Mechanical models for malaria infected erythtocytes 2

2.1.2 Experimental Works on Probing Membrane Mechanical Property Change a Micropipette Aspiration Micropipette aspiration has been applied to measure the membrane elasticity of differe

Chapter 2 Literature Review

Both experimental techniques and computational models have been used to study the mechanical properties of biological cells In this chapter, several experimental works on probing the mechanical properties of malaria infected erythrocytes will be introduced Some of the existing developed computational models for both healthy and malaria infected erythrocytes will also be presented

2.1 Experimental Works on Probing Mechanical Property

Changes in Malaria Infected Erythrocytes

Since the emergence of cell mechanics in the 1960s (Fung et al 1968; Fung

1990; Fung 1993; Fung 1997), the experimental techniques for investigating the mechanical properties of biological cells have developed very rapidly (Merkel 2001;

Missirlis et al 2002; Van Vliet et al 2003; Huang et al 2004) To investigate the

deformability of cells, tools and techniques such as cell filtration, rheoscope, microfluidics, micropipette aspiration and optical tweezers have been applied In this chapter, some experimental works carried out to investigate the mechanical properties

of erythocytes, especially malaria-infected erythrocytes will now be introduced

2.1.1 Experimental Works on Probing Overall Cell Deformability

2.1.1.1 Cell Filtration

Flow characteristics of cells through polycarbonate sieves can be studied using the constant-pressure method of cell filtration (Gregersen et al 1967) A polycarbonate sieve with a mean pore diameter of 5µm is placed in a filter holder connected to a pressure transducer and a syringe, which is driven by an infusion pump (Miller et al 1971) The pressure-time curve can be recorded to study the filtration pressure under a constant flow rate

Cell filtration has been applied in studying the effect of Plasmodium knowlesi

on the flow of infected erythrocytes through small sieves The researchers plotted the relationship between pressure and time for Ringer’s solution and suspension of infected and uninfected erythrocytes, and showed that the maturation of parasite influenced the flow of erythrocytes through the polycarbonate sieves

2.1.1.2 Rheoscope

Rheoscope is able to discriminate between the erythrocyte deformability

distributions (RBC-DD) of density separated cells (Dobbe et al 2002) The basic

setup is shown in Figure 2.1 (a) Between two counter-rotating parallel plates, the red blood cells are deformed by simple shear flow within the plate-plate chamber Rheoscope has been applied in studying diseases which contain anomalous fraction of less deformable or rigid cells, such as sickle cell disease, malaria tropica and hereditary elliptocytosis Figure 2.1 (b & c) plots the deformability distribution of

Figure 2.1 The basic setup of an automated rheoscope and the deformability 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

The mean deformability index may decrease due to either the presence of a small portion of less deformable cells or the slight overall deformability decrease Comparing to other measuring techniques which only provide the mean deformability

which is helpful in determining the abnormal mechanical property of the cell population

2.1.1.3 Microfluidics

Figure 2.2 Schematic illustrating the geometry of the microchannel The white arrow

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

Shelby et al used microfluidics to mimic the capillary blood flow and test the cell stiffness effect on capillary blockage by malaria-infected erythrocytes (Shelby et

al 2003). Due to their ability to replicate and control micro-environments efficiently, microfluidics have been largely applied in biology, medicine and biochemistry To meet the need for mimicking capillary micro-environment, capillary-like channel

systems have been designed in different substrates such as silicon (Sutton et al 1997), glass (Cokelet et al 1993), and PDMS (Duffy et al 1998) Figure 2.2 illustrates the

geometry of a microchannel made of PDMS, mimicking capillaries between 2 to 8

microns in diameter (Shelby et al 2003) The elastic modulus of PDMS can be

adjusted to provide a good approximation of the material properties of capillaries

(McDonald et al 2002)

Figure 2.3 Passage of malaria-infected erythrocytes at different stages through 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

Being able to control pressure, temperature and flow rate, microfluidics can closely mimic the physiological micro-environment This can be realized by integrating the system into comprehensive testing platforms One of the works which was closely related to our research objective was the microfluidic device which was fabricated out of PDMS to study malaria infected erythrocytes, as shown in Figure 2.3

It was shown that late stage malaria infected erythrocytes had difficulty in passing through extremely narrow channels

2.1.2 Experimental Works on Probing Membrane Mechanical Property

Change

(a) Micropipette Aspiration

Micropipette aspiration has been applied to measure the membrane elasticity

of different cells, such as leukocytes and erythrocytes (Rand et al 1964; Dong et al 1988; Hochmuth 2000; Shao et al 2004) It has also been used to measure the deformability of single erythrocytes at different stages of malaria progression (Zhou

et al 2004; Lim et al 2006) When the cell is aspirated into a micropipette,

measuring of the leading edge of the aspirated portion of the cell can be used in evaluating the elastic modulus of the cell Micropipette aspiration has its advantage of exerting a wide range of aspiration pressure to induce deformation on a specific section of the cell membrane The deformability of erythrocytes was found to decrease progressively as the parasite matures within the cell, as shown in Figure 2.4

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

In Chapter 1, malarial proteins on the membrane surface of the erythrocytes are introduced, such as KAHRP, PfEMP1 and PfEMP2 Another protein is PfEMP3, which is associated with the spectrin network of erythrocyte (Pasloske et al 1993; Kyes et al 1999) In order to analyze the effect of KAHRP or PfEMP3 on the

decreased deformability of malaria infected erythrocytes, Glenister et al used

micropipette aspiration to measure the membrane shear elastic modulus of normal, uninfected and parasitized cells (Glenister et al 2002), as shown in Figure 2.5 It is suggested that the decrease in deformability of uninfected erythrocytes may be due to the exo-antigens released by mature parasites KAHRP and PfEMP3 knockouts exhibit less stiffening effects on the cell membrane when compared to normal infected erythrocytes The uninfected erythrocytes referred to the cells that were cultured together with malaria infected cells but not invaded by parasites

Figure 2.5 Effect of proteins KAHRP and PfEMP3 on the membrane shear elastic modulus

of erythrocytes infected by mature stages of P falciparum, reprinted from (Glenister et al

2002) with permission

(b) Optical Tweezers Stretching

Optical tweezers were first developed by Arthur Ashkin and his co-workers in the early 1970s, and applied in a wide range of experiments, from cooling and

trapping of neutral atoms to manipulating live bacteria and viruses (Ashkin et al 1985; Ashkin et al 1987) It has also been used to manipulate a single cell and probe the elasticity of cellular components such as cytoskeleton (Lenormand et al 2003) and membrane (Sleep et al 1999) The mechanical properties of DNA (Bustamante et al 2003) as well as that of whole cells (Henon et al 1999; Barjas-Castro et al 2002; Brandao et al 2003; Lim et al 2004), and protein-protein interaction forces (Litvinov

et al 2002) has also been studied

Optical tweezers has been applied in testing force-displacement responses of

P.f infected erythrocytes (Lim et al 2006), as shown in Figure 2.6 Two silica beads

were attached to the opposite ends of the cell surface Laser beams were used to control the silica beads to stretch the cell

The results showed variations in axial and transverse diameters of erythrocyte over the erythrocytic developmental stages of the parasite development: healthy, uninfected, ring stage, trophozoite and schizont (H-RBC, Pf-U-RBC, Pf-R-pRBC, Pf-T-pRBC and Pf-S-pRBC) in PBS solution at room temperature The ability of the parasitized cell to deform in both the axial and transverse directions is progressively reduced with erythrocytic development of the parasite The experimental results indicated significant stiffening of the erythrocyte with the maturation of the parasite from the ring to the schizont stage

Figure 2.6 Sketch and optical image of optical tweezers stretching experiment, (a) 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

2.2 Modelling of Erythrocytes

Vast literatures are available on computational modeling in single cell

mechanics They served well to quantitatively evaluate the mechanical properties as

well as to investigate the cellular responses to external stimulations Different

approaches have been used while all have a single common objective, which is how

(c)

best to fit the experimentally observed phenomena with suitable choices of computational models and/or material properties

2.2 1 Mechanical Models of Living Cells

(a) Overview

The mechanical models for living cells developed by various researchers were

reviewed by Lim et al (Lim et al 2006), as shown in Figure 2.7 Either the

continuum approach or the micro/nanostructural approach can be used to develop the models The continuum approach regards a living cell as comprising a continuum material of which the constitutive material model and related parameters are derived from experiments While the micro/nanostructural approach regards the cytoskeleton

as the main structural component In comparison, the micro/nanostructural approach can probe a more detailed molecular mechanical change in the cell membrane and spectrin network, but the continuum approach provides an overall distribution of stress and strain on the cell, which can improve the micro/nano structural model by computing the distribution and transmission of the induced forces to cytoskeletal and subcellular level

Figure 2.7 Overview of the mechanical models developed for living cells, reprinted from

(Lim et al 2006) with permission

(b) The Liquid Drop Models

The liquid drop model consists of a cortical shell and a liquid core The

existing models can be categorized into the Newtonian liquid drop model (Evans et al 1989), the compound Newtonian liquid drop model (Dong et al 1991; Hochmuth et

al 1993), the shear thinning liquid drop model (Tsai et al 1993) and the Maxwell liquid drop model (Dong et al 1988), amongst which the Newtonian liquid drop

model is shown in Figure 2.8 The membrane is assumed to be an anisotropic viscous fluid layer with static tension and no bending resistance The interior is assumed to be

a homogeneous Newtonian viscous liquid

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

The constitutive relations of the cortical shell and the liquid core have been derived by Evans et al.(Evans et al 1989) The constitutive relations of the cortical

membrane, derived using membrane theory, are given as,

,3,

2/)(

,2/2

/)(

2 1

0 2

T

V T T

,3

1,

v x

v x

u x

u

i j

j

i ij i j

j

i ij

kk ij ij ij

ij ij

This model has been applied in studying white blood cell’s deformability in large deformation The advantage is its compatibility with various kinds of experimental conditions

Figure 2.9 Structure of a compound liquid drop model, reprinted from (Lim et al 2006)

with permission

The compound Newtonian liquid drop model is shown in Figure 2.9 Based on the fact that the cytoplasm is softer than the enclosed nucleus, this model has thr ee layers The outer layer consists of the plasma membrane and the cytoplasm The middle layer is the fluid-like endoplasm The core layer consists of the nucleus and

surrounding cytoskeleton (Dong et al 1991; Hochmuth et al 1993)

The shear thinning liquid drop model was first proposed (Tsai et al 1993) to

simulate micropipette aspiration of cells, taking into account the fact that when the aspiration pressure increases, the apparent viscosity of cytoplasm decrease by a power-law relationship:

b c m c



where c is the characteristic viscosity at characteristic shear rate,  c  is the mean m

shear rate and b is the power The instantaneous shear rate is given by

ij ij



If the shear rate remains constant, the equation above can be simplified as

b c



)(



as shown in Figure 2.10

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

The shear thinning liquid drop model is more suitable for simulating large deformation rather than small ones, while the Maxwell liquid drop model serves to study small deformation and recovery behavior It’s constitutive equation is given by

(Dong et al 1988)

ij ij ij

k 



where  is the viscous constant and k is the elastic constant, as shown in Figure 2.11

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

(c) Solid Models

Solid models regard the whole cell as homogeneous After certain loading, convergence can usually be achieved The material models include the linear viscoelastic solid model and the linear elastic solid model, which simplifies the viscoelastic model by neglecting the time factor

The linear elastic model is given by

ij

where G is the shear modulus, given by E2(1)G where E is the Young’s

modulus and  is the Poisson’s ratio

The viscoelastic solid model regards the whole cell as a homogeneous viscoelastic standard linear solid (SLS) The constitutive relation of the SLS model is described by (Schmid-Schonbein et al 1981; Sato et al 1990)

ij ij

ij ij

k

k k

2

where µ is the viscous constant, k1 and k2 are two elastic constants, represented by the dashpot and springs as shown in Figure 2.12