Phase transitions in phospholipid monolayers: theory versus

Subscriber access provided by Iowa State University | Library

New Concepts at the Interface: Novel Viewpoints and Interpretations, Theory and Computations

Phase transitions in phospholipid monolayers: theory versus experiments Francisco de Oliveira, and Mário Noboru Tamashiro Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03244 • Publication Date (Web): 25 Jan 2019 Downloaded from http://pubs.acs.org on January 29, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Phase transitions in phospholipid monolayers: theory versus experiments F. O. de Oliveira∗,†,‡ and M. N. Tamashiro∗,† †Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas (UNICAMP), Rua Sérgio Buarque de Holanda, 777, Cidade Universitária, Campinas SP, 13083-859, Brazil ‡Present address: Instituto de Física, Universidade de São Paulo (USP), Rua do Matão, 1371, Cidade Universitária, São Paulo SP, 05508-090, Brazil E-mail: [email protected]; [email protected] Abstract The Doniach lattice gas (DLG) represents a ternary-mixture statistical model, whose components, water molecules (w), ordered-chain lipids (o), and disordered-chain lipids (d) — the latter carrying a high degenerescence ω 1 — are located at each site of a two-dimensional lattice. The DLG model was introduced to describe phospholipid Langmuir films at the air-water interface and can be mapped into a spin-1 model, with the single-site states si = 0, +1, −1 representing the three types of molecules in the system (w,o,d), respectively. The model allows lipid-density fluctuations and has been analyzed at the mean-field approximation, 1 as well as at the pair approximation. 2 In this work, we focus on performing an explicit comparison of the theoretical predictions obtained for the DLG model at the pair approximation with isothermal monolayer compression experiments, 3 for the two most commonly studied saturated zwitterionic phospholipids, DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) and DPPC (1,2dipalmitoyl-sn-glycero-3-phosphocholine). The model parameters obtained by fitting

1

ACS Paragon Plus Environment

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

to the experimental data yield phase diagrams that are qualitatively consistent with the observed phase transitions on DMPC and DPPC monolayers, with absence of a low-density gas phase. Quantitative agreement, however, was less significant, partially due to the challenging reproducibility of Langmuir monolayer compression experiments, claimed in the literature to be influenced by kinetic effects.

Introduction Amphiphile molecules are ubiquitous in biological systems. The cell membrane of all living organisms is made, essentially, of an amphiphile bilayer, most notably phospholipids, that possess a polar head attached to two fatty-acid (hydrocarbonic) chains, or tails. These are commonly between 14 and 22 carbons long each, but can be observed to be between 2 and 36 in plants and animals, or even 80 in some bacteria. 4 On both external sides of the membrane, the polar heads form hydrogen bonds with water molecules, while the tails from the inner and outer leaflets stay on the inside of the membrane, away from the water molecules and, thus, passively preventing a wide range of substances from crossing into or out of the cell, particularly water and ions which tend strongly to remain in the polar milieu. 5 Because the bilayer is thermodynamically stable, the cell does not need to spend much energy maintaining it, which is probably why we see this kind of structure (or analogous ones) in every living organisms we know of. It is notable that the membrane thickness is of the order of the length of the fatty-acid chains, orders of magnitude smaller than the overall membrane area, making it approximately a two-dimensional fluid. A very commonly investigated experimental system that mimetizes the cell membrane is an amphiphile Langmuir monolayer, 6,7 that is, a single layer of phospholipids or other amphiphiles that resides on the air-water interface, with the polar heads in contact with the water and hydrocarbon chains in the air. In this configuration the monolayer tends to spread, yielding very high specific areas and exerting a surface pressure if one tries to constrain it. By compressing or expanding the available area for the monolayer or, equivalently, changing 2


Page 2 of 36

Page 3 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

its temperature whilst keeping the area constant, it may go through a series of phase transitions. 6 Albeit different substances generally present different transitions, there are two that are particularly common: a low-density to a medium-density transition, referred sometimes as the gas-liquid expanded (G-LE) transition, and a medium-density to a high-density transition, referred as the liquid expanded-liquid condensed (LE-LC), or main, transition. After years of controversy about the nature of these two phase transitions, 8–11 it is currently recognized that they would be indeed discontinuous (first-order) transitions, 6,7 displaying thus an associated latent heat. Upon increased compression, a second-order transition, liquid condensed-solid crystalline (LC-SC), may be observed, which will not be considered in this work. For higher surface pressures, eventually the monolayer expands to three dimensions and may even collapse. 12 The actual bilayer formed in aqueous lipid suspensions is tensionfree and kept stable by hydrophobicity alone, while the monolayer can expand freely and is only kept stable by applied external lateral pressure, but we are able to draw significant parallels between bilayer and monolayer behavior. 13 While generally the lipid tails can be asymmetric pairs of different lengths and/or chemical insaturations, lowering thus the overall molecule symmetry and potentially giving rise to more complex behavior, such as interdigitation and chirality, we will restrict ourselves to consider fully saturated fatty-acid chains of equal length, for the sake of simplicity. Similarly, while the head-groups may, depending on the local chemical-environment conditions, acquire a net electric charge, we will consider only the zwitterionic case, when there is a local, but permanent, charge imbalance that creates a short-ranged dipolar interaction, but the overall lipid charge still remains neutral. There have been many different approaches to the theoretical modelling of phase transitions in phospholipids. 14,15 The first groundbreaking successful model was introduced by John Nagle. 16 Some approaches are more phenomenological, based on Landau-like expansions of a free-energy functional, and some are more microscopically detailed, and try to obtain the thermodynamical properties of the system through solving an interaction Hamil-

3


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

tonian, frequently at a mean-field approximation (MFA) level. Nevertheless, the popularity of the dimer mapping obtained by Nagle was hampered by the distinct mindsets of the physics and the chemical/biochemical communities, with the mathematical complexity of that approach being somewhat mystifying for some members of the latter. The approach taken by Doniach 17 was to simplify the formulation of the lipid problem, mapping it into the more familiar two-dimensional spin-1/2 ferromagnetic Ising model, 18–21 with one of the spin states carrying an intrinsic disorder in the form of a degenerescence ω 1 that drives a thermotropic first-order phase transition ending at a critical point. The disorder is caused by the excitation of internal degrees of freedom, namely the departure from a closed-packed, low-entropy, all-trans stretched lipid-tail state to a configuration where there can be one or many cis carbon bonds on a chain, leading to highly entropic inefficient packing caused by steric effects. In order to take into account the observed lipid-density gap that accompanies the order-disorder transition, Doniach took a step forward over previous work, 22 by arbitrarily introducing different areas for the two lipid-chain states. The Doniach model 17 has the advantage of simplicity and borrows results from the long-studied spin-1/2 ferromagnetic Ising model, 18–21 but also carries many drawbacks, including the omission of the well-known low-density gas phase in its construction. It also condenses all the exact results for the entropy obtained by Nagle 16 into a single parameter for the degenerescence ω of the melted state, loosing much of the subtlety involved in the highly cooperative steric-induced melting of the membrane, which makes it first order for a range of conditions. However, it has inspired the formulation of similar statistical lattice models that were successful in describing some aspects of the main transition of zwitterionic lipids. 23–25 A recent proposal 1 has been made to extend the Doniach model to include vacancies, hence it being designated as the Doniach lattice gas (DLG) model. A shortcoming of the Doniach model is the fact that, by the model definition, the lipid area is linearly dependent on the chain order parameter, being thus unable to describe lipid-density fluctuations, a feature that represents a severe limitation of the model to describe the behavior of ionic-

4


Page 4 of 36

Page 5 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

lipid systems. 26,27 While the DLG model still being a very simple lattice model, it obviously presents a more complex phase diagram than the Doniach model and has the potential to better capture some of the overall richness of phospholipids phases. Within this diluted model, there is no need to define an a priori area for the disordered-chain lipid, as proposed in the original Doniach model. 17 Instead, this area arises naturally from the average lipid surface density. The DLG, or diluted Doniach model, can be cast in the form of an extension of the spin-1 Blume–Emery–Griffiths (BEG) 28 model, with bilinear, biquadratic and cubic spin-spin couplings, 29–32 supplemented then by linear and quadratic temperature-dependent fields, due to the inclusion of the degenerescence ω of the disordered state. 1 In the original DLG model proposal, 1 the authors settled for a general MFA analysis of the model predictions, together with Monte Carlo simulations for some points of interest in the phase diagram. While some combinations of parameters result in phase diagrams with topology akin to the lipid systems, the authors did not explore model parameters that imply the existence of a critical point at the end of the LE-LC first-order transition line, which is, however, observed experimentally for the two specific lipids to be considered here, DMPC and DPPC. Therefore, we have decided to investigate the DLG model further: (i) instead of solving the model at MFA, in an accompanying paper 2 we treated it at a pair-approximation level, which is more accurate than the fully-connected MFA Hamiltonian to describe critical behavior, in order to check and confirm the robustness of the MFA predictions; (ii) we then seek for model parameters whose associated predictions may indeed be directly compared to results obtained from isothermal monolayer compression experiments for the two most commonly studied saturated zwitterionic phospholipids, DMPC and DPPC. 3 For that reason, the purpose of the current study is to reanalyze the DLG model, reviewed in the ‘Theoretical Section’, under the framework of the pair approximation, implemented via the Bethe-Gujrati 33 lattice scheme, whose main results are summarized in the Appendix: the BL grand-potential density, the BL equations of state, and the associated critical conditions. The comparison between measurements of isothermal monolayer compression experiments 3

5


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 36

and our theoretical predictions of the DLG model at the pair approximation is detailed in the ‘Results and Discussion’ section. Some final comments are presented in the ‘Conclusions’ section.

Theoretical Section: Definition of the DLG model In the DLG model, 1 the Nt sites of a two-dimensional lattice with lattice parameter a0 and coordination z can be occupied either by a phospholipid head-group or water. We hereby define those sites as being occupied and vacant sites, respectively. The occupied sites can be in one of two states: an ordered state, or singlet, characterized by a vanishing entropy of the molecular internal degrees of freedom, and a disordered state, or multiplet, characterized by a degenerescence ω 1, that replaces all different possible trans/gauche configurations of the double fatty-acid chains, not necessarily truly degenerated, by a single weighted, average state. As shown in Figure 1, we will label the ordered, disordered and vacant sites as o, d and w. Besides the single-site intramolecular energies −x , x ∈ (o, d, w), nearest-neighbors sites interact by short-ranged attractive pairwise interactions −xy , x ∈ (o, d, w), y ∈ (o, d, w), as shown in Figure 2. The total energy E of the model system is a sum of single-site intramolecular energies, to which intermolecular nearest-neighbors interactions are added

E=−

X

x Nx −

x=o,d,w

X

xy Nxy ,

(1)

x,y=o,d,w

where Nx is the number of sites on the state x and Nxy is the number of nearest-neighbors xy pairs. Although the original DLG model 1 missed the first contribution of eq 1 arising from the intramolecular energies, we have included it for completeness. However, it will only shift the effective chemical potential and add a term to the field parameter H of the effective Hamiltonian. In order to proceed with the statistical analysis of the model, eq 1 can be conveniently recast in terms of a spin-1 model, where the (o, d, w) single-site states at lattice site i = 6


Page 7 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

o

air

d w

w 1111 0000

0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 00000000000000000000 11111111111111111111 0000 1111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 0000 1111 00000000000000000000 11111111111111111111 water 00000000000000000000 11111111111111111111 0000 1111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 0000 1111

+ 0 − 0 a0

(b)

(a)

Figure 1: (a) Schematic representation of a small portion of a Langmuir film consisting of a lipid monolayer at the air-water interface. In order to avoid overcrowding of the figure, only the back row is filled with molecules. There are two water molecules (w), as well as two lipid molecules: an ordered-chain lipid (o) and a disordered-chainchicotheory lipid (d). The gray sites, with coordination number z = 6 (number of interacting nearest-neighbor sites), represent the hypothesized regular hexagonal lattice of lipid molecules, reflecting the observed sixfold local orientational order of neighboring molecules. 6 (b) Two-dimensional representation of the underlying triangular lattice (lattice parameter a0 ) associated with the local lipid ordering. The spin-1 representation, to be used in the statistical model, is also shown for the upper/back row. 1, · · · , Nt are represented by spin variables si = +1, −1, 0, respectively,

E({si }) = −J

X (i,j)

si sj − K

X (i,j)

s2i s2j −

X X LX si sj (si + sj ) + D s2i − H si − Ew , (2) 2 i i (i,j)

where the coefficients (J, K, L, D, H) and the reference energy Ew are linear combinations of

7


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 36

the lower-level intra- and intermolecular energies (x , xy ), 1 (oo + dd − 2od ) , 4 1 K = (oo + dd + 4ww + 2od − 4wo − 4wd ) , 4 1 L = (oo − dd − 2wo + 2wd ) , 2 z 1 D = (2ww − wo − wd ) + (2w − o − d ) , 2 2 z 1 H = (wo − wd ) + (o − d ) , 2 2 z ww + w . Ew = Nt 2 J=

(3) (4) (5) (6) (7) (8)

The first three terms of eq 2 involve sums that run over all pairs of nearest-neighbor sites, the fourth and fifth single-site terms are sums over all Nt lattice sites, while the last term Ew is a constant, associated with the energy of the full water-filled state (si = 0, ∀i = 1, · · · , Nt ). Equation 2 is the most general energy expression for spin-1 systems that includes only nearest-neighbors interactions, and, apart the inclusion of the degenerescence parameter ω, it has been studied before in relation to model-systems phase diagrams exhibiting tricritical behavior. 29–32 It represents an extension of the spin-1 Blume–Emery–Griffiths (BEG) model, 28 originally proposed to describe phase transitions in 3 He–4 He mixtures. To relate the BEG-based and DLG models, one should notice that the magnetization hsi i is a measure of the average disorganization of the fatty-acid tails, while the quadrupole moment hs2i i is a measure of the average lipid surface density, irrespective of the lipid-chain ordering. It is possible to see that the attractive bilinear coupling J > 0, akin to a van der Waals-like interaction, favors segregation between the ordered-chain and disordered-chain lipid states, whilst the biquadratic interaction K > 0 favors segregation between occupied and vacant states. The cubic term L > 0 favors the appearance of the ordered state at low temperatures. Later, the crystal-field term D will be incorporated into the effective chemical potential when changing to the grand-canonical ensemble, in order to simplify the statistical analysis of the

8


Page 9 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(a) possible single−site states and energies − εo

− εd

ordered−chain lipid

disordered−chain lipid

− εw

water

(b) possible pair configurations and energies − εoo

− εod

− εdd

− εwo

− εwd

− εww

Figure 2: Possible single-site states and pair configurations, and their associated energies. problem. It also allows to easily obtain the quadrupole moment per site hs2i i, in analogy with the magnetic system, through a partial derivative of the grand potential. Finally, the H-dependent Zeeman term will be kept in order to obtain the magnetization per site hsi i by a partial derivative, in the same way the crystal-field D term can be used to obtain hs2i i. It will be postulated that the only difference between the ordered-chain and the disordered-chain lipid states is of entropic nature, through ω, so that wo = wd and o = d , implying thus H = 0, which will be assumed in the numerical calculations. Although this hypothesis is quite reasonable, the paper that introduced the DLG model 1 overlooked any explanation as to why the linear field H is considered null. Here we see that this is not really imperative, but the result of an unstated assumption that simplifies the analysis of the model. It is worth noting that, in real systems, the hydration of the polar head is usually different between 9


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 36

the ordered-chain and disordered-chain lipid states, implying thus wo 6= wd . However, we can disregard this difference as being small to actually impact the results at the level of approximation we are working with. The spin-1 system representation allows one to readily obtain the number Nx of molecules of each type x by suitable sums of the state variables over all Nt lattice sites, 1X si (si + 1), 2 i 1X Nd = si (si − 1), 2 i X Nlip = No + Nd = s2i , No =

(9) (10) (11)

i

Nw = Nt − Nlip =

X

(1 − s2i ),

(12)

i

where Nlip is the total number of lipid molecules, irrespective of whether the associated fattyacid chains are ordered or disordered. On the other hand, the trace over all 3Nt microstates {si } is more easily performed in the grand-canonical ensemble, where the constraint of fixed number of lipids Nlip and water molecules Nw no longer exists. Therefore, it is convenient to introduce the chemical potentials of lipids µlip and water molecules µw , and incorporate them into an effective Hamiltonian

H({si }) ≡ E({si }) − µlip Nlip − µw Nw X X X X LX si sj (si + sj ) − µeff s2i − H s i − E0 , = −J si sj − K s2i s2j − 2 i i (i,j)

(i,j)

(i,j)

(13) with the effective chemical potential µeff and shifted reference energy E0 given by

µeff = µlip − µw − D, E 0 = Ew + N t µ w = N t

10

(14) z 2

ww + w + µw .


(15)

Page 11 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

One should notice that, because of the additional constraint in the DLG model of fixed total area A = Nt a0 , only the chemical potential difference ∆µ = µlip − µw is relevant as a thermodynamic field in µeff . We will consider the water chemical potential µw and the crystal-field parameter D as fixed, and take the effective chemical potential µeff as a proxy for the lipid chemical potential µlip . The associated grand-canonical partition function then reads

Ξ(T, A = Nt a0 , µeff , H) =

X

ω Nd ({si }) exp [−βH({si })] ,

(16)

{si }

with β ≡ (kB T )−1 , kB is the Boltzmann constant, and T the absolute temperature. As already said before, the general form of the effective Hamiltonian, eq 13, we have just obtained is the most general spin-1 model with nearest-neighbor interactions and single-site fields, and has been applied to describe the tricritical behavior of simple fluids, 30 binary 31,32 and ternary mixtures. 29,32 Analogously as the distinction between the original spin-1/2 ferromagnetic Ising 18–21 and Doniach 17 models, the crucial difference between these abovementioned BEG-based models and the DLG model is the inclusion of the degenerescence ω for the disordered-chain lipid states. Although it would be possible, in principle, to map the general results obtained via BEG-based models for the DLG model by considering a formalism of temperature-dependent effective fields, Heff (T ) = H − 12 kB T ln ω and Deff (T ) = D − 12 kB T ln ω, in a similar way as done by Doniach 17 in connection with the Ising model, this is not feasible for the DLG model, because the multicritical behavior of the BEG-like models is much more complex than of the spin-1/2 ferromagnetic Ising model, which displays only a first-order transition line at H = 0, ending at a critical point. Therefore, we follow the approach adopted by the original DLG paper, 1 i.e., the degenerescence ω for the disordered-chain lipid states is taken explicitly into account, when summing up over microstates to obtain the grand-partition function. One should also mention that there are a few spin-1 models proposed in the literature to describe lipid monolayers, 34–39 which, how-

11


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 36

ever, predict second-order (continuous) phase transitions, in disagreement with the current hypothesis that they represent, in fact, discontinuous (first-order) transitions. 8–11 This undesirable prediction of continuous transitions, however, cannot be attributed to an inadequacy of MFA, but it represents a faulty feature of the proposed models themselves. It will be convenient to introduce the dimensionless parameters

j ≡ βJ,

k ≡ βK,

l ≡ βL,

µ ≡ βµeff ,

h ≡ βH,

(17)

in terms of which one may write the grand-canonical partition function

Ξ = eβE0

X hY {si }

i

i nX o 1 l 2 jsi sj + ks2i s2j + si sj (si + sj ) . ω 2 si (si −1) eµsi +hsi exp 2

(18)

(i,j)

Henceforth the reference energy will be set to E0 ≡ 0, by taking the full water-filled state as the reference state. This procedure is harmless, since later we will be only interested on the surface pressure Π ≡ γw − γ, defined as the difference of the surface tensions between the reference state γw and a particular lipid-monolayer state γ. The solution of the grandcanonical partition function, eq 18, at the pair (or Bethe–Peierls) approximation (BPA) is presented in Appendix.

Results and Discussion: Theory versus experiments To generate surface-pressure versus temperature (Π × T ) phase diagrams, we remind that the grand potential Ψ = Nt ψ is related to the surface tension (lateral pressure) γ conjugated

12


Page 13 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

to the total area A = Nt a0 by the Euler relation

Ψ(T, A, µeff , H) = −kB T ln Ξ = γA,

(19)

ψ(T, µeff , H) = γa0 , ∂Ψ Ψ ψ γ≡ = = . ∂A T,µeff ,H A a0

(20) (21)

In fact, one does only need to obtain the surface pressure Π, defined as the difference between the surface tensions in the absence and presence of the thin lipid monolayer,

Π ≡ γw − γ =

E0 E0 ψ −γ = − . A A a0

(22)

Therefore, if the constant term E0 is neglected in eqs 13 and 18, one obtains directly the surface pressure by Π = −ψ/a0 , that being the reason why we could safely ignore this constant. Also, the area per lipid a(T, µeff , H) is directly related to the lipid surface density σ(T, µeff , H) = q0 = hs20 i,

a(T, µeff , H) ≡

a0 A = . hNlip i σ(T, µeff , H)

(23)

To present the numerical results, instead of using the previously defined dimensionless parameters (j, k, l, µ, h), it is more convenient to use dimensionless parameters scaled to the bilinear coupling J,

t≡

1 1 = , j βJ

K k k¯ ≡ = , J j

¯l ≡ L = l , J j

µ ¯≡

µeff µ = , J j

¯ ≡ H = h = 0. h J j (24)

For comparison of our BPA theoretical predictions with experimental measurements of the LE-LC phase transition, we chose the two most commonly studied saturated zwitterionic phospholipids, DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) and DPPC (1,2-

13


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

dipalmitoyl-sn-glycero-3-phosphocholine). Before we proceed with our comparison, it is important to bear in mind that the reproducibility of Langmuir monolayer compression experiments are not always satisfactory, as discussed for the specific case of DPPC in Ref. 40. For DPPC, although horizontal coexistence-pressure plateaus can be observed in careful experiments performed at very slow compression rates, 8 in general this does not represent the rule. Furthermore, usually hysteresis occurs, depending whether the system is compressed or expanded. 40–42 It has been argued that, for DPPC, these effects have a domain-growth kinetic origin, 11 a feature that can not be described by investigation of equilibrium properties. Variation among different monolayer compression experiments and nonhorizontal coexistence plateaus also occur for DMPC, 3,43–45 which may be attributed, in addition to kinetic effects, due to trace amounts of impurities and uncontrolled humidity. For DMPC, we selected and extracted three sets of isothermal compression data from the literature. 3,44,45 After a consistency analysis (Figure 3), we chose to use the set 3 that presented the critical data (temperature and pressure) closer to the most commonly accepted values. One can see that the chosen set 3 was overall more consistent and also denser. For DPPC, we compared in Figure 4 two sets of isothermal compression data. 3,46 Although the data sets could be said to be in somewhat of an agreement, the second study 46 was carried out with pulmonary surfactant, which represents mostly DPPC, but not pure DPPC. For that reason, we compared our model predictions again to the first data set. 3 The two chosen experimental data sets for comparison, both extracted from Ref. 3, are presented in Tables 1 and 2. Table 3 summarizes some numerical parameters chosen to perform the calculations, as well as results obtained a posteriori from the fitting procedure to be described later. Below we justify our choice of the parameters (z, ω, a0 , J): • As the LC phase exhibits a sixfold (hexagonal-like) ordering, 6 we take the lattice coordination z = 6; • In the context of the spin-1/2 Doniach model, 17 it would be possible to estimate the 14


Page 14 of 36

Page 15 of 36

ì

15

ì

15

DMPC

æ æ

10 5 0

(a)

Da HÞ²L

Da HÞ²L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

æ ò ò æ ò ò æ ò ì ò ò ò ì ò ì

8 10 12 14 16 18 20

DMPC

æ æ

òò æ ò

10

ì

5 0

(b)

òæ æ òò òò ì ì

ò

10 15 20 25 30 35 40

PHm Nm L

T H°CL

Figure 3: Lipid-area difference, ∆a ≡ aLE − aLC , along the LE-LC phase transition in DMPC monolayers: (a) as a function of temperature T ; and (b) as a function of coexistence surface pressure Π. Experimental data extracted from: Ref. 3 (s); Ref. 44 (l); Ref. 45 (u). In the first set, 3 we consider aord = a0 = 46.9˚ A2 . For the second set, 44 area differences are given directly. The third paper 45 provides only the areas for the LE phase, but it was possible to estimate the area of the LC phase through the area-surface pressure isotherms. The value obtained is around 46˚ A2 , which is in agreement with the literature on the subject. The lines joining the data points are only a guide for the eyes.

Table 1: Experimental data of the LE-LC first-order transition, extracted from the surfacepressure isotherms of Ref. 3, for DMPC monolayers. The last line on the table refers to the critical-point data (ac , Πc , Tc ). DMPC data 2

a (Å ) Π (mN/m) 57.4372 21.7682 56.9653 22.9849 56.0887 25.4179 55.2798 27.4935 53.8639 31.2864 53.3245 32.7892 52.6503 34.5783 52.0437 36.0100 critical point 48.0704 43.3160

15

T (◦ C) 12 13 14 15 16 17 18 19 20


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

PHm Nm L

Langmuir

40

Page 16 of 36

DPPC

à à

30 à

20 0

ò

ò

à

10

ò

àò à ò

20

25

30

35

40

T H°CL Figure 4: Comparison between experimental coexistence surface pressures and transition temperatures for the LE-LC phase transition in DPPC monolayers. Experimental data extracted from: Ref. 3 (s); Ref. 46 (n). The lines joining the data points are only a guide for the eyes. degenerescence factor ω by calorimetric measurements of the latent heat of melting Q = 12 kB T ∗ ∆m0 ln ω, where ∆m0 is the magnetization jump at the first-order LE-LC phase-transition temperature T ∗ [see eq 14 of Ref. 17, and eq 2.20 of Ref. 24], but the relative experimental errors are quite sizable and we chose to use a simpler expression ω = 6 × 3M −6 , where M is the lipid-chain length (M = 14 for DMPC, M = 16 for DPPC), extracted from Table 3 of Ref. 24; • The estimated areas per lipid of the ordered phase a0 = aord were obtained from Ref. 3 (DMPC) and Ref. 11 (DPPC); • All interaction parameters in the theoretical model are scaled to the bilinear coupling J, and eventually, the critical grand potential ψc , which relates to the critical surface pressure Πc , requires knowledge of the actual energy scale J. At this point, we try to improve the accuracy of this estimate and, possibly, the accuracy of the surface pressure-temperature phase diagrams and surface pressure-area compression isotherms, reminding that the LE-LC critical point that we are investigating occurs between two 16


Page 17 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

dense phases. This means that, even though the lipid order parameter m0 = hs0 i have opposite signs for each phase, the surface density σ = q0 = hs20 i ≈ 1 for both phases. Under these circumstances, we have almost no vacancies and the DLG model can be mapped back to the Doniach model, 17 which, in turn, can be understood in terms of the spin-1/2 ferromagnetic Ising model. This is of interest, because we know the exact critical temperature for the two-dimensional triangular-lattice Ising model and its relation to the pair-approximation critical temperature. Therefore, we will use the ratio (4) = ln 3/[2 ln (3/2)] ≈ 1.35476, as (z = 6)/texact of these two temperatures, φ˜ = tBPA c c ˜ a correction factor for the surface pressure, Π = −φψ/a 0 , to hopefully obtain a closer numerical agreement between the calculated model results and the experimental data. The raw energy scales obtained would be J = kB Tc /tc ≈ 7.71 × 10−22 J/mol for DMPC and J = kB Tc /tc ≈ 8.25×10−22 J/mol for DPPC. After multiplying them by the lattice ˜ we coordination z and further applying the above spin-1/2 Ising correction factor φ, obtain values compatible with the estimates in the original DLG article 1 for DMPC, JMFA ≈ 6 × 10−21 J/mol, as well as with the MFA values reported in Table 3 of Ref. 24, which are also listed in Table 3. In order to obtain the remaining free model parameters to reproduce the experimental data presented in Tables 1 and 2, one should bear in mind that we have to solve five ¯ m, q): simultaneous equations in six unknowns (t, µ ¯, ¯l, k, ¯ solve the fixed points (m, q) of the two recurrence relations, eqs 26 and • given (t, µ ¯, ¯l, k), 27; ¯ m, q); • solve the critical conditions, eqs 35 and 36, for (t, µ ¯, ¯l, k, • adjust the measured experimental critical surface pressure Πc to the theoretical value ˜ c /a0 , where the grand potential is given by eq 25, and it depends on Πc = −φψ ¯ m, q). (t, ¯l, k,

17


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

In other words, by just taking into account the critical-point data, given in the last rows of Tables 1 and 2, one condition is still missing to fully solve the six unknowns. Therefore, to fulfill the missing condition, we have decided to arbitrarily choose a particular isotherm, distinct from the critical one, and fit its associated coexistence pressure to the one calculated by ¯ are then adjusted simultaneously to reproduce the critical the model. The parameters (¯l, k) data (temperature and surface pressure) and the coexistence data (temperature and surface pressure) at an arbitrarily chosen point of the experimental data set, that we pointed out in Table 3. At the end of Table 3, we display the parameters and results obtained from this fitting. Once we have all the parameters corresponding to the specific phospholipid in consideration, we are able to obtain the temperature versus effective chemical potential T × µ ¯/z phase diagrams (Figure 5), the coexistence surface pressure versus transition temperature Π × T phase diagrams (Figures 6 and 8), and the Π × a isotherms (Figures 7 and 9). Although the DLG model may predict, depending on the interaction parameters, the occurrence of a gas (G) phase, in Figure 5 we can see that, for the fitted parameters corresponding to both analyzed phospholipids, this G phase is consistently absent. Also, there is only a LE-LC first-order transition line that ends at a critical point. Even though we selected the experimental data sets that were overall more consistent, 3 this choice, specially for DPPC, led to a very inaccurate coexistence Π × T phase diagram (Figure 8), extraordinarily large areas for the LE phase (Figure 9) and very inaccurate intermediary pressures. It is possible that these results could be improved, from the theoretical side, with a finer adjustment of the degenerescence ω, as well as analysis with refined experimental isothermal compression data, displaying horizontal coexistence-pressure plateaus. Unfortunately, the few experimental investigations that address this issue 8,11 do not cover the full range of temperatures and surface pressures, including the critical point. In general there is a better agreement for the DMPC coexistence Π × T phase diagram (Figure 6) and compression isotherms (Figure 7), but again the experimental nonhorizontal coexistence-pressure plateaus, attributed to kinetic effects, 11 can not be explained in the

18


Page 18 of 36

Page 19 of 36

Table 2: Experimental data of the LE-LC first-order transition, extracted from the surfacepressure isotherms of Ref. 3, for DPPC monolayers. The last line on the table refers to the critical-point data (ac , Πc , Tc ). DPPC data

40

0.90

DPPC 0.86 t z

LE LC

T H°CL

25 20 15 0.86 10 0.84 (a) 5 0.82 0 -7.5 -7. -6.5 - 6. -5.5 - 5. Μ/z 0.88 DMPC

T (◦ C) 20 25 30 35

LE LC

0.82

(b)

0.78

-11. -10.5 -10. -9.5 -9. Μ/z

50 40 30 20 10 0

T H°CL

2

a (Å ) Π (mN/m) 79.8887 2.85714 70.1051 7.63975 61.3892 16.4596 55.2305 24.472 critical point 48.3669 32.9814

t z

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Figure 5: Temperature T versus effective chemical potential µ ¯/z phase diagram obtained for the DLG model under BPA using the numerical parameters from: (a) the first column of Table 3, obtained by fitting experimental DMPC isothermal compression data; (b) the second column of Table 3, obtained by fitting experimental DPPC isothermal compression data. The dashed lines represent the LE-LC first-order transition, ending at a critical point (l).

19


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 3: Numerical data for the LE-LC first-order phase transition on the two analyzed saturated diacyl PC lipids, DMPC and DPPC. † Lattice coordination z to reproduce the sixfold lipid headgroup orientational order; 6 ‡ estimated degenerescence ω = 6 × 3M −6 from Table 3 of Ref. 24; § estimated areas per lipid of the ordered phase a0 = aord from Ref. 3 (DMPC) and Ref. 11 (DPPC); ∗ data selected and extracted from the surface-pressure isotherms of Ref. 3; ] MFA energy scale J0 from Table 3 of Ref. 24, listed only for comparison ˜ with the calculated bilinear-coupling raw value J and the φ˜ factor-adjusted, zJ φ(z). Lipid chain length M 14 (DMPC) 16 (DPPC) input parameters † z 6 6 ‡ 4 ω 4 × 10 3.5 × 105 2 § a0 (Å ) 46.9 48 ∗ lower (T, Π) input data for fitting T (◦ C) 12 25 Π (mN/m) 21.7682 7.63975 ∗ critical (Tc , Πc ) input data for fitting Tc (◦ C) 20 40 Πc (mN/m) 43.3160 32.9814 fitted parameters k¯ 6.51200 10.20970 ¯l 9.30161 11.17619 calculated critical parameters and energy scales J, J0 2 ac (Å ) 47.3645 48.5833 tc /z 0.8751 0.8736 µ ¯c /z −5.30818 −8.9758 mc −0.7337 −0.8069 −22 J (Joules) 7.709 × 10 8.248 × 10−22 ˜ zJ φ(z) (Joules) 6.27 × 10−21 6.70 × 10−21 ] −21 J0 (Joules) 6.36 × 10 7.34 × 10−21

20


Page 20 of 36

45 40 DMPC 35 30 LC 25 LE 20 15 6 8 10 12 14 16 18 20

T H°CL Figure 6: Temperature × coexistence surface-pressure phase diagram for DMPC. The dashed line represents the LE-LC first-order transition, ending at a critical point (l), calculated from the DLG model under BPA using the parameters from the first column of Table 3. The experimental data from Table 1 is represented by (4). The critical point and the lowest-temperature experimental point were used to simultaneously adjust the parameters ¯ ¯l}. {k,

PHmNmL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

PHm Nm L

Page 21 of 36

DMPC 40 30 20 10 0 40 50 60 70 80 90 100 aHÞ²L

Figure 7: DMPC surface-pressure isotherms versus lipid area obtained from the DLG model under BPA using the parameters from the first column of Table 3, from the lowest temperature of 12◦ C, increasing in 1◦ C steps to the critical temperature of 20◦ C, from bottom to top. These calculated isotherms should be compared with the experimental profiles shown in Figure 1 of Ref. 3.

21


Langmuir

PHm Nm L

40

DPPC

30 20

LC

10 0 0

10

20

LE 30

40

T H°CL Figure 8: Temperature × coexistence surface-pressure phase diagram for DPPC. The dashed line represents the LE-LC first-order transition, ending at a critical point (l), calculated from the DLG model under BPA using the parameters from the second column of Table 3. The experimental data from Table 2 is represented by (4). The critical point and the second lowest-temperature experimental point were used to simultaneously adjust the parameters ¯ ¯l}. {k,

40

PHm Nm L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 36

30

DPPC

20 10 0 40

60

80

100

120

a HÞ²L Figure 9: DPPC surface-pressure isotherms versus lipid area obtained from the DLG model under BPA using the parameters from the second column of Table 3, corresponding to the temperatures of 20◦ C to 40◦ C in 5◦ C steps, from bottom to top. These calculated isotherms should be compared with the experimental profiles shown in Figure 2 of Ref. 3.

22


Page 23 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

context of equilibrium first-order transitions. Notice that the experimental DMPC coexistence data in the Π × T phase diagram (Figure 6) displays a nonmonotonic behavior, with a slight inconsistent S shape. The DMPC LE areas and intermediary isotherms are closer to the experimental data, but still significantly larger at lower temperatures and smaller near the critical point. This is not much of a surprise, given that the critical exponents for the pair approximation are still in the mean-field realm, while the real system exhibits a distinct behavior. In fact, authors of Ref. 3 — see their Figure 4 — argued that their experimental data are consistent with a power-law behavior close to the critical exponent β = 1/8, associated with the universality class of the two-dimensional Ising model. We can see, from the almost vertical slope at the left side of the theoretical isotherms, that the predicted isothermal compressibility κ ≡ −(1/a)(∂a/∂Π)T of the LC phase is almost null: the condensed-phase density is very near the maximum density of the model, and the LC area corresponds closely to the lattice parameter a0 . This is in contrast to the experimental data, where it is clear that the LC phase is still somewhat compressible. At this point, one should keep in mind that the DLG model does not distinguish the LC phase from a possible solid crystalline phase at higher lipid densities, since it neglects the translational entropy, attributing the LE-LC transition mainly to the lipid-chain melting, controlled by the degenerescence parameter ω, in analogy to the original Doniach model. 17 Furthermore, in a real system, this might be also related to the onset of three-dimensional structures and buckling out of the monolayer plane upon further compression, 12 which were not taken into account in the simplified two-dimensional DLG model.

Conclusions We have compared the theoretical predictions of the DLG model at the pair-approximation level 2 with experimental data of isothermal compressions on DMPC and DPPC monolayers, 3 in order to obtain the model parameters corresponding to the description of each phospho-

23


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

lipid. With these fitted parameters, it is possible to calculate the corresponding predicted phase diagrams and compression isotherms and compare them with the experimental data that was not constrained by the fitting, such as the variation in area for the coexisting phases with increasing temperature, and the intermediary compression isotherms. Although the DLG model predicts, depending on the interaction parameters, the occurrence of a gas (G) phase, we have found that, for the fitted parameters corresponding to both investigated phospholipids, this phase is consistently absent. We suggest that further investigation seeking for an improved agreement between theoretical predictions and experimental data relies, from the experimental side, on obtaining horizontal coexistence-pressure plateaus, in order to be able to describe them as truly equilibrium first-order transitions. From the theoretical side, removal of the imposed restriction of vanishing field parameter H, as well as a better adjustment of the degenerescence parameter ω through the Clausius-Clapeyron condition Q = 12 kB T ∗ ∆m0 ln ω, may also play a role.

Appendix: DLG model on the Bethe lattice and Gujrati’s method In this Appendix, the model properties derived from the grand-partition function, eq 18, will be evaluated at the pair-approximation level by solving the model on the so-called Bethe lattice (BL), 47,48 which yields approximated solutions of the equivalent problem formulated on a regular lattice, generally identical to the traditional Bethe-Peierls approximation (BPA). 49–52 It is well known that BPA leads to better quantitative agreement than the usual MFA, specially near criticality. The Cayley tree (CT) is a cycle-free connected graph without closed loops, as shown in Figure 10 by N = 3 successive generations of a tree with coordination z = 6. To obtain the BL equations of state, one should consider only the central region of a large CT, in order to avoid the pathologies originated from the surface sites of a CT. 53–55 This is achieved by formulating the problem as a two-dimensional discrete map and 24


Page 24 of 36

Page 25 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

associating the fixed points of the recurrence relations with the thermodynamic solutions on the BL. Finally, Gujrati’s method 33 was employed to obtain the thermodynamically consistent BL grand-potential density ψ, from which the BL equations of state can be derived. A detailed presentation of the derivation of the BL equations of state and the corresponding analysis have been given elsewhere. 2 Next we recapitulate only the main results essential to our comparison with experiments, performed in the ‘Results and Discussion’ section.

n =0 n =1 n=2

n=N =3

Figure 10: A Cayley tree of coordination z = 6, comprised of a zeroth (n = 0) site surrounded by N = 3 generations of sites. Each surface site (at generation n = N = 3) has a single nearest neighbor lying at the preceding generation n = 2, while all interior sites (n < N ) have equally z nearest neighbors. Application of standard statistical mechanical methods and Gujrati’s method 33 yields the following pair-approximation BL grand-potential density per site 1 ΞN 1 lim ln z−1 = ln(1 − q) + (z − 2) 2 N →∞ ΞN −1 2 × ln 1 − q 2 + 14 ej+k+l (q + m)2 + 14 ej+k−l (q − m)2 + 21 ek−j (q 2 − m2 ) ,

βψ(T, µ, h) = −

25


(25)

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 36

where ΞN is the grand-canonical partition function of a whole CT with N generations, and (m, q) are the fixed points (m, q) = (mn , qn ) = (mn−1 , qn−1 ) of the recurrence relations eh X z−1 (mn , qn ) − ωe−h Y z−1 (mn , qn ) , e−µ + eh X z−1 (mn , qn ) + ωe−h Y z−1 (mn , qn ) eh X z−1 (mn , qn ) + ωe−h Y z−1 (mn , qn ) qn−1 = −µ , e + eh X z−1 (mn , qn ) + ωe−h Y z−1 (mn , qn ) 1 1 X(mn , qn ) = 1 − qn + ej+k+l (qn + mn ) + ek−j (qn − mn ) , 2 2 1 j+k−l 1 k−j (qn − mn ) . Y (mn , qn ) = 1 − qn + e (qn + mn ) + e 2 2 mn−1 =

(26) (27) (28) (29)

One should remark that the fixed points (m, q) of the recurrence relations do not represent the bona fide BL order parameters (m0 , q0 ), evaluated as thermal averages at the root of the CT, eh X z (m, q) − ωe−h Y z (m, q) m0 ≡ hs0 i = −µ =− e + eh X z (m, q) + ωe−h Y z (m, q) 2

j+k+l

∂βψ ∂h

j+k−l

T, µ 2

4m (1 − q) + e (q + m) − e (q − m) , 2 2 j+k+l j+k−l 4 (1 − +e (q + m) + e (q − m) + 2ek−j (q 2 − m2 ) ∂βψ eh X z (m, q) + ωe−h Y z (m, q) 2 =− q0 ≡ hs0 i = −µ e + eh X z (m, q) + ωe−h Y z (m, q) ∂µ T, h =

q2)

4q (1 − q) + ej+k+l (q + m)2 + ej+k−l (q − m)2 + 2ek−j (q 2 − m2 ) = . 4 (1 − q 2 ) + ej+k+l (q + m)2 + ej+k−l (q − m)2 + 2ek−j (q 2 − m2 )

(30)

(31)

These latter order parameters are related to the fractions of sites filled with ordered-chain and disordered-chain lipids in the system, φo and φd , and the lipid surface density σ for the DLG model under the BPA, hNo i 1 = (q0 + m0 ), Nt 2 hNd i 1 φd ≡ = (q0 − m0 ), Nt 2 hNlip i hNo + Nd i hNlip ia0 σ≡ = = = q0 . Nt Nt A φo ≡

26


(32) (33) (34)

Page 27 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

The spinodal-line and critical conditions can be written as

η η

(1)

(2)

≡ ≡

∂h ∂m

∂ 2h ∂m2

= 0 → hm µq − hq µm = 0 → κ2 e2k + κ1 ek + κ0 = 0,

(35)

µ,T

= 0 → (2hq µmq + h2q µm − hm µ2q ) hm = (2hmq µm + hq µ2m − h2m µq ) hq , µ,T

(36) where, for the sake of a clean notation, the partial derivatives of the fields (h, µ) were denoted through subscripts,

∂hq ∂hm hm ≡ , hq ≡ , h2m ≡ , h2q ≡ , hmq ≡ , ∂q m ∂q m q m q ∂µ ∂µm ∂µq ∂µm ∂µ , µq ≡ , µ2m ≡ , µ2q ≡ , µmq ≡ , µm ≡ ∂m q ∂q m ∂m q ∂q m ∂q m ∂h ∂m

∂h ∂q

∂hm ∂m

(37) with the fields h(m, q) and µ(m, q) given by

e

2h

z−1 ω (q + m) Y (m, q) = , q−m X(m, q)

e

−2µ

4ω (1 − q)2 [X(m, q)Y (m, q)]z−1 , = 2 2 q −m

(38)

and the coefficients {κj } κ2 = e−j [(z − 1)q + 2 − z] [ze−2j − e2j (z − 2)](q 2 − m2 ) + el (q + m)2 + e−l (q − m)2 , (39) n κ1 = (1 − q) el (q + m) m(z − 1)2 − z[(z − 4)q + 2] − 3q + 4 − e−l (q − m) o × m(z − 1)2 + z[(z − 4)q + 2] + 3q − 4 − 2e−2j m2 (z − 1)2 + (1 − z 2 )q 2 − 2q , (40) κ0 = 4(1 − q)2 el [(z − 1)q + 1].

(41)

These critical-condition equations represent an extension, to the DLG model treated at the 27


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

BPA level, of previous general results reported in the literature. 29,56 Notice, however, that here (m, q) are not the BL order parameters, as discussed in the Appendix A of Ref. 2. Depending on the model system parameters (j, k, l, µ, h, ω, z), different types of phase diagrams may be obtained, 2 which exhibit phase transitions between distinct thermal phases, that would correspond to the G, LE and LC phases. An additional staggered (Stg) phase, overlooked in the MFA analysis, 1 was also found in our BL calculations, 2 which required to consider the system as a bipartite lattice, in analogy to the treatment of the ferromagnetic BEG model with repulsive biquadratic interactions. 57 Our aim in this work is to compare predicted thermodynamic model features with experimental results for the LE-LC phase transition measured on DMPC and DPPC monolayers. 3

Acknowledgement The authors thank H. S. Guidi and V. B. Henriques for fruitful discussions. F. O. de Oliveira acknowledges financial support from the Brazilian agencies CAPES (Coordination for the Improvement of Higher Education Personnel, Master fellowship Grant No. 1186359/2013) and CNPq (National Council for Scientific and Technological Development, PhD fellowship Grant No. 142249/2017-3). The National Institute of Science and Technology Complex Fluids (INCT-FCx) is also acknowledged, sponsored by the Brazilian agencies CNPq (National Council for Scientific and Technological Development, Grant No. 465259/2014-6), FAPESP (São Paulo Research Foundation, Grant No. 2014/50983-3) and CAPES (Coordination for the Improvement of Higher Education Personnel, Grant No. 88887.136373/2017-00).

References (1) Guidi, H. S.; Henriques, V. B. Lattice Solution Model for Order-Disorder Transitions in Membranes and Langmuir Monolayers. Phys. Rev. E 2014, 90, 052705, DOI: 10. 1103/PhysRevE.90.052705. 28


Page 28 of 36

Page 29 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(2) de Oliveira, F. O.; Tamashiro, M. N. Phase Transitions in Phospholipid Monolayers: Statistical Model at the Pair Approximation. To appear in Phys. Rev E. (3) Nielsen, L. K.; Bjørnholm, T.; Mouritsen, O. G. Thermodynamic and Real-Space Structural Evidence of a 2D Critical Point in Phospholipid Monolayers. Langmuir 2007, 23, 11684–11692, DOI: 10.1021/la7016352. (4) Mouritsen, O. G. Life — As a Matter of Fat: The Emerging Science of Lipidomics; The Frontiers Collection; Springer-Verlag: Berlin, 2005; Chap. 2, p. 24. (5) Singer, S. J.; Nicolson, G. L. The Fluid Mosaic Model of the Structure of Cell Membranes. Science 1972, 175, 720–731, DOI: 10.1126/science.175.4023.720. (6) Kaganer, V. M.; Möhwald, H.; Dutta, P. Structure and Phase Transitions in Langmuir Monolayers. Rev. Mod. Phys. 1999, 71, 779–819, DOI: 10.1103/RevModPhys.71.779. (7) Gragson, D. E.; Beaman, D.; Porter, R. Using Compression Isotherms of Phospholipid Monolayers to Explore Critical Phenomena. A Biophysical Chemistry Experiment. J. Chem. Educ. 2008, 85, 272–275, DOI: 10.1021/ed085p272. (8) Hifeda, Y. F.; Rayfield, G. W. Evidence for First-Order Phase Transitions in Lipid and Fatty Acid Monolayers. Langmuir 1992, 8, 197–200, DOI: 10.1021/la00037a036. (9) Pallas, N. R.; Pethica, B. A. First-Order Phase Transitions and Equilibrium Spreading Pressures in Lipid and Fatty Acid Monolayers. Langmuir 1993, 9, 361–362, DOI: 10. 1021/la00025a068. (10) Denicourt, N.; Tancrède, P.; Teissié, J. The Main Transition of Dipalmitoylphosphatidylcholine Monolayers: A Liquid Expanded to Solid Condensed High Order Transformation. Biophys. Chem. 1994, 49, 153–162, DOI: 10.1016/0301-4622(93)E0066-E. (11) Arriaga, L. R.; López-Montero, I.; Ignés-Mullol, J.; Monroy, F. Domain-Growth Kinetic Origin of Nonhorizontal Phase Coexistence Plateaux in Langmuir Monolayers: 29


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Compression Rigidity of a Raft-like Lipid Distribution. J. Phys. Chem. B 2010, 114, 4509–4520, DOI: 10.1021/jp9118953. (12) Lee, K. Y. C. Collapse Mechanisms of Langmuir Monolayers. Annu. Rev. Phys. Chem. 2008, 59, 771–791, DOI: 10.1146/annurev.physchem.58.032806.104619. (13) Marsh, D. Lateral Pressure in Membranes. Biochim. Biophys. Acta 1996, 1286, 183– 223, DOI: 10.1016/S0304-4157(96)00009-3. (14) Mouritsen, O. G. Theoretical Models of Phospholipid Phase Transitions. Chem. Phys. Lipids 1991, 57, 179–194, DOI: 10.1016/0009-3084(91)90075-M. (15) Myer, B.; Evans, E.; Mouritsen, O. G. Physical Properties of the Fluid Lipid-Bilayer Component of Cell Membranes: A Perspective. Quart. Rev. Biophys. 1991, 24, 293– 397, DOI: 10.1017/S0033583500003735. (16) Nagle, J. F. Theory of Biomembrane Phase Transitions. J. Chem. Phys. 1973, 58, 252–264, DOI: 10.1063/1.1678914. (17) Doniach, S. Thermodynamic Fluctuations in Phospholipid Bilayers. J. Chem. Phys. 1978, 68, 4912–4916, DOI: 10.1063/1.435647. (18) Ising, E. Beitrag Zur Theorie Des Ferromagnetismus [Contribution to the Theory of Ferromagnetism]. Z. Phys. 1925, 31, 253–258, DOI: 10.1007/bf02980577. (19) Onsager, L. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev. 1944, 65, 117–149, DOI: 10.1103/PhysRev.65.117. (20) Yang, C. N. The Spontaneous Magnetization of a Two-Dimensional Ising Model. Phys. Rev. 1952, 85, 808–816, DOI: 10.1103/PhysRev.85.808. (21) Lee, T. D.; Yang, C. N. Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model. Phys. Rev. 1952, 87, 410–419, DOI: 10.1103/ PhysRev.87.410. 30


Page 30 of 36

Page 31 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(22) Caillé, A.; Rapini, A.; Zuckermann, M. J.; Cros, A.; Doniach, S. A Simple Model for Phase Transitions in Monolayers and Bilayers of Lipid Molecules. Can. J. Phys. 1978, 56, 348–357, DOI: 10.1139/p78-044. (23) Zuckermann, M. J.; Pink, D. The Correlation Length and Lateral Compressibility of Phospholipid Bilayers in the Presence of Thermodynamic Density Fluctuations. J. Chem. Phys. 1980, 73, 2919–2926, DOI: 10.1063/1.440463. (24) Caillé, A.; Pink, D.; de Verteuil, F.; Zuckermann, M. J. Theoretical Models for QuasiTwo-Dimensional Mesomorphic Monolayers and Membrane Bilayers. Can. J. Phys. 1980, 58, 581–611, DOI: 10.1139/p80-083. (25) Mouritsen, O. G.; Boothroyd, A.; Harris, R.; Jan, N.; Lookman, T.; MacDonald, L.; Pink, D. A.; Zuckermann, M. J. Computer Simulation of the Main Gel-Fluid Phase Transition of Lipid Bilayers. J. Chem. Phys. 1983, 79, 2027–2041, DOI: 10.1063/1. 445987. (26) Tamashiro, M. N.; Barbetta, C.; Germano, R.; Henriques, V. B. Phase Transitions and Spatially Ordered Counterion Association in Ionic-Lipid Membranes: A Statistical Model. Phys. Rev. E 2011, 84, 031909, DOI: 10.1103/PhysRevE.84.031909. (27) Henriques, V. B.; Germano, R.; Lamy, M. T.; Tamashiro, M. N. Phase Transitions and Spatially Ordered Counterion Association in Ionic-Lipid Membranes: Theory Versus Experiment. Langmuir 2011, 27, 13130–13143, DOI: 10.1021/la202302x. (28) Blume, M.; Emery, V. J.; Griffiths, R. B. Ising Model for the λ Transition and Phase Separation in He3 -He4 Mixtures. Phys. Rev. A 1971, 4, 1071–1077, DOI: 10.1103/ PhysRevA.4.1071. (29) Mukamel, D.; Blume, M. Ising Model for Tricritical Points in Ternary Mixtures. Phys. Rev. A 1974, 10, 610–617, DOI: 10.1103/PhysRevA.10.610.

31


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(30) Lajzerowicz, J.; Sivardière, J. Spin-1 Lattice-Gas Model. I. Condensation and Solidification of a Simple Fluid. Phys. Rev. A 1975, 11, 2079–2089, DOI: 10.1103/PhysRevA. 11.2079. (31) Sivardière, J.; Lajzerowicz, J. Spin-1 Lattice-Gas Model. II. Condensation and Phase Separation in a Binary Fluid. Phys. Rev. A 1975, 11, 2090–2100, DOI: 10.1103/ PhysRevA.11.2090. (32) Sivardière, J.; Lajzerowicz, J. Spin-1 Lattice-Gas Model. III. Tricritical Points in Binary and Ternary Fluids. Phys. Rev. A 1975, 11, 2101–2110, DOI: 10.1103/PhysRevA.11. 2101. (33) Gujrati, P. D. Bethe or Bethe-like Lattice Calculations Are More Reliable Than Conventional Mean-Field Calculations. Phys. Rev. Lett. 1995, 74, 809–812, DOI: 10.1103/PhysRevLett.74.809. (34) Baret, J. F.; Firpo, J. L. A Spin-1 Model to Describe Amphiphile Monolayer Phase Transition. J. Colloid Interface Sci. 1983, 94, 487–496, DOI: 10.1016/0021-9797(83) 90288-6. (35) Legré, J. P.; Firpo, J. L.; Albinet, G. Mean-Field Simulation in a RenormalizationGroup Procedure and Application to Amphiphilic Monolayers. Phys. Rev. A 1985, 31, 1703–1706, DOI: 10.1103/PhysRevA.31.1703. (36) Banville, M.; Caillé, A.; Albinet, G. Description of Monolayers of Discotic Molecules at Air-Water Interface with Spin One Models Including Vacancies and Nesting of Pairs. J. Phys. France 1985, 46, 101–107, DOI: 10.1051/jphys:01985004601010100. (37) Pękalski, A. Phase Diagrams for a Model of a Lipid Monolayer. Eur. Biophys. J. 1988, 16, 39–44, DOI: 10.1007/BF00255324.

32


Page 32 of 36

Page 33 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(38) Buzano, C.; Evangelista, L. R. The Phase Transition in Amphiphilic Monolayers: Isotherms in the Cluster Variation Method. J. Phys.: Condens. Matter 1994, 6, 5323– 5334, DOI: 10.1088/0953-8984/6/28/009. (39) Lavis, D. A.; Bell, G. M. Statistical Mechanics of Lattice Systems, Vol. 1: ClosedForm and Exact Solutions; Springer-Verlag: Berlin, 1999; Sec. 7.4: Phase Transitions in Amphipatic Monolayers, pp. 181–187. (40) Duncan, S. L.; Larson, R. G. Comparing Experimental and Simulated Pressure-Area Isotherms for DPPC. Biophys. J. 2008, 94, 2965–2986, DOI: 10.1529/biophysj.107. 114215. (41) von Tscharner, V.; McConnell, H. An Alternative View of Phospholipid Phase Behavior at the Air-Water Interface. Microscope and Film Balance Studies. Biophys. J. 1981, 36, 409–419, DOI: 10.1016/S0006-3495(81)84740-6. (42) Crane, J. M.; Putz, G.; Hall, S. B. Persistence of Phase Coexistence in Disaturated Phosphatidylcholine Monolayers at High Surface Pressures. Biophys. J. 1999, 77, 3134– 3143, DOI: 10.1016/S0006-3495(99)77143-2. (43) Albrecht, O.; Gruler, H.; Sackmann, E. Polymorphism of Phospholipid Monolayers. J. Phys. France 1978, 39, 301–313, DOI: 10.1051/jphys:01978003903030100. (44) Blume, A. A Comparative Study of the Phase Transitions of Phospholipid Bilayers and Monolayers. Biochim. Biophys. Acta 1979, 557, 32–44, DOI: 10.1016/0005-2736(79) 90087-7. (45) Steppich, D.; Griesbauer, J.; Frommelt, T.; Appelt, W.; Wixforth, A.; Schneider, M. F. Thermomechanic-Electrical Coupling in Phospholipid Monolayers Near the Critical Point. Phys. Rev. E 2010, 81, 061123, DOI: 10.1103/PhysRevE.81.061123.

33


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(46) Discher, B. M.; Maloney, K. M.; Grainger, D. W.; Sousa, C. A.; Hall, S. B. Neutral Lipids Induce Critical Behavior in Interfacial Monolayers of Pulmonary Surfactant. Biochemistry 1999, 38, 374–383, DOI: 10.1021/bi981386h. (47) Thompson, C. J. Local Properties of an Ising Model on a Cayley Tree. J. Stat. Phys. 1982, 27, 441–456, DOI: 10.1007/BF01011085. (48) Baxter, R. J. Exactly Solved Models in Statistical Mechanics, 3rd printing; Dover Books on Physics Series; Dover: Mineola/NY, 2007; Chap. 4: Ising Model on the Bethe Lattice, pp. 47–59. (49) Huang, K. Statistical Mechanics, 2nd ed.; Wiley: New York, 1987; Sec. 14.5: The Bethe–Peierls Approximation, pp. 357–361. (50) Pathria, R. K. Statistical Mechanics, 2nd ed.; Butterworth-Heinemann: Oxford, 1996; Sec. 11.6: Ising Model in the First Approximation, pp. 328–334. (51) Lavis, D. A.; Bell, G. M. Statistical Mechanics of Lattice Systems, Vol. 1: Closed-Form and Exact Solutions; Springer-Verlag: Berlin, 1999; Sec. 7.3: The Pair Approximation and the Ising Model, pp. 178–181. (52) Salinas, S. R. A. Introduction to Statistical Physics; Springer-Verlag: New York, 2001; Sec. 13.4: The Bethe–Peierls Approximation, pp. 268–271. (53) Runnels, L. K. Phase Transition of a Bethe Lattice Gas of Hard Molecules. J. Math. Phys. 1967, 8, 2081–2087, DOI: 10.1063/1.1705123. (54) Eggarter, T. P. Cayley Trees, the Ising Problem, and the Thermodynamic Limit. Phys. Rev. B 1974, 9, 2989–2992, DOI: 10.1103/PhysRevB.9.2989. (55) Müller-Hartmann, E.; Zittartz, J. New Type of Phase Transition. Phys. Rev. Lett. 1974, 33, 893–897, DOI: 10.1103/PhysRevLett.33.893.

34


Page 34 of 36

Page 35 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(56) Bartis, J. T. Thermodynamic Equations for Tri- or Third Order Critical Points. J. Chem. Phys. 1973, 59, 5423–5430, DOI: 10.1063/1.1679892. (57) Ercule, A.; Tamashiro, M. N. Multicritical Behavior of the Ferromagnetic BlumeEmery-Griffiths Model with Repulsive Biquadratic Couplings. Phys. Rev. E 2018, 97, 062145, DOI: 10.1103/PhysRevE.97.062145.

35


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 36

Graphical TOC Entry o

air

d w

w 11111 00000

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 00000 11111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 00000 11111 0000000000000000000000 1111111111111111111111 water 0000000000000000000000 1111111111111111111111 00000 11111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 00000 11111

+ 0 − 0 a0

Π

36


Phase transitions in phospholipid monolayers: theory versus

Recommend Documents