Malaria is a mosquito-borne infectious disease caused by a protozoan parasite of the genus Plasmodium. Prior to transmission, the parasite proceeds through both asymptomatic and symptomatic developmental stages in the host (Miller et al., 2002; Cowman et al., 2012; White et al., 2014). After an asymptomatic development stage within the liver, merozoites are released into the bloodstream. They have an egg-like shape with a typical size of approximately 1.5µm (Bannister et al., 1986b; Dasgupta et al., 2017; Dasgupta et al., 2014b). During the blood stage of infection, which is a clinically symptomatic stage, parasites invade healthy red blood cells (RBCs) and multiply inside them. This process aids parasites to evade the immune response. The total life-cycle within each infected RBC lasts for about 48 hr, after which the cell membrane is ruptured and new merozoites are released into the bloodstream.

Invasion of RBCs by parasites is a complex process that involves the following steps: (i) initial random attachment, (ii) reorientation (or alignment) of the apex toward cell membrane, and (iii) formation of a tight junction followed by the final invasion (Koch and Baum, 2016; Cowman and Crabb, 2006). The parasite’s apex contains the required machinery for the invasion process, and thus, apex alignment toward the cell membrane is a necessary step for a successful invasion to follow. Merozoite adhesion to a RBC is facilitated by proteins at the parasite surface which can bind to the cell membrane (Bannister et al., 1986b; Gilson et al., 2006; Beeson et al., 2016). Recent optical tweezers experiments provide an estimation for the force required to detach a parasite adhered to RBC membrane to be in the range of 10–40 pN (Crick et al., 2014). Other experiments (Dvorak et al., 1975; Gilson and Crabb, 2009; Glushakova et al., 2005; Crick et al., 2013) demonstrate that the parasite is dynamic at the RBC membrane, and induces considerable membrane deformation during alignment. Furthermore, there is a positive correlation between such deformations and parasite alignment. The time required for the parasite to align is found to be on the order of 16 s (Weiss et al., 2015). Our recent investigation of the parasite alignment with adhesion modeled by a homogeneous interaction potential has confirmed the importance of membrane deformations for proper alignment, but the alignment times were found to be significantly less than 1 s (Hillringhaus et al., 2019). The main shortcoming of this model is that it produces only static membrane deformations and the parasite exhibits very little dynamics at the RBC surface. This model has been extended by including realistic bond-based adhesion interactions between the parasite and RBC membrane (Hillringhaus et al., 2020), which results in alignment times consistent with the experimental measurements.

A typical merozoite has an egg-like shape with the apical complex sitting on the pointed edge. Our previous work (Hillringhaus et al., 2020) suggests that parasite alignment occurs due to RBC deformability and stochastic fluctuations in bond dynamics. Stochastic fluctuations and consequent rolling-like (or rotational) motion of the parasite at the membrane surface are especially important at low adhesion strengths, as they facilitate alignment toward pointed apex. The egg-like shape naturally adheres to RBC membrane with its less curved side, as this adhesion state corresponds to the largest contact area. Then, a rotational motion of the parasite toward the apex is required to establish an apex-membrane contact. If parasite adhesion interactions with a membrane are strong, merozoite mobility is significantly suppressed, and the alignment is mainly facilitated through wrapping of the parasite by cell membrane, emphasizing the importance of RBC deformability. These are two major mechanisms for the alignment of an egg-like merozoite.

Even though most types of Plasmodium merozoites have an egg-like shape, the merozoite of Plasmodium yoelii changes its shape from an oval to a spherical shape right before its attachment followed by alignment and invasion (Yahata et al., 2012). The alignment time for Plasmodium yoelii is also reported to be longer than for Plasmodium falciparum (Yahata et al., 2012). It is not clear why the Plasmodium yoelii parasite adapts its shape before the alignment process at the RBC membrane. This raises a question whether the parasite shape has important advantages/disadvantages in the alignment process or simply results from the structural organization of its internal elements. Therefore, it is important to understand the effect of parasite shape in the alignment process.

In this article, the role of parasite shape in the alignment process at the RBC membrane is studied by mesoscopic computer simulations. In particular, we show that basic dynamical properties, such as parasite mobility, RBC membrane deformation, and the number of adhesion bonds, are significantly affected by different parasite shapes. In turn, these are tightly coupled to parasite alignment characteristics that determine its alignment success. In general, parasite shapes with large aspect ratios (e.g. oblate and long prolate ellipsoid) are disadvantageous for alignment, as these shapes result in a significant reduction of parasite mobility at the membrane. A spherical parasite is more mobile than an egg-like merozoite, which is advantageous in cases with low adhesion interactions or increased membrane stiffness. However, the spherical shape is disadvantageous for strong adhesion interactions, when parasite mobility is suppressed, as parasite alignment by membrane wrapping is often unsuccessful because the apex may not be within the adhesion area. As a result, the egg-like shape exhibits an alignment performance that is generally not worse than of other studied shapes, but more robust for disparate conditions in parasite adhesion strength and RBC membrane deformability.

To investigate the role of parasite’s shape in the alignment process, five different shapes with varying aspect ratios are chosen: $(i)$ an egg-like (EG) shape that is the typical shape of Plasmodium falciparum merozoite, $(ii)$ a sphere (SP), $(iii)$ a short ellipsoid (SE) whose dimensions are similar to the egg-like shape, $(iv)$ a long ellipsoid (LE), and $(v)$ an oblate (OB) shape, see Figure 1(a). The corresponding maximum and minimum dimensions for these shapes are ${\displaystyle r_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.5\mu \text{m}}$ & ${\displaystyle r_{\mathrm{m}\mathrm{i}\mathrm{n}}=1.08\mu \text{m}}$ for EG, ${\displaystyle r_{\mathrm{m}\mathrm{a}\mathrm{x}}=r_{\mathrm{m}\mathrm{i}\mathrm{n}}=1.2\mu \text{m}}$ for SP, ${\displaystyle r_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.6\mu \text{m}}$ & ${\displaystyle r_{\mathrm{m}\mathrm{i}\mathrm{n}}=1.02\mu \text{m}}$ for SE, ${\displaystyle r_{\mathrm{m}\mathrm{a}\mathrm{x}}=2.4\mu \text{m}}$ & ${\displaystyle r_{\mathrm{m}\mathrm{i}\mathrm{n}}=0.76\mu \text{m}}$ for LE, and ${\displaystyle r_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.5\mu \text{m}}$ & ${\displaystyle r_{\mathrm{m}\mathrm{i}\mathrm{n}}=0.64\mu \text{m}}$ for OB. All shapes are selected such that they have approximately the same surface area and the same number of vertices or equivalently the same density of adhesion receptors. The fraction of receptors that can form long bonds is kept at 0.4 similarly to our previous study (Hillringhaus et al., 2020), while the fraction of receptors for short bonds is equal to 0.6. Other parasite-RBC interaction parameters are calibrated through the displacement of an egg-shaped parasite at RBC membrane (see Figure 1(b) and Video 1) against available experimental data (Weiss et al., 2015). In this calibration procedure, kinetic rates and strength of the bonds are adjusted, so that simulated and experimental translational displacements of the parasite match well, see Table 1 and the Materials and methods section for details. Note that the translational displacement of the parasite results from its stochastic rolling-like motion. The set of calibrated parameters in Table 1 will be referred to as the reference parameter set. All other parameters including bond rates are kept same unless stated otherwise.

Figure 1

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Video 1

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Dynamical properties of different shapes

Figure 1(c-e) presents basic dynamical measures of merozoites with different shapes. These include [Figure 1(c)] fixed-time displacements $\mathrm{\Delta }d$ traveled by the parasite over fixed time intervals of ${\displaystyle \mathrm{\Delta }t=1\mathrm{s}}$ and normalized by an effective parasite diameter $D_{\text{p}}=\sqrt{A_{\text{p}}/\pi }$ (${\displaystyle A_{\text{p}}\simeq 4.6\mu {\text{m}}^2}$ is the parasite membrane area for all shapes), [Figure 1(d)] change in membrane total energy $\mathrm{\Delta }E/k_{\mathrm{B}}T$ due to deformation induced by parasite adhesion, and [Figure 1(e)] the number of bonds $n_{\mathrm{b}}/N_{\mathrm{para}}$. The spherical shape is most mobile (i.e. has the largest $\mathrm{\Delta }d/D_{\text{p}}$, see Video 2), while the oblate shape is slowest with the lowest $\mathrm{\Delta }d$. Intuitively, a shape with a lower local curvature (e.g. OB shape) should form a larger adhesion area, and thus be less dynamic or mobile. This is in agreement with our results in Figure 1(e), where the SP (OB) shape has the smallest (largest) number of bonds, which is directly proportional to the adhesion area. Note that the egg-like and short-ellipsoid shapes show very similar dynamic characteristics, as these shapes are very close to each other. Fixed-time displacement of the long ellipsoid (see Videos 2 and 3) has values between those for the SE and OB shapes, which is consistent with the number of bonds (or equivalently the adhesion area) in Figure 1(e).

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The RBC deformation energy $E/k_{\mathrm{B}}T$ can generally be expected to be proportional to the adhesion area or the number of formed bonds. This is true for the spherical shape that induces the lowest deformation energy in Figure 1(d). Furthermore, both oblate and long ellipsoid shapes result in a large deformation energy. However, the LE shape has a slightly larger value of $\mathrm{\Delta }E$ than the OB shape, even though the oblate shape has a larger adhesion area. This can be rationalized by the fact that the adhesion of the oblate shape to RBC membrane induces a lower deformation than the LE shape, as the OB shape has a lower curvature at its flat side (see Video 3). Furthermore, the exact position where parasite adheres to RBC membrane is important, as local curvature at the membrane (e.g. negative curvature in the dimple areas) can oppositely match the curvature at the parasite surface. This is the main reason why a successful alignment occurs more frequently in the concave areas of RBC dimples than at the convex rim of the membrane (Hillringhaus et al., 2020). Note that the RBC deformation energy in Figure 1(d) displays opposite trends in its shape dependence than the parasite mobility or fixed-time displacement in Figure 1(c). As a result, parasite shapes with a large asphericity are less dynamic at the RBC surface, while the EG, SP, and SE shapes show comparable dynamical characteristics.

Parasite alignment characteristics

To characterize parasite alignment, we introduce two quantities: $(i)$ apex distance d_apex and $(ii)$ alignment angle θ (Hillringhaus et al., 2020) given by

(1) $d_{\mathrm{apex}}=\underset{i}{\min }(|{𝐫}_{\mathrm{apex}}-{𝐫}_i|),\theta =\arccos (𝐧\cdot {𝐧}^{\mathrm{face}}),$

where ${\mathbf{𝐫}}_{\mathrm{apex}}$ is the apex position, ${\mathbf{𝐫}}_i$ is the position of vertex $i$ at the membrane, $\mathbf{𝐧}$ is the parasite’s directional vector pointing from its back to the apex, and ${\mathbf{𝐧}}^{\mathrm{face}}$ is the normal vector of a RBC membrane triangular face whose center of mass is closest to the parasite’s apex. Both the apex distance $d_{\text{apex}}$ and the alignment angle θ are schematically depicted in Figure 2(a). The directional vector is defined for all shapes by selecting two opposite vertices along the shape axis, which represent the apex and the back. A perfect alignment is achieved when the apex distance is equal to the distance at which the repulsive interaction vanishes (i.e. at $\simeq 2^{1/6}\sigma$) and the alignment angle θ is equal to π. Due to limitations in the discretization of both the RBC and the parasite (Hillringhaus et al., 2020), a successful parasite alignment can be characterized by the criteria

(2) $d_{\mathrm{apex}}\le 2^{1/6}\sigma +r_{\mathrm{junc}}\mathit{\hspace{1em}}\&\mathit{\hspace{1em}}\theta \ge 0.8\pi ,$

 where ${\displaystyle r_{\mathrm{j}\mathrm{u}\mathrm{n}\mathrm{c}}=10}$ nm defines the junctional interaction range of the parasite’s apex (Bannister et al., 1986b).

Figure 2

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Figure 2 presents apex-distance and alignment-angle distributions for different parasite shapes, where the alignment criteria from Eq (2) are indicated by the dashed lines. Even though alignment of the SE shape is similar to the EG shape, it is slightly worse for the SE shape as the d_apex distribution in Figure 2(a) is shifted further away from the alignment criterion for d_apex than that for the EG shape. This is due to the fore-aft asymmetry of the EG shape, whose largest-adhesion configuration corresponds to a slightly tilted orientation of the parasite with its apex closer to the membrane than its back. Figure 2(b) and (d) compares the alignment characteristics of the egg-like shape with oblate and long-ellipsoid shapes. The key advantage of the egg-like shape in comparison to LE and OB shapes is that the EG shape has a reduced adhesion area due to a larger local curvature, which allows the EG parasite to fluctuate more around its directional vector, leading to wider distributions of its alignment characteristics. Since the OB shape forms the largest adhesion area in comparison to the EG and LE shapes, it has the narrowest distributions (i.e. lowest fluctuations) for both the apex distance and alignment angle. A further advantage of the EG shape is the aforementioned fore-aft asymmetry, which results in a shift of the apex-distance and alignment-angle distributions toward a better alignment in comparison to those for the OB and LE shapes.

Clearly, parasite alignment characteristics for the spherical shape are qualitatively different from all other (ellipsoid-like) shapes. At first glance, d_apex and θ distributions in Figure 2(d) and (f) seem to suggest that the SP shape might be best for parasite alignment. However, the membrane deformability breaks the ’up-down’ symmetry of these distributions, leading to two distinct cases: $(i)$ the apex is not within the parasite-membrane adhesion area [Figure 2(b)] for which the alignment characteristics are very poor and (ii) the apex is within the adhesion area [Figure 2(c)] resulting in good alignment. Hence, alignment performance of the spherical parasite is more subtle than indicated by the distributions in Figure 2 and will be discussed further below.

Alignment of a spherical parasite

Figure 3 shows apex-distance and alignment-angle distributions of the spherical parasite for different values of the off-rate k_off. Both d_apex and θ distributions generally display a sharp peak near the alignment criteria and a long tail with a wider distribution for non-aligned parasite orientations. Thus, the sharp peak represents parasite orientations when its apex is within the membrane-parasite contact area $A_{\mathrm{m}}$, as schematically illustrated in Figure 2(c). Correspondingly, the long tail characterizes orientations when the parasite’s apex is not within the adhesion area $A_{\mathrm{m}}$, as sketched in Figure 2(b).

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At large values of the off-rate (e.g. $k_{\mathrm{off}}/k_{\mathrm{on}}^{\mathrm{long}}=4.0$), the spherical parasite is very mobile at the membrane surface (i.e. has a large effective rotational diffusion) and induces nearly no membrane deformations. In such cases, there is no significant wrapping of the parasite by the membrane, resulting in wide d_apex and θ distributions in Figure 3. As k_off is decreased, the merozoite becomes partially wrapped by the membrane, leading to the development of the sharp peak in both distributions. At the smallest off-rate of $k_{\mathrm{off}}/k_{\mathrm{on}}^{\mathrm{long}}=0.5$, the alignment properties in Figure 3 seem to be qualitatively different from those for larger off-rates. Note that for small off-rates, the parasite forms a large number of bonds with the membrane, resulting essentially in its arrest with nearly zero rotational diffusion. Therefore, these simulations are too short to fully capture d_apex and θ distributions, which are also expected to have the sharp peak characterized by the wrapped area $A_{\mathrm{m}}$.

To rationalize apex-distance and alignment-angle distributions for a spherical parasite, we use a simple model of a sphere with radius $R$ partially wrapped by the membrane, as illustrated in Figure 2(b) and (c). Since the adhered parasite is mobile, the parasite’s directional vector can point toward any possible direction when sampled over times longer than a characteristic time of parasite rotational motion. Therefore, the probability of alignment can be approximated as $A_{\mathrm{m}}/A_{\mathrm{p}}$, where $A_{\mathrm{m}}$ and $A_{\mathrm{p}}$ are the adhesion area and the total surface area of a sphere, respectively. As a result, the sharp peak in d_apex and θ distributions near the alignment criteria must increase with an increase in adhesion strength or a decrease in k_off. A theoretical model for sphere wrapping based on energy minimization results in Hillringhaus et al., 2019,

(3) $\frac{A_{\mathrm{m}}}{A_{\mathrm{p}}}=2\sqrt{\frac{2}{Y}\left(\frac{\mathrm{\Delta }{U}_{\mathrm{b}}}{A_{\mathrm{c}}}-\frac{2\kappa }{R^2}\right)},$

where $Y$ is the Young’s modulus of the membrane, κ is the bending rigidity, and $\mathrm{\Delta }{U}_{\mathrm{b}}\sim k_{\mathrm{B}}T\ln (k_{\mathrm{on}}/k_{\mathrm{off}})$ is the energy gained through single-bond association per area $A_{\mathrm{c}}$, which can be considered as the effective area of a single bond. Therefore, $A_{\mathrm{m}}$ increases with a decrease in k_off. Furthermore, for small deformations, $A_{\mathrm{m}}$ is essentially governed by the competition of bending and adhesion energies, while for strong adhesion, stretching elasticity of the membrane also becomes important.

When the parasite is not aligned, the apex distance can be approximated by a height of the parasite’s apex with respect to the flat part of the membrane, see Figure 2(b). Note that this assumption becomes strictly valid for a flat membrane without wrapping or a weak adhesion of the merozoite. Then, the probability distribution for the apex distance d_apex is characterized by the area of a spherical segment (or frustum) with height $\mathrm{\Delta }h$ as

(4) $P(d_{\mathrm{apex}})\mathrm{\Delta }h=\frac{A_{\mathrm{\Delta }h}}{A_{\mathrm{p}}}=\frac{2\pi R\mathrm{\Delta }h}{A_{\mathrm{p}}}=\frac{1}{2R}\mathrm{\Delta }h,$

where $\mathrm{\Delta }h$ is an infinitesimally small interval around d_apex. Therefore, $P(d_{\mathrm{apex}})=1/(2R)$ is independent of the apex distance, which is consistent with nearly flat d_apex distributions for $k_{\mathrm{off}}/k_{\mathrm{on}}^{\mathrm{long}}\ge 2.0$ in Figure 3(a) and Figure 2(d). Similarly, the probability distribution $P(\theta )$ can be approximated using segment areas $A_{\mathrm{\Delta }\theta }=2\pi R^2\left(\cos (\theta )-\cos (\theta +\mathrm{\Delta }\theta )\right)\approx 2\pi R^2\sin (\theta )\mathrm{\Delta }\theta$, resulting in

(5) $P(\theta )\approx \frac{\sin (\theta )}{2}$

for not aligned parasite orientations. This approximation is consistent with the data in Figure 3(b) and Figure 2(f). In summary, such unique distributions of alignment properties for the SP shape are possible due to the spherical symmetry. For non-spherical parasite shapes, a sharp peak disappears because parasite adhesion to the membrane favors a specific parasite orientation.

Effect of adhesion strength on parasite alignment time

Figure 4(a) and (b) show apex-distance and alignment-angle properties for different parasite shapes and various off-rates which is used to control the strength of merozoite adhesion to the membrane. The apex distance decreases when the off-rate is decreased or the strength of adhesion is increased. Similarly, the alignment angle increases toward the alignment criterion in Equation (2), as the adhesion strength is increased. For all non-spherical shape cases, successful alignment is generally achieved at low enough k_off values, which imply strong membrane deformations and a significant wrapping of the parasite by the membrane. This is consistent with deformation energies shown in Figure 4(c), which significantly increase with decreasing k_off. The main difference for the spherical parasite is that the best alignment is achieved for intermediate values of off-rates (e.g. $k_{\mathrm{off}}/k_{\mathrm{on}}^{\mathrm{long}}\approx 1.0$). As mentioned before, small values of k_off significantly suppress parasite mobility, which is required for successful alignment of the spherical parasite because its apex may not be immediately within the parasite-membrane contact area after initial adhesion. Interestingly, the OB shape results in a significantly lower deformation energy than other merozoite shapes for $k_{\mathrm{off}}/k_{\mathrm{on}}^{\mathrm{long}}<2.0$. Here, the magnitude of local curvature has a pronounced effect, such that the OB shape forms a large adhesion area over its nearly flat part with very low curvature, while close to the rim, where the curvature is large, adhesion interactions are too weak to induce membrane wrapping and deformation. For the other shapes, the adhesion strength is still sufficient to induce partial wrapping of the parasite by the membrane over moderate curvatures.

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To compute alignment times, we employ Monte Carlo simulations based on $(d_{\mathrm{apex}},\theta )$ probability maps constructed from approximately 10 independent direct simulation trajectories for each parameter set (Hillringhaus et al., 2020). Figure 4(d) shows alignment times ${\tau }_{\mathrm{n}}$ for different parasite shapes and off-rates, where all times are normalized by the alignment time of an egg-like shape for the reference parameter set (Hillringhaus et al., 2020). In some cases, the bars are missing in the plot, indicating that the alignment has not occurred in direct simulations whose maximum time length is about 26 s. Alignment times of the spherical parasite are very long at small off-rates and become comparable with those of the egg-like shape at intermediate and high values of k_off. The SE, LE, and OB shapes generally align very fast at small off-rates, but often do not align at all when adhesion becomes weak. This means that these spheroidal shapes require substantial membrane deformation for a successful alignment.

Alignment at a rigid RBC

To understand the importance of RBC deformability in the alignment process for different parasite shapes, we have simulated parasite alignment at a rigid RBC. Figure 5(a) and (b) presents fixed-time displacement and alignment time for different parasite shapes and two off-rates. Generally, a small fixed-time displacement (or low mobility at the membrane) results in a long alignment time and vise versa. Both long-ellipsoid and oblate shapes do not align or have a very long alignment time at a rigid membrane, as they require considerable amount of membrane deformation for the alignment. Both egg-like shape and short ellipsoid exhibit similar fixed-time displacements and alignment times. However, for the egg-like shape, an increase in off-rate (i.e. more mobility) improves the alignment whereas for the short ellipsoid an opposite trend is observed. The spherical parasite shows the fastest alignment in comparison to the egg-like and short ellipsoid shapes due to its increased mobility. Thus, at a rigid RBC, the spherical shape shows best alignment properties, at least for intermediate off-rate values.

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We have investigated the importance of merozoite shape for its alignment at the RBC membrane which is a prerequisite for the invasion process. This study is a continuation of our previous work (Hillringhaus et al., 2020) where the alignment of an egg-like parasite, a natural shape of merozoite, was investigated. Motivated by experimental observations by Bannister et al., 1986a, adhesion between the RBC membrane and the parasite is implemented by discrete bonds of two different types, with long and short interaction ranges. The density of both long and short bonds, their kinetic rates and extensional rigidities are calibrated through fixed-time displacement of an egg-like shaped parasite against available experimental data (Weiss et al., 2015). Alignment times from two independent experiments are found to be 16 s (Weiss et al., 2015) and 7-44 s (Yahata et al., 2012), respectively. For the egg-like shape, an average alignment time of ${\displaystyle 10s}$ was obtained in our simulations (Hillringhaus et al., 2020).

To study the effect of parasite shape on alignment, five different parasite shapes, including the original egg-like (EG) shape, short ellipsoid (SE), sphere (SP), long ellipsoid (LE), and oblate (OB) shapes, were considered. The question ‘Which parasite shape performs best for apex alignment and potential invasion?’ does not have a unique answer, as parasite performance also depends on the membrane properties and the characteristics of the adhesion bonds, including their dynamics. In general, some parasite shapes are advantageous when the binding kinetics are slow or large RBC membrane deformations take place, while other shapes are advantageous in case of fast binding kinetics or small membrane deformations.

One of our key results is that the spherical parasite exhibits apex-distance and alignment-angle distributions different from those for non-spherical shapes. Distributions of alignment characteristics for the SP shape generally have a sharp peak near the alignment criteria representing parasite orientations with its apex within the adhesion area. At small off-rates or for strong adhesion interactions, there is a considerable parasite wrapping by the membrane and parasite mobility is suppressed. In this case, a successful alignment of the spherical parasite occurs only if the apex ends up directly within the wrapped part of the membrane. This means that the SP shape exhibits ‘all or nothing’ alignment behavior at small off-rates. In contrast, the egg-like shape adheres with its side to the membrane, and is able to establish a direct membrane-apex contact due to significant parasite wrapping. On the other hand, the SP shape performs better than the EG shape at large values of off-rates when membrane deformation is almost negligible. The spherical symmetry of the SP shape results in its faster mobility in comparison with the egg-like shape. Furthermore, a fluctuation of the EG parasite toward successful alignment due to adhesive dynamics is associated with a larger energetic barrier in comparison to the SP shape for which all directions of motion are statistically equivalent. This is also the main reason why the spherical shape leads to the fastest alignment at a rigid membrane in comparison to all other shapes. Interestingly, even though most types of Plasmodium merozoites have an egg-like shape, Plasmodium yoelii transforms into a spherical shape from an egg-like shape after the egress from an infected RBC (Yahata et al., 2012). This shape transition seems to be essential for the successful invasion. A plausible hypothesis is that Plasmodium yoelii exhibits a rather weak adhesion to RBCs in comparison to Plasmodium falciparum, so that the spherical shape becomes advantageous for the alignment process. However, we cannot exclude the possibility that this shape transition is just a result of some internal processes such as cytoskeleton rearrangement, in preparation of the merozoite for a subsequent invasion.

Short ellipsoid geometrically resembles the EG shape except that the egg-like shape has asymmetric ends along the cylindrical axis. Therefore, the SE shape shows alignment characteristics that are closest to the EG shape. However, the alignment of the SE shape is slightly worse than that of the egg-like parasite, as the alignment-angle distribution for short ellipsoid is shifted further away from the alignment criterion in comparison to the egg-like shape. This is due to the asymmetry of EG shape along the cylindrical axis, which favors an adhesion orientation tilted toward the apex. At low enough bond off-rates or strong adhesion interactions, alignment of the SE shape is faster than the egg-like shape, as the wrapping of more curved apex region of the EG parasite is slightly less energetically favorable than that of a symmetric SE shape. Furthermore, alignment of oblate and long-ellipsoid shapes proceeds only through significant wrapping of the parasite by the membrane, which occurs only at low off-rates or for strong adhesion interactions. This is also evident from simulations at a rigid RBC, where alignment times of the LE and OB shapes are either very long or no alignment potentially occurs. Despite the fact that the egg-like shape has some advantages over the investigated spheroidal shapes, it is not clear whether this asymmetry exists simply due to the internal parasite structure (e.g. placement of essential organelles of different sizes) or has some functional importance.

Finally, apical alignment at the RBC membrane is followed by parasite invasion, which requires the formation of a tight junction. During invasion, the tight junction is formed at the apical end and moves toward the back of the parasite with the aid of the actomyosin machinery (Keeley and Soldati, 2004; Robert-Paganin et al., 2019; Cowman and Crabb, 2006). Even though the invasion includes mainy mechanochemical processes, parasite shape must play an important role, as it significantly affects the energy required to deform RBC membrane. For instance, particles with a larger aspect ratio such as oblate and long ellipsoids require a larger energy for complete wrapping (or uptake) (Bahrami et al., 2014; Dasgupta et al., 2014a; Dasgupta et al., 2017). From a dynamical perspective, fluctuations that are important for particle uptake also depend on the geometry of a particle (Frey et al., 2019). Note that particle uptake studies are performed majorly for vesicles, while RBCs possess shear elasticity in addition to membrane bending rigidity. For instance, Hillringhaus et al., 2019 show that for small interaction strengths, bending energy has a dominant contribution to membrane deformation energy, while for strong interactions shear elastic energy exhibits a more dominant contribution. Different aspects related to the performance of various parasite shapes for RBC invasion clearly require further investigations.

All data generated or analysed during this study are included in the manuscript and supporting files. Source data for all figures are provided.

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Acceptance summary:

This manuscript studies the alignment of malaria parasites (merozoites) at the surface of red blood cells (RBCs), a key element of their reproduction cycle during the blood stage of the disease. Building on a computational model the authors developed previously that incorporates the stochastic nature of RBC deformations and adhesive bonds between the merozoite and RBC, it is demonstrated that parasite shape plays a key role in its alignment dynamics. The authors shed new light on the egg-like shape typically observed in Plasmodium merozoites, which has important implications for how effectively the parasite can survive and multiply.

Decision letter after peer review:

Thank you for submitting your article "Effect of malaria parasite shape on its alignment at erythrocyte membrane" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Suzanne Pfeffer as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Michael Gomez (Reviewer #1).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1. Figure 1a: It would be helpful to plot the egg-like shape here too, so it can be easily compared to the other shapes (including the short-ellipsoid, to which it is most similar). Also, highlighting the apex on each shape would be helpful. (There is also the typo "elliposid".)

2. Line 106: It would be good to refer to the details of the computational model in the Methods section at this point.

3. Line 110 (and Figures 1c, 2a-b, 3a, 4a, 5a): Is there a particular reason why the effective RBC diameter D₀ is used to normalize the fixed-time displacement Δd and apex distance d_apex when presenting the results? Since the RBC is much larger than the parasite, this means the normalized values are all much smaller than unity. A more informative choice might be to normalize by the effective merozoite diameter, equal to the square root of A_s/π where A_s is the typical merozoite surface area (which is precisely 2R in the case of a sphere). The normalized values of Δd would then give a better indication of how much the merozoites move relative to their size, and the normalized values of d_apex would lie in the range [0,1].

4. Lines 131-140: It would be helpful to have a schematic illustrating the quantities n, n_face, d_apex and θ, similar to Figure 3a in Hillringhaus et al., 2020. There should also be a brief description of where the alignment criteria in Equation 2 come from (in addition to referencing Hillringhaus et al., 2020), including the meaning of the 2^1/6σ term and that d_apex cannot obtain values below the repulsion length.

5. Lines 145-146: The authors should be more precise here as to what features of the alignment-angle distributions make the egg-like shape align better than the LE and OB shapes. The LE and OB shapes have a narrower distribution with a peak at θ/π ≈ 0.6, which presumably corresponds to the configuration of largest adhesion area in which the apex is pointing almost tangentially to the membrane. The tapering of the egg-like shape breaks the fore-aft symmetry and tilts the apex towards the membrane in the configuration of largest adhesion area.

6. Lines 146-150 and Figures 2a,c: It is worth emphasising here that membrane deformability is what breaks the rotational symmetry for a spherical shape, so that the alignment-angle distribution is not uniform: if the apex is within the contact area A_m, then the deformation of the membrane will push θ closer to π.

7. Line 156: The reference to the inset of Figure 3a here is confusing, since the situation sketched there (with the apex away from the contact region) is not what is being discussed in the text. It might be helpful to have a sketch of both scenarios (i.e. apex inside and outside the contact area), which could be combined with the new schematic suggested in comment 4 above.

8. Lines 177-189: The area being described in Equation 3 is not quite clear: it is worth mentioning the area is what is called a spherical segment/frustum, whose curved surface area depends only on the sphere radius and the height h of the segment (not its slanted length).

9. Figure 4a: The label on the y-axis should be d_apex/D₀.

10. Lines 260-261: Is it possible to speculate as to why the merozoite of Plasmodium yoelii changes its shape from oval to spherical prior to attachment to a RBC membrane? For example, could it be that it has much weaker adhesive bonds (consistent with the observation that the alignment times are longer than those of Plasmodium falciparum), so that it is always in the high-mobility/low-adhesion regime in which a spherical shape is advantageous?

11. A primary concern with this manuscript is the lack of detail on what is done in the simulations. While we know that the authors have published a detailed description of the method elsewhere, it is not sufficient to just cite that, requiring a reader to read a separate paper to even get a basic understanding of what goes on in the simulations. While we understand that the authors do not want to provide the full description here, they need to provide enough description that a reader is aware of the basic implementation and can look to the other publication to fill in details, if needed.

12. As is, the authors state that the RBC and the merozoite are handled using triangulated meshes, that adhesion molecules are located at the vertices of these meshes, and that there is an external fluid that is simulated using dissipative particle dynamics. How is the elasticity of the RBC handled? How is the rigidity of the merozoite handled? We would imagine that the alignment time, etc depends on where on the RBC the merozoite is, since the RBC membrane does not have constant curvature. Are all simulations started from the same location on the RBC surface or are these randomly assigned? How might either of these choices affect the results? Basic parameters of the DPD should also be included.

Reviewer #1:

This manuscript studies the invasion of red blood cells (RBCs) by malaria parasites (merozoites), a key element of their reproduction cycle during the blood stage of the disease. For successful invasion to occur, the merozoite must first align its apex almost perpendicularly to the RBC membrane. In a previous study (reference Hillringhaus et al., 2020), the authors developed a computational model that incorporates stochastic deformations of the RBC membrane and the discrete nature of the adhesive bonds between the merozoite and RBC (arising from filaments on the merozoite surface); together these effects enable partial wrapping of the membrane around the merozoite to aid alignment. This manuscript builds upon this framework to examine the influence of the parasite shape on the alignment dynamics, using five different reference shapes: the egg-like shape typical of Plasmodium merozoites, a sphere, and ellipsoids of varying aspect ratio. By exploring the influence of various parameters such as the bond kinetics and RBC membrane stiffness, they demonstrate that the parasite shape plays a key role in its alignment dynamics. In particular, the egg-like shape is found to be more robust to different adhesion strengths and membrane deformability: it is relatively mobile compared to the ellipsoidal shapes and, unlike a sphere, does not easily become arrested in the high-adhesion limit due to its lack of spherical symmetry.

The manuscript is excellently written and discusses the simulation results clearly and succinctly. The resolution of the simulations is very impressive and yields unprecedented insight into the effect of merozoite shape on alignment dynamics, which has important implications for how effectively the parasite can survive and multiply. The conclusions reached by the authors are certainly justified by the simulation data. In particular, the authors are careful not to draw conclusions beyond the limits of their study, and acknowledge other factors which may influence the merozoite shape, such as internal structural constraints and the energy of invasion following successful alignment.

Regarding weaknesses of the manuscript, some of the explanations of the trends observed in the simulation data could be expanded slightly, to help gain a deeper understanding of the competition between adhesion and RBC deformability underlying the alignment dynamics. These are described in more detail below.

1. Line 114 and lines 120-129: The discussion here of the trends observed in Figure 1 (including why the LE shape has a larger energy compared to the OB shape despite having a smaller adhesion area) is somewhat vague and should be developed further. For example, currently there is only a video showing the egg-like shape and a second video comparing the LE shape to a spherical shape – it would be helpful to have a further video comparing the LE and OB shapes and the different RBC deformations they cause. Moreover, the explanation of the energy/mobility of each shape in terms of curvatures (e.g. the OB shape having "lower curvature at its flat side") could be made more precise. I would expect that the adhesion area depends on how close the principal curvatures of the merozoite surface are to being equal and opposite to the natural curvatures of the RBC, since this determines the bending energy associated with wrapping the merozoite and forming short bonds. This would explain why the spherical shape is most mobile (its principal curvatures are constant so there is no region where at least one is relatively small), and why alignment is most likely to occur in the dimple of the RBC where the membrane is naturally concave-outward. For a given adhesion area, the deformation energy should depend on the difference in principal curvatures in the contact region, with a larger difference causing more bending of the RBC membrane. This difference is larger for the LE shape, since one principal curvature remains large at each point on the surface, compared to the OB shape whose principal curvatures are both small on the 'flat side' where contact is most likely to occur.

2. Lines 175-176: Given that the ratio A_m/A_s (adhesion area to total surface area) plays a key role in the probability of alignment, the authors should be more quantitative at this point. How does the ratio A_m/A_s (as measured directly, or indirectly e.g. by the area under the probability distributions inside the alignment region in figures 3a,b) scale with the system parameters, such as the adhesion strength and the off-rate k_off? Can it be estimated from an energy balance between RBC bending/stretching and the average adhesion energy?

3. Line 197-198 and Figure 4c: Why is the deformation energy associated with the OB shape much lower than all other shapes for values of ${\displaystyle k_{\mathrm{o}\mathrm{f}\mathrm{f}}/k_{on}^{long}<2}$?4. Alignment requires that the distance between the merozoite apex and RBC membrane is very small, and the alignment criteria necessitate examining small changes in the apex angle \theta from \pi. Can the authors comment on how sensitive are the results to the numerical discretisation used?

Reviewer #2:

This manuscript seeks to determine the role that malarial shape plays in the ability of this parasite to infect red blood cells. The authors use computational modeling to explore the dynamics of different parasite shapes and the effect of adhesion strength in getting the malaria parasite to bind into the correct orientation for invasion into the red blood cell.

A major strength of the results is that it investigates an unstudied problem in malarial pathogenesis. The results pertaining to adhesion strength may be informative for preventing the organism from invading red blood cells. A primary weakness is that there is too little detail provided in the methods for this reviewer to adequate assess the computational method. Secondly, the results are somewhat inconclusive. While the egg-shape performs better than certain other shapes, there is no clear final understanding why this shape is preferred over the spherical or short ellipsoidal shapes. However, this possibly provides some clues as to why a certain malarial species does actively adopt a spherical shape during red blood cell binding and invasion.

Overall, the authors achieved their aims by quantitatively assessing the effect of parasite shape and adhesion strength on cell alignment, which is a proxy for invasion. The discussion at the end of the manuscript provides an accurate evaluation of the results that puts them into the context of invasion.

While to some extent the results presented here are inconclusive, I do think that this paper achieves an important goal for its field. This is an understudied area pertinent to a major disease. This manuscript has the potential to bring questions of the biophysics of malarial invasion out to the broader community, specifically introducing these questions to biophysicists as well as microbiologists. Furthermore, the results naturally lead to new questions. If the spherical and egg shapes do not confer a strong advantage, then these specific shapes must also play a role in other processes. The authors do suggest some possibilities in the Discussion. That their remain interesting questions is a great spur for future work.

Essential revisions:

1. Figure 1a: It would be helpful to plot the egg-like shape here too, so it can be easily compared to the other shapes (including the short-ellipsoid, to which it is most similar). Also, highlighting the apex on each shape would be helpful. (There is also the typo "elliposid".)

We have modified Figure 1 and included the egg-like shape along with the other shapes. The position of the apex is also indicated now for all shapes. We thank the reviewer for spotting the typo, which has been corrected.

2. Line 106: It would be good to refer to the details of the computational model in the Methods section at this point.

We have added a reference to the Methods section there.

3. Line 110 (and Figures 1c, 2a-b, 3a, 4a, 5a): Is there a particular reason why the effective RBC diameter D₀ is used to normalize the fixed-time displacement Δd and apex distance d_apex when presenting the results? Since the RBC is much larger than the parasite, this means the normalized values are all much smaller than unity. A more informative choice might be to normalize by the effective merozoite diameter, equal to the square root of A_sπ where A_s is the typical merozoite surface area (which is precisely 2R in the case of a sphere). The normalized values of Δd would then give a better indication of how much the merozoites move relative to their size, and the normalized values of d_apex would lie in the range [0,1].

We agree with the reviewer that a normalization with the parasite size might be more intuitive for these quantities. Therefore, both the translational displacement and apex distance are normalized now with the merozoite’s diameter instead of RBC diameter and the corresponding figures and text were modified. We have used the surface area of an egg-like shape for the normalization of both quantities for all shapes, as they have approximately the same surface area. As expected, now the major part of d_apex distributions is mainly within the range of [0,1]. Note that d_apex normalized with the merozoite diameter can take values slightly larger than unity.

4. Lines 131-140: It would be helpful to have a schematic illustrating the quantities n, n_face, d_apex and θ, similar to Figure 3a in Hillringhaus et al., 2020. There should also be a brief description of where the alignment criteria in Equation 2 come from (in addition to referencing Hillringhaus et al., 2020), including the meaning of the 2^1/6σ term and that d_apex cannot obtain values below the repulsion length.

We have added both a sketch and the corresponding text, providing more details on the alignment characteristics and criterion in the manuscript.

5. Lines 145-146: The authors should be more precise here as to what features of the alignment-angle distributions make the egg-like shape align better than the LE and OB shapes. The LE and OB shapes have a narrower distribution with a peak at θ/π ≈ 0.6, which presumably corresponds to the configuration of largest adhesion area in which the apex is pointing almost tangentially to the membrane. The tapering of the egg-like shape breaks the fore-aft symmetry and tilts the apex towards the membrane in the configuration of largest adhesion area.

We completely agree with the reviewer about this argument. We have improved the discussion of figures 2 (b) and (d) to make this point clear.

6. Lines 146-150 and Figures 2a,c: It is worth emphasising here that membrane deformability is what breaks the rotational symmetry for a spherical shape, so that the alignment-angle distribution is not uniform: if the apex is within the contact area A_m, then the deformation of the membrane will push θ closer to π.

Thank you for emphasizing this. The relevant text has been added to the manuscript.

7. Line 156: The reference to the inset of Figure 3a here is confusing, since the situation sketched there (with the apex away from the contact region) is not what is being discussed in the text. It might be helpful to have a sketch of both scenarios (i.e. apex inside and outside the contact area), which could be combined with the new schematic suggested in comment 4 above.

We have added two sketches of a spherical parasite, interacting with the membrane and representing the two cases of apex orientation, namely within the adhesion area and away from it. These sketches are combined with the sketch that the referee suggested in question 4.

8. Lines 177-189: The area being described in Equation 3 is not quite clear: it is worth mentioning the area is what is called a spherical segment/frustum, whose curved surface area depends only on the sphere radius and the height h of the segment (not its slanted length).

We mention it now explicitly in the text.

9. Figure 4a: The label on the y-axis should be d_apex/D₀.

We have corrected it.

10. Lines 260-261: Is it possible to speculate as to why the merozoite of Plasmodium yoelii changes its shape from oval to spherical prior to attachment to a RBC membrane? For example, could it be that it has much weaker adhesive bonds (consistent with the observation that the alignment times are longer than those of Plasmodium falciparum), so that it is always in the high-mobility/low-adhesion regime in which a spherical shape is advantageous?

We are not aware any data which would directly support a lower adhesion of Plasmodium yoelii in comparison to Plasmodium falciparum. Nevertheless, we think that this is an interesting idea and we have added discussion about a possible advantage of the spherical shape for Plasmodium yoelii.

11. A primary concern with this manuscript is the lack of detail on what is done in the simulations. While we know that the authors have published a detailed description of the method elsewhere, it is not sufficient to just cite that, requiring a reader to read a separate paper to even get a basic understanding of what goes on in the simulations. While we understand that the authors do not want to provide the full description here, they need to provide enough description that a reader is aware of the basic implementation and can look to the other publication to fill in details, if needed.

We have significantly expanded the Materials and methods section, which should now contain all necessary model details.

12. As is, the authors state that the RBC and the merozoite are handled using triangulated meshes, that adhesion molecules are located at the vertices of these meshes, and that there is an external fluid that is simulated using dissipative particle dynamics. How is the elasticity of the RBC handled? How is the rigidity of the merozoite handled? We would imagine that the alignment time, etc depends on where on the RBC the merozoite is, since the RBC membrane does not have constant curvature. Are all simulations started from the same location on the RBC surface or are these randomly assigned? How might either of these choices affect the results? Basic parameters of the DPD should also be included.

The modeled RBC has an in-plane shear elasticity and bending resistance, while the merozoite is treated as a rigid body. We have added necessary details to the Materials and methods section. Details about the DPD method have also been added to the Materials and methods section.

For all simulations, we initially place the parasite close to RBC surface (at the side) with the back side of the parasite (i.e. the apex is oriented away from the membrane). The alignment time indeed depends on the initial placement, which has been discussed in our previous article Hillringhaus et al. eLife (2020). In this manuscript, our main focus was to look at the effect of the parasite shape, which already leaves us with a relatively large parameter space.

Reviewer #1:

al.[…] Regarding weaknesses of the manuscript, some of the explanations of the trends observed in the simulation data could be expanded slightly, to help gain a deeper understanding of the competition between adhesion and RBC deformability underlying the alignment dynamics. These are described in more detail below.

1. Line 114 and lines 120-129: The discussion here of the trends observed in Figure 1 (including why the LE shape has a larger energy compared to the OB shape despite having a smaller adhesion area) is somewhat vague and should be developed further. For example, currently there is only a video showing the egg-like shape and a second video comparing the LE shape to a spherical shape – it would be helpful to have a further video comparing the LE and OB shapes and the different RBC deformations they cause. Moreover, the explanation of the energy/mobility of each shape in terms of curvatures (e.g. the OB shape having "lower curvature at its flat side") could be made more precise. I would expect that the adhesion area depends on how close the principal curvatures of the merozoite surface are to being equal and opposite to the natural curvatures of the RBC, since this determines the bending energy associated with wrapping the merozoite and forming short bonds. This would explain why the spherical shape is most mobile (its principal curvatures are constant so there is no region where at least one is relatively small), and why alignment is most likely to occur in the dimple of the RBC where the membrane is naturally concave-outward. For a given adhesion area, the deformation energy should depend on the difference in principal curvatures in the contact region, with a larger difference causing more bending of the RBC membrane. This difference is larger for the LE shape, since one principal curvature remains large at each point on the surface, compared to the OB shape whose principal curvatures are both small on the 'flat side' where contact is most likely to occur.

We have expanded the discussion of these results to make it clearer. Furthermore, a new video was generated to visually see differences between different shapes.

2. Lines 175-176: Given that the ratio A_m/A_s (adhesion area to total surface area) plays a key role in the probability of alignment, the authors should be more quantitative at this point. How does the ratio A_m/A_s (as measured directly, or indirectly e.g. by the area under the probability distributions inside the alignment region in figures 3a,b) scale with the system parameters, such as the adhesion strength and the off-rate k_off? Can it be estimated from an energy balance between RBC bending/stretching and the average adhesion energy?

A change in A_m as a function of adhesion strength can be estimated analytically for a sphere, as was done in Hillringhaus et al. Biophys. J. 117:1202, 2019. For small deformations, there is essentially a competition of bending and adhesion energies, while for strong adhesion, stretching-elasticity contribution becomes important. We have included this theoretical result into the manuscript and discuss its implications.

3. Line 197-198 and Figure 4c: Why is the deformation energy associated with the OB shape much lower than all other shapes for values of ${\displaystyle k_{\text{off}}/k_{\text{on}}^{\text{long}}<2}$?

For ${\displaystyle k_{\text{off}}/k_{\text{on}}^{\text{long}}<2}$, the magnitude of local curvature has a pronounced effect. For the OB shape, a large adhesion area is formed over the area with very low curvature, and close to the rim where the curvature is large, the adhesion strength may not be strong enough to induce membrane wrapping and deformation. For other shapes, the adhesion strength is large enough to lead to partial wrapping of the parasite by the membrane over moderate curvatures. As a result, the integrated deformation energy is significantly lower for the OB shape than for the other shapes in this regime of adhesion strengths. We have added this clarification to the manuscript.

4. Alignment requires that the distance between the merozoite apex and RBC membrane is very small, and the alignment criteria necessitate examining small changes in the apex angle \theta from \pi. Can the authors comment on how sensitive are the results to the numerical discretisation used?

The discretization length does affect the tightness of the alignment criteria. In our simulations, the average discretization length of the RBC membrane is about l₀=0.2 μm. The half circumference length of a parasite (corresponding to angle π) is πR, which is equal to about 12 l₀ for R=0.75 μm, such that our angle resolution with respect to the parasite size is 0.1π. Therefore, we use 0.2π for the alignment criteria, which is large enough to avoid strong discretization effects. Simulations with a finer discretization are possible, but they become very expensive computationally.

Reviewer #2:

This manuscript seeks to determine the role that malarial shape plays in the ability of this parasite to infect red blood cells. The authors use computational modeling to explore the dynamics of different parasite shapes and the effect of adhesion strength in getting the malaria parasite to bind into the correct orientation for invasion into the red blood cell.

A major strength of the results is that it investigates an unstudied problem in malarial pathogenesis. The results pertaining to adhesion strength may be informative for preventing the organism from invading red blood cells. A primary weakness is that there is too little detail provided in the methods for this reviewer to adequate assess the computational method. Secondly, the results are somewhat inconclusive. While the egg-shape performs better than certain other shapes, there is no clear final understanding why this shape is preferred over the spherical or short ellipsoidal shapes. However, this possibly provides some clues as to why a certain malarial species does actively adopt a spherical shape during red blood cell binding and invasion.

We thank the reviewer for a positive judgment of our manuscript. We have significantly expanded the methods section, so it should contain now all necessary simulation details. We agree with the reviewer that the conclusions about shape advantages/disadvantages are equivocal to some extent, but this is exactly what our simulation data show. However, from our data it is clear that the two shapes (i.e. egg-like and sphere) stand out, and they also correspond to real examples of merozoite shapes. As the reviewer points out, we do discuss some clues for the importance of parasite shape in the alignment process.

Overall, the authors achieved their aims by quantitatively assessing the effect of parasite shape and adhesion strength on cell alignment, which is a proxy for invasion. The discussion at the end of the manuscript provides an accurate evaluation of the results that puts them into the context of invasion.

While to some extent the results presented here are inconclusive, I do think that this paper achieves an important goal for its field. This is an understudied area pertinent to a major disease. This manuscript has the potential to bring questions of the biophysics of malarial invasion out to the broader community, specifically introducing these questions to biophysicists as well as microbiologists. Furthermore, the results naturally lead to new questions. If the spherical and egg shapes do not confer a strong advantage, then these specific shapes must also play a role in other processes. The authors do suggest some possibilities in the Discussion. That their remain interesting questions is a great spur for future work.

Thank you for emphasizing the importance of multidisciplinarity. We also hope that our work will ignite interest in different communities, as only a multidisciplinary effort can bring us much closer to understanding of parasite alignment and invasion, which clearly include a combination of different mechanical and biochemical processes.

Author details

1. Anil K Dasanna

   Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   Contribution

   Software, Formal analysis, Investigation, Visualization, Methodology, Writing - original draft

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-5960-4579

2. Sebastian Hillringhaus

   Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   Contribution

   Software, Formal analysis, Investigation, Methodology

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-0100-9368

3. Gerhard Gompper

   Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   Contribution

   Conceptualization, Project administration, Writing - review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-8904-0986

4. Dmitry A Fedosov

   Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   Contribution

   Conceptualization, Supervision, Project administration, Writing - review and editing

   For correspondence

   d.fedosov@fz-juelich.de

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-7469-9844

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