INTRODUCTION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL FORMULATION RESULTS AND DISCUSSION APPENDIX REFERENCES

The behavior of a living cell is dictated by its ability to integrate information about its changing environment, a complex biochemical process known as signal transduction. In animal cells, the plasma membrane plays a central role in organizing early signaling events, because it is accessible to intracellular and extracellular molecules. Indeed, signaling pathways are most often initiated by transmembrane receptors, which typically possess separate domains for engaging extracellular ligands and intracellular proteins. A well-characterized example is the class of receptor tyrosine kinases, which includes all growth factor receptors. Upon ligand binding and activation of their intrinsic tyrosine kinase domains, these receptors are autophosphorylated on multiple residues, which then participate in specific binding interactions with cytosolic enzymes (van der Geer et al., 1994; Pawson, 1995). The formation of such receptor-enzyme complexes is the first step in the activation of specific signaling pathways by receptor tyrosine kinases and many other signaling receptors.

When recruited by an activated receptor, a cytosolic protein is transiently confined within a thin layer adjacent to the plasma membrane. This leads to an interesting physical situation, because receptor-bound signaling enzymes often act upon specific lipid or lipid-anchored protein substrates that diffuse laterally in the membrane. For example, this is true of the three signaling cascades that have received the most attention over the past decade, the Ras, phospholipase C (PLC), and phosphoinositide (PI) 3-kinase pathways. Ras is a small GTPase most noted for its role in cell proliferation, which is inactive for signaling when bound to GDP and active when bound to GTP. It is anchored in the membrane by lipid modifications, and its nucleotide-binding state is regulated by cytosolic enzymes (Wittinghofer, 1998). PLC isoforms and type I PI 3-kinases are enzymes that modify a common lipid substrate, PI(4,5)P₂. PLC-mediated hydrolysis of PI(4,5)P₂ has varying functions in different cell types, including cell migration and gene expression, whereas phosphorylation of the lipid by PI 3-kinases has been implicated in chemotaxis and cell survival (Rhee and Bae, 1997; Rameh and Cantley, 1999). In each of these three major signaling pathways, recruitment of the relevant enzyme to the membrane is expected to have a significant impact on its observed activity. Indeed, enzymes isolated from lysates of stimulated cells often show little or no change in activity, whereas modifying these enzymes for stable membrane insertion is generally sufficient for activating downstream signaling in cells (Buday and Downward, 1993; Aronheim et al., 1994; Quilliam et al., 1994; Klippel et al., 1996).

The importance of enzyme localization in intracellular signaling suggests that significant insights could be gained through a combined biochemical and biophysical approach, whereby concentrations of membrane components, reaction-rate constants, and diffusion coefficients are quantitatively determined. The ability to accurately predict measured signaling pathway fluxes in cells would validate such an approach, but this would require an appropriate theoretical framework. In this paper, a generalized continuum model is formulated to describe various signaling pathways in which receptor-enzyme complexes modify laterally diffusing membrane substrates, with the primary goal of relating steady-state reaction rates to physical and kinetic constants.

The influence of translational diffusion on molecular interactions in two-dimensional (2D) domains has been the subject of numerous theoretical studies. With the mean capture time (MCT) approach, individual absorbers are placed at the centers of independent, circular domains, and molecules are reflected at the boundary to balance diffusion into and out of the domain (Adam and Delbrück, 1968; Berg and Purcell, 1977). Thus, the MCT approach introduces an ad hoc boundary condition and imposes that absorbers are evenly spaced, a nonrandom distribution. In contrast, the mean field (or effective medium) approximation smears absorbers throughout the domain, accounting for the depletion of molecules available to one absorber by each of the others (Wiegel and DeLisi, 1982; Goldstein et al., 1988; Khakhar and Agarwal, 1993). The problem of 2D diffusion in a random array of absorbers is mathematically analogous to the Brinkman equation describing fluid flow past an array of disks; for this system, it was found that the effective medium approximation is accurate for disk area fractions less than 0.3 (Howells, 1974; Dodd et al., 1995). Finally, an analysis of linearized statistical fluctuations about the average concentration has also been used to calculate diffusion-limited reaction rates (Keizer et al., 1985). Like the mean field approach, this theory allows the average concentration to be reached at infinite separation from an absorber.

Considering intracellular signaling interactions in particular, the effect of membrane localization on enzyme action has been estimated previously. Diffusion and reaction rate limitations have been considered simultaneously for both cytosolic and receptor-bound enzyme pools, using the MCT equation for the diffusion-limited contribution in the membrane (Haugh and Lauffenburger, 1997). A similar analysis considered that both pools experience either diffusion- or reaction-limited behavior, and the time-dependent diffusive flux to a single absorber in a semi-infinite membrane was used (Kholodenko et al., 2000). A different signaling mechanism, activation of heterotrimeric G proteins by direct interaction with ligated receptors, has been analyzed using Monte Carlo simulations on a 2D lattice (Mahama and Linderman, 1994). These simulations described, for the first time, how the average lifetime of receptor-ligand complexes can significantly affect steady-state G-protein activation rates in the diffusion limit. Indeed, Shea et al. (1997) used similar simulations to show that major discrepancies exist between effective rate constants obtained using Monte Carlo techniques and the various analytical approaches described above. However, the extent to which fundamental differences between lattice and continuum models affect the values of computed rate constants is unclear.

The results of the simulations performed by Linderman and colleagues reinforce the fact that the effective rate constant for a membrane reaction depends on all parameters that influence the substrate concentration profile in the membrane. The model presented here is therefore designed to be sufficiently general, including reactions that compete with, reverse the action of, or supply substrate to a receptor-recruited enzyme. Limitations imposed by diffusion and reaction on the action of this enzyme are considered simultaneously. Perhaps most importantly, it is demonstrated for the first time that a continuum approach can be used to describe receptor-enzyme complexes of finite average lifetime. Following the formulation of the general model, the implications of these various effects on the substrate concentration profile will be examined, with an emphasis on the Ras and PLC signaling pathways.

	MATHEMATICAL MODEL FORMULATION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL FORMULATION RESULTS AND DISCUSSION APPENDIX REFERENCES

Model considerations

To construct a realistic model with as few adjustable parameters as possible, three major assumptions are invoked:

1.	There is a random distribution of cell surface receptors active for signaling within the membrane domain of interest. Although certainly reasonable as a default assumption, the theory may not be accurate in certain regions of the membrane, such as near clathrin-coated pits when internalization of receptors is diffusion-limited (Goldstein et al., 1988).
2.	The enzymatic reaction in the membrane follows a second-order rate law, with a rate proportional to the concentrations of substrate and receptor-enzyme complexes in the membrane. Thus, in terms of Michaelis-Menten kinetics, the Michaelis constant KM is significantly greater than the membrane concentrations of either species. The resulting second-order rate constant is given by the kcat/KM ratio of the enzyme at the membrane, where kcat is the first-order catalytic rate constant.
3.	The mobility of substrate molecules relative to receptor-enzyme complexes is described by 2D Fickian diffusion in a semi-infinite domain, with a constant diffusion coefficient. In making this assumption, nonidealities associated with diffusion in cellular membranes, and complex media in general, must be acknowledged (Sheetz, 1993; Feder et al., 1996; Pralle et al., 2000). However, the motion of the lipid or lipid-anchored substrate is expected to dominate the diffusion coefficient in this case. Unlike many transmembrane receptors, such molecules exhibit diffusion coefficients close to theoretical values and essentially complete fluorescence recovery after photobleaching (Schlessinger et al., 1977; Niv et al., 1999).

In the sections that follow, models describing two distinct signaling-pathway types will be presented (Fig. 1). Major signaling pathways adequately simulated by each model are discussed, and the appropriate equations are formulated. It is then shown that a generalized model encompasses both pathway types, and analytical solutions are derived for the effective enzymatic rate constant at steady state. Finally, the theory is extended to account for the finite mean lifetime of receptor-enzyme complexes.

View larger version (32K): [in this window] [in a new window]   FIGURE 1   Pathway types described by the general model. (a) Two-state mechanism. A membrane-anchored substrate is converted back and forth between inactive and activated states. Activation and inactivation occur in the bulk membrane with observed first-order rate constants ka and ki, respectively. Substrate activation is enhanced by receptor-enzyme complexes, which act upon inactive substrate with second-order rate constant kRE. (b) Regulated supply and turnover. The concentration of a membrane-anchored substrate is determined by the relative rates of consumption and transfer to the membrane. The first-order rate constant describing basal consumption is kc, and RT,0 is the basal rate of substrate transfer. Both the turnover and supply of the substrate are modulated by activated receptors. Receptor-enzyme complexes consume substrate with second-order rate constant kRE, whereas receptor-transfer protein complexes deliver substrate with first-order rate constant kRT.

Pathway type 1: reversible, two-state mechanism

In signaling pathways of this type (Fig. 1 a), a membrane-associated substrate is converted to an activated state by a receptor-recruited enzyme. The inactive substrate is later recovered by a first-order reaction, such that the total number of molecules in the active and inactive states is conserved on the time scale of interest. Regulation of the membrane-anchored GTPase Ras, which is inactive in the GDP-bound state and active in the GTP-bound state, is well described by this two-state model. In numerous cell types, an increase in Ras-GTP is mediated by the recruitment of guanine nucleotide exchange factors (GEFs) from the cytosol to the plasma membrane, in complex with adaptor proteins that link them to signaling receptors (Medema et al., 1993; Buday and Downward, 1993). GEFs catalyze the dissociation of bound nucleotides from Ras in cells; the subsequent rebinding of nucleotides is extremely rapid, favoring uptake of the more abundant GTP (Lenzen et al., 1998). To recover Ras-GDP, the GTPase activity of Ras hydrolyzes bound GTP, a reaction that is significantly accelerated by GTPase-activating proteins (GAPs). These GAP activities help maintain the majority of Ras in the GDP-bound state prior to cell stimulation (Wittinghofer et al., 1997). The two-state mechanism also serves as an idealized model of G-protein activation. Ligand-bound G protein-coupled receptors interact directly with heterotrimeric G proteins, which are composed of , , and  subunits (Hamm and Gilchrist, 1996). The GDP-bound  subunit is thereby converted to the active GTP-bound state and liberated from the subunits; following the GTPase reaction and recovery of the GDP-bound state, the  and subunits rapidly reassociate.

In this model pathway, substrate in the inactive state is conserved by

(1)

where n_S is the 2D concentration of the inactive substrate as a function of radial distance r from a receptor-enzyme complex and time t. The lateral diffusion coefficient of the substrate relative to receptor-enzyme complexes, D, is assumed to have the same value for the inactive and active states. The first-order rate constants k_a and k_i characterize constitutive activation and inactivation steps, respectively; the latter acts upon the active form of the substrate, which has a 2D concentration n_S*. The mean field approximation is introduced through the inclusion of a rate term that accounts for consumption of substrate by each of the neighboring receptor-enzyme complexes (2D concentration n_RE), with effective second-order rate constant k. Substituting the relation n_S + n_S* = n_S,tot = constant, one finds that

(2)

Eq. 2 is subject to two boundary conditions. At the enzyme-substrate encounter distance (r = s, interpreted as the sum of the associating molecules' radii), the flux of inactive substrate to each receptor-enzyme complex is balanced by the rate of the enzymatic reaction. The true second-order rate constant of the reaction is k_RE. The second boundary condition maintains a finite value of n_S at infinite separation from the receptor-enzyme complex; this value of n_S is equivalent to its average in the membrane, defined as _S,

(3)

Finally, the enzymatic reaction rate, k, and k_RE are related by

(4)

It follows that the effective rate constant k deviates from k_RE when the substrate is not homogeneously distributed, and so the magnitude of k can be used to assess the influence of spatial effects on the reaction rate.

Pathway type 2: regulated substrate supply and turnover

In signaling pathways of this type (Fig. 1 b), a membrane-associated substrate is consumed by receptor-enzyme complexes, but here the action of the enzyme cannot be reversed by another reaction in the membrane. Rather, the substrate is delivered to the membrane continuously by a cytosolic transfer protein. Like the enzyme, a fraction of the transfer protein is engaged by activated receptors, and it is assumed that the enzyme and transfer protein bind to independent receptor sites. The formation of receptor-transfer protein complexes enhances substrate delivery, allowing higher reaction rates through the receptor-enzyme complexes, and the availability of substrate is determined by the relative extents to which its supply and consumption are modulated.

The action of receptor-recruited PLC, which hydrolyzes the membrane lipid PI(4,5)P₂, is well described by this model. The products of PI(4,5)P₂ hydrolysis are inositol (1,4,5)-trisphosphate, released into the cytosol, and (1,2)-diacylglycerol, which remains in the membrane. These products are metabolized and then recombined to form phosphatidylinositol in the endoplasmic reticulum, and this lipid precursor must be transferred to the plasma membrane for reconstitution of the substrate PI(4,5)P₂ (Hsuan and Tan, 1997; Toker, 1998). To prevent depletion of PI(4,5)P₂, its supply to the plasma membrane is also positively modulated by receptor stimulation (Batty et al., 1998; Willars et al., 1998). The PI(4,5)P₂ delivery rate is probably influenced by direct interactions between receptors and proteins involved in phosphatidylinositol transfer and phosphorylation (Kauffmann-Zeh et al., 1994, 1995). The regulated supply and turnover model is also suitable for the PI 3-kinase pathway, in which PI(4,5)P₂ is phosphorylated to form the lipid second messenger PI(3,4,5)P₃ (Vanhaesebroeck and Waterfield, 1999). Though this reaction can be reversed in the membrane by different routes, the action of PI 3-kinase is expected to be influenced by the aforementioned regulation of the PI(4,5)P₂ concentration.

The substrate conservation equation for the regulated supply and turnover model is

(5)

Again, n_S(r, t) and D are the concentration and relative diffusion coefficient of the substrate, n_RE is the concentration of receptor-enzyme complexes, and k is the effective second-order rate constant for enzyme action at the membrane. The first-order rate constant k_c describes basal substrate consumption, and R_T,0 is the basal rate of substrate delivery to the membrane. Substrate delivery is accelerated by receptor-transfer protein complexes (2D concentration n_RT), which insert substrate with first-order rate constant k_RT. The boundary condition at the enzyme-substrate encounter distance (r = s) is given by

(6)

where k_RE is the second-order rate constant of the enzymatic reaction. The second term on the right-hand side of Eq. 6 accounts for the possibility that the receptor engaging the enzyme may be bound to a transfer protein as well, in which case substrate will be delivered at this location. This probability is given by n_RT/n_R, where n_R is the total concentration of activated receptors in the membrane. The boundary condition at r =  and the relationship between k and k_RE are the same as in the two-state model.

General model and solution for stable receptor-enzyme complexes

A conservation equation and encounter-distance boundary condition that encompass both of the pathway models illustrated in Fig. 1 can be posed,

(7)

Here, the identities of the rate constants k₀ and k_v and the source term V₀ will depend on the pathway type. In the two-state model, k₀  k_a + k_i, V₀  k_in_S,tot, and k_v = 0; in the regulated supply and turnover model, k₀  k_c, V₀  R_T,0, and k_v  (n_RT/n_R)k_RT. To reduce the number of constant parameters, Eq. 7 is nondimensionalized,

(8)

The dimensionless parameter Da is a Damköhler number comparing the rates of basal consumption and diffusion of the substrate. The effective and actual second-order enzymatic rate constants are scaled to the diffusion coefficient D to yield  and , respectively. The dimensionless density of activated receptors is given by _R, and _RE is the corresponding density of activated receptors bound to enzyme. The enhancement of the substrate delivery rate by activated receptors is characterized by . The identities of these dimensionless parameters and their estimated value ranges are summarized in Table 1.

When the lifetime of a receptor-enzyme complex is much longer than the time required for the substrate profile to evolve around it, the following steady state solution is obtained:

(9)

where K_i are modified Bessel functions of order i, and _ss is the dimensionless analog of _S at steady state. Finally, an implicit solution relating the steady-state effective rate constant  to the other parameters is found,

(10)

A possible modification of Eq. 8 is to distinguish receptor-enzyme complexes that are contributing to substrate delivery from those that are not with separate boundary conditions. It was confirmed that such an approach, with an overall  computed as a weighted sum of the  values for the two receptor pools, yields the same result as in Eq. 10 (not shown).

At this stage, it is instructive to review limiting cases of Eq. 10 and compare these with previous analytical theories. In the limit of low receptor-enzyme density, the term in Eq. 8 invoking the mean field approximation vanishes, and Eq. 10 becomes an explicit function of constant parameters (Da* = Da). For the case of  = 0, this low-density limit was derived previously to describe the Langmuir-Hinshelwood surface reaction mechanism with reactant absorption/desorption (Freeman and Doll, 1983). In the limit of very high receptor-enzyme density (Da* Da) and  = 0, the result of Wiegel and DeLisi (1982) describing binding of ligands to cell receptors through nonspecific membrane adsorption and diffusion is obtained. The effective rate constant at any receptor-enzyme density, again with  = 0, also agrees with results derived for other systems. Goldstein et al. (1988) described diffusion-limited capture of cell surface receptors by internalizing traps, and Khakhar and Agarwal (1993) extended the work of Freeman and Doll to describe the Langmuir-Hinshelwood mechanism for higher reactant densities.

General model solution for receptor-enzyme complexes with finite mean lifetime

In general, an individual enzyme-receptor complex may dissociate before the substrate concentration can achieve the profile predicted from Eq. 9, though a constant concentration of such complexes will be maintained, on average, at steady state. Hence, the mean field approach was used to calculate the impact of receptor-enzyme complex stability on the effective enzymatic rate constant. It should be noted that the receptor-dependent source term, characterized by , might involve a receptor-transfer protein complex; this interaction is assumed to be stable, because its lifetime is not considered here.

Eq. 8 is used to derive the evolution of the substrate concentration profile surrounding an individual enzyme-receptor complex during its lifetime (the "enzyme-on" phase), with the average density of receptor-enzyme complexes at steady state reflected in the effective rate term. The transient solution is given by

(11)

where _ss() is the steady-state substrate profile in the infinite lifetime limit (Eq. 9), (, 0) is the initial substrate profile, and J_i and Y_i are Bessel functions of order i. Eq. 11 is obtained using either Fourier-like integral transform or Laplace transform (Carslaw and Jaeger, 1940) solution methods. The dimensionless mean lifetime of the complex, inversely proportional to the dissociation rate constant of the receptor-enzyme interaction, is defined as _RE. The effective enzymatic rate constant is then found using the implicit relation,

(12)

After the complex dissociates, the substrate profile around the free activated receptor homogenizes. The transient for this "enzyme-off" phase is found by analogy to Eq. 11, with  = 0. The initial condition is given by the substrate profile at the end of the enzyme-on phase, calculated from Eq. 11. The dimensionless average duration of the enzyme-off state is defined as _R, and

(13)

This process is important because the substrate profile at the end of the enzyme-off phase is the initial condition for the next enzyme-on phase, and so on. This implicit condition is required to completely specify the problem, but the solution simplifies greatly for the interesting limiting cases. As _R vanishes (_RE/_R  1), it is readily shown that  = _ss; the finite complex lifetime does not matter, because the cytosolic concentration of the enzyme is high enough to saturate all activated receptors. As _R becomes large (_RE/_R 1), it is apparent that (, 0) = _ss at the beginning of each enzyme-on phase, and the implicit relation for the effective enzymatic rate constant becomes

(14)

Eq. 14 was obtained from Eq. 12 by specifying the initial condition as the homogeneous _ss and evaluating (1, ) from Eq. 11. The integral is evaluated numerically. In the purely diffusion-limited regime (  ), Eq. 14 further simplifies to

(15)

A satisfying result is the presence of three clearly separated contributions to the effective rate constant in this regime. The first term describes a receptor-enzyme complex of infinite lifetime, the second describes the contribution of receptor-mediated substrate delivery (nonzero ), and the third contains the influence of a finite lifetime _RE.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL FORMULATION RESULTS AND DISCUSSION APPENDIX REFERENCES

Two-state mechanism

Influences of substrate inactivation and neighboring receptor-enzyme complexes on the effective enzymatic rate constant

View larger version (23K): [in this window] [in a new window]   FIGURE 2   Effective enzymatic rate constant, two-state mechanism: stable receptor-enzyme complexes. (a) The effective rate constant  is computed as a function of the dimensionless rate constant  = kRE/D, for the indicated values of Da = (ka + ki)s2/D and stable receptor-enzyme complexes at low density (RE  0, RE  ). (b) The effective rate constant  is computed as a function of the dimensionless receptor-enzyme density RE = s2nRE for the indicated values of Da, diffusion-limited enzyme action, and stable receptor-enzyme complexes (  , RE  ). The dashed line is the prediction of the MCT approach (Eq. 16).

(16)

Influence of receptor-enzyme complex lifetime on the enzymatic rate constant and comparison with Monte Carlo simulations

1

3

View larger version (22K): [in this window] [in a new window]   FIGURE 3   Effective enzymatic rate constant, two-state mechanism: receptor-enzyme complexes with finite mean lifetime. (a) The effective rate constant  is computed as a function of the dimensionless receptor-enzyme lifetime RE = DtRE/s2 for the indicated values of Da, diffusion-limited enzyme action, and receptor-enzyme complexes at low density (  , RE  0). (b) The effective rate constant  is computed as a function of the dimensionless receptor-enzyme density RE for the indicated values of RE, Da = 103, RE/R 1, and diffusion-limited enzyme action (  ).

Predicting the fraction of substrate in the activated state

(17)

View larger version (25K): [in this window] [in a new window]   FIGURE 4   Fraction of substrate in the activated state, two-state mechanism. The activated substrate fraction is computed as a function of the receptor-enzyme density RE and constant rate parameters from Eq. 17, with ka = 0 and diffusion-limited enzyme action (  ). (a) Stable receptor-enzyme complexes (RE  ). The value of  in Eq. 17 was calculated using: solid lines, the full mean field theory; dotted lines, the low RE limit ( constant); dashed lines, the MCT approach (Eq. 16). (b) Receptor-enzyme complexes with finite mean lifetimes. Plots are for RE R, Da = 103, and various RE values (, 100, 10, and 1), with curves for lower RE shifted to the left.

Regulated supply and turnover

Receptor-mediated substrate delivery affects the enzymatic rate constant, even when enzyme action is reaction limited

4

4

View larger version (24K): [in this window] [in a new window]   FIGURE 5   Effective enzymatic rate constant, regulated supply and turnover. In this pathway type, the dimensionless rate constant Da = kcs2/D, and the parameter , describing receptor-mediated receptor transfer, comes into play. (a) The effective rate constant  is computed as a function of  for the indicated values of Da and stable receptor-enzyme complexes at low density (RE  0, RE  ). Solid lines and closed symbols,  = 0; dot-dashed lines and open symbols,  = 500. (b) The effective rate constant  is computed as a function of the dimensionless receptor-enzyme density RE for the indicated values of Da and , diffusion-limited enzyme action, and stable receptor-enzyme complexes (  , RE  ). Closed symbols,  = 0; open symbols,  = 500. Solid lines, RE = R; dotted lines, RE = 0.1R.

4

4

4

Prediction of enzymatic reaction rates and comparison with PI(4,5)P₂ hydrolysis data