Membrane shrinkage and cortex remodelling are predicted to work in harmony to retract blebs
Numerous cell types undergo an oscillatory form of dynamics known as blebbing, whereby pressure-driven spherical protrusions of membrane (known as blebs) expand and contract over the cell's surface. Depending on the cell line, blebs play important roles in many different phenomena including mitosis and locomotion. The expansion phase of cellular blebbing has been mathematically modelled in detail. However, the active processes occurring during the retraction phase are not so well characterized. It is thought that blebs retract because a cortex reforms inside, and adheres to, the bleb membrane. This cortex is retracted into the cell and the attached bleb membrane follows. Using a computational model of a cell's membrane, cortex and interconnecting adhesions, we demonstrate that cortex retraction alone cannot account for bleb retraction and suggest that the mechanism works in tandem with membrane shrinking. Further, an emergent hysteresis loop is observed in the intracellular pressure, which suggests a potential mechanism through which a secondary bleb can be initiated as a primary bleb contracts.
1. Introduction
Many animal cells have the ability to produce large, dynamic protrusions such as lamellae, filopods, microspikes and pseudopods [1]. Each of these four protrusion types rely on the polymerization of actin filaments in order to push the cell membrane outwards. Often, cells use these membrane extensions to undergo motility [2]; however, they can also be involved in a number of other different phenomena, such as mitosis [3].
Here, we are interested in a specific protrusion type known as a cellular bleb, which plays an important role in the locomotion of tumour cells, embryonic cells and stem cells [4–7]. Unlike the previously mentioned ‘actin-driven protrusions’, blebs do not extend because of actin filaments pushing the membrane outwards. Instead, blebs occur when a cell's lipid membrane bilayer delaminates from its actin cortex. If a cell's internal pressure is higher than the external pressure, then the pressure difference induces a flow of the cell's cytosol driving the membrane away from the cell and into a spherical protrusion, known as a bleb [8]. This localized swelling requires additional membrane to cover the bleb, but it is not currently known how this extra membrane can be produced and removed as quickly as observed. It has been hypothesized that extra membrane stems from high levels of wrinkling in the cellular surface [9]. Alternatively, it has been suggested that endo- and exo-cytosis processes could account for the membrane activity through localized recruitment [10]. In both cases, the growth of the membrane can be modelled as an increase in reference configuration, which is the approach taken here.
After approximately 10–30 s, the bleb expansion stops and an actin cortex reforms inside the bleb. Over a longer timescale of 1–2 min, the cell retracts the newly formed cortex within the bleb (which is coupled to the membrane), causing the bleb to shrink back into the cell, and allowing the process to begin again (figure 1). It is thought that myosin motors contract the cortex, causing it to shrink and, potentially, thicken [10]. It is this retraction phase that we are interested in modelling in this paper because bleb retraction is required for cells to move efficiently.
Figure 1. Confocal microscopy for a uniform timecourse showing a single bleb on a muscle stem cell being retracted over, approximately, 1 min. Used with permission from the Skeletal Muscle Development Group, University of Reading. The fluorescence highlights polymerized actin. Scale bar, 1.5 μm.
One should note that these ‘pressure-driven’ bleb protrusions, differ in mechanism, character and behaviour from actin-driven protrusions. For example, it has been shown that cellular motion dependent on blebs is much faster than many forms of lamellipodial movement. Further, blebbing cells are able to change the direction of their migration much quicker [11,12]. Owing to the fast-moving properties of the blebs over small spatial scales, it can be difficult to generate experimental insights into the physical processes that couple together to make blebbing possible, motivating the use of mathematical models to investigate the blebbing process.
Although blebbing is an extremely complex behaviour, the structural shape of a cell is thought to depend on three components: a flexible lipid bilayer membrane, a stiff actin cortex and adhesion proteins that couple these two structures [8]. Mathematical modelling offers a framework within which hypotheses can be tested, generating new predictions concerning the underlying mechanisms that control the blebbing expansion and retraction cycle. The hypothesis currently present in the biological literature is that blebs shrink simply due to the retraction of the bleb's reformed cortex, implying that the membrane is slave to the dynamics of the cortex [10]. Here, we test the biophysical and mechanical plausibility of experimentally suggested mechanisms that induce bleb retraction. However, an absence of molecular-level knowledge entails that we implement a phenomenological model of membrane and cortex retraction which captures the concept that myosin motors induce cortex shrinkage, but without detailed dynamics.
In terms of previous work, there are a number of different theoretical frameworks available considering diverse aspects of blebbing. Although some groups focus on using very high-level models to capture the entire blebbing expansion–retraction cycle [13,14], the majority deal with only the expansion phase of blebbing [15–19]. Our aim is to extend our previous model [20] to include mechanisms that will describe the retraction of spherical protrusions in order to investigate the expansion and retraction of blebbing in a unified mechanical model.
In our framework, the cell's membrane is an extensible, axisymmetric, elastic shell. The adhesion proteins that link the cortical cytoskeleton to the plasma membrane are thought to be members of the highly conserved ezrin–radixin–moesin (ERM) family [21]. Based on the work of Liu et al. [22], we model the adhesion molecules as piecewise neo-Hookean springs, in that their retraction force is a nonlinear function of their extension, up until a critical length. Beyond this critical extension, the adhesion molecules detach from the membrane, leaving the cortex and membrane no longer connected. Finally, the cortex is represented simply as a stiff elastic structure in which the adhesions are fixed.
Other frameworks for the cell membrane do exist; for example, it can be treated as a highly viscous fluid. However, we encompass these features within the solid mechanics framework as viscosity can be represented by a membrane with an evolving reference configuration. Equally, the growth of the membrane through a change in the arc-length kinematically captures all possible internal effects such as growth by addition of new material, resorption and fluid-like properties.
We begin in §2 by reproducing the key equations of the previously presented shell model of a bleb [15,16,20] and extend it to include the production of a new cortex in the bleb, the retraction of this new cortex and membrane shrinking. The initial results in §3 demonstrate that cortex retraction cannot produce bleb retraction on its own. Cortex retraction is then coupled to membrane shrinkage and it is observed that we are able to reproduce the observed bleb retraction, as well as produce membrane wrinkling depending on the ratio of timescales between the cortex retraction and membrane shrinking mechanism. Finally, in §4, we summarize the results and suggest how bleb retraction may lead to the initiation of further blebs, thus allowing cells to undergo self-consistent cyclical bleb dynamics.
2. Mathematical framework
The geometry and shell mechanics [23] are defined as in previous articles [16] and the pertinent equation system coupling the membrane, adhesions and cortex is briefly recapitulated here and explained below. A brief overview of the variables can be found in table 1 and further detail can be found in appendix A. The equations are
Table 1.
Reference table for the variables and parameters in system (2.1)–(2.5). See text for further details.
Equation set (2.1) defines the axisymmetrical geometry of the shell around the axis of rotational symmetry, here taken to be the z-axis (figure 2). As the two-dimensional shell is axisymmetric, we only need to consider a one-dimensional cross-section, at which point the radial coordinates can be related to the standard rectilinear Cartesian coordinates. Specifically, a reference configuration,
Figure 2. Definition of geometric variables concerning the coupling of the membrane, (z,y), the reference configuration,
Initially, the reference configuration is a sphere of radius ρ, thus σ=σ0∈[0,ρπ]. The bleb production simulations are initiated by removing adhesions at the front of the cell, around σ=0. After the initial ablation of adhesions, the membrane deformation arises through reference configuration remodelling. This means that the arc length and profile of the reference configuration are able to update according to some postulated evolution rule. Critically, we only remodel the reference configuration within the unadhered region,
The solution configuration represents the shape that the reference configuration takes once it has been pressurized and is defined by the horizontal and vertical coordinates (z,y). Furthermore, s measures the arc length of the solution configuration and θ is the outward pointing normal angle of the membrane measured anticlockwise from the z-axis. Finally, in order to complete the geometric definition, we define κs and κϕ through equation set (2.2), respectively, to be the principal curvatures of an axisymmetric surface.