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Microscale modeling of coupled water transport and mechanical
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deformation of fruit tissue during dehydration
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Solomon Workneh Fanta1, Metadel K. Abera1, Wondwosen A. Aregawi1, Quang Tri Ho1,
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Pieter Verboven1, Jan Carmeliet3,4, Bart M. Nicolai1,2
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BIOSYST-MeBios, KU Leuven, Willem de Croylaan 42, B-3001, Leuven, Belgium
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Flanders Centre of Postharvest Technology (VCBT), Willem de Croylaan 42, B-3001,
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Leuven, Belgium
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Strasse 15, 8093 Zürich, Switzerland
Building Physics, Swiss Federal Institute of Technology Zurich (ETHZ), Wolfgang-Pauli-
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Testing and Research (Empa), Überlandstrasse 129, 8600 Dübendorf, Switzerland
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* Corresponding author: Bart M. Nicolai
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MeBioS, KU Leuven
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Willem de Croylaan 42
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B-3001 Leuven
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BELGIUM
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Tel. +32 (0)16 322375
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Fax. +32 16 322955
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Email: [email protected]
Laboratory for Building Science and Technology, Swiss Federal Laboratories for Materials
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Keywords: structure, turgor, diffusion, mechanics, cell wall
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1
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ABSTRACT
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Water loss of fruit typically results in fruit tissue deformation and consequent quality loss. To
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better understand the mechanism of water loss, a model of water transport between cells and
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intercellular spaces coupled with cell deformation was developed. Pear (Pyrus communis L.
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cv. Conference) was chosen as a model system as this fruit suffers from shriveling with
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excessive water loss. A 2D geometric model of cortex tissue was obtained by a virtual fruit
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tissue generator that is based on cell growth modeling. The transport of water in the
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intercellular space, the cell wall network and cytoplasm was predicted using transport laws
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using the chemical potential as the driving force for water exchange between different
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microstructural compartments. The different water transport properties of the microstructural
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components were obtained experimentally or from literature. An equivalent microscale model
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that incorporates the dynamics of mechanical deformation of the cellular structure was
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implemented. The model predicted the apparent tissue conductivity of pear cortex tissue to
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be 9.42 ± 0.40  10-15 kg.m-1.s-1.Pa-1, in the same range as those measured experimentally.
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The largest gradients in water content were observed across the cell walls and cell
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membranes. A sensitivity analysis of membrane permeability and elastic modulus of the wall
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on the water transport properties and deformation showed that the membrane permeability
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has the largest influence. The model can be improved further by taking into account 3-D
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connectivity of cells and intercellular pore spaces. It will then become feasible to evaluate
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measures to reduce water loss of fruit during storage and distribution using the microscale
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model in a multiscale modeling framework.
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2
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List of symbols
b
cψ
D
E
F
J
Keff
k
l
L
mi
Pm
R
T
Vw
v
x
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50
Damping factor
Water capacity
Diffusion coefficient
Young’s modulus of elasticity
Total force acting upon the node
Water flux
Water conductivity
Spring constant
Cell wall length
Thickness of the simulated tissue
Mass of the vertex
Permeability of the membrane
Universal gas constant (8.314)
Temperature
Molar volume of water (18*10-6)
Velocity
Position
kg.kgDM-1Pa -1
m2.s-1
Pa
N
kg.m-2
kg m-1Pa-1 s-1
N.m-1
m
m
kg
m.s-1
J.mol−1.K−1
K
m3.mol−1
m.s-1
m
Strain
Density of water
Dry matter density
m.m-1
kg.m-3
kg DM.m-3

Stress
Water potential
Pa
Pa
Subscripts
a
c
i
m
s
T
w
Air
Cell
Node
Cell membrane
Solute
Total
Cell wall
Greek symbols

w
 DM

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3
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1 Introduction
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Fresh fruits are mostly composed of water, the unique universal solvent that is fundamentally
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important in all life processes. Water loss equates to loss of saleable weight, and thus means a
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direct loss in revenue, as well as affects overall fruit quality. Measures that minimize water
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loss after harvest will usually enhance profitability. Loss in weight of only 5 per cent will
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cause many perishable commodities to appear wilted or shriveled (Wills et al., 1998). Texture
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is one of the most important quality attributes of fruit and vegetables. Most plant materials
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contain a significant amount of water and other liquid-soluble materials surrounded by a
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semi-permeable membrane and cell wall. The texture of fruits and vegetables is dependent on
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the turgor pressure, and the composition of individual plant cell walls and the middle lamella,
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which “glues” individual cells together (Barrett et al., 2010). Cell walls are accepted as the
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main structural component affecting the mechanical properties of fruits and vegetables
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(Zdunek and Konstankiewicz, 2004; Bourne, 2002; Waldron et al., 2003; Vanstreels et al.,
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2005). Also the turgor pressure, cell size and shape, volume of vacuole and volume of
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intercellular spaces, chemical composition have a major influence on tissue strength and
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macroscopic fruit firmness (Oey et al., 2007).
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Shrinkage is one of the major physical changes that occur during the dehydration process. It
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results from the collapse of cells during water evaporation, which has a negative impact on
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the quality of dehydrated product. At first, shrinkage causes changes in the shape of the
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product. These changes are due to the stresses developed while water is removed from the
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material. (Rajchert and Rzace, 2009). Shrinkage during dehydration can be classified in three
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different types (Gekas, 1992): one-dimensional when the volume change follows the
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direction of diffusion; (2) isotropic or three-dimensional; and (3) anisotropic or arbitrary.
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Volume reduction patterns for fruits and vegetables are often of type 3 and in a less extent of
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type 2. Shrinkage of apple parenchyma, for example, was found to be highly anisotropic
4
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(Mavroudis et al., 1998; Moreira et al., 2000). Cellular shrinkage during dehydration has been
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observed during osmotic dehydration of parenchymatic pumpkin tissue (Mayor et al., 2008),
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apple (Lewicki & Porzecka Pawlak, 2005) and convective drying of grapes (Ramos et al.
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2004).
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With respect to modeling mechanical deformation of fruit tissue, most models are based on
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continuum mechanics. It is often assumed that the biologic material behaves as a nonlinear
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viscoelastic continuum. Recent work has allowed better understanding and modeling the
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nonlinear shrinkage of fruit tissue at the macroscale (Aregawi et al., 2012; Defraeye et al.,
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2013). Most microscopic works on the deformation are based on single cell analysis. Feng
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and Yang (1973) considered the problem of the deformation and the consequential stresses in
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an inflated, non-linear elastic, gas-filled spherical membrane compressed between two
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frictionless rigid plates. Lardner and Pujara (1980) extended this model further by
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considering the sphere to be filled with an incompressible liquid rather than gas. Their model
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was able to predict accurately the deformation of sea urchin eggs, as previously reported by
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Yoneda (1973). Liu et al. (1996) improved the computational algorithm, and applied the
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model to data on microcapsules. None of these studies allowed for water loss from the
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sphere. Smith et al. (1998) created a finite element model in which volume loss was included,
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and applied this to compression data from yeast cells (Smith et al., 2000). Using a finite
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element method, it was possible in principle to consider any cell wall material constitutive
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equation, although in practice Smith et al. (2000) only considered the linear elastic case. The
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more recent work is by Dintwa et al. (2011) who developed a finite element model to
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simulate the compression of a single suspension-cultured tomato cell, using data from Wang
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et al. (2004). The model could serve as a basic building block for more complex models for
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tissue deformation under mechanical loading. The model was limited to mechanical loading
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and not to the deformation due to water loss and also was applied for single cell and not for a
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real tissue. We have recently developed a cell growth algorithm that generates representative
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in silico fruit tissue geometries from increasing cell turgor in and cell wall generation by the
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individual cells in a tissue (Abera et al., 2013a). Using this algorithm in the reversed sense,
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it becomes possible to perform simulations of the deformation mechanics of tissue as a result
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of hydrostatic stress occurring during water loss.
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We previously also modeled water diffusion in pear fruit tissue samples taking into account
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the cellular structure of the tissue (Fanta et al., 2013). Shrinkage was, however, taken into
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account in a static manner by considering different equilibrium states at different water
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contents, using a global shrinkage coefficient the model lacks to incorporate dynamic
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deformation due to water loss. Using the cell mechanics algorithm, however, the simulation
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of the deformation of individual cells in the tissue is possible.
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The aim of the present work was to combine and apply the microscale transient water
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transport model with the cell mechanics model for predicting cell and tissue deformation due
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to water loss in the actual microstructural architecture of the tissue. The model was also used
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to calculate the apparent water conductivity of the tissue. Pear fruit (Pyrus communis L. cv.
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conference) was used as a model system. Pears quickly deform resulting in shriveling as a
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consequence of water loss during low temperature storage (Nguyen et al., 2006).
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2 Model formulation
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2.1 Microscale model of water transport coupled with deformation
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2.1.1 Microscale water transport model
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Cortex tissue of pear consists of an agglomerate of cells and intercellular spaces of different
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shapes and sizes (Verboven et al., 2008). To take into account this microstructure, we have
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introduced the microscopic layout into the modeling as the computational geometry of the
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model.
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The transport of water in the intercellular space, the cell wall network and cytoplasm were
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modeled using diffusion laws and irreversible thermodynamics (Nobel, 1991). The full
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derivation of the diffusion equation for the tissue compartment (cell, cell wall and
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intercellular space) can be found in our previous work (Fanta et al., 2013). For the cells, the
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unsteady-state model of water transport reads:
(  DM ,c  xc
 DM ,c
xc
)c ,c
 DM ,c c ,c
 c
   Dc (
) c
t
1  xc
(1)
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where  DM ,c is the dry matter density of the cell (kg DMm-3),
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content (kgkg DM-1), Dc the water diffusion coefficient inside cells (m2s-1), c ,c the water
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capacity (kgkg
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unsteady-state diffusion model is given by:
DM
-1
xc the dry matter base water
Pa-1), and  c the water potential of the cell (Pa). For the cell wall, the
c , w  DM , w
 w
    DM , w Dw c , w w
t
(2)
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where c ,w is the water capacity of the wall (kgkg DM-1 Pa-1),  DM , w the dry matter density
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(kg DMm-3),  w the water potential (Pa), and Dw the water diffusion coefficient (m2s-1).
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Unsteady diffusion in the air phase is described by:
c , a  DM , a
 a
    DM ,a Da c ,a  a
t
(3)
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c ,a
is the water capacity of the air (kgkg
-1
Pa-1),  DM , a the dry matter density
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where
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(kg DMm-3),  a the water potential (Pa), and Da the water diffusion coefficient (m2s-1).
DM
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A simple flux law was applied at the cell membrane (Nobel, 1991):
J
w PmVw
( c  w )
RT
(4)
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with J the water flux (kg m-2),  w density of water (kg m-3), Pm the membrane permeability
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(m s-1), Vw the molar volume of water (1810-6 m3mol-1), R the universal gas constant (8.314
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J mol-1K-1 ), T the temperature (K),  c the water potential at the cytoplasma (interior) side of
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the membrane (Pa), and
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membrane (Pa).
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2.1.2 Microscale mechanics model
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A cell micromechanics model was used to calculate the deformation as a result of turgor loss
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of the individual cells in the tissue as a result of water transport (Abera et al., 2013a). In the
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model, the cell is represented as a closed thin walled structure, maintained in tension by
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turgor pressure. The cell boundary is represented as a set of walls (modeled as springs)
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connected at points called vertices. The cell walls of adjacent cells are modeled here as
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parallel and linearly elastic elements which obey Hooke's law, an approach similar to that
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taken in other plant tissue models (Rudge and Haseloff, 2005; Dupuy et al., 2008, 2010). The
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shrinkage mechanics is modeled by considering Newton’s law. The following system of
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equations is solved for the velocity and position of the vertices i of the cell wall network:
 w the water potential at the (exterior) cell wall side of the
mi
dv i
 FT ,i
dt
dxi
 vi
dt
(5)
(6)
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where mi is the mass of the vertex (kg) which is assumed to be unity in order to simplify the
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model , which makes the rate of change of velocity (acceleration) of the vertices equal to the
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net force acting on the vertex; xi (m) and vi (ms) are the position and velocity of node i,
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respectively, and FT ,i is the total force acting upon this node (N). Cell shrinkage or growth is
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then the result from the action of forces caused by a decrease or increase, respectively, of
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turgor pressure acting on the cell wall. The water potential of each cell from water transport
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simulation can be converted to turgor pressure using the relationship outlined below. The
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resultant force on each vertex, the position of each vertex, and, thus, the shape of the cells is
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then computed as follows. The total force acting on a vertex is given by the formula
FT 
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F
wW
w
 Fd
(7)
where Fw are forces contributed by the set of walls w incident to the vertex, and
Fd  bv
(8)
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is a damping force, expressed as a product of a damping factor b and the vertex velocity v .
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The force Fw is the resultant of the net turgor pressure force Fturgor between the two adjacent
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cells working normal to the wall and the force associated with it.
Fw  Fturgor  Fs
(9)
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The net turgor force on the vertex is calculated by taking the difference in turgor pressure
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Pcell of the two adjacent cells multiplied by half the length of the wall as it is divided by the
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two incident vertices defining the wall:
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(10)
1

Fturgor   ( Pcell ,1  Pcell ,2 )l  n
2

with n the unit normal vector to the cell wall, while l is the actual cell wall length at the
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current time. The force Fs acts along the wall and its magnitude is determined by Hooke's
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law
Fs  ku
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(11)
where u the net vector of the cell wall,
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u  l  ln
(12)
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with ln the natural length of the unpressurised cell wall (m) and k the spring constant (N/m).
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The latter is calculated from the Young's modulus of elasticity E (Pa) by dividing the tensile
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stress  (Pa) by the tensile strain  (mm-1) in the elastic (initial, linear) portion of the
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stress-strain curve
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(13)
 Fs A0 Fs l0



u l0
A0u
where Fs is the force exerted on an object; A0 is the original cross-sectional area through
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which the force is applied (m2) ; and l0 is the original length of the object (m). Hooke's law
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can then be derived from this formula, which describes the stiffness of an ideal spring:
E
Fs 
EA0
u   ku
l0
(14)
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so that
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(15)
EA0
l0
To find the positions of each vertex of all cell walls of every single cells and, thus, the shape
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of the cells with time, a system of differential equations for the positions and velocities of
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each vertex were established and solved using a Runge-Kutta fourth and fifth order (ODE45)
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method.
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2.1.3 Model coupling
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The transient water transport model is solved for certain time steps and the water loss results
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in loss of water potential in the cells. This change in water potential of the cells results in loss
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of turgor pressure. This is only true for the high range of equilibrium relative humidity values
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of the cells during dehydration until turgor drops to zero. When tissue drying proceeds at
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lower values of relative humidity, cell protoplasts will detach from the cell walls creating air
k
10
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spaces inside the cell wall and collapse of the cell wall, effectively changing the water
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transport and deformation mechanisms. This process is not calculated here. The dehydration
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is thus carried out in the relative humidity range of 99 to 97.7%. Below this value of relative
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humidity the turgor pressure is zero and the osmotic potential will be equal to the water
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potential. The osmotic potential  s can be obtained from Eq. 16.
 s  Cs RT  
w
mw
ns RT  
 w ns
xc ms
RT
(16)
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The turgor pressure is then equal to
 p   c  s
(17)
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The relationship between turgor pressure and water potential is shown in Figure 1 for the
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range considered.
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2.2 Initial tissue geometry
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The initial geometry of the tissue was generated in silico using the microscale mechanics
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model outlined in section 2.1.2. A Voronoi tessellation was used to generate a start topology
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of the cells. Anisotropic cell expansion then resulted from turgor pressure acting on the
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yielding cell wall material until full turgor (1 MPa) was reached. The size of the initial
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geometry was 200×200 μm and contained 60 Voronoi cells. The resulting geometry was
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750×550 μm and had an average cell area of 8.28±0.97 μm2 and a porosity of 6.42%. More
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details can be found in Abera et al. (2013a) who showed that this procedure yields tissue
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geometries that correspond well to actual ones visualised by synchrotron tomography.The
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cell wall was defined by shrinking the cell geometry until the desired cell wall thickness was
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obtained. The cells, the pores and the cell walls were then exported as separate bodies so that
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different material properties could be specified.
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2.3 Model parameters
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The water transport model parameters were obtained from our previous work (Fanta et al.,
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2012; Fanta et al., 2013) and are listed in table 1. The water content of the cell wall and
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cytoplasm was calculated from their moisture isotherms (Fanta et al., 2012; Fanta et al.,
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2013). The mechanical properties were determined by Abera et al. (2013a) and are listed in
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table 2. We have assumed the cross sectional area of the cell wall to be 1 µm2, and the
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average of the initial resting length of the cell walls obtained from the Voronoi tessellation
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was used to calculate the k value.
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The nominal values in the tables were used for the simulations, while also a sensitivity
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analysis was conducted with respect to the main parameters, as explained below.
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2.4 Implementation details
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The geometric models of pear cortex tissue constructed by Abera et al. (2013) were imported
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into Comsol Multiphysics 3.5a (Comsol AB, Stockholm, SE) for numerical computation of
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the water exchange using the model equations outlined above. Meshing was performed
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automatically by the Comsol mesh generator and produced 303,824 quadratic elements with
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triangular shape by the automatic Comsol mesh generator.
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The procedure is outlined in Fig. 2. A sequence of time steps was considered for solving the
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coupled moisture transport and mechanical deformation model. In every time step the
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transient moisture potential field was solved. The non-linear coupled model equations were
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discretised over the discretisation mesh using the finite element method. A direct solver was
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applied for solving the resulting set of ordinary differential equations with an accuracy
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threshold less than 10-6. The results of the simulations were the water potential and water
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content distribution in the tissue samples as well as the water flux through the sample for a
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given water potential gradient across the sample. The sample was the entire tissue geometry
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used in the computations. Subsequently the corresponding mechanical equilibrium was
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calculated using a dedicated Matlab code (Matlab 7.6.0, The Mathworks, Natick, MA). The
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whole system of equations was numerically solved using a Runge-Kutta method of order 4
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and 5. The simulation was iterated until a mechanical equilibrium state was reached. This
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equilibrium was assumed once the velocity of all points was below a given threshold, as the
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velocities would go to zero only when the system would be at a steady state. The resulting
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tissue geometry was then introduced and meshed again in Comsol and the next time step was
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initiated. In total four time steps of 50 s or eight time steps of 25 s were required to reach
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equilibrium.
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The initial water transport calculation was done on the initial pear cortex tissue geometry that
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was obtained using virtual fruit tissue generator with simulation time of 50 seconds. From
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this simulation, the water potential of each cells was obtained and converted in to turgor
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pressure using the relation shown in Fig. 1. Then these set of turgor pressures are used in the
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shrinkage mechanics (presented in section 2.2.2) to find the new equilibrium configuration of
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the cells.
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Computation time was 2 hours for the unsteady state water transport simulation in each time
263
step on a 8 GB of RAM quad-core PC, and 25 seconds for the mechanical equilibrium
264
calculations.
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2.5 Apparent water conductivity of pear cortex tissue
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In silico analysis was carried out to study microscale water exchange in pear fruit tissue. A
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difference in relative humidity of 99/97.7% was applied to the top and bottom of the tissue
268
geometry, respectively, while the other two lateral boundaries were defined to be insulated.
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The macroscopic water conductivity K eff (kg m-1Pa-1 s-1) of the tissue sample was computed
270
at steady state from
13
K eff   J
L

(18)
271
272
with J (kg m-2 s-1) the total flux through the fruit tissue,  (Pa) the assigned water
273
potential difference between the two opposite sides and L (m) the thickness of the simulated
274
tissue. The minus sign indicates that the water diffuses from high to low potential.
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2.6 Sensitivity analysis
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A sensitivity analysis was performed to study how sensitive a particular predicted model
277
output was with respect to changes in model parameters. First, the effect of tissue geometry
278
was evaluated. Hereto different tissues were randomly generated using the cell growth
279
algorithm (Abera et al., 2013a). Then, effects of variability in tissue properties were
280
quantified. The membrane permeability (Pm) was expected to have the highest influence on
281
the tissue conductivity (Fanta et al., 2013), while the elastic modulus of the cell wall (E) had
282
the largest influence on the mechanics of the tissue (Abera et al., 2013a). Here, we thus
283
investigated the effects of both parameters on the dynamic dehydration process. The
284
perturbation of the parameters was taken to be 10 times of the nominal value of Pm which has
285
been shown to vary in a large range (2.5-3000 µ m s-1, Fanta et al., 2013) and 10% of the
286
nominal value of E.
287
3 Results
288
3.1 Water potential and water content profile of pear tissue during dehydration
289
Model simulations that incorporate the dynamics of mechanical deformation of the cellular
290
structure were performed on the fruit cortex tissue samples, where a difference of RH (99-
291
97.7%) across the tissue samples was applied (Fig. 3) such that the bottom of the sample is
292
dehydrated to completely remove the cell turgor (1 MPa). Cells at the same position in the
14
293
gradient tend to have similar and uniform water potential, which is logical due to the high
294
water conductivity inside the cells. Gradients mainly exist from one cell to another in the
295
direction of the applied gradient.
296
The water content of the cells remains relatively uniform and constant during the dehydration
297
process. This is logical as the cells shrink upon water loss, but do not change drastically their
298
water content.
299
3.2 Deformation of pear tissue during dehydration
300
During the dehydration process, there is a relatively large shrinkage at the early time and no
301
significant change after 200s of simulation which indicates that steady state has been
302
achieved. This could be indeed the case, considering the small sample size considered (0.6
303
mm thickness). The high shrinkage at the early time was also observed on dehydration of
304
grape tissue by Ramos et al. (2004).
305
The deformation of the individual cells in the tissue can be seen in Fig. 3. Deformation occurs
306
in all dimensions as the turgor loss in the cells relaxes the entire cell wall, but locally depends
307
on the turgor difference between adjacent cells Shrinkage of the cells is more severe on the
308
drying bottom side of the sample than on the top equilibrium side. As the tissue is a dynamic
309
assembly of pressurised cells that are in mechanical equilibirum, deformation of one cell will,
310
however, affect the shape and size of other cells as well. Therefore, all cells in the tissue will
311
deform to some extent. The reduction in the overall cell size distribution is evident from Fig.
312
4.
313
The relative deformation resulting from simulations with time steps of 25s and 50s is shown
314
in Fig. 5. The relative error is less for the final time and higher for initial time this is because
315
moisture loss is higher at early times and at the end of 200 s steady state will be achieved.
15
316
The relative average cell area in steady state is around 60%, and identical for the two
317
simulations with the different time steps.
318
3.3 Validation of the overall shrinkage of the sample
319
As a validation we calculated the actual water loss of pear from the desorption isotherm that
320
we measured previously (Nguyen et al., 2004) at 99% and 98.25% (average RH for the
321
simulation) and found it equal to 35.62%. By assuming that shrinkage is proportional to water
322
loss this thus corresponds to a volume loss of 35.62%. This value is comparable with the
323
proportional volume loss of 38.47% that was calculated from the simulation.
324
3.4 Average profile of the water potential and water content
325
The average water potential and water content is calculated based on the potential and water
326
content distributions across the tissue. Fig. 6a shows how the water potential decreases with
327
time and reaches an equilibrium at 200 s. The drastic decrease in water potential is in line
328
with Fig. 3.
329
Fig. 6b displays the average water content change, which in relative terms is much smaller
330
than the change in water potential. This clearly indicates that early stages of slow dehydration
331
processes are not necessarily associated with large changes in water content, but rather will
332
result in deformation due to loss of turgor. Fig. 6c shows the average water content change of
333
the wall with time.
334
3.5Deformation of different tissue structures
335
The macroscopically observed variation in water conductivity was linked to variability in the
336
microstructure of the tissue. Simulations for three different pear cortex tissue structures
337
randomly generated in this study showed that differences in porosity (6.42%, 5.94%, 5.39%),
338
connectivity and cell distribution affected the water transport in the tissue to a very minor
339
extent (Fig. 7). The difference in conductivity between the different tissue samples is small
16
340
(9.44 10-15, 9.35 10-15and 9.19 10-15 kg.m-1.s-1.Pa-1). It also had a minor effect on the
341
average water potential and water content (Fig. 6). A similar result has been obtained for a
342
microscale model of gas exchange in fruit tissue (Ho et al., 2009).
343
3.6 Sensitivity analysis
344
Effect of membrane permeability
345
The sensitivity of the tissue deformation, water potential and water content to the membrane
346
permeability is shown in Figure 8. In previous work, this parameter was identified to have a
347
major effect on tissue conductivity. It is seen that the membrane permeability during dynamic
348
water loss also has a large influence on the relative cell average area, water potential and
349
water content. Reported values of membrane permeability cover several orders of magnitude.
350
Segui et al. (2006) reported values higher to 3000 µm s-1, while other authors obtained values
351
that were much lower, from 2.5 to 500 µ m s-1 on different plant tissues (Ramahaleo et al.,
352
1999; Ferrando et al., 2002; Suga et al., 2003; Moshelion et al., 2004; Murai-Hatano and
353
Kuwagata, 2007; Volkov et al., 2007). Dedicated experiments will be needed to better
354
understand the effect of membrane permeability of fruit cortex cells.
355
Effect of elastic modulus of the cell wall
356
The sensitivity of water loss and deformation to the elastic modulus of the wall for simulation
357
coupled with dynamic deformation is shown in Fig. 9. It is seen that the elastic modulus of
358
the wall has less influence on the water potential and water content, but a more significant
359
influence on the deformation.
360
3.7 Prediction of tissue conductivity with deformation
361
The effective conductivity of the entire tissue in steady state was calculated from the obtained
362
fluxes for the RH levels of 99-97.7% and temperature of 25°C. The values correspond well
17
363
with the range of measured values of tissue conductivity at 25°C as can be seen in Table 3.
364
From the previous work of (Fanta et al., 2013), the effective conductivity is increasing
365
towards the high range of experimental effective conductivity, probably due to the larger
366
deformation than estimated in our previous work. The smaller deviation of predicted effective
367
conductivity values may be attributed to the fact that the 2D model is unable to take into
368
account the 3D connectivity of the pore space (Ho et al., 2009). Measured values of the
369
effective water conductivity of pear tissue showed a large variation (Table 3).
370
4 Discussion
371
Microscale water transport in fruit tissue was coupled with dynamic deformation using
372
mechanistic models, and used to calculate cell deformation as well as effective tissue
373
properties such as tissue conductivity as a basis for a multiscale model of water transport in
374
fruit (Ho et al., 2011). Compared to previous work on deformation of single cells (Feng and
375
Yang, 1973; Lardner and Pujara, 1980; Smith et al., 2000; Dintwa et al., 2011) the present
376
work allows to compute deformation of cells in a tissue under water loss. The tissue context
377
is relevant because loss of turgor of one cell will affect neighboring cells as well. Indeed, cell
378
wall tension and extension is a result of the balance of pressures in the two neighboring cells
379
of the cell wall. As a consequence, when turgor is lost, the cell wall relaxes deforming all
380
touching cells.
381
The deformation also occurs in all dimensions because the turgor acts on all cell walls
382
surrounding the cell. Therefore, even if the water transport is essentially one directional (in
383
the case studied here), deformation of the tissue will also occur in directions perpendicular to
384
that of the global water flux through the tissue. This will be a main reason why fruit shrivel
385
upon water loss (Nguyen et al., 2006; Veraverbeke et al., 2003): the cuticle can be regarded
386
as a relatively inelastic material that will maintain its surface area during water loss, while the
18
387
underlying cell layers tend to shrink in every dimension. The present model predicts this
388
behavior.
389
The water transport modeling approach presented here is similar to that presented by Esveld et
390
al. (2012a and b), who used a microscale model to investigate effective water transport
391
properties of complex porous foods. While the work of Esveld and coworkers considered
392
vapour transport and sorption into the solid matrix, here we considered dehydration from
393
plant cells under turgor confined by a cell wall matrix. In addition, deformation was included.
394
In our approach, the changing structure of the tissue directly affects also the water loss, which
395
has not been achieved in any previous work to our knowledge.
396
The value of the effective tissue conductivity obtained from the simulation was closer to the
397
experimental average of late picked pears compared to those obtained from the previous work
398
(Fanta et al., 2013) that incorporated deformation in a static manner. The values from static
399
and dynamic simulations are significantly different indicating the difference in results of the
400
static approach and more physically correct dynamic approach that includes a mechanic
401
deformation model.
402
The water content of the wall decreased faster compared to the average water content of the
403
tissue. The dry weight basis equilibrium water content of the cell is initially high as observed
404
in sorption isotherm experiments (Fanta et al., 2013). It appears that the cell wall is an
405
important component to quickly equilibrate the tissue system to a new water status. The
406
consequences of changes in water content on the cell wall mechanics were not included in
407
this work. However, for the high relative humidity range considered in this work this should
408
not present a major issue, as shown in the sensitivity analysis. Understanding cell wall
409
structure and mechanics remains a major area of plant research as evidenced in recent
410
literature (Yi and Puri, 2012; Dyson and Jensen, 2012; Park and Cosgrove, 2012).
19
411
The sensitivity of the simulation results to cell membrane permeability showed that it has a
412
large influence on the water transport and deformation. Correct understanding and
413
measurement of cell membrane permeability is still an active area of research (Ramahaleo et
414
al., 1999; Ferrando et al., 2002; Suga et al., 2003; Moshelion et al., 2004; Segui et al., 2006;
415
Murai-Hatano and Kuwagata, 2007; Volkov et al., 2007)
416
The 2D microscale model coupled with deformation lacks to incorporate connectivity of cell
417
walls and air spaces that can affect considerably the transport phenomena (Ho et al., 2009;
418
Ho et al., 2011). Further advances require that 3D modeling of water transport and cell
419
mechanics in the microstructure of the tissue is targeted to explain the effect of
420
interconnectivity on the macroscopic water transport of a tissue. Thereto, 3D imaging of the
421
cellular tissue with advanced image analysis or 3D tissue modeling will be required to
422
provide representative 3D cellular models. The latter is currently under progress based on the
423
2D cell growth algorithm (Abera et al., 2013b). 3D imaging of fruit tissue is possible using
424
X-ray microtomography; however, the image processing to obtain 3D models is still
425
cumbersome (Verboven et al., 2008; Ho et al., 2011; Herremans et al., 2013).
426
5 CONCLUSIONS
427
A microscale water transport model coupled with mechanical model in pear cortex tissue was
428
presented and incorporated water transport and mechanical properties of the pores, cell walls,
429
cell and cell membranes and the representative tissue morphology. The model predicted the
430
effective tissue conductivity of pear cortex tissue in the same range as those measured
431
experimentally but was significantly different from other modeling approaches that ignored
432
modeling the deformation mechanics. The model presents a major step towards better
433
understanding of the changes in tissues during dehydration that result in complex phenomena
434
such as shriveling during storage or large scale deformations during drying. Next steps
20
435
include elaborating the model in 3D, application to different tissue types and extending the
436
model formulation to cope with more severe dehydration relevant to drying processes.
437
ACKNOWLEDGEMENTS
438
Financial support by the Flanders Fund for Scientific Research (project FWO G.0603.08),
439
KU Leuven (project OT 08/023, project OT/12/055) and the EC (project InsideFood FP7-
440
226783) is gratefully acknowledged. The opinions expressed in this document do by no
441
means reflect their official opinion or that of its representatives.
442
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