5.8: Bond Stability and Thermal Motion (a non-biological moment)

Molecules do not exist out of context. In the real, or at least the biological world they do not sit alone in a vacuum. Most biologically-relevant molecular interactions occur in aqueous solution. That means, biological molecules are surrounded by other molecules, mostly water molecules. As you may already know from physics there is a lowest possible temperature, known as absolute zero (0 K, −273.15 ºC, −459.67 °F). At this, biologically irrelevant temperature, molecular movements are minimal, but not apparently absent all together159. When we think about a system, we inevitably think about its temperature. Temperature is a concept that makes sense only at the system level. Individual molecules do not have a temperature. The temperature of a system is a measure of the average kinetic energy of the molecules within it. The average kinetic energy is:

$E_k = \frac{1}{2} (\text{average mass}) \times (\text{average velocity})^2$

It does not matter whether the system is composed of only a single type of molecule or many different types of molecules, at a particular temperature the average kinetic energy of all of the different molecules has one value. This is not to say that all molecules have the same kinetic energy, they certainly do not; each forms a distribution that is characterized by its average energy, this distribution is known as the Boltzmann (or Maxwell-Boltzmann) distribution. The higher the temperature, the more molecules will have a higher kinetic energy.

In a gas we can largely overlook the attractive intermolecular interactions between molecules because the average kinetic energies of the molecules of the system are sufficient to disrupt those interactions - that is, after all, why they are a gas. As we cool the system, we remove energy from it, and the average kinetic energy of the molecules decreases. As the average kinetic energy gets low enough, the molecules will form a liquid. In a liquid, the movement of molecules is not enough to completely disrupt the interactions between them. This is a bit of a simplification, however. Better to think of it more realistically. Consider a closed box partially filled with a substance in a liquid state. What is going on? Assuming there are no changes in temperature over time, the system will be a equilibrium. What we will find, if we think about it, is that there is a reaction going on, that reaction is:

$\text{Molecule}_{(gas)} \rightleftharpoons \text{Molecule}_{(liquid)}.$

At the particular temperature, the liquid phase is favored, although there will be some molecules in the system’s gaseous phase. The point is that at equilibrium, the number of molecules moving from liquid to gas will be equal to the number of molecules moving from the gas to the liquid phase. If we increase or decrease the temperature of the system, we will alter this equilibrium state, that is, the relative amounts of molecules in the gaseous versus the liquid states will change. The equilibrium is dynamics, in that different molecules may be in gaseous or the liquid states, even though the level of molecules will be steady.

In a liquid, while molecules associate with one another, they can still move with respect to one another. That is why liquids can be poured, and why they assume the shape of the (solid) containers into which they are poured. This is in contrast to the container, whose shape is independent of what it contains. In a solid the molecules are tightly associated with one another and so do not translocate with respect to one another (although they can rotate and jiggle in various ways). Solids do not flow. The cell, or more specifically, the cytoplasm, acts primarily as a liquid and many biological processes take place in the liquid phase: this has a number of implications. First molecules, even very large macromolecules, move with respect to one another. Driven by thermal motions, molecules will move in a Brownian manner, a behavior known as a random walk.

Thermal motion will influence whether and how molecules associate with one another. We can think about this process in the context of an ensemble of molecules, let us call them A and B; A and B interact to form a complex, AB. Assume that this complex is held together by van der Waals interactions. In an aqueous solution, the A:B complex is colliding with water molecules. These water molecules have various energies (from low to high), as described by the Boltzmann distribution. There is a probability that in any unit of time, one or more of these collisions will deliver energy greater than the interaction energy hold them together; this will lead to the disassociation of the AB complex into separate A and B molecules. Assume we start with a population of 100% AB complexes, the time it takes for 50% of these molecules to dissociate into A and B is considered the half life of the complex. Now here is the tricky part, much like the situation with radioactive decay, but subtly different. While we can confidently conclude that 50% of the AB complexes will have disassembled into A and B at the half-life time, we can not predict which of these AB complexes will have disassembled and which will remain intact. Why? Because we cannot predict exactly which collisions will provide sufficient energy to disassociate a particular AB complex160. This type of process is known as a stochastic process, since it is driven by random events. Genetic drift is another form of a stochastic process, since in a particular drifting population it is not possible to predict which alleles will be lost and which fixed, or if and when fixation will occur. A hallmark of a stochastic process is that they are best understood in terms of probabilities.

Stochastic processes are particularly important within biological systems because, generally, cells are small and may contain only a small number of molecules of a particular type. If, for example, the expression of a gene depends upon a protein binding (reversibly) to specific sites on a DNA molecule, and if there are relatively small numbers of that protein and (usually) only one or two copies of the gene (that is, the DNA molecule) present, we will find that whether or not a copy of the protein is bound to a specific region of the DNA is a stochastic process161. If there are enough cells, then the group average will be predictable, but the behavior of any one cell will not be. In an individual cell, sometimes the protein will be bound and the gene will be expressed and sometimes not, all because of thermal motion and the small numbers of interacting components involved. This stochastic property of cells can play important roles in the control of cell and organismic behavior. It can even transform a genetically identical population of organisms into subpopulations that display two or more distinct behaviors, a property with important implications, that we will return to.

Questions to answer & to ponder:

• Explain why the Boltzmann distributions is not symmetrical around the highest point.
• Based on your understanding of various types of intermolecular and intramolecular interactions, propose a model for why the effect of temperature on covalent bond stability is not generally significant in biological systems?
• How does temperature influence intermolecular interactions? How might changes in temperature influence molecular shape (particularly in a macromolecule)?
• Why are some liquids more viscous (thicker) that others? Draw a picture of your model.
• In considering generating a graph that describes radioactive decay or the dissociation of a complex (like the AB complex discussed above) as a function of time, why does population size matter?

References

160 It should be noted that, in theory at least, we might be able to make this prediction if we mapped the movement of every water molecule. This is different from radioactive decay, where it is not even theoretically possible to predict the behavior of an individual radioactive atom.

161 This is illustrated here (https://phet.colorado.edu/en/simulat...ression-basics) and we will return to this type of behavior later on.

Contributors

• Michael W. Klymkowsky (University of Colorado Boulder) and Melanie M. Cooper (Michigan State University) with significant contributions by Emina Begovic & some editorial assistance of Rebecca Klymkowsky.