Mechanical models for malaria infected erythtocytes 1

The effect of shear modulus and bending stiffness on the erythrocyte deformability was analyzed using the two-component model.. The model was able to predict the cell deformation in both

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MECHANICAL MODELS FOR MALARIA INFECTED

ERYTHROCYTES

JIAO GUYUE

(B.Sc., FUDAN UNIVERSITY)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009

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Summary

Mechanical properties play an important role in the physiology of cells For example, human erythrocytes play a crucial role in oxygen exchange in the human body and an important property of normal human erythrocyte is that it can undergo extremely large deformation while passing through small capillaries However, when erythrocytes

are infected by malaria parasites such as Plasmodium falciparum (P.f.), it is extremely

deadly due to its ability to cause cerebral malaria It is believed that the increase in

stickiness and decrease in deformability of the malaria (P.f.) infected erythrocytes will

lead to capillary blockage and obstruction of blood flow Hence in this thesis, a component model and a multi-component model were developed to quantitatively

two-investigate the decrease in deformability of malaria (P.f.) infected erythrocytes

The elastic rigidity of the cell membrane is associated to the change in free energy caused by both the stretching and bending of the erythrocyte membrane Finite element analysis was done using the finite element program ABAQUS to simulate erythrocyte deformation in micropipette aspiration and optical tweezers stretching The effect of shear modulus and bending stiffness on the erythrocyte deformability was analyzed using the two-component model The model was able to predict the cell deformation in both micropipette aspiration and optical tweezers stretching By comparing simulation results with experimental results, the model was also able to quantify the increase of shear modulus with the progression of disease stage The numerical results were found to be

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insensitive to the mesh parameter changes The effect of bending stiffness on the deformation of the cell, ranged from 3.3 x 10 -20 J to 1.5 x 10 -18 J, which covered the range of bending stiffness reported by other researchers, was also quantified using the two-component model

The two-component model proposed in this thesis was also able to study the validity of the commonly used hemispherical cap model The hemispherical cap model was popular in analyzing cell deformation in micropipette aspiration, due to its simplicity

in calculating membrane shear modulus The model and method proposed in this thesis allowed us to test the validity of hemispherical cap model by applying the model to analyze the simulation curves with known values of membrane shear modulus, and to obtain the valid range of pipette radius used in micropipette aspiration as a function of cell radius

The multi-component model proposed in this thesis was developed based on the two-component model It was able to quantitatively analyze the effect of parasitophorous vacuole membrane (PVM) which enclosed the parasite within the host cell, on cell deformation in micropipette aspiration and optical tweezers stretching

It is hoped that the computational models proposed in this thesis will contribute to

more accurate evaluations of the mechanical properties of malaria (P.f.) infected

erythrocytes and a better understanding of the underlying pathophysiology

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Acknowledgements

This thesis involves the joint efforts of many people First, I would like to thank my supervisors Prof Lim Chwee Teck and Associate Prof Sow Chorng Haur for their help in improving my scientific thinking and writing They introduced me to the research field in biomechanics, and provided me with ample freedom in pursuing

my own research interest

I am also grateful to all the people from Nano Biomechanics Lab: Zhou Enhua,

Li Ang, Fu Hongxia, Vedula S.R.K., Eunice Tan, Hairul N.B.R., Kelly Low, Tan Lee Ping, Chong Ee Jay, Li Qingsen, Shi Hui, Ng Sin Yee, Anthony Lee, Cheng Tien Ming (National Taiwan University), Zhong Shaoping, Zhang Yousheng, Sun Wei, Yuan Jian, Tan Swee Jin, Hou Han Wei, Earnest Mendoz, Yow Soh Zeom and Lim Tong Seng Without their kind encouragement and support, I could not have finished the work in this thesis I would like to especially thank Rosemary Zhang and Lee Yeong Yuh for sharing their experimental data

My sincere thank also goes to all the people in SMART Infectious Diseases group, especially to Dr Ming Dao who instructed me and offered great help in my research works I also want to express my heartfelt thanks to Sandra Ho and Lena Lui for their warm friendship and advice in scientific writing

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I would also like to thank all the staff from Dr Kevin S.W Tan’s lab in Department of Microbiology, especially Yin Jing, Alvin Chong and Vivien Loon for their help in cell culture

I’m also gratitude to NUS and NUSNNI for their research scholarship, which financially supported me in my research life I’m also grateful to Singarpoe-MIT Alliance and the SMART Center for their funding support to our project

Last but not least, to my parents, I really appreciate your constant love, support and understanding

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Table of Contents

Acknowledgements i

Table of Contents iii

Summary viii

List of Tables x

List of Figures xi

List of Symbols xix

Chapter 1 Introduction 1

1.1 Background 1

1.1.1 The Infectious Human Disease—Malaria 1

1.1.2 Structure and Functions of an Erythrocyte 1

1.1.3 Life Cycle of the Malaria Parasite 2

1.1.4 Connections between Cell Mechanics & the Pathogenesis of Malaria 3

1.1.5 Structural Changes of the Malaria Infected Erythrocytes 5

1.1.6 Proteins Secreted to the Infected Erythrocyte’s Membrane 6

1.2 Objectives and scope of work 7

1.3 Thesis Organization 8

Chapter 2

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Literature Review 10

2.1 Experimental Works on Probing Mechanical Property Changes in Malaria-Infected Erythrocytess 10

2.1.1 Experimental Works on Probing Overall Cell Deformability 11

2.1.2 Experimental Works on Probing Membrane Mechanical Property Change 15

2.2 Modelling of Erythrocytes 18

2.2.1 Mechanical Models of Living Cells 19

2.2.2 Modeling of Normal Erythrocytes 26

2.2.3 Modeling of Plasmodium falciparum Infected Erythrocytes 34

2.2.4 Modeling of Optical Tweezers Stretching of Erythrocytes 36

2.2.5 Modeling of Cell Entrance into Capillaries 37

2.2.6 Multicomponent Models of Cells 38

2.3 Deformation Models for Micropipette Aspiration 40

2.3.1 Hemispherical Cap Model 40

2.3.2 Homogenous Half-space Model 42

2.3.3 Comments 42

Chapter 3 A Two-Component Model of Malaria Infected Erythrocytes 43

3.1 Introduction 43

3.2 Material constitutive relations 44

3.3 Simulation of Micropipette Aspiration 48

3.3.1 Geometric Description of Micropipette Aspiration 49

3.3.2 Boundary and Loading Conditions 51

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3.3.3 Finite Element Mesh 52

3.3.4 Finite Element Analysis Using ABAQUS 52

3.3.5 Comparison between Two Different Computational Models 56

3.4 Simulation of Optical Tweezers Stretching 61

3.4.1 Geometric Description 61

3.4.4 Finite Element Analysis using ABAQUS 65

3.4.5 Comparison of Deformed Cell Shapes between Simulation and Experiments 66

3.4.6 Computational Results and Comparison with Other Works 73

3.5 Conclusions 74

Chapter 4 Effect of Bending Stiffness on the Deformation of Malaria Infected Erythrocytes ……… … 76

4.1 Introduction 76

4.2 Material Constitutive Relations 77

4.3 Simulation of Micropipette Aspiration of Erythrocytes 79

4.3.1 Geometry, Boundary and Loading Conditions 79

4.3.3 Discussion 84

4.4.1 Geometric Description 85

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4.4.3 Finite Element Analysis using ABAQUS 86

4.4.4 Discussion 92

4.5 Conclusions 94

Chapter 5 Study of the Valid Range of Pipette Radius Used in Micropipette Aspiration of Erythrocytes 96

5.1 Introduction 96

5.3 Methodology 98

5.4 Geometric Description 99

5.5 Discussion 107

5.6 Membrane Shear Modulus of Malaria Infected Erythrocytes Calculated Using Hemispherical Cap Model 111

5.7 Conclusions……….……….112

Chapter 6 A Multi-component Model for the Malaria Infected Erythrocyte 114

6.1 Introduction 114

6.3 Simulation of Micropipette Aspiration 116

6.3.1 Geometric Description of Micropipette Aspiration 116

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6.3.5 Effect of Enclosed Fluid 121

6.3.6 Effect of Parasite Location 122

6.3.7 Effect of PVM Rigidity 127

6.4.1 Geometric Description of Optical Tweezers Stretching 130

6.4.4 Finite Element Analysis using ABAQUS for Optical Tweezers Stretching 132

6.4.5 Effect of Erythrocyte Membrane Stiffness, Interaction between Erythrocyte Membrane and PVM, and PVM Sizes on the Cell Deformation Undergoing Optical Tweezers Stretching 134

6.4.6 Effect of PVM Stiffness on the Erythrocyte Deformation undergoing Optical Tweezers Stretching 137

6.5 Conclusions 138

Chapter 7 Conclusions and Future Work 140

7.1 Conclusions 140

7.2 Future Works 142

References……….144

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erythrocytes are infected by malaria parasites such as Plasmodium falciparum (P.f.), it

is extremely deadly due to its ability to cause cerebral malaria It is believed that the

increase in stickiness and decrease in deformability of the malaria (P.f.) infected

erythrocytes will lead to capillary blockage and obstruction of blood flow Hence in this thesis, a two-component model and a multi-component model were developed to

quantitatively investigate the decrease in deformability of malaria (P.f.) infected

erythrocytes

The elastic rigidity of the cell membrane is associated to the change in free energy caused by both the stretching and bending of the erythrocyte membrane Finite element analysis was done using the finite element program ABAQUS to simulate erythrocyte deformation in micropipette aspiration and optical tweezers stretching The effect of shear modulus and bending stiffness on the erythrocyte deformability was analyzed using the two-component model The model was able to predict the cell deformation in both micropipette aspiration and optical tweezers stretching By comparing simulation results with experimental results, the model was also able to quantify the increase of shear modulus with the progression of disease stage The

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numerical results were found to be insensitive to the mesh parameter changes The effect of bending stiffness ranged from 3.3 x 10 -20 J to 1.5 x 10 -18 J, which covered the range of bending stiffness reported by other researchers, was also quantified using the two-component model

The two-component model proposed in this thesis was also able to study the validity of the commonly used hemispherical cap model The hemispherical cap model was popular in analyzing cell deformation in micropipette aspiration, due to its simplicity in calculating membrane shear modulus The model and method proposed

in this thesis allowed us to test the validity of hemispherical cap model by applying the model to analyze the simulation curves with known values of membrane shear modulus, and to obtain the valid range of pipette radius used in micropipette aspiration as a function of cell radius

The multi-component model proposed in this thesis was developed based on the two-component model It was able to quantitatively analyze the effect of parasitophorous vacuole membrane (PVM) which enclosed the parasite within the host cell, on cell deformation in micropipette aspiration and optical tweezers stretching

It is hoped that the computational models proposed in this thesis will

contribute to more accurate evaluations of the mechanical properties of malaria (P.f.)

infected erythrocytes and a better understanding of the underlying pathophysiology

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List of Tables

Table 5.1 The difference between shear modulus calculated using hemispherical cap model and the initial shear modulus of the material………109

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List of Figures

transmembrane proteins, the lipid bilayer, and the spectrin network, reprinted from (Dao et al 2003) with permission 2

Figure 1.2 Life cycle of the malaria parasites, reprinted from (Miller et al 2002)

with permission 3

Figure 1.3 Development of parasites in a malaria-infected erythrocyte, reprinted

from (Marti et al 2005) with permission 5

Figure 1.4 Distribution of P.f proteins on the surface of an infected erythrocyte

membrane RBCM, red blood cell membrane; EDM, electron-dense material; PfEMP, P.f erythrocyte membrane protein; PfHRP, P.F histidine-rich protein, also known as knob-associated histidine-rich protein (KAHRP); RESA, ring-infected erythrocyte surface antigen, reprinted from (Ho et al 1999) with permission 7

distribution of erythrocytes in a blood sample cultured with malaria tropica (a) A dilute erythrocyte suspension is subjected to the shear flow between two counter-rotating plates The upper glass plate is below a water bath which controls the temperature of the suspension The erythrocytes are observed to be elongated with the inverted microscope (b) Control sample and infected blood sample (c) Fraction

of parasitized cells δ: deformability index; f: probability density; n: number of red blood cells, reprinted from (Dobbe et al 2002) with permission 12

arrow represents the direction of fluid flow, reprinted from (Shelby et al 2003) with permission ……….13

microfluidic channels (A-D) Ring stage infected erythrocytes could pass through channels of all the 4 sizes (E-L) early and late trophozoite stage infected erythrocytes could only pass through the 6 and 8µm channels but not the 2 and 4µm channels (M-P) Schizont stage infected erythrocytes could only pass through the 8µm channel The flow direction is indicated by the white arrows, reprinted from (Shelby et al 2003) with permission 14

Figure 2.4 Mechanical probing of the various disease states of a malaria-infected

erythrocyte using the micropipette aspiration technique, reprinted from (Lim et al 2006) with permission 15

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modulus of erythrocytes infected by mature stages of P falciparum, reprinted from (Glenister et al 2002) with permission 16

original shape and (b) deformed shape of the erythrocytes, the axial and transverse diameters of the cell were compared between simulation and experiments as a function of stretching force (c) Optical image of malaria-infected erythrocytes at different stages stretched using optical tweezers at room temperature, reprinted from (Lim et al., 2006) with permission 18

from (Lim et al 2006) with permission 20

Figure 2.8 The Newtonian liquid drop model consisting of a cortical layer with

constant tension and a Newtonian liquid droplet (a) The structure of the model; (b) The creep response, reprinted from (Lim et al 2006) with permission 21

2006) with permission 22

Figure 2.10 Shear thinning liquid drop model (a) The structure of the model; (b)

Simple shear creep response of a power law fluid, reprinted from (Lim

et al 2006) with permission 24

Figure 2.11 Maxwell liquid drop model (a) The structure of the model (b) The

creep response of a Maxwell liquid, reprinted from (Lim et al 2006) with permission ………24

Figure 2.12 The homogeneous SLS model (a) The homogeneous linear viscoelastic

solid model (b) The creep response (γ) of a standard linear viscoelastic solid subjected to a stress τ ( = , µ=10 , τ= ), reprinted from (Lim et al 2006) with permission ……….26

Figure 2.13 (a) Unstressed shape of a normal erythrocyte (Fung et al 1968) (b)

Average normal erythrocyte shape measured (Evans et al 1972), reprinted from (Evans et al 1972) with permission 27

Figure 2.14 Cell shapes show streamlines of membrane motion associated with

tank-treading (a) 5μm tube; (b) 6μm tube; (c) 7μm tube, reprinted from (Secomb et al 1989) with permission 28

Figure 2.15 Hyperelastic constitutive response used in some of the computational

models (a) Uniaxial stress-strain relationships (b) Membrane shear modulus as a function of shear strain in the first order hyperelastic model (c) Membrane shear modulus as a function of shear strain in a

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higher order hyperelastic model Reprinted from (Mills et al 2004) with permission 32

Figure 2.16 Comparison between experimental results and values of shear modulus

µ and bending modulus B calculated in finite-temperature dynamics simulation, where k is the spring constant, reprinted from ( Marcelli et al., 2005) with permission….………33

particle-Figure 2.17 The axisymmetric model of micropipette aspiration of a

malaria-infected erythrocyte, reprinted from (Zhou et al 2004) with permission 35

Figure 2.18 Erythrocyte shape at different optical tweezers stretch forces simulated

using spectrin level energetic, reprinted from (Li et al 2005) with permission 37

Figure 2.19 Meridian plane of the simulation setup There is an 90 degree arc

connecting two cylindrical tubes, reprinted from (Zhou et al 2007) with permission ……… 38

Figure 2.20 Multicomponent model: the cell monolayer is in light green and the

nucleus is in dark green The arrow indicates the direction of the flow above the monolayer, reprinted from (Ferko et al 2007) with permission 39 Figure 2.21 Schematic diagram of the hemispherical cap model 41

Figure 3.1 (a) The stress-strain relationship of the hyperelastic thin membrane

under uniaxial stretch, & (b) the relationship between membrane shear modulus and shear strain, the shear strain illustrated in the figure is expressed in term of 2γs 48

measured by Evans and Fung (1972) 49

ring stage and trophozoite stage infected erythrocytes Rcell is the radius of erythrocyte and Rp is the radus of pipette 50

Figure 3.4 Schematic geometry for the micropipette aspiration of a schizont stage

infected erythrocyte 51

aspiration of erythrocyte 52 Figure 3.6 Schematic diagram of micropipette aspiration 53

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