energetics of cholesterol-modulated membrane...

UPTEC X 11 003

Examensarbete 30 hpJanuari 2011

Energetics of cholesterol-modulated membrane permeabilities. A simulation study

Christian Wennberg

Molecular Biotechnology Programme

Uppsala University School of Engineering

UPTEC X 11 003

Date of issue 2011-02

Author

Christian Wennberg

Title (English)


Title (Swedish)

Abstract Molecular dynamics simulations were used to study the permeation of four different solutes through different cholesterol containing lipid bilayers. In all bilayers the limiting permeation barrier shifted towards the hydrophobic core, as the cholesterol concentration was increased. Cholesterols reducing effect on the permeation rate was observed, but under certain conditions results indicating an increased permeation rate with increasing cholesterol concentration were also obtained.

Keywords Molecular dynamics, permeability, cholesterol, lipid bilayer, umbrella simulation, potential of mean force

Supervisors

David van der Spoel Uppsala University

Scientific reviewer

Johan Åqvist Uppsala University

Project name

Sponsors

Language

English

Security

ISSN 1401-2138

Classification

Supplementary bibliographical information Pages

34

Biology Education Centre Biomedical Center Husargatan 3 Uppsala Box 592 S-75124 Uppsala Tel +46 (0)18 4710000 Fax +46 (0)18 471 4687


Christian Wennberg

Populärvetenskaplig sammanfattning

Skiftande förhållanden i en cells omgivning gör att den måste ha ett försvar som skyddar den från skadliga ämnen. En viktig del i detta försvar är cellens cellmembran som bland annat verkar som ett skyddande skal kring cellen. Cellmembranet skapar en barriär som skadliga ämnen i cellens omgivning måste ta sig igenom, och därmed får cellen ett naturligt skydd mot dessa ämnen. Dock finns en dualitet i detta eftersom de ämnen cellen behöver för att överleva också finns i dess omgivning vilket innebär att cellen måste kunna reglera vilka molekyler som tar sig in och ut. När det gäller stora och/eller laddade molekyler sköts denna reglering av membranbundna transportproteiner som ser till att transportera dessa molekyler genom membranet. Små oladdade molekyler kan dock ta sig genom membranet utan hjälp av transportproteiner. Reglering av denna passiva genomsläpplighet är möjligt för cellen genom inkorporering av kolesterol i membranet. Kolesterol gör membranet stelare vilket försvårar för ämnen att ta sig igenom det. De exakta mekanismerna med vilket detta sker är dock inte fullt kartlagda än, men den stora ökning i datorkraft som skett senaste decenniet har gjort närmare analys av interaktionerna i cellmembran möjliga. I detta arbete har kolesterols effekt på fyra olika membran simulerats. Fyra olika ämnen, utvalda för deras olika egenskaper, har via dator-simuleringar studerats vid sin transport genom de olika membranen. Kolesterolkoncentrationen i membranen har varierats från 0 till 50 % och från simuleringarna har energibarriärer för transporten genom membranen beräknats.

Examensarbete 30hp Civilingenjörsprogrammet Molekylär bioteknik

Uppsala universitet februari 2011

1

Contents: Page

1. Introduction

1.1 The cell membrane 2 1.2 Phospholipids 2

1.3 Cholesterol 3 1.4 Membrane permeation 4

1.5 Aim 5

2. Molecular dynamics 2.1 Basic principles 6

2.2 Force fields 7 2.3 Algorithms 8

2.4 Potential of mean force 10 2.5 Umbrella sampling 10

2.6 Weighted histogram analysis method 11

3. Methodology 3.1 Lipids and solutes 12

3.2 Simulation procedure 13

4. Results 4.1 Lipid patches 15

4.2 PMFs 15 4.3 Umbrella simulations 15

5. Discussion

5.1 Tail region 17 5.2 Head region 18

5.3 Solutes 19

6. Conclusion 19

7. Acknowledgements 20

8. References 21

Appendix A 23 Appendix B 27

Appendix C 31

2

1. Introduction 1.1 The cell membrane

A cell’s membrane defines the outer limit of the cell and protects it from the surrounding

environment by providing a physical barrier for molecules trying to enter or exit the cell’s interior. This allows the cell to maintain certain properties such as pH or ion concentration in its cytoplasm

despite environmental changes. The membrane also serves as an anchor point for the cells cytoskeleton, thereby also aiding in the maintenance of the cells shape.

Incorporated into the cell membrane are also membrane bound proteins. These proteins manage a vast number of functions, such as in- and outflow of solutes from the cell’s interior, cell

communication and production of molecules essential for the cell (e.g. ATP). The cell membrane’s basic structure is built up by phospholipids, which gives the membrane its

characteristic properties with a hydrophobic core separated from the interior and exterior environment by its hydrophilic surfaces.

Other lipids existing in the membrane are cholesterol and glycolipids, that aid in the membrane stability (cholesterol) or serve as cell markers (glycolipids).

1.2 Phospholipids As the major component in cell membranes phospholipids play a crucial role in

protecting the cells from the outside environment. Phospholipids are built up by a hydrophilic head region which, depending on the type of lipid, contains a

different number of charged groups of which one always is a phosphate group. The head group is attached to a tail region that consists of long fatty chains,

making them very hydrophobic (figure 1), and it is this amphipathic character of the phospholipids that enables them to form bilayers in water, where the

hydrophobic tails group together on the inside of the layer with the head groups outwards forming the boundary to the surrounding water.

Looking at the various lipids found in cell membranes all but one seems to have glycerol as the “backbone” molecule. The one left out is sphingomyelin,

which instead is built up from the amino alcohol spingosine. The composition of lipids in the cell membrane differs between the outer and

inner leaflets. The outer leaflet tends to be populated by phosphatidyl-choline (PC) and spingomyelin while the inner leaflet's major components are

phosphatidyl-ethanolamine (PE) and phosphatidyl-serine1. The reason for this

contrast in composition is still unclear, but it has been found that changes in

this asymmetry are a source for different diseases. Since cells invest energy in trying to maintain this difference in composition between the leaflets the

subject has a high physiological relevance2.

Figure 1. Lipid molecule.

Carbons shown in grey,

Phosphate, oxygens and

nitrogen are coloured

according to atom type.

Hydrogens attached to carbons not shown.

3

1.3 Cholesterol Looking further into the composition of lipid membranes, cholesterol is an

important structural molecule, as it regulates membrane fluidity and thereby also the permeability of solutes across the membranes

3.

Cholesterol consists of four rings fused together, a hydrocarbon tail in one end and a hydroxyl group bound at the other. This gives cholesterol an amphiphilic

character that enables it to position itself in the membrane with the hydroxyl group interacting with the head region and the hydrophobic ring structure

interacting with the tails of the lipid molecules4 (figure 2). Another important

feature of the cholesterol molecule is the fact that the two methyl groups bound

to the ring structure are located on the same side, thereby creating two distinct sides of the molecule denoted the �- and �-face (methyl groups bound to the �-

face). Molecular dynamics (MD) - simulations have shown that the lipid tails prefer

interaction with the smooth �-face but interestingly a removal of the methyl groups from the �-face will not increase the effect of cholesterol on the

membranes but rather decrease it, indicating that the structure of cholesterol has been shaped by evolution to interact in an optimal way with the lipids in

membranes5. Experiments have also shown that cholesterol prefers interaction

with saturated lipids compared to unsaturated due to the more favorable van der

Waals interactions obtained with the saturated tails4,6

. Increasing the cholesterol concentration in the membrane is known to alter the

behavior of the lipid tails by inducing a phase transition. A pure lipid bilayer can exist in either the gel-phase (L�) where the tails are elongated as much as possible, or in the

more fluid liquid-crystalline (L�) phase which exists at higher temperatures. Cholesterol reduces the transition temperature of the L� to L� phase transition. At sufficiently high cholesterol

concentrations the bilayer enters a phase that resembles both the L� and the L� phase in that it is rigid as the L� phase but it retains fluidity from the L� phase. This state is commonly referred to as

the liquid ordered (Lo) phase, and the L� phase is usually called the liquid disordered phase7.

From MD simulations this ordering effect caused by cholesterol can be quantified in a variety of

ways. Some examples are the decreased amount of gauche rotamers in the lipid tails or the average tilt angle of the chains relative to the membrane normal

8, and visualization of the bilayers

is an easy way to see the difference between the different phases (figure 3). The rigidifying effect of cholesterol has also been seen through its ability to greatly reduce the

amounts of lipids that flip-flop between the two leaflets in bilayers9. This indicates that cholesterol

could have a role in maintaining the asymmetry between leaflets, found in cell membranes.

Figure 2. Lipid with

cholesterol.

�-face of cholesterol pointing towards reader.

Figure 3. Snapshots from two different

simulations.

Left: Lipid bilayer in L� phase.

Right: Lipid bilayer with 40% cholesterol

(cholesterol removed from image for clearer

view) in the Lo phase

4

1.4 Membrane permeation In order for a cell to maintain functionality, various solutes have to be transported across the

membrane to the cell’s interior. This transport is mediated by either membrane bound transport proteins, passive permeation through the membrane by the solutes themselves or by pore

formation in the membrane. Of these mechanisms the transport by proteins governs mostly ions and other charged molecules, whereas small uncharged molecules can permeate the membrane by

themselves at a reliable pace. Overton

10 and Meyer

11 described this passive permeability through lipid membranes for different

solutes at the end of the 19th century. They stated that the permeability of a solute through a membrane is directly proportional to the solutes’ solubility in the membrane (or oil/water partition

coefficient). From this, the so-called solubility-diffusion model, in which the solutes first partition from water into the membrane and then diffuse through it, was developed

12. It can be expressed by

a simple equation:

� ��

�

Here P is the permeability coefficient, K is the partition coefficient of the solute between the membrane and water, D is the diffusion constant in the membrane and d is the thickness of the membrane. Despite its simplicity the solubility-diffusion model correctly describes the behavior of

many solutes, but exceptions have been found and stirred up a debate whether they violate the model or not

13. The solubility-diffusion model does however over-simplify the relatively complex

system that is a lipid bilayer and therefore more advanced models have been constructed. Examples of this are the 4-region model by Marrink and Berendsen

14, which divides the

membrane into four regions with different characteristics. And also the model constructed by Nagle et al, which divides the membrane into three layers and then computes the total

permeability by combining the permeation through each of those layers15

. The first MD simulations of lipid bilayers was performed by Kox et al.

16, who managed to

simulate a phase transition in a lipid membrane despite using a very simplified system in which water was excluded. Later on, simulations with water included started to emerge

17, and also solute

diffusion in the membranes were studied using MD18

. These studies revealed that the diffusion constant of solutes in the membrane was not constant but rather depended upon the position of the

solute within the membrane. This difference in diffusion was proposed to exist because the diffusion process was not linear. Instead the solutes “jump” between different pockets of free

volume within the membrane, which occurs due to kinks in the lipid tails. In following studies it was observed that the amount of kinks, and thereby also the amount of free volume, was reduced

as cholesterol was incorporated into the membrane21

. This was proposed as a reason for the lowered permeation of solutes through membranes containing cholesterol.

During recent years more comprehensive studies containing many different lipid bilayer and solutes have been done. Mathai et al.

19 did experimental studies on seven different lipid

membranes and found that water permeability was strongly correlated with the surface area occupied by the lipids and poorly correlated with the thickness of the membrane, which later on

resulted in the 3-layer model by Nagle mentioned earlier. Bemporad et al.20

did all-atom simulations on eight small molecules from which they discovered that the size of the solute had

less impact on diffusion through the membrane than was previously thought. The partitioning of the solute into the membrane was however proposed to be more size dependent than previously

thought.

5

1.5 Aim In this work a comprehensive study has been done on the effect of cholesterol on solute

permeation through lipid membranes. The permeation of four different solutes through four different lipid membranes at five different cholesterol concentrations has been studied. The aim of

the study was to investigate how certain structural properties in the lipids would change the influence of cholesterol on the permeation event, and to see if the different characteristics of the

solutes had a notable impact on their permeation. These studies were enabled through calculations of the potentials of mean force (PMF) for the permeation events.

6

2. Molecular dynamics 2.1 Basic principles Molecular dynamics (MD) is a computational method that simulates a system of choice by

integration of Newton’s law of motion. The reliability of MD to correctly recreate the real systems that it tries to simulate lies in the accuracy of the three main approximations MD is based on:

• Born-Oppenheimer approximation The Born-Oppenheimer approximation separates the motions of the electrons and nuclei of

an atom and assumes that the motion of the electrons are so much faster than the nuclei that they instantaneously adjust themselves to the motion of the nuclei. Therefore only the

motion of the nuclei is accounted for in the MD simulations.

• Classical treatment Solving the Schrödinger equation for all the nuclei in the systems one desires to study with MD is not possible. Therefore it is assumed that the nuclei can be treated as classical

particles that behave according to Newton’s second law.

• Energy surface Exactly as in the case with the nuclei, solving the Schrödinger equation for all the electrons

in the system in order to obtain the potential energy surface for the molecules is impossible for large (>100 atoms) systems. Therefore the interactions in the system are approximated

by empirical models (force fields) designed to mimic the real energy surface as closely as possible.

Using these three approximations MD converts an extremely complex system to a relatively

simple one, basically representing all particles and molecules with a ball-and-stick model. Starting from Newton’s second law and the simple relation between the particles position and their

velocities, the core equations of MD can be set up:

��

��

� ��

�� (1)

��

�� (2)

Here U is the potential energy, F the force acting on particle i, m the mass of particle i and vi the particles velocity. From these equations the positions, accelerations and velocities of all the

particles in the system can be followed as the system evolves in time. This is done step-by-step following the methodology outlined below:

1. Calculate forces on each particle based on current positions

2. Calculate new velocities 3. Calculate new positions by evolving the system in time

4. Repeat from 1

This scheme is then repeated until the desired simulation time has been reached, but in order to get useful output a certain amount of criteria has to be met. First of all the potential function, commonly denoted “force field”, has to reproduce the “real” system in a fairly accurate way in

order to get the correct forces acting on all particles. Then one has to choose a time step, used to calculate the new positions, that is long enough to get a time evolution that makes it possible to

7

study the wanted properties of the system, but short enough to account for the fastest motion in the

system in order to avoid numerical and integration errors due to collisions that should not occur throughout the simulation. The “good choice” of a time step is hard to decide but a reasonable

choice when using flexible molecules is one order of magnitude lower than the fastest motion in the system

23. This can severely limit the simulation speed in respect to time evolution, and if this

is the case one can sometimes constrain the fastest motions in order to increase the time step (the fastest motions tends to be bond vibrations which are not of great interest for the properties one

would like to extract in most cases). One large problem when doing simulations on systems of finite size is boundary effects created at

the boundaries of the simulation box. This is solved in MD by the use of periodic boundary conditions (PBC), which removes the boundaries in a simulation of a system containing a small

amount of particles compared to what would be required in order to remove boundary effects if PBC was not used. The way PBC works is basically by replicating the simulation box in all

directions so that if a molecule exits the box through one of the sides, an identical particle enters on the opposite side. This obviously also puts some limitations on the type of simulation box we

can use since in order for PBC to work it has to be formed so that it can be replicated in three dimensions without leaving any gaps between the boxes (A sphere is an example of a shape that

can not be used with PBC).

2.2 Force fields In order to calculate the forces acting on each particle in a system a force field describing the interactions between particles is required. The force fields utilized in MD is often constructed in

the same way with different expressions for the energies governing bonded and non-bonded interactions:

� � � ��

Bonded interactions are composed by three parts describing bonds, angles and dihedrals.

��

��

� ��

��

� ��

��

��

Here the bond stretching potential is described by the simple harmonic potential:

� ��

But it could also be expressed by the more accurate Morse potential:

� ��

This potential, however, is computationally more expensive, since it involves an exponential and also requires more parameters for each bond (dissociation energy De and also the constant a). The

harmonic potential also agrees well with the Morse potential at short distances from the equilibrium value.

The dihedral term describes the rotation around bonds, where mi gives an indication of the barrier height for the rotation around the bond, �i is the torsion angle, n is the number of minima when

rotating the bond 360 degrees and �0 describes the minimum values for the torsion angle (other expressions using multiple cos-terms can also be used).

The non-bonded part can be divided into Lenard-Jones (LJ) and electrostatic interactions

8

��

��

��

��

��

�

��

��

Where � is the collision parameter (zero energy) of two particles, � is the interaction energy in the LJ-potential, and the electrostatic interaction is just the Coulomb energy of two point charges

interacting with each other. Non-bonded interactions only account for pairwise interactions and do not incorporate many-

body effects. Many-body potentials would increase the computational cost drastically and instead a majority of the many-body effects are incorporated effectively into the pairwise potentials,

which are often called “effective pair potentials”.

Calculation of the non-bonded energies takes significantly more computer power than the bonded energies since the non-bonded interactions are calculated between all the particles in the system

and thereby scale as N2, whereas the bonded interactions grow proportionally to N. In order to

reduce the cost of non-bonded interactions a cut-off is often utilized which sets all interactions

with particles outside of this cut-off with particle i to zero. This works well for the LJ interactions since they decrease rapidly with distance, but using a cut-off for electrostatic interactions requires

care since these decrease very slowly and often other methods are used to deal with these. The one often used in MD is particle mesh Evald (PME), which divides the electrostatic interaction

potential into a short-ranged term and a long-ranged term. The short-ranged term is quick to calculate in real space, but this is not the case for the long-range part. So this is solved by

calculating the long range part in the Fourier space instead, which makes the computational cost scale as O(NlogN) instead of O(N

2) as in the real space calculations.

2.3 Algorithms Analytical solutions to the equations obtained in (1) and (2) cannot be found for the large systems that are used in MD, due to the complexity of the potential energy function U. Instead numerical

solution methods have to be utilized, namely finite difference methods. Of these the two most common ones are the Verlet and Leapfrog algorithms.

Verlet The Verlet-algorithm used in molecular dynamics is constructed from a third order Taylor

expansion of the position r(t) at time t+�t and also for t-�t. This results in the following expressions:

� ��

��

�

�

��

��

��

�

�

��

��

��

�� (3)

� ��

��

�

�

��

��

��

�

�

��

��

��

�� (4)

From this one notices that��

�� and

��

�� is the velocity and acceleration of a particle at time t.

The third derivative,��

��, is eliminated by adding the two equations and thereby obtaining:

� � � �� (5)

9

Where a(t) denotes the acceleration at time t. The acceleration can be obtained from the potential energy using equation (1) The error associated with truncating the Taylor expansion after the third term is of the order �t

4

making the Verlet algorithm a fairly accurate method. One problem however is that the velocities are not obtained directly from the equations, but they are needed to calculate important features of

the system such as the kinetic energy. An easy way to obtain them would be to truncate after the second term and subtract (4) from (3):

� � ��

��

�� (6)

But this gives velocities at time t, as opposed to the positions, which are obtained at time t+�t.

And another notable thing is also that the error is proportional to �t2 instead of �t4.

In order to get velocities and positions at the same time one could instead use the so called velocity Verlet algorithm which is constructed by a second order Taylor expansion of the position

r(t):

� � � ��

��

��

�� (7)

And also a second order expansion of the velocity:

� ��

�

��

��

��

�� (8)

Using the forward difference approximation ��

��

� ��

�� one obtains:

� ��

��

�� (9)

This enables us to calculate the velocities and positions of the particles at time t+�t using the

values from time t. The acceleration at time t is calculated using equation (1).

Leapfrog algorithm The other algorithm often used is the leapfrog algorithm, in which the velocities and position “leap” over each other as they are calculated at different time steps:

� ��

�� (10)

� ��

��

�

�� (11)

The leapfrog algorithm is equivalent to the Verlet algorithm, something that is realized by looking at r(t) evaluated in a previous step:

� � � � ��

�� (12)

10

Arranging (12) and inserting into (11) gives:

� ��

��

�

�� (13)

And then (13) and (10) gives:

� � � ��

� (14) Which is identical to (5).

The advantage of the leapfrog method over the Verlet is that the algorithm explicitly calculates the velocities, but the drawback is that they are calculated at a different time.

2.4 Potential of mean force A key concept in computational studies of biological systems is the potential of mean force

(PMF), G(r), which describes the average force acting upon a particle along a certain reaction coordinate (e.g. rotation around bond, binding events). It is defined as:

� � � � ��

��

��

Where �(r) is the average distribution function along the coordinate and G(r*) and �(r*) are arbitrary constants. It can be shown that the PMF is in fact a free energy profile along the reaction

coordinate and therefore is closely related to the distribution function along this coordinate24

. Extracting the distribution function from simulations is however not trivial and the method used in

this thesis will be described below. 2.5 Umbrella sampling First introduced by Torrie and Valleau in 1976

25, umbrella sampling is

a technique that is used to sample a reaction coordinate of a system, which has an energy landscape that makes it hard to use conventional

MD-sampling for this purpose. This is illustrated in figure 4 where the topmost system will probably only yield configurations either to the

left or right of the energy barrier since no barrier crossing will occur, but the bottom one is easily sampled by conventional MD.

The way umbrella sampling solves this problem is by introducing a new potential:

�� (15)

Here W(r) is the introduced potential, and it is designed to bias the simulation around some region in configurational space and thereby

generate a distribution that is non-Boltzmann. Using figure 4 as an example the potential W(r) is designed in such a way that it reduces (or removes) the barrier in the middle, and thereby enables full sampling of the configurational space.

Figure 4. Umbrella sampling.

Top: Sampling of the configurational

landscape along a reaction coordinate

Q in the presence of an energy barrier

Bottom: Umbrella sampling of the

same area will sample configurations over all space

11

This means that the system sampled is not the real system, but despite this, Torrie and Valleau showed that it is possible to retrieve the “true” value of the sought thermodynamic property (here

denoted as S) through the formula:

� �

��

��

(16)

Where ��indicates that the average is taken over the

distribution created by (15).

2.6 Weighted histogram analysis method (WHAM) A more specific use of the method of umbrella sampling is when utilizing a harmonic biasing potential:

� � � �� (17)

Which will result in a relatively local sampling of the

configurational space around the coordinate r0. By doing many simulations with different values of r0 a large set of simulations are obtained, each one centered on a unique r0-value. Recombination of these simulations (or “windows” as they are

called) can then be done in order to calculate the desired properties of the system (illustrative example can be seen in

figure 5). A more thorough description of the way of combining the

windows can be read in reference 26, but the final results are these two equations:

��

��

�

��

��

�

��(18)

��

��

��

�� (19)

Where �(r) is the “real” distribution function, N the number of umbrella simulations, ni and nj the number of independent data points used to construct the biased distributions, �� is the ith

biased distribution function and F represents the free energy associated with the introduction of the potential W(r). As can be seen, equation (18) needs the constants Fj obtained from equation (19) in order to calculate the unbiased distribution function. But equation (19) demands the distribution function

from (18) in order to calculate them, so an initial guess of the constants F is needed and then the equations are solved self-consistently until the changes in F and p(r) are sufficiently small.

Figure 5. Schematic overview of the

WHAM-method.

Top: The unknown potential G(�) along the

coordinate �. Middle: Umbrella simulations centered on different �-values will create a set of

distributions each one influenced by the

local shape of the potential G(�).

Bottom: Recombination of the different

umbrella windows will yield an estimate of

the unknown potential G(�).

Picture taken and modified, with permission, from reference 24

12

3. Methodology

3.1 Lipids and solutes The lipids chosen for the project were:

• 1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE)

• 1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC)

• 1,2-Dimyristoyl-sn-glycero-3-phosphocholine (DMPC)

• 1,2-Dipalmitoyl-sn-glycero-3-phosphocholine (DPPC)

POPE and POPC have the same structural tail region with one unsaturated and one saturated tail. The head region of POPC is however larger than in POPE due to the three methyl groups which

have replaced the hydrogens in the amine-group of POPE. DPPC and DMPC have the same head region as POPC but contain two saturated tails. The

lengths of the tails in DPPC are 16 carbons and those in DMPC are 14 carbons. The different structures of the four lipids enabled studies of how cholesterol would influence the

permeation across lipid membranes with structural differences:

• Head groups: small: POPE large: POPC, DPPC, DMPC

• Tails: saturated: DPPC, DMPC unsaturated: POPE, POPC

long: DPPC short: DMPC

13

Solutes were chosen to be nitric oxide (NO), propane, benzene and neopentane in order to study

the cholesterol effect on hydrophobic (propane, benzene, neopentane) and amphiphilic (NO) solutes. Earlier data from hydrophilic solutes (ammonia in this case) was already available, and is

included in the analysis. The three different hydrophobic solutes also enabled studies of how bulkiness would effect the permeation across the membrane.

3.2 Simulation procedure

Gromacs The simulations performed in this work were done using the Gromacs

31 simulation package.

Gromacs is a MD-software designed for biochemical simulations and it is also the fastest software

available since it handles the non-bonded interactions extremely well.

Membranes The membranes were built by first putting lipids at evenly distributed positions in a quadratic grid, and then copying and inverting this grid in order to get a bilayer. For the membranes containing

cholesterol, lipids at random positions in the grid were exchanged for cholesterol molecules in order to obtain four different cholesterol concentrations for each membrane type in addition to the

pure membranes. Table 1 shows a schematic over the composition of each system. The OPLS (Optimized Potentials for Liquid Simulations) united atom force field

32 and TIP4P

33

model was used for the solutes and water molecules. All lipids except DPPC used the Berger force field

34. For DPPC the parameters from Ulmschneider et al.

35 was used instead. The time step

utilized was 2 fs. Bonds were constrained using the LINCS36

(Linear Constraint Solver) algorithm. Electrostatic interactions were calculated by particle-mesh Ewald and dispersion interactions were

described by a Lennard-Jones potential with a 1 nm cut-off. Three-dimensional periodic boundary conditions were used in all simulations.

Equilibration The membranes were solvated in a box of water and equilibrated for 200 ns (20-40 ns for the pure lipid membranes). The temperature was kept at 300 K (323 K for DPPC in order to avoid a phase

transition to the gel-phase) using velocity rescale, and pressure was kept at 1 bar by the Berendsen barostat.

Table 1. Simulated systems. Composition of each bilayer at each concentration of cholesterol.

X/Y notation shows the number of lipids (X) and the number of cholesterol molecules (Y) in each system. Percentages is based on number of molecules.

Nitric oxide Propane Benzene Neopentane

14

After the equilibration, random snapshots where extracted from the last 20 ns in the trajectory

(last 10 ns for the pure membranes). These were then used as starting structures in the umbrella simulations.

Umbrella simulations The umbrella simulations were done with a spacing of 0.25 Å

between umbrella windows using a harmonic umbrella potential with a force constant of 1000 kJ/mol/nm

2. Temperature was

kept at 300 K by a nose-hoover thermostat, and the pressure was set to 1 bar by a Parrinello-Rahman barostat.

In order to increase the sampling, four solute molecules were constrained in each step, at a distance of one half of the

simulation box away from each other. This ensured that they did not interact with each other. Each simulation also contained 4-5

“slices” with different umbrella windows in order to reduce the total simulation time (example in figure 6). These slices where

put 15 Å apart for the small solutes (propane, NO) and 20 Å apart for the larger ones (benzene, neopentane).

The constrained atoms in the umbrella simulations for each

solute was as follows:

• NO: A dummy atom centered between the oxygen and nitrogen atom

• Propane: The central carbon atom

• Benzene: A dummy atom placed at the center of the benzene ring

• Neopentane: The central carbon atom The small solutes were inserted into the membrane at their respective z-positions making sure that

they are not placed within 1.2 Å of a lipid molecule, in order to avoid bad interactions. The larger solutes could not be inserted in this way so instead they were placed within the membrane using a

smaller tolerance radius (0.3 Å). Thereafter, a short free energy perturbation (FEP) -simulation was done in order to “grow” the solute into the membrane.

Equilibration was done using a conjugated gradient energy minimization, followed by a 1 ns MD simulation.

PMFs The first 200 ps of the umbrella simulations were removed for equilibration purposes and then the

PMFs were calculated by the WHAM-method using Gromacs inbuilt g_wham-tool37

.

Figure 6. Snapshot from umbrella simulation.

Pure POPE bilayer simulated with propane

(black) in 5 different umbrella windows (see

WHAM-section), water molecules are colored blue.

15

4. Results 4.1 Lipid patches The final equilibration of the lipid patches were verified to be converged through the stabilized

total energy plotted against time. All these plots can be seen in appendix A (DPPC with 30% cholesterol and DMPC with 40% cholesterol shown here in figure 7 & 8). Almost all patches seem

to have converged at the end of the equilibration. The exception to this is DPPC with 30% cholesterol, which still shows a downward slope in the energy plot indicating a slow convergence

and that a longer equilibration time is needed. The data from these simulations will be presented in the plots despite this, so the reader is asked to remember that the 30% cholesterol data for

DPPC-membranes could look different if a longer equilibration is done.

4.2 PMFs The PMFs computed for each respective bilayer were put together in plots as seen in

figure 9 (which shows propane in POPE). A plot contains all the PMFs for one solute at

each cholesterol concentration in one bilayer. The center of the bilayer is at zero and the

two distinct regions on each side including the barriers represent the head group region

(at around +/- 2 on the z-axis) and the tail region (further towards the center).

4.3 Umbrella simulations The PMFs calculated from the umbrella

simulations can be found in appendix B, and in order to do a more qualitative analysis the output were divided into data from the head

Figure 8. Equilibration of DMPC.

Plot of total energy from equilibration of DMPC with 40% cholesterol

Figure 7. Equilibration of DPPC.

Plot of total energy from equilibration of DPPC with 30 %

cholesterol

Figure 9. Permeation of propane through POPE.

PMFs for the permeation of propane through POPE. Definition of head- and tail-

region highlighted by boxes. Vertical line shows how values for tail-region-peaks at lower concentrations were approximated for the plots in appendix C.

16

group and tail regions of the lipids. From each of these regions, the highest point of the barrier was

plotted against the cholesterol concentration and these plots can be found in appendix C. As can be seen in figure 9 the tail region at the lower cholesterol concentrations does not have a

clear peak, so in these cases the “highest point” was chosen at the plateau-like region or in the cases where this was not possible the point were chosen by just drawing a vertical line as seen in

figure 9. This of course introduces an uncertainty in the results but since the data analysis is more focused on the profile of the plots rather than the absolute values of each point this should not

impact the end result drastically. In order to get an idea of how the overall permeability is influenced by the cholesterol presence

the highest point of the PMF, the limiting barrier in the permeation, was plotted against cholesterol concentration for all the solutes in all the membranes (figure 10).

Figure 10. Permeation barriers in all membranes.

Plots showing the highest peak in the PMFs plotted against the cholesterol concentration for each solute, in each membrane. Errorbars not shown

in order to get an easy overview.

17

5. Discussion 5.1 Tail region Comparison of the different results from the POPC,

DMPC and DPPC simulations (figure 10) shows that the main difference is how the barrier height

changes with increasing cholesterol content. In the patches with saturated tails (DMPC and DPPC)

there seems to be a fairly constant increase in barrier height with increasing cholesterol concentration, whereas in the unsaturated POPC tail decrease in

barrier height is seen at a cholesterol concentration of 30%. From the plots in appendix C (as an example plots for propane is shown in figure 11) it can be

seen that it is after this cholesterol concentration that the “tail-barrier” overcomes the barrier in the head group region and becomes the rate-limiting barrier that the solute has to cross. In DMPC this

barrier continues to increase as one adds more cholesterol, but in POPC the increase in barrier height is slower when increasing the cholesterol concentration above 30%. Comparing this with

phase transition diagrams for DMPC27

and POPC28

the explanation for the faster increase in DMPC could be a phase transition between the L� and the Lo phase of DMPC that occurs

approximately around 30% cholesterol, whereas the transition in POPC occurs at 50 % cholesterol. A phase transition to the more rigid Lo would mean that the solute has less free

volume available during the permeation and thereby we see this continuing increase in the barrier in DMPC compared to POPC. In DPPC, which undergo a phase transition at approximately 20 %

cholesterol29

, the effect of this phase transition can be seen clearly in figure 10 when one looks at the permeation barrier for ammonia. The unsaturation of one of the tails in POPC also means that

the kink in this tail introduces more free volume for the solutes to use during their diffusion in the membrane.

Figure 11. Head- and tail-barriers for propane

Barrier height at the head- and tail-region plotted against cholesterol content

in the membrane.

Solute: propane

18

5.2 Head region Comparison between POPE and POPC showed no large difference in the barrier at the tail region

(as one should expect since their tail-regions are identical) but a striking difference is the continuous reduction of the barrier in the head region of POPE. The same region in the bilayers

containing PC-head groups showed an increase in the barrier until ~30% cholesterol where it starts to decrease. The larger PC head groups of POPC allow more space between the tails for

cholesterol to position itself within. POPE on the other hand has a rather small head group and therefore the space between the tails is more constricted. An analogy would be to see POPC as a

football with two ropes attached, as tails, and POPE would then be the same construction but with a handball instead.

Inserting cholesterol into the POPE bilayer would then mean that the head groups would be pushed apart from each other as more and more cholesterol are added. In POPC the cholesterol

has additional space between the head groups, that it can populate before “saturation” is achieved and the head groups are pushed apart. This increase in the barrier at the head region, which is seen

in the PC head groups, is due to the condensing effect that cholesterol has on the membrane. This effect can be seen in figure 12, which shows the area per lipid plotted against cholesterol

concentration in the membrane. The area per lipid for a pure membrane is calculated by dividing the total area of the bilayer with the number of lipids in one leaflet. When cholesterol is present

this is however not as simple, since the area occupied by cholesterol has to be accounted for. This is solved by subtracting the area occupied by cholesterol from the total area of the membrane. The

average area of cholesterol in bilayers does however vary a lot between different studies17

, but the values calculated for cholesterol in bilayers vary between 22-29 Å

2, which does not change the

main features of the plot. So for this analysis the average value of 27 Å2 calculated by Hofsäß et

al.30

was used. To see the different behavior of the PC head group compared to the PE head group

figure 12 also contains the area per lipid without correction for the cholesterol in the membrane in order to see the swelling of the membrane as cholesterol is added. POPE starts to swell directly as

cholesterol is added whereas the PC-containing lipids show an increased “resistance” towards swelling as cholesterol is incorporated.

Figure 12. Area per lipid in all membranes. Left: Area per lipid for all membranes with correction for the area occupied by cholesterol. Right: Area per lipid

without correction for cholesterol.

19

5.3 Solutes Comparing the different PMFs for all the solutes (appendix A) the largest difference between the

hydrophilic ammonia and the hydrophobic solutes was the single barrier over each leaflet for ammonia, whereas the hydrophobic solutes starts out with a barrier over the head group region and

then, at higher cholesterol concentrations, actually have the rate limiting barrier located in the hydrophobic tail region. Ammonia has the largest barrier at all cholesterol concentrations in all the

membranes except in pure POPE where the head group barrier for neopentane is even higher (figure 10). This is quite interesting since it shows that the hydrophilic region at the lipid-water

interface can actually be as rate limiting for hydrophobic solutes as the hydrophobic interior is to hydrophilic solutes.

Propane and benzene behaves roughly the same in all membranes except DMPC where the reduced free volume, due to the phase transition, at higher cholesterol concentrations seemed to

influence the larger benzene molecule more than propane. This is however not the case in DPPC where they again behave roughly similar, with benzene having some large outliers.

The permeation barriers for neopentane is however larger than those for benzene and propane in almost all simulations (one large outlier in POPC with 40% cholesterol), and the overall behavior

for the different cholesterol concentrations is the same. This similarity between propane and benzene, but dissimilarity when comparing to neopentane is interesting. It indicates that benzene

despite its more bulky structure still manages to permeate the membrane with relative ease compared to propane. Neopentane shows a similar permeation barrier in the pure PC-membranes,

but with added cholesterol the barrier directly increases more than those of propane and benzene. The reason for this should be that neopentane, being even more bulky than benzene, is directly

affected of the more condensed head group regions created by the addition of cholesterol, something that can be seen in the plots in appendix C. This is also illustrated by neopentane’s

permeation barrier in pure POPE, which has a fairly condensed head group region in the pure membrane compared to the PC-membranes.

The small amphipathic nitric oxide has an overall low permeability barrier, as could be expected since it should be able to cross both the head and tail region with relative ease. Comparing it to the

hydrophobic solutes the only real difference is the permeation through pure POPE, were NO has a much smaller permeation barrier. This is due to the hydrophilic barrier created by POPE, which is

larger than those of the PC-bilayers.

6. Conclusion Four different solutes were used in simulations in four different bilayers at five different

cholesterol concentrations. The effect of cholesterol on the permeation through the lipid bilayers was studied through the calculated PMFs of the permeation event. Incorporation of cholesterol

was found to shift the rate-limiting barrier of permeation for the hydrophobic solutes; a barrier at the tail region replaced the hydrophilic barrier at the head groups and became the rate-limiting

factor. In the three PC-bilayers, incorporation of cholesterol increased the permeation barrier for hydrophobic solutes by condensing the membranes and thereby reduce the amount of free volume

available. In POPE, where the head groups are much smaller than in PC-bilayers, an opposite effect was observed as the permeation barrier was decreased upon addition of up to 30%

cholesterol. This shows that the head group size is a deciding factor of how cholesterol will change the behavior of a lipid bilayer. It also demonstrates that cholesterol can have an increasing

effect on the permeability of hydrophobic solutes across specific lipid bilayers, which is very interesting since research often is focused towards cholesterols ability to decrease permeability

across membranes. Lipid bilayers using a single type of lipids can be utilized to study cell permeability to some

20

extent, but the membrane in a cell consists of a variety of lipids, distributed in an asymmetric way

between the outer and inner leaflet. Future simulations focused on membranes with an asymmetric lipid composition are of interest, since the effect of cholesterol on those membranes could give

some new insights to why this asymmetry is maintained by the cells.

7. Acknowledgements I would like to thank Dr. Jochen Hub and Prof. David van der Spoel for their guidance and help throughout the whole project.

21

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Appendix A Shown here is the total energy during the equilibration of the lipid patches. Energy is shown in

kJ/mol and is plotted against the simulation time in ps. Running average over 1000 data points shown in red.

Pure POPE: POPE with 20% cholesterol

POPE with 30% choelsterol POPE with 40% cholesterol

POPE with 50% cholesterol

24

Pure POPC POPC with 20% cholesterol

POPC with 30% cholesterol POPC with 40% cholesterol

POPC with 50% cholesterol

25

Pure DMPC DMPC with 20% cholesterol

DMPC with 30% cholesterol DMPC with 40% cholesterol

DMPC with 50% cholesterol

26

Pure DPPC DPPC with 20% cholesterol

DPPC with 30% cholesterol DPPC with 40% cholesterol

DPPC with 50% cholesterol

27

Appendix B PMFs for all solutes in all membranes. Each plot shows five PMFs for each solute in each

membrane, one for each cholesterol concentration. Center of the membrane is located at z=0, and head group regions at approximately +/- 2.

POPE

Nitric oxide Propane

Benzene Neopentane

Ammonia

28

POPC


Benzene Neopentane

Ammonia

29

DMPC

Propane Nitric oxide

Benzene Neopentane

Ammonia

30

DPPC


Benzene Neopentane

Ammonia

31

Appendix C Plots showing changes in the permeation barrier in the head- and tail-region of the lipid patches,

for each solute, at different cholesterol concentrations. Head group region shown in orange, tail region in yellow.

Nitric oxide

32

Propane

33

Benzene

34

Neopentane

energetics of cholesterol-modulated membrane...

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