Kinetic analysis of p53 gene network with time delays and PIDD

p53 kinetics plays a key role in regulating cell fate.Based on the p53 gene regulatory network composed by the core regulatory factors ATM,Mdm2,Wipl,and PIDD,the effect of the delays in the process of transcription and translation of Mdm2 and Wipl on the dynamics of p53 is studied theoretically and numerically.The results show that these two time delays can affect the stability of the positive equilibrium.With the increase of delays,the dynamics of p53 presents an oscillating state.Further,we also study the effects of PIDD and chemotherapeutic drug etoposide on the kinetics of p53.The model indicates that(i)PIDD low-level expression does not significantly affect p53 oscillatory behavior,but high-level expression could induce two-phase kinetics of p53;(ii)Too high and too low concentration of etoposide is not conducive to p53 oscillation.These results are in good agreement with experimental findings.Finally,we consider the infuence of internal noise on the system through Binomial r-leap algorithm.Stochastic simulations reveal that high-intensity noise completely destroys p53 dynamics in the deterministic model,whereas low-intensity noise does not alter p53 dynamics.Interestingly,for the stable focus,the internal noise with appropriate intensity can induce quasi-limit cycle oscillations of the system.Our work may provide the useful insights for the development of anticancer therapy.

Application of low-dimensional finite element method to fractional diffusion equation

Classical finite element method(FEM)has been applied to solve some fractional differential equations,but its scheme has too many degrees of freedom.In this paper,a low-dimensional FEM,whose number of basis functions is reduced by the theory of proper orthogonal decomposition(POD)technique,is proposed for the time fractional diffusion equation in two-dimensional space.The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved.Moreover,error estimation of the method is obtained.Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation

In this paper,we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation.In order to avoid using higher order elements,we introduce an intermediate variableσ=∆u and translate the fourth-order derivative of the original problem into a second-order coupled system.We discretize the fractional time derivative terms by using the L1-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula.In the fully discrete scheme,we implement the finite element method for the spatial approximation.Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained.Numerical experiments are carried out to demonstrate our theoretical analysis.

Fully discrete two-step mixed element method for the symmetric regularized long wave equation

A numerical method based on the explicit two-step method in time direction and the mixed finite element method in spatial direction is presented for the symmetric regularized long wave(SRLW)equation.The optimal a priori error estimates(O((∆t)^(2)+h^(m+1)+h^(k+1)))for fully discrete explicit two-step mixed scheme are derived.Moreover,a numerical example is provided to confirm our theoretical results.

A NEW CHARACTERISTIC EXPANDED MIXED METHOD FOR SOBOLEV EQUATION WITH CONVECTION TERM

In this paper,a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term.The hyperbolic part d(x)∂u/∂t+c(x,t)·∇u is handled by the characteristic method and the diffusion term∇·(a(x,t)∇u+b(x,t)∇ut)is approximated by the new expanded mixed method,whose gradient belongs to the simple square integrable(L^(2)(Ω))^(2)space instead of the classical H(div;Ω)space.For a priori error estimates,some important lemmas based on the new expanded mixed projection are introduced.An optimal priori error estimates in L^(2)-norm for the scalar unknown u and a priori error estimates in(L^(2))^(2)-norm for its gradientλ,and its fluxσ(the coefficients times the negative gradient)are derived.In particular,an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

In this paper,a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element(MFE)method in space combined with L1-approximation and implicit second-order backward difference scheme in time.The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived.Finally,some numerical tests are shown to verify our theoretical analysis.