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I have approximately 10 papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "methods of variational calculus" to derive the following: $$\Delta p - 2\lambda H + k(2H+c_0)(2H^2-2K-c_0H)+2k\nabla^2H=0$$ But I'm having a lot of trouble tracking down the original derivation. The guy who did it first was Helfrich, and here's his and Ou-yang's paper deriving it: . However, they don't show an actual derivation, instead saying "the derivation will appear in a full paper by the authors" or something like that. Yet everybody cites the paper I just linked for a derivation. Does anybody know a source that can derive this, or can give me some hints to figure it out myself? To be honest I can't even figure out how to find the first variation.

Edit: So, after some careful thought and hours and hours of work and learning, I realized that the answer that got the bounty was wrong. The author stopped replying to my messages after I gave him bounty.... thanks guys. That said, I've almost got it all figured out (in intense detail) and will post a pdf of my own notes once I'm done!

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4 Answers 4

up vote 7 down vote accepted

Edit: Note that I am doing only the first variation, and I am not doing each and every step, mainly those pertinent in understanding how the general shape equation is determined. If you want to see the full derivation, you will need to understand the Geometric Mathematic Primer discussed in Sections 2 and 3 of the book.

  • Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases by Zhong-Can Ou-Yang, Ji-Xing Liu, Yu-Zhang Xie, Xie Yu-Zhang

$c_{0}$: Spontaneous curvature of the membrane surface

$k_{c}$: Bending rigidity of the vesicle membrane

$H$: Mean curvature of the membrane surface at any point $P$

$K$: Gaussian curvature of the membrane surface at any point $P$

$dA$: Area element of the membrane

$dV$: Volume element enclosed by the closed bilayer

$\lambda$: Surface tension of the bilayer, or the tensile strength acting on the membrane

$\Delta p$: Pressure difference between the inside and outside of the membrane.

The shape energy of a vesicle is given by:

$$ F = F_{c} + \Delta p \int dV + \lambda \int dA $$


$$ F_{c}=\frac{k_{c}}{2}(2H-c_{0})^{2} = \frac{k_{c}}{2}(c_{1}+c_{2}-c_{0})^{2} $$

The variation of $dA$ and $dV$ are needed, refer to the book to locate those.

Next we'll calculate the first variation of $F$. And we can break this into components by starting with the first variation $F_{c}$.

$$ \delta ^{(1)}F_{c} = \frac{k_{c}}{2}\oint (2H+c_{0})^{2} \delta ^{(1)}(dA) + \frac{k_{c}}{2}\oint 4(2H+c_{0})^{2}(\delta ^{(1)}H)dA $$

Where the first order variation of $\psi$ gives us:

$$ \delta ^{(1)}dA = -2H\psi g^{1/2}dudv $$ $$ \delta ^{(1)}dV = \psi g^{1/2}dudv $$ $$ \delta ^{(1)}H = (2H^{2}-K))\psi + (1/2)g^{ij}(\psi _{ij}-\Gamma _{ij}^{k}\psi_{k}) $$

Note: $\Gamma_{ij}^{k}$ is the Christoffel symbol defined by (for reference):

$$ \Gamma_{ij}^{k} = \frac{1}{2}g^{kl}(g_{il,j} + g_{jl,i} - g_{ij,l}) $$

And we plug those into the variation of $F_{c}$:

$$ \delta ^{(1)}F_{c} = k_{c}\oint [(2H+c_{0})^{2}((2H^{2}-K)\psi + (1/2)g^{ij}(\psi_{ij}-\Gamma_{ij}^{k}\psi_{k}))] $$ $$ = k_{c}\oint [(2H+c_{0})(2H^{2}-c_{0}H-2K)\psi + (1/2)g^{ij}(2H+c_{0})\psi_{ij} - g^{ij}\Gamma_{ij}^{k}(2H+c_{0})\psi_{k}]g^{1/2}dudv $$

And there are two relations ($i,j = u,v$)

$$ \oint f\phi_{i}dudv = -\oint f_{i}\phi dudv $$ $$ \oint f\phi_{ij}dudv = \oint f_{ij}\phi dudv $$

So then we have:

$$ \delta ^{(1)}F_{c} = k_{c}\oint \left \{ (2H+c_{0})(2H^{2}-c_{0}H-2K)g^{1/2} + [g^{1/2}g^{ij}(2H+c_{0})]_{ij} + [g^{1/2}g^{ij}(2H+c_{0})\Gamma_{ij}^{k}]_{k} \right \}\psi dudv $$

And we can re-write:

$$ [g^{1/2}g^{ij}(2H+c_{0})]_{ij} = [(g^{1/2}g^{ij})_{j}(2H+c_{0})]_{i} + [g^{1/2}g^{ij}(2H+c_{0})\Gamma_{ij}^{k}]_{k} \psi dudv $$

And for functions $f(u,v)$, where $u,v = i,j$, we can directly expand:

$$ [(g^{1/2}g^{ij})_{j}f]_{i} = -(\Gamma_{ij}^{k}g^{1/2}g^{ij}f)_{k} $$

A Laplacian operator for these surfaces is defined in the book, and is given as:

$$ \bigtriangledown^{2} = g^{1/2}\frac{\partial }{\partial i}(g^{1/2}g^{ij}\frac{\partial }{\partial j}) $$

So then we have:

$$ [g^{1/2}g^{ij}(2H+c_{0})_{j}]_{i} = g^{1/2}\bigtriangledown^{2}(2H+c_{0}) $$

Using these methods in the first variation:

$$ \delta^{(1)}F_{c} = k_{c}\oint [(2H+c_{0})(2H^{2}-c_{0}H-2K) + \bigtriangledown^{2}(2H+c_{0})]\psi g^{1/2}dudv $$

And now we want the variation of $F$.

$$ \delta^{(1)}F = \delta^{(1)}F_{c} + \delta^{(1)}(\Delta p\int dV) + \delta^{(1)}(\lambda\int dA) $$

Which gives us:

$$ \delta^{(1)}F = \oint [\Delta p-2\lambda H + k_{c}(2H+c_{0})(2H^{2}-c_{0}-2K) + k_{c}\bigtriangledown^{2}(2H+c_{0})]\psi g^{1/2}dudv $$

And since $\psi$ is a very small, well smooth function of $u$ and $v$, the vanishing of the first variation of $F$ requires that:

$$ \Delta p = 2\lambda H + k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K) + k_{c}\bigtriangledown^{2}(2H+c_{0}) = 0 $$

Which is the general shape equation of the vesicle membrane. $c_{0}$ is a constant unless the symmetry effect of the membrane and its environment varies between each point (we assume it doesn't) otherwise $c_{0}$ becomes a function of $u$ and $v$. So we can reduce to:

$$ \Delta p = 2\lambda H + k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K) + 2k_{c}\bigtriangledown^{2}H = 0 $$

Hope this helps. Again I would locate that book to see the full derivations. I don't know if the visible section of the book on Google shows you everything that you need to know, but I surely hope this points you in the right direction to understanding the problem.

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I'm actually a math undergrad doing a biophysics summer research project... so I guess I'm a mini-mathematician, and I definitely do like to focus on the derivation haha, especially when it's so unclear. It boggles me that people cite things without knowing where they came from. My school library doesn't seem to have anything useful sadly. Thanks for trying, I hope you or somebody can figure something out! – Samuel Reid Jun 6 '13 at 17:54
I'm going to try working this backwards while learning a little of the subject matter. It helps for me to make assumptions about the derivation. By reading however, the second equation you describe is just one of the boundary conditions. Are you wanting to figure out the derivation for all of the boundary conditions or just that one? – Signus Jun 6 '13 at 21:35
Actually I think those 4 boundary conditions are for open lipid membranes. So you need only the one for the closed membrane. I'm trying to figure out with what respect to take the integral, and from there I can get a characteristic equation that I think I can expand out into the form above. – Signus Jun 7 '13 at 17:19
@Signus The book you cite actually contains all the details of the calculation(s), as far as I remember. See my answer for more details. The paper cited by AJK also contains plenty details, see for her/his answer. – FraSchelle Jun 12 '13 at 17:12
@BiancaDeSanctis It's about the description of the membrane. It's a surface after all, right ? One can describe surface in multiple way. One of these, easy to visualise, is called Monge parametrisation/patch/.... It consists in describing a 2D surface in a 3D environment by supposing the third dimension to depend on the first two ones $z=f(x,y)$, that's what I noted $z(u,v)$. Now, you want to deform this surface to know the energy associated to the deformation, right ? You can do that by defining $z=\Psi + \psi$ where $\Psi$ verifies the Helfrich equation without deformation. Does-it help ? – FraSchelle Jun 14 '13 at 6:20

I studied this problem long time ago (almost ten years) during a training period. It's a really interesting problem which, as far I remember, is not so complicated to understand. It's a basic variational method applied to the curvature. In your notation, $H$ and $K$ are the mean and Gaussian curvature, the two invariants on a (curved) surface.

Perhaps a good knowledge of Stokes theorem (how to translate volume to surface) and variational calculation is a first step in the understanding. Tensor notation in the theory of surface is also a prerequisite. I was particularly enjoying a book by Do Carmo Differential Geometry of Curves and Surfaces Pearson (1973) at the time of this training period, even if I would no more consider it as a modern text-book.

The strategy is to express the curvatures $c_1$ and $c_2$ in your notation as some differential elements of the metric tensor onto which you can apply usual variation calculus. I think you should have something like $c_{1}+c_{2}=2H$, the mean curvature, since I believe $c_{1,2}$ are the curvatures along the principal axis of the surface, am I correct ? You also need to express the change in the infinitesimal volume and the infinitesimal area in some differential formulas (in term of the metric tensor) that you can vary. Then you should be full of Christophell, and you should try to recognise the Gaussian and mean curvatures of the final result. They may be simplifying physical assumption, like conservation of the volume, which enters the game, too.

An other way is to express everything in term of the normal vector to the surface. Then, you express the curvature in term of the fundamental form of first and second kind, and make the variational calculation with them. This second method is equivalent to the first one of course, since the fundamental forms can be given in term of the metric. The paper you discuss uses the metric approach if I remember correctly.

I may eventually give you my notes of this period. The only embarrassing point is that they are in French, and I'm really lazy to translate in English :-( The project was to calculate the Green function of a deformed sphere. So it's simpler than yours as believe.

All the details of the calculation can be found in the book by the author:

with some introductory material about variational calculation and differential geometry of surface (called manifold in modern text-book).

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See also the answer by AJK which gives a detailed calculation paper, too. The results of this paper are obviously reproduced in the book I cited. – FraSchelle Jun 12 '13 at 6:53
See also the answer by Signus who gave the same book reference as an edit. – FraSchelle Jun 12 '13 at 17:10
Thank you!! Do you have any idea where I might find this book online? I'm an undergrad and definitely don't have 77$ to spend on it... – Samuel Reid Jun 14 '13 at 5:27
@BiancaDeSanctis Please see my comment after Signus answer. This book might not be available online. As far as I check, it is not. See the answer by AJK for the reference of a detailed paper available on the APS server, too. I would suggest you to contact the next library if you're at some university. The librarians know how to find the book. – FraSchelle Jun 14 '13 at 5:43
Found it, it's being sent from another library and should be mine within 7 business days :). I didn't even know you could do that, thanks for the suggestion! – Samuel Reid Jun 14 '13 at 18:23

The paper you're looking for is probably:

"Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders" Ou-Yang Zhong-can and Wolfgang Helfrich Phys. Rev. A 39, 5280–5288 (1989)

Hopefully you have university access - I couldn't find a free copy anywhere. This is actually a more complicated question than most physics papers like to make it appear! Good luck with your project. Membranes are a very cool subject.

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This is close to what I needed, but was insanely confusing because before yesterday I didn't know what a Christoffel symbol was, let alone all of their crazy definitions. Looking back on it now, if I'd been up to speed on all my prereqs for this question, that would have been really useful. Thanks! – Samuel Reid Jun 14 '13 at 21:46

In the paper by Lin et al. (2003) Progress in Theoretical Physics they mention in the abstract that they extend the work of Ou-yang and Helfrich by expanding the bending energy to fourth order. That means you should be able to work out the lower order solutions from their paper as well.

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