4532
J. Phys. Chem. B 2000, 104, 4532-4536
A Model of Diffusion-Influenced Enzyme Activation Andrzej Molski† Department of Physical Chemistry, Adam Mickewicz UniVersity, Grunwaldzka 6, 60-780 Poznan´ , Poland ReceiVed: October 7, 1999; In Final Form: February 8, 2000
The kinetics of a model of diffusion-influenced enzyme activation is studied in two spatial dimensions. Enzyme molecules diffuse and collide with an immobile, spherical, uniformly reactive receptor-ligand complex. Enzymes do not interact with each other, but interact with the complex via an interparticle potential. No hydrodynamic interactions are included. Inactive enzymes get activated upon collisions with the complex. Enzyme deactivation is modeled as a first-order kinetic process independent of diffusion and collisions. Exact expressions are derived for the steady-state number of active enzymes and for the Laplace transform of the time-dependent activation rate coefficient. The activation kinetics in the time domain can be calculated by numerical inversion of the kinetic expressions in Laplace space.
1. Introduction Cellular signal transduction is initiated when a ligand (L) binds to a receptor (R) in the plasma membrane to form a receptor-ligand complex (RL):1
R + L T RL
(1)
The receptor-ligand complex may then start a sequence of processes that eventually triggers a cellular response. To gain insight into the mechanisms of signal transduction, one needs to understand the kinetics of the processes following the binding of a ligand to a receptor.1 Linderman et al.2-5 studied a model of enzyme activation defined by the following scheme:
RL + E f RL + E*
(2)
E* f E
(3)
where E is an inactive enzyme molecule and the star on E* denotes an active form of the enzyme. Process 2 represents diffusion-mediated bimolecular activation occurring in the plasma membrane, and process 3 represents unimolecular deactivation. Note the cyclic nature of enzyme activation in processes 2 and 3, and that a steady state can be achieved without adding or removing enzyme molecules from the system. Linderman et al.2-5 used Monte-Carlo simulations to study various aspects of processes 1-3 at the steady state, and found that no available theory adequately described the model. Since the complexity of processes 1-3 makes it impossible to find their exact kinetics, we have to search for a limiting situation where an exact solution is feasible. In this paper we consider a case where the total enzyme concentration is higher than that of the receptor-ligand complexes, so that the competition of the ligand-receptor complexes for enzymes can be neglected. We further limit ourselves to the situation where ligands bind tightly to the receptors so that the back process in eq 1 can be neglected. In this limit, enzyme activation and deactivation are described by processes 2 and 3, where process 2 represents a single receptor-ligand complex surrounded by many enzymes ready to be activated. The purpose of this work †
E-mail:
[email protected].
is to report on the exact kinetics of processes 2 and 3 for the case where the receptor-ligand complexes are immobile and the enzyme molecules diffuse. The case studied here is rather limited, and we are not aware of a specific enzyme-substrate system that meets the required conditions. Nevertheless, the present solution may serve as a test case for more general, and therefore more elaborate theories. 2. Model We are concerned with activation and deactivation processes 2 and 3 coupled with diffusion in two spatial dimensions. A receptor is placed at the origin of a planar coordinate system. At time t ) 0 the receptor binds a ligand to form a receptorligand complex. The central RL complex is modeled as a uniformly reactive sphere. The dissociation of the complex is neglected. The complex serves as an activation center for N enzyme molecules diffusing in an area S. The total concentration of enzymes is [E]T ) N/S. Enzymes diffuse and interact with the complex via an interparticle potential U(r), but do not interact with each other. No hydrodynamic interactions are included. Initially all enzyme molecules are inactive. When an inactive enzyme (E) molecule collides with the central complex (RL), the enzyme can get activated (process 2). Active enzymes (E*) deactivate with the first-order rate constant k- (process 3) independent of their collisions with the complex. The evolution of the system is expressed in terms of the joint probability function PN ) PN(r1,g1,...,rN,gN,t), where ri is the position of the ith enzyme molecule and gi is its state which is either inactive, gi ) 0, or active, gi ) 1. The function PN changes in time due to diffusion, activation, and deactivation. We assume the same diffusion coefficient D for each enzyme molecule irrespective of its state (active or inactive), so that the PN evolves according to
∂PN(r1,g1,...,rN,gN,t)/∂t ) N
[Lr + Lg + Rg (ri)]PN(r1,g1,...,rN,gN,t) ∑ i)1 i
i
i
(4)
The contribution of diffusion in eq 4 is described by the operator
10.1021/jp9935844 CCC: $19.00 © 2000 American Chemical Society Published on Web 04/15/2000
A Model of Diffusion-Influenced Enzyme Activation
J. Phys. Chem. B, Vol. 104, No. 18, 2000 4533
Lri ) ∇riDe-βU(ri)‚∇rie-βU(ri)
(5)
where U(ri) is the interaction potential between the RL complex and the ith enzyme. We assume the reflective boundary condition at contact
n‚∇riPN(r1,g1,...,rN,gN,t)|ri)a ) 0
[
]
p(r,1,0) ) 0
N
∫ d r1 ∑ ... ∫ d rN ∑PN(r1,g1,...,rN,gN,t) ∑δg 1 2
2
g1
gN
i)1
∫
) [E]T d2r p(r,1,t) (7)
+k-PN(...,ri,1,...,t) for gi ) 0 (8) -k-PN(...,ri,1,...,t) for gi ) 1
The activation is described by the operator
[
-κ(ri) 0 Rgi(ri) ) +κ(ri) 0
]
(9)
From eqs 17 and 19 it follows that
{
-κ(ri) PN(...,ri,0,...,t) for gi ) 0 Rgi(ri) PN(...,ri,gi,...,t) ) +κ(ri) PN(...,ri,0,...,t) for gi ) 1 (10) We focus on the case where the distance-dependent reactivity κ(r) represents activation at contact
κ(r) ) (k+/2πa) δ(r - a)
(in 2D)
(11)
where k+ is the intrinsic activation rate constant and a is the complex-enzyme collision radius. When combined with a reflective boundary condition at contact, eq 11 is equivalent to using the Collins-Kimball boundary condition.6 Initially all enzymes are inactive and in equilibrium with respect to diffusion, which is expressed as
(20)
where the activation rate coefficient k(t) is defined as
k(t) )
∫d2r κ(r) p(r,0,t) ) k+p(a,0,t)
(21)
The initial condition for the rate eq 20 is 〈N1〉 ) 0. Equation 20 can be solved to give
〈N1〉 ) [E]T
whose action on PN is defined as
i
(19)
d〈N1〉/dt ) -k-〈N1〉 + k(t)[E]T
{
(18)
We are interested in the average 〈N1〉 of the number N1 ) N δgi1 of active enzymes ∑i)1
〈N1〉 )
whose action on PN is defined as
LgiPN(...,ri,gi,...,t) )
p(r,0,0) ) e-βU(r)
(6)
where n is a vector normal to the surface of the complex. The deactivation is described by the operator
0 +kLgi ) 0 -k-
supplemented with the initial conditions
∫0t dt′ e-k (t-t′) k(t′) -
(22)
At long times the activation rate coefficient reaches a steadystate value kss, and the average number of active enzymes 〈N1〉ss becomes
〈N1〉ss ) kss[E]T/k-
(23)
Equations 14, 20, and 21 define the kinetics of the model. Equation 20 is the rate equation involving the time-dependent rate coefficient k(t). Equation 21 defines k(t) in terms of the function p(r,0,t), which can be obtained by solving eq 14. Given k(t), the number of active enzymes can be calculated from eq 22 and, at the steady state, from eq 23. In the next section we show that the Laplace transform of k(t) is related to the Laplace transform of the well-known Smoluchowski rate coefficient, so that the solution to the present activation kinetic problem reduces to inverting a rate expression in Laplace space.
N
PN(r1,g1,...,rN,gN,0) )
[δg 0e-βU(r )/S] ∏ i)1 i
i
(12)
where δgi0 is a Kronecker delta: δgi0 ) 1 when gi ) 0, and δgi0 ) 0 otherwise. With this initial condition the solution of eq 4 factorizes
[p(ri,gi,t)/S] ∏ i)1
(13)
where p(r,g,t) satisfies
∂p(r,g,t)/∂t ) [Lr + Lg + Rg(r)]p(r,g,t)
(14)
with the initial condition -βU(r)
p(r,g,0) ) δg0e
Let GD(r,t|r′) denote the Green function defined as a solution of the equation
∂GD(r,t|r′)/∂t ) LrGD(r,t|r′)+ δ(t) δ(r-r′)
(24)
and supplemented by reflective boundary conditions
N
PN(r1,g1,...,rN,gN,t) )
3. Activation Rate Coefficient k(t)
(15)
Equation 14 is equivalent to the set of two coupled equations
∂p(r,0,t)/∂t ) Lrp(r,0,t) + k-p(r,1,t) - κ(r) p(r,0,t)
(16)
∂p(r,1,t)/∂t ) Lrp(r,1,t) - k-p(r,1,t) + κ(r) p(r,0,t)
(17)
n‚∇rGD(r,t|r′)|r)a ) 0
(25)
The Green function GD(r,t|r′) gives the probability density of finding an enzyme molecule at position r at time t, given that it was at position r′ at time t ) 0. Similarly, let G(r,g,t|r′,g′) denote the Green function defined as a solution of the equation
∂G(r,g,t|r′,g′)/∂t ) [Lr + Lg]G(r,g,t|r′,g′) + δ(t) δ(r-r′) δgg′ (26) The Green function G(r,g,t|r′,g′) gives the probability density of finding an enzyme molecule at position r and in state g at time t, given that it was at position r′ and state g′ at time t ) 0, and that the only allowed change of state is deactivation as described by the operator Lg. Since deactivation is independent
4534 J. Phys. Chem. B, Vol. 104, No. 18, 2000
Molski
of diffusion, the two Green’s functions satisfy
kˆ S(z) )
G(r,0,t|r′,0) ) GD(r,t|r′)
(27)
e-βU(a) zG ˆ D(a,z|a)
(37)
whose explicit form in 2D and in the absence of the interaction potential, U(r) ) 0, is given by6
and
G(r,0,t|r′,1) ) (1 - e-k-t)GD(r,t|r′)
(28)
kˆ S(z) πa2
Moreover one has
∫ d2r′ G(r,0,t|r′,0) e-βU(r′) ) e-βU(r)
(29)
since e-βU(r) is an equilibrium distribution of the diffusion operator Lr. In terms of Green’s function G(r,g,t|r′,g′) eq 14 can be formally solved to give
p(r,g,t) )
∫ d2r′ ∑ G(r,g,t|r′,g′) p(r′,g′,0) +
∫ dt′ d r′ ∑ G(r,g,t-t′|r′,g′) Rg′(r′) p(r′,g′,t′)
kˆ dc(z) )
-
(31)
where
GD(a,t|a) )
1 1 1 ) + zkˆ (z) k(0) zkˆ dc(z)
(32)
Equation 31 can be solved to obtain the Laplace transform of the rate coefficient k(t)
k+e-βU(a) z[1 + k+ G ˆ D(a,z+k-|a)]
(33)
where the Laplace transform of a function f(t) is denoted ˆf(z) ) ∫∞0 dt exp(-zt) f(t). Multiplying eq 33 by the Laplace variable z and taking the short time limit (z f ∞), one recovers the initial, equilibrium value of the rate coefficient
k(0) ) k+e-βU(a)
(34)
as expected. Equation 33 gives the Laplace transform of the rate coefficient k(t) in terms of the quantity GD. We now show that the Laplace transform of k(t) is related to the Laplace transform of the Smoluchowski rate coefficient kS(t). When k- ) 0, eq 16 reduces to
∂p(r,0,t)/∂t ) [Lr - κ(r)]p(r,0,t)
(35)
which in combination with eq 21 is nothing but the definition of the Collins-Kimball rate coefficient kCK(t).6 Thus, in the special case k- ) 0 eq 33 gives the Laplace transform of kCT(t)
kˆ CK(z) )
k+e-βU(a) z[1 + k+G ˆ D(a,z|a)]
(38)
(39)
(36)
The limit of infinite intrinsic reactivity (k+ f ∞) of eq 36 gives the Laplace transform of the Smoluchowski rate coefficient6
(40)
Equation 40 gives an explicit expression for kˆ dc(z) in terms of the Smoluchowski rate coefficient kˆ S(z). Equations 33, 34, and 40 allow one to write the following relationship among the Laplace transform of the diffusioninfluenced rate coefficient k(t), its initial, equilibrium value k(0), and the Laplace transform of the diffusion-controlled limit of k(t):
∫ d2r d2r′ (2πa)-2δ(r-a) δ(r′-a) GD(r,t|r′)
kˆ (z) )
e-βU(a) zG ˆ D(a,z+k-|a)
kˆ dc(z) ) (1 + k-/z)kˆ S(z+k-)
(30)
Using eq 30 in connection with eq 21, one finds
∫0t dt′ GD(a,t-t′|a)e-k (t-t′) k(t′)
(za2/D)1/2K0[(za2/D)1/2]
so that kˆ dc(z) can be expressed in terms of kˆ S(z) as follows:
g′
k(t) ) k+e-βU(a) - k+
2K1[(za2/D)1/2]
where K0 and K1 are the modified Bessel functions of the second kind and zero and first order, respectively. The diffusion-controlled (k+ f ∞) limit of eq 33 is
g′
2
)
(41)
Interestingly, similar relationships hold for the Collins-Kimball rate coefficient,6 and for its generalization to the time-dependent rate coefficient describing binding of small molecules to a stochastically gated macromolecule performing translational and rotational motion.8 Taking the long time limit (z f 0) of eq 41, one finds for steady states
1 1 1 ) + ss ss k(0) k kdc
(42)
ˆ S(k-) kss dc ) k-k
(43)
where
Equations 40-43 are the main results of this section. Equations 40 and 41 provide an exact expression for the Laplace transform kˆ (z) of the activation rate coefficient k(t) in terms the Laplace transform of the Smoluchowski rate coefficient. Equations 40 and 41 can be inverted numerically to give the rate coefficient k(t) in the time domain. Equations 42 and 43 provide explicit exact expressions for the steady-state rate coefficient kss. 4. Illustrative Calculations: Diffusion-Controlled Activation Activation process 2 creates a depletion zone near the central complex where the concentration of the inactive enzymes is lower than [E]T. When diffusion is fast, it can offset the effect of activation, and the concentration of inactive enzymes is uniform around the complex. In this case the overall activation rate is determined by the intrinsic rate of activation, and does not depend on diffusion. In the opposite, diffusion-controlled limit, activation is fast (k+ f ∞) compared to diffusion, and the
A Model of Diffusion-Influenced Enzyme Activation
Figure 1. Dimensionless rate coefficient k/πD as a function of the dimensionless time t/τD calculated by numerically Laplace inverting eq 40 for the values of the parameter x ) k-a2/D as indicated.
Figure 2. Average number of active enzymes 〈N1〉 as a function of the dimensionless time t/τD calculated by numerically inverting the Laplace transform of eq 20 in connection with eq 40, for [E]T ) (πa2)-1 and the values of the parameter x ) k-a2/D as indicated.
effects of diffusion on the overall kinetics are most pronounced. In this section we consider the time dependence and steady states for the diffusion-controlled limit of the activation kinetics. For the sake of presentation it is convenient to consider the dimensionless rate coefficient k/πD as a function of the dimensionless time t/τD. Figure 1 shows the time dependence of k/πD, parametrized by the dimensionless quantity x ) k-a2/ D. Parameter x can be interpreted as the ratio x ) τD/τ, of the characteristic time for diffusion τD ) a2/D and the lifetime of the excited enzyme molecule τ ) k-1 - . Large values of x correspond to fast deactivation. At t ) 0 the rate coefficient k(0) is infinite. The case x ) 0 (no deactivation) corresponds to the Smoluchowski kinetics, where the rate coefficient in two spatial dimensions goes to zero at long times. For x > 0 the rate coefficient reaches a nonzero steady state whose amplitude gets higher when x is larger. The steady-state kss/πD is approached faster when deactivation is faster. The average number of active enzymes 〈N1〉 as a function of the dimensionless time t/τD can be calculated by numerically inverting the Laplace transform of eq 22 in connection with eq 41.9 The average number 〈N1〉 is proportional to the total enzyme concentration [E]T. Figure 2 shows the time dependence of 〈N1〉 for [E]T ) (πa2)-1, parametrized by x ) k-a2/D. Except for x ) 0, 〈N1〉 reaches a steady state, whose value is smaller when the deactivation is faster. Figure 3 shows how the steady state of the average number of active enzymes, 〈N1〉ss, and the steady-state rate coefficient, kss/πD, depend on x ) k-a2/D. Note that those two curves carry the same information since 〈N1〉ss ) kss/πDx for [E]T ) (πa2)-1. 5. Summary and Comments In this paper we have studied a model of diffusion-influenced enzyme activation in two spatial dimensions, where enzyme
J. Phys. Chem. B, Vol. 104, No. 18, 2000 4535
Figure 3. Long time limits of the average number of active enzymes 〈N1〉ss (eq 23 for [E]T ) (πa2)-1), and of the dimensionless rate coefficient kss/πD (eq 43) as function of x ) k-a2/D.
molecules diffuse and interact with an immobile receptor-ligand complex. Inactive enzymes get activated upon collisions with the complex, and deactivate according to first-order kinetics. We have derived expressions for the steady-state number of active enzymes (eqs 42 and 43) and for the Laplace transform of the time-dependent activation rate coefficient (eqs 40 and 41). The activation kinetics in the time domain can be calculated by numerical inversion of the kinetic expressions in Laplace space. Illustrative calculations were presented in section 4 for the diffusion-controlled regime, where diffusion is slow compared to activation. Recently there has been much interest in the effect of concurrent processes on the rate of diffusion-influenced reactions. Some of those complex processes can be related to the present problem of cyclic enzyme activation. In the gated substrate problem,7,8,10 a macromolecule processes substrate molecules which diffuse and undergo an interconversion between reactive and nonreactive states. Only reactive molecules can interact with the macromolecule. In the gated substrate problem the change of state (reactive T nonreactive) is an intramolecular first-order kinetic process, whereas here activation occurs upon bimolecular diffusion-mediated interactions with the receptor-ligand complex. A related problem concerns fluorescence quenching where excited fluorophores are generated according to the first-order excitation kinetics and are quenched through a bimolecular diffusion-mediated process.11,12 It is well known that irreversible steady states of diffusioninfluenced reactions depend on the way the particles are inserted into the system.13 One way of looking at processes 2 and 3 is to view the bimolecular activation 2 as a diffusion-mediated removal of inactive enzymes (E) from the system. Similarly, the unimolecular deactivation 3 can be viewed as a generation of inactive enzymes. Those two processes are spatially and temporally correlated since active enzymes (E*) have a finite lifetime. Correlated particle insertion mechanisms analogous to those defined by eqs 2 and 3 were studied by Bergling14 and in ref 15. Although this work has been motivated by the work of Linderman et al.,2-5 it is not intended to interpret the simulation in refs 2-5. First, we consider only the limiting case of pseudofirst-order kinetics of processes 2 and 3 when the dissociation of the complex is ignored. In refs 2-5 the concentrations of enzymes and receptors were comparable, and process 1 was allowed for. Second, we consider continuous diffusion whereas in refs 2-5 enzymes diffuse on a lattice. Consequently, noting that the dependence of k(t) on k- in Figure 1 resembles that in Figure 3 of ref 2, we do not make any quantitative comparisons. Processes 1-3 form a complex reaction scheme where one of the steps (process 2) is diffusion-controlled. It seems that, to
4536 J. Phys. Chem. B, Vol. 104, No. 18, 2000 solve the kinetics of coupled processes 1-3, one needs to resort to approximations. In this paper we have solved a limiting case, which may serve as a benchmark for more elaborate but approximate theories of the full scheme, eqs 1-3. Acknowledgment. This work was supported by the MEN/ NSF-98-329 grant from the Maria Skłodowska-Curie Fund. References and Notes (1) Lauffenburger, D. A.; Linderman, J. J. Receptors, Models for Binding, Trafficking, and Signaling; Oxford University Press: New York, 1993. (2) Shea, L. D.; Omann, G. M.; Linderman, J. J. Biophys. J. 1997, 73, 2949.
Molski (3) Shea, L. D.; Linderman J. J. Biochem. Pharmacol. 1997, 53, 519. (4) Mahama, P. A.; Linderman J. J. Anal. Biomed. Eng. 1995, 23, 299. (5) Mahama, P. A.; Linderman J. J. Biophys. J. 1994, 67, 1345. (6) Szabo, A. J. Phys. Chem. 1989, 93, 6929. (7) Zhou, H.-X.; Szabo, A. J. Phys. Chem. 1996, 100, 2597. (8) Zhou, H.-X.; Szabo, A. Biophys. J. 1996, 71, 2440. (9) Stehfest, H. Commun. ACM 1970, 13, 47. (10) Spouge J. L. J. Phys. Chem. B 1997, 101, 5206. (11) Yang, M.; Lee. S.; Shin, K. J. J. Chem. Phys. 1998, 108, 117. (12) Naumann, W. J. Chem. Phys. 1999, 110, 3926. (13) Clement, E.; Sander, L. M.; Kopelman, R. Phys. ReV. 1989, 39, 6466, 6472. (14) Bergling, S. Biophys. Chem. 1995, 56, 227. (15) Molski, A., Keizer, J. J. Chem. Phys. 1996, 104, 3567.
