Hot Metal Bridge, crossing the Monongahela river on a dedicated pedestrian/bike path, Pittsburgh.

Closeup.

This bird just briefly visited me on my window sill. Is this a baby cardinal?

🎇 Happy anniversary, America! May there be happy years in your future! 🎆

Thanks for the alt text. Current image reliability on Bluesky leaves a lot to be desired…

That’s awkward—but only for a short while! Once the average nonlinear solution has relaxed, the time dependence in the coefficients vanishes and we go back to an exactly solvable Ornstein-Uhlenbeck process. And, importantly: the elastic coupling remains in the equation! 23/26

Our paper walks you through the tricky but beautiful derivation of what is called a “system size expansion”, or “linear noise approximation.” The upshot is, we get a much simpler PDE, which is close to an Ornstein-Uhlenbeck process—except its coefficients are time dependent! 22/26

For those interested in funky animations—here’s an illustration of the time evolution of P(N+,t) for the case N=100, with an initial N+ of 10. Observe that the variance converges faster than the mean, so that the asymptotic motion of the probability density is a “sideways slide.” 20/26

Note that for the “ideal gas case” (i.e., constant rate r0) this Fokker-Planck equation HUGELY simplifies, and we get a very elegant analytical solution: a Gaussian with a time-dependent mean and variance! This is called an “Ornstein-Uhlenbeck process”: diffusion in a quadratic potential! 19/26

This equation is easy to write down, but generally hard to solve—especially analytically. But if we’re OK with a CONTINUUM APPROXIMATION, we can (via the so-called “Kramers-Moyal expansion”) derive a partial differential equation (PDE) for P(N+,t): the FOKKER-PLANCK EQUATION. 17/26

In our paper we derive, via a sequence of standard arguments, “equations of motion” for the probability distribution P(N+,t) of having N+ lipids in the upper leaflet at time t. The most intuitive one is the “master equation,” shown and briefly motivated in the attached image. 16/26

Example: here are two stochastic trajectories (created by the so-called “Gillespie algorithm”), and a “red haze” of 500 more. Obviously, the (blue) solution to our beloved rate equation only describes the AVERAGE time evolution. Question: what can we say about the FLUCTUATIONS around it? 15/26

Now that lipids see each other, let’s have different types! Say, 300 A-lipids and 200 B-lipids, with the A’s initially balanced, but not the B’s. We get a rapid relaxation of ABUNDANCE (which messes up the original A-balance) and a much slower relaxation of COMPOSITION (which restores it). 10/26

Example time: 300 lipids, initially 175 in the upper leaflet. (That’s a hefty 17% initial relative abundance asymmetry.) The ideal and stress-corrected decay to the equilibrium 150 lipids per leaflet is shown here (for α=19). Observe the massively faster decay driven by elastic stresses! 9/26

Two things are notable. First, for typical systems α=19, so this parameter is NOT small! Second, if the rates are state-dependent, our good old flip-flop rate equation becomes nonlinear! 😬 8/26

In our paper we explain that the ratio between this parameter and the thermal energy yields a new dimensionless “elastic coupling parameter,” α, which determines how the abundance asymmetry affects flip-flop rates. Namely: exponentially! 7/26

If for whatever reason the two leaflets are unevenly packed, each one experiences an elastic energy that quadratically penalizes the difference between the actual and the optimal area. The key parameter is the product of area expansion modulus and area per lipid. 6/26

We can define a relative abundance asymmetry as the normalized number difference between the upper and lower leaflet. It follows a simple differential equation that describes the exponential relaxation from some initial asymmetry to the equilibrium steady state. 4/26

The canonical approach uses linear first-order rate equations. If N+ and N– = N – N+ are the lipid numbers in the upper (+) and lower (–) leaflet, then each leaflet loses lipids proportional to its current number, times some rate. And it gains what the other one loses. 3/26

First off—what is this about? We know that lipids can “flip-flop” between the two leaflets of a membrane. But we are NOT interested in the tortuous contortions a single one of them has to do to do so. Instead, we want to describe the many-lipid long-time stochastic fluxes that ensue. 2/26

While others are fascinated by different types of explosive revelations.

What in the fucking fuck is this supposed to mean? The mental rot is brain melting. What’s the goal here? “Sure, I have no health insurance, can’t pay for food and gas, and will never own a home. But thank GOD my representative can name check all obscure wide receivers from the early 2000s.”

In my parents’ garden 😍

I’ve just been at a screening of “Star Wars: The Empire Strikes Back,” with live music by the Pittsburgh Symphony Orchestra. And let me tell you: there is no better way to watch this movie or listen to this score. What a phenomenal show. What an amazing orchestra. Unforgettable. 🥰

And his name will be Sam Lowry.

This is in many ways a very good movie.

I’m sorry. This is a lengthy and whiny hot take. He makes a bunch of generalizations for which we rightfully call out modern columnists. Language changes, people change, modes of communication change—times change. Old people struggle to keep up and feel it’s a profound sign of doom.

📣 To celebrate Biophysics Week, the Membrane Structure and Function Subgroup (MSAF) of @biophysicalsoc.bsky.social will host a webinar on March 24 at 11am EDT. Our speakers are @mejohnson81.bsky.social, @ibudin.bsky.social, and Cecília Leal (not on 🦋). 🧪 👀 More info in the alt text! Please share! 🔄