# Phosphatidylcholine Membrane Fusion Is pH-Dependent

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## Abstract

**:**

## 1. Introduction

## 2. Results

_{B}—Bolzmann constant; T—absolute temperature (k

_{B}T ~ 4⋅10

^{−21}J); ν—characteristic frequency of attempts of the system to cross the energy barrier; the experimentally determined dependence of the average waiting time to fusion corresponds to change of the energy barrier approximately by 3.5 k

_{B}T. Indeed, if we designate the height of the barrier and the average waiting time at pH = 4.1 as E

_{4.1}and τ

_{4.1}, respectively, and at pH = 7.5, as E

_{7.5}and τ

_{7.5}, respectively, then:

_{7.5}− E

_{4.1}≈ k

_{B}Tln(35) ≈ 3.5 k

_{B}T.

_{B}T, which clearly cannot explain the observed experimental dependence on the environmental pH . Thus, it would appear that change of pH should primarily modify the bending modulus [36,37] and spontaneous curvature of lipid monolayers.

_{f}, d), as a function of half-distance between the bulges occurring on the membranes in the fluctuation mode, d, and the radius of the hydrophobic spots, r

_{f}, occurring on the surfaces of the bulges due to lateral displacement of the polar lipid heads from the area of tight contact between the monolayers. Figure 4 shows isolines of energy W(r

_{f}, d), calculated for DOPC:DOPE = 5:1 membranes at pH = 7.5 (Figure 4a,c) and at pH = 4.1 (Figure 4b,d). The energy is minimal at large distances between the membranes and zero radius of the hydrophobic spot, which corresponds to the equilibrium state of the membranes brought together by hydrostatic pressure. Besides that, when the distance between the membranes is small, and the radius of the hydrophobic spot is large, the energy is also minimal because of hydrophobic attraction between the spots. The energy grows abruptly at large d and large r

_{f}(hydrophobic surfaces are exposed into the bulk water on a large surface area), as well as at small d and smal r

_{f}(the hydration-induced repulsion of the membrane closely juxtaposed on a large area) (Figure 4).

_{f}

^{s}, d

^{s}} on the energy surface, determined by the following conditions:

_{7.5}= 39.5 k

_{B}T; at pH = 4.1—E

_{4.1}= 36 k

_{B}T, and thus, E

_{7.5}− E

_{4.1}= 3.5 k

_{B}T, in good agreement with the dependence of the average lag time of the monolayer fusion upon pH probed experimentally Equation (2), Figure 2. According to the estimates made in [17], the energy barrier of the height of ~40 k

_{B}T is to be crossed at the expense of thermal fluctuations of lipids over time of the order of 1 min, which is in good agreement with the measured average waiting time for monolayer fusion at pH = 7.5.

_{B}T (data not shown; the energy surface is visually almost indistinguishable from that shown in Figure 4a,c). Thus, increase of the bending modulus by 20% at the constant spontaneous curvature of lipid monolayers insignificantly affects the height of the energy barrier and the trajectory of the fusion process.

_{f}, d), performed for the elastic parameters corresponding to DOPC at pH = 7.5 revealed that the energy in the saddle point equals E

_{DOPC}= 44.5 k

_{B}T, and hence the average lag time for the membranes consisting of DOPC has to amount to:

## 3. Discussion

_{B}T, and that is still much lower than the observed effect of pH change on the membrane mechanics.

## 4. Materials and Methods

#### 4.1. Experimental Section

_{app}~ 10 Pa was applied between the peripheral and the central compartments of the cell by means of plunging Teflon pistons into the peripheral compartments in order to form bubbles facing each other. Thereafter, the tips of the cones were brought in tight plane–parallel contact. According to [16], in order to reproduce the initial conditions, we maintained the diameter of the BLM contact area in the range of 0.25–0.35 mm. The parameter measured in the experiment was the monolayer fusion lag time, i.e., the time between bringing the membranes into plane–parallel contact and the time when the hemifusion diaphragm was formed. Fusion was detected as an abrupt increase of the observed electrical capacitance of the contact area.

#### 4.2. Theoretical Section

**n**. This field is defined on a certain surface inside the lipid monolayer, known as the dividing surface. The shape of the dividing surface is determined by the field of unit normals to it,

**N**. We took credit for the following deformations of the membrane: (1) bending, characterized by divergence of the director along the dividing surface, div(

**n**); (2) tilt, characterized by deviation of the director from the normal, (

**n**–

**N**); (3) lateral compression/expansion, characterized by the local change of the area of the dividing surface, α = (a − a

_{0})/a

_{0}, where a and a

_{0}—are the current and the initial area of the dividing surface; (4) saddle-like bending, in a certain Cartesian system of coordinates {x, y} defined as $K=\frac{\partial {n}_{x}}{\partial x}\frac{\partial {n}_{y}}{\partial y}-\frac{\partial {n}_{x}}{\partial y}\frac{\partial {n}_{y}}{\partial x}$, where n

_{x}, n

_{y}are the respective projections of the director. Besides, we took into account that Muller–Rudin membranes are connected to a lipid reservoir, and hence certain lateral tension, 2σ

_{0}, is inevitably present. The neutral surface, where by definition the bending and lateral tension/compression deformations are independent of each other in terms of energy, was used as a dividing surface. It was experimentally determined that such a surface lies near the area of junction of polar heads and carbohydrate tails of the lipids at the depth of ~0.7 nm from the outer polar surface of the monolayer [48]. The deformations were assumed to be small, and hence the energy was calculated to the second order of deformations:

_{t}, K

_{a}, K

_{G}are the moduli of bending, tilt, lateral stretch/compression, saddle-like bending, respectively; J

_{0}—spontaneous curvature of lipid monolayer; integration is performed over the neutral surface of the monolayer; A

_{0}—initial area of the neutral surface of non-deformed monolayer. It is assumed that the bulges formed through fluctuations on the juxtaposed membranes have local polar symmetry. Let us introduce a polar coordinate system Ozr, origin O and axis Or of which are located in the neutral surface plane of the distal monolayer of one of the membranes, the lower one, for the sake of precision, the Oz axis coincides with the axis of rotational symmetry of the system. Due to polar symmetry, all the deformations depend exclusively on the coordinate r, i.e., the system is unidimensional. Thus, all the vectors can be replaced with their projections on the Or axis:

**n**→ n

_{r}= n,

**N**→ N

_{r}= N, whereas divergence of the director, with the adopted degree of accuracy, equals div(

**n**) = n′ + n/r, where prime sign stands for derivative by r. Hereafter, we call the projection of the director and normal upon Or simply “director” and “normal”. The hydrophobic part of the lipid bilayer can be considered volumetrically incompressible [49,50], i.e., the volume of the monolayer element is not affected by deformations. The local incompressibility condition can be written as [47]:

_{c}—is the current thickness of the monolayer, h—monolayer thickness in the undeformed state. The values related to the upper (contact) and the lower (distal) monolayer will be designated by “u” and “d” indexes, respectively. The state of each membrane is characterized by seven functions: (1) directors of the upper (contact) and the lower (distal) monolayers, n

_{u}(r) and n

_{d}(r); (2) relative changes of the area of neutral surfaces of monolayers, α

_{u}(r) and α

_{d}(r); (3) distance from the Or plane to the intermonolayer surface, m(r); (4) distance from the Or plane to the neutral surface of the upper and the lower monolayers, H

_{u}(r) and H

_{d}(r). In this notation, the local incompressibility condition, Equation (6), reads as follows:

_{u}= H′

_{u}(r), N

_{d}= −H′

_{d}(r), and:

_{a}/K

_{t}, σ = σ

_{0}/K

_{t}, k

_{G}= 2K

_{G}/K

_{t}. The first integral is taken over the neutral surface of the upper monolayer, the second over the neutral surface of the lower monolayer. The functional variation, Equation (9), with respect to the functions n

_{u}, n

_{d}, m, α

_{u}, α

_{d}, yields five Euler-Lagrange differential equations. The equations are quite cumbersome, and therefore not presented here. However, they can be solved analytically, the general solution of the system being as follows:

_{u}, the lower—to n

_{d}, and

_{0}, I

_{1}, K

_{0}, K

_{1}, Y

_{0}, Y

_{1}, J

_{0}, J

_{1}are the respective Bessel functions—of order zero and one; c

_{1}, c

_{2}, ..., c

_{7}—complex coefficients, which need to be determined from the boundary conditions. We imposed on the solutions (10)–(13) of Euler–Lagrange equations the following boundary conditions: (1) all the functions must be real for any real argument r; (2) all the functions must be limited for any r; (3) the director is continuous everywhere; (4) neutral surfaces of monolayers determined from Equation (7) are continuous everywhere besides the hydrophobic spots on the tops of the membrane bulges (Figure 3); (5) the director of contact monolayer on the boundary of the hydrophobic surface spot, r = r

_{f}, equals n

_{u}(r

_{f}) = −r

_{f}/h; (6) away from the bulges; the distance between the neutral surfaces of contact monolayers equals 2Z

_{0}≈ 6 nm, which is determined from the condition of equilibrium of the planar membranes brought into proximity by hydrostatic pressure P

_{app}≈ 10 Pa; (7) distance between the neutral surfaces of contact monolayers at the boundary of the hydrophobic spot (r = r

_{f}) equals 2d. These conditions allow the determination of some of the coefficients; other free constants are determined by minimization of elastic energy. Ultimately, we obtained the free energy of the system as a function of two parameters: hydrophobic spot radius, r

_{f}, and half-distance between the hydrophobic spots in two membranes, d. The final expression for the free energy is extremely cumbersome and is omitted here.

_{W}is the macroscopic surface tension on the boundary between water and hydrocarbon chains of the lipids; ξ

_{f}—characteristic length of hydrophobic interactions. The energy associated with the hydration repulsion between the membranes is found according to the expression:

_{h}—characteristic length of hydration interaction; P

_{0}is the wedging pressure determining the amplitude of hydration repulsion [52,53]; integration is performed over the hydrophilic surface of the contact monolayers. In order to qualitatively define the value of the integral in Equation (15), we use the Derjaguin–Landay–Verwey–Overbeek theory, according to which integration in Equation (15) can be restricted to the area, in which the distance between the membranes is increased by a value smaller than or equal to ξ

_{h}, having replaced the deformed hydrophilic surfaces of the contact monolayers on the horizontal planes.

_{B}T [43]; tilt modulus (per monolayer) K

_{t}= 40 mN/m [47,54]; lateral stretch/compression modulus (per monolayer) K

_{a}= 100 mN/m [43]; saddle-like bending modulus (per monolayer) K

_{G}= −0.5B ≈ −4 k

_{B}T [55]; monolayer thickness h = 2 nm [43]; characteristic length of hydrophobic interactions ξ

_{f}= 1 nm [51]; surface tension of water/lipid hydrocarbon chains macroscopic boundary σ

_{W}= 40 mN/m [17,54]; characteristic length of hydration repulsion ξ

_{h}= 0.35 nm [53]; wedging pressure P

_{0}= 6⋅10

^{8}Pa [53], lateral tension of the membrane 2σ

_{0}= 1.5 mN/m. Spontaneous curvature of monolayers was calculated as the spontaneous curvature of lipid components averaged with the weighing factors proportional to their molar concentrations. As was demonstrated earlier [56], additivity of the spontaneous curvature can be violated if saturated lipids are combined with cholesterol. We used lipid components with identical unsaturated (oleic) hydrocarbon chains; therefore there is no reason to consider the spontaneous curvature non-additive. It was assumed that at neutral pH the spontaneous curvature of DOPC J

_{DOPC}= −0.091 nm

^{−1}[57]; for DOPE J

_{DOPE}= −0.399 nm

^{−1}[57]; and for DOPC:DOPE = 5:1 mixture—J

_{0}= 5/6J

_{DOPC}+ 1/6J

_{DOPE}= −0.142 nm

^{−1}. We also assumed that change of pH from 7.5 to 4.1 results in a change of the bending modulus and spontaneous curvature of a monolayer by 20%, i.e., they become equal to 1.2B and 1.2J

_{0}, respectively.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BLM | Bilayer Lipid Membrane |

DOPC | Dioleoylphosphatidylcholine |

DOPE | Dioleoylphosphatidylethanolamine |

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**Figure 1.**Schematics of a possible trajectory of the membrane fusion process. The membranes are shown in gray, the water volumes surrounded by them in blue and yellow. (

**a**) Convergence of membranes with formation of tight contact. The contact location is designated by the red rectangle. (

**b**) Stalk, a structure, in which the contact monolayers of membranes have already fused, while the distal ones have not yet done so. (

**c**) Hemifusion diaphragm: during stalk expansion, lipids of the distal monolayers are brought into contact, thus forming a bilayer. (

**d**) During diaphragm poration, the originally isolated aqueous compartments start mixing. (

**e**) Fusion pore.

**Figure 2.**Dependence of the average waiting time for monolayer fusion of model lipid membranes upon pH of the solution. Each point represents a value averaged over 10 measurements. For points at pH = 4.1 and 4.9 the error bars are directed inside the representing circles, as the statistical error for these points is smaller than the size of the circle; thus, the circles appear as gray (partially filled).

**Figure 3.**Top—Bilayer lipid membranes thermally fluctuate around the equilibrium distance between the monolayers, determined by the applied hydrostatic pressure and hydration repulsion. On the fluctuation-induced bulges facing each other (surrounded by red rectangle), lipids can displace laterally with the formation of hydrophobic spots (shown in yellow) at the expense of the hydration-induced repulsion.

**Figure 4.**Contour plot (equal energy lines) of W(r

_{f}, d), calculated for dioleoylphosphatidylcholine (DOPC):dioleoylphosphatidylethanolamine (DOPE) = 5:1 membranes at pH = 7.5 (

**a**,

**c**) and at pH = 4.1 (

**b**,

**d**). For plots (

**a**,

**b**) blue color corresponds to 0, red—to 120 k

_{B}T; the distance between the isolines is 2 k

_{B}T. Red circles outline the saddle points of the energy surface, defining the heights of the energy barriers along the optimal trajectories of the fusion process (shown by yellow lines). The energy in the saddle points amounts to the following values: at pH = 7.5 (

**a**)—39.5 k

_{B}T; at pH = 4.1 (

**b**)—36 k

_{B}T. The plot (

**c**) presents the zoomed vicinity of the saddle point of the plot (

**a**) (pH = 7.5); the plot (

**d**)—vicinity of the saddle point of the plot (

**b**) (pH = 4.1). For plots (

**c**,

**d**) blue color corresponds to 30 k

_{B}T, red—to 45 k

_{B}T.

**Figure 5.**Schematic representation of the experimental cell used for investigation of the model bilayer lipid membranes. G is the generator; A1 and A2 are operational amplifiers. Green arrows represent the directions of the movement of the cell parts.

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**MDPI and ACS Style**

Akimov, S.A.; Polynkin, M.A.; Jiménez-Munguía, I.; Pavlov, K.V.; Batishchev, O.V.
Phosphatidylcholine Membrane Fusion Is pH-Dependent. *Int. J. Mol. Sci.* **2018**, *19*, 1358.
https://doi.org/10.3390/ijms19051358

**AMA Style**

Akimov SA, Polynkin MA, Jiménez-Munguía I, Pavlov KV, Batishchev OV.
Phosphatidylcholine Membrane Fusion Is pH-Dependent. *International Journal of Molecular Sciences*. 2018; 19(5):1358.
https://doi.org/10.3390/ijms19051358

**Chicago/Turabian Style**

Akimov, Sergey A., Michael A. Polynkin, Irene Jiménez-Munguía, Konstantin V. Pavlov, and Oleg V. Batishchev.
2018. "Phosphatidylcholine Membrane Fusion Is pH-Dependent" *International Journal of Molecular Sciences* 19, no. 5: 1358.
https://doi.org/10.3390/ijms19051358