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Forces are measured in Newtons (N); one Newton corresponds typically to the weight of a cup of tea. Of course, forces at the molecular scale are far smaller. They turn out to be on the order of picoNewtons (pN). If you divide a Newton by one million you get a microNewton; if you divide a microNewton by a million, you get a picoNewton (1pN = 10 ^-12 N).

The strongest forces encountered at the molecular scale are those required to break the covalent bond. Covalent bonds occur between two atoms when they form a molecule. Chemists measure the strength of a chemical bond by the amount of energy Eb involved in this binding. Strictly speaking this energy is not a force, but both are related by a length-scale l : E_b = F_b.l. The typical energy of a covalent bond is 1 electron-Volt (1 eV = 1.6 .10^-19 Joules, or 24 kcalories/mole) and the typical distance l over which the bond persists is typically 0.1 nanometer. Thus the force required to break a covalent bond is on the order of 1 eV/0.1 nm ~ 1600 pN.

Noncovalent bonding forces are very important in biology: the vast majority of bonds that hold together our cells and our body are noncovalent. For example, proteins have been selected via evolution for their ability to bind selectively yet noncovalently to another protein. The immune system relies on antibodies which are proteins designed to bind strongly, selectively and yet noncovalently to an invading external agent such as a virus. Cells in a tissue are glued to each other by a set of noncovalent ligand/receptor proteins. These weak bonds are numerous enough to make us physically strong, but they are not as strong as covalent bonds: this explains why our body is weaker than a metallic object, which is made up of covalently bonded atoms. The secret behind the noncovalent bond’s power is that strength comes in numbers. By placing in parallel a very large number of these weak bonds, their additive effect gives strength to the material, be it a cell membrane or a living tissue. Breaking a noncovalent bond involves modifications of the molecular structure on a nanometer scale: this requires breaking and rearranging many van der Waals, hydrogen, or ionic bonds as well as stretching covalent bonds. The energies involved are typical bond energies, again on the order of an electron-volt, but the bond extends over a larger characteristic distance l = 1 nm. The typical rupture force of a noncovalent bond is thus of order eV/nm = 160 pN. These are typically the forces which are experimentally required to break receptor/ligand bonds \\\\\\\\\\\\\\\\cite{flor94,moy94,will96,damm95,lee94} or to deform the internal structure of a molecule \\\\\\\\\\\\\\\\cite{cluz96,smit96,lebr96}.

Weak bonds are noncovalent bonds corresponding to the typical interaction between two small molecules such as ions. In water, salt dissociates into positively-charged sodium ions and negatively-charged chlorine ions. Although water significantly screens (reduces) the electrostatic force between ions, the ions nevertheless experience a weak electrostatic bonding force. The strength of this bond is sufficiently low that molecular Brownian motion (thermal jostling) often bumps one ion out of its binding with its counterpart. We just discussed the ionic nature of weak bonds, but there exist other kinds of weak interactions like hydrogen bonding or the hydrophobic interaction. All these interactions have in common that their strength is comprable to that of thermal molecular agitation. The paradigm of such a weak force is the base-pairing interaction which holds together complementary DNA bases and involves either two or three hydrogen bonds. The strength of a weak bond is given by its typical energy which is on the oder of k_BT. This is the thermal energy available at the absolute temperature T, where k_B, the Boltzmann factor, is equal to R/Na (R is the perfect gas factor derived from PV = nRT and N_a is Avogadro’s number N_a = 6.02.10 ²³. T, measured in units of degrees Kelvin, is 300 K at room temperature.

We have said that Brownian motion is a big issue at the molecular level, and at the cellular level (on the scale of microns), this phenomenon is still quite strong. If, for instance, we consider an E. coli bacteria, its Brownian motion arises from the random bombardment of water molecules banging into the bacteria’s membrane. If we imagine a plane separating the bacteria into two equal parts, one could argue that the same number of particules should hit both sides of the plane producing a perfect cancellation of this molecular bombardment and thus no net motion. This arguement is true but only statistically, that is if we average on a huge number of molecular collisions. Consider what occurs in a tiny slice of time: Nr is the number of molecules pushing the bacteria to the right, Nl is the number of molecules pushing it to the left. Nr and Nl have an average value of N and a standard deviation equal to its square root, N^1/2. In other words, they are not perfectly equal for any given tiny slice of time. N is proportional to the cross-section of the object: the larger the object, the more water molecules will bombard it. The small mismatch between Nr and Nl will produce a force and thus a displacement of the bacteria in a random direction. This random force is called the Langevin force, and its strength is thus related to the size of the object. Note, however, that this strength must be compared to that of the viscous drag experienced by the object in the fluid. Since the viscous drag force is proportional to the size, the relative effect of the Brownian motion scales like N^1/2/N and increases drastically when the size of the object (and thus N) is reduced.

The exact computation of the Langevin force F_L acting on a spherical object of diameter d in a medium with viscosity η and within a time slice Δt=1/Δf is given by F_L = [12 π k_BT η d Δf]^1/2. This is not a real force but rather a noise density of force; it’s units is N.Hz^-1/2. In other words, the magnitude of this force depends on the actual bandwidth Δf of the measuring device, or equivalently the time slice Δt = 1/Δf over which the random force is allowed to try to average (i.e. cancel) itself out. For a micron-size object such as an E. coli bacteria, the radom Langevin force obtained after one second of averaging time reaches 0.01 pN. It is interesting to compare this number with the weight of E. coli. Imagine pulling the poor bacteria out of the water and weighing it with a scale, its typical weight will be on the order of 0.01 pN. To get an idea of the violence of Brownian motion at the scale of a bacteria, imagine that you are swimming in a pool where every second you recieve a random blow whose strength is comparable to your own weight!

Just above these random bombardment forces lie the entropic forces that result from a reduction of the number of possible configurations of the system consisting of the molecule (e.g. protein, DNA) and its solvent (water, ions). As an example, a free DNA molecule in solution adopts a random coil configuration that maximizes its configurational entropy \\\\cite{pgdg79}. Upon stretching, the molecular entropy is reduced so that at full extension there is but one configuration left: a straight polymer linking both ends. To reach that configuration, work against entropy has to be performed, i.e. a force has to be applied. Entropic forces are rather weak. Since the typical energies involved are thermal and of order k_BT and the typical lengths are of the order of a nanometer, entropic forces are of order k_BT/nm = 4 pN (4 x 10^-12 N). These are typically the forces exerted by molecular motors, such as myosin on actin \\\\cite{fine94}, the force necessary to stretch a DNA molecule to its contour length \\\\cite{smit92} or to unzip the two strands of the molecule \\\\cite{esse97,bock97}.

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