Concept of Molecular Dynamics Simulations 

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Biological membranes

Biological membranes are one of the most important constituents of prokaryotic and eu-karyotic cells. The very beginning of life can be related to the formation of a biological compartments protecting the very first organisms from their environment. This was pos-sible only with the existence of biological membranes – physical barriers between cells and their exterior. A following step in the evolution of cells was a further compartmentaliza-tion of cells, leading to the formation of organelles. In this way, biological membranes allowed various microenviornments to coexist within cells, which increased their ability to adapt to di↵erent environments. A biological membrane consists of lipid molecules forming mono- and bi- layers. These layers, in cellular conditions, always form closed structures because of hydrophobic e↵ects resulting from interactions between lipids and water molecules [Berg et al. [2005]]. The reason is that a lipid molecule has a hydrophobic part – a water-repulsive tail, and a hydrophilic one – a water-attractive headgroup. In order to shield hydrophobic regions from water, lipids organize themselves in an aggre-gated phase. A biological membrane contains also many other diverse types of molecules and actually is a very complex and heterogeneous system [Alberts et al. [2005]]. This is why biological membranes play also an important role in the bidirectional transport of molecules across the bilayer (or monolayer), in relaying signals, in the adhesion of cells, and in many enzymatic processes.

General Introduction

Lipids are a very diverse group of chemical compounds that are either hydrophobic or amphipathic (soluble in both – water and fats) [Berg et al. [2005]]. The lipids can refer to amassed fatty acids in a volume, more complex steroids, or phospholipids. Table 1.1 shows that cholesterol, cardiolipin and phospoholipids are the most abundant lipids in animal cell membranes. The percentage of the total masses in di↵erent tissues indicates that the most abundant are phosphatidylcholine and phosphatidylethanolamine (phospholipids) and cholesterol for erythrocytes.
Phospholipids have three main moieties (Fig. 1.2):
1. A hydrophilic headgroup, which consists of the phosphate group and additional groups, e.g. a N(CH)3 group. The latter is called a choline group and is a part of phosphatidylcholines. Another example for a headgroup is the ethanolamine rest in phosphatidylethanolamine.
2. A backbone, consisting mostly of derivatives of glycerol, but in case of sphingomye-lines, the base is a sphingosine, which is an amino alcohol with unsaturated fatty acid rest.
3. Tail(s) consisting of hydrocarbon chains that are derivatives of fatty acids. They can be either saturated (all the carbon atoms in the chains are connected by single bonds) or mono- and poli- unsaturated (at least one double bond).
The lipid composition of membranes presented in Table 1.2 for various organelles shows again that the most common ones are phosphatidylcholines and phosphatidylethanolamines. The abundance of the first does not fall below 40% of the total weight of the membrane for all the presented organelles and is the lowest for mitochondria, which exhibits on the other hand the highest concentration of cardiolipin and phosphatidylethanolamine. The latter is the second most abundant type of lipid, with the lowest abundance of 20% in a plasma membrane. According to these findings, phosphatidylethanolamine and in partic-ular phosphatidylcholine are the basic lipids constituting biological membranes in cells. They stand for 51% of the total weight of all membranes in a tissue. The comparison above referred only to the properties of the headgroup moieties. The other di↵erences concern the lengths of the hydrocarbon chains (tails) and their level of saturation (number of double bonds). In Table 1.3 the content of diverse fatty acid tails is presented for six animal species: rat, pig, duck, horse, herring and seal. The nomencla-ture in the first column should be read as – the total number of carbon atoms in a chain : number of double bonds (“20 : 1” means for example a tail consisting of twenty carbon atoms with one double bond). As one can easily see, the most abundant tails are the saturated palmitoleic fatty acid rests (sixteen carbon atoms) and oleic acid rests with one double bond (eighteen carbon atoms). This result suggests that the most common lipid would be a phosphocholine with palmitoleic and oleic tails, which would be a 1-palmitoyl-2- oleoyl-sn-glycero-3-phosphocholine (POPC). An interesting observation is that higher order lipids with the unsaturated hydrocarbon chains of twenty and more carbon atoms are found only in aquatic animals leaving in cold environment, here herring and seal.
The surface area of a membrane is huge in comparison to macromolecules, so the prob-ability of finding a specific membrane protein is low. Many processes in cells involve several proteins, and for that reason macromolecules taking part in the same reaction must stay localized and close to each other in biological membranes [Berg et al. [2005]]. To raise the probability of finding the necessary proteins in the reaction volume, nature developed several strategies. One of them is a confinement of proteins within physical barriers, which are formed by the cytoskeleton attached to the membrane. Another strategy, more interesting from the perspective of this work, is to slow down the di↵usion of the lipids and proteins, keeping them more localized in the bilayer. In this case, there is no need for physical barriers, since the very nature of the di↵usion of molecules would already raise the probability of keeping them in the same place. One of the examples is the self-organization of lipids into domains (so-called rafts), which seem to attract pro-teins [Risselada & Marrink [2008]; Silva et al. [2007]]. These proteins di↵use with the lipids in the raft.

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Transport of molecules in cells and membranes

The di↵usion of molecules observed in living cells deviates in many cases from normal di↵usion (the definition and its generalization is described in more details in the next section), where normality is defined by Einstein’s di↵usion law [Einstein [1905]], which predicts that the mean square displacement of the di↵using particle grows linearly with time. In biological membranes one observes instead an anomalous sublinear growth.
Anomalous di↵usion has been observed in cells by various spectroscopic experimental methods. Cells, including biological membranes, are crowded with a huge number of di↵erent organelles and particles, as proteins, lipids, DNA, RNA and many more [Berg et al. [2005]]. Moreover, organelles are also internally crowded with diverse molecules. Very useful methods to study di↵usion in such crowded systems are fluorescence-based techniques, such as the fluorescence correlation spectroscopy (FCS) and single particle tracking methods (SPT). The former measures the concentration fluctuations of fluores-cent particles and the latter, the movements of a single fluorescent particle [Bronstein et al. [2009]; Owen et al. [2009]; Schwille et al. [1999b]].
SPT revealed for example that chromosomal loci in bacteria are moving subdi↵usively through the cytosole[Weber et al. [2010]]. This finding is significant for the dynamical organization chromosomes, in particular for a possible contribution to the preservation of the chromosomal territories during the segregation process. Further results from di↵erent studies on chromosomal movements, which have been obtained again with SPT, show that the telomeres in eucaryotic nucleus stay localized and their movements exhibit a transient subdi↵usion [Bronstein et al. [2009]]. This could explain the strong conservation of the genome organization. From a general point of view of chromosomal transport within nuclei, subdi↵usion also leads to a preservation of the position without physical constraints.
Fluorescence Correlation Spectroscopy (FCS) studies of the di↵usion of the DNA-binding protein Lacl (lac repressor) show that this protein di↵uses normally between bindings [Elf et al. [2007]]. This would suggest that most of the time the lacl protein is di↵using freely along DNA strands and is non-specifically bound. The same conclusions can be drawn from other FCS and SPT measurements for di↵erent small molecules in cells, whose di↵usion constants are several times smaller in denser media, like cytoplasm (in comparison to water solution), but di↵use normally [Dix & Verkman [2008]].
The aforementioned methods are also extensively used in studying the dynamics of molecules within biological membranes. The FCS studies of Schwille et al. [1999b] on living cells, revealed subdi↵usive movements of dilauroyl-sn-glycero-3-phosphocholine lipids (DLPC) in the cell membrane. In contrast, the di↵usion in lipid granules in yeast cells display tran-sient anomalous di↵usion for shorter times (SPT), but for longer times becomes normal [Jeon et al. [2012]]. A very similar observation has been published using quasi-elastic neu-tron scattering method (QENS) for the dimyristoyl-sn-glycero-3-phosphocholine (DMPC) in the fluid phase at the pico- to nano-second time scale [Armstrong et al. [2011]]. In the latter work the di↵usion is found to be normal for larger inter-lipid distances, with a value of di↵usion coe cient consistent with other findings. But for smaller lipid-lipid lateral distances (0.23 nm) the nature of the movements are ballistic (motions without collisions with other molecules). Other QENS results show that phospholipids might dif-fuse together with their neighbors [Busch et al. [2010]]. This experiment of pico- to nano-second time scale suggests a dynamical clustering of lipids moving together in plane of a membrane.
As biological bilayers are vivid mosaics containing not only lipids but also proteins, the scope of experimentalists also focused on the di↵usion of the latter. FCS studies of the Golgi resident membrane proteins [Weiss et al. [2003]] tracked the subdi↵usion of these molecules in the endoplasmic reticulum. Other findings using SPT on potassium membrane channels, imply anomalous di↵usion of these proteins [Weigel et al. [2011]]. Movements which are supposedly a↵ected by the actin cytoskeleton network.

Table of contents :

1 General Introduction 
1.1 Biological membranes
1.2 Transport of molecules in cells and membranes
1.3 Modeling anomalous di↵usion
1.4 Recent simulation work
2 Concept of Molecular Dynamics Simulations 
2.1 Classical Molecular Dynamics Simulations
2.1.1 Concept and potential function
2.1.2 Periodic boundary conditions and treatment of electrostatic interactions
2.2 Treatment of thermodynamic conditions
3 Simulated systems and simulations 
3.1 System setup
3.1.1 All-atom OPLS force-field
3.1.2 Coarse-grained MARTINI force-field
3.2 Membrane simulations
3.2.1 OPLS-AA force-field
3.2.1.1 System equilibration
3.2.1.2 Short production run – 15 ns
3.2.1.3 Extended simulation – 150 ns
3.2.2 MARTINI force-field
3.2.2.1 System equilibration
3.2.2.2 600 ns production run in the canonical ensemble
3.2.2.3 600 ns production run in the NApzT ensemble
3.3 Bulk water reference simulation
4 Results 
4.1 Single molecule dynamics in a lipid bilayer
4.1.1 Mean Square Displacements
4.1.1.1 Atom-detailed simulations
4.1.1.2 Coarse-grained force-field results
4.1.1.3 Comparison between OPLS-AA and MARTINI force-fields
4.1.1.4 Bulk water comparison with membrane hydration water
4.1.2 Velocity autocorrelation function analysis
4.1.3 Density of States (DOS)
4.1.3.1 DOS as Fourier transform of VACF
4.1.3.2 Density of the States from the Autoregressive Model
4.1.4 Conclusions
4.2 Collective motions of lipids
4.2.1 Pair Correlation Function
4.2.2 Visualization of the cage e↵ect
4.2.3 Conclusions
5 Conclusions 
6 Publications 
References

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