Molecular mechanics: an introduction
As theoretical biophysicists, we are ultimately interested in the properties of biomolecules. We want to know their shapes, energies, diffusivity, etc. We approach these problems through modeling; we presume that we can take a real system in the world and use math and physics to approximate our observations. Our model doesn't have to be 100% correct, and indeed it won't be, but it needs to be accurate enough to reliably answer any questions we might have. Modeling, as a descriptive tool and method of prediction, is important in many other fields of science aside from molecular biology. Models allow us to ask many questions. For instance, where around the Sun will the Earth be in 14 days? What will my bank account balance be in a year if my interest rate doubles, and I stop forking money over to the bank? How many licks does it take to get to the center of a Tootsie Pop? The necessity to model is ubiquitous. No matter what, we want to create a formal description of problems to probe underlying phenomena.
The specific question we have here is how to describe the behavior of a biomolecule. We aren't interested in planetary motion, finance, or consumer goods — just proteins, lipids, and DNA. Let's say that our biomolecules are collections of atoms in space. If we want, we can call this the resolution of our system. This language is similar to talking about the resolution of your TV or computer screen; it is assessing the fineness versus the graininess. We could go smaller down to subatomic particules, but at that resolution things get a bit trickier. We will stick with atoms. With time, our atoms vibrate, interact with each other, and ultimately diffuse through space. To simplify our problem further, let's focus only on the interactions that occur within our biomolecular system. We also can simplify our problem by specifying the lengths of time we are interested in. Let's say we aren't interested in any events that occur shorter than a picosecond; really we are after larger reconfigurations of our molecules of interest.
This latter choice is critical as it leads us into the realm of classical as opposed to quantum mechanics. In other words, we are saying that our atoms behave in a way consistent with Newton's laws of motion. They are ballistic and follow a deterministic trajectory, much like what happens when we shoot a cannonball out of a canon or a baseball player hits a grand slam. This question of time scale is related to the speed of the properties we are interested in. If we were interested in really fast transitions of molecules, say how atoms might wiggle in place with the increase in temperature, we would have to revisit our assumption that classical mechanics will suffice. Truly, classical mechanics here is a model in and of itself. It is a surprisingly reliable way of estimating where stuff will end up given where it has been, assuming that the lengths of time we are interested in are long enough. There are, of course, more precise models but computing these might be overkill. Mankind, for instance, made it to the moon on classical mechanics. At time scales larger than a picosecond, we can model some cool phenomena of biomolecules such as the formation or relaxation of protein secondary structure, the folding of proteins, the interaction of proteins with other molecules of interest, and aggregation events that are putatively responsible for some diseases like Alzheimer's.
It's our goal to model the way atoms interact, and we want to do this in as simple a way as possible. We will do this by reducing the problem as far as we can. We have this great idea that we can represent the way atoms interact by simple mechanical interactions. We are being reductionistic in the scope of the interactions we consider. Let's think about this. If we have two atoms, they can be bonded covalently or non-bonded. If we have three atoms, they can be bonded in a chain forming an angle. If we have four atoms, we can define two such angles that actually form two planes; we can also compute the angle between these two planes. Can we have five atoms in a chain? Sure, but we can define this in terms of the interactions we've already defined. Five atoms in a chain is just a collection of two-, three-, and four-bonded interactions. The essence of molecular modeling is to make this reductionistic assumption, that using simple equations we can represent the bonded nature of molecules. But how do these equations actually look?
The first equation to think about is the one that defines the interactions between two bonded atoms. Using our idea that we can use mechanical equations to model the behavior of atoms, we will turn to Hooke's law. In other words, we can say that the energy of our bond is
Here, Ubond is the energy of our covalent bond, k is the force constant, r is the distance between our two atoms, and r0 is the equilibrium distance of atoms. The equilibrium distance is the distance of our atoms at energetic minimum.
Moving on, we can also define an angle potential also based on a spring. In the equation below, we specify this potential as Uangle, where k is again a force constant, θ is the angle between three bonded atoms, and θ0 is the angle at energetic minimum. The equation is
The third and last bonded energy term we have is related to the angle between planes formed by four atoms in a row. This is called the torsional or dihedral term. Depending on the nature of these bonds (single, double, triple), there can be certain periodicity of our torsional angles. This means that there can be multiple minimum, or multiple values, our four atoms can find themselves in. We are going to devise an equation that takes periodicity n into account, as well as a stiffness (force constant) k, and the actual angle φ versus ideal angle φ0. This funky equation is
What we've neglected thus far is talking about non-bonded energy contributions. These are interactions that occur between atoms regardless if they are bonded or not. The physical relevance of these is related to apolar effect (van der Waals) or Coulombic (charge-charge) interactions. In fact, following our desire to be simplistic, we can represent van der Waals and charged interactions with the introduction of two new equations. The first we will talk about takes care of van der Waals. The famous potential to mention is the one by John Lennard-Jones from the 1920s. This equation produces an energy function that is a function of the distance r between any two atoms. There is an affinity term ε that is the energetic minimum of our two atoms at some ideal distance σ. The full equation is
Last, we have to consider charge interactions. There is a classical equation from Coulomb that will do this for us. We will call the electrostatic energy Uelec and specify Coulomb's constant C, the charges qi and qj of two atoms i and j, the dielectric constant ε0, and the distance r between our two atoms. This equation looks like
Together, these equations can take us far. Now that we can approximate the energy of biomolecules of interest, we can evaluate the relative stability of the same molecules in different folds. We can also compute the force equal to the negative gradient of the potential energy. This drives us toward molecular dynamics simulations, where we can evaluate trajectories of biomolecules as they evolve over time. There are also different approaches to specifying the parameters (force constants, ideal distances/angles). The collection of a set of parameters is called a force field, and we will discuss this another day.