The cell membrane is a richly inhabited landscape. Its undulating and dynamic terrain is peppered with proteins regulating what enters and leaves the cell. Various classes of membrane proteins interact with different environmental signals to determine when to allow molecular species such as ions to pass through the membrane. For example, voltage-gated ion channels are sensitive to millivolt-scale transmembrane electric potentials and respond to these voltages by undergoing a conformational change that allows selected ions to pass.
A growing body of work [1
] suggests that the properties of the membrane influence the gating behavior of channels. In other words, the bilayer is not a passive bystander in membrane-protein function. This is demonstrated in the context of mechanosensitive channels whose function is acutely sensitive to properties of the surrounding bilayer such as lipid tail length, spontaneous curvature, and tension [2
]. Previous theoretical work has focused on the observed connection between channel function and membrane elastic properties by examining membrane deformations at the protein-lipid interface [8
Sensitivity to membrane mechanical properties is not unique to mechanosensitive channels. Voltage-gated ion channels also demonstrate sensitivity to applied membrane tension [3
] and intrinsic elastic properties such as membrane stiffness, which has been shown to be correlated to deactivation of voltage-dependent sodium channels [7
]. Therefore it is well established that the physical properties of the membrane influence channel gating, and in this work we exploit the channel-membrane interaction in the hope of learning about the structural changes of the channel itself. Our models are “coarse-grained” in the sense that the channel-membrane interactions are represented by different classes of membrane boundary conditions that replace the complex details of atomic-level motions.
An equivalent mindset that has proven useful for considering the connection between ion-channel function and membrane properties is the use of the so-called lateral pressure profile [19
]. These profiles have been shown to be related to stability of conformational states of membrane-bound proteins. In our work, we consider the complementary elastic-deformation models, in which we assume that the important contributions from the lateral pressure are accounted for in their lowest order moments, in the form of membrane parameters such as bilayer and leaflet spontaneous curvature and tension.
A. Structure and function of voltage-gated ion channels
As an example of the type of problem this work addresses, we consider voltage-gated ion channels as a case study. Although the crystal structure of the well studied Shaker
channel Kv1.2 is available in the open conformation [22
], no voltage-gated channel structure has thus far been determined in both the closed and open conformations. The mechanism by which voltage-gated ion channels open and close in response to changing electric potentials remains uncertain; the goal of this paper is to explore the implications of different classes of structural models for membrane-protein interactions. The comparison is based on the channel’s sensitivity to bulk membrane mechanical properties.
All the channel mechanisms we explore contain two critical features: a pore region responsible for selectively blocking and passing ions across the membrane, and sensor regions that confer voltage sensitivity to the pore region. The voltage sensing motif is highly conserved across voltage-gated channels and consists of a bundle of four transmembrane helices [23
]. At every third position on the fourth helix (named S4) there is a charged arginine or lysine residue that is responsible for voltage sensitivity [24
]. In Shaker
family channels, for example, these charged residues contribute 12 positive elementary charges per tetrameric channel [27
], or three for each subunit. The conformational change to the conducting state decreases the electrostatic potential energy of these charged residues by a mechanism that remains uncertain. The charges either move through the electric potential or the channel manipulates and changes the electric field around them.
From the point of view of membrane deformations, the differences between the channel gating models are best described in terms of how the sensor regions move during opening and closing to modify the electrostatic environment of the charged residues. They may swing across the plane of the membrane as a paddle [28
], or they may undergo a more subtle motion like a helical screw [29
]. Some models do not rely upon the sensor domain actively transporting the charges across the membrane, but rather propose that its motion creates crevices that control how far the surrounding ionic solution penetrates into the protein, thus manipulating the electrostatic field itself. For a thorough description of various gating models and comparisons to experimental results, see Refs. [23
The energy associated with changing the electrostatic environment of the residues is the voltage-dependent part of the gating energy, which we estimate using values for the Shaker family K+ channels. Assuming simple two-state Boltzmann statistics in which all four channel subunits occupy the same state at any given time, the probability that a channel is in the open state is given by 1
is Boltzmann’s constant and T
is temperature. In the absence of deactivation, the conductance of the membrane is proportional to Popen
. We define ΔGtot
, the total free energy difference between closed and open states. We can write ΔGtot
, where the terms represent the change in electrostatic gating energy, internal protein conformation free energy, and membrane-deformation free energy, respectively. We use a two-state model where “open” and “closed” describe the activation state of the pore. This is sufficient provided the conformations of the sensors and membrane are tightly correlated to that of the pore. Biological channels have many transition states which we assume have insignificant thermodynamic weight. Therefore the channel spends little time in those states, and we do not include them in our equilibrium model.
The energies calculated below also inform the kinetics of opening and closing. The free energy barriers for the kinetic transitions include membrane deformation energies of the transient states. The kinetics will therefore inherit membrane-parameter dependence through the membrane deformation energies. However, the many transition states may all have different transient membrane deformations, and extracting the membrane-parameter dependence of any one transition would be difficult. Equilibrium statistics, however, would not depend appreciably on these transient states.
We estimate the electrostatic energy using experimental results. The transmembrane voltage at which Popen
= 0.5 is defined as V0.5
, where this half-activation voltage is typically negative and on the order of tens of millivolts. From Eq. (1)
, this voltage coincides with Gopen
= 0, implying that at V0.5
the electrostatic gating energy, ΔGelec
, balances the sum of membrane deformation and protein free energies: ΔGelec
). In other words, the internal energies of the system balance the energy supplied by the external electric field. Direct measurements of gating current give the effective charge per channel as Q
], where eo
is one positive elementary charge (i.e., eo
= 1.6 × 10−19
C). We estimate an upper bound on the total electrostatic gating energy as ΔGelec
, assuming the charges move across the full potential. With V0.5
= −35 mV as a typical transmembrane voltage at half activation for a potassium channel [27
], one finds that ΔGelec
. Therefore an upper bound on the combined membrane and protein contribution to the gating free energy that balances the electrostatic contribution is −16kBT
. For comparison, measurements of conductance-voltage curves for wild-type Shaker
channels have yielded electrostatic gating energies of 4 – 6kBT
per channel [32
]. Kinetics measurements on potassium channels yield activation energies of about 20kBT
per channel [34
]. Our calculated upper bound will serve as a benchmark against which we will compare membrane energetic contributions.
B. Conformation changes during gating
To focus on how the channel protein causes membrane deformation, we consider a coarse-grained model in which the protein is an axially symmetric shape that dictates the lipid-protein interface and creates deformations in the membrane. As the channel switches between open and closed states the membrane deforms and relaxes. We consider three types of deformations, as described in . Any small membrane deformation can be expressed as a combination of the three types. The deformed membrane may be associated with either the open or closed channel, so we commit to neither case and investigate both possibilities.
FIG. 1 Models of gating in terms of three types of deformation induced in the membrane. We discard all molecular details of the channel and focus solely on how it deforms the membrane. The types of deformation are (a) bending of the midplane; (b) normal compression (more ...)
A relaxed membrane, with no spontaneous curvature, lies flat when undisturbed. The first type of deformation, called midplane bending
, bends the bilayer away from the relaxed planar configuration. This deformation is induced by an effective protein shape with sloped sides  [35
]. Such a shape could arise from a noncylindrical protein structure, such as a truncated cone. The next type of deformation compresses or stretches the membrane leaflets from their equilibrium thickness. This is referred to as compression deformation
and is induced by dictating a nonequilibrium bilayer thickness at the membrane-protein boundary . A difference in hydrophobic thickness between protein and lipid bilayer, called hydrophobic mismatch, causes this type of deformation [11
]. The last type of deformation accompanies changes in the cross-sectional area of the protein. As the channel opens and closes, its areal footprint in the membrane may change, thus yielding to or pulling against the mechanical tension in the lipid bilayer. We refer to this as footprint dilation
. These three scenarios make the implicit assumption that the membrane shape is enslaved to the protein conformation. Although not mandatory, it is clear that some amount of frustration accompanied by an energy cost will result from a mismatch between the protein conformation and the natural lipid order. Our assumptions frame the simplest way to investigate the effect of this frustration on the channel activity.
C. Modeling strategy
The effective protein shape is, in principle, related to the atomistic details of the protein. It is determined by the geometry of the protein boundary and the locations and orientations of the hydrophobic and hydrophilic residues. However, the atomistic detail of the protein in the closed state is one of the unresolved issues, and we therefore avoid those details. Instead, we focus exclusively on the membrane deformations outlined in and ask (i) which deformation types contribute an energy that is relevant in the total free energy budget? and (ii) how does gating couple to membrane parameters? We will demonstrate that those parameters which can be tuned experimentally, such as mechanical bi-layer tension and thickness, can be used as tools to determine if there is a dominant mode of deformation during gating.
Such a coarse-grained approach to investigating lipid-protein interactions has clear advantages over atomistic approaches, such as simplicity, broad applicability to a wide range of systems, and the ability to classify proteins based upon a generic shape deformation. However, such an approach has natural limitations as well. Elastic theory may poorly describe membrane mechanics at short length scales, and in this analysis, we do not probe noncircularly symmetric deformations. Furthermore, we assume only one deformation type is significant. If multiple types exist and the energetic contributions are comparable, the deformations would be difficult to identify. This work is a starting point, and to obtain detailed descriptions of gating for specific proteins, it must be supplemented with other data from structural studies and simulations.
Thus far, we have proposed a simple model of the protein. In Sec. II we model the bilayer as a continuous elastic sheet and describe deformations in terms of functions giving the height and thickness of the bilayer. This formulation allows us to concentrate on estimating the energy while discarding the details of individual lipids and their interactions. Section II A discusses the energy functionals for the deformations. In Sec. III we utilize analytic methods to find the deformation profiles that minimize the energy functionals subject to the boundary conditions imposed by the protein shapes in each deformation type. We then use these results and membrane-parameter values from the literature to compute equilibrium energies. Section IV interprets these results in terms of membrane-parameter dependence in V0.5, and makes predictions for a new set of experiments.