Stability of discoidal high-density lipoprotein particles Mohsen Maleki and Eliot Fried * Motivated by experimental and numerical studies revealing that discoidal high-density lipoprotein (HDL) particles may adopt flat elliptical and nonplanar saddle-like configurations, it is hypothesized that these might represent stabilized configurations of initially unstable flat circular particles. A variational description is developed to explore the stability of a flat circular discoidal HDL particle. While the lipid bilayer is modeled as two-dimensional fluid film endowed with surface tension and bending elasticity, the apoA-I belt is modeled as one-dimensional inextensible twist-free chain endowed with bending elasticity. Stability is investigated using the second variation of the underlying energy functional. Various planar and nonplanar instability modes are predicted and corresponding nondimensional critical values of salient dimensionless parameters are obtained. The results predict that the first planar and nonplanar unstable modes occur due to in-plane elliptical and transverse saddle-like perturbations. Based on available data, detailed stability diagrams indicate the range of input parameters for which a flat circular discoidal HDL particle is linearly stable or unstable. 1. Introduction The packaging and transport of water-insoluble cholesterol in the bloodstream are mediated by lipoprotein particles. In “reverse cholesterol transport,” high-density lipoprotein (HDL) particles scavenge cholesterol from tissues and other types of lipoprotein particles and deliver it to the liver for excretion into bile or other use. A comprehensive understanding of the biophysical basis for the vasculoprotective functionalities of HDL particles is essential to developing effective strategies to prevent, diagnose, and treat atherosclerosis. However, as Vuorela et al. 1 observe: “The functionality of HDL has remained elusive, and even its structure is not well understood.” During reverse cholesterol transport, an HDL particle sustains shape transitions that are accompanied by changes in the conformation of its apolipoprotein building block apoA-I. Davidson & Silva 2 explain that the functionality of apoA-I is linked to its conformational variations and emphasize the need to understand the diverse range of conformations that it adopts in its lipid-free and lipid-bound forms. A discoidal HDL particle consists of a lipid bilayer bound by an apoA-I chain. Camont et al. 3 argue that the low lipid content and high surface uidity of discoidal HDL particles induces conformational changes of apoA-I that result in enhanced exposure to its aqueous surroundings and, thus, in an increased capacity to acquire blood lipids. Using all-atom molecular dynamics (MD) simula- tions, Catte et al. 4 predict that assembling a at circular HDL particle from a lipid-free apoA-I chain involves the formation of intermediate nonplanar, twisted, saddle-like particles. Coarse- grained molecular dynamics simulations of Shih et al. 5,6 and experiments of Silva et al., 7 Miyazaki et al., 8 and Huang et al. 9 conrm this prediction. In addition, experimental results of Skar-Gislinge et al. 10 reveal that HDL particles exhibit an intrinsic tendency to adopt planar, elliptical congurations. Simulations have provided valuable insight regarding the molecular interactions that govern the assembly and dynamics of discoidal HDL particles. However, the small time steps needed to correctly capture the highest frequency of molecular vibrations and preserve numerical accuracy make it difficult to access time scales long enough to determine equilibria or draw conclusions regarding stability. For these purposes, continuum models provide a valuable complement to simulations. In particular, continuum models have been used with remarkable success to determine equilibria and study stability in bio- membranes and biomolecules. Inspired by the aforementioned experiments and simula- tions, a continuum mechanical model for the equilibrium and stability of a at circular HDL particle is presented. Guided by prevalent continuum models of biomembranes and biomole- cules, the bilayer is treated as a two-dimensional uid lm endowed with surface tension and resistance to bending and the apoA-I chain as a one-dimensional inextensible, twist-free, elastic lament endowed with resistance to bending. The bilayer and apoA-I chain are required to be perfectly bonded, in which case the boundary of the uid lm and the elastic lament must have the same shape. A variational description of the equilib- rium of a discoidal HDL particle is provided. A at, circular shape is chosen as a reference conguration. To study the linear stability of the reference shape, innitesimal perturbations involving both planar and transverse components are consid- ered. Such perturbations can be caused by thermal uctuations Department of Mechanical Engineering, McGill University, Montr´ eal, QC, Canada H3A 0C3. E-mail: [email protected]Cite this: DOI: 10.1039/c3sm51955d Received 17th July 2013 Accepted 5th September 2013 DOI: 10.1039/c3sm51955d www.rsc.org/softmatter This journal is ª The Royal Society of Chemistry 2013 Soft Matter Soft Matter PAPER Published on 06 September 2013. Downloaded by University of Aberdeen on 24/09/2013 14:39:51. View Article Online View Journal
Received 17th July 2013Accepted 5th September 2013
DOI: 10.1039/c3sm51955d
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This journal is ª The Royal Society of
Stability of discoidal high-density lipoprotein particles
Mohsen Maleki and Eliot Fried*
Motivated by experimental and numerical studies revealing that discoidal high-density lipoprotein (HDL)
particles may adopt flat elliptical and nonplanar saddle-like configurations, it is hypothesized that these
might represent stabilized configurations of initially unstable flat circular particles. A variational
description is developed to explore the stability of a flat circular discoidal HDL particle. While the lipid
bilayer is modeled as two-dimensional fluid film endowed with surface tension and bending elasticity,
the apoA-I belt is modeled as one-dimensional inextensible twist-free chain endowed with bending
elasticity. Stability is investigated using the second variation of the underlying energy functional. Various
planar and nonplanar instability modes are predicted and corresponding nondimensional critical values
of salient dimensionless parameters are obtained. The results predict that the first planar and nonplanar
unstable modes occur due to in-plane elliptical and transverse saddle-like perturbations. Based on
available data, detailed stability diagrams indicate the range of input parameters for which a flat
circular discoidal HDL particle is linearly stable or unstable.
1. Introduction
The packaging and transport of water-insoluble cholesterol inthe bloodstream are mediated by lipoprotein particles. In“reverse cholesterol transport,” high-density lipoprotein (HDL)particles scavenge cholesterol from tissues and other types oflipoprotein particles and deliver it to the liver for excretion intobile or other use. A comprehensive understanding of thebiophysical basis for the vasculoprotective functionalities ofHDL particles is essential to developing effective strategies toprevent, diagnose, and treat atherosclerosis. However, asVuorela et al.1 observe: “The functionality of HDL has remainedelusive, and even its structure is not well understood.”
During reverse cholesterol transport, an HDL particlesustains shape transitions that are accompanied by changes inthe conformation of its apolipoprotein building block apoA-I.Davidson & Silva2 explain that the functionality of apoA-I islinked to its conformational variations and emphasize the needto understand the diverse range of conformations that it adoptsin its lipid-free and lipid-bound forms. A discoidal HDL particleconsists of a lipid bilayer bound by an apoA-I chain. Camontet al.3 argue that the low lipid content and high surface uidityof discoidal HDL particles induces conformational changes ofapoA-I that result in enhanced exposure to its aqueoussurroundings and, thus, in an increased capacity to acquireblood lipids. Using all-atom molecular dynamics (MD) simula-tions, Catte et al.4 predict that assembling a at circular HDLparticle from a lipid-free apoA-I chain involves the formation ofintermediate nonplanar, twisted, saddle-like particles. Coarse-
ill University, Montreal, QC, Canada H3A
Chemistry 2013
grained molecular dynamics simulations of Shih et al.5,6 andexperiments of Silva et al.,7 Miyazaki et al.,8 and Huang et al.9
conrm this prediction. In addition, experimental results ofSkar-Gislinge et al.10 reveal that HDL particles exhibit anintrinsic tendency to adopt planar, elliptical congurations.
Simulations have provided valuable insight regarding themolecular interactions that govern the assembly and dynamicsof discoidal HDL particles. However, the small time stepsneeded to correctly capture the highest frequency of molecularvibrations and preserve numerical accuracy make it difficult toaccess time scales long enough to determine equilibria or drawconclusions regarding stability. For these purposes, continuummodels provide a valuable complement to simulations. Inparticular, continuum models have been used with remarkablesuccess to determine equilibria and study stability in bio-membranes and biomolecules.
Inspired by the aforementioned experiments and simula-tions, a continuum mechanical model for the equilibrium andstability of a at circular HDL particle is presented. Guided byprevalent continuum models of biomembranes and biomole-cules, the bilayer is treated as a two-dimensional uid lmendowedwith surface tension and resistance to bending and theapoA-I chain as a one-dimensional inextensible, twist-free,elastic lament endowed with resistance to bending. The bilayerand apoA-I chain are required to be perfectly bonded, in whichcase the boundary of the uid lm and the elastic lament musthave the same shape. A variational description of the equilib-riumof adiscoidalHDLparticle is provided. Aat, circular shapeis chosen as a reference conguration. To study the linearstability of the reference shape, innitesimal perturbationsinvolving both planar and transverse components are consid-ered. Such perturbations can be caused by thermal uctuations
of the lipid bilayer or the apoA-I chain or by interactions betweenthe HDL particle and its environment. Closed-form analyticalsolutions for the linearized equilibrium conditions are obtainedand stability is explored via the second-variation condition. Inaddition, available values of the physical parameters that enterthemodel are used to determine the range of inputs underwhicha at, circular HDL particle is linearly stable or unstable. Lastly,connections between our result and previous experimentalmeasurements and numerical simulations are made.
2. Energetics of a discoidal HDL particle
Geometrically, a discoidal HDL particle is treated as a smooth,orientable surface S with boundary C ¼ vS . The interior andboundary of S correspond, respectively, to the bilayer and apoA-I components of the particle. Following convention, H and Kdenote the mean and Gaussian curvatures of S and k denotesthe curvature of C .
To capture the energetics of the bilayer, S is endowed with auniform surface tension s and an areal bending-energy density
j ¼ 1
2mH2 þ mK ; (1)
of the type put forth by Canham11 and Helfrich,12 where m >0 and �m are the splay and saddle-splay moduli. The relevance ofspontaneous curvature, which ordinarily appears in the Can-ham–Helfrich model, to discoidal HDL particles has yet to beinvestigated and, thus, is omitted from (1).
To capture the energetics of the apoA-I chain, C is endowedwith a lineal bending-energy density 4 depending on thecurvature k of C and its arc length derivative k0. The latterdependence is included to account for the energetic cost oflarge, localized curvature variations associated with kinks onthe apolipoprotein chain discussed by Brouillette et al.13 andKlon et al.14 For simplicity, it is assumed that
4 ¼ 1
2ak2 þ 1
2bðk0Þ2; (2)
where a > 0 is the constant exural rigidity of C and b $ 0 is ahigher-order generalization thereof. Since C is closed,including a quadratic coupling term proportional to 2kk0 ¼ (k2)0
in 4 would not alter the net potential energy and no generality islost by neglecting such a contribution. The particular choice (2)of 4 is a special case of a general expression for the lineal free-energy density of a polymer chain proposed by Zhang et al.,15
who allow for arbitrary dependence on k, k0, and the torsion s ofC . Granted the foregoing assumption and that external forcesassociated with gravity, van der Waals interactions, or ow-related forces are negligible, the net potential-energy of adiscoidal HDL particle is given by
E ¼ðSðsþ jÞ þ
ðC4: (3)
As a surface with boundary, S has Euler characteristic equalto unity. On using (1) and (2) in (3) and applying the Gauss–Bonnet theorem (taking into consideration that C is assumed tobe smooth), the net potential-energy E becomes
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E ¼ E a + E l + 2p�m, (4)
where
E a ¼ðS
�sþ 1
2mH2
�(5)
and
E l ¼ðC
�1
2ak2 þ 1
2bðk0Þ2 � mkg
�; (6)
denote the effective areal and lineal potential energies, with kg
being the geodesic curvature of C . Without loss of generality,the additive constant 2p�m in (4) is disregarded hereaer.
The assumed inextensibility of the apoA-I chain is imposedby working with the augmented net potential-energy
F ¼ E a þ E l þðCl; (7)
where l is an unknown Lagrange multiplier.
3. Parameterization andnondimensionalization
Let D ¼ {(r, q) ˛ R2: 0 # r # R, 0 # q # 2p} denote the disk of
radius R. The surface and boundary of a discoidal HDL particlecan then be described by a smooth function
x: D / R3. (8)
Due to the inextensibility of C , x must satisfy
|xq(R, q)| ¼ R, 0 # q # 2p. (9)
With this choice, the bilayer and apoA-I chain are repre-sented by
x(r, q), 0 # r < R, 0 # q # 2p, (10)
and
x(R, q), 0 # q # 2p. (11)
On determining expressions for the geometrical objects H, k,k0, and kg consistent with the parametrization (10) and (11), theaugmented net potential-energy F dened in (7) can beexpressed as a functional of x.
It is convenient to present results in dimensionless form viathe change of variables
x(r, q) ¼ Rx(r, q), r ¼ Rr, (12)
in which case the dimensionless reference domain is a disk ofradius unity denoted by R . In addition, it is convenient tointroduce the following group of dimensionless quantities
ðH ; h; h; n; ı; 3Þ :¼�FR
a;mR
a;mR
a;sR3
a;lR2
a;
b
aR2
�: (13)
In particular, the dimensionless counterpart H of theaugmented net potential energy dened (7) takes the form
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Fig. 1 Schematic of a slightly perturbed discoidal HDL particle and its flat circularreference configuration (in grays). The transverse deformation is exaggerated forillustrative purposes.
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H ¼ð2p0
ð10
�nþ 1
2hR2H2
�|xr � xq| drdq
þð2p0
�1
2R2k2 þ 1
23R4ðk0Þ2 � hRkg þ ı
�| xq|r¼1
dq: (14)
For brevity, the adjective ‘dimensionless’ is droppedhereaer.
To obtain linearized equilibrium equations and study thestability of a discoidal HDL particle, it suffices to use an inn-itesimal displacement approximation in which the position of ageneric point on S is given by (12), with
where o indicates the origin of the reference disk and u, v, and ware the components of x in er, eq, and ez directions, respectively(Fig. 1). The inextensibility condition (9) becomes
|xq(1, q)| ¼ 1, 0 # q # 2p. (16)
The expansion (15) can be used to express (14) component-wise. The linearized equilibrium conditions arise on expandingall terms in (14) up to the second order of u, v, w, including theirpartial derivatives. For brevity, the intermediate calculations aresuppressed.
4. Equilibrium conditions
At equilibrium, the rst variation _H of the functional H in (14)vanishes. Notice that a superposed dot indicates the rst vari-ation. Imposing the requirement _H ¼ 0 yields the partial-differential equation
D2w � z2Dw ¼ 0 on R , (17)
with D the Laplacian on R and z ¼ 2ffiffiffiffiffiffiffiffin=h
p, and boundary
conditions��ıþ n� 1
2
�þ nðuþ vqÞ þ 1
2ð2uþ 5uqq þ 2uqqqq � vqÞ
�3ðuqq þ 2uqqqq þ uqqqqqqÞ ��ıðuq � vÞ�
q
�r¼1
¼ 0;
(18)
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�ıq þ
�ıþ n� 1
2
�ðuq � vÞ
�r¼1
¼ 0; (19)
�� h
4ðDwÞr þ nwr þ 1
2ð3wqq þ 2wqqqqÞ
�hðwqq � wrqqÞ � ðıwqÞq�r¼1
¼ 0; (20)
hh4Dwþ hðwr þ wqqÞ
ir¼1
¼ 0: (21)
Moreover, the linearized version of the inextensibilitycondition (16) requires that
(u + vq)r¼1 ¼ 0. (22)
The equilibrium condition (17), which governs the localgeometry of the lipid bilayer, is the linearized version of theshape equation familiar from works on vesicles. The boundaryconditions (18)–(20) express force balance on C in the er, eq, andez directions, respectively. The remaining boundary condition(21) involves the slope of the edge in the er-direction and, thus,expresses moment balance on C .
Up to the order considered, the partial-differential equation(17) imposes no restrictions on the in-plane displacements uand v. Hence, w and the in-plane components u and v arecoupled only on the boundary of R . Also, u and v are absentfrom the boundary conditions (20) and (21). Thus, (17) and theassociated boundary conditions (20) and (21) may be used todetermine w, independently. Satisfaction of (17) and (18)–(21) atthe trivial solution (u ¼ v ¼ w ¼ 0) results in a relation,
ı ¼ 1
2� n; (23)
for the Lagrange multiplier l which is analogous to a resultobtained by Giomi & Mahadevan16 in their work on soap lmsbound by inextensible, elastic laments. Next, using (23) in (19)yields
ı ¼ constant. (24)
The eq-component of force balance on the boundary of adiscoidal HDL particle therefore requires that the Lagrangemultiplier ı be uniform.
5. Solving the system of equations
Along with conditions (23) and (24), the partial-differentialequation (17) and boundary conditions (20) and (21), suffice tocompletely determine the transverse displacement w. In addi-tion, using (23) in (18) yields
Eqn (25) is an ordinary-differential equation with constantcoefficients. In view of the periodicity of u (i.e., u(1, q) ¼ u(1, q +2pk), ck ˛ Z), (25) admits a representation of the form
u(1, q) ¼ Usin(mq) (m ˛ Z). (26)
Substitution of (26) in (25) yields a characteristic equation
(m2 � 1)[(m2 � 1) + m2(m2 � 1)3 � n] ¼ 0. (27)
One solution of (27) is m2 ¼ 1, which corresponds to theplanar rigid body translation and is of no physical interest.Otherwise, (27) yields a critical value,
nim ¼ m2 � 1 + m2(m2 � 1)3, (28)
of n for each planar mode m. Granted that 3 $ 0, the lowestcritical value of n corresponds to m ¼ 2 and is given by
nic ¼ ni2 ¼ 3 + 123. (29)
The value nic¼ 3 arising for 3¼ 0 is consistent with the resultsobtained by Chen & Fried17 for a circular soap lm bound by aninextensible, elastic lament.
5.2 Transverse displacement
Modulo a rigid translation, the general solution of the partial-differential equation (17) is
wðr; qÞ ¼ a0I0ðzrÞ þXNn¼1
ðcnrn þ anInðzrÞÞcosðnqÞ
þXNn¼1
ðdnrn þ bnInðzrÞÞsinðnqÞ; (30)
where Ii, i ˛ N, is a modied Bessel function of the rst kind.Substituting (30) into the boundary conditions (20) and (21) andinvoking (23) and (24) results in an eigenvalue problem leadingto the dispersion relationh� h
4
�z3I 000n ðzÞ þ z2I 00n ðzÞ
�þ h4þ n
zI 0nðzÞ þ
h4� h
n2zI 0nðzÞ
þn4InðzÞ �nþ 1� hþ h
2
n2InðzÞ
i½hnð1� nÞ�
�h� h
4nðn� 1Þ2 þ
h4þ n
nþ
h4� h
n3 þ n4
�nþ 1� hþ h
2
n2ihh4z2I 00n ðzÞ þ
h4þ h
�zI 0nðzÞ � n2InðzÞ
�i ¼ 0:
(31)
The terms involving c1 and d1 in (30) represent rigid bodyrotations about the diameter of domain R and, thus, arephysically irrelevant. In addition, the requirements a0 ¼ a1 ¼ b1¼ 0must be met to satisfy the boundary conditions (20) and (21)for n ¼ 0 and n ¼ 1. Thus, n ¼ 2 is the rst nontrivial mode ofthe transverse deformation w. Due to its complexity, (31) will bestudied numerically and discussed in Section 7. Whereas n istreated as a control parameter, h and �h are treated as knowninput parameters. The solution of (31), which distinguishes the
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critical surface tension for each transverse mode n, is denotedby ntn. Thus, w can be written as
wðr; qÞ ¼XNn¼1
unðrÞQnðqÞ; (32)
with
unðrÞ ¼ InðzrÞ þ gnrn;
QnðqÞ ¼ an cos nqþ bn sin nq;
�(33)
and
gn ¼h
4z2I 00n ðzÞ þ
h4þ h
�zI 0nðzÞ � n2InðzÞ
�h nðn� 1Þ : (34)
6. Stability of a flat circular HDL particle
The stability of the equilibrium conguration can be addressedby checking the sign of the second variation H€ of the functionalH . Consistent with the notation for the rst variation, asuperposed double dot indicates the second variation. Thequantity H€can be decomposed into a sum
The decoupling of the planar and transverse displacements in(35) enables separate studies of the stability of a discoidal HDLparticle to planar and transverse perturbations.
Planar stability requires that
H€ i . 0: (38)
Using a Fourier expansion, the in-plane variation _u may beexpressed as
_uð1; qÞ ¼XNm¼1
fm sin mq; (39)
which, on substitution into (38), yieldsXNm¼1
fm2�m2 � 1
���m2 � 1
�þm2�m2 � 1
�3� n
. 0: (40)
Since the variation _u is arbitrary, the coefficients fm in (39)are independent and each term of the summand in (40) must
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separately satisfy the inequality (40). In response to planarperturbations, a at, circular HDL particle therefore obeys
n\nim : stable;n. nim : unstable;
�(41)
with nim given in (28).Transverse stability requires that
H€t . 0: (42)
Similar to the strategy used to investigate stability withrespect to planar perturbations, a general transverse variation _wcan be expanded in a Fourier series. However, since the coeffi-cients of each mode in the Fourier expansion of _w are inde-pendent, it is, without loss of generality, possible to consider
_w(r, q) ¼ cn(r)cos(nq), n ˛ N, (43)
with cn being an arbitrary function. Determination of condi-tions necessary and sufficient to ensure (42) for cn arbitraryappears to be challenging. An alternative approach invokes theRayleigh–Ritz variational method, in which cn is approximatedby sum of known functions multiplied by unknown coefficients.The known functions must satisfy the geometrical boundaryconditions but may otherwise be chosen arbitrarily. Guided bythe structure of the general solution (30), consider the Ansatz
cn(r) ¼ gnrn + hnIn(zr), (44)
where gn and hn are independent unknown coefficients.Substitution of (44) in (43), and subsequently in (42), andevaluating the relevant integrals yields
H€t ¼ p
2½V �u½M�½V �. 0; (45)
for each n, with [V] ¼ [gn hn]┬ and [M] a 2 � 2 matrix provided in
the Appendix. The condition necessary and sufficient for (45) to besatised is that [M] be positive-denite, namely that its compo-nents obey
M11 > 0, M11M22 > (M12)2. (46)
As a consequence of (46)1, it follows that
n < n(n + 1) � 2n�h. (47)
Due to its complexity, (46)2 will be studied numerically anddiscussed in Section 7.
6.2 Onset of instability
The onset of instability corresponds to the vanishing of thesecondvariationH€ ofH . It canbeshownthat thecritical values ninand ntnof the surface tensioncorrespond, respectively, to theonsetof the planar and transverse instability. For n¼ nin and n¼ ntn, thesolutions (26) and (32) can be used in (36) and (37), respectively.Specically, for each m and n, _u and _w can be expressed as
_uð1; qÞ ¼ _UsinðmqÞ;_wðr; qÞ ¼ unðrÞ _an cos nqþ _bn sin nq
;
9>=>; (48)
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where _U, _an, and _bn are the variations of the coefficients U, an,and bn. Regarding (48)1 and (25), it is readily observed that H€ivanishes identically. Thus, (27) determines the planar insta-bility requirement and delivers the critical surface tension nim
given in (28). Also, in view of (48)2 and the equilibrium condi-tions (17), (20), and (21), H€t vanishes. Thus, the dispersionrelation (31) furnishes the condition necessary for the onset oftransverse instability.
The connection between the stability conditions (46) and thecritical value ntn at the onset of instability will be discussed in thenext section.
7. Numerical results and discussion
Results from numerical studies based on the model aredescribed next. Regarding the various input parameters, itseems reasonable to x some of them. In particular, the splaymodulus m and the bounding loop bending stiffness a are keptxed, unless mentioned otherwise. Due to the lack of data forthe bending stiffness or persistence length of apoA-I, existingdata for the persistence length of apolipoprotein C-II chains,which are another common component of lipoprotein particles,are used. The input parameters are merely used to illustrate theprimary features of the problem; modest deviations from theirexact values should not signicantly affect the nature of thestability. Hatters et al.18 report that the persistence length ofapolipoprotein C-II is approximately 36 nm, which correspondsto a bending stiffness of (36 nm)kBT, with kB Boltzmann'sconstant and T the absolute temperature. Assuming a double-belt apolipoprotein structure for the bounding loop yields a �(70 nm)kBT. A representative value m z 0.5 � 10�19 J is used forthe splay modulus of a lipid bilayer19 and it is assumed20–25 thatthe reference HDL particle has diameter 2R z 10 nm. Withthese choices, (13)2 yields hz 0.83. It thus seems reasonable touse h ¼ 1.
Fig. 2 depicts the transverse stability of a at circular HDLparticle for different values of the surface tension n and thesaddle-splay modulus �h. Only the rst four modes are consid-ered. The solid lines indicate the variation of the critical surfacetension ntn with �h for each mode, obtained from the dispersionrelation (31). The numerical technique used to solve thedispersion relation (31) involves systematically checking thesign of its le-hand side for wide ranges of the input parameters�h and n. Careful numerical checks have been performed toensure the accuracy of the solution. For various values of n and �h
in the stability plane, the second variation condition (46) hasbeen used to carefully determine the nature of stability indifferent regions of the (n, �h)-plane. Whereas the necessarycondition (46)1 limits the stable domain to the region below thedashed lines, (46)2 limits the stable domain exactly into theregion enclosed by each solid line. The intersection of (46)1 and(46)2 determines the shaded region enclosed by each solid lineas the domain where a at circular HDL particle is stable undera transverse perturbation with mode n. Outside each enclosedregion, the particle is unstable under a perturbation with moden. Evidently, the solid lines correspond to the onset of insta-bility, namely the point at which an exchange of stability occurs.
Fig. 2 Stability plane showing the domains where a flat circular discoidal HDLparticle is stable or unstable under transverse perturbations. Solid lines show ntn,namely the solution of the dispersion equation (31). Regions below the dashedlines are the domains where the stability requirement (46)1 is met.
Fig. 3 Stability plane for a flat circular HDL particle subjected to transverse(n ¼ 2) and planar (m ¼ 2) perturbations, including four distinct regions (a)–(d).While the solid line shows the variation of nt2, the dashed line shows ni2 ¼ 3 for3 ¼ 0 (as provided in (29)). Also depicted are side and top views of post-buckledconfigurations corresponding to the regions (a)–(d).
Fig. 4 Effect of h on the variation of the critical surface tension nt2 with �h.Requirement (46)1 is met in the region below the dashed line.
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Interestingly, for each n, the stable region for mode n is con-tained in the stable region of mode n + 1. It is therefore evidentthat within the stable region for n ¼ 2 the particle is stable withrespect to all higher modes. It is also found that, within thestable region enclosed by each solid line, stability is enhancedby negative values of the saddle-splay modulus �h, as the stabledomain for �h < 0 is larger than that for �h > 0. Finally, it isnoteworthy that, while the dispersion relation (31) has two rootsfor sufficiently large negative �h, it otherwise has only one root.
Fig. 3 depicts the stability plane of a at circular HDLparticle under transverse (saddle-like) and planar (elliptical)perturbations. While the solid line corresponds to nt2, thedashed line corresponds to ni2 ¼ 3 for 3 ¼ 0 (given in (29)). Forother values of 3 $ 0, the dashed line is merely shied upwardwhile remaining straight and horizontal. According to (41), inthe region below the dashed line, a planar discoidal HDLparticle is stable. The intersection of the transverse and planarstable and unstable regions determines four distinct regions.In region (a), a discoidal HDL particle is stable under bothtransverse and planar perturbations. Thus, a at, circularparticle should be observable only for values of n and �h inregion (a). In region (b), a discoidal HDL particle is stableunder planar perturbations but is destabilized by transversesaddle-like perturbations. In region (c), a discoidal HDLparticle is stable under transverse perturbations but isunstable to planar perturbations. Thus, for values of n and �h inregion (c), a noncircular at HDL particle should be observed.Lastly, in region (d), a discoidal HDL particle is unstable underboth transverse and planar perturbations. Hence, for values ofn and �h in region (d), at and saddle-like HDL particles withcircular projections onto the reference plane are not observ-able. To aid in visualizing the various possibilities, schematiccongurations of a discoidal HDL particle in regions (a)–(d) arealso provided in Fig. 3. However, it should be emphasized that
Soft Matter
the linear analysis presented here is unable to exactly predictnal post-buckled congurations involving large distortions.The schematic congurations provided in Fig. 3 are merelybased on the stability/instability of a circular at HDL particleunder elliptic and transverse saddle-like perturbations.
So far, it has been assumed that the dimensionless param-eter h is xed while allowing the other dimensionless parame-ters n and �h to vary. Regarding (13)2, if the radius R is held xed,the constancy of h requires that the ratio m/a of the splaymodulus m of the lipid bilayer and the bending rigidity a of theapoA-I chain to be constant. However, to have a more completepicture of the results, considering different values of h reveals
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the inuence of m or a on the stability of discoidal HDL parti-cles. Particularly, the effect of a, due to lack of information onthe bending modulus of apoA-I, seems essential. The variationof the critical surface tension nt2 with �h has been obtained fordifferent values of h and is plotted in Fig. 4. On increasing h, theregion conned between each curve and the horizontal axis ismagnied and extends toward more negative values of �h. Forlarger values of m or smaller values of a, the domain of stabilityfor a discoidal HDL particle therefore grows.
8. Concluding remarks
Simulations of Catte et al.4 reveal that gradually removing lipidmolecules from discoidal HDL particles induces a transitionfrom planar circular to nonplanar saddle-like congurations.Since the length of the apoA-I chain does not change during thedepletion of lipid molecules from the bilayer of an HDL particle,decreasing the number of lipid molecules while keeping thesurface area of HDL particle xed should increase the averagespacing between neighboring lipid molecules and, hence, thetension on the surface of particle. This is analogous toincreasing the distance between the lipid molecules in eachleaet of the bilayer by imposing an areal stretch. It is evidentfrom the results of Fig. 2 and 3 that increasing the surfacetension n diminishes the range of stable values for the saddle-splay modulus �h and favors instability. For n > nic, with nic given in(29), a at circular HDL particle loses its shape under in-planeperturbations. Similarly, for values of the surface tension n > ntn,a at circular HDL particle becomes unstable to transverseperturbations. To reiterate, the rst planar and nonplanarunstable modes correspond respectively to planar elliptical andnonplanar saddle-like shapes.
Although the linear analysis performed here is incapable ofspecifying the nal shape that a discoidal HDL particle mightadopt, our results, the simulations of Catte et al.4 and Shihet al.,5,6 and the experimental observations of Silva et al.,7
Miyazaki et al.,8 Huang et al.,9 and Skar-Gislinge et al.,10 suggestthat the observed planar elliptical and nonplanar saddle-likeshapes of discoidal HDL particles might represent stabilizedpost-buckled congurations of initially at circular particleswhich have become unstable due to identical types of pertur-bation—i.e., the planar elliptic (mode m ¼ 2) and nonplanarsaddle-like (mode n ¼ 2). This hypothesis is based on a long-standing tradition of analogous observations in structuralmechanics, a tradition wherein linearized stability analysispredicts the critical or buckling conditions under which astructure adopts a nontrivial conguration, usually a congu-ration with the same mode shape of the driving perturbation.The linearized analysis presented in this paper determinesconditions necessary for instability of a at circular HDLparticle. Nevertheless, a comprehensive understanding of theequilibrium and stability of discoidal HDL particles requires anonlinear analysis capable of determining nontrivial congu-rations involving large distortions.26
An important next step would be to compare the predictionsof the linearized analysis to results from experiments ornumerical simulations. The numerical results presented here
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predict the stability of a at circular HDL particle for tentativevalues of the salient dimensionless parameters. Although thequalitative features of these results should not change signi-cantly for modest deviations from parameter values, access tomore realistic values for these parameters would naturally allowfor more accurate predictions. For example, the bending rigiditya of the apoA-I chain is unavailable and has been approximatedby the a reported value for apolipoprotein C-II. Also, the surfacetension s (and perhaps the bending moduli m and �m) may varywith the composition of the lipid bilayer and the numberdensity of lipid molecules, which are sensitive to experimentalconditions and assumptions underlying simulations. Theexisting simulations4–6 and experiments7–10 do not providequantitative information guidelines for comparisons. Furtherexperiments or simulations designed to evaluate the underlyingphysical parameters and to determine the conditions underwhich at noncircular and saddle-like congurations arisetherefore seem worthy of the required effort.
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