Interplay between membrane fluctuations and the kinetics of a membrane-anchored receptor

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The main topic of this dissertation is the interplay between a bendable thin sheet (membrane) and its highly localized pinning. The most general formulation of the problem allows for the reversible breaking and formation of the membrane pinning, with the kinetic rates depending on the dynamics of the membrane. In order to break the problem into more manageable parts, we first investigate the effect of a permanent pinning on the static and dynamic properties of the pinned membrane, such as its shape and fluctuations. After that we explore how are the kinetic rates for the pinning breaking and formation affected by membrane fluctuations. A paradigmatic example that motivates investigation of such problems is a biological membrane anchoring receptors which interact with environmental ligands. The receptor-ligand ”lock-key” interactions are effective only when receptors and their corresponding ligands are in close proximity for a certain amount of time, which in turn depends on the fluctuations of the receptor-anchoring membrane. On the other hand, the formed ligand-receptor bond pins the membrane locally, affecting membrane dynamics and consequently the rate for bond rupture. Of course, the described problem of a bendable sheet locally pinned by a stochastic pinning is more general than the biological case and can be applied to many different systems. However, we will frame our investigations mostly through the biological perspective. Along the way we will build on the corresponding literature which already offers many useful attempts at a general descriptive model. In the General Introduction, we discuss the theoretical frameworks that define the problems we aim to solve. We first review the continuum Canham-Helfrich model of the membrane energetics [1, 2], which describes the membrane by only two coarsegrained parameters; bending rigidity and tension. We then describe the extensions of the model that account for the non-specific interactions of the membrane with the environment, and introduce the Hamiltonian that accounts for the local, specific interactions. After that, we present the Langevin formalism used for modeling membrane dynamics in a hydrodynamic surrounding [3], together with the obtainable analytic solutions for the case of non-specific interactions. Finally, we consider the available models for the receptor-ligand interactions in situations in which at least one of the interacting molecules is anchored to a fluctuating membrane. We start with the frequently used phenomenological models [4, 5] and end with the coarse-grained kinetic rates that account for the effect of membrane fluctuations [6]. Publication P1 treats the classical Canham-Helfrich model of the membrane extended by two additional terms; one representing the non-specific interactions with the environment, and the other representing a localized pinning of the membrane modeling specific receptor-ligand interaction. We assume that the pinning is permanent and immobile and explore its consequences on the thermal equilibrium properties of the membrane, such as its mean shape, spatial two-point correlation function and shape fluctuations. Interestingly, we find that the correlation function of a pinned membrane is proportional to the free-membrane correlation, and consequently that the correlation length of a pinned membrane does not depend on the pinning properties. However, we find a non-monotonous dependence of the correlation length on the membrane tension, indicating that the effect of the pinning cannot be accounted for by a constant effective tension. We explore the corresponding tension regimes of the correlation length and show its universally exponential decay. We use these results to gain insight into the membrane-mediated interactions between two pinnings and find that interactions are present even in the absence of the mean shape deformation, solely due to the effect of thermal fluctuations of the membrane. Publication P2 extends the treatment of the pinned membrane by exploring the membrane dynamics in the context of the Langevin formalism accounting for the hydrodynamic effects of the surrounding solution. We resolve the membrane dynamics by analytically calculating the Green’s function of the differential equation defining the problem. We use this result to focus on the case of thermal forces and calculate the power spectral density (PSD) of a thermally agitated pinned membrane in a hydrodynamic surrounding. We validate the correctness of the analytical result with explicit numerical simulations of the membrane dynamics. We then propose several experimental protocols that can use the derived theoretical results for the extraction of system parameters from the measurements of membrane spatio-temporal dynamics. In publication P3 we tackle the problem of the reversible ligand-receptor (LR) interactions, with the aim to account for the effect of membrane fluctuations on these interactions by developing effective LR reaction rates. First, we give a firstprinciple derivation of the otherwise phenomenologically introduced rates with familiar Bell-Dembo properties [4, 5]. We then assume that the receptor is anchored on a fluctuating membrane and account for the effect of membrane fluctuations by calculating the expected (un)binding rates with respect to the time-independent probability distribution of membrane fluctuations. Based on the fluctuation measurements in non-activated and activated human macrophages and red blood cells, we construct a general model that captures both Gaussian and non-Gaussian fluctuations exhibited by both cell types. Thus, the introduced fluctuation model, which is a convolution of Gaussian and exponential distributions, enables us to model the effect of active fluctuations in a cell-type-independent way. Finally, we show that even the calculated non-Gaussian LR rates have a Bell-Dembo structure in the biologically relevant regime in which the receptor is much stiffer than the membrane. This result emphasizes the robustness of the Bell-Dembo assumptions and gives a framework that unifies the treatment of the effective LR rates under passive and active membrane fluctuations. Work in this dissertation builds a theoretical framework for the investigation of the interplay between membrane fluctuations and the kinetics and affinity of the membrane-anchored receptors. The framework is based on a few simple principles and as such offers well-defined metrics for testing the validity of the corresponding models. Furthermore, it offers a way to bridge the gap between the models of passive and active fluctuations, as well as between different fluctuation sources in the active case, thus tying separate models into a single, coherent picture. All of this should facilitate analytical modeling of complex systems that were out of reach until now, as well as increase the spatio-temporal scales on which the systems can be effectively simulated