The interplay between membrane viscosity and ligand-binding receptor kinetics in lipid bilayers

Early findings assumed the eukaryotic cell membranes as a bi-dimensional assembly of lipids organized in a fluid bilayer where transmembrane proteins can laterally diffuse [1]. Lipids self-assemble in a \(\sim 5 nm\) thick bilayer [2] and, once in vesicles, achieve an areal stretch of the order of \(5 \%\) [3]. Phospholipids can move in the planar direction and, so, plasma membranes are characterized by quasi-fluid deformable surfaces that express solid–fluid-like behavior, resulting in systems wherein in-plane fluidity and elasticity may simultaneously emerge [4]. Such fluidity is measured through the viscosity, whose available literature data are, however, highly experiment dependent, sometimes varying by orders of magnitude [5]. A possible explanation for this huge variability could be that membrane surface viscosity is a macroscopic quantity modeled at scales where the bilayer is assumed to behave like a 2-dimensional quasi-incompressible fluid. For this reason, micro- or nano- scale measurements may not be sufficient to catch the effective continuum viscosity but, rather, the so-called "microviscosity". The latter is a local quantity influenced by the environment [6]. Membrane fluidity is therefore associated with the high molecular mobility inside the lipid bilayer, enabling for a lateral diffusion of the embedded proteins [7]. Hence, viscosity results to be measured through the estimation of lipid diffusion coefficient [5]. It is indeed confirmed that the ligand-binding of receptors –as for example the G-Protein Coulped Receptors (GPCRs)– requires the presence of molecules that are able to move within the membrane [8]. In this regard, it has been established the difference, in terms of viscosity, among the resistance to flow under an applied shear stress and the capability of molecules to move and diffuse inside the membrane [9]. In the latter, it has been demonstrated that high diffusion mobility could be linked to a finite macroscopic shear viscosity, however discussing many cases of gel-phase of single saturated phospholipids or solid ceramide lipids that are able to pack themselves into a solid structure with high shear stiffness and viscosity. Quantitative stability analyses of viscoelastic lipid bilayers with properties deduced by [9], have been provided in [10]. Furthermore, in complex bio-membranes gel domains may coexist with fluid ones, thus promoting regions with vastly distinct viscosities [11]. Actually, evidences show that the mammalian cell membrane has a time-varying force response as nonlinear function of strain, so behaving as a visco-elastic or non-Newtonian fluid [12]. Related to this phenomenology, one can recall that lipid bilayers undergo various stages at which they may experience area expansion, thereby responding with compression and shear moduli [9]. Such a variation in the local mechanical properties seems to be responsible for the majority of cellular processes [13].

Several experimental strategies have been used to quantify the dynamical visco-elasticity of lipid systems [14, 15]. Recently, AFM measurements were performed to capture both the elastic and viscous properties of lipid systems that resulted to affect the propagation or attenuation of mechano-signaling across the cell membrane [16]. Also, high frequency experiments, modeled through a continuum mechanical theory, revealed that the plasma membrane displays a visco-elastic behavior [17]. In particular, it has been estimated that the cell surface responds like an elastic material on short time scales of around 1s, while exhibiting properties of a viscous body on longer time scales \(\sim 10-100\,s\) [18]. Bulk membrane viscosity and transverse stiffness are therefore correlated but also influenced by lipid packing density [19].

Modulation of membrane behavior has been demonstrated to be fundamental in various diseases [20‐24]. For instance, it is indeed confirmed that changes in membrane viscosity influence the evolution of the metastatic progression of cancerous cells [25, 26]. In [27] it is shown that the latter are softer than healthy cells and that they are also characterized by a more fluid membrane. For these reasons, the measure of membrane visco-elasticity leads to the possibility of discriminating between normal and cancerous cells through the application of multi-frequency vibrations [17].

Lipid rafts have been demonstrated to be involved in cardiovascular signaling as determinant regulators of vascular endothelial and smooth muscle cells, and in particular in signal transduction across the plasma membrane, of primary importance to many functional activities. At present, little is known about the specific role of lipid rafts in cardiac function and dysfunction, increasing attention focusing on their contribution to the pathogenesis of several structural and functional processes including cardiac hypertrophy and heart failure, as well as atherosclerosis, ischemic injury and different myocardial functions [28]. Lipid rafts in cardiomyocyte membranes are enriched in signaling molecules and ion channel regulatory proteins, therefore contributing to calcium handling and Ca2+ entry that control excitation-contraction of heart muscle cells. Thus, they can actively participate in differential cardiomyocyte ion channel targeting and regulation [28, 29].

Ordered microdomains result fundamental to stabilize signal transduction activities required for angiogenesis. In fact, it has been observed that VEGF receptor-2 (VEGFR-2), which stimulates angiogenic signaling, co-localizes with lipid rafts to regulate its activation. Also, long-term VEGFR2 relocation closely depends on lipid raft integrity, disruption of lipid rafts directly causing receptors’ depletion and inefficacy. In this sense, therapeutic strategies are more and more oriented towards the possible modulation of lipid rafts to control cells’ sensitivity to VEGF expression [30, 31]. Also, GPCRs have a primary influence in cardiac remodeling. Activation of epidermal growth factor receptors is in fact mediated by a large repertoire of GPCRs in the heart, and promotes cardiomyocyte survival, thus suggesting innovative therapeutic scenarios based on their targeting [32, 33].

Despite available pure mechanical descriptions of the lipid bilayers [34, 35] or purely diffusive approaches where the influence of micro-mechanical stimuli is neglected [36], there is still no modeling approach that takes into account the synergistic influence of membrane viscosity on transmembrane proteins activation and mobility and/or viceversa the role of proteins and lipids in membrane fluidity. Actually, it is well known that physical and chemical events act together to form the complexity of processes responsible for cell functions [37]. Therefore, a multiphysics analysis becomes manifest to provide new insights into the very complex world of plasma membranes. In this regard, mathematical production provided in Carotenuto et al. [38] confirmed the common knowledge that active receptors prefer to cluster on the so-called lipid rafts –wherein high cholesterol concentration increases bilayer rigidity [39]– through a chemo-mechanical coupled model. In [38], the model was regulated by the coupling of the membrane remodeling and its energetics dependent on the active proteins involved in the system, i.e. \(\beta 2-\)adrenergic receptors. Moreover, recent findings [40] highlighted the effects produced by the receptors and transporters on raft formation and coalescence through Cahn-Hilliard-type dynamics in a two-dimensional hyper-elastic framework.

Neverthless, as aforementioned the lipid bilayer is characterized by viscous properties and so, in order to obtain a more faithful solid–liquid description of this kind of system, a visco-hyperelastic model should be considered. This may provide an explicit interaction between the characteristic time evolution of the populations of transmembrane proteins and the relaxation time of the lipid bilayer. This is because, at the microscopic level, single protein re-arrangement and configurational changes are known to occur within milliseconds and are likely to locally produce elastic pressures at the membrane-protein interfaces [41, 42]. This can be extended at the population level through the presented continuum approaches, in which the dynamics of entire protein clusters is followed in response to the ligand time-varying precipitation stimulus. The morpho-elastic reconfiguration of the membrane thus can produce maps of heterogeneous stress and deformation that could project at the continuum scale the instantaneous packing of lipids and protein activation occurring within the ordered phase.

All this considered, the aim of the present study is to enrich well-grounded hyper-elastic models [38, 43‐45] of cell membranes by incorporating a material viscous component in the constitutive model. This provides an explicit interaction between the characteristic time evolution of the population of transmembrane proteins and the relaxation time of the lipid bilayer, by so calling into play a possible competition between the pseudo-viscous and the characteristic viscous terms.

It is well established that the plasma membrane undergoes a thickness change due to an ordered-disordered phase transition occurring at the lipid scale. This thickness variation is mainly caused by the lipid re-arrangement that, in assuming an ordered configuration, have straightened tails and appear tightly packed together as it occurs in functional micro-domains of the lipid membrane denoted as raft phase [46]. Several approaches have been adopted to analyze the mechanical behavior of membrane systems when experience phase transition based on either molecular dynamics simulations or, at the continuum scale, phase separation and elasticity models [47‐50]. Recently, a nonlinear hyperelastic response of the plasma membrane has been used to build up a fully-coupled framework describing the membrane’s macroscopic remodeling and functional reorganization as regulated by the leading biochemical events occurring among interacting protein species in forming lipid raft domains [38]. In the subsequent work by Bernard et al. [40], this evolutionary approach has been further enriched by Cahn-Hilliard energetics and kinetics for the involved species, thereby accounting for rafts nucleation and coalescence. The time-varying nature of the involved biological species associated to configurational remodeling terms gave to the system a pseudo-visco-elastic nature (with eventual dissipation), the rate of the internal species kindling a viscous-type (chemical) stress. However, in [40] the explicit role of intrinsic visco-elasticity of the lipid membrane and the possible influence of the fluid component of the bilayer on raft development was not considered. To this purpose, we here analyze a two-dimensional system capable to experience a lipid phase separation and manifest raft coarsening within a visco-elastic environment. The whole phenomenon will be the result of the coupling between the conformational remodeling guided by the presence of the active protein species and the energetics of the membrane. In particular, the elastic part of the membrane response –in line with well-established literature [51‐53]– is modeled by assuming a neo-Hookean type behavior [40], by neglecting for now the spontaneous trends of lipids to reorganize themselves in co-existing phases (this can be accounted for not convex energy terms [54]). At the molecular scale, the activation of a single transmembrane protein within the lipid environment provokes a re-arrangement of its sub-units, which induces a stress in the surrounding membrane in the form of an in-plane pressure. This, inevitably, calls into play the adaptation of the neighboring lipids. In the absence of any viscous component, the adaptation of the lipid membrane is entirely dictated by the dynamics of the protein populations. In this sense, at the macroscopic scale the overall deformation and morphological remodeling of the lipid membrane is seen as the averaged result of the overall behavior of protein densities. The latter will pass to their active state asynchronously by introducing delays and by exchanging (positive or negative) chemical feedbacks. These give rise to more complex spatial and temporal patterns of the membrane heterogeneity. Noteworthy, the characteristic times of the membrane evolution do not simply follow the activation times of single units (of the order of few milliseconds). Rather, instead ensue the collective dynamics of active resident proteins and their progressive recruitment. Indeed, lipid and proteins’ clusters have a much larger life-span (from seconds to several minutes [55‐57]). In this sense, the micro- and macro- scopic scales of the ordered macro-islands could potentially describe multi-scale kinematics in a cascade manner. Through the above described mechanisms, in [40] an interspecific protein dynamics, enriched with a Cahn-Hilliard energetics and kinetics phenomena, has been adopted to successfully trace back the complex spatio-temporal adaptation of the membrane. Of course, the chemo-mechanical coupling becomes absolutely crucial to theoretically explain how protein density dynamics affects the structural remodeling of the membrane, leading to the nucleation of raft domains. The heterogeneity noticed in lipid bilayers has to be indeed addressed to the coexistence of disordered and ordered lipid phases [58]. To this end, well-grounded observations show the formation of zones with different concentration levels [59]. In particular, regions with high concentration of proteins have been recognized in lipid rafts [60], where the clustering phenomena give rise to the initiation of most of cellular processes [61‐63]. For this reason, the introduction of a phase-separation diffusive model able to predict coalescence of different species becomes apparent. Within this framework, the Cahn-Hilliard equation is typically used to describe two-phase separation problems [64‐66] that are mathematically described by a diffusion equation for the species concentration [67]. In this respect, the theoretical model proposed in [40] described the evolution of protein species through Cahn-Hilliard-like energetics and kinetics wherein reaction interspecific terms account for the mutual influence among protein populations, i.e. the above mentioned GPCRs and their antagonist the Multidrug Resistance Proteins (MRPs), while non-local species momenta are enriched by strain-dependent morphotaxis terms. The latter enable the movement of protein species along the gradients of lipid order distribution, so promoting the tendency of signaling proteins to reside on raft domains by favoring spatial co-localization of such species on raft islands. When the viscous component of the membrane is introduced and a visco-elastic behavior of the membrane is considered, the above described dynamics can be altered by the direct competition between both the characteristic adaptation and the intrinsic bilayer relaxation times. Indeed, it is expected that viscosity may affect the membrane deformation triggered by proteins through creep-associated effects in raft emergence, thus so influencing its chemical stability and persistence. On the other hand, stress relaxation phenomena could occur as well by redistributing internal stresses with effect on the residual stress-induced stiffness and membrane tension. However, rough estimations of the visco-elastic and lipid raft characteristic times –respectively of microseconds and tens of seconds– would suggest that these phenomena would minimally concur together in determining the structural re-organization of the membrane. More important effects could be rather produced by the synergy of protein dynamics with nonlinear deformations and viscous response, which could lead instead to more significant changes into the material remodeling of membrane properties. This would meet some experimental evidences showing that rafts are highly viscous and stiff zones of the membrane. To do this, in what follows we present the governing equations of the coupled model within a visco-elastic framework. This will enable to investigate how membrane fluidity is influenced by the dynamical re-organization. In particular, we will initially consider the effects of a constant (i.e. linear) viscous term on raft persistence. While afterwords a strain-level dependent viscosity will be considered to explore if the increase of viscosity of heterogeneous lipid membranes plays a key influence on co-evolving with lipid rafts.

Within the framework of membrane visco-elasticity, we here present numerical results that permit to observe the viscosity landscape of the phase-separated domains, by focusing on possible differences in terms of raft lifespan and heterogeneity. To this aim, sensitivity analyses will be carried out to map the evolution of an initially (geometrically and materially) homogeneous membrane, by observing how raft domains and viscosity change. This will be mainly investigated as a function of the membrane’s (elastic and viscous) tangent properties and initial protein distributions. In the light of the pivotal role of mechanics in the spatio-temporal dynamics of the raft-associated proteins, we analyze protein-induced adaption processes. Indeed, conformational changes of GPCR and MRP populations are capable to induce the overall remodeling of the bilayer at the membrane scale. With this in mind, in order to trigger the activation dynamics, we consider the realistic situation in which extracellular molecules randomly precipitate on the domain. This is done by assigning a random distributions to the ligand precipitation rate functions used in (23) and by modulating the amount of precipitating ligand to induce differential receptor responses, thus orienting the membrane dynamics towards various patterns.

In numerical analyses, we start from studying the effects of a constant viscosity on the spatio-temporal behavior of the ordered phase. To then investigate more in depth the material adaptation of the bilayer in terms of the evolution of viscous properties of the rafts through a strain-sensitive viscosity term. This enrichment allows to follow the strain-induced remodeling of the lipid phase. In particular, this is done by meeting wide literature evidences demonstrating that viscosity of ordered clusters tends to increase as the phase order increases [94]. Starting from the initial Newtonian hypothesis, sensitivity analyses are carried out by varying the viscosity over a range compatible with literature data. In this respect, surface shear viscosity seems to exhibit a large variability depending on the particular composition of the mixed lipid system, on the specific conditions in which tests are performed as well as on the adopted experimental methods. Typical values of tangent viscosity for the most of biological membranes result of the order of \(10^{-3}-10^2 Pa.s\) [5, 9, 10, 89, 95]. Fewer cases were found to instead exhibit significantly higher tangent viscosities ranges of \(10^5-10^6 Pa.s\) [9, 89], up to peaking to unusual values \(10^9 Pa.s\) in case of the so-called tough visco-elastic systems [91, 92]. However, it is worth highlighting that these experimental observations report significant differences when cholesterol is introduced in the mixed lipid systems. In particular, cholesterol highly affects the stiffening and the viscosity increase of the membranes and it has a direct impact on raft stabilization as well [89, 96, 97]. In the present model, we limit our analyses to pure and mixed lipid systems, for now excluding the explicit modeling of cholesterol as a structural component of the membrane medium, which could be instead taken into account through the suitable determination of homogenized material properties depending on the extent of cholesterol fraction.

Following a recent theoretical formulation describing the mechanobiology of lipid membrane remodeling and raft formation carried out in [38, 40], the current study aims at investigating the dynamic visco-elastic response of plasma membranes to chemo-mechanical stimuli. Through in silico analyses accounting for viscous-associated terms in the constitutive model, the multiphysics coupling between chemical events and mechanical adaptation highlights how the solid–fluid behavior of the bilayer evolves with the activity of the membrane. The evolved processes are strongly influenced by the dynamics of the transmembrane proteins activation and their interaction with the lipid medium. By considering both the cases of a Newtonian shear viscosity and a strain-sensitive viscosity, in this present paper we investigate the relationship between the reconfiguration of an initially inactive membrane micro-environment as a function of the competition between the internal viscous dissipation and the kinetics of phase transitions governing the emergence of lipid islands.

Numerical outcomes allowed one to observe that the shear viscosity varies in phase-separated membranes resulting in higher values for ordered-phase domains, i.e. lipid rafts. Hence, this provides a mechanically-based explanation of a well-known phenomenon highlighted by a large number of biophysical studies by means of various experimental methods. The synergy between active protein regions and raft emergence leads the system to re-organize itself by creating thicker and more viscous domains. Also, sensitivity analyses revealed how the visco-elastic behavior is influenced by the intra-cellular pressure applied at the boundary. That alters the mechanical properties of the membrane, and the volume fraction of the liquid-disordered phase. Hence, our visco-elastic approach enriches the existing studies regulating the mechanisms on the lipid membrane’s behavior. This could help to earn some insights in characterizing the role of lipid rafts in membrane mechanics and in mediating important cellular biochemical processes.

By refining the modeling of species inter-specificity, one would have the opportunity to include some other agents influencing membrane dynamics in the analysis. This may allow one to enlarge the complex multi-species environment under exam, as well as to further enrich the membrane constitutive framework. To this aim, the self-reconfiguration of lipids could be studied by considering non-convex terms in the elastic strain energy (see e.g. [34, 38] and reference cited therein). Moreover, enriched coupling terms may be considered in the model in order to have deeper insights into the influence of the mechanical stress on the interspecific dynamics. In fact, through ad hoc mechanical feedback functions, it would be possible to better investigate the processes of cell mechano-sensing and mechano-trasduction, that inevitably involve the mediation of membrane selectivity during cell-environment communication. Also, as emerged from the presented analyses, one of the main components that can be included to further refine and enrich the description of membrane visco-elastic adaptation could be the cholesterol. This has a direct responsibility for lipid rafts stabilization and bilayer lateral diffusion, GPCRs re-configuration and activity, besides its participation to determine the membrane effective properties. For this significant reason, this will be object of future investigations.