Effective cell membrane tension protects red blood cells against malaria invasion

A critical step in how malaria parasites invade red blood cells (RBCs) is the wrapping of the membrane around the egg-shaped merozoites. Recent experiments have revealed that RBCs can be protected from malaria invasion by high membrane tension. While cellular and biochemical aspects of parasite actomyosin motor forces during the malaria invasion have been well studied, the important role of the biophysical forces induced by the RBC membrane-cytoskeleton composite has not yet been fully understood. In this study, we use a theoretical model for lipid bilayer mechanics, cytoskeleton deformation, and membrane-merozoite interactions to systematically investigate the influence of effective RBC membrane tension, which includes contributions from the lipid bilayer tension, spontaneous tension, interfacial tension, and the resistance of cytoskeleton against shear deformation on the progression of membrane wrapping during the process of malaria invasion. Our model reveals that this effective membrane tension creates a wrapping energy barrier for a complete merozoite entry. We calculate the tension threshold required to impede the malaria invasion. We find that the tension threshold is a nonmonotonic function of spontaneous tension and undergoes a sharp transition from large to small values as the magnitude of interfacial tension increases. We also predict that the physical properties of the RBC cytoskeleton layer—particularly the resting length of the cytoskeleton—play key roles in specifying the degree of the membrane wrapping. We also found that the shear energy of cytoskeleton deformation diverges at the full wrapping state, suggesting the local disassembly of the cytoskeleton is required to complete the merozoite entry. Additionally, using our theoretical framework, we predict the landscape of myosin-mediated forces and the physical properties of the RBC membrane in regulating successful malaria invasion. Our findings on the crucial role of RBC membrane tension in inhibiting malaria invasion can have implications for developing novel antimalarial therapeutic or vaccine-based strategies.

where s max is the maximum membrane arclength that adheres to the merozoite.Additionally, for axisymmetric coordinates, the principal extension ratios can then be written as where (.) ′ = d(.)ds and s 0 is the arclength along the undeformed shape of the axisymmetric skeleton mapping to an unknown position s on the deformed shape.
The egg shape of an archetypal merozoite in an axisymmetric coordinate can be parametrized as [1] ( where where 0 < ϕ < 2π and 0 < θ < π.Using Eq.S4, the radius of merozoite (R) as a function of angle θ can be written as Having the radius of merozoite, we can find the radial distance from the axis of rotation (r) and the elevation from the reference plane (z) given by Using Eq.S6 and the definition of axisymmetric coordinates, we have where k 1 and k 2 are the surface principal curvatures and ds dθ = (dr/dθ) 2 + (dz/dθ) 2 .Eq. S7 allows us to find the mean curvature H along the merozoite surface as a function of θ given as where ( .)= d(.)dθ .The integral over the adhered area (Eq.S1) and the extension ratios (Eq.S2) can also be calculated as a function of θ where θ max is the maximum wrapping angle.Assuming that actomyosin motors apply forces tangentially along the membrane surface, in axisymmetric coordinates, the net radial force (f r = f cos(θ)) is zero.Thus, only the axial component of actomyosin forces (f z = f sin(θ)) pushes the merozoite forward and the work on the membrane (Eq. 3) is simplified as Using Eqs.S1, S9a, and S10, the change in the energy of bilayer/cytoskeleton due to the adhesion of merozoite and deformation bilayer/cytoskeleton (Eq.7) can be written as a function θ where Entropic energy of spectrin filaments orientations Here, in Eq.S11, we assumed that the force density applied by the actomyosin motor (f ) is constant all along the area of adhered merozoite.

Incompressible bilayer and cytoskeleton
Let us assume that a flat circular patch of a lipid bilayer and relaxed cytoskeleton with radius (s 0 ) deformed to fit the merozoite contour in the adhesive region.Thus, for an incompressible bilayer/cytoskeleton, the area conservation can be written as Eq. S16 allows us to find s 0 and calculate the extension ratios using Eq.S9b, which simplifies as which is consistent with zero local area strain (α = λ 1 λ 2 − 1 = 0) for an incompressible bilayer/cytoskeleton [2].Additionally, for an incompressible cytoskeleton, the shear modulus µ is simplified as [3] where

Numerical implementation
For an egg shape merozoite parametrized by Eq.S3, we numerically calculate the change in the energy of the bilayer/cytoskeleton as a function of wrapping angle θ (Eq.S11).Then, for any given set of constant parameters, we find an angle θ * at which the invasion state becomes an energy minimum.

Incompressible bilayer and cytoskeleton
Let us assume that a flat circular patch of a lipid bilayer and relaxed cytoskeleton with radius (s 0 ) deformed to fit the merozoite contour in the adhesive region.Thus, for an incompressible bilayer/cytoskeleton, the area conservation can be written as Eq. S16 allows us to find s 0 and calculate the extension ratios using Eq.S9b, which simplifies as which is consistent with zero local area strain (α = λ 1 λ 2 − 1 = 0) for an incompressible bilayer/cytoskeleton Additionally, for an incompressible cytoskeleton, the shear modulus µ is simplified as where

Numerical implementation
For an egg shape merozoite parametrized by Eq.S3, we numerically calculate the change in the energy of the bilayer/cytoskeleton as a function of wrapping angle θ (Eq.S11).Then, for any given set of constant parameters, we find an angle θ * at which the invasion state becomes an energy minimum.

Analytical approximations
In this section, we explore the analytical solution for the minimum energy state, ignoring the effects of membrane cytoskeleton energy and modeling the merozoite as a spherical particle with radius a.In this condition, the change in the energy of the system (Eq.S11) can be written as where y = 1 − cos(θ).By taking ∂∆E ∂y = 0, we have Considering our definition for a completely wrapped state (θ * > π/2), we can find the transition condition to the completely wrapped state by setting y = 1 in Eq.S20.Below, we simplified Eq.S20 for different conditions.

• Case 1: Relationship between lipid bilayer tension and adhesion strength
Considering the condition that γ = 0, f = 0, and H 0 = 0, Eq.S20 simplifies as Eq. S21 suggests that the particle can get fully wrapped with increasing adhesion strength.
• Case 2: Relationship between lipid bilayer tension and spontaneous curvature Considering the condition that γ = 0 and f = 0, Eq.S20 gives Based on Eq.S22, when the induced spontaneous curvature is smaller than the curvature of the particle (H 0 < 1/a), the induced spontaneous curvature assists the progress of complete particle wrapping.However, larger spontaneous curvatures (H 0 > 1/a) impede the complete wrapping transition.
• Case 3: Relationship between lipid bilayer tension and interfacial forces As can be seen, Eq.S20 has a symmetric barrier at θ = π/2 (the line tension term vanishes for y = 1).Thus, to find the analytical approximation, we set y = 1 ± ϵ, where ϵ is a small number, and expanded the Eq.S20 until the first order for the case that f = 0 and H 0 = 0 Based on Eqs.S23, in the first half of wrapping (θ < π/2), a line tension prevents the complete membrane wrapping process.However, once the equator is passed (θ > π/2), a line tension accommodates the particle encapsulation.It should be mentioned that with no line tension (γ = 0), a non-wrapped state (θ * = 0) is always a local minimum of ∆E(θ).However, the line tension and actomyosin force energy terms scale as √ y and their derivatives diverge at θ = 0.This means that a line tension can create an energy barrier with no minimum energy state between 0 ⩽ θ ⩽ π in which the particle even does not adhere to the membrane.
• Case 4: Motor forces required for a complete wrapping as a function of membrane physical properties To calculate the minimum motor forces that are required for a complete particle wrapping (based on our definition θ * > π/2), we substitute y = 1 + ϵ in Eq.S20 and find the force density (f ) as The total force in the z direction (F z ) is obtained as Substituting Eq.S24 into Eq.S25 for a complete wrapping condition (y = 1 + ϵ), we have Based on Eq.S26, for a tensionless membrane (σ = 0), with no adhesion energy (ω = 0), no line tension (γ = 0), and no spontaneous curvature (H 0 = 0), a minimum force of F z = 3 pN is required for a complete wrapping of a spherical particle.This is consistent with the calculated magnitude of actomyosin forces required for a merozoite invasion by Dasgupta et al [1].It should be mentioned that Eq.S26 is derived for the minimum axial force needed for a complete invasion.This means if the right hand side of Eq.S26 becomes negative, the physical forces are enough to push the merozoite into the RBC and thus F z = 0.The effects of physical properties of the cytoskeleton on the efficiency of malaria invasion, σ bilayer = 0.63 pN/nm.(A) A discontinuous transition from a completely to a partially wrapped state followed by a continuous transition from a partially to a non-wrapped wrapped state with increasing L 0 from 25 nm to 85 nm.σ = 0.1 pN/nm, ω = 0.8 pN/nm, p = 25 nm, and L max = 200 nm.(B) A continuous transition from a partially to a completely wrapped state followed by a discontinuous transition from a partially to a completely wrapped state with increasing L max from 180 nm to 210 nm.σ = 0.1 pN/nm, ω = 0.8 pN/nm, p = 25 nm, and L 0 = 35 nm.(C) A discontinuous transition from a partially wrapped to a completely wrapped state with increasing the persistence length of spectrin p. σ = 0.1 pN/nm, ω = 0.8 pN/nm, L 0 = 35 nm, and L max = 200 nm.

Figure A :
Figure A: The change in the energy of the RBC bilayer with no cytoskeleton layer as a function of wrapping angle (θ) for a fixed ω = 2.5 pN/nm and three different bilayer tension.The change in the energy is minimized in a completely wrapped state (θ * = π) independent of the magnitude of the bilayer tension.

Figure
Figure B: θ * as a function of interfacial tension for wrapping of an egg-shaped merozoite without the cytoskeleton layer.(A) A discontinuous transition from θ * ∼ 5π/6 to a full wrapped state (θ * = π) with an increase in the magnitude of interfacial tension, ω = 1 pN/nm and σ bilayer = 0.6 pN/nm.(B) A discontinuous transition from θ * ∼ 5π/9 to a non-adhered state with an increase in the magnitude of interfacial tension, ω = 0.4 pN/nm and σ bilayer = 0.6 pN/nm.(C) A discontinuous transition from a partially wrapped state to a non-adhered state with an increase in the magnitude of interfacial tension, ω = 0.5 pN/nm and σ bilayer = 1 pN/nm.

Figure D :
Figure D: Minimum axial force (F z ) required for a complete merozoite entry as a function of (A) bilayer tension, (B) spontaneous tension, (C) adhesion strength, and (D) interfacial tension.p = 25 nm, L 0 = 35 nm, and L max = 200 nm.The gray circles show the results that we obtained from the energy minimization (Eq.S11).The dotted line represents the fitted curves and the solid blue line indicates the analytical approximation for the motor-driven force (Eq.S26).The green arrow demonstrates the increase in the magnitude of the axial force compared to the analytical approximations because of the cytoskeleton resistance against deformation.(A) F z increases as a linear function of bilayer tension.The dashed line shows the linear dependence on the bilayer tension by fitting to a line (Aσ bilayer +B), where A = 2.48 and B = 4.73 with R 2 = 0.99.(B) F z varies as a linear function of spontaneous tension.The dashed line shows a linear dependence on the spontaneous tension by fitting to a line (Aσ spon +B), where A = 5.7, B =-0.09 with R 2 = 0.99.(C) F z decreases as a linear function of adhesion strength.The dashed line shows the linear dependence on the adhesion strength by fitting to the line (Aω+B), where A = -2.93 and B = 7.4 with R 2 = 0.99.(D) Switch-like increases in axial force from F z = 0.52 nN to F z = 0.7 nN with increasing the magnitude of interfacial tension.