Give an Example of Van Der Waals Forces

Use of Molecular Docking as a Decision-Making Tool in Drug Discovery

Azizeh Abdolmaleki , ... Jahan B. Ghasemi , in Molecular Docking for Computer-Aided Drug Design, 2021

2.1.2 Van der Waals potential in protein–ligand complexes

Van der Waals force is a factor that contributes to the formation of protein–ligand complexes. This force can be computed by physical models considering the potential of Lennard-Jones which can accurately predict van der Waals interactions with a complex calculation, which enable us for its usage to simulations of molecular docking and virtual screening of large ligands' database. One of the most challenging topics in structural biology is predicting the molecular binding of protein–ligand or protein–protein pairs ( Fahmy & Wagner, 2002). The main problems are the following:

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Reliable estimation of the binding free energies for docked states

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Take into account the possible docking orientations in a high-resolution situation

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Consider structural relocations and movement of the docking surfaces upon interaction

Fahmy and Wagner have proposed TreeDock algorithm by minimizing the van der Waals energies for protein docking. It focuses on the problems in a rigid body docking search by exploring enough number of docking orientations and indicating molecules as multidimensional binary search trees (Fahmy & Wagner, 2002). The traditional mechanism of action for a noncovalent inhibitor is commonly reliant on the small inhibitor that binds to a protein–target noncovalently and the consequent improvement of selectivity and potency. This mechanism is usually made by refining the shape and noncovalent interfaces (van der Waals interactions, hydrogen bonds, salt bridges, etc.) between the inhibitor and target binding site residues.

Some empirical scoring functions use approximate van der Waals interactions with the Lennard-Jones potential to estimate protein–ligand interactions for a specific biological system. Docking processes are different for the covalent bonding from the noncovalent bonding, even using the same software. Developing the mainstream docking techniques has been emphasized on the operational estimate of the binding modes of noncovalent inhibitors (Aljoundi et al., 2020).

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Interfacial Phenomena

E. Hadjittofis , ... J.Y.Y. Heng , in Developing Solid Oral Dosage Forms (Second Edition), 2017

8.2.1 Van der waals forces

Van der Waals forces can be categorized, on the ground of the molecules involved in the interaction, to: Keesom forces, Debye forces, and London (dispersion) forces, summarized schematically in Fig. 8.1. For Keesom forces, two polarized molecules interact because of the inherent difference in charge distribution. In the case of Debye forces, a molecule with a permanent dipole induces charge redistribution to neighboring molecules with no dipole moments. Finally, London forces arise in molecules without permanent dipoles. The fluctuations on the electron cloud lead to temporary changes in the charge distribution inducing a charge redistribution to neighboring molecules. The mathematical formulation of all three components has the general form:

Figure 8.1. Schematic representation of the three components of van der Waals forces.

(8.2) $U\left(r\right)=-\frac{C}{r^6}$

where C is a constant changing slightly for each component and r is the intermolecular distance.

Apart from van der Waals forces, there exists an interaction between electron poor and electron rich atoms, via the sharing of a lone pair of free electrons from the latter to the former, called dipole-dipole interactions. When a hydrogen atom is involved in this interaction, the interaction is called hydrogen bond. Due to its nature, it is characterized by directionality and short-range action, meaning that the electron poor and the electron rich sites should "face" one another at a close distance. Directionality leads to the formation of weak structures of molecules held together via the bonds formed through the sharing of lone pairs of free electrons. The effect of these bonds is affected by the proximity of the molecules. Thus, it changes depending on the state of the matter. In particular, as matter moves from vapor to liquid or solid states, the importance of them increases. The physical chemical property of ice is the most striking manifestation of the effects of these forces.

Additionally, Coulombic forces develop between charged atoms or ions. The nature (attractive or repulsive) and magnitude of the interaction depend on the sign (positive or negative), the ionic charge of the contributing elements, the distance, and the medium separating them. For instance, two oppositely charged ions exhibit attractive interactions, the magnitude of which is given by the following relation:

(8.3) $U\left(r\right)=\frac{Q_1{Q}_2}{4\pi {\epsilon }_0{r}^2}$

where Q ₁ and Q ₂ are the charges, r is the distance between them, ε ₀ is the dielectric constant of the medium separating these ions.

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Structure, Synthesis, and Application of Nanoparticles

Ashok K. Singh PhD , in Engineered Nanoparticles, 2016

Appendix 2: Van der Waals Forces

Van der Waals forces are weak electrostatic forces that attract neutral molecules to one another. Particles in liquid or air vibrate and move constantly. Thus, they collide with other particles, including the media's particles such as water molecules—the process known as Brownian motion ( Figure 50). When these neutral particles reach within a threshold distance, electrons from one particle are pulled towards the nucleus of the other particle, thus causing transient polarization (electron-rich domain: δ⁻ and electron-deficient domain: δ⁺). The δ⁺ side of one particle attracts the δ⁻ side of another (London dispersion interaction). Particle vibration or motion breaks the interparticle interaction and the particles separate.

Figure 50. Nanoparticles in perpetual Brownian motion (A) thus they collide with the dispersion phase particles. Randomly, two particles come close so that their electrons localize (because of their electrostatic repulsing) and develop transient dipole–dipole interaction (B). Vibration breaks the weak bond and the electrons are delocalized (C). Simultaneously, other particles come close and form transient bonds (D). Thus, even the noble metals exhibit weak transient bond formations.

VDW interactions have an attractive interaction between the atoms, which results from the induced dipoles, and a repulsive interaction, which results from overlap of the electron clouds of the two atoms, when they get too close to each other. The total energy of VDW interactions can be approximated by the Lennard-Jones expression (Figure 51).

Figure 51. The van der Waals Potential is a function of the distance between the centers of two particles.

When two nonbonding particles are in an infinite distance apart, the possibility of them coming together and interacting is minimal. As the distance of separation decreases, the probability of interaction increases. The particles come closer together until they reach a region of separation where the two particles become bound and their bonding potential energy decreases from zero to become a negative quantity. While the particles are bound, the distance between their centers will continue to decrease until the particles reach an equilibrium, which is specified by the separation distance at which the minimum potential energy is reached. Now, if we keep pushing the two bound particles together past their equilibrium distance, repulsion begins to occur, as particles are so close to each other that their electrons are forced to occupy each other's orbitals. Therefore, repulsion occurs as each particle attempts to retain the space in their respective orbitals. Despite the repulsive force between both particles, their bonding potential energy rises rapidly as the distance of separation between them decreases below the equilibrium distance.

The π-electron interaction occurs between two organic molecules having CC π electrons or between an organic molecule having CC π electrons and another polar or nonpolar molecule (Matthews et al., 2014). As shown in Figure 52(A), benzene ring contains six CC π-electrons that, when delocalized, form two bands of electron-rich regions (δ⁻) separated by a positively charged region (δ⁺) due to the H atoms (Figure 52(A)). Two benzene rings form a π-π interaction in stacked, T-shaped, and parallel modes. Figure 52(B) shows an interaction between δ⁺ and δ⁻ charges between two molecules. Figure 52(C–F) shows some of the examples of different types of π-electron interaction.

Figure 52. Van der Waals interaction involving π electrons.

(A) Benzene ring contains six p-orbital electrons that forms two rings of delocalized charges (δ⁻ and δ⁺). (B) Two benzene rings interact via π–π bonds on different configurations. (C) Water and benzene rings form polar H-bonds between δ⁺ of water's H atoms and δ⁻ of the benzene tings. (D) VDW interaction between a metal and an π electrons of benzene. (E) Stacking VDW bond formed between fluoro benzene and benzene. (F) Hydrophobic π–π interaction between two hydrophobic groups.

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Computer-Assisted Drug Design

A.R. Leach , in Comprehensive Medicinal Chemistry II, 2007

4.05.2.11 van der Waals' Interactions

van der Waals' forces constitute the final contribution to the general force field introduced in eqn [1] and are named in honor of van der Waals, who quantified the deviations from ideal gas behavior that are demonstrated by systems such as the noble gases. The conundrum is that in such systems there are no permanent multipole moments and so the electrostatic interactions described above cannot be used to invoke the deviations from ideal-gas behavior. If the interaction energy between two rare gas atoms is studied experimentally (using, for example, a molecular beam) then two distinct regions are observed. At infinite distance their interaction energy is zero; as they approach each other the energy decreases (i.e., they attract). The curve passes through a minimum and then rapidly increases (Figure 5).

Figure 5. Schematic illustration of the Lennard-Jones potential, showing the steep repulsion as atoms approach each other, the minimum, and the asymptotic approach to zero potential energy as the atoms move further apart.

The attractive contribution arises from dispersive forces (sometimes called London forces ⁷¹ ). These are due to instantaneous dipoles, which then induce dipoles in neighboring atoms, giving rise to an attractive effect. The repulsive contribution has a quantum mechanical origin in the Pauli principle, which forbids any two electrons in a system from having the same set of quantum numbers. This has the effect of reducing the electron density in the internuclear region as two atoms approach closely. This reduced electron density leads to repulsion between the incompletely shielded nuclei.

The mathematical models used to reproduce this behavior usually involve two terms, one corresponding to the attractive and one to the repulsive parts of the potential. The Lennard-Jones 12–6 function is perhaps the best known of these, with the following functional form:

[18] $E=4\epsilon \left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]$

There are two adjustable parameters in this equation: the collision diameter σ (the separation for which the energy is zero) and the well depth ε. An equivalent expression writes the equation in terms of the separation at which the energy passes through a minimum, r_m :

[19] $E=\epsilon \left[{\left(\frac{r_m}{r}\right)}^{12}-2{\left(\frac{r_m}{r}\right)}^6\right]=\frac{A}{r^{12}}-\frac{C}{r^6}$

The r ⁻⁶ contribution has a reasonable theoretical basis that is demonstrated in some of the simple models of the dispersive interactions. By contrast, the r ⁻¹² term has no such theoretical basis and is used primarily to facilitate rapid calculation (by squaring the r ⁻⁶ term). Various alternative formulations of the van der Waals' interaction have been suggested. Some of these involve the use of a different power (e.g., 9 or 10) for the repulsive part of the potential in order to give a less steep curve. Halgren ^72–74 proposed the use of a 'buffered 14–7' potential based on the following general functional form:

[20] $E=e_{\mathit{ij}}{\left(\frac{1+\delta }{{\rho }_{\mathit{ij}}+\delta }\right)}^{n-m}\left(\frac{1+\gamma }{{\rho }_{\mathit{ij}}^m+\gamma }-2\right)$

In this equation ρ _ij =r_ij /r_m,ij and δ and γ are constants. This equation returns the Lennard-Jones expression for n=12, m=6, δ=γ=0. In Halgren's 14–7 potential n=14, m=7, δ=0.07, and γ=0.12. The rationale for using such a seemingly complicated expression includes: the desire for the potential to have a finite value as the interatomic distance approaches zero (to eliminate the problems that can arise when trying to calculate the energy of very strained systems); to give a more accurate representation of the dispersion interaction; and to provide the flexibility to reduce the repulsive component without significantly changing the distance at which the potential crosses zero or the depth of the energy minimum (by modifying the value of the parameter δ).

Two alternative options for modeling the van der Waals' contribution are the Buckingham potential that has three adjustable parameters and in which the r ⁻¹² term is replaced with a theoretically more reasonable (but computationally more expensive) exponential term:

[21] $E=\epsilon \left[\frac{6}{\alpha -6}\exp \left[-a\left(r/r_m-1\right)\right]-\frac{\alpha }{\alpha -6}{\left(\frac{r_m}{r}\right)}^6\right]$

and the Hill potential, ⁷⁵ which is an exponential-6 potential with just two parameters; the minimum energy radius r_m and the well depth ε:

[22] $E=-2.25\epsilon {(r_m/r)}^6+8.28\times {10}^5\epsilon \exp (-r/0.0736r_m)$

Determining the most appropriate and accurate parameters for the van der Waals' forces is often difficult and time consuming. In contrast to the other terms in the force field, it is common for the van der Waals' parameters to be dependent solely upon the atomic number and not upon the atom type (e.g., the same van der Waals' parameters are used for all carbon atoms, irrespective of hybridization). A further common approximation is to assume that the parameters for interactions between unlike atoms can be derived from the parameters for the pure atoms using 'mixing rules.' Commonly used are the Lorentz–Berthelot mixing rules, in which the collision diameter σ _AB for the A-B interaction equals the arithmetic mean for the two pure species, and the well depth ε _AB is the geometric mean:

[23] ${\sigma }_{\mathit{AB}}=0.5({\sigma }_{\mathit{AA}}+{\sigma }_{\mathit{BB}});{\epsilon }_{\mathit{AB}}=\sqrt{{\epsilon }_{\mathit{AA}}{\epsilon }_{\mathit{BB}}}$

The collision diameter is sometimes derived as the geometric mean √(σ _AA σ _BB ) rather than the arithmetic mean, for example, in the OPLS (optimized parameters for liquid simulations) force field.

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Biocompatibility, Surface Engineering, and Delivery of Drugs, Genes and Other Molecules

T.F. Moriarty , ... R.G. Richards , in Comprehensive Biomaterials II, 2017

4.8.4.2.1 Lifshitz–van der Waals: Apolar molecular forces

Lifshitz–van der Waals forces arise from the attraction or repulsion of molecules due to the unequal distribution of electrons between bound atoms. ⁸⁷ When electrons are unequally distributed between atoms, one atom will gain a net negative charge by having extra electrons, while the other atom will be positively charged due to the excess charge of the nucleus resulting from the loss of electrons. When this arrangement of electrons occurs the atoms are called a dipole. An example of two dipoles interacting is illustrated in Fig. 9.

Fig. 9. An example of Lifshitz–van der Waals forces between two molecules of hydrogen chloride. The net positive charge of the hydrogen atom attracts the net negative charge of the chlorine atom.

Dipoles can exist due to a number of different reasons 87 :

1. In a molecule such as hydrogen chloride, a dipole is formed due to the higher positive charge of the chlorine nucleus than that of the hydrogen nucleus, this leads to the migration of electrons toward the chlorine nucleus leaving the hydrogen positively charged 2. Alternatively, molecules without this predisposition to form dipoles may temporarily gain a charge differential due to the varying probability of the arrangement of orbiting electrons. This can happen in any atom and creates a very weak- and short-lived intermolecular force 3. A dipole may also be induced by the proximity of another charged molecule. Induced dipoles occur due to the displacement of electrons of one molecule brought about by the attractive or repulsive affect of another. The dipole in this case is formed when a permanent dipole, for example, HCl, interacts with any other atom

Altogether the sum effect of these interactions is called the Lifshitz–van der Waals force. The Lifshitz–van der Waals forces are collectively defined as the apolar energy component. Lifshitz–van der Waals forces are relatively long range and inversely proportional to the distance between the two components. Due to the range of these forces they are often believed to initiate bacterium–biomaterial interactions by either attracting or repelling the bacterial cell at a relatively large distance.

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Functionalized Transition Metal Dichalcogenide-Based Nanomaterials for Biomedical Applications

Priyadarshi Kumar , ... Swee Liang Wong , in Biomedical Applications of Functionalized Nanomaterials, 2018

2 Basic Properties of Transition Metal Dichalcogenides

Owing to weak interlayer Van der Waals forces, TMDCs can be easily thinned down to a single molecular plane made up of three atoms along the out-of-plane axis (∼0.7  nm height; Fig. 10.1E). Because of its planar characteristic, TMDCs exhibit a very large surface-to-volume ratio, which provides straightforward access for large area functionalization. The relatively large surface area also allows maximal interaction with the target biomaterial for increased efficiency and sensitivity in the case of sensing and imaging (Fig. 10.2). Furthermore, lack of dangling bonds on its intrinsic basal surface allows high stability in both air and liquids as it does not readily react with ambient chemicals (Chhowalla et al., 2013). This stability is retained even when their lateral dimensions are reduced to the order of tens of nanometers (Tan et al., 2015). Such stability allows these materials to be easily introduced into biological systems for biomedical purposes or exposed to their associated environments for external diagnostics.

Semiconducting TMDCs possess an electronic band gap that increases with decreasing thicknesses (Wang et al., 2012). Coupled with considerable charge carrier mobility, it enables TMDCs to be used in field-effect transistors as biosensors where changes in conductance when exposed to the target analytes allow monitoring of their concentrations. Compared with graphene, which has no electronic band gap, higher sensitivities can be reached because of a larger current on–off ratio, which enhances the signal-to-noise ratio (SNR). In addition, as the layered crystal is thinned down to a single monolayer, it experiences a transition from an indirect electronic band gap to a direct electronic band gap (Wang et al., 2012). This results in a significant increase in photoluminescence (PL) yield of monolayer TMDC as compared to its thicker counterparts. As 2D TMDCs are atomically thick, they are highly sensitive to external elements, and the variation in PL strength in the presence of specific environments can be exploited for these materials to act as biosensors (Kalantar-zadeh and Ou, 2016; Kalantar-zadeh et al., 2015; Yang et al., 2015a; Chen et al., 2015; Pumera and Loo, 2014; Wang et al., 2014c; He and Tian, 2016). Electrochemical responses of these materials are also very sensitive to external perturbations and therefore can also function as a parameter for biosensing (Huang et al., 2014a,b; Wang et al., 2014b; Huang et al., 2013a).

Because of their unique electronic band structure, TMDCs exhibit strong optical absorption in the near-infrared (NIR) region (≈800   nm), higher than that of graphene oxide, and its extinction coefficient is also better than that of gold nanorods (Chou et al., 2013). These properties allow TMDCs to be efficient photothermal agents for use in drug delivery or therapeutic purposes as NIR lasers can efficiently penetrate through tissues with depth of several centimeters while having low tissue absorption. Presence of such photothermal agents enhances the selectivity of the excitation laser, allowing generation of heat in targeted areas (e.g., at tumor tissue) for therapeutic or drug release purposes while lowering the power density of laser required (Chen et al., 2015). Suspensions of TMDCs such as MoS₂ with low concentrations of 100   s of ppm can reach temperatures greater than 80°C when exposed for several minutes to NIR radiation (≈800   nm, 0.8   W/cm²) (Chen et al., 2015).

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Atomic Force Microscopy in Cell Biology

Daniel J. Müller , Andreas Engel , in Methods in Cell Biology, 2002

A Contact Mode Imaging

In a simplified model, electrostatic and van der Waals forces govern the tip sample interactions in aqueous solution. Hydrophilic surfaces of biological specimen are charged in water, leading to long-range electrostatic double-layer (EDL) interactions, which can be attractive or repulsive, depending on the surface charges which, in turn, depend on the pH. These interactions can conveniently be monitored by approaching the stylus to the sample and withdrawing it periodically. Attractive or repulsive forces are revealed by the deflection of the cantilever ( Butt et al., 1995; Heinz and Hoh, 1999). The interaction length of these forces can be controlled by screening the surface charges with electrolytes. Since the AFM tip (silicon nitride, Si₃N₄) is negatively charged at neutral pH and protein layers are often negatively charged as well, the electrostatic forces are frequently repulsive. In biological systems, van der Waals interactions do not depend on the ionic strength, decay rapidly, and are always attractive. The DLVO theory (Israelachvili, 1991) quantitatively describes these forces and allows the interactions between a spherical stylus and a planar sample to be modeled, providing critical information for the optimization of imaging conditions (Müller, Fotiadis et al., 1999).

Although suppliers specify AFM tip radii of 10–50   nm, topographs of flat biological surfaces that exhibit a resolution of better than 1   nm have been acquired routinely (Fotiadis et al., 2000; Mou et al., 1996; Müller and Engel, 1999, Müler, Sass et al., 1999, Müller, Schabert et al., 1995; Schabert et al., 1995; Scheuring, Müller et al., 1999; Scheuring, Ringler et al., 1999; Seelert et al, 2000, Walz et al., 1996). Therefore, the AFM tips employed most likely had a single nanometer-sized asperity that protruded sufficiently to contour the finest surface structures. However, such a small asperity would be expected to exert a prohibitively high pressure on the underlying structure inducing its deformation and reducing the resolution (Fig. 1A). Only contact of the sample with the large area of the whole tip would reduce the pressure on the macromolecules to a reasonable level (Shao et al., 1996). Developing this model further, electrolytes can be used to adjust the tip–sample interactions, provided the electrostatic forces are repulsive. Because of their long-range interaction, the electrostatic double-layer force does not contribute to submolecular structures observed by the high-resolution AFM topograph (Fig. 2). Ideally, the scanning AFM tip then surfs on a cushion of the long-range electrostatic repulsion while the small asperity is in contact contouring fragile details of the biological surface (Fig. 1B) (Müller, Fotiadis et al., 1999).

Fig. 1. Contact mode AFM imaging of native membrane proteins at subnanometer resolution. (A) Periplasmic surface of native porin OmpF simultaneously recorded in trace and in retrace direction. Imaged in 50   mM MgCl₂, 50   mM KCl (10   mM Tris–HCl, pH 7.6) the electrostatic double-layer (EDL) interaction was not eliminated. Correlation averages (insets; n  =   157) of each direction show structural differences. (B) Periplasmic surface recorded after adjusting the EDL damping. when imaging in 300   mM KCl (10   mM Tris–HCl, pH 7.6) the EDL interaction between AFM stylus and periplasmic surface exhibited a long range (≈5–10   nm) repulsive force of close to 0.1   nN. The scarming AFM stylus was thus damped by the long-range EDL forces allowing the soft protein structure to be reproducibly contoured at subnanometer resolution (Fig. 2). Structural differences between correlation averages (insets; n  =   104) of both scanning directions are minimized. While surface structures recorded in (A) do not correlate well with the atomic structure of OmpF porin (see Fig. 12) topographs observed in (B) show excellent agreement. Both topographs were imaged using contact mode at applied forces of 0.1   nN and at scan frequencies of 15.4   Hz. Vertical brightness range of topographs corresponds to 1.2   nm (raw data) and 1   nm (insets). Topographs (A) and (B) are displayed as reliefs tilted by 5°.

Fig. 2. Forces interacting between AFM tip and biological sample in buffer solution. The electrostatic double-layer (EDL) force interacts via long-range forces with a relatively large area of the macromolecular assembly. In contrast, the shott range van der Waals attraction and Pauli repulsion interact between individual microscopic protrusions of the AFM tip and the biological sample. The force effectively interacting at the AFM tip apex is a composite of all interacting forces. If the EDL force is negligible (F _el ≈ 0) or eliminated, the effective force is equal to the sum of the applied force and of the attractive van der Waals force |F _eff|   =   |F _appl  + F _vdw| < |F _appl|. Being the opposite of sign, a sufficiently high EDL force will partially compensate the applied force. Thus, under these conditions, the effective force is smaller than the applied force |F _eff|   =   |F _appl  + F _el  + F _vdw| < |F _appl|. At minimized interacting forces, possible structural deformation of the flexible biological sample will be minimized (Müller, Fotiadis et al., 1999). Additionally, minimized forces reduce the contact area between AFM tip and sample which will enhance the lateral and vertical resolution (Weihs et al., 1991).

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Structure and Dynamics of Membranes

R. Lipowsky , in Handbook of Biological Physics, 1995

5.3 Attractive interactions and unbinding transitions

Two identical membranes experience attractive interactions arising from Van der Waals forces. If the membranes carry no electric charges and the solvent contains no macromolecules, this van der Waals attraction dominates the direct interaction V(l) for large membrane separations l.

The precise form of this Van der Waals interaction has been discussed in section 3.2. For large separations, the form as given by (3.9) leads to ∆V(l) ~ 1/l ⁴. Since ∆V(l) ≪ 1/l ² for large l, such an interaction belongs to the universality class of short-ranged interactions. The simplest example for this universality class is provided by the square-well potential with

(5.11) $\mathrm{\Delta }V\left(l\right)=\{\begin{array}{cc}U& \text{for 0}<l<l_{\text{v}},\\ 0& \text{for}{l}_{\text{v}}<l,\end{array}$

which depends only on two parameters: the potential depth U < 0 and the potential range l _v. In general, any attractive interaction can be characterized by an effective potential depth and by an effective potential range. Such an interaction may then be approximated by a square-well potential with the appropriate values for U and l _v.

If one now considers the superposition, ∆F(l) = V _hw(l) + ∆V(l) + V _fl(l), one finds that the qualitative form of ∆F(l) depends on the temperature T as schematically shown in fig. 9(a). For low T, the function ∆F(l) exhibits its global minimum at small l which represents the bound state of the two membranes. For high T, the global minimum of ∆F(l) is at l = ∞ which corresponds to the unbound state of the membranes. At the characteristic temperature $T_{*}=\sqrt{\kappa \left|U\right|l_{\text{v}}^2/c_{\text{fl}}}$ , the bound and the unbound state have the same free energy. Therefore, the superposition predicts that the membranes undergo an unbinding transition at this temperature which proceeds in a discontinuous manner [3, 94].

Fig. 9. Unbinding transition as obtained (top) from a simple superposition of fluctuation induced and direct forces, and (bottom) from functional renormalization. The unbinding temperature is denoted by T_*.

The parameter dependence for T_* as obtained within the superposition approach is confirmed by more systematic methods. This approach fails, however, as far as the character of the transition is concerned since the unbinding transition proceeds, in fact, in a continuous fashion. This was first shown by renormalization group (RG) methods [3]. The corresponding RG transformations are displayed in fig. 9(b). As shown in this figure, the unbinding transition is governed by a critical fixed point of the RG transformation.

It is easy to understand the continuous character of the unbinding transition by analogy with strings in two dimensions. As mentioned, the separation of these strings corresponds to the spatial coordinate of a quantum-mechanical particle moving in the interaction potential V(l). The probability distribution $\mathcal{P}\left(l\right)$ for the string separation l is governed by the ground state within this potential well. This ground state is localized at low temperature but becomes delocalized in a continuous manner as the unbinding temperature is approached from below.

The unbinding transition for square well potentials has also been studied by Monte Carlo simulations [79]. From these simulations, one finds that the unbinding temperature depends strongly on the small-scale cutoff for the bending undulations.

As an example, consider two relatively stiff membranes with k ₁ = k ₂ = 2k= 10⁻¹⁹ J which interact with a square well potential with the relatively small potential range l _v = a_⊥ /10 where a_⊥ denotes the membrane thickness as before. For a lipid bilayer, one typically has a_⊥ ≃ 4 nm which implies the small potential range l _v ≃ 0.4 nm. In this case, the unbinding temperature T_* has been determined for several choices of the small-scale cutoff a _‖. As a result, one finds T_*/T _room ≃ 0.7, 1 and 1.5 for a _|| /a _⊥ = 1, 1.6, and 2, respectively [79]. Thus, the unbinding temperature T _* is roughly proportional to the small-scale cutoff a _|| for the bending undulations. Extrapolation to zero a _|| leads to the estimate T _*/T _room = 0.4 ± 0.1 for this particular interaction potential.

As far as the critical behavior is concerned, RG calculations, MC simulations and the analogy with strings show that the mean separation ℓ and the roughness ${\xi }_{\perp }=\sqrt{〈{(l-\ell )}^2〉}$ scale as ℓ ~ ξ_⊥ ~ (T/k)¹/²ξ_|| and diverge as

(5.12) $\ell \sim 1/{|T-T_{*}|}^{\psi }\text{with}\psi =1$

as the transition temperature T_* is approached from below. The same critical behavior applies to all short-ranged interaction potentials ∆V(l) which decay faster to zero than ~ 1/l ² for large l as follows from scaling arguments, RG calculations and the analogy with strings [86, 91]. These interactions belong to the so-called strong-fluctuation regime which is characterized by universal critical behavior at the unbinding transition [6]. On the other hand, interaction potentials ∆V(l) which decay as ~ 1/l ² for large l belong to the so-called intermediate fluctuation regime which is characterized by rather complex critical behavior [6, 88, 89].

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Give an Example of Van Der Waals Forces

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