1. Physics of Living Systems
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Stochastic bond dynamics facilitates alignment of malaria parasite at erythrocyte membrane upon invasion

  1. Sebastian Hillringhaus
  2. Anil K Dasanna
  3. Gerhard Gompper  Is a corresponding author
  4. Dmitry A Fedosov  Is a corresponding author
  1. Theoretical Physics of Living Matter, Institute of Biological Information Processing and Institute for Advanced Simulation, Forschungszentrum Jülich, Germany
Research Article
Cite this article as: eLife 2020;9:e56500 doi: 10.7554/eLife.56500
9 figures, 3 videos, 2 tables and 1 additional file

Figures

Sketch of parasite and RBC models.

(a) Two-dimensional sketch of a parasite with a directional vector 𝐧 from the parasite’s back at rx=1.5μm to its apex at rx=0. (b) Three-dimensional triangulated surfaces of a RBC (red) and a parasite (blue). Bonds between the parasite and RBC can form within the contact zone which is illustrated by a magnified view, where discrete receptor-ligand interactions (or bonds) are sketched. A bond can form with a constant on-rate kon and break with a constant off-rate koff.

Calibration of parasite adhesion parameters.

(a) A time instance of parasite motion at RBC membrane from an experimental video (Weiss et al., 2015) (top) and simulation (bottom), see also Video 1. To obtain the distribution of merozoite fixed-time displacements, the marked parasite (red circle) is tracked over the course of its interaction with the RBC membrane. (b) Comparison between experimental (20 samples) and simulated (100 samples) fixed-time displacements (Δd) of the parasite at RBC membrane, which is normalized by the effective RBC diameter D0=A0/π calculated from the membrane area A0. By adapting the interaction parameters, the displacement distribution in simulations is calibrated against the experimental distribution. The resulting reference parameters for our model can be found in Table 2. (c) Mean squared displacement (MSD) of a parasite from simulations as a function of time. The black solid line marks a diffusive regime with MSDt. Note the subdiffusive dynamics for short times, less than about 1s.

Figure 3 with 1 supplement
Parasite adhesion to a deformable RBC.

(a) Sketch of apex distance dapex and alignment angle θ. The apex distance dapex is defined as a distance (magenta line) between the parasite’s apex and the closest vertex of RBC membrane. The alignment angle θ corresponds to the angle between the parasite’s directional vector (black arrow) and the normal vector 𝐧face (green arrow) of a triangular face whose center is closest to the apex. Note that the angle π-θ is drawn in the plot. (b and c) Probability distributions of the apex distance dapex/D0 and the alignment angle θ/π. Data are obtained for parameters shown in Table 2, and accumulated starting from an initial adhesion contact (i.e., formation of a few bonds). The dashed line in the apex distance distribution indicates the cutoff 21/6σ of repulsive LJ interactions. Note that a good parasite alignment requires small values of dapex/D0 and values of θ/π close to unity.

Figure 3—figure supplement 1
Parasite adhesion to a rigid RBC (see section ‘Effect of RBC rigidity’).

Probability distributions of (a) the apex distance dapex/D0, and (b) the alignment angle θ/π. The dashed line in the apex distance distribution indicates the cutoff 21/6σ of repulsive LJ interactions.

Comparison of alignment times obtained from direct DPD simulations and MC sampling.

(a) Two-dimensional probability map as a function of dapex and θ. Each bin represents a single alignment state and the color corresponds to probability of that state. The dark green area (dapex/D00.036 and θ/π0.8, compare with Equation 4) represents the criteria for a successful alignment. The black dashed line corresponds to the cutoff 21/6σ of repulsive LJ interactions. (b) Distribution of alignment times ta obtained from 86 statistically independent DPD simulations. ta is defined as a time interval starting from an initial adhesive contact (i.e., formation of a few bonds) to the instance when the alignment criteria for dapex and θ in Equation 4 are met. The average alignment time is equal to ta9.53 s. (c) Alignment time distribution from MC sampling using the probability map in (a). The alignment time is defined as a number of MC steps needed to satisfy the alignment criteria, as the MC procedure does not have an inherit timescale. Note that the sample size in MC modeling (8000 trajectories) is much larger than that in (b).

Figure 5 with 2 supplements
Variations in stretching ΔEsp and bending ΔEbend energies, the number of bonds nb, the head distance dapex, and the alignment angle θ as a function of time for the default parameter set given in Table 2.

Temporal changes in the number of bonds are shown for both long and short bond types. The dashed lines in the bottom plot correspond to the alignment criteria in Equation 4. For all quantities, the corresponding averages and variances represented by box plots are depicted on the right.

Figure 5—figure supplement 1
Dependence of parasite wrapping on the position at RBC membrane.

(a) Average total number of bonds between the merozoite and RBC as a function of the distance dcm between their centers of mass. (b) Illustration of parasite adhesion at the RBC rim (marked by I) and in the dimple (II). The parasite forms more bonds in the dimples (position II) than at the RBC rim (position I).

Figure 5—figure supplement 2
Different alignment characteristics, including (a) deformation energy, (b) number of bonds, (c) apex distance, (d) alignment angle, and (e) fixed-time displacement, for several values of parameter σ which determines the effective membrane thickness.
Figure 6 with 1 supplement
Effect of the off-rate koff on (a) the parasite’s fixed-time displacement, (b) RBC deformation energy, and (c) alignment time.

Since the off-rate controls the lifetime of bonds, a smaller off-rate results in a stronger adhesion, a lower parasite displacement, and a faster alignment time.

Figure 6—figure supplement 1
Effect of the off-rate koff on (a) the apex distance, (b) alignment angle, and (c) the number of bonds.
Figure 7 with 1 supplement
Effect of the extensional bond rigidities on parasite alignment.

(a) RBC deformation energy and (b) the number of short and long bonds as a function of λ/λrefλref corresponds to the reference case with parameters given in Table 2. Note that both λlong and λshort are changed by the same factor with respect to their λref values. Here, the bond kinetic rates are konshort=290.3τ-1, konlong=36.3τ-1, and koff=72.6τ-1.

Figure 7—figure supplement 1
Effect of the extensional bond rigidities on (a) the apex distance, (b) alignment angle, and (c) fixed-time displacement of the parasite.
Figure 8 with 2 supplements
Effect of the density of long ligands ρlong on parasite alignment.

(a) Number of short and long bonds and (b) parasite alignment times as a function of ρlong/ρpara. Note that ρlong+ρshort=ρpara remains constant in all simulations. Here, the bond kinetic rates are konshort=290.3τ-1, konlong=36.3τ-1, and koff=72.6τ-1. In case of ρlong/ρpara=0.1, parasite alignment time could not be computed through the MC sampling, since merozoite alignment has never occurred in direct simulations.

Figure 8—figure supplement 1
Effect of the density of long ligands ρlong on (a) deformation energy, (b) fixed-time displacement, (c) apex distance, and (d) alignment angle.
Figure 8—figure supplement 2
Alignment results of simulations with only long ligands, i.e. for ρlong/ρpara=1.

(a) Deformation energy, (b) the number of bonds, (c) apex distance, (d) alignment angle, and (e) fixed-time displacement of the merozoite for the three cases: (1) ρlong/ρpara=1 and koff/konlong=2, (2) ρlong/ρpara=0.4 and koff/konlong=2 (the reference case), (3) ρlong/ρpara=1 and koff/konlong=0.25.

Effect of RBC membrane rigidity on (a) alignment time and (b) parasite fixed-time displacement for different off-rates koff.

Note that for a rigid RBC with koff/konlong=1, parasite alignment time could not be computed through the MC sampling, as the alignment criteria have never been met in direct simulations.

Videos

Video 1
Parasite motion at the membrane of a deformable RBC for the reference RBC-parasite interactions from Table 2.

koff/konlong=2. See Figure 2a.

Video 2
Parasite adhesion and dynamics on a deformable RBC for a reduced off-rate koff.

koff/konlong=1.

Video 3
Parasite dynamics at the surface of a rigid RBC for the reference RBC-parasite interactions from Table 2.

koff/konlong=2.

Tables

Table 1
Simulation parameters given in both model and physical units.

The effective RBC diameter D0=A0/π sets a basic length, the thermal energy kBT defines an energy scale, and RBC relaxation time τ=ηD03/κ sets a time scale in the simulated system, where A0 is the RBC surface area, κ is the bending rigidity, and η is the fluid dynamic viscosity. The values of bending rigidity κ, 2D shear µ and Young’s Y moduli are chosen such that they correspond to average properties of a healthy RBC. Parameters σ and ϵ correspond to RBC-parasite excluded-volume interactions represented by the purely repulsive LJ potential in Equation 11.

ParameterSimulation valuePhysical value
A0133.5133.5μm2
D0A0/π=6.56.5μm
kBT0.014.282×10-21J
τηD03/κ=725.80.92 s
η1.851×103Pas
κ70kBT3.0×10-19J
µ4.6×104kBT/D024.8μN/m
Y1.82×105kBT/D0218.9μN/m
Npara1230
Nrbc3000
σ0.031D00.2μm
ϵ1000kBT4.282×10-18J
Table 2
List of bond parameters that are used to calibrate displacement of the parasite at the RBC membrane in simulations (see Video 1) against available experimental data (Weiss et al., 2015), as shown in Figure 2b.

The parameter values in simulations are given in terms of the length scale D0, energy scale kBT, and timescale τ=ηD03/κ. The densities of long and short ligands are given in terms of parasite vertex density ρpara270μm2. Note that ρlong+ρshort=ρpara in all simulations.

ParameterSimulation valuePhysical value
efflong0.0154D0100nm
effshort0.0031D020nm
ρlong0.4 ρpara107μm2
ρshort0.6 ρpara161μm2
konlong36.3τ-139.6s1
konshort290.3τ-1317.0s1
koff72.58τ-179.2s1
λlong25.3×105kBT/D020.264pN/nm
λshort8.45×105kBT/D020.0882pN/nm

Data availability

All data generated or analysed during this study are included in the manuscript and supporting files. Source data files have been provided for Figures 2-9, including all figure supplements. Figure 1 is a model schematic, which does not contain any data.

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