The relation between thermal fluctuations and the mechanical response of a free membrane has been explored in great detail, both theoretically and experimentally. However, understanding this relationship for membranes, locally pinned by proteins, is significantly more challenging. Given that the coupling of the membrane to the cell cytoskeleton, the extracellular matrix and to other internal structures is crucial for the regulation of a number of cellular processes, understanding the role of the pinning is of great interest. In this manuscript we consider a single protein (elastic spring of a finite rest length) pinning a membrane modelled in the Monge gauge. First, we determine the Green's function for the system and complement this approach by the calculation of the mode coupling coefficients for the plane wave expansion, and the orthonormal fluctuation modes, in turn building a set of tools for numerical and analytic studies of a pinned membrane. Furthermore, we explore static correlations of the free and the pinned membrane, as well as the membrane shape, showing that all three are mutually interdependent and have an identical long-range behaviour characterised by the correlation length. Interestingly, the latter displays a non-monotonic behaviour as a function of membrane tension. Importantly, exploiting these relations allows for the experimental determination of the elastic parameters of the pinning. Last but not least, we calculate the interaction potential between two pinning sites and show that, even in the absence of the membrane deformation, the pinnings will be subject to an attractive force due to changes in membrane fluctuations.

Adhesion domains forming at the membrane interfaces between two cells or a cell and the extracellular matrix commonly involve multiple proteins bridges. However, the physical mechanisms governing the domain structures are not yet fully resolved. Here we present a joint experimental and theoretical study of a mimetic model-system, based on giant unilammelar vesicles interacting with supported lipid bilayers, with which the underlying physical effects can be clearly identified. In our case, adhesion is induced by simultaneous action of DNA linkers with two different lengths. We study the organization of bridges into domains as a function of relative fraction of long and short DNA constructs. Irrespective of the composition, we systematically find adhesion domains with coexisting DNA bridge types, despite their relative differences in length of 9 nm. However, at short length scales, below the optical resolution of the microscope, simulations suggest the formation of nanodomains by the minority fraction. The nano-aggregation is more significant for long bridges, which are also more stable, even though the enthalpy of membrane insertion is the same for both species.Cell adhesion is a fundamental biological process that relies on the formation of complex macromolecular assemblies, involving a number of proteins with different binding affinities, elastic properties and lengths 1 . These proteins may mediate adhesion in either a sequential manner, as it is the case of weak selectin adhesion preceding strong integrin adhesion in rolling leukocytes 2-4 , or simultaneously, for example during the formation of immune synapses. In this case T-cell receptors and integrins segregate 5,6 , presumably because of the differences in the length of their respective adhesive constructs 7 . However, the full understanding of the length-induced separation is still missing.One proven approach to study these mechanisms is using cell mimetic systems consisting of functionalized giant unilamelar vesicles (GUVs) adhering to supported lipid bilayers (SLB). So far, most mimetic adhesion systems involved only one protein pair, or + contributed equally mixtures between binders and repellers 8,9 , while the design of systems consisting of more than one binding pair is still challenging, and an active field of studies 10 .