Dynamics of active filaments in porous media

TL;DR

This paper investigates how active polymers move in porous media, revealing that flexibility and activity levels cause transitions from localized to diffusive motion, with stiff chains moving freely and flexible ones becoming trapped.

Contribution

It introduces a model explaining the transition from localization to diffusion based on polymer stiffness and activity, highlighting the role of a continuous time random walk with heavy-tailed waiting times.

Findings

Stiff chains exhibit near-unhindered movement.

Flexible chains tend to spiral and get trapped.

Heavy-tailed waiting time distributions lead to subdiffusive behavior.

Abstract

The motion of active polymers in a porous medium is shown to depend critically on flexibilty, activity and degree of polymerization. For given Peclet number, we observe a transition from localisation to diffusion as the stiffness of the chains is increased. Whereas stiff chains move almost unhindered through the porous medium, flexible ones spiral and get stuck. Their motion can be accounted for by the model of a continuous time random walk with a renewal process corresponding to unspiraling. The waiting time distribution is shown to develop heavy tails for decreasing stiffness, resulting in subdiffusive and ultimately caged behaviour.

Equations

$$U_{b}=\frac{K_{a}}{2}\sum_{i=2}^{M-1}\big{(}\theta_{i}-\pi\big{)}^{2},\qquad\cos{\theta_{i}}=\frac{\mathbf{r}_{i,i-1}\cdot\mathbf{r}_{i+1,i}}{{r}_{i,i-1}{r}_{i+1,i}}$$

$$U_e=\frac{1}{2}{K}_e\sum _{i\ne j}(r_{i,j}-2R{)}^2\theta (2R-r_{i,j})$$

$$\gamma {\dot{\mathbf{r}}}_i=F_{act}{\mathbf{t}}_i-{\nabla }_{i}U+{\mathbf{\eta }}_i$$

$${\dot{\mathbf{R}}}_{CM}=\frac{F_{act}}{\gamma L}{\mathbf{R}}_{\mathbf{e}}+\frac{1}{M}\sum _{i=1}^M{\mathbf{\eta }}_i$$

$$<(\mathbf{R}(t)$$

$$+$$

$$<(\mathbf{R}(t)-\mathbf{R}(0){)}^2>=<n(t)>\Delta R^2$$

$$<n(s)>=\frac{\psi (s)}{s(1-\psi (s))}$$

$$<(\mathbf{R}(t)-\mathbf{R}(0){)}^2>=\frac{\Delta R^2}{A\Gamma (1+\alpha )}{t}^{\alpha }.$$

Full text

Understanding the motion of biological agents in porous media is essential for a wide range of biological, medical and industrial processes. Mucus, a hydrogel which coats the stomach and other intestinal walls, serves as an important defense against invading bacteria cornick2015roles ; laux2005role ; cohen199524 ; celli2009helicobacter . In fighting cancer, bacteria are now engineered to sense the porous environment of a tumor anderson2006environmentally ; felfoul2016magneto . Other bacteria such as Myxobacteria glide in the porous environment of soil and synthesize a number of biomedically useful chemicals dawid2000biology . Yet another exmaple of active motion in a porous medium is the crowded world of a cell, in which active filaments move in a polymeric gel. Technical applications involve the motion of bacteria in porous media in the context of oil recovery brown2010microbial , water purification and decomposition of contaminants trapped in the ground ginn2002processes ; simon2002groundwater .

It is known that the dynamics of biological agents in porous media is strongly influenced by their shape, size, and properties of the media such as the pore sizes spagnolie2013locomotion ; fu2010low ; leshansky2009enhanced . In spite of many experimental studies on the motion of active elongated agents in porous media croze2011migration ; creppy2019effect ; brown2016swimming ; wolfe1989migration ; martinez2014flagellated ; sosa2017motility ; raatz2015swimming , theoretical approaches are sparse barton1997mathematical ; bertrand2018optimized . In contrast, the flow of passive polymers in porous media has been studied extensively. Whereas flexible chains can be represented by a sequence of blobs Yamakov1997 whose size is determined by the cavities of the porous media, the dynamics of stiff polymers was found to follow the reptation picture Nam2010 with, however, different kinetic exponents.

Bacteria and many other microorganism have an elongated shape and resist deformations with a finite stiffness. Their flexibility plays an important role in their dynamics: While stiff active filaments such as microtubules form large coherently moving bundles sanchez2011cilia ; lam2016cytoskeletal ; inoue2015depletion , active agents with flexible bodies, such as bacteria, have been found to form slowly diffusing spirals lin2014dynamics . Analytical and numerical studies of active filaments have revealed that bundles and spirals form in different regions of phase space, solely determined by the activity and stiffness of the chains liverpool2003anomalous ; isele2015self ; prathyusha2018dynamically . Flexibility is even more important in a crowded environment: a large number of biological agents maximize their transport by deforming their shape depending on the environment and interaction with other objects menzel2012soft ; ohta2009deformable . The interest in active elongated agents is further driven by new developments to prepare them in vitro schaller2010polar ; sumino2012large ; kraikivski2006enhanced and synthesize them, e.g. by means of bonding several Janus colloidal spheres through electric fields yan2016reconfiguring , and also by immersing chains of passive colloidal particles in an active bath, where directed transport has been observed sasaki2014colloidal .

In this work we analyse the motion of self propelled filaments in a porous medium with help of numerical simulations, supported by analytical arguments. Our main result is a phase diagram with a diffusive and a localised phase, whose intuitive interpretation is based on the competition between the activity driving the polymers through narrow channels and the flexibility favouring a spiraling, or more generally folded state of the filaments which then are caged in the porous medium.

Our model consists of three parts: a standard model for a semiflexible chain, overdamped dynamics of the chain including active beads and an ensemble of static obstacles. We consider a wormlike chain, consisting of $M$ active beads of radius $R$ at positions $\mathbf{r}_{i},i=1,2...M$. The potential energy $U_{pol}=U_{s}+U_{b}+U_{e}$ of the chain has three contributions: The connectivity of the chain is modelled by springs with spring constant $K_{s}$ and rest length $b$. The bending energy is given by

[TABLE]

with $\mathbf{r}_{i,i-1}=\mathbf{r}_{i}-\mathbf{r}_{i-1}$ and bending stiffness $K_{a}$. The excluded volume is modeled as a contact potential

[TABLE]

We are interested in approximately hard particles of fixed bond length and hence take both, $K_{s}$ and $K_{e}$, to be large. To avoid chain crossings, the bond length $b=2.1R$ is chosen only slightly larger than the diameter of the beads. This leaves us with one free parameter, namely the bond stiffness, $K_{a}$, or rather the persistence length $\xi/L=2bK_{a}/(Lk_{B}T)$ relative to the contour length $L=Mb$.

The dynamics of the chain is assumed to be overdamped

[TABLE]

The active force has amplitude $F_{act}$ and points along the tangent of the polymer contour, $\mathbf{t}_{i}=(\mathbf{r}_{i,i-1}+\mathbf{r}_{i+1,i})/|\mathbf{r}_{i,i-1}+\mathbf{r}_{i+1,i}|$. The potential $U$ includes $U_{pol}$ as well as the interactions with the obstacles, yet to be specified. The damping constant is denoted by $\gamma$ and the random noise is chosen in accordance with the fluctuation dissipation theorem: $<\boldsymbol{\eta}_{i}(t)\cdot\boldsymbol{\eta}_{j}(t^{\prime})>=4k_{B}T\delta_{ij}\gamma\delta(t-t^{\prime})$. It is convenient to work with dimensionless equations. Hence we measure length in units of particle radius $R$ energies in units of $k_{B}T$ and time in units of $t_{0}=R^{2}\gamma/(k_{B}T)$. In these units the active force is given by $(F_{act}R)/(k_{B}T)$ and represents the second control parameter besides the persistence length. Actually it is more convenient to multiply this quantity by the dimensionless factor $L^{2}/(bR)$ in order to obtain the Peclet number $\rm{Pe}=(F_{act}L^{2})/(bk_{B}T)$, which is defined as the ratio of the convective transport to the diffusive transport.

So far our model is the same as the one used in ref.(isele2015self ). However, here we are interested in the dynamics of semiflexible chains in a crowded environment. We introduce N static obstacle of radius $R_{o}$ randomly into a two-dimensional square box of size $\ell$. The interaction between particles and obstacles are taken to be elastic collisons, reversing the normal component of the relative velocity Mokhtari2017 . We impose the constraint that obstacles do not overlap, so that their packing fraction is given by $\phi=N\pi R_{o}^{2}/\ell^{2}$. Of particular interest is a porous medium with a typical pore size, which is modelled with help of an additional constraint: the relative distance between any two obstacles is at least $2R_{o}+2.5R$, allowing single beads to pass in between two obstacles. The central questions of our study are the following: Are the polymers free to move through the medium or are they localised? How does their dynamics depend on chain length, stiffness and Peclet number? A similar porous medium has been set up in a recent experiment creppy2019effect , where the motion of bacteria in the presence of randomly placed pillars in microfluidic chips has been studied.

Eq. 3 was integrated using HOOMD-blue anderson2008general ; glaser2015strong with an in-house modification to include the active force along the tangent of the polymer contour. Simulations were performed on graphical processing units (GPUs). We set $K_{s}=K_{e}=500k_{B}T/R^{2}$ and $M=30$ unless said otherwise, and explore a large parameter space for $\rm{Pe}$ and $\xi/L$ by varying $F_{act}$ and $K_{a}$. Measurements are carried out after an initial time lapse, and for as long as $10^{6}t_{0}$.

The central topic of our paper is the selective localisation of stiff and flexible polymers in an obstructed medium. Before addressing this subject in detail, we comment on the diffusion of semiflexible polymers in an unobstructed environment which has been discussed in recent literature isele2015self ; prathyusha2018dynamically . One of the most prominent results is the spiraling phase, where flexible polymers form persistent spirals. We of course also observe these states, sometimes the flexible polymers spiral around an obstacle, provided their contour length is larger than the circumference of the obstacle. These results are to be expected and a sample configuration is shown in Fig. 1.

The stiff polymers tend to follow the obstacles only for short sections of their contour length, compromising the cost of bending energy and activity $F_{act}L$. This qualitatively different behaviour is at the heart of the selective transport, discussed in detail in the following sections.

For a single polymer, obeying the dynamics of Eq.(3), we can compute the centre of mass diffusion (without obstacles), because all interactions are pairwise and do not contribute to the CM motion:

[TABLE]

Here $\mathbf{R}_{e}$ denotes the end-to-end vector. Since the activity points along the contour, the polymer motion is well approximated by “railway motion” isele2015self , allowing for the computation of the end-to end vector within the Kratky-Porod model. The mean square displacement (MSD) then follows

[TABLE]

The activity induced rotational diffusion constant is given by $D_{r}=F_{act}/(\xi\gamma)$ isele2015self and $f(x)=2(e^{-x}-1+x)/x^{2}$. The dynamics is diffusive for short times with bare diffusion constant $D_{t}=(k_{B}T)/(\gamma M)$, ballistic for intermediate times and diffusive again at long times with, however, an enhanced diffusion constant $D_{eff}=D_{t}+F_{act}\xi f(L/\xi)/(2\gamma)$, due to the activity of the chain.

Our focus here is on the dynamics of active polymers in a porous medium. As a first step we have computed the MSD for various values of the parameters for filling fraction $\phi=0.6$. An example is shown in Fig.2 for $M=30$ and varying stiffness. For $\xi/L<0.2$, we observe 3 different regimes: ballistic due to activity at small $t$, diffusive at intermediate $t$ and saturated or caged for the largest $t$. For larger $\xi/L$, the polymers are seen to move through the whole sample; for $\xi/L=0.33$, one can still observe the crossover from ballistic to diffusive motion, whereas for very large $\xi/L$ the motion is ballistic almost up to system size. Some trajectories of the very stiff polymers are depicted in the left panel of Fig.3 (see also SI).

If not all flexible polymers are localised, then the mobile ones will dominate the MSD for long times. Hence we need a better indicator for localistion, e.g. the fraction of polymers which have moved less than a given distance $d$, comparable to the system size. We define $Q(t)=<\theta(d-|\mathbf{R}(t_{0}+t)-\mathbf{R}(t_{0}))>$ where the average is over many different polymers. We show $Q(t)$ for $d=\ell/8$ in Fig.(4) for the same data as in Fig.2. One clearly observes a time-persistent part for the smallest values of $\xi/L$, indicating that a finite fraction of the polymers is localised in good agreement with the data from the $\rm{MSD}$. Furthermore the relaxation time of $Q(t)$ grows with decreasing $\xi/L$ and diverges at the critical $\xi$. The inset in Fig.4 displays the time $\tau_{Q}$, when more than $20\%$ of the particles have moved by more than $d$.

Flexible polymers are localised in the porous medium, because they tend to spiral or fold into a dense state which does not allow them to pass through the narrow channels. To quantify this statement, we have computed the radius of gyration for the above set of parameters, i.e. $\phi=0.6$ and $M=30$. The result is plotted in the inset of Fig.5 ; a clear transition is oberved from the dense to the extended state at around $\xi/L=0.2$. To pass through the narrow channels, the polymers have to unspiral, which is a rare event, even in a system without obstacles and more so in the presence of obstacles. This can be demonstrated by monitoring the fraction of polymers which have not unspiralled once in time $t$. We denote this fraction by $Y(t)$ and plot it for various values of the stiffness in Fig.5. One clearly observes a time persistent part for $\xi/L\leq 0.2$.

How do these results depend on Peclet number? Spiraling of flexible polymers is enhanced by high activity or high Peclet number, as evidenced by the strong spiraling regime in ref.(isele2015self ). Hence increasing the Peclet number implies a stronger tendency to localisation, as displayed in the phase diagram shown in Fig.6: even stiffer chains get localised with increasing $\rm{Pe}$. On the other hand high activity favours directed motion in the extended state. For $\rm{Pe}\geq 3000$ we observe an increased fraction of time in the extended state and hence also an increase in the size of the random steps taken in the extended state, giving rise to diffusion for the largest $\rm{Pe}$ (see SI). For these very large $\rm{Pe}$, finite size effects become important and possibly shift the phase boundary to larger $\rm{Pe}$. Localisation also depends on the degree of polymerization, $M$. Whereas the dynamics of the very stiff ones is hardly effected, spiraling and hence localisation can only occur for sufficiently long chains (see SI).

Flexible polymers have to unwind before they can move through the narrow channels of the obstructed environment. Our simulations show that a flexible chain is most of the time in a dense state, unspirals infrequently, makes a sudden jump in the extended state and immediately curles up again (see right panel of Fig.3 and SI). In a simple model, we neglect the time span in the stretched state so that the dynamics can be modelled by instanteneous jumps of typical size $\Delta R$, separated by random time intervals $\tau.$ The unspiraling of the flexible polymers then corresponds to the renewal process of a continuous time random walk (CTRW) montroll1965random . The “renewals” are the unspiraling events, allowing the polymer to make a displacment. The central quantity is the distribution of waiting times $\psi(\tau)$, where $\tau$ denotes the difference between two unspiralling events.The mean square displacement

[TABLE]

is expressed in terms of $<n(t)>$, the mean number of jumps in time $t$. It can be calculated from $\psi(\tau)$:

[TABLE]

Here we have introduced the Laplace transform of $n(s)=\int_{0}^{t}dtn(t)e^{-st}$ and similarly for $\psi(s)$. The low frequency behaviour of $\psi(s)$ determines the long time behaviour of $<n(t)>$ and hence the MSD. If $\psi(s)\sim 1-s<\tau>$ is regular for samll $s$, then the MSD displays ordinary diffusion. If, on the other hand, $\psi(s)\sim 1-As^{\alpha}$ with $\alpha<1$, then the MSD is given by

[TABLE]

Hence subdiffusive bahaviour should occur and saturation for $\alpha=0$. We have measured $\psi(\tau)$ for several values of stiffnes $K_{a}$. An example is shown in Fig.(7). One clearly observes a heavy tail with an algebraic decay approximately like $t^{-1.3}$ which implies subdiffusive behaviour $<(\mathbf{R}(t)-\mathbf{R}(0))^{2}>\sim t^{0.3}$ for the MSD. Due to limited simulation time, we are missing the very long times, which would lower the exponent to even smaller values. In the inset, we compare the MSD from our simulations to the result of CTRW as given in Eq.6.

In conclusion, active polymers in porous media show unexpected, rich behaviour. Whereas stiff chains are able to cross the porous medium from one side to the other, flexible chains spiral and are caged. Thus, a localisation transition occurs as a function of chain stiffness and Peclet number. We have worked out the phase in the $\xi/L-\rm{Pe}$ plane and estimated the effects of chain length. Localisation of flexible chains occurs due to the persistence of the spiraling state. Identifying unspiraling as a renewal event, we can model the dynamics of flexible chains as a continuous time random walk. The distribution of waiting times between the renewal events was shown to develop a heavy tail, as the localisation transition is approached, giving rise to subdiffusive behaviour and ultimately caging.

Many extensions of our work lie ahead: Porous media exist in a variety of geometries and topologies, giving rise to correspondingly diverse dynamics of active polymers. An example is the interplay of pore size, filling fraction and chain length in a dense gel. Another possible extension are more general models of a crowded environment, which is not strictly static but moving slowly as compared to the active agent. Other mechanisms of activity are also of interest, such as a dragged chain or the inclusion of a tumbling component licata2016diffusion .