"… I seem […] only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me". – Isaac Newton.
Last year, after a series of unsuccessful attempts and acquiring three incubators across two countries, my youngest son’s unwavering determination finally paid off. From a batch of twelve mixed quail eggs, seven hatched successfully, marking the start of our new venture into farm animal husbandry. Currently, we’ve settled for manageable pets like a Siberian hamster, an aquarium, and pond fish, plus several rounds of stick insects, mantises, and spiders, along with their grasshopper and locust food supplies. However, quail care is more demanding. While our sons’ happiness is undoubtedly the most important reward, the delicious eggs produced by our farm breeding activity are equally rewarding for the whole family. It’s particularly satisfying collecting every evening the two expected eggs from the punctual quail hens and admiring their different sizes and pigmentation like beautiful little gems.
If you’re still reading, you’ve probably guessed the main topics of my traditional Easter blog: quail eggs and their shapes and patterns.
The Smoluchowski diffusion equation describes the time evolution of the probability density function (PDF) of a particle undergoing overdamped Brownian motion in a potential energy landscape. It is a central equation in statistical mechanics, soft matter physics, and chemical physics.
Its origins trace back to the early 20th century, in the context of the theoretical understanding of Brownian motion. Following the seminal work of Albert Einstein in 1905, who provided a statistical description of diffusion and established a quantitative link between microscopic fluctuations and macroscopic transport, further developments aimed to incorporate external forces and interactions. In 1916, Marian Smoluchowski extended Einstein’s framework by considering particles subjected to systematic forces arising from a potential field. His formulation led to what is now known as the Smoluchowski equation, effectively describing diffusion in the overdamped (high-friction) limit where inertial effects can be neglected. This marked a crucial step toward connecting stochastic motion with deterministic drift. A complementary perspective emerged through the work of Paul Langevin (1908), who introduced a stochastic differential equation for particle motion, explicitly incorporating random forces. The equivalence between the Langevin description and the corresponding evolution equation for probability densities—later formalized as the Fokker–Planck equation—provided a deep and unifying framework. The general mathematical structure of such evolution equations was further clarified by Adriaan Fokker and Max Planck in the early 20th century, leading to the modern formulation of the Fokker–Planck equation. The Smoluchowski equation can be viewed as a specific limit of this more general framework. Later, in the 1940s, Hendrik Anthony Kramers applied these ideas to chemical reaction rates, analyzing barrier crossing in potential landscapes. His work revealed how transition rates depend exponentially on the energy barrier height, establishing the foundation of what is now known as Kramers’ theory—an essential concept for understanding metastability and rare events.
In this article, we consider the one-dimensional (1D) case, where a particle moves along a coordinate under the influence of a potential of mean force .
Important structural components of proteins, like linker loops and intrinsically disordered regions, are highly flexible and constantly change shape in solution. These flexible protein regions — especially those containing glycine- and serine-rich segments — do not behave like neatly folded proteins. They fluctuate, breathe, and explore broad conformational landscapes. These motions can often be central to biological function. But capturing them consistently, both structurally and dynamically, remains challenging. To understand the physics of this flexibility, we often turn to short model peptides that isolate the essential ingredients of chain dynamics. In an earlier work, we explored glycine- and serine-rich octapeptides using molecular dynamics (MD) simulations in combination with concepts from FRET (Förster Resonance Energy Transfer) spectroscopy. The goal was to understand how flexible chains fluctuate and how these fluctuations are reflected in experimentally measurable distances.
In a new publication in The Journal of Physical Chemistry B [1], we have built directly on that foundation, but push the idea further towardquantitative integration between simulation and experiment. At the center of both studies is a small but powerful fluorescent probe: 2,3-diazabicyclo[2.2.2]oct-2-ene (DBO). Paired with tryptophan, DBO enables measurements of extremely short intramolecular distances. Because it is compact and minimally perturbing, it is particularly well suited for probing flexible peptides that would be difficult to characterize using larger fluorophores. In the earlier work, the focus was primarily on understanding conformational ensembles and distance distributions.
In this new study, the Dbo model has been upgraded to the more recent version of the GROMOS force field (54A7), and using extensive MD simulations, we have verified whether the new model can more quantitatively reproduce both structural and kinetic FRET experimental observables. In particular, we combined time-resolved FRET experiments with microsecond-scale MD simulations to study model peptides of the form Trp–(GS)n–Dbo and Trp–(PP)n–Dbo with n=0,1,2,3, where the glycine–serine sequences represent highly flexible chains, and the polyproline sequences provide a more rigid reference.
The results of the simulations showed:
Simulated end-to-end distances agree with FRET-derived experimental values within 5% for the flexible (GS)_npeptides.
Contact formation kinetics (looping rates) quantitatively match experiment once solvent viscosity is properly accounted for.
The relationship between chain flexibility and fluorophore separation is systematically captured.
Beyond equilibrium averages, we also analyzed time correlations and dynamical fluctuations, linking conformational free-energy landscapes to experimentally observable FRET signals.
Instead, it demonstrates that combining equilibrium FRET distances and time-resolved kinetic data provides a stringent benchmark for simulation models of flexible peptides. Furthermore, this integrated FRET–MD framework with the improved Dbo model can be applied to:
Flexible linkers in multidomain proteins
Intrinsically disordered protein segments
Small proteins undergoing conformational adaptation
REFERENCE
[1] D. Roccatano . Quantitative Integration of FRET and Molecular Dynamics for Modeling Flexible Peptides. J. Phys. Chem. B, (2026-02-27) doi: https://doi.org/10.1021/acs.jpcb.5c08148
I have recently written, for WIREs Computational Molecular Science, a review article on the use of Principal Component Analysis (PCA) in the study of dynamical systems, with a particular focus on molecular dynamics (MD) simulations of biomolecules [1]. The aim of this work is to provide a clear and practical overview of how PCA has become a central tool for extracting meaningful collective motions from high-dimensional simulation data, and how modern methodological extensions continue to expand its capabilities.
AGGIORNAMENTO 2: Un primo articolo sulla Rapendula è stato pubblicatol’8 gennaio 2026 [1] sul The European Journal of Physics .
AGGIORNAMENTO1: Nel maggio 2025, il progetto ha ottenuto riconoscimento vincendo il 1° premio nel concorso Instructables All Things Pi. Un grande grazie al team di Instructables!
Qualche giorno fa ho pubblicato un nuovo progetto educativo sul mio sito Instructables. Il dispositivo, che ho battezzato RaPenduLa (dalle iniziali in inglese di RaspPi Pendulum Laboratory), è stato ribattezzato in italiano CAMPO (Computer Analisi Moto Pendolare Oscillante) grazie a un suggerimento di ChatGPT. Ma, come direbbe Shakespeare, ‘What’s in a name? That which we call a rose by any other name would smell as sweet’: il cuore del progetto è infatti una piattaforma video per lo studio delle oscillazioni meccaniche. Utilizzando un Raspberry Pi Zero W2 dotato di modulo fotocamera, il sistema registra ad alta velocità il movimento dei pendoli. Poi, con un’analisi video basata su Python e OpenCV, RaPenduLa è in grado di tracciare il percorso preciso della punta del pendolo, visualizzandone il comportamento oscillatorio in 2D.
It has become a recurrent habit for me to write a blog on the shape of eggs to wish you a Happy Easter. Not repeating oneself and finding a new interesting topic is a brainstorming exercise of lateral thinking and a systematic search in literature to find an interesting connection. This year, I wanted to explore an idea that has been lurching in my mind for some time for other reasons: billiards.
I used to play snooker from time to time with some old friends. I am a far cry from being even an amateur in the billiard games, but I had a lot of fun verifying the laws of mechanics on a green table. I soon discovered that studying the dynamics of bouncing collision of an ideal cue ball in billiards of different shapes keeps brilliant mathematicians and physicists engaged in recreational academic studies and important theoretical implications.
UPDATE2: On 8 January 2026, a paper on Rapendula was published in the European Journal of Physics [1].
UPDATE 1: In May 2025, the project achieved recognition by winning first prize in the Instructables contest “All Things Pi.” A big thank you to the Instructables teams!
I have recently published another educational project on my Instructables website. I called the device RaPenduLa for the RaspPi Pendulum Laboratory, and it is a video platform for studying mechanical oscillations. It uses a Raspberry Pi Zero W2 equipped with a camera module to record the motion of pendulums at high speed. The interesting part happens through video analysis: using Python and the fantastic OpenCV library, RaPenduLa can track the precise path of a pendulum’s tip and help visualize its oscillatory behavior in two dimensions.
I am pleased to announce the publication of the second edition of my book chapter: “A Short Introduction to the Molecular Dynamics Simulation of Nanomaterials” [1] in Micro and Nanomanufacturing, Volume II, edited by W. Ahmed and M. J. Jackson, Springer, 2025. This new edition reflects both the rapid evolution of molecular dynamics (MD) simulations over the past decade and their growing role in nanoscience.
Molecular dynamics simulations have become a cornerstone of modern nanoscience. They allow us to observe matter at the atomic scale, following the motion of thousands—or millions—of atoms in time, effectively turning the computer into a virtual microscope. From nanoparticles and nanotubes to polymers, membranes, and bio–nano interfaces, MD simulations provide insights that are often inaccessible to experiments alone. They help us understand:
Structural organization at the nanoscale
Dynamic processes such as adsorption, diffusion, and self-assembly
Thermodynamic and mechanical properties relevant to material design
This chapter is written with the explicit goal of making these ideas accessible, without sacrificing physical rigor.
What if an antique scientific instrument could be reimagined to inspire modern classrooms, foster creativity, and even be used to create form of dynamic art ? Meet the KaleidoPhoneScope, a contemporary twist on Wheatstone’s classic Kaleidophone [1]. By integrating 3D printing, laser technology, and microcomputers, this revamped device transforms the teaching of physics, engineering, and even mathematics into an engaging and interactive experience.
Figure 1: The KaleidoPhoneScope. A) Diagram of the KaleidophoneScope with indications of the different parts described in the text. B) Photo of the final apparatus with the horizontal cantilever wire. C) variant with the all-flexible Γ wire.Continue reading →
Surfactants are everywhere in protein science — from biochemical laboratories to industrial detergents. Among them, sodium dodecyl sulfate (SDS) is perhaps the most famous (or infamous), widely used for its ability to bind, deactivate, and often denature proteins. Despite decades of experimental and theoretical work, the molecular details of how surfactants bind to protein surfaces are still not fully understood. In my recent study, “Binding Dynamics of Linear Alkyl-sulfates of Different Chain Lengths on a Protein Surface” [1], I have explored this problem using molecular dynamics (MD) simulations, focusing on how the length of the surfactant’s hydrocarbon chain influences protein binding.
under the influence of a potential of mean force 
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