Dynamic model of hormonal systems coupled by negative feedback.

Most hormone concentrations in the body are regulated by negative feedback mechanisms in which the production and release of hormones are regulated according to the concentration of related species. Also, it has been observed that several hormones are released in a variety of pulsatile patterns. In most cases, the mechanism driving these complex patterns is not well understood. Our model of two cells coupled through negative feedback to their external products demonstrates periodic, aperiodic and chaotic oscillations. The coupling between the cells seems to be responsible for these dynamic behaviors. The variety of dynamic behaviors observed in the model demonstrates that a simple physiological feedback loop mimicking the coupling between circulatory hormones and production centers could be the source of complex hormone release patterns observed in vivo.

Steady state approximation in the minimal model of the alternative pathway of complement.

Complement is a response mechanism of the immune system. Two initiation pathways have been characterized for complement. The classical pathway is antibody mediated while the alternative pathway is not. Since the alternative pathway is independent of antibodies, it is always active. For the alternative pathway we have previously developed a minimal model. Using parameters within physiological bounds, the model showed complex behavior also within physiological bounds. Thus the model seems to be an appropriate representation of the alternative pathway response. By applying a steady state assumption to the Michaelis Menten step of the minimal model, we reduce the number of variables from six to five. A comparison between the dynamics of the minimal and contracted models reveals that the two descriptions may not be compatible. Although both systems show chaotic behavior it occurs in different regions of parameter space.

Mixed-mode oscillations in a self-replicating dimerization mechanism.

Recently, self-replicating molecules have been synthesized in the laboratory by Rebek. Given the importance of such molecules, we proposed a simple model of a self-replicating dimer, which works as a template for its own formation. Here we consider a three variable model. For the model, we obtain mixed-mode and chaotic oscillations. Also, we find coexistence between two periodic attractors as well as a periodic and a chaotic attractor.

Chaos in a minimal model of the alternative pathway of the complement system.

In previous work, we introduced a minimal model of the alternative pathway of the complement. We also limited our analysis to a reduced set of parameter values because, for some parameters, experimentally supported estimates were not found. On the other hand, changes in value of some parameters may be a result of a pathological condition. Therefore, here we extend our analysis and include a wider range of values of five of the physiologically relevant parameters. For all the parameters considered, we observe chaotic oscillations, and we construct bifurcation diagrams using Poincaré sections of local maxima.

Effect of linear reinsertion of receptor on the distribution of receptors around coated pits.

We consider a linear receptor reinsertion step in our kinetic model of the primary steps occurring in receptor-mediated endocytosis. In contrast with our previous zeroth order receptor reinsertion assumption, here we consider a first order process, and we study the effect of receptor diffusion on the trapping rate constant (k+) and the radial distribution of receptors around coated pits (grp). Using experimental data for low density lipoproteins (LDL) receptors on fibroblast cells, we find that the trapping of receptors by coated pits is diffusion-controlled for any value of the escaping rate constant (k-). This result is significantly different from our previous findings. In fact, for a zeroth order process, we find that either diffusion has no effect on k+ or, at the most, receptor trapping is 84% diffusion-controlled. Moreover, we find values for the receptor reinsertion rate constant (kappa), which range between 15% of the pit's invagination rate constant, lambda, and three halves lambda. In addition, the ratio kappa/lambda is equal to the ratio of the concentration of receptors in pits with respect to the internalized receptors. Comparison between the experimental radial distribution of receptors around pits and the theoretical should provide an indication of the diffusion effect on k+.

A theoretical model of LDL-receptor trapping on a spherical cell.

The kinetics of the trapping of LDL-receptor complexes by coated pits on the surface of fibroblasts is examined in this paper. We have recently developed a mathematical formalism to extend Keizer's non-linear, non-equilibrium fluctuation-dissipation theory to the kinetics of chemical systems constrained to a spherical surface. Keizer's theory is ideally suited to the study of open biological systems. In the past it has been used to investigate endocytosis on fibroblasts. However, these applications have modeled the cell membrane with an infinite plane. As such, the finite size of the cellular membrane, as well as its precise symmetry, could not be incorporated into the previous studies. Thus in this paper we use our recently developed methodology to reexamine the trapping step in endocytosis on spherical cells. For cell surface processes, the theoretical consideration of a spherical symmetry or an infinite plane, in model calculations, will depend on the experimental or in vivo conditions of the processes of interest. For a spherical symmetry, we find that the finite size of the cell surface does not significantly affect the rate of the trapping step given the empirically determined values for the relevant parametes on fibroblasts. This result supports the approximation used in the previous investigation. However, this and other analyses indicate that the finitie size of the biological surface probably is an important parameter for processes which occur on smaller biological surfaces such as those found on organelles.

The effect of diffusion on the trapping of membrane-bound receptors by localized coated pits.

Localized coated pits are considered in the primary steps of receptor-mediated endocytosis. According to the pit reinsertion mechanism, we have modified our previous kinetic model and studied the effect of diffusion on the trapping rate constant (k+). Using experimental data for low density lipoprotein (LDL) receptors on fibroblast cells, we found that the binding of receptors to coated pits is not totally diffusion controlled. For example, the process is less than 78% diffusion controlled if receptors are not allowed to escape the coated pits. However, due to the large uncertainties in the experimental parameters, a diffusion-controlled process cannot be ruled out. The greatest differences between localized and random reinsertion were found when the escaping rate constant (k-) is much greater than the rate constant for invagination of the pits (lambda 1). Under this condition, k+ for localized reinsertion is no less than 39% diffusion controlled, while k+ for random reinsertion shows no diffusion effect at all.

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+.