Biology for quants, again. Required reading, Part 2.

“Proteins don’t know biology” is one of those things I’m overly fond of saying. Fortunately, it’s true, and it gives quantitative folks a foot in the door of the magical world of biology. And it’s not only proteins that are ignorant of their role in the life of a cell, the same goes for DNA, RNA, lipids, etc. None of these molecules knows anything. They can only follow physical laws.

Is this just a physicist’s arrogance along the lines of, “Chemistry is just a bunch of special cases, uninteresting consequences of quantum mechanics”? I hope not. To the contrary, you should try to see that cells employ basic physics, but of a different type than what we learned (most of us, anyway) in our physical sciences curricula. This cell biophysics is fascinating, not highly mathematical, and offers a way of understanding numerous phenomena in the cell, which are all ‘special cases’ … but special cases of what?

Hopefully, you’ve already gotten a basic sense of the cell’s business by reading Franklin Harold’s wonderful book, discussed in an earlier post. Harold makes explicit reference to the physical driving forces at play.

Here we’re going to take a peek at a book that makes the connection between physics and cell function more explicit, with a focus on the cell’s “molecular machines.” These macromolecular complexes perform incredible tasks like using stored free energy to pump a molecule of interest against its concentration gradient – e.g., importing nutrients into a cell or exporting toxins. Pumps can use free energy stored in the chemical potential difference of an ion on two sides of a cell membrane or using the free energy stored in ATP. Other machines perform even more dramatic feats such as “motors” which can change cell shape or contract muscle and the remarkable machinery which transcribes and translates DNA to RNA to protein along with error correction! Such topics are so incredible, they inspired me to write a second book, which is available online: Physical Lens on the Cell.

The book of interest now, which describes the fairly elementary physics needed to understand all this, is Terrell Hill’s Free Energy Transduction and Biochemical Cycle Kinetics. Perhaps this title is a little off-putting, but the book is absolutely a must-read.

Why do we want to know about cycles in the first place? Simply put, cycles are the main way a cell does its business at the molecular level. Think about one of the transporter machines I mentioned earlier, one that imports glucose into the cell. As with a human-scale machine, a molecular machine is something that needs to be used over and over. If the cell had to re-synthesize a new protein for every glucose molecule imported, that wouldn’t pay for itself. Instead, like in a factory, molecular machines are used repeatedly.

Plasma Membrane Bound Transporter

Let’s consider the simplest concrete example, sketched above, of a plasma-membrane bound transporter that can bind an ion (“N” – think sodium) a substrate (“S” – think sugar) in one of two conformations, outward-facing (“O”) and inward-facing (“I”). Outward-facing here means facing the exterior of a cell membrane, while inward-facing faces the cytoplasm or interior of the cell. This system has eight possible states, as shown in the cube diagram.

cube diagram

In the cube schematic, most of the (un)binding events are not shown explicitly, but are implicit: for example, from state OS to ONS, an ion N binds the outward-facing transporter (which already has substrate S bound); from IN to I, an ion unbinds from the inward-facing conformation.

Remarkably, this simple state space can describe transporters which lead to substrate import (from outside to inside, following the blue cycle) or to substrate export (from inside to outside, orange cycle). In fact, there are four possible cycles of each type! Can you find them? Each represents a possible process for a working machine. It’s a good bet that nature uses more than one of each kind.

Not only does Hill explain machine mechanisms using elegant diagrams similar to this one, but he covers the simple non-equilibrium physics underlying such cycles. The physics is non-equilibrium because it’s driven – in this case, by the lower chemical potential of N inside compared to outside. In equilibrium, there would be no net transport.

Hill teaches us how to think about non-equilibrium processes using elementary chemical kinetics, based on a simple mass action picture, and also how to quantify the corresponding free energies which arise from ideal gas/solution theory. In the case of our transporter, the free energy stored in the ion gradient must exceed the opposing substrate free energy difference in order for the processes to occur in the “forward” directions shown in blue and orange. (Blue and orange cycles would be found in entirely separate machines – it’s just that they’re shown in the same space.)

What’s remarkable, and obvious from the cube schematic, is that if the gradients are switched, these processes can be run in reverse. (Indeed, the molecular machines don’t “know” what they’re supposed to do, they can only follow the cycles allowed by their state spaces.) What’s more, the gradient of substrate could even be used to drive ion transport under the right conditions.

Pretty amazing stuff. So learn about these ideas, and how to quantify them, from Hill’s great book. It’s the perfect way to appreciate that simple physics can teach deep lessons about biology.

By the way, my advice is to focus on Hill’s first chapter. The second is largely devoted to technical methods which are cool, but much less valuable to a non-expert, in my view. (Caveat:Section 8 of Chapter 2 gives a nice discussion of the Hill relation connecting the rate and mean first-passage time.) The third and final chapter is quite interesting again, showing a more practical application of the theory to muscle contraction.

If you like what you read in Hill’s book, also check out my online book Physical Lens on the Cell, which goes into more detail on some of the issues and shows further applications of the ideas. But I want to be clear that 100% of the inspiration for my book comes from Hill.

So you want to learn biology? Required reading, Part 1.

You’re a quantitative person and you want to learn biology.  My friend, you are in a difficult situation.  If you really want to learn how biology works in a big-picture sense, as opposed to cutting yourself a very narrow slice of the great biological pie, then you have a challenging road ahead of you.  Fortunately, many have walked it before you, and I want to give you some advice based on my own experiences.  I should say at the outset that my own learning has focused mostly on the cell-biology part of the pie – not physiology, zoology, ecology, … and so my comments here refer to learning cell biology.

The scary thing is that I have been at this for almost 20 years (very part-time admittedly) and I would never dare to call myself a cell biologist.  But I think it’s fair to say that by now I have a decent sense of what I know and what I don’t know.  I will never be able to draw out the Krebs cycle, but I have a qualitative sense of its purpose and importance, as well as of general principles of cycles and catalyzed reactions in biochemistry.  Not that impressive, I know, but I’m proud of it anyway.

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So you want to do some path sampling…

Basic strategies, timescales, and limitations

Key biomolecular events – such as conformational changes, folding, and binding – that are challenging to study using straightforward simulation may be amenable to study using “path sampling” methods.  But there are a few things you should think about before getting started on path sampling.  There are fairly generic features and limitations that govern all the path sampling methods I’m aware of.

Path sampling refers to a large family of methods that, rather than having the goal of generating an ensemble of system configurations, attempt to generate an ensemble of dynamical trajectories.  Here we are talking about trajectory ensembles that are precisely defined in statistical mechanics.  As we have noted in another post, there are different kinds of trajectory ensembles – most importantly, the equilibrium ensemble, non-equilibrium steady states, and the initialized ensemble which will relax to steady state.  Typically, one wants to generate trajectories exhibiting events of interest – e.g., binding, folding, conformational change.

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FAQ on Trajectory Ensembles

Q: What is a trajectory?

A trajectory is the time-ordered sequence of system configurations which occur as all the coordinates evolve in time following some rules – hopefully rules embodying reasonable physical dynamics, such as Newton’s laws or constant-temperature molecular dynamics.

Q: What is a trajectory ensemble?

It’s a set of independent trajectories that together characterize a particular condition such as equilibrium or a non-equilibrium steady state.  That is, the trajectories do not interact in any way, but statistically they describe some condition because of how they have been initiated – and when they are observed relative to their initialization … see below.

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More is better: The trajectory ensemble picture

The trajectory ensemble is everything you’ve always wanted, and more.  Really, it is.  Trajectory ensembles unlock fundamental ideas in statistical mechanics, including connections between equilibrium and non-equilibrium phenomena.  Simple sketches of these objects immediately yield important equations without a lot of math.  Give me the trajectory-ensemble pictures over fancy formalism any day.  It’s harder to make a mistake with a picture than a complicated equation.

A trajectory, speaking roughly, is a time-ordered sequence of system configurations.  Those configurations could be coordinates of atoms in a single molecule, the coordinates of many molecules, or whatever objects you like.  We assume the sequence was generated by some real physical process, so typically we’re considering finite-temperature dynamics (which are intrinsically stochastic due to “unknowable” collisions with the thermal bath).  The ‘time-ordered sequence’ of configurations really reflects continuous dynamics, so that the time-spacing between configurations is vanishingly small, but that won’t be important for this discussion.

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Everything is Markovian; nothing is Markovian

The Markov model, without question, is one of the most powerful and elegant tools available in many fields of biological modeling and beyond.  In my world of molecular simulation, Markov models have provided analyses more insightful than would be possible with direct simulation alone.  And I’m a user, too.  Markov models, in their chemical-kinetics guise, play a prominent role in illustrating cellular biophysics in my online book, Physical Lens on the Cell.

Yet it’s fair to say that everything is Markovian and nothing is Markovian – and we need to understand this.

If you’re new to the business, a quick word on what “Markovian” means.  A Markov process is a stochastic process where the future (i.e., the distribution of future outcomes) depends only on the present state of the system.  Good examples would be chemical kinetics models with transition probabilities governed by rate constants or simple Monte Carlo simulation (a.k.a. Markov-chain Monte Carlo).  To determine the next state of the system, we don’t care about the past: only the present state matters.

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“Proof” of the Hill Relation Between Probability Flux and Mean First-Passage Time

The “Hill relation” is a key result for anyone interested in calculating rates from trajectories of any kind, whether molecular simulations or otherwise.  I am not aware of any really clear explanation, including Hill’s original presentation.  Hopefully this go-around will make sense.

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Why Hair Gel Matters to Statistical Biophysicists

I was worried that a discussion of hair gel would have a certain bias toward men, but my wife assures me that women are just as likely to use a leave-in hair product.  I’m going to rely on that unstatistical assurance and roll right on.

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A hello: The point of this blog

Statistical physics governs the behavior of biological systems from the molecular scale (think protein stability and fluctuations) to the cellular scale (including heterogeneity and stochasticity of cellular behavior).  This is not a claim that understanding statistical physics, a.k.a. statistical mechanics, implies an understanding of cell biology.  But I do claim that cell biology cannot be understood without statistical physics.

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