Article Information

• PDF (152 kb)
• Full Text with Large Figures

PubMed

• Articles by Chinlin Guo
• Articles by Herbert Levine

Related Articles

• Effects of Receptor Interaction in Bacterial Chemotaxis

  Effects of Receptor Interaction in Bacterial Chemotaxis Biophysical Journal, Volume 87, Issue 3, 1 September 2004, Pages 1578-1595 Bernardo A. Mello, Leah Shaw and Yuhai Tu Abstract Signaling in bacterial chemotaxis is mediated by several types of transmembrane chemoreceptors. The chemoreceptors form tight polar clusters whose functions are of great biological interest. Here, we study the general properties of a chemotaxis model that includes interaction between neighboring chemoreceptors within a receptor cluster and the appropriate receptor methylation and demethylation dynamics to maintain (near) perfect adaptation. We find that, depending on the receptor coupling strength, there are two steady-state phases in the model: a stationary phase and an oscillatory phase. The mechanism for the existence of the two phases is understood analytically. Two important phenomena in transient response, the overshoot in response to a pulse stimulus and the high gain in response to sustained changes in external ligand concentrations, can be explained in our model, and the mechanisms for these two seemingly different phenomena are found to be closely related. The model also naturally accounts for several key in vitro response experiments and the recent in vivo fluorescence resonance energy transfer experiments for various mutant strains. Quantitatively, our study reveals possible choices of parameters for fitting the existing experiments and suggests future experiments to test the model predictions. Abstract | Full Text | PDF (302 kb)

• The Influence of Polymer Molecular Weight in Lamellar Gels B...

  The Influence of Polymer Molecular Weight in Lamellar Gels Based on PEG-Lipids Biophysical Journal, Volume 75, Issue 1, 1 July 1998, Pages 272-293 Heidi E. Warriner, S.L. Keller, Stefan H.J. Idziak, Nelle L. Slack, Patrick Davidson, Joseph A. Zasadzinski and Cyrus R. Safinya Abstract We report x-ray scattering, rheological, and freeze-fracture and polarizing microscopy studies of a liquid crystalline hydrogel called L. The hydrogel, found in DMPC, pentanol, water, and PEG-DMPE mixtures, differs from traditional hydrogels, which require high MW polymer, are disordered, and gel only at polymer concentrations exceeding an “overlap” concentration. In contrast, the L uses very low-molecular-weight polymer-lipids (1212, 2689, and 5817g/mole), shows lamellar order, and requires a lower PEG-DMPE concentration to gel as water concentration increases. Significantly, the L contains fluid membranes, unlike L gels, which gel via chain ordering. A recent model of gelation in L phases predicts that polymer-lipids both promote and stabilize defects; these defects, resisting shear in all directions, then produce elasticity. We compare our observations to this model, with particular attention to the dependence of gelation on the PEG MW used. We also use x-ray lineshape analysis of scattering from samples spanning the fluid-gel transition to obtain the elasticity coefficients and B; this analysis demonstrates that although B in particular depends strongly on PEG-DMPE concentration, gelation is uncorrelated to changes in membrane elasticity. Abstract | Full Text | PDF (1045 kb)

• Protein-Cofactor Interactions in Bacterial Reaction Centers ...

  Protein-Cofactor Interactions in Bacterial Reaction Centers from Rhodobacter sphaeroides R-26: II. Geometry of the Hydrogen Bonds to the Primary Quinone QA⋅⁡− by H and H ENDOR Spectroscopy Biophysical Journal, Volume 92, Issue 2, 15 January 2007, Pages 671-682 M. Flores, R. Isaacson, E. Abresch, R. Calvo, W. Lubitz and G. Feher Abstract The geometry of the hydrogen bonds to the two carbonyl oxygens of the semiquinone in the reaction center (RC) from the photosynthetic purple bacterium R-26 were determined by fitting a spin Hamiltonian to the data derived from H and H ENDOR spectroscopies at 35GHz and 80K. The experiments were performed on RCs in which the native Fe (high spin) was replaced by diamagnetic Zn to prevent spectral line broadening of the due to magnetic coupling with the iron. The principal components of the hyperfine coupling and nuclear quadrupolar coupling tensors of the hydrogen-bonded protons (deuterons) and their principal directions with respect to the quinone axes were obtained by spectral simulations of ENDOR spectra at different magnetic fields on frozen solutions of deuterated in HO buffer and protonated in DO buffer. Hydrogen-bond lengths were obtained from the nuclear quadrupolar couplings. The two hydrogen bonds were found to be nonequivalent, having different directions and different bond lengths. The H-bond lengths are 1.73±0.03Å and 1.60±0.04Å, from the carbonyl oxygens O and O to the NH group of Ala M260 and the imidazole nitrogen N of His M219, respectively. The asymmetric hydrogen bonds of affect the spin density distribution in the quinone radical and its electronic structure. It is proposed that the H-bonds play an important role in defining the physical properties of the primary quinone, which affect the electron transfer processes in the RC. Abstract | Full Text | PDF (281 kb)

• …more

Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 77, Issue 5, 2358-2365, 1 November 1999

doi:10.1016/S0006-3495(99)77073-6

Biophysical Theory and Modeling

A Thermodynamic Model for Receptor Clustering

Chinlin Guo and Herbert Levine^,  

Department of Physics, University of California, San Diego, La Jolla, California 92093-0319 USA

Address reprint requests to Dr. Herbert Levine, Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0319. Tel.: 858-534-4844; Fax: 858-534-7697.

Cell growth, differentiation, migration, and apoptosis are regulated in part by extracellular polypeptide growth factors or cytokines (Heldin, 1995,Stuart and Jones, 1995). As these molecules are unable to pass through the hydrophobic cell membrane, they have to bind to the extracellular domains of specific surface receptors to exert their effects. Much effort has gone into investigating the fundamental question of how the ligand-receptor interaction can trigger the proper intracellular signals. One popular hypothesis is that ligand-induced “clustering” of ligand-receptor complexes can be a key element in the proper activation of downstream signals. (Ashkenazi and Dixit, 1998,Bray et al,Heldin, 1995,Germain, 1997,Lemmon and Schlessinger, 1994,Lemmon and Schlessinger, 1998,Reich et al,Sakihama et al).

As an example of this line of reasoning, we consider the signaling cascade mediated by the binding of tumor necrosis factor (TNF) to the receptor TNF-R1. Internally, the cytoplasmic domain of TNF-R1 is “sensed” by a variety of adaptor proteins, namely TRADD, FADD, TRAF2, and RIP; this sensing leads eventually to NF-κB/JNK/SAPK activation and apoptosis. To accomplish the downstream signaling, an oligomerization of these adaptor proteins is required (Ashkenazi and Dixit, 1998). One way to facilitate oligomerization is via construction of a molecular scaffolding by TNF-induced TNF-R1 clustering. It is known that TNF-R1 will not aggregate in the absence of TNF; this is due to the association of an inhibitor, “silencer of death domain” (SODD), which normally attaches to TNF-R1 cytoplasmic domains and prevents receptor aggregation (Jiang et al), or, alternatively, is due to the receptor extracellular domains, inasmuch as spontaneous association of TNF-R1 has been observed in cells that express truncated receptors (Boldin et al,Vandevoorde et al). TNF treatment, however, can bring two or more receptors into proximity via its multiple binding capacity (Jones et al,Jones et al). This “proximity” might “squeeze” out SODD (Jiang et al), expose the cytoplasmic “death” domains to adaptor proteins, and thereby stabilize receptor clusters. Thus, a molecular scaffold/nuclei is generated to initiate signaling.

Over a longer time scale, the signaling messages can provide feedback to modify the capability of surface receptor clustering (Humphries, 1996,Wyszynski et al). This leads to a complex dynamical process involving both the intracellular signaling cascades as well as the surface receptor clustering. The self-organization made possible by these feedback processes has been intensively discussed for signaling cascades (see, e.g., Jafri and Keizer, 1995,Barkai and Leibler, 1997). Much less is understood, however, regarding the role of receptor clustering. It is clear, though, that given the hypothesis that cellular signaling relies on the formation of receptor clusters, the temporal and spatial characteristics of clustering would certainly affect the process of signaling transduction. Thus, modeling the physical properties of receptor clustering is as important as modeling signaling cascades.

Because clustering is due to an interaction between nearest-neighbor receptors, it is obviously a cooperative process. From a physics perspective a system with this type of cooperativity can exhibit a first-order phase transition, corresponding to a jump in the surface density of ligand-receptor complexes. In the coexistence region of this transition, the surface will spontaneously segregate into two phases, dilute and dense. This first-order phase transition endows the signal transduction process with the ability to produce a digital signal in an analog world; this is independent of the details of intracellular cascades, arising instead from the intrinsic cooperativity in ligand-receptor interaction. This has not been adequately addressed in the few models studied to date (Goldstein and Wiegel, 1983,Goldstein and Perelson, 1984,Riley et al,Coutsias et al,Shea et al).

The purpose of this work is to introduce a phenomenological model for the TNF-TNFR1 system to describe the onset of receptor clustering (phase separation). Specifically, we assume that clustering can be described by the statistical mechanics of a simple lattice Hamiltonian, incorporating the fundamental mechanism of a multimeric binding capacity for the ligand. We will calculate (via mean-field theory) a phase diagram and show that clustering will be thermodynamically favored for some range of ligand and receptor densities. Finally, we will do a simple Monte Carlo simulation of this system, showing that receptor diffusion will lead rapidly to cluster formation in the relevant parameter range. We neglect the possibility that there exist long-time feedback processes that modify the clustering capacity, and we ignore some inessential details of the receptor-ligand interaction. More detailed models including these effects, as well as applications to other signaling systems, will be presented in the future.

In our model, we treat the cell surface as a lattice with a spacing on the order of a few nm; this is the closest that neighboring receptors can get to each other. Each lattice site i has either one or zero receptor molecules, denoted as n_i=1 or 0. Our receptor has only two states: liganded or unliganded, and the interaction between receptor molecules is determined by their states. This “two-state” model is oversimplified, yet we will see that it gives reasonable predictions for the phase diagram. A “state” label, t_i=1 or 2, to represent unliganded or liganded, then, can be assigned to each occupied receptor. We will further assume that the only ligands on the surface are those bound to receptors. If we let the chemical potential of the ligand be μ_L and that of the receptor be μ_R, we then get a contribution to the effective Hamiltonian of the system

(1)

We should clarify the relationship between the parameters used here and those in real experiments. Using standard ideas (Changeux et al), we notice that with only this term, the partition function can be factorized and reduced to a single site problem,

(2)

We next add a receptor-receptor interaction term. This takes the general form

(3)

Display large version of this figure

Figure 1

Interleaved sublattices labeled as filled/open circles on a one-dimensional and a two-dimensional (square/honeycomb) space. On a square lattice, j_1,2,3,4 is the nearest neighbor to site i.

a(1, 2) and a(2, 1) are the interaction energies, for which we will use an effective binding strength g_E on the order of g_L/10, arising via one or two hydrogen bonds between receptors. It is important to realize that our simplified model does not treat explicitly the formation of multimers via multimeric binding. Instead, it arbitrarily assigns the one ligand (e.g., binding two receptors into a dimer) to one of the receptors and describes the dimeric binding as an attraction between a bound and an unbound receptor. Because of this, the model cannot distinguish between this relatively strong interaction and the subsequent much weaker interaction between the dimers. In future work, we will show that this complication does not alter the basic picture presented here.

As discussed above, in the TNF system there is probably a short-range and nonspecific “excluding” interaction between two unliganded or two liganded (with different ligand molecules) receptors. For the sake of simplicity, we will assume that the repulsive energy is on the same order of magnitude as the associative one, i.e., a(1, 1) ≈ a(2, 2) ≈ −g_E. This assumption is not necessary, yet it greatly simplifies the mathematical task for analysis.

The symmetry of a(t_i, t_j) allows us to introduce a simple matrix notation for the total Hamiltonian H₁+H₂. If we use two-component vectors for the state labeling: τ_i=[₀¹] for t_i=1, and τ_i=[₁⁰] for t_i=2, then the Hamiltonian can be rewritten as

(4)

(5)

(6)

If we define a new notation u_i=n_iσ_i, our model would be very similar to a spin-1 antiferromagnetic (AFM) BEG model (Blume et al),

(7)

We should point out that this negative cooperation is not universal. In the case of an EGF-R (epidermal growth factor receptor) system, a ferromagnetic (FM) behavior (“positive cooperation”) is more likely, because there clustering requires two or more liganded receptors (Lemmon et al). Thus the higher the ligand concentration, the more the EGF-R cluster can be formed, and the EGF-EGFR signaling response behaves in a sigmoidal rather than a window-like pattern. It is clear that in both EGF-R and TNF-R (and hGH-R, EPO-R) systems, the ligand multiple binding capacity is the essential ingredient for inducing clustering (of course one should consider the effect of the receptor cytoplasmic domain as well). Which kind of cooperation (negative or positive) one should consider depends on the details of the receptor-receptor interaction (also including the chemical modifications on receptor cytoplasmic domains) and needs to be established experimentally. But, the essential feature of a first-order transition-like behavior in receptor clustering is independent of the sign of this additional cooperativity.

To see if our model can generate clustering, we perform a Monte Carlo simulation on a square lattice with the standard Metropolis scheme. For simplicity, we fix the number of liganded and unliganded receptors and do not allow these to fluctuate. Given the rather strong binding, this is not an important constraint. Furthermore, we allow motion only for individual receptors and do not explicitly allow a cluster to move as a whole; this might not be the case in reality. The “jumping” probability that a receptor will move to another lattice site is determined by the Hamiltonian and obeys the detailed balance law. In detail, we pick a receptor at random and try to move it in a randomly chosen direction. The move is accepted if it lowers the energy, and the move is accepted with probability e^−βΔH if the energy increases.

From Fig. 2, we immediately see that for a given receptor density, changing the ligand concentration moves the system from a nonclustering to a clustering phase. In this figure, the open and filled circles indicate liganded and unliganded receptor molecules, respectively. Note that the open and filled circles are arranged in an alternative way to form the cluster (i.e., inside a cluster, the nearest neighbors of the open circles must be filled circles, and vice versa). This implies that the equilibrium state (which must be translationally invariant) can be described by dividing the system into two interleaved sublattice systems: one sublattice is occupied by one species of receptor molecule (liganded or unliganded), and all of its nearest neighbors belong to the alternative sublattice, which is occupied by another species.

Display large version of this figure

Figure 2

Monte Carlo simulation with Metropolis scheme. Here we test the model under a fixed receptor density but different ligand concentrations. In both upper and lower panels, the left figure represents initial conditions, and the right figures are results after 10⁸ Monte Carlo steps. The open and filled circles indicate liganded and unliganded receptor molecules, respectively. There is no stable cluster formation in the upper snapshot, whereas the clustering is stable in the lower one. Here we use g_E=6k_BT, density of liganded receptor: upper plane, 0.001; lower plane, 0.03; and density of unliganded receptor: upper plane, 0.059; lower plane, 0.03.

To obtain more insight into the conditions where receptor clustering can take place, we next analyze the partition function via the mean-field approximation.

To proceed, we decouple the quadratic term in the Hamiltonian by introducing an auxiliary Gaussian field and employing the standard Hubbard-Stratonovich/Gaussian transformation (see, e.g., Amit, 1993,Parisi, 1988) (Eq. (A3)). The benefit of this transformation is to decouple the quadratic terms into linear terms such that we can sum over the ensemble configuration ({n_i, σ_i}) at each lattice site i independently. This yields (see Appendix for details)

(8)

In mean-field theory, we try to determine a “homogeneous” saddle point approximation for the partition function. For our system, the negative cooperation (i.e., the AFM nature) suggests that the system might prefer having neighboring sites in oppositely liganded states. Thus, we separate the lattice into two interleaved sublattice systems: all nearest neighbors of a lattice site belong to the alternate sublattice (Fig. 1). We then assign two “uniform” order parameters, ϕ_±, to each sublattice. After this assumption, the exponent of the Boltzmann factor in the partition function (Eq. (8)) now becomes → (N/2)[βg_EDϕ+ϕ₋+S(ϕ+, ϕ₋)], where N is the number of total lattice sites, S(ϕ+, ϕ₋)=Σ_k=± ln[1+2z cosh(β[g_EDϕ_k+y])], and D is the number of nearest neighbors, which depends on the structure of the lattice. For instance, a square lattice yields D=4, whereas a honeycomb lattice yields D=3.

Next, we minimize the free energy by varying ϕ_±. The variation yields the “saddle point” equation

Working this out explicitly, we find a self-consistent equation for _±,

(9)

(10)

(11)

There is no closed-form solution for Eq. (9). To get some analytical information, we define _±=m±ϵ and, with

we have

(12)

(13)

To proceed, let us assume that ϵ is small and solve Eqns. (12) to order ϵ²:

(14)

(15)

(16)

We must consider separately the cases where the denominator of Eq. (16) is positive or negative. Let us first imagine it is positive. Then the existence of a nontrivial solution of Eq. (16) requires that {βg_ED[〈n〉−m₀²]−1}>0. At small 〈n〉 this condition will clearly fail, and we will have only the trivial solution. Furthermore, this condition will fail at 〈n〉 close to 1 for large enough |y|. We can see this by comparing the equation for m₀ with the expression for 〈n〉. Note that if y is large enough such that the hyperbolic functions can be replaced by exponentials, we have |m₀|=〈n〉, and the above expression can be replaced by {βg_ED[〈n〉−〈n〉²]−1}; this is negative for the stated condition. As we cross a line in parameter space such that this factor changes sign to positive, there will be new solutions at nonzero ϵ², and the one at ϵ=0 becomes a local maximum of the free energy. This emergence of a double-well structure with a continuous growth of the nonzero ϵ² solution, indicates that the system exhibits a second-order phase transition.

We must next take into account the possibility that {m₁U⁽²⁾(m₀) −1/6U⁽³⁾(m₀)}<0. Having the denominator cross zero gives rise in our current approximation to a large value of ϵ, which thus invalidates the neglect of higher-order terms. Typically, the higher-order terms will stabilize the system at some finite value of ϵ, which thus appears “spontaneously” as some parameter threshold is crossed. This is a first-order phase transition, or equivalently, a triple-well structure for the free energy. If the local minima (for zero and nonzero ϵ²) have equally low free energy densities, the system can exist in a mixture of the two phases. As we will see, the two coexisting phases differ in their receptor densities. Finally, the points where both {βg_ED[〈n〉−m₀²−1]=0} and {m₁U(2)(m₀) −1/6U⁽³⁾(m₀)}=0 are “critical end-points” points, because they correspond to places where a second-order transition line ends at a first-order line. A diagram of this behavior, generated by the numerical solution of the mean-field equations, is given in Fig. 3.

Display large version of this figure

Figure 3

Numerically computed phase diagram, showing that there is a pair of second-order lines, each of which ends on the first-order transition curve (C = critical point; C_E=critical endpoint). The phase to the lower right has ϵ ≠ 0. Here we used g_E=6k_BT and D=3. To show the symmetry, we plot the ligand concentration in logarithm units, normalized with respect to the dissociation constant K_d^tnf=0.59 nM. Here the chemical potential μ_R is related to receptor density. In Fig. 4, we convert the receptor chemical potential into the molecular density.

For a given ligand concentration, we can find the phase coexistence lines arising because of the first-order phase transition. This is done by finding two solutions (solved with differing values of ϵ²) of the mean-field equations and then fixing z (as a function of y) by requiring that they have equal free energy,

(17)

Display large version of this figure

Figure 4

The phase diagram, now shown as a function of the receptor density 〈n〉 related to the ligand concentration. To show the symmetry, we plot the ligand concentration in logarithm units, normalized with respect to the dissociation constant K_d^tnf=0.59 nM. The region inside the solid lines is the coexistence region where states of high and low density coexist. As the ligand concentration is altered so as to cross one of these lines, the receptors will spontaneously cluster and thereby allow signaling to occur. 〈n〉_min^(d) is the minimal receptor density for clustering. For this set of parameters, clustering will occur even for very small overall receptor density. Here a₀ is the length scale for the lattice spacing. For surface receptor molecules such as TNF-R1, we might take a₀≈ 1nm.

As can be seen from the figure, the “clustering” window will cease to exist below some minimal receptor density, as we never enter the phase coexistence region. By symmetry, this minimal density can be found by solving the mean field equations for y=0 where m=0. This leads after some algebra to the self-consistent equations

(18)

Display large version of this figure

Figure 5

The variation of 〈n〉_min^(d) as a function of the association energy g_E. We find that 〈n〉_min^(d) rapidly approaches zero once g_ED≥15k_BT. If we assign D=3–4, this energy scale corresponds to a single hydrogen bond. Here a₀ is the length scale for the lattice spacing. For a surface receptor molecule such as TNF-R1, we might take a₀≈ 1nm.

We have presented a simple model for signal transduction via receptor clustering, based loosely on the TNF-TNFR1 system. Our basic idea is simple. The interaction between receptors can lead to a first-order phase transition with a discontinuous jump in the receptor density as a function of the receptor chemical potential and/or the ligand concentration. Turning this around, this implies that the receptor system will spontaneously phase separate for a range of ligand concentrations. This fact about the thermodynamic equilibrium state will lead under reasonable kinetic assumptions to the rapid formation of receptor clusters. Assuming that these clusters are necessary for the signal to proceed downstream has the immediate consequence that the system exhibits a strong robust response independent of any details of the intracellular signaling cascade. This might provide a simple solution to the problem faced by biological evolution of how to get a digital response in an analog world.

From a physics perspective, there is nothing very surprising about our phase diagram findings. The idea of a “lattice” Hamiltonian with intrinsic “cooperativity” has been proposed before (Changeux et al), and on general grounds models of this sort can be expected to have first-order phase transitions. What is new here is the connection of the transition to signaling via the idea of receptor clustering. This connects nicely with increasing evidence that clustering is “universal” among many types of receptor classes.

In our model, we have ignored more-than-two receptor interaction, and relevant internal chemical degrees of freedom (such as the dissociation of SODD in the TNF-R1 system). We do not expect these detailed considerations to change the overall picture, but a more sophisticated model will be needed to make more quantitative estimates of ligand thresholds, cluster structures, and, most interestingly, clustering dynamics. We hope to report on these issues in the future, as well as on the extension of our models to other ligand-receptor systems.

Finally, it would be important to extend our work to later-stage dynamics, as that would allow the consideration of processes such as adaptor protein-mediated receptor internalization, cytoskeleton-assisted cluster stabilization, receptor affinity regulation, receptor cross-talk, and adaptation (Barkai and Leibler, 1997,Hahn et al,Humphries, 1996,Holsinger et al,Luo and Lodish, 1997,Stewart et al,Sundberg and Rubin, 1996,Valitutti et al,Wyszynski et al). Other possible extensions might involve the inclusion of spatial fluctuations, the explicit treatment of external perturbations (Shoyab and Todaro, 1981), the local heterogeneity of the microenvironment (Bean et al,Ward and Hammer, 1992), or fluctuations of ligand concentration; all of these issues have been neglected here.

Appendix

The Gaussian transformation

The identity

(A1)

with g>0, i=√-1, and C as some constant, can be generalized to

(A2)

as long as J is a symmetrical positive definite matrix. Thus, Eq. (8) can be obtained by

(A3)

where i=ϕ_k=iθ_k, S_j({ϕ})=ln[1+2z cosh(β[g_EΣ_kJ_jkϕ_k+y])], and C_θ, C_ϕ are integral constants.