New Phytologist Supporting Information Figs S1–S3, Table S1 and Methods S1
Article title: Pressure volume curves: Revisiting the impact of negative turgor during cell collapse by
literature review and simulations of cell micro-mechanics.
Authors: Yiting Ding, Yanxiang Zhang, Quan-Shui Zheng and Melvin T. Tyree
Article acceptance date: 30 March 2014
The following Supporting Information is available for this article:
Fig. S1 The structure of a capsule in our models: a mesh of volume elements and XYZ coordinates.
Fig. S2 Simulation of a pressure-volume curve of a single cell the size of an Allium cell (radius 25 µm,
length 156 µm, wall thickness 1 µm); the value of E was adjusted 300MPa to yeild ε=9.11MPa.
Fig. S3 The water potential isotherm calculations of Allium cell derived from Fig. S2. Actual values are
the values of Osmotic Pressure πc and Turgor Pt from the simulations. The derived values are what can be
computed from classical PV-curve analysis.
Table S1 Sensitivity analysis of output = percent change in πc. MAL = maximum arc length increment
Inputs
MAL
Element size
Imperfection
Young’s Modulus
Thickness
Output change
caused by 5% change of input
-0.02%
0.08%
-0.26%
5.10%
18.785%
Output change
caused by 50% change of input
-0.24%
-0.15%
-2.31%
52.43%
-91.949%
Notes S1 Notes regarding Table S1 Sensitivity analysis.
ABAQUS is considered to be the most reliable commercial Finite Element Method (FEM) software to
simulate nonlinear problems with large deformation. It is difficult to obtain nonlinear static equilibrium
solutions for post-buckling analysis where the load may decrease as the solution evolves. The modified
Riks method is an algorithm that allows effective solutions for such problems. That is why ABAQUS and
Riks algorithm are chose here.
During the execution of the FEM the input turgor pressure, Pt, was used to calculate model
outputs: cell volume, vc, cell osmotic pressure, πc (Eqn 2), and simulated balance pressure PB = c - Pt.
Then conventional pressure-volume curve analysis (of 1/PB vs 1-RWC) was used to compute ‘derived’
values of cell osmotic pressure, 𝜋𝑐∗, cell turgor pressure, 𝑃𝑡∗ , and bulk modulus of elasticity, ε*. It is
important to remember that the outputs (c and ε) are not necessarily equal to the derived values (𝜋𝑐∗ and
ε*) and also the input Pt will not necessarily equal the derived 𝑃𝑡∗ . The sensitivity analysis was used to
calculate percentage change in c =𝜋𝑐∗ − 𝜋𝑐 after a 5% or 50% change of input parameters.
Inputs of the model included geometry parameters (radius r, half cylinder length a, thickness δ),
material parameter (Young’s modulus, E, and Poisson’s ratio, ), and ABAQUS simulation parameters
(volume element size, defect size and step size). The defect size is the imperfection induced to the capsule
at the beginning of the simulation. In the Riks algorithm (for buckle and post buckle analysis) the
independent variable was the arc length; the magnitude of arc length was automatically selected by the
ABAQUS. However we can change Maximum Arc Length increment (MAL) to test the effect of arc
length increment size (step size). Pressure was applied to the structure and the displacement and vc was
calculated. The effects of r and a have been discussed in detail in the main draft. All inputs used in the
sensitivity analysis were the Robinia inputs (r = 3 μm, δ = 1 μm, a = 10 μm, E = 55 MPa,  = Poisson’s
ratio, scale of imperfection in 0.02 μm, element size = 0.2 m, and MAL = 1) were changed by 5% and
50%) and the output were calculated relatively and the changes in percentage are shown in Table S1.
Young’s modulus, E, and thickness, δ, have biggest compact on c. But E was adjusted to make
the bulk modulus, ε*, equal to literature values, but we held δ constant for all simulations.
Program S1 The Matlab programme package named ‘SupportingInformationProgamS1.rar’ included
eight *.m files. The name and usage of each file is explained in detail in Method S1 section.
Methods S1 Protocol of FEM modeling (Including Fig. S2-1~22)
1
Preparation:
1.1
Install software ABAQUS 6.12-1 (there will be error if you run with a lower version) and
MATLAB R2011b.
1.2
Confirm install path of ABAQUS, the folder of commands, e.g.
ABAQUS path = C:\SIMULIA\Abaqus\Commands
1.3
Confirm working directory of ABAQUS, e.g.
ABAQUS work path = D:\TEMP\
1.4
Confirm working directory of MATLAB, make sure the MATLAB working directory name is
without spaces, it is better to create a folder under a hardware partition root directory, e.g.
MATLAB work path = D:\buck\, unzip buckle.rar and copy the 8 *.m files (aba_gen.m,
aba_riks.m, main.m, picknd.m, setpressure.m, volcal_surf.m, volucal2.m, volucal_riks2.m) to
this directory.
2
Protocol:
2.1
ABAQUS modeling: (Important: do not change the file names!)
2.1.1 Create 3D structure:
Module Part：Create part named Part-1, 3D deformable revolution solid as shown in Fig.
S2-1. We choose ABAQUS unit sets with length unit mm and pressure unit MPa.
Approximate size set as 0.05mm i.e. 50μm. Sketch the section for revolution body in x-y
plane as shown in Fig. S2-2: The center point for the arc is (0,0.01), inner arc radius
0.002, outer arc radius 0.003, height of straight line is 0.01. Close two ends with straight
lines. Create oblique line as Y-axis. Finish sketch, select the oblique line as axis of
symmetry, revolve angle set as 90°. The revolved part is shown as Fig. S2-3. Save as
Rob-Model.cae.
Fig. S2-1
Fig. S2-2
Fig. S2-3
2.1.2 Create analytical rigid plane:
Module Part: Create part named Part-2, 3D Analytical rigid body, Extruded shell,
approximate size 0.05. We just need to draw a line on Y-axis as shown in Fig. S2-5, the
line needs to be a little longer than Part-1, here for example, two ends set as (0, -0.001)
and (0, 0.015). This rigid body is used to prevent over deformation of Part-1, so it needs
to be longer than deformed structure, we need to estimate how much the deformation will
be. Finish sketch, the depth of the sketch should be bigger than the outer radius if the arc
in Part-1, set as 0.005 here. We get a rigid shell as shown in Fig. S2-6.
Fig. S2-4
Fig. S2-5
2.1.3 Define Material Property:
Fig. S2-6
Module Property:
Create material named Material-1, ‘General’-‘Density’ set 1e+12 kg/mm3, which is
1000kg/m3; ‘Mechanical’-‘Elasticity’-‘Elastic’, isotropic, Young’s Modulus 55MPa,
Poission’s ratio 0.49, as shown in Fig. S2-7.
Create section named Section-1, Solid Homogeneous, material set as Material-1.
Assign section: assign Part-1 to Section-1.
Fig. S2-7
2.1.4 Assemble parts to instance:
Module Assembly: Create Instance, induce Part-1 once and Part-2 twice, independent,
auto off set from others, this will result in Fig. S2-8. We call the first induced Part-2
‘RB1’ short for ‘Rigid Body 1’, the second induced Part-2 ‘RB2’ short for ‘Rigid Body
2’. Translate RB1, the tip point to (0,0,0); Rotate RB2 around Y-axis for 90 degrees, and
translate tip point to (0,0,0), this will result in the structure as shown in Fig. S2-9.
Fig. S2-8
Fig. S2-9
2.1.5 Create modes analyze step:
Module Step: Create step named ‘Pressure’, insert after initial, step type set as a linear
perturbation - buckle, Eigensolver choose ‘Subspace’, Number of eigenvalues requested
set 3, for the other parameters use the default value. Maximum number of iterations could
be extended if necessary, as shown in Fig. S2-10.
Fig. S2-10
2.1.6 Interaction：
Module Interaction: Create interaction property ‘IntProp-1’, Contact, MechanicalTangential Behavior-Frictionless; Mechanical - Normal Behavior, ‘Hard’ contact, others
use default, shown as Fig. S2-11.
Fig. S2-11
Create interaction ‘Int-1’, Step-Initial, Types: Surface-to-surface contact (Standard).
Master surface choose RB1, select the side attached with Part-1; for the slave surface
choose the inner surface of Part-1, NOT the contact surface, type is surface, as shown in
Fig. S2-12; Sliding formulation: Finite sliding; for the contact interaction property choose
IntProp-1, others use default, as shown in Fig. S2-13. Create interaction ‘Int-2’ all the
same except use RB2 as master surface.
If the surfaces are not connected to each other in the beginning, there will be warning of
contacted pair failed to build. Just ignore the warning.
Fig. S2-12
Fig. S2-13
2.1.7 Constraints:
Module Interaction: Create Reference Point (RP), name RP1 at tip point of RB1, create
RP2 at tip point of RB2, as shown in Fig. S2-14.
Create constraint named ‘Constraint-1’, type - Rigid Body, Region type – Analytical
Surface, pick region: pick RB1, choose inner side, Reference point pick RP-1, shown in
Fig 2-14. Create constraint name ‘Constraint-2’, pick inner side of RB2 and RP-2 as
Analytical Surface and Reference Point.
Fig. S2-14
2.1.8 Load:
Module Load: Create Load named ‘inner pressure’, Step: ‘Pressure’, Category:
Mechanical, Types for Selected Step: Pressure, Region: Select surface: Select inner
surface of Part-1, set Magnitude 0.001MPa, as shown in Fig. S2-15.
Create Load named ‘out pressure’, Step: ‘Pressure’, Category: Mechanical, Types for
Selected Step: Pressure, Select surface: Select out surface of Part-1, set Magnitude 1MPa.
Fig. S2-15
2.1.9 Boundary condition:
Module Load：Create Boundary Condition named ‘BC-1’, Step: ‘Pressure’, Category:
Mechanical, Types for Selected Step: Symmetry / Antisymmetry / Encastre. Select the
surface on Part-1 perpendicular to X-axis, and choose XSYMM, as shown in Fig. S2-16.
Create ‘BC-2’ similarly, select the surface perpendicular to Y axis, choose YSYMM;
‘BC-3’, surface perpendicular to Z axis, choose ZSYMM.
Create ‘BC-4’, Step: ‘Initial’, Category: Mechanical, Types for Selected Step:
Symmetry/Antisymmetry/Encastre. Select RP1, choose ENCASTRE; Create ‘BC-5’
similar with BC-4 select PR2, choose ENCASTRE.
Finally we get the Boundary condition as shown in Fig. S2-17.
Fig. S2-16
Fig. S2-17
2.1.10 Mesh:
Module Mesh: Partition Cell: cut the cell with the plane divide the 1/8 of sphere and 1/4
of cylinder, thus they are both able to mesh with hex elements. Mesh type choose
structured hex.
Seed part instance, choose to seed Part-1, approximate global size set 0.0002, i.e. 0.2μm,
as shown in Fig. S2-18.
Mesh part instance, choose Part-1, you should get 6500 elements.
Fig. S2-18
Assign Element Type: choose Part-1, Element Type: Standard, Quadratic, 3D stress,
Hybride formulation, Reduced integration. Finally you should get element type
C3D80RH, as shown in Fig. S2-19.
Fig. S2-19
2.1.11 Output Set:
Module Mesh: Tools – Set – Create, create set named ‘Set-1’, Tyep – Node, select nodes
by angle, select all the nodes on the inner surface of Part-1, as shown in Fig. S2-20.
Fig. S2-20
2.2
ABAQUS Analysis
2.2.1 Write input
Module Job: Create Job named ‘***-buck’ (*** is the name of the Model), ‘Rob-Modelbuck’ here for example, Source Model-1. Full analyze. Others set as default, could
change if necessary. Then write input.
2.2.2 Open the working directory of ABAQUS (ABAQUS work path in section 1.3), find the
file ‘Rob-Model-buck.inp’ and open it with txt tools. Find the line ‘Output Request’, add
two lines as marked in red below, close file and save the change.
** OUTPUT REQUESTS
**
*Node file
U
*Restart, write, frequency=0
2.2.3 Create new job named ‘Rob-Model-buckle’, source from Input file, select the input file
‘Rob-Model-buck.inp’. As shown in Fig. S2-21. Continue and others set default. Submit
this job to get the bulk modes of the structure. Meanwhile, because of the two lines added
in last step, a file named ‘Rob-Model-buckle.fil’ has been created and the Nodes
displacement is outputted into that file.
Fig. S2-21
2.2.4 After the job finished, we can see the results in Module Visualization. The three bulk
modes are presented. We can have a basic idea of how they look like. We will introduce
the first mode as imperfections to the structure in the analysis later.
2.2.5 Choose Module Step, select step ‘Pressure’ in Step Manager. Replace this step with New
procedure type ‘General’ – ‘Static, General’ , use default setting.
Module Job, create job named ‘Rob-Model-gen’, default setting, write input.
2.2.6 Choose Module Step, select step ‘Pressure’ in Step Manager. Replace this step with New
procedure type ‘General’ – ‘Static, Riks’ , Edit step – Basic- Nlgeom – On; Increment:
Maximum number of increments: 30, Arc length increment, Initial 0.001, Minimum 1E005, Maximum 1; Estimated total arc length 1. As shown in Fig. S2-22.
Module Job, create job named ‘Rob-Model-riks’, default setting, write input.
Fig. S2-22
2.3
Prepare ABAQUS files for MATLAB
Open the working directory of ABAQUS, find the file ‘Rob-Model-gen.inp’ and ‘RobModel-riks.inp’. Make sure the two lines before node coordinate at the beginning of the
file exactly the same with following: nothing else is allowed between these lines, or there
will be error reading data.
*Instance, name=Part-1-1, part=Part-1
*Node
1,
0., 0.00999999978, 0.00200000009
2.3.1 Make three changes to each files and save as ‘Rob-Model_gen.inp’ and ‘RobModel_riks.inp’:

Induce imperfection: find line ‘STEP’ in the file, add the red lines as below.
For Rob-Model_gen.inp
** STEP: Pressure
**
*imperfection, file=Rob-Model-buckle, step=1
1, 2e-5
*Step, name=Pressure
*Static
1., 1., 1e-05, 1.
For Rob-Model_riks.inp
*Step:
** STEP: Pressure
**
*imperfection, file=Rob-Model-buckle, step=1
1, 2e-5
*Step, name=Pressure, nlgeom=YES, inc=30
*Static, riks
0.001, 1., 1e-05, 1., ,

For both ‘_gen.inp’ and ‘_riks.inp’ files, find ‘LOADS’ add four spaces in front of
the value of inner pressure and out pressure, i.e. 0.001 and 1 as shown in red below,
and add at least 10 spaces after each value. ** LOADS
**
** Name: inner presure Type: Pressure
*Dsload
_PickedSurf40, P,
0.001
** Name: out pressure Type: Pressure
*Dsload
_PickedSurf41, P,

1.
For both ‘_gen.inp’ and ‘_riks.inp’ files, find ‘OUTPUT REQUSTS’ and add the
lines marked in red for output to .dat file.
** OUTPUT REQUESTS
**
*Node print
U
*Restart, write, frequency=0
2.3.2 Copy 12 files from ABAQUS working directory to MATLAB working directory, where
the .m files has been stored. Those file names are:
Rob-Model_riks.inp
Rob-Model_gen.inp
Rob-Model-buckle.* (include all ten files with this name)
2.4
Run program in MATLAB
2.4.1 Open MATLAB, change Current Folder to MATLAB work path which include all the .m
files and ABAQUS files；
2.4.2 Open ‘aba_gen.m’ and ‘’aba_riks.m’, change followed lines:
system('path=C:\SIMULIA\Abaqus\Commands');
cmd=strcat('call abaqus job=',jobname,' input=D:\bulk\',inp,' int ');
path= ABAQUS path in section 1.2, input= MATLAB work path in section 1.4.
2.4.3 If the naming of model does not change, run ‘main.m’. If the model name changed, open
‘main.m’ and edit several parameters:
names=***，the model names, ‘Rob-Model’ here. Several models can run once, you need to
input a series of model names;
nn= , the size of matrix names, means how many models you are going to run;
The following parameters do not need to be changed, unless you really know what they mean
and want to tune:
p0=0.001;q0=1; q0 is Initial Osmosis, usually set as 1MPa; p0 is initial out pressure, we use
0.001MPa, to mimic almost pure water in cell wall；
p1=1;q1=0.0001; Reference out pressure and inner pressure set for Riks step, in fact we are
calculating the pressure difference, the out and inner pressure did not present the real water
potential or osmosis, q1 should be 0 here, we use 0.0001 to prevent possible errors；
m= 30, number of increment in Riks step, we use 30 here, if we change that in ABAQUS, we
need to change this number here in MATLAB at the same time.
2.4.4 The analysis may take several minutes or several hours depend on your computer. After
the program finished, ***.mat, here Rob-Model.mat file is saved with all the values
calculated. Variable pf stands for out pressure calculated (we can understand it as water
potential), qf stands for inner pressure (i.e. osmosis), tf stands for pressure difference,
here means turgor calculated, positive stands for total pressure outward and negative
means pressure inward. vf is the volume inside the shell, v0 is initial volume inside shell.
Figure of pf, qf, tf, change with (vf-v0)/v0 is presented at the end of program. The units
are the units of ABAQUS, here mm for length and MPa for pressure.
3
NOTICE
3.1
The geometry can be changed, but several feature names are used in MATLAB file, so it could
not be changed or you have to manipulate the .m file. Those names include: part name ‘Part1’, material name ‘Material-1’, load name ‘inner pressure’ and ‘out pressure’, set name ‘Set1’.
3.2
The volume of cell is calculated in MATLAB from the inner surface node displacement as
input. The method to calculate volume in current ‘.m’ file is designed for this shape, its usage
is limited. It calculates 1/8 of shape, so you should draw 1/8 of the structure, or you have to
divide the result relatively. It can solve all the convex shapes, it can also solve concave
structure with at most one dent in 90 degrees structure. If there are more than one concave the
volume calculation might be wrong.
3.3
If Reference Points number are other than 2, let’s say nRP, we need to change lines in
volucal2.m and volucal_riks2.m:
nodeID=str2num(ss(1:indx(1)-1))+2;
change +2 to +nRP.