Enrique Peacock-López

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.

Bifurcation diagrams and Turing patterns in a chemical self-replicating reaction-diffusion system with cross diffusion.

Chemical self-replication of oligonucleotides and helical peptides exhibits the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzymatic sink. For this generalized model, we consider the role of cross diffusion in pattern formation, and we obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, Autocatalator, and Templator models as specific cases.

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+.

Turing patterns in a self-replicating mechanism with a self-complementary template

A variety of nonlinear chemical models, such as the Selkov–Schnakenberg, exhibit Turing patterns. The Templator, which is based on a minimal autocatalytic monomer–dimer system, is a newer two-variable model also able to show Turing patterns. Here we find that the dynamic behavior of the Templator is quite similar to other models with cubic nonlinearities. This is demonstrated through a series of computer simulations in two dimensions utilizing the cellular automata approach. The selection of parameter values is based on linear stability analysis, which provides a relatively simple method of predicting Turing pattern formation. The simulations reveal spot, labyrinth, and striped patterns, in agreement with the predictions of the analysis. Other behaviors, such as honeycomb patterns, are also observed. For some parameter values, we study transient spot replication. Our findings strongly suggest that the Templator may belong to the same class of models previously studied by Pearson.

Complex dynamics in a cross-catalytic self-replication mechanism

The authors consider a minimal cross-catalytic self-replicating system of only two cross-catalytic templates that mimics the R3C ligase ribozyme system of Dong-Eu and Joyce [Chem. Biol. 11, 1505 (2004)]. This system displays considerably more complex dynamics than its self-replicating counterpart. In particular, the authors discuss the Poincare-Andronov-Hopf bifurcation, canard transitions, excitability, and hysteresis that yield birhythmicity between simple and complex oscillations.

Chemical oscillations and Turing patterns in a generalized two-variable model of chemical self-replication

Chemical self-replication of oligonucleotides and helical peptides show the so-called square root rate law. Based on this rate we extend our previous work on ideal replicators to include the square root rate and other possible nonlinearities, which we couple with an enzimatic sink. Although the nonlinearity is necessary for complex dynamics, the nature of the sink is the essential feature in the mechanism that allows temporal and spatial patterns. We obtain exact general relations for the Poincare-Adronov-Hopf and Turing bifurcations, and our generalized results include the Higgins, autocatalator, and templator models as specific cases.

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.

One‐dimensional reactive systems: The effect of diffusion on rapid bimolecular processes

Bimolecular reactions in one dimension are studied using the fluctuation–dissipation theory. In particular, we calculate the rate constants for three reactive systems. First, for an infinite straight line, we consider the reaction A+A→A+P, and the isomerization 2A→A2. In the first case, we obtain a linear dependence between rate constant and the reactant concentration. In the second, we obtain the same linear dependence only when the characteristic dimerization time τd is much greater than the characteristic pumping time τp. On the other hand if τp≫τd, we find a rate constant which is independent of the reactant concentration. Our results are in qualitative agreement with recent computer simulations of these reactions. Second, we consider trapping, A+S→S, by static sinks on a ring. For the cases of interest, we find negligible corrections, due to curvature, to the functional form of the infinite straight line rate constant.

On the “best Fokker-Planck equation” for systems driven by colored noise

Abstract The “best Fokker-Planck equation” for systems driven by colored noise has been criticized because it may lead to probability distributions that have a finite region of support. Herein we argue that sometimes this behavior might point to problems in the model used to represent the physical system. We illustrate this contention in the context of a stochastic model of a type frequently used to describe intensity fluctuations in a dye laser. A “parent” model that is bounded throughout the entire range of the variable leads to a probability density obtained with the BFPE technique that is well-behaved over the entire phase space. We compare the results of the BFPE with direct computer simulations of the stochastic differential equation, and suggest a combination of parameters whose value characterizes the applicability of the BFPE for this model.

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.