Subdiffusive Behavior in Crowded Environments: Impact of Obstacle Mobility and Spatial Restrictions
Polanowski Piotr 1, Sikorski Andrzej 2*
1 Lodz University of Technology
Department of Molecular Physics
ul. Żeromskiego 116, 90-924 Łódz, Poland2 University of Warsaw
Faculty of Chemistry
ul. Pasteura 1, 02-093 Warsaw, Poland
∗E-mail: sikorski@chem.uw.edu.pl
Received:
Received: 7 August 2023; in final form: 27 August 2023; accepted: 28 August 2023; published online: 18 September 2023
DOI: 10.12921/cmst.2023.0000020
Abstract:
Biological systems are generally dense reaction-diffusion systems. Therefore, determining the mechanism of motion in such systems is of crucial importance in understanding their dynamics. Subdiffusive behavior is very common in biological systems but its origin usually does not have a clear explanation. One attempt to explain this behavior is the presence of randomly placed stationary obstacles in a medium filled with molecules of a certain medium. With an appropriate concentration of obstacles, the molecules of the medium cease to perform classic Brownian motions and motion becomes subdiffusive. This mechanism seems to be well documented in both simulations and experiments. The question arises whether a similar effect can be obtained in systems where obstacles are not stationary, but their mobility is drastically reduced comparing to medium molecules, or the reduction in mobility is combined with a limitation in movement (the movement of obstacles resembles, for example, the Orestein-Ulhenbeck movement). Is it possible to observe subdiffusion behavior in such a situation? We try to answer this question on the basis of Monte Carlo simulations based on the Dynamic Lattice Liquid (DLL) model. Based on the concept of cooperative movements, this model has a unique feature that allows one to take into account the correlation of movements between the elements that make up the examined system, which is important in the case of high densities due to the strict correlation of movements between the moving elements. The tests concern systems where obstacles were single beads whose mobility was changed with additional restrictions imposed on the displacement. It was shown that no entrapment of medium molecules was observed and a slight deviation from normal diffusion was also shown.
Key words:
anomalous diffusion, lattice model, macromolecular crowding, Monte Carlo method
References:
[1] S. Havlin, D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 51, 187–292 (2002).
[2] J. Szymanski, M. Weiss, Elucidating the origin of anomalous diffusion in crowded fluids, Phys. Rev. Let. 103, 038102 (2009).
[3] F. Höfling, K.-U. Bamberg, T. Franosch, Anomalous transport resolved in space and time by fluorescence correlation spectroscopy, Soft Matter 7, 1358–1563 (2011).
[4] I.M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043–9052 (2012).
[5] F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76, 046602 (2013).
[6] E. Montroll, G. Weiss, Random walks on lattices. 2, J. Math. Phys. (N.Y.) 6, 167–181 (1965).
[7] S. Condamin, V. Tejedor, R. Voituriez, O. Bénichou, J. Klafter, Probing microscopic origins of confined subdiffusion by first-passage observables, Proc. Natl. Acad. Sci. USA 105, 5675–5680 (2008).
[8] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339, 1–77 (2010).
[9] E. Barkai, Y. Garini, R. Metzler, Strange kinetics of single molecules in living cells, Phys. Today 65, 29–35 (2012).
[10] W.S. Trimble, S. Grinstein, Barriers to the free diffusion of proteins and lipids in the plasma membrane, J. Cell. Biol. 208, 259–271 (2015).
[11] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractals and disordered systems, Cambridge University Press, Cambridge (2000).
[12] F. Höfling, T. Franosch, E. Frey, Localization transition of the three-dimensional Lorentz model and continuum percolation, Phys. Rev. Lett. 96, 165901 (2006).
[13] T. Bauer, F. Höfling, T. Munk, E. Frey, T. Franosch, The localization transition of the two-dimensional Lorentz model, Eur. Phys. J. Special Topics 189, 103–118 (2010).
[14] T.O.E. Skinner, S.K. Schnyder, D.G.A.L. Aarts, J. Horbach, R.P.A. Dullens, Localization dynamics of fluids in random confinement, Phys. Rev. Lett. 111, 128301 (2013).
[15] L.F. Elizondo-Aguilera, M. Medina-Noyola, Localization and dynamical arrest of colloidal fluids in a disordered matrix of polydisperse obstacles, J. Chem. Phys. 142, 224901 (2015).
[16] F. Camboni, A. Koher, I.M. Sokolov, Diffusion of small particles in a solid polymeric medium, Phys. Rev. E 88, 022120 (2013).
[17] P. Polanowski, A. Sikorski, Diffusion of small particles in polymer films, J. Chem. Phys. 147, 014902 (2017).
[18] P. Polanowski, A. Sikorski, Simulation of diffusion in a crowded environment, Soft Matter 10, 3597–3607 (2014).
[19] P. Polanowski, A. Sikorski, Simulation of molecular transport in systems containing mobile obstacles, J. Phys. Chem. B 120, 7529–7537 (2016).
[20] I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Tal-Shav, E. Barkai, Y. Garini, Transient anomalous dffusion of telomeres in the nucleus of mammalian cells, Phys. Rev. Lett. 103, 018102 (2009).
[21] S.C. Weber, A.J. Spakowitz, J.A. Theriot, Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm, Phys. Rev. Lett. 104, 238102 (2010).
[22] H. Berry, H. Chate, Anomalous diffusion due to hindering by mobile obstacles undergoing Brownian motion or Orstein-Uhlenbeck processes, Phys. Rev. E 89, 022708 (2014).
[23] P. Polanowski, T. Pakula, Studies of polymer conformation and dynamics in two dimensions using simulations based on the Dynamic Lattice Liquid (DLL) model, J. Chem. Phys. 117, 4022–4029 (2002).
[24] P. Polanowski, T. Pakula, Studies of mobility, interdiffusion, and self-diffusion in two-component mixtures using the Dynamic Lattice Liquid model, J. Chem. Phys. 118, 11139–11146 (2003).
[25] P. Polanowski, T. Pakula, Simulation of polymer–polymer interdiffusion using the dynamic lattice liquid model, J. Chem. Phys. 120, 6306–6311 (2004).
[26] B.J. Sung, A. Yethiraj, Lateral diffusion of proteins in the plasma membrane: Spatial tessellation and percolation theory, J. Phys. Chem. B 112, 143–149 (2008).
[27] J. Kurzidim, D. Coslovich, G. Kahl, Single-particle and collective slow dynamics of colloids in porous confinement, Phys. Rev. Lett. 103, 138303 (2009).
[28] B.J. Alder, T.E. Wainwright, Studies in molecular dynamics. I. General method, J. Chem. Phys. 31, 459–466 (1959).
[29] J.A. Barker, D. Henderson, What is “liquid”? Understanding the states of matter, Rev. Mod. Phys. 48, 587–672 (1976).
[30] A. Rahman, Correlation in the motion of atoms in liquid argon, Phys. Rev. 136, A405–A411 (1964).
[31] R. Metzler, J.-H. Jeon, A.G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: Non-stationary, non-ergodicity, and ageing at the centenary of Single Particle Tracking, Phys. Chem. Chem. Phys. 16, 24128–24164 (2014).
Biological systems are generally dense reaction-diffusion systems. Therefore, determining the mechanism of motion in such systems is of crucial importance in understanding their dynamics. Subdiffusive behavior is very common in biological systems but its origin usually does not have a clear explanation. One attempt to explain this behavior is the presence of randomly placed stationary obstacles in a medium filled with molecules of a certain medium. With an appropriate concentration of obstacles, the molecules of the medium cease to perform classic Brownian motions and motion becomes subdiffusive. This mechanism seems to be well documented in both simulations and experiments. The question arises whether a similar effect can be obtained in systems where obstacles are not stationary, but their mobility is drastically reduced comparing to medium molecules, or the reduction in mobility is combined with a limitation in movement (the movement of obstacles resembles, for example, the Orestein-Ulhenbeck movement). Is it possible to observe subdiffusion behavior in such a situation? We try to answer this question on the basis of Monte Carlo simulations based on the Dynamic Lattice Liquid (DLL) model. Based on the concept of cooperative movements, this model has a unique feature that allows one to take into account the correlation of movements between the elements that make up the examined system, which is important in the case of high densities due to the strict correlation of movements between the moving elements. The tests concern systems where obstacles were single beads whose mobility was changed with additional restrictions imposed on the displacement. It was shown that no entrapment of medium molecules was observed and a slight deviation from normal diffusion was also shown.
Key words:
anomalous diffusion, lattice model, macromolecular crowding, Monte Carlo method
References:
[1] S. Havlin, D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 51, 187–292 (2002).
[2] J. Szymanski, M. Weiss, Elucidating the origin of anomalous diffusion in crowded fluids, Phys. Rev. Let. 103, 038102 (2009).
[3] F. Höfling, K.-U. Bamberg, T. Franosch, Anomalous transport resolved in space and time by fluorescence correlation spectroscopy, Soft Matter 7, 1358–1563 (2011).
[4] I.M. Sokolov, Models of anomalous diffusion in crowded environments, Soft Matter 8, 9043–9052 (2012).
[5] F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76, 046602 (2013).
[6] E. Montroll, G. Weiss, Random walks on lattices. 2, J. Math. Phys. (N.Y.) 6, 167–181 (1965).
[7] S. Condamin, V. Tejedor, R. Voituriez, O. Bénichou, J. Klafter, Probing microscopic origins of confined subdiffusion by first-passage observables, Proc. Natl. Acad. Sci. USA 105, 5675–5680 (2008).
[8] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339, 1–77 (2010).
[9] E. Barkai, Y. Garini, R. Metzler, Strange kinetics of single molecules in living cells, Phys. Today 65, 29–35 (2012).
[10] W.S. Trimble, S. Grinstein, Barriers to the free diffusion of proteins and lipids in the plasma membrane, J. Cell. Biol. 208, 259–271 (2015).
[11] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractals and disordered systems, Cambridge University Press, Cambridge (2000).
[12] F. Höfling, T. Franosch, E. Frey, Localization transition of the three-dimensional Lorentz model and continuum percolation, Phys. Rev. Lett. 96, 165901 (2006).
[13] T. Bauer, F. Höfling, T. Munk, E. Frey, T. Franosch, The localization transition of the two-dimensional Lorentz model, Eur. Phys. J. Special Topics 189, 103–118 (2010).
[14] T.O.E. Skinner, S.K. Schnyder, D.G.A.L. Aarts, J. Horbach, R.P.A. Dullens, Localization dynamics of fluids in random confinement, Phys. Rev. Lett. 111, 128301 (2013).
[15] L.F. Elizondo-Aguilera, M. Medina-Noyola, Localization and dynamical arrest of colloidal fluids in a disordered matrix of polydisperse obstacles, J. Chem. Phys. 142, 224901 (2015).
[16] F. Camboni, A. Koher, I.M. Sokolov, Diffusion of small particles in a solid polymeric medium, Phys. Rev. E 88, 022120 (2013).
[17] P. Polanowski, A. Sikorski, Diffusion of small particles in polymer films, J. Chem. Phys. 147, 014902 (2017).
[18] P. Polanowski, A. Sikorski, Simulation of diffusion in a crowded environment, Soft Matter 10, 3597–3607 (2014).
[19] P. Polanowski, A. Sikorski, Simulation of molecular transport in systems containing mobile obstacles, J. Phys. Chem. B 120, 7529–7537 (2016).
[20] I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Tal-Shav, E. Barkai, Y. Garini, Transient anomalous dffusion of telomeres in the nucleus of mammalian cells, Phys. Rev. Lett. 103, 018102 (2009).
[21] S.C. Weber, A.J. Spakowitz, J.A. Theriot, Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm, Phys. Rev. Lett. 104, 238102 (2010).
[22] H. Berry, H. Chate, Anomalous diffusion due to hindering by mobile obstacles undergoing Brownian motion or Orstein-Uhlenbeck processes, Phys. Rev. E 89, 022708 (2014).
[23] P. Polanowski, T. Pakula, Studies of polymer conformation and dynamics in two dimensions using simulations based on the Dynamic Lattice Liquid (DLL) model, J. Chem. Phys. 117, 4022–4029 (2002).
[24] P. Polanowski, T. Pakula, Studies of mobility, interdiffusion, and self-diffusion in two-component mixtures using the Dynamic Lattice Liquid model, J. Chem. Phys. 118, 11139–11146 (2003).
[25] P. Polanowski, T. Pakula, Simulation of polymer–polymer interdiffusion using the dynamic lattice liquid model, J. Chem. Phys. 120, 6306–6311 (2004).
[26] B.J. Sung, A. Yethiraj, Lateral diffusion of proteins in the plasma membrane: Spatial tessellation and percolation theory, J. Phys. Chem. B 112, 143–149 (2008).
[27] J. Kurzidim, D. Coslovich, G. Kahl, Single-particle and collective slow dynamics of colloids in porous confinement, Phys. Rev. Lett. 103, 138303 (2009).
[28] B.J. Alder, T.E. Wainwright, Studies in molecular dynamics. I. General method, J. Chem. Phys. 31, 459–466 (1959).
[29] J.A. Barker, D. Henderson, What is “liquid”? Understanding the states of matter, Rev. Mod. Phys. 48, 587–672 (1976).
[30] A. Rahman, Correlation in the motion of atoms in liquid argon, Phys. Rev. 136, A405–A411 (1964).
[31] R. Metzler, J.-H. Jeon, A.G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: Non-stationary, non-ergodicity, and ageing at the centenary of Single Particle Tracking, Phys. Chem. Chem. Phys. 16, 24128–24164 (2014).
