Summary1. Performing sensitivity analyses for deterministic multi-stage, multi-parameter population models that include nonlinearity presents a significant computational challenge. 2. We implement a standard forward sensitivity analysis to evaluate partial derivatives of population state variables with respect to parameters at all times for which the solution is known. We present a hybrid Matlab/Maple software package, sensai, which automates the steps required to calculate sensitivities and elasticities. We focus on sensitivities of transient states of populations. Our analysis therefore differs from the more common analysis of linear systems where sensitivities are computed for the asymptotic stable-stage distribution assuming unbounded population growth. The method generalizes the matrix calculus methods for nonlinear, stage-structured matrix models previously developed by Caswell (2009, Journal of Difference Equations and Applications, 15, 349-369). 3. We present an example of a nonlinear, discrete-time population model in the form of a stagestructured Lefkovitch matrix, and another multi-stage, continuous-time SIR disease model in the form of a system of ordinary differential equations. In addition, we extend the framework for composite nonlinear maps of population demography to include single-locus genetics with natural selection discriminating among genotypes. 4. SENSAI allows for the analysis of a quantity of interest (an arbitrary function of variables and parameters, e.g. the proportion of infected individuals in a disease model), and sensitivity to combinations of model parameters. For example, we can compute sensitivities with respect to parameters and initial conditions at all times during an outbreak in a disease model, before a steady state of disease prevalence is attained. Similarly, in population genetic models, we can compute sensitivity of a response to natural selection (e.g. the change in allele frequencies) in relation to fitness differences among genotypes. 5. We illustrate the capabilities of sensai through a series of canonical models with relatively few variables and parameters. However, sensai has the capability to analyse significantly more complex models.
This work presents a mathematical model for the localization of multiple species of diffusion molecules on membrane surfaces. Morphological change of bilayer membrane in vivo is generally modulated by proteins. Most of these modulations are associated with the localization of related proteins in the crowded lipid environments. We start with the energetic description of the distributions of molecules on curved membrane surface, and define the spontaneous curvature of bilayer membrane as a function of the molecule concentrations on membrane surfaces. A drift-diffusion equation governs the gradient flow of the surface molecule concentrations. We recast the energetic formulation and the related governing equations by using an Eulerian phase field description to define membrane morphology. Computational simulations with the proposed mathematical model and related numerical techniques predict (i) the molecular localization on static membrane surfaces at locations with preferred mean curvatures, and (ii) the generation of preferred mean curvature which in turn drives the molecular localization.
Abstract. Simulating protein-membrane interactions is an important and dynamic area of research. A proper definition of electrostatic forces on membrane surfaces is necessary for developing electromechanical models of protein-membrane interactions. Here, we model the bilayer membrane as a continuum with general continuous distributions of lipids charges on membrane surfaces. A new electrostatic potential energy functional is then defined for this solvated protein-membrane system. Key geometrical transformation properties of the membrane surfaces under a smooth velocity field allow us to apply the Hadamard-Zolésio structure theorem, and the electrostatic forces on membrane surfaces can be computed as the shape derivative of the electrostatic energy functional.
Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate the membrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.
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