Abstract
Efficient navigation through disordered, porous environments poses a major challenge for swimming microorganisms and future synthetic cargocarriers. We perform Brownian dynamics simulations of active stiff polymers undergoing runreverse dynamics, and so mimic bacterial swimming, in porous media. In accord with experiments of Escherichia coli, the polymer dynamics are characterized by trapping phases interrupted by directed hopping motion through the pores. Our findings show that the spreading of active agents in porous media can be optimized by tuning their run lengths, which we rationalize using a coarsegrained model. More significantly, we discover a geometric criterion for the optimal spreading, which emerges when their run lengths are comparable to the longest straight path available in the porous medium. Our criterion unifies results for porous media with disparate pore sizes and shapes and for runandtumble polymers. It thus provides a fundamental principle for optimal transport of active agents in denselypacked biological and environmental settings.
Introduction
Microorganisms display agile motility features to optimize their survival strategies and efficiently navigate through their natural disordered and porous habitats^{1,2,3,4,5}. While locomotion by swimming represents the most prominent, inevitable transport feature of many microorganisms, sudden changes of their swimming direction are also an essential tool for their efficient search for nutrients^{6} or escape from harmful environments^{7}. These reorientation events are generated by intrinsic biophysical mechanisms and generate different swimming modes, such as the runandtumble motion of Escherichia coli^{3} or Bacillus subtilis^{8}, runreverse(flick) patterns of diverse bacteria^{9,10}, sharp turns in swimming algae^{11}, and runreverse behavior of different species of archaea^{12}. In unconfined media these transport features lead to trajectories reminiscent of a random walk, yet their consequences for the navigation through real, porous environments, characteristic of a wide variety of biological, biomedical, and environmental contexts, such as biological gels and tissues or environmental soils and sediments, remain largely unexplored. Understanding the underlying physical mechanisms is thus paramount for revealing fundamental microbiological processes, such as biofilm formation and community ecology^{13,14}, and has significant potential to enable novel nanotechnological applications^{15,16}.
Engineering the propulsive mechanisms of microorganisms has proven to be a promising route towards the design and development of smart, selfpropelled cargocarriers^{17,18} that overcome several limitations of their passive counterparts (e.g., ordinary colloids). Yet, their ability to selfpropel might not suffice to make them generally suitable for performing complex tasks in biomedical and environmental settings, where, for example, they may be expected to deliver drugs to a specific target^{19}, penetrate the porous structure of tumors^{20,21}, or find and induce degradation of contaminants^{16,22}. In fact, randomizing the swimming direction of these autonomous agents could be an efficient strategy for reaching a target. To date, however, experimental realizations of controlled reorientation of selfpropelled synthetic agents are sparse^{23,24}.
Experiments of biological microswimmers^{25,26,27,28} and synthetic, active agents^{29,30} in confined, disordered environments are often concerned with nearsurface motility. These studies display a range of unusual phenomena, ranging from the circular swimming motion of bacteria near walls^{25,26}, hydrodynamic trapping^{29,30}, and enhancement of bacterial transport near surfaces due to the presence of obstacles^{27}. Similarly, theoretical studies on active transport in crowded environments mainly focus on 2D models for active, pointlike particles moving in periodic structures^{31}, on a lattice with obstacles^{32}, or disordered environments^{33}. Accounting for elongated shapes, Brownian dynamics simulations of selfpropelled flexible polymers have revealed subdiffusive motion in 2D porous media^{34}. Quantitative studies of active transport in 3D porous media, however, are sparse and it was only recently that the hop and trap mechanism of individual E. coli cells moving in a 3D porous structure was identified^{4,5}.
While such studies shed light on how porescale confinement influences bacterial motility, it is still unclear what motility patterns are optimal for spreading in porous media. A clue comes from the seminal work of Wolfe and Berg^{1}, who studied the spreading of engineered bacteria, which lacked the ability to sense chemical gradients and whose tumbling rate could be controlled chemically. Their experiments indicated that smooth swimming strains of E. coli get stuck in the porous structure of the semisolid agar, similarly to incessantly tumbling cells, while at an intermediate tumbling rate bacterial transport appeared more efficient. A nonmonotonic transport behavior as a function of the tumbling rate has been found also theoretically in 2D systems^{32,35,36}, where the effects of pore shape^{36} and mobile obstacles^{32} have been addressed. However, the underlying optimal transport mechanism, dictated by the interplay of swimming characteristics and geometric features of the 3D porous medium, remains an open question.
In this work, we elucidate the spreading of selfpropelled stiff polymers, as model systems for elongated microorganisms, in porous media by performing Brownian dynamics simulations and developing a coarsegrained theory. We demonstrate that reorientation mechanisms are indispensable for efficient dispersion through porous media, as intrinsic reversals enhance the overall diffusivities by up to two orders in magnitude. We identify a competition between the pore length, a direct measurement of straight pathways available in the pore space, and the run length of selfpropelled polymers. In particular, the hopping lengths of rarely and frequently reversing polymers are constrained by the pore length and their intrinsic run length, respectively. Most importantly, maximal spreading occurs when the intrinsic run length of the polymers is comparable to the longest pore length of the porous medium, which allows us to introduce a simple but robust geometric criterion for optimal transport. Subsequently, we rationalize the nonmonotonic transport behavior in terms of a renewal theory.
While such a nonmonotonic behavior is predicted in refs. ^{32,35,36}, our study unravels the underlying mechanism and demonstrates that this largescale nonmonotonic behavior persists irrespective of the pore shapes and reorientation mechanism of the polymers and is dictated by the maximal pore length only. These findings together with our geometric criterion provide fundamental, physical insight into earlier experimental observations^{1} and thereby should guide the future design of synthetic cargocarriers, applicable in biomedical and environmental settings.
Results
Model: runreverse polymer in a porous environment
We model the elongated shape of a bacterial cell by a stiff polymer with aspect ratio L/σ = 5, where L denotes the polymer length and σ the diameter of the individual monomers. We employ Brownian dynamics simulations of the discretized polymer, where the monomers are connected via stiff springs according to the wellestablished beadspring model (see Fig. 1a and Methods). The stiffness of the polymer is characterized by a persistence length ℓ_{p}, which exceeds its contour length L, ℓ_{p}/L ≫ 1. The selfpropulsion of the polymer is modeled by active forces of magnitude F_{p} acting on each monomer in the direction tangential to its backbone^{37} (Fig. 1a). The velocity along the contour of the polymer is then determined by the friction coefficient ζ of a single monomer, v_{c} = F_{p}/ζ, and each monomer is subject to translational diffusion with diffusivity D_{0} = k_{B}T/ζ, where k_{B} is the Boltzmann constant and T the temperature. The diffusive time scale of a single monomer is τ_{0} = σ^{2}/D_{0}.
In addition, the active polymer randomly reverses its swimming direction at exponentially distributed times with reversal rate λ (Fig. 1b). At these ‘pseudo’ ‘reversal events’ the polymer either instantaneously changes its swimming direction and moves along the opposite direction or continues swimming along the same direction. The run length of the polymer is then defined as the length the polymer moves before it can reverse: ℓ_{run} ≡ v_{c}/λ. A runreverse mechanism is employed by several microorganisms^{9,12}, yet for studying the largescale spreading of active agents in a porous environment we anticipate that it can be used to model runandtumble bacteria^{3,8} since often the only way to escape narrow pores is by reversing the swimming direction^{4,5}. We demonstrate this by complementing our findings for runreverse polymers by runandtumble polymers, which change their swimming direction randomly^{3,8} (see Methods).
The swimming characteristics of runreverse polymers can be described by two dimensionless parameters: (1) the Péclet number Pe ≡ v_{c}L/D_{0}, which measures the selfpropulsion strength relative to diffusion, and (2) the reversal rate λ with respect to the characteristic diffusive time scale of a single monomer τ_{0}, i.e., λτ_{0}.
To model the porous medium, we generate a disordered, monodisperse, porous structure composed of N overlapping spheres of size 4σ within a cubic box of length 30σ [Figs. 1c and d for N = 10^{3}]. The microarchitecture of the porous medium can be characterized by the distribution of the straight paths, referred to as chord lengths, available in the medium, \({\varphi }_{{L}_{c}}(\ell )\)^{38}. It is shown for different N in Fig. 1e. We further introduce the maximal chord length \({L}_{c,\max }\) via \({\varphi }_{{L}_{c}}({L}_{c,\max })=1{0}^{5}/\sigma\). Throughout the manuscript we mainly focus on very densely packed environments and hence N = 10^{3}, unless stated otherwise. For this case, the maximal chord length is \({L}_{c,\max }\simeq 20\sigma\) and the medium is characterized by narrow channels of average pore diameter σ_{p} comparable in size to the individual monomer σ_{p}/σ ≃ 3.5 (or the polymer σ_{p}/L ≃ 0.7), as is typical of many natural environments^{39} (see Methods).
Dynamics: meansquare displacement
To investigate the dynamics of a selfpropelled polymer in a porous environment, we measure the meansquare displacement (MSD) of the center monomer \(\langle {{{\Delta }}{{{{{{{{\bf{r}}}}}}}}}_{c}(t)}^{2}\rangle\) with Δr_{c}(t) = r_{c}(t) − r_{c}(0) and \({{\Delta }}{{{{{{{{\bf{r}}}}}}}}}_{c}(t)={[{{\Delta }}{x}_{c}(t),{{\Delta }}{y}_{c}(t),{{\Delta }}{z}_{c}(t)]}^{T}\). We keep the Péclet number fixed, Pe = 50, and tune the rate of reversal events, λ (Fig. 2). At short times \(t\lesssim {\tau }_{{{{{\mbox{diff}}}}}}\equiv {D}_{0}/{v}_{c}^{2}\), the MSDs of active agents with different reversal rates λ collapse and display a linear increase reflecting their diffusive motion at short times, which remains independent of the porous structure of the environment. At intermediate times, the directed motion of the polymers dominates and the MSDs exhibit a superdiffusive increase (for λτ_{0} ≳ 5 · 10^{−4}), which varies for different λ. The MSDs eventually cross over to a linear regime, which characterizes the effective diffusive behavior of the runreverse polymers in a porous medium.
In contrast, nonreversing polymers can only reorient due to the interplay of thermal fluctuations, activity, and conformational chain dynamics^{37}. The corresponding rotational relaxation time of the polymer is denoted by \({\tau }_{\,{{{{\mbox{rot}}}}}\,}^{0}\) and can be extracted by measuring the fluctuation of the endtoend direction of the polymer, e(t) = [r_{5}(t) − r_{1}(t)]/∣r_{5}(t) − r_{1}(t)∣, which evolves as \(\langle  {{\Delta }}{{{{{{{\bf{e}}}}}}}}(t){ }^{2}\rangle =22\exp (t/{\tau }_{\,{{{{\mbox{rot}}}}}\,}^{0})\)^{40}. The MSD for rarely reversing polymers, \(\lambda\tau_0=5\cdot10^{4}\) (and polymers with λτ_{0} = 5 ·10^{−3}), displays a subdiffusive behavior at intermediate times, \(t \sim {\tau }_{\,{{{{\mbox{rot}}}}}\,}^{0}\), where the effect of the porous environment becomes apparent as the polymer slows down. At long times, the polymer has merely moved a distance comparable to its own length over the whole simulation time, which is a fingerprint of confined transport and indicates that motion in tight narrow spaces becomes hindered if active agents cannot reverse efficiently.
Moreover, we find that the crossover time to longtime diffusion is determined by the faster of the two times: the reorientation time due to reversing 1/λ and the orientational relaxation time \({\tau }_{\,{{{{\mbox{rot}}}}}\,}^{0}\) (indicated in Fig. 2). To quantify the longtime behavior, we extract the effective diffusivities via
Our findings demonstrate that the longtime effective diffusivities can be enhanced by more than two orders of magnitude (over the whole simulation time of 10^{3}τ_{0}) upon introducing a reversal mechanism, \({D}_{\,{{{{\mbox{eff}}}}}}^{{{{{\mbox{sim}}}}}}(\lambda {\tau }_{0}=0.5)/{D}_{{{{{\mbox{eff}}}}}}^{{{{{\mbox{sim}}}}}\,}(\lambda {\tau }_{0}=0.0005)\simeq 4\cdot 1{0}^{2}\). This is in stark contrast to the motion of runreverse polymers in a dilute environment, whose long time transport becomes suppressed upon increasing the reversal rate, \({D}_{\,{{{{\mbox{eff}}}}}}^{{{{{\mbox{sim}}}}}}(\lambda {\tau }_{0}=0.5)/{D}_{{{{{\mbox{eff}}}}}}^{{{{{\mbox{sim}}}}}}(\lambda {\tau }_{0}=0.0005)\simeq 3\cdot 1{0}^{1}\).
Nonmonotonic transport behavior: effective diffusion
Most prominently, the MSDs indicate that the longtime diffusivities \({D}_{\,{{{{\mbox{eff}}}}}}^{{{{{\mbox{sim}}}}}\,}\) of runreverse polymers display a nonmonotonic behavior with respect to the reversal rate λ. We further introduce the scaled path length\({{\Lambda }}={L}_{c,\max }/{\ell }_{{{{{\mbox{run}}}}}}\) = \({L}_{c,\max }\lambda /{v}_{c}\), which characterizes the tradeoff between the maximal pore length of the medium and the run length of the polymer. We find that the nonmonotonic behavior persists for a broad range of Péclet numbers Pe = 25, …75 (Fig. 3a filled symbols and solid lines). Most significantly, the effective diffusivities for all Péclet numbers display a prominent maximum, where the scaled path length is
Hence, we propose that equation (2) serves as geometric criterion for optimal transport of active agents in porous media, which occurs when the run length ℓ_{run} is comparable to the maximal pore length \({L}_{c,\max }\) characteristic of the porous the environment.
Further, we found that the nonmonotonic transport behavior persists for porous media with fewer obstacles, N = 500, 750, but becomes rather weak for dilute environments, N ≲ 500, where it approaches the monotonic behavior of active polymers in an unconfined environment, see Fig. 3b. The chordlength distributions, \({\varphi }_{{L}_{c}}(\ell )\) (Fig. 1e), show a strong decay for densely packed environments.
Furthermore, we have addressed the effect of runandtumble motion on the overall spreading of active polymers, where, instead of reversing, the swimming direction after the tumbling event is random. Our results (Fig. 3c) demonstrate that our geometric criterion remains valid. We observe that the overall diffusivities increase for N = 1000 with respect to those of runreverse polymers, as 3D tumbling allows polymers to spread further. Most importantly, in dense porous environments (N = 1200) the effective diffusivities of runandtumble polymers collapse with those of runreverse polymers. This indicates that the overall spreading in dense porous environments is characterized by hopandtrap dynamics, which are fully determined by the geometry, irrespective of the reorientation mechanism. It further emphasizes that indeed the longest pore length of the environment is the characteristic length scale dictating the transport behavior of active polymers and furthermore confirms our geometric criterion.
In contrast to previous work^{36}, we have also found nonmonotonic spreading of active agents in porous environments with concave pore shapes (Fig. 5 in Methods). This highlights that solely the interplay of length scales dictates this intricate behavior and not the pore shape.
Our findings offer an interpretation for the observations of the seminal experimental work by Wolfe and Berg^{1}, who used engineered strains of nonchemotactic E. coli. The tumbling rate and thus the run length of these mutants can be tuned by varying the concentration of an external inducer (isopropyl βDthiogalactoside (IPTG)) in the medium. Monitoring the flagella bundles of cells tethered to a surface suggested an increase of the tumbling rate upon increasing the concentration of IPTG. Qualitatively, the experiments of Wolfe and Berg revealed that bacterial swarms that tumble at an intermediate rate spread more than either incessantly tumbling cells or smooth swimming, i.e., nontumbling, strains. However, the dynamical behavior of individual cells has not been quantified. Therefore, we anticipate that our simulations, which qualitatively confirm these experimental findings, enable us to elucidate the physics of the nonmonotonic behavior and provide predictions for the optimal spreading.
Individual trajectories
To elucidate this nonmonotonic behavior of the diffusivities, we investigate individual trajectories for Pe = 50 (Fig. 4a). For rarely reversing agents, λτ_{0} = 5 · 10^{−4}, the trajectories in the xy plane indicate highly confined motion, whereas the trajectories of a polymer reversing at higher rates, λτ_{0} = 0.5 and 15, cover significantly larger areas including many pores [Fig. 4a (inset)]. By inspecting the associated time evolution of the displacements, we observe that the motion of rarely reversing polymers λτ_{0} = 5 · 10^{−4}, i.e., the black curve in Fig. 4, is characterized by fast hopping events, at which the polymer moves through the pore space, and long, extended phases of trapping, where the polymer is trapped inside a pore. This motility pattern agrees with experimental findings of wildtype E. coli moving in a porous structure^{4,5}. We further observe that upon increasing the reversal rate the trapping events become shorter, see blue curve for λτ_{0} = 0.5 in Fig. 4a. In fact, they disappear for large reversal rates, λτ_{0} = 15, where the trajectory is dominated by hopping events, i.e., the red curve in Fig. 4a.
Now, we explain the mechanism for the trapping and hopping, which identifies the main features necessary for an active agent to explore a porous medium. Suppose the active agent is exploring the porous environment with an intrinsic run time τ_{run} ~ 1/λ and after a while it enters a deadend pore at a time t. Consequently, the agent will be trapped within the deadend pore for the remaining time interval [t, τ_{run}]. This implies that the trapping time of a selfpropelled agent can potentially be reduced by increasing the reversal rate λ. The exemplary trajectory for λτ_{0} = 0.5, which corresponds to the optimal reversal rate, exhibits the largest displacements with few trapping events (Fig. 4a). Moreover, we observe that despite the sparse and short trapping events of frequently reversing polymers, \(\lambda\tau_0=15\), their hops become significantly shorter.
We quantify this behavior by extracting the distributions (1) φ_{T}(t) for the trapping time, i.e., the time the polymer spends trapped inside a pore, and (2) \({\varphi }_{{\ell }_{H}}(\ell )\) for the hop length, i.e., the distance the polymer moves from one trapping event to another or from one trapping event to the next reversal, from the individual trajectories (see Methods).
Trapping time distribution
We find that the trapping time distributions for active polymers with Pe = 50 exhibit a powerlaw scaling at long times with an exponent that increases from 2 to 7/2 for increasing reversal rates λ (Fig. 4b). To rationalize this behavior, we extend the concept of an ‘entropic trap’ model introduced in the study of diffusing long polymers escaping small outlets^{41} and recently for E. coli bacteria moving in a porous medium^{4,5}. In this model, the escape of an active polymer from an obstruction in the porous medium is determined by the number of orientations (Ω_{t}) keeping it trapped inside the pore and the number of orientations (Ω_{e}) allowing the polymer to leave it. In short (see Methods), our model predicts a powerlaw trapping time distribution φ_{T}(t) ~ t^{−(1+β)} at long times. Here, the exponent β = X/C_{0} is related to the active energy X, which has dimensions of F_{p}L, and the average depth of the entropic trap C_{0}, which can be quantified by the average free energy difference between the two states: \({C}_{0}={k}_{B}T\langle {{{{{{\mathrm{ln}}}}}}}\,({{{\Omega }}}_{t}/{{{\Omega }}}_{e})\rangle\), where the brackets correspond to the ensemble average over all pores.
In accord with this phenomenological model, we find that the exponent β increases for increasing reversal rate λ (Fig. 4b). Specifically, at a fixed Pe (corresponding to X = const.), a larger reversal rate increases the probability Ω_{e} of leaving the (e.g., deadend) pores and leads to a relatively lower trap depth C_{0}, and thus a larger exponent β = X/C_{0}. Further, we note that for β≤1, corresponding to the case where the active energy becomes smaller than the trap depth X ≤ C_{0}, the entropic trap model predicts a divergence of the mean trapping duration τ_{T} ~ (β−1)^{−1}. This becomes evident in the monotonic increase of the mean trapping duration by orders of magnitude with decreasing reversal rates (Fig. 4b (inset) open circles).
Hopping length distribution
We further extract the distributions of the hopping length of the agents \({\varphi }_{{\ell }_{H}}(\ell )\) (Fig. 4c for Pe = 50) and rationalize that the hop lengths are determined by the interplay of the intrinsic run length of the active polymers ℓ_{run} ≡ v_{c}/λ and the pore geometry. In particular, we find that the chord length distribution (Fig. 4c black solid line) is approached by the hopping length distribution of rarely reversing polymers, λτ_{0} = 5 · 10^{−4} (and 5 · 10^{−3}). These agents have an intrinsic run length of ℓ_{run} ≳ 10^{3}σ (and 10^{2}σ), which is much longer than the longest available straight pathways, \({L}_{{{c}},\max }\simeq 20\sigma\), and therefore their hopping motion is fully determined by the chord lengths of the porous medium (Fig. 4c). The distributions also show longer hops, however, the probabilities are rather low. Further, the average hop length is comparable to the average chord length for small λ, \(\langle\ell_H\rangle\sim\langle L_c\rangle\) [Fig. 4c (inset)]. For large reversal rates λτ_{0} ≳ 5 the distribution decays faster than that of polymers with an intermediate reversal rate λτ_{0} = 0.5 and the average hop length approaches the run length of the polymer, 〈ℓ_{H}〉 → ℓ_{run}.
Our findings further show that polymers with an intermediate reversal rate, e.g., λτ_{0} = 0.5, can follow the straight path available in the porous structure and continue to explore another successive pore without getting trapped, which leads to hopping lengths longer than the longest chord length. Furthermore, we find that the probability for longer hops becomes larger at an intermediate reversal rate than the chordlength distribution, which indicates that the polymer explores these more often than the shorter pores of the medium.
Our results also demonstrate that the probability for long hops and also the average hop length 〈ℓ_{H}〉 [Fig. 4c (inset)] vary nonmonotonically with λτ_{0}: they are lowest for λτ_{0} = 5 · 10^{−4} and λτ_{0} = 15 and highest for λτ_{0} = 0.5. This strong nonmonotonic variation of the average hop length could explain the optimal spreading and thus the maximal longtime effective diffusivity of runreverse polymers in porous media (Fig. 3a). In particular, agents with Péclet number Pe = 50 and reversal rate λτ_{0} = 0.5 have an intrinsic run length of ℓ_{run} = 20σ, comparable to the longest pore length \({L}_{{{c}},\max }\). Our findings reveal that the active polymer continuously explores the pore space without getting trapped too often (Fig. 4a), which suggests the qualitative picture: the agent moves through the pores until it reaches an obstruction, where subsequent reversals allow it to continue to explore the pore space.
Coarsegrained dynamics
To quantitatively characterize the longtime effective diffusivities, we develop a coarsegrained model for the 3D dynamics of the alternating hopping and trapping phases. During the hopping phase the agent, modeled as a point particle, moves straight at effective velocity v and reverses its swimming direction at rate λ with propagator \({{\mathbb{P}}}_{H}({{\Delta }}{{{{{{{\bf{r}}}}}}}},t)\), which denotes the probability that the particle displaces Δr during time t while hopping. During the trapping phase the particle cannot move and after the trapping event it hops along a new, random direction. The time the particle spends in a hopping or trapping phase is determined by the hopping and trapping time distributions extracted from the simulations (Fig. 4b, c). As the swim speed of the polymer during the hopping events remains roughly constant, the hopping time, i.e., the duration of a hopping phase of length ℓ, t ~ ℓ/v, also follows an exponential distribution \({\varphi }_{H}(t)=\exp (t/{\tau }_{H})/{\tau }_{H}\) with mean duration τ_{H}. The trapping time distribution obeys a powerlaw behavior φ_{T}(t) = β(1+t/τ)^{−1−β}/τ with mean duration τ_{T} = τ(β−1)^{−1}, where τ is a characteristic time scale for trapping. This powerlaw behavior suggests that the dynamics are nonMarkovian and therefore we use a renewal theory^{42,43}.
The probability density P(Δr, t) for the particle to have displaced Δr during lag time t is the sum of the probability densities for the particle to be in a hopping or a trapping phase: P(Δr, t) = P_{H}(Δr, t) + P_{T}(Δr, t). We further introduce the probability densities (per time) for the particle to start a hopping phase and a trapping phase at displacement Δr and at lag time t by H(Δr, t) and T(Δr, t), respectively. Then the probability P_{T}(Δr, t) that the particle is trapped at Δr for lag time t is obtained as the sum of the probability of never having hopped before \({P}_{T}^{(0)}({{\Delta }}{{{{{{{\bf{r}}}}}}}},t)\) and the probability of having hopped at least once before getting trapped at Δr and at an arbitrary earlier time \(tt^{\prime}\):
Here, \({\varphi }_{T}^{(0)}(t)=\int\nolimits_{t}^{\infty }dt^{\prime} \,{\varphi }_{T}(t^{\prime} )={[\tau /(\tau +t)]}^{\beta }\) denotes the probability that the trapping time exceeds t. The probability to become trapped at Δr and time t reads
where T^{(1)}(Δr, t) denotes the probability for the first trapping event. The second term corresponds to the sum over the probabilities that the agent has escaped a trap and started to hop at Δr − ℓ at an earlier time, \(tt^{\prime}\), until it gets trapped again at Δr and t. Similar equations hold for P_{H}(Δr, t) and H(Δr, t) (see equations (11)–(12) in Methods). We can derive an analytic solution for the probability density in Fourier–Laplace space, \(P(k,s)=\int_{0}^{\infty }\ dt\,{e}^{st}{\int}_{{{{{{{\mathbb{R}}}}}}}^{3}}\ d{{\Delta }}{{{{{{{\bf{r}}}}}}}}\,{e}^{i{{{{{{{\bf{k}}}}}}}}\cdot {{\Delta }}{{{{{{{\bf{r}}}}}}}}}\,P({{\Delta }}{{{{{{{\bf{r}}}}}}}},t)\equiv \int\nolimits_{0}^{\infty }\ \ dt\,{e}^{st}\langle {e}^{i{{{{{{{\bf{k}}}}}}}}\cdot {{\Delta }}{{{{{{{\bf{r}}}}}}}}}\rangle\)^{42}. By isotropy, it can be expanded for small wave numbers k up to \({{{{{{{\mathcal{O}}}}}}}}({k}^{4})\) via \(P(k,s)\simeq \int\nolimits_{0}^{\infty } dt\,{e}^{st} \left[1 {k}^{2}\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(t){ }^{2}\rangle /3!\right]={s}^{1}{k}^{2}\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(s){ }^{2}\rangle /3!,\) which allows for an analytical derivation of the Laplace transform of the meansquare displacement \(\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(s){ }^{2}\rangle.\)
Finally, we obtain the longtime transport behavior by taking the limit of \(\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(s){ }^{2}\rangle\) for s → 0. We find \(\mathop{\lim }\limits_{s\to 0}\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(s){ }^{2}\rangle \simeq 6{D}_{\,{{{{\mbox{eff}}}}}}^{{{{{\mbox{theo}}}}}\,}{s}^{2}\), which corresponds to a longtime diffusive behavior \(\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(t){ }^{2}\rangle \simeq 6{D}_{\,{{{{\mbox{eff}}}}}}^{{{{{\mbox{theo}}}}}\,}t\) for t → ∞ with effective diffusivity:
It is determined by the fraction of time spent hopping between traps, τ_{H}/(τ_{H} + τ_{T}), the hopping time, τ_{H}, the effective velocity, v, and the tumbling rate λ. The tumbling angle for runreverse motion, as employed here, is^{9} ϑ_{0} = − π. The expression [equation (5)] depends on the exponent β of the trapping time distribution via the average trapping time τ_{T}. We note that for a powerlaw distribution truncated at rate γ, \({\varphi }_{T}(t) \sim \exp (\gamma t){(1+t/\gamma )}^{1\beta }\), which may be more appropriate to describe our data (Fig. 4b), the effective diffusivity assumes the same form as in equation (5) with average trapping time τ_{T} ≡ τ_{T}(γ, τ, β) depending on the parameters γ, τ, and β (see Methods).
To compare the simulation data with our coarsegrained theory, we extract the average hopping and trapping times, τ_{H} and τ_{T}, from the individual trajectories and obtain the effective diffusivity from the hopandtrap model [equation (5)]. We observe that our model captures the nonmonotonic trend of the simulation results, \({D}_{\,{{{\mbox{eff}}}}}^{{{{\mbox{sim}}}}} {\sim} {D}_{{{{\mbox{eff}}}}}^{{{{\mbox{theo}}}}}\) (Fig. 3a dashed lines). It is interesting that the hopping and trapping times, used as inputs for the model, can semiquantitatively predict the nonmonotonic trend of the effective diffusivities of a complex system. The deviations at small reversal rates, corresponding to the scaled path length Λ ≲ 10^{−1}, are expected because at such small reversal rates the polymers remain trapped most of the time. Therefore, the extracted mean trapping times may be underestimated and even longer simulations would be required.
We rationalize the nonmonotonic behavior by inspecting the fraction of time spent hopping, p = τ_{H}/(τ_{H} + τ_{T}) (Fig. 4b inset). As expected, for rare reversal events the fraction of time spent hopping is small p ≪ 1/2. This corresponds to τ_{H} ≪ τ_{T} and therefore equation (5) reduces to \({D}_{{{{{\mbox{eff}}}}}}^{{{\mbox{theo}}}}={v}^{2}{\tau }_{H}^{2}/(3{\tau }_{T})\to 0\), as large trapping times suppress transport. In contrast, the trapping times for frequently reversing polymers are negligible compared to the hopping times, τ_{T} ≪ τ_{H} as p ≫ 1/2. Therefore, the behavior of the effective diffusivity is dominated by the reversal rate and equation (5) simplifies to \(D_{{{\mbox{eff}}}}^{{{\mbox{theo}}}}\sim v^2/\lambda\to 0\), which vanishes as the reversal rate increases. Hence, maximal transport occurs at an intermediate reversal rate λ, in agreement with our simulation results for the longtime effective diffusivities and the average hopping lengths.
Our results indicate that the nonmonotonic behavior of the effective diffusivities is a clear signature of hopandtrap dynamics. This behavior vanishes for dilute environments, where the dynamics are solely determined by the runreverse motion of the polymers. The transition between both motility modes is shown in Fig. 3b.
Discussion
To rationalize the seminal experiments of Wolfe and Berg^{1}, which characterized spreading in a porous environment of bacteria with different tumbling rates, we have performed Brownian dynamics simulations of runreverse stiff polymers in a porous environment and find a nonmonotonic transport behavior as a function of the reversal rate. We introduce, for the first time, a geometric criterion for the optimal spreading of active agents in a porous environment, which occurs when the intrinsic run length is comparable to the longest straight path available in the environment. In particular, we demonstrate that this criterion remains valid in dense environments for different reorientation mechanisms of the active polymers. We further show that individual polymer trajectories exhibit a hopandtrap mechanism, in accord with recent experiments of E. coli^{4,5}. Our results suggest that optimal transport at an intermediate reversal rate is characterized by a maximal average hop length. In contrast, the motion of rarely and frequently reversing polymers is set by the pore length or the runlength, respectively. We corroborate these findings using a renewal theory for the coarsegrained hopandtrap dynamics.
Our results demonstrate that this nonmonotonic transport behavior of active agents in a porous environment persists irrespective of the details of the reorientation mechanism and the shape of the pores. These findings indicate that the ‘size of the pores, not their shape, matters’ for this largescale spreading and we therefore anticipate that this behavior is universal for densely packed environments. Nevertheless, this nonmonotonic behavior fades for low packing fraction, which corroborates earlier predictions for dilute 2D porous environments with concave pore shapes^{36}.
In the future, it will be interesting to study the effect of swimmer shape anisotropy on active transport in a porous environment. In particular, recent work has shown that the nonmonotonic behavior persists even for point particles on a 2D lattice with obstacles^{32}. However, the interplay between pore geometry and run length remains to be explored. Most importantly, our study and recent experiments^{4,5} predict powerlaw distributions for the trapping times, which could be amplified by particle shape. Taking into account this memory dependence in a coarsegrained description goes beyond earlier theoretical predictions^{32,35}.
We anticipate that our theoretical findings can be tested in various biological systems, such as bacteria^{3,8,9,10}, algae^{11}, or archaea^{12}. This could shed light on fundamental microbiological processes, which include the adaption of the tumbling behavior under varying environmental conditions. Experimental observations of Bacillus subtilis^{44} have shown that these cells can reverse their swimming direction once they encounter an obstacle and, similarly, Spirochetes^{45} display an increase of reversal rate while invading a heterogeneous fibrous medium. It would be interesting to elucidate if these spatially varying reversal mechanisms allow them to optimize their motion in a 3D porous medium. Similarly, our results could guide the design of future synthetic swimmers with, e.g., specific magnetic properties^{46}. Their spreading could be enhanced by reorienting them via externally applied magnetic fields. On the macroscale, our findings could find application in instructing robots during search and rescue operations in disaster zones^{36}.
Beyond porous media, our results lay the foundation for studying transport of active semiflexible polymers in dynamically rearranging environments composed of other polymers, such as the interior of cells^{40,47}. The strongly interacting, crowded environment could lead to a relaxation of the extended trapping events of the active agents and entail complex entanglement effects of the individual constituents.
Methods
Brownian dynamics simulations of a stiff, runreverse polymer in a porous environment
We model the active polymer chain in terms of the wellknown beadspring model in 3D^{37}. In particular, the polymer is a chain of contour length L composed of N_{p} spherical monomers with diameter σ that are connected by springs (Fig. 1a). The monomers have positions r_{i} and the distance between two monomers is denoted by r_{i,j} = ∣r_{j} − r_{i}∣, where i, j = 1, …N_{p}. We further introduce the tangent vector between two monomers by \({{{{\bf{t}}}}}_{i,i+1}=\left({{{{\bf{r}}}}}_{i+1}{{{{\bf{r}}}}}_{i}\right)/{r}_{i,i+1}\). The dynamics of the semiflexible polymer are described by the equations of motion for each monomer,
with friction coefficient ζ, active forces \({{{{{{{{\bf{F}}}}}}}}}_{p}^{(i)}={F}_{p}({{{{{{{{\bf{t}}}}}}}}}_{i1,i}+{{{{{{{{\bf{t}}}}}}}}}_{i,i+1})\), and stochastic forces \({{{{{{{{\bf{F}}}}}}}}}_{r}^{(i)}\) characterized by zero mean \(\langle {{{{{{{{\bf{F}}}}}}}}}_{r}^{(i)}\rangle ={{{{{{{\bf{0}}}}}}}}\) and variance \(\langle {F}_{r,j}^{(i)}(t){F}_{r,k}^{(i)}(t^{\prime} )\rangle =2{k}_{B}T\zeta {\delta }_{jk}\delta (tt^{\prime} )\). The interaction energy, U, is characterized by the interaction between neighboring beads and the interaction between nonneighboring beads of the polymer chain itself and with the porous environment (see later section on details about the porous environment).
The elasticity of the chain is characterized by the wellestablished wormlike chain (WLC) model pioneered by Kratky and Porod^{48}. The discretized interaction energy is
where we have introduced the persistence length ℓ_{p} of a semiflexible polymer. It is a measure of the decay length of its tangenttangent correlations and allows distinguishing between flexible, ℓ_{p}/L ≲ 1, semiflexible, ℓ_{p}/L ≃ 1, and stiff polymers, ℓ_{p}/L ≳ 1.
Interactions between neighboring beads are modeled by the finitely extensible nonlinear elastic (FENE) potential,
which ensures a finite, maximal distance between two monomers, L/N_{p} + δ with δ = σ/4. For monomer–monomer distances larger than L/N_{p} + δ the interaction potential diverges, U^{FENE} = ∞. Consequently, the entire polymer chain has a maximal length of L + N_{p}δ. The interactions between nonneighboring polymer beads and the interactions between the polymer and the surrounding, porous environment are modeled using the Weeks–Chandler–Anderson (WCA) potential^{49},
Here, d = σ for the interaction of two monomers and d = (σ + σ_{s})/2 for the interaction between a monomer and an obstacle of the porous environment with diameter σ_{s} (see later section on the Porous environment). The full contribution is the sum over all monomers and beads of the porous environment \({U}^{{{{{\mbox{WCA}}}}}}={\sum }_{i,j;i\ne j}{U}_{ij}^{{{{{\mbox{WCA}}}}}\,}\).
The total interaction energy is then U = U^{WLC} + U^{FENE} + U^{WCA}, which includes (amongst others) three important dimensionless parameters: the persistence length of the polymer chain relative to the contour length ℓ_{p}/L, and ϵ_{FENE} and ϵ_{WCA}, which measure the relative importance of the interaction energies with respect to thermal energy. In the present study, we simulate stiff polymers with ℓ_{p}/L = 50. To assure minimal polymer extension and prevent overlap of monomers and the environment, we set ϵ_{FENE} = 10^{3} and ϵ_{WCA} = 5, respectively. Furthermore, we use a Brownian time step of τ_{B} = 10^{−6}τ_{0} to integrate equation (6) using a modified version of LAMMPS^{50} with τ_{0} the diffusive time scale of a monomer and wait for 10^{9}τ_{B} before taking measurements.
In addition to selfpropulsion, the polymer performs a pseudo runreverse motion, as sketched in Fig. 1b. The reversal events occur at randomly drawn times, t, which follow an exponential distribution \(\lambda \exp (\lambda t)\) with reversal rate λ. At the runreverse event, the active forces, acting along the polymer chain, randomly change sign, \({{{{{{{{\bf{F}}}}}}}}}_{p}^{(i)}\to \alpha {{{{{{{{\bf{F}}}}}}}}}_{p}^{(i)}\) with α randomly chosen from { − 1, 1}. In our framework the reversal events occur instantaneously, so that the polymer does not stop before moving into the new direction.
Porous environment
The porous structure of the environment is generated by randomly distributed obstacles of diameter σ_{s}, which are allowed to overlap. By varying the diameter of the obstacles, we tune the average pore diameter, σ_{p}, of the medium, inspired by the experimental setup of refs. ^{4,5}. The porous environment is characterized by the average pore diameter and the chord length. For ensemble averaging, we use 20 statistically independent structures.
We extract the average pore diameter of the medium by monitoring the transport behavior of Brownian particles with radius σ_{B} in the porous environment. Therefore, we measure the meansquare displacement, \(\langle {{{\Delta }}{{{{{{{{\bf{r}}}}}}}}}_{B}(t)}^{2}\rangle\) with Δr_{B}(t) = r_{B}(t) − r_{B}(0), and calculate the local exponent, \(\alpha (t)=d{{{{{{\mathrm{ln}}}}}}}\,\left[\langle  {{\Delta }}{{{{{{{{\bf{r}}}}}}}}}_{B}(t){ }^{2}\rangle \right]/d{{{{{{\mathrm{ln}}}}}}}\,t\), as a function of time. The local exponent displays a minimum at an intermediate time \({\tau }_{\min }\), which allows introducing the pore diameter by \({\sigma }_{p}={\sigma }_{B}+{\langle  {{\Delta }}{{{{{{{{\bf{r}}}}}}}}}_{B}({\tau }_{\min }){ }^{2}\rangle }^{1/2}\). For simplicity, we choose σ_{B} = σ.
To measure the chord length distribution \({\varphi }_{{L}_{c}}(\ell )\), we image several two dimensional planes randomly passing through the simulated porous medium (Fig. 1d). These images provide a map of the pore space. We then binarize these images where two phases represent solid obstacles and open pores, respectively. We calculate the distribution of chords of length ℓ, which fit within each binarized pore space image. This protocol yields a direct measurement of straight pathways available in the pore space.
Effect of pore shape
We have addressed the effect of pore shape on the largescale spreading of active agents. In particular, we have replaced the WCA potential [equation (9)], corresponding to pores with convex boundaries, and modeled the interaction between the polymer and concave pores by the interaction potential:
where R_{i} is the distance between monomer i of the polymer and the nearest obstacle. We choose ϵ_{C} = 100 and a = 3.7, corresponding to concave voids of diameter ~ 7.4σ, as illustrated in Fig. 5c. We find that also for pores with concave walls the longtime behavior of the meansquare displacement is diffusive, see Fig. 5a. Extracting the longtime effective diffusivities shows that the nonmonotonic behavior persists as a function of the reversal rate (Fig. 5b). The maximal diffusivity occurs at a reversal rate of λτ_{0} = 0.5 and a run length of ℓ_{run} = 20σ. This optimal behavior agrees with the findings for a porous environments with convex pore shapes, as discussed in the main text. Overall, the effective diffusivities are smaller than for a convex environment, which may indicate that the polymer explores the individual pores for a longer time and should depend on the energy depth ϵ_{C}.
Effect of the reorientation mechanism
We further study the effect of the reorientation mechanism of active polymers on their spreading in a porous environment. To complement our findings for runreverse polymers, we consider a runandtumble polymer, which randomly changes its swimming direction at exponentially distributed tumbling events with tumbling rate λ (Fig. 6a). At the tumbling event the active forces may change sign F_{p} → αF_{p} with α randomly chosen from { − 1, 1} and the polymer is subject to tumbling torques. In particular, only at the tumbling event (hence during one time step) opposite forces \({{{{{{{{\bf{F}}}}}}}}}_{t}^{0,1}\) and \({{{{{{{{\bf{F}}}}}}}}}_{t}^{3,4}\) are applied to the zeroth and first and third and forth monomers, respectively (Fig. 6b), so that the polymer gets a random ‘kick’ leading to its reorientation.
These ‘random’ tumbling forces are chosen (on average) perpendicular to the backbone of the polymer: \({{{{{{{{\bf{F}}}}}}}}}_{t}^{0,1}={F}_{t}({{{{{{{{\bf{n}}}}}}}}}_{0,1}+{{{{{{{{\bf{n}}}}}}}}}_{1,2})/2\) and \({{{{{{{{\bf{F}}}}}}}}}_{t}^{3,4}={F}_{t}({{{{{{{{\bf{n}}}}}}}}}_{2,3}+{{{{{{{{\bf{n}}}}}}}}}_{3,4})/2\), where the force magnitude is F_{t} = 2 ⋅ 10^{5}k_{B}T/σ ≫ F_{p}. As the polymer is stiff, we note that the forces approximately balance \({{{{{{{{\bf{F}}}}}}}}}_{t}^{0,1}+{{{{{{{{\bf{F}}}}}}}}}_{t}^{3,4}\simeq {{{{{{{\bf{0}}}}}}}}\).
To obtain the component of the force direction n_{0,1} (and similarly n_{1,2}, n_{2,3}, n_{3,4}), we first define the plane normal to the tangent vector t_{0,1}, which is spanned by the unit normal vectors \({{{{{{{{\bf{n}}}}}}}}}_{0,1}^{(1)}\) and \({{{{{{{{\bf{n}}}}}}}}}_{0,1}^{(2)}={{{{{{{{\bf{t}}}}}}}}}_{0,1}\times {{{{{{{{\bf{n}}}}}}}}}_{0,1}^{(1)}\) (Fig. 6b). Then we choose n_{0,1} as a random direction in this plane via \({{{{{{{{\bf{n}}}}}}}}}_{0,1}=\cos (\beta ){{{{{{{{\bf{n}}}}}}}}}_{0,1}^{(1)}+\sin (\beta ){{{{{{{{\bf{n}}}}}}}}}_{0,1}^{(2)}\) with β drawn from a uniform distribution \({{{{{{{{\mathcal{U}}}}}}}}}_{[0,2\pi ]}\).
To illustrate our algorithm, we show typical trajectories for polymers with vanishing translational diffusivity, D_{0} = 0, and runlength ℓ_{run}/σ = 20, in Fig. 6c. The trajectories (solid lines) demonstrate that the polymer changes its swimming direction randomly at tumbling events and tumbles in 3D. We note that in this case reversing polymers only move back and forth along a straight line (dotted lines).
Data analysis of the individual trajectories
We extract the distributions for the trapping time, hopping time, and the hopping length from the individual trajectories. We note that the hopping time is defined as the time between hopping from one trap to the next or from one trapping event to the next reversal and the hopping length is the length the polymer moved during this time. To extract these quantities, we follow the approach from refs. ^{4,5} and first measure the average velocity of a nontumbling polymer in a free environment, 〈v〉. Then we calculate the instantaneous velocities of the center monomer, v_{i} = ∣r_{c}(t_{i+1}) − r_{c}(t_{i})∣/(t_{i+1} − t_{i}), where t_{i} corresponds to the ith time step, of individual particle trajectories. Thus, we can classify a hopping phase by v_{i}≥〈v〉/3 and a trapping phase by v_{i} < 〈v〉/3, which allows extracting the trapping and hopping time distributions, φ_{T}(t) and φ_{H}(t) with mean durations, τ_{T} and τ_{H}. In addition, we keep track of the reversal times, which are input to our simulations, and compare it with the hopping duration. This provides the hopping length distribution, \({\varphi }_{{\ell }_{H}}(t)\) with mean hopping length 〈ℓ_{H}〉. In particular, if the reversal time since the last trapping event t_{λ} is shorter than the hopping phase t_{hop}, the hopping length corresponds to the length displaced until the reversal event. For t_{λ} > t_{hop} the hopping length is the displacement from one trapping event to the next. For the comparison of the effective diffusivities extracted from simulations, \({D}_{\,{{{{\mbox{eff}}}}}}^{{{{{\mbox{sim}}}}}\,}\), to the predictions of the hopandtrap model [equation (5)], we further use as effective velocity the cutoff velocity, v = 〈v〉/3. We can fully recover the nonmonotonic behavior and explain the simulation data up to a constant prefactor.
To test our approach, we have varied the cutoff velocity between 〈v〉/5 to 〈v〉/2 and found that it does not change our conclusions: the powerlaw behavior of the trapping time distributions, the behavior of the hopping length distributions, and the semiquantitative agreement of the effective diffusivities of the coarsegrained model and the simulations remain preserved.
Entropic trap model for the trapping time distribution
Motivated by other disordered media^{4,5,41}, the probability for the entropic trap C is assumed to follow \(P(C)={C}_{0}^{1}\exp (C/{C}_{0})\) with average trap depth C_{0}. By analogy to equilibrium physics, the probability of an active polymer to escape a trap of depth C is assumed to obey an Arrheniuslike relation. Then the trapping duration is given by \(t=\tau (\exp (C/X)1)\), where X characterizes the active energy due to its swimming motion and τ corresponds to a characteristic time scale for trapping. In particular, the trapping duration vanishes, t → 0, for small entropic traps, i.e., C/X → 0. For a passive polymer the active energy X is replaced by the thermal energy k_{B}T. The probability distribution of the trapping times can be obtained as φ_{T}(t) = P(C)(∂t/∂C)^{−1} = β(1+t/τ)^{−1−β}/τ with β = X/C_{0}.
Renewal theory for the hopandtrap dynamics
The probability density for a particle to be in a hopping phase follows:
Here, \({P}_{H}^{(0)}({{\Delta }}{{{{{{{\bf{r}}}}}}}},t)\) denotes the probability that the particle has never been trapped before and the second term corresponds to the sum over all hopping phases, which started after at least one trapping event. Further, \({\varphi }_{H}^{(0)}(t)=\int\nolimits_{t}^{\infty }dt^{\prime} \,{\varphi }_{H}(t^{\prime} )\) is the probability that the hopping time exceeds t. The probability density (per time) that a new hopping phase starts obeys the equation of
where H^{(1)}(Δr, t) is the probability that the particle starts the first hop. After a Fourier transform of the probability densities, Δr → k, and by the convolution theorem, the renewal equations [Eqs. (3), (4), (11), (12)] simplify to
We further need to specify the probability densities
with p = τ_{H}/(τ_{H} + τ_{T}), which account for the fact that the system starts in a stationary state^{42}. In particular, the probability to have never hopped before, \({P}_{T}^{(0)}(k,t)\), depends on the probability that the particle is in a trapped state, 1 − p, and on the time integral, which represents the probability that the trapping phase exceeds time t. It can be rationalized as follows: The probability density that the trapping phase is of length \(t^{\prime}\) is given by \(t^{\prime} {\varphi }_{T}(t^{\prime} )/{\tau }_{T}\). The probability that after lag time t the particle is still trapped is \((t^{\prime} t){{\Theta }}(t^{\prime} t)/t^{\prime}\), where Θ( ⋅ ) denotes the Heaviside function. Then the probability that the particle has not yet started a hopping phase at time t is obtained by integrating over all durations \(t^{\prime}\): \(\int\nolimits_{t}^{\infty }dt^{\prime} \,{\varphi }_{T}(t^{\prime} )(t^{\prime} t)/{\tau }_{T}\). Similarly, we can derive \({P}_{H}^{(0)}(k,t)\).
Moreover, the probability densities (per time) for the first trapping and hopping event are
Here, the probability for the first trapping event T^{(1)}(k, t) depends on the probability that the particle is in a hopping state p and has hopped for a time t with propagator \({{\mathbb{P}}}_{H}(k,t)\). Further, the probability density for a hopping phase to be of length \(t^{\prime}\) is given by \(t^{\prime} {\varphi }_{H}(t^{\prime} )/{\tau }_{H}\). The probability for the lag time t = 0 to be in the same interval is uniformly distributed, \(1/t^{\prime}\). Then the probability density that the trapping phase starts at t given a hopping phase of length \(t^{\prime}\) is obtained by the integral over all possible \(t^{\prime}\), \(\int\nolimits_{t}^{\infty }dt^{\prime} \ {\varphi }_{H}(t^{\prime} )/{\tau }_{H}\). Similar considerations hold for H^{(1)}(k, t).
Finally, we specify the propagator in the hopping phase \({{\mathbb{P}}}_{H}(k,t)\). Since we are interested in terms up to \({{{{{{{\mathcal{O}}}}}}}}({k}^{2})\), we expand it in k: \({{\mathbb{P}}}_{H}(k,t)\simeq 1{k}^{2}{\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(t){ }^{2}\rangle }_{{{{{\mbox{RR}}}}}}/3!\), where the meansquare displacement of a runreverse particle is \({\langle  {{\Delta }}{{{{{{{\bf{r}}}}}}}}(t){ }^{2}\rangle }_{{{{{\mbox{RR}}}}}}=2{v}^{2}/{\tilde{\lambda }}^{2}(\exp (\tilde{\lambda }t)+\tilde{\lambda }t1)\). The effective rate depends on the turning angle ϑ_{0} via \(\tilde{\lambda }=\lambda (1\cos {\vartheta }_{0})\)^{9}. Subsequently, we perform a Laplace transform, t → s, of equation (13a)–(13d) and use the convolution theorem to derive an analytical solution for the renewal equations in Fourier–Laplace space^{42}. The formal solution has been presented elsewhere^{42}. We insert the expansion of \({{\mathbb{P}}}_{H}\) and the hop and trapping time distributions, φ_{H} and φ_{T}, into the theoretical predictions in Fourier–Laplace space and keep only terms up to \({{{{{{{\mathcal{O}}}}}}}}({k}^{4})\). Using the expansion from the main text, we derive an analytical solution for the meansquare displacement in Laplace space, 〈∣Δr(s)∣^{2}〉. An analytical backtransform to time space is not possible, however, we can extract the longtime (corresponding to s → 0) behavior, see main text.
We further note that the longtime effective diffusivity can also be derived analytically for a truncated powerlaw distribution, \({\varphi }_{T}(t)=\exp (\gamma t){(1+t/\tau )}^{1\beta }f(\gamma \tau ,\beta )\) with \(f(\gamma \tau ,\beta )={\left[{e}^{\gamma \tau }\beta {{{{{{{\rm{ExpIntE}}}}}}}}(1+\beta ,\gamma \tau )\right]}^{1}\), where ExpIntE( ⋅ , ⋅ ) denotes the generalized exponential integral function^{51}. It assumes the same form as equation (5) with average trapping time \({\tau }_{T}=\int\nolimits_{0}^{\infty }t{\varphi }_{T}(t)\ {{{{{{{\rm{d}}}}}}}}t=\tau \left[{{{{{{{\rm{ExpIntE}}}}}}}}(\beta ,\gamma \tau )/{{{{{{{\rm{ExpIntE}}}}}}}}(1+\beta ,\gamma \tau )1\right]\). We note that the average trapping time of the truncated powerlaw distribution reduces for γ = 0 to that of the powerlaw distribution used in our manuscript. Details of the distribution become apparent in the shorttime behavior of the meansquare displacement, but do not affect our data analysis for the longtime effective diffusivities.
Data availability
Source data are provided with this paper in the Source Data file (Source_Data.zip). Source data are provided with this paper.
Code availability
The computer code used for simulations is available from the corresponding authors upon reasonable request.
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Acknowledgements
Work by C.K. was funded by the Austrian Science fund (FWF) via the Erwin Schrödinger fellowship (Grant No. J4321N27). The work was further supported by the NSF grant MCB1853602 (H.A.S.). Work by S.M. and H.L. was supported by the Deutsche Forschungsgemeinschaft (Grant No. LO 418/23). Work by T.B. and S.S.D. was supported by NSF grant CBET1941716, the Project X Innovation Fund, the Eric and Wendy Schmidt Transformative Technology Fund, a distinguished postdoctoral fellowship from the Andlinger Center for Energy and the Environment at Princeton University to T.B., and in part by funding from the Princeton Center for Complex Materials, a Materials Research Science and Engineering Center supported by NSF grants DMR1420541 and DMR2011750. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
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C.K., S.M., H.L., S.S.D., and H.A.S. designed the overall study. C.K. and S.M. performed the simulations, analyzed the data, and developed the coarsegrained theory. T.B. measured chord length distributions. S.S.D. and T.B. helped to design the analysis of hopping and trapping statistics. All authors discussed the results and implications. C.K. and S.M. wrote the manuscript with the help of all other authors.
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Kurzthaler, C., Mandal, S., Bhattacharjee, T. et al. A geometric criterion for the optimal spreading of active polymers in porous media. Nat Commun 12, 7088 (2021). https://doi.org/10.1038/s41467021269420
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DOI: https://doi.org/10.1038/s41467021269420
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