Molecular Dynamics simulations of lipid bilayers

Molecular Dynamic Simulations to

Probe Interactions of Buffer Molecules

with Lipid Bilayers

A PROJECT REPORT

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF ENGINEERING

BY

MOHAMMAD SIRAJUDDIN

DEPARTMENT OF CHEMICAL ENGINEERING,

INDIAN INSTITUTE SCIENCE, BANGALORE

JUNE-2015

i

Declaration

I verify that all chapters in the report were written by myself, and during the course of

writing the report:

i. Simulations results presented have been obtained by me without any bias,

modification and alteration. One can obtain the identical results using the

information provided in the report to draw similar conclusion.

ii. I have not copied or edited any form of content or chapter from published or

unpublished sources. However figures, theory or any other material which was

essential during report writing and was not derived, obtained or written by me

was given due credit by citing it in text of the report and details have been shared

in the reference section.

iii. Ethical issues have been taken into consideration while formatting and shaping

the report so as to accord with the departmental guidelines.

Mohammad Sirajuddin

ii

Acknowledgements

Research work presented in this report is the outcome of invaluable guidance, suggestions

and kind support of many adroit and elite peoples from my department. First and foremost

I express sincere and enormous gratitude to my advisor Prof. K Ganapathy Ayappa, whose

adept knowledge and dexterous advice enabled me to relish and explore the diverse aspects

of my project. His affable and cheerful personality attest instrumental during my project

course to outreach a trite milestone. I further extend my appreciation to astounding lab

members Ayush, Pradeep, Vadhana, Rajshekhran, Subbarao and Rajat for pleasant and

informal conversation during numerous lunches and meetings. My special thanks to Rajat

for his continuous assessment and umpteen support during my project in the form of

discussion, chats and constructive critics which sculpted my project to its final shape in

due time.

I am grateful to all the faculty members from my department for their enlightenment and

professionalism during entire program. Many thanks to my class mates for their

cooperation and amity which moulded the meticulous and burdensome Master’s program

into adorable and zealous environment. Extra special thanks to my camaraderie

Satyaghosh, Shubhashish and Jitendra for nurturing and weaving golden memories during

my stay at IISc. Jitendra’s companionship from my very first question “where is chemical

engineering department?” till leaving department has imprinted unexplainable memories.

Innocence of Satyaghosh with crazy Subhashish’s quest for haunting new restaurants on

iii

every weekend added blend of flavours and textures to my M.E. tenure. I owe my every

achievements to you nutty guys!

I’d like to convey my heartfelt thanks to all colleagues in the department for wonderful

trips and peaceful stay. My acknowledgement to AICTE for scholarship during M.E.

program, adequate to pay off my bills and hunger. Finally, I express intense gratitude

towards the hallowed portals of IISc, which manifested sheer and obscure arena, buzzing

with soothing sound of birds and breezing through the shadows of green garment, entitled

my inner self to invoke “Yes… Heaven in here!”

To the end, I express deepest appreciation to all my near and dear who believed this naive

person with uttermost encouragement which propelled my research project to fruition.

iv

Abstract

Buffers are very widely used in biophysical and biochemical studies to maintain a desired

pH. The effect of buffers on the lipid bilayers is often ignored or assumed to be negligible,

since buffers which are generally hydrophilic are not expected to interact strongly with the

membranes. However recent literature has shown that buffers in presence of other salt

molecules can interact with lipid bilayers through van der Waals forces, electrostatic and

hydration forces and alter the bending rigidity of the bilayer membrane [4, 5].

In order to obtain a molecular understanding of these interactions and their influence on

membrane properties, we carried out a molecular dynamic simulation of Hydroxy ethyl

piperazine ethane sulfonic acid (HEPES) buffer on a 1,2-dimyristoyl-sn-glycero-3-

phosphocholine (DMPC) lipid bilayer. Initially we carried out ab-initio calculations at the

Hartree–Fock (HF) level of theory to construct an electrostatic potential map and derived

charges using the restrained electrostatic potential (RESP) fitting procedure. These charges

are used in classical force fields to carry out molecular dynamic simulations. The structural

properties like radial distribution function, density distribution are presented at different

buffer concentrations.

iii

Contents

Chapter 1 ........................................................................................................................... 1

Introduction ....................................................................................................................... 1

1.1 Cell ............................................................................................................................ 1

1.2 Cell Membrane.......................................................................................................... 2

1.3 Lipid Bilayer ............................................................................................................. 3

1.4 Role of buffers in cell membrane.............................................................................. 5

1.4 Problem definition .................................................................................................... 6

Chapter 2 ........................................................................................................................... 8

Molecular Dynamics ......................................................................................................... 8

2.1 Introduction ............................................................................................................... 8

2.2 Simulations … How it works? .................................................................................. 8

Chapter 3 ......................................................................................................................... 14

Molecular dynamics simulations of DMPC .................................................................. 14

3.1 Introduction ............................................................................................................. 14

3.2 Simulation methods and details .............................................................................. 16

3.3 Results and discussion ............................................................................................ 17

Chapter 4 ......................................................................................................................... 22

Ab-initio calculations and simulation of HEPES .......................................................... 22

4.1 Introduction ............................................................................................................. 22

4.1.1 HEPES ................................................................................................................. 23

4.1.2 Charge derivation for HEPES .............................................................................. 24

4.1.3 Molecular geometry optimization and partial charge .......................................... 29

4.2 Simulation of HEPES in water ............................................................................... 33

4.2.1 Analysis................................................................................................................ 34

Chapter 5 ......................................................................................................................... 37

Simulation of DMPC in presence of buffer molecules ................................................. 37

5.1 Introduction ............................................................................................................. 37

5.2 Simulation Method.................................................................................................. 37

5.3 Results and Discussion ........................................................................................... 38

5.3.1 Mass density......................................................................................................... 38

5.3.2 Hydrogen bonds ................................................................................................... 41

5.3.3. Area per lipid ...................................................................................................... 46

5.3.4 Radial distribution function ................................................................................. 48

Chapter 6 ......................................................................................................................... 50

Umbrella sampling .......................................................................................................... 50

6.1 Introduction ............................................................................................................. 50

6.2 Steps in Umbrella techniques.................................................................................. 53

6.3 Results and discussion ........................................................................................... 54

Chapter 7 ......................................................................................................................... 59

Conclusions and suggestions for future work .............................................................. 59

7.1 Conclusions ............................................................................................................. 59

7.2 Future work ............................................................................................................. 60

References………………………………………………………………………………61

Appendix A…………………………………………………………………………….64

Appendix B……………………………………………………………………………65

Appendix C……………………………………………………………………………...70

List of Figures

1.1 Description of typical eukaryotic cell .....................................................................2

1.2 Structure of phospholipids ......................................................................................4

2.1 Bonded and non-bonded interaction terms in force field .......................................10

3.1 Structure of Dimyristoyl phosphatidylcholine (DMPC) ........................................15

3.2 Snapshot of DMPC in VMD .................................................................................18

3.3 Membrane thickness variation as a function of simulation time ...........................19

3.4 Order parameter comparision from simulations and experiments .........................20

4.1 Chemical structure of HEPES ................................................................................24

4.2 Comparison of optimised and experimental bond length,angles and dihedrals for

HEPES molecule ................................................................................................................33

4.2 HEPES structure after geometry optimisation .....................................................33

5.3 Density distribution of the buffer molecules at different concentrations .............41

6.3.1 Snapshots from the simulation showing the pulling of HEPES molecule ............54

6.3.2 Umbrella histograms .............................................................................................56

6.3.3 Free energy profile for insertion of HEPES from aqueous phase to lipid phase .57

List of Tables

3.1 Physical properties of DMPC obtained from Avanti polar lipid………………......15

4.1 Physical properties of HEPES………...…………………………………………...24

4.2 List of some commonly used Good’s buffers……………………………………..26

5.3.2 Average number of different hydrogen bonds in the system ……………………45

5.3.3 Area per lipid and bilayer thickness of DMPC ……….……...………………….47

5.3.4 Coordination number at different buffer concentrations………………………....49

1

iv

Chapter 1

Introduction

1.1 Cell

Cells are the primary building blocks of living organisms. Biologists have divided cells

into two primary types: eukaryotic and prokaryotic cells. Eukaryotic cells are characterised

by the presence of a well-defined nucleus. The nucleus, which houses DNA, is contained

within a membrane, separated from other cellular structures. Prokaryotic cells however

have no true nucleus. DNA in a prokaryotic cell is not separated from the rest of the cell

but coiled up in a region called the nucleoid. Nevertheless independent of its type, cells are

the basic structural, functional and biological unit of all known living organisms including

viruses which can replicate independently.

Every cell is enclosed by a membrane which gives structure to the cell and allows for the

passage of nutrients and metabolic wastes into and out of the cell. The cell membrane is

made up of the bilayer which is predominantly made up of lipids with their hydrocarbon

tails facing inwards. The cell membrane typically contains other molecules such as

carbohydrates and proteins, which serve as receptor sites for other messenger molecules.

Interactions with the cell membrane allows molecular signalling events which

communicate with processes that occur within the cell [5].

1.2 Cell Membrane

Cell membranes are rich in two classes of molecules: lipids and proteins. Proteins serve as

enzymes, transport molecules, and provide the membrane with distinctive functional

properties while lipids provide the structural integrity to the cell as discussed in Figure 1.1.

Lipids found in the cell membrane consist of two parts: hydrophilic (water soluble) and

hydrophobic (water insoluble). The hydrophobic portion of the lipids is the non-polar long

Figure 1.1: Structure of eukaryotic cell membrane showing different types of proteins and phospholipids.

Both the exterior and interior region of the cell membrane are multicomponent made up of a large number

of different types of molecules. [1]

hydrocarbon chains of two fatty acids. The fatty acids are present as esters bonded to

glycerol. The third-OH group on glycerol is ester bonded to phosphate hence the term

phospholipid. The phosphate ester portion of the molecule is polar or even ionic and hence

is water soluble.

1.3 Lipid Bilayer

At the macroscopic (cellular) level, the membrane are modelled as two dimensional layers,

covering a cell and appears as a “fluid mosaic model”, rich in complexity along with a

heterogeneous distribution of lipids forming a matrix in which proteins are embedded.

The bilayer structure which has a thickness of 3-4 nm results spontaneously from lipid self-

assembly, driven by the hydrophobic effect. Arrangement of lipid bilayers in cell

membranes can be derived from X-Ray diffraction data. Fig 1.2 shows the chemical

structure of phospholipids in the lipid bilayer. Animal cells are arranged as a bilayer

stacked with the non-polar hydrocarbon chains pointed inward while the polar ends act as

the external surface facing the surrounding aqueous environment. The hydrophobic layer

of the cell membrane acts as a barrier for ionic and polar molecules from directly entering

inside of the cell.

Eukaryotic cell are characterised by the presence of sterol (eg. cholesterol), inserted

between the non-polar chains, and makes up about 20% of the molecules of the membrane.

This helps to make the membrane more rigid and adds to its physical strength. Apart from

this, sterols and glycerophospholipids account for about 85-90% of the total lipids in

membrane with sphingolipids (eg. sphingomyelin) accounts for majority of the remaining

fraction.

Figure 1.2: Structure of a typical phospholipid in the lipid bilayer. Figure shows chemical structure of

hydrophilic and hydrophobic tail which is produced from esterification of fatty acids and glycerol [2] .

The prominent function played by lipids can be categorised into three parts [6]. First they

provide the barrier for passive diffusional motion of small polar solutes like ions, sugar and

low molecular weight metabolites as well as all macromolecules such as proteins, nucleic

acid and other poly saccharides.

Secondly they provide a unique solvation environment for trans membrane protein. Finally

third and less studied is their role in the internal organisation of the cell, which accounts

the major fraction of cell function. Thus regulating various chemical processes that occur

in the cell.

1.4 Role of buffers in cell membrane

Buffers plays a crucial role in maintaining pH of a medium by resisting changes in H+ ions

and OH- ions concentrations. In the human body, blood plasma has a buffer mixture of

carbonic acid and hydrogen carbonate ions which keeps the pH constant and avoid acidosis

or alkalosis, a condition resulting due to an increase or decrease in the pH. Moreover in

the laboratory, buffers are used to maintain the pH during cell culture or bacterial growth.

In biophysical and biochemical studies of lipid bilayers the influence of buffer is often

ignored or assumed to be negligible on membrane structure, elasticity, or other physical

properties. However, experimental observations on giant unilamellar vesicles [7] suggest

that buffering molecules may considerably affect the bending rigidity of

phosphatidylcholine bilayers. Furthermore, a synergistic effect on the bending modulus is

observed in the presence of both salt and buffer molecules, which serves as a warning to

experimentalists while data interpretation of their studies, since typical lipid bilayer studies

contain buffer and ion molecules.

1.4 Problem definition

In this project work we focused our attention towards the specific buffer molecules and

their interactions with the lipid bilayers by using atomistic molecular dynamic simulations.

We choose zwitterionic buffer molecules that are commonly used as buffering agents while

carrying out experiments and the DMPC lipid bilayer as a prototypical lipid bilayer. All

atom molecular dynamics simulations are performed to study effect of the buffer on bilayer

structural properties such as radial distribution functions and density distribution as a

function of buffer concentrations In addition we also carry out potential of mean force

calculations to investigate the binding affinity of the buffer molecules with the bilayer

membrane. All simulations were performed using GROMACS molecular dynamics

simulation package which is an open source molecular dynamics code under GNU Public

License (GPL).

The chapters in the report are organized in the following manner. Chapter 1 introduce

reader with the cell and its role in our body followed by research problem definition. In

Chapter 2, we describe molecular dynamics simulation methods and introduce the different

methods for solving Newton’s equation of motion. In Chapter 3, we simulate the pure

DMPC lipid bilayers in order to test our simulation protocols. Preliminary results of these

simulations are presented to validate our method. In Chapter 5 we present simulations of

the DMPC lipid bilayer in presence of different buffer molecules at various concentrations.

Density distribution at different buffer concentration, hydrogen bonding and radial

distribution functions are also presented. In Chapter 6, we performed free energy

calculation for the penetration of buffer molecules inside the lipid bilayers. A brief theory

for biased umbrella sampling is presented in this chapter. Free energy profile along bilayer

normal (z-direction) is computed to obtain the free energy landscape of the buffer-bilayer

interactions, enabling us to comment on the thermodynamically most favourable state.

Chapter 2

Molecular Dynamics

2.1 Introduction

Lipid bilayers dynamics vary over large length and time scale. A typical bilayer thickness

in cell membrane ranges from 3 to 4 nm with bilayer undulations ranging from 4 Å to 0.25

mm. Bond vibrations occur on the femtosecond time scales and lipid lateral diffusion

occurs on the nanosecond time scale. Various undulation modes range from nanoseconds

to milliseconds time scales. Hence the lipid dynamics occurring at these length and time

scale not only elude bare eyes but also ordinary microscope. Therefore power of molecular

dynamics simulation comes into picture, which has an ability to scan out the atomic level

description and orientations of atoms and their interaction. In this chapter the method of

molecular dynamics is described along with some commonly used force fields which are

used in all atom molecular dynamic simulations.

2.2 Simulations … How it works?

Molecular dynamics simulations involves numerically integrating Newton’s equation of

motion [8] thereby generating trajectories obtained from solving forces derived from an

appropriate interaction potential, assuming that the molecules follow classical mechanics.

The equations of motion for an assembly of N particles with positions 𝑟𝑖, momenta 𝑝𝑖 and

masses 𝑚𝑖 can be expressed as follows:

𝜕𝑟𝑖

𝜕𝑡=

𝑝𝑖

𝑚𝑖= 𝑣𝑖 (2.1)

𝜕𝑝𝑖

𝜕𝑡= 𝐹𝑖 = −

𝜕𝑈(𝑟𝑖)

𝜕𝑟𝑖 𝑤ℎ𝑒𝑟𝑒 𝑖 = 1,2 … . . 𝑁 (2.2),

and 𝐹i is the total conservative force on 𝑖th particle due to all other particles.

The non-bonded interaction potential for soft spheres is the Lennard-Jones 12-6 potential,

𝑢(𝑟𝑖𝑗) = 4휀 [(𝜎

𝑟)

12

− (𝜎

𝑟)

6

] (2.3)

The total potential energy 𝑈 can be written as

𝑈 = ∑ ∑ 𝑢(𝑟𝑖𝑗) (2.4)

𝑁

𝑗=1𝑖>𝑗

which is the sum of interaction energies on the 𝑖th particle due to all other particles, 𝑟𝑖𝑗 is

the distance between the two particles, ε and σ are the depth of the potential well and

distance at which the inter-particle potential is zero respectively. 𝐹𝑖 is related to 𝑈 by the

equation 2.2. The interaction potential 𝑈 contains parameters which are taken from

experiments and/or quantum mechanical calculations.

There are various algorithms for solving these equations of motion including Verlet, leap-

frog, Velocity-Verlet algorithms etc. One of the popular methods for the numerical

integration is the Velocity-Verlet algorithms, which is described below,

𝑟𝑖(𝑡 + 𝛿𝑡) = 𝑟𝑖(𝑡) + 𝛿𝑡𝑣𝑖(𝑡) +𝛿𝑡2

2𝑚𝑖 𝐹𝑖(𝑡) (2.5)

𝑣𝑖(𝑡 + 𝛿𝑡) = 𝑣𝑖(𝑡) +𝛿𝑡

2𝑚𝑖 [𝐹𝑖(𝑡) + 𝐹𝑖(𝑡 + 𝛿𝑡)] (2.6)

Figure 3.1: Bonded and non-bonded interaction terms in the force field. Right figure

shows the non-bonded interaction potential and consist of columbic interaction (bottom

right) and Lennard-Jones potential (top right). Left figure shows the bonded interaction

which consist of bond angle, length (bottom left) and dihedral terms (top left) [4] .

Here 𝑣𝑖(𝑡) is the velocity of the ith particle and a trajectory of velocities is obtained with an

increment of 𝛿𝑡 time interval. For more complex molecules and environments, like lipid

bilayer with embedded proteins in addition to intermolecular forces there will be

intramolecular force as well. Figure 3.1 shows the potential energy for bonded and non-

bonded interactions which can be written as a sum of different interaction energies as

follows:

𝐸 = 𝐸𝐿𝐽 + 𝐸𝑐𝑜𝑢𝑙𝑜𝑚𝑏 + 𝐸𝑏𝑜𝑛𝑑𝑙𝑒𝑛𝑔𝑡ℎ + 𝐸𝑏𝑜𝑛𝑑𝑎𝑛𝑔𝑙𝑒 + 𝐸𝑡𝑜𝑟𝑠𝑖𝑜𝑛𝑎𝑙 (2.7)

𝐸𝐿𝐽 is the soft sphere Lennard-Jones non bonded interaction potential and 𝐸𝑐𝑜𝑢𝑙𝑜𝑚𝑏 is the

electrostatic non bonded interaction between fixed charges.

The functional form can be described as follows:

𝐸𝐿𝐽 = 4 ∑ 휀𝑖𝑗

𝑖,𝑗

[(𝜎𝑖𝑗

𝑟𝑖𝑗)

12

− (𝜎𝑖𝑗

𝑟𝑖𝑗)

6

] (2.8)

where, in this equation 휀𝑖𝑗 gives the strength of interaction between non bonded

atoms and 𝜎𝑖𝑗 is a distance parameter at which potential is zero and 𝑟𝑖𝑗 is the distance

between atoms i and j.

𝐸𝑐𝑜𝑢𝑙𝑢𝑚𝑏 = ∑𝑞𝑖𝑞𝑗

4ᴨ휀0𝑟𝑖𝑗 (2.9)

𝑖,𝑗

where 𝑞𝑖 and 𝑞𝑗 are the effective charges on the atoms and 휀0 is the free space permittivity

Similarly the functional form of the bonded interaction can be described as follows:-

𝐸𝑏𝑜𝑛𝑑𝑙𝑒𝑛𝑔𝑡ℎ = ∑𝑘𝑏

2(𝑟𝑖𝑗−𝑙𝑖𝑗)2

𝑖,𝑗

(2.10)

where, 𝑘𝑏 is spring constant and 𝑙𝑖𝑗 is a equilibrium value of length between atoms

i and j which are bonded.

The bond-angle term is a three body potential and can be written as a sum of harmonic

potentials between three atoms.

𝐸𝑏𝑜𝑛𝑑𝑎𝑛𝑔𝑙𝑒 = ∑𝑘𝛩

2(𝛩𝑖𝑗𝑘−𝛩0

𝑖𝑗𝑘)2

𝑖,𝑗,𝑘

(2.11)

where 𝛩𝑖𝑗𝑘 is the angle formed by i,j and k atoms and 𝛩0𝑖𝑗𝑘 is the equilibrium value of

angle.

Torsional or dihedral angle potential describes the rotational flexibility of the chains and

one of the popular functional form is given below.

𝐸𝑡𝑜𝑟𝑠𝑖𝑜𝑛𝑎𝑙 = ∑ ∑𝑣𝑛

2(1 ± cos (𝑛𝜏𝑖𝑗𝑘𝑙))2 (2.12)

𝑛𝑖𝑗𝑘𝑙

where 𝜏𝑖𝑗𝑘𝑙 is the angle between the planes formed by atoms i, j and k and atoms j, k and l.

The first sum is over all dihedral angles. Integer n is typically a value between one and

three.

It is worth mentioning that there are no unique force fields describing all kinds of lipids

bilayer systems. In fact depending on our working environment and the types of lipids we

need to carefully select the force fields for our system. To name some of the well-known

atomistic force fields in literature are CHARMM, AMBER, OPLS-AA, OPLS,

GROMACS87, GROMACS96, GROMOS.

In these CHARMM, AMBER and OPLS-AA are all-atom force fields and consider

hydrogen atoms explicitly in the simulations. Others are united atom force fields and do

not consider non-polar hydrogen’s attached to aliphatic carbon atoms due to their light

weight. Both types of force fields are known as atomistic force fields and are used

depending on the nature of problem.

To eliminate edge effects and mimic a macroscopic system, we take a help of a simulation

box having a fixed number of molecules replicated by using three-dimensional periodic

boundary conditions (PBCs). In the case of a membrane simulation, application of PBCs

to a small membrane portion generates an infinite multi lamellar system. The constant

pressure and temperature (NPT) or isothermal-isobaric ensemble is particularly useful for

membranes because it gives the possibility for validating simulations results by checking

some experimentally reported parameters. For example the phase behaviour can be

analysed from order parameters and dynamic structure factors which can then be compared

with published literature results.

Chapter 3

Molecular dynamics simulations of

DMPC

3.1 Introduction

For decades, biophysicists have used dimyristoyl phosphatidylcholine (DMPC) bilayers as

model system to study the structural and dynamics of the lipid membrane as well as

understanding the interaction of small molecules and proteins with the lipid membrane. For

the most part, DMPC’s popularity as a model membrane system can be attributed to the

fact that it is stable, inexpensive, and easy to obtain. Importantly, the bilayers that it forms

have physical properties not dissimilar to those found in biological membranes (e.g., liquid

crystalline order, hydrophobic thickness, etc.). In this chapter our focus is to perform

molecular dynamic simulations with pure DMPC lipid bilayers and to compare the results

with the experimental studies.

DMPC lipid bilayers have a net zero charge with a positive charge on nitrogen atom and a

negative charge on phosphate group. The zwitterionic structure of DMPC molecule is

shown in Figure 3.1 and its physical properties are given in table 3.1. In this study we have

simulated DMPC bilayer at 370C which is well above its transition temperature of 240C. In

order to confirm it liquid crystalline phase (Lα) we have computed structural properties

like acyl tail order parameter and the area per head group which shows good agreement

with experimental values.

Figure 3.1: Chemical structure of dimyristoyl phosphatidylcholine (DMPC)

Molecular formula C36H72NO8P

Molar mass 677.933 g mol−1

Transition Temperature 24oC

Percent composition (w/w%) C-63.78%,H-10.70%,N-2.07%,O-18.88%,P 4.57%

Table 3.1 Physical properties of DMPC lipid bilayers obtained from Avanti polar lipid

3.2 Simulation methods and details

Simulations were carried out in the constant pressure, temperature and number of

molecules (NPT) ensemble with periodic boundary condition in all the three direction using

the GROMACS molecular simulation package [9].

The DMPC bilayer was taken from Stockholm Lipids (S-Lipid) [10]website which consist

of 128 lipid molecules. We have maintained 30 waters per lipid so as to mimic fully

hydrated system. Hydration was carried out using the TIP3P (transferable intermolecular

potential 3P) water model. The total system size was 63.2 × 63.2 × 63.6 Å. Dynamics were

propagated using a leapfrog integrator with time steps of 2 femtoseconds and the

coordinates were saved after every 2 ps. MD Simulation were performed in three stages:

first energy minimisation in order to avoid any repulsive or steric contacts followed by

NVT and NPT ensemble. We have chosen a temperature of 310 K to ensure system

temperature is well above its transition temperature of 297 K. The system temperature was

maintained at 310 K using the Nosé-Hoover thermostat and pressure was maintained at 1

atm using Parrinello-Rahman barostat with semi isotropic pressure coupling. Particle mesh

Ewald summation was used to account for the conditionally convergent long-range

electrostatic interactions. During simulations all bonds were constrained using LINKS

algorithms [11] in order to achieve longer run.

3.3 Results and discussion

Figure 3.2 shows snapshot of simulation box which consists of 128 DMPC lipid molecules

and 3840 water molecules. We have maintained 30waters per lipid molecules in order to

confirm bilayer is well above its critical hydration number of 18 water per lipid [12]. In a

simulation box with three dimensional periodic boundary conditions, bilayer d-spacing is

simply the box dimension in the direction perpendicular to the bilayer membrane. As the

system under study contain single lipid bilayers in a box, the total length of box in the

direction of bilayer normal can be considered as the d-spacing. Thus 𝐿𝑧 is the d-spacing in

our study.

The properties of a lipid bilayer are commonly described using the area per lipid, bilayer

thickness and acyl chain order parameters. These properties also help us understand the

phase of the lipid at a given temperature.

The average area per lipid is computed by taking mean value of the XY box vectors and

dividing by the total number of lipids. This is found to be 61.58 Å2 which is in good

Figure 3.2 Snapshot of simulation box which is drawn from VMD (Virtual Molecular

dynamics) simulation package. The simulation is carried out with 128 DMPC lipid

molecules with 64 in the upper leaflet and 64 in the bottom leaflet. Total number of water

molecules in a box are 3840 in order to ensure bilayer is well above its critical hydration

number. Figure illustrate the d-spacing (Lz) which is length of z-coordinate in a

simulation box with periodic system.

agreement with experimental value of 60.6 Å2 obtained from X-ray diffraction experiments

at 310 K [13]

Thickness of the bilayer is not uniform and shows thermal fluctuations in XY plane of

bilayer. In GROMACS membrane thickness is obtain by tagging a reference atom in the

head group usually phosphorus atom both in upper and lower leaflet and using the script

g_dist tool to obtain the distance between them. Figure 3.3 shows the variation of bilayer

thickness with average value of 3.016 nm which is in good agreement of reported value of

3.67 nm [13].

Figure 3.3: Membrane thickness variation as a function of simulation time with a mean

value of 3.016 nm.

Lipids in a fluid bilayer are highly dynamic. Many movements on different timescales take

place: rotation around chemical bonds and Trans/gauche isomerisation (picoseconds),

rotation (axial diffusion) around the lipid axis (nanoseconds), lateral diffusion

(microseconds), flip-flop across the bilayer (millisecond) and undulation of the membrane

(milliseconds to seconds). Most of these movements influence the order parameters of the

acyl chains

Lipid acyl chain order parameters are obtained easily from deuterium NMR experiments

and can be compared with simulation. In simulation lipid order parameters are a defined as

Figure 3.4: figure 3.4(a) shows the experimentally measured order parameter and figure

3.4(b) to the right shows corresponding results for simulati on. Series1 and Series2 are

values for SN1 and SN2 tails of DMPC respectively.

𝑆 = ⟨3 cos 𝜃2 − 1

2⟩ (3.1)

where 𝜃 is the (time dependent) angle between the C–H bond vector and a reference axis

which Z coordinate in our case. Figure 3.4 shows both simulated and reported values for

order parameter. S = 1 means perfect alignment with the bilayer normal, S= −0.5 anti-

alignment, and S = 0 random orientation.

Chapter 4

Ab-initio calculations and simulation of

HEPES

4.1 Introduction

Buffer solutions are essential to maintain the desire pH in organisms to function properly.

Many enzymes and proteins work only under very precise pH conditions; if the pH moves

outside a narrow range, enzyme activity can be slowed or stopped completely, protein

unfolding can occur leading to denaturing the cell. In many studies pH is adjusted outside

the working range in order to bring denaturation or cell-lysis. Until 1966 biologists used

buffers made up of carbonic acid (H2CO3) and bicarbonate (HCO3−) which were not very

effective in maintaining a neutral pH due to their low pKa values [14].

The rapid development of molecular biology prompted biologists to search for effective

buffering agents, which resulted in synthesis of a number of different buffer compounds

that can effectively maintain the physiological pH range of 7.0–8.0. A set of 12 of them

were described by Norman Good and co-workers in 1966 as suitable for biological

applications and since then have been described as the ‘Good’ buffers. Table 5.2 describes

a set of some commonly used Good’s buffesr. Good’s buffers apart from maintaining

desired pH, exhibit other experimentally useful properties like resistance to enzymatic

degradation, lack of UV absorbance, lack of interference with biological assays and very

limited cell-wall permeability. In our study we have first chosen HEPES buffer and

monitored its effect on a DMPC lipid bilayer[14].

In this Chapter we will discuss the derivation of partial charges for HEPES molecule from

ab-initio calculations. We have validated the molecular structure of the HEPES molecule

from experimental data and carry out molecular dynamic simulations in order to check its

stability in water.

4.1.1 HEPES

HEPES (HydroxyEthylPiperazineEthaneSulfonic acid) is a zwitterionic organic buffering

agent and included in one of the twenty mentioned Good's buffers. HEPES is commonly

used in cell culture because of its ability in maintaining physiological pH during cellular

respiration as a result of the variation in carbon dioxide concentration. Figure 4.1 and table

4.1 describes the chemical structure and physical properties of HEPES. During simulations

we have maintained a neutral pH in order to keep pKa and pH corresponding to

experimental conditions where the buffer performance is optimal.

4.1.2 Charge derivation for HEPES

In order to perform simulations on any given molecule, it is necessary to obtain values for

partial charges on each atom, its geometrical parameters such as bond lengths, angles and

dihedrals. Moreover a reliable force field has to be selected which involves bond, angle,

dihedrals, bonded and non-bonded interaction parameters. In what follows we briefly

describe the quantum mechanical basis for the charge derivation procedure.

Molecular formula C8H18N2O4S

Molar mass 238.30 gmol−1

Melting point 238oC

Appearance White crystalline powder

pKa (250C) 7.5

pH range 6.8 to 8.2

Table 4.1 Physical properties of HEPES

Figure 4.1 Chemical structure of HEPES

Matter is composed of atomic nuclei and electrons, and complex interaction of these atomic

particles is responsible for all intrinsic characteristics of the material. To explain electronic

structure of the material we need to perform quantum mechanics calculations and it was

known more than hundred years ago that solving the many body Schrodinger wave

equation can in principle yield all the material properties. The general form of time-

independent Schrödinger wave equation is,

Buffer Effective

pH range Molecular structure

MES

(MorpholinoEthaneSulfonic acid)

5.5–7.7

BES

(Bis2-hydroxyethyl)-2-

aminoEthaneSulfonic acid)

6.4–7.8

MOPS

(MorpholinoPropanSulfonic acid)

6.5–7.9

TES

(Trishydroxymethyl]-2-

aminoEthaneSulfonic acid)

6.8–8.2

HEPES (2-Hydroxyethyl)-1-

PiperazineEthaneSulfonic acid) 6.8–8.2

Table 4.2 List of some commonly used Good’s buffers.

𝐸𝜓 = Ĥ𝜓 (4.1)

where 𝜓 is the wave function and 𝐸 is the total energy of the system corresponding to that

wave function and Ĥ is the Hamiltonian operator which characterises the total energy of

any given wave function. The total Hamiltonian of a system containing NI atomic nuclei

and Ne electrons can be written as,

Ĥ = Ĥ𝑛𝑛+ Ĥ𝑒𝑒+ Ĥ𝑛𝑒

= {− ∑ ħ2

2𝑀𝐼𝛻𝐼

2 + 𝑉𝑛𝑛 𝑁𝐼

𝐼=1} + {− ∑

ħ2

2𝑚𝛻𝑖

2 + 𝑉𝑒𝑒 𝑁𝑒

𝑖=1} (4.2)

where the subscript ‘I’ indicates ions and the subscript ‘i’ indicates electrons, 𝑀𝐼 is the

ionic mass, m is the electron mass, ħ =h/2π where h is the Planck’s constant. The first term

in Eq. 4.2 represent the Hamiltonian Ĥ𝑛𝑛 for the nuclear coordinates, the second term

corresponds to the Hamiltonian for the electronic coordinates Ĥ𝑒𝑒 and the last term Ĥ𝑛𝑒

corresponds to the interactions between the nuclei and the electrons. 𝐻𝑛𝑒 describes the

interactions between electrons and nuclei. Once the Hamiltonian is known one can write

the many body Schrodinger wave equation as,

{Ĥ𝑛𝑛+ Ĥ𝑒𝑒+ Ĥ𝑛𝑒} 𝜓𝑡𝑜𝑡 = 𝐸𝜓𝑡𝑜𝑡 , (4.3)

where 𝜓𝑡𝑜𝑡 is the wave function of the total system comprising of nuclei and associated

electrons and E is the corresponding energy.

The apparent simplicity of the equations however belies the actual complexity involved in

solving the problem. It is generally impossible to find the true solution for many body total

wave function𝜓𝑡𝑜𝑡, even for very small system involving few ions and electrons. Hence in

order to obtain the solution involving a large number of ions and electrons it is often

necessary to introduce suitable approximations and reformulations of the above equations.

Here we state three simplifications in order to handle three contribution to the Hamiltonian

accurately and efficiently.

1. The Born-Oppenheimer approximation for separating nuclear degrees of freedom from

the electronic degrees of freedom

2. The density functional theory for handling ground state electronic interactions.

3. Mean field approximation where the system comprising of large number of small

interacting individuals components in which the effect of all other individuals on any given

individual is approximated by a single averaged effect, thus reducing a many body problem

to one body problem.

We will not go in details and functional form of the above equations and restrict ourselves

at this point. Next part of this chapter is concerned with calculating partial charges on

molecule. Various quantum mechanical packages are available for carrying out these

calculations; GAMESS, Gaussian, Jaguar, and Quantum ESPRESSO. We use Guassian-

09 package for our calculations.

Initial structure for HEPES molecule was built using GUASSVIEW and the energy

minimisation and optimisation of the structure was carried out in Gaussian using the

B3LYP basis set. Details are provided in the Appendix A.

4.1.3 Molecular geometry optimization and partial charge

Geometry optimization for HEPES has been achieved by energy minimization, using DFT

at the B3LYP level, employing the split valence basis set 6-311G (d). Initial structure of

the molecule was built using the Gauss view program then optimized in steps to obtain

local minima on the potential energy surface. The optimized molecular structures thus

obtained along with the numbering scheme of the atoms are shown in Fig. 4.1. We

compared the values for bond length, bond angle, and dihedrals from CIF (Crystallographic

Information File) file for HEPES molecule and found the results to be in good agreement

as shown in fig 4.2 and 4.3.

In second step we perform HF (Hartree-Fock) calculations employing 6-311G (d) basis set

to get electrostostatic points (ESP) for molecule. Since we are using GAFF (Generalized

Amber Force Field) for HEPES, we need to obtain RESP (Restrained electrostatic

potential) charges as discussed in the GAFF computational procedure [15]. Hence we input

ESP data into Antechamber package which is design to perform two main task; first assign

the atom types from GAFF and find any missing force field parameters of the molecule.

Second to calculate and assign partial charges based on the RESP procedure. Finally we

use ACPYPE tool to generate topologies for the molecule based on GAFF as a necessary

input requirement to GROMACS. Fig 4.2 shows RESP partial charges on atoms (hydrogen

atoms are not shown for visual simplicity).

Figure 4.2 (b)

Figure 4.2 (a)

Figure 4.2 (c)

Figure 4.2: Comparison of optimized bond length with experimental values (a) for all

thirty three bonds in HEPES molecule. Figure 4.2(b) and 4.2(c) shows comparison of

some selected bond angle and dihedra ls with experimental values . Experimental data are

obtained from crystallographic information file for HEPES.

4.2 Simulation of HEPES in water

Initial structure for HEPES molecule which is generated in Section 4.1.3; is simulated in a

cubic box in presence of water molecules under constant pressure, temperature and number

of molecules (NPT ensemble) for 10 nanoseconds. Protonated structure of HEPES was

hydrated using TIP3P water model in GROMACS with genbox tool. The total system size

is 3.0 × 3.0 × 3.0 Å. Leap frog integrator with a time step of 2 femtoseconds is used for

integrating equations of motion and the coordinates are saved after every 2 picosecond.

Fig 4.3 HEPES structure after geometry optimization. Numbers on each atom shows

RESP (Restrained Electrostatic Potential) partial charges whic h are derived from

Hartree-Fock basis set. (Hydrogen atoms are not shown for visual simplicity)

The system temperature is maintained at 310 K using the Nosé-Hoover thermostat and

pressure is maintained at 1 atm using Parrinello-Rahman barostat with isotropic pressure

coupling.

4.2.1 Analysis

The average number of hydrogen bonds ⟨NHB ⟩ per molecule for each saved frame was

determined based on a geometrical criterion with a cut-off donor−acceptor (DA) distance

of 0.35 nm and a cut-off donor−hydrogen−acceptor (DHA) angle of 30°.The donor-

acceptor (DA) criteria adopted here gives the simulated DHA angle distribution and the

Figure 4.4: Hydrogen bond distribution for HEPES in water (a) after 10 nanoseconds of simulation. Total

11.11 number of hydrogen bonds are formed using donor acceptor distance of 0.35 nm and

donor−hydrogen−acceptor (DHA) cut off angle of 30°. Figure 4.4(b) shows RMSD plot for HEPES

molecule with respect to simulation time. HEPES molecule shows the structure stability during simulation

time of 10 ns with fluctuations around 0.1 nm.

Figure 4.4 (a) Figure 4.4 (b)

Nu

mb

er o

f h

yd

rogen

bond

DA distance in water similar to the experimental values [16]. Average number of hydrogen

bond ⟨NHB ⟩ are reproduced by using the standard tool g_hbond implemented in

GROMACS.

In the protonated form HEPES buffer has two pKa. The first dissociation constant (pKa1)

refers to the dissociation of the sulfonic group, and the second dissociation is due to

dissociation of the protonated amino group (pKa2). Thus in aqueous solutions, HEPES

molecule possesses both negatively charged sulfonic group (SO3-) and a positively charged

amino groups (NH3+) and becomes a zwitterionic molecule [14]. Due to donor acceptor

sites, HEPES provides a number of possibilities for formation of hydrogen bonds with the

solvent. Fig 4.4 (a) shows the distribution of hydrogen bonds as a function of distance. The

average number of H-bonds for HEPES molecule ⟨NHB ⟩ with water molecules is 11.11

which is compared with the reported value of 11.94 [17].

We have verified the stability of HEPES structure by computing the RMSD (root mean

square deviation) as shown in Figure 4.4 (b).Using GROMACS tool g_rms, each structure

from trajectory is compared to a reference structure for all time frames. RMSD is calculated

using following formula:

𝑅𝑀𝑆𝐷 = √∑ (𝑥𝑖−𝑥𝑟𝑒𝑓)2 𝑛

𝑡=1

𝑁 (4.4)

where 𝑥𝑖 is the coordinate of the molecules at time 𝑡, 𝑥𝑟𝑒𝑓 is the reference structure

coordinate and 𝑁 is the total number of atoms in the molecule.

Initial structure as described in section 4.2 for HEPES is the reference structure and we

monitor the displacement of molecules from the centre of mass position with respect to the

reference structure. For a 10 nanosecond simulation, the HEPES molecules shows

structural stability in water with fluctuations around 0.1 nm (Figure 4.4b). We did not

observe any distortion in the geometry during this time

Chapter 5

Simulation of DMPC in presence of

buffer molecules

5.1 Introduction

Several Good’s buffers which fulfills the selection criteria based on pKa, solubility, low

membrane permeability and ease of preparation are commonly used in laboratories. In this

study we investigate the effect of three different buffer molecules on the DMPC lipid

bilayer, namely HydroxyEthyl Piperazine Ethane Sulfonic acid (HEPES), Morpholino

Ethane Sulfonic acid (MES), Piperazine Ethane Sulfonic acid (PIPES). We have discussed

the interactions with lipid bilayers by calculating number of hydrogen bonds, radial

distribution functions and density distributions.

5.2 Simulation Method

The equilibrated DMPC bilayer consisting of 128 lipid molecules with box dimensions of

64.0 × 64.0 × 70.0 Å is obtained from the previous simulations in Chapter 3. The geometry

of the simulation box is such that the bilayer surface is in the x,y-plane and the bilayer

normal is along the z-axis. Buffer molecules are first placed on top of bilayer with 1.5 nm

distance from the head group. Water molecules are then added to the system and the

required concentration of buffer is obtained by increasing the number of buffer molecules

and decreasing the number of water molecules. The system is further equilibrated in the

NPT ensemble for 5ns.

We have chosen a temperature of 310 K to ensure the system temperature is well above the

bilayer melting transition temperature of 297 K. The temperature is maintained at 310 K

using a Nosé-Hoover thermostat and the pressure is maintained at 1 bar using the

Parrinello-Rahman barostat with semi isotropic pressure coupling. Both the temperature

and pressure of the solvent and lipid are controlled independently. The particle mesh Ewald

summation was used to account for the long-range electrostatic interactions. During the

simulations all bonds were constrained using the LINKS algorithm. We derived the partial

charges and optimised the structure for MES and PIPES buffer molecules in the similar

fashion as outlined for the HEPES molecule in Chapter 4.

5.3 Results and Discussion

5.3.1 Mass density

Density is computed by dividing the simulation box into number of slices along the z-

direction. Mass of the reference molecule in each individual slice is computed over all

frames which is then divided by the volume of each slice to obtain the local density in each

slice. The plot of the mass densities for the three simulation systems under study are shown

in the Figure 5.3. The density of buffer molecules indicates that buffer molecules are

uniformly distributed in the aqueous region and the non-zero value next to the bilayer head

groups indicate that there is a weak interaction of these molecules with the bilayer. In the

case of HEPES we observe that the buffer density penetrates into the bilayer to a greater

extent when the buffer concentration is increased and a weak maxima is observed in the

vicinity of the head groups. Similar trends are observed with MES and PIPES. However in

all cases, the density drops towards the centre of the bilayer indicating that the buffer

molecules predominantly interact with the zwitterionic head group region of the DMPC

bilayer. An important result is that the buffer molecules shows a weak electrostatic

attraction towards the oppositely charged head group molecules in the bilayer.

5.3.2 Hydrogen bonds

Calculation of hydrogen bonds involves identifying donor and acceptors as follows; A

hydrogen atom attached to a relatively electronegative atom will play the role of hydrogen

bond donor while an electronegative atom such as fluorine, oxygen or nitrogen will act as

the hydrogen bond acceptor, irrespective of whether it is bonded to hydrogen or not. In the

Figure 5.3: density distribution of the buffer molecules at different concentrations.

Average density of buffer molecules have been amplified by multiplying a factor of 8

and 10.Maximum density for buffer molecules lies near the head group region showing

a weak attraction towards the lipid head groups.

donor molecule, the electronegative atom attracts the electron cloud from the hydrogen

nucleus by decentralising the electron cloud, giving rise to a partial positive charge on the

hydrogen atom and a partial negative charge on the electronegative atom.

In our study we calculated the hydrogen bonds based on the following criteria; the distance

between the water oxygen (OW) and the DMPC oxygen is less than or equal to 3.25Å and

the hydrogen bond angle 𝜃, which is the angle between the DMPC oxygen and H-OW

bonds of water is less than or equal to 350 as proposed by Raghavan et.al. [18].

In this study several hydrogen bonding forms have been explored; between buffer

molecules and the lipids, between water and the lipids as well as between water and buffer

molecules. The results are summarized in Table 5.3.2 which shows that all the buffer

molecules form hydrogen bonds with the DMPC lipids. Although the presence of buffer

molecules for the concentrations examined in this study has a weak influence on the DMPC

bilayer some differences can be observed. In the case of HEPES the number of water-

DMPC hydrogen bonds shows a small decrease when the number of HEPES molecules is

increased to 10. However the water-DMPC hydrogen bonds for MES and PIPES do not

show any changes with the buffer concentration. Similar number of hydrogen bonds are

obtained on comparing the values for pure lipids hydrogen bond with water from the study

[16].

Table 5.3.2 (a)

5.3.3. Area per lipid

One of the important quantities in describing the behaviour of lipid bilayers is the average

area per lipid. We calculated the average area per lipid molecule by multiplying the time

average XY dimensions of the simulation box and dividing the result by the number of

lipid molecules present in one leaflet of the bilayer.

Table 5.3.3 presents the area per lipid and bilayer thickness under different buffer and

concentration. For reference DMPC system the calculated area per lipid is 62.59 Å2 , which

is close to the experimental value of 61.2 Å2 at 300 K and 1 bar, and is in good agreement

with the values obtained in previous simulations [13]. From the results we can conclude

that the buffer molecules do not influence the area per head group of the bilayers.

5.3.4 Radial distribution function

Radial distribution function (RDF) is a useful tool to describe the structure of a system. It

gives the value of local density of a particle in a shell at a distance r from the reference

molecule. We have plotted RDF for different concentrations of buffer molecules in order

to obtain better insight into probability distribution of water molecules around the lipid

bilayers through its coordination number. To calculate RDF from simulations, the

neighbours around each atom or molecule are sorted into bins. Finally the number of

neighbours in each bin is averaged over the entire simulation time.

We define the coordination number as the number of neighbours in its first hydration shell

and is obtained by integrating the RDFs curve up to its first minimum and is presented is

table 5.3.4. A slight decrease in the coordination number at higher concentrations of buffer

molecule is observed and this is consistent with the decrease in the HBs observed earlier

for the case of the HEPES molecules.

System

Number

of buffer

molecule

Coordination number

for

Lipid methyl groups

and water oxygen

Coordination number

for Lipid non

esterified oxygen and

water oxygen

DMPC + water

(reference) 0 7.5765 @ 4.65 2.2945 @ 3.25

DMPC + water +

HEPES 2 7.5555 @ 4.65 2.2936 @ 3.25

DMPC + water +

HEPES 10 7.4685 @ 4.65 2.2768 @ 3.25

DMPC + water +

MES 2 7.5420 @ 4.65 2.2846 @ 3.25

DMPC + water +

MES 8 7.5231 @ 4.65 2.2794 @3.25

DMPC + water +

PIPES 2 7.5700 @ 4.65 2.2878 @ 3.25

DMPC + water +

PIPES 8 7.4972 @ 4.65 2.2889 @ 3.25

Table 5.3.4: coordination number at different buffer concentrations

Chapter 6

Umbrella sampling

6.1 Introduction

Calculation of free energy differences is one of the most important thermodynamic

parameter which helps us in the thermodynamically most favourable states. It also helps

us to estimate reaction rates or free energy barriers for a given system. In a finite time

simulation it is generally impossible to sample all the possible configurations for a given

ensemble because of different energy barriers on the free energy landscape. Hence there

is a need for biased sampling where we can force the system at a specific reaction

coordinate in order to sample enough configurations which are necessary to get free energy

landscape at that reaction coordinate. In our study we have estimated the free energy

landscape for insertion of HEPES molecule by first keeping it in aqueous phase (state a)

and then bringing it inside the middle of the bilayer (state b). In the following discussion

we have presented a short description of umbrella sampling methods adopted from the

study [19].

In a canonical ensemble the free (Helmholtz) energy of a system is computed as:

𝐴 = −1

𝛽 ln(𝑄𝑁𝑉𝑇) (6.1)

Where

𝑄𝑁𝑉𝑇 =1

ℎ3𝑁𝑁!∬ exp [−𝛽𝐻(𝑥, 𝑝𝑥)] 𝑑𝑥 𝑑𝑝𝑥 (6.2)

Here 𝐻(𝑥, 𝑝𝑥), is the Hamiltonian of the system. 𝑥 and 𝑝𝑥 are 3N-dimentional vectors

containing the atomic coordinates and momenta of the particles in the system. ℎ is the

Planck constant. The probability to find the system in a particular microscopic

configuration 𝐻(𝑥, 𝑝𝑥) is defined as

𝑃(𝑥, 𝑝𝑥) =exp [−𝛽𝐻(𝑥, 𝑝𝑥)]

∬ exp [−𝛽𝐻(𝑥′, 𝑝′𝑥)] 𝑑𝑥′ 𝑑𝑝′𝑥

(6.3)

we are interested in a free energy difference between two states as function of reaction co-

ordinate which possible to obtain from biased simulations such as Umbrella sampling.

In Umbrella sampling a bias potential, an additional energy term is applied to the system

to ensure efficient sampling along the whole reaction coordinate. We are using harmonic

potential which allows the forces to vary according to the nature of interactions of HEPES

and DMPC lipid i.e. forces will build up until certain critical interactions are broken. The

functional form of harmonic biasing potential can be written as:

𝑤𝑖(𝑧) = (𝐾/2)(𝑧 − 𝑧𝑖)2 (6.4)

If 𝑈(𝑹, 𝑧) be the potential energy of the system as a function of position and reaction

coordinate then the biased potential energy of the system becomes:

𝑈𝑏(𝑹, 𝑧) = 𝑈𝑢(𝑹, 𝑧) + 𝑤𝑖(𝑧) (6.5)

Hence the unbiased probability distribution along z coordinate according to equation 6.3

will become

𝑃𝑢(𝑧𝑖) =∫ 𝛿(𝑧−𝑧𝑖)exp [−𝛽𝑈(𝑹,𝑧)]𝑑𝑧𝑑𝑹

∫ exp [−𝛽𝑈(𝑹,𝑧)]𝑑𝑧𝑑𝑹 (6.6)

Similarly, the biased probability distribution from equation 6.5 will become

𝑃𝑏(𝑧𝑖) =∫ 𝛿(𝑧 − 𝑧𝑖) exp [−𝛽𝑈(𝑹, 𝑧) + 𝑤𝑖(𝑧) ]𝑑𝑧𝑑𝑹

∫ exp [−𝛽𝑈(𝑹, 𝑧) + 𝑤𝑖(𝑧)]𝑑𝑧𝑑𝑹 (6.7)

Which leads to,

𝑃𝑢(𝑧𝑖) = 𝑃𝑏(𝑧𝑖) exp[𝛽𝑤𝑖(𝑧)] < exp[−𝛽𝑤𝑖(𝑧)] > (6.8)

Now the free energy as a function of reaction coordinate z can be defined as:

𝐴(𝑧𝑖) = −1

𝛽ln ∫ 𝛿(𝑧 − 𝑧𝑖) exp[−𝛽𝑈(𝑹, 𝑧)] 𝑑𝑧𝑑𝑹 (6.9)

From equation 6.6 and 6.7, we get

𝐴(𝑧𝑖) = − (1

𝛽) 𝑃𝑏(𝑧𝑖) − 𝑤𝑖(𝑧) + 𝐹𝑖 (6.10)

this can be obtain from MD simulations.

Here 𝐹𝑖 = − (1

𝛽) ln < exp[−𝛽𝑤𝑖(𝑧)] > (6.11)

6.2 Steps in Umbrella techniques

Generate a series of configuration along reaction coordinate using constant velocity

steered molecular dynamic simulation (cv-SMD)

Extract frames from trajectory generated above by dividing the reaction coordinate

into windows with desired COM spacing

Run umbrella sampling on in individual window by restraining the molecule within

window corresponding to desire COM distance

Estimate 𝑃𝑏(𝑧𝑖) for each simulation

Combine results from all simulations using Weighted Histogram Analysis Method

(WHAM)

6.3 Results and discussion

HEPES molecule is kept at a distance of 3.5 nm from the centre of mass distance

of DMPC lipid bilayer and the values of 𝑧𝑖 ranged from 3.5 to 0 nm with ∆𝑧𝑖 = 0.2

nm

However depending upon the overlap between windows we have to save

configurations more often, or sufficient to save configurations less often. The idea

is to get sufficient overlap between windows in the umbrella histograms to get

continuous energy profile.

During constant velocity Steered molecular dynamic simulation (cv-SMD) HEPES

molecule was moved with a pull rate of 0.01 nm per ps using a dummy atom that

is attached to the centre of mass of the HEPES via a virtual spring with a spring

constant k, of 1000 kJ mol-1nm-2

The dummy atom was moved at a constant velocity in the direction perpendicular

to the membrane plane (z- axis) with restrictions on x and y co-ordinates.

We are monitoring free energy change of HEPES molecule between two

thermodynamic states: one above membrane (state a) and one in the middle of

membrane (state b) with z-axis as a reaction coordinate.

In each window we simulated the system for 10 ns. The work done to pull the

nanotube across the lipid bilayer using the cv-SMD simulations was used to

calculate the potential of mean force .The PMF curves was generated after

combining all the results from each window using the weighted histogram analysis

method (WHAM).

At room temperature a molecule with N number of atoms will have 3N degrees of freedom.

For HEPES molecule with restrains along X and Y coordinates, it will have one

translational, three rotational (nonlinear molecule) and 95 vibrational (3N-4) degrees of

freedom. Rotational and translation degree of freedom contributes 0.5 RT of total energy

per molecule and RT of total energy per molecule for vibrational degree of freedom which

is occurring at temperature more than 5000C. Hence at room temperature HEPES molecule

has 2.0 RT of energy per molecule in order to access rotational and translational degrees

of freedom. In our special case even if we consider only one translational mode (along z-

direction) and ignore vibrational and rotational mode; HEPES molecule will have 0.308

Figure 6.3.3: a) free energy profile for insertion of HEPES from aqueous phase to lipid phase

b) average and standard deviation generated from the WHAM procedure [3]

∆𝐺𝑎→𝑏 = 15.74 𝑘𝑐𝑎𝑙𝑚𝑜𝑙−1

Aqueous phase

(HEPES+water) Lipid bilayers

Figure 6.3.3 Figure 6.3.3

𝑘𝑐𝑎𝑙 𝑚𝑜𝑙−1 of translational energy which is not sufficient to cross an energy barrier of

15.74 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙−1 at temperature of 310 K. Hence we can confidently state that at

temperature of 310K it is not possible for HEPES molecule to insert inside the DMPC lipid

bilayer.

During the course of biased simulations, HEPES molecule is penetrating in the lipid bilayer

by creating a hydrophobic defect and exposing the hydrophobic membrane core to water.

HEPES insertion is accompanied by the breaking of Hydrogen bonds between lipid head

groups and water and between the lipid molecules themselves. When HEPES molecule was

placed in the middle of bilayer (state b), there is strong repulsion between hydrophobic

core and charged atom on HEPES molecule which give rise to highest energy in the PMF

curve.

Values on the PMF curve gets less and less negative as we move HEPES molecule from

state a to state b with its lowest value of 15.74 𝑘𝑐𝑎𝑙𝑚𝑜𝑙−1 at 2.5 nm from COM distance

of lipid. Finally a conclusion can be drawn that in order to penetrate HEPES molecule in

the middle of bilayer an energy barrier of 15.74 𝑘𝑐𝑎𝑙𝑚𝑜𝑙−1 has to be crossed and from the

above discussion, it is not feasible for HEPES molecule to be in state b.

Chapter 7

Conclusions and suggestions for future

work

7.1 Conclusions

Molecular dynamic simulations of DMPC lipid bilayers and buffer molecules are carried

out using GROMACS simulation package. Simulation results for a DMPC and water

system are verified with the experimentally reported values and found to be in good

agreement. Partial charges on buffer molecules are derived from ab-initio calculations at

the Hartree-Fock level of theory to account for the columbic interactions. Simulations of

DMPC lipid bilayers with buffers molecules shows a weak interactions of buffer molecules

toward lipid head groups. Similar results are observed for density distribution of buffer

molecules at different concentrations with maximum density near the lipid head groups.

We observed hydrogen bond formation between buffer molecule and lipid head groups

however area per lipid and bilayer thickness remains unaffected during simulation time.

Finally free energy profile of HEPES molecule along reaction coordinate (z-axis) are

obtained using umbrella sampling techniques. Results from the plots suggests that at

temperature of 310 K, energy possess by HEPES molecule is not sufficient to cross the

energy barrier of 15.74 𝑘𝑐𝑎𝑙𝑚𝑜𝑙−1. Thus it will remain in the aqueous phase with a

maximum HEPES density near the lipid head groups.

7.2 Future work

In future it is useful to incorporate salt mixtures along with buffer molecules to investigate

the effect on bilayer properties. Different lipids having net positive, negative and neutral

charge can be studied in presence of various buffer molecules to figure out extent of

coulomb interactions and its role in lipid bilayers properties such as bending rigidity,

bilayer swelling and phase transition.

References

1. Available from http: // www. rsc.org/ Education/ Teachers/ Resources/ cfb/

cells.htm;2015

2. Available from: http://free-stock-illustration. com/ phospholipids +structure

+properties ;2015

3. Shankar Kumar, D.B., Robert H. Swendsen, Peter A. K and John M. Rosenberg,

The Weighted Histogram Analysis Method for Free-Energy Calculations on

Biomolecules. I. The Method. Computational Chemistry, 1992. 13(8): p. 1011-

1021.

4. Brand, E.G., Molecular dynamic simulation of fluid lipid membrane, in KTH

Engineering Science. 2011, Royal Institute of Technology,Stockholm,Sweden. p.

103.

5. Available from: http://chemistry.elmhurst.edu/ ;2015

6. Fennell, E.D. and W. Hakan, The Colloidal Domain: Where Physics, Chemistry,

Biology, and Technology Meet. Second Edition ISBN: 978-0-471-24247-5 p. 672

7. Bouvrais, H., L. Duelund, and J.H. Ipsen, Buffers affect the bending rigidity of

model lipid membranes. Langmuir, 2014. 30(1): p. 13-6.

8. Daan Frenkel and Berend Smit, Understanding Molecular Simulation; From

Algorithms to Applications Second Edition. ISBN: 978-0-12-267351-1. 2002.

9. Erik Lindahl, D.v.d.S., Berk Hess. Available from: http://www.gromacs.org/.

http://www.rsc.org/Education/Teachers/Resources/cfb/cells.htm

http://www.rsc.org/Education/Teachers/Resources/cfb/cells.htm

http://chemistry.elmhurst.edu/

http://www.gromacs.org/

10. Lyubartsev, P.A.; Available from: http:// people .su.se /~jjm/ Stockholm_Lipids

/Downloads.html.

11. Berk Hess, H.B., Herman J. C. Berendsen,Johannes G. E. M. Fraaije, LINCS: A

Linear Constraint Solver for Molecular Simulations. Computational Chemistry,

1997. 18: p. 1463-1472.

12. Pohle, W., et al., Lipid hydration: headgroup CH moieties are involved in water

binding. Biopolymers, 2004. 74(1‐2): p. 27-31.

13. Norbert Kučerkaa, M.-P.N., John Katsarasa, Fluid phase lipid areas and bilayer

thicknesses of commonly used phosphatidylcholines as a function of temperature.

Biochimica et Biophysica Acta (BBA) - Biomembranes, 2011. 1808(11): p. 2761-

2771.

14. Śledź, P., et al., An experimental charge density of HEPES. Acta Crystallographica

Section B: Structural Science, 2010. 66(4): p. 482-492.

15. Wang, J., et al., Development and testing of a general amber force field. J Comput

Chem, 2004. 25(9): p. 1157-74.

16. Carlos F. Lopez, S.O.N., and Michael L. Klein, Hydrogen Bonding Structure and

Dynamics of Water at the Dimyristoylphosphatidylcholine Lipid Bilayer Surface

from a Molecular Dynamics Simulation. Phys. Chem. B, 2004. 108: p. 6603-6610.

17. Taha, M., I. Khoiroh, and M.-J. Lee, Phase behavior and molecular dynamics

simulation studies of new aqueous two-phase separation systems induced by

HEPES buffer. The Journal of Physical Chemistry B, 2013. 117(2): p. 563-582.

18. Marta Pasenkiewicz-Gierula, Y.T., Hiroo Miyagawa, Kunihiro Kitamura, and

Akihiro Kusumi, Hydrogen Bonding of Water to Phosphatidylcholine in the

Membrane As Studied by a Molecular Dynamics Simulation: Location, Geometry,

and Lipid-Lipid Bridging via Hydrogen-Bonded Water. Phys. Chem, 1996. 101: p.

3677-3691.

19. Kastner, J., Umbrella sampling. Wiley Interdisciplinary Reviews: Computational

Molecular Science, 2011. 1(6): p. 932-942.

Appendix A

Input file for Gaussian charge calculation

%chk=charge.chk

#p hf/6-31g* geom=connectivity iop(6/33=2,6/42=17,6/41=10) pop=mk

Title Card Required

0 1

C 1.83718600 1.72023200 0.68732700

C 0.32581100 1.52051500 0.88957200

N -0.25971800 0.65354300 -0.19548300

C 0.53301800 0.77993200 -1.44947200

C 1.97236600 0.27769800 -1.24965500

N 2.40278800 0.49700200 0.13479200

H 2.03708400 2.60878700 0.05842300

H -0.21466600 2.47138700 0.87887100

H 0.01239100 0.22607100 -2.23353600

H 2.03745700 -0.79316100 -1.46305900

C -0.49095400 -0.78509600 0.22339000

C -1.80299900 -0.91701500 0.99534100

C 3.83994300 0.35727500 0.33257600

C 4.38040200 -1.00731800 -0.11400900

S -3.20120400 -0.27672900 -0.01544300

O -3.12646000 -1.05536300 -1.26985700

O -4.40923800 -0.38955600 0.80732800

O -2.74098800 1.17587300 -0.21443400

O 3.61329500 -2.10716000 0.34521500

H 3.55475100 -2.05139700 1.31179300

H 2.29584700 1.92050800 1.66083900

H 0.12292800 1.02696900 1.84121200

H 0.51205900 1.83886700 -1.72306600

H 2.61819000 0.80400200 -1.97842500

H 4.37570400 -1.08729400 -1.20526100

H 5.43281400 -1.07656400 0.20495000

H 4.03573800 0.48971400 1.40575100

H 4.42121100 1.14163100 -0.19449700

H -0.57276400 -1.36753700 -0.69702300

H 0.38303600 -1.12175200 0.78531400

H -1.99379400 -1.97337000 1.20163600

H -1.80041800 -0.37304500 1.94507200

H -1.30129200 1.02232600 -0.36349900

Appendix B

C.I.F. file for HEPES

##############################################################################

### ###

### Electronic paper (Acta Crystallographica Section E) ###

### ###

##############################################################################

# #

# This CIF contains the data in a paper accepted for publication in Acta #

# Crystallographica Section E. It conforms to the requirements of Notes #

# for Authors for Section E, and has been peer reviewed under the auspices #

# of the IUCr Commission on Journals. #

# #

# Full details of the Crystallographic Information File format #

# are given in the paper "The Crystallographic Information File (CIF): #

# a New Standard Archive File for Crystallography" by S. R. Hall, F. H. #

# Allen and I. D. Brown [Acta Cryst. (1991), A47, 655-685]. #

# #

# The current version of the core CIF dictionary is obtainable from #

# ftp://ftp.iucr.org/pub/cif_core.dic. The current version number is 2.4. #

# #

# Software is freely available for graphical display of the structure(s) in #

# this CIF. For information consult the CIF home page http://www.iucr.org/ #

# cif/home.html #

# #

# This file may be used for bona fide research purposes within the #

# scientific community so long as proper attribution is given to the journal #

# article from which it was obtained. #

# #

##############################################################################

data_I

_audit_creation_method

'HKL-3000SM automatic completion and interactive editing'

_audit_conform_dict_name cif_core.dic

_audit_conform_dict_version 2.3

_audit_block_code sm

_chemical_name_systematic

;

2-[4-(2-Hydroxyethyl)piperazin-1-ium-1-yl]ethanesulfonate

;

_chemical_name_common HEPES

_chemical_formula_moiety 'C8 H18 N2 O4 S1'

_chemical_formula_sum 'C8 H18 N2 O4 S1'

_chemical_formula_iupac 'C8 H18 N2 O4 S1' _chemical_formula_weight 238.31

_chemical_melting_point ?

_symmetry_cell_setting orthorhombic

;

_geom_special_details

; loop_

_geom_bond_atom_site_label_1

_geom_bond_atom_site_label_2

_geom_bond_site_symmetry_2

_geom_bond_distance

_geom_bond_publ_flag

S1 O1 . 1.4525(3) ? S1

O3 . 1.4532(2) ?

S1 O2 . 1.4771(2) ?

S1 C1 . 1.7874(3) ?

N2 C4 . 1.4719(3) ?

N2 C5 . 1.4724(3) ?

N2 C7 . 1.4736(3) ?

N1 C6 . 1.4971(3) ?

N1 C3 . 1.4984(3) ?

N1 C2 . 1.5008(3) ?

N1 H1N . 0.827(8) ?

O4 C8 . 1.4226(4) ?

O4 H1O4 . 0.848(10) ?

C1 C2 . 1.5239(4) ?

C1 H1A . 0.939(9) ?

C1 H1B . 0.950(8) ?

C5 C6 . 1.5162(3) ?

C5 H5A . 0.995(8) ?

C5 H5B . 0.963(8) ?

C3 C4 . 1.5190(3) ?

C3 H3B . 0.951(8) ?

C3 H3A . 0.949(8) ?

C6 H6A . 0.898(9) ?

C6 H6B . 0.987(8) ?

C4 H4B . 0.981(8) ?

C4 H4A . 0.988(8) ?

C2 H2A . 1.016(9) ? C2

H2B . 0.948(8) ?

C8 C7 . 1.5250(4) ?

C8 H8B . 0.946(9) ?

C8 H8A . 0.935(8) ?

C7 H7B . 0.988(8) ? C7

H7A . 1.037(7) ?

loop_

_geom_angle_atom_site_label_1



_geom_angle_site_symmetry_1

_geom_angle_site_symmetry_3

_geom_angle _geom_angle_publ_flag

O1 S1 O3 . . 114.856(17) ?

O1 S1 O2 . . 111.907(17) ?

O3 S1 O2 . . 111.966(15) ?

O1 S1 C1 . . 106.317(13) ?

O3 S1 C1 . . 105.650(15) ?

O2 S1 C1 . . 105.296(12) ?

C4 N2 C5 . . 108.376(18) ?

C4 N2 C7 . . 112.22(2) ?

C5 N2 C7 . . 108.719(19) ?

C6 N1 C3 . . 109.504(18) ?

C6 N1 C2 . . 113.10(2) ?

C3 N1 C2 . . 110.809(19) ?

C6 N1 H1N . . 109.3(5) ?

C3 N1 H1N . . 107.8(5) ?

C2 N1 H1N . . 106.1(5) ?

C8 O4 H1O4 . . 107.0(6) ?

C2 C1 S1 . . 110.620(17) ?

C2 C1 H1A . . 113.1(6) ?

S1 C1 H1A . . 106.9(6) ?

C2 C1 H1B . . 113.9(5) ? S1

C1 H1B . . 106.2(5) ?

H1A C1 H1B . . 105.5(8) ?

N2 C5 C6 . . 111.782(19) ?

N2 C5 H5A . . 110.8(4) ?

C6 C5 H5A . . 108.7(4) ?

N2 C5 H5B . . 109.5(5) ?

C6 C5 H5B . . 107.3(5) ?

H5A C5 H5B . . 108.7(7) ?

N1 C3 C4 . . 110.055(19) ?

N1 C3 H3B . . 104.6(5) ?

C4 C3 H3B . . 113.0(5) ?

N1 C3 H3A . . 107.7(4) ?

C4 C3 H3A . . 111.1(5) ?

H3B C3 H3A . . 110.1(7) ?

N1 C6 C5 . . 110.17(2) ?

N1 C6 H6A . . 107.9(6) ?

C5 C6 H6A . . 109.1(6) ?

N1 C6 H6B . . 105.6(4) ?

C5 C6 H6B . . 113.7(4) ?

H6A C6 H6B . . 110.1(7) ?

N2 C4 C3 . . 111.07(2) ?

N2 C4 H4B . . 107.2(5) ?

C3 C4 H4B . . 109.3(5) ?

N2 C4 H4A . . 111.4(4) ?

C3 C4 H4A . . 108.3(4) ?

H4B C4 H4A . . 109.5(6) ?

N1 C2 C1 . . 111.129(19) ?

N1 C2 H2A . . 107.6(5) ?

C1 C2 H2A . . 109.6(5) ? N1

C2 H2B . . 107.6(5) ?

C1 C2 H2B . . 112.4(5) ?

H2A C2 H2B . . 108.4(7) ?

O4 C8 C7 . . 114.27(2) ?

O4 C8 H8B . . 106.2(5) ?

C7 C8 H8B . . 111.5(5) ?

O4 C8 H8A . . 110.3(5) ?

C7 C8 H8A . . 108.4(5) ?

H8B C8 H8A . . 105.8(7) ?

N2 C7 C8 . . 114.71(2) ?

N2 C7 H7B . . 107.7(5) ?

C8 C7 H7B . . 110.3(5) ?

N2 C7 H7A . . 110.5(4) ?

C8 C7 H7A . . 110.7(4) ?

H7B C7 H7A . . 102.1(5) ? loop_

_geom_torsion_atom_site_label_1




_geom_torsion_site_symmetry_1




_geom_torsion

_geom_torsion_publ_flag

O1 S1 C1 C2 . . . . -59.26(2) ?

O3 S1 C1 C2 . . . . 178.27(2) ?

O2 S1 C1 C2 . . . . 59.63(2) ?

C4 N2 C5 C6 . . . . 59.21(3) ?

C7 N2 C5 C6 . . . . -178.55(2) ?

C6 N1 C3 C4 . . . . -56.75(3) ?

C2 N1 C3 C4 . . . . 177.80(2) ?

C3 N1 C6 C5 . . . . 55.97(3) ?

C2 N1 C6 C5 . . . . -179.911(19) ?

N2 C5 C6 N1 . . . . -58.33(3) ?

C5 N2 C4 C3 . . . . -59.66(2) ?

C7 N2 C4 C3 . . . . -179.726(19) ?

N1 C3 C4 N2 . . . . 59.63(3) ?

C6 N1 C2 C1 . . . . 62.57(3) ?

C3 N1 C2 C1 . . . . -174.03(2) ?

S1 C1 C2 N1 . . . . 159.105(18) ?

C4 N2 C7 C8 . . . . -68.69(3) ?

C5 N2 C7 C8 . . . . 171.44(2) ?

O4 C8 C7 N2 . . . . 76.07(3) ? loop_

;

University of Virginia

Department of Molecular Physiology & Biological Physics

1340 Jefferson Park Avenue

Charlottesville, VA 22908

USA

;

_publ_contact_author_email [email protected]

_publ_contact_author_fax +1-434-9821616

_publ_contact_author_phone +1-434-2430033

_publ_section_title

;

2-[4-(2-Hydroxyethyl)piperazin-1-ium-1-yl]ethanesulfonate at 100 K

; loop_

_publ_author_name

_publ_author_address

'Pawel Sledz'

;





USA

;

'Thomas Minor'

;





USA

;

'Maksymilian Chruszcz'

;




Charlottesville, VA 22908,USA

Appendix C

.mdp file for pure DMPC in water simulation

title = Production run for DMPC in water

; Run parameters

integrator = md ; leap-frog integrator

nsteps = 5000 ; 2 * 500000 = 1000 ps

dt = 0.002

; Output control

; 2 fs

nstxout = 25000 ; save coordinates every 50.0 ps

nstvout = 25000 ; save velocities every 1.0 ps

nstenergy = 25000 ; save energies every 1.0 ps

nstlog = 25000 ; update log file every 1.0 ps

; Bond parameters continuation = yes ;

Restarting after NVT constraint_algorithm = lincs ;

holonomic constraints

constraints = all-bonds ; all bonds (even heavy atom-H bonds)

constrained

; Neighborsearching

cutoff-scheme = Verlet

ns_type = grid ; search neighboring grid cells

nstlist = 10 ; 20 fs, largely irrelevant with Verlet scheme

rcoulomb = 1.0 ; short-range electrostatic cutoff (in nm)

coulomb-modifier = Potential-shift-Verlet

vdwtype = Cut-off

rvdw = 1.0 ; short-range van der Waals cutoff (in nm)

vdw-modifier = Potential-shift-Verlet

; Electrostatics

coulombtype = PME ; Particle Mesh Ewald for long-range electrostatics

pme_order = 4 ; cubic interpolation

fourierspacing = 0.12 ; grid spacing for FFT

optimize-fft = yes table-extension = 1 ; Temperature coupling is on

tcoupl = Nose-Hoover ; modified Berendsen thermostat tc-grps

= DMPC MOL_SOL ; two coupling groups - more accurate tau_t = 0.1 0.1

; time constant, in ps ref_t = 310 310 ; reference temperature,

one for each group, in K

; Pressure coupling is on

pcoupl = Parrinello-Rahman ; Pressure coupling on in NPT

pcoupltype = semiisotropic ; uniform scaling of box vectors

tau_p = 10.0 10.0 ; time constant, in ps

ref_p = 1.013 1.013 ; reference pressure, in bar

compressibility = 4.5e-5 4.5e-5 ; isothermal compressibility

of water, bar^-1

; Periodic boundary conditions

pbc = xyz

; Dispersion correction

; 3-D PBC

DispCorr = EnerPres ;

Velocity generation

; account for cut-off vdW scheme

gen_vel = no ; Velocity generation is off

; COM motion removal

; These options remove motion of the bilayer relative to the

solvent/hepes nstcomm = 1

comm-mode = Linear comm-grps = DMPC MOL_SOL

Documents

Molecular Dynamics simulations of lipid bilayers