Modelling the mechanical properties of yeast cells

Chemical Engineering Science 64 (2009) 1892 -- 1903

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Modelling the mechanical properties of yeast cells J.D. Stenson, C.R. Thomas ∗ , P. Hartley School of Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

A R T I C L E

I N F O

Article history: Received 16 April 2008 Received in revised form 17 December 2008 Accepted 13 January 2009 Available online 20 January 2009 Keywords: Cellular biology and engineering Yeast Mechanical properties Elasticity Mathematical modelling Solid mechanics

A B S T R A C T

An analytical model has been developed to describe the compression of a single yeast cell between parallel flat surfaces. Such cells were considered to be thin walled, liquid filled, spheres. Because yeast cells can be compressed at high deformation rates, time dependent effects such as water loss during compression and visco-elasticity of the cell wall could be and were neglected in the model. As in previously published work, a linear elastic constitutive equation was assumed for the material of the cell walls. However, yeast compression to failure requires large deformations, with high wall strains and associated rotations. New model equations appropriate to such high strains with rotations were therefore developed, based on work-conjugate Kirchhoff stresses and Hencky strains. This is an improvement on the earlier use of infinitesimal strains, and on the alternative of Green strains and 2nd Piola–Kirchhoff stresses. It is shown that the choice of stress and strain definition has a significant influence on model predictions for given wall material properties, and will affect estimates of the wall elastic modulus or other wall material property parameters obtained by fitting experimental data. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The disruption of microbial cells is important in bioprocessing, as a means to release the cellular contents. In high-pressure homogenisation, the extent of release depends on the local fluid flow fields in the processing equipment and on the mechanical properties of the cells (Kleinig and Middelberg, 1996, 1998). In other types of disruption, there may be different types of stress generation but the mechanical properties of the cells will still be important. Improvement in disruption has been the motivation of recent studies into the mechanical properties of yeast cells (Mashmoushy et al., 1998; Smith et al., 1998, 2000a–c). A common method to characterise the mechanical properties of cells is by the compression experiment, using micromanipulation (Zhang et al., 1991; Liu, 1995; Mashmoushy et al., 1998; Blewett et al., 2000; Peeters et al., 2003; Liu and Zhang, 2004; Wang et al., 2004). In such experiments the cell is compressed between flat parallel plates, and by measuring the force being imposed on the cell during compression, whole cell force–deformation data up to cell bursting can be generated. In order to extract useful information about the cell wall material properties, such data must be mathematically modelled.

∗ Corresponding author. Tel.: +44 121 4145355; fax: +44 121 4145324. E-mail address: [email protected] (C.R. Thomas). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.01.016

Approaches to modelling the compression of hollow spheres are often based on analyses of Feng and Yang (1973)and Lardner and Pujara (1980). Both assumed the wall is thin enough to be treated as a (mechanical) membrane, and while the former concerned gas-filled spherical shells, Lardner and Pujara (1980) considered such shells filled with incompressible liquid. The liquid volume was assumed to be constant, which implied impermeability of the wall material. Because the wall is treated as a mechanical membrane, the stresses are expressed as wall tensions, and it is presumed the wall could not support out-of-plane shear stresses or bending moments. This situation is described as plane stress, as the only non-zero stresses are in the plane of the cell wall. Feng and Yang (1973) and Lardner and Pujara (1980) solved their analytical equations numerically. Cheng (1987a) described the alternative computational approach of finite element analysis, and applied this to previously published data on sea urchin eggs (Cheng, 1987b). Smith et al. (1998) also used a finite element approach, but introduced the possibility of permeable cell walls, i.e. a changing liquid volume inside the shell. In the absence of better information, it is generally assumed that the material of the cell wall is homogeneous and isotropic. For the analysis of force–deformation data from compression experiments it is also necessary to use an appropriate constitutive equation, relating the stresses and strains that are generated in the cell wall. Suitable constitutive equations can be obtained by assuming that the cell wall is hyperelastic, in which case the stress components are derived from a strain energy function, usually presented as depending on strain invariants. Strain energy is the energy stored in a body that

J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

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Table 1 Strain energy functions used to interpret compression testing data. Source

Cell type

Skalak et al. (1973)

Red blood cells

Skalak et al. (1973)

Red blood cells

Lardner and Pujara (1980)

Sea-urchin eggs

Cheng (1987a)

Sea-urchin eggs

Liu (1995) Smith (1999) Wang et al. (2004)

Tomato cells Yeast cells Tomato cells

Strain energy function   C B 1 2 I + I1 − I2 + I22 where I1 = 2(e1 + e2 ) and I2 = I1 + 4e1 e2 and B W= 4 2 1 8 and C are material constants. This is the STZC strain energy function 2 2 2 −2 −2 −2 W = C1 (I1 − 3) + C2 (I2 − 3) where I1 = 1 + 2 + 3 and I2 = 1 + 2 + 3 , and C1 and C2 are material constants. This is the Mooney-Rivlin strain energy function Mooney–Rivlin and STZC functions Eh0 (2 + 22 + 21 2 ) W= 2(1 − 2 ) 1 where  is Poisson's ratio Mooney–Rivlin function Cheng (1987a) function Cheng (1987a) function

W: strain energy per unit initial volume; W: strain energy per unit initial area, for (mechanical) membranes; I strain invariant; e Green strain;  infinitesimal strain, subscript1 meridian direction, 2 longitudinal direction; E elastic modulus; h0 initial wall thickness.

results from an elastic deformation, and is equal to the work required to produce the strain in the body (assuming that any temperature changes and other losses can be ignored). Strain invariants depend on the principal strains and have values independent of the choice of coordinate system. For work on biological cells, the assumption of a suitable cell wall constitutive equation or a strain energy function is necessary because the constitutive equation is not usually or ever known, a priori. There are many choices of strain energy function that might be appropriate to cell walls, and a summary of those that have been used to interpret data from compression testing (of cells) is provided in Table 1. This table excludes models where the tension in the wall was assumed to be uniform (Zhang et al., 1992; Kleinig, 1997). In some cases (Skalak et al., 1973; Lardner and Pujara, 1980; Liu, 1995) the wall was assumed to be incompressible, whereas other workers have proposed strain energy functions incorporating Poisson's ratio  to allow for compressibility (Cheng, 1987a; Smith, 1999; Wang et al., 2004). Even in these cases it is common to assume incompressibility i.e.  = 0.5, when fitting the model to experimental data, because it is very difficult to determine a value for this parameter or to find a value by fitting a model to experimental data. Once a constitutive equation or strain energy function has been assumed, fitting experimental data should allow characterisation of cell wall material properties. However, Smith et al. (1998) showed that unique determination of material parameters from compression testing still requires accurate and precise measurements of some geometrical parameters. Firstly, cells are usually already inflated at the start of the compression (because of a difference between internal and external pressures) and the initial stretch ratio is therefore important, as are the (uninflated) cell wall thickness and the volume loss caused by any liquid loss through the cell wall during compression. Even with an assumed linear elastic constitutive equation, at least two of these geometrical parameters should be known to obtain a unique value for the elastic modulus. However, if the constitutive equation is unknown, all three measurements are necessary, because two different constitutive relationships can give the same force–displacement curves. Even then, the resulting constitutive equation should ideally be evaluated by tests involving different types of mechanical testing. For yeast cells, the subject of this paper, alternatives to compression testing are not readily available, mainly because yeasts cells are microscopic and generally strong. It was decided therefore that a generalised form of Hooke's Law should be assumed, that various ways of deriving a strain energy function should then be considered, and that simulations based on the resulting strain energy functions should be compared with each other and an illustrative set of experimental data. It was also assumed that any water loss from a cell

during compression could be neglected in modelling, because yeast cells can be compressed at high deformation rates by micromanipulation, as described later. Hooke's law implies a linear relation between stress and strain, i.e. a constant elastic modulus, and it is believed to be the best practical choice here because, (a) there is only one material parameter, the elastic modulus; (b) it might in any case be difficult to identify more material parameters from limited experimental data; and (c) the material model is simple and readily understood by workers in other fields. Smith et al. (1998) suggested a linear constitutive equation of this form be adopted, and data from Smith et al. (2000b) suggest it may be appropriate for yeast cells. It should be noted that simulated force–deformation curves will be non-linear even if a constant elastic modulus is assumed, because of changes in the geometry of the cell as it is deformed. Although the use of a constant (or average) elastic modulus has been assumed before (it underlies the choice of the strain energy function of Cheng, 1987a, see Table 1), it is still important that the equations used to derive the strain energy function are properly formulated. Alternative definitions of strain are in common use, and each has an appropriate and corresponding stress definition. The linear constitutive equations resulting from these alternative definitions are different. Furthermore, the equations of Feng and Yang (1973) and Lardner and Pujara (1980) are expressed in terms of stretch ratios, which are an alternative to strains as a measure of deformation, and are often used for materials that can support high deformations before failure. A stretch ratio in a given direction is the length of the deformed material divided by its original length. As such, a stretch ratio is directly dependent on the geometry of the compressed cell. As local stretch ratios cannot be measured, they must be estimated using a model, and the formulation of the model will determine the resulting values. Each strain definition leads to a different relationship between the stretch ratio and strain, which must be allowed for in modelling.

2. Strain energy functions For material deformation that is very small, as in the case of elastic deformation of most metals, the infinitesimal strain approach is a valid approximation. In this case principal strain components (i ) may be related to the principal stretch ratios (i ) by the equation

i = (i − 1)

(1)

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J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

where the subscript i indicates a principal direction. Principal stretch ratios are defined and measured in directions along which there is no shear, only stretching or compression. When loading conditions lead to greater material deformation, alternative measures of strain must be considered. For the case of non-infinitesimal strain (i.e. finite strain), the most common definition is Green (or Lagrangian) strain. The principal components of Green strain (Ei ) are related to the principal stretch ratios by the equation 2

Ei = 12 (i − 1)

(2)

This provides a measure of strain that is rotationally invariant, and therefore avoids the errors introduced by the infinitesimal strain model when the deformation involves large rotations. There are, however, limitations to its use. When large deformation is accompanied by large strain, a more appropriate measure to use is Hencky strain (Hi ; often referred to as true, logarithmic or natural strain). This is related to the principal stretch ratios by the equation Hi = ln i

(3)

The immediate question this raises concerns the appropriate measure of strain for a particular deformation. There are no absolute measures and the choice of strain model may be guided by the level of accuracy deemed acceptable, or by the ease of its inclusion in an analytical model. In very broad terms, infinitesimal strain is suitable for very small deformations where the strain may be of the order of 0.1%; for larger strains of the order of 1–5%, especially where material rotation is involved, then Green strain should be used, and for any magnitude of strain (in particular for even larger strains) the most accurate model is that using Hencky strain. However, these are merely approximate guidelines and in some circumstances (for example where the principal stress axes do not rotate during deformation) infinitesimal strain may provide a reasonable first approximation even where the strain reaches 5%, and Green strain will give results very close to Hencky strain if the analysis is performed in small increments of deformation. For this reason, Green strain has proved popular in computational finite-element models of material deformation. This paper uses all three strain models to demonstrate and compare their effectiveness in the analytical approach adopted here, in which there is only a single step from the uninflated state of a cell to each required level of deformation. The advantage of this approach is that a picture of the deformation history is constructed without using an accumulation of deformation increments. For each strain, it is necessary to use an appropriate measure of stress. If it is assumed that the material of the cell wall is hyperelastic, then a linear constitutive equation may be applied across any magnitude of deformation (Xiao and Chen, 2002; Malvern, 1969), subject to the stress and strain being work conjugate. This implies the stresses may be determined by differentiation of an appropriate strain energy function with respect to the chosen strain component. The relevant work-conjugate stress–strain pairs for the measures of strain mentioned above, written in terms of the principal components, are Cauchy stress i –infinitesimal strain i , so i =

jW ji

Piola–Kirchhoff 2 (PK2) stress Si –Green strain Ei , so Si = Kirchhoff stress i –Hencky strain Hi , so i =

jW jHi

where W is the chosen strain energy function.

(4)

jW j Ei

(5)

(6)

If the modelling were only to be based on the limiting case of infinitesimal strain, then the constitutive equation could be written using indicial notation in the form of Malvern (1969):

ij = ij kk + 2 ij

(7)

ij is the Cauchy stress in the jth direction, on the plane normal to the ith direction, and ij is the corresponding strain. A repeated subscript implies a summation over all its values (1–3 for stress and strain tensors).  and are Lamé's constants and ij is the Kronecker delta (=1 if i = j and 0 if i
j). Lamé's constants are defined as (Lemaitre and Chaboche, 1994)

=

E (1 + )(1 − 2)

(8a)

and 2 =

E 1+

(8b)

where E is the elastic modulus and  is Poisson's ratio. The axisymmetric deformation of the yeast cell (see Fig. 1), together with the assumption of plane stress conditions (i.e. the only non-zero stresses are in the plane of the cell wall), means that the meridian and longitudinal stresses and strains are the principal components, in which case Eq. (7) may be written specifically for these components as

i = v + 2 i

(9)

where i indicates the principal component, and v is the bulk or volumetric strain (i.e. the sum of the three principal strain components). For each definition of strain, a strain energy function is needed that can be differentiated in accordance with Eq. (4), (5) or (6), to give an expression that is consistent with that obtained from Eq. (9) or its equivalents for the other stress–strain pairs. Using the infinitesimal strain case as an example, the 1 component from Eq. (9) is

1 =

E E (1 + 2 + 3 ) + 1 (1 + )(1 − 2) (1 + )

(10)

Although the cell wall is approximated here as a two-dimensional plane stress problem, there will exist a strain in the normal direction, given by Lemaitre and Chaboche (1994) as

3 = −



1−

(1 + 2 )

(11)

This allows Eq. (10) to be reduced to

1 =

E [1 + 2 ] (1 − 2 )

(12)

For this infinitesimal strain case, a suitable strain energy function that can be differentiated with respect to both 1 and 2 is 1 1   = (1 1 + 2 2 ) 2 i i 2  E E 1 1 (1 + 2 ) + 2 (2 + 1 ) = 2 2 2 (1 −  ) (1 −  ) E 2 2 ( + 2 + 21 2 ) = 2(1 − 2 ) 1

W=

(13)

This is the strain energy function derived by Cheng (1987a); see Table 1. A similar analysis can be followed for the other stress–strain pairs (Eqs. (5) and (6)). To do this, it is of course necessary to assume that Hooke's law (Eqs. (7) and (9) in the infinitesimal case) can be generalised to any work-conjugate stress–strain pair, and that Poisson's ratio is constant for all strains, or that some average value is a

J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

good approximation across the deformation range. For the uniaxial, infinitesimal strain case, Poisson's ratio is defined by

=−

trans axial

(14)

where trans is the transverse strain caused by an imposition of an axial strain axial . It is presumed here that this is an infinitesimal strain limit of a general large strain equivalent, which is presumed to apply for all deformations. With this additional assumption, it is possible to extend Eq. (13) to large strains by replacing infinitesimal strain directly by Green strain E W= (E2 + E22 + 2E1 E2 ) 2(1 − 2 ) 1

(15)

and for Hencky strain W=

E (H2 + H22 + 2H1 H2 ) 2(1 − 2 ) 1

(16)

Regardless of the measure of stress used in modelling the compression of yeast cells, the final outcome of the model must be the true or Cauchy stress, defined as the ratio of the current force and current area. For infinitesimal strain, any change in area is normally assumed to be negligible and Cauchy stress is equated with the nominal stress (current force divided by initial area). For other definitions of strain the differences between the original and final dimensions cannot be ignored. In the case of Kirchhoff stress, the Cauchy stress is equal to this stress divided by the ratio of the final volume divided by the initial volume, so that

i = Ji

(17)

where J = 1 2 3

(18)

Note that J =1 for an incompressible material. The PK2 stress is often interpreted as the initial force divided by the original area. In this case, the true (Cauchy) stress may be obtained through a mapping between the original and deformed configurations. This is the basis of the general relationship between the Cauchy stress (ij ) and the PK2 stress (Skl ), i.e. 1 J

ij = Skl

jxi jxj jXk jXl

1895

in part by using the mixed strain approach in the analytical model. In the present work the equations are reformulated, and the significance of the reformulation is investigated. Whether or not the cell wall is assumed to be incompressible (J = 1), the cell wall thickness may decrease during stretching of the cell wall in the compression experiment. Several models ignore this change, and in some circumstances it may indeed be negligible. Nevertheless, cell wall thinning will be incorporated here for completeness. 3. Model of the compression The equations governing the deformation of a liquid-filled spherical membrane were derived by Feng and Yang (1973). Fig. 1 shows the geometry, and it can be seen that there are parts of the cell wall that contact the compressing surfaces, and parts that do not (noncontact regions). There are separate groups of governing equations for contact and non-contact regions. The independent variable in these equations is , which relates the position of any point on the boundary of the cell back to the original position of that point in the inflated but uncompressed cell.  is the angular position of the point measured from the vertical axis of symmetry. Contact region:

1 d1 =− d 2 sin 



f3 f1



 −

1 − 2 cos  sin 



f2 f1

d2 1 − 2 cos  = d sin 

 (21)

(22)

where f1 , f2 and f3 are functions of the principal tensions and are defined later. Non-contact region: d1 = d



 cos  − sin  sin2 



f2 f1

 −

 f  3



f1

(23)

where

 = 2 sin 

(24)

(19)

in which jxi / jXk , jxj / jXl are the components of the deformation gradient tensor (Malvern, 1969). Since the deformation here only involves plane stresses, the deformation gradients are equivalent to the principal stretch ratios, and therefore 1 J

(20a)

1 J

(20b)

1 = S1 21 and

2 = S2 22

Because the deformations required to burst yeast cells are large (Mashmoushy et al., 1998), large strain assumptions may well be essential, and this has never been done in an analytical model. For example, Smith (1999), Wang et al. (2004) and (for analysis of the compression of pollen grains) Liu and Zhang (2004) used mixed low and high strain assumptions, by introducing an infinitesimal definition of strain (Eq. (2)) into the large strain relations. Smith et al. (1998) validated their finite element code against the analytical equations found in Smith (1999). There were discrepancies, which were attributed by Smith et al. (1998) to the inability of the analytical model to account for wall thinning, but may have been caused

Fig. 1. Schematic diagram of the geometry of the cell where  is the angular position of a point on cell wall from vertical axis of symmetry prior to compression,

identifies the angle of the point on the edge of the contact region between the compression surface and the cell following compression.  and  are the horizontal and vertical coordinates, ¯ is the distance between the compression surface and the equatorial plane, and z is half the distance that the cell has been compressed.

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J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

and

Because a constant Poisson's ratio has been assumed, these expressions for the tensions may be used in Eqs. (28a)–(28c), to give

d d   sin  −  cos  d2 = d sin2 

=

(25)

2



T2 T1

 −

1 (21 − 2 )1/2 Pr0 T1

(27)

jT1 j1

(28a)

jT1 f2 = j2

(28b)

and f3 = T1 − T2

(28c)

where T1 , T2 are the tensions in the meridian and circumferential directions, respectively. The tensions link the geometry (Eqs. (21) –(27)) to the cell wall constitutive equation. Since the compression of the cell consists of axisymmetric deformation, T1 and T2 are principal tensions, which are defined with respect to the current deformed shape of the cell. 1 and 2 are therefore the principal stretches in the meridian and circumferential directions. In using Eqs. (28a)–(28c), the stretch ratios must be related to an appropriate choice of strain measure (Eqs. (1)–(3)), and in each case the corresponding definition of stress (Eqs. (4)–(6)) must be used. The final stage of the modelling is to use the strain energy functions of Eqs. (13), (15) and (16) to derive stresses from which tensions can be derived for use in Eqs. (28a)–(28c).

For the limiting case of infinitesimal strain, the strain energy function for the cell wall is that given in Eq. (13). This gives rise to the familiar plane-stress equations: E (1 + 2 ) (1 − 2 )

(29a)

jT1 Eh0 = j2 (1 − 2 )

(32b)

f3 = T1 − T2 =

Eh0 (1 − 2 ) Eh0 (1 − 2 )(1 − ) = (1 + ) (1 − 2 )

E (2 + 1 ) 2 = (1 − 2 )

(29b)

Since in this model, large deformation, and hence finite (Green) strains are assumed, any strain energy functions must be formulated in terms of the 2nd Piola–Kirchhoff (PK2) stresses (Malvern, 1969). The true (Cauchy) stress may be obtained through a mapping between the original and deformed configurations using Eqs. (20a) and (20b). The principal PK2 stresses may be determined from the strain energy function given in Eq. (15), and using the definition of Green strain given in Eq. (2), these may be written in terms of the principal stretch ratios as S1 =

1 jW 1 j1

(33a)

1 jW 2 j2

(33b)

and S2 =

The Cauchy stresses are then

1 jW J j1

(34a)

2 jW J j2

(34b)

and

2 =

As earlier, the principal Cauchy tensions in the wall are defined as

(30a)

and T2 = 2 h

(30b)

where h is the (current) wall thickness. Since an infinitesimal strain approach is adopted, any change in the initial uninflated cell wall thickness (h0 ) and final cell wall thickness (h) is assumed to be negligible, so h ≈ h0 . Then, from Eqs. (1), (29) and (30): Eh0 {(1 − 1) + (2 − 1)} T1 = 1 − 2

(31a)

Eh0 {(2 − 1) + (1 − 1)} 1 − 2

(31b)

This approach was adopted by Wang et al. (2006) as a simplification of an earlier model published by Wang et al. (2004).

(30a)

and T2 = 2 h

The principal Cauchy tensions in the wall may be defined as T1 = 1 h

(32c)

3.2. Finite (Green) strain

T1 = 1 h

and

T2 =

f2 =

1 =

3.1. Infinitesimal strain

1 =

(32a)

For incompressible deformation these equations may be simplified by substituting  = 0.5.

The analysis gives f1 =

jT1 Eh0 = j1 (1 − 2 )

(26)

The turgor pressure (P) inside the cell (i.e. the hydrostatic pressure difference across the wall) can be related to using d d1 ( − 2 ) = + 1 d d  1 

f1 =

(30b)

where h is the current membrane wall thickness. Since finite strains are assumed here, the change in cell wall thickness might be important and the true stresses (or tensions) must be defined in terms of the current dimensions, rather than assuming h ≈ h0 . From Eq. (18), noting that

3 =

h h0

(35)

it can be seen that h=

Jh0

1 2

(36)

From Eqs. (30), (34) and (36), expressions or T1 and T2 may be found. T1 =

h0 jW 2 j1

(37a)

J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

and T2 =

h0 jW 1 j2

(37b)

The appropriate expression for W can be found from Eq. (15), using the definition of Green strain from Eq. (2) E 2 2 2 2 (( − 1)2 + (2 − 1)2 + 2(1 − 1)(2 − 1)) W= 8(1 − 2 ) 1

(38)

In this case, it may be shown that f1 =

jT1 Eh0 2 2 {3 + 2 − (1 + )} = j1 2(1 − 2 ) 1

(39a)

f2 =

jT1 1 2 Eh0 2 = { − 1 + (1 + )} j2 2(1 − 2 ) 22 2

(39b)

f3 = T1 − T2 =

Eh0 2(1 − 2 )



1 2 2 2 [ − (1 + )] − [ − (1 + )] 2 1 1 2

(39c)

and if required, these equations can be reduced to the incompressible case by substituting  = 0.5. 3.3. Hencky (logarithmic) strain Using this measure of strain, most appropriate for large deformations and large strains, the required stress is the Kirchhoff stress. This is related to the Cauchy stress by

i = Ji

(17)

The principal components are again given by Eqs. (34a) and (34b), and the tensions are given in Eqs. (37a) and (37b). In terms of principal stretch ratios, using the definition of Hencky strain from Eq. (3), the strain energy function in Eq. (16) becomes W=

E ((ln 1 )2 + (ln 2 )2 + 2 ln 1 ln 2 ) 2(1 − 2 )

(40)

This leads to Eqs. (41a)–(41c): f1 =

jT1 Eh0 {1 − ln(1 ) −  ln(2 )} = j1 21 2 (1 − 2 )

(41a)

f2 =

jT1 Eh0 { −  ln(2 ) − ln(1 )} = j2 1 22 (1 − 2 )

(41b)

f3 = T1 − T2 =

 1 Eh0 ln 1 2 (1 + ) 2

(41c)

f1 =

jT1 2Eh0 2 = {2 − ln(1 2 )} j1 321 2

(42a)

f2 =

jT1 2Eh0 2 = {1 − ln(1 2 )} j2 31 22

(42b)

 1 2Eh0 ln 3 1  2 2

(42c)

The expressions in (32), (39) and (41) (or their equivalents for the incompressible case) provide the functions required to solve Eqs. (21)–(27), using the following boundary conditions (Feng and Yang, 1973):

 = 0,

1 = 2 = 0

 = ,

1 (contact region) = 1 (non-contact region)

2 (contact region) = 2 (non-contact region)

 = ,

 = ¯

 = ,

 = 0 or = 1

=

 2

,

=0

where identifies the angle of the points on the edge of the contact region between the compression surface and the cell,  and  are the horizontal and vertical coordinates and ¯ is the distance between the compression surface and the equatorial plane (see Fig. 1). In this work, it was decided to assume incompressibility of the cell walls as there is no information available on this parameter, and this follows the approach of most previous workers, as described earlier. However, the more general Eqs. (32), (39) and (41) would allow this assumption to be revisited if better information on Poisson's ratio should become available. The significance of this incompressibility assumption to the determination of the elastic modulus is discussed later. 4. Numerical simulation The governing equations were solved by the Runge–Kutta method using the MATLAB (MathWorks Inc.) ode45 solver. The method for solving these equations has been outlined in Wang et al. (2004). The model requires a number of initial modelling parameters to be defined which include the initial stretch ratio (s ), elastic modulus (E), cell radius (r0 ) and the cell wall thickness (h0 ). Of these parameters, the cell radius can be measured directly from experimental force–deformation data for a specific cell under compression testing, and from this, the cell wall thickness could be determined using a previously determined (population mean) cell wall thickness to cell radius ratio. The value of this was 0.037 (Stenson, 2008), which is the value used by Smith et al. (2000a) for similar cells. In essence, the cell wall thickness to cell radius ratio was an assumed value for modelling the behaviour of this cell; the population mean value was the best available estimate. This means that only two parameters (E and s ) had to be identified by fitting model simulations to experimental data. 5. Materials and methods

For the incompressible case used in this work

f3 = T1 − T2 =

 = ,

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A typical set of force–deformation data for compression of a single yeast cell (the test set) was chosen to illustrate how the model could fit experimental data, and to demonstrate the importance of the correct formulation of the model equations. The method for producing the test data is described briefly below. Dried bakers yeast (Fermipan Red, DSM Bakery Ingredients, Dordrecht, Holland) was resuspended in Isoton II solution (Coulter Electronics Ltd, Hertfordshire, UK) at 2 g L−1 for 24 h at 4 ◦ C to give a reproducible suspension of late stationary phase cells (Kleinig, 1997). Prior to use the suspension of cells was warmed to room temperature. The method of compression testing by micromanipulation is described in detail elsewhere (Blewett et al., 2000). In this technique, an individual cell is fully compressed between two parallel and rigid platens until it ruptures. The compression speed was 68 m s−1 . The compression experiment produces voltage time data that can be converted to force–displacement data using the methods outlined by Mashmoushy et al. (1998). The micromanipulation experiments from which the test set was chosen were performed at room temperature under Isoton II solution which has an osmotic pressure under these conditions of 0.8 MPa. At a speed of 68 m s−1 , compression of a yeast cell took less than 0.1 s. This was a much higher speed of compression than those used

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by Smith et al. (2000a) i.e. 1.03, 2.70 and 7.68 m s−1 . The intention was to obviate the effects of water loss or other time dependent effects, such as viscoelasticity, reducing the number of parameters to be fitted. Using the data of Smith et al. (2000b) for Fermipan yeast hydraulic conductivity (ca. 0.3 × 10−12 m s−1 Pa−1 ), and turgor pressure (up to 1 MPa at a stretch ratio of about 1.08 in an osmotic pressure test), and assuming spherical geometry, an estimate can be made of the time constant for water loss from a yeast cell of diameter about 5 m. This time constant is about 2 s, which suggests that the impermeable wall assumption is reasonable for compressions of the order of 0.1 s. This is discussed later.

5.1. Matching simulations to experimental data Numerical simulations produce force–deformation curves whose magnitude and shape are determined by the initial model parameters including the elastic modulus (E) and the initial stretch ratio (s ). These parameters can be determined by fitting the force–deformation curves obtained from the numerical simulations to the experimental data. For the different models numerical simulations were produced with the initial stretch ratio ranging from 1.00 to 1.11. This was done as independent tests on the effects of changes of osmotic pressure on cell size have shown that the initial stretch ratio in a suspending solution of 0.8 MPa is about 1.04 (Stenson, 2008) but might be as high as 1.08 (Meikle et al., 1988). In the process of matching simulations to experimental data, comparisons were made of the simulated and experimental forces in 2 dimensionless form. The dimensionless force (F/Eh0 r0 s )) and fractional deformation (X = z/(r0 s )), where z represents half the displacement of the probe, were defined by Lardner and Pujara (1980). For each set of experimental force–deformation data, including the test set described in this paper, different combinations of the elastic modulus (E) and the initial stretch ratio (s ) could be used to give a reasonable fit. The best fit in each case was chosen by finding the combination of these parameters that maximised the coefficient of determination between the numerical simulation and the experimental data. It was not necessary to fit the permeability

of the cell wall (Smith et al., 1998) because it had been assumed that water loss during compression might be neglected in modelling, given the fast compression speeds attainable by micromanipulation. 6. Results and discussion As large strains and rotations may be present in yeast cell walls during cell compression, it was likely that Hencky strains would be the most appropriate strain measure. Fig. 2 demonstrates a fit of the Hencky strain numerical simulation to typical experimental data i.e. the test set. With an initial stretch ratio of 1.04 and an elastic modulus of 172 MPa the model fits the data for this specific cell very well up to the point of cell bursting, with a coefficient of determination of 0.998. Failure was at a deformation of about 67%, which was typical. Fig. 3 shows a surface plot of coefficients of determination generated by the fitting process for Hencky strain numerical simulations. The best fit at E = 172 MPa, s = 1.04 was clearly identifiable. Similar plots were obtained for all the models. When the data shown in Fig. 2 were also fitted using the infinitesimal strain model, the best fit led to an elastic modulus of 80 MPa, and an initial stretch ratio of 1.11. The coefficient of determination was found to be 0.996. For the finite strain model the elastic modulus was 60 MPa, the initial stretch ratio 1.11, and the coefficient of determination 0.995. It is clear that all the models were able to fit the data well, and the small changes in coefficient of determination may not be sufficient to justify choosing one model over another, although the preferred Hencky strain model is marginally the best. Nevertheless, the choice of strain measure is important, as it significantly affects the value of the elastic modulus determined by modelling. The elastic modulus is a significantly higher value for the Hencky model because it gives lower strains for given stretch ratios. This conclusion is independent of any specific experimental data. The Hencky model elastic modulus of 172 MPa for the data of Fig. 2 (a single cell value) is reasonable compared to the sample mean value of 150 ± 15 MPa of Smith et al. (2000b), also acquired using Fermipan, with the same cell wall thickness to cell radius ratio assumption in both cases. Smith et al. (2000b) used finite element

Fig. 2. Typical example of the Hencky strain model fitting experimental data up to the point of cell failure. (Parameters for the model: elastic modulus of the cell wall 172 MPa, Poisson's ratio 0.5, initial stretch ratio 1.04, cell radius 2.28 m). Coefficient of determination 0.998.

J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

analysis to determine the elastic modulus, and it is believed this was also based on Hencky strains. It should be noted that the modulus value depends on the choice of wall thickness, and it might be best to treat Eh0 as a single parameter to be determined by modelling. It is likely that this combination of parameters would also appear in any application of the data. Fig. 4 compares simulations using the models with the parameters that were determined for Hencky strains (s = 1.04, E = 172 MPa,

Point of best fit 1.00

Coefficient of determination (R2)

0.98 0.96 0.94 0.92 0.90 0.88

0.86 1.02 1.04 Initial 1.06 stretch ratio 1.08 1.10

170

180

190

160 Elastic modulus (MPa)

Fig. 3. Surface plot showing an example of data generated by the fitting process. The black point shows the best fit (E = 172 MPa, s = 1.04, coefficient of determination 0.998).

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r0 = 2.28 m, h0 = 84 nm). Again, it can be seen that the choice of strain measure is important, especially for the higher deformations needed to induce cell breakage. This is independent of the choice of experimental data to fit; it is a simulation result. As mentioned earlier, there is only a single step from the uninflated state of a cell to each required level of deformation in the analytical approach described here. In such an approach, infinitesimal strains are really suitable only for small deformations where the strain is approximately 0.1%. At higher strains, the infinitesimal strain model does not allow properly for rigid body movements, i.e. rotation and translation. Such movements lead to incorrect estimates of force at higher deformations, leading to underestimation of the elastic modulus. The finite strain model is better, as it is able to take into account rotations, although it should not be applied when large deformations are accompanied by large strains. However, if this model is used over small increments of deformation, as in finite element analysis, it should give results close to that of the Hencky strain. The simulations also provided profiles of the compressed cell, with a typical example shown in Fig. 5. The profiles are axisymmetric about the axis of compression and symmetrical across the equatorial plane, allowing them to be fully characterised with just one quadrant. In these simulations, the same force and deformation was used, to match a specific experimental condition. The profiles are very similar, which unfortunately means that independent verification of the correctness of the Hencky strain model using image analysis of the cell shape is impractical. Using any of the models, the strains in the meridian and the circumferential directions at any location on the cell boundary can be calculated, for any deformation. In particular, the values at the equator were found, and for deformations above 30%, these were always the largest strains. This result confirms similar observations of Cheng (1987b) and Smith et al. (2000b). Fig. 6a shows the equatorial strains using the preferred Hencky strain model. It can be seen that the circumferential strain can be up to fivefold greater than the meridian strain. Fig. 6b shows the true (Cauchy) stress components at the equator; the circumferential stress is similarly higher than the

Fig. 4. Numerical simulations produced using the models derived with the different definitions of strain i.e. Eqs. (32), (39) and (42) with Poisson's ratio set to 0.5. (Model parameters: elastic modulus 172 MPa, initial stretch ratio 1.04, cell radius 2.28 m.)

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Fig. 5. Influence of the different choices of strain definition on the cell wall boundary at the same deformation (62%) and applied force (71 N). The infinitesimal strain profile was almost identical to that of the Hencky strain and is not plotted.

Fig. 6. (a) Strain components at the equator of the cell obtained from the numerical simulations during compression of the cell. Circumferential strain at failure =0.45. Hencky strain model parameters: elastic modulus of the cell wall 172 MPa, Poisson's ratio 0.5, initial stretch ratio 1.04, cell radius 2.28 m. (b) True (Cauchy) stress components at the equator of the cell obtained from the numerical simulations during compression of the cell. Circumferential stress at failure = 112 MPa.

J.D. Stenson et al. / Chemical Engineering Science 64 (2009) 1892 -- 1903

stress in the meridian direction. Clearly the wall stresses are far from uniform. Because the models fit the experimental data well up to failure, it is possible to estimate the stresses and strains at failure. Smith et al. (2000b) proposed using a von Mises failure criterion for this purpose, but it is not certain that this is the most appropriate choice. It is not even clear whether the mode of failure of the walls is ductile or brittle, and because of the small size of yeast cells, it has proved impossible to confirm the mechanism of failure experimentally. Since the relative sizes of the circumferential and meridian strains at the equator, the practical choice for a criterion of strain at failure is the circumferential Hencky strain, which was 0.45 for the cell being considered here. It would seem reasonable to characterise the stress at

Dimensionless force (F/Ehoroλs2)

4

3

No water loss 68 μms-1 - water loss 8 μm s-1 - water loss

2

1

0 0.0

0.2

0.4 Fractional Deformation

0.6

Fig. 7. Comparison of simulations with waterloss and without waterloss. E=100 MPa, s = 1.04, r0 = 2.5 m, h0 = 92.5 nm, Lp = 0.3 × 10−12 m s−1 Pa−1 , compression speeds = 8 and 68 m s−1 .

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failure as the corresponding circumferential Kirchhoff stress, or the Cauchy stress for the incompressible case (112 MPa in the present case). Another possibility is the strain energy at failure, on the assumption that any cell breakage in processing would require the energy stored in the cell wall to exceed some critical level. The strain energy at failure Wf for the incompressible case is given by Wf =

2E 2 2 (H1f + H2f + H1f H2f ) 3

(47)

where H1f and H2f are the Hencky strains at failure in the meridian and circumferential direction, respectively. The value for the data shown in Fig. 2 is 28 MJ m−3 . A key assumption of this work is that the compressions are fast enough that water loss from the cells can be ignored in the analysis. This allows fitting of just two parameters (E and s ). This assumption can be tested by using simulations in which the cell wall was taken to be permeable, following the method of Smith et al. (1998) in which the volume of the cell was adjusted for water loss during the compression. Fig. 7 shows that the simulations for no water loss and for a compression speed of 68 m s−1 are very close and that both differ significantly from the behaviour at a lower compression speed of 8 m s−1 . This confirms that it was reasonable to assume negligible water loss during compression for speeds of 68 m s−1 (and above). Furthermore, Stenson (2008) has shown by experiment that estimates of the elastic modulus are not dependent on compression speed for speeds of 68 m s−1 and above, at least on a population mean basis. This provides further confirmation that the approach taken here is acceptable. It is not possible at present to test the assumption of a constant Poisson's ratio for all strains (Eq. (14)), but it is possible to simulate the effect of choosing different values of this parameter on the force–deformation curves. Fig. 8 shows such simulations for given values of all the other parameters, and it would seem that the effect is significant for a sensible range of Poisson ratios. If the best fit to the experimental data is found for each Poisson ratio, the values of the elastic modulus are 238 MPa ( = 0.3) and 205 MPa ( = 0.4), compared to the value for the incompressible case

Fig. 8. The effect of the Poisson's ratio on the force–deformation curves obtained from the Hencky strain numerical simulation. (Model parameters: elastic modulus 172 MPa, initial stretch ratio 1.04, cell radius 2.28 m).

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(172 MPa). The elastic modulus found by fitting this or any other models (including finite element analysis) to experimental data are clearly dependent on the choice of Poisson's ratio as well as the cell wall thickness, neither of which are particularly amenable to experimental determination. It is possible some alternative method of characterising the mechanical properties of yeast cells would resolve these issues, and such a method should be sought. In any case, the dependence on Poisson's ratio is not great, especially at lower strains. Furthermore, the modulus values found here were determined in a consistent manner. They might therefore be used in other studies, for example in investigating how cell wall material properties depend on cell wall composition and architecture.

Wf xi

7. Conclusion

 ij axial , trans

A Hencky strain model with constant elastic modulus has been developed that is capable of extracting intrinsic material properties from compression experiments performed on single suspended yeast cells. It is believed that this is a more correctly formulated analytical model for this situation than those that have been used previously. It should also apply to cells other than yeast cells, and to microcapsules. The model should be valuable in studying the mechanisms of cell disruption in bioprocessing, and in improving current understanding of yeast cell wall mechanics.

Notation B C C1 , C2 e1 , e2 E Ei f1 , f2 , f3 F h h0 Hi H1f , H2f

I1 , I2 J Lp P r0 Si Sij

T1 , T2 W W

material constant in the STZC strain energy function, J m−2 material constant in the STZC strain energy function, J m−2 material constants in the Mooney–Rivlin strain energy function, Pa Green strain in the STZC strain energy function, dimensionless elastic modulus, Pa principal component of Green strain in direction i, dimensionless functions of the principal tensions defined in Eqs. (28a)–(28c), N m−1 force on cell, N cell wall thickness during compression, m initial (uninflated) cell wall thickness, m principal component of Hencky strain in direction i, dimensionless Hencky strains at failure in the meridian and circumferential direction, respectively, dimensionless strain invariants in the STZC and Mooney–Rivlin strain energy function, dimensionless Jacobian determinant, dimensionless hydraulic conductivity, m s−1 Pa−1 turgor pressure, Pa uninflated cell wall radius, m principal component of 2nd Piola–Kirchhoff (PK2) stress in direction i, Pa second Piola–Kirchhoff (PK2) stress in the jth direction, on the plane normal to the ith direction, Pa tension in the meridian and circumferential directions respectively, N m−1 strain energy function, Pa strain energy function, J m−2

jxi / jXk , jxj / jXl X Xi z

strain energy at failure, Pa ith co-ordinate of a material particle in the deformed (or final) configuration, m components of the deformation gradient tensor, dimensionless fractional deformation, dimensionless ith co-ordinate of a material particle in the undeformed (or initial) configuration, m half the displacement of the probe, m

Greek letters

i ij kk v  ¯  i s    i ij i 

the angle of the point on the edge of the contact region between the compression surface and the cell following compression, rad  = 2 sin  (Eq. (24)), dimensionless Kronecker delta, dimensionless axial strain, transverse strain in uniaxial, infinitesimal strain, dimensionless principal infinitesimal strain component in direction i, dimensionless infinitesimal strain in the jth direction, on the plane normal to the ith direction, dimensionless summation of the three strain components (equivalent to volumetric strain), dimensionless bulk or volumetric strain, dimensionless vertical coordinate of cell wall, m distance between the compression surface and the equatorial plane, m derivative of  with respect to , m rad−1 principal stretch ratio in direction i, dimensionless initial stretch ratio, dimensionless Lamé's constant, Pa Lamé's constant, Pa Poisson's ratio, dimensionless horizontal coordinate of cell wall, m Cauchy stress in direction i, Pa Cauchy stress in the j th direction, on the plane normal to the ith direction, Pa Kirchhoff stress in direction i, Pa the angular position of a point on the cell wall from the vertical axis of symmetry prior to compression, rad derivative of  with respect to , dimensionless

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