Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets

An integrated understanding of molecular and developmental biology must consider the large number of molecular species involved and the low concentrations of many species in vivo. Quantitative stochastic models of molecular interaction networks can be expressed as stochastic Petri nets (SPNs), a mathematical formalism developed in computer science. Existing software can be used to define molecular interaction networks as SPNs and solve such models for the probability distributions of molecular species. This approach allows biologists to focus on the content of models and their interpretation, rather than their implementation. The standardized format of SPNs also facilitates the replication, extension, and transfer of models between researchers. A simple chemical system is presented to demonstrate the link between stochastic models of molecular interactions and SPNs. The approach is illustrated with examples of models of genetic and biochemical phenomena where the UltraSAN package is used to present results from numerical analysis and the outcome of simulations.

Extension of the thermodynamics of small systems to open metastable states: An example

By using a simplified model of small open liquid-like clusters with surface effects, in the gas phase, it is shown how the statistical thermodynamics of small systems can be extended to include metastable supersaturated gaseous states not too far from the gas–liquid equilibrium transition point. To accomplish this, one has to distinguish between mathematical divergence and physical convergence of the open-system partition function.

On the controllability of distributed systems

To “control” a system is to make it behave (hopefully) according to our “wishes,” in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed—i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our “wish” is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any “neighborhood” of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is “unstable” (chaotic, turbulent) the “simplest,” or the “cheapest,” it is to achieve exact or approximate controllability. When the PDEs are the Navier–Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.

Complex dynamics of multilocus systems subjected to cyclical selection.

Earlier we have shown that oscillations with a long period ("supercycles") may arise in two-locus systems experiencing cyclical selection with a short period. However, this mode of complex limiting behavior appeared to be possible for narrow ranges of parameters. Here we demonstrate that a multilocus system subjected to stabilizing selection with cyclically moving optimum can generate ubiquitous complex limiting behavior including supercycles, T-cycles, and chaotic-like phenomena. This mode of multilocus dynamics far exceeds the potential attainable under ordinary selection models resulting in simple behavior. It may represent a novel evolutionary mechanism increasing genetic diversity over long-term time periods.

On signal sequence polymorphisms and diseases of distribution.

We report a previously unappreciated property of the signals that target organelle-specific proteins to their subcellular sites of action. Such targeting sequences are shown to be polymorphic. We discovered this polymorphism when we cloned the mitochondrial manganese-containing superoxide dismutase from cell lines of normal individuals and patients with genetic diseases of premature aging and compared their sequences to each other and to those previously reported. The polymorphism consists of a single nucleotide change in the region of the DNA that encodes the signal sequence such that either an alanine or valine is present. Subsequently, eight cell lines were analyzed and all three possible combinations of the two signal sequences were observed. Such signal sequence polymorphisms could result in diseases of distribution, where essential proteins are not properly targeted, thereby leading to absolute or relative deficiencies of critical enzymes within specific cellular compartments. Progeria and related syndromes may be diseases of distribution.

Stability of Coulomb systems in a magnetic field.

I study N electrons and M protons in a magnetic field. It is shown that the total energy per particle is bounded below by a constant independent of M and N, provided the fine structure constant is small. Here, the total energy includes the energy of the magnetic field.