Emergent programmable behavior and chaos in dynamically driven active filaments

Significance

Single-celled protozoa display remarkable animal-like behaviors without the aid of neurons. Mechanistically understanding how this behavior emerges from underlying physical and biochemical components remains a challenge. In this work, inspired by the rapid search and hunting behavior of Lacrymaria olor using its slender neck-like protrusion, we develop an active filament model toward understanding cell behavior from first principles. Using our model, we reveal how filament shape dynamics (“behavior”) emerge from time sequences of activity patterns (“programs”). We explore this idea further by designing simple activity motifs that lead to useful functions like homing and search. Finally, to close the loop, we measure the statistical properties of biological programs in L. olor and use these to directly compare model outputs and experiments.

Abstract

How the behavior of cells emerges from their constituent subcellular biochemical and physical parts is an outstanding challenge at the intersection of biology and physics. A remarkable example of single-cell behavior occurs in the ciliate Lacrymaria olor, which hunts for its prey via rapid movements and protrusions of a slender neck, many times the size of the original cell body. The dynamics of this cell neck is powered by a coat of cilia across its length and tip. How a cell can program this active filamentous structure to produce desirable behaviors like search and homing to a target remains unknown. Here, we present an active filament model that allows us to uncover how a “program” (time sequence of active forcing) leads to “behavior” (filament shape dynamics). Our model captures two key features of this system—time-varying activity patterns (extension and compression cycles) and active stresses that are uniquely aligned with the filament geometry—a “follower force” constraint. We show that active filaments under deterministic, time-varying follower forces display rich behaviors including periodic and aperiodic dynamics over long times. We further show that aperiodicity occurs due to a transition to chaos in regions of a biologically accessible parameter space. We also identify a simple nonlinear iterated map of filament shape that approximately predicts long-term behavior suggesting simple, artificial “programs” for filament functions such as homing and searching space. Last, we directly measure the statistical properties of biological programs in L. olor, enabling comparisons between model predictions and experiments.

The behavior of biological systems emerges due to the interplay of parts across molecular, cellular, and tissue scales (1, 2). Yet, despite the notion of behavior as a complex phenomenon, recent efforts that quantify behavior in diverse systems have found a common theme of underlying simplicity: Even seemingly complex behaviors can be described fruitfully using the dynamics in a relatively low-dimensional space, as compared to the total possible degrees-of-freedom (3, 4). However, mapping the emergence and dynamics of these finite “behavioral modes” to physical, biochemical, genetic, or epigenetic features of the organism has remained elusive, primarily due to the complex intervening layers involved (5).

Unicellular ciliates, protozoans with a large number of cilia, display remarkable, rapid, animal-like behaviors such as swimming, walking, jumping, and hunting without the aid of a neuromuscular system (6–9). Thus, studying ciliates presents a unique opportunity to understand emergence of complex behavior in a system with no neurons. In this reductionist approach, multiple layers of complexity of a neuromuscular system are replaced by the relative “simplicity” of mechanical and biochemical interactions between constituent subcellular parts (10).

A striking example of complex behavior in a unicellular protist occurs in the ciliate Lacrymaria olor, which utilizes a dynamic, ciliated, slender protrusion that can stretch as large as ∼10 to 20 times the cell body size (which is only ≈50 $\mathrm{\mu }\mathrm{m}$), to search and hunt for prey. During these hunting bouts, the protrusion or cell neck adopts a striking range of shapes allowing the tip/head to rapidly and exhaustively sample the cell’s surroundings over a time scale of minutes, even while the cell body remains attached to a substrate (Fig. 1A) (7, 11). When compared to better-studied cellular protrusions like filopodia which typically change length at rates of ∼10 μm/min (12), protrusions in ciliates are much more dynamic, characterized by length and shape changes occurring over millisecond and subsecond time scales (7). The speed and reactive dynamics of these movements rule out any genetic mechanisms and pave the road to search for mechanochemical feedback loops responsible for the diversity of behaviors.

Fig. 1.

Active elastohydrodynamics of filaments under dynamic follower forces. (A) The phenomena of rapid morphodynamics to search space and hunt for prey in the single-celled ciliate Lacrymaria olor using a long cellular protrusion. Cilia along the protrusion and specialized tip cilia provide active stresses, which are constrained by the cell’s membrane and cytoskeleton and hence follow the geometry of the cell, thus giving rise to a so-called follower-force system. Changes in ciliary beat direction alternate the force between compression and extension. (B) Active filament model to study filament behavior under dynamic active follower-force stresses. Model filaments consist of a string of colloids which can be either active (i.e., can apply active stresses on the surrounding fluid; shown in red) or passive (green). In our work, we start by considering the simplest case where only the distal colloid is active (red), and also the more general case where all colloids are active (not shown). The activity profile can be parametrized by the activity strength $\mathcal{A}$ and time scale for one compression + extension cycle ${\tau }_a$. Elastic potentials for intercolloid distance and equilibrium angle constrain the filament shape along with short-ranged repulsive potentials to prevent overlap of colloids and filament self-intersection.

A key feature of protrusion dynamics exhibited by Lacrymaria olor is that the active stresses are constrained by the local geometry of the neck since the cilia are directly anchored to the cell membrane and cytoskeleton. Thus the direction of forces generated by these anchored cilia change dramatically with the orientation vector of the cell surface—a classical condition in mechanics often described as a “follower-force” constraint (13–15). First described to characterize the stability of rocket engines (16), follower forces create a unique nonconservative condition wherein the force “follows” the geometry of the system, and occurs naturally in the biological context of self-driven filaments (13–15, 17).

Our recent work studying L. olor experimentally under lab conditions has identified key components that produce active stresses in the system—including cilia in the neck and tip of the protrusion and contractile centrin-like proteins (7). Furthermore, the unique cytoskeletal structure of the neck imparts nonlinear mechanical properties to the neck (7, 11). Although the source of active stresses and mechanical constraints has been mapped in this system, currently, we still lack a first-principles understanding of how this complex behavior emerges. Specifically, how do time-varying activity patterns due to ciliary reversals, follower-force constraints, and the mechanical properties of the cell lead to the observed complex shape dynamics and behavioral repertoires? What class of dynamical systems does this particular cell behavior fall into?

Toward this goal, here, we present a filament subject to time-varying follower forces as a toy model system to understand complex behavior of L. olor. To capture the essence of the behavioral complexity observed in cells like L. olor, we consider the simplest model with two key features: 1) Time-varying activity patterns, wherein the active stresses generated by the filament can periodically change in magnitude or direction (specifically between compression and extension of the filament). This directly captures the effects of ciliary reversals which are very common across ciliates, and have previously been studied in relation to avoidance reactions in Paramecium (18); 2) Follower-force activity: we constrain the active stresses to be along the local tangent vector of the filament to capture the follower-force constraint that arises due to the constraints on the ciliary beat plane, relative to the cell membrane and cytoskeleton (Fig. 1B).

Using this active filament model, we seek to leverage the single-cell context of this behavior and the relative simplicity of the filament geometry to seek an explicit, deterministic framework for behavior consisting of an input, map, and output, with no hidden layers: Dynamic activity patterns (inputs) acting via nonconservative follower-force mechanics (map) lead to the observed time-varying shape dynamics of the cell (behavioral output)—all integrated together with functional benefits to the organism at large (Fig. 1B).

1. Results

Active Filament Model and Simulation Method.

We consider an active filament framework (14, 19), wherein the filament consists of a series of N connected colloids of radius a that can be either active or passive (Fig. 1B). The active colloids can exert a surface slip velocity or stress on the ambient fluid (20). The material response of the filament is captured using potentials that penalize departures from equilibrium bond length, bond angles and prevent self-intersection of the colloids. This potential is given by:

$$\begin{array}{cc}\hfill U=& {\displaystyle \sum _{m=1}^{N-1}}{U}^C({\mathit{R}}_m,{\mathit{R}}_{m+1})+{\displaystyle \sum _{m=2}^{N-1}}{U}^E({\mathit{R}}_{m-1},{\mathit{R}}_m,{\mathit{R}}_{m+1})\hfill \\ \hfill & +{\displaystyle \sum _{m\ne n}}{U}^s({\mathit{R}}_m,{\mathit{R}}_n),\hfill \end{array}$$

where R_m is the position of the mth colloid, U^C(R, R′)=k(r − b₀)²/2, with r = |R − R′| and R, R′ correspond to the positions of two distinct colloids, b₀ the equilibrium bond length, k the axial spring stiffness, and U^E = κ(1 − cosϕ)/b₀, with κ the bending stiffness, ϕ the angle between consecutive colloidal bonds (Fig. 1B). Near-field spherical and spherocylindrical Lennard-Jones type repulsive interactions are specified by U^S and prevent overlap of the colloids and also prevent self-intersection of the filament (19). The resulting constraint force on the nth colloid is given by F_n^B = −∇_nU.

We consider the inertialess (force and torque-free), and non-Brownian limit of the colloids corresponding to the scale of single cells (> 10 μm). The active colloid framework used captures both the local as well as many-body hydrodynamic interactions due to both passive and active surface stresses on the colloids, and results in the following equations of motion of the colloids (20, 21):

$$\begin{array}{c}\hfill {\dot{\mathit{R}}}_n={\mathit{\mu }}_{\mathit{nm}}^{\mathit{TT}}·{\mathit{F}}_m^B+{\mathit{\mu }}_{\mathit{nm}}^{\mathit{TR}}·{\mathit{T}}_m^B+{\mathit{\mu }}_{\mathit{nm}}^{T,3t}·{\mathit{V}}_m^{(3t)},\end{array}$$ [1]

where the dot signifies temporal derivatives, ${\mathit{\mu }}_{\mathit{nm}}^{\mathit{TT}}$ and ${\mathit{\mu }}_{\mathit{nm}}^{\mathit{TR}}$ are the many-body forms of the well-known translational and rotational mobility tensors for spheres in an unbounded fluid (22), and μ^{T, 3t} is the propulsion tensor corresponding to the active slip on the colloids, where V_m^(3t) specifies the active slip due to a potential dipole of strength d₀. The principal axis of this active slip is along the local tangent vector of the colloids such that ${\mathit{V}}_n^{(3t)}=a^2{d}_0{\mathit{t}}_n$ where t_n is the tangent vector at the nth colloid (14, 23). The expressions for the mobility and propulsion tensors are given in SI Appendix, section 1.4. The angular degrees of freedom of the colloids are constrained to be along the local tangent vector of the filament and are hence not true degrees-of-freedom of the system (19), which is equivalent to neglecting the influence of constraint torques T_m^B on the colloids. The filament boundary conditions are clamped at the base and free at the tip, such that the equilibrium shape is a straight filament (Fig. 1B).

Of the family of models possible within the above active filament framework, we start with the simplest one where activity is confined to only the distal colloid and is due to a potential dipole of strength d₀: the simplest hydrodynamic representation of a finite-sized, force, and torque-free swimmer. Importantly, the orientation of this potential dipole activity is constrained to be along the local tangent at the filament tip. This so-called tip follower force is inspired by ciliates like L. olor wherein the cilia are anchored to the cell membrane and underlying cytoskeleton and thus their beat-plane, and hence thrust direction is constrained by the local orientation of the organism. While the angle between the active stresses and the filament tangent can have nonzero values and is indeed dynamic in L. olor, we consider the simplest case of where the stresses are tangential at all times and only switch direction to model ciliary reversals. Beyond tip activity, we also consider the more general follower-force case of activity distributed along the filament length to more closely model L. olor (7), with the dipole axis at each location being parallel to the local tangent.

To model the dynamics of the active stresses, we start by considering a deterministic activity profile such that the potential dipole changes sign periodically at a fixed time scale τ_a, switching between phases of compression and extension of the filament (Fig. 1B). This leads to a square-wave profile for the activity strength given by:

$$\begin{array}{c}\hfill d_N(t)=\left\{\begin{array}{cc}-d_0\hfill & 0\le t<\alpha {\tau }_a\text{:compression}\hfill \\ d_0\hfill & \alpha {\tau }_a\le t<{\tau }_a\text{:extension}.\hfill \end{array}\right.\end{array}$$ [2]

Here 0 ≤ α ≤ 1 is the duty cycle corresponding to the duration of the compressive phase relative to the total activity time scale. The square-wave activity profile chosen has a biophysical basis in the brief transitory phase between the much longer forward and reversed beat phases of the cilia (7). We later generalize the temporal activity profile to include more complex patterns both artificial and inspired by biological experiments.

Our active filament model thus consists of N colloids, where N_active are active while the remaining N − N_active are passive, giving a filament of equilibrium length L = (N − 1)b₀. Thus, tip activity corresponds to N_active = 1, while distributed activity corresponds to N_active = N. The system can be made dimensionless using the colloid radius a as the length scale, the stretch relaxation time scale ηL/k, velocity scale a²d₀, and the force and moment scales of ηa³d₀ and ηa³d₀L, respectively, where η is the fluid viscosity. We can also define a dimensionless activity strength which is the ratio of active and elastic restoring moments given by: 𝒜 = ηa³N_actived₀L²/κ (15). All quantities in the rest of this work are presented in dimensionless terms unless otherwise indicated.

We validated our model using two case studies: one static and one dynamic (SI Appendix, section 1.3). In the first, we compared the steady-state shape of a filament under a fixed transverse force to theoretical predictions for a Euler–Bernoulli beam and find good agreement (SI Appendix, section 1.3). In the second, we simulated the filament dynamics for the cases of a constant compressive tip and distributed follower force and compared the resulting “flapping” frequency with the scaling law predicted by ref. 13, and again find good agreement with the predicted −4/3 power-law (SI Appendix, section 1.3). This flapping time scale can be regarded as a natural active time scale of the system for a given set of filament parameters, and for our system can be written as

$$\begin{array}{c}\hfill {\tau }_{\mathit{flapping}}=C\left(\frac{\eta L^4}{\kappa }\right){\mathcal{A}}^{-4/3},\end{array}$$ [3]

where ηL⁴/κ is the equilibrium time scale for the relaxation of bending perturbations of the filament, and C is a constant that depends on the activity distribution (SI Appendix, section 1.3).

Periodically Driven Active Filaments Display Rich Behaviors.

To explore the space of filament behaviors, we evolve the dynamics of filaments starting from various straight initial shapes, over hundreds of activity cycles, with a deterministic activity profile (given by Eq. 2), and for a range of activity strengths $\mathcal{A}$ and times ${\tau }_a$. For weak activity strengths, or short activity times, there is no buckling during the compressive phase, and the filament remains straight and the only dynamics is a small change in filament length in phase with the driving signal. For higher activity strengths and durations, the filament dynamics during an activity cycle follows a typical pattern wherein the compressive phase causes the filament to buckle in-plane and the tip of the filament to decorrelate in orientation and also rapidly change position (Fig. 2A). Note that this buckling is not the classical Euler buckling of an elastic beam under constant load, but is due to the so-called tip follower force since the direction of compressive force is determined by the tip orientation itself (Figs. 1B and 2 B and C) (14, 15). This is made apparent by visualizing the flow around the filament during the compressive and extensional phases, with a strong coupling observed between the filament geometry and the active flow and hence, stress generated by the tip (Fig. 2 B and C and Movie S1). The subsequent extensional phase largely preserves the tip orientation, but the filament as a whole now points in a new direction (Fig. 2A). As the filament extends, the “swimmer” reaches the end of its tether leading to lower tip speeds and filament displacements during the extensional phase, but faster flows around the filament (Fig. 2 A and C). Each cycle of compression and extension thus leads to a change in filament shape and orientation. With regard to the filament tip, the filament behavior for high enough activity strength resembles a rapid orientation decorrelation, followed by correlated movement of the tip analogous to the tumbles and runs in bacterial motility (24) (SI Appendix, Fig. S3).

Fig. 2.

Emergent behavior in active filaments under dynamic follower forces. (A) Filament center line shape dynamics during a single cycle of compression and extension. The filament center lines are colored by activity phase (red: compression, blue:extension). Filament tips are colored by the dimensionless tip speed. (B and C) The instantaneous flow around the filament during compression and extension phases, respectively, with color denoting the dimensionless flow speed (log-scale). (D–G) Long-time filament dynamics (over 500 activity cycles) for varying activity strengths ($\mathcal{A}$) and τ_a= 1,744, showing both periodic (D and E) and aperiodic behavior (F and G). Filament center lines shown at the end of compression (red) and extension (black) phases for the last 100 activity cycles. In (F and G), the unique locations sampled by the filament tip are shown colored by the first-passage cycle (points are not to scale). The activity strength is given by 𝒜 = ηa³L²d₀/κ, where η is the fluid viscosity, a the filament/colloid radius, L the filament rest-length, d₀ the potential-dipole strength of the tip colloid and κ the bending rigidity. (H) Long-term filament behavioral state diagram for varying activity strength and time scale over 500 activity cycles. The dashed lines and shaded regions are a guide to the eye. Each symbol represents simulations from one initial condition (3 initial conditions per parameter value pair: a total of 540 simulations run over 500 cycles). (I) Search cloud of filament tip positions over 500 activity cycles for 𝒜 = 147.6 (aperiodic dynamics), colored by the local density of points. The red dot represents the base of the filament. Higher probabilities correspond to more sampling of the region by the filament tip.

Over many iterative activity cycles, we find a rich space of filament behaviors (Fig. 2D–H). We observe period-doubling dynamics where the filament shapes repeat every 2, 4, 6, … cycles, as well as period-1 cycles where the filament repeatedly samples a particular angular location away from the symmetry axis (Fig. 2D, E, and H and Movie S2). Intriguingly, above a critical activity strength, we observe aperiodic dynamics where the filament shapes never repeat even for long times (Fig. 2F–I and Movie S2). The sequence of filament shapes appears unpredictable and random for the aperiodic cases: For instance, even the half-plane that the filament tip occupies after each cycle (Left vs. Right) appears to be a random sequence (Movie S3). At higher activity strengths, we also observe that filaments can “escape” from the half-space, which we define as when the angle between the first two colloids at the base and the equilibrium angle becomes greater than π/2 (Fig. 2H).

Searching Space as a Functional Consequence of Aperiodic Filament Behaviors.

These aperiodic filament dynamics have an important consequence for how the tip of the filament samples space, a function of interest for biological systems including L. olor (7, 25–28). In striking contrast to periodic dynamics where the filament tip samples only a small number of locations repeatedly, aperiodic dynamics leads to the tip sampling ever newer locations even for long times (Figs. 2F, G, and I and 3B). Further, the tip exploration of space is not systematic in nature but appears “stochastic” with the tip darting between points on either side of the symmetry axis (Movie S3).

Fig. 3.

Filament search dynamics for tip and distributed activity. (A) Filaments driven by periodic activity cycles confined to the filament tip. (B) Long-term filament tip search clouds showing marked difference in spatial sampling due to periodic and aperiodic filament behaviors. The red circle marks the filament base. (C) The dynamics of unique locations sampled by the filament tip (symbols: simulations) and lines (theoretical fit). The dynamics follow a kinetic equation of the form n_unique = n_satt/(t + τ_1/2), where n_sat is the saturation value, t is the time, and τ_1/2 is the time to reach n_sat/2 sites. The results are rescaled by n_max, the theoretical maximum unique locations available to the filament tip n_max = L²/a². (D) Variation of n_sat/n_max(%) and τ_1/2/<τ_a> with activity number. Aperiodic dynamics lead to effective search behaviors while periodic dynamics lead to much fewer unique locations sampled. (E) Filaments driven by activity uniformly distributed along their length and with periodic forcing in time. (F) Similar to tip forcing case above, distributed activity also leads to both periodic and aperiodic behaviors over long times, with corresponding changes to the tip search clouds. (G) Search dynamics of the filament tip and (H) variation of search metrics with activity also show a qualitatively similar trend to tip forcing.

Interestingly, we find that the dynamics of unique locations (to within a colloid radius) sampled by the filament-tip in our simulations n_unique(t), are well-approximated by a Michaelis–Menten-type kinetic equation of the form:

$$\begin{array}{c}\hfill n_{\mathit{unique}}=\frac{n_{\mathit{sat}}t}{t+{\tau }_{1/2}},\end{array}$$ [4]

where n_sat is the maximum saturation value for the unique locations visited, and can be interpreted as a “search range” of the filament, and τ_1/2 is the time taken to reach 50% of the n_sat locations, or a characteristic “search time scale” (Fig. 3). This search range can be further rescaled by the maximum number of locations accessible to the filament, which is constrained by the filament length and is given by n_max = L²/a², where L is the filament length and a the colloid radius. This gives the fractional search range $n_{\mathit{unique}}/n_{\mathit{max}}(\%)=100\frac{n_{\mathit{sat}}}{n_{\mathit{max}}}\frac{t}{t+{\tau }_{1/2}}$. Thus, fitting the simulation data to this kinetic equation allows us to describe the filament search metrics using just two dimensionless parameters the search range n_sat/n_max(%) and time scale τ′_1/2 = τ_1/2/<τ_a>, where <τ_a> is the characteristic time scale of an activity cycle. Interestingly, both periodic and aperiodic tip dynamics follow this simple kinetic equation, however, periodic dynamics lead to very small values of n_sat and a very limited search range, after ignoring transients (Fig. 3 C and D). Strikingly for aperiodic dynamics the filament can cover as much as ≈40% of the maximum available locations with a τ′_1/2 ≈ 40 cycles (Fig. 3 C and D).

Distributed Activity Leads to Qualitatively Similar Dynamics as Tip Activity.

To more closely tie our model to biological systems like L. olor which have cilia distributed all along their slender necks (7, 11), we next considered the effects of distributed activity on the observed filament dynamics (Fig. 3E). We considered two types of spatially distributed activity 1) uniform activity along the filament, that changes sign between compressive and extensional stress and 2) “Lacrymaria-like” distribution which even more closely follows the distribution in cells, wherein the activity is distributed along the filament, but the tip activity is stronger than the rest of the filament and is further inactive during compression (SI Appendix, Table S2).

For distributed activity and periodic driving, we once again find periodic behaviors as well as aperiodic behaviors above a critical activity strength (Fig. 3F and Movie S4). The search behavior of the filament tip is again well approximated by Eq. 4, and aperiodic behaviors lead to typical search-ranges of ≈25% of the maximum available locations for high activity strengths. We find qualitatively similar trends for the Lacrymaria-like activity distribution (SI Appendix, Fig. S5). Overall, we find that the qualitative features of the filament dynamics remain the same for tip concentrated and distributed activity, and we mainly consider tip activity for further elucidating the origins of this rich behavioral space.

Overall even when driven by simple, periodic activity patterns, the combination of compressive and extensional follower forces leads to a very rich behavioral space and a functional search behavior of the filament tip—in a manner that is qualitatively similar for both tip and distributed activity (Fig. 2H). Next, we explore the emergence of these behaviors further using tools from nonlinear dynamics.

A Transition to Chaos Underlies Aperiodic Filament Dynamics and Dense Spatial Sampling.

Are the observed aperiodic behaviors a signature of underlying chaotic dynamics? To answer this question, we created a reduced-order representation of the filament dynamics using principal component analysis (PCA). We computed the eigenvectors corresponding to the covariance matrix C_ij of the tangent angle representation of the filament center lines θ_t(s, t), where θ_t is the local tangent angle at the arc length location s and at time t, drawn from simulations over long times (500 activity cycles). We further used the participation ratio of the eigenvectors $D_{\mathit{PCA}}={(\sum _{n=1}^N{\lambda }_n)}^2/(\sum _{n=1}^N{\lambda }_n^2)$ (29) to estimate the dimensionality of the dynamics, where λ_n is the nth eigenvalue of the covariance matrix C_ij. We find that the dynamics is low-dimensional (D_PCA ≤ 3) over a large range of activity strengths spanning no-bucking, periodic and aperiodic filament behaviors (Fig. 4F). By projecting the filament shapes onto the first three eigenmodes, we found the time-varying amplitudes that describe filament dynamics. Using this representation, the filament dynamics for low and moderate activity becomes a trajectory in an ℝ³ “shape space,” with the origin corresponding to the equilibrium, straight-filament shape (Fig. 4 C and D).

Fig. 4.

A transition to chaos underlies aperiodic filament behavior. (A and B) Evolution of 10 closely spaced initial filament shapes (pairwise euclidean distance ≤10⁻¹²), driven by deterministic activity profiles. Initial and final shapes are plotted, along with time evolution of the filament base-tip angle. For the periodic case (A), the final shapes all lie on top of each other and are indistinguishable at the scale of the filament radius. In striking contrast, for the aperiodic case (B), closely spaced filament shapes evolve into distinct final shapes over just 50 cycles. (C and D) Visualization of filament shape dynamics using a lower-dimensional representation by projecting onto an orthogonal set of eigenvectors (shape modes). For both periodic and aperiodic cases, the filament dynamics settles onto an attractor (black trajectory). The projections of this trajectory along the principal coordinate axes are shown as red, green, and blue curves. (C) Periodic dynamics results in closed loops in shape space, while (D) aperiodic dynamics leads to trajectories on a strange-attractor. (C and D) Evolution of closely spaced initial conditions (green dots) into final shapes (red triangles) for the periodic and aperiodic cases are overlaid. (E) Maximum Lyapunov exponents as a function of activity strength calculated from the growth/decay rate of small perturbations. Positive values are a feature of chaotic dynamics, while negative values correspond to periodic dynamics. Exponents are calculated over 100 unique initial conditions chosen on the attractor, for each of which we evolved 10 closely spaced initial shapes. (F) Participation ratio of the eigenvalues to estimate the dimensionality of the dynamics as a function of the activity strength.

As expected, periodic dynamics leads to a limit cycle in shape–space, but interestingly aperiodic dynamics leads to flow on a more complex, but finite-sized attractor (Fig. 4 C and D). We estimated the dimensionality of this attractor by calculating the correlation dimension and found that it has a fractal dimension between 1 and 2 for aperiodic dynamics (SI Appendix, Fig. S7) (30). To test whether the dynamics was chaotic, we chose 100 random points along this attractor and seeded 10 nearby initial conditions by adding small transverse perturbations of amplitude δ₀ = 10⁻¹² to the filament shape at that point. We then computed the pairwise euclidean distance between filament shapes evolved from these initial conditions. The pairwise euclidean distance between filament shape m and n is defined as ${\delta }_{m,n}(t)=\sum _{i=1}^N||{\mathit{R}}_i^m-{\mathit{R}}_i^n{||}_2$, where R_i^m is the position of the ith colloid on filament m. We find that the pairwise distances grow exponentially for aperiodic dynamics, leading to the nearby trajectories ending up spread across the attractor and leading to distinct filament shapes even over 50 cycles (Fig. 4 B and D and Movie S6). In contrast, periodic dynamics does not lead to exponential growth, and nearby trajectories remain close (to within 1% of the filament radius) even for long times (Fig. 4 A and C and Movie S5).

We further estimated the growth or decay rate of these perturbations or the maximum Lyapunov exponent by fitting a function of the form δ(t)=δ₀ e^λ_maxt to the pairwise distance trajectories starting at 100 initial locations spread across the attractor. Calculated for varying activity strengths, we find that at low enough activity strengths ($\mathcal{A}$ ​ < 50) the dynamics is always periodic corresponding to λ_max < 0 (Fig. 4E). We also find that beyond a critical activity, we can have λ_max > 0, corresponding to aperiodicity (Fig. 4E). Interestingly, we also find patches of negative growth rates (λ_max < 0) for higher activities corresponding to windows of periodic dynamics (for instance see the window around $\mathcal{A}$ ≈ 100 in Fig. 4E). Based on the typical values of computed positive growth rates, we find that practically indistinguishable initial filament shapes (to within 10⁻¹² in pairwise distance) become completely distinct over 𝒪(50) activity cycles (Fig. 4B, D, and E and Movie S6).

Overall, we find that the filament dynamics above a critical activity strength is aperiodic for long times, deterministic, and highly sensitive to initial conditions (positive Lyapunov exponent), thus satisfying all the conditions for chaos (31). This chaotic dynamics underlies the unpredictable sampling behavior observed for the filament (Fig. 2 D and E), and also leads the filament to sample ever newer shapes and tip locations, thus minimizing spatial resampling. The above finding of chaotic dynamics places the behavior of L. olor in a new light, since exploiting these dynamics can offer unique advantages in rapidly sampling the extracellular environment, and might explain the highly dynamic shape changes of the neck. At the same time, chaotic dynamics will place strong constraints on the control of these behaviors in a biological context where noise is an important factor. To explore control of filament dynamics using dynamic activity and possible design strategies for programming filament behaviors, we now explore the fundamental nonlinearities inherent in the follower-force compression and extension operation that underpin periodic and chaotic long-term dynamics.

A 1D Iterated Map of Filament Shape to Predict and Design Long-Term Filament Behavior.

How do periodic and chaotic behaviors emerge from simple, deterministic follower-force activity profiles? And how do the emergent long-term behaviors depend on the underlying properties of the filament and activity dynamics? We next present an analysis that seeks to explain the observed behaviors by exploring how nonlinearities arise in this simple system and how they are connected to underlying system parameters.

Overall our strategy is to postulate that the filament dynamics can be interpreted as a map acting iteratively on a reduced representation of the filament shape, and to check this postulate a posteriori. We then analyze the fixed points of this map (which correspond to periodic, limit cycle filament dynamics) and their linear stability to predict filament behaviors from just the shape of the map. We now describe our analysis in more detail.

We interpreted the dynamics as a discrete iterated map operating on the filament shape (Fig. 5A). We considered the simplest version of this map that involves only the filament orientation angle θ_n, defined as the angle between a ray connecting the base of the filament to its tip and the filament’s equilibrium orientation, after n activity cycles (Fig. 5A). We plotted this map (i.e., θ_{n + 1} vs. θ_n) using data from full numerical simulations and observed that it showed a lot of structure even for the case of aperiodic filament dynamics (Fig. 5 D and E). This observation underscores the fact that these dynamics are chaotic but not stochastic (32). Next, we computed the qualitative shape and features of this map, approximated as a function, f(θ), where θ₁ = f(θ₀), by simulating a single cycle of compression and extension and computing the final angles θ₁ given a range of initial angles θ₀ ∈ [0, π/2]. Note that this does not approximate the underlying dynamics, but assumes the filament is straight, and prestretched (corresponding to an extensional cycle at the appropriate activity strength and time scale) at the beginning of the simulated activity cycle. We found that for both varying activity strength and time scale, the predicted f(θ) closely follows the overall shape and qualitative features of the map from full simulations over many hundreds of cycles (Fig. 5 D and E).

Fig. 5.

A nonlinear 1D map predicts long-term filament behavior. (A) Filament dynamics during an activity cycle (compression + extension) as a 1D iterated map of the filament base-tip angle θ. Long-term filament dynamics can then be regarded as iterations of this map, i.e., θ_{n + 1} = f(θ_n), where n is the activity cycle number. (B and C) The shape of the map f(θ) from the single-cycle predictions plotted for varying activity strength (B) and time scale (C). The fixed points of f(θ)=θ (red circles) and f(θ)= − θ (blue circles), correspond to the existence and the stability of period-1 and even period limit cycles, respectively. Filled circles: stable fixed-points and empty circles: unstable fixed-points. (D and E) Predicted iterated maps vs. simulation results for different activity strengths (D) and time scales (E). The dots correspond to full numerical simulations, and the solid lines to the predicted map over a single cycle. (F and H) Bifurcation diagram of the filament base-tip angles sampled at the end of the extension phase, for full numerical simulations for (F) varying activity strengths and (H) and timescales. (G and I) Predicted bifurcation diagram from the single-cycle analysis for varying (G) activity strengths and (I) timescales. Filled circles: stable fixed points; crosses: unstable fixed points. In (F–I), dashed vertical lines mark the predicted and actual transitions from no-buckling to periodic dynamics, and solid vertical lines mark the transitions from periodic to chaotic dynamics.

Next, we analyzed two types of fixed points of the map f(θ) as well as their stability to predict long-term filament behavior. First, we find that due to the symmetry of the system about θ = 0, f(θ) must be odd, i.e., f(−θ)= − f(θ). This implies that the fixed points θ^* of f(θ)= − θ are fixed points of any even-iterate map f^2m(θ), which corresponds to 2n-period limit cycles of the filament dynamics, where n ∈ ℕ (SI Appendix, section 1.5 for a simple proof). The stability of both types of fixed points can be evaluated by the eigenvalue that governs the growth rate of perturbations about that fixed point/ limit cycle, given by λ = |f′(θ^*)|, where the ′ signifies a derivative. The fixed point is linearly stable/unstable for λ < 1 and > 1, respectively (SI Appendix, section 1.5). The fixed points and their stability for the single-cycle iterated maps computed for varying activity strength and time scale are shown in Fig. 5 B and C, respectively.

Using this stability analysis, we predicted the long-term filament dynamics. We visualized the dynamics as a bifurcation diagram in the observed filament orientation angles at the end of each extension phase (after ignoring transients) as a function of activity strength or timescale. Overall, we found good agreement between results from full numerical simulations and with the fixed points and their stability predicted using the single-cycle analysis above (Fig. 5 F and G). As seen in Fig. 5G, at low activity, we predict no buckling, followed by dynamic equilibrium in the form of period-1 dynamics where the origin (straight filament) remains a fixed point. At higher activity strengths, we see the first period-doubling bifurcation and eventually the onset of chaos, which is due to a cascade of period-doubling bifurcations (Fig. 5F). The single-cycle prediction closely matches the observed onset of the first period-doubling bifurcation, as well as the onset of chaotic dynamics (Fig. 5 F and G).

We also used the above stability analysis to predict behaviors for a fixed activity strength and increasing activity timescale. Here, we predict a bifurcation from no buckling to an off-axis period-1 cycle, which matches results from full simulations (Fig. 5 H and I). We also predict the onset of aperiodic dynamics at a critical activity time scale which is consistent with full simulations (Fig. 5 H and I). However, the stability analysis does not predict certain periodic windows observed in full simulations at higher activity strength or time scales (Fig. 5F–I). This breakdown is likely because the filament orientation angle stops being a good approximation for filament shape at these parameter values, due to the dynamics becoming higher dimensional (Fig. 4F).

The shape and fixed points of the single-cycle iterated map allow us to infer, at a glance, how long-term filament dynamics emerges due to periodic follower-force actuation, making these curves an important design tool. At low activity strengths and/or activity time scales, the map is almost diagonal, since the activity is too weak or short-lived to effect a large change in the filament orientation (Fig. 5 B and C). With increasing activity strength/time scale, the map deviates from the f(θ)=θ diagonal, but is still nearly linear, with the origin as the only stable fixed point, which corresponds to a period-1 cycle (Fig. 5 B and F, period-1 cycles). With further increase in activity strength or time scale, we see the first bifurcation to a period-2 dynamics happen as the map develops a valley and intersects the negative diagonal at a stable fixed point, since f′(θ^*)< 1 (Fig. 5 B and C). The location of this fixed point also gives us the angular location that the filament tip repeatedly returns to (Fig. 5 G and I, period-2m cycles). With further increase in activity strength or time scale, the valley in the map becomes steeper such that the fixed point of the even iterate map eventually loses stability when |f′(θ^*)| > 1 via a flip-bifurcation (Fig. 5 B and C) (31). Interestingly, as shown above, this implies that all even iterate maps f^2m(θ), and hence, all period-2m cycles, are now unstable. This then is the transition point to chaotic dynamics (Fig. 5F–I). The analysis above allows us a ready means, by analyzing the two major classes of fixed points, to infer long-term dynamics from the shape of f(θ).

We note that our simplified analysis above is only a first step with a goal of highlighting the key nonlinearities in the follower-force compression and extension operation, and is not a replacement for full numerical simulations. A more detailed analysis, that accounts for more degrees-of-freedom (33), will be necessary to predict the existence of other interesting features observed in our simulations such as periodic windows within the chaotic regions of parameter space (Fig. 5 F and H), transient chaos, as well as exotic odd-periodic (3, 5, etc.) cycles (Fig. 2H).

We now use insights from these predicted nonlinear maps to explore how filament behaviors may be programmed by modulated activity signals.

Programming Filament Behavior Using Modulated Follower-Force Activity.

Here, we briefly demonstrate how functional filament behaviors can be programmed through modulated activity signals by combining results from the nonlinear dynamics of follower force compression and extension that we explored in the previous section. These examples serve as a first step in understanding the space of possible behaviors in a biologically accessible parameter space including amplitude and frequency modulation of the activity signals as well as the effects of noise in the signals.

A useful control capability for active filaments is the ability to quickly return the filament to its equilibrium orientation. We show that such a shape “reset” can be performed by modulating the activity pulses that drive the filament. We use the iterated-map profiles in Fig. 5 B and C as a design tool, and leverage the existence of a “dynamic equilibrium” of the straight filament state as a fixed point of the dynamics at low activity strengths or time scales. Thus by driving the filament at a low activity strength level or fast time scale, we expect to drive the filament to a straight shape, at a much faster rate than due to equilibrium bending forces (Fig. 6A). We confirmed this using simulations starting from 100 randomly chosen filament shapes and showed that these could be quickly driven close to the straight filament shape in ≈50 cycles for frequency modulation and even faster in ≈10 cycles for amplitude modulation (Fig. 6A and Movie S7). In comparison, the equilibrium return to the straight filament would take a time t ∼ ηL⁴/κ, equivalent to ≈300 cycles at the activity time scale considered.

Fig. 6.

Artificial and biological programs for controlling filament behaviors. (A) Demonstration of filament homing using (i) amplitude or (ii) frequency-modulated signals. Overlay of filament center lines at the start (green) and end (red) of 50 cycles of activity starting from 100 random filament shapes. Time evolution of the filament base-tip angle for (iii) amplitude and (iv) frequency modulation. (B) Programming local and global sampling of the filament tip using alternating periods of low- and high-frequency activity. (i) Filament shapes during low- and high-frequency states shown in red and blue, respectively, with the tip trajectory (dashed line). (ii) Plotting the tip speed reveals that the filament dwells in a local region during high-frequency states and rapidly moves across space during the low-frequency state. (iii) Zoomed in view of a local sampling trajectory. (C) Biological activity “programs” extracted from experimental data of L. olor as a time sequence of extension and compression phases (Top, sample time trace). (i) Representative images of L. olor during compression and extension phases. (ii) Distributions of extension and compression phases for N = 7 cells and 200 activity cycle events. The line shows a log-normal fit of the underlying data. (iii) Simulation search clouds of filaments with a Lacrymaria-like activity distribution and log-normal activity profile. (iv) Experimental search clouds for L. olor from ref. 7. (v and vi) Search dynamics of the filament tip for simulations and experiments. In (v), symbols are simulations, and lines are theoretical fits to Eq. 4. In (v) and (vi) cycle counts are estimated by rescaling time by the mean cycle duration <τ_a> in simulations and experiments, respectively.

Next, we show that more complex filament behaviors can be programmed by combining sequences of activity pulses. Specifically, we show how one can use alternating periods of low- and high-frequency activity pulses to perform global and local sampling of space by the filament tip (Fig. 6B and Movie S7). Low-frequency pulses drive follower-force bending in the nonlinear regime wherein there is a large change in the filament orientation thus transporting the tip to a new location in space. On the other hand, high-frequency pulses drive the filament dynamics in the linear regime, causing the filament orientation to be mostly correlated and the tip to explore a small cone of space near the current filament orientation (Fig. 6B and Movie S7). We also find that the tip speeds are lower during the high-frequency phase and higher during the low-frequency phase (Fig. 6B). Overall such activity profiles allow us to implement the equivalent of an “intermittent search” strategy through the filament dynamics [a behavior typically found in foraging behaviors of higher animals (25)] using just modulated activity patterns. Herein the filament tip has a higher chance of locating a target when searching locally, due to the slower tip speeds, but can then quickly relocate to a new region of space to minimize resampling.

A natural way to connect our model to biological systems is to ask what temporal activity profiles do cells like L. olor use? Toward this, we quantified long-term behavioral data of L. olor and segmented the behavior into compression and extension phases using image processing of the cell’s slender protrusion/neck, which is a bona fide readout of the activity state of the cilia (Fig. 6C, i and SI Appendix, section 1.2 and Movie S8). This analysis results in a time series of extensional and compression phases, a sample of which is shown in Fig. 6C, Top. We find that on average extension phases are longer than compression phase (τ_ext = 1.45 ± 1.12 s, τ_comp = 0.92 ± 0.61 s; N = 7 cells, 200 activity cycles). Further, on quantifying the distribution of extension and compression durations we find them to be well-approximated by a log-normal distribution (Kolmogorov–Smirnov test; compression, ks-statistic = 0.042 P-value = 0.88; extension, ks-statistic = 0.072, P-value = 0.27; SI Appendix, section 1.2) This is an intriguing finding since log-normal distributions are common in natural systems and are thought to arise due to multiplicative noise (in contrast to normal distributions that arise due to additive noise as per the central limit theorem) (34).

We next used this log-normal activity profile derived from experimental data as an input to our filament simulations and asked what dynamics and search behavior one observes. For a median activity duration comparable to the active flapping timescale of the filament, most activity phases drawn from a log-normal distribution are short compared to the filament’s intrinsic time scale. This short activity only leads to a small change in filament orientation and thus a “local” search about the current orientation (Movie S9). In contrast, samples from the tails of the distribution, while less frequent, correspond to long compressive phases and lead to large changes in filament orientation and thus a more “global” search (Movie S9). Taken together, this means that a log-normal distribution, which occurs in L. olor, leads to the encoding of an intermittent search of the kind we found earlier for synthetic activity motifs (Movies S7 and S9). Next, we compared the search clouds of filaments driven by log-normal activity profiles to experimental data from L. olor and find that the simulations qualitatively reproduce certain features of the experimental search clouds including the regions of increased sampling near the base of the filament/neck (Fig. 6C, iii and iv). Further, we find that, in both simulations and experiments, these hotspots correspond to long compressive phases where the filament “flaps” in a characteristic fashion due to distributed follower forces (Fig. 6C, iii and iv and Movie S9).

Comparing Search Behavior of the Active Filament Model to Experiments with L. olor.

We now compare and contrast the search performance predicted by our model, under different activity profiles, and spatial distributions, to that observed in cells of L. olor. From experimental measurements over 25 hunting events, we find that the search dynamics obeys a Michaelis–Menten-type saturation curve Eq. 4, which is consistent with our observations from simulations (Fig. 6C, v and vi) (7). Further, these experiments show a search range (n_sat/n_max (%)≈65% and a τ_1/2 ≈ 93.3 ± 4.1s or equivalently ≈40 cycles (Fig. 6C, vi) (7). Note that for the experimental data, the cycle count is an approximate one based on a measured average activity cycle duration of <τ_a> = 2.37 s. Interestingly, our model filament with just tip activity and purely deterministic drive achieves a peak search range of ≈40%, or 66% of the observed search range of real cells, with a comparable τ_1/2 of 40 cycles (Fig. 2). For spatially distributed activity and a log-normal activity profile as found above for cells, our model again predicts typical search ranges of ≈40% (Fig. 6C, v), and a peak search range of ≈60% (SI Appendix, Fig. S6). However, L. olor was able to sample a comparable search range faster than what we observed in simulations. We find τ_1/2|_exp ≈ 40 cycles vs. a typical saturation time scale in simulations τ_1/2|_sim ≈ 200 cycles, indicating that while the search ranges are comparable, the extent of resampling by the filament tip is lesser in experiments. This is consistent with our observation that the tangential tip autocorrelation function for real cells decays faster than simulations across a range of activity strengths (SI Appendix, Fig. S4), implicating additional scrambling mechanisms in real cells.

Overall, even strong stochasticity in the input drive in the form of a log-normal distribution only leads to a 1.5× increase in peak search range and a comparable typical search range when compared to purely deterministic activity. Further, the search metrics for purely deterministic, tip-concentrated activity already compares favorably with the search performance of real cells, both in terms of search range and time scale. Taken together, these results indicate that the inherent nonlinearity in the follower-force mechanics strongly influences the search behavior of the filament, and is likely a key mechanism driving the search behavior in L. olor, with stochasticity in the drive only acting on top of this intrinsic scrambling mechanism.

2. Discussion

In this work, inspired by the remarkable behaviors of L. olor, we have described a toy model consisting of an active filament driven by time-varying follower forces and characterized the rich space of behaviors including the prediction of chaotic dynamics. This dynamical behavior is in striking contrast to other active elastohydrodynamic systems such as cilia whose typical dynamics are periodic (35). In contrast to earlier active filament models (13) that require stochasticity/noise to achieve aperiodicity and mixing, our work points to a simple scrambling recipe involving just periodic forcing that results in chaos even at non-Brownian length scales (deterministic limit). By generalizing the driving activity to have time-varying phases of extension and compression, we also introduced the notion of programming, wherein a “program” (time sequence of activity) is directly converted to “behavior” (dynamics of filament shapes) through a nonlinear mapping function which can be derived from first-principles of nonvariational mechanics. Using a simple iterated map–based stability analysis, we explicitly derive an approximation of this nonlinear map and show how it leads to this rich behavioral space. This map serves as a useful design tool which we use to suggest simple artificial programs for functions such as homing a filament or performing intermittent searches.

Biological programs, however, are likely noisy, and through experimental measurements, we make the interesting finding that activity phase durations in L. olor cells are log-normally distributed. Implementing this biological program in our filament model allows direct comparisons with experiments, with the biologically relevant function of spatial search performance as one of the comparison metrics. Surprisingly, chaotic dynamics arising from simple periodic programs already account for a sizeable fraction of the search performance observed in real cells (for e.g., model filaments driven by tip-activity and periodic forcing achieve 66% of the observed search range in cells and sample it in a comparable time). Stochasticity, as expected, completely precludes any periodic dynamics and, along with distributed activity, pushes the search performance of our model closer to experiments. However, real cells sample their available search range faster than model predictions with log-normal activity, thus pointing to other mechanisms that may reduce spatial resampling of the filament tip in cells. From experimental observations of L. olor (7), we speculate that these may include noise in the ciliary forcing, and torsional degrees-of-freedom, leading to rotational diffusion of the filament tip; nonlinear/hysteretic variations in the material properties of the neck due to the underlying cytoskeleton geometry, among others. Exploring these additional scrambling mechanisms would be an interesting future investigation. While our current model neglects these additional mechanisms, our results still indicate that the shape scrambling resulting from the intrinsic nonlinearity of the follower-force compression and extension operation is a key mechanism underlying the shape dynamics and emergent search behavior of L. olor.

The movement patterns of organisms that allow them to search their environment, and how these behaviors are controlled, is a problem at the interface of ecology, physics, mathematics, and neurobiology (25–28, 36, 37). Interestingly, chaotic dynamics have been observed in animal behaviors, including ants and mud-snails, specifically in the context of searching and foraging, but the underlying generative processes leading to these dynamics are unclear (38, 39). In the unicellular context, nonequilibrium flux loops and time-irreversibility have been observed in behavioral state dynamics of flagellates (40), and ciliates (41), for which mechanosensitivity and mechanical mediation through the cytoskeleton have been implicated, respectively. Our work offers an interesting example of how nonlinear dynamics, chaos, and emergent search behaviors can result directly from the follower-force mechanics of an active filament: nonlinear mechanics being one of the key generative processes for behavior. In this way, active filaments with time-varying follower-force activity are a simple, mechanical embodiment of “intermittent search behavior” in unicellular organisms like L. olor, a behavior that is typically observed in higher animals possessing nervous systems (25, 27, 28, 37).

Models such as ours that include time-varying activity coupled to geometry of filaments and surfaces could serve as a starting point for understanding behavior in other ciliates where force generators (cilia) are anchored directly to cell membranes and the cytoskeleton, whose geometry is in turn a function of active stresses. These cilia-covered cell surfaces are a hallmark of one of the largest groups of protists that show a rich class of behaviors (42). The simplicity of the proposed active filament system and driving signals suggests the possibility of experimental realizations (at both macroscales and microscales) and theory-experiment dialogue to fully map out the space of filament behaviors, and test the predictions made here. From an engineering context, our work expands the functional space of active filaments by leveraging nonlinear, chaotic dynamics, thus suggesting a distinct class of self-driven, soft, microscale robotic arms to perform useful functions. From a biological perspective, it will be interesting to explore how sensing and control strategies have evolved to shape behavior under the nonlinear constraints of active filament dynamics identified here. Our work, though inspired by L. olor, is likely more broadly applicable to the design of active matter/filament systems including microscale robots, wherein a physical task/computation, an algorithmic representation of the task, and its actual physical implementation can all be analyzed and explored systematically—with a direct analogy to Marr’s level of analysis (developed in the context of information processing systems) applied to single-celled protist behavior (43).

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Flow around the filament during a cycle of compressional and extensional follower-force activity.

Movie S2.

Examples of periodic filament dynamics including 2, 6 and 10 periods.

Movie S3.

Examples of aperiodic filament behaviors for long times.

Movie S4.

Effects of distributed activity on filament dynamics.

Movie S5.

Periodic dynamics: Visualizing the evolution of nearby initial conditions in a reduced-order state-space.

Movie S6.

Chaotic dynamics: Visualizing the rapid divergence of nearby initial conditions as dynamics in a reduced-order state-space.

Movie S7.

Programming filament behaviors including (1) Homing and (2) Local vs Global search using frequency and amplitude modulated activity.

Movie S8.

Example of a temporally-segmented video showing L. olor alternating between compression and extension activity phases.

Movie S9.

Biological activity profiles and their effects on filament dynamics.

Data, Materials, and Software Availability

Simulation data will be made available in the Dryad data repository (https://doi.org/10.6078/D12T52) (44). The code used for simulations, analysis, and figure generation can be accessed at the following GitHub repository (https://github.com/deepakkrishnamurthy/PyFilaments) (45).