Molecular Forces

A. The Origin and Measurement of Molecular Forces

We divide these forces into two broad categories, both of which can be either attractive or repulsive. First, there are interactions that are connected with fields emanating from sources within or on the macromolecules themselves (16) (e.g., electrostatic fields pointing from the fixed-charge distributions on macromolecules into the surrounding space, fields of connectivity of hydrogen bond networks extending from the macromolecular surfaces into the bulk solution that are seen in hydration interactions). Second, there are forces due to fluctuations that originate either in thermal Brownian motion or quantum jitter (15). Consequent interactions include the van der Waals or dispersion forces that originate from

Figure 2 Highly ordered assemblies, ubiquitous among biological structures, can be explained through the properties of a very small number of fundamental forces acting between macromolecules. On the left-hand side, electron micrograph of a part of a human eye rod cell showing multilamellar bilayer aggregate. (From Kessel RG, Kardon RH. Tissues and Organs. San Francisco: W.H. Freeman and Co, 1979.) In the middle, electron micrograph of an in vivo cholesteric phase of a wild-type E. coli DNA. (Adapted from D. Frankiel-Krispin et al. EMBO J 20 (2001) 1184-1191.) For comparison we show the same type of structure for DNA in vitro below. (Adapted from A. Leforestier and F. Livolant, Biophys. J 65 (1993) 56-72.) On the right-hand side, cryomicrographs and computer-processed images of T7 phage heads showing ordered DNA spooling within the viral heads. (From Ref. 13.)

Figure 2 Highly ordered assemblies, ubiquitous among biological structures, can be explained through the properties of a very small number of fundamental forces acting between macromolecules. On the left-hand side, electron micrograph of a part of a human eye rod cell showing multilamellar bilayer aggregate. (From Kessel RG, Kardon RH. Tissues and Organs. San Francisco: W.H. Freeman and Co, 1979.) In the middle, electron micrograph of an in vivo cholesteric phase of a wild-type E. coli DNA. (Adapted from D. Frankiel-Krispin et al. EMBO J 20 (2001) 1184-1191.) For comparison we show the same type of structure for DNA in vitro below. (Adapted from A. Leforestier and F. Livolant, Biophys. J 65 (1993) 56-72.) On the right-hand side, cryomicrographs and computer-processed images of T7 phage heads showing ordered DNA spooling within the viral heads. (From Ref. 13.)

thermal as well as quantum mechanical fluctuations of electromagnetic fields in the space between and within the interacting molecules; conformation-fluctuation forces from thermal gyrations by the macromolecule when thermal agitation pushes against the elastic energy resistance of the molecule and confinement imposed by neighboring macromolecules (16).

There are many ways to detect interactions between macromolecules. Here we consider only macromolecules interacting in ordered arrays that are particularly relevant for investigations of the packing and energetics of DNA-lipid complexes.

A fundamental concept in macromolecular arrays is that of osmotic pressure (Fig. 3). It is equal to the pressure needed to hold a macromolecular array together against the forces acting between its constituent macromolecules. It can be applied either mechanically across a semipermeable membrane or via the osmotic stress of a high molecular weight (e.g., PEG (poly-ethyleneglycol), PVP (polyvinylpyrrolidone), dextrane) polymer solution. At chemical equilibrium, the osmotic pressure of one solution (macromolecular array) balances that of another (the bathing polymer solution). The chemical equilibrium can be maintained either via a semipermeable membrane or simply because the bathing polymer solution phase separates from the macromolecular array, as is many times the case with PEGs, PVP, and dextrane. This osmotical balancing of different molecular solutions is the basis of the ''osmotic stress method'' of measuring the equation of state of macromolecular arrays (18).

The equation of state of a macromolecular solution is defined as the dependence of its osmotic pressure on the density of the array (Fig. 4). By equilibrating the macromolecular array vs. a solution of high molecular weight polymer with a known osmotic pressure, one can set the osmotic pressure in the macromolecular array itself (18). If in addition the concurrent density of the macromolecular array is measured, either via X-ray scattering or direct densitometry, one gets the dependence of the osmotic pressure of the array on its density (i.e., its equation of state). This is the essence of the osmotic stress method.

1. Hydration Force

The hydration force is connected with a simple observation that it takes increasing amounts of work to remove water from

Figure 3 Osmotic pressure in macromolecular arrays. Dissolved polymers such as PEG exert an osmotic pressure on the the part of the solution from which they are excluded (shown schematically by the weight). Instead of exerting directly a pressure on the macromolecular subphase such as DNA or lipid arrays (small circles), one can equilibrate it with a solution of PEG at a set concentration (what amounts to the same thing: a set osmotic pressure) and PEG itself will exert osmotic stress on the macromolecular subphase. Osmotic weighing of polymers one against the other (the one with the known, set osmotic pressure against the unknown one) is the essence of the osmotic stress technique of measuring interactions in macromolecular solutions. See the color insert for a color version of this figure.

Figure 3 Osmotic pressure in macromolecular arrays. Dissolved polymers such as PEG exert an osmotic pressure on the the part of the solution from which they are excluded (shown schematically by the weight). Instead of exerting directly a pressure on the macromolecular subphase such as DNA or lipid arrays (small circles), one can equilibrate it with a solution of PEG at a set concentration (what amounts to the same thing: a set osmotic pressure) and PEG itself will exert osmotic stress on the macromolecular subphase. Osmotic weighing of polymers one against the other (the one with the known, set osmotic pressure against the unknown one) is the essence of the osmotic stress technique of measuring interactions in macromolecular solutions. See the color insert for a color version of this figure.

between electrically neutral lipids in multilamellar arrays, or from between ordered arrays of polymers at large polymer concentrations (18). Direct measurements of this work strongly suggest that it increases exponentially with the diminishing separation between colloid surfaces with a certain decay length that depends as much on the bulk properties of the solvent as on the detailed characteristics of the interacting surfaces. There is nevertheless some profound universality in the interactions between macromolecular surfaces at close distances (Fig. 5), whether they are charged, zwitterionic, or uncharged, that strongly suggest that water is essential in maintaining the stability of biological matter at high densities.

Hydration forces can be understood in different terms with no consensus yet on mechanism (11). Marcelja and coworkers (19) first proposed the idea that colloid surfaces perturb the vicinal water and that the exponential decay of the hydration force is due to the weakening of the perturbation of the solvent as a function of the distance between the interacting surfaces (Fig. 6). They introduced an order parameter P(z) as a function of the transverse coordinate z, between the surfaces located at z = D/2 and z = — D/2, that would capture the local condition, or local ordering of solvent molecules between the surfaces. The detailed physical nature of this order parameter is left unspecified, but because the theory builds on general principles of symmetry and perturbation expansions molecular details are not needed. All one needs to know about P is that within the bulk water P = 0 and close to a macromolecular surface P remains nonzero. As a mnemonic, one can envision P as an arrow associated with each water molecule. In the bulk, the arrows point in all directions with equal probability.

Close to a bounding macromolecular surface, they point preferentially toward or away from the surface (Fig. 6), depending on the surface-orienting fields.

If we envisage solvent molecules between two perturbing surfaces, we can decompose the total free energy F of their configuration into its energy W and entropy S parts via the well-known thermodynamic definition F = W —TS, where T is the temperature. Energetically it would be most favorable for the surface-induced order to persist away from the surfaces, but that would create conflict between the apposing surfaces (Fig. 3). Entropy fights any type of ordering and wants to eliminate all orderly configurations between the two surfaces, creating a homogeneous state of molecular disorder characterized by P = 0. Energy and entropy compromise to create a nonuniform profile of the order parameter between the surfaces; surface-induced order propagates but progressively decreases away from the surfaces.

From the free energy, we can derive the repulsive hydration osmotic pressurep acting between the surfaces because by definition it is proportional to the derivative of the free energy with respect to the separation D. Osmotic pressure between two ap-posed lipid surfaces has been measured extensively for different lipids (20) and has been measured to have the formp = p0 exp( — D/\h), consistent with previously theoretically derived form of the hydration free energy if one assumes that p0 ~ P2(z = D/2). Here \H is the hydration decay length of 0.1-0.4 nm measuring the spatial extent of water perturbation. From these experiments, one can deduce the magnitude of the prefactor p0, which for a great variety of lipids and lipid mixtures can be found within an interval 1012 to 1010 dynes/cm2. This ratio also pen)

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Figure 4 The osmotic stress method (18). DNA liquid crystals are equilibrated against solutions of a neutral polymer (e.g., PEG or PVP, depicted as disordered coils). These solutions are of known osmotic pressure, pH, temperature and ionic composition (54). Equilibration of DNA under the osmotic stress of external polymer solution is effectively the same as exerting mechanical pressure on the DNA subphase with a piston that passes water and small solutes but not DNA. After equilibration under this known stress, DNA separation is measured either by X-ray scattering, if the DNA subphase is sufficiently ordered, or by densito-metry (55). DNA density and osmotic stress thus determined immediately provide an equation of state (osmotic pressure as a function of the density of the DNA subphase) to be codified in analytical form over an entire phase diagram. See the color insert for a color version of this figure.

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