Gheorghe Craciun

Multiple Equilibria in Complex Chemical Reaction Networks: I. The Injectivity Property

The capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the reaction network. Examples show that there is a very delicate relationship between reaction network structure and the possibility of multiple equilibria. We suggest a new method for discriminating between networks that have the capacity for multiple equilibria and those that do not. Our method can be implemented using standard computer algebra software and gives answers for many reaction networks for which previous methods give no information.

MULTIPLE EQUILIBRIA IN COMPLEX CHEMICAL REACTION NETWORKS: II. THE SPECIES-REACTION GRAPH ∗

For mass action kinetics, the capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the structure of the underlying network of chemical reactions. We suggest a new graph-theoretical method for discriminating between complex reaction networks that can admit multiple equilibria and those that cannot. In particular, we associate with each network a species-reaction graph, which is similar to reaction network representations drawn by biochemists, and we show that, if the graph satisfies certain weak conditions, the differential equations corresponding to the network cannot admit multiple equilibria {\em no matter what values the rate constants take}. Because these conditions are very mild, they amount to powerful (and quite delicate) necessary conditions that a network must satisfy if it is to have the capacity to engender multiple equilibria. Broad qualitative results of this kind are especially apt, for individual reaction rate constants are rare...

Toric dynamical systems

Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.

Product-form stationary distributions for deficiency zero chemical reaction networks.

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg’s deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.

A life-span extending form of autophagy employs the vacuole-vacuole fusion machinery.

While autophagy is believed to be beneficial for lifespan extension, it is controversial which forms or aspects of autophagy are the responsible ones. We addressed this by analyzing the lifespan of yeast autophagy mutants under caloric restriction, a longevity manipulation. Surprisingly, we discovered that the majority of proteins involved in macro-autophagy and several forms of micro-autophagy were dispensable for lifespan extension. The only autophagy protein that is critical for lifespan extension was Atg15p, a lipase that is located in the endoplasmic reticulum (ER) and transported to vacuoles for disintegrating membranes of autophagic bodies. We further found that vacuole-vacuole fusion was required for lifespan extension, which was indicated by the shortened lifespan of mutants missing proteins (ypt7Δ, nyv1Δ, vac8Δ) or lipids (erg6Δ) involved in fusion. Since a known function of vacuole-vacuole fusion is the maintenance of the vacuole membrane integrity, we analyzed aged vacuoles and discovered that aged cells had altered vacuolar morphology and accumulated autophagic bodies, suggesting that certain forms of autophagy do contribute to longevity. Like aged cells, erg6Δ accumulated autophagic bodies, which is likely caused by a defect in lipase instead of proteases due to the existence of multiple vacuolar proteases. Since macro-autophagy is not blocked by erg6Δ, we propose that a new form of autophagy transports Atg15p via the fusion of vacuoles with vesicles derived from ER, and we designate this putative form of autophagy as secretophagy. Pending future biochemical studies, the concept of secretophagy may provide a mechanism for autophagy in lifespan extension.

MULTIPLE EQUILIBRIA IN COMPLEX CHEMICAL REACTION NETWORKS: SEMIOPEN MASS ACTION SYSTEMS ∗

In two earlier articles, we provided sufficient conditions on (mass action) reaction network structure for the preclusion of multiple positive steady states in the context of what chemical engineers call the continuous flow stirred tank reactor. In such reactors, all species are deemed to be present in the effluent stream, a fact which played a strong role in the proofs. When certain species are deemed to be entrapped within the reactor, the questions that must be asked are more subtle, and the mathematics becomes substantially more difficult. Here we extend results of the earlier papers to semiopen reactors and show that very similar results obtain, provided that the network of chemical reactions satisfies certain weak structural conditions; weak reversibility is sufficient but not necessary.

Persistence and Permanence of Mass-Action and Power-Law Dynamical Systems

Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering and are often used to describe the dynamics in interaction networks. We prove that two-species mass-action systems derived from weakly reversible networks are both persistent and permanent, for any values of the reaction rate parameters. Moreover, we prove that a larger class of networks, called endotactic networks, also give rise to permanent systems, even if the reaction rate parameters vary in time (to allow for the influence of external signals). These results also apply to power-law systems and other nonlinear dynamical systems. In addition, ideas behind these results allow us to prove the global attractor conjecture for three-species systems.

A combinatorial H4 tail library for exploring the histone code.

Histone modifications modulate chromatin structure and function. A posttranslational modification-randomized, combinatorial library based on the first 21 residues of histone H4 was designed for systematic examination of proteins that interpret a histone code. The 800-member library represented all permutations of most known modifications within the N-terminal tail of histone H4. To determine its utility in a protein binding assay, the on-bead library was screened with an antibody directed against phosphoserine 1 of H4. Among the hits, 59 of 60 sequences were phosphorylated at S1, while 30 of 30 of those selected from the nonhits were unphosphorylated. A 512-member version of the library was then used to determine the binding specificity of the double tudor domain of hJMJD2A, a histone demethylase involved in transcriptional repression. Global linear least-squares fitting of modifications from the identified peptides (40 hits and 34 nonhits) indicated that methylation of K20 was the primary determinant for binding, but that phosphorylation and acetylation of neighboring sites attenuated the interaction. To validate the on-bead screen, isothermal titration calorimetry was performed with 13 H4 peptides. Dissociation constants ranged from 1 mM to 1 microM and corroborated the screening results. The general approach should be useful for probing the specificity of any histone-binding protein.

Multigraph Conditions for Multistability, Oscillations and Pattern Formation in Biochemical Reaction Networks

We represent interactions among biochemical species using a directed multigraph, which is a generalization of a more commonly used digraph. We show that network properties that are known to lead to multistability or oscillations, such as the existence of a positive feedback cycle, can be generalized to ldquocritical subnetworksrdquo that can contain several cycles. We also derive corresponding graph-theoretic conditions for pattern formation for the respective reaction-diffusion models. We present as an example a model for cell cycle and apoptosis along with bifurcation diagrams and sample solutions that confirm the predictions obtained with the help of the multigraph network conditions.

Homotopy methods for counting reaction network equilibria

Dynamical system models of complex biochemical reaction networks are usually high-dimensional, non-linear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfied. We describe methods based on homotopy invariance of degree which allow us to determine the number of equilibria for complex biochemical reaction networks and how this number depends on parameters in the model.