Authordmz

A “proof” of the discretized Hill Relation

This is yet another one of those things where, after reading this, you’re supposed to say, “Oh, that’s obvious.” And I admit it is kind of obvious … after you think about it for a few minutes! So spend those few minutes now to learn one more cool thing about non-equilibrium trajectory physics.

In non-equilibrium calculations of transition processes, we often wish to estimate a rate constant, which can be quantified as the inverse of the mean first-passage time (MFPT). That is, one way to define a rate constant is just reciprocal of the average time it takes for a transition. The Hill relation tells us that probability flow per second into a target state of interest (state “B”, defined by us) is exactly the inverse MFPT … so long as we measure that flow in the A-to-B steady state based on initializing trajectories outside state B according to some distribution (state “A”, defined by us) and we remove trajectories reaching state B and re-initialize them in A according to our chosen distribution.

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Are our “stories” fiction? Can we tell right from wrong?

Here’s a true story from a number of years ago. A postdoc in the group comes to me in frustration. He has built a cool “semi-atomistic” coarse-grained protein model that has generated disappointing results. An alpha helix that’s clearly resolved in the X-ray structure of his protein completely unravels. Disappointment. But playing the optimistic supervisor, I ask, “Are we sure you’re wrong? Could that helix be marginally stable?” Further digging revealed an isoform of the protein where the helix in question was not resolvable via X-ray. Relief! I was pretty pleased with myself, I must say.

But now I’m disappointed that I was pleased.

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Rules of Thumb for Sampling Assessment

Some quick guidance for analyzing molecular dynamics (MD) or Markov-chain Monte Carlo (MC) data in hard-to-sample systems – e.g., biomolecules. I can summarize the advice this way: Ask not how to compute error bars. Ask first whether error bars are even appropriate. A meaningless error bar is more dangerous (to you and the community) than no error bar at all. This guidance is essentially abstracted from our recent Best Practices paper, and I hope it will set in context some of the theory discussed in an earlier post.

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Absolutely the simplest introduction to Bayesian statistics

I realized that I owe you something. In a prior post, I invoked some Bayesian ideas to contrast with boostrapping analysis of high-variance data. (More precisely, it was high log-variance data for which there was a problem, as described in our preprint.) But the Bayesian discussion in my earlier post was pretty quick. Although there are a number of good, brief introductions to Bayesian statistics, many get quite technical.

Here, I’d like to introduce Bayesian thinking in absolutely the simplest way possible. We want to understand the point of it, and get a better grip on those mysterious priors.

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Learn non-equilibrium physics now! It’s easy

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.

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Recovering from bootstrap intoxication

I want to talk again today about the essential topic of analyzing statistical uncertainty – i.e., making error bars – but I want to frame the discussion in terms of a larger theme: our community’s often insufficiently critical adoption of elegant and sophisticated ideas. I discussed this issue a bit previously in the context of PMF calculations. To save you the trouble of reading on, the technical problem to be addressed is statistical uncertainty for high-variance data with small(ish) sample sizes.

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