Joel Keizer

The effect of diffusion on the binding of membrane-bound receptors to coated pits.

We have formulated a kinetic model for the primary steps that occur at the cell membrane during receptor-mediated endocytosis. This model includes the diffusion of receptor molecules, the binding of receptors to coated pits, the loss of coated pits by invagination, and random reinsertion of receptors and coated pits. Using the mechanistic statistical theory of nonequilibrium thermodynamics, we employ this mechanism to calculate the two-dimensional radial distribution of receptors around coated pits at steady state. From this we obtain an equation that describes the effect of receptor diffusion on the rate of binding to coated pits. Our equation does not assume that ligand binding is instantaneous and can be used to assess the effect of diffusion on the binding rate. Using experimental data for low density lipoprotein receptors on fibroblast cells, we conclude that the effect of diffusion on the binding of these receptors to coated pits is no more than 84% diffusion controlled. This corresponds to a dissociation rate constant for receptors on coated pits (k-) that is much less than the rate constant for invagination of the pits (lambda = 3.3 X 10(-3)/s) and a correlation length for the radial distribution function of six times the radius of a coated pit. Although the existing experimental data are compatible with any value of k-, we obtain a lower bound for the value of the binding constant (k+) of 2.3 X 10(-2)(micron)2/s. Comparison of the predicted radial distributions with experiment should provide a clear indication of the effect of diffusion on k+.

Nonequilibrium Steady States

In several places in this book we have encountered the notion of a nonequilibrium steady state. Because of statistical fluctuations a steady state, like an equilibrium state, should not be thought of as the state of a single system, but rather as the state of an ensemble. For example, in Section 4.7 we examined the coupled chemical reactions A + X → 2X and 2X → E. Using the canonical theory we discovered that, on the average, there are two densities of X which do not change as a function of time. One of these was ρx (1) = 0, which was found to be unstable to small perturbations, and the other was ρx (2) = k1/2k2, which is stable. The stable density is like an equilibrium density in that it supports a stationary probability distribution. In other words, associated with the time-independent average density ρ(2) x is a unique, stationary probability distribution that characterizes single-time averages in the steady-state ensemble. This situation turns out to be relatively common. Indeed, it has already arisen in our treatment of electrochemical reactions in Section 5.7, in our discussion of reaction-diffusion fluctuations in Section 6.6, and in the calculation of the light scattering spectrum from a temperature gradient in Section 6.8. In this chapter we consider the statistical thermodynamic description of stable nonequilibrium steady states in a more general setting. We begin in this section by characterizing the average statistical state

Irreversible Processes: The Onsager and Boltzmann Pictures

The primary objective of this book is to develop a mathematical picture of measurable quantities that can be used to understand macroscopic observations of matter. As we have discussed in Chapter 1, that picture is necessarily stochastic and involves ensembles of systems that are prepared in similar ways. In Chapter 1 we outlined some of the techniques of the theory of stochastic processes that are necessary for understanding physical ensembles. Although we used Brownian motion to illustrate the physical relevance of stochastic processes, the stochastic point of view is essential for understanding all kinds of macroscopic observations. Fluctuations are inherent in all matter because of its molecular constitution. Indeed, one of the lessons of Brownian motion is that these fluctuations are observable and that they are closely related to the irreversible processes caused by molecular motion.

Elementary Processes and Fluctuations

To understand matter on a macroscopic scale, it is necessary to pay attention to its molecular makeup. That is one of the lessons to be learned from the kinetic theory of Maxwell and Boltzmann. Another lesson from kinetic theory is that the dynamical equations, which describe the change of macroscopic systems, have thermodynamics built into them. Indeed, this is the basic content of Boltzmann’s H-theorem. This is also apparent in the Onsager theory which is based on the observation that thermodynamic forces are responsible for the relaxation to equilibrium. In the Onsager theory molecular fluctuations are also related to thermodynamic quantities: The stationary equilibrium fluctuations are determined by the second differential of the entropy, δ2S, and the random noise in the thermodynamic fluxes is determined by the dynamic coupling matrix, Lij. All this suggests that both thermodynamics and fluctuations are imbedded in a deeper formalism—a formalism which, like the Boltzmann equation, is based on a picture of molecular events.

Ensembles and Stochastic Processes

The study of matter which is large with respect to molecular size is one of the oldest of the sciences. It originated in the early metallurgy of China and the Middle East and permeates modern research in both the physical and biological sciences. From a theoretical point of view, the reversible or mechanical properties of matter were the first to be understood. These are the properties characterized by Newton’s laws of motion which relate forces, velocities, and spatial positions. These properties are termed reversible because if one reverses all velocities, the magnetic field, and the time, the resulting motion is the reverse of what was observed up to that time—like seeing a movie run backwards. This reversibility of the mechanical theory suggested for many years that any observable motion should have a twin reversed motion which can also be observed. The consequences of reversibility, however, seem contrary to our intuition: Although salt spontaneously dissolves in water, no one has ever seen salt precipitate from an unsaturated solution.

The Hydrodynamic Level of Description

The hydrodynamic level of description is an extension of the thermodynamic level that takes into account the dependence of the extensive variables on spatial coordinates. Even an ensemble in which the systems are spatially uniform on the average involves fluctuations that differ from one position to another. Consider, for example, the particle mass density, ρ(r, t), in a simple fluid like water. In an equilibrium ensemble the average density, ρ e , will be constant in the absence of an external field. Because of the molecular nature of water it is clear that at a given time t and position r different members of the ensemble will possess different values of the number density. We have already encountered this at the Boltzmann level of description in Sections 3.2 and 3.3. There it was necessary to keep track of the number of particles with a given range of positions and momenta. The hydrodynamic level is intermediate between the Boltzmann and thermodynamic levels and adds the momentum to the basic extensive thermodynamic variables. At the hydrodynamic level one has a closed description of the spatial dependence of the densities of extensive variables throughout a system.

Thermodynamic-Level Description of Chemical, Electrochemical, and Ion Transport Mechanisms

One of the few chemical systems for which concentration fluctuations have been measured is the association-disassociation reaction of beryllium and sulfate ions in aqueous solution. Earlier, using conventional fast-reaction techniques, the mechanism of association was deduced to consist of the two elementary reactions n n

Nonstationary Processes: Transients, Limit Cycles, and Chaotic Trajectories

{text{Be}}_{{text{aq}}}^{2 + } + {text{SO}}_{4{text{aq}}}^{2 - } rightleftarrows {text{Be}}_{{text{aq}}}^{2 + } {text{SO}}_{4{text{aq}}}^{2 - }