Imaginary chemical potentials and the phase of the fermionic determinant

A numerical technique is proposed for an efficient numerical determination of the average phase factor of the fermionic determinant continued to imaginary values of the chemical potential. The method is tested in QCD with eight flavors of dynamical staggered fermions. A direct check of the validity of analytic continuation is made on small lattices and a study of the scaling with the lattice volume is performed.Comment: 6 pages, 6 figure

Graph-theoretic analysis of multistationarity using degree theory

Biochemical mechanisms with mass action kinetics are often modeled by systems of polynomial differential equations (DE). Determining directly if the DE system has multiple equilibria (multistationarity) is difficult for realistic systems, since they are large, nonlinear and contain many unknown parameters. Mass action biochemical mechanisms can be represented by a directed bipartite graph with species and reaction nodes. Graph-theoretic methods can then be used to assess the potential of a given biochemical mechanism for multistationarity by identifying structures in the bipartite graph referred to as critical fragments. In this article we present a graph-theoretic method for conservative biochemical mechanisms characterized by bounded species concentrations, which makes the use of degree theory arguments possible. We illustrate the results with an example of a MAPK network

Switching in mass action networks based on linear inequalities

Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e. a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our point of view this approach is too restrictive, as, one the one hand, due to predominant parameter uncertainty numerical methods are generally difficult to apply to realistic models originating in Systems Biology. And on the other hand switching already arises with the occurrence of a saddle type steady state (characterized by a Jacobian where exactly one Eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and parameters where the Jacobian derived from a mass action network has a defective zero eigenvalue so that -- under certain genericity conditions -- a saddle-node bifurcation occurs. Our conditions are applicable to general mass action networks involving at least one conservation relation, however, they are only sufficient (as infeasibility of linear inequalities does not exclude defective zero eigenvalues).Comment: in revision SIAM Journal on Applied Dynamical System

A global convergence result for processive multisite phosphorylation systems

Multisite phosphorylation plays an important role in intracellular signaling. There has been much recent work aimed at understanding the dynamics of such systems when the phosphorylation/dephosphorylation mechanism is distributive, that is, when the binding of a substrate and an enzyme molecule results in addition or removal of a single phosphate group and repeated binding therefore is required for multisite phosphorylation. In particular, such systems admit bistability. Here we analyze a different class of multisite systems, in which the binding of a substrate and an enzyme molecule results in addition or removal of phosphate groups at all phosphorylation sites. That is, we consider systems in which the mechanism is processive, rather than distributive. We show that in contrast with distributive systems, processive systems modeled with mass-action kinetics do not admit bistability and, moreover, exhibit rigid dynamics: each invariant set contains a unique equilibrium, which is a global attractor. Additionally, we obtain a monomial parametrization of the steady states. Our proofs rely on a technique of Johnston for using "translated" networks to study systems with "toric steady states", recently given sign conditions for injectivity of polynomial maps, and a result from monotone systems theory due to Angeli and Sontag.Comment: 23 pages; substantial revisio

On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle

Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study four simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parameterize the steady state and Hurwitz matrices