I don’t about you but I grew up on equilibrium statistical mechanics. The beauty of a partition function, an ensemble, the ability to understand thermodynamic principles from microscopic rules. I love that stuff.
But what if we want to understand biology? Is a partition function really the most important object? This Fall, I’m going to lecture on biophysics for an assortment of biology and biomedical engineering students for just a few weeks; and for the first time in my teaching career, I’m planning to omit a partition-function based description of molecular behavior. I’m just not convinced it’s important enough for an abbreviated set of lectures.
Instead, I will cover discrete-state kinetics in a Markovian picture, emphasizing non-equilibrium behavior from the start. A lot of this material will be taken from my online book, Physical Lens on the Cell. I will discuss ideas of transience – relaxation from an initial distribution toward a steady-state distribution (possibly equilibrium but usually not). Time-dependent transient behavior will be contrasted with steady-state non-equilibrium behavior, and both of those from equilibrium. We’ve seen this in a configurational/trajectory context, but the ideas are easier to understand for discrete systems.
[Wait a minute! Is this post by the same Zuckerman who has voiced concerns about Markov state models for analyzing molecular simulations?? Yes, it is, and I still have those concerns, hinted at in an earlier post and to be elaborated upon in future posts and manuscripts. To be clear, my view is not that good MSMs don’t exist, but rather that they are extremely challenging to construct from typical molecular dynamics datasets – and further that ‘good enough’ MSMs are much more limited in value for mechanistic interpretation than people realize. But here, the issue is selecting the best educational tool for teaching non-equilibrium principles of physics and biology. For that purpose, I know of no better framework than discrete-state Markov models.]
What’s so great about the discrete-state picture? For starters, it can be applied from molecular to cellular scales. At the molecular scale, think of machines like transporters and motors. The key thing about these machines is that they can tap into stored free energy, which could be in the form of a concentration difference (really, chemical potential difference) of an ion across a membrane, or it could be from an activated molecule like ATP or GTP.
Discrete-state models enable us to understand how free energy is stored rather simply. Consider a reversible process such as phosphorylation of ADP (ADP + Pi ATP) or an ion I moving from one side of a membrane to another (I_in I_out). Each of these processes has an equilibrium point, when forward reactions balance reverse, that unfortunately stores no free energy. Instead, the cell will use the energy it generates from food molecules to push these reactions continually out of equilibrium in one direction, and it is the tendency to relax back to equilibrium that provides usable free energy.
Basically, the cell can capture usable energy from the flows that result as different processes try to relax toward equilibrium. Think of it as the microscopic analog of harvesting hydroelectric power. Just as the sun’s energy causes water to evaporate, which leads to rain and ultimately powers hydroelectricity based on the earth’s water cycle, so the cell’s consumption of energetic molecules (e.g., glucose) continually renews those microscopic energy stores of ATP, ion gradients, etc.
With stored free energy as context, discrete-state models can provide a very strong understanding of the physics of molecular machines. One of my favorite examples is the ribosome, which not only translates mRNA into protein but does so with built-in error correction known as proofreading. This proofreading is only possible because of the use of free energy stored in the activated molecule GTP … or more accurately because GTP is out of equilibrium with its hydrolysis products. And discrete-state models show directly how it all works.
Moving up in scale, discrete-state models are commonly used to describe signaling networks. Signaling, after all, consists of only a few types of events: binding, catalysis, and conformational change. It may not always be possible to enumerate all the states of a signaling network, let alone know all the rate constants, but discrete descriptions have proven very useful. For example, in as-yet-unpublished work with colleagues performing live-cell microscopy, we were able to use measured rate constants and populations of discrete diffusional states to demonstrate directly that there is a net flow through the signaling network.
All the way at the whole-cell scale, recent experimental work has shown that certain types of cells undergo spontaneous transitions from one phenotype to another. This is thought to be the case in populations of tumor cells, with implications for how therapies are scheduled. And guess what simple formulation can describe this behavior? That’s right – discrete-state Markov models.
So even if you’ve mastered partition functions, don’t delay learning non-equilibrium physics. I have developed a lot of introductory material on Physical Lens on the Cell, mostly using Markovian discrete-state models but also some continuum theory based on Fokker-Planck/Smoluchowski ideas. Check it out.
- Physical Lens on the Cell, Daniel M. Zuckerman
- A first course in systems biology, Eberhard Voit, Garland Science, 2017.
- Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells, Gupta, P. B., Fillmore, C. M., Jiang, G., Shapira, S. D., Tao, K., Kuperwasser, C., & Lander, E. S. (2011). Cell, 146:633-644 (2011).
- Accurate estimation of protein folding and unfolding times: beyond Markov State models. Suárez, Ernesto, Joshua L. Adelman, and Daniel M. Zuckerman, Journal of chemical theory and computation 12:3473-3481 (2016).