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Heltberg ML, Miné-Hattab J, Taddei A, Walczak AM, Mora T. Physical observables to determine the nature of membrane-less cellular sub-compartments. eLife 2021; 10:69181. [PMID: 34677123 PMCID: PMC8598233 DOI: 10.7554/elife.69181] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 10/21/2021] [Indexed: 12/19/2022] Open
Abstract
The spatial organization of complex biochemical reactions is essential for the regulation of cellular processes. Membrane-less structures called foci containing high concentrations of specific proteins have been reported in a variety of contexts, but the mechanism of their formation is not fully understood. Several competing mechanisms exist that are difficult to distinguish empirically, including liquid-liquid phase separation, and the trapping of molecules by multiple binding sites. Here, we propose a theoretical framework and outline observables to differentiate between these scenarios from single molecule tracking experiments. In the binding site model, we derive relations between the distribution of proteins, their diffusion properties, and their radial displacement. We predict that protein search times can be reduced for targets inside a liquid droplet, but not in an aggregate of slowly moving binding sites. We use our results to reject the multiple binding site model for Rad52 foci, and find a picture consistent with a liquid-liquid phase separation. These results are applicable to future experiments and suggest different biological roles for liquid droplet and binding site foci.
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Affiliation(s)
- Mathias L Heltberg
- Laboratoire de physique de l'École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris, France.,Institut Curie, CNRS, PSL University, Sorbonne Université, Paris, France
| | - Judith Miné-Hattab
- Institut Curie, CNRS, PSL University, Sorbonne Université, Paris, France
| | - Angela Taddei
- Institut Curie, CNRS, PSL University, Sorbonne Université, Paris, France
| | - Aleksandra M Walczak
- Laboratoire de physique de l'École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris, France
| | - Thierry Mora
- Laboratoire de physique de l'École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris, France
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Khokhlova SS, Agmon N. Green's function for reversible geminate reaction with volume reactivity. J Chem Phys 2013; 137:184103. [PMID: 23163360 DOI: 10.1063/1.4764357] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
Abstract
The kinetics of a diffusing particle near a reversible trap may be described by an extension of the Feynman-Kac equation to the case of reversible binding, which can occur within a finite reaction sphere. We obtain the Green's function solution for the Laplace transform of this equation when the particle is initially either bound or unbound. We study the solution in the time-domain by either inverting the Laplace transform numerically or propagating the partial differential equation in the time-domain. We show that integrals of this solution over the reaction sphere agree with previously obtained solutions.
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Affiliation(s)
- Svetlana S Khokhlova
- The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
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Rojo F, Budde CE, Wio HS, Budde CE. Enhanced transport through desorption-mediated diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012115. [PMID: 23410291 DOI: 10.1103/physreve.87.012115] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2012] [Indexed: 06/01/2023]
Abstract
We present a master equation approach to the study of the bulk-mediated surface diffusion mechanism in a three-dimensional confined domain. The proposed scheme allowed us to evaluate analytically a number of magnitudes that were used to characterize the efficiency of the bulk-mediated surface transport mechanism, for instance, the mean escape time from the domain, and the mean number of distinct visited sites on the confined domain boundary.
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Affiliation(s)
- Félix Rojo
- Fa.M.A.F., Universidad Nacional de Córdoba, Ciudad Universitaria, X5000HUA Córdoba, Argentina
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Rojo F, Wio HS, Budde CE. Narrow-escape-time problem: the imperfect trapping case. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:031105. [PMID: 23030864 DOI: 10.1103/physreve.86.031105] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2012] [Revised: 08/14/2012] [Indexed: 06/01/2023]
Abstract
We present a master equation approach to the narrow escape time (NET) problem, i.e., the time needed for a particle contained in a confining domain with a single narrow opening to exit the domain for the first time. We introduce a finite transition probability, ν, at the narrow escape window, allowing the study of the imperfect trapping case. Ranging from 0 to ∞, ν allowed the study of both extremes of the trapping process: that of a highly deficient capture and situations where escape is certain ("perfect trapping" case). We have obtained analytic results for the basic quantity studied in the NET problem, the mean escape time, and we have studied its dependence in terms of the transition (desorption) probability over (from) the surface boundary, the confining domain dimensions, and the finite transition probability at the escape window. Particularly we show that the existence of a global minimum in the NET depends on the "imperfection" of the trapping process. In addition to our analytical approach, we have implemented Monte Carlo simulations, finding excellent agreement between the theoretical results and simulations.
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Affiliation(s)
- Félix Rojo
- Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina
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Ziener CH, Kampf T, Herold V, Jakob PM, Bauer WR, Nadler W. Frequency autocorrelation function of stochastically fluctuating fields caused by specific magnetic field inhomogeneities. J Chem Phys 2008; 129:014507. [DOI: 10.1063/1.2949097] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Margolin G, Barkai E. Aging correlation functions for blinking nanocrystals, and other on-off stochastic processes. J Chem Phys 2007; 121:1566-77. [PMID: 15260705 DOI: 10.1063/1.1763136] [Citation(s) in RCA: 96] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Following recent experiments on power law blinking behavior of single nanocrystals, we calculate two-time intensity correlation functions I(t)I(t+t') for these systems. We use a simple two state (on and off) stochastic model to describe the dynamics. We classify possible behaviors of the correlation function and show that aging, e.g., dependence of the correlation function on age of process t, is obtained for classes of the on time and off time distributions relevant to experimental situation. Analytical asymptotic scaling behaviors of the intensity correlation in the double time t and t' domain are obtained. In the scaling limit I(t)I(t+t('))-->h(x), where four classes of behaviors are found: (i) finite averaged on and off times x=t' (standard behavior); (ii) on and off times with identical power law behaviors x=t/t' (case relevant for capped nanocrystals); (iii) exponential on times and power law off times x=tt' (case relevant for uncapped nanocrystals); (iv) for defected off time distribution we also find x=t+t'. Origin of aging behavior is explained based on simple diffusion model. We argue that the diffusion controlled reaction A+B <==>AB, when followed on a single particle level exhibits aging behavior.
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Affiliation(s)
- G Margolin
- Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46556, USA
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Cohen J, Kim K, King P, Seibert M, Schulten K. Finding gas diffusion pathways in proteins: application to O2 and H2 transport in CpI [FeFe]-hydrogenase and the role of packing defects. Structure 2005; 13:1321-9. [PMID: 16154089 DOI: 10.1016/j.str.2005.05.013] [Citation(s) in RCA: 117] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2005] [Revised: 05/17/2005] [Accepted: 05/23/2005] [Indexed: 10/25/2022]
Abstract
We report on a computational investigation of the passive transport of H2 and O2 between the external solution and the hydrogen-producing active site of CpI [FeFe]-hydrogenase from Clostridium pasteurianum. Two distinct methodologies for studying gas access are discussed and applied: (1) temperature-controlled locally enhanced sampling, and (2) volumetric solvent accessibility maps, providing consistent results. Both methodologies confirm the existence and function of a previously hypothesized pathway and reveal a second major pathway that had not been detected by previous analyses of CpI's static crystal structure. Our results suggest that small hydrophobic molecules, such as H2 and O2, diffusing inside CpI, take advantage of well-defined preexisting packing defects that are not always apparent from the protein's static structure, but that can be predicted from the protein's dynamical motion. Finally, we describe two contrasting modes of intraprotein transport for H2 and O2, which in our model are differentiated only by their size.
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Affiliation(s)
- Jordi Cohen
- Department of Physics and Beckman Institute, University of Illinois, Urbana, IL 61801, USA
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Re MA, Budde CE. Diffusion-mediated reactions with a time-dependent absorption rate. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:1110-1120. [PMID: 11046381 DOI: 10.1103/physreve.61.1110] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/1999] [Indexed: 05/23/2023]
Abstract
Diffusion-mediated reactions models are particularly useful for the characterization of physical, chemical, and biological problems. In this paper we present a theoretical study of the absorption probability density, survival probability, and reaction rate for diffusion-mediated reactions models with a time-dependent finite absorption rate (an extension of a model usually referred to as the "imperfect trap model"). The results are obtained by means of the formalism of continuous time random walk on a lattice and considering a general reaction dynamics upon encounter of the reactives. First jump probability densities are included to take initial conditions into account. Previous results presented by Collins and Kimball [J. Colloid. Sci. 4, 425 (1949)] and Noyes [J. Chem. Phys. 22, 1349 (1954)] are reobtained for the particular case of a time-independent absorptivity. Short and long time behaviors are analyzed resulting, in particular, in that the long time behavior of the absorption probability density exhibits the same time dependence as the first passage time density. The results obtained are illustrated by considering a one-dimensional model with consequent discussion.
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Affiliation(s)
- MA Re
- Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5010 Cordoba, Argentina
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Kurzynski M, Palacz K, Chelminiak P. Time course of reactions controlled and gated by intramolecular dynamics of proteins: predictions of the model of random walk on fractal lattices. Proc Natl Acad Sci U S A 1998; 95:11685-90. [PMID: 9751726 PMCID: PMC21701 DOI: 10.1073/pnas.95.20.11685] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Computer simulations of random walk on the Sierpinski gasket and percolation clusters demonstrate that the short, initial condition-dependent stage of protein involving reactions can dominate the progress of the reaction over the main stage described by the standard kinetics. This phenomenon takes place if the intramolecular conformational transition dynamics modeled by the stochastic process is slow enough and the initial conformational substate of the protein already belongs to the transition state of the reaction. Both conditions are realized in two kinds of experiments: small ligand rebinding to protein after laser flash photolysis and direct recording of single protein channel activity. The model considered suggests simple analytical formulae that can explain the time behavior of the processes observed and its variation with temperature. The initial condition-dependent stage, and not the stage described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.
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Affiliation(s)
- M Kurzynski
- Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
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Nadler W, Huang T, Stein DL. Random walks on random partitions in one dimension. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4037-4047. [PMID: 9965552 DOI: 10.1103/physreve.54.4037] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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