Nonequilibrium Dynamics in an Active Viscoelastic Bath: Particles and Polymers

Abstract

In this thesis we investigate the dynamics of various systems that reside in an active viscoelastic bath. What makes these baths particularly interesting is that they are both complex and not in equilibrium. The complexity of the baths is found in their viscoelastic nature that, other than in viscous baths, hinder persistent motion. This characteristic is achieved by memory effects that influence both the friction and the thermal agitation that the bath inflicts on the system. These effects often yield anomalous dynamics, which implies a deviation from normal diffusion. A solely viscoelastic bath obeys the fluctuation-dissipation relation. The system will, in such a bath, always thermalise. The baths we consider, however, are also active, making them a nonequilibrium environment. This active forcing is modelled by stochastic forces with a time dependent correlation. They are assumed to push in a particular, yet random, direction for some characteristic time-scale. After this time, they change their direction randomly. To describe the evolution of the system, we propose a generalised Langevin equation that governs its reduced dynamics. The motivation to study active viscoelastic baths is their ability to model the interior fluid of a biological cell. This fluid is called the cytosol and it is very complex and certainly not in equilibrium. The first system we study is the one-dimensional particle in a potential. We consider three potentials. The first is a constant potential resulting in a free particle. We provide the exact solution to its generalised Langevin equation and conclude that the free particle displays both super- and subdiffusion. We apply these results on experiments of tracer particles inside living cells and find that they can, al least qualitatively, explain the observed dynamics. The second potential is a quadratic one, constituting to a harmonic oscillator. Again, we compute the exact time-evolution of its dynamics, but we also apply our numerical scheme on this system to check the algorithm's validity. We find that the initial dynamics of the harmonic oscillator is identical to those of the free particle. For long times however, the system reaches a steady state with an interesting dependence on the viscoelasticity. The results of the harmonic oscillator are used to predict how the behaviour of molecular motors is altered in a complex, nonequilibrium environment. We find that their mobility is increased. The third potential is a double well, i.e. two local minima separated by an energy barrier. The generalised Langevin equation of this system can not be solved analytically, we therefore relied on our algorithm to study its dynamics. On the basis of empirical arguments, we propose a possible expression for the escape rate in an equilibrated viscoelastic bath. When the bath is active, we observe an interesting dependency of the steady state probability distribution on the viscoelasticity. We also discuss the possible existence of an effective potential. The escape rate in this nonequilibrium bath proved to be non-intuitive. We use the double well as a model for the folding dynamics of DNA-hairpins. The second system we study is the Rouse chain, it is a simple model for a polymer. It consists of beads, linearly connected by harmonic springs. After discussing how a Rouse chain behaves in an equilibrated viscoelastic bath (the beads display three subdiffusive regimes), we expose it to two different nonequilibrium processes. In the first, we apply a constant force on one end of the chain. Because this force is conservative, the chain will thermalise after some time. But shortly after the activation of the force, the chain will experience a nonequilibrium transient phase. It is mainly this transient phase we investigate. We observe how the rotational symmetry of the chain is broken in favour of an elongated shape. We derive the evolution of both the average length and the particular trumpet shape. We also investigate how the force-front propagates the chain. In the second, the bath delivers active fluctuations to the Rouse chain. Since this is nonconservative forcing, the chain will remain out-of- equilibrium indefinitely. We find that the diffusion profile of the beads acquires several new regimes compared to the purely viscoelastic case, these new regimes are both super- and subdiffusive. Due to these active forces the chain will swell, this swelling is heavily dependent on the viscoelasticity of the bath. We qualitatively compare our results of the Rouse chain to recent experiments. To conclude, we propose some potentials to increase the realism of the Rouse chain. These include self-avoidance, finite extendible springs and bending rigidity. To investigate the properties of these more complex chains we use the algorithm since an analytic treatment is not at hand. We find that the length-force relation for a self-avoiding, finite extendible chain is no longer linear and displays two distinct regimes. A semi-flexible chain in an active viscoelastic bath has different exponents for its anomalous diffusion (which can also be partly predicted using an approximated theory). Apart from swelling, a semi-flexible chain can also shrink in these baths. To conclude, we construct the generalised Langevin equation for the middle bead of a Rouse chain. The reduced dynamics this equation entails originates from the projection of the degrees-of-freedom of the surrounding viscous heat bath and the rest of the chain on the middle bead. We first review the derivation of this generalised Langevin equation when the heat bath is in equilibrium. Thereafter, we introduce two driving agents that will pull the system away from equilibrium. Firstly, a constant force on the first bead. We find the expression for the effective force on the middle bead resulting from this perturbation. Secondly, we consider the heat bath to be driven by stochastic active forces. The inclusion of these active forces produces a frenetic contribution to the second fluctuation-dissipation relation. When the active forces are exponentially correlated, a closed form for the expression of the frenetic contribution is derived. This expression is analysed for some specific cases.