Mathematical description of the interactions of CycE/Cdk2, Cdc25A,and P27Kip1 in a core cancer subnetwork

Recent studies have shown that the initiation of human cancer is due to the malfunction of some genes (such as E2F, CycE, CycD, Cdc25A, P27^Kip1, and Rb) at the R‐checkpoint during the G₁‐to‐S transition of the cell cycle. Identifying and modeling the dynamics of these genes provide new insight into the initiation and progression of many types of cancers. In this study, a cancer subnetwork that has a mutual activation between phosphatase Cdc25A and the CycE/Cdk2 complex and a mutual inhibition between the Cdk inhibitor P27^Kip1 and the CycE/Cdk2 complex is identified. A new mathematical model for the dynamics of this cancer subnetwork is developed. Positive steady states are determined and rigorously analyzed. We have found a condition for the existence of positive steady states from the activation, inhibition, and degradation parameter values of the dynamical system. We also found a robust condition that needs to be satisfied for the steady states to be asymptotically stable. We determine the parameter value(s) under which the system exhibits a saddle–node bifurcation. We also identify the condition for which the system exhibits damped oscillation solutions. We further explore the possibility of Hopf and homoclinic bifurcations from the saddle–focus steady state of the system. Our analytic and numerical results confirm experimental results in the literature, thus validating our model. Copyright © 2016 John Wiley & Sons, Ltd.     

The Role of Pressure During Shearing, Dissipative Collapse

We analyse a new family of solutions to the Einstein field equations describing the collapse of a fluid sphere in the presence of heat flux and shear. These solutions ensure that the collapsing fluid is accelerating and provide a generalisation of the geodesic fluid models studied in earlier treatments. In particular, we demonstrate the role played by pressure in the dynamics of the collapse process.     

On a Framework for Generating Nonstatic Solutions of the Einstein Field Equations

In this paper we present a simple generalisation of all spherically symmetric static solutions. We present a framework to obtain both anisotropic and isotropic models with and without a barotropic equation of state of the form p = . The nonstaticity in some models necessarily requires a nonzero heat flux which dictates the thermodynamics of our models.     

Lattice statics calculations for a vacancy in magnesium

Using the Green function method of lattice statics, the lattice distortion in the neighborhood of a vacancy in magnesium has been calculated in three different axially-symmetric (AS) host lattice models and also in a model derived from an empirical interatomic pair potential. The variations in the lattice distortion and the relaxation energy of the vacancy are studied as the size of the defect space is allowed to vary from two to four surrounding neighbors. The perfect static lattice Green functions are computed up to 19 neighbors in the 4 models and the values obtained are shown to be not very sensitive to the model chosen. The lattice relaxation is found to be negligible in all models except in a four-neighbor AS model. The relaxation energies in these four models are computed to be 0.0027 eV, 0.034 eV, 0.28 eV and 0.0069 eV respectively. Results for the monovacancy formation energy, the elastic dipole tensor and the volume change of the crystal due to the vacancy are also presented.     

First integrals for charged perfect fluid distributions

We study the evolution of shear-free spherically symmetric charged fluids in general relativity. We find a new parametric class of solutions to the Einstein-Maxwell system of field equations. Our charged results are a generalisation of earlier treatments for neutral relativistic fluids. We regain the first integrals found previously for uncharged matter as a special case. In addition an explicit first integral is found which is necessarily charged.