In the past half century, biologists have made great strides in understanding the intricate structure of the cell and the relationship between this structure and cellular function. The most convincing evidence against such a simplistic picture is provided by the complex and organized structure of the cell itself.
Mechanics and structure
These experiments directly observed the processive, “hand over hand” movement of the motor along actin filaments [ 9 ]. As we have discussed above, many of the most important processes in the cell are structurally complex.
Mechanotransduction and MscL
Road map for the MscL Chapters
Chapter 4 is a detailed development of the framework for studying membrane-protein interactions that was used in the previous chapter. The membrane-protein interaction framework developed in this chapter can also be applied to studies of the function of other transmembrane proteins.
The high-curvature mechanics of DNA
Road map for the DNA mechanics chapters
Chapter 7 outlines the limitations of the breakable Wormlike Chain model described in the previous section. At sufficiently short contour lengths, the Wormlike Chain model fails to describe the chain statistics of the general models.
![Figure 1.2: Does the Wormlike Chain model fail? The cyclization J factor measured by Cloutier and Widom [20] (red) is compared to theoretical prediction of the Wormlike Chain model (blue) in the left-hand panel](https://thumb-ap.123doks.com/thumbv2/123dok/11220180.0/22.918.125.761.352.557/wormlike-cyclization-measured-cloutier-compared-theoretical-prediction-wormlike.webp)
Introduction to mechanotransduction
Biological membranes
Although this biological membrane is sometimes called a lipid bilayer membrane, this is a misnomer as biological membranes contain a very large number of constituent proteins in addition to lipid molecules. In the test tube, lipid bilayer membranes can self-assemble in polar solutions due to the hydrophobic interaction of the tail groups of the lipid molecules, which aggregate into bilayers, exposing their polar head groups.1 (See Figure 4.3.3.) Some constituent proteins can also the artificial membrane is reconstituted.
Mechanosensation and mechanosensitive channels
The channel is mechanically linked to both the cell's cytoskeleton and an extracellular anchor. Once the mechanosensitive channel opens, ions pass through the channel and generate an electrochemical signal, which is then amplified and transmitted through other voltage-gated channels [4, 6].
Mechanosensation in prokaryotic cells
MscL is tension gated
The free energy associated with the conformational changes of the channel with respect to pressure is of order. The ratio between the free energy due to tension and that due to pressure is [1].
Patch clamp experiments
For MscL, the current is measured as a function of the applied pressure difference across the membrane. The most striking feature of the difference in free energy between the open and closed states is this. it depends linearly on the membrane voltage.
The MscL crystal structure
The sensitivity of the channel, the slope of the free energy with respect to the voltage, is the surface difference between the states ∆AOC = 6.5 nm2. This measurement of area change is large for a channel protein.3 This large area change is due to the opening of a 3 nm pore in the channel core and must correspond to a very significant conformational change in the protein.
The importance of membrane-protein interactions
An analogy to nucleation
Therefore, these data show a dramatic increase in the input voltage (voltage at which PO = 0.5) as a function of bilayer thickness (acyl chain length). The energetic contribution from the line voltage scales as the interface size (R) and the contribution due to the voltage scales as the channel area (R2).
Summary of MscL Chapters
Analytical estimates of the forces and free energies generated by bilayer deformation reveal a compelling and intuitive model for the MscL channel transition analogous to phase two nucleation. We argue that the competition between hydrophobic mismatch and tension results in a surprisingly rich story that can provide quantitative comparison with open tension measurements for MscL when reconstituted in bilayers of different thicknesses and qualitative insights into the channel function of MscL and other transmembrane proteins.
Introduction
In a forthcoming paper, we will show that the mechanics of the bilayer must play an integral role in mechanotransduction and channel function. These analytical calculations reveal a compelling and intuitive model for the MscL channel gate that is the subject of this current article.
The energy landscape of the bilayer
In fact, we will treat the bilayer deformation energy as an external potential with respect to the conformational energy landscape of the protein. The strain energy of the bilayer is plotted as a function of radius for different values of the applied voltage.
Results
If the double layer deformation dominates the free energy of the states, the ephemeral nature of the substates is a natural consequence of their intermediate radii. In section 3.2, we presented some rather general arguments about the shape of the two-layer deformation energy landscape.
Discussion
In the above argument, we focused on the radial dependence of the bilayer deformation free energy and fixed the other components of the state vector X. Our model can also naturally explain the stabilization of the open state by LPC [6] as well as the short-lived nature of the substates [5].
Acknowledgments
Appendix: Lipids
Supporting Methods
Lipids
Line Tension
If the inclusion interface is relatively flat and the top and bottom surfaces of the bilayer are normal to the inclusion interface as depicted in Fig. We therefore conclude that the mechanics of the bilayer play a significant role in determining the conformation and function of the channel.
Introduction
Specifically, we calculated the free energy due to the deformation of the bilayer due to the presence of a membrane protein using an analytical model developed for the study of bilayer mechanics (Canham, 1970; Helfrich, 1973; Evans, 1974 ). We find that the free energy of the double-layer deformation can be of the same order as the free energy differences between conformational states of the MscL channel, as measured by Sukharev et al.
Free Energy of the Bilayer-Inclusion System
- The Calculation of the Bilayer Free Energy
- Connection between H ± 0 and Channel Geometry
- Forces, Torques, and Tensions
- Relation between pressure gradients and generalized forces
The energy cost for changing the thickness of the bilayer is the modulus of compression and extension, KA. Minimization of the effective free energy (Huang, 1986) gives two separate differential equations (Fournier, 1999) for the equilibrium configuration in the fields su(~x) and h(~x), namely,.
Free Energy Estimates and Physical Interpretation
- Areal Deformation
- Gaussian Curvature
- Spontaneous Curvature
- Bilayer Interface Energy
- Mid-Plane Deformation
- Thickness Deformation
- Saturation of Thickness Deformation
The dominant term in the midplane strain energy scales linearly with R, since the area of double-layer strain is roughly proportional to the perimeter. The thickness deformation energy is also roughly linear in R, since the area of the deformed double layer is roughly proportional to the channel perimeter.
Application to MscL Gating
Opening Probabilities for Two State System
Note that this expression is independent of the free energies of other states as a result of working with the ratio of open and closed probabilities. This observed compensation behavior is indirect evidence that the difference in area between the closed and open states is roughly independent of the bilayer parameters, implying that the open and closed states are quite well defined, at least radially.
Mismatch and Gating
The theoretical free energy difference compared to the experimental data of Perozo et al. 2002a) for different choices of the geometrical parameters characterizing the open state thickness. However, it is very difficult to reconcile such small values of the line voltage with the measured free energy differences in Perozo.
Spontaneous Curvature and Gating
Remember that the relative difference in voltage and opening pressure will be the same if the radius of slicing is approximately constant.). We cannot rule out that more subtle mechanisms are responsible for the changes in input sensitivity.
Conclusion
Finally, we suggest the need for a detailed test of the static conformational approach by carefully measuring the area change between states as a function of both the applied stress and the acyl chain length.
Acknowledgments
Lipid selectivity of calcium- and magnesium-dependent adenosine triphosphatase studied by fluorescence quenching with brominated phospholipid. Loss-of-function mutations in the funnel margin of the mechanosensitive channel MscL.Biophys.
Appendix
- Units and Conversions
- Bilayer Parameter Model
- Effective free energy density
- Equilibrium equations and solutions
- Calculation of Free Energy
- Saturation of Thickness Deformation
- Details of the Perozo vs Powl comparison
The Gaussian bending energy is therefore exact. where θ is the angle of the bilayer away from horizontal at the interface. The Gaussian curvature contribution causes bending of the protein to relieve the bending of the bilayer.
Translating genetic code into biological function
DNA packaging
The mean square end-to-end distance is approximately the square of the physical length of the polymer in solution. The end of the genome probably has a radius of curvature of only a few nanometers.
Transcriptional regulation
Panel C shows a schematic representation of the effective concentration of the auxiliary operator (blue dot) when the primary operator is bound to the repressor (pin). Consequently, the behavior of gene regulatory circuits is sensitively dependent on the base pair spacing of the operators.
DNA chain statistics and the wormlike chain model
- The random walk model
- The wormlike chain model
- The mechanical limit
- Chain Statistics
- The equivalence of the WLC model to a free quantum particle
- What is the persistence length?
The length scale above which DNA is efficiently bent by thermal forces is the persistence length (ξ∼50 nm). The data in this figure comes from Thijn van der Heijden, Fernando Moreno and Cees Dekker in Delft.). The persistence length is the correlation length of the polymer tangent (Eq. 5.23), but is also equal to the flexural modulus divided by kT.
Effective concentration
Chain statistics and the J Factor
The DNA is fixed at one end, resulting in effective operator concentration (blue dot). We will denote the generalized coordinates that describe the relative displacement and orientation of the end {qi}.
Ligation, kinetics, and the cyclization assay
Note also that the bands corresponding to the linear sequences run at the correct speed relative to the base pair scale on the right side of the gel. 2 The shelf life of unbound products is not long enough for the gel to work.
A startling discovery?
In the plot above, the relative cyclization free energy is compared to the relative nucleosome formation free energy for 94 bp sequences. The WLC model does not include the twist modulation of the J factor, which is evident in the continuous sets of experimental data and discussed in Sect.5.3.1.
What’s wrong with the wormlike chain model
Recent DNA cyclization experiments by Cloutier and Widom have provided evidence that the linear-elastic bending theory fails to describe the highly curved mechanics of DNA. Motivated by this recent experimental work, we develop a simple and exact theory of the statistical mechanics of linear-elastic polymer chains that can undergo a bending transition.
Introduction
This model is an extension of the well-known Wormlike Chain (WLC); we call it the Kinkable Wormlike Chain (KWLC). In section 6.5 we precisely calculate the Fourier-Laplace transform of the spatial propagator and discuss several limitations of these results.
Kinkable Wormlike Chain Model
We show that kinking causes an exact renormalization of the tangent persistence length, and we write exact expressions for the mean squared end distance and radius of gyration. When σi = 0, the vertex is kink-like and the energy is independent of the bending angle at that vertex, but there is an energy penalty for realizing the kink state.
Partition functions
The convergence of the partition function assures us that the continuum limit is well defined. Expanding the partition function in a power series shows that the twist number distribution is also correct.
Tangent partition function and propagator
Moment-Bend & kink number
We calculate the constitutive relation in the usual way for the deviation angle θ. In the short length limit, the WLC chain moment is simply linear in deflection: τ =−θξ/L.
Persistence length
At least in principle it is therefore possible to determine the bending stability from higher order moments of the distribution. From an experimental perspective, this corresponds to scattering experiments at a large wavenumber, power expansion for large power, or circuit experiments for short contour length.
Tangent-spatial and spatial propagators
Wave number limits
By expanding the exponential in the definition of the Fourier transform, it can be shown that this result is equivalent to showing that R2 is exactly renormalized. The rigid rod propagator is useful when discussing the limiting behavior of the J-factor at short contour lengths and is discussed in more detail in the Appendix.
Partition function in an external field and force-extension charac- teristicteristic
To calculate the power-expansion ratio, we need to calculate the partition function in an external field which can be concisely written in terms of the spatial partition function. Essentially, the kink modes freeze out and measurement of the elongation versus force measures the bending stability rather than the effective persistence length of the KWLC polymer chain.
Structure factor
For comparison, we have shown the structure factor for WLC with the same contour length for identical bending strength lengths (dashed) and identical effective strength lengths (dotted). The structure factor can be rewritten in terms of the Laplace-Fourier transform propagator. where L−1 is the inverse Laplace transform which can be calculated numerically.
Cyclized chains and the J factor
The looping J factor
We define the cycle J factor as the ratio of cycle to dimerization equilibrium constants. The KWLC loopingJ factor, JL∗, can be written in terms of the spatial propagator as.
The cyclization J factor
As discussed in the text, our theory does not include the twist-induced 10.5 bp modulation of the J factor. Our derivation of the cyclized partition function implies that the KWLC tangent-spatial propagator is known for one special case.
Topologically induced kinking
At large contour length L, the cyclization constraint is irrelevant, but as the arc length shrinks to roughly a persistence length, the bending energy required to cycle the chain becomes significant and there is a dramatic transition to the two-kink state, which dominates at short contour length. The dramatic transition from the zero-crack state to the two-crack state at short contour length is evident in fig. 6.6.3.
Discussion
The small contour length divergence is due to the two kink joint, which can close a loop without elastic bending regardless of the contour length, via the creation of 180-degree kinks. Its advantage is that it offers a simple way to characterize rigid biopolymers and a quantitative guide to the mesoscale effects of kinking.
Acknowledgments
We can consider a number of modifications to the theory to fix this problem: bend angle discontinuities, bends that are not fully flexible, etc. Physical properties of DNA in vivo as probed by the length dependence of the bending process of the lac operator. Biochemistry von Hippel , and J¨org Langowski.
Fourier & Laplace Transforms & convolution theorems
Circular convolutions
By circular we mean that the end points are identified so that the arc length position of the chain points is not at an end point of the propagator. In this case it is convenient to redefine the arc length coordinate system to be zero at the position of the first kink, L1, and sum over the position of the chain points as depicted in Fig.6.9.1.
Numerical inverse transforms
J factor computation
It is therefore convenient to use the extended definition of K∗ in terms of theK(m)(eq6.62). Numerical calculations of the inverse transforms of powers of ˜K are still problematic in the L→0 limit since, while they are more convergent than K, they must be integrated with large k where it is difficult to calculate accurate Laplace transforms.
Kinkable rigid rod
The buckling rigid bar theory, derived above, provides a very useful analytical check of the KWLC model at short contour length. As we have illustrated above, the calculation of the dominant two kink contribution is straightforward in this limit.
Gaussian limit
Summary of notation
This work would eventually lead me to the Netherlands to get to the bottom of some strange AFM data. This chapter informally describes this experimental odyssey and the evolution in our thinking about DNA bending leading up to the final chapter on generalized models of polymer statistics.
Does DNA kink?
The kinking of macroscopic rods
For mean curvatures in the range where the bending energy density is nonconvex (κ1,κ2), the curvature is localized to kinks (regions of κ2). However, for mean curvatures in the range over which the bending energy density is non-convex, the minimum energy configurations consist of regions of curvature κ1 and κ2.
Beyond kinking
- AFM measurements of the bending energy of DNA
- Trip to Delft
- Surprising results from Vologodskii
- New data from Jon Widom
- What have we learned?
- General models for stiff polymers
Cees Dekker and colleagues had published a measurement of the bending energy as a function of the deflection angle for 5 nm DNA segments. These new bending energy measurements were astonishingly linear in deflection angle.
![Figure 7.3: The bending energy of DNA on short length scales? The tangent distribution function is measured as a function of deflection angle for 5 nm sections of long sequences of DNA absorbed to mica via AFM [11]](https://thumb-ap.123doks.com/thumbv2/123dok/11220180.0/184.918.269.563.90.303/distribution-function-measured-function-deflection-sections-sequences-absorbed.webp)
Introduction
Confronting two world views
First, to put the possible failure of the WLC model into perspective, it is useful to consider the bending of macroscopic rods. Regardless of the complicated structure of the theory at short length scales, the Renormalization Group guarantees that the long length scale chain statistics will be described only by a theory of renormalizable operators.
Summary
Violations of linear elastic theory analogous to those observed in macroscopic bending are therefore predicted in experiments that probe DNA bending at short lengths: experiments such as the cyclization experiments of Cloutier and Widom. Finally, we discuss the results of this paper in the context of recent cyclization measurements by Vologodsky et al [7] and new preliminary AFM measurements of the tangent distribution function.
Defining discrete link theories
Statistical mechanics
The systematic study of the length scale dependence of statistical mechanics theories is called the Renormalization Group [8]. This assumption implies that the partition function for a chain of N links splits into N−1 factors of the one-link partition function:. where ti+1 is the outgoing tangent of the vertex of link i.
Sub-Elastic Chain Model: A toy model
