see (33). Under these assumptions, the Green’s function is 1 H1 ðxÞ Y0 ðxÞ Y2 ðxÞ 2 þ 2 dij Gij ðxÞ ¼ H0 ðxÞ 4hm x 2 px 1 2H1 ðxÞ 4 ri rj þ Y2 ðxÞ þ 2 ; H0 ðxÞ 4hm x px r2 (14) where x is the position vector normalized by the SaffmanDelbr€uck length, with magnitude x ¼ jx j . Also, H is the Struve function and Y is the Bessel function of the second kind. The longitudinal component of this Green’s function is 1 g ðxÞ; 4phm
(15a)
pH1 ðxÞ pðY0 ðxÞ þ Y2 ðxÞÞ 2 2: x 2 x
(15b)
GL ðxÞ ¼
with
g ðxÞh
An effective diffusion coefficient can be defined from Eqs. 13 and 15 by Deff ðrÞhD½1 dD g ðkrÞ;
(16)
where we have defined dD hkB T=ð2phm DÞ. Deff depends on the distance between the particles. Since gðxÞ is a
Reaction Rates in Membranes
monotonic function that decays to zero as x/N, the diffusion is unaffected at large distances, whereas as x/0, gðxÞ increases and hence Deff decreases. Since GL ðxÞ of Eq. 15 is positive for any x, the main effect of adding hydrodynamic interactions is to reduce the effective diffusion coefficient. These results are summarized in Fig. 2, which presents the effective diffusion normalized by D for both a membrane and a regular 3D fluid. In 3D, the effective diffusion coefficient is given by Deff ðrÞhD½1 3aR aT = ðr ðaR þ aT ÞÞ (9). In both cases, as the particles draw nearer, Deff decreases, which is a result of hydrodynamic interactions, but in 3D, we note that Deff differs from D only at much shorter distances. The kinetic rate is related to the time it takes particles to diffuse from far away up to the target. It will therefore depend on the diffusion coefficient at all distances from L up to r0 . The results in Fig. 2 indicate that the rate coefficient for membranes is hindered more than its 3D counterpart by hydrodynamic interactions. We show that this is indeed the case and discuss this result further in Conclusions. To calculate the kinetic rate in quasi-2D suspensions with hydrodynamic interactions, we must first find the joint probability, nðrÞ. Substituting Eq. 15 in Eq. 13 and solving for nðrÞ, assuming nðr0 Þ ¼ 0, gives Z r dr 0 nðrÞ ¼ AIðkrÞ; where IðkrÞ ¼ : 0 0 r0 r ð1 dD gðkr ÞÞ (17) The coefficient A is determined by the boundary conditions at r ¼ L, similar to Eq. 1: the particle density at L equals the unperturbed value, nðLÞ ¼ 1, giving A ¼
1 : IðkLÞ
(18)
From Eqs. 12 and 17, the kinetic rate constant is K ¼ 2pr0 Jðr0 Þ ¼ 2pDð1 dD gðkr0 ÞÞ
dn ¼ 2pDA: dr r0 (19)
The limit of high target concentration, kL 1
We now turn to finding the rate constant, K, analytically in different asymptotic limits. For high target concentration, the distance between two targets, 2L, is much smaller than the Saffman-Delbr€uck length, L 1=k, which is typically 1 mm for membranes. (L must still be much larger than the size of reactants or targets, 10 nm.) The hydrodynamic response in that limit is strictly 2D, i.e., no coupling to the surrounding fluid, with a cutoff given by 1=k. The Green’s function of Eq. 14 simplifies to 2 kr 1 1 gðkrÞ ! g þ log : 2 kr
(20)
In this limit, Eq. 17 can be integrated analytically. Substituting the result in Eq. 19 gives
"
kL 1
K ! hm log
kB T kL 2 þ dD 1 þ 2g þ 2log 2 kr0 2 þ dD 1 þ 2g þ 2log 2
#:
(21)
Whereas the kinetic rate constant without hydrodynamic interactions, K0 of Eq. 5, depends logarithmically on target separation, L, Eq. 21 exhibits a much weaker dependence, only through logðlogðkLÞÞ. The limit of low target concentration, kL [ 1
Let us now examine the opposite limit of large separation between targets, kL [ 1. For large distances, kr [ 1, and the hydrodynamic response is mostly 3D in nature. In this 3D point force limit, kr [ 1
gðkrÞ !
FIGURE 2 Effective diffusion coefficient, Deff, normalized by the bare diffusion, D, as a function of normalized distance, kr (see Eq. 16), for a membrane (solid line) and an unbounded 3D fluid (dashed line). Plots are for aR ¼ aT ha, and taking ka ¼ 1=1000. In both cases, the diffusion coefficient tends to D at large distances and is suppressed at shorter distances due to hydrodynamic interactions. Notice that in 3D, the deviation of Deff from D occurs at much shorter distances.
2 : kr
(22)
In general, the coefficient A in Eq. 18, and thus also the reaction rate, is determined by an integration over all distances from r0 up to L. Consequently, it is not possible to simply use the Green’s function at large distances. Even so, it is still possible to use Eq. 22 to find the dependence of the rate coefficient on L. To do so, we will rewrite the denominator in Eq. 18R as a sum of two definite integrals, L IðkLÞ ¼ IðkRÞ þ R dr 0 =ðr 0 ð1 dD gðkr 0 ÞÞÞ, where R is a number chosen such that kR [ 1. The second term only involves the flow response at large distances and therefore is
Biophysical Journal 113, 440–447, July 25, 2017 443
Oppenheimer and Stone
vn ¼ DV2 n þ q0 : vt
found analytically using the Green’s function in the limit kr [ 1 of Eq. 22. The first term, which does not depend on L, is calculated numerically. We thus have kL [ 1
K !
2pD ; logðkL 2dD Þ þ aðkRÞ
(23)
with aðkRÞ ¼ IðkRÞ log ðkR 2dD Þ. In Fig. 3, we chose R ¼ 100k (but any value >10 converges to the same result), and ka ¼ 1=1000, which produces a ¼ 25. In this limit the kinetic rate constant depends logarithmically on L, which is the same as the kinetic rate without hydrodynamics, K0 of Eq. 5, but here the cutoff of the logarithmic divergence is 1=k rather than r0 . In Fig. 3 A, we plot the reaction rate with hydrodynamic interactions, K, relative to K0 —the rate lacking hydrodynamic coupling between particles. The ‘‘’’ symbols are the result of the numerical integration of Eq. 18. The blue line corresponds to the limit of small separation between targets kL 1 (see Eq. 21) and the green line corresponds to the opposite limit of Eq. 23. Fig. 3 B is the result of the numerical integration of K=K0 for different values of the SaffmanDelbr€ uck length, while still maintaining the limit ka 1.
We will simplify the problem, as before, by assuming radial symmetry: a reactant at L has equal probability of reaching r0 or a target a distance 2L away ðL=r0 [ 1Þ. The second assumption is a simplification, as the lack of radial symmetry poses a more difficult problem to solve. We expect the nature of our results to hold even in systems lacking such symmetry, though prefactors may vary. The boundary conditions are therefore nðr0 Þ ¼ 0 and nð2LÞ ¼ 0. At steady state the source term balances the flux to the sinks. Solving Eq. 24, we find the flux of particles at the reaction radius, r0 . This gives the kinetic rate constant, 2 L r02 4 r2 K0 ¼ 2pq0 0 : (25) log ð2L=r0 Þ 2 Introducing hydrodynamic interactions, as done in Hydrodynamic interactions in membranes, we find I1 ð2kLÞ q0 r02 K ¼ 2pD (26a) Ið2kLÞ 2D and I1 ðkrÞ ¼
Constant flux from 3D For some reactions in the cell, reactants undergo a 3D diffusion before adhering to the membrane. For this physical situation, and under the assumption that reactants are well mixed in the bulk such that there are no concentration gradients, a more appropriate condition on the membrane transport equation is to include a source term, q0 , coming from the outer fluid (see Fig. 4 for a schematic representation). Working in the relative frame of the reactants and targets (see Simple diffusion in 2D) and ignoring center-of-mass motion, i.e., motion in the X direction, this problem is defined by A
(24)
q0 2Dk2
Z
kr
kr0
xdx ; ð1 dD gðxÞÞ
(26b)
where IðkrÞ is given in Eq. 17, dD is defined by Eq. 16, and gðxÞ is defined in (15). In the limit of small separation between targets, kL 1, the rate constant is 1 0 B 2 Ei½f ðkLÞ Ei½f ðkr0 =2Þ r02 C kL 1 h i K ; C !2pq0 B @k2 f ðkLÞ 2A z e log f ðkr0 =2Þ
(27a)
B
FIGURE 3 The ratio of the hydrodynamic-corrected rate constant (K) to the rate constant without hydrodynamic interactions ðK0 Þ as a function of the distance between targets, L, normalized by the Saffman-Delbr€uck length, 1=k. (A) 1=k ¼ 1000a and r0 ¼ 2a. Black crosses are given by numerical integration of Eqs. 18 and 19 divided by Eq. 5. The dashed line is the theoretical value for a small distance between targets, kL 1, as determined by Eq. 21. The solid line (gray) is the opposite limit of kL [ 1, as given by Eq. 23. (B) Numerical integration of K=K0 for four different values of the Saffman-Delbr€uck length, 1=ðkaÞ ¼ 100; 200; 500; 1000 (colors go respectively from light to dark). To see this figure in color, go online.
444 Biophysical Journal 113, 440–447, July 25, 2017
Reaction Rates in Membranes
FIGURE 4 Schematic representation of the system where there is a constant flux from 3D. Reactants are small red dots adsorbing on the membrane at a fixed rate, q0 , from the bulk, and targets are larger green dots. The distance between targets is 2L, as before. The targets act as sinks, i.e., the concentration of reactants is zero at r ¼ r0 and at r ¼ 2L. To see this figure in color, go online.
f ðxÞ ¼ z þ 2logðxÞ;
(27b)
where zh2=dD 1 þ 2g, and Ei is the exponential integral function. In the opposite limit of a low concentration of targets, kL [ 1, the kinetic rate coefficient is
K ! log ð2kL2dD ÞþaðkRÞ 4dD2 log ð2kL 2dD Þ þ 2kLðkL þ 2dD Þ þ a1 ðkRÞ pq0 r02 ; kL [ 1
pq0 =k2
(28)
where a is defined after Eq. 23 and a1 ðkRÞ ¼ I1 ðkRÞ 4dD2 log ðkR 2dD Þ R2 k2 =2 2kRdD . Here, R is chosen such that kR [ 1. We calculated IðkRÞ; I1 ðkRÞ numerically. In Fig. 5, we used kR ¼ 100, which for ka ¼ 1=1000 gives a ¼ 25 and a1 ¼ 1:3. The kinetic rate constant with hydrodynamic interactions relative to the kinetic rate without hydrodynamics, K=K0 , as calculated numerically, is displayed in Fig. 5. In addition, we provide the asymptotic limits of small and large separation between targets given by Eqs. 27 and 28. Unlike results in the previous section, the results of this section indicate a minimum with respect to target separation, L. Surprisingly, when examining the ratio K=K0 for large values of kL, i.e., for large separation between targets, the results for this and the previous sections converge (as can be seen in Fig. 5, inset), even though the rates themselves differ. Assumptions The theory presented here provides a first approximation to the reduction in the membrane kinetic rate due to hydrodynamic interactions. We have neglected several effects. First, the calculation is valid for low area fractions of reactants and targets, and we thus neglected hydrodynamic interac-
FIGURE 5 The ratio of the hydrodynamic-corrected kinetic rate constant, K, to the rate constant without hydrodynamic interactions, K0 , as a function of the distance between targets, L, normalized by the SaffmanDelbr€uck length, 1=k, in the case of a source of reactants coming from the 3D fluid. The ratio K=K0 is independent of the source term, q0 . Plots are for values of ka ¼ 1=1000 and r0 ¼ 2a. Black dots are given by numerical integration of Eq. 26. The dashed line is the theoretical value for kL 1 as given by Eq. 27. The solid line (gray) is the opposite limit of low concentration of targets, or kL [ 1, as determined by Eq. 28. The inset compares K=K0 for the current boundary condition of a source from 3D (black line) to the results in the previous section, where the source is coming from within the membrane (gray line). The two plots converge for large values of kL, i.e., in the limit of low concentration.
tions between more than two particles. Second, interactions between the reactants were neglected as well. We assumed that the distribution of targets is radially symmetric, whereas in practice it is random. Finally, the Green’s function of Eq. 14 is valid only for large distances, but to find the kinetic rate, an integration was performed up to the interaction radius. CONCLUSIONS In this work, we found that the kinetic rate constant in membranes can be reduced by almost an order of magnitude when accounting for hydrodynamic interactions between reactants and targets. The effect is stronger when the concentration of targets is high enough, such that the distance between them is smaller than the Saffman-Delbr€uck length, which is the length over which flows become 3D in nature. For typical membranes, the Saffman-Delbr€uck length is 1 mm; we therefore expect the high concentration limit to be relevant biologically. In this limit, the dependence on the outer boundary condition is practically nonexistent, Kflog ðlog ðkLÞÞ, whereas it is described by K0 flog ðL=r0 Þ when no hydrodynamic interactions are considered (compare Eqs. 5 and 21). Let us consider an example. For an area fraction of targets of just 0:01%, the distance between targets, L, is 100 nm and kL 0:1. The result in Fig. 3 indicates that the kinetic rate for these values is seven times slower than the kinetic rate without hydrodynamics. When comparing this result to the 3D case, where the reduction is by a factor of
Biophysical Journal 113, 440–447, July 25, 2017 445
Oppenheimer and Stone TABLE 1
Summary of the Results for the Kinetic Rate Constant in a Membrane in the Different Limits Summary of Membrane Kinetic Rate Constants in the Various Regimes
2D source
no H.I. H.I.
Eq. 5 kL 1
Eq. 21
K/ hm log
kL [ 1 Bulk source
no H.I.
H.I.
2pD log ðL=r0 Þ
K¼
kB T
kL 2 þ dD 1 þ 2g þ 2 log 2 kr0 2 þ dD 1 þ 2g þ 2 log 2
Eq. 23
2pD K/ logðkL 2dD Þ þ aðkRÞ
Eq. 25
K ¼ 2pq0
kL 1
Eq. 27
kL [ 1
Eq. 28
0
L2 r02 =4 r02 logð2L=r0 Þ 2 1
B 2 Ei½f ðkLÞ Ei½f ðkr0 =2Þ r02 C C ; f ðxÞ ¼ z þ 2 logðxÞ; z ¼ 2=dD 1 þ 2g K/2pq0 B @k2 f ðkLÞ 2A ez log f ðkr0 =2Þ pq0 =k2 ½4dD2 log ð2kL 2dD Þ þ 2kLðkL þ 2dD Þ þ a1 ðkRÞ pq0 r02 K/ log ð2kL 2dD Þ þ aðkRÞ
The upper three rows correspond to the initial conditions of Hydrodynamic interactions in membranes, in which there is a source of reactants within the membrane. The bottom three rows correspond to the boundary conditions of Constant flux from 3D, where reactants come from the bulk fluid. H.I., hydrodynamic interactions. The forms of the numerical factors aðkRÞ and a1 ðkRÞ are given below Eqs. 23 and 28, respectively. For ka ¼ 1=1000 and kR ¼ 100, we have a ¼ 25 and a1 ¼ 1:3.
2 (8), we see that the effect is stronger in 2D due to the long-range nature of 2D flows. For a low concentration of targets, the dependence on separation between targets, L, is logarithmic. However, the cutoff is no longer r0, the reaction radius, but 1=k, the Saffman-Delbr€ uck length. Even at a very low concentration of targets, kL [ 1, there is still a reduction of the kinetic rate: it saturates to a factor of 2 (i.e., K=K0 x1=2), comparable to the 3D case, since in this limit there are flows in the surrounding fluid. Notice, however, that this limit is not expected to be relevant biologically. Fig. 3 indicates that saturation occurs for values of kL > 104 , which is a few hundreds of microns at best, much more than the size of a typical cell. Table 1 summarizes our findings for the reaction rate constant, considering different scenarios and various limiting cases. We are hopeful that these different estimates will be useful for understanding new experiments.
REFERENCES
AUTHOR CONTRIBUTIONS
10. Berg, H. C., and E. M. Purcell. 1977. Physics of chemoreception. Biophys. J. 20:193–219.
N.O. and H.A.S. designed research, performed research, and wrote the manuscript.
11. Keizer, J. 1985. Theory of rapid bimolecular reactions in solution and membranes. Acc. Chem. Res. 18:235–241.
ACKNOWLEDGMENTS
12. Adam, G., and M. Delbruck. 1968. Reduction of dimensionality in biological diffusion processes. In Structural Chemistry and Molecular Biology. A. Rich and N. Davidson, editors. W. H. Freeman, pp. 198–215.
We thank Itay Budin, Matan Yah Ben Zion, and Haim Diamant for helpful conversations. We gratefully acknowledge the National Science Foundation for partial support of this research via grant DMS-1219366.
13. Shea, L. D., G. M. Omann, and J. J. Linderman. 1997. Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking. Biophys. J. 73:2949–2959.
1. Creighton, T. E. 1984. Proteins: Structures and Molecular Properties. W.H. Freeman, New York. 2. Vance, D. E., and J. E. Vance. 2008. Biochemistry of Lipids, Lipoproteins and Membranes. Elsevier, Amsterdam, the Netherlands. 3. Trumpower, B. L. 1982. Function of Quinones in Energy Conserving Systems. Academic Press, New York. 4. von Smoluchowski, M. 1916. Drei vortrage uber diffusion. Brownsche bewegung und koagulation von kolloidteilchen. Phys. Z. 17:557–585. 5. Periasamy, N., S. Doraiswamy, ., G. R. Fleming. 1988. Diffusion controlled reactions: experimental verification of the time-dependent rate equation. J. Chem. Phys. 89:4799–4806. 6. Arita, T., O. Kajimoto, ., Y. Kimura. 2004. Experimental verification of the Smoluchowski theory for a bimolecular diffusion-controlled reaction in liquid phase. J. Chem. Phys. 120:7071–7074. 7. Nakatani, H., and H. B. Dunford. 1979. Meaning of diffusioncontrolled association rate constants in enzymology. J. Phys. Chem. 83:2662–2665. 8. Deutch, J. M., and B. U. Felderhof. 1973. Hydrodynamic effect in diffusion controlled reaction. J. Chem. Phys. 59:1669–1671. 9. Rice, S. A. 1985. Diffusion Limited Reactions. Elsevier, New York.
446 Biophysical Journal 113, 440–447, July 25, 2017
Reaction Rates in Membranes 14. Lauffenburger, D. A., and J. J. Linderman. 1993. Receptors: Models for Binding, Trafficking, and Signaling. Oxford University Press, New York.
24. Levine, A. J., T. B. Liverpool, and F. C. MacKintosh. 2004. Dynamics of rigid and flexible extended bodies in viscous films and membranes. Phys. Rev. Lett. 93:038102.
15. Saffman, P. G., and M. Delbr€uck. 1975. Brownian motion in biological membranes. Proc. Natl. Acad. Sci. USA. 72:3111–3113.
25. Levine, A. J., and F. C. MacKintosh. 2002. Dynamics of viscoelastic membranes. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66:061606.
16. Hughes, B. D., B. A. Pailthorpe, and L. R. White. 1981. The translational and rotational drag on a cylinder moving in a membrane. J. Fluid Mech. 110:349–372. 17. Fournier, J.-B. 2015. On the hydrodynamics of bilayer membranes. Int. J. Non-linear Mech. 75:67–76. 18. Camley, B. A., and F. L. H. Brown. 2013. Diffusion of complex objects embedded in free and supported lipid bilayer membranes: role of shape anisotropy and leaflet structure. Soft Matter. 9:4767–4779.
26. Komura, S., S. Ramachandran, and K. Seki. 2012. Anomalous lateral diffusion in a viscous membrane surrounded by viscoelastic media. Europhys. Lett. 97:68007.
19. Sigurdsson, J. K., and P. J. Atzberger. 2016. Hydrodynamic coupling of particle inclusions embedded in curved lipid bilayer membranes. Soft Matter. 12:6685–6707. 20. Evans, E., and E. Sackmann. 1988. Translational and rotational drag coefficients for a disk moving in a liquid membrane associated with a rigid substrate. J. Fluid Mech. 194:553–561. 21. Stone, H. A., and A. Ajdari. 1998. Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth. J. Fluid Mech. 369:151–173. 22. Seki, K., and S. Komura. 1993. Brownian dynamics in a thin sheet with momentum decay. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics. 47:2377–2383. 23. Fischer, T. M. 2004. The drag on needles moving in a Langmuir monolayer. J. Fluid Mech. 498:123–137.
27. Gov, N. S. 2006. Diffusion in curved fluid membranes. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73:041918. 28. Rahimi, M., A. DeSimone, and M. Arroyo. 2013. Curved fluid membranes behave laterally as effective viscoelastic media. Soft Matter. 9:11033–11045. 29. Zimmermann, E., and U. Seifert. 2015. Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91: 022709. 30. Russel, W. B., D. A. Saville, and W. R. Schowalter. 1989. Colloidal Dispersions. Cambridge University Press, Cambridge. 31. Zwanzig, R. 1969. Langevin theory of polymer dynamics in dilute solution. Adv. Chem. Phys. 15:325–331. 32. Van Kampen, N. G. 2007. Stochastic Processes in Physics and Chemistry, Third Edition. Elsevier, Amsterdam. 33. Oppenheimer, N., and H. Diamant. 2009. Correlated diffusion of membrane proteins and their effect on membrane viscosity. Biophys. J. 96:3041–3049.
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