Equilibrium Download Movies
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Time-lapse microscopy imaging provides direct access to the dynamics of soft and living systems. At mesoscopic scales, such microscopy experiments reveal intrinsic thermal and non-equilibrium fluctuations. These fluctuations, together with measurement noise, pose a challenge for the dynamical analysis of these Brownian movies. Traditionally, methods to analyze such experimental data rely on tracking embedded or endogenous probes. However, it is in general unclear, especially in complex many-body systems, which degrees of freedom are the most informative about their non-equilibrium nature. Here, we introduce an alternative, tracking-free approach that overcomes these difficulties via an unsupervised analysis of the Brownian movie. We develop a dimensional reduction scheme selecting a basis of modes based on dissipation. Subsequently, we learn the non-equilibrium dynamics, thereby estimating the entropy production rate and time-resolved force maps. After benchmarking our method against a minimal model, we illustrate its broader applicability with an example inspired by active biopolymer gels.
In addition to thermal effects, active processes can strongly impact the stochastic dynamics of a system8,9,10,11,12. Recently, there has been a growing interest in quantifying and characterizing the non-equilibrium nature of the stochastic dynamics in active soft and living systems13,14,15,16,17,18,19,20,21,22,23,24,25. In cells, molecular-scale activity, powered for instance by ATP hydrolysis, controls mesoscale non-equilibrium processes in assemblies, such as cilia26,27, flagella28, chromosomes29, protein droplets30, or cytoskeletal networks31,32,33,34. The irreversible nature of such non-equilibrium processes can lead to measurable dissipative currents in a phase space of mesoscopic degrees of freedom9,17,18,35,36,37,38. Such dissipative currents can be quantified by the entropy production rate39, which is a measure of the irreversibility of the dynamics40. New approaches have been developed to measure this rate in real systems22,24, shedding light onto the structure of dissipative processes19 and their impact on the dynamics of living matter20. However, it remains an outstanding challenge to accurately infer the entropy production rate by analyzing Brownian movies of such systems.
Thus, with sufficient information, we can use our approach to accurately predict at any instant of time the physical force fields in image space from the Brownian movie, even if the system is out of equilibrium. Moreover, the results for this simple two-bead system demonstrate the validity of our approach: we reliably infer the non-equilibrium dynamics of this system. Arguably, direct tracking of the two beads is, in this case, a more straightforward approach. However, this changes when considering more general soft assemblies comprised of many degrees of freedom.
DCA relies on the measurement of an intuitive trajectory-based non-equilibrium quantity: the area enclosing rate (AER) matrix \\(\\dot{{\\boldsymbol{A}}}\\) associated to a general set of coordinates y. The elements of the AER matrix, in Itô convention, are defined by24,38,60,61,62
In non-equilibrium systems our DCA-based method infers non-zero entropy production rates, even with poor optical resolution (Fig. 3f, Supplementary Note 6, and Supplementary Fig. 4) and with strong measurement noise (Supplementary Note 7 and Supplementary Fig. 5). At the same time we measure no dissipation in equilibrium systems. Thus, this example illustrates the potential applicability of our approach to real experiments on biological assemblies.
We considered the dynamics of movies of time-lapse microscopy data. Under the assumptions outlined in the first section of the Results, these movies undergo Brownian dynamics in image space: the image-field obeys an overdamped Langevin equation of the form of Eq. (3). Rather than tracking selected degrees of freedom, we propose to analyze the Brownian movie as a whole.
P.R. and C.P.B. conceived the project. F.S.G. wrote the new codes developed in this manuscript and performed all simulations and Brownian movies analysis. P.R. provided support for the SFI analysis and G.G. largely developed the derivations underlying the DCA analysis. All authors contributed conceptually to developing the Brownian movie analysis and DCA frameworks, interpreting the results, and writing the paper.
A Beautiful Mind won the Academy Award for best picture in 2001 and it chronicles the life of John Nash who is noted for his pioneering work on general equilibrium theory. This scene is a great way to discuss self-interest and contrast it with the social optimum. Nash's work has augmented Adam Smith's \"invisible hand\" theory by extending how self-interest is modeled. This scene is must viewing even though the scene itself contains flawed economics (explaining why the scene is NOT a Nash equilibrium helps students learn the concept in greater depth).
Christian Bale makes this 10x better. I just find it insanely watchable seeing his character gradually feel emotion while undergoing an internal conflict. It's best not to take this too seriously. It's a slice of early 2000s dystopian cheesy sci-fi goodness and it knows it. If this got a higher budget, a bigger studio behind it and a more prominent director, this could've been on the level of other beloved sci-fi movies.
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Transient two-dimensional infrared spectroscopy (T2D-IR) has been used to observe in real time the non-equilibrium structural dynamics of intramolecular hydrogen bond breaking in a small cyclic disulfide-bridged peptide.
Abstract: The structures of two constructs of the measles virus (MeV) phosphoprotein (P) multimerization domain (PMD) are reported and are compared with a third structure published recently by another group [Communie et al. (2013), J. Virol. 87, 7166-7169]. Although the three structures all have a tetrameric and parallel coiled-coil arrangement, structural comparison unveiled considerable differences in the quaternary structure and unveiled that the three structures suffer from significant structural deformation induced by intermolecular interactions within the crystal. These results show that crystal packing can bias conclusions about function and mechanism based on analysis of a single crystal structure, and they challenge to some extent the assumption according to which coiled-coil structures can be reliably predicted from the amino-acid sequence. Structural comparison also highlighted significant differences in the extent of disorder in the C-terminal region of each monomer. The differential flexibility of the C-terminal region is also supported by size-exclusion chromatography and small-angle X-ray scattering studies, which showed that MeV PMD exists in solution as a dynamic equilibrium between two tetramers of different compaction. Finally, the possible functional implications of the flexibility of the C-terminal region of PMD are discussed.
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