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We propose an approach to study small limit cycle bifurcations on a center manifold in analytic or smooth systems depending on parameters. We then apply it to the investigation of limit cycle bifurcations in a model of calcium oscillations in the cilia of olfactory sensory neurons and show that it can have two limit cycles: a stable cycle appearing after a Bautin (generalized Hopf) bifurcation and an unstable cycle appearing after a subcritical Hopf bifurcation.

This paper discusses rationale for a theory in biology: what exactly is a theory in biology? Is it of a mathematical nature? How to conceive an integrative theory and why? Replies to these questions are offered for subsequent discussions as concerns the mathematical theory of integrative physiology (MTIP) proposed by the author. It is shown that such a theory is a theoretical framework built on a representation in terms of hierarchical functional interactions and a specific formalism, the S-Propagator, to traverse the levels of organization. As for all natural theories, the MTIP is based on a general principle specific to biology, the principle of auto-associative stabilization (PAAS). In this framework, two models are revisited for a novel interpretation: the first addresses the dynamics of biochemical networks, the second addresses the selection of groups of neurons (TSGN) as suggested by Edelman.

We describe Dizzy, a software tool for stochastically and deterministically modeling the spatially homogeneous kinetics of integrated large-scale genetic, metabolic, and signaling networks. Notable features include a modular simulation framework, reusable modeling elements, complex kinetic rate laws, multi-step reaction processes, steady-state noise estimation, and spatial compartmentalization.

Whilst data on biochemical networks has increased several-fold, our comprehension of the underlying molecular biology is incomplete and inadequate. Simulation studies permit data collation from disparate time points and the imputed trajectories can provide valuable insights into the molecular biology of complex biochemical systems. Although, stochastic simulations are accurate, each run is an independent event and the data that is generated cannot be directly compared even with identical simulation times. This lack of robustness will preclude a biologically meaningful result for the metabolite(s) of concern and is a significant limitation of this approach. “TemporalGSSA” or temporal Gillespie Stochastic Simulation Algorithm is an R-wrapper which will collate and partition SSA-generated datasets with identical simulation times (trials) into finite sets of linear models (technical replicates). Each such model (time step of a single run, absolute number of molecules for a metabolite) computes several coefficients (slope, intercept, etc.). These coefficients are averaged (mean slope, mean intercept) across all trials of a technical replicate and along with an imputed time step (mean, median, random) is incorporated into a linear regression equation. The solution to this equation is the number of molecules of a metabolite which is used to compute the molar concentration of the metabolite per technical replicate. The summarized (mean, standard deviation) data of this vector of technical replicates is the outcome or numerical estimate of the molar concentration of a metabolite and is dependent on the duration of the simulation. If the SSA-generated dataset comprises runs with differing simulation times, “TemporalGSSA” can compute the time-dependent trajectory of a metabolite provided the trials-per technical replicate constraint is complied with. The algorithms deployed by “TemporalGSSA” are rigorous, have a sound theoretical basis and have contributed meaningfully to our comprehension of the mechanism(s) that drive complex biochemical systems. “TemporalGSSA”, is robust, freely accessible and easy to use with several readily testable examples.