Eindhoven University of Technology BACHELOR Protein-mediated instability of curved membranes Klaasse, R.L. Award date: 2011 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
Transcript
Eindhoven University of Technology
BACHELOR
Protein-mediated instability of curved membranes
Klaasse, R.L.
Award date:2011
Link to publication
DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Studenttheses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the documentas presented in the repository. The required complexity or quality of research of student theses may vary by program, and the requiredminimum study period may vary in duration.
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
Protein-Mediated Instability of Curved Membranes
Ralph Leonard Klaasse
August 30, 2011Supervisors: prof.dr. M.A. Peletier, dr. C. Storm
Bachelor thesisEindhoven University of Technology
Department of Mathematics and Computer Science and Department of Applied Physics
Abstract
The process of endocytosis in living cells is not completely well understood from amodelling point of view. Instability of lipid bilayer membranes due to protein concentra-tions result in budding processes presenting during endocytosis. In this thesis we studythe interplay between the geometry of such membranes and the presence of additionalprotein concentrations. Working on a Monge patch with radial symmetry, we derive dy-namic equations using an energy gradient flow approach. This approach combines bothL2 and Wasserstein metrics to form a gradient flow of a Canham-Helfrich energy withan entropic protein contribution. We furthermore look at geometric instability of the flatmembrane state for arbitrary protein energy contributions. We derive criteria for insta-bility on both the protein energy contribution and the size of the membrane. Lastly wediscuss the influence of several modelling choices on the behavior of the system.
This thesis is written as a conclusion of the bachelor’s degrees in both Applied Physicsand Industrial and Applied Mathematics at Eindhoven University of Technology (TU/e).Below we describe its subject and content, first in a less formal way regarding physicalmotivation, and then more mathematically.
1.1 Endocytosis and the cell
One could say that the basic building block of every living organism is the cell. How-ever, physicists can hardly claim a cell to be simple enough to be described by elementaryphysics. Indeed, within a cell countless processes take place at varying speed and complex-ity. The membrane surrounding a cell, protecting it from unwanted external influences,consists of a lipid bilayer. These lipid proteins consist of a hydrophilic head and two hy-drophobic tails. Because both the cell’s surroundings and interior contain large amountsof water, the lipids want to arrange themselves such that their tails come into contactwith water as little as possible. This immediately leads to the lipids arranging themselvesinto the bilayer structure, where lipids pair up with their tails facing each other, huddlingside to side to ensure their hydrophobic tails do not touch water. See figure 1(a) for aschematic view of such a lipid bilayer membrane structure.
However, a cell cannot survive entirely on its own. It needs resources from the outsideworld as well as a place to transport waste to. While smaller molecules are able to slipthrough the membrane structure by means of diffusion, larger molecules cannot. Theseinstead have to be transported through the membrane, and nature has found an ingeniousway of doing this: endocytosis. Endocytosis is the name given to the process in whichan external molecule comes in the vicinity of a cell and by doing so forces a structuralchange in the membrane to let the molecule through. The name is fitting, as ‘endon’ isGreek for ’within’, while ‘kytos’ means ‘hollow vessel’ or ‘cell’.
This process can roughly be described as follows: without the molecule, the membranehas taken on some most energy favorable state and shape. By introducing the molecule,proteins from within the cytoplasm, a thick liquid inside of the cell, are recruited tothe inner side of the membrane. As these proteins diffuse, they change of the shapeof the membrane. In this sense they are the energy carriers making the shape changesof the membrane possible. The membrane will change such that locally an indentationwill appear around the molecule. As the molecule moves closer to the membrane, it ismore and more enveloped into this indentation. When finally the molecule is completely
surrounded, the indentation closes up at the top, leaving the molecule trapped by themembrane lipids. The bottom portion of this trap, the initial indentation part, can thenbe pinched off allowing the molecule into the cell without the cell ever having its mem-brane’s defenses broken. A schematic view of this process can be found in figure 1(b).The reverse process of this, where a molecule moves out of the cell, is done completelyanalogous to the above and is called exocytosis. Similarly, ‘exos’ is Greek for ‘outside’.
As such there is a struggle between the membrane lipids wanting to arrange themselvessuch that their tails are in contact with water as little as possible, and the attractedproteins wanting to be close together. This struggle will be described by associating anenergy scalar to the combined process. As a closed system will want to minimize thisenergy, in doing so one can order possible states of the system in terms of how favorablethey are, where having a lower energy means being more favorable. The system will goto a state with at least locally minimal energy, where the best possible compromise hasbeen made between these two opposing processes.
1.2 Mathematical introduction
The cell membrane we are interested in is mathematically described by a two-dimensionalsurface M. While surfaces can be defined and exist without any surrounding ambientspace, we will always see M as being explicitly embedded into R3: this embedding isimportant because of the need to be able to define two types of curvature.
OnM we will introduce a protein concentration φ, which diffuses on the surface as well aschange the surface’s shape. The manner of diffusion depends on the current shape ofM.This interplay betweenM and φ gives rise to dynamic behavior, described by a system ofpartial differential equations. Our goal is to show that given certain starting conditions abifurcation process from the initially stable shape of the membrane will take place. Thiswould then indicate that indeed the introduction of proteins on the membrane can forceit out of its spherical shape and start a budding process. We are not so much interested inwhat shapes become stable after sufficiently large periods of time; the main goal is to showthat instabilities arise. Do note that we will only look at the local dynamical propertiesof the cells, in the sense that we will restrict ourselves to a small portion of the membrane.
Our approach will be through gradient flow using an energy functional F . We will use theconcept of energy to quantize the changes in the membrane’s shape and the distributionof proteins on the membrane. The functional F will be the sum of an adjusted Canham-Helfrich functional and an entropic term,
F =
∫M
2κ(φ)H2 + κ(φ)K + η(φ) dA.
Here H and K are the mean and Gaussian curvatures with their accompanying bendingrigidities, while η(φ) = φ log φ is the proteins’ entropy density. By positing a gradientflow structure on the system we determine dynamic equations for its behavior, and showthat the ground state consisting of a flat membrane with constant protein concentration isunstable. As will be explained, the inner product necessary to define our gradient flow is acombination of a standard L2 one derived from Stokes’ law, and the Wasserstein distanceW2.
In section 2 we define the notion of curvature for surfaces and derive the shape of theCanhma-Helfrich functional, as well as state the Gauss-Bonnet theorem. We then moveon in section 3 to gradient flows as a means of describing dynamical behavior. In section
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Protein-Mediated Instability of Curved Membranes
4 we discuss the modelling choices made. Then in section 5 we derive dynamic equationsfor the system we are studying, and in section 6 show it to be unstable under certainconditions. Lastly, in section 7 one can find some background mathematics used in therest of the thesis. The main results of this thesis are Theorem 5.12 regarding the dynamicequations, and Theorems 6.4 and 6.8 regarding instability.
1.3 Acknowledgements
I would like to thank my supervisors prof.dr. M.A. Peletier and dr. C. Storm for theirguidance and help in writing this thesis. They both have shown great support and patiencewith me.
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Protein-Mediated Instability of Curved Membranes
2 Curvature
In this section we will introduce the notion of curvature of surfaces and give its mathemat-ical definition. We will give an intuitive description as well as our motivation for wantingto introduce it in the context of the membrane system. For this we will first discuss thenotion of curvature for curves and from it deduce a usable notion for two-dimensionalsurfaces. This exposition follows that in [7].
The notion of curvature is important for us because with it we can express the effect ofprotein concentrations locally changing the shape of the surface. The curvature of a spacemeasures how much the space deviates from being flat. Indeed, it tells us how curved aspace is. The canonical example of a flat space is the Euclidean plane R2. We will seethat for a two-dimensional surface there are two separate notions of curvature, both ofwhich are of interest to us. One of these curvatures is intrinsic to the surface, while theother requires a chosen ambient space to be determinable.
2.1 Curvature of curves
A curve in the plane is defined as a continuous function x from some interval [a, b] to R2.The function x is also called the position vector of the curve, with a specific argumentvalue t ∈ [a, b] describing a particular point x(t) on the curve. It is clear that one candescribe the same curve using multiple parametrizations, say by replacing [a, b] by [0, 1]and mapping t 7→ 1
b (t − a). The derivative dxdt =: e is called the tangent vector of the
curve at a certain point, e(t) corresponding to the point x(t). The length L of the curveis defined using the standard Euclidean inner product by
L =
b∫a
√e · edt.
The integrand is equal to the norm of the rate of change of position with respect to t,giving the rate of change of length of the curve. The arc length s(t) of the curve up to acertain point x(t) is given by
s(t) =
t∫a
√e · edu.
From this we getds
dt=
∥∥∥∥dx
dt
∥∥∥∥ = ‖e‖. (1)
From here on we will parametrize the curve in terms of its arc length, as then the normof the tangent vector is equal to unity. This can be seen from equation (1) upon writingx = x(s), i.e. taking t = s.
It is clear that prescribing a tangent vector to every point along the curve gives a vectorfield restricted to the image x([a, b]). At points where nearby tangent vectors vary greatly,intuitively one would want to say that the curve is curved at that point. To describe this,decompose as
de
ds= κ(s)n(s),
where κ : [0, L] → R is the vector’s magnitude, and the unit vector n : [0, L] → R2+ is
called the normal vector, given by rotating e by an angle π2 clockwise with respect to the
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Protein-Mediated Instability of Curved Membranes
origin and positive x-axis. Now, n is perpendicular to e by construction, but also becausee is a unit vector:
0 =d
ds‖e‖2 =
d
ds(e · e) = 2e · de
ds,
from which e · n = 0 follows. The quantity κ(s) is called the local curvature of the curve.
In three dimensions, a curve x : [a, b]→ R3 enjoys one additional degree of freedom. Thisis described using the binormal vector b(s), given by
b(s) = e(s)× n(s),
where × denotes the cross product. At x(s) we now have a basis for the tangent space,given by the vectors e, n and b. As n is a unit vector, its derivative is perpendicular toit, and as such we can write for some α, τ
dn
ds= αe+ τb.
From e · n = 0 we see by differentiating that α = −κ. Similarly, from differentiating b wesee that τ = − db
ds · n. The quantity τ is called the torsion of the curve. We are now done:an elegant summary of the above is given by the Frenet-Serret equations, viz.
d
ds
enb
=
0 κ 0−κ 0 τ0 −τ 0
enb
.
2.2 Curvature of surfaces
Similar to curves in the previous section, a surface in three dimensions is described in termsof coordinates of R3. The surface itself is two-dimensional, which means we will need twoparameters to parametrize it. However, there does not exist a natural parametrizationfor these surfaces as we had in one dimension using the arc length. As such, any properdefinition of curvature should be invariant under changes of parametrization.
Given a surface S ⊂ R3, let it be described by the set S = x(u, v) |u, v ∈ U wherex : U → R3 and U ⊂ R2. Similarly to that of curves, we define two tangent vectors euand ev by taking partial derivatives
eu =∂x
∂u, ev =
∂x
∂v.
As before, we wish to construct a basis for the tangent space at a certain point x(u, v), butnow cannot use the convenient arc length parametrization. Indeed, in general eu and evdo not necessarily have unit norm, nor are they necessarily perpendicular. However, theyare necessarily linearly independent. To get a full basis for the tangent space consistingof three vectors, we take the normalized cross product of the two tangent vectors,
n =eu × ev‖eu × ev‖
,
defining the unit normal n to the surface. For the above definition, it is necessary that thesurface be orientable. Orientability means exactly that it is possible to make a consistentchoice of surface normal vector. As most household surfaces such as spheres and torii areorientable, we can safely assume this is the case. The second fundamental form II is aquadratic form on the tangent plane of S. It is given by
II = Ldu2 + 2M dudv +N dv2,
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Protein-Mediated Instability of Curved Membranes
where the coefficients L,M,N are given by
L = xuu · n, M = xuv · n, N = xvv · n,
i.e. they are given at every point in the parametric tangent plane of S by projecting thesecond partial derivatives of the parametrization onto the line normal to S. In matrixform the second fundamental form becomes
CM =
(L MM N
),
where CM is called the curvature matrix. The principal curvatures c1 and c2 are theeigenvalues of CM. Two invariants of this matrix are its determinant det and trace Tr.Both stay the same regardless of choice of basis, and so are invariant under changes inparametrization. This suggests the following definitions.
Definition 2.1. The mean curvature H and the Gaussian curvature K of a surface Sare defined in terms its principal curvatures c1, c2 by
H :=1
2(c1 + c2) , K := c1c2.
We have H = 12 Tr (CM) and K = det (CM), and so they are indeed coordinate invariant.
Remark. There is a more illuminating interpretation of the principal curvatures at acertain point. Given a unit normal vector n and tangent vector p, let the normal plane Nbe that plane containing both n and p. The plane N cuts the surface along a plane curve.To this plane curve we can associate a curvature as we did in the previous section. Theprincipal curvatures are the minimum and maximum of such curvatures over all tangentvectors p. As the principal curvatures are the eigenvalues of the curvature matrix, we seethat these tangent directions are the corresponding normalized eigenvectors.
2.3 Canham-Helfrich functional
In the previous section we have shown that the only two curvature-dependent geometricproperties that are invariant under choice of parametrization of the surface are the meanand Gaussian curvature. It is clear then that all polynomials in the mean and Gaussiancurvature are invariant as well. In this section we use this to find an expression for theenergy associated to a certain shape of a membrane surface. We will see that symmetrygreatly restrict the form of this energy.
Because this energy must not depend on any particular parametrization of the surface,this implies that the energy functional F can only depend on H and K. In fact, assumingboth principal curvatures are small, we have the following theorem.
Theorem 2.2. (Canham-Helfrich functional) The only energy functional F de-pending solely on the curvature of a surfaceM, is given up to first order in the magnitudeof its principal curvatures by
F [H,K] = 2κH2 + κK.
F is known as the Canham-Helfrich functional.
Proof Assuming that F = F [H,K] is sufficiently smooth such that it has a convergingTaylor series approximation, we can develop F by
for some real constants ai, κ and κ, the latter two being called the Gaussian and meanbending rigidities respectively. Note that a constant term is not required as only differ-ences in energy are meaningful. Assuming the principal curvatures c1 and c2 of the surfaceare both small and of the order ε we see that we have
H =c1 + c2
2= O(ε), K = c1c2 = O(ε2).
This allows us to order the terms in the Taylor expansion as in equation (2) not in termsof powers of H and K but in terms of nth order in ε. Doing this gives that
F [H,K] = a1H + κK + 2κH2 +O(ε3). (3)
We argue that a1 must be zero, which is the same as showing that F must be an evenfunction of H. Note that the ambient space R3 is isotropic. This means that mirroringthe surface around one coordinate-axis should not change its energy. By doing this, wesee that we must have ∫
M
F [H,K] dA =
∫M
F [−H,K] dA,
It is here that we assume to only consider a single Monge patch and not an entire mem-brane, because there must not be an inside of the membrane for this to be valid. Lookingat equation (3) we see that this implies that
∫M a1H dA = 0, for all possible H. This can
be accomplished either by having a1 = 0, or by having∫MH dA = 0 for every choice of
H. The latter is not possible, from which we conclude that instead a1 must be zero.
This functional is named after Canham [1] and Helfrich [6], who first used it in theirdescription of membranes and in particular the red blood cell. We see that the onlypossible form of the energy up to lowest order is one quadratic in the mean curvature andlinear in the Gaussian curvature. Of course, this derivation assumed restricting ourselvesto small principal curvatures. This is not a problem, as the membrane surfaces we areinterested in are only slightly curved.
2.4 Gauss-Bonnet Theorem
Gaussian curvature is a property intrinsic to the surface, meaning that it can be definedwithout embedding the surface in some ambient space. There is a theorem, the Gauss-Bonnet theorem, which says that the Gaussian curvature integrated over the entire surfaceis a purely topological property of that surface. Below we give a particular case of a moregeneral statement.
Theorem 2.3. (Gauss-Bonnet) Let S ⊂ R3 be a closed compact two-dimensionalsurface. Then we have that ∫
S
K dA = 2πχ(S),
where χ(S) is the Euler characteristic of S.
Proof Omitted. See for example [5].
Looking at the shape of the Canham-Helfrich functional, we see that if the Gaussianrigidity is assumed constant through space, the term including the Gaussian curvature isconstant for homotopically equivalent surfaces. The Euler characteristic is defined purelyin terms of the topology of the surface and so is a topological constant. When one is onlyperturbing the system from a certain ground state, all dynamic behavior changing theshape of the surface will still leave its topology unchanged.
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Protein-Mediated Instability of Curved Membranes
Corollary 2.4. The Euler characteristic of a surface is a homotopic invariant. Becauseof this, we have for constant κ∫
κK dA = κ 2πχ(S) = c ∈ R.
This means that when wanting to find the surface minimizing the Canham-Helfrich func-tional, it would suffice to only look at the term including the mean curvature. This isbecause one can achieve this by considering perturbations around some stationary solu-tion, say the flat solution with zero curvature everywhere. However, in this thesis we arelooking at surfaces where the Gaussian rigidity is not constant. It is instead a functionof the local concentration of proteins. Because of this, we will not be able to use theGauss-Bonnet theorem to simplify our energy functional by disregarding the Gaussianterm.
2.5 Monge patch
In this thesis we will discuss a section of membrane described via a Monge patch.
Definition 2.5. A two-dimensional Monge patchM consists of a connected open subsetU ∈ R2 together with a map x : U → R3 of the form
x(u, v) := (u, v, h(u, v)).
Here, h : U → R is called the height function, indicating the height of the patch at anypoint in U . Note that x is automatically injective. We will assume that h is sufficientlysmooth, say C2, to ensure additional structure such as curvature can be imposed onM = x(U).
Remark. The reader might note that a height function cannot describe budding surfacesbecause h must be single-valued. This is of course true. However, if one wanted to describesuch a surface, one can always partition it into small enough parts such that overlappingMonge patches do suffice. Nevertheless the Monge patch description presupposes smallcurvatures, but that is exactly the situation we will be in.
We now derive the explicit expressions for both the Gaussian and mean curvature in thecase of the surfaceM being a two-dimensional Monge patch with radial symmetry. Moreprecisely, the surface M⊂ R3 is described by
M =
x =
r cosφr sinφh(r)
: r > 0, φ ∈ [0, 2π)
.
This implies that the two tangent vectors to the surface become
er =∂x
∂r=
cosφsinφ
h(r)
, eφ =∂x
∂φ=
−r sinφr cosφ
0
.
The metric tensor becomes
g =
(1 + h2 0
0 r2
),
because gab = ea · eb. The unit normal vector becomes
n =er × eφ|er × eφ|
=1√
1 + h2
−h cosφ
−h sinφ1
,
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Protein-Mediated Instability of Curved Membranes
which means the second fundamental form becomes
II =
h√1+h2
0
0 rh√1+h2
,
giving rise to the curvature matrix CM given by CM = g−1II
CM =
h
(1+h2)3/2 0
0 h
r√
1+h2
.
From this we see that the mean and Gaussian curvature become
2H(r) = Tr (CM) =h+ h3 + rh
r(
1 + h2)3/2
=h
r√
1 + h2+
h(1 + h2
)3/2,
K(r) = det (CM) =hh
r(
1 + h2)2 .
Lastly, the square root of the determinant of the metric tensor, g, is given by
g =√
det g = r(
1 + h2)1/2
.
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Protein-Mediated Instability of Curved Membranes
3 Gradient flow
In this section we discuss the concept of a gradient flow. We will see that many ofthe well-known equations of motion arise naturally as gradient flows. This concept isuseful to us in describing the interplay between the membrane wanting to be flat, and theproteins wanting to maximize their entropy. This section closely follows lecture notes byPeletier [10], which in turn uses ideas from Otto [8] regarding Wasserstein gradient flows.
Example. Consider a one-particle system in one dimension. Let x(t) denote the positionof the particle as a function of time and let m be the particle’s mass. We assume thesystem to be conservative. In that case, the energy Hamiltonian E = T + V splits, whereT is the kinetic energy of the particle and V = V (x) is some conservative energy potential.In the one-dimensional case we have T = 1
2mx2, where the overdot denotes differentiation
with respect to time. Because the Hamiltonian is conserved, i.e. the total energy of thesystem is constant, we have
∂tE = 0, giving x (mx+ V ′(x)) = 0. (4)
This gives us the equation of motion for the particle in the form of Newton’s second law
mx = −V ′(x), F = ma,
where a = x is the particle’s acceleration and F = −V ′(x) is the force exerted on it. If weforego the system being conservative and introduce a dissipative friction force Fw = −ξx,the total contribution of this force to the energy, i.e. the work done by this force, is equalto ∫
R
Fw dx = −∫T
ξxdx
dtdt = −
∫T
ξx2 dt,
which means that this friction contributes −ξx2 to the energy density. Using this, ourearlier equation becomes
∂tE = −ξx2,
which using equation (4) more explicitly reads
x (mx+ V ′(x)) = −ξx2, (5)
and so gives the standard equation for a particle in a potential with friction
mx+ ξx+ V ′(x) = 0. (6)
This again is Newton’s second law, with the total force being equal to −V ′(x)− ξx. Letus take m = 0 for a moment. Then equation (6) turns into
ξx+ V ′(x) = 0,
which we will write more suggestively as
x = −1
ξ∇V.
This is our first example of a gradient flow. The time derivative of the system’s orderparameter is given by minus the gradient of some energy, in this case V/ξ. We will nowgive a more formal definition, but for this we will first need some definitions.
Definition 3.1. Given a normed space V , a functional F : V → R is said to be Frechetdifferentiable at v ∈ V if there exists a ξ ∈ V ′ such that
F [v + w]− F [v]− 〈ξ, w〉+ o (‖w‖V ) as w → 0.
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Protein-Mediated Instability of Curved Membranes
The topological dual of V , denoted by V ′, is a normed space given by
V ′ = ξ : V → R, linear and continuous ,
and has canonical norm
‖ζ‖V ′ := sup06=v∈V
〈ξ, v〉‖v‖V
.
Here 〈ξ, v〉 = V ′〈ξ, v〉V is equal to the value ξ(v) and is called the dual pairing betweenV and V ′. Given a Hilbert space H, a complete vector space with inner product (·, ·)Hand norm ‖z‖2H := (z, z)H , one can map dual vectors to normal vectors bijectively, as thefollowing lemma shows.
Lemma 3.2. (Riesz representation lemma) Let H be a Hilbert space. For eachξ ∈ H ′ there exists a unique w ∈ H such that
H′〈ξ, v〉H = (v, w)H for all v ∈ H.
Furthermore, ‖w‖H = ‖ζ‖H′ .Proof Omitted. See for example [2].
Note that the vector w depends on what inner product is given to H. Similarly to thedefinition of the directional or Gateaux derivative of a normal function, we then have
limt→0
F [v + tw]− F [v]
t= 〈ξ, w〉.
The dual vector ξ ∈ V ′ is called the Frechet derivative of F at v, and we write ξ = F ′[v].Other names for the same concept are first variation, and functional derivative. Indeed,the first variation or functional derivative δF of a functional F is the linear functional (sodual vector) such that w ∈ V is mapped to
δF [v](w) = limt→0
F [v + tw]− F [v]
t=
d
dtF [v + tw]
∣∣∣∣t=0
.
This can be seen to be identical to the above by writing δF [v](w) = 〈F ′[v], w〉. Given aFrechet differentiable functional F on some Hilbert space H, the gradient gradF at somev ∈ H is the dual vector found through Lemma 3.2 by
(gradF [v], w)H = H′〈F ′[v], w〉H for all w ∈ H.
When working in Rn we usually write ∇F for gradF . Note that while being Frechetdifferentiable or not is independent of the choice of inner product on the space, the gradientclearly does depend on the inner product.
Definition 3.3. Abstractly, a gradient flow on a spaceM is the dynamical system givenby the differential equation
dρ
dt= − gradF [ρ],
for ρ ∈M, where F is a functional on M.
For this to make sense, a metric tensor g or inner product must be specified to turn Minto a Riemannian space, or else there is no concept of a gradient. As such a gradientflow is constructed by providing three things:
• A space M, say a differentiable manifold;
• A metric tensor g to turn (M, g) into a Riemannian space, or an inner product onthe tangent space;
14
Protein-Mediated Instability of Curved Membranes
• A functional F on M.
The gradient is constructed using g by the following. Take the differential diff F of F ,which is a cotangent vector field, and turn it into the gradient gradF of F , which is atangent vector field, by the appropriate form of Riesz’ lemma,
g(gradF,w) = diff F · w for all vector fields w on M. (7)
Example. In the case of our one-dimensional example the chosen space was R, and sothe tangent space is Tp(R) ∼= R. As inner product we chose a weighted variation of thestandard Euclidean inner product, namely (v, w)ξ = ξvw for v, w ∈ Tp(R), so that we canrecast equation (5) as
∂tH = −(x, x)ξ.
Note that we can write the gradient flow equation in weak formulation as
gρ
(dρ
dt, w
)+ gρ (gradF,w) = 0 for all vector fields w along ρ,
the subscript in gρ indicating that it is the metric at ρ. Using equation (7) we see thatthis means
gρ
(dρ
dt, w
)+ diff F · w = 0, (8)
and using this we get
d
dtF [ρ] = diff Fρ ·
dρ
dt= −gρ
(dρ
dt,
dρ
dt
)< 0,
from which we see that indeed energy is being dissipated.
Remark. There are various other models which can be described as a gradient flow,notably Model A (Glauber) and Model B (Cahn-Hilliard), following the nomenclature ofHohenberg and Halperin [9]. Given a Hamiltonian functional E = E(φ, t) where φ is thesystem’s order parameter, Model A and B without noise term are given respectively bythe choices
∂tφ = −ΓδEδφ
and ∂tφ = λ∇2 δEδφ.
Both of these models can be seen to be gradient flows of E with respect to two differentinner products producing the gradients, namely
∂tφ = −Γ gradL2(E) and ∂tφ = λ gradH−10
(E).
Here H−10 is a Sobolev space, but we will not discuss this further. We see that mathemat-
ically these two models are the same. The only difference is the choice of inner producton the tangent space, leading to different gradients.
3.1 Wasserstein gradient flow
Denote by P(X) the set of all Borel probability measures on X, where (X, d) is a metricspace. The Wasserstein distance will be defined on the set of probability measures withfinite second moment,
P2(X) =
µ ∈ P(X) :
∫X
|x|2 µ(dx) <∞.
15
Protein-Mediated Instability of Curved Membranes
Definition 3.4. Given µ, ν ∈ P2(X), the L2-Wasserstein distance between µ and ν isdefined as
W2(µ, ν) := infπ∈Γ(µ,ν)
∫X×X
d(x, y)2dπ(x, y)
1/2
,
where Γ(µ, ν) ⊂ P2(X ×X) denotes the set of couplings of µ and ν.
Here Γ(µ, ν) contains all probability measures which have µ and ν as their marginals.More specifically, elements γ of Γ(µ, ν) satisfy the condition that for all Borel sets A ⊂ X
γ(A×X) = µ(A) and γ(X ×A) = ν(A).
This can be seen to be equivalent to the condition that for every (φ, ψ) ∈ L1µ(X)×L1
ν(X)we have ∫
X×X
[φ(x) + ψ(y)] dγ(x, y) =
∫X
φ(x) dµ(x) +
∫X
ψ(x) dν(x).
Note that W2(µ, ν) is finite. The γ achieving this infimum exists and is called the op-timal transport plan, as it describes the optimal plan of transporting mass from config-uration µ to ν, interpreting both as denoting mass concentrations. The cost functionc(x, y) = d(x, y)2 describes the cost of moving a unit of mass from position x to positiony. See [11] for an introduction on this subject.
Note that it is called the L2-Wasserstein distance because the distance used inside theintegral is the standard L2 one. The Wasserstein distance can be better understood bymoving to discrete probability measures, and then later considering an arbitrary proba-bility measure to be the limit of discrete ones. Consider a system of N particles in a spaceX, their positions denoted by x1, . . . , xN ∈ X. From this we create a probability measurepN (x) by
pN (x) :=1
N
N∑i=1
δxi .
Given two such probability measures µ = N−1∑i δxi and ν = N−1
∑i δyi , the natural
distance between the two is the one inherited from X, namely
d2(µ, ν) = inf
1
N
N∑i=1
d(xi, yσ(i))2 : σ a permutation of 1, . . . , N
.
We see from this that the distance d2 exactly describes multi-particle Euclidean geometry,modulo permutations of the particles. This process of taking the infimum over all permu-tations σ is replaced in the more general case by taking the infimum over all couplings ofthe measures µ and ν.
Recall that given functional F defined on L2(Rn), and a vector v ∈ L2(Rn) and tangentvector w ∈ L2(Rn), the definition of the derivative of F at v in the direction of w is givenby the Frechet derivative or L2-gradient,
limt→0
F [v + tw]− F [v]
t= L2(Rn)′〈F ′[v], w〉L2(Rn) = (gradL2 F [v], w)L2(Rn) .
A very useful property for determining similar derivatives for gradients on P2(Rn) is thatthe sets L2(Rn) and P2(Rn) share a common subset of nonnegative functions in C∞c (Rn).
16
Protein-Mediated Instability of Curved Membranes
At least formally, we have for a functional E on P2(Rn), a point ρ ∈ C∞c (Rn) and atangent s ∈ Tanρ ∩ C∞c (Rn) that
limt→0
E[ρ+ ts]− E[ρ]
t= (gradL2 E[ρ], s)L2(Rn) .
This can only be done formally, as problems may arise because for example ρ + ts maynot be nonnegative for any t 6= 0. However, we will not discuss this further.
Definition 3.5. The Wasserstein gradient gradW2of some functional E : P2(Rn)→ R
is the distribution on Rn given by
gradW2E[µ] := −div [ρ∇ gradL2 E[µ]] .
With this we can define the Wasserstein gradient flow of a functional E.
Definition 3.6. The Wasserstein gradient flow of a functional E is given by
∂tρ = − gradW2E[ρ].
4 Modelling choices
In this section we describe how we use a gradient flow approach to describe both ourmembrane M and the proteins φ thereon. Here again we use ideas from Otto [8]. Thechoice of energy functional F will need to describe both the curvature energy coming fromthe membrane’s shape, and the energy contribution from the protein distribution. Forthis we will use the adjusted Canham-Helfrich functional
F =
∫M
2κ(φ)H2 + κ(φ)K + η(φ) dA.
The bending rigidities κ and κ are now assumed to be dependent on the particle densityφ: the proof of Theorem 2.2 on the Canham-Helfrich functional used Taylor expansionwith respect to curvature only. Now proteins are involved this means that the coefficientscoming from this Taylor expansion must now be seen as functions of φ. The function ηdescribes the energy contribution from the protein concentration. The interpretation ofthe φ-dependent rigidities is that as more proteins gather at a certain position, they willcurve the membrane, making it more rigid. This then is reflected by an increase in en-ergy. A minimization of F will necessarily include a compromise between the membrane’scurvature energy and the protein energy contribution. Surely κ will be nonnegative, forelse the membrane will spontaneously curve.
4.1 The phase space PNext, we need to determine the space of admissible states P and its tangent space Tp(P).To do this, we will describe our membrane by means of a radially symmetric Monge patch.LetM be described by some open set U ⊆ R2 and a height function h : U → R. For easeof notation we will extend U to all of R2, setting h equal to zero outside U . Using thisexplicit representation of M, the energy F of the membrane-protein system is given by
F [h, φ] =
∫2κ(φ)H2(h) + κ(φ)K(h) + η(φ) dA.
The height function is given by h = h(r), where r is the radial distance from the origin.The assumption that the patch is radially symmetric is reflected in the height function
17
Protein-Mediated Instability of Curved Membranes
being merely a function of the radial distance from the origin. As such, the boundaryconditions on h are h(0) = 0 and h = 0 on the boundary of U . Note that we choose φ/gto be the protein density on the membrane surface, which means that φ/r is the proteindensity on U , and φ is the density on the half real line [0,∞). With this, we can defineP and its tangent space TpP.
Definition 4.1. The phase space P is given by the cartesian product P = (h, φ) : h ∈ H,φ ∈ Φ =H × Φ, where H and Φ are given respectively by
H = h : [0,∞)→ R, h(0) = h(∞) = 0 and Φ =
φ : [0,∞)→ R+,
∞∫0
φ = 1
.
Definition 4.2. The tangent space TpP of P is given by the cartesian product TpP =TpH × TpΦ, where TpH and TpΦ are given respectively by
TpH =
(~v, ψ) : ~v : [0,∞)→ R3
and TpΦ =
ψ : [0,∞)→ R,∞∫
0
ψ = 0
.
4.2 Entropic energy contribution
Our description of the protein concentration does not allow for a continuous concentrationfunction at the local level. Because of this, we cannot use a Dirichlet ∼ (∇φ)
2functional
with L2 inner product to arrive at the diffusion equation through gradient flow. Instead,the relevant form of local energy is given by the entropy of the protein configuration. Oneis led to the following energy density η,
η(φ) = φ log φ,
reminiscent and indeed derived from the discrete concept of entropy, as is shown below.With this energy density, it is also possible to have the diffusion equation arise as a gra-dient flow. Instead of the L2 metric, we will use the Wasserstein metric from Definition 3.4.
Consider a lattice of N sites, on each of which exactly one particle some type i is present.Denote the total amount of particles of type i by ni. The amount of ways Ω in which onecan arrange the system such that there are ni particles of type i is equal to
Ω =N !∏i
ni!.
Let us now consider log Ω is when N is large. For this we use Stirling’s formula
log n! =
n∑k=1
log k = n log n− n+ o(n) as n→∞,
to estimate
log Ω = logN !−∑i
log ni!
= N logN −N −∑i
(ni log ni − ni) + o(N)
= −N∑i
(φi log φi) + o(N) as N →∞,
18
Protein-Mediated Instability of Curved Membranes
where we have φi := ni/N and we used∑i ni = N . This means that
log Ω = −N∑i
φi log φi.
From this we see that a possible interpretation of the entropy is that it counts the numberof realizations of a given configuration,
η(φ) = − limN→∞
1
Nlog Ω,
where Ω is the number of realizations of the fractional densities φi. A perhaps moreconvincing argument for why the entropy then is a natural characterization of the proteinenergy is the following. Recall the discussion in the previous section on the interpretationof the Wasserstein distance through discrete probability measures. Given an empiricalmeasure pN = 1
n
∑Ni=1 δxi describing N particles in a space X, Sanov’s theorem states
that pN follows the large deviation principle,
P (pN ≈ p) ∼ e−N∫ dp
dµ log dpdµ dµ
for some particle distribution p. Here µ is for example the Haussdorff surface measureH2, if X is taken to be a two-dimensional surface. We see that the relative entropy of pwith respect to µ describes the probability of observing some state p. It turns out thatthe Wasserstein distance together with the entropy are the natural ingredients to describea discrete system of particles undergoing Brownian motion.
Having made this choice, it is now possible to see that the diffusion equation is theWasserstein gradient flow of the entropy integral.
Corollary 4.3. The diffusion equation ∂tρ = ∆ρ is the Wasserstein gradient flow ofE[µ] =
∫ρ log ρdx, where ρ is the Lebesgue density of µ.
Proof We have thatgradL2 E[ρ] = log ρ+ 1,
and so using definitions 3.5 and 3.6 we get that
∂tρ = − gradW2E[ρ] = div (ρ gradL2 E[ρ]) = ∆ρ.
4.3 Introducing dissipation
We now introduce dissipation of F by means of an inner product (·, ·) on TpP, where ‖ ·‖2is the derived norm. This inner product can be split up as
where (·, ·)c and (·, ·)c′ are the inner products on TpH and TpΦ respectively. The dynamicsof the system are then given by the gradient flow
DtF = −‖(~v, ψ)‖2.
This however must be read in weak formulation
F ′[h, φ] · (~v2, ψ2) = −((~v1, ψ1), (~v2, ψ2)),
for all (~v2, ψ2) ∈ TpP. Here some state (h, φ) and tangent vector (v1, ψ1) are given. Theexpression F ′[h, φ] is also called the first variation of F . To motivate our choice of inner
19
Protein-Mediated Instability of Curved Membranes
product, we first take a detour. Consider a spherical particle with some radius R andvelocity ~v moving through a viscous fluid with viscosity η. Stokes’ law then states thatthe frictional force ~f on this particle is given by
~f = c~v.
In three dimensions the coefficient of friction c is given by c = 6πRη, assuming the fluidis isotropic. We see from this that the energy dissipation due to this frictional force isgiven by
~f · ~v = c‖~v‖2.
Inspired by this result, the standard choice of energy dissipation for a membrane sur-rounded by some viscous fluid, usually water, is the following: suppose the membrane israre enough such that it can be regarded as being made up of a network of individualprotein strands. If we assume the coefficient of friction to be isotropic, the total frictionon the membrane surface S is given by∫
S
c‖~v‖2 dA,
where ~v is the displacement velocity vector at each point on the membrane surface. Collo-quially, the membrane is considered to be a collection of spherical particles, and the totalfriction is just given by Stokes’ law integrated over the entire surface. The assumption ofa rare membrane is typically false in the case of a lipid bilayer membrane, but still thistype of friction is commonly used. As such, we choose as inner product (·, ·)c on TpH theexpression
(~v1, ~v2)c =
∫M
c~v1 · ~v2 dA.
The choice of dissipation for the protein rearrangement is somewhat more elaborate. Letψ ∈ TpΦ be characterized by a velocity wτ . To define this velocity, we use the surfacegradient ∇Γ which is given by
∇Γu = ∇u− ~n (~n · ∇u) ,
for scalar fields u, where ~n is the surface normal and ∇ is the usual gradient operator.With this, with the surface divergence being given by divΓG = ∇Γ ·G for vector fields G,we define wτ by
ψ
g= divΓ
φ
gwτ .
Remark. This correspondence is not unique, as one can add to some wτ any w withdivΓ
φg w = 0. Supposing that ψ = divΓ ω for some ω is done without loss of generality
because ψ has zero integral.
The norm on TpΦ then is defined using this velocity wτ as
‖ψ‖2 = inf
∫M
c′φ
g|wτ |2 gds
∣∣∣∣∣∣ ψg = divΓφ
gwτ
,
where c′ is a coefficient of friction. One can show that the wτ achieving said infimum isa gradient, so wτ = −∇p, where p is a pressure. Indeed, the minimizer satisfies∫
v · wτ = 0 for all vector fields v on Rn with ∇ · v = 0,
20
Protein-Mediated Instability of Curved Membranes
which means there is some function p on Rn with wτ = −∇p. Then the norm on ψ canbe simplified to
‖ψ‖2 =
∫M
c′φ (∇p)2dA.
This also shows that the part (·, ·)c′ of the inner product on the second half of the tangentspace of P is given by
(ψ1, ψ2)c′ =
∫M
c′φ∇p1∇p2 dA,
where the ψi and pi are related through ψi = −div (φ∇pi). This pressure interpretationof p comes from Darcy’s law in fluid mechanics, where the vector field u on Rn describingthe average velocity of a gas is given by
u = −M∇p,
where p is the pressure of the gas and the matrix M describes the mobility of the gasthrough some medium. By non-dimensionalizing we get u = −∇p.
Definition 4.4. The inner product (·, ·) on TpP at some point (h, φ) ∈ P is given by
((~v1, ψ1), (~v2, ψ2)) =
∫M
c~v1 · ~v2 + c′φ∇p1∇p2 dA
for (~v1, ψ1), (~v2, ψ2) ∈ TpP, where the ψi and pi are related through ψi = −div (φ∇pi).
Remark. In the radially symmetric two-dimensional case the expression for wτ will re-duce to ψ = ∂r (φwτ ), or
wτ (r) =1
φ(r)
r∫0
ψ(s) ds.
Written differently, one can also see TpΦ to be isomorphic to p : Rn → R / ∼, where ∼denotes equivalence up to an additive constant. This is the same freedom as expressedearlier by the ability to add any w. The isomorphism is given by the equation
−div (φ∇p) = ψ.
The corresponding metric tensor g for Φ is given by
gφ (ψ1, ψ2) =
∫φ∇p1 · ∇p2,
where the ψi and pi are related using the above isomorphism. Using partial integration,we see that we get
gφ (ψ1, ψ2) =
∫ψ1 p2. (9)
Remark. The quantity∫c′φ |wτ |2 has a physical meaning. Namely, it is the rate of dis-
sipation of kinetic energy due to friction as the proteins move with velocity wτ throughthe membrane. This is completely analogous to
∫c‖~v‖2 for the membrane itself.
It is now easy to see that the diffusion equation is the Wasserstein gradient flow of theentropy.
21
Protein-Mediated Instability of Curved Membranes
Theorem 4.5. With the choices η(φ) = φ log φ and gφ (ψ1, ψ2) =∫φ∇p1 · ∇p2 with
−div (φ∇pi) = ψi, the gradient flow
dφ
dt= − grad η(φ)
reduces to the diffusion equation∂tφ = ∆φ.
Proof Using the framework developed above, we see that
diff η(φ) · w =
∫(log φ+ 1)w,
which using equation (9) turns equation (8) into∫∂tφ p+
∫(log φ+ 1)ψ = 0.
We now substitute the relation between ψ and p to get∫∂tφ p−
∫(log φ+ 1) div (φ∇p) = 0,
which after integrating by parts twice becomes∫ (∂tφ−∇2φ
)p = 0,
which is the diffusion equation in weak form
∂tφ = ∆φ.
22
Protein-Mediated Instability of Curved Membranes
5 Dynamic equations
In this section we derive the dynamic equations of the membrane-protein system in thecase of a two-dimensional membrane patch using the gradient flow approach describedin section 3. As was stated in section 4, for simplicity it is assumed that this patch isradially symmetric around some point called the origin, and assumed to be infinite.
We wish to perturb a state (h, φ) ∈ P additively by a pair (h, φ), and then calculate thefirst variation F ′[h, φ] of the energy functional F by means of the formula
F ′[h, φ] · (h, φ) =d
d∆tF[h+ ∆t h, φ+ ∆t φ
], (10)
where the meaning of the inner product · on the left hand side is not made precise yet.However, it will not be convenient to work with purely additive perturbations of (h, φ), so
instead we will use a perturbation pair (~v, ψ) ∈ TpP and consider h and φ to be functionsof that pair, writing
F ′[h, φ] · (~v, ψ) =d
d∆tF[h+ ∆t h(~v, ψ), φ+ ∆t φ(~v, ψ)
].
Recall that the perturbation ~v will be a radially symmetric vector field ~v = ~v(r) : [0,∞)→R3 with no tangential component. For notational convenience, we will furthermore define
~v :=(~v‖, v⊥
), vr := ~v · (er, 0) ,
where er = er(~x) = ~x/‖~x‖ ∈ R2 for ~x ∈ R2 is the two-dimensional unit vector pointingradially outwards. Note furthermore that this implies that
~v‖ = vr er.
As we will see, we will have no use for the full vectors (~v‖, 0) and (0, v⊥), instead onlyusing vr and v⊥. Furthermore, recall that the perturbation ψ = ψ(r) will be a radiallysymmetric protein concentration field defined by
ψ : [0,∞)→ R.
Note that ψ can take on negative values as it describes the moving around of proteinsfrom one position to another, as a result of them wanting to minimize their entropy. Wealso must have that ψ has zero integral, i.e.∫
[0,∞)
ψ dr = 0.
We now choose how the pair (~v, ψ) perturbs a state (h, φ). The finite difference equationsfor the height and concentration function, describing their value a time step ∆t ahead,are given by
h(r, t+ ∆t) = h(r, t) + ∆t v⊥(r), (11)
φ(r + ∆t vr, t+ ∆t) =φ(r, t) + ∆t ψ(r, t)
1 + ∆t ∂rvr(r). (12)
The change in h is merely an additive one, increasing h by an amount ∆t v⊥. We readilysee that this means that ∂th = v⊥. The change in φ is somewhat more intricate. Thereare two effects in the right hand side of the equation governing the protein concentration.The first is the addition of an amount ∆t ψ to the local concentration. The second is the
23
Protein-Mediated Instability of Curved Membranes
change due to change of the membrane’s shape, with the right hand side being dividedby the Jacobian of the transformation
r 7→ r + ∆t vr(r),
in response to moving an amount vr(r) radially away from some point ~x. This stems fromthe need to rescale the concentration as the area changes: the same amount of proteinmolecules on a more stretched out area will result locally in a lower concentration. Wesee that the Jacobian of this transformation is given by
1 + ∆t ∂rvr(r).
Remark. As said, φ is the protein concentration on the half real line [0,∞). We see thatthe protein mass mφ(A) on some annulus A = A× S1 on U with A = [r1, r2] is given by
mφ(A) =
∫A
φ.
To determine the first variation of F , we will need to take a detour. First, let us notethat the right hand side of equation (10) can also be written as
d
d∆tF[h+ ∆t h, φ+ ∆t φ
]= Dt
[∫2κ(φ)H2(h) + κ(φ)K(h) + η(φ)
].
Because the operator Dt is linear, we will split F into three functionals Fi for i = 1, 2, 3.
Definition 5.1. The functionals Fi for i = 1, 2, 3 are defined by
F1[h, φ] :=
∫M
2κ (φ) H2(h) dA, F2[h, φ] :=
∫M
κ (φ) K(h) dA, F3[h, φ] :=
∫M
η (φ) dA.
This allows us to write F as the sum of the three Fi, viz.
F [h, φ] = F1[h, φ] + F2[h, φ] + F3[h, φ].
To then find the derivative of each of these Fi, we note the following. Each Fi can bewritten as
Fi =
∫M
fi (φ) Gi(h) dA =
∞∫0
fi
(φ
g
)Gi(h) gdr,
for some functions fi and Gi. This is done because for any functions f and G we have
Dt
∫M
f (φ)G(h) dA = Dt
∞∫0
f
(φ
g
)G(h) gdr (13)
=
∞∫0
[f ′
gDtφ− φDtg
gG+ f (DtgG+ gDtG)
]dr.
To this end, we define functions Gi and fi for i = 1, 2, 3.
Definition 5.2. The functions Gi and fi for i = 1, 2, 3 are defined by
G1 := H2 G2 := K, G3 := 1,
f1 := 2κ, f2 := κ, f3 := η.
24
Protein-Mediated Instability of Curved Membranes
Inspired by equation (13), we search for an expression for the time derivative Dtφ.
Lemma 5.3. We have that
Dtφ = −∂r (vrφ) + ψ,
from which one can see that ψ is purely the additive (non-diffusive) term in the timeevolution of φ.
Proof Because our finite difference equation for φ gives us the value for φ not just atime step ∆t ahead, but also a position step ∆t vr further, our calculation is somewhatelaborate. Write r = r′ + ∆t vr(r
′), viewing r as being the image of some r′ after addinga step ∆t vr(r
′).
Note that ∂tφ is defined as
∂tφ = lim∆t→0
φ(r, t+ ∆t)− φ(r, t)
∆t.
Let ζ(r, t) be some test function. We then have∫(∂tφ) ζ rdr = lim
∆t→0
∫φ(r, t+ ∆t)− φ(r, t)
∆tζ(r, t) rdr.
We now wish to manipulate the right hand side of this equation. Let us forego taking thelimit ∆t→ 0 for the moment. Note that dr = (1 + ∆t ∂rvr(r
′)) dr′. Then the right handside can be written as∫
φ(r, t+ ∆t)− φ(r, t)
∆tζ(r, t) rdr
r=r′+∆t vr(r′)=
1
∆t
∫φ(r′ + ∆t vr(r
′), t+ ∆t)ζ(r′ + ∆t vr(r′), t) (r′ + ∆t vr(r
′)) (1 + ∆t ∂rvr(r′)) dr′
− 1
∆t
∫φ(r, t)ζ(r, t) rdr
equation (11)=
1
∆t
∫[φ(r′, t) + ∆tψ(r′, t)] ζ(r′ + ∆t vr(r
′), t) (r′ + ∆t vr(r′)) dr′
− 1
∆t
∫φ(r, t)ζ(r, t) rdr.
Now we can relabel r′ 7→ r in the first integral again to see that the previous is equal to
1
∆t
∫φ(r, t) [ζ(r + ∆t vr(r), t)− ζ(r, t)] rdr
+
∫ψ(r, t)ζ(r + ∆t vr(r), t) (r + ∆t vr(r)) dr
+1
∆t
∫φ(r, t)ζ(r + ∆t vr(r), t) ∆t vr(r) dr.
Upon letting ∆t→ 0 we see that this is equal to∫φ(r, t) ∂rζ(r, t) vr(r) rdr +
∫ψζ rdr +
∫φζ vr dr,
and by partial integration we have∫φ∂rζ vr rdr = −
∫ζ ∂r (φvrr) dr.
25
Protein-Mediated Instability of Curved Membranes
Combining these results, we see that∫(∂tφ) ζr =
∫ζ [−∂r (φvrr) + ψr + φvr] =
∫ζ [−r∂r (φvr) + ψr] ,
as we have −∂r (φvrr) = −r∂r (φvr)− φvr. Because this must hold for each test functionζ, we conclude using Lemma 7.3 that
∂tφ = −∂r (φvr) + ψ.
Looking back, we see that we used the technique of integrating with a test function ζ tofirst make a shift from r to r′, and later swap the differentiation with respect to r from ζto the rest of the integrand. This allowed us to determine φ(r, t+ ∆t) when only knowingφ(r + ∆t vr, t+ ∆t).
Remark. Using the above Lemma, we can calculate the change of protein mass on someannulus A on U :
∂tmφ(A) = ∂t
∫A
φ =
∫A
∂tφ =
∫A
−∂r (vrφ) + ψ =
∫A
ψ − [vrφ]∂A ,
using the divergence theorem. We see that the change of protein mass is given by thedifference of the diffusive protein transport and the protein flux through the boundary ofA. In particular when we take A = U , because the boundary terms then vanish, we havethat
∂tmφ(U) =
∫[0,∞)
ψ = 0.
Next we turn to the time derivative Dtg. Here we do not have the same problem as withDtφ, and can use the normal rules for differentiation to compute.
Lemma 5.4. We have that
Dtg = vr
√1 + h2 +
rh√1 + h2
(h vr + v⊥
).
Proof We will use both h and ∂rh to denote the derivative of h with respect to r. Notethat ∂tr = vr. We have
Dtg = ∂rg ·dr
dt+ ∂tg
=
((1 + h2
)1/2
+ r ∂r
(1 + h2
)1/2)· vr + r ∂t
(1 + h2
)1/2
.
Now, a term ∂x
(1 + h2
)1/2
is given by
∂x
(1 + h2
)1/2
=(
1 + h2)−1/2
h(∂xh
).
Now using that ∂th = v⊥ and that the operators ∂t and ∂r commute, i.e. ∂t∂r = ∂r∂t, wesee that our earlier expression is equal to
Dtg =
((1 + h2
)1/2
+ r(
1 + h2)−1/2
hh
)· vr + r
(1 + h2
)−1/2
h v⊥
= vr
√1 + h2 +
rh√1 + h2
(h vr + v⊥
).
26
Protein-Mediated Instability of Curved Membranes
5.1 Explicit coordinates
We now turn to explicit coordinates to find F ′. We will first introduce universal notationfor the terms in the explicit expressions. This Γ-notation will prove to be a useful tool incomputing F ′.
Definition 5.5. For m,n ∈ N0 and k ∈ Q, the quantity Γm,nk is defined as
Γm,nk :=hmhn(
1 + h2)k .
Lemma 5.6. The derivative DtΓm,nk is given by
DtΓm,nk = mv⊥ Γm−1,n
k + nv⊥ Γm,n−1k − 2kv⊥ Γm+1,n
k+1 .
Proof See appendix A.1.
Lemma 5.7. Two terms Γm,nk , Γm′,n′
k′ satisfy the following multiplication property
Γm,nk Γm′,n′
k′ = Γm+m′,n+n′
k+k′ .
Proof Trivial.
Lemma 5.8. Using Γ-notation, the Gi are given by
G1 =1
4
(1
r2Γ2,0
1 +2
rΓ1,1
2 + Γ0,23
), G2 =
1
rΓ1,1
2 , G3 = 1.
Furthermore, g and Dtg are given by
g = r Γ0,0
− 12
,
Dtg = v⊥r Γ1,012
+ vr
(Γ0,0
− 12
+ r Γ1,112
).
Proof Using the expressions for the mean and Gaussian curvature derived in section2.5, we see that the Gi in explicit form become
G1(h) =
h+ h3 + rh
2r(
1 + h2)3/2
2
, G2(h) =hh
r(
1 + h2)2 , G3(h) = 1. (14)
We can expand the expression for G1 by writing
G1 =
h+ h3 + rh
2r(
1 + h2)3/2
2
=h2
4r2(
1 + h2) +
hh
2r(
1 + h2)2 +
h2
4(
1 + h2)3 .
Recall that g is given explicitly by
g = r(
1 + h2)1/2
.
For the last part, use Lemma 5.4.
27
Protein-Mediated Instability of Curved Membranes
Lemma 5.9. The expressions DtgGi + gDtGi are given by
i = 1 : v⊥
(1
2rΓ1,0
32
(1 +
1
2Γ2,0
0
)+
1
2Γ0,1
32
(1− 3Γ2,0
1
)− 5r
4Γ1,2
72
)+ vr
(−1
4r2Γ2,0
12
+1
4rΓ3,1
32
+1
2Γ0,2
52
(2 + Γ2,0
0
)+ rΓ1,3
72
)+ v⊥
(1
2Γ1,0
32
+r
2Γ0,1
52
);
i = 2 : v⊥Γ0,132
(1− 3Γ2,0
1
)+ v⊥ Γ1,0
32
+ vr Γ2,152
;
i = 3 : v⊥r Γ1,012
+ vr
(Γ0,0
− 12
+ r Γ1,112
).
Proof See appendix A.1.
Lemma 5.10. The expression gDtφ−φDtgg is given explicitly by
gDtφ− φDtg
g= ψ − ∂r (vrφ)− φ
(v⊥ Γ1,0
1 + vr
(1
r+ Γ1,1
1
)).
Proof By using Lemma’s 5.3 and 5.4 we have
gDtφ− φDtg
g= Dtφ− φ
Dtg
g
= ψ − ∂r (vrφ)− φ(v⊥ Γ1,0
1 + vr
(1
r+ Γ1,1
1
)).
We now have all the necessary ingredients for the explicit form of F ′[h, φ] · (v, ψ). Recallthe right hand side of equation (13), restated here for convenience in slightly altered form
DtFi =
∞∫0
[f ′i
gDtφ− φDtg
gGi + fi (DtgGi + gDtGi)
]dr.
Define Ai and Bi by
Ai :=
∞∫0
f ′igDtφ− φDtg
gGi dr, Bi :=
∞∫0
fi (DtgGi + gDtGi) dr,
from which it follows thatDtFi = Ai +Bi.
Theorem 5.11. The first variation F ′[h, φ] of F is given by
F ′[h, φ] · (~v, ψ) =
∞∫0
[f ′i
gDtφ− φDtg
gGi + fi (DtgGi + gDtGi)
]dr.
Proof The correctness of the formula is clear. For an explicit expression, see Definition5.2 together with Lemma’s 5.9, 5.10 and equation (14).
28
Protein-Mediated Instability of Curved Membranes
With this we are done; we have found the dynamic equations arising from the gradientflow using F . The above results are summarized in the following theorem.
Theorem 5.12. (Dynamic equations arising from gradient flow) The dynamicequations for P arising from the gradient flow of the energy functional F , its structurebeing, and F being given by, and the inner product (·, ·) on TpP given by
DtF [h, φ] = −‖(~v, ψ)‖,
F [h, φ] =
∫M
2κ(φ)H2(h) + κ(φ)K(h) + η(φ) dA,
((~v1, ψ1), (~v2, ψ2)) =
∫M
c~v1 · ~v2 + c′φ∇p1∇p2 dA,
where the ψi and pi are related through ψi = − div (φ∇pi), are given in weak formulationby
In this section we will turn our attention to determining the stability of a state (h0, φ0) ∈P. We are only interested in the flat membrane state, where h0 is constant, and φ0
describes a concentration which is constant on the membrane surface, and so also on thebase set of the Monge patch. Because of our radial symmetry assumption, this meansthat φ0 = c0r for some constant c0 > 0. Furthermore, we will let η be an arbitrary energycontribution, not necessarily the entropy η(φ) = φ log φ chosen for the derivation of thedynamic equations.
Lemma 6.1. A state (h0, φ0) is stable when F ′′[v, v] > 0 for all v ∈ TpP. Similarly, itis unstable when F ′′[v, v] < 0 for some v ∈ TpP.
Proof By theorem 7.14, this amounts to determining whether all of the eigenvaluesλi of F ′′ are positive. Note that F ′′ is symmetric due to the commutativity of partialderivatives. Because of this, λi > 0 for all i is equivalent to F ′′[v, v] > 0 for all v ∈ TpP,because for the smallest eigenvalue λ1 we have the Rayleigh quotient
λ1 = inf
F ′′[v, v]
‖v‖2: v ∈ TpP
.
It is clear that we will need to determine F ′′ explicitly. For this we will further linearizethe expression for F ′ given in theorem 5.11. Note that most of the work in this sectionhas already been done during the computations for the dynamic equations. Indeed, thefirst variation of F with respect to an additive perturbation is found by setting vr = 0 inthe result of theorem 5.11. In that case, v⊥ and ψ form a pair of additive perturbationsto (h, φ). Let us instead of (v⊥, ψ) use the notation v = (vh, vφ).
Theorem 6.2. The second variation F ′′[v, w] of F at (h0, φ0) for tangent vectors v =(vh, vφ), w = (wh, wφ) explicitly reads
but a term Γm,nk (h0) is equal to zero unless both m and n are zero, in which case it isequal to unity. This means that Γm,nk (h) up to first order in ε becomes
Γm,nk (h) =
1 if m = n = 0,
εwh if m = 1, n = 0,
εwh if m = 0, n = 1,
0 else.
This will greatly reduce the amount of relevant Γm,nk terms, as there are quite a few havingtoo high m and n indices. In fact we can, as far as this calculation is concerned, replaceour earlier expressions for DtgGi + gDtGi by
i = 1 : vh
(1
2rΓ1,0
32
+1
2Γ0,1
32
)+ vh
(1
2Γ1,0
32
+r
2Γ0,1
52
)= ε
1
2(vh + rvh)
(1
rwh + wh
),
i = 2 : vhΓ0,132
+ vhΓ1,032
= ε (vhwh + vhwh) ,
i = 3 : vhrΓ1,012
= εvhrwh.
Note in particular that there are no longer any zeroth order terms in ε. Because of this,
30
Protein-Mediated Instability of Curved Membranes
the first order terms of the linearized Bi are equal to
B1 =
∞∫0
κ(c0)1
r(vh + rvh) (wh + rwh) dr,
B2 =
∞∫0
κ(c0) (vhwh + vhwh) dr,
B3 =
∞∫0
η(c0)rvhwh dr.
We can immediately note that B2 vanishes, because we have
∞∫0
(vhwh + whvh) dr =
∞∫0
d
dr(vhwh) dr = 0,
as by assumption the boundary terms vanish. To determine the linearized Ai, note firstlythat Gi(h0) and G′i(h0) are both zero for i = 1, 2. Because of this, the linearized A1 andA2 are also both zero. Continuing, note that we have
gDtφ− φDtg
g= vφ − εφ0vhwh +O(ε2).
Due to this, the first order term of the linearized A3 is given by
A3 =
∞∫0
(η′′(c0)vφwφ − η′(c0)c0rvhwh) dr.
With this we have found the second variation F ′′ of F at (h0, φ0), because we have
F ′′[v, w] = A3 +B1 +B3.
From this we easily get an explicit expression for F ′′[v, v].
Corollary 6.3. The diagonal part F ′′[v, v] for a tangent vector v = (vh, vφ) explicitlyreads
F ′′[v, v] =
∞∫0
κ(c0)
r(vh + rvh)
2+ η′′(c0)v2
φ + (η(c0)− c0η′(c0)) rv2h dr.
Proof Merely take w = v in theorem 6.2.
Remark. We see that there is no contribution from the Gaussian curvature term of Fto F ′′, and furthermore that only κ in the ground state is important. From this we canconclude that making κ and κ dependent on φ has no effect. In hindsight we could alsohave found out at an earlier stage that the Gaussian term is not important. Namely, asthe ground state has principal curvatures c1,0 = c2,0 = 0, perturbing these by εc1,p andεc2,p respectively gives together with κ = κ(c0) + εκ′(c0)
vφr that∫
M
κ(φ)K dA = ε2
∫M
κ(c0)c1,pc2,p dA+O(ε3).
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Protein-Mediated Instability of Curved Membranes
However, the right hand side is constant up to second order by Gauss-Bonnet, Theorem2.3, and can be disregarded.
We can now state and prove our main result regarding instability of the ground state(h0, φ0).
Theorem 6.4. (Criteria for instability) The ground state (h0, φ0) is unstable whenat least one of the following conditions is satisfied:
• η′′(c0) < 0;
• η(c0)− c0η′(c0) < 0.
Furthermore, when η′′(c0) > 0 and η(c0) − c0η′(c0) > 0, the ground state (h0, φ0) is
stable.
Proof To show (h0, φ0) to be unstable, by Lemma 6.1 we must show that we haveF ′′[v, v] < 0 for some v ∈ TpP. Note that vh and vφ are completely independent.
Case η′′(c0) < 0: We would be done if we could find a v = (0, vφ) with F ′′[v, v] < 0. Notethat we have
F ′′[(0, vφ), (0, vφ)] = η′′(c0)
∞∫0
v2φ dr.
Because η′′(c0) < 0, it is clear that there are such v.
Case η(c0)− c0η′(c0) < 0: We can restrict ourselves to v = (vh, 0). Define w := rvh anddefine the constant α by
α :=η(c0)− c0η′(c0)
κ(c0).
Because ∂rw = vh + rvh, we then have
F ′′[(vh, 0), (vh, 0)] = κ(c0)
∞∫0
1
r
((∂rw)
2+ αw2
)dr.
Now define wλ(r) := w(λr) and the nonnegative constants µ and ν by
µ :=
∞∫0
1
r(∂rw)
2(r) dr and ν :=
∞∫0
1
rw2(r) dr.
We can then compute with some abuse of notation
F ′′[wλ, wλ] = κ(c0)
∞∫0
1
r
((∂rwλ)
2(r) + αw2
λ(r))
dr
= κ(c0)
∞∫0
1
λr
(λ2 (∂rwλ)
2(λr) + αw2
λ(λr))
dλr
= κ(c0)
∞∫0
1
r′
(λ2 (∂rw)
2(r′) + αw2(r′)
)dr′,
from which we conclude that
F ′′[wλ, wλ] = κ(c0)(λ2µ+ αν
).
32
Protein-Mediated Instability of Curved Membranes
From this we see that indeed there are v such that F ′′[v, v] < 0 in this case. Namely, anynonzero perturbation (vh, 0) works, if rescaled by an amount λ such that
λ2µ+ αν < 0.
Because α < 0, we can always find a small enough λ such that this is true.
The fact that (h0, φ0) is stable when η′′(c0) > 0 and η(c0)− c0η′(c0) > 0 is clear, becauseevery integrand in the expression of F ′′ is quadratic and thus nonnegative.
When the energy contribution η is chosen to be the entropy η = φ log φ, we can nowconclude the following.
Corollary 6.5. Using the entropy η = φ log φ, the ground state (h0, φ0) is unstable.
Proof In the specific case where η(φ) = φ log φ, we have
By theorem 6.4, the ground state (h0, φ0) is unstable.
Remark. For the proof of theorem 6.4 to work in the second case, we need to havethat our Monge patch is infinite, so that we can always rescale perturbations. This is anassumption we made earlier to simplify notation, but now we see it conceals an importantobservation: namely that the conditions under which the ground state is unstable dependon the dimensions of the Monge patch. To analyze this further, we will need the latter ofthe following two inequalities.
Lemma 6.6. (Poincare-Wirtinger inequality) There exists a D > 0 such that forall functions φ ∈ C1 ([0, a]) with φ(0) = φ(a) = 0, where a is some positive real number,we have
a∫0
φ2(x) dx ≤ Da2
a∫0
φ′2(x) dx.
In particular, the smallest such D is equal to π−2.
Proof Omitted. A standard proof uses Fourier theory and Plancherel’s theorem.
Lemma 6.7. There exists a C > 0 such that for all functions φ ∈ C1 ([0, a]) withφ(0) = φ(a) = 0, where a is some positive real number, we have
a∫0
1
xφ2(x) dx ≤ Ca2
a∫0
1
xφ′2(x) dx.
In particular, the smallest such C is equal to j2(1,1), where j(1,1) is the first zero of the first
Bessel function of the first kind.
Proof See Appendix A.2. Numerically, C is approximately equal to 14.6819.
We will now assume the Monge patch has some finite radius R > 0. Any integral from0 to ∞ is then replaced by one from 0 to R, and similarly the boundary conditions atinfinity are replaced by those at R. Define the nonnegative constants µR and νR by
µR :=
R∫0
1
r(∂rw)
2(r) dr and νR :=
R∫0
1
rw2(r) dr.
33
Protein-Mediated Instability of Curved Membranes
Theorem 6.8. (Criteria for instability, finite case) Assuming a Monge patch offinite radius R, the ground state (h0, φ0) is unstable when at least one of the followingconditions is satisfied:
• η′′(c0) < 0;
• η(c0)− c0η′(c0) < κ(c0)j2(1,1)
1R2 .
Furthermore, when η′′(c0) > 0 and η(c0) − c0η′(c0) > 0, the ground state (h0, φ0) is
stable.
Proof The only thing that needs to be proven is the case where η(c0) − c0η′(c0) <κ
j(1,1)1R2 , the rest follows directly from theorem 6.4.
In the restricted case v = (vh, 0), using the same notation as in the proof of theorem 6.4we have
F ′′[v, v] = κ(c0) (µR + ανR) .
The condition F ′′[v, v] < 0 becomes µR + ανR < 0, while using Lemma 6.7 we see that
νR ≤ CR2µR,
and equality is attained. This means that the limiting case where F ′′[v, v] < 0 is no longerpossible is when νR = CR2µR, or
1 + αCR2 = 0.
Unravelling the notation we see that the condition for instability becomes
η(c0)− c0η′(c0) <κ(c0)
j2(1,1)
1
R2.
Remark. Note that Theorem 6.8 reduces to Theorem 6.4 when R→∞.
When the energy contribution η is chosen to be the entropy η = φ log φ, we can nowconclude the following.
Corollary 6.9. Assuming a Monge patch of finite radius R and using the entropy
η = φ log φ, the ground state (h0, φ0) is unstable when c0 >κ(c0)j2(1,1)
1R2 .
Proof In the specific case where η(φ) = φ log φ, we have
By theorem 6.8, the ground state (h0, φ0) is unstable whenever
−c0 = η(c0)− c0η′(c0) <κ(c0)
j2(1,1)
1
R2,
which is the same as
c0 >κ(c0)
j2(1,1)
1
R2.
34
Protein-Mediated Instability of Curved Membranes
Remark. We see from Theorems 6.4 and 6.8 that the condition for instability boils downto the concavity or convexity of the protein energy contribution η. Clearly the conditionη′′(c0) < 0 implies that η is concave, whereas we can write η(x)− xη′(x) as
η(x)− xη′(x) = −x∫
0
sη′′(s) ds.
When η′′(s) > 0 on this interval [0, x], so when η is convex there, we immediately haveη(x) − xη′(x) < 0 as required. This second condition clearly is weaker than convexity,and instead says that η must be on average convex. The expression η(x) − xη′(x) < 0
rearranged to η(x)x − η
′(x) shows that this is the difference between chord from the originto the point (x, η(x)) and the slope η′(x) of the tangent at that point.
Lastly, our results will undoubtedly have been affected by the assumption that our Mongepatch is radially symmetric. Nevertheless, Theorems 6.4 and 6.8 still give sufficient con-ditions for instability, even in the case of a general Monge patch.
35
Protein-Mediated Instability of Curved Membranes
7 Background mathematics
In this section we develop some background mathematics on the subjects of variationalcalculus and linear stability analysis. Results in this section are sometimes called uponfrom the main text. We will be brief regarding proofs and motivation, leaving furtherelaboration to the references provided.
7.1 Variational calculus
In traditional calculus, problems in which one wants to maximize or minimize a certainfunction are considered. One wants to find, for a certain function f : X → R say, a pointxmax ∈ X such that f(xmax) > f(x) for every other x ∈ X. Variational calculus, however,deals with finding a function which maximizes or minimizes a given integral with it beingpart of the integrand. Variational problems arise naturally in physics.
Definition 7.1. A function is said to be of class Ck if it is k-times continuously differ-entiable.
For example, the class of C0 functions consists of continuous functions, whereas the classC∞ consists of infinitely differentiable or smooth functions. Usually the domain andcodomain are also given, with Ck(M,N) being the set of k-times continuously differen-tiable functions from M to N . If M = N one simply writes Ck(M).
Definition 7.2. A functional is a function from a function space F to R. The space offunctionals F ′ associated with F , also called the dual of F , is given by the set
F ′ := F : F → R .
We can write for some F ∈ F ′ that F = F [f ] with f ∈ F . Analogously to traditionalcalculus, one now wants to find a function fmax ∈ F such that F [fmax] > F [f ] for everyother f ∈ F . Similarly to traditional calculus, issues of existence and uniqueness arise.The field of variational calculus has a rich history; for this and a more extensive treatmentwe refer the reader to either [3] or [4].
The fundamental lemma of variational calculus is a useful tool to convert problems fromtheir weak formulation to the appropriate strong formulation.
Lemma 7.3. (Fundamental lemma of variational calculus) Let f ∈ Ck ([a, b]) begiven and assume that
b∫a
f(x)h(x) dx = 0
for every h ∈ Ck ([a, b]) with h(a) = h(b) = 0. Then f = 0 identically on [a, b].
Proof Define on [a, b] the function r by r(x) := −(x−a)(x− b). Note that r is smoothand r ≥ 0 with r(a) = r(b) = 0. Then define h(x) := r(x)f(x). It is clear that h is Ck
with h(a) = h(b) = 0. It follows by hypothesis that
0 =
b∫a
f(x)h(x) dx =
b∫a
r(x)f2(x) dx.
The right integral vanishes and the integrand is nonnegative, so the integrand must in factvanish. Because r is positive on (a, b), f must vanish on (a, b). By continuity f vanisheson [a, b].
36
Protein-Mediated Instability of Curved Membranes
The weak formulation of the strong f(x) = 0 is exactly the hypothesis of Lemma 7.3. Nowthat we have proven this fundamental lemma, we can state and prove a theorem on theEuler-Lagrange equation. This is a differential equation whose solutions are stationaryfunctions of a certain functional.
Theorem 7.4. (Euler-Lagrange equation) Given a functional F defined by
F [f ] :=
b∫a
L(x, f(x), f ′(x)) dx,
where L : [a, b] × C0[a, b] × C1[a, b] → R and it is assumed that the functions f satisfyf(a) = xa and f(b) = xb for some specified xa, xb ∈ R, the functions which extremize Fsatisfy the differential equation given by
∂L
∂f− d
dx
∂L
∂f ′= 0.
This differential equation is called the Euler-Lagrange equation corresponding to F .
Proof If a function f is to extremize F , any slight perturbation by εφ of f will eitherincrease or decrease the value of F . Define gε(x) := f(x)+εφ(x) to be such a perturbation.Here, φ is any continuously differentiable function on [a, b] satisfying φ(a) = φ(b) = 0,whereas ε > 0 is some small quantity. Define the auxiliary function Jε by
Jε(ε) :=
b∫a
L(gε(x), g′ε(x), x) dx.
We now calculate the total derivative of Jε with respect to ε. We have
dJεdε
=d
dε
b∫a
L(gε(x), g′ε(x), x) dx =
b∫a
d
dεL(gε(x), g′ε(x), x) dx.
The derivative can be taken inside the the integral as we have uniform continuity of theintegrand on every interval [a+ δ, b− δ] for δ > 0. Note that
dL
dε=
dgεdε
∂L
∂gε+
dg′εdε
∂L
∂g′ε+
dx
dε
∂L
∂x,
which turns intodL
dε= φ(x)
∂L
∂gε+ φ′(x)
∂L
∂g′ε.
This implies that
dJεdε
=
b∫a
[φ(x)
∂L
∂f+ φ′(x)
∂L
∂f ′
]dx.
Note that when ε = 0 we have gε = f . Furthermore, since f is an extreme value of F itfollows that dJε
dε (0) = 0. Using integration by parts we get
0 =dJεdε
(0) =
b∫a
[∂L
∂f− d
dx
∂L
∂f ′
]φ(x) dx+
[φ(x)
∂L
∂f ′
]ba
.
Using the boundary conditions on φ, the latter term vanishes. We can now apply Lemma7.3 to arrive at the Euler-Lagrange equation
0 =∂L
∂f− d
dx
∂L
∂f ′.
37
Protein-Mediated Instability of Curved Membranes
Note that the theorem does not state that all solutions to the Euler-Lagrange equationsnecessarily extremize the functional F . Next, consider the case when the Lagrangian Ldoes not explicitly depend on x, i.e. ∂L
∂x = 0. A result that is often useful in this case isthe Beltrami identity, stating that the Euler-Lagrange equation can be simplified.
Corollary 7.5. (Beltrami identity) Given a functional F defined by
F [f ] :=
b∫a
L(f(x), f ′(x)) dx,
where L : C1[a, b]×C0[a, b]→ R and it is assumed that the functions f satisfy f(a) = xaand f(b) = xb for some specified xa, xb ∈ R, the functions which extremize F satisfy thedifferential equation given by
L− f ′ ∂L∂f ′
= C,
where C is some constant.
Proof This can be seen to hold by first performing a Legendre transformation on Lwith respect to f and then using ∂L
∂x = 0. Intuitively, the proof amounts to integrating
with respect to f after writing ddx = d
dfdfdx .
The Euler-Lagrange equation can be expanded to allow for more complicated functionals,namely those also using higher derivatives of the argument function. We then have thefollowing generalization of theorem 7.4, which is proven analogously through induction.
Theorem 7.6. (Euler-Lagrange equation with higher derivatives) Given somenatural n and a functional F defined by
F [f ] :=
b∫a
L(x, f(x), f ′(x), . . . , f (n)x) dx,
where L : [a, b]×C0[a, b]× . . .×Cn[a, b]→ R and it is assumed that the functions f satisfyf(a) = xa and f(b) = xb for some specified xa, xb ∈ R, the functions which extremize Fsatisfy the Euler-Lagrange differential equation given by
n∑i=0
(−1)idi
dxi
(∂iL
∂f (i)
)= 0.
Note that we do not yet know which of the critical points of the functional are actualminimizers. Relevant concepts to this discussion are convexity, lower semi-continuity andcoerciveness of the functional. We will not discuss these topics here, and instead refer thereader to for example [3].
7.2 Linear stability analysis
Linear stability analysis is a method used to determine the behavior of a system aroundits stationary solutions. Within our scope, a system will be a system of partial differentialequations describing the time evolution of certain quantities.
Definition 7.7. A system is a function x : Rn → Rn for some natural n, together witha functional F ∈ Ck(Rn)′.
We will restrict ourselves to the case n = 1. Here k is generally large, or at least largeenough. The time evolution of the system is given by the gradient flow
x = −F ′[x], (15)
38
Protein-Mediated Instability of Curved Membranes
where F ′ is the Frechet derivative of F with respect to the standard Euclidean innerproduct, see section 3. Such a system may or may not have a stationary solution x0
satisfying x0 = 0. Note furthermore that there may be multiple solutions to any onesystem.
Lemma 7.8. Given a system (x, F ) with stationary solution x0, a perturbation εφsatisfies the differential equation
φ = −F ′′[x0]φ.
Proof More precisely, let φ be a smooth enough function and let ε > 0 be small. Weconsider the perturbation x0 + εφ. According to equation (15) we have
d
dt(x0 + εφ) = −F ′[x0 + εφ].
Expanding F ′ in a Taylor series around x0, up to first order in ε, this becomes
x0 + εφ = −(F ′[x0] + F ′′[x0]εφ+O(ε2)
).
By assumption x0 = F ′[x0] = 0. Using this and dividing by ε, ignoring higher order termsin ε, this simplifies to
φ = −F ′′[x0]φ.
We have now found an expression for the time evolution of a small perturbation arounda stationary solution. This result can be used to deduce the stability of that stationarysolution. That is to say, it is now possible to determine whether or not perturbations willgrow larger in time. To see this, we look at the norm ‖φ‖ of the induced perturbation.
Lemma 7.9. Given a system (x, F ) with stationary solution x0, the norm ‖φ‖ of aperturbation εφ satisfies the differential equation
d
dt‖φ‖2 = −F ′′[x0] ‖φ‖2.
Proof Using Lemma 7.8, we have
d
dt‖φ‖2 =
(φ, φ
)= (−F ′′[x0]φ, φ) = −F ′′[x0] ‖φ‖2.
We see that the sign of F ′′[x0] determines the stability of the stationary solution x0.
Theorem 7.10. Given a system (x, F ) with stationary solution x0, when F ′′[x0] > 0,x0 is a stable stationary solution. When F ′′[x0] < 0, x0 is unstable. Special care must begiven to the case where F ′′[x0] = 0.
Proof This is clear from Lemma 7.9, as when F ′′[x0] is positive or negative respectively,the norm of the perturbation will decrease or increase in time.
We can in fact give an explicit expression for the time evolution of the norm of theperturbation.
Corollary 7.11. Given a system (x, F ) with stationary solution x0, the norm of aperturbation εφ is given by
‖φ‖2 = ‖φ0‖2e−F′′[x0]t,
where φ0 = φ(0).
39
Protein-Mediated Instability of Curved Membranes
Proof Let g(x) := ‖x‖2. We see from Lemma 7.9 after dividing both sides by ‖φ‖2that
g
g= −F ′′[x0] giving ln g = −F ′′[x0]t+ C,
where C is some constant. Exponentiating and using the initial condition φ(0) = φ0, weconclude that
‖φ‖2 = ‖φ0‖2e−F′′[x0]t.
From this one can also see that it is the sign of F ′′[x0] which determines stability of astationary solution x0.
7.2.1 Coupled systems
We now consider a coupled system of partial differential equations describing the changesof two or more quantities. We will restrict ourselves to case of two such quantities, but itis clear that the results below can easily be generalized.
Definition 7.12. A coupled system is given by two functions x and y, and two func-tionals F and G such that
x = −F ′[x, y],
y = −G′[x, y].(16)
To perform linear stability analysis on such a system, we again will perturb around sta-tionary solutions. Therefore, assume there are x0, y0 such that F [x0, y0] = G[x0, y0] = 0.Define the matrix M by
M :=
(Fx FyGx Gy
).
Lemma 7.13. Given a coupled system (x, y, F,G) with stationary solution (x0, y0), aperturbation ε (φ, ψ) satisfies the differential equation(
φ
ψ
)= −M
(φψ
).
Proof The proof of this lemma is very similar to that of Lemma 7.8. Again, let (φ, ψ) besmooth enough and let ε > 0 be small. We consider the perturbation (x0 + εφ, y0 + εψ).Expanding F as a Taylor series up to first order in ε we have
F [x, y] = F [x0, y0] + ε (Fx[x0, y0]φ+ Fy[x0, y0]ψ) .
Using this together with the assumptions that 0 = x0 = F ′[x0, y0] = G′[x0, y0] = y0 gives(φ
ψ
)= −
(Fx FyGx Gy
)(φψ
).
40
Protein-Mediated Instability of Curved Membranes
It is important to note that this is a linear equation. We know that in one dimension, givensome scalar M , the equation φ = Mφ has solution φ = φ0e
−Mt. In the two-dimensionalcase where M is a matrix, the solution is similar. An equation of the form ∂t ~v = M ~vcan be approached as follows. Let ~wi be a basis consisting of eigenvectors of M , withcorresponding eigenvalues λi. Now write ~v as ~v =
∑i (~v, ~wi) ~wi.
Theorem 7.14. Given a system (x, y, F,G) with stationary solution (x0, y0), whenλi > 0 for all i, (x0, y0) is a stable stationary solution. When λi < 0 for some i, (x0, y0)is unstable.
Proof Let ~v = (φ, ψ) and use the decomposition into eigenvectors ~v =∑i (~v, ~wi) ~wi.
Using Lemma 7.13 and linearity we get∑i
(~v, ~wi) ∂t ~wi = −∑i
(~v, ~wi)M ~wi.
Because M ~wi = λi ~wi, this implies that for each i we must have ∂t ~wi = −λi ~wi. This inturn gives ~wi = ~w0
i e−λit, where ~w0
i = ~wi(0). The condition for linear stability then isthat all eigenvalues be positive, i.e. λi > 0 for all i. Conversely, if we can show even oneeigenvalue to be negative, the stationary solution (x0, y0) is unstable.
41
Protein-Mediated Instability of Curved Membranes
References
[1] P.B. Canham. The Minimum Energy of Bending as a Possible Explanation of theBiconcave Shape of the Human Red Blood Cell. J. Theoret. Biol. 26, 61-81, 1970.
[2] J.B. Conway. A course in functional analysis. Springer, 1990.
[3] B. Dacorogna. Introduction to the Calculus of Variations. Imperial College Press,2004.
[4] L.C. Evans. Partial Differential Equations: Second Edition. Graduate Studies inMathematics, American Mathematical Society, 1998.
[5] T. Frankel. The Geometry of Physics: An Introduction. Cambridge University Press,2007.
[6] W. Helfrich. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments.Z. Naturforsch. 28c, 693-703, 1973.
[7] T. Idema. Structure, shape and dynamics of biological membranes. PhD thesis,Eindhoven University of Technology, 2009.
[8] F. Otto. The Geometry of Dissipative Evolution Equations: the Porous MediumEquation. Commun. in Partial Differential Equations, 26 (1&2), 101-174, 2001.
[9] B.I. Halperin P.C. Hohenberg. Theory of dynamic critical phenomena. Reviews ofModern Physics, Vol. 49, No. 3, 1977.
[10] M.A. Peletier. Variational Modelling. Lecture notes, 2010.
[11] C. Villani. Topics in optimal transportation. American Mathematical Society, 2003.
42
Protein-Mediated Instability of Curved Membranes
A Appendix
In this appendix one can find proofs of three lemma’s which would otherwise interruptthe flow of the main text, as well as a popular description of the subject of this thesis.
A.1 Proofs of expressions for DtΓm,nk and DtgGi + gDtGi
In this section we prove Lemma’s 5.6 and 5.9 used in the main text, which give explicitexpressions for DtΓ
m,nk and DtgGi + gDtGi.
Lemma 1.1. The derivative DtΓm,nk is given by
DtΓm,nk = mv⊥ Γm−1,n
k + nv⊥ Γm,n−1k − 2kv⊥ Γm+1,n
k+1 .
Proof To determine the derivative DtΓm,nk , we look at how Γm,nk changes when h is
perturbed by ∆t v⊥. Recall that
(a+ εb)m
= am + εmam−1b+O(ε2),
(a+ εb) (c+ εb) = ac+ ε (bc+ ad) +O(ε2).
This means that we have(h+ ∆t v⊥
)m= hm + ∆tmhm−1v⊥ +O(∆t2),(
h+ ∆t v⊥
)n= hn + ∆t nhn−1v⊥ +O(∆t2).
Combining these two results we get(h+ ∆t v⊥
)m (h+ ∆t v⊥
)n= hmhn + ∆t
(mhm−1hnv⊥ + nhmhn−1v⊥
)+O(∆t2),
1(1 +
(h+ ∆t v⊥
)2)k =
1(1 + h2
)k(
1−∆t2kv⊥h
1 + h2
)+O(∆t2).
We can now combine these results to arrive at(h+ ∆t v⊥
)m (h+ ∆t v⊥
)n(
1 +(h+ ∆t v⊥
)2)k =
hmhn(1 + h2
)k + ∆t
mhm−1hnv⊥ + nhmhn−1v⊥(1 + h2
)k − 2kv⊥hm+1hn(
1 + h2)k+1
+O(∆t2).
From this we see that DtΓm,nk is given by
DtΓm,nk =
hmhn(1 + h2
)k(mv⊥
h+nv⊥
h− 2kv⊥h
1 + h2
),
which we can rewrite to
DtΓm,nk = mv⊥ Γm−1,n
k + nv⊥ Γm,n−1k − 2kv⊥ Γm+1,n
k+1 .
43
Protein-Mediated Instability of Curved Membranes
Lemma 1.2. The expressions DtgGi + gDtGi are given by
i = 1 : v⊥
(1
2rΓ1,0
32
(1 +
1
2Γ2,0
0
)+
1
2Γ0,1
32
(1− 3Γ2,0
1
)− 5r
4Γ1,2
72
)+ vr
(−1
4r2Γ2,0
12
+1
4rΓ3,1
32
+1
2Γ0,2
52
(2 + Γ2,0
0
)+ rΓ1,3
72
)+ v⊥
(1
2Γ1,0
32
+r
2Γ0,1
52
);
i = 2 : v⊥Γ0,132
(1− 3Γ2,0
1
)+ v⊥ Γ1,0
32
+ vr Γ2,152
;
i = 3 : v⊥r Γ1,012
+ vr
(Γ0,0
− 12
+ r Γ1,112
).
Proof To determine the DtGi, we can write using Lemma 5.8, noting that ∂tr = vr,
DtG1 =1
4
(1
r2DtΓ
2,01 − 2vr
r3Γ2,0
1 +2
rDtΓ
1,12 − 2vr
r2Γ1,1
2 +DtΓ0,23
),
DtG2 =1
rDtΓ
1,12 − vr
r2Γ1,1
2 ,
DtG3 = 0.
Using Lemma 5.6, we see that for specific choices (m,n, k) we get
(2, 0, 1) : DtΓ2,01 = 2v⊥ Γ1,0
1 − 2v⊥ Γ3,02 = 2v⊥Γ1,0
1
(1− Γ2,0
1
)= 2v⊥ Γ1,0
2 ,
(1, 1, 2) : DtΓ1,12 = v⊥ Γ0,1
2 + v⊥ Γ1,02 − 4v⊥ Γ2,1
3 = v⊥
(Γ0,1
2 − 4Γ2,13
)+ v⊥ Γ1,0
2 ,
(0, 2, 3) : DtΓ0,23 = 2v⊥ Γ0,1
3 − 6v⊥ Γ1,24 .
This now implies that the DtGi become
DtG1 =1
4
(1
r22v⊥ Γ1,0
2 − 2vrr3
Γ2,01 +
2
r
(v⊥
(Γ0,1
2 − 4Γ2,13
)+ v⊥ Γ1,0
2
)− 2vr
r2Γ1,1
2 +(
2v⊥ Γ0,13 − 6v⊥ Γ1,2
4
))= v⊥
(1
2r2Γ1,0
2 +1
2r
(Γ0,1
2 − 4Γ2,13
)− 3
2Γ1,2
4
)+ v⊥
(1
2rΓ1,0
2 +1
2Γ0,1
3
)+ vr
(− 1
2r3Γ2,0
1 − 1
2r2Γ1,1
2
)DtG2 = v⊥
1
r
(Γ0,1
2 − 4Γ2,13
)+ v⊥
1
rΓ1,0
2 + vr−1
r2Γ1,1
2 ,
DtG3 = 0.
44
Protein-Mediated Instability of Curved Membranes
Again using Lemma 5.8, the expressions DtgGi + gDtGi are then given by
i = 1 :[v⊥r Γ1,0
12
+ vr
(Γ0,0
− 12
+ r Γ1,112
)] [1
4
(1
r2Γ2,0
1 +2
rΓ1,1
2 + Γ0,23
)]+[r Γ0,0
− 12
] [v⊥
(1
2r2Γ1,0
2 +1
2r
(Γ0,1
2 − 4Γ2,13
)− 3
2Γ1,2
4
)+ v⊥
(1
2rΓ1,0
2 +1
2Γ0,1
3
)+ vr
(− 1
2r3Γ2,0
1 − 1
2r2Γ1,1
2
)]
= v⊥
(1
2rΓ1,0
32
(1 +
1
2Γ2,0
0
)+
1
2Γ0,1
32
(1− 3Γ2,0
1
)− 5r
4Γ1,2
72
)+ vr
(−1
4r2Γ2,0
12
+1
4rΓ3,1
32
+1
2Γ0,2
52
(2 + Γ2,0
0
)+ rΓ1,3
72
)+ v⊥
(1
2Γ1,0
32
+r
2Γ0,1
52
);
i = 2 :[v⊥r Γ1,0
12
+ vr
(Γ0,0
− 12
+ r Γ1,112
)] [1
rΓ1,1
2
]+[r Γ0,0
− 12
] [v⊥
1
r
(Γ0,1
2 − 4Γ2,13
)+ v⊥
1
rΓ1,0
2 + vr−1
r2Γ1,1
2
]
= v⊥Γ0,132
(1− 3Γ2,0
1
)+ v⊥ Γ1,0
32
+ vr Γ2,152
;
i = 3 : v⊥r Γ1,012
+ vr
(Γ0,0
− 12
+ r Γ1,112
).
A.2 Proof of Poincare-like inequality
In this section we prove Lemma 6.7 used in the main text.
Lemma 1.3. There exists a C > 0 such that for all functions φ ∈ C1 ([0, a]) withφ(0) = φ(a) = 0, where a is some positive real number, we have
a∫0
1
xφ2(x) dx ≤ Ca2
a∫0
1
xφ′2(x) dx.
In particular, the smallest such C is equal to j2(1,1), where j(1,1) is the first zero of the first
Bessel function of the first kind.
Proof It is clear that by transforming x′ = ax we can restrict ourselves to the casea = 1. To find the optimal constant C, note that it can be written as
C = inf
(∫φ′2
1
xdx
)(∫φ2 1
xdx
)−1
: φ ∈ C1([0, 1]), φ(0) = φ(1) = 0
= inf
φCφ.
Setting the derivative of this quotient in the direction of some ψ equal to zero gives
0 = 2
∫φ2 1
xdx
∫φ′ψ′
1
xdx− 2
∫φ′2
1
xdx
∫φψ
1
xdx.
45
Protein-Mediated Instability of Curved Membranes
This reduces to ∫φ′ψ
1
xdx = Cφ
∫φψ
1
xdx,
which we rewrite using partial integration to
−∫ψ
(φ′
x
)′dx = Cφ
∫ψφ
xdx.
The above equation must hold for all ψ. Using Lemma 7.3 we get the differential equation(φ′
x
)′+ Cφ
φ
x= 0,
with boundary conditions φ(0) = φ(1) = 0. By multiplying by x we see that this isequivalent to
φ′′ − φ′
x+ Cφφ = 0.
This equation has as general solution
φ(x) = a1xJ1
(√Cφx
)+ a2xY1
(√Cφx
),
where J1 and Y1 are the Bessel functions of the first and second kind respectively, anda1, a2 are arbitrary constants. Due to the condition φ(0) = 0 we must have a2 = 0. Fordetermining Cφ, we can clearly choose a1 = 1, which leaves
φ(x) = xJ1
(√Cφx
).
The condition φ(1) = 0 then dictates that√Cφ is a zero of J1. In particular, it is the
smallest positive zero, also denoted by j(1,1), which numerically is about 3.8317. Withthis the optimal constant C is equal to
C = Cφ = j2(1,1) ≈ 14.6819.
This can also be verified by numerical integration of 1/xφ2 and 1/xφ′2, giving about0.0811 and 1.1098 respectively.
46
Protein-Mediated Instability of Curved Membranes
A.3 Popular description
This section contains a copy of the ‘A4 populair’ written in Dutch accompanying thisthesis, explaining in layman’s terms what this thesis is about. It is written for third tofourth year high school (VWO) students.
Endocytose: membraankromming door eiwitten
Ralph Klaasse, 2 augustus 2010
Technische Wiskunde en Technische Natuurkunde, TU/e
In dieren en mensen zitten een heleboel cellen.Een levende cel is in zekere zin een afgeslotenwereldje, ook al werken de cellen natuurlijksamen om jou en mij te vormen. De cellenzijn omringd door een membraan (omhulsel)van lipiden; dat zijn vetzuren die net als zeepeen kop en een staart hebben. Deze lipidenvormen in paren (met de staarten ofwel dehydrofobe delen naar elkaar) het membraan.Zie figuur 1 voor een schematische weergavevan de structuur van het membraan.
Hoewel de cel door dit membraan afgeslotenwordt van de buitenwereld, kan hij natuurlijkniet geheel op zichzelf overleven. Waar hetvia zogeheten diffussieprocessen mogelijk is omkleine stofjes door het membraan te bewegen,is dit niet mogelijk bij grotere moleculen. Omtoch deze grotere moleculen in en uit de celte kunnen transporteren heeft de natuur ietsanders bedacht: endocytose.
Endocytose (en zijn tegenhanger exocytose) ishet proces van het vormen van een instulping (ofuitstulping) in het membraan. Het membraantrekt steeds verder naar binnen toe totdat het dete transporteren deeltjes omvat. Dan sluit hetmembraan aan de bovenkant weer en zitten dedeeltjes gevangen. Dit alles duurt maar een paarminuten. Exocytose is precies dit proces, maardan andersom, ofwel transport naar buiten toe.Zie ook figuur 2 voor een schematische weergavevan het endocytoseproces.
Figuur 1: Schematische weergave membraan.
Figuur 2: Schematische weergave endocytose.
Om endocytose te laten plaatsvinden moethet celmembraan dus een aantal drastischevormveranderingen ondergaan. Om dezeveranderingen te beschrijven is de wiskundebij uitstek geschikt. Ik heb in mijn projecteen aantal nieuwe gereedschappen ontwikkeldom dit te doen. Ik heb een wiskundig modelgemaakt van de cel met daarin de voor ditproces relevante eigenschappen. Ik wil met ditmodel het proces uit de natuur nadoen. Vragendie ik wil beantwoorden zijn: hoe werkt hetprecies? Wat heb je er allemaal voor nodig?Kost het erg veel moeite (ofwel kost het energie)of gaat het vanzelf?
Het blijkt dat dit vervormen van het membraaninderdaad energie kost. Waar komt die energievandaan? De cel brengt eiwitten aan op hetmembraan die in zekere zin energiedragers zijn:ze zitten het liefst bij elkaar en doordat ze op hetmembraan naar elkaar toe bewegen wordt hetmembraan vervormd. Vervolgens kunnen weons afvragen hoeveel van deze eiwitten je nodighebt om het membraan voldoende te vervormen.
Het onderwerp van mijn project valt in de cate-gorie genaamd biofysica. Het is bij uitstek eenproject waar natuurkunde en wiskunde samenkomen en elkaar helpen bij het begrijpen van denatuur.
