Model a living cell (walled tube under pressure)

[Anja]

I am biologist and know NOTHING about mechanical engineering or finite element analysis - so please forgive my amateur approach.
I would like to model the stiffness of a single plant cell, a pollen tube. It could be described as a fire hose kind of pipe (length up to several hundred micrometers), diameter ca. 6 micrometers, with an apical, half-sphere shaped dome. The pipe is filled with cytoplasm (for simplicity: homogeneous liquid). The resistance to lateral deformation consists of 1. the turgor pressure (internal pressure of a living cell), 2. the cell wall (thickness ca. 100-200 nm).
Hypothetical case: The plasticity and thickness of the cell wall is identical at the apex and the remaining part of the cell. How can I calculate the force I would need to compress the apical part of the cell (i.e. cylindrical part directly adjacent to the half-sphere), compared to the more distal part (i.e. the purely cylindrical part further down the pipe).
Any kind of input, also reference to basic engineering literature would be VERY welcome!!

 

[JAE]

Whew...what a challenging, but interesting problem. I don't have any personnal experience with this sort of calculation, but I would think a good analogy would be the condition of ductile iron pipes buried in the ground. They take external pressure, with a fluid within, and there may be some research out there that could be applied to your "micro" situation.

Try:

http://www.dipra.com/

 

[prex]

You should better describe the kind of force you want to determine.
May it be modeled as a circumferential line load uniform along circumference? Or is it closer to a uniform pressure distributed all around and over a given pipe length? In the latter case a guess for that length should be given.
Another point that should be clarified is the thickness of the apical sphere: may it be taken the same as that of cylinder?
Once all the necessary data are provided, formulae exist in the literature that can solve your problem. However you should bear in mind that the result will be in the form of a force needed to obtain a given radial displacement at the load point of application, and this result will depend not only on geometry (e.g. diameter and thicknesses), but also on material's modulus of elasticity (that I suppose you can't even guess). If you can accept a result in the form of the ratio between an apical force and an intermediate force to obtain the same displacement, then this will be simpler, as should not depend on the modulus. However the result will still depend on the assumed geometry (and a factor of 2 on thickness makes a lot of difference!).
If you can complete your assumptions, I'll be more exhaustive in a subsequent message.


prex
motori@xcalcs.com

http://www.xcalcs.com

Online tools for structural design

 

[Anja]

Hi JAE and Prex,
thanks for getting back to me so quickly. I will provide more details on the subject, as required. Remember, I am a TOTAL amateur regarding physics and engineering, so I probably use the wrong vocabulary, but I hope you will be able to convert this into proper engineering talk. Okay, so here are the details:
1. The geometry of the pollen tube: The dome (sphere)-shaped apex has the same diameter as the shank (tube) part of the cell. (Actually, not totally true, it has more of a parabola form, but let's just assume for simplicity, it's a half-sphere). Imagine it as the end of an ideal saussage or tube-shaped balloon.
2. What I did is the following: I used an indenter with a cylindrical probe with the diameter of around 4 micrometers to indent the cell ar various locations. Since the diameter of the probe (4 micron) is almost as big as the diameter of the cell (around 6 micron), this resulted in squeezing the cell laterally (radial force?) on a length of 4 micron. I found that the apical part of the cell (I measured as close to the (growing) tip of the cell as possible without the indenting probe wiggling away because of the round shape of the apex) is less stiff than the more distal, cylindrical part of the cell. This is great, since it was to be expected due to the presumed difference in chemical composition of the cell wall at the apex, but here we go into biology. Back to physics. What I would like to show by modelling the thing is, that the apex is NOT softer because of the geometry of the cell at this part, but that this difference (or at least part of it) is due to the higher plasticity of the cell wall. Therefore I need to have a model for the force I would need to indent this cell, if the cell wall was identical all around. A ratio (between apex and distal part) as a result would be good enough!
3. The numbers. I can provide some numbers, which would make the thing easier. In order to indent the cell in the way mentioned above (with the 4 micron cylindrical probe) I need, say 300 millidyne/micron in the tube part (far away from the apex) and only 150 millidyn/micron at the apex. This is in the turgid cell, so both cell wall AND pressure play a role. I indented also empty cells, i.e. presumably not under pressure, i.e. only measuring cell wall resistance to deformation, and that value was around 10 to 20 millidyne/micron in the cylindrical part.

Now, I hope this information is detailed enough. I am very much looking forward to your comments!!!
Anja

 

[Anja]

PS: Forgot to mention, ideas and calculations that contribute substantially to the eventual model of my pollen tube might be rewarded with a co-authorship on a scientific paper. Promised!

 

[Qshake]

A very interesting problem indeed! I recommend that you hit the libraries (assuming that you haven't already) with searches for "cell mechanics" or "cellular engineering". I did a search of a large midwestern medical research library and found several promising leads. At least all were in the right ballpark.

Also, you may want to post this in the Finite Element forum as finite element technology readily lends itself to analysis of bioengineering elements. Especially when the subject matter will involve some inelastic, non-linear behavior. Most everything structural engineers deal with is in the elastic, linear range of stress-strain.

The geometry and material properties of your situation are not within our realm. For very small loads (nano-loads even) the cell wall, it seems, will undergo large displacements and the geometry of the cell will change from that of a circular element to an elliptical element thereby changing the internal pressure distribution. Of course, there may be a case to be made for stiffer cell walls and really, really light loads (that "relative" thing again) so that one could take advantage of the linear, elastic behavior, but I woulsn't count on it without research.

Good Luck

 

[engrx2]

Post this in the Finite element forum. Also, look into research regarding material composites. Finite Element Analyses (FEA) have been perfomed on exotic materials before, so there has to be some sort of relationship to your model. Consider the fact that when new pipe materials are developed (ex. composite), they go through research to develop load capacities. Structural engineers dealing with the design of pressure vessels and such can also be of assistance. I don't know if this is helpful, but I thought I'd try.

 

[prex]

I'll propose the following way of reasoning.
First I shall assume that the material constituting the cell wall behaves like a membrane (in structural engineering this means it resists only to stretching, not to bending).
Now take a long cylinder and squeeze it along a diameter in the middle, as you indeed do on the cell. Over a certain length its section will take a like elliptical shape: for reasons that are known to structural engineers, the change in shape can be taken as affecting a length of the cylinder not very different from its diameter (the rest of the cylinder being taken as undeformed). The change in shape of that portion of the cylinder would tend to change it's volume, but this is resisted by the inside liquid under pressure (liquids are quite incompressible), so the wall will be obliged to stretch (elongate) in order to accomodate that change in shape without change in volume.
The force required to obtain the deformation is a measure of this stretching, and will be somehat proportional to the change in surface of the affected portion of the cylinder.
Now take the case with the load applied at the junction of the cylinder to the domed end, again radially with respect to cylinder. A length of the cylinder equal to a half diameter is now affected plus the emisphere.
The problem is to determine the change in surface of a sphere squeezed along a diameter to take the shape of an ellipsoid of rotation without change in volume. I don't know a simple formula for that (the length of an ellipse and the surface of an ellipsoid cannot be expressed by compact formulae), but it is simple to solve the problem by numerical experimentation in a worksheet: the result is that the sphere will change its surface by an amount that is exactly half that of a cylinder with a length of one diameter (this result must have some relationship with the fact that the required thickness of a sphere under pressure is half that of a cylinder, but I don't see the path at the moment).
Coming back to the force applied at the junction, the conclusion is that the force required to obtain the same deformation is 75% of that for the cylinder. If you apply your force at the apex of the sphere (along cylinder axis) then the force would be much closer to 50%.
Note that the above reasoning required some simplifications, as the liquid, being not exactly incompressible, will participate in the phenomenon, and also the inside pressure should do some work that collaborates to the stiffness of the cell: however I think that what I said above is not far from reality, at least as far as the orders of magnitude are concerned.
You already see my conclusion: the difference of 50% that you measure can be fully explained by the elastic behaviour of the cell wall.
In my opinion (no experience in bioengineering!) a difference in chemical composition (presumably not very strong) would hardly explain a steep change in elastic behaviour (NOTE: the correct term for expressing the wall ability to stretch is elasticity, not plasticity, as for engineers plasticity refers to a permanent deformation and elasticity to a deformation that disappears when the load is removed): however I'll leave such an evaluation to you.


prex
motori@xcalcs.com

http://www.xcalcs.com

Online tools for structural design

 

[Anja]

Thanks Prex and the others for your ideas.
I need to add two things:
1. The cell wall is probably not just a membrane, since it does not collapse upon removal of turgor. It stays cylindrical, even though the resistance to deformation decreases by a factor 20. So I probably have to calculate walled tube and membraneous properties, don't I?
2. The cylindrical part is indeed probably purely elastic. However, the apical dome, from what we know from its chemical composition, has probably something of a plastic material (biol. explanation: the cell grows at this part, i.e. it expands, probably involving stretching of the wall simultaneous to insertion of new cell wall material). In fact, when I indent at this apical position, it looks in the microscope, as if the cell suddenly grows somewhat longer. More than would be expected from the pure squeezing (and therefore flattening) of the cell.
3. I got one more observation: when I arrest cell growth (and presumably the apical part assumes an elastic, stiffer behavior, since no new cell wall material is added), this is indeed what I measure: the apical part has about the same resistance to deformation than the cylindrical tube. Even though gemetry hasn't changed, only wall composition. If the cell was to be modelled as membranous tube, than this could be only explaned with the apical cell wall being even stiffer than the wall in the cylindrical tube, which is not what we would expect from what we know about the cell. Hmmmmm.......

 

[Gourile]

Hi!
There are many ways to consider your problem. The simplest one is to define it as a thin walled tube which is well described in any elementary 'Mechanics of Solids' or 'Strength of Materials' book. For instance try E.Popov. The formulla in the simplest form is as follows:
s=pr/2t
s= stress (force per unit area)due to internal pressure in the wall of the cylinder. It is a tensile stress along culinder axis.
p=internal pressure
r=average diameter of the cylinder
t= wall thickness of the cylinder
In the dome apex, however, stresses are different. The stress is circumferential to the spherical surface. (The formula is not on top of my head. I can write a complete essay for you if the above argument seems helpful to you.) The stesses on the spherical apex will be higher than that on cylindrical one for the same wall thickness. In addition, the deflection of the dome shaped end and the cylinder will not completely consistent, even when the dome thickness is increased so that the stresses become equal to the cylindrical part.
Was that any help? If you need more of me, here is my email and don't hesitate to send me messages. I'm not visiting this site very frequently:
a_bahrami_uk@yahoo.co.uk

 

[prex]

To Anja
I'll comment your points 1 to 3:
1)No surprise if the cell wall is not a pure membrane, however the decrease in resistance to squeeze with no pressure is so important, that the bending resistance appears to be negligible, at least for a first approximation.
2)I can't understand exactly what you mean (should have a look into your microscope, but it's not easy). More specifically I don't see why the cell growth should lower the squeezing force; if the direction of growth is towards outside, this should stiffen it. One more question: does this growth take place also when you squeeze on the distal part, or is it a phenomenon somewhat due to the squeezing itself? And what about the liquid turgor after the growth: if the cell is closed so there's no feed of liquid, the pressure should sharply decrease.
3)If the model I proposed is correct (and of course this is far from being proved) you should have an equivalent stiffness at the apex only if the thickness of the spherical dome is higher than that of the cylinder or the dome is formed out of a stiffer material(as the growth takes place there, this seems not to be excluded). Can you indent on the side but right at the dome to cylinder junction and measure the force? Both cylinder and dome would participate and one would expect a result midway of the two others.


prex
motori@xcalcs.com

http://www.xcalcs.com

Online tools for structural design

 

[Anja]

Answer to prex:
Comments to points 1 to 3:
1. You are probably right, that the stiffness of the wall is comparably small to the pressure factor. Still ..... the fact that I can notably influence the stiffness of the apical part tells me the the wall plays a role. Actually, the biology is even more complicated: The cell is not just a bag filled with water, it contains a so called cytoskeleton. This consists of structural proteins that provide the cell with a skeleton from the inside. I don't know how much that contributes to stability. To make it complicated, the cytoskeletal structure is different between apex and tubular part.
2. I'll try to explain more thoroughly, I hope this helps. The cell grows in one direction (the direction of the long axis), imagine a tubular balloon that you blow up and that gets longer and longer. So when I indent the cell, because of the squeezing (I indent the cell by about half its diameter) the view from above shows a somewhat increased diameter of the tube in perpendicular direction to the long axis (because it is flattened by the action). Now, when I indent close to the tip, this increase in diameter is perpendicular and parallel to the long axis (since it's the dome) and thus contributes to total tube length of course. However, it seems as if this increase is much more than would be expected from the flattening. It seems as if I cause the cell to make a sudden growth pulse. Imagine our cylindrical rubber balloon that was not totally blown up and now elongates a little bit more.
Now, what happens during normal growth (=elongation at the tip by the own cellular machinery), the cell regulates its turgor by taking up water from the environment and regulating its solute concentration. Very sophisticated system, but the turgor does not drop normally, unless the cell is heavily disturbed in its function, or the osmotic value of the surrounding liquid is altered.
Furthermore I should add that the indenting at the distal part probably does not cause a considerable pressure change anywhere. The reason is that the total volume of the total cell is very big. So even if liquid is not compressible, the reservoir is that big, that a small change would not matter or cause an additional elongation pulse in the tip. (For the biologically interested: this tubular cell we are talking about, the pollen tube, is only a protrusion of the pollen grain, which is a big sphere shaped cell, diameter up to 50 micron, thus representing a big reservoir).
3. If your calculation is correct, then indeed, this should be the case. Puzzling though, but not totally unreal. The cell wall COULD get thicker at the apex as a result of the drug I used to arrest growth. I definitely need to do more control experiments. Especially with other drugs.
The dome to cylinder junction is actually where I found the values to be half of the values for cylinder measurements. I wasn't able to measure the dome only, since the indenter would wiggle away because of the round shape.

Now, I still have a problem with the apex being calculated less stiff than the cylinder by all you guys (Thanks, Gourile, for your comments as well!!). If I look, say, at a finger of a rubber glove, it seems to me to require a higher force to squeeze-flatten the finger tip (i.e. apical dome) than the cylindrical part. How does this fit? Perhaps I should fill my glove with water to test the pressurized situation.....