Investigation of Membrane Curvature Dependency on Cytochrome c Binding to Cardiolipin

Cytochrome c (Cyt c), an efficient electron transport protein in cellular respiration that makes biochemical energy ATP, is recently found to take part in initiating apoptosis (programmed cell death) through first oxidizing a lipid called cardiolipin, and then dissociating from the inner membrane of mitochondria to trigger the apoptosis cascade. If cell apoptosis is inhibited, it can cause cancer. Regulation of Cyt c in cardiolipin binding on the mitochondrial membranes potentially enables regulation of the intrinsic pathway of apoptosis. Cardiolipin has four hydrocarbon chains and a negatively charged head group which can interact with anionic site A on Cyt c that contains positively charged lysine amino acids. It is believed that the electrostatic interactions between anionic site A and CL on the inner membrane of a mitochondria lead to protein binding and partial unfolding. In this experiment, we isolate anionic site A, and use cardiolipin liposomes, a spherical sac formed artificially that has a lipid bilayer, to trap Cyt c as a mimic of the concave curvature of the cristae of the mitochondrial inner membrane. Circular dichroism spectroscopy is used to monitor the amount of trapped Cyt c. Previous studies have examined Cyt c-CL binding but using a convex surface that is not physiologically relevant other than it is composed of lipid CL. By comparing to previous similar studies, we can find out whether lipid curvature affects Cyt c-CL binding affinity. The understanding of apoptosis can be used toward novel therapies that can be developed to specifically engage apoptosis in cancer treatments

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 23(3), 840--865 (2001)] and [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM's family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.Comment: 27 pages, 9 figures; Accepted by SCIENCE CHINA Mathematics on January 17, 202

Nonmonotone local minimax methods for finding multiple saddle points

In this paper, by designing a normalized nonmonotone search strategy with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang--Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai--Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai--Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.Comment: 32 pages, 7 figures; Accepted by Journal of Computational Mathematics on January 3, 202

Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems

In this paper, we develop an efficient spectral-Galerkin-type search extension method (SGSEM) for finding multiple solutions to semilinear elliptic boundary value problems. This method constructs effective initial data for multiple solutions based on the linear combinations of some eigenfunctions of the corresponding linear eigenvalue problem, and thus takes full advantage of the traditional search extension method in constructing initials for multiple solutions. Meanwhile, it possesses a low computational cost and high accuracy due to the employment of an interpolated coefficient Legendre-Galerkin spectral discretization. By applying the Schauder's fixed point theorem and other technical strategies, the existence and spectral convergence of the numerical solution corresponding to a specified true solution are rigorously proved. In addition, the uniqueness of the numerical solution in a sufficiently small neighborhood of each specified true solution is strictly verified. Numerical results demonstrate the feasibility and efficiency of our algorithm and present different types of multiple solutions.Comment: 23 pages, 7 figures; Chinese version of this paper is published in SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-143

Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes

The use of accelerated gradient flows is an emerging field in optimization, scientific computing and beyond. This paper contributes to the theoretical underpinnings of a recently-introduced computational paradigm known as second-order flows, which demonstrate significant performance particularly for the minimization of non-convex energy functionals defined on Sobolev spaces, and are characterized by novel dissipative hyperbolic partial differential equations. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal and spatial discretization. We prove the convergence to stationary points of such schemes in the semi-discrete setting. Further, we establish their convergence to time-continuous solutions as the time-step tends to zero, and perform a comprehensive error analysis in the fully discrete case. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of non-convex variational models in applied sciences, such as the Ginzburg-Landau energy in phase-field modeling and a specific case of the Landau-de Gennes energy of the Q-tensor model for liquid crystals.Comment: 37 pages, 4 figure

Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes

The use of accelerated gradient flows is an emerging field in optimization, scientific computing and beyond. This paper contributes to the theoretical underpinnings of a recently-introduced computational paradigm known as second-order flows, which demonstrate significant performance particularly for the minimization of non-convex energy functionals defined on Sobolev spaces, and are characterized by novel dissipative hyperbolic partial differential equations. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal and spatial discretization. We prove the convergence to stationary points of such schemes in the semi-discrete setting. Further, we establish their convergence to time-continuous solutions as the time-step tends to zero, and perform a comprehensive error analysis in the fully discrete case. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of non-convex variational models in applied sciences, such as the Ginzburg-Landau energy in phase-field modeling and a specific case of the Landau-de Gennes energy of the Q-tensor model for liquid crystals