Lipolysis on Lipid Droplets: Mathematical Modelling and Numerical Discretisations

TL;DR

This work develops a first PDE-based model of lipolysis on lipid droplets by representing surface-bound enzymatic activity in a thin active region connected to a reservoir through an interface, and by tracking TG and DG concentrations in two subdomains with a constant ATGL influence. The authors prove well-posedness of the parabolic problem, establish existence and uniqueness of a positive equilibrium for a fixed total mass via a fixed-point argument, and demonstrate exponential convergence to equilibrium using entropy methods. They implement two numerical discretizations (finite elements and isogeometric analysis) to solve both the parabolic and elliptic systems, validating convergence and showing the entropy decays exponentially in time. A realistic extension with ATGL clustering reveals that heterogeneous enzyme distributions can significantly slow lipolysis, highlighting the importance of spatial organization for lipid metabolism and offering a framework for quantitative investigations into enzyme localization effects.

Abstract

Lipolysis is a life-essential metabolic process, which supplies fatty acids stored in lipid droplets to the body in order to match the demands of building new cells and providing cellular energy. In this paper, we present a first mathematical modelling approach for lipolysis, which takes into account that the involved enzymes act on the surface of lipid droplets. We postulate an active region near the surface where the substrates are within reach of the surface-bound enzymes and formulate a system of reaction-diffusion PDEs, which connect the active region to the inner core of lipid droplets via interface conditions. We establish two numerical discretisations based on finite element method and isogeometric analysis, and validate them to perform reliably. Since numerical tests are best performed on non-zero explicit stationary state solutions, we introduce and analyse a model, which describes besides lipolysis also a reverse process (yet in a physiologically much oversimplified way). The system is not coercive such that establishing well-posedness is a non-standard task. We prove the unique existence of global and equilibrium solutions. We establish exponential convergence to the equilibrium solutions using the entropy method. We then study the stationary state model and compute explicitly for radially symmetric solutions. Concerning the finite element methods, we show numerically the linear and quadratic convergence of the errors with respect to the $H^{1}$- and $L^{2}$-norms, respectively. Finally, we present numerical simulations of a prototypical PDE model of lipolysis and illustrate that ATGL clustering on lipid droplets can significantly slow down lipolysis.

Figures (17)

• Figure 1: Lipolysis is a three-step process: ATGL hydrolyses TGs producing DGs and a first free FA; HSL hydrolyses DGs producing MGs and a second free FA; finally, MGL hydrolyses MGs releasing the backbone glycerol and a third free FA.
• Figure 2: ATGL also catalyses transacylation of two DG molecules into one TG and one MG.
• Figure 3: (Left) A cytoplasmic lipid droplet in a cultured hepatoma cell adopted from farese2009. (Right) Structural components in a lipid droplet adopted from onal2017. The lipid droplet is composed of several components; the triglycerides are found inside the droplet and are surrounded by a layer of phospholipids.
• Figure 4: (Left) The reservoir region $\Omega_{1}$ is the ball with radius $R > 0$ and outer unit normal $\eta$. The active region $\Omega_{2}$ is the annulus with thickness $\delta > 0$. The boundary of the reservoir region coincides with the interface$\Gamma$ towards the active region. The outer boundary of the active region is the boundary $\partial\Omega$. (Right) A chemical reaction network for TGs and DGs in the reservoir and active regions. The constants $\mu, \, \nu > 0$ denote the interface flux rate for TG and DG, $\kappa > 0$ models the feedback rate of DG into TG at the reservoir region, and $f$ is the Michaelis-Menten reaction term which governs the hydrolytic reaction between TG and ATGL.
• Figure 5: Variational crime in $\Omega_{2}$. We have that $V_{2,h} \not \subset H^{1}(\Omega_{2})$ due to the nonconvexity of the active region $\Omega_{2}$. Here, we have $R = 10$ and $\delta = 5$.

Theorems & Definitions (19)

• Lemma 3.1: Continuity of $\mathcal{A}$
• proof
• Lemma 3.2: Gårding Inequality
• proof
• Theorem 3.3: Well-posedness of the Parabolic System
• proof
• Theorem 3.4: Global $\mathbf{L}^{2}$ bounds
• proof
• Theorem 3.5: Regularity
• proof