This allows the mechanical properties of the bilayer to influence the conformation of embedded proteins. The shape of these potentials depends on the type of deformation and the shape of the interacting proteins.
Introduction
As a result of these studies, a number of ideas have been proposed to explain the different ways in which external forces can couple to membrane protein conformation. In the next section, we discuss experimental considerations that will help establish a closer link between theory and experimental techniques, with specific reference to how double-layer mechanics could be used to better understand structural changes during channel gating.
Statistical Mechanics of Mechanosensitive Channels
Lipid Bilayer vs. Protein Internal Degrees of Freedom
When the channel opens, these weights are lowered and the potential energy of the charging device is reduced. Given these definitions, we see that the change in energy of the charging device is given by
Bilayer Free Energy and Channel Gating
The Case Study of MscL
First, each membrane leaflet resists angle changes between adjacent lipid molecules, leading to membrane bending stiffness. These parameters depend on the elastic properties of the bilayer, in particular the flexural modulus of the bilayer (κb), the elongation modulus of the bilayer surface (KA) and the hydrophobic leaflet thickness (l).
Bilayer Deformation, Free Energy and the Role of Tension
- Midplane Deformation
- Thickness Deformations
- Approximating Bilayer Deformation: The Variational Approach
- Variational Approach for Midplane Deformations
- Variational Approach for Membrane Thickness Deformations
- Distilling the Design Principles
The midplane deformation energy scales approximately linearly with the radius of the protein and is unfavorable. Using this trial function, the free energy can be written as a simple expression of the form.
Experimental Considerations
Other Experimental Clues
In addition to the aforementioned analysis of MscL, other avenues of research have shown interesting connections between the function of membrane proteins and the lipid environment. One such pathway is the effects of membrane doping (by toxins, lipids, or cholesterol) on channel activity, as schematically.
Interaction between Transmembrane Proteins
- Enthalpic Interactions
- Entropic Interactions
- Protein Conformations Affected by Interaction
- Interaction and Protein Density in Biological Membranes
The Casimir force between two membrane proteins arises because the available spectrum of oscillations of the membrane midplane depends on the distance between the two proteins. Lateral fluctuations in lipid density in the membrane also lead to entropic forces between proteins.
Overview and Concluding Remarks
Alternatively, mutagenesis can be used to explore the same effect by varying the hydrophobic thickness of the protein. Due to the elastic deformations induced by mechanosensitive channels in the lipid bilayer, nearby channels can communicate their conformational state, resulting in cooperative coupling.
Acknowledgments
Regardless of the problems highlighted above, it is clear that the emergence of an increasing number of structures of ion channels together with functional studies of these proteins has raised the bar for what should be expected from theoretical models of channel function. Happy is he who discovered the causes of things and threw all fears, inevitable fate and the noise of the consuming Underworld under his feet." – Virgil.
Introduction
Background
In the second part we investigate the differences in gating behavior of two MscL proteins when held at a fixed distance. In the third part we investigate the conformational and spatial behavior of diffusing MscL proteins as a function of surface density.
Elastic Deformation Induced by Membrane Proteins
At the protein-lipid interface (r = ro), the hydrophobic regions of the protein and the bilayer must match, i.e. finally, the slope of the bilayer at the protein–lipid interface is set to zero (i.e.
Gating Behavior of Two Interacting Channels
As mentioned earlier, the interaction component of the free energy between two channels is a function of their states (s1 and s2), their edge separation (d), and voltage. At these separations, elastic interactions change the critical gate voltage and change the voltage sensitivity of the channel (see Figure 2.4).
Interactions between Diffusing Proteins
As the areal density increases, the more beneficial open-open interaction (see Fig. 2.3) shifts the open probability to lower voltages and reduces the response range (dashed lines) while increasing the maximum sensitivity, indicating that the density of the area can change the functional characteristics of a transmembrane protein. b) The probability that exactly one channel is open (P1 - solid lines) is shown with a low (blue) and high (red) density. As the areal density decreases further, the open–open interaction is no longer strong enough to overcome the entropy.
Further Discussion
- Comparison to Other Interactions
- Role of Elastic Boundary Conditions
- Effects of Spontaneous Curvature
- Implications of Bilayer Volume Conservation
- Experimental Examples of Bilayer Mediated Interactions
In the scenario where the energy is minimized with respect to the boundary slope, the position of the maximum slope is given by. If volume is conserved, the change in area of the membrane will be proportional to the compression.
Computational Methods
As another example, recent FRET studies showed that rhodopsin oligomerization is driven by exactly these types of elastic interactions and exhibits a marked dependence on the severity of deformation modulated by the thickness of the bilayers [90]. In summary, we have demonstrated that leaflet deformations are one of the main mechanisms of bilayer-mediated protein-protein interactions.
Accession Number and Acknowledgments
Summary of Bilayer-Mediated Interactions
Non-Specific Interactions from Crowding
- Ideal Crowding Tension
- Slightly Non-Ideal Crowding Tension
- Proteins that Change Area
- Choosing an Ensemble
- Excluded Area Effects
The total surface area of the proteins is then simply Ap =N α, where α is the surface area per protein, so that the total surface area of the vesicle is Atot=A`+Ap. So on the time scale of protein conformational change, the area of the lipids added, compared to the area change of the protein, is small.
Crowding in a Lipid Domain
For values of χ greater than this critical value, there is an energy barrier for proteins to join the domain. Although above the critical value of χ there is a critical protein concentration below which addition of the proteins to the domain is inhibited.
Concluding Remarks
Do thermal fluctuations of the water and dissolved molecules affect the flow properties of the channel? On the experimental side, we will discuss in detail how to measure the ensemble current properties of the bacterial large conductance mechanosensitive channel (MscL), using both wild-type and mutant forms, to gain a better understanding of both the lipid-related gating behavior and the nanoscopic flow through a membrane.
Gating and Conductance Properties of MscL
In the microbial environment, the pressure gradient across the inner membrane couples to the radius of curvature of the membrane to produce a lateral tension on the membrane surface. This gives us confidence that the behavior of the channel is well described by the relatively simple dose-response curve shown in Figure 1.
Modeling Hypo-Osmotic Shock
Then the volume change of the vesicle as a function of time is given by the differential equation dV. The second, more objectionable assumption is that the volume flux through the pore in the vesicle is independent of the flux of osmolytes through the pore, down the concentration gradient and out of the vesicle.
Measuring Ensemble Channel Flow
Bilayer Permeability and Proof of Concept
In (a) and (b) the pressure gradient is p2−p3 = 3000 Pa. c) Channel opening probability and volume normalized V/Vo as a function time for the gain-of-function mutant channel (∆G= 14kBT). Channel opening probability and normalized volume V /Voas a function time for the mutant channel. f) Membrane voltage as a function of time for the same physical scenario as (e).
Using Gain-of-Function Mutants
Fortunately, mutational studies of MscL have shown that by changing the hydration energy of the channel pore, the free energy of gating associated with channel structure, ∆Gprot, can be reduced [34,33]. Using our estimates of the reduction in hydration energy of the threonine and aspartic point mutants, the gate voltage is reduced to ~ 1.98kBT /nm2 and ~ 1.78kBT /nm2, respectively.
Controlling for Channel Density
Thus, within the constraints of this estimate, hydration of valines in the pore raises the free energy by ~3.9 kBT, while hydration of threonine and aspartic acid lowers the free energy by ~2.1 kBT and ~3.6 kBT, respectively. Thus, given the change in transition free energy from wild type to one of these mutants, we estimate that the transition free energy is lowered by ~6kBT for the threonine point mutation and by ~7.5kBT for the aspartic acid mutation.
Speculations
Possible Effects of Temperature
The pressure drop across the membrane acts to push water and osmolytes from areas of high pressure to areas of low pressure, performing thermodynamic work in the typical PV manner. Since the pore contains ~18 nm3 fluid volume at any given time, moving that volume across the membrane corresponds to a change in free energy from ~0.005kBT at lower pressures to ~1kBT at higher pressures seen in the actual physiological setting .
Interesting Possibilities
For comparison, consider the free energy change of a typical monovalent ion moving down an electrochemical gradient. A typical transmembrane potential is ~ 50 mV [1], which means a change in free energy for an ion moving from one side of the membrane to the other ~ 0.05 eV ' 2kBT.
Effects of Glass-Bilayer Binding in Electrophysiology
The Zero Pressure Limit
Patches Under Pressure
Additionally, this shows that for a given pressure and pipette radius, stiffer membranes have higher stresses. If we further assume that the membrane area is conserved, i.e. κ→ ∞, we find something analogous to the Laplace-Young relation.
Determining γ from Patch Curvature
One can easily derive a rather complicated expression for ζ, where this peculiar property of ρ is accompanied by ζ taking on complex values, as shown by the vertical dashed line in Fig.3.12c. Analytical results for stress (red) inkBT/nm2, dimensionless delamination (green) and dimensionless curvature (blue) as a function of pressure in mmHg for KA = 55kBT/nm2, approximately the value for DOPC.
Concluding Remarks
These forces, which are dependent on domain morphology, play an important role in regulating lipid domain size and in the lateral organization of lipids in the membrane. In the second main theme of this chapter, we use a simplified model mixture of lipids and cholesterol to investigate the interplay between lipid phase separation and bilayer morphology.
Dilute Domain Interactions
- The Elastic Model and Morphological Transitions
- Elastic Interactions of Dimpled Domains
- Discussion of Dilute Domain Interactions
- Summary of Dilute Domain Interactions
Inset: Equilibrium phase diagrams for bending moduli ratios of σ = 0.5(red), σ = 1(green) and σ = 2(blue) (the dashed lines are the approximation of eq.. b) Energy difference between the flat and dimpled states, normalized by the bending modulus κb , for domains with and without spontaneous curvature (υo = 0 → thin black line; With the single domain boundary slope, , set by the energy minimization of the previous section (i.e. eq. 4.6 ), the pairwise energy at each domain spacing, d, is minimized by ∂G/∂φ = 0 to find the domain tilt angle that minimizes the strain energy (see Section 4.4 for details).
Dimpled Domain Organization and Budding
Spatial Organization of Dimpled Domains
A clear example of the difference in domain growth rate, with and without elastic interactions, is shown in Fig.4.7 (d and c), respectively, and [214]. This can be understood in terms of the free energy of the entire set of domains.
The Budding Transition
Free energy of a blooming domain as a function of line tension (χ) and roll angle (θ) for domain sizeα= 10. At low line tension (before the blue line), flat and blooming morphologies are stable, but the flat ones state has a lower elastic free energy and there is an energy barrier between the two stable states.
The ‘Algebra’ of Morphology
The next transition is a flat and dimple domain that come together to form a dimple domain, as depicted in Fig. 4.14(c). The next transition is a flat and a knob domain that come together to form a dimple domain, as depicted in Fig. 4.14(g).
Discussion of Domain Morphology
A time series of spontaneous tube formation, nucleated by a domain (as indicated by the white arrow). At low voltage, the pitted domains still exhibited significantly slower coupling kinetics, with an interaction length scale of hundreds of nanometers (data not shown).
Summary of Domain Organization and Morphology
Our results indicate that charge is not a major player simply because membrane tension is the distinction between fast, flat-domain confluence and slow, pit-domain confluence, as shown in Figs. We have qualitatively observed the motions of dimple domains in the presence of high salt (∼300mM), where the Debye length is on the order of ten nanometers.
Detailed Analysis of Theory and Experiment
Overview of Materials and Methods
Without precise control of the membrane tension or the exact initial conditions (i.e., the exact number and size distribution of domains), many vesicles had to be sampled to see transitions. Often a slight increase in temperature (∼2C) was used to increase the available membrane surface area, thereby lowering the membrane tension sufficiently to induce morphological transitions.
Calculating Membrane Curvature and Area
On the surface defined by , the unit normal vector field is the gradient normalized by the size of the small part of the area associated with the unit vector at the pointer, respectively. The small green lines indicate the directions of the major curvatures, while their lengths indicate the magnitude of the major curvatures at those points.
Conservation of Domain Area
Zones whose principal curvatures have the same sign are colored in red (positive Gaussian curvature), while zones whose principal curvatures have opposite signs are colored in blue (negative Gaussian curvature). a) is adapted from an illustration by Eric Gaba under CCL licensor. For simplicity, let us consider the case where the domain is flat and thus the membrane voltage and phase boundary line voltage directly compete with each other - this is also the scenario where we would expect the largest potential area change.
The Small Gradient Limit
Let us assume for the moment that the complete solution of the fourth-order equation is the summation of the solutions of each of these second-order equations. The only remaining component of the free energy is the elastic contribution of spontaneous curvature in the domain.
Gaussian Curvature
Finally, with all contributions accounted for, we can sum the free energy of the system with contributions. The trace of this tensor is the sum of the principal curvatures, while the determinant is the Gaussian curvature [57].
Equilibrium Domain Shapes
If we approach the phase boundary from a large domain area or a large line voltage, the boundary slope is → 0 at a critical value of the membrane parameters, and therefore we can ignore the terms O(2) in the above equations. In the regime where the dimensionless domain area is small, the relationship between these three equations simplifies to 5.
Scaling and the Critical Exponent
The first two equations can be solved numerically to find the critical value βc at the phase boundary as a function of α, and then the critical line voltage for pits isχc = (σβ2c + 1)p. The red lines are forn= 1, while the blue lines are for n= 2, the black lines correspond to the flat state. e).
Divergence Theorem Solution for the Deformation Energy
We write the integrand of the functional axis. 4.135) With subsequent application of the divergence theorem we obtain a relationship similar to region 1. 4.137). Now we see that the total elastic free energy is a measure of the change in curvature at the boundary between the two regions.
Spherical Domain Budding
The second interesting line in the phase diagram is when it becomes equally favorable to be in the flat or nascent state, defined by G|θ=π = G|θ=0 = 0, which simply yields χbud/2. Figure 4.18 shows the bifurcation diagrams resulting from different scenarios of ∂G/∂θ = 0 in the spherical emerging model, including cases with spontaneous curvature.
Vesicle Tension and Entropy
Constructing a Thermal Ensemble
The Equation of State
Simulating Membrane Conformations
The 1D Interaction Potential
Interactions of Asymmetric Domains
Effects of Domain Size Asymmetry
Corrections from Size Asymmetry
Coarse Control of Membrane Tension and Inducing Phase Separation
Error Introduced by the Curved Vesicle Surface
In vitro Selection and Representative Data
Domain Tracking and Data Analysis
The Fictitious Confining Potential
The Geometric Derivation
Plasma Membrane Domains
Experimental Methods
Plasma Membrane Domain Interactions
Concluding Remarks
Acknowledgments
Physiology of L1 Binding
The Adhered Vesicle Shape
Shape Constraints
Calculating Surface Area
Calculating Volume
Protein Conservation and Surface Energy
Vesicle Bending Energy
Free Energy of Adhesion
Connection to Experiment
Simulating Adhered Vesicle Shape
Note on Linearity of Tension and Pressure
Tether-Based Assay of Vesicle Adhesion
Tethered Vesicle Area
Tethered Vesicle Volume
Bending Energy of a Tethered Vesicle
Free Energy of Tethered Vesicle Adhesion
Connection with Experiments
Experimental Setup
Controls and Vesicle-Vesicle Binding
Azimuthally Symmetric Vesicle Adhesion
Possible Extensions and Complications
Concluding Remarks
Derivation of the Cahn-Hilliard Equation
Deriving a Natural Length and Time Scale
Kinetics in Fourier Space
