# Determinants of Lipid Domain Size

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**α**, spontaneous monolayer curvature J

_{0}is the primary determinant of domain size [9]. In contrast, hypothesis

**β**prioritizes line tension, λ [10].

**α**envisions the curvature stress for domains with large J

_{0}values in compositional symmetric bilayers [11]. The reasoning is that domain registration from the two monolayers results in a flat bilayer (Figure 1). In an asymmetric bilayer fragment consisting of an LDD monolayer on one side and a LOD monolayer on the other side, curvature frustration may be smaller as the resulting bilayer is not necessarily flat. It acquires bilayer curvature J

_{B}[12] if J

_{0}differs for LDD monolayers and LOD monolayers. In a bilayer with several such fragments, the differently curved areas must alternate [13] to maintain a flat membrane geometry. The division between the two scenarios of symmetric and asymmetric bilayer fragments is not strict, as thermal undulations are supposed to cause transient local asymmetry [14]. In any of the above cases, the domain size decreases with absolute values of J

_{0}. For membrane compositions with large positive J

_{0}values, the equilibrium domain size is as small as 10 nm [11].

**α**is at odds with the observation that domains are always registered, even in asymmetric bilayers [8]. The formation of locally asymmetric fragments is energetically unfavorable because stiffed regions from the two leaflets attract each other [15]. Stiffer lipid domains tend to be distributed into areas with lower monolayer curvature. Consequently, membrane undulations naturally align domains in the opposing monolayers [16]. The corresponding driving force scales with membrane area. It becomes insufficient for the alignment of tiny domains. Accordingly, for smaller domains, another driving force is dominant: λ. It drives LDD and LOD registration [17]. The free energy of registered LODs and LDDs is smaller than that of anti-registered LODs and LDDs [18].

**β**recognizes the vital role of λ for domain formation and registration. λ originates from the hydrophobic mismatch between thinner LDDs and thicker LODs [19,20]. It reflects the energy penalty associated with elastic lipid deformations at the LOD–LDD boundary. Along with this mechanical component, λ also contains a chemical contribution that reflects the difference in lipid concentrations in two contacting phases. The total boundary energy E

_{TB}increases with the interphase boundary’s perimeter [21,22]. As the state of minimal E

_{TB}corresponds to an LOD/LDD system with minimal total boundary length, the domains tend to fuse, giving rise to the macroscopically large domains observed in artificial bilayers. Nevertheless, nanometer-sized domains may still exist because line-active components decrease λ, thereby averting domain fusion [23,24,25].

**α**or

**β**. Therefore, we change the J

_{0}of lipids entering the composition of preformed domains by exposure to light. Domain dissipation due to increased absolute J

_{0}values would prove hypothesis

**α**. Light-triggered conformational lipid changes may also affect λ, provided that the photo-sensitive lipid adopts a localization at the phase boundary or modifies the elastic properties of bulk phases. For this case, hypothesis

**β**predicts changes in domain size evolution.

## 2. Results

_{0}[27,28] (Figure 2).

**α**, this intervention should allow LDDs to increase in size. However, many LDDs dissolved (Figure 3). We used fluorescence laser scanning microscopy for visualization (see Section 4). Lipid domains of all sizes were visible as the dyes partitioned preferentially into LDDs. Domain tracing enabled us to calculate the diffusion coefficients of LODs and LDDs, allowing us to assign domain size [8]. The method works well also for domains that are smaller than the diffraction limit.

_{0}(Table 1), we utilized the phase diagram for DPhPC:DPPC:Chol mixtures [29] and assumed (i) trans-PhoDAG-1′s packing density to be similar to DPPC, (ii) cis-PhoDAG-1′s packing density to be similar to DPhPC, and (iii) equal distribution coefficients between LODs and LDDs for (a) trans-PhoDAG-1 and DPPC and (b) cis-PhoDAG-1 and DPhPC. We calculated the following compositions of LODs: 3.5:28.5:39.5:28.5 DPhPC:DPPC:Chol:DAG in PhoDAG-1′s trans-state; 5:39:54:2 DPhPC:DPPC:Chol:DAG in PhoDAG-1′s cis-state. We arrived at the following compositions of LDDs: 53:9:29:9 DPhPC:DPPC:Chol:DAG in PhoDAG-1′s trans-state; 45:8:25:22 DPhPC:DPPC:Chol:DAG in PhoDAG-1′s cis-state.

**α**, which predicts a decrease in domain size for increasing absolute J

_{0}values (compare Figure 1). Photoswitching produced only minute J

_{0}changes for LODs (Table 1). Accordingly, the failure of the bulky LOD phase to dissipate into smaller LOD domains cannot be used to refute or confirm

**α**’s predictions.

**β**. The hydrophobic thickness of the longer trans-PhoDAG-1 [27] matches that of DPPC lipids, while the hydrophobic thickness of the shorter cis-PhoDAG-1 [27] matches that of DPhPC lipids. Accordingly,

**β**predicts the longer trans lipids’ assembly into LODs, and the ordered phase’s total area increase upon cis-to-trans photoswitching (Figure 3). For the same reason, trans-to-cis photoswitching drives the LDD assembly within macroscopic LODs (Figure 3).

**α**and

**β**fundamentally differ in their predictions about the time-dependence of domain sizes:

**α**envisages a curvature frustration-driven dissolution of large LDDs. At the same time,

**β**foresees continuous domain growth driven by the tendency to minimize the total boundary energy E

_{TB}. We were able to observe micrometer-wide, i.e., macroscopically large, LDDs that did not dissolve after we switched the photo lipid to its cis state (Figure 4). This observation disproved hypothesis

**α**.

**β**may provide a rationale for both observations—the dependence on (i) the state of PhoDAG-1 and (ii) domain size—we performed a theoretical analysis based on membrane elasticity theory [30] and the corresponding theoretical framework [17,31].

_{t}, and K

_{a}. The energy required for bending depends on J

_{0}. A lipid reservoir (torus) subjects the membrane to lateral tension, 2σ

_{0}. We minimized elastic energy with respect to deformation fields at a fixed distance between the domain boundaries for a discreet set of distances.

_{TB}as a function of the distance between domain boundaries showed a maximum E

_{B}at a distance D ~4–6 nm (Figure 7a). E

_{B}resulted from the mechanical work bestowed on the lipids between two approaching domains (Figure 7b). That is, the intervening lipids departed from the conformation they had adjacent to an isolated domain to adapt to their new environment that may differ in hydrophobic thickness, as well as in splay and tilt of the adjacent lipids. As their initial state served to maintain the energetic minimum for the isolated domain, the new conformation must represent a state of increased energy.

_{BO}= 0.03 k

_{B}T per 1 nm of contact boundary length. To obtain energy barrier E

_{i}that opposes domain mergers, the specific energy E

_{B}is multiplied by the size of the boundary l

_{R}involved in domain repulsion. For circular domains, one may use the Derjaguin approximation [32], according to which the effective length of the domain boundary entangled in the interaction is ${l}_{R}=2\sqrt{2{l}_{c}{R}_{0}}$, where l

_{c}is the characteristic decay length of deformation (~1 nm) and R

_{0}is the domain radius. The Boltzmann factor ${e}^{-{E}_{i}/{k}_{B}T}={e}^{-{l}_{R}{E}_{B}/{k}_{B}T}$ governs the probability of domain mergers. The prefactor A accounts for the frequency of the merging attempts. Thus, we can write Equation (1) for the logarithm of the merger probability p:

_{i}and E

_{B}, as well as the probabilities of domain fusion events (Table 1), assuming a constant frequency of fusion attempts, A. The assumption seems justified as the A governing the domain diffusion coefficient depends but weakly on domain radius [33]. Similarly, the other A-determining factor, the characteristic frequency of thermal domain boundary fluctuations, is independent of the domain size, as governed by membrane viscosity. l

_{R}is roughly equal to 8 nm for domains of ~8 nm in radius according to the Derjaguin approximation. For micrometer-sized domains, l

_{R}increased to about 60 nm, yielding E

_{i}values of several k

_{B}T. In contrast, the experimentally observed ratios of fusion probabilities p

_{cis}/p

_{trans}were close to unity (1.2–2.8, Table 1), suggesting that only a tiny part of any domain boundary interacts with the border of the opposite domain at any time, i.e., l

_{R}≈ 8 nm works for domains of all sizes.

_{io}≈ 0.24 k

_{B}T for trans-PhoDAG-1-containing LODs. Similar calculations for LDDs yielded E

_{Bd}= 0.02 k

_{B}T/nm (E

_{id}≈ 0.16 k

_{B}T), which is compatible with the observed probability of domain mergers (Table 1). For cis-PhoDAG-1, the energy barriers for LOD–LOD and LDD–LDD interactions were equal to E

_{Bo}= 0.06 k

_{B}T/nm and E

_{Bd}= 0.16 k

_{B}T/nm, respectively. The corresponding total interaction energies were E

_{io}= 0.47 k

_{B}T and E

_{id}= 1.26 k

_{B}T (Table 1).

## 3. Discussion

_{0}′s domain-size-governing role. The observed domain diameters always spanned an extensive range, varying from ~100 nm to 10 s of µm. Curvature stress due to high J

_{0}values did not lead to domain dissipation. This observation indicates that curvature stress is either independent of domain radius or negligibly small. A simple model for the energy density F of a membrane corroborated the conclusion (Equation (2)). It describes F as depending quadratically on J

_{B}and local principal membrane curvatures C

_{1}and C

_{2}:

_{G}is the Gaussian rigidity. In flat (C

_{1}= C

_{2}= 0) symmetric bilayers, F is equal to zero as the curving propensities of the individual monolayers counteract each other, i.e., J

_{B}is equal to zero.

_{M}of a flat monolayer:

_{M}is a function of total domain area A

_{D}:

_{M}does not increase or decrease if a large domain dissipates into small ones, as long as A

_{D}remains invariant.

**α**[11]. The following assumptions could be responsible: (i) LDDs and LODs have similar elastic parameters, and (ii) LODs have significant positive J

_{0}values. However, assumption (i) artificially decreases E

_{TB}. Assumption (ii) is invalid because naturally occurring LODs generally have negative or only slightly positive J

_{0}values. Cholesterol’s J

_{0}amounts to −0.37 nm

^{−1}[34], DPPC’s reaches +0.1 nm

^{−1}[35], and DOPC’s or POPC’s J

_{0}equals −0.091 nm

^{−1}or −0.022 nm

^{−1}, respectively [36]. Attaining the assumed J

_{0}values of +2 nm

^{−1}would have required membrane-destabilizing amounts of short-tail lysophosphatidylcholine [37].

_{0}does not matter, as curvature frustration per se may be sufficient for

**α**to work [12]. Yet, vanishing line tension at the domain boundary was the cornerstone of hypotheses

**α**, and it is unclear how negatively curved monolayers may produce such negligible λ values. Only positive J

_{0}values may decrease the hydrophobic mismatch at the LOD–LDD boundary (Figure 1). On the contrary, two negatively curved monolayers’ hydrophobic interaction compresses the lipids at the midpoint of the arches (Figure 8). The hydrophobic thicknesses at the domain interface remain unaltered. The persisting hydrophobic mismatch between LODs and LDDs incurs considerable λ values, the minimization of which will govern domain size. Thus, we conclude that the sign of J

_{0}matters. Realistic J

_{0}values do not support the idea of curvature-controlled domain sizes.

_{0}) governs domain size. As the energy stored in the bilayer’s rim decreases with the border’s length, domains tend to fuse. For two domains to approach, the intervening lipids must be released from the strain that tilts or bends them (Figure 7b). Accordingly, this pre-existing lipid deformation may give rise to a kinetic trap that opposes neighboring domains from coming close to each other. It is caused by a local energy maximum, E

_{B}, that can be calculated as a function of the distance between domain boundaries (Figure 7a). Our calculations predicted that E

_{B}for two interacting LODs might differ from E

_{B}for interacting LDDs. The prediction has been confirmed by experimental observations (Table 1).

_{1}notably alters raft morphology even when present in fractions of a molar percent [24,25].

## 4. Materials and Methods

#### 4.1. Materials

#### 4.2. Planar Membrane Formation

^{−8}S cm

^{−2}and ≥0.7 µF/cm², respectively. The overlay of the fluorescence excited at 561 nm and 633 nm showed domain registration (Figure 3, Figure 4, Figure 5 and Figure 6).

#### 4.3. Light Scanning Microscopy (LSM) and Polychrome V Monochromator

#### 4.4. Theoretical Model

**n**. The field of directors is set at some surface lying inside the lipid monolayer, referred to as the dividing surface. The shape of the dividing surface is described by a vector field

**N**of the unit normal to it. We take into account the following deformations of the membrane: (1) splay, characterized by splay modulus, B, and by the divergence of the director along the dividing surface, div(

**n**); (2) tilt, characterized by tilt modulus, K

_{t}, and by tilt-vector

**t**, equal to the deviation in the director from the normal at a given point of the dividing surface,

**t**=

**n**−

**N**; (3) lateral compression/stretching, characterized by stretching modulus, K

_{a}, and by the relative change in the area of the dividing surface, α = (a − a

_{0})/a

_{0}, where a and a

_{0}are current and initial areas per molecule at the dividing surface, respectively. The membrane may be subjected to lateral tension, 2σ

_{0}, set by a lipid reservoir. The deformations are set at the specific dividing surface, at which the splay and the lateral compression/stretching deformations are energetically decoupled. This surface is referred to as the neutral surface; it was shown to lie in the region of the junction of polar heads and hydrophobic tails [42]. We assumed that the domain radius significantly exceeds the characteristic length of deformation decay, which usually amounts to several nanometers [17,22,31]. In the case of a translational symmetry along the domain boundary, the system becomes effectively unidimensional. As a result, Gaussian curvature makes zero contribution to the elastic energy.

_{0}is the spontaneous curvature of the lipid monolayer, H is the distance from the reference plane to the monolayer’s neutral surface measured along the normal to the reference plane, and α

_{0}= σ/K

_{a}is the equilibrium stretching of the monolayer caused by the lateral tension. The membrane is assumed flat and parallel to the Oxy plane, far away from the boundary. The elastic energy functional, Equation (5), was minimized with respect to deformation fields at a fixed distance between the domain boundaries for a discreet set of distances. The resulting dependence of the energy on the distance between domains gives the interaction energy profile.

#### 4.5. Parameters of the System

_{o}= 20 k

_{B}T (k

_{B}T ~4 × 10

^{−21}J), B

_{d}= 10 k

_{B}T, K

_{a}= 30 k

_{B}T/nm

^{2}= 120 mN/m [43], σ = 0.1 k

_{B}T/nm

^{2}= 0.4 mN/m, h

_{d}= 1.3 nm, h

_{o}= 1.8 nm. We used different elastic moduli for LODs and LDDs as their splay moduli may differ 2–5 fold [44]. In contrast, the tilt modulus is almost independent of the membrane composition [30]. Generally, J

_{0}depends on the composition (and correspondingly on the particular membrane domain). We calculated J

_{0}values (Table 1) as the weighted concentration average of the components’ J

_{0}using the phase diagram of DPPC/DPhPC/Chol [29]: J

_{chol}= −0.37 nm

^{−1}[34]; J

_{DPPC}= 0.07 nm

^{−1}[35]; J

_{DPhPC}= −0.2 nm

^{−1}; J

_{DAG-trans}= −0.2 nm

^{−1}; J

_{DAG-cis}= −1 nm

^{−1}[28].

_{0}of a lipid mixture may be calculated as the weighted concentration average of the components’ J

_{0}is commonly accepted. It has been used to determine most values of individual lipid’s (DPPC, DPhPC, cholesterol, etc.) spontaneous curvatures available today, as the underlying experiments were conducted in binary lipid mixtures [36,37]. Molecular dynamics simulations arguing for nonadditive compositional curvature energetics of lipid bilayers [35] have thus far not been confirmed experimentally. Importantly, molecular dynamics only allow the calculation of the product formed from lipid mixture’s average spontaneous curvature and average bending modulus. They cannot provide values for the spontaneous curvature as such [35].

_{0}values (Table 1) of LODs for cis-PhoDAG-1 (−0.2 nm

^{−1}) and trans-PhoDAG-1 (−0.26 nm

^{−1}), as well as of LDD for trans-PhoDAG-1 (−0.25 nm

^{−1}), did not significantly differ from each other, whereas J

_{0}of LDD for cis-PhoDAG-1 (−0.4 nm

^{−1}) differed significantly from the other J

_{0}values.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Determinants of lipid domain size. According to alternative hypotheses

**α**and

**β**, the equilibrium domain size is either governed by spontaneous monolayer curvature, J

_{0}= 1/R

_{optimal}(top), or line tension, λ (bottom).

**α**. As the bending propensities of the upper and lower monolayers are identical, the resulting LOD is nearly flat. That is, the radius R of the geometrical curvature of its monolayer surface is much larger than the radius, R

_{optimal}, corresponding to the spontaneous monolayer curvature. Dissipation of the large domain into smaller domains of equal total area releases the curvature stress. R

_{optimal}governs the radius R

_{0}of the new LODs.

**β**. LOD–LOD fusion releases part of the energy spent to deform lipids at the height-mismatched LOD/LDD boundary (red line) as boundary length decreases. The gain in energy scales with line tension λ. Accordingly, λ determines the rate of fusion and thus domain size in a nonequilibrated system.

**Figure 2.**Photolipids and branched lipids. The photoswitchable PhoDAG-1 has two conformations: Left-trans-isomer and middle-cis-isomer. Illumination at 365 nm induces azobenzene’s trans-to-cis isomerization; cis-to-trans isomerization occurs upon illumination at 474 nm. The right panel shows the chemical structure of DPhPC.

**Figure 3.**Photoswitching PhoDAG-1 affects phase separation. Illumination at 474 nm (top row, left three frames) dissolves LDDs (bright spots). The numbers in the left upper corners of the images indicate the time (in seconds) after setting the wavelength to 474 nm. The wavelength change at t = 159 s to 365 nm results in LDD reassembly. The 2nd number in the fourth frame of the top row indicates how long we illuminated the membrane at 365 nm. The bottom row illustrates the gradual dissolution of a particular domain (marked by a white arrow). The scale bar has a length of 50 µm. The experiment was carried out at an elevated temperature of 30 °C.

**Figure 4.**Domain time evolution.

**Top row**: The bulky and bright LDD domain covers most of the membrane surface. Due to its content of cis-PhoDAG-1, J

_{0}is very high. Nevertheless, the LDD does not dissipate into small LDDs. The large LOD (large dark circle in the fourth quadrant) grows only ~2% between the first and the last image, indicating delayed LOD–LOD fusion.

**Bottom row**: the bulk of the membrane lipids are in the ordered state. The diameter (>10 µm) of the two visible bright LDDs in the center does not change between the first and last image, indicating a hampered LDD–LDD merger. The same conclusion can be drawn from the minor decrease in the total domain number (in an arbitrarily chosen square 25 × 25 μm in the center of the images of the bottom row, the number of domains was 18, 18, 17, 17 from left to right frame). The time (in seconds) after photoinducing cis-PhoDAG-1 is in the left upper corner of the images. The scale bar has a length of 50 µm.

**Figure 5.**Size evolution of LODs and LDDs. The top two rows reflect representative events, eventually leading to LOD growth. Unsuccessful fusion events (domain bouncing) and successful fusion events are discernible, as depicted in the first and the second rows. The third and fourth rows show attempted LDD fusion events (bouncing) and a successful LDD merger (bottom row), respectively. Green circles mark the domains of interest. The numbers in the left upper corners of the images indicate the time (in seconds) after setting the wavelength to 474 nm, i.e., all pictures were obtained for trans-PhoDAG-1. The scale bar has a length of 50 µm.

**Figure 6.**The number of trans-PhoDAG-1-containing domains decreases with time. Small LODs (dark, top row) and LDDs (light, bottom row) easily merge, which results in a fast decrease in domain number at an approximately constant area of the corresponding phase. The time (in seconds) after photoinducing trans-PhoDAG-1 is in the left upper corner of the images. The scale bar has a length of 50 µm.

**Figure 7.**Membrane-mediated domain interaction. (

**a**) Energy profiles of domain interactions in the presence of trans-PhoDAG-1 (dashed lines) and cis-PhoDAG-1 (solid lines). Black and blue plots present the interaction energy (elastic energy of the membrane) as a function of the distance between two LODs and two LDDs, respectively. The energy barrier toward domain fusion, E

_{i}, is illustrated. (

**b**) The energy barrier to domain fusion arises from the competition between two domain boundaries to orient the intervening lipids. The lipids at a certain distance from the domain boundary were tilted and bent (top) to minimize the energy stored in an isolated LOD boundary. However, an adjacent second domain forces them to depart from that energy-minimizing orientation (bottom). The new, nontilted orientation minimizes the total elastic energy for the system consisting of two LODs. Yet, it signifies increased E

_{TB}for each of the two LODs. Thus, elastic deformations may kinetically stabilize nanodomains.

**Figure 8.**Negative monolayer curvature cannot govern LOD size. The mutual attraction of two registered LODs would lead to the compression of the lipids at the midpoint of the arches instead of decreasing the hydrophobic mismatch at the domains’ edges. Nonnegligible line tension is the consequence. The scheme assumes that the actual inverse monolayer curvature 1/J

_{0}is close to optimal radius R.

**Table 1.**PhoDAG 1′s effect on the fusion of large domains, the fusion-opposing energy barrier E

_{i}, the fusion-opposing energy barrier per unit length of domain boundary, E

_{b}, the probability of merger, p, and the calculated spontaneous curvatures, J

_{0}. N is the number of events; A is the frequency of merger attempts (see Equation (1)).

PhoDAG-1 State | trans | cis |
---|---|---|

LODs | ||

N, merged | 30 | 9 |

N, bounced | 19 | 10 |

Merging rate, events/s | 30/200 = 0.09 | 9/210 = 0.025 |

p | 61 ± 14% | 47 ± 22% |

E_{b}, k_{B}T/nm | 0.03 | 0.06 |

E_{i}, k_{B}T | 0.24 | 0.47 |

k_{B}T × (ln(A) − ln(p)), k_{B}T | 0.16 ± 0.2 | 0.43 ± 0.4 |

J_{0}, nm^{−1} | −0.26 | −0.2 |

LDDs | ||

N, merged | 53 | 21 |

N, bounced | 42 | 86 |

Merging rate, events/s | 53/80 = 0.4 | 21/60 = 0.2 |

p | 56 ± 10% | 20 ± 8% |

E_{b}, k_{B}T/nm | 0.02 | 0.16 |

E_{i}, k_{B}T | 0.16 | 1.26 |

k_{B}T × (ln(A) − ln(p)), k_{B}T | 0.25 ± 0.17 | 1.27 ± 0.34 |

J_{0}, nm^{−1} | −0.25 | −0.4 |

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**MDPI and ACS Style**

Saitov, A.; Kalutsky, M.A.; Galimzyanov, T.R.; Glasnov, T.; Horner, A.; Akimov, S.A.; Pohl, P.
Determinants of Lipid Domain Size. *Int. J. Mol. Sci.* **2022**, *23*, 3502.
https://doi.org/10.3390/ijms23073502

**AMA Style**

Saitov A, Kalutsky MA, Galimzyanov TR, Glasnov T, Horner A, Akimov SA, Pohl P.
Determinants of Lipid Domain Size. *International Journal of Molecular Sciences*. 2022; 23(7):3502.
https://doi.org/10.3390/ijms23073502

**Chicago/Turabian Style**

Saitov, Ali, Maksim A. Kalutsky, Timur R. Galimzyanov, Toma Glasnov, Andreas Horner, Sergey A. Akimov, and Peter Pohl.
2022. "Determinants of Lipid Domain Size" *International Journal of Molecular Sciences* 23, no. 7: 3502.
https://doi.org/10.3390/ijms23073502