## Correlation function of Brownian particles

We now proceed to calculate the normalised time correlation function g(2) (Q, t) (Equation 40) for particles undergoing Brownian motion. We have already derived the normalisation factor (J(Q))2 N2 (Equation 12) and will thus first concentrate on the time correlation function G(2) (Q, t) EEE E (e rj(0)-rk( )+ri(t)-rm(T. (47) We assume that the particles are independent. Then, as in the preceding text (Equation 43), considering only pairs of j, fc, l and m that are equal, we obtain j k l m T...

## Array

Apart from the difference between elastic energies of a vortex line vs. a flexible polymer, the two cases are analogous. The interaction Hamiltonian of an oriented DNA polymer array with hexagonal local symmetry can be written in the form 2 H Jdz( d (z)) + JdzV r (n'm)(z) r (n''m')(z)) , where r (n,m) (z) is the local displacement of a polymer chain at the (n, m) lattice position perpendicular to the long axis, z, while V r (n'm)(z) r (n''m')(z) is the interaction potential between...

## Resonant trapping

A disadvantage of optical tweezers in comparison, for example, to magnetic manipulation is that in a crowded environment, such as the inside of a cell, force is exerted in discriminately. Another limitation is the maximum force available with optical tweezers while avoiding damage to the specimen and to conventional optics by heating. Optical tweezers operating under 1 W of average laser power can exert forces of up to about 100 pN enough to stall mechanoenzymes and stretch DNA (Ashkin 1997)....

## Micromanipulation techniques

Many techniques have been developed in the past ten years to micro-manipulate single biomolecules. Common to all these set-ups is the binding of the molecule of interest to a fixed surface at one extremity and to a force sensor at the other. The sensor (or the surface) is often displaced and the resulting force and molecular extension are measured. Different force sensors have been implemented atomic force (AFM) cantilevers (Florin etal. 1994) and microfibers (Ishijima etal. 1991,Cluzel etal....

## First passage times an exact result

Now, we set about using the tools developed in the last section to tackle a more interesting potential. Consider a potential U (x) with a single minimum at x 0 (the mean position of the diffusers). We want to calculate the average first passage time of a particle through position s > 0, given that it is introduced at x 0 at t 0. This is equivalent to the mean lifetime of the particle if an absorbing barrier is placed at x s. To solve this problem, we consider introducing, as before, a steady...

## Equations of equilibrium and shape of interfaces

The first equation of equilibrium of an interface is due to Laplace (1806) and has a well-known form relating the interfacial tension y, the total curvature J and the difference of pressure AP between the homogeneous phases separated by the interface, This equation perfectly describes interfaces with considerable tension. However, it proved insufficient for analysis of strongly curved interfaces with small or vanishing tensions. The equations of equilibrium for an interface with arbitrarily low...

## Crystallisation near a critical point

Proteins are notoriously difficult to crystallise. The experiments indicate that proteins only crystallise under very specific conditions (McPherson 1982, Durbin and Feher 1996, Rosenberger 1996). Moreover, the conditions are often not known beforehand. As a result, growing good protein crystals is a time-consuming business. Interestingly, there seems to exist a similarity between the phase diagram of globular proteins and of colloids with a short-ranged attractive interactions (Rosenbaum et...

## Statistical potentials

The idea of statistical potentials was pioneered by Scheraga (Tanaka and Scheraga 1976) and popularised by Miyazawa and Jernigan (Miyazawa and Jernigan 1984). It is probably the most widely used functional form of an energy function in the protein-folding field. At this point it is useful to introduce some probabilistic arguments to motivate the following computational approach. As argued above, the free energy is directly related to the probability of finding the system at a particular state....

## Self assembly 31 Aggregation

Bilayers are only one of several self-assembled structures formed from phospholipids. Self assembled objects are usually either rod-like (one dimensional), plate-like (2D), spherical (3D) or more complex (3D periodic or random structures such as those formed in diblock copolymers (Hamley 1998) or possibly the Golgi). Self assembly is ubiquitous in biology examples of one-dimensional self assembly include G actin into F actin (Howard 2001), tubulin into microtubules and misfolded proteins into...

## DLVO Potential

Combining Equation 13 and Equation 16, we obtain the DLVO potential that describes the interaction between charged colloids (see further Andelman's chapter in this volume) V _ Q exp(kR) X2 exp(-Kr) A f 2R2 + 2R2 + r2 4R2 1 1 + kR ) er 6 r2 4R2 r2 r2 J Note that, at short distances, the dispersion forces always win. This would suggest that the dispersion interaction will always lead to colloidal aggregation. However, the electrostatic repulsion usually prevents colloids from getting close enough...

## Polymer dynamics

The previous two sections focussed on the equilibrium properties of floppy polymers, either as individual chains, or as many chains together in a solution. In this section, we visit some of the concepts that are used to understand the dynamics of floppy polymers. Let us start by recapitulating some aspects of Brownian motion. A colloid particlejiggles about in a fluid due to the fluctuating pressure field. The mean square displacement grows linearly with time, (r2) 6Dt where D is the classical...

## Stretching DNA

As mentioned earlier, the forces of interest span a range between tens of femtonewtons to hundreds of piconewtons. The combination of the different set-ups described earlier allows one to study the elastic behaviour of DNA over this wide range of forces. The first measurements of the entropic elasticity of a single DNA molecule were Figure 6. Force vs. extension for dsDNA and polymer models for DNA. (a) Experimental force extension curve for one dsDNA of 16 m (squares). The fit using the...

## Heating in optical tweezers

In order to obtain forces on the order of tens or hundreds of pN, laser powers of typically tens to hundreds of mW are used, leading to focal intensities exceeding MW cm2 (for comparison, the intensity of bright sunlight on the surface of the earth is on the order of 100 mW cm2). The potential of thermal and non-thermal damage caused by these high intensities to (biological) samples has been a matter of concern and investigation (Ashkin and Dziedzic 1987, Barkalow and Hartwig 1995, Allen et al....

## Neutron scattering from proteins

Neutron scattering is widely used to probe picosecond-nanosecond timescale dynamics of condensed-phase molecular systems (Bee 1988, Lovesey 1987). In hydrogen-rich molecules like proteins the scattering will be dominated, due to the high incoherent scattering length of hydrogen atoms, by incoherent scattering. The basic quantity measured is the incoherent dynamic structure factor S(Q, w), where E is the energy transfer and Q kf - k , with k and kf being the incident and final wave vector of the...

## DNA under torsion

Most polymers are insensitive to torsion because their monomers are linked by single covalent bonds about which they are free to rotate. This property is lost when the polymer possesses no single covalent bond about which torsion can be relaxed. This is the case of the double-helical structure of a DNA molecule with no nicks (no break in one of the strands). This particular feature has very important biological implications. First, from a structural point of view, twisted DNA provides an...

## Signaltonoise ratio and resolution

To determine the position of a known object, the classical diffraction limit (Born and Wolf 1989) of a microscope for the distinction of two unknown objects is not relevant. The fundamental limitation for the amplitude of motion that can be detected in a given time comes from the 'shot noise' of photon counting, i.e., the standard deviation of N1 2 associated with counting N random events in a sampling time interval (Reif 1965). With a few milliwatts of laser power focused on a micron-sized...

## Depletion interaction

One of the most surprising effects of the solvent on the interaction between colloids is the so-called depletion interaction. Unlike the forces that we discussed up to this point, the depletion force is not a solvent-induced modification of some pre-existing force between the colloids. It is a pure solvent effect. It is a consequence of the fact that the colloidal particles exclude space to the solvent molecules. To understand it, return to Equation 12 Let us consider a system of hard particles...

## Depletion Flocculation

Let us next consider a slightly more realistic example of an entropy-driven phase separation in a binary mixture, namely polymer-induced flocculation of colloids. Experimentally, it is well known that the addition of a small amount of free, non-adsorbing polymer to a colloidal suspension induces an effective attraction between the colloidal particles and may even lead to coagulation. This effect has been studied extensively and is theoretically well understood (Asakura and Oosawa 1958, Vrij...

## Colloid dynamics

For the computer simulator, the study of colloid dynamics is a challenge. The reason is that colloid dynamics span a wide range of timescales. No single simulation can cover all timescales simultaneously. In the following, I shall discuss two aspects of colloid dynamics that clearly illustrate the timescale problem. The first is colloidal hydrodynamics. The second is homogeneous nucleation of a new phase from a metastable phase. 3.1 Hydrodynamic effects in colloidal suspensions Colloid dynamics...

## Statistical physics of many chains

In the previous section, we discussed the properties of an isolated chain, using the random walk model as a basis. In this section, we discuss the properties of many chains together in other words, the properties of polymer solutions and polymer melts. Again, many of the fundamental ideas are due to Flory and his coworkers. We start with Flory-Huggins theory, which is a model for the free energy of a polymer solution or melt. It is traditionally based on a lattice model, where the lattice...

## Statistical physics of single chains

The fundamental model of a floppy polymer molecule is a random walk. Let us consider a random walk of N steps, illustrated in Figure 1 (left). The end-end vector R is a useful characteristic of the size. For any particular conformation, it is given by R J2li where the are the individual steps. The mean and variance of the end-end vector distribution follow easily from this < R) T(li> 0, (R2> (li lj> < l > Nl2. (1) In writing this, we assume that there is no preferred direction, thus...

## Poisson Boltzmann equation in spherical coordinates Charged colloids

Dispersion of small submicron particles in a liquid solution is called a colloidal suspension. The suspension can be stabilised against van der Waals attractive forces by several means. In aqueous solutions if the particles are charged, the competition between the electrostatic repulsion and the van der Waals attraction can stabilise the suspension. This is the idea behind the famous DLVO theory of Deryagin, Landau, Verwey and Overbeek Deryagin and Landau 1941, Verwey and Overbeek 1948 , where...

## The Grahame equation and the Contact theorem

The PB equation can be integrated once and leads to a relation known as the Grahame equation and also as the Contact theorem Grahame 1947, Israelachvili 1992 . This is a relation between the surface charge density a and the limiting value of the ionic density profile at the boundary, n z 0 . ekeT n 0 n- 0 2no ekeT n 0 2no For large n 0 n_ 0 exp 2e s fcBT gt 1, and n_ 0 is neglected in the above equation. For example, for a surface charge density of one electronic charge per 25 A2 as in Figure 3...