Force Spectroscopy

The di↵erence in stretching behavior between di↵erent polymers proved extremely informative and gave rise to the experimental method called

“force spectroscopy”. For instance, the data of C. Bustamante and his co-workers shown in the Figure 7.10 are considered the experimental proof of the fact that dsDNA is a worm-like chain. Strictly speaking, formula (7.40) is not enough to make this conclusion, for it only describes the situation at rather large force; one needs a more sophisticated formula that connects smoothly between the universal behavior at small forces (7.33) and the blow-up at large ones.

We will not derive general formulas here, but just write them down for both freely-jointed and worm-like chains — in case our reader wants to play with them. For the freely-jointed chain the formula expresses end-to-end distance as a function of force and reads

R = N ` this formula is called after P. Langevin (1872–1946) (he discovered the for-mula while studying the magnetization dependence on the applied magnetic field — yet another physical analogy). For the worm-like chain the corre-sponding formula is called after J. Marko and E. Siggia (who first used it to interpret Bustamante data), it expresses force in terms of the extension and reads

The reader should check that these formulas have proper limiting behavior at small and large forces.

Worm-like chain model is good not only for dsDNA, but for a number of other polymers. The notable example is so-called F-actin. Strictly speak-ing, calling F-actin a polymer is a little bit of a stretch: it does not have a covalent backbone; it is actually a chain-like assembly of protein globules (called G-actin). The diameter of F-actin “chain” is about 5 nm, and the chain is rather sti↵, its e↵ective segment is close to 30 µm, about three

orders of magnitude larger than for dsDNA. Actin fibers are important components of cytoskeleton supporting shapes of eukaryotic cells.

The success of worm-like chain model for dsDNA made this model and formula (7.42) fashionable among the scientists (somewhat surprising fact of life is that there is such thing as fashion in science!); nowadays formula (7.42) is often used to fit the data for which it is not at all the most appro-priate. We sincerely hope that Langevin formula (7.41) and its underlying freely-jointed model, as well as rotational isomers and other models will soon regain their rights and will be used where appropriate.

Meanwhile, every meaningful scientific theory, from very large (like rela-tivity) to rather small (like worm-like chain model) has its limits of applica-bility. Finding these limits is not a penny less important than formulating the theory itself. Of course, worm-like chain model does not work for many polymers (having joints of some kind). Furthermore, if we pull on DNA stronger and stronger, we can eventually start deforming the double he-lix itself — which means we can start testing the energetic component of the DNA elasticity in addition to the entropic part. Indeed, under usual conditions of temperature and ionic strength, something serious happens to DNA at the force about 65 pN (see below Section 8.2 about the estimates of forces; you will see that 65 pN is actually a huge force). Definitely worm-like chain model fails above this force, but what exactly happens is not clear, and the debate continues whether double helix unwinds or undergoes some other transformation.

Unfortunately, this book is not the place to describe all the beautiful tools employed in force spectroscopy experiments, such as optical tweezers, magnetic tweezers, atomic force microscope, and several others. But the fact of the matter is that force spectroscopy methods which were born and gained popularity in the experiments on dsDNA are used with increasing success. People pull on proteins to learn how they fold and unfold (Section 5.7 and Chapter 10); on DNA to study its translocation (Figure C5.7), or examine helix-coil transition (called unzipping in this context), or to see how DNA tail is being packaged in the virus head (Figure C9.11); researchers pull on molecular motors (see Section 5.8) to find how strong they are; and the list of applications continues to grow rapidly.

Chapter 8

The Problem of Excluded Volume

I hate to tell you, but there ain’t any chance for but one of us. Bolivar, he’s plenty tired, and he can’t carry double.

O. Henry, The Roads We Take

8.1 Linear Memory and Volume Interactions

What are the chances that one or another theoretical study will be a suc-cess? As history shows, it greatly depends on whether theorists can think of a nice, manageable model idealizing the real world. Of course, there are no ideally simple systems in nature. However, we can use our imagination and invent an ideal gas (whose molecules do not interact at all), an ideal crystal (with no defects at all to the regular atomic structure), and so on.

As a matter of fact, you can say that all these models are ideal indeed, meaning that they are the best for physicists. This is because they are the simplest — but they are simultaneously the most basic ones. So one has to master them first, before moving any further in either statistical mechanics, or hydrodynamics, or solid state physics, or whatever chapter of physics.

How crude are the results we might get from such “ideal” models? Are there some cases where they work well, and some where they fail? There is a special trick that often helps us to decide. It involves finding some dimensionless parameters, either large or small, which describe the system.

For example, a gas can be characterized by the fraction of volume which is taken up by the molecules. If this parameter is much less than one,

147

then the molecules are typically very far away from each other, and the gas can be treated as ideal. Similarly, a crystal is nearly ideal if the dimen-sionless fraction of incorrectly occupied sites of the lattice is small. Notice also that these small parameters emphasize the connection between “ide-alizations” and “approximations”: under certain real circumstances, when proper parameter is small (or its inverse is large), a real system might be well approximated by an idealized model.

What sort of large or small dimensionless parameters can describe a polymer? One of them we have actually used already, and not just once. It is the large number of monomer units (N 1) in a chain. We have shown that a huge N can account for many things. It explains, for example, the low concentration of monomers in a coil (see (6.14)), the existence of semi-dilute solutions (Figure 4.7 c), and the high elasticity of polymers.

Another special polymer parameter comes from the hierarchy of in-teractions. The energy E1 of a covalent bond between two neighboring monomers in a chain is normally about 5 eV 0.8 10 ¹⁸J. This is much higher than the typical energy E2 of any other interactions (say, between the polymer and the solvent, or between monomers which are not nearest neighbors along the chain, etc.) Roughly E2 0.1 eV 1.6 10 ²⁰ J.

Therefore, the ratio E2/E1 1 is just the type of small parameter we were seeking. It allows us to introduce an ideal polymer chain approximation.

Indeed, let’s see what happens near room temperature (kBT 2.6 10 ²eV 0.41 10 ²⁰ J). This region is the most interesting one as far as polymer properties are concerned. Covalent bonds cannot be broken due to thermal fluctuations, since E1/kBT 200 1). This means that the sequence of units is “cemented” into the chain by the high energies of the backbone covalent bonds. Each unit “remembers” its own number which it acquired when the chain was formed. To put it briefly, a polymer chain has a fixed linear memory.

Having sorted out the covalent bonds between the neighbors, we can now concentrate on all the other interactions. These are frequently referred to as “volume interactions”. As we have said, they have a typical energy E2, and are much weaker than those in charge of the linear memory. In the crudest theory, we may completely neglect them. Then we shall end up with exactly what is called an ideal polymer chain. This is just how we handled all the calculations in the previous chapters. It worked fairly well, and we coped with quite a number of problems. We described how a chain rolls up into a loose coil, and we revealed the peculiar entropic nature of the high elasticity of polymers.

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Nevertheless, the ideal polymer chain approximation — just as well as ideal gas or ideal liquid or any other idealization in science — proves not to be enough for many purposes. The properties of real polymers are much richer and more diverse than idealization predicts. If you are not convinced, think back to Chapter 4. There we talked about the various physical states of polymers. In order to understand fully how all those states are formed, and why, we need to allow for volume interactions. These include, in par-ticular, interactions between di↵erent macromolecules. Monomers in the same macromolecule will also interact, even if they are not close neighbors along the chain, but somehow come close to each other in space due to the chain bending and wiggling. In this chapter, we shall look at interactions of both types — between monomers belonging to the same or to di↵erent macromolecules.

What can we say in general about interaction between two monomers?

We will discuss it in some further details in the next Section 8.2, but it will be handy to make some preliminary arguments here. The interaction certainly depends on the type of chain, and on the solvent too. However, we can roughly sketch the potential energy of this interaction u(r), as a function of the distance r between the monomers (Figure 8.1 a). (In general, the potential energy does not only depend on r, but also on the mutual orientation of the monomers, and bulky monomers can have some flexibility of their own to a↵ect the u — see, e.g., Figure 4.2. We do not take this into account directly, since the main qualitative features are well enough represented by the simplified Figure 8.1 a.) Qualitatively, main features of u(r) are quite common for all types of molecules and monomers: If r is small, u(r) is positive and very large. This is because the monomers cannot penetrate into each other. In other words, the volume taken up by each monomer is automatically excluded from that available to any other one (hence the phrase “excluded volume”). As r becomes larger, monomers usually start to attract each other. This is the region on the right-hand side of the minimum in Figure 8.1 a. Usually, the crossover distance r0

between the two regimes (corresponding to the minimum) should have the same magnitude as the size of a monomer unit, i.e. r0 1 nm = 10 ⁹ m.

What is the physical meaning of u(r)? To bring two monomers together, as close as r, some work has to be done. This work is stored in u(r). It is done against the solvent molecules, as they need to be squeezed out of the way. Hence, the potential energy u(r) represents the e↵ective interaction of monomers through the solvent. It should depend, therefore, on the contents and state of the solvent, as well as on the temperature.

By the way, to describe interaction of monomers through the solvent in terms of potential u(r) is also an idealization of sorts. It works well, but not always, and more ideas have to be brought to bear if it fails — in cases such as strongly elongated particles, when dependence on orientation is important; charged monomers, because Coulomb interaction is strong and long-ranged and involves many particles collectively; monomers forming covalent chemical bonds in addition to those along the chain backbone, etc.

We will touch quite a few of those special cases later in the book, but the good practice in science (and, we believe, in life in general) is to examine simple things first, to gain insight and intuition which we can later bring to bear on complications.

8.2 Four Forces in Molecular World; Scales and Units

Before we move to our next subject, which is polymers with excluded volume, it is prudent to digress and review the forces operating between molecules and monomers in a more specific way. We realized already and will see more all the time that properties of polymeric substances are dic-tated by the interactions between monomers or between di↵erent chains.

What are those interactions?

The reader may have heard about the four fundamental forces in nature (gravity, electromagnetic, weak, and strong are their names), but this is not what we are talking about. In molecular world all the relevant forces are fundamentally various forms of interplay between electromagnetism and quantum mechanics. But these can masquerade in di↵erent costumes, tra-ditionally grouped into four categories, as illustrated in Figure 8.1: panels b through e represent, respectively, van der Waals interactions (see, e.g., Section 9.3), hydrophobic interactions (Section 5.1), Coulomb attraction or repulsion between charged groups (Section 2.5.3), and hydrogen bonds (Section 5.1). Van der Waals forces are the most generic, they are always present. Panel a represents a sketch of characteristic potential profile, in-cluding repulsion at short distances and attraction at longer ones. This arises typically of combination of several types of forces, including the in-teractions with the solvent.

To be quantitative, we should indicate the characteristic scale of these forces. This is a delicate matter, because we have to choose the units.

Indeed, hundred dollar bill is inconvenient to operate a public telephone, while change or small coins are equally inconvenient to buy a pair of shoes;

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Fig. 8.1 This cartoon illustrates four types of forces operational in molecular world, such as (b) van der Waals interactions (which include both repulsion at short distances and attraction at longer ones), hydrophobic interactions (c) between nonpolar molecules and groups in water medium, Coulomb forces (d), which might be very strong and require special attention, but present only when some groups are charged, and hydrogen bonds (e), which are directional and saturate (unlike others). Panel (a) represents a typical potential of interaction between two monomers in the solvent which usually results from the interplay of several of the above mentioned interactions: repulsion at short distances is always present due to the van der Waals component, while other aspects might be influenced by the solvent and other circumstances.)

similarly, the common units (meters, kilograms, etc), chosen for their con-gruency to the human body scale, are inconvenient for the molecular world.

The natural scale of energy in molecular world, as we already saw, is kBT , because polymers and biopolymers usually exist under the conditions in which absolute temperature does not really change that much: for instance, the whole interval between 0 C and 100 C in terms of absolute temperature corresponds to room temperature plus or minus 10 or 20%. Therefore, for the purposes of rough estimates the T is always close to 300 K.

In terms of this natural energy unit, the typical strengths of the inter-actions are as follows. To begin with, covalent bond is as strong as about 100 or 200kBT ; this is why covalent bonds do not break on their own. This is also why covalent bonds are not included among four types of forces in molecular world: unbreakable covalent bonds are always among the deter-mining circumstances of all non-covalent interactions. Among the latter,

hydrogen bond (Figure 8.1 e) is usually an order of magnitude weaker than covalent, about 10kBT . Van der Waals interaction for atoms is even weaker, a fraction of kBT — but one has to keep in mind that when two molecules or two monomers of a polymer approach each other, then several atoms come into contact; therefore, typical van der Waals energy for the atomic groups like monomers (which is " in Figure 8.1 a) is typically several kBT , close to hydrogen bond. Characteristic energies of hydrophobic and Coulomb interactions are more dependent on circumstances, and we will touch upon them below.

Last but not least, the energy kBT allows us to estimate also the char-acteristic values of forces involved. Indeed,

kBT 4.1 nm pN at room temperature. (8.1) Notice that there is no powers of ten involved, in the chosen units kBT is just about 4. So what are these convenient units? It is nanometer (10 ⁹m) times picoNewton (10 ¹¹N), distance times force. Nanometer is a natural unit for distance, because, for instance, atom size is about 0.1 in these units, while the DNA diameter is 2. Therefore, given that kBT is the natural scale of energy, and nanometer is natural for distances, we discover that picoNewtons is the natural scale of forces. The reader will be well advised to keep this in mind.

8.3 Excluded Volume — Formulating the Problem

Let’s discuss how interactions of the type shown in Figure 8.1 a might influence the shape of an isolated polymer chain in a dilute solution (Figure 4.7 a). First of all, would volume interactions make the coil swell or shrink?

This, it turns out, depends on the temperature of the solution.

Suppose the characteristic energy of attraction " (Figure 8.1 a) is much greater than the thermal energy kBT . Then attraction will dominate. As a result, the macromolecule will shrink to become more compact than an ideal coil. This is a special polymer state, called a polymer globule. We shall come back to it in the next chapter.

It is not the same story if " is smaller than kBT . In this case, attraction is not too important. Repulsion at shorter distances between monomers is the prevailing form of interaction. It makes the coil swell. Such swelling is called the excluded volume e↵ect. (You presumably understand where the name comes from. As we have already said, the repulsion at short distances occurs because the volume of each monomer is excluded for all the others.)

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a b

Fig. 8.2 A random walk (a) and a self-avoiding path (b) on the lattice in two dimensions (on the plane). Notice that random walk frequently retraces its own path, while self-avoiding walk never does so.

In this chapter, we are going to tackle the problem of excluded volume, that is, we shall try to picture how a polymer coil swells.

For an isolated polymer chain, the problem is purely geometrical. In-deed, the spatial shape of an ideal chain resembles the path of a randomly wandering Brownian particle (see Chapter 6). What new features will the shape of the chain acquire, if we allow for the excluded volume? Clearly, since the “private space” of each monomer is not available to the rest, the chain cannot possibly cross itself at any stage. This sort of behavior can be described as self-avoiding. For example, if there were an equivalent Brownian particle, it would not be allowed to cross its own track. A two-dimensional version of such a trajectory is sketched in Figure 8.2. Thus, we have made it a purely geometrical problem of self-avoiding random walks.

This problem can be quite successfully approached by computer simu-lation. The simplest way to set it up is to use a random number generator to try out various trajectories of a polymer chain (just as described in Sec-tion 2.4). Then whenever we obtain a trajectory with a self-crossing, we merely ignore it. Thus, we only keep the self-avoiding paths, and when we

This problem can be quite successfully approached by computer simu-lation. The simplest way to set it up is to use a random number generator to try out various trajectories of a polymer chain (just as described in Sec-tion 2.4). Then whenever we obtain a trajectory with a self-crossing, we merely ignore it. Thus, we only keep the self-avoiding paths, and when we