Delayed nucleation in lipid particles
Guy Jacoby, Irina Portnaya, Dganit Danino, Haim Diamant, Roy Beck
DD ELAYED NUCLEATION IN LIPID PARTICLES
A P
REPRINT
Guy Jacoby
Department of Condensed Matter,The Raymond and Beverly SacklerSchool of Physics and Astronomy,Tel-Aviv University,Ramat Aviv, Tel Aviv 6997801, Israel [email protected]
Irina Portnaya
Department of Chemical EngineeringTechnion-Israel Institute of TechnologyHaifa 3200003, Israel
Dganit Danino
Department of Chemical EngineeringTechnion-Israel Institute of TechnologyHaifa 3200003, Israel
Haim Diamant
The Raymond and Beverly School of ChemistryTel-Aviv University,Ramat Aviv, Tel Aviv 6997801, Israel
Roy Beck
Department of Condensed Matter,The Raymond and Beverly SacklerSchool of Physics and Astronomy,Tel-Aviv University,Ramat Aviv, Tel Aviv 6997801, Israel [email protected]
September 17, 2019 A BSTRACT
Metastable states in first-order phase-transitions have been traditionally described by classical nucle-ation theory (CNT). However, recently an increasing number of systems displaying such a transitionhave not been successfully modelled by CNT. The delayed crystallization of phospholipids uponsuper-cooling is an interesting case, since the extended timescales allow access into the dynamics.Herein, we demonstrate the controllable behavior of the long-lived metastable liquid-crystalline phaseof dilauroyl-phosphatidylethanolamine (DLPE), arranged in multi-lamellar vesicles, and the ensuingcooperative transition to the crystalline state. Experimentally, we find that the delay in crystallizationis a bulk phenomenon, which is tunable and can be manipulated to span two orders of magnitude intime by changing the quenching temperature, solution salinity, or adding a secondary phospholipid.Our results reveal the robust persistence of the metastability, and showcase the apparent deviationfrom CNT. This distinctive suppression of the transition may be explained by the resistance of themulti-lamellar vesicle to deformations caused by nucleated crystalline domains. Since phospholipidsare used as a platform for drug-delivery, a programmable design of cargo hold and release can be ofgreat benefit.
Classical nucleation theory (CNT), although decades old, is still the prevalent theory for many systems undergoingfirst-order phase-transitions. Only a few key assumptions are needed for CNT: dominant short range interactions,smooth interfaces (capillarity approximation), and that the nucleus has similar properties to the final bulk phase [1].The dynamics are then described as a single stochastic excitation process that governs the transition. Mineral crystal a r X i v : . [ c ond - m a t . s o f t ] S e p PREPRINT - S
EPTEMBER
17, 2019formation [2, 3, 4], virus capsid assembly [5], and protein nucleation [6, 7] are examples of phase transition dynamicsthat can be successfully modeled by CNT. Despite its simplifying assumptions and basic description of the interactions,when applicable, CNT can properly capture the quantitative features of the nucleation process of many researchedsystems.However, there is an increasing number of systems displaying nucleation processes that do not conform to this classicalpicture. Complex dynamics can arise due to intermediate states leading to multi-step nucleation [8, 9, 10], or long-rangeinteractions that can result in macroscopic nucleation [11] or a cooperative delayed transition [12]. Such dynamicsmay require an extension of the classical theory or in some cases a comprehensive revision [13]. Examples of suchcomplex dynamics can be found in self-assembled amphiphilic systems, which display long-lived metastable phasesupon temperature change.Amphiphiles are molecules that contain hydrophilic and hydrophobic chemical groups. The key characteristic associatedwith amphiphilic molecules is their ability to spontaneously self-assemble into macro-molecular structures [14].Biological amphiphiles, such as lipids, self-assemble into a wide variety of mesophases. Most notably, lipids constitutethe membranes of cells and organelles, and are involved in many important biological functions. Alongside basicresearch into their physical and biochemical properties, self-assembly is also utilized for designing modern biomedicalapplications such as drug delivery [15].In particular, the phospholipid amphiphiles have several predominant lamellar phases, such as the disordered liquid-crystalline phase ( L α ), gel phase ( L β ) and ordered crystalline phase ( L c ), which differ by their degree of spatialsymmetry. Transitions between these phases can be induced by changing the temperature, but the pathways dependon the physicochemical properties of the molecules and their thermal history. Previously, the phospholipid DLPEwas shown to have long-lived metastable phases with lifetimes on the order of hours or even days [16, 17, 18]. Thesetime-scales are orders of magnitude longer than the rapid transitions as in the case of melting. The reports were basedmostly on X-ray scattering and differential scanning calorimetry (DSC), both very useful techniques for phase detectionand characterization. However, they were performed as static measurements at different points in time, separated bylong periods of unrecorded incubation. These limited observations only allowed for qualitative descriptions of thedynamics.Herein, we present our experimental investigation of the metastable L α to L c phase-transition using time-resolvedsolution X-ray scattering (SXS) and DSC measurements. We demonstrate the cooperative and controllable behaviorof the transition dynamics. We highlight and discuss the deviations from CNT, and rationalize them based on thefree-energy cost of deforming the L α vesicles by crystal nucleation. ◦ C for 3 hours, andhomogenized using a vortexer every 25 minutes. Samples were then placed in quartz capillaries, containing about 100 µl , and centrifuged for 5 minutes at 3000 rpm, to create a pellet of lipids. Samples at 30 mg/ml lipid concentration were measured in 1.5 mm diameter sealed quartz capillaries. Measurementswere performed using an in-house solution X-Ray scattering system, with a GeniX (Xenocs) low divergence
Cu K α radiation source (wave length of 1.54 Å) and a scatter-less slits setup [19]. Two-dimensional scattering data with a q range of . − Å − at a sample-to-detector distance of about 230 mm were collected on a Pilatus 300K detector(Dectris), and radially integrated using MATLAB (MathWorks) based procedures (SAXSi). Background scattering datawas collected from buffer solution alone. The background-subtracted scattering correlation peaks were fitted using aGaussian with a linearly sloped baseline. For each sample, time-resolved correlation peaks position, intensity and widthwere extracted. 2 PREPRINT - S
EPTEMBER
17, 2019
DSC experiments were performed using a VP-DSC micro calorimeter (MicroCal Inc., Northampton, MA). Calorimetricdata analysis was done with the Origin 7.0 software. Degassed systems of pure DLPE and 90:10 DLPE:DLPG (mole%) at a total concentration of 30 mg/ml were placed in the sample cell (0.5 ml), and MES buffer (20 mM MES + 130mM NaCl at pH 6.7) in the reference cell. DSC thermograms were recorded during multiple heating-cooling cycles atvarious scan rates and different pre- and post-scan periods. First, each sample was heated from 25 ◦ C to 60 ◦ C with at arate of 90 ◦ C/hour and 15 minute pre- and 3 hour post-scan periods. Then the sample was cooled from 60 ◦ C to 37 ◦ Cwith a scan rate of 90 ◦ C/hour, and identical 15 minute pre- and post-scan periods. The additional scans (presented inthe text) were carried out at a very slow rate of 0.1 ◦ C/hour for heating and 0.43 ◦ C/hour for cooling, with 1 minutepre- and post-scan periods. In this manner, we were using the DSC in a quasi-isothermal mode, where we just wait forthe metastability to end in an exothermic transition to the crystalline state. Additionally, on the same instrument DSCmeasurements of 10-fold diluted samples (3 mg/ml) of pure DLPE, and 97:3; 95:5; 92:8; 90:10, 85:15 DLPE:DLPG(mole %) were performed, to measure the transition temperature and transition enthalpy. The buffer composition wasthe same. DSC thermograms were recorded during double heating-cooling cycles (between 25 ◦ C and 60 ◦ C) with ascan rate of 90 ◦ C/hour, and identical 15 min pre- and post-scan periods. Calorimetric data analysis was done with theOrigin 7.0.
A fresh sample of DLPE in solution is found in the low energy L c phase at temperatures below 43 ◦ C. When heatedabove 43 ◦ C, the molecules undergo a melting transition T c → α to the L α phase. Seddon et al. showed that if the samesample is then cooled to a temperature T Q below T c → α there are two possible pathways back to the equilibrium L c phase [16]. If T Q < T β → α (30 ◦ C), the gel-to-liquid crystalline transition, the system will rapidly transition to the L β phase, which will subsequently become metastable until returning to equilibrium. However, if T β → α < T Q < T c → α ,the L α phase will become metastable and directly transition to the L c phase.Here, the metastable L α to L c phase-transition was recorded by time-resolved SXS. In the experiments, the X-raybeam illuminating the sample has a cross sectional area of approximately 0.64 mm , which produces a bulk-averagedscattering signal. The scattering from a sample that has not been pre-heated is first recorded at T Q ( T β → α < T Q EPTEMBER 17, 2019 (a) (b) Figure 1: (a) Time-resolved scattering spectra of lipid system containing 93:7 DLPE:DSPG (mole %). Initially, at T Q = 37 ◦ C (black spectrum), the crystalline phase is indicated by the presence of wide-angle correlation peaks. Whenheated to 60 ◦ C (red spectra) these peaks disappear with the loss of in-plane order. After cooling back to T Q (bluespectra), the L α phase remains metastable for τ = 47 hours, and then phase-transitions (green spectra) back into thecrystalline phase (black spectra). (b) Scattering intensity of lamellar correlation peak (001) and mixed wide-anglepeak (206) used to measure the order parameters and extract the temporal features of the dynamics. Inset: schematicillustration of the lipid conformation in both the liquid-crystalline ( L α ) and the crystalline ( L c ) phases.which are orders of magnitude larger than the melting transition times. Moreover, in all cases there is no detectablescattering at wide-angle prior to the transition, which implies a collective bulk transition rather than stochastic events ofcrystallization within the macroscopic illuminated area.We set out to explore the different system parameters that can affect the dynamics of the transition. By changing systemparameters such as the lipid stoichiometry and chemical structure, salinity and the quenching temperature ( T Q ) wefound that we were able to manipulate the metastability in a pre-determined and controllable fashion.In previous studies, the metastability was examined in samples of pure DLPE. However, we found that the addition ofa secondary phospholipid not only preserves the metastable phase, but also extends its lifetime (Fig. 2). Moreover,the delay time is sensitive to changes in the hydrocarbon chain length and headgroup. Phosphatidylglycerols (PGs)were chosen as a secondary charged lipid due to the stabilizing effect they have on PE bilayers [20], and specifically,DLPG was previously used along with DLPE as the building blocks for a drug delivery system [21, 22, 23]. PGs with12 (DLPG), 14 (DMPG), 16 (DPPG) or 18 (DSPG) carbons in their saturated hydrocarbon chains were chosen as chainlength variants. In addition, the zwitterionic dilauroyl-phosphatidylcholine (DLPC) was chosen as a headgroup variant.The delay time seems to increase as a function of chain length for the PGs, and it is greatly increased when the PGheadgroup is swapped with a PC.Since electrostatics are known to have a central role in stabilizing lipid lamellar systems, and the delay time is observedto increase with the fraction of charged PG lipids in our system, we tested the effect of the solution salt concentrationon the delay time of samples with different DLPE:DLPG ratios. As shown in Fig. 3, changing the average membranecharge density produces two features in the salt dependence: (a) there seems to be a minimum of the delay time atapproximately 150 mM, splitting the dependency into two regimes, and (b) the delay time increases with the fraction ofDLPG at a given salt concentration.We propose that these two regimes originate from two different phenomena. At low salt concentrations (<150 mM), thedecrease in delay time towards the minimum can be attributed to the decrease in the electrostatic screening length. At150 mM the electrostatic screening length is comparable to the DLPG headgroup diameter ( ≈ Å) [24]. Segregation ofnon-DLPE lipids, which is essential for recovering the homogeneous DLPE crystals, is facilitated by the screening ofinteracting charged PG headgroups. On the contrary, high salt concentrations (>300 mM) can lead to adsorption of ionson the charged membrane. This can lead to an increase in τ , since ions must evacuate from between the lamellae, yetion transport across membranes is unfavored. The adsorption of charges can result in an increase in the membrane’sbending rigidity, which in turn can strengthen the metastability (see Sec. 4.2). The results show that the delay time of4 PREPRINT - S EPTEMBER 17, 2019Figure 2: Delay time ( τ ) as a function of the molar fraction (mole %) of the secondary lipid, labelled in the figure, i.e. (100 - DLPE) (mole %) . An increase in chain length results in an increase in delay time at a lower fraction.Dashed lines represent the minimal delay time for samples that did not transition within 180 hours. Samples withhigher concentrations of DPPG and DSPG were measured but omitted from the results due to an alteration of the finalcrystalline form.Figure 3: Delay time as a function of monovalent salt concentration, at different DLPE:DLPG ratios (mole %). At lowsalt concentrations (< 150 mM), samples that contain DLPG (black and blue) show an increase in the delay time assalt concentration is decreased. At high salt concentrations (>300 mM) the opposite trend occurs. However, when thesample contains pure DLPE, there is no significant effect of the salt concentration on the delay time.5 PREPRINT - S EPTEMBER 17, 2019samples containing charged headgroups responded to changes in salt concentration, yet no significant dependence wasobserved in samples containing only DLPE. We would like to accentuate the extended lifetime of the metastable phaseat the highest concentration measured (500 mM), which exceeded 500 hours in the case of 90:10 DLPE:DLPG (mole%).The lifetime of the metastable phase depends on the strength of the thermodynamic force driving the transition. Closeto the transition temperature the energy barrier that the system must overcome is high, which results in a small rate ofnucleation. As the temperature is lowered, the barrier becomes smaller and the rate increases. This is demonstrated hereby the increasing persistence of the metastable L α phase, closer to T c → α . Figure 4 shows an exponential increase in theaverage delay time as a function of the quenching temperature, T Q . Remarkably, for most quenching temperatures thespread of experimental results is very small, further supporting our claim that τ is an intrinsic property of the system’sdynamics set by its macroscopic parameters. However, due to larger fluctuations closer to the critical temperature( T c → α ) we notice a large spread of τ at T Q = 41 ◦ C. There, two samples did not transition within the duration ofthe experiment (800 hours). In addition, at T Q = 31 ◦ C the L α phase rapidly transitioned to L β , which then becamemetastable. Evidently, the delay time for the L α phase at T Q = 32 ◦ C is shorter than at T Q = 31 ◦ C for the L β gelphase, as the liquid phase is expected to be more labile [25].Figure 4: Delay time as a function of quenching temperature. Empty squares represent measurements of individualcapillaries at each quenching temperature. Solid red curve is the fit using Eq. (2), excluding the measurements at T Q = 31 ◦ C (empty stars represent the L α → L β transition) and T Q = 41 ◦ C due to the high variance caused byfluctuations. Red vertical dashed-line indicates the estimated T c → α transition temperature.The structural study of the delayed nucleation phenomenon, using time-resolved SXS, does not directly report onthe thermodynamic processes. To address this, calorimetric measurements are commonly used to investigate thethermodynamics of lipid systems, mostly in the form of differential scanning calorimetry (DSC). However, since weare investigating a time-delayed transition at a fixed temperature, we employed DSC in a non-trivial quasi-isothermalmanner. After samples were incubated at 60 ◦ C for 3 hours, the temperature was lowered to T Q = 37 ◦ C, and thesamples were scanned back-and-forth between 36 and 37 ◦ C at a very slow rate (0.1 ◦ C/hour on heating, 0.43 ◦ C/houron cooling).In Fig. 5 we compare the quasi-isothermal energy flux measurements performed on a sample of pure DLPE and asample containing 90:10 DLPE:DLPG (mole %). The results in both cases show an exothermic signature, as expectedfor the transition back to the low-energy crystalline phase, with similar timescales to those in our X-ray scattering6 PREPRINT - S EPTEMBER 17, 2019 (a) (b) Figure 5: Time-resolved quasi-isothermal DSC measurements of two samples: (a) Pure DLPE, (b) 90:10 DLPE:DLPG(mole %). The exothermic peak seen in both occurs at similar times as in the scattering experiments. The red curves(shown in both) are heating scans and the blue curves shown only in (a) are cooling scans. There is a mismatch betweencooling and heating scan data possibly due to the different scanning rates (see experimental section), thus the coolingscans were shifted in (a) and omitted from (b), for clarity. Raw data are shown in supplementary information.measurements (Fig. 2). The sample with pure DLPE shows a broad and slow change in the excess heat capacity, peakingat 28 hours, while the mixed sample shows a much narrower peak, centered at 55 hours. The corresponding averagedelay times in the X-ray scattering experiments are 20 and 45 hours, respectively. In addition, a DSC measurementwas performed to determine the enthalpy of transition from the L c to the L α phase for a pure sample of DLPE, whichyielded h c = 11 . kcal/mole.In the process of sample measurement and analysis of the phase-transition dynamics, the extracted temporal parametersrepresent the transformation occurring in the illuminated volume. However, this volume includes only a portion of thelipid pellet at the bottom of the capillary. If one follows a nucleation and growth framework, it is important to assesswhether the transformation initiates concurrently throughout the sample or propagates successively from a startingpoint. To test this, we prepared a sample with a large pellet at the bottom of the capillary ( ≈ mm ), and measured thedelay time at different locations along the vertically held capillary. The capillary size and experimental protocol (SXS)remained as previously described. Figure 6a shows the delay time as a function of the spatial coordinate along thecapillary. The transition seems to propagate outward from a certain location, with neighboring locations transitioning atlater times.A similar experiment was performed on a horizontally held capillary. There, the transition began at the water-pelletinterface and propagated at a steady velocity of approximately 100 µ m/hour towards the end of the pellet (Fig. 6b). Inboth experiments a new feature in the scattering spectra could be observed: a coordinated increase in lamellar scatteringover a period of time prior to the transition (Fig. 6c, d). We denote the beginning of this period by τ B , the point in timefrom which a slow increase in scattering culminates in a sharp drop of intensity. Only after the drop in the lamellarscattering intensity is there a detectable change in wide-angle scattering. Therefore, the metastable state remains duringthe build-up period. Surprisingly, this structural reorganization is coordinated over several millimeters in the sample(Fig. 6c, d).Metastable phases are often very sensitive to energy fluctuations, as even minute inputs of energy can result in atransition to the stable phase. Since the lipid metastable phase is stable against various changes in system parameters,we tested its stability against external inputs of energy by subjecting lipid dispersions to mechanical agitation in theform of rigorous pipetting. A lipid dispersion of approx. 1.5 ml, at 30 mg/ml, was prepared as a bulk dispersion fromwhich samples would be pipetted out, and measured intermittently. It was incubated at 37 ◦ C for one hour, followedby 3 hours at 60 ◦ C, as performed regularly with the SXS samples. The incubator was then set to T Q = 37 ◦ C and asample was drawn from the bulk dispersion after t = 1, 2, 3, 5.5 and 19.5 hours by pipetting out approximately 100 µl and placing into a capillary. The capillary was then placed in the SXS temperature chamber, pre-heated to 37 ◦ C,and measured after ∆ t minutes (Fig. 7). The control sample, taken from the bulk dispersion before it was placed inthe incubator, underwent the regular temperature procedure in the SXS temperature chamber, and transitioned after7 PREPRINT - S EPTEMBER 17, 2019 (a) (b)(c) (d) Figure 6: The delay time occurs at different times for different locations, but structural changes to the lipid particlesare coordinated over millimeters. (a) Delay time vs. spatial coordinate in vertically held capillary. (b) τ vs. spatialcoordinate in horizontally held capillary. (c) Time-resolved scattering intensity of (001) as a function of time. Error barsin (a) and (b) represent τ ∗ , the duration of the transition. Numeric tags in (c) and (d) correspond to spatial coordinate in(a) and (b), respectively. Inset in (d) shows τ B , the beginning of the build up period.8 PREPRINT - S EPTEMBER 17, 2019Figure 7: The metastable state’s lifetime is significantly shortened by applying mechanical agitation in the form ofrigorous pipetting. From left column to right, samples that were taken from a bulk reservoir t = 1, 3, 5.5 and 19.5 hoursafter temperature quenching from T = 60 ◦ C to T Q = 37 ◦ C. Each sample is subjected to pipetting, and measuredimmediately after extraction and ∆ t minutes afterwards. Blue background indicates the sample is still in the metastable L α phase, and green that the sample is mid-transition. τ = 19 . hours. The samples from the bulk dispersion, taken during first few hours, transitioned approximately an hourafter being pipetted out. The sample taken after 5.5 hours was in the middle of transitioning when measured initially( ∆ t = 0 min ), and the 19.5 hour sample had already transitioned (Fig. 7). This experiment demonstrates that thelifetime of the metastable phase is significantly shortened by mechanical agitation applied after thermal incubation.Lastly, phospholipids are utilized in bio-medicine as building blocks of vesicles designed for specific targeting andcontrolled release. When designing such drug-delivery systems, it is crucial to assert the stability of the carrier withits cargo. Since DLPE and DLPG have been used as the lipid components of such systems [23, 21, 22], we tested thestability of the metastable phase in the presence of cargo. Figure 8 shows the delay time of samples containing 90:10DLPE:DLPG (mole %) and the hydrophobic drug Prednisolone; an established and commercially available steroidused to treat a wide range of conditions and illnesses. The results show that the addition of the drug had a large impacton the delay time, shortening it by almost an order-of-magnitude. Since lipid systems continue to serve as appealingingredients in drug delivery systems, controlled delayed nucleation may serve as a novel designing factor to depositcargo in a predetermined timing. Nonetheless, the effect of cargo on transition dynamics should not be overlookedwhen designing such systems. The findings presented in this work are in clear contrast to those expected from a system described by CNT. Instead of asingle stochastic process, which would produce a single timescale for the transition, we present multiple experimentalevidence of coordinated delayed nucleation and multiple timescales ( τ , τ ∗ and τ B ), representing the complexity of thedynamics. These timescales are orders of magnitude larger than the typical microscopic timescales associated with lipidsystems. To emphasize the separation of timescales and highlight the collective behavior of the transition, we re-scale9 PREPRINT - S EPTEMBER 17, 2019Figure 8: The delay time decreases in the presence of prednisolone, a commercial hydrophobic drug, but the delay intransition still persists.our entire data set of lamellar-scattering peak intensities by the delay time τ (Fig. 9). The time-dependent intensitiescollapse onto roughly the same sigmoidal shape, with slight variations in width representing the variations in τ ∗ .We present additional findings that we would like to discuss in the context of deviations from CNT. Firstly, it isimportant to mention that at no point in the preparation of the samples was there any effort to homogenise theparticles’ sizes. And yet, despite this heterogeneity, the delay time was shown to be reproducible with a peak in theprobability at a non-zero value [18]. Secondly, our results shown in Fig. 6 demonstrate that the transition occursat different times at different locations, yet the onset of the structural change in samples with large pellets, markedby τ B , is macroscopically coordinated over millimeters. This length-scale is orders of magnitude larger than anymicroscopic length-scale associated with lipid self-assembly. Lastly, in heterogeneous nucleation impurities areconsidered preferential nucleation sites due to a lower surface energy penalty compared to the homogeneous case. Inour system, the inclusion of a secondary lipid only served to hinder crystallization.During the incubation time at the high temperature ( ≥ ◦ C) water molecules and ions enter in between the membranes[18]. Concurrently, secondary lipid molecules enter the liquid membrane and disrupt its homogeneity. Upon cooling,the water molecules and ions must evacuate, and lipid segregation must occur to re-form the network of connections asin the initial homogeneous crystal. Here, we demonstrated that the persistence of the metastable phase is sensitive to theproperties of the secondary lipid. Not only does the metastability with its features persist, but the chemical structure ofthe lipid has a large impact on the change of the delay time (Fig. 2).A recent study experimentally showed long-range interlayer alignment of phase-separated intralayer domains, acrosshundreds of lamellae in multi-component supported lipid membranes [26]. A follow-up study proposed a theoreticalexplanation to the interlayer correlation between phase-separated domains [27]. Using a model of stacked 2D Isingspins to represent the stacked lipid membranes, they showed that the system forms a continuous columnar structure inequilibrium, for any finite interaction across adjacent layers. Such an interlayer interaction should be a key componentin cooperative nucleation in MLVs, for which a mechanism is proposed below. We would like now to examine a possible mechanism for the exceptional metastability of the L α phase and the strongcooperativity of the transition.Because of the rigidity of the crystalline phase, the formation of a crystalline domain in a membrane flattens that region,which affects adjacent membranes and thus locally deforms the MLV. The free-energy penalty per unit area due tothe deformation is proportional to the effective surface tension γ = √ BK of the MLV, arising from its compression( B ) and bending ( K ) moduli [28]. This penalty makes the free-energy of the crystalline phase effectively higher,shifting the transition from T (0)c for an isolated membrane to a lower temperature T c for a membrane in a curvedMLV. The change in the free-energy per unit area between the two phases is ∆ g ( T c ) (cid:39) γ/ . Using the relation ∆ g (cid:39) ( h c /a )( T (0)c − T ) /T (0)c , the temperature shift can be estimated from the measured enthalpy of transition10 PREPRINT - S EPTEMBER 17, 2019Figure 9: The lamellar correlation peak intensity over time can be re-scaled by τ to highlight the cooperative nature ofthe transition, regardless of the conditions changed in the experiments. The residual variation in the time-dependentscattering curves is due to τ ∗ . The data set presented here consists of 85 different experiments.( h c = 11 . kcal/mole), the area per lipid ( a (cid:39) . nm ), and assuming γ larger than . mN/m. We get a decrease inthe transition temperature, proportional to γ , of more than 10 degrees.Thus, under conditions where an isolated membrane would crystallize, a single membrane in the MLV would not. Onthe other hand, if all the membranes in the MLV were to crystallize, the total free-energy would inevitably decrease.Hence, the MLV must ultimately crystallize, but it can do so only through a multi-membrane cooperative process. Thiscooperativity is essential for departing from a single Poisson process, typical to CNT.The compression and/or bending moduli of the L α phase, and therefore also γ , increase with increasing lipid chainlength and increasing membrane charge. Such modifications, by increasing γ , should deepen the metastability. This isconsistent with the experimental observations reported here.Let us consider the collective metastability described above in slightly more detail. Since the deformation of themembrane stack is localized [28], its free-energy penalty is intensive in the number of membranes, whereas the free-energy gain due to crystallization is extensive. Hence, there is a critical number of membranes, n c ∼ γ/ (∆ g ) , beyondwhich the suppression of the transition is overcome. The multi-membrane critical nucleus has the size R c ∼ λn c , where λ is the MLV’s periodicity. It corresponds to a multi-membrane nucleation barrier, F c ∼ σR ∼ σγ λ / (∆ g ) , where σ is an effective surface tension; a combination of γ and the line tension of intra-membrane crystalline domains.To sum up this analysis, we obtain the multi-membrane nucleation barrier as F c ∼ ¯ σ λ a h (cid:18) T c T c − T (cid:19) , (1)where ¯ σ = ( σγ ) / is an effective surface tension. This implies a delay time of the form τ = τ e F c / ( k B T ) = τ e b/ ( T c − T ) , (2)where the coefficient b is extracted from Eq. (1), and τ will depend on faster effects not addressed here. Using theexperimental values of λ , a , h c , and T c , and taking ¯ σ ∼ mN/m, we get b of order K . This value is sensitive to thevalue of ¯ σ and should be regarded just as a consistency check.Figure 4 shows the fit of the function in Eq. (2) to the data, where T c was fixed to 43 ◦ C. The fitting parameter b yieldsa value of ≈ K , consistent with the qualitative estimate shown above. Changing T c to be 42 or 44 ◦ C changes b tobe approx. 8 or 44, respectively. A detailed theory along the lines presented in this section will be presented elsewhere. Our results herein show a deterministic and controllable behavior of the lipid metastable phase. Our experimentalresults demonstrate the sensitivity of the delay time to a variety of system parameters. Furthermore, the long timescales11 PREPRINT - S EPTEMBER 17, 2019and robustness against changes allow one to access the complex dynamics, which could otherwise be a difficult obstacleto overcome. Finally, the proposed mechanism is consistent with the experimental results and might account for thecooperative nature of the transition. A better understanding of the mechanism for delayed nucleation can benefit boththe fundamental physics of nucleation processes and applications harnessing it for releasing cargo in a timely manner. References [1] J. J. De Yoreo. Principles of Crystal Nucleation and Growth. Reviews in Mineralogy and Geochemistry , 54(1):57–93, jan 2003.[2] Jens Baumgartner, Archan Dey, Paul H. H. Bomans, Cécile Le Coadou, Peter Fratzl, Nico A. J. M. Sommerdijk,and Damien Faivre. Nucleation and growth of magnetite from solution. Nature Materials , 12(4):310–314, apr2013.[3] Anthony J. Giuffre, Laura M. Hamm, Nizhou Han, James J. De Yoreo, and Patricia M. Dove. Polysaccharidechemistry regulates kinetics of calcite nucleation through competition of interfacial energies. Proceedings of theNational Academy of Sciences , 110(23):9261–9266, jun 2013.[4] L. M. Hamm, A. J. Giuffre, N. Han, J. Tao, D. Wang, J. J. De Yoreo, and P. M. Dove. Reconciling disparateviews of template-directed nucleation through measurement of calcite nucleation kinetics and binding energies. Proceedings of the National Academy of Sciences , 111(4):1304–1309, jan 2014.[5] Roya Zandi, Paul van der Schoot, David Reguera, Willem Kegel, and Howard Reiss. Classical Nucleation Theoryof Virus Capsids. Biophysical Journal , 90(6):1939–1948, mar 2006.[6] Sathish V. Akella, Aaron Mowitz, Michael Heymann, and Seth Fraden. Emulsion-Based Technique To MeasureProtein Crystal Nucleation Rates of Lysozyme. Crystal Growth & Design , 14(9):4487–4509, sep 2014.[7] Mike Sleutel, Jim Lutsko, Alexander E.S. Van Driessche, Miguel A. Durán-Olivencia, and Dominique Maes.Observing classical nucleation theory at work by monitoring phase transitions with molecular precision. NatureCommunications , 5(1):5598, dec 2014.[8] Mike Sleutel and Alexander E. S. Van Driessche. Role of clusters in nonclassical nucleation and growth of proteincrystals. Proceedings of the National Academy of Sciences , 111(5):E546–E553, feb 2014.[9] James J. De Yoreo, P. U. P. A. Gilbert, N. A. J. M. Sommerdijk, R. Lee Penn, Stephen Whitelam, Derk Joester,Hengzhong Zhang, Jeffrey D. Rimer, Alexandra Navrotsky, Jillian F. Banfield, Adam F. Wallace, F. Marc Michel,Fiona C. Meldrum, H. Colfen, and Patricia M. Dove. Crystallization by particle attachment in synthetic, biogenic,and geologic environments. Science , 349(6247):aaa6760–aaa6760, jul 2015.[10] N. Duane Loh, Soumyo Sen, Michel Bosman, Shu Fen Tan, Jun Zhong, Christian A. Nijhuis, Petr Král, PaulMatsudaira, and Utkur Mirsaidov. Multistep nucleation of nanocrystals in aqueous solution. Nature Chemistry ,9(1):77–82, 2017.[11] Masamichi Nishino, Cristian Enachescu, Seiji Miyashita, Per Arne Rikvold, Kamel Boukheddaden, and FranaçoisVarret. Macroscopic nucleation phenomena in continuum media with long-range interactions. Scientific Reports ,1:1–5, 2011.[12] D. A. Neumann, D. B. McWhan, P. Littlewood, G. Aeppli, J. P. Remeika, and R. G. Maines. Nucleation near thetricritical point of BaTiO3. Physical Review B , 32(3):1866–1868, aug 1985.[13] P. Chandra. Nucleation in the presence of long-range interactions. Physical Review A , 39(7):3672–3681, apr 1989.[14] Jacob Israelachvili. Intermolecular and Surface Forces . Elsevier, 2011.[15] Bhushan S. Pattni, Vladimir V. Chupin, and Vladimir P. Torchilin. New Developments in Liposomal DrugDelivery. Chemical Reviews , 115(19):10938–10966, 2015.[16] J. M. Seddon, K. Harlos, and D. Marsh. Metastability and polymorphism in the gel and fluid bilayer phasesof dilauroylphosphatidylethanolamine. Two crystalline forms in excess water. Journal of Biological Chemistry ,258(6):3850–3854, mar 1983.[17] Hua Chang and Richard M. Epand. The existence of a highly ordered phase in fully hydrated dilauroylphos-phatidylethanolamine. Biochimica et Biophysica Acta (BBA) - Biomembranes , 728(3):319–324, mar 1983.[18] Guy Jacoby, Keren Cohen, Kobi Barkan, Yeshayahu Talmon, Dan Peer, and Roy Beck. Metastability in lipidbased particles exhibits temporally deterministic and controllable behavior. Scientific Reports , 5(1):9481, aug2015. 12 PREPRINT - S EPTEMBER 17, 2019[19] Youli Li, Roy Beck, Tuo Huang, Myung Chul Choi, and Morito Divinagracia. Scatterless hybrid metal–single-crystal slit for small-angle X-ray scattering and high-resolution X-ray diffraction. Journal of Applied Crystallog-raphy , 41(6):1134–1139, dec 2008.[20] Ana Tari and Leaf Huang. Structure and function relationship of phosphatidylglycerol in the stabilization ofphosphatidylethanolamine bilayer. Biochemistry , 28(19):7708–7712, sep 1989.[21] Ilia Rivkin, Keren Cohen, Jacob Koffler, Dina Melikhov, Dan Peer, and Rimona Margalit. Paclitaxel-clusterscoated with hyaluronan as selective tumor-targeted nanovectors. Biomaterials , 31(27):7106–7114, sep 2010.[22] Gideon Bachar, Keren Cohen, Roy Hod, Raphael Feinmesser, Aviram Mizrachi, Thomas Shpitzer, Odelia Katz,and Dan Peer. Hyaluronan-grafted particle clusters loaded with Mitomycin C as selective nanovectors for primaryhead and neck cancers. Biomaterials , 32(21):4840–4848, jul 2011.[23] Keren Cohen, Rafi Emmanuel, Einat Kisin-Finfer, Doron Shabat, and Dan Peer. Modulation of Drug Resistancein Ovarian Adenocarcinoma Using Chemotherapy Entrapped in Hyaluronan-Grafted Nanoparticle Clusters. ACSNano , 8(3):2183–2195, mar 2014.[24] Jianjun Pan, Frederick A. Heberle, Stephanie Tristram-Nagle, Michelle Szymanski, Mary Koepfinger, JohnKatsaras, and Norbert Kuˇcerka. Molecular structures of fluid phase phosphatidylglycerol bilayers as determined bysmall angle neutron and X-ray scattering. Biochimica et Biophysica Acta (BBA) - Biomembranes , 1818(9):2135–2148, sep 2012.[25] Hui Xu, Frances A. Stephenson, Hai-nan Lin, and Huang Ching-hsien. Phase metastability and supercooledmetastable state of diundecanoylphosphatidylethanolamine bilayers. Biochimica et Biophysica Acta (BBA) -Biomembranes , 943(1):63–75, aug 1988.[26] Lobat Tayebi, Yicong Ma, Daryoosh Vashaee, Gang Chen, Sunil K. Sinha, and Atul N. Parikh. Long-rangeinterlayer alignment of intralayer domains in stacked lipid bilayers. Nature Materials , 11(12):1074–1080, oct2012.[27] Takuma Hoshino, Shigeyuki Komura, and David Andelman. Correlated lateral phase separations in stacks of lipidmembranes. Journal of Chemical Physics , 143(24), 2015.[28] Pierre-Gilles de Gennes and Jacques Prost.