Statistical thermodynamics of association colloids : the equilibrium structure of micelles, vesicles, and bilayer membranes
The aim of the present study was to unravel the general equilibrium physical properties of lipid bilayer membranes. We consider four major questions:1. What determines the morphology of the association colloids (micelles, membranes, vesicles) in general?2. Do the apolar tails of the lipids in the bilayer organise themselves more like matches in a box or rather like hot spaghetti in a pan?3. How does this membrane organisation depend on temperature?4. How do additives like surfactants or polymers interact with the bilayer?These four questions cover a wide range of topics currently subject to intensive research. Each one of them calls for a rigorous answer. We believed that it would be possible to design one single theory covering the whole field. The development of such a theory is undertaken in the present thesis.Recently, the statistical thermodynamics of homopolymers at interfaces has been worked out by Scheutjens and Fleer (SF). This theory is an extension of the Flory Huggins (FH) theory for polymers in solution in the sense that it allows for inhomogeneities in one dimension. In the other two dimensions a mean field, i.e., an average segment density, assumption is applied. One of the strong points of this theory is that, by using a Markov-type approximation, all possible conformations of the chains are considered with a minimum of computational effort. The SF theory can be extended to describe copolymers at interfaces.For well-chosen amphipolar molecules the theory is able to deal with local phase separation phenomena. Preliminary calculations on surfactant bilayers showed that the SF theory needed some modifications in order to be relevant to the four topics given above. The main reason for this is that for the very small surfactant molecules the Markov-type approximation is not very accurate. Five extensions of the theory are presented in this thesis:1. For the chain statistics the Markov-type approximation is extended to the so called rotational isomeric state scheme. This scheme prevents backfolding in chain sections of five consecutive segments. The improvement allowed us to adjust the stiffness of the chain as a function of temperature.2. The theory is generalised for arbitrary geometries. With this extension the polymorphism of association colloids could be studied.3. The theory is extended to account for branched chain molecules. This has been used to simulate lipid molecules with two apolar tails and one polar head group.4. In the SF theory the statistical weight of each conformation is found by Boltzmann statistics. The potential of each conformation depends on segment-segment interactions, hard core contact potentials, and the number of gauche bonds in the chain. A new weighting factor is introduced which accounts for the average orientation of the molecules. The statisticalweight of a conformation is increased when its bond directions match with those of the surrounding molecules. With this molecular orientational field co-operative phenomena like crystallisation can be studied.5. Allowing for inhomogeneities in two dimensions enables us to study membrane-"protein" interactions.The properties of the theory with these new features are thoroughly examined in five chapters. A short summary of the results and main conclusions of each chapter is given below.Chapter 1 dealt with the morphology of association colloids. In this chapter we prove that the formation of micelles is a first order transition. However, the theoretical critical micelle concentration is not observed very sharply, because it is very low. We showed that, with Increasing concentration of bipolar molecules, the micelles first grow and eventually change their shapes. Lecithin-like molecules prefer lamellar aggregates over globular ones.In chapter 2 the rotational isomeric state scheme is presented and details of the statistics of branched chain molecules is given. We present an overview of the behaviour of the membranes as a function of the four energy parameters. There is no need to restrict the molecules to pre-assigned positions in the system. The membrane thickness adjusts itself. The equilibrium membrane is free of tension. Its excess free energy per surface area is very small. When fluctuations and long range Van der Waals attractions are neglected the excess free energy is essentially zero.Vesicle systems are studied in chapter 3. We show that the excess free energy of curvature per vesicle is constant for vesicles composed of one type of lipid, irrespective of the radius of the vesicle. This remains true for bi-lamellar and hence for multilamellar vesicles. We show that as a rule, the thicker a membrane is the more energy it costs to bend it. Adding surfactants to a system containing vesicle is disastrous for the vesicle structures. Increasing the surfactant/ lipid ratio causes the vesicles to brake up in micelles. When vesicles are formed by two compatible lipid molecules, the free energy of curvature varies linearly with their composition. If the two bipolar molecules do not mix, they partition themselves over the two membrane sides and the excess free energy of curvature shows, at constant vesicle radius, a minimum as a function of composition. For a given composition the vesicle adopts an optimal vesicle radius.The membrane structure predicted by the theory significantly improves when the orientational dependent molecular field is applied. We derive the partition function for this SCAF (Self-Consistent Anisotropic Field) theory in chapter 4. Among other things, the order parameter profiles now show the well known plateau along the lipid tails. In agreement with experiments, we find a first order phase transition which transforms the membrane from a high temperature liquid into a low temperature gel state. In the gel phase the lipid tails are virtually in a all trans conformation. Because of this, the density in the gel membrane is higher than in the liquid phase. For the model membrane we observed two possible gel phases. One gel phase was about twice as thick as the other. The thin, intercalated, gel membrane was found in the case that the membranes were isolated, i.e., when they did not interact with each other, while the other gel phase, obviously with non-intercalated membranes, was found in the concentrated regime.In the final chapter we studied two cases of the interaction of long copolymers ("proteins") with a model membrane. In the first example the molecule is in a trans membrane configuration. In the second example a group of four molecules is clustered together and forms a hydrophilic pore, through which polar molecules can pass the membrane. In this chapter we also study the boundary region between two areas of lipid molecules which do not mix (lateral phase separation). It is characteristic for membrane system, that the lipids in the membrane are very efficient in camouflaging the inhomogeneities in the boundary layers. No big differences in solvent profiles are observed along the boundary layers. This ability of the lipid molecules to compensate Irregularities explains why membranes are not easily disrupted.It is the first time that a statistical thermodynamical theory is presented that can deal with association phenomena without the requirement to fix the head groups to pre-assigned positions. We showed that this theory does give a very detailed insight into equilibrium membrane properties. The correspondence with experimental data is satisfactory. The theory can be easily extended to incorporate more details in the calculations and better quantitative agreement with experimental data Is well feasible.
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| Format: | Doctoral thesis biblioteca |
| Language: | English |
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Landbouwuniversiteit Wageningen
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| Subjects: | colloids, cum laude, dispersion, entropy, membranes, statistics, thermodynamics, colloïden, dispersie, entropie, membranen, statistiek, thermodynamica, |
| Online Access: | https://research.wur.nl/en/publications/statistical-thermodynamics-of-association-colloids-the-equilibriu |
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| Summary: | The aim of the present study was to unravel the general equilibrium physical properties of lipid bilayer membranes. We consider four major questions:1. What determines the morphology of the association colloids (micelles, membranes, vesicles) in general?2. Do the apolar tails of the lipids in the bilayer organise themselves more like matches in a box or rather like hot spaghetti in a pan?3. How does this membrane organisation depend on temperature?4. How do additives like surfactants or polymers interact with the bilayer?These four questions cover a wide range of topics currently subject to intensive research. Each one of them calls for a rigorous answer. We believed that it would be possible to design one single theory covering the whole field. The development of such a theory is undertaken in the present thesis.Recently, the statistical thermodynamics of homopolymers at interfaces has been worked out by Scheutjens and Fleer (SF). This theory is an extension of the Flory Huggins (FH) theory for polymers in solution in the sense that it allows for inhomogeneities in one dimension. In the other two dimensions a mean field, i.e., an average segment density, assumption is applied. One of the strong points of this theory is that, by using a Markov-type approximation, all possible conformations of the chains are considered with a minimum of computational effort. The SF theory can be extended to describe copolymers at interfaces.For well-chosen amphipolar molecules the theory is able to deal with local phase separation phenomena. Preliminary calculations on surfactant bilayers showed that the SF theory needed some modifications in order to be relevant to the four topics given above. The main reason for this is that for the very small surfactant molecules the Markov-type approximation is not very accurate. Five extensions of the theory are presented in this thesis:1. For the chain statistics the Markov-type approximation is extended to the so called rotational isomeric state scheme. This scheme prevents backfolding in chain sections of five consecutive segments. The improvement allowed us to adjust the stiffness of the chain as a function of temperature.2. The theory is generalised for arbitrary geometries. With this extension the polymorphism of association colloids could be studied.3. The theory is extended to account for branched chain molecules. This has been used to simulate lipid molecules with two apolar tails and one polar head group.4. In the SF theory the statistical weight of each conformation is found by Boltzmann statistics. The potential of each conformation depends on segment-segment interactions, hard core contact potentials, and the number of gauche bonds in the chain. A new weighting factor is introduced which accounts for the average orientation of the molecules. The statisticalweight of a conformation is increased when its bond directions match with those of the surrounding molecules. With this molecular orientational field co-operative phenomena like crystallisation can be studied.5. Allowing for inhomogeneities in two dimensions enables us to study membrane-"protein" interactions.The properties of the theory with these new features are thoroughly examined in five chapters. A short summary of the results and main conclusions of each chapter is given below.Chapter 1 dealt with the morphology of association colloids. In this chapter we prove that the formation of micelles is a first order transition. However, the theoretical critical micelle concentration is not observed very sharply, because it is very low. We showed that, with Increasing concentration of bipolar molecules, the micelles first grow and eventually change their shapes. Lecithin-like molecules prefer lamellar aggregates over globular ones.In chapter 2 the rotational isomeric state scheme is presented and details of the statistics of branched chain molecules is given. We present an overview of the behaviour of the membranes as a function of the four energy parameters. There is no need to restrict the molecules to pre-assigned positions in the system. The membrane thickness adjusts itself. The equilibrium membrane is free of tension. Its excess free energy per surface area is very small. When fluctuations and long range Van der Waals attractions are neglected the excess free energy is essentially zero.Vesicle systems are studied in chapter 3. We show that the excess free energy of curvature per vesicle is constant for vesicles composed of one type of lipid, irrespective of the radius of the vesicle. This remains true for bi-lamellar and hence for multilamellar vesicles. We show that as a rule, the thicker a membrane is the more energy it costs to bend it. Adding surfactants to a system containing vesicle is disastrous for the vesicle structures. Increasing the surfactant/ lipid ratio causes the vesicles to brake up in micelles. When vesicles are formed by two compatible lipid molecules, the free energy of curvature varies linearly with their composition. If the two bipolar molecules do not mix, they partition themselves over the two membrane sides and the excess free energy of curvature shows, at constant vesicle radius, a minimum as a function of composition. For a given composition the vesicle adopts an optimal vesicle radius.The membrane structure predicted by the theory significantly improves when the orientational dependent molecular field is applied. We derive the partition function for this SCAF (Self-Consistent Anisotropic Field) theory in chapter 4. Among other things, the order parameter profiles now show the well known plateau along the lipid tails. In agreement with experiments, we find a first order phase transition which transforms the membrane from a high temperature liquid into a low temperature gel state. In the gel phase the lipid tails are virtually in a all trans conformation. Because of this, the density in the gel membrane is higher than in the liquid phase. For the model membrane we observed two possible gel phases. One gel phase was about twice as thick as the other. The thin, intercalated, gel membrane was found in the case that the membranes were isolated, i.e., when they did not interact with each other, while the other gel phase, obviously with non-intercalated membranes, was found in the concentrated regime.In the final chapter we studied two cases of the interaction of long copolymers ("proteins") with a model membrane. In the first example the molecule is in a trans membrane configuration. In the second example a group of four molecules is clustered together and forms a hydrophilic pore, through which polar molecules can pass the membrane. In this chapter we also study the boundary region between two areas of lipid molecules which do not mix (lateral phase separation). It is characteristic for membrane system, that the lipids in the membrane are very efficient in camouflaging the inhomogeneities in the boundary layers. No big differences in solvent profiles are observed along the boundary layers. This ability of the lipid molecules to compensate Irregularities explains why membranes are not easily disrupted.It is the first time that a statistical thermodynamical theory is presented that can deal with association phenomena without the requirement to fix the head groups to pre-assigned positions. We showed that this theory does give a very detailed insight into equilibrium membrane properties. The correspondence with experimental data is satisfactory. The theory can be easily extended to incorporate more details in the calculations and better quantitative agreement with experimental data Is well feasible. |
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