level of resolution and therefore permits immediate comparison with atomic level
experimental data.
2 Theory and Simulation of Lipid Bilayers
To provide a basis for both the theoretical ideas and the computational techniques
that we will discuss in this chapter, we start by reminding the reader of some
essential concepts. Section 2.1 reviews some basic aspects of the Helfrich Hamiltonian. Section 2.2 introduces three coarse-grained membrane models that will be
used in the remainder of this chapter. In Sects. 2.3 and 2.4, we discuss the bending
moduli and the surface tension of membranes in more detail, and finally comment
on multicomponent membranes in Sect. 2.5.
2.1 Basic Continuum Theory Concepts
2.1.1 Continuum Elasticity of Lipid Membranes
Lipid molecules are amphipathic: they consist of a hydrophilic head group and
typically two hydrophobic (fatty acid) tails. Yet, despite their amphipathic nature,
lipid molecules dissolved in water have an extremely low critical aggregate concentration (nanomolar or even smaller [1]), and thus under most common conditions lipids spontaneously aggregate. Because the roughly cylindrical shape of
lipids leads to two-dimensional self-assembly, thermodynamic considerations [2]
show that – in contrast to the finite size of spherical and wormlike micelles – a
single macroscopic aggregate containing almost all of the lipids will form: a
two-dimensional bilayer membrane. Its lateral dimensions can exceed its thickness
by several orders of magnitude.
2.1.2 The Helfrich Hamiltonian
If lipid membranes are subjected to lateral tension, they typically rupture at stresses
of several millinewtons per meter (mN/m), with a remarkably low rupture strain of
only a few percent [3]. At large scales and moderate tensions it is hence an excellent
approximation to consider membranes as largely unstretchable two-dimensional
surfaces. Their dominant soft modes are not associated with stretching but with
bending [4–6]. Within the well-established mathematical framework developed by
Helfrich [5], the energy of a membrane patch P, amended by a contribution due to
its boundary ∂P [7], is expressible as:
Computational Studies of Biomembrane Systems: Theoretical Considerations. . .
239
experimental data.
2 Theory and Simulation of Lipid Bilayers
To provide a basis for both the theoretical ideas and the computational techniques
that we will discuss in this chapter, we start by reminding the reader of some
essential concepts. Section 2.1 reviews some basic aspects of the Helfrich Hamiltonian. Section 2.2 introduces three coarse-grained membrane models that will be
used in the remainder of this chapter. In Sects. 2.3 and 2.4, we discuss the bending
moduli and the surface tension of membranes in more detail, and finally comment
on multicomponent membranes in Sect. 2.5.
2.1 Basic Continuum Theory Concepts
2.1.1 Continuum Elasticity of Lipid Membranes
Lipid molecules are amphipathic: they consist of a hydrophilic head group and
typically two hydrophobic (fatty acid) tails. Yet, despite their amphipathic nature,
lipid molecules dissolved in water have an extremely low critical aggregate concentration (nanomolar or even smaller [1]), and thus under most common conditions lipids spontaneously aggregate. Because the roughly cylindrical shape of
lipids leads to two-dimensional self-assembly, thermodynamic considerations [2]
show that – in contrast to the finite size of spherical and wormlike micelles – a
single macroscopic aggregate containing almost all of the lipids will form: a
two-dimensional bilayer membrane. Its lateral dimensions can exceed its thickness
by several orders of magnitude.
2.1.2 The Helfrich Hamiltonian
If lipid membranes are subjected to lateral tension, they typically rupture at stresses
of several millinewtons per meter (mN/m), with a remarkably low rupture strain of
only a few percent [3]. At large scales and moderate tensions it is hence an excellent
approximation to consider membranes as largely unstretchable two-dimensional
surfaces. Their dominant soft modes are not associated with stretching but with
bending [4–6]. Within the well-established mathematical framework developed by
Helfrich [5], the energy of a membrane patch P, amended by a contribution due to
its boundary ∂P [7], is expressible as:
Computational Studies of Biomembrane Systems: Theoretical Considerations. . .
239